LMFDB - Embedded newform 75.9.c.c.26.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,9,Mod(26,75)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 9, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("75.26");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	75.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$30.5533957546$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-14}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 14 $ x^2 + 14
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 3)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.2
Root			$3.74166i$ of defining polynomial
Character	$\chi$	$=$	75.26
Dual form			75.9.c.c.26.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+22.4499i q^{2} +(-45.0000 - 67.3498i) q^{3} -248.000 q^{4} +(1512.00 - 1010.25i) q^{6} +1750.00 q^{7} +179.600i q^{8} +(-2511.00 + 6061.48i) q^{9} +O(q^{10})$	\(q+22.4499i q^{2} +(-45.0000 - 67.3498i) q^{3} -248.000 q^{4} +(1512.00 - 1010.25i) q^{6} +1750.00 q^{7} +179.600i q^{8} +(-2511.00 + 6061.48i) q^{9} -6959.48i q^{11} +(11160.0 + 16702.8i) q^{12} -25730.0 q^{13} +39287.4i q^{14} -67520.0 q^{16} +74893.0i q^{17} +(-136080. - 56371.8i) q^{18} +18938.0 q^{19} +(-78750.0 - 117862. i) q^{21} +156240. q^{22} -470461. i q^{23} +(12096.0 - 8081.98i) q^{24} -577637. i q^{26} +(521235. - 103651. i) q^{27} -434000. q^{28} -460897. i q^{29} -351478. q^{31} -1.46984e6i q^{32} +(-468720. + 313177. i) q^{33} -1.68134e6 q^{34} +(622728. - 1.50325e6i) q^{36} -1.33517e6 q^{37} +425157. i q^{38} +(1.15785e6 + 1.73291e6i) q^{39} -1.87547e6i q^{41} +(2.64600e6 - 1.76793e6i) q^{42} +3.52615e6 q^{43} +1.72595e6i q^{44} +1.05618e7 q^{46} -4.08104e6i q^{47} +(3.03840e6 + 4.54746e6i) q^{48} -2.70230e6 q^{49} +(5.04403e6 - 3.37019e6i) q^{51} +6.38104e6 q^{52} -6.60177e6i q^{53} +(2.32697e6 + 1.17017e7i) q^{54} +314299. i q^{56} +(-852210. - 1.27547e6i) q^{57} +1.03471e7 q^{58} -1.37149e7i q^{59} +753602. q^{61} -7.89066e6i q^{62} +(-4.39425e6 + 1.06076e7i) q^{63} +1.57128e7 q^{64} +(-7.03080e6 - 1.05227e7i) q^{66} -2.26889e6 q^{67} -1.85735e7i q^{68} +(-3.16855e7 + 2.11707e7i) q^{69} -1.70220e7i q^{71} +(-1.08864e6 - 450974. i) q^{72} -2.76728e7 q^{73} -2.99745e7i q^{74} -4.69662e6 q^{76} -1.21791e7i q^{77} +(-3.89038e7 + 2.59937e7i) q^{78} -2.29810e7 q^{79} +(-3.04365e7 - 3.04408e7i) q^{81} +4.21042e7 q^{82} -4.63952e7i q^{83} +(1.95300e7 + 2.92298e7i) q^{84} +7.91619e7i q^{86} +(-3.10414e7 + 2.07404e7i) q^{87} +1.24992e6 q^{88} -7.26152e7i q^{89} -4.50275e7 q^{91} +1.16674e8i q^{92} +(1.58165e7 + 2.36720e7i) q^{93} +9.16191e7 q^{94} +(-9.89937e7 + 6.61429e7i) q^{96} -1.47271e8 q^{97} -6.06665e7i q^{98} +(4.21848e7 + 1.74753e7i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 90 q^{3} - 496 q^{4} + 3024 q^{6} + 3500 q^{7} - 5022 q^{9}+O(q^{10}) $ 2 * q - 90 * q^3 - 496 * q^4 + 3024 * q^6 + 3500 * q^7 - 5022 * q^9	$ 2 q - 90 q^{3} - 496 q^{4} + 3024 q^{6} + 3500 q^{7} - 5022 q^{9} + 22320 q^{12} - 51460 q^{13} - 135040 q^{16} - 272160 q^{18} + 37876 q^{19} - 157500 q^{21} + 312480 q^{22} + 24192 q^{24} + 1042470 q^{27} - 868000 q^{28} - 702956 q^{31} - 937440 q^{33} - 3362688 q^{34} + 1245456 q^{36} - 2670340 q^{37} + 2315700 q^{39} + 5292000 q^{42} + 7052300 q^{43} + 21123648 q^{46} + 6076800 q^{48} - 5404602 q^{49} + 10088064 q^{51} + 12762080 q^{52} + 4653936 q^{54} - 1704420 q^{57} + 20694240 q^{58} + 1507204 q^{61} - 8788500 q^{63} + 31425536 q^{64} - 14061600 q^{66} - 4537780 q^{67} - 63370944 q^{69} - 2177280 q^{72} - 55345540 q^{73} - 9393248 q^{76} - 77807520 q^{78} - 45961964 q^{79} - 60872958 q^{81} + 84208320 q^{82} + 39060000 q^{84} - 62082720 q^{87} + 2499840 q^{88} - 90055000 q^{91} + 31633020 q^{93} + 183238272 q^{94} - 197987328 q^{96} - 294542020 q^{97} + 84369600 q^{99}+O(q^{100}) $ 2 * q - 90 * q^3 - 496 * q^4 + 3024 * q^6 + 3500 * q^7 - 5022 * q^9 + 22320 * q^12 - 51460 * q^13 - 135040 * q^16 - 272160 * q^18 + 37876 * q^19 - 157500 * q^21 + 312480 * q^22 + 24192 * q^24 + 1042470 * q^27 - 868000 * q^28 - 702956 * q^31 - 937440 * q^33 - 3362688 * q^34 + 1245456 * q^36 - 2670340 * q^37 + 2315700 * q^39 + 5292000 * q^42 + 7052300 * q^43 + 21123648 * q^46 + 6076800 * q^48 - 5404602 * q^49 + 10088064 * q^51 + 12762080 * q^52 + 4653936 * q^54 - 1704420 * q^57 + 20694240 * q^58 + 1507204 * q^61 - 8788500 * q^63 + 31425536 * q^64 - 14061600 * q^66 - 4537780 * q^67 - 63370944 * q^69 - 2177280 * q^72 - 55345540 * q^73 - 9393248 * q^76 - 77807520 * q^78 - 45961964 * q^79 - 60872958 * q^81 + 84208320 * q^82 + 39060000 * q^84 - 62082720 * q^87 + 2499840 * q^88 - 90055000 * q^91 + 31633020 * q^93 + 183238272 * q^94 - 197987328 * q^96 - 294542020 * q^97 + 84369600 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			22.4499i			1.40312i	0.712610	+	0.701561i	$0.247513\pi$
$2$			22.4499i			1.40312i	−0.712610	+	0.701561i	$0.752487\pi$
$3$	−45.0000	−	67.3498i	−0.555556	−	0.831479i
$3$	−45.0000	−	67.3498i	−0.555556	−	0.831479i
$4$	−248.000			−0.968750
$5$		0			0
$5$		0			0
$6$	1512.00	−	1010.25i	1.16667	−	0.779512i
$7$	1750.00			0.728863			0.364431	−	0.931230i	$-0.381263\pi$
$7$	1750.00			0.728863			0.364431	+	0.931230i	$0.381263\pi$
$8$			179.600i			0.0438475i
$9$	−2511.00	+	6061.48i	−0.382716	+	0.923866i
$10$		0			0
$11$		−	6959.48i		−	0.475342i	−0.971346	−	0.237671i	$-0.923616\pi$
$11$		−	6959.48i		−	0.475342i	0.971346	−	0.237671i	$-0.0763840\pi$
$12$	11160.0	+	16702.8i	0.538194	+	0.805496i
$13$	−25730.0			−0.900879			−0.450439	−	0.892807i	$-0.648733\pi$
$13$	−25730.0			−0.900879			−0.450439	+	0.892807i	$0.648733\pi$
$14$			39287.4i			1.02268i
$15$		0			0
$16$	−67520.0			−1.03027
$17$			74893.0i			0.896697i	0.893859	+	0.448348i	$0.147988\pi$
$17$			74893.0i			0.896697i	−0.893859	+	0.448348i	$0.852012\pi$
$18$	−136080.	−	56371.8i	−1.29630	−	0.536997i
$19$	18938.0			0.145318			0.0726590	−	0.997357i	$-0.476852\pi$
$19$	18938.0			0.145318			0.0726590	+	0.997357i	$0.476852\pi$
$20$		0			0
$21$	−78750.0	−	117862.i	−0.404924	−	0.606035i
$22$	156240.			0.666963
$23$		−	470461.i		−	1.68117i	−0.541678	−	0.840586i	$-0.682211\pi$
$23$		−	470461.i		−	1.68117i	0.541678	−	0.840586i	$-0.317789\pi$
$24$	12096.0	−	8081.98i	0.0364583	−	0.0243597i
$25$		0			0
$26$		−	577637.i		−	1.26404i
$27$	521235.	−	103651.i	0.980796	−	0.195038i
$28$	−434000.			−0.706086
$29$		−	460897.i		−	0.651647i	−0.945431	−	0.325823i	$-0.894359\pi$
$29$		−	460897.i		−	0.651647i	0.945431	−	0.325823i	$-0.105641\pi$
$30$		0			0
$31$	−351478.			−0.380585			−0.190292	−	0.981727i	$-0.560944\pi$
$31$	−351478.			−0.380585			−0.190292	+	0.981727i	$0.560944\pi$
$32$		−	1.46984e6i		−	1.40175i
$33$	−468720.	+	313177.i	−0.395237	+	0.264079i
$34$	−1.68134e6			−1.25817
$35$		0			0
$36$	622728.	−	1.50325e6i	0.370756	−	0.894995i
$37$	−1.33517e6			−0.712409			−0.356205	−	0.934408i	$-0.615929\pi$
$37$	−1.33517e6			−0.712409			−0.356205	+	0.934408i	$0.615929\pi$
$38$			425157.i			0.203899i
$39$	1.15785e6	+	1.73291e6i	0.500488	+	0.749062i
$40$		0			0
$41$		−	1.87547e6i		−	0.663704i	−0.943331	−	0.331852i	$-0.892327\pi$
$41$		−	1.87547e6i		−	0.663704i	0.943331	−	0.331852i	$-0.107673\pi$
$42$	2.64600e6	−	1.76793e6i	0.850340	−	0.568157i
$43$	3.52615e6			1.03140			0.515700	−	0.856769i	$-0.327532\pi$
$43$	3.52615e6			1.03140			0.515700	+	0.856769i	$0.327532\pi$
$44$			1.72595e6i			0.460488i
$45$		0			0
$46$	1.05618e7			2.35889
$47$		−	4.08104e6i		−	0.836333i	−0.908370	−	0.418167i	$-0.862673\pi$
$47$		−	4.08104e6i		−	0.836333i	0.908370	−	0.418167i	$-0.137327\pi$
$48$	3.03840e6	+	4.54746e6i	0.572374	+	0.856651i
$49$	−2.70230e6			−0.468759
$50$		0			0
$51$	5.04403e6	−	3.37019e6i	0.745585	−	0.498165i
$52$	6.38104e6			0.872726
$53$		−	6.60177e6i		−	0.836675i	−0.908292	−	0.418337i	$-0.862613\pi$
$53$		−	6.60177e6i		−	0.836675i	0.908292	−	0.418337i	$-0.137387\pi$
$54$	2.32697e6	+	1.17017e7i	0.273663	+	1.37618i
$55$		0			0
$56$			314299.i			0.0319589i
$57$	−852210.	−	1.27547e6i	−0.0807323	−	0.120829i
$58$	1.03471e7			0.914340
$59$		−	1.37149e7i		−	1.13184i	−0.824461	−	0.565919i	$-0.808521\pi$
$59$		−	1.37149e7i		−	1.13184i	0.824461	−	0.565919i	$-0.191479\pi$
$60$		0			0
$61$	753602.			0.0544280			0.0272140	−	0.999630i	$-0.491336\pi$
$61$	753602.			0.0544280			0.0272140	+	0.999630i	$0.491336\pi$
$62$		−	7.89066e6i		−	0.534007i
$63$	−4.39425e6	+	1.06076e7i	−0.278948	+	0.673372i
$64$	1.57128e7			0.936554
$65$		0			0
$66$	−7.03080e6	−	1.05227e7i	−0.370535	−	0.554566i
$67$	−2.26889e6			−0.112594			−0.0562969	−	0.998414i	$-0.517929\pi$
$67$	−2.26889e6			−0.112594			−0.0562969	+	0.998414i	$0.517929\pi$
$68$		−	1.85735e7i		−	0.868675i
$69$	−3.16855e7	+	2.11707e7i	−1.39786	+	0.933985i
$70$		0			0
$71$		−	1.70220e7i		−	0.669849i	−0.942245	−	0.334925i	$-0.891289\pi$
$71$		−	1.70220e7i		−	0.669849i	0.942245	−	0.334925i	$-0.108711\pi$
$72$	−1.08864e6	−	450974.i	−0.0405093	−	0.0167812i
$73$	−2.76728e7			−0.974454			−0.487227	−	0.873275i	$-0.661992\pi$
$73$	−2.76728e7			−0.974454			−0.487227	+	0.873275i	$0.661992\pi$
$74$		−	2.99745e7i		−	0.999597i
$75$		0			0
$76$	−4.69662e6			−0.140777
$77$		−	1.21791e7i		−	0.346459i
$78$	−3.89038e7	+	2.59937e7i	−1.05103	+	0.702246i
$79$	−2.29810e7			−0.590011			−0.295006	−	0.955496i	$-0.595322\pi$
$79$	−2.29810e7			−0.590011			−0.295006	+	0.955496i	$0.595322\pi$
$80$		0			0
$81$	−3.04365e7	−	3.04408e7i	−0.707057	−	0.707157i
$82$	4.21042e7			0.931257
$83$		−	4.63952e7i		−	0.977599i	−0.872396	−	0.488799i	$-0.837435\pi$
$83$		−	4.63952e7i		−	0.977599i	0.872396	−	0.488799i	$-0.162565\pi$
$84$	1.95300e7	+	2.92298e7i	0.392270	+	0.587096i
$85$		0			0
$86$			7.91619e7i			1.44718i
$87$	−3.10414e7	+	2.07404e7i	−0.541831	+	0.362026i
$88$	1.24992e6			0.0208426
$89$		−	7.26152e7i		−	1.15736i	−0.815555	−	0.578679i	$-0.803568\pi$
$89$		−	7.26152e7i		−	1.15736i	0.815555	−	0.578679i	$-0.196432\pi$
$90$		0			0
$91$	−4.50275e7			−0.656617
$92$			1.16674e8i			1.62864i
$93$	1.58165e7	+	2.36720e7i	0.211436	+	0.316448i
$94$	9.16191e7			1.17348
$95$		0			0
$96$	−9.89937e7	+	6.61429e7i	−1.16553	+	0.778751i
$97$	−1.47271e8			−1.66353			−0.831764	−	0.555129i	$-0.812669\pi$
$97$	−1.47271e8			−1.66353			−0.831764	+	0.555129i	$0.812669\pi$
$98$		−	6.06665e7i		−	0.657726i
$99$	4.21848e7	+	1.74753e7i	0.439152	+	0.181921i
$100$		0			0
$101$			1.03545e8i			0.995045i	0.867451	+	0.497522i	$0.165757\pi$
$101$			1.03545e8i			0.995045i	−0.867451	+	0.497522i	$0.834243\pi$
$102$	7.56605e7	+	1.13238e8i	0.698986	+	1.04615i
$103$	1.66064e8			1.47545			0.737726	−	0.675100i	$-0.235899\pi$
$103$	1.66064e8			1.47545			0.737726	+	0.675100i	$0.235899\pi$
$104$		−	4.62110e6i		−	0.0395013i
$105$		0			0
$106$	1.48209e8			1.17396
$107$		−	2.25540e7i		−	0.172063i	−0.996292	−	0.0860316i	$-0.972581\pi$
$107$		−	2.25540e7i		−	0.172063i	0.996292	−	0.0860316i	$-0.0274186\pi$
$108$	−1.29266e8	+	2.57055e7i	−0.950146	+	0.188943i
$109$	−1.09975e8			−0.779091			−0.389546	−	0.921007i	$-0.627368\pi$
$109$	−1.09975e8			−0.779091			−0.389546	+	0.921007i	$0.627368\pi$
$110$		0			0
$111$	6.00826e7	+	8.99235e7i	0.395783	+	0.592354i
$112$	−1.18160e8			−0.750928
$113$			2.87748e8i			1.76481i	0.470490	+	0.882405i	$0.344077\pi$
$113$			2.87748e8i			1.76481i	−0.470490	+	0.882405i	$0.655923\pi$
$114$	2.86343e7	−	1.91321e7i	0.169538	−	0.113277i
$115$		0			0
$116$			1.14303e8i			0.631283i
$117$	6.46080e7	−	1.55962e8i	0.344781	−	0.832291i
$118$	3.07899e8			1.58811
$119$			1.31063e8i			0.653569i
$120$		0			0
$121$	1.65924e8			0.774050
$122$			1.69183e7i			0.0763692i
$123$	−1.26312e8	+	8.43961e7i	−0.551856	+	0.368724i
$124$	8.71665e7			0.368691
$125$		0			0
$126$	−2.38140e8	−	9.86507e7i	−0.944822	−	0.391397i
$127$	−2.75994e8			−1.06092			−0.530462	−	0.847708i	$-0.677982\pi$
$127$	−2.75994e8			−1.06092			−0.530462	+	0.847708i	$0.677982\pi$
$128$		−	2.35290e7i		−	0.0876523i
$129$	−1.58677e8	−	2.37486e8i	−0.573000	−	0.857588i
$130$		0			0
$131$		−	2.89118e8i		−	0.981725i	−0.871237	−	0.490862i	$-0.836682\pi$
$131$		−	2.89118e8i		−	0.981725i	0.871237	−	0.490862i	$-0.163318\pi$
$132$	1.16243e8	−	7.76678e7i	0.382886	−	0.255826i
$133$	3.31415e7			0.105917
$134$		−	5.09365e7i		−	0.157983i
$135$		0			0
$136$	−1.34508e7			−0.0393180
$137$		−	2.07562e8i		−	0.589205i	−0.955620	−	0.294602i	$-0.904813\pi$
$137$		−	2.07562e8i		−	0.589205i	0.955620	−	0.294602i	$-0.0951872\pi$
$138$	−4.75282e8	−	7.11337e8i	−1.31049	−	1.96137i
$139$	−1.42668e8			−0.382180			−0.191090	−	0.981573i	$-0.561202\pi$
$139$	−1.42668e8			−0.382180			−0.191090	+	0.981573i	$0.561202\pi$
$140$		0			0
$141$	−2.74857e8	+	1.83647e8i	−0.695394	+	0.464630i
$142$	3.82143e8			0.939880
$143$			1.79067e8i			0.428226i
$144$	1.69543e8	−	4.09271e8i	0.394302	−	0.951835i
$145$		0			0
$146$		−	6.21252e8i		−	1.36728i
$147$	1.21604e8	+	1.82000e8i	0.260422	+	0.389763i
$148$	3.31122e8			0.690147
$149$			8.19236e8i			1.66213i	0.556179	+	0.831063i	$0.312267\pi$
$149$			8.19236e8i			1.66213i	−0.556179	+	0.831063i	$0.687733\pi$
$150$		0			0
$151$	4.23861e8			0.815296			0.407648	−	0.913139i	$-0.366349\pi$
$151$	4.23861e8			0.815296			0.407648	+	0.913139i	$0.366349\pi$
$152$			3.40126e6i			0.00637184i
$153$	−4.53963e8	−	1.88056e8i	−0.828428	−	0.343180i
$154$	2.73420e8			0.486124
$155$		0			0
$156$	−2.87147e8	−	4.29762e8i	−0.484848	−	0.725654i
$157$	7.59851e8			1.25063			0.625316	−	0.780371i	$-0.284970\pi$
$157$	7.59851e8			1.25063			0.625316	+	0.780371i	$0.284970\pi$
$158$		−	5.15922e8i		−	0.827857i
$159$	−4.44628e8	+	2.97079e8i	−0.695678	+	0.464819i
$160$		0			0
$161$		−	8.23307e8i		−	1.22534i
$162$	6.83394e8	−	6.83297e8i	0.992227	−	0.992087i
$163$	−6.68160e8			−0.946520			−0.473260	−	0.880923i	$-0.656923\pi$
$163$	−6.68160e8			−0.946520			−0.473260	+	0.880923i	$0.656923\pi$
$164$			4.65116e8i			0.642963i
$165$		0			0
$166$	1.04157e9			1.37169
$167$			1.96306e8i			0.252387i	0.992006	+	0.126194i	$0.0402761\pi$
$167$			1.96306e8i			0.252387i	−0.992006	+	0.126194i	$0.959724\pi$
$168$	2.11680e7	−	1.41435e7i	0.0265731	−	0.0177549i
$169$	−1.53698e8			−0.188417
$170$		0			0
$171$	−4.75533e7	+	1.14792e8i	−0.0556156	+	0.134254i
$172$	−8.74485e8			−0.999168
$173$			1.02319e9i			1.14228i	0.820852	+	0.571141i	$0.193499\pi$
$173$			1.02319e9i			1.14228i	−0.820852	+	0.571141i	$0.806501\pi$
$174$	−4.65620e8	−	6.96877e8i	−0.507966	−	0.760255i
$175$		0			0
$176$			4.69904e8i			0.489732i
$177$	−9.23696e8	+	6.17170e8i	−0.941100	+	0.628799i
$178$	1.63021e9			1.62391
$179$			1.28895e9i			1.25552i	0.778408	+	0.627759i	$0.216028\pi$
$179$			1.28895e9i			1.25552i	−0.778408	+	0.627759i	$0.783972\pi$
$180$		0			0
$181$	4.71707e8			0.439499			0.219749	−	0.975556i	$-0.429476\pi$
$181$	4.71707e8			0.439499			0.219749	+	0.975556i	$0.429476\pi$
$182$		−	1.01086e9i		−	0.921314i
$183$	−3.39121e7	−	5.07550e7i	−0.0302378	−	0.0452558i
$184$	8.44946e7			0.0737153
$185$		0			0
$186$	−5.31435e8	+	3.55080e8i	−0.444016	+	0.296670i
$187$	5.21217e8			0.426238
$188$			1.01210e9i			0.810198i
$189$	9.12161e8	−	1.81390e8i	0.714866	−	0.142156i
$190$		0			0
$191$			1.61787e8i			0.121565i	0.998151	+	0.0607827i	$0.0193597\pi$
$191$			1.61787e8i			0.121565i	−0.998151	+	0.0607827i	$0.980640\pi$
$192$	−7.07075e8	−	1.05825e9i	−0.520308	−	0.778725i
$193$	1.58840e9			1.14480			0.572401	−	0.819974i	$-0.306012\pi$
$193$	1.58840e9			1.14480			0.572401	+	0.819974i	$0.306012\pi$
$194$		−	3.30623e9i		−	2.33413i
$195$		0			0
$196$	6.70171e8			0.454110
$197$		−	5.37769e8i		−	0.357052i	−0.983935	−	0.178526i	$-0.942867\pi$
$197$		−	5.37769e8i		−	0.357052i	0.983935	−	0.178526i	$-0.0571328\pi$
$198$	−3.92319e8	+	9.47046e8i	−0.255257	+	0.616184i
$199$	6.47586e8			0.412938			0.206469	−	0.978453i	$-0.433803\pi$
$199$	6.47586e8			0.412938			0.206469	+	0.978453i	$0.433803\pi$
$200$		0			0
$201$	1.02100e8	+	1.52809e8i	0.0625521	+	0.0936194i
$202$	−2.32457e9			−1.39617
$203$		−	8.06570e8i		−	0.474961i
$204$	−1.25092e9	+	8.35806e8i	−0.722285	+	0.482597i
$205$		0			0
$206$			3.72812e9i			2.07024i
$207$	2.85169e9	+	1.18133e9i	1.55318	+	0.643412i
$208$	1.73729e9			0.928152
$209$		−	1.31799e8i		−	0.0690758i
$210$		0			0
$211$	5.81104e7			0.0293173			0.0146586	−	0.999893i	$-0.495334\pi$
$211$	5.81104e7			0.0293173			0.0146586	+	0.999893i	$0.495334\pi$
$212$			1.63724e9i			0.810529i
$213$	−1.14643e9	+	7.65990e8i	−0.556966	+	0.372139i
$214$	5.06336e8			0.241426
$215$		0			0
$216$	1.86157e7	+	9.36136e7i	0.00855195	+	0.0430055i
$217$	−6.15086e8			−0.277394
$218$		−	2.46893e9i		−	1.09316i
$219$	1.24527e9	+	1.86376e9i	0.541363	+	0.810238i
$220$		0			0
$221$		−	1.92700e9i		−	0.807815i
$222$	−2.01878e9	+	1.34885e9i	−0.831144	+	0.555332i
$223$	−4.40200e9			−1.78004			−0.890021	−	0.455920i	$-0.849310\pi$
$223$	−4.40200e9			−1.78004			−0.890021	+	0.455920i	$0.849310\pi$
$224$		−	2.57222e9i		−	1.02168i
$225$		0			0
$226$	−6.45992e9			−2.47624
$227$			3.53592e9i			1.33168i	0.746095	+	0.665839i	$0.231926\pi$
$227$			3.53592e9i			1.33168i	−0.746095	+	0.665839i	$0.768074\pi$
$228$	2.11348e8	+	3.16317e8i	0.0782094	+	0.117053i
$229$	−1.86569e9			−0.678420			−0.339210	−	0.940711i	$-0.610160\pi$
$229$	−1.86569e9			−0.678420			−0.339210	+	0.940711i	$0.610160\pi$
$230$		0			0
$231$	−8.20260e8	+	5.48059e8i	−0.288074	+	0.192477i
$232$	8.27770e7			0.0285731
$233$		−	2.72132e9i		−	0.923328i	−0.887055	−	0.461664i	$-0.847253\pi$
$233$		−	2.72132e9i		−	0.923328i	0.887055	−	0.461664i	$-0.152747\pi$
$234$	3.50134e9	+	1.45045e9i	1.16781	+	0.483769i
$235$		0			0
$236$			3.40129e9i			1.09647i
$237$	1.03414e9	+	1.54777e9i	0.327784	+	0.490582i
$238$	−2.94235e9			−0.917037
$239$			2.27461e9i			0.697132i	0.937284	+	0.348566i	$0.113331\pi$
$239$			2.27461e9i			0.697132i	−0.937284	+	0.348566i	$0.886669\pi$
$240$		0			0
$241$	−1.74667e9			−0.517778			−0.258889	−	0.965907i	$-0.583356\pi$
$241$	−1.74667e9			−0.517778			−0.258889	+	0.965907i	$0.583356\pi$
$242$			3.72500e9i			1.08609i
$243$	−6.80540e8	+	3.41973e9i	−0.195177	+	0.980768i
$244$	−1.86893e8			−0.0527272
$245$		0			0
$246$	−1.89469e9	−	2.83571e9i	−0.517365	−	0.774321i
$247$	−4.87275e8			−0.130914
$248$		−	6.31253e7i		−	0.0166877i
$249$	−3.12471e9	+	2.08778e9i	−0.812853	+	0.543110i
$250$		0			0
$251$		−	1.37549e9i		−	0.346547i	−0.984874	−	0.173274i	$-0.944566\pi$
$251$		−	1.37549e9i		−	0.346547i	0.984874	−	0.173274i	$-0.0554345\pi$
$252$	1.08977e9	−	2.63068e9i	0.270230	−	0.652329i
$253$	−3.27417e9			−0.799132
$254$		−	6.19605e9i		−	1.48861i
$255$		0			0
$256$	4.55069e9			1.05954
$257$		−	7.93672e9i		−	1.81932i	−0.415356	−	0.909659i	$-0.636343\pi$
$257$		−	7.93672e9i		−	1.81932i	0.415356	−	0.909659i	$-0.363657\pi$
$258$	5.33154e9	−	3.56228e9i	1.20330	−	0.803988i
$259$	−2.33655e9			−0.519249
$260$		0			0
$261$	2.79372e9	+	1.15731e9i	0.602034	+	0.249396i
$262$	6.49068e9			1.37748
$263$			3.22555e8i			0.0674187i	0.999432	+	0.0337093i	$0.0107320\pi$
$263$			3.22555e8i			0.0674187i	−0.999432	+	0.0337093i	$0.989268\pi$
$264$	−5.62464e7	−	8.41819e7i	−0.0115792	−	0.0173302i
$265$		0			0
$266$			7.44025e8i			0.148614i
$267$	−4.89062e9	+	3.26769e9i	−0.962319	+	0.642977i
$268$	5.62685e8			0.109075
$269$		−	3.47314e9i		−	0.663304i	−0.943402	−	0.331652i	$-0.892394\pi$
$269$		−	3.47314e9i		−	0.663304i	0.943402	−	0.331652i	$-0.107606\pi$
$270$		0			0
$271$	−1.44216e9			−0.267385			−0.133693	−	0.991023i	$-0.542683\pi$
$271$	−1.44216e9			−0.267385			−0.133693	+	0.991023i	$0.542683\pi$
$272$		−	5.05678e9i		−	0.923843i
$273$	2.02624e9	+	3.03259e9i	0.364787	+	0.545964i
$274$	4.65976e9			0.826726
$275$		0			0
$276$	7.85800e9	−	5.25035e9i	1.35418	−	0.904798i
$277$	−3.38046e9			−0.574192			−0.287096	−	0.957902i	$-0.592690\pi$
$277$	−3.38046e9			−0.574192			−0.287096	+	0.957902i	$0.592690\pi$
$278$		−	3.20289e9i		−	0.536245i
$279$	8.82561e8	−	2.13048e9i	0.145656	−	0.351609i
$280$		0			0
$281$		−	4.02262e9i		−	0.645184i	−0.946538	−	0.322592i	$-0.895446\pi$
$281$		−	4.02262e9i		−	0.645184i	0.946538	−	0.322592i	$-0.104554\pi$
$282$	−4.12286e9	−	6.17053e9i	−0.651932	−	0.975722i
$283$	1.04253e10			1.62533			0.812666	−	0.582730i	$-0.198016\pi$
$283$	1.04253e10			1.62533			0.812666	+	0.582730i	$0.198016\pi$
$284$			4.22146e9i			0.648917i
$285$		0			0
$286$	−4.02006e9			−0.600853
$287$		−	3.28207e9i		−	0.483749i
$288$	8.90943e9	+	3.69078e9i	1.29503	+	0.536473i
$289$	1.36679e9			0.195935
$290$		0			0
$291$	6.62720e9	+	9.91868e9i	0.924183	+	1.38319i
$292$	6.86285e9			0.944002
$293$			1.03927e10i			1.41012i	0.709146	+	0.705061i	$0.249081\pi$
$293$			1.03927e10i			1.41012i	−0.709146	+	0.705061i	$0.750919\pi$
$294$	−4.08588e9	+	2.72999e9i	−0.546885	+	0.365403i
$295$		0			0
$296$		−	2.39796e8i		−	0.0312374i
$297$	−7.21360e8	−	3.62753e9i	−0.0927099	−	0.466213i
$298$	−1.83918e10			−2.33216
$299$			1.21050e10i			1.51453i
$300$		0			0
$301$	6.17076e9			0.751749
$302$			9.51565e9i			1.14396i
$303$	6.97372e9	−	4.65951e9i	0.827359	−	0.552803i
$304$	−1.27869e9			−0.149717
$305$		0			0
$306$	4.22185e9	−	1.01914e10i	0.481524	−	1.16238i
$307$	2.99309e9			0.336951			0.168476	−	0.985706i	$-0.446116\pi$
$307$	2.99309e9			0.336951			0.168476	+	0.985706i	$0.446116\pi$
$308$			3.02042e9i			0.335632i
$309$	−7.47286e9	−	1.11843e10i	−0.819696	−	1.22681i
$310$		0			0
$311$		−	6.44832e9i		−	0.689295i	−0.938732	−	0.344647i	$-0.887998\pi$
$311$		−	6.44832e9i		−	0.689295i	0.938732	−	0.344647i	$-0.112002\pi$
$312$	−3.11230e8	+	2.07949e8i	−0.0328445	+	0.0219452i
$313$	−3.27737e7			−0.00341467			−0.00170733	−	0.999999i	$-0.500543\pi$
$313$	−3.27737e7			−0.00341467			−0.00170733	+	0.999999i	$0.500543\pi$
$314$			1.70586e10i			1.75479i
$315$		0			0
$316$	5.69928e9			0.571573
$317$		−	1.17797e10i		−	1.16653i	−0.812282	−	0.583264i	$-0.801775\pi$
$317$		−	1.17797e10i		−	1.16653i	0.812282	−	0.583264i	$-0.198225\pi$
$318$	−6.66942e9	−	9.98187e9i	−0.652198	−	0.976120i
$319$	−3.20761e9			−0.309755
$320$		0			0
$321$	−1.51901e9	+	1.01493e9i	−0.143067	+	0.0955907i
$322$	1.84832e10			1.71931
$323$			1.41832e9i			0.130306i
$324$	7.54825e9	+	7.54931e9i	0.684961	+	0.685058i
$325$		0			0
$326$		−	1.50002e10i		−	1.32808i
$327$	4.94888e9	+	7.40680e9i	0.432829	+	0.647798i
$328$	3.36833e8			0.0291018
$329$		−	7.14182e9i		−	0.609573i
$330$		0			0
$331$	−1.20100e10			−1.00053			−0.500265	−	0.865872i	$-0.666764\pi$
$331$	−1.20100e10			−1.00053			−0.500265	+	0.865872i	$0.666764\pi$
$332$			1.15060e10i			0.947049i
$333$	3.35261e9	−	8.09311e9i	0.272651	−	0.658171i
$334$	−4.40705e9			−0.354130
$335$		0			0
$336$	5.31720e9	+	7.95806e9i	0.417182	+	0.624381i
$337$	−1.59214e10			−1.23441			−0.617207	−	0.786801i	$-0.711736\pi$
$337$	−1.59214e10			−1.23441			−0.617207	+	0.786801i	$0.711736\pi$
$338$		−	3.45051e9i		−	0.264372i
$339$	1.93798e10	−	1.29486e10i	1.46740	−	0.980451i
$340$		0			0
$341$			2.44611e9i			0.180908i
$342$	−2.57708e9	−	1.06757e9i	−0.188375	−	0.0780354i
$343$	−1.48174e10			−1.07052
$344$			6.33295e8i			0.0452243i
$345$		0			0
$346$	−2.29706e10			−1.60276
$347$		−	4.94792e9i		−	0.341275i	−0.985334	−	0.170638i	$-0.945417\pi$
$347$		−	4.94792e9i		−	0.341275i	0.985334	−	0.170638i	$-0.0545828\pi$
$348$	7.69826e9	−	5.14361e9i	0.524899	−	0.350713i
$349$	−7.29567e9			−0.491772			−0.245886	−	0.969299i	$-0.579079\pi$
$349$	−7.29567e9			−0.491772			−0.245886	+	0.969299i	$0.579079\pi$
$350$		0			0
$351$	−1.34114e10	+	2.66695e9i	−0.883578	+	0.175706i
$352$	−1.02293e10			−0.666311
$353$		−	6.93875e9i		−	0.446871i	−0.974719	−	0.223436i	$-0.928273\pi$
$353$		−	6.93875e9i		−	0.446871i	0.974719	−	0.223436i	$-0.0717272\pi$
$354$	−1.38554e10	−	2.07369e10i	−0.882282	−	1.32048i
$355$		0			0
$356$			1.80086e10i			1.12119i
$357$	8.82706e9	−	5.89782e9i	0.543429	−	0.363094i
$358$	−2.89368e10			−1.76164
$359$		−	1.60096e10i		−	0.963838i	−0.876216	−	0.481919i	$-0.839940\pi$
$359$		−	1.60096e10i		−	0.963838i	0.876216	−	0.481919i	$-0.160060\pi$
$360$		0			0
$361$	−1.66249e10			−0.978883
$362$			1.05898e10i			0.616670i
$363$	−7.46660e9	−	1.11750e10i	−0.430028	−	0.643607i
$364$	1.11668e10			0.636098
$365$		0			0
$366$	1.13945e9	−	7.61325e8i	0.0634994	−	0.0424273i
$367$	−1.36364e10			−0.751686			−0.375843	−	0.926683i	$-0.622647\pi$
$367$	−1.36364e10			−0.751686			−0.375843	+	0.926683i	$0.622647\pi$
$368$			3.17655e10i			1.73207i
$369$	1.13681e10	+	4.70930e9i	0.613173	+	0.254010i
$370$		0			0
$371$		−	1.15531e10i		−	0.609821i
$372$	−3.92249e9	−	5.87065e9i	−0.204829	−	0.306559i
$373$	2.44062e10			1.26085			0.630427	−	0.776248i	$-0.282880\pi$
$373$	2.44062e10			1.26085			0.630427	+	0.776248i	$0.282880\pi$
$374$			1.17013e10i			0.598063i
$375$		0			0
$376$	7.32953e8			0.0366712
$377$			1.18589e10i			0.587055i
$378$	4.07219e9	+	2.04780e10i	0.199463	+	1.00304i
$379$	−1.98392e10			−0.961542			−0.480771	−	0.876846i	$-0.659643\pi$
$379$	−1.98392e10			−0.961542			−0.480771	+	0.876846i	$0.659643\pi$
$380$		0			0
$381$	1.24197e10	+	1.85881e10i	0.589403	+	0.882137i
$382$	−3.63211e9			−0.170571
$383$		−	1.51133e10i		−	0.702366i	−0.936307	−	0.351183i	$-0.885780\pi$
$383$		−	1.51133e10i		−	0.702366i	0.936307	−	0.351183i	$-0.114220\pi$
$384$	−1.58467e9	+	1.05880e9i	−0.0728811	+	0.0486957i
$385$		0			0
$386$			3.56594e10i			1.60630i
$387$	−8.85416e9	+	2.13737e10i	−0.394733	+	0.952875i
$388$	3.65232e10			1.61154
$389$			1.79991e10i			0.786056i	0.919527	+	0.393028i	$0.128572\pi$
$389$			1.79991e10i			0.786056i	−0.919527	+	0.393028i	$0.871428\pi$
$390$		0			0
$391$	3.52342e10			1.50750
$392$		−	4.85332e8i		−	0.0205539i
$393$	−1.94720e10	+	1.30103e10i	−0.816284	+	0.545403i
$394$	1.20729e10			0.500987
$395$		0			0
$396$	−1.04618e10	−	4.33386e9i	−0.425429	−	0.176236i
$397$	2.35673e10			0.948739			0.474370	−	0.880326i	$-0.342676\pi$
$397$	2.35673e10			0.948739			0.474370	+	0.880326i	$0.342676\pi$
$398$			1.45383e10i			0.579403i
$399$	−1.49137e9	−	2.23207e9i	−0.0588428	−	0.0880678i
$400$		0			0
$401$		−	1.37692e10i		−	0.532515i	−0.963902	−	0.266257i	$-0.914213\pi$
$401$		−	1.37692e10i		−	0.532515i	0.963902	−	0.266257i	$-0.0857871\pi$
$402$	−3.43056e9	+	2.29214e9i	−0.131359	+	0.0877682i
$403$	9.04353e9			0.342861
$404$		−	2.56791e10i		−	0.963950i
$405$		0			0
$406$	1.81075e10			0.666428
$407$			9.29209e9i			0.338638i
$408$	6.05284e8	+	9.05906e8i	0.0218433	+	0.0326921i
$409$	3.58480e10			1.28107			0.640533	−	0.767931i	$-0.278714\pi$
$409$	3.58480e10			1.28107			0.640533	+	0.767931i	$0.278714\pi$
$410$		0			0
$411$	−1.39793e10	+	9.34031e9i	−0.489912	+	0.327336i
$412$	−4.11838e10			−1.42934
$413$		−	2.40011e10i		−	0.824955i
$414$	−2.65207e10	+	6.40203e10i	−0.902785	+	2.17930i
$415$		0			0
$416$			3.78191e10i			1.26281i
$417$	6.42007e9	+	9.60868e9i	0.212322	+	0.317775i
$418$	2.95887e9			0.0969217
$419$		−	2.23996e10i		−	0.726750i	−0.931643	−	0.363375i	$-0.881624\pi$
$419$		−	2.23996e10i		−	0.726750i	0.931643	−	0.363375i	$-0.118376\pi$
$420$		0			0
$421$	−1.49535e10			−0.476008			−0.238004	−	0.971264i	$-0.576493\pi$
$421$	−1.49535e10			−0.476008			−0.238004	+	0.971264i	$0.576493\pi$
$422$			1.30457e9i			0.0411357i
$423$	2.47372e10	+	1.02475e10i	0.772660	+	0.320078i
$424$	1.18567e9			0.0366861
$425$		0			0
$426$	−1.71964e10	−	2.57373e10i	−0.522156	−	0.781491i
$427$	1.31880e9			0.0396706
$428$			5.59339e9i			0.166686i
$429$	1.20602e10	−	8.05804e9i	0.356061	−	0.237903i
$430$		0			0
$431$			6.40436e10i			1.85595i	0.372640	+	0.927976i	$0.378453\pi$
$431$			6.40436e10i			1.85595i	−0.372640	+	0.927976i	$0.621547\pi$
$432$	−3.51938e10	+	6.99854e9i	−1.01049	+	0.200943i
$433$	5.22954e9			0.148769			0.0743843	−	0.997230i	$-0.476301\pi$
$433$	5.22954e9			0.148769			0.0743843	+	0.997230i	$0.476301\pi$
$434$		−	1.38087e10i		−	0.389218i
$435$		0			0
$436$	2.72738e10			0.754745
$437$		−	8.90959e9i		−	0.244305i
$438$	−4.18412e10	+	2.79563e10i	−1.13686	+	0.759598i
$439$	4.34801e10			1.17066			0.585332	−	0.810793i	$-0.300964\pi$
$439$	4.34801e10			1.17066			0.585332	+	0.810793i	$0.300964\pi$
$440$		0			0
$441$	6.78548e9	−	1.63800e10i	0.179402	−	0.433070i
$442$	4.32610e10			1.13346
$443$		−	3.78737e10i		−	0.983383i	−0.870770	−	0.491691i	$-0.836379\pi$
$443$		−	3.78737e10i		−	0.983383i	0.870770	−	0.491691i	$-0.163621\pi$
$444$	−1.49005e10	−	2.23010e10i	−0.383415	−	0.573843i
$445$		0			0
$446$		−	9.88246e10i		−	2.49761i
$447$	5.51754e10	−	3.68656e10i	1.38202	−	0.923403i
$448$	2.74973e10			0.682620
$449$		−	2.95505e10i		−	0.727076i	−0.931579	−	0.363538i	$-0.881569\pi$
$449$		−	2.95505e10i		−	0.727076i	0.931579	−	0.363538i	$-0.118431\pi$
$450$		0			0
$451$	−1.30523e10			−0.315486
$452$		−	7.13614e10i		−	1.70966i
$453$	−1.90737e10	−	2.85469e10i	−0.452942	−	0.677902i
$454$	−7.93813e10			−1.86851
$455$		0			0
$456$	2.29074e8	−	1.53057e8i	0.00529806	−	0.00353991i
$457$	2.02181e10			0.463529			0.231764	−	0.972772i	$-0.425550\pi$
$457$	2.02181e10			0.463529			0.231764	+	0.972772i	$0.425550\pi$
$458$		−	4.18847e10i		−	0.951905i
$459$	7.76277e9	+	3.90369e10i	0.174890	+	0.879476i
$460$		0			0
$461$		−	7.01826e10i		−	1.55391i	−0.629556	−	0.776955i	$-0.716763\pi$
$461$		−	7.01826e10i		−	1.55391i	0.629556	−	0.776955i	$-0.283237\pi$
$462$	−1.23039e10	−	1.84148e10i	−0.270069	−	0.404202i
$463$	−4.16009e9			−0.0905271			−0.0452635	−	0.998975i	$-0.514413\pi$
$463$	−4.16009e9			−0.0905271			−0.0452635	+	0.998975i	$0.514413\pi$
$464$			3.11198e10i			0.671374i
$465$		0			0
$466$	6.10935e10			1.29554
$467$			2.88138e10i			0.605806i	0.953021	+	0.302903i	$0.0979558\pi$
$467$			2.88138e10i			0.605806i	−0.953021	+	0.302903i	$0.902044\pi$
$468$	−1.60228e10	+	3.86786e10i	−0.334006	+	0.806282i
$469$	−3.97056e9			−0.0820654
$470$		0			0
$471$	−3.41933e10	−	5.11758e10i	−0.694796	−	1.03988i
$472$	2.46319e9			0.0496283
$473$		−	2.45402e10i		−	0.490268i
$474$	−3.47472e10	+	2.32165e10i	−0.688346	+	0.459921i
$475$		0			0
$476$		−	3.25036e10i		−	0.633145i
$477$	4.00165e10	+	1.65770e10i	0.772975	+	0.320209i
$478$	−5.10649e10			−0.978161
$479$		−	4.47149e10i		−	0.849395i	−0.905335	−	0.424698i	$-0.860380\pi$
$479$		−	4.47149e10i		−	0.849395i	0.905335	−	0.424698i	$-0.139620\pi$
$480$		0			0
$481$	3.43539e10			0.641795
$482$		−	3.92127e10i		−	0.726505i
$483$	−5.54496e10	+	3.70488e10i	−1.01885	+	0.680747i
$484$	−4.11493e10			−0.749861
$485$		0			0
$486$	−7.67727e10	−	1.52781e10i	−1.37614	−	0.273857i
$487$	−5.72836e10			−1.01839			−0.509195	−	0.860651i	$-0.670057\pi$
$487$	−5.72836e10			−1.01839			−0.509195	+	0.860651i	$0.670057\pi$
$488$			1.35347e8i			0.00238654i
$489$	3.00672e10	+	4.50005e10i	0.525845	+	0.787012i
$490$		0			0
$491$			7.25262e10i			1.24787i	0.781477	+	0.623934i	$0.214467\pi$
$491$			7.25262e10i			1.24787i	−0.781477	+	0.623934i	$0.785533\pi$
$492$	3.13255e10	−	2.09302e10i	0.534611	−	0.357202i
$493$	3.45180e10			0.584330
$494$		−	1.09393e10i		−	0.183688i
$495$		0			0
$496$	2.37318e10			0.392106
$497$		−	2.97885e10i		−	0.488228i
$498$	−4.68706e10	−	7.01495e10i	−0.762050	−	1.14053i
$499$	−2.64368e10			−0.426389			−0.213195	−	0.977010i	$-0.568387\pi$
$499$	−2.64368e10			−0.426389			−0.213195	+	0.977010i	$0.568387\pi$
$500$		0			0
$501$	1.32212e10	−	8.83376e9i	0.209855	−	0.140215i
$502$	3.08797e10			0.486248
$503$			7.52828e10i			1.17604i	0.808845	+	0.588022i	$0.200093\pi$
$503$			7.52828e10i			1.17604i	−0.808845	+	0.588022i	$0.799907\pi$
$504$	−1.90512e9	−	7.89205e8i	−0.0295257	−	0.0122312i
$505$		0			0
$506$		−	7.35048e10i		−	1.12128i
$507$	6.91640e9	+	1.03515e10i	0.104676	+	0.156665i
$508$	6.84465e10			1.02777
$509$			6.45184e10i			0.961197i	0.876941	+	0.480599i	$0.159581\pi$
$509$			6.45184e10i			0.961197i	−0.876941	+	0.480599i	$0.840419\pi$
$510$		0			0
$511$	−4.84273e10			−0.710243
$512$			9.61394e10i			1.39901i
$513$	9.87115e9	−	1.96295e9i	0.142527	−	0.0283426i
$514$	1.78179e11			2.55272
$515$		0			0
$516$	3.93518e10	+	5.88964e10i	0.555094	+	0.830788i
$517$	−2.84019e10			−0.397544
$518$		−	5.24554e10i		−	0.728569i
$519$	6.89119e10	−	4.60437e10i	0.949783	−	0.634601i
$520$		0			0
$521$			7.65146e10i			1.03847i	0.854632	+	0.519235i	$0.173783\pi$
$521$			7.65146e10i			1.03847i	−0.854632	+	0.519235i	$0.826217\pi$
$522$	−2.59816e10	+	6.27189e10i	−0.349932	+	0.844727i
$523$	8.46771e10			1.13177			0.565886	−	0.824483i	$-0.308534\pi$
$523$	8.46771e10			1.13177			0.565886	+	0.824483i	$0.308534\pi$
$524$			7.17012e10i			0.951046i
$525$		0			0
$526$	−7.24133e9			−0.0945966
$527$		−	2.63232e10i		−	0.341269i
$528$	3.16480e10	−	2.11457e10i	0.407202	−	0.272073i
$529$	−1.43023e11			−1.82634
$530$		0			0
$531$	8.31326e10	+	3.44381e10i	1.04567	+	0.433173i
$532$	−8.21909e9			−0.102607
$533$			4.82558e10i			0.597917i
$534$	−7.33594e10	−	1.09794e11i	−0.902175	−	1.35025i
$535$		0			0
$536$		−	4.07492e8i		−	0.00493696i
$537$	8.68104e10	−	5.80026e10i	1.04394	−	0.697510i
$538$	7.79717e10			0.930696
$539$			1.88066e10i			0.222821i
$540$		0			0
$541$	1.43470e11			1.67483			0.837415	−	0.546568i	$-0.184066\pi$
$541$	1.43470e11			1.67483			0.837415	+	0.546568i	$0.184066\pi$
$542$		−	3.23765e10i		−	0.375174i
$543$	−2.12268e10	−	3.17694e10i	−0.244166	−	0.365434i
$544$	1.10081e11			1.25695
$545$		0			0
$546$	−6.80816e10	+	4.54889e10i	−0.766053	+	0.511841i
$547$	−1.64171e11			−1.83378			−0.916892	−	0.399134i	$-0.869311\pi$
$547$	−1.64171e11			−1.83378			−0.916892	+	0.399134i	$0.869311\pi$
$548$			5.14755e10i			0.570792i
$549$	−1.89229e9	+	4.56795e9i	−0.0208305	+	0.0502842i
$550$		0			0
$551$		−	8.72847e9i		−	0.0946961i
$552$	−3.80226e9	−	5.69070e9i	−0.0409529	−	0.0612928i
$553$	−4.02167e10			−0.430037
$554$		−	7.58912e10i		−	0.805661i
$555$		0			0
$556$	3.53817e10			0.370237
$557$			1.54420e11i			1.60429i	0.597130	+	0.802145i	$0.296308\pi$
$557$			1.54420e11i			1.60429i	−0.597130	+	0.802145i	$0.703692\pi$
$558$	4.78291e10	+	1.98135e10i	0.493351	+	0.204373i
$559$	−9.07278e10			−0.929166
$560$		0			0
$561$	−2.34547e10	−	3.51039e10i	−0.236799	−	0.354408i
$562$	9.03075e10			0.905271
$563$			1.54622e11i			1.53900i	0.638646	+	0.769500i	$0.279495\pi$
$563$			1.54622e11i			1.53900i	−0.638646	+	0.769500i	$0.720505\pi$
$564$	6.81646e10	−	4.55444e10i	0.673663	−	0.450110i
$565$		0			0
$566$			2.34047e11i			2.28054i
$567$	−5.32638e10	−	5.32714e10i	−0.515348	−	0.515420i
$568$	3.05714e9			0.0293712
$569$			1.15380e11i			1.10073i	0.834925	+	0.550364i	$0.185511\pi$
$569$			1.15380e11i			1.10073i	−0.834925	+	0.550364i	$0.814489\pi$
$570$		0			0
$571$	−1.63410e11			−1.53722			−0.768608	−	0.639720i	$-0.779050\pi$
$571$	−1.63410e11			−1.53722			−0.768608	+	0.639720i	$0.779050\pi$
$572$		−	4.44087e10i		−	0.414844i
$573$	1.08963e10	−	7.28041e9i	0.101079	−	0.0675363i
$574$	7.36823e10			0.678759
$575$		0			0
$576$	−3.94548e10	+	9.52427e10i	−0.358434	+	0.865250i
$577$	−7.42282e10			−0.669678			−0.334839	−	0.942275i	$-0.608682\pi$
$577$	−7.42282e10			−0.669678			−0.334839	+	0.942275i	$0.608682\pi$
$578$			3.06844e10i			0.274920i
$579$	−7.14779e10	−	1.06978e11i	−0.636001	−	0.951879i
$580$		0			0
$581$		−	8.11916e10i		−	0.712535i
$582$	−2.22674e11	+	1.48780e11i	−1.94078	+	1.29674i
$583$	−4.59449e10			−0.397707
$584$		−	4.97002e9i		−	0.0427274i
$585$		0			0
$586$	−2.33315e11			−1.97857
$587$		−	6.72877e10i		−	0.566739i	−0.959011	−	0.283369i	$-0.908548\pi$
$587$		−	6.72877e10i		−	0.566739i	0.959011	−	0.283369i	$-0.0914523\pi$
$588$	−3.01577e10	−	4.51359e10i	−0.252283	−	0.377583i
$589$	−6.65629e9			−0.0553059
$590$		0			0
$591$	−3.62187e10	+	2.41996e10i	−0.296881	+	0.198362i
$592$	9.01507e10			0.733977
$593$		−	2.36444e10i		−	0.191210i	−0.995419	−	0.0956048i	$-0.969521\pi$
$593$		−	2.36444e10i		−	0.191210i	0.995419	−	0.0956048i	$-0.0304785\pi$
$594$	8.14378e10	−	1.61945e10i	0.654154	−	0.130083i
$595$		0			0
$596$		−	2.03170e11i		−	1.61018i
$597$	−2.91414e10	−	4.36148e10i	−0.229410	−	0.343350i
$598$	−2.71756e11			−2.12507
$599$		−	3.03370e10i		−	0.235649i	−0.993034	−	0.117825i	$-0.962408\pi$
$599$		−	3.03370e10i		−	0.235649i	0.993034	−	0.117825i	$-0.0375921\pi$
$600$		0			0
$601$	3.37911e10			0.259003			0.129501	−	0.991579i	$-0.458662\pi$
$601$	3.37911e10			0.259003			0.129501	+	0.991579i	$0.458662\pi$
$602$			1.38533e11i			1.05480i
$603$	5.69718e9	−	1.37528e10i	0.0430914	−	0.104022i
$604$	−1.05117e11			−0.789818
$605$		0			0
$606$	1.04606e11	+	1.56560e11i	0.775649	+	1.16089i
$607$	3.82366e10			0.281660			0.140830	−	0.990034i	$-0.455023\pi$
$607$	3.82366e10			0.281660			0.140830	+	0.990034i	$0.455023\pi$
$608$		−	2.78359e10i		−	0.203700i
$609$	−5.43224e10	+	3.62957e10i	−0.394920	+	0.263867i
$610$		0			0
$611$			1.05005e11i			0.753435i
$612$	1.12583e11	+	4.66380e10i	0.802539	+	0.332456i
$613$	−1.08066e11			−0.765330			−0.382665	−	0.923887i	$-0.624994\pi$
$613$	−1.08066e11			−0.765330			−0.382665	+	0.923887i	$0.624994\pi$
$614$			6.71948e10i			0.472783i
$615$		0			0
$616$	2.18736e9			0.0151914
$617$		−	4.72538e9i		−	0.0326059i	−0.999867	−	0.0163029i	$-0.994810\pi$
$617$		−	4.72538e9i		−	0.0326059i	0.999867	−	0.0163029i	$-0.00518962\pi$
$618$	2.51088e11	−	1.67765e11i	1.72136	−	1.15013i
$619$	2.29845e10			0.156557			0.0782786	−	0.996932i	$-0.475058\pi$
$619$	2.29845e10			0.156557			0.0782786	+	0.996932i	$0.475058\pi$
$620$		0			0
$621$	−4.87639e10	−	2.45221e11i	−0.327893	−	1.64889i
$622$	1.44764e11			0.967165
$623$		−	1.27077e11i		−	0.843555i
$624$	−7.81780e10	−	1.17006e11i	−0.515640	−	0.771739i
$625$		0			0
$626$		−	7.35768e8i		−	0.00479119i
$627$	−8.87662e9	+	5.93094e9i	−0.0574351	+	0.0383754i
$628$	−1.88443e11			−1.21155
$629$		−	9.99949e10i		−	0.638815i
$630$		0			0
$631$	−1.01892e11			−0.642722			−0.321361	−	0.946957i	$-0.604140\pi$
$631$	−1.01892e11			−0.642722			−0.321361	+	0.946957i	$0.604140\pi$
$632$		−	4.12737e9i		−	0.0258705i
$633$	−2.61497e9	−	3.91372e9i	−0.0162874	−	0.0243767i
$634$	2.64453e11			1.63678
$635$		0			0
$636$	1.10268e11	−	7.36757e10i	0.673938	−	0.450294i
$637$	6.95302e10			0.422295
$638$		−	7.20106e10i		−	0.434624i
$639$	1.03179e11	+	4.27422e10i	0.618851	+	0.256362i
$640$		0			0
$641$		−	1.17803e11i		−	0.697791i	−0.937162	−	0.348896i	$-0.886557\pi$
$641$		−	1.17803e11i		−	0.697791i	0.937162	−	0.348896i	$-0.113443\pi$
$642$	−2.27851e10	−	3.41016e10i	−0.134125	−	0.200740i
$643$	2.62680e10			0.153668			0.0768339	−	0.997044i	$-0.475519\pi$
$643$	2.62680e10			0.153668			0.0768339	+	0.997044i	$0.475519\pi$
$644$			2.04180e11i			1.18705i
$645$		0			0
$646$	−3.18413e10			−0.182836
$647$			3.10527e11i			1.77208i	0.463612	+	0.886038i	$0.346553\pi$
$647$			3.10527e11i			1.77208i	−0.463612	+	0.886038i	$0.653447\pi$
$648$	5.46715e9	−	5.46638e9i	0.0310071	−	0.0310027i
$649$	−9.54486e10			−0.538010
$650$		0			0
$651$	2.76789e10	+	4.14260e10i	0.154108	+	0.230648i
$652$	1.65704e11			0.916942
$653$			3.48345e10i			0.191583i	0.995401	+	0.0957914i	$0.0305382\pi$
$653$			3.48345e10i			0.191583i	−0.995401	+	0.0957914i	$0.969462\pi$
$654$	−1.66282e11	+	1.11102e11i	−0.908940	+	0.607311i
$655$		0			0
$656$			1.26632e11i			0.683796i
$657$	6.94863e10	−	1.67738e11i	0.372939	−	0.900265i
$658$	1.60333e11			0.855304
$659$		−	3.77120e10i		−	0.199957i	−0.994990	−	0.0999787i	$-0.968123\pi$
$659$		−	3.77120e10i		−	0.199957i	0.994990	−	0.0999787i	$-0.0318775\pi$
$660$		0			0
$661$	−1.39619e11			−0.731372			−0.365686	−	0.930738i	$-0.619166\pi$
$661$	−1.39619e11			−0.731372			−0.365686	+	0.930738i	$0.619166\pi$
$662$		−	2.69623e11i		−	1.40386i
$663$	−1.29783e11	+	8.67149e10i	−0.671682	+	0.448786i
$664$	8.33256e9			0.0428653
$665$		0			0
$666$	1.81690e11	+	7.52659e10i	0.923494	+	0.382562i
$667$	−2.16834e11			−1.09553
$668$		−	4.86838e10i		−	0.244500i
$669$	1.98090e11	+	2.96474e11i	0.988912	+	1.48007i
$670$		0			0
$671$		−	5.24468e9i		−	0.0258719i
$672$	−1.73239e11	+	1.15750e11i	−0.849510	+	0.567603i
$673$	1.29783e11			0.632642			0.316321	−	0.948652i	$-0.397552\pi$
$673$	1.29783e11			0.632642			0.316321	+	0.948652i	$0.397552\pi$
$674$		−	3.57434e11i		−	1.73203i
$675$		0			0
$676$	3.81171e10			0.182529
$677$		−	3.40648e11i		−	1.62163i	−0.585306	−	0.810813i	$-0.699025\pi$
$677$		−	3.40648e11i		−	1.62163i	0.585306	−	0.810813i	$-0.300975\pi$
$678$	2.90696e11	+	4.35075e11i	1.37569	+	2.05895i
$679$	−2.57724e11			−1.21248
$680$		0			0
$681$	2.38144e11	−	1.59117e11i	1.10726	−	0.739821i
$682$	−5.49149e10			−0.253836
$683$			1.02876e11i			0.472750i	0.971662	+	0.236375i	$0.0759593\pi$
$683$			1.02876e11i			0.472750i	−0.971662	+	0.236375i	$0.924041\pi$
$684$	1.17932e10	−	2.84685e10i	0.0538776	−	0.130059i
$685$		0			0
$686$		−	3.32650e11i		−	1.50208i
$687$	8.39562e10	+	1.25654e11i	0.376900	+	0.564092i
$688$	−2.38086e11			−1.06262
$689$			1.69863e11i			0.753742i
$690$		0			0
$691$	3.58259e11			1.57140			0.785698	−	0.618611i	$-0.212304\pi$
$691$	3.58259e11			1.57140			0.785698	+	0.618611i	$0.212304\pi$
$692$		−	2.53752e11i		−	1.10659i
$693$	7.38234e10	+	3.05817e10i	0.320082	+	0.132595i
$694$	1.11081e11			0.478851
$695$		0			0
$696$	−3.72496e9	−	5.57501e9i	−0.0158740	−	0.0237580i
$697$	1.40459e11			0.595141
$698$		−	1.63787e11i		−	0.690016i
$699$	−1.83280e11	+	1.22459e11i	−0.767728	+	0.512960i
$700$		0			0
$701$			2.49323e11i			1.03250i	0.856438	+	0.516250i	$0.172673\pi$
$701$			2.49323e11i			1.03250i	−0.856438	+	0.516250i	$0.827327\pi$
$702$	−5.98729e10	−	3.01085e11i	−0.246537	−	1.23977i
$703$	−2.52854e10			−0.103526
$704$		−	1.09353e11i		−	0.445183i
$705$		0			0
$706$	1.55775e11			0.627015
$707$			1.81203e11i			0.725251i
$708$	2.29077e11	−	1.53058e11i	0.911691	−	0.609149i
$709$	−3.88874e11			−1.53895			−0.769474	−	0.638678i	$-0.779482\pi$
$709$	−3.88874e11			−1.53895			−0.769474	+	0.638678i	$0.779482\pi$
$710$		0			0
$711$	5.77052e10	−	1.39299e11i	0.225807	−	0.545091i
$712$	1.30417e10			0.0507473
$713$			1.65357e11i			0.639829i
$714$	1.32406e11	+	1.98167e11i	0.509465	+	0.762497i
$715$		0			0
$716$		−	3.19659e11i		−	1.21628i
$717$	1.53195e11	−	1.02357e11i	0.579651	−	0.387296i
$718$	3.59416e11			1.35238
$719$		−	3.35735e10i		−	0.125626i	−0.998025	−	0.0628132i	$-0.979993\pi$
$719$		−	3.35735e10i		−	0.125626i	0.998025	−	0.0628132i	$-0.0200072\pi$
$720$		0			0
$721$	2.90611e11			1.07540
$722$		−	3.73228e11i		−	1.37349i
$723$	7.86002e10	+	1.17638e11i	0.287654	+	0.430521i
$724$	−1.16983e11			−0.425764
$725$		0			0
$726$	2.50878e11	−	1.67625e11i	0.903058	−	0.603381i
$727$	−2.53113e11			−0.906102			−0.453051	−	0.891485i	$-0.649664\pi$
$727$	−2.53113e11			−0.906102			−0.453051	+	0.891485i	$0.649664\pi$
$728$		−	8.08692e9i		−	0.0287911i
$729$	2.60942e11	−	1.08053e11i	0.923920	−	0.382586i
$730$		0			0
$731$			2.64084e11i			0.924853i
$732$	8.41020e9	+	1.25872e10i	0.0292929	+	0.0438416i
$733$	−2.07602e11			−0.719144			−0.359572	−	0.933117i	$-0.617077\pi$
$733$	−2.07602e11			−0.719144			−0.359572	+	0.933117i	$0.617077\pi$
$734$		−	3.06137e11i		−	1.05471i
$735$		0			0
$736$	−6.91504e11			−2.35659
$737$			1.57903e10i			0.0535205i
$738$	−1.05724e11	+	2.55214e11i	−0.356407	+	0.860357i
$739$	4.15014e11			1.39151			0.695753	−	0.718281i	$-0.255071\pi$
$739$	4.15014e11			1.39151			0.695753	+	0.718281i	$0.255071\pi$
$740$		0			0
$741$	2.19274e10	+	3.28179e10i	0.0727300	+	0.108852i
$742$	2.59366e11			0.855653
$743$		−	3.30690e11i		−	1.08509i	−0.840027	−	0.542545i	$-0.817461\pi$
$743$		−	3.30690e11i		−	1.08509i	0.840027	−	0.542545i	$-0.182539\pi$
$744$	−4.25148e9	+	2.84064e9i	−0.0138755	+	0.00927095i
$745$		0			0
$746$			5.47918e11i			1.76913i
$747$	2.81224e11	+	1.16498e11i	0.903170	+	0.374143i
$748$	−1.29262e11			−0.412918
$749$		−	3.94695e10i		−	0.125411i
$750$		0			0
$751$	−4.77978e11			−1.50262			−0.751308	−	0.659952i	$-0.770577\pi$
$751$	−4.77978e11			−1.50262			−0.751308	+	0.659952i	$0.770577\pi$
$752$			2.75552e11i			0.861652i
$753$	−9.26390e10	+	6.18970e10i	−0.288147	+	0.192526i
$754$	−2.66231e11			−0.823709
$755$		0			0
$756$	−2.26216e11	+	4.49847e10i	−0.692526	+	0.137714i
$757$	8.95066e10			0.272566			0.136283	−	0.990670i	$-0.456484\pi$
$757$	8.95066e10			0.272566			0.136283	+	0.990670i	$0.456484\pi$
$758$		−	4.45390e11i		−	1.34916i
$759$	1.47337e11	+	2.20514e11i	0.443962	+	0.664462i
$760$		0			0
$761$		−	5.71473e11i		−	1.70395i	−0.523581	−	0.851976i	$-0.675404\pi$
$761$		−	5.71473e11i		−	1.70395i	0.523581	−	0.851976i	$-0.324596\pi$
$762$	−4.17303e11	+	2.78822e11i	−1.23775	+	0.827004i
$763$	−1.92456e11			−0.567851
$764$		−	4.01231e10i		−	0.117766i
$765$		0			0
$766$	3.39292e11			0.985504
$767$			3.52884e11i			1.01965i
$768$	−2.04781e11	−	3.06488e11i	−0.588634	−	0.880986i
$769$	2.56194e11			0.732596			0.366298	−	0.930498i	$-0.380625\pi$
$769$	2.56194e11			0.732596			0.366298	+	0.930498i	$0.380625\pi$
$770$		0			0
$771$	−5.34537e11	+	3.57152e11i	−1.51273	+	1.01073i
$772$	−3.93923e11			−1.10903
$773$		−	1.00523e11i		−	0.281543i	−0.990042	−	0.140772i	$-0.955042\pi$
$773$		−	1.00523e11i		−	0.281543i	0.990042	−	0.140772i	$-0.0449584\pi$
$774$	−4.79838e11	−	1.98775e11i	−1.33700	−	0.553859i
$775$		0			0
$776$		−	2.64498e10i		−	0.0729417i
$777$	1.05145e11	+	1.57366e11i	0.288472	+	0.431745i
$778$	−4.04080e11			−1.10293
$779$		−	3.55176e10i		−	0.0964482i
$780$		0			0
$781$	−1.18464e11			−0.318408
$782$			7.91007e11i			2.11521i
$783$	−4.77727e10	−	2.40236e11i	−0.127096	−	0.639132i
$784$	1.82459e11			0.482950
$785$		0			0
$786$	−2.92081e11	−	4.37146e11i	−0.765266	−	1.14535i
$787$	2.15788e11			0.562508			0.281254	−	0.959633i	$-0.409250\pi$
$787$	2.15788e11			0.562508			0.281254	+	0.959633i	$0.409250\pi$
$788$			1.33367e11i			0.345894i
$789$	2.17240e10	−	1.45150e10i	0.0560572	−	0.0374548i
$790$		0			0
$791$			5.03559e11i			1.28631i
$792$	−3.13855e9	+	7.57637e9i	−0.00797679	+	0.0192558i
$793$	−1.93902e10			−0.0490331
$794$			5.29083e11i			1.33120i
$795$		0			0
$796$	−1.60601e11			−0.400034
$797$		−	6.69766e11i		−	1.65993i	−0.557815	−	0.829966i	$-0.688360\pi$
$797$		−	6.69766e11i		−	1.65993i	0.557815	−	0.829966i	$-0.311640\pi$
$798$	5.01099e10	−	3.34811e10i	0.123570	−	0.0825636i
$799$	3.05641e11			0.749938
$800$		0			0
$801$	4.40156e11	+	1.82337e11i	1.06924	+	0.442940i
$802$	3.09118e11			0.747183
$803$			1.92588e11i			0.463199i
$804$	−2.53208e10	−	3.78967e10i	−0.0605973	−	0.0906938i
$805$		0			0
$806$			2.03027e11i			0.481075i
$807$	−2.33915e11	+	1.56291e11i	−0.551524	+	0.368502i
$808$	−1.85966e10			−0.0436303
$809$		−	9.93417e10i		−	0.231920i	−0.993254	−	0.115960i	$-0.963006\pi$
$809$		−	9.93417e10i		−	0.231920i	0.993254	−	0.115960i	$-0.0369944\pi$
$810$		0			0
$811$	−2.71549e11			−0.627718			−0.313859	−	0.949470i	$-0.601622\pi$
$811$	−2.71549e11			−0.627718			−0.313859	+	0.949470i	$0.601622\pi$
$812$			2.00029e11i			0.460119i
$813$	6.48974e10	+	9.71295e10i	0.148547	+	0.222325i
$814$	−2.08607e11			−0.475150
$815$		0			0
$816$	−3.40573e11	+	2.27555e11i	−0.768156	+	0.513246i
$817$	6.67782e10			0.149881
$818$			8.04785e11i			1.79749i
$819$	1.13064e11	−	2.72934e11i	0.251298	−	0.606626i
$820$		0			0
$821$		−	5.63227e11i		−	1.23968i	−0.784727	−	0.619842i	$-0.787197\pi$
$821$		−	5.63227e11i		−	1.23968i	0.784727	−	0.619842i	$-0.212803\pi$
$822$	−2.09689e11	−	3.13834e11i	−0.459292	−	0.687406i
$823$	4.42272e11			0.964028			0.482014	−	0.876164i	$-0.339905\pi$
$823$	4.42272e11			0.964028			0.482014	+	0.876164i	$0.339905\pi$
$824$			2.98249e10i			0.0646950i
$825$		0			0
$826$	5.38823e11			1.15751
$827$			4.54543e11i			0.971745i	0.874030	+	0.485873i	$0.161498\pi$
$827$			4.54543e11i			0.971745i	−0.874030	+	0.485873i	$0.838502\pi$
$828$	−7.07220e11	−	2.92969e11i	−1.50464	−	0.623305i
$829$	−1.32535e11			−0.280615			−0.140308	−	0.990108i	$-0.544809\pi$
$829$	−1.32535e11			−0.280615			−0.140308	+	0.990108i	$0.544809\pi$
$830$		0			0
$831$	1.52121e11	+	2.27674e11i	0.318996	+	0.477429i
$832$	−4.04290e11			−0.843722
$833$		−	2.02383e11i		−	0.420334i
$834$	−2.15714e11	+	1.44130e11i	−0.445877	+	0.297914i
$835$		0			0
$836$			3.26861e10i			0.0669172i
$837$	−1.83203e11	+	3.64312e10i	−0.373276	+	0.0742286i
$838$	5.02871e11			1.01972
$839$		−	4.87205e10i		−	0.0983250i	−0.998791	−	0.0491625i	$-0.984345\pi$
$839$		−	4.87205e10i		−	0.0983250i	0.998791	−	0.0491625i	$-0.0156552\pi$
$840$		0			0
$841$	2.87820e11			0.575357
$842$		−	3.35705e11i		−	0.667897i
$843$	−2.70923e11	+	1.81018e11i	−0.536457	+	0.358435i
$844$	−1.44114e10			−0.0284011
$845$		0			0
$846$	−2.30056e11	+	5.55348e11i	−0.449109	+	1.08414i
$847$	2.90368e11			0.564176
$848$			4.45751e11i			0.862004i
$849$	−4.69138e11	−	7.02141e11i	−0.902962	−	1.35143i
$850$		0			0
$851$			6.28145e11i			1.19768i
$852$	2.84314e11	−	1.89965e11i	0.539561	−	0.360509i
$853$	7.24501e11			1.36849			0.684247	−	0.729250i	$-0.260131\pi$
$853$	7.24501e11			1.36849			0.684247	+	0.729250i	$0.260131\pi$
$854$			2.96071e10i			0.0556626i
$855$		0			0
$856$	4.05068e9			0.00754455
$857$		−	4.16662e11i		−	0.772433i	−0.922408	−	0.386216i	$-0.873782\pi$
$857$		−	4.16662e11i		−	0.772433i	0.922408	−	0.386216i	$-0.126218\pi$
$858$	1.80902e11	+	2.70750e11i	0.333807	+	0.499597i
$859$	7.54443e11			1.38565			0.692826	−	0.721105i	$-0.256366\pi$
$859$	7.54443e11			1.38565			0.692826	+	0.721105i	$0.256366\pi$
$860$		0			0
$861$	−2.21047e11	+	1.47693e11i	−0.402227	+	0.268750i
$862$	−1.43778e12			−2.60413
$863$		−	2.06295e10i		−	0.0371916i	−0.999827	−	0.0185958i	$-0.994080\pi$
$863$		−	2.06295e10i		−	0.0371916i	0.999827	−	0.0185958i	$-0.00591957\pi$
$864$	−1.52351e11	−	7.66133e11i	−0.273395	−	1.37483i
$865$		0			0
$866$			1.17403e11i			0.208741i
$867$	−6.15057e10	−	9.20533e10i	−0.108853	−	0.162916i
$868$	1.52541e11			0.268726
$869$			1.59936e11i			0.280457i
$870$		0			0
$871$	5.83785e10			0.101433
$872$		−	1.97515e10i		−	0.0341612i
$873$	3.69798e11	−	8.92681e11i	0.636659	−	1.53688i
$874$	2.00020e11			0.342789
$875$		0			0
$876$	−3.08828e11	−	4.62212e11i	−0.524446	−	0.784918i
$877$	−5.98158e11			−1.01115			−0.505577	−	0.862782i	$-0.668720\pi$
$877$	−5.98158e11			−1.01115			−0.505577	+	0.862782i	$0.668720\pi$
$878$			9.76126e11i			1.64259i
$879$	6.99945e11	−	4.67670e11i	1.17249	−	0.783401i
$880$		0			0
$881$		−	1.62306e11i		−	0.269420i	−0.990885	−	0.134710i	$-0.956990\pi$
$881$		−	1.62306e11i		−	0.269420i	0.990885	−	0.134710i	$-0.0430103\pi$
$882$	3.67729e11	+	1.52334e11i	0.607650	+	0.251722i
$883$	6.06459e11			0.997605			0.498803	−	0.866716i	$-0.333773\pi$
$883$	6.06459e11			0.997605			0.498803	+	0.866716i	$0.333773\pi$
$884$			4.77895e11i			0.782571i
$885$		0			0
$886$	8.50262e11			1.37981
$887$			4.65904e11i			0.752666i	0.926484	+	0.376333i	$0.122815\pi$
$887$			4.65904e11i			0.752666i	−0.926484	+	0.376333i	$0.877185\pi$
$888$	−1.61502e10	+	1.07908e10i	−0.0259733	+	0.0173541i
$889$	−4.82989e11			−0.773269
$890$		0			0
$891$	−2.11852e11	+	2.11822e11i	−0.336141	+	0.336094i
$892$	1.09170e12			1.72442
$893$		−	7.72867e10i		−	0.121534i
$894$	8.27631e11	+	1.23868e12i	1.29565	+	1.93915i
$895$		0			0
$896$		−	4.11757e10i		−	0.0638865i
$897$	8.15267e11	−	5.44723e11i	1.25930	−	0.841407i
$898$	6.63408e11			1.02018
$899$			1.61995e11i			0.248007i
$900$		0			0
$901$	4.94426e11			0.750243
$902$		−	2.93023e11i		−	0.442666i
$903$	−2.77684e11	−	4.15600e11i	−0.417638	−	0.625064i
$904$	−5.16794e10			−0.0773826
$905$		0			0
$906$	6.40877e11	−	4.28204e11i	0.951178	−	0.635533i
$907$	−7.47111e11			−1.10397			−0.551983	−	0.833855i	$-0.686129\pi$
$907$	−7.47111e11			−1.10397			−0.551983	+	0.833855i	$0.686129\pi$
$908$		−	8.76909e11i		−	1.29006i
$909$	−6.27635e11	−	2.60001e11i	−0.919288	−	0.380820i
$910$		0			0
$911$		−	9.01051e11i		−	1.30821i	−0.756406	−	0.654103i	$-0.773046\pi$
$911$		−	9.01051e11i		−	1.30821i	0.756406	−	0.654103i	$-0.226954\pi$
$912$	5.75412e10	+	8.61198e10i	0.0831763	+	0.124487i
$913$	−3.22887e11			−0.464694
$914$			4.53896e11i			0.650387i
$915$		0			0
$916$	4.62692e11			0.657219
$917$		−	5.05956e11i		−	0.715543i
$918$	−8.76375e11	+	1.74274e11i	−1.23401	+	0.245392i
$919$	−1.14699e12			−1.60804			−0.804021	−	0.594601i	$-0.797310\pi$
$919$	−1.14699e12			−1.60804			−0.804021	+	0.594601i	$0.797310\pi$
$920$		0			0
$921$	−1.34689e11	−	2.01584e11i	−0.187195	−	0.280168i
$922$	1.57560e12			2.18032
$923$			4.37976e11i			0.603453i
$924$	2.03424e11	−	1.35919e11i	0.279071	−	0.186462i
$925$		0			0
$926$		−	9.33937e10i		−	0.127020i
$927$	−4.16985e11	+	1.00659e12i	−0.564679	+	1.36312i
$928$	−6.77447e11			−0.913447
$929$			4.36054e11i			0.585433i	0.956199	+	0.292716i	$0.0945592\pi$
$929$			4.36054e11i			0.585433i	−0.956199	+	0.292716i	$0.905441\pi$
$930$		0			0
$931$	−5.11762e10			−0.0681191
$932$			6.74887e11i			0.894474i
$933$	−4.34293e11	+	2.90174e11i	−0.573135	+	0.382942i
$934$	−6.46869e11			−0.850019
$935$		0			0
$936$	2.80107e10	+	1.16036e10i	0.0364939	+	0.0151178i
$937$	−1.98881e11			−0.258009			−0.129004	−	0.991644i	$-0.541178\pi$
$937$	−1.98881e11			−0.258009			−0.129004	+	0.991644i	$0.541178\pi$
$938$		−	8.91388e10i		−	0.115148i
$939$	1.47482e9	+	2.20731e9i	0.00189704	+	0.00283923i
$940$		0			0
$941$		−	1.28069e12i		−	1.63338i	−0.577079	−	0.816689i	$-0.695807\pi$
$941$		−	1.28069e12i		−	1.63338i	0.577079	−	0.816689i	$-0.304193\pi$
$942$	1.14889e12	−	7.67637e11i	1.45907	−	0.974883i
$943$	−8.82335e11			−1.11580
$944$			9.26030e11i			1.16610i
$945$		0			0
$946$	5.50926e11			0.687905
$947$		−	1.35658e12i		−	1.68673i	−0.537337	−	0.843367i	$-0.680570\pi$
$947$		−	1.35658e12i		−	1.68673i	0.537337	−	0.843367i	$-0.319430\pi$
$948$	−2.56468e11	−	3.83846e11i	−0.317541	−	0.475251i
$949$	7.12020e11			0.877865
$950$		0			0
$951$	−7.93358e11	+	5.30084e11i	−0.969945	+	0.648072i
$952$	−2.35388e10			−0.0286574
$953$		−	5.01575e11i		−	0.608085i	−0.952658	−	0.304042i	$-0.901664\pi$
$953$		−	5.01575e11i		−	0.608085i	0.952658	−	0.304042i	$-0.0983365\pi$
$954$	−3.72153e11	+	8.98368e11i	−0.449292	+	1.08458i
$955$		0			0
$956$		−	5.64103e11i		−	0.675347i
$957$	1.44342e11	+	2.16032e11i	0.172086	+	0.257555i
$958$	1.00385e12			1.19180
$959$		−	3.63234e11i		−	0.429450i
$960$		0			0
$961$	−7.29354e11			−0.855155
$962$			7.71244e11i			0.900516i
$963$	1.36711e11	+	5.66330e10i	0.158963	+	0.0658514i
$964$	4.33175e11			0.501597
$965$		0			0
$966$	−8.31744e11	−	1.24484e12i	−0.955171	−	1.42957i
$967$	−1.10988e11			−0.126932			−0.0634661	−	0.997984i	$-0.520215\pi$
$967$	−1.10988e11			−0.126932			−0.0634661	+	0.997984i	$0.520215\pi$
$968$			2.98000e10i			0.0339402i
$969$	9.55239e10	−	6.38246e10i	0.108347	−	0.0723924i
$970$		0			0
$971$			2.95380e11i			0.332280i	0.986102	+	0.166140i	$0.0531304\pi$
$971$			2.95380e11i			0.332280i	−0.986102	+	0.166140i	$0.946870\pi$
$972$	1.68774e11	−	8.48092e11i	0.189078	−	0.950119i
$973$	−2.49669e11			−0.278557
$974$		−	1.28601e12i		−	1.42892i
$975$		0			0
$976$	−5.08832e10			−0.0560758
$977$		−	6.09091e11i		−	0.668503i	−0.942484	−	0.334252i	$-0.891516\pi$
$977$		−	6.09091e11i		−	0.668503i	0.942484	−	0.334252i	$-0.108484\pi$
$978$	−1.01026e12	+	6.75007e11i	−1.10427	+	0.737824i
$979$	−5.05365e11			−0.550141
$980$		0			0
$981$	2.76147e11	−	6.66612e11i	0.298171	−	0.719776i
$982$	−1.62821e12			−1.75091
$983$		−	1.84012e12i		−	1.97076i	−0.170384	−	0.985378i	$-0.554501\pi$
$983$		−	1.84012e12i		−	1.97076i	0.170384	−	0.985378i	$-0.445499\pi$
$984$	−1.51575e10	−	2.26857e10i	−0.0161677	−	0.0241975i
$985$		0			0
$986$			7.74927e11i			0.819885i
$987$	−4.81000e11	+	3.21382e11i	−0.506847	+	0.338651i
$988$	1.20844e11			0.126823
$989$		−	1.65892e12i		−	1.73396i
$990$		0			0
$991$	−5.72336e11			−0.593412			−0.296706	−	0.954969i	$-0.595888\pi$
$991$	−5.72336e11			−0.593412			−0.296706	+	0.954969i	$0.595888\pi$
$992$			5.16617e11i			0.533485i
$993$	5.40449e11	+	8.08870e11i	0.555850	+	0.831920i
$994$	6.68750e11			0.685044
$995$		0			0
$996$	7.74928e11	−	5.17770e11i	0.787451	−	0.526138i
$997$	1.44614e12			1.46363			0.731813	−	0.681505i	$-0.238674\pi$
$997$	1.44614e12			1.46363			0.731813	+	0.681505i	$0.238674\pi$
$998$		−	5.93504e11i		−	0.598276i
$999$	−6.95937e11	+	1.38392e11i	−0.698728	+	0.138947i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.9.c.c.26.2		2
3.2	odd	2	inner	75.9.c.c.26.1		2
5.2	odd	4		75.9.d.b.74.2		4
5.3	odd	4		75.9.d.b.74.3		4
5.4	even	2		3.9.b.a.2.1	✓	2
15.2	even	4		75.9.d.b.74.4		4
15.8	even	4		75.9.d.b.74.1		4
15.14	odd	2		3.9.b.a.2.2	yes	2
20.19	odd	2		48.9.e.b.17.1		2
40.19	odd	2		192.9.e.f.65.2		2
40.29	even	2		192.9.e.e.65.1		2
45.4	even	6		81.9.d.d.26.2		4
45.14	odd	6		81.9.d.d.26.1		4
45.29	odd	6		81.9.d.d.53.2		4
45.34	even	6		81.9.d.d.53.1		4
60.59	even	2		48.9.e.b.17.2		2
120.29	odd	2		192.9.e.e.65.2		2
120.59	even	2		192.9.e.f.65.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.9.b.a.2.1	✓	2	5.4	even	2
3.9.b.a.2.2	yes	2	15.14	odd	2
48.9.e.b.17.1		2	20.19	odd	2
48.9.e.b.17.2		2	60.59	even	2
75.9.c.c.26.1		2	3.2	odd	2	inner
75.9.c.c.26.2		2	1.1	even	1	trivial
75.9.d.b.74.1		4	15.8	even	4
75.9.d.b.74.2		4	5.2	odd	4
75.9.d.b.74.3		4	5.3	odd	4
75.9.d.b.74.4		4	15.2	even	4
81.9.d.d.26.1		4	45.14	odd	6
81.9.d.d.26.2		4	45.4	even	6
81.9.d.d.53.1		4	45.34	even	6
81.9.d.d.53.2		4	45.29	odd	6
192.9.e.e.65.1		2	40.29	even	2
192.9.e.e.65.2		2	120.29	odd	2
192.9.e.f.65.1		2	120.59	even	2
192.9.e.f.65.2		2	40.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	75.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+22.4499i q^{2} +(-45.0000 - 67.3498i) q^{3} -248.000 q^{4} +(1512.00 - 1010.25i) q^{6} +1750.00 q^{7} +179.600i q^{8} +(-2511.00 + 6061.48i) q^{9} +O(q^{10})\)	\(q+22.4499i q^{2} +(-45.0000 - 67.3498i) q^{3} -248.000 q^{4} +(1512.00 - 1010.25i) q^{6} +1750.00 q^{7} +179.600i q^{8} +(-2511.00 + 6061.48i) q^{9} -6959.48i q^{11} +(11160.0 + 16702.8i) q^{12} -25730.0 q^{13} +39287.4i q^{14} -67520.0 q^{16} +74893.0i q^{17} +(-136080. - 56371.8i) q^{18} +18938.0 q^{19} +(-78750.0 - 117862. i) q^{21} +156240. q^{22} -470461. i q^{23} +(12096.0 - 8081.98i) q^{24} -577637. i q^{26} +(521235. - 103651. i) q^{27} -434000. q^{28} -460897. i q^{29} -351478. q^{31} -1.46984e6i q^{32} +(-468720. + 313177. i) q^{33} -1.68134e6 q^{34} +(622728. - 1.50325e6i) q^{36} -1.33517e6 q^{37} +425157. i q^{38} +(1.15785e6 + 1.73291e6i) q^{39} -1.87547e6i q^{41} +(2.64600e6 - 1.76793e6i) q^{42} +3.52615e6 q^{43} +1.72595e6i q^{44} +1.05618e7 q^{46} -4.08104e6i q^{47} +(3.03840e6 + 4.54746e6i) q^{48} -2.70230e6 q^{49} +(5.04403e6 - 3.37019e6i) q^{51} +6.38104e6 q^{52} -6.60177e6i q^{53} +(2.32697e6 + 1.17017e7i) q^{54} +314299. i q^{56} +(-852210. - 1.27547e6i) q^{57} +1.03471e7 q^{58} -1.37149e7i q^{59} +753602. q^{61} -7.89066e6i q^{62} +(-4.39425e6 + 1.06076e7i) q^{63} +1.57128e7 q^{64} +(-7.03080e6 - 1.05227e7i) q^{66} -2.26889e6 q^{67} -1.85735e7i q^{68} +(-3.16855e7 + 2.11707e7i) q^{69} -1.70220e7i q^{71} +(-1.08864e6 - 450974. i) q^{72} -2.76728e7 q^{73} -2.99745e7i q^{74} -4.69662e6 q^{76} -1.21791e7i q^{77} +(-3.89038e7 + 2.59937e7i) q^{78} -2.29810e7 q^{79} +(-3.04365e7 - 3.04408e7i) q^{81} +4.21042e7 q^{82} -4.63952e7i q^{83} +(1.95300e7 + 2.92298e7i) q^{84} +7.91619e7i q^{86} +(-3.10414e7 + 2.07404e7i) q^{87} +1.24992e6 q^{88} -7.26152e7i q^{89} -4.50275e7 q^{91} +1.16674e8i q^{92} +(1.58165e7 + 2.36720e7i) q^{93} +9.16191e7 q^{94} +(-9.89937e7 + 6.61429e7i) q^{96} -1.47271e8 q^{97} -6.06665e7i q^{98} +(4.21848e7 + 1.74753e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 90 q^{3} - 496 q^{4} + 3024 q^{6} + 3500 q^{7} - 5022 q^{9}+O(q^{10}) \) 2 * q - 90 * q^3 - 496 * q^4 + 3024 * q^6 + 3500 * q^7 - 5022 * q^9	\( 2 q - 90 q^{3} - 496 q^{4} + 3024 q^{6} + 3500 q^{7} - 5022 q^{9} + 22320 q^{12} - 51460 q^{13} - 135040 q^{16} - 272160 q^{18} + 37876 q^{19} - 157500 q^{21} + 312480 q^{22} + 24192 q^{24} + 1042470 q^{27} - 868000 q^{28} - 702956 q^{31} - 937440 q^{33} - 3362688 q^{34} + 1245456 q^{36} - 2670340 q^{37} + 2315700 q^{39} + 5292000 q^{42} + 7052300 q^{43} + 21123648 q^{46} + 6076800 q^{48} - 5404602 q^{49} + 10088064 q^{51} + 12762080 q^{52} + 4653936 q^{54} - 1704420 q^{57} + 20694240 q^{58} + 1507204 q^{61} - 8788500 q^{63} + 31425536 q^{64} - 14061600 q^{66} - 4537780 q^{67} - 63370944 q^{69} - 2177280 q^{72} - 55345540 q^{73} - 9393248 q^{76} - 77807520 q^{78} - 45961964 q^{79} - 60872958 q^{81} + 84208320 q^{82} + 39060000 q^{84} - 62082720 q^{87} + 2499840 q^{88} - 90055000 q^{91} + 31633020 q^{93} + 183238272 q^{94} - 197987328 q^{96} - 294542020 q^{97} + 84369600 q^{99}+O(q^{100}) \) 2 * q - 90 * q^3 - 496 * q^4 + 3024 * q^6 + 3500 * q^7 - 5022 * q^9 + 22320 * q^12 - 51460 * q^13 - 135040 * q^16 - 272160 * q^18 + 37876 * q^19 - 157500 * q^21 + 312480 * q^22 + 24192 * q^24 + 1042470 * q^27 - 868000 * q^28 - 702956 * q^31 - 937440 * q^33 - 3362688 * q^34 + 1245456 * q^36 - 2670340 * q^37 + 2315700 * q^39 + 5292000 * q^42 + 7052300 * q^43 + 21123648 * q^46 + 6076800 * q^48 - 5404602 * q^49 + 10088064 * q^51 + 12762080 * q^52 + 4653936 * q^54 - 1704420 * q^57 + 20694240 * q^58 + 1507204 * q^61 - 8788500 * q^63 + 31425536 * q^64 - 14061600 * q^66 - 4537780 * q^67 - 63370944 * q^69 - 2177280 * q^72 - 55345540 * q^73 - 9393248 * q^76 - 77807520 * q^78 - 45961964 * q^79 - 60872958 * q^81 + 84208320 * q^82 + 39060000 * q^84 - 62082720 * q^87 + 2499840 * q^88 - 90055000 * q^91 + 31633020 * q^93 + 183238272 * q^94 - 197987328 * q^96 - 294542020 * q^97 + 84369600 * q^99

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