LMFDB - Embedded newform 48.9.e.b.17.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,9,Mod(17,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("48.17");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	48.e (of order $2$, degree $1$, not 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:	$19.5541732829$
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:	$\Z[a_1, \ldots, a_{5}]$
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			17.1
Root			$3.74166i$ of defining polynomial
Character	$\chi$	$=$	48.17
Dual form			48.9.e.b.17.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-45.0000 - 67.3498i) q^{3} +224.499i q^{5} +1750.00 q^{7} +(-2511.00 + 6061.48i) q^{9} +O(q^{10})$	\(q+(-45.0000 - 67.3498i) q^{3} +224.499i q^{5} +1750.00 q^{7} +(-2511.00 + 6061.48i) q^{9} +6959.48i q^{11} +25730.0 q^{13} +(15120.0 - 10102.5i) q^{15} -74893.0i q^{17} -18938.0 q^{19} +(-78750.0 - 117862. i) q^{21} -470461. i q^{23} +340225. q^{25} +(521235. - 103651. i) q^{27} -460897. i q^{29} +351478. q^{31} +(468720. - 313177. i) q^{33} +392874. i q^{35} +1.33517e6 q^{37} +(-1.15785e6 - 1.73291e6i) q^{39} -1.87547e6i q^{41} +3.52615e6 q^{43} +(-1.36080e6 - 563718. i) q^{45} -4.08104e6i q^{47} -2.70230e6 q^{49} +(-5.04403e6 + 3.37019e6i) q^{51} +6.60177e6i q^{53} -1.56240e6 q^{55} +(852210. + 1.27547e6i) q^{57} +1.37149e7i q^{59} +753602. q^{61} +(-4.39425e6 + 1.06076e7i) q^{63} +5.77637e6i q^{65} -2.26889e6 q^{67} +(-3.16855e7 + 2.11707e7i) q^{69} +1.70220e7i q^{71} +2.76728e7 q^{73} +(-1.53101e7 - 2.29141e7i) q^{75} +1.21791e7i q^{77} +2.29810e7 q^{79} +(-3.04365e7 - 3.04408e7i) q^{81} -4.63952e7i q^{83} +1.68134e7 q^{85} +(-3.10414e7 + 2.07404e7i) q^{87} -7.26152e7i q^{89} +4.50275e7 q^{91} +(-1.58165e7 - 2.36720e7i) q^{93} -4.25157e6i q^{95} +1.47271e8 q^{97} +(-4.21848e7 - 1.74753e7i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 90 q^{3} + 3500 q^{7} - 5022 q^{9}+O(q^{10}) $ 2 * q - 90 * q^3 + 3500 * q^7 - 5022 * q^9	$ 2 q - 90 q^{3} + 3500 q^{7} - 5022 q^{9} + 51460 q^{13} + 30240 q^{15} - 37876 q^{19} - 157500 q^{21} + 680450 q^{25} + 1042470 q^{27} + 702956 q^{31} + 937440 q^{33} + 2670340 q^{37} - 2315700 q^{39} + 7052300 q^{43} - 2721600 q^{45} - 5404602 q^{49} - 10088064 q^{51} - 3124800 q^{55} + 1704420 q^{57} + 1507204 q^{61} - 8788500 q^{63} - 4537780 q^{67} - 63370944 q^{69} + 55345540 q^{73} - 30620250 q^{75} + 45961964 q^{79} - 60872958 q^{81} + 33626880 q^{85} - 62082720 q^{87} + 90055000 q^{91} - 31633020 q^{93} + 294542020 q^{97} - 84369600 q^{99}+O(q^{100}) $ 2 * q - 90 * q^3 + 3500 * q^7 - 5022 * q^9 + 51460 * q^13 + 30240 * q^15 - 37876 * q^19 - 157500 * q^21 + 680450 * q^25 + 1042470 * q^27 + 702956 * q^31 + 937440 * q^33 + 2670340 * q^37 - 2315700 * q^39 + 7052300 * q^43 - 2721600 * q^45 - 5404602 * q^49 - 10088064 * q^51 - 3124800 * q^55 + 1704420 * q^57 + 1507204 * q^61 - 8788500 * q^63 - 4537780 * q^67 - 63370944 * q^69 + 55345540 * q^73 - 30620250 * q^75 + 45961964 * q^79 - 60872958 * q^81 + 33626880 * q^85 - 62082720 * q^87 + 90055000 * q^91 - 31633020 * q^93 + 294542020 * q^97 - 84369600 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$.

$n$	$17$	$31$	$37$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−45.0000	−	67.3498i	−0.555556	−	0.831479i
$3$	−45.0000	−	67.3498i	−0.555556	−	0.831479i
$4$		0			0
$5$			224.499i			0.359199i	0.983740	+	0.179600i	$0.0574802\pi$
$5$			224.499i			0.359199i	−0.983740	+	0.179600i	$0.942520\pi$
$6$		0			0
$7$	1750.00			0.728863			0.364431	−	0.931230i	$-0.381263\pi$
$7$	1750.00			0.728863			0.364431	+	0.931230i	$0.381263\pi$
$8$		0			0
$9$	−2511.00	+	6061.48i	−0.382716	+	0.923866i
$10$		0			0
$11$			6959.48i			0.475342i	0.971346	+	0.237671i	$0.0763840\pi$
$11$			6959.48i			0.475342i	−0.971346	+	0.237671i	$0.923616\pi$
$12$		0			0
$13$	25730.0			0.900879			0.450439	−	0.892807i	$-0.351267\pi$
$13$	25730.0			0.900879			0.450439	+	0.892807i	$0.351267\pi$
$14$		0			0
$15$	15120.0	−	10102.5i	0.298667	−	0.199555i
$16$		0			0
$17$		−	74893.0i		−	0.896697i	−0.893859	−	0.448348i	$-0.852012\pi$
$17$		−	74893.0i		−	0.896697i	0.893859	−	0.448348i	$-0.147988\pi$
$18$		0			0
$19$	−18938.0			−0.145318			−0.0726590	−	0.997357i	$-0.523148\pi$
$19$	−18938.0			−0.145318			−0.0726590	+	0.997357i	$0.523148\pi$
$20$		0			0
$21$	−78750.0	−	117862.i	−0.404924	−	0.606035i
$22$		0			0
$23$		−	470461.i		−	1.68117i	−0.541678	−	0.840586i	$-0.682211\pi$
$23$		−	470461.i		−	1.68117i	0.541678	−	0.840586i	$-0.317789\pi$
$24$		0			0
$25$	340225.			0.870976
$26$		0			0
$27$	521235.	−	103651.i	0.980796	−	0.195038i
$28$		0			0
$29$		−	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	$-0.439056\pi$
$31$	351478.			0.380585			0.190292	+	0.981727i	$0.439056\pi$
$32$		0			0
$33$	468720.	−	313177.i	0.395237	−	0.264079i
$34$		0			0
$35$			392874.i			0.261807i
$36$		0			0
$37$	1.33517e6			0.712409			0.356205	−	0.934408i	$-0.384071\pi$
$37$	1.33517e6			0.712409			0.356205	+	0.934408i	$0.384071\pi$
$38$		0			0
$39$	−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	$-0.107673\pi$
$42$		0			0
$43$	3.52615e6			1.03140			0.515700	−	0.856769i	$-0.327532\pi$
$43$	3.52615e6			1.03140			0.515700	+	0.856769i	$0.327532\pi$
$44$		0			0
$45$	−1.36080e6	−	563718.i	−0.331852	−	0.137471i
$46$		0			0
$47$		−	4.08104e6i		−	0.836333i	−0.908370	−	0.418167i	$-0.862673\pi$
$47$		−	4.08104e6i		−	0.836333i	0.908370	−	0.418167i	$-0.137327\pi$
$48$		0			0
$49$	−2.70230e6			−0.468759
$50$		0			0
$51$	−5.04403e6	+	3.37019e6i	−0.745585	+	0.498165i
$52$		0			0
$53$			6.60177e6i			0.836675i	0.908292	+	0.418337i	$0.137387\pi$
$53$			6.60177e6i			0.836675i	−0.908292	+	0.418337i	$0.862613\pi$
$54$		0			0
$55$	−1.56240e6			−0.170742
$56$		0			0
$57$	852210.	+	1.27547e6i	0.0807323	+	0.120829i
$58$		0			0
$59$			1.37149e7i			1.13184i	0.824461	+	0.565919i	$0.191479\pi$
$59$			1.37149e7i			1.13184i	−0.824461	+	0.565919i	$0.808521\pi$
$60$		0			0
$61$	753602.			0.0544280			0.0272140	−	0.999630i	$-0.491336\pi$
$61$	753602.			0.0544280			0.0272140	+	0.999630i	$0.491336\pi$
$62$		0			0
$63$	−4.39425e6	+	1.06076e7i	−0.278948	+	0.673372i
$64$		0			0
$65$			5.77637e6i			0.323595i
$66$		0			0
$67$	−2.26889e6			−0.112594			−0.0562969	−	0.998414i	$-0.517929\pi$
$67$	−2.26889e6			−0.112594			−0.0562969	+	0.998414i	$0.517929\pi$
$68$		0			0
$69$	−3.16855e7	+	2.11707e7i	−1.39786	+	0.933985i
$70$		0			0
$71$			1.70220e7i			0.669849i	0.942245	+	0.334925i	$0.108711\pi$
$71$			1.70220e7i			0.669849i	−0.942245	+	0.334925i	$0.891289\pi$
$72$		0			0
$73$	2.76728e7			0.974454			0.487227	−	0.873275i	$-0.338008\pi$
$73$	2.76728e7			0.974454			0.487227	+	0.873275i	$0.338008\pi$
$74$		0			0
$75$	−1.53101e7	−	2.29141e7i	−0.483876	−	0.724199i
$76$		0			0
$77$			1.21791e7i			0.346459i
$78$		0			0
$79$	2.29810e7			0.590011			0.295006	−	0.955496i	$-0.404678\pi$
$79$	2.29810e7			0.590011			0.295006	+	0.955496i	$0.404678\pi$
$80$		0			0
$81$	−3.04365e7	−	3.04408e7i	−0.707057	−	0.707157i
$82$		0			0
$83$		−	4.63952e7i		−	0.977599i	−0.872396	−	0.488799i	$-0.837435\pi$
$83$		−	4.63952e7i		−	0.977599i	0.872396	−	0.488799i	$-0.162565\pi$
$84$		0			0
$85$	1.68134e7			0.322093
$86$		0			0
$87$	−3.10414e7	+	2.07404e7i	−0.541831	+	0.362026i
$88$		0			0
$89$		−	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
$92$		0			0
$93$	−1.58165e7	−	2.36720e7i	−0.211436	−	0.316448i
$94$		0			0
$95$		−	4.25157e6i		−	0.0521981i
$96$		0			0
$97$	1.47271e8			1.66353			0.831764	−	0.555129i	$-0.187331\pi$
$97$	1.47271e8			1.66353			0.831764	+	0.555129i	$0.187331\pi$
$98$		0			0
$99$	−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	$0.834243\pi$
$102$		0			0
$103$	1.66064e8			1.47545			0.737726	−	0.675100i	$-0.235899\pi$
$103$	1.66064e8			1.47545			0.737726	+	0.675100i	$0.235899\pi$
$104$		0			0
$105$	2.64600e7	−	1.76793e7i	0.217687	−	0.145448i
$106$		0			0
$107$		−	2.25540e7i		−	0.172063i	−0.996292	−	0.0860316i	$-0.972581\pi$
$107$		−	2.25540e7i		−	0.172063i	0.996292	−	0.0860316i	$-0.0274186\pi$
$108$		0			0
$109$	−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
$112$		0			0
$113$		−	2.87748e8i		−	1.76481i	−0.470490	−	0.882405i	$-0.655923\pi$
$113$		−	2.87748e8i		−	1.76481i	0.470490	−	0.882405i	$-0.344077\pi$
$114$		0			0
$115$	1.05618e8			0.603876
$116$		0			0
$117$	−6.46080e7	+	1.55962e8i	−0.344781	+	0.832291i
$118$		0			0
$119$		−	1.31063e8i		−	0.653569i
$120$		0			0
$121$	1.65924e8			0.774050
$122$		0			0
$123$	−1.26312e8	+	8.43961e7i	−0.551856	+	0.368724i
$124$		0			0
$125$			1.64075e8i			0.672053i
$126$		0			0
$127$	−2.75994e8			−1.06092			−0.530462	−	0.847708i	$-0.677982\pi$
$127$	−2.75994e8			−1.06092			−0.530462	+	0.847708i	$0.677982\pi$
$128$		0			0
$129$	−1.58677e8	−	2.37486e8i	−0.573000	−	0.857588i
$130$		0			0
$131$			2.89118e8i			0.981725i	0.871237	+	0.490862i	$0.163318\pi$
$131$			2.89118e8i			0.981725i	−0.871237	+	0.490862i	$0.836682\pi$
$132$		0			0
$133$	−3.31415e7			−0.105917
$134$		0			0
$135$	2.32697e7	+	1.17017e8i	0.0700576	+	0.352301i
$136$		0			0
$137$			2.07562e8i			0.589205i	0.955620	+	0.294602i	$0.0951872\pi$
$137$			2.07562e8i			0.589205i	−0.955620	+	0.294602i	$0.904813\pi$
$138$		0			0
$139$	1.42668e8			0.382180			0.191090	−	0.981573i	$-0.438798\pi$
$139$	1.42668e8			0.382180			0.191090	+	0.981573i	$0.438798\pi$
$140$		0			0
$141$	−2.74857e8	+	1.83647e8i	−0.695394	+	0.464630i
$142$		0			0
$143$			1.79067e8i			0.428226i
$144$		0			0
$145$	1.03471e8			0.234071
$146$		0			0
$147$	1.21604e8	+	1.82000e8i	0.260422	+	0.389763i
$148$		0			0
$149$			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	$-0.633651\pi$
$151$	−4.23861e8			−0.815296			−0.407648	+	0.913139i	$0.633651\pi$
$152$		0			0
$153$	4.53963e8	+	1.88056e8i	0.828428	+	0.343180i
$154$		0			0
$155$			7.89066e7i			0.136706i
$156$		0			0
$157$	−7.59851e8			−1.25063			−0.625316	−	0.780371i	$-0.715030\pi$
$157$	−7.59851e8			−1.25063			−0.625316	+	0.780371i	$0.715030\pi$
$158$		0			0
$159$	4.44628e8	−	2.97079e8i	0.695678	−	0.464819i
$160$		0			0
$161$		−	8.23307e8i		−	1.22534i
$162$		0			0
$163$	−6.68160e8			−0.946520			−0.473260	−	0.880923i	$-0.656923\pi$
$163$	−6.68160e8			−0.946520			−0.473260	+	0.880923i	$0.656923\pi$
$164$		0			0
$165$	7.03080e7	+	1.05227e8i	0.0948569	+	0.141969i
$166$		0			0
$167$			1.96306e8i			0.252387i	0.992006	+	0.126194i	$0.0402761\pi$
$167$			1.96306e8i			0.252387i	−0.992006	+	0.126194i	$0.959724\pi$
$168$		0			0
$169$	−1.53698e8			−0.188417
$170$		0			0
$171$	4.75533e7	−	1.14792e8i	0.0556156	−	0.134254i
$172$		0			0
$173$		−	1.02319e9i		−	1.14228i	−0.820852	−	0.571141i	$-0.806501\pi$
$173$		−	1.02319e9i		−	1.14228i	0.820852	−	0.571141i	$-0.193499\pi$
$174$		0			0
$175$	5.95394e8			0.634822
$176$		0			0
$177$	9.23696e8	−	6.17170e8i	0.941100	−	0.628799i
$178$		0			0
$179$		−	1.28895e9i		−	1.25552i	−0.778408	−	0.627759i	$-0.783972\pi$
$179$		−	1.28895e9i		−	1.25552i	0.778408	−	0.627759i	$-0.216028\pi$
$180$		0			0
$181$	4.71707e8			0.439499			0.219749	−	0.975556i	$-0.429476\pi$
$181$	4.71707e8			0.439499			0.219749	+	0.975556i	$0.429476\pi$
$182$		0			0
$183$	−3.39121e7	−	5.07550e7i	−0.0302378	−	0.0452558i
$184$		0			0
$185$			2.99745e8i			0.255897i
$186$		0			0
$187$	5.21217e8			0.426238
$188$		0			0
$189$	9.12161e8	−	1.81390e8i	0.714866	−	0.142156i
$190$		0			0
$191$		−	1.61787e8i		−	0.121565i	−0.998151	−	0.0607827i	$-0.980640\pi$
$191$		−	1.61787e8i		−	0.121565i	0.998151	−	0.0607827i	$-0.0193597\pi$
$192$		0			0
$193$	−1.58840e9			−1.14480			−0.572401	−	0.819974i	$-0.693988\pi$
$193$	−1.58840e9			−1.14480			−0.572401	+	0.819974i	$0.693988\pi$
$194$		0			0
$195$	3.89038e8	−	2.59937e8i	0.269062	−	0.179775i
$196$		0			0
$197$			5.37769e8i			0.357052i	0.983935	+	0.178526i	$0.0571328\pi$
$197$			5.37769e8i			0.357052i	−0.983935	+	0.178526i	$0.942867\pi$
$198$		0			0
$199$	−6.47586e8			−0.412938			−0.206469	−	0.978453i	$-0.566197\pi$
$199$	−6.47586e8			−0.412938			−0.206469	+	0.978453i	$0.566197\pi$
$200$		0			0
$201$	1.02100e8	+	1.52809e8i	0.0625521	+	0.0936194i
$202$		0			0
$203$		−	8.06570e8i		−	0.474961i
$204$		0			0
$205$	4.21042e8			0.238402
$206$		0			0
$207$	2.85169e9	+	1.18133e9i	1.55318	+	0.643412i
$208$		0			0
$209$		−	1.31799e8i		−	0.0690758i
$210$		0			0
$211$	−5.81104e7			−0.0293173			−0.0146586	−	0.999893i	$-0.504666\pi$
$211$	−5.81104e7			−0.0293173			−0.0146586	+	0.999893i	$0.504666\pi$
$212$		0			0
$213$	1.14643e9	−	7.65990e8i	0.556966	−	0.372139i
$214$		0			0
$215$			7.91619e8i			0.370478i
$216$		0			0
$217$	6.15086e8			0.277394
$218$		0			0
$219$	−1.24527e9	−	1.86376e9i	−0.541363	−	0.810238i
$220$		0			0
$221$		−	1.92700e9i		−	0.807815i
$222$		0			0
$223$	−4.40200e9			−1.78004			−0.890021	−	0.455920i	$-0.849310\pi$
$223$	−4.40200e9			−1.78004			−0.890021	+	0.455920i	$0.849310\pi$
$224$		0			0
$225$	−8.54305e8	+	2.06227e9i	−0.333336	+	0.804665i
$226$		0			0
$227$			3.53592e9i			1.33168i	0.746095	+	0.665839i	$0.231926\pi$
$227$			3.53592e9i			1.33168i	−0.746095	+	0.665839i	$0.768074\pi$
$228$		0			0
$229$	−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
$232$		0			0
$233$			2.72132e9i			0.923328i	0.887055	+	0.461664i	$0.152747\pi$
$233$			2.72132e9i			0.923328i	−0.887055	+	0.461664i	$0.847253\pi$
$234$		0			0
$235$	9.16191e8			0.300410
$236$		0			0
$237$	−1.03414e9	−	1.54777e9i	−0.327784	−	0.490582i
$238$		0			0
$239$		−	2.27461e9i		−	0.697132i	−0.937284	−	0.348566i	$-0.886669\pi$
$239$		−	2.27461e9i		−	0.697132i	0.937284	−	0.348566i	$-0.113331\pi$
$240$		0			0
$241$	−1.74667e9			−0.517778			−0.258889	−	0.965907i	$-0.583356\pi$
$241$	−1.74667e9			−0.517778			−0.258889	+	0.965907i	$0.583356\pi$
$242$		0			0
$243$	−6.80540e8	+	3.41973e9i	−0.195177	+	0.980768i
$244$		0			0
$245$		−	6.06665e8i		−	0.168378i
$246$		0			0
$247$	−4.87275e8			−0.130914
$248$		0			0
$249$	−3.12471e9	+	2.08778e9i	−0.812853	+	0.543110i
$250$		0			0
$251$			1.37549e9i			0.346547i	0.984874	+	0.173274i	$0.0554345\pi$
$251$			1.37549e9i			0.346547i	−0.984874	+	0.173274i	$0.944566\pi$
$252$		0			0
$253$	3.27417e9			0.799132
$254$		0			0
$255$	−7.56605e8	−	1.13238e9i	−0.178940	−	0.267813i
$256$		0			0
$257$			7.93672e9i			1.81932i	0.415356	+	0.909659i	$0.363657\pi$
$257$			7.93672e9i			1.81932i	−0.415356	+	0.909659i	$0.636343\pi$
$258$		0			0
$259$	2.33655e9			0.519249
$260$		0			0
$261$	2.79372e9	+	1.15731e9i	0.602034	+	0.249396i
$262$		0			0
$263$			3.22555e8i			0.0674187i	0.999432	+	0.0337093i	$0.0107320\pi$
$263$			3.22555e8i			0.0674187i	−0.999432	+	0.0337093i	$0.989268\pi$
$264$		0			0
$265$	−1.48209e9			−0.300533
$266$		0			0
$267$	−4.89062e9	+	3.26769e9i	−0.962319	+	0.642977i
$268$		0			0
$269$		−	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	$-0.457317\pi$
$271$	1.44216e9			0.267385			0.133693	+	0.991023i	$0.457317\pi$
$272$		0			0
$273$	−2.02624e9	−	3.03259e9i	−0.364787	−	0.545964i
$274$		0			0
$275$			2.36779e9i			0.414012i
$276$		0			0
$277$	3.38046e9			0.574192			0.287096	−	0.957902i	$-0.407310\pi$
$277$	3.38046e9			0.574192			0.287096	+	0.957902i	$0.407310\pi$
$278$		0			0
$279$	−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	$-0.104554\pi$
$282$		0			0
$283$	1.04253e10			1.62533			0.812666	−	0.582730i	$-0.198016\pi$
$283$	1.04253e10			1.62533			0.812666	+	0.582730i	$0.198016\pi$
$284$		0			0
$285$	−2.86343e8	+	1.91321e8i	−0.0434017	+	0.0289990i
$286$		0			0
$287$		−	3.28207e9i		−	0.483749i
$288$		0			0
$289$	1.36679e9			0.195935
$290$		0			0
$291$	−6.62720e9	−	9.91868e9i	−0.924183	−	1.38319i
$292$		0			0
$293$		−	1.03927e10i		−	1.41012i	−0.709146	−	0.705061i	$-0.750919\pi$
$293$		−	1.03927e10i		−	1.41012i	0.709146	−	0.705061i	$-0.249081\pi$
$294$		0			0
$295$	−3.07899e9			−0.406555
$296$		0			0
$297$	7.21360e8	+	3.62753e9i	0.0927099	+	0.466213i
$298$		0			0
$299$		−	1.21050e10i		−	1.51453i
$300$		0			0
$301$	6.17076e9			0.751749
$302$		0			0
$303$	6.97372e9	−	4.65951e9i	0.827359	−	0.552803i
$304$		0			0
$305$			1.69183e8i			0.0195505i
$306$		0			0
$307$	2.99309e9			0.336951			0.168476	−	0.985706i	$-0.446116\pi$
$307$	2.99309e9			0.336951			0.168476	+	0.985706i	$0.446116\pi$
$308$		0			0
$309$	−7.47286e9	−	1.11843e10i	−0.819696	−	1.22681i
$310$		0			0
$311$			6.44832e9i			0.689295i	0.938732	+	0.344647i	$0.112002\pi$
$311$			6.44832e9i			0.689295i	−0.938732	+	0.344647i	$0.887998\pi$
$312$		0			0
$313$	3.27737e7			0.00341467			0.00170733	−	0.999999i	$-0.499457\pi$
$313$	3.27737e7			0.00341467			0.00170733	+	0.999999i	$0.499457\pi$
$314$		0			0
$315$	−2.38140e9	−	9.86507e8i	−0.241875	−	0.100198i
$316$		0			0
$317$			1.17797e10i			1.16653i	0.812282	+	0.583264i	$0.198225\pi$
$317$			1.17797e10i			1.16653i	−0.812282	+	0.583264i	$0.801775\pi$
$318$		0			0
$319$	3.20761e9			0.309755
$320$		0			0
$321$	−1.51901e9	+	1.01493e9i	−0.143067	+	0.0955907i
$322$		0			0
$323$			1.41832e9i			0.130306i
$324$		0			0
$325$	8.75399e9			0.784644
$326$		0			0
$327$	4.94888e9	+	7.40680e9i	0.432829	+	0.647798i
$328$		0			0
$329$		−	7.14182e9i		−	0.609573i
$330$		0			0
$331$	1.20100e10			1.00053			0.500265	−	0.865872i	$-0.333236\pi$
$331$	1.20100e10			1.00053			0.500265	+	0.865872i	$0.333236\pi$
$332$		0			0
$333$	−3.35261e9	+	8.09311e9i	−0.272651	+	0.658171i
$334$		0			0
$335$		−	5.09365e8i		−	0.0404436i
$336$		0			0
$337$	1.59214e10			1.23441			0.617207	−	0.786801i	$-0.288264\pi$
$337$	1.59214e10			1.23441			0.617207	+	0.786801i	$0.288264\pi$
$338$		0			0
$339$	−1.93798e10	+	1.29486e10i	−1.46740	+	0.980451i
$340$		0			0
$341$			2.44611e9i			0.180908i
$342$		0			0
$343$	−1.48174e10			−1.07052
$344$		0			0
$345$	−4.75282e9	−	7.11337e9i	−0.335487	−	0.502110i
$346$		0			0
$347$		−	4.94792e9i		−	0.341275i	−0.985334	−	0.170638i	$-0.945417\pi$
$347$		−	4.94792e9i		−	0.341275i	0.985334	−	0.170638i	$-0.0545828\pi$
$348$		0			0
$349$	−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
$352$		0			0
$353$			6.93875e9i			0.446871i	0.974719	+	0.223436i	$0.0717272\pi$
$353$			6.93875e9i			0.446871i	−0.974719	+	0.223436i	$0.928273\pi$
$354$		0			0
$355$	−3.82143e9			−0.240609
$356$		0			0
$357$	−8.82706e9	+	5.89782e9i	−0.543429	+	0.363094i
$358$		0			0
$359$			1.60096e10i			0.963838i	0.876216	+	0.481919i	$0.160060\pi$
$359$			1.60096e10i			0.963838i	−0.876216	+	0.481919i	$0.839940\pi$
$360$		0			0
$361$	−1.66249e10			−0.978883
$362$		0			0
$363$	−7.46660e9	−	1.11750e10i	−0.430028	−	0.643607i
$364$		0			0
$365$			6.21252e9i			0.350023i
$366$		0			0
$367$	−1.36364e10			−0.751686			−0.375843	−	0.926683i	$-0.622647\pi$
$367$	−1.36364e10			−0.751686			−0.375843	+	0.926683i	$0.622647\pi$
$368$		0			0
$369$	1.13681e10	+	4.70930e9i	0.613173	+	0.254010i
$370$		0			0
$371$			1.15531e10i			0.609821i
$372$		0			0
$373$	−2.44062e10			−1.26085			−0.630427	−	0.776248i	$-0.717120\pi$
$373$	−2.44062e10			−1.26085			−0.630427	+	0.776248i	$0.717120\pi$
$374$		0			0
$375$	1.10505e10	−	7.38339e9i	0.558798	−	0.373363i
$376$		0			0
$377$		−	1.18589e10i		−	0.587055i
$378$		0			0
$379$	1.98392e10			0.961542			0.480771	−	0.876846i	$-0.340357\pi$
$379$	1.98392e10			0.961542			0.480771	+	0.876846i	$0.340357\pi$
$380$		0			0
$381$	1.24197e10	+	1.85881e10i	0.589403	+	0.882137i
$382$		0			0
$383$		−	1.51133e10i		−	0.702366i	−0.936307	−	0.351183i	$-0.885780\pi$
$383$		−	1.51133e10i		−	0.702366i	0.936307	−	0.351183i	$-0.114220\pi$
$384$		0			0
$385$	−2.73420e9			−0.124448
$386$		0			0
$387$	−8.85416e9	+	2.13737e10i	−0.394733	+	0.952875i
$388$		0			0
$389$			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
$392$		0			0
$393$	1.94720e10	−	1.30103e10i	0.816284	−	0.545403i
$394$		0			0
$395$			5.15922e9i			0.211931i
$396$		0			0
$397$	−2.35673e10			−0.948739			−0.474370	−	0.880326i	$-0.657324\pi$
$397$	−2.35673e10			−0.948739			−0.474370	+	0.880326i	$0.657324\pi$
$398$		0			0
$399$	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	$-0.0857871\pi$
$402$		0			0
$403$	9.04353e9			0.342861
$404$		0			0
$405$	6.83394e9	−	6.83297e9i	0.254010	−	0.253974i
$406$		0			0
$407$			9.29209e9i			0.338638i
$408$		0			0
$409$	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
$412$		0			0
$413$			2.40011e10i			0.824955i
$414$		0			0
$415$	1.04157e10			0.351153
$416$		0			0
$417$	−6.42007e9	−	9.60868e9i	−0.212322	−	0.317775i
$418$		0			0
$419$			2.23996e10i			0.726750i	0.931643	+	0.363375i	$0.118376\pi$
$419$			2.23996e10i			0.726750i	−0.931643	+	0.363375i	$0.881624\pi$
$420$		0			0
$421$	−1.49535e10			−0.476008			−0.238004	−	0.971264i	$-0.576493\pi$
$421$	−1.49535e10			−0.476008			−0.238004	+	0.971264i	$0.576493\pi$
$422$		0			0
$423$	2.47372e10	+	1.02475e10i	0.772660	+	0.320078i
$424$		0			0
$425$		−	2.54805e10i		−	0.781001i
$426$		0			0
$427$	1.31880e9			0.0396706
$428$		0			0
$429$	1.20602e10	−	8.05804e9i	0.356061	−	0.237903i
$430$		0			0
$431$		−	6.40436e10i		−	1.85595i	−0.372640	−	0.927976i	$-0.621547\pi$
$431$		−	6.40436e10i		−	1.85595i	0.372640	−	0.927976i	$-0.378453\pi$
$432$		0			0
$433$	−5.22954e9			−0.148769			−0.0743843	−	0.997230i	$-0.523699\pi$
$433$	−5.22954e9			−0.148769			−0.0743843	+	0.997230i	$0.523699\pi$
$434$		0			0
$435$	−4.65620e9	−	6.96877e9i	−0.130039	−	0.194625i
$436$		0			0
$437$			8.90959e9i			0.244305i
$438$		0			0
$439$	−4.34801e10			−1.17066			−0.585332	−	0.810793i	$-0.699036\pi$
$439$	−4.34801e10			−1.17066			−0.585332	+	0.810793i	$0.699036\pi$
$440$		0			0
$441$	6.78548e9	−	1.63800e10i	0.179402	−	0.433070i
$442$		0			0
$443$		−	3.78737e10i		−	0.983383i	−0.870770	−	0.491691i	$-0.836379\pi$
$443$		−	3.78737e10i		−	0.983383i	0.870770	−	0.491691i	$-0.163621\pi$
$444$		0			0
$445$	1.63021e10			0.415722
$446$		0			0
$447$	5.51754e10	−	3.68656e10i	1.38202	−	0.923403i
$448$		0			0
$449$		−	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
$452$		0			0
$453$	1.90737e10	+	2.85469e10i	0.452942	+	0.677902i
$454$		0			0
$455$			1.01086e10i			0.235856i
$456$		0			0
$457$	−2.02181e10			−0.463529			−0.231764	−	0.972772i	$-0.574450\pi$
$457$	−2.02181e10			−0.463529			−0.231764	+	0.972772i	$0.574450\pi$
$458$		0			0
$459$	−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	$-0.283237\pi$
$462$		0			0
$463$	−4.16009e9			−0.0905271			−0.0452635	−	0.998975i	$-0.514413\pi$
$463$	−4.16009e9			−0.0905271			−0.0452635	+	0.998975i	$0.514413\pi$
$464$		0			0
$465$	5.31435e9	−	3.55080e9i	0.113668	−	0.0759476i
$466$		0			0
$467$			2.88138e10i			0.605806i	0.953021	+	0.302903i	$0.0979558\pi$
$467$			2.88138e10i			0.605806i	−0.953021	+	0.302903i	$0.902044\pi$
$468$		0			0
$469$	−3.97056e9			−0.0820654
$470$		0			0
$471$	3.41933e10	+	5.11758e10i	0.694796	+	1.03988i
$472$		0			0
$473$			2.45402e10i			0.490268i
$474$		0			0
$475$	−6.44318e9			−0.126569
$476$		0			0
$477$	−4.00165e10	−	1.65770e10i	−0.772975	−	0.320209i
$478$		0			0
$479$			4.47149e10i			0.849395i	0.905335	+	0.424698i	$0.139620\pi$
$479$			4.47149e10i			0.849395i	−0.905335	+	0.424698i	$0.860380\pi$
$480$		0			0
$481$	3.43539e10			0.641795
$482$		0			0
$483$	−5.54496e10	+	3.70488e10i	−1.01885	+	0.680747i
$484$		0			0
$485$			3.30623e10i			0.597538i
$486$		0			0
$487$	−5.72836e10			−1.01839			−0.509195	−	0.860651i	$-0.670057\pi$
$487$	−5.72836e10			−1.01839			−0.509195	+	0.860651i	$0.670057\pi$
$488$		0			0
$489$	3.00672e10	+	4.50005e10i	0.525845	+	0.787012i
$490$		0			0
$491$		−	7.25262e10i		−	1.24787i	−0.781477	−	0.623934i	$-0.785533\pi$
$491$		−	7.25262e10i		−	1.24787i	0.781477	−	0.623934i	$-0.214467\pi$
$492$		0			0
$493$	−3.45180e10			−0.584330
$494$		0			0
$495$	3.92319e9	−	9.47046e9i	0.0653459	−	0.157743i
$496$		0			0
$497$			2.97885e10i			0.488228i
$498$		0			0
$499$	2.64368e10			0.426389			0.213195	−	0.977010i	$-0.431613\pi$
$499$	2.64368e10			0.426389			0.213195	+	0.977010i	$0.431613\pi$
$500$		0			0
$501$	1.32212e10	−	8.83376e9i	0.209855	−	0.140215i
$502$		0			0
$503$			7.52828e10i			1.17604i	0.808845	+	0.588022i	$0.200093\pi$
$503$			7.52828e10i			1.17604i	−0.808845	+	0.588022i	$0.799907\pi$
$504$		0			0
$505$	−2.32457e10			−0.357419
$506$		0			0
$507$	6.91640e9	+	1.03515e10i	0.104676	+	0.156665i
$508$		0			0
$509$			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
$512$		0			0
$513$	−9.87115e9	+	1.96295e9i	−0.142527	+	0.0283426i
$514$		0			0
$515$			3.72812e10i			0.529981i
$516$		0			0
$517$	2.84019e10			0.397544
$518$		0			0
$519$	−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	$0.826217\pi$
$522$		0			0
$523$	8.46771e10			1.13177			0.565886	−	0.824483i	$-0.308534\pi$
$523$	8.46771e10			1.13177			0.565886	+	0.824483i	$0.308534\pi$
$524$		0			0
$525$	−2.67927e10	−	4.00997e10i	−0.352679	−	0.527842i
$526$		0			0
$527$		−	2.63232e10i		−	0.341269i
$528$		0			0
$529$	−1.43023e11			−1.82634
$530$		0			0
$531$	−8.31326e10	−	3.44381e10i	−1.04567	−	0.433173i
$532$		0			0
$533$		−	4.82558e10i		−	0.597917i
$534$		0			0
$535$	5.06336e9			0.0618050
$536$		0			0
$537$	−8.68104e10	+	5.80026e10i	−1.04394	+	0.697510i
$538$		0			0
$539$		−	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	$0.184066\pi$
$542$		0			0
$543$	−2.12268e10	−	3.17694e10i	−0.244166	−	0.365434i
$544$		0			0
$545$		−	2.46893e10i		−	0.279849i
$546$		0			0
$547$	−1.64171e11			−1.83378			−0.916892	−	0.399134i	$-0.869311\pi$
$547$	−1.64171e11			−1.83378			−0.916892	+	0.399134i	$0.869311\pi$
$548$		0			0
$549$	−1.89229e9	+	4.56795e9i	−0.0208305	+	0.0502842i
$550$		0			0
$551$			8.72847e9i			0.0946961i
$552$		0			0
$553$	4.02167e10			0.430037
$554$		0			0
$555$	2.01878e10	−	1.34885e10i	0.212773	−	0.142165i
$556$		0			0
$557$		−	1.54420e11i		−	1.60429i	−0.597130	−	0.802145i	$-0.703692\pi$
$557$		−	1.54420e11i		−	1.60429i	0.597130	−	0.802145i	$-0.296308\pi$
$558$		0			0
$559$	9.07278e10			0.929166
$560$		0			0
$561$	−2.34547e10	−	3.51039e10i	−0.236799	−	0.354408i
$562$		0			0
$563$			1.54622e11i			1.53900i	0.638646	+	0.769500i	$0.279495\pi$
$563$			1.54622e11i			1.53900i	−0.638646	+	0.769500i	$0.720505\pi$
$564$		0			0
$565$	6.45992e10			0.633919
$566$		0			0
$567$	−5.32638e10	−	5.32714e10i	−0.515348	−	0.515420i
$568$		0			0
$569$			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	$-0.220950\pi$
$571$	1.63410e11			1.53722			0.768608	+	0.639720i	$0.220950\pi$
$572$		0			0
$573$	−1.08963e10	+	7.28041e9i	−0.101079	+	0.0675363i
$574$		0			0
$575$		−	1.60063e11i		−	1.46426i
$576$		0			0
$577$	7.42282e10			0.669678			0.334839	−	0.942275i	$-0.391318\pi$
$577$	7.42282e10			0.669678			0.334839	+	0.942275i	$0.391318\pi$
$578$		0			0
$579$	7.14779e10	+	1.06978e11i	0.636001	+	0.951879i
$580$		0			0
$581$		−	8.11916e10i		−	0.712535i
$582$		0			0
$583$	−4.59449e10			−0.397707
$584$		0			0
$585$	−3.50134e10	−	1.45045e10i	−0.298958	−	0.123845i
$586$		0			0
$587$		−	6.72877e10i		−	0.566739i	−0.959011	−	0.283369i	$-0.908548\pi$
$587$		−	6.72877e10i		−	0.566739i	0.959011	−	0.283369i	$-0.0914523\pi$
$588$		0			0
$589$	−6.65629e9			−0.0553059
$590$		0			0
$591$	3.62187e10	−	2.41996e10i	0.296881	−	0.198362i
$592$		0			0
$593$			2.36444e10i			0.191210i	0.995419	+	0.0956048i	$0.0304785\pi$
$593$			2.36444e10i			0.191210i	−0.995419	+	0.0956048i	$0.969521\pi$
$594$		0			0
$595$	2.94235e10			0.234761
$596$		0			0
$597$	2.91414e10	+	4.36148e10i	0.229410	+	0.343350i
$598$		0			0
$599$			3.03370e10i			0.235649i	0.993034	+	0.117825i	$0.0375921\pi$
$599$			3.03370e10i			0.235649i	−0.993034	+	0.117825i	$0.962408\pi$
$600$		0			0
$601$	3.37911e10			0.259003			0.129501	−	0.991579i	$-0.458662\pi$
$601$	3.37911e10			0.259003			0.129501	+	0.991579i	$0.458662\pi$
$602$		0			0
$603$	5.69718e9	−	1.37528e10i	0.0430914	−	0.104022i
$604$		0			0
$605$			3.72500e10i			0.278038i
$606$		0			0
$607$	3.82366e10			0.281660			0.140830	−	0.990034i	$-0.455023\pi$
$607$	3.82366e10			0.281660			0.140830	+	0.990034i	$0.455023\pi$
$608$		0			0
$609$	−5.43224e10	+	3.62957e10i	−0.394920	+	0.263867i
$610$		0			0
$611$		−	1.05005e11i		−	0.753435i
$612$		0			0
$613$	1.08066e11			0.765330			0.382665	−	0.923887i	$-0.375006\pi$
$613$	1.08066e11			0.765330			0.382665	+	0.923887i	$0.375006\pi$
$614$		0			0
$615$	−1.89469e10	−	2.83571e10i	−0.132445	−	0.198226i
$616$		0			0
$617$			4.72538e9i			0.0326059i	0.999867	+	0.0163029i	$0.00518962\pi$
$617$			4.72538e9i			0.0326059i	−0.999867	+	0.0163029i	$0.994810\pi$
$618$		0			0
$619$	−2.29845e10			−0.156557			−0.0782786	−	0.996932i	$-0.524942\pi$
$619$	−2.29845e10			−0.156557			−0.0782786	+	0.996932i	$0.524942\pi$
$620$		0			0
$621$	−4.87639e10	−	2.45221e11i	−0.327893	−	1.64889i
$622$		0			0
$623$		−	1.27077e11i		−	0.843555i
$624$		0			0
$625$	9.60656e10			0.629575
$626$		0			0
$627$	−8.87662e9	+	5.93094e9i	−0.0574351	+	0.0383754i
$628$		0			0
$629$		−	9.99949e10i		−	0.638815i
$630$		0			0
$631$	1.01892e11			0.642722			0.321361	−	0.946957i	$-0.395860\pi$
$631$	1.01892e11			0.642722			0.321361	+	0.946957i	$0.395860\pi$
$632$		0			0
$633$	2.61497e9	+	3.91372e9i	0.0162874	+	0.0243767i
$634$		0			0
$635$		−	6.19605e10i		−	0.381083i
$636$		0			0
$637$	−6.95302e10			−0.422295
$638$		0			0
$639$	−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	$-0.113443\pi$
$642$		0			0
$643$	2.62680e10			0.153668			0.0768339	−	0.997044i	$-0.475519\pi$
$643$	2.62680e10			0.153668			0.0768339	+	0.997044i	$0.475519\pi$
$644$		0			0
$645$	5.33154e10	−	3.56228e10i	0.308045	−	0.205821i
$646$		0			0
$647$			3.10527e11i			1.77208i	0.463612	+	0.886038i	$0.346553\pi$
$647$			3.10527e11i			1.77208i	−0.463612	+	0.886038i	$0.653447\pi$
$648$		0			0
$649$	−9.54486e10			−0.538010
$650$		0			0
$651$	−2.76789e10	−	4.14260e10i	−0.154108	−	0.230648i
$652$		0			0
$653$		−	3.48345e10i		−	0.191583i	−0.995401	−	0.0957914i	$-0.969462\pi$
$653$		−	3.48345e10i		−	0.191583i	0.995401	−	0.0957914i	$-0.0305382\pi$
$654$		0			0
$655$	−6.49068e10			−0.352635
$656$		0			0
$657$	−6.94863e10	+	1.67738e11i	−0.372939	+	0.900265i
$658$		0			0
$659$			3.77120e10i			0.199957i	0.994990	+	0.0999787i	$0.0318775\pi$
$659$			3.77120e10i			0.199957i	−0.994990	+	0.0999787i	$0.968123\pi$
$660$		0			0
$661$	−1.39619e11			−0.731372			−0.365686	−	0.930738i	$-0.619166\pi$
$661$	−1.39619e11			−0.731372			−0.365686	+	0.930738i	$0.619166\pi$
$662$		0			0
$663$	−1.29783e11	+	8.67149e10i	−0.671682	+	0.448786i
$664$		0			0
$665$		−	7.44025e9i		−	0.0380453i
$666$		0			0
$667$	−2.16834e11			−1.09553
$668$		0			0
$669$	1.98090e11	+	2.96474e11i	0.988912	+	1.48007i
$670$		0			0
$671$			5.24468e9i			0.0258719i
$672$		0			0
$673$	−1.29783e11			−0.632642			−0.316321	−	0.948652i	$-0.602448\pi$
$673$	−1.29783e11			−0.632642			−0.316321	+	0.948652i	$0.602448\pi$
$674$		0			0
$675$	1.77337e11	−	3.52648e10i	0.854249	−	0.169874i
$676$		0			0
$677$			3.40648e11i			1.62163i	0.585306	+	0.810813i	$0.300975\pi$
$677$			3.40648e11i			1.62163i	−0.585306	+	0.810813i	$0.699025\pi$
$678$		0			0
$679$	2.57724e11			1.21248
$680$		0			0
$681$	2.38144e11	−	1.59117e11i	1.10726	−	0.739821i
$682$		0			0
$683$			1.02876e11i			0.472750i	0.971662	+	0.236375i	$0.0759593\pi$
$683$			1.02876e11i			0.472750i	−0.971662	+	0.236375i	$0.924041\pi$
$684$		0			0
$685$	−4.65976e10			−0.211642
$686$		0			0
$687$	8.39562e10	+	1.25654e11i	0.376900	+	0.564092i
$688$		0			0
$689$			1.69863e11i			0.753742i
$690$		0			0
$691$	−3.58259e11			−1.57140			−0.785698	−	0.618611i	$-0.787696\pi$
$691$	−3.58259e11			−1.57140			−0.785698	+	0.618611i	$0.787696\pi$
$692$		0			0
$693$	−7.38234e10	−	3.05817e10i	−0.320082	−	0.132595i
$694$		0			0
$695$			3.20289e10i			0.137279i
$696$		0			0
$697$	−1.40459e11			−0.595141
$698$		0			0
$699$	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	$0.827327\pi$
$702$		0			0
$703$	−2.52854e10			−0.103526
$704$		0			0
$705$	−4.12286e10	−	6.17053e10i	−0.166895	−	0.249785i
$706$		0			0
$707$			1.81203e11i			0.725251i
$708$		0			0
$709$	−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
$712$		0			0
$713$		−	1.65357e11i		−	0.639829i
$714$		0			0
$715$	−4.02006e10			−0.153818
$716$		0			0
$717$	−1.53195e11	+	1.02357e11i	−0.579651	+	0.387296i
$718$		0			0
$719$			3.35735e10i			0.125626i	0.998025	+	0.0628132i	$0.0200072\pi$
$719$			3.35735e10i			0.125626i	−0.998025	+	0.0628132i	$0.979993\pi$
$720$		0			0
$721$	2.90611e11			1.07540
$722$		0			0
$723$	7.86002e10	+	1.17638e11i	0.287654	+	0.430521i
$724$		0			0
$725$		−	1.56809e11i		−	0.567569i
$726$		0			0
$727$	−2.53113e11			−0.906102			−0.453051	−	0.891485i	$-0.649664\pi$
$727$	−2.53113e11			−0.906102			−0.453051	+	0.891485i	$0.649664\pi$
$728$		0			0
$729$	2.60942e11	−	1.08053e11i	0.923920	−	0.382586i
$730$		0			0
$731$		−	2.64084e11i		−	0.924853i
$732$		0			0
$733$	2.07602e11			0.719144			0.359572	−	0.933117i	$-0.382923\pi$
$733$	2.07602e11			0.719144			0.359572	+	0.933117i	$0.382923\pi$
$734$		0			0
$735$	−4.08588e10	+	2.72999e10i	−0.140003	+	0.0935432i
$736$		0			0
$737$		−	1.57903e10i		−	0.0535205i
$738$		0			0
$739$	−4.15014e11			−1.39151			−0.695753	−	0.718281i	$-0.744929\pi$
$739$	−4.15014e11			−1.39151			−0.695753	+	0.718281i	$0.744929\pi$
$740$		0			0
$741$	2.19274e10	+	3.28179e10i	0.0727300	+	0.108852i
$742$		0			0
$743$		−	3.30690e11i		−	1.08509i	−0.840027	−	0.542545i	$-0.817461\pi$
$743$		−	3.30690e11i		−	1.08509i	0.840027	−	0.542545i	$-0.182539\pi$
$744$		0			0
$745$	−1.83918e11			−0.597034
$746$		0			0
$747$	2.81224e11	+	1.16498e11i	0.903170	+	0.374143i
$748$		0			0
$749$		−	3.94695e10i		−	0.125411i
$750$		0			0
$751$	4.77978e11			1.50262			0.751308	−	0.659952i	$-0.229423\pi$
$751$	4.77978e11			1.50262			0.751308	+	0.659952i	$0.229423\pi$
$752$		0			0
$753$	9.26390e10	−	6.18970e10i	0.288147	−	0.192526i
$754$		0			0
$755$		−	9.51565e10i		−	0.292854i
$756$		0			0
$757$	−8.95066e10			−0.272566			−0.136283	−	0.990670i	$-0.543516\pi$
$757$	−8.95066e10			−0.272566			−0.136283	+	0.990670i	$0.543516\pi$
$758$		0			0
$759$	−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	$-0.324596\pi$
$762$		0			0
$763$	−1.92456e11			−0.567851
$764$		0			0
$765$	−4.22185e10	+	1.01914e11i	−0.123270	+	0.297570i
$766$		0			0
$767$			3.52884e11i			1.01965i
$768$		0			0
$769$	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
$772$		0			0
$773$			1.00523e11i			0.281543i	0.990042	+	0.140772i	$0.0449584\pi$
$773$			1.00523e11i			0.281543i	−0.990042	+	0.140772i	$0.955042\pi$
$774$		0			0
$775$	1.19582e11			0.331480
$776$		0			0
$777$	−1.05145e11	−	1.57366e11i	−0.288472	−	0.431745i
$778$		0			0
$779$			3.55176e10i			0.0964482i
$780$		0			0
$781$	−1.18464e11			−0.318408
$782$		0

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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	48.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-45.0000 - 67.3498i) q^{3} +224.499i q^{5} +1750.00 q^{7} +(-2511.00 + 6061.48i) q^{9} +O(q^{10})\)	\(q+(-45.0000 - 67.3498i) q^{3} +224.499i q^{5} +1750.00 q^{7} +(-2511.00 + 6061.48i) q^{9} +6959.48i q^{11} +25730.0 q^{13} +(15120.0 - 10102.5i) q^{15} -74893.0i q^{17} -18938.0 q^{19} +(-78750.0 - 117862. i) q^{21} -470461. i q^{23} +340225. q^{25} +(521235. - 103651. i) q^{27} -460897. i q^{29} +351478. q^{31} +(468720. - 313177. i) q^{33} +392874. i q^{35} +1.33517e6 q^{37} +(-1.15785e6 - 1.73291e6i) q^{39} -1.87547e6i q^{41} +3.52615e6 q^{43} +(-1.36080e6 - 563718. i) q^{45} -4.08104e6i q^{47} -2.70230e6 q^{49} +(-5.04403e6 + 3.37019e6i) q^{51} +6.60177e6i q^{53} -1.56240e6 q^{55} +(852210. + 1.27547e6i) q^{57} +1.37149e7i q^{59} +753602. q^{61} +(-4.39425e6 + 1.06076e7i) q^{63} +5.77637e6i q^{65} -2.26889e6 q^{67} +(-3.16855e7 + 2.11707e7i) q^{69} +1.70220e7i q^{71} +2.76728e7 q^{73} +(-1.53101e7 - 2.29141e7i) q^{75} +1.21791e7i q^{77} +2.29810e7 q^{79} +(-3.04365e7 - 3.04408e7i) q^{81} -4.63952e7i q^{83} +1.68134e7 q^{85} +(-3.10414e7 + 2.07404e7i) q^{87} -7.26152e7i q^{89} +4.50275e7 q^{91} +(-1.58165e7 - 2.36720e7i) q^{93} -4.25157e6i q^{95} +1.47271e8 q^{97} +(-4.21848e7 - 1.74753e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 90 q^{3} + 3500 q^{7} - 5022 q^{9}+O(q^{10}) \) 2 * q - 90 * q^3 + 3500 * q^7 - 5022 * q^9	\( 2 q - 90 q^{3} + 3500 q^{7} - 5022 q^{9} + 51460 q^{13} + 30240 q^{15} - 37876 q^{19} - 157500 q^{21} + 680450 q^{25} + 1042470 q^{27} + 702956 q^{31} + 937440 q^{33} + 2670340 q^{37} - 2315700 q^{39} + 7052300 q^{43} - 2721600 q^{45} - 5404602 q^{49} - 10088064 q^{51} - 3124800 q^{55} + 1704420 q^{57} + 1507204 q^{61} - 8788500 q^{63} - 4537780 q^{67} - 63370944 q^{69} + 55345540 q^{73} - 30620250 q^{75} + 45961964 q^{79} - 60872958 q^{81} + 33626880 q^{85} - 62082720 q^{87} + 90055000 q^{91} - 31633020 q^{93} + 294542020 q^{97} - 84369600 q^{99}+O(q^{100}) \) 2 * q - 90 * q^3 + 3500 * q^7 - 5022 * q^9 + 51460 * q^13 + 30240 * q^15 - 37876 * q^19 - 157500 * q^21 + 680450 * q^25 + 1042470 * q^27 + 702956 * q^31 + 937440 * q^33 + 2670340 * q^37 - 2315700 * q^39 + 7052300 * q^43 - 2721600 * q^45 - 5404602 * q^49 - 10088064 * q^51 - 3124800 * q^55 + 1704420 * q^57 + 1507204 * q^61 - 8788500 * q^63 - 4537780 * q^67 - 63370944 * q^69 + 55345540 * q^73 - 30620250 * q^75 + 45961964 * q^79 - 60872958 * q^81 + 33626880 * q^85 - 62082720 * q^87 + 90055000 * q^91 - 31633020 * q^93 + 294542020 * q^97 - 84369600 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more