LMFDB - Embedded newform 75.10.b.f.49.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,10,Mod(49,75)]

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([0, 1]))

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("75.49");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	75.b (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:	$38.6276877123$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{241})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 121x^{2} + 3600 $ x^4 + 121*x^2 + 3600
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.3
Root			$-8.26209i$ of defining polynomial
Character	$\chi$	$=$	75.49
Dual form			75.10.b.f.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.78626i q^{2} -81.0000i q^{3} +451.374 q^{4} +630.687 q^{6} -1839.88i q^{7} +7501.08i q^{8} -6561.00 q^{9} +O(q^{10})$	\(q+7.78626i q^{2} -81.0000i q^{3} +451.374 q^{4} +630.687 q^{6} -1839.88i q^{7} +7501.08i q^{8} -6561.00 q^{9} +44385.9 q^{11} -36561.3i q^{12} +136584. i q^{13} +14325.8 q^{14} +172698. q^{16} +253591. i q^{17} -51085.7i q^{18} -85435.7 q^{19} -149030. q^{21} +345600. i q^{22} -979409. i q^{23} +607588. q^{24} -1.06348e6 q^{26} +531441. i q^{27} -830473. i q^{28} -2.58640e6 q^{29} +8.94787e6 q^{31} +5.18523e6i q^{32} -3.59526e6i q^{33} -1.97452e6 q^{34} -2.96147e6 q^{36} -1.56064e7i q^{37} -665225. i q^{38} +1.10633e7 q^{39} +2.44893e7 q^{41} -1.16039e6i q^{42} +1.27592e7i q^{43} +2.00346e7 q^{44} +7.62593e6 q^{46} +6.16764e7i q^{47} -1.39886e7i q^{48} +3.69685e7 q^{49} +2.05408e7 q^{51} +6.16504e7i q^{52} +5.70418e6i q^{53} -4.13794e6 q^{54} +1.38011e7 q^{56} +6.92029e6i q^{57} -2.01384e7i q^{58} -8.35095e7 q^{59} +1.48622e8 q^{61} +6.96704e7i q^{62} +1.20714e7i q^{63} +4.80479e7 q^{64} +2.79936e7 q^{66} +1.68003e8i q^{67} +1.14464e8i q^{68} -7.93321e7 q^{69} +2.10986e8 q^{71} -4.92146e7i q^{72} -1.43534e8i q^{73} +1.21515e8 q^{74} -3.85635e7 q^{76} -8.16646e7i q^{77} +8.61416e7i q^{78} +4.55960e8 q^{79} +4.30467e7 q^{81} +1.90680e8i q^{82} -3.55106e8i q^{83} -6.72683e7 q^{84} -9.93465e7 q^{86} +2.09498e8i q^{87} +3.32942e8i q^{88} +4.24540e8 q^{89} +2.51297e8 q^{91} -4.42080e8i q^{92} -7.24777e8i q^{93} -4.80228e8 q^{94} +4.20003e8 q^{96} -1.19905e9i q^{97} +2.87846e8i q^{98} -2.91216e8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 1082 q^{4} - 5022 q^{6} - 26244 q^{9}+O(q^{10}) $ 4 * q - 1082 * q^4 - 5022 * q^6 - 26244 * q^9	$ 4 q - 1082 q^{4} - 5022 q^{6} - 26244 q^{9} - 43024 q^{11} - 923328 q^{14} + 774530 q^{16} + 191792 q^{19} - 2286144 q^{21} + 4233546 q^{24} - 10838332 q^{26} + 5356424 q^{29} + 21564864 q^{31} - 11445244 q^{34} + 7099002 q^{36} + 3934008 q^{39} + 52120744 q^{41} + 170859944 q^{44} + 34190784 q^{46} - 146565860 q^{49} + 25426872 q^{51} + 32949342 q^{54} + 484908480 q^{56} + 70989328 q^{59} + 682994680 q^{61} + 410240942 q^{64} + 470048184 q^{66} - 119112768 q^{69} + 420572128 q^{71} - 250480468 q^{74} - 437024728 q^{76} + 49510080 q^{79} + 172186884 q^{81} + 1838386368 q^{84} - 1746292744 q^{86} + 855278232 q^{89} - 2253718656 q^{91} - 3295087792 q^{94} + 1037208726 q^{96} + 282280464 q^{99}+O(q^{100}) $ 4 * q - 1082 * q^4 - 5022 * q^6 - 26244 * q^9 - 43024 * q^11 - 923328 * q^14 + 774530 * q^16 + 191792 * q^19 - 2286144 * q^21 + 4233546 * q^24 - 10838332 * q^26 + 5356424 * q^29 + 21564864 * q^31 - 11445244 * q^34 + 7099002 * q^36 + 3934008 * q^39 + 52120744 * q^41 + 170859944 * q^44 + 34190784 * q^46 - 146565860 * q^49 + 25426872 * q^51 + 32949342 * q^54 + 484908480 * q^56 + 70989328 * q^59 + 682994680 * q^61 + 410240942 * q^64 + 470048184 * q^66 - 119112768 * q^69 + 420572128 * q^71 - 250480468 * q^74 - 437024728 * q^76 + 49510080 * q^79 + 172186884 * q^81 + 1838386368 * q^84 - 1746292744 * q^86 + 855278232 * q^89 - 2253718656 * q^91 - 3295087792 * q^94 + 1037208726 * q^96 + 282280464 * 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 $

$2$	7.78626i	0.344107i	0.985088	+	0.172054i	$0.0550403\pi$
$2$	7.78626i	0.344107i	−0.985088	+	0.172054i	$0.944960\pi$
$3$	− 81.0000i	− 0.577350i
$3$	− 81.0000i	− 0.577350i
$4$	451.374	0.881590
$5$	0	0
$5$	0	0
$6$	630.687	0.198671
$7$	− 1839.88i	− 0.289633i	−0.989459	−	0.144816i	$-0.953741\pi$
$7$	− 1839.88i	− 0.289633i	0.989459	−	0.144816i	$-0.0462591\pi$
$8$	7501.08i	0.647469i
$9$	−6561.00	−0.333333
$10$	0	0
$11$	44385.9	0.914066	0.457033	−	0.889450i	$-0.348912\pi$
$11$	44385.9	0.914066	0.457033	+	0.889450i	$0.348912\pi$
$12$	− 36561.3i	− 0.508986i
$13$	136584.i	1.32634i	0.748470	+	0.663169i	$0.230789\pi$
$13$	136584.i	1.32634i	−0.748470	+	0.663169i	$0.769211\pi$
$14$	14325.8	0.0996648
$15$	0	0
$16$	172698.	0.658791
$17$	253591.i	0.736399i	0.929747	+	0.368199i	$0.120026\pi$
$17$	253591.i	0.736399i	−0.929747	+	0.368199i	$0.879974\pi$
$18$	− 51085.7i	− 0.114702i
$19$	−85435.7	−0.150400	−0.0752001	−	0.997168i	$-0.523960\pi$
$19$	−85435.7	−0.150400	−0.0752001	+	0.997168i	$0.523960\pi$
$20$	0	0
$21$	−149030.	−0.167220
$22$	345600.i	0.314537i
$23$	− 979409.i	− 0.729775i	−0.931052	−	0.364887i	$-0.881108\pi$
$23$	− 979409.i	− 0.729775i	0.931052	−	0.364887i	$-0.118892\pi$
$24$	607588.	0.373816
$25$	0	0
$26$	−1.06348e6	−0.456403
$27$	531441.i	0.192450i
$28$	− 830473.i	− 0.255337i
$29$	−2.58640e6	−0.679054	−0.339527	−	0.940596i	$-0.610267\pi$
$29$	−2.58640e6	−0.679054	−0.339527	+	0.940596i	$0.610267\pi$
$30$	0	0
$31$	8.94787e6	1.74017	0.870085	−	0.492901i	$-0.164064\pi$
$31$	8.94787e6	1.74017	0.870085	+	0.492901i	$0.164064\pi$
$32$	5.18523e6i	0.874164i
$33$	− 3.59526e6i	− 0.527736i
$34$	−1.97452e6	−0.253400
$35$	0	0
$36$	−2.96147e6	−0.293863
$37$	− 1.56064e7i	− 1.36897i	−0.729026	−	0.684486i	$-0.760026\pi$
$37$	− 1.56064e7i	− 1.36897i	0.729026	−	0.684486i	$-0.239974\pi$
$38$	− 665225.i	− 0.0517538i
$39$	1.10633e7	0.765761
$40$	0	0
$41$	2.44893e7	1.35347	0.676735	−	0.736227i	$-0.263394\pi$
$41$	2.44893e7	1.35347	0.676735	+	0.736227i	$0.263394\pi$
$42$	− 1.16039e6i	− 0.0575415i
$43$	1.27592e7i	0.569135i	0.958656	+	0.284568i	$0.0918500\pi$
$43$	1.27592e7i	0.569135i	−0.958656	+	0.284568i	$0.908150\pi$
$44$	2.00346e7	0.805832
$45$	0	0
$46$	7.62593e6	0.251121
$47$	6.16764e7i	1.84365i	0.387607	+	0.921825i	$0.373302\pi$
$47$	6.16764e7i	1.84365i	−0.387607	+	0.921825i	$0.626698\pi$
$48$	− 1.39886e7i	− 0.380353i
$49$	3.69685e7	0.916113
$50$	0	0
$51$	2.05408e7	0.425160
$52$	6.16504e7i	1.16929i
$53$	5.70418e6i	0.0993006i	0.998767	+	0.0496503i	$0.0158107\pi$
$53$	5.70418e6i	0.0993006i	−0.998767	+	0.0496503i	$0.984189\pi$
$54$	−4.13794e6	−0.0662235
$55$	0	0
$56$	1.38011e7	0.187528
$57$	6.92029e6i	0.0868336i
$58$	− 2.01384e7i	− 0.233668i
$59$	−8.35095e7	−0.897226	−0.448613	−	0.893726i	$-0.648082\pi$
$59$	−8.35095e7	−0.897226	−0.448613	+	0.893726i	$0.648082\pi$
$60$	0	0
$61$	1.48622e8	1.37435	0.687177	−	0.726490i	$-0.258850\pi$
$61$	1.48622e8	1.37435	0.687177	+	0.726490i	$0.258850\pi$
$62$	6.96704e7i	0.598806i
$63$	1.20714e7i	0.0965443i
$64$	4.80479e7	0.357985
$65$	0	0
$66$	2.79936e7	0.181598
$67$	1.68003e8i	1.01854i	0.860606	+	0.509272i	$0.170085\pi$
$67$	1.68003e8i	1.01854i	−0.860606	+	0.509272i	$0.829915\pi$
$68$	1.14464e8i	0.649202i
$69$	−7.93321e7	−0.421336
$70$	0	0
$71$	2.10986e8	0.985349	0.492675	−	0.870214i	$-0.336019\pi$
$71$	2.10986e8	0.985349	0.492675	+	0.870214i	$0.336019\pi$
$72$	− 4.92146e7i	− 0.215823i
$73$	− 1.43534e8i	− 0.591566i	−0.955255	−	0.295783i	$-0.904420\pi$
$73$	− 1.43534e8i	− 0.591566i	0.955255	−	0.295783i	$-0.0955805\pi$
$74$	1.21515e8	0.471074
$75$	0	0
$76$	−3.85635e7	−0.132591
$77$	− 8.16646e7i	− 0.264744i
$78$	8.61416e7i	0.263504i
$79$	4.55960e8	1.31706	0.658529	−	0.752555i	$-0.271179\pi$
$79$	4.55960e8	1.31706	0.658529	+	0.752555i	$0.271179\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	1.90680e8i	0.465739i
$83$	− 3.55106e8i	− 0.821309i	−0.911791	−	0.410655i	$-0.865300\pi$
$83$	− 3.55106e8i	− 0.821309i	0.911791	−	0.410655i	$-0.134700\pi$
$84$	−6.72683e7	−0.147419
$85$	0	0
$86$	−9.93465e7	−0.195844
$87$	2.09498e8i	0.392052i
$88$	3.32942e8i	0.591830i
$89$	4.24540e8	0.717239	0.358620	−	0.933484i	$-0.383248\pi$
$89$	4.24540e8	0.717239	0.358620	+	0.933484i	$0.383248\pi$
$90$	0	0
$91$	2.51297e8	0.384151
$92$	− 4.42080e8i	− 0.643362i
$93$	− 7.24777e8i	− 1.00469i
$94$	−4.80228e8	−0.634413
$95$	0	0
$96$	4.20003e8	0.504699
$97$	− 1.19905e9i	− 1.37520i	−0.726092	−	0.687598i	$-0.758665\pi$
$97$	− 1.19905e9i	− 1.37520i	0.726092	−	0.687598i	$-0.241335\pi$
$98$	2.87846e8i	0.315241i
$99$	−2.91216e8	−0.304689
$100$	0	0
$101$	−1.77085e9	−1.69331	−0.846654	−	0.532144i	$-0.821387\pi$
$101$	−1.77085e9	−1.69331	−0.846654	+	0.532144i	$0.821387\pi$
$102$	1.59936e8i	0.146301i
$103$	3.03322e8i	0.265544i	0.991147	+	0.132772i	$0.0423878\pi$
$103$	3.03322e8i	0.265544i	−0.991147	+	0.132772i	$0.957612\pi$
$104$	−1.02453e9	−0.858763
$105$	0	0
$106$	−4.44142e7	−0.0341701
$107$	1.95414e8i	0.144121i	0.997400	+	0.0720607i	$0.0229575\pi$
$107$	1.95414e8i	0.144121i	−0.997400	+	0.0720607i	$0.977042\pi$
$108$	2.39879e8i	0.169662i
$109$	−2.31494e9	−1.57080	−0.785401	−	0.618988i	$-0.787543\pi$
$109$	−2.31494e9	−1.57080	−0.785401	+	0.618988i	$0.787543\pi$
$110$	0	0
$111$	−1.26412e9	−0.790377
$112$	− 3.17743e8i	− 0.190808i
$113$	1.31945e9i	0.761270i	0.924725	+	0.380635i	$0.124294\pi$
$113$	1.31945e9i	0.761270i	−0.924725	+	0.380635i	$0.875706\pi$
$114$	−5.38832e7	−0.0298801
$115$	0	0
$116$	−1.16743e9	−0.598648
$117$	− 8.96126e8i	− 0.442113i
$118$	− 6.50227e8i	− 0.308742i
$119$	4.66576e8	0.213285
$120$	0	0
$121$	−3.87842e8	−0.164483
$122$	1.15721e9i	0.472925i
$123$	− 1.98363e9i	− 0.781426i
$124$	4.03884e9	1.53412
$125$	0	0
$126$	−9.39914e7	−0.0332216
$127$	− 3.18960e9i	− 1.08798i	−0.839093	−	0.543988i	$-0.816914\pi$
$127$	− 3.18960e9i	− 1.08798i	0.839093	−	0.543988i	$-0.183086\pi$
$128$	3.02895e9i	0.997349i
$129$	1.03350e9	0.328590
$130$	0	0
$131$	−5.40115e9	−1.60238	−0.801190	−	0.598410i	$-0.795799\pi$
$131$	−5.40115e9	−1.60238	−0.801190	+	0.598410i	$0.795799\pi$
$132$	− 1.62281e9i	− 0.465247i
$133$	1.57191e8i	0.0435608i
$134$	−1.30811e9	−0.350488
$135$	0	0
$136$	−1.90220e9	−0.476796
$137$	− 1.19903e7i	− 0.00290795i	−0.999999	−	0.00145398i	$-0.999537\pi$
$137$	− 1.19903e7i	− 0.00290795i	0.999999	−	0.00145398i	$-0.000462815\pi$
$138$	− 6.17701e8i	− 0.144985i
$139$	−9.15482e8	−0.208010	−0.104005	−	0.994577i	$-0.533166\pi$
$139$	−9.15482e8	−0.208010	−0.104005	+	0.994577i	$0.533166\pi$
$140$	0	0
$141$	4.99578e9	1.06443
$142$	1.64279e9i	0.339066i
$143$	6.06239e9i	1.21236i
$144$	−1.13307e9	−0.219597
$145$	0	0
$146$	1.11760e9	0.203562
$147$	− 2.99445e9i	− 0.528918i
$148$	− 7.04432e9i	− 1.20687i
$149$	−5.00462e9	−0.831827	−0.415913	−	0.909404i	$-0.636538\pi$
$149$	−5.00462e9	−0.831827	−0.415913	+	0.909404i	$0.636538\pi$
$150$	0	0
$151$	3.20554e9	0.501770	0.250885	−	0.968017i	$-0.419278\pi$
$151$	3.20554e9	0.501770	0.250885	+	0.968017i	$0.419278\pi$
$152$	− 6.40860e8i	− 0.0973794i
$153$	− 1.66381e9i	− 0.245466i
$154$	6.35862e8	0.0911002
$155$	0	0
$156$	4.99368e9	0.675088
$157$	4.63430e8i	0.0608745i	0.999537	+	0.0304373i	$0.00968998\pi$
$157$	4.63430e8i	0.0608745i	−0.999537	+	0.0304373i	$0.990310\pi$
$158$	3.55023e9i	0.453210i
$159$	4.62038e8	0.0573312
$160$	0	0
$161$	−1.80199e9	−0.211367
$162$	3.35173e8i	0.0382342i
$163$	− 1.27948e10i	− 1.41968i	−0.704363	−	0.709840i	$-0.748767\pi$
$163$	− 1.27948e10i	− 1.41968i	0.704363	−	0.709840i	$-0.251233\pi$
$164$	1.10538e10	1.19320
$165$	0	0
$166$	2.76495e9	0.282619
$167$	1.85699e10i	1.84750i	0.382992	+	0.923752i	$0.374894\pi$
$167$	1.85699e10i	1.84750i	−0.382992	+	0.923752i	$0.625106\pi$
$168$	− 1.11789e9i	− 0.108269i
$169$	−8.05063e9	−0.759171
$170$	0	0
$171$	5.60544e8	0.0501334
$172$	5.75917e9i	0.501744i
$173$	4.90746e9i	0.416533i	0.978072	+	0.208266i	$0.0667821\pi$
$173$	4.90746e9i	0.416533i	−0.978072	+	0.208266i	$0.933218\pi$
$174$	−1.63121e9	−0.134908
$175$	0	0
$176$	7.66536e9	0.602179
$177$	6.76427e9i	0.518014i
$178$	3.30558e9i	0.246807i
$179$	1.28930e10	0.938678	0.469339	−	0.883018i	$-0.344492\pi$
$179$	1.28930e10	0.938678	0.469339	+	0.883018i	$0.344492\pi$
$180$	0	0
$181$	2.66233e10	1.84378	0.921889	−	0.387453i	$-0.126645\pi$
$181$	2.66233e10	1.84378	0.921889	+	0.387453i	$0.126645\pi$
$182$	1.95667e9i	0.132189i
$183$	− 1.20384e10i	− 0.793483i
$184$	7.34663e9	0.472506
$185$	0	0
$186$	5.64331e9	0.345721
$187$	1.12558e10i	0.673117i
$188$	2.78391e10i	1.62534i
$189$	9.77786e8	0.0557399
$190$	0	0
$191$	−2.72802e10	−1.48319	−0.741595	−	0.670848i	$-0.765930\pi$
$191$	−2.72802e10	−1.48319	−0.741595	+	0.670848i	$0.765930\pi$
$192$	− 3.89188e9i	− 0.206683i
$193$	− 9.65442e9i	− 0.500862i	−0.968134	−	0.250431i	$-0.919428\pi$
$193$	− 9.65442e9i	− 0.500862i	0.968134	−	0.250431i	$-0.0805723\pi$
$194$	9.33612e9	0.473215
$195$	0	0
$196$	1.66866e10	0.807636
$197$	− 1.21091e10i	− 0.572812i	−0.958108	−	0.286406i	$-0.907539\pi$
$197$	− 1.21091e10i	− 0.572812i	0.958108	−	0.286406i	$-0.0924606\pi$
$198$	− 2.26748e9i	− 0.104846i
$199$	−1.89741e10	−0.857673	−0.428837	−	0.903382i	$-0.641076\pi$
$199$	−1.89741e10	−0.857673	−0.428837	+	0.903382i	$0.641076\pi$
$200$	0	0
$201$	1.36082e10	0.588056
$202$	− 1.37883e10i	− 0.582680i
$203$	4.75866e9i	0.196676i
$204$	9.27161e9	0.374817
$205$	0	0
$206$	−2.36174e9	−0.0913757
$207$	6.42590e9i	0.243258i
$208$	2.35878e10i	0.873779i
$209$	−3.79214e9	−0.137476
$210$	0	0
$211$	−8.91928e9	−0.309784	−0.154892	−	0.987931i	$-0.549503\pi$
$211$	−8.91928e9	−0.309784	−0.154892	+	0.987931i	$0.549503\pi$
$212$	2.57472e9i	0.0875424i
$213$	− 1.70898e10i	− 0.568892i
$214$	−1.52154e9	−0.0495933
$215$	0	0
$216$	−3.98638e9	−0.124605
$217$	− 1.64630e10i	− 0.504010i
$218$	− 1.80248e10i	− 0.540524i
$219$	−1.16263e10	−0.341541
$220$	0	0
$221$	−3.46364e10	−0.976714
$222$	− 9.84275e9i	− 0.271975i
$223$	− 4.42755e10i	− 1.19892i	−0.800403	−	0.599462i	$-0.795381\pi$
$223$	− 4.42755e10i	− 1.19892i	0.800403	−	0.599462i	$-0.204619\pi$
$224$	9.54018e9	0.253187
$225$	0	0
$226$	−1.02735e10	−0.261959
$227$	5.40045e10i	1.34994i	0.737847	+	0.674968i	$0.235843\pi$
$227$	5.40045e10i	1.34994i	−0.737847	+	0.674968i	$0.764157\pi$
$228$	3.12364e9i	0.0765516i
$229$	−5.80250e9	−0.139430	−0.0697149	−	0.997567i	$-0.522209\pi$
$229$	−5.80250e9	−0.139430	−0.0697149	+	0.997567i	$0.522209\pi$
$230$	0	0
$231$	−6.61483e9	−0.152850
$232$	− 1.94008e10i	− 0.439667i
$233$	5.41865e10i	1.20445i	0.798326	+	0.602226i	$0.205719\pi$
$233$	5.41865e10i	1.20445i	−0.798326	+	0.602226i	$0.794281\pi$
$234$	6.97747e9	0.152134
$235$	0	0
$236$	−3.76940e10	−0.790986
$237$	− 3.69328e10i	− 0.760404i
$238$	3.63288e9i	0.0733930i
$239$	−7.91761e10	−1.56965	−0.784826	−	0.619716i	$-0.787248\pi$
$239$	−7.91761e10	−1.56965	−0.784826	+	0.619716i	$0.787248\pi$
$240$	0	0
$241$	−6.14920e10	−1.17420	−0.587100	−	0.809514i	$-0.699730\pi$
$241$	−6.14920e10	−1.17420	−0.587100	+	0.809514i	$0.699730\pi$
$242$	− 3.01984e9i	− 0.0565998i
$243$	− 3.48678e9i	− 0.0641500i
$244$	6.70841e10	1.21162
$245$	0	0
$246$	1.54451e10	0.268894
$247$	− 1.16691e10i	− 0.199481i
$248$	6.71187e10i	1.12671i
$249$	−2.87636e10	−0.474183
$250$	0	0
$251$	2.89319e10	0.460093	0.230046	−	0.973180i	$-0.426112\pi$
$251$	2.89319e10	0.460093	0.230046	+	0.973180i	$0.426112\pi$
$252$	5.44873e9i	0.0851125i
$253$	− 4.34719e10i	− 0.667062i
$254$	2.48350e10	0.374381
$255$	0	0
$256$	1.01633e9	0.0147896
$257$	− 1.22388e11i	− 1.75001i	−0.484114	−	0.875005i	$-0.660858\pi$
$257$	− 1.22388e11i	− 1.75001i	0.484114	−	0.875005i	$-0.339142\pi$
$258$	8.04706e9i	0.113070i
$259$	−2.87139e10	−0.396499
$260$	0	0
$261$	1.69694e10	0.226351
$262$	− 4.20548e10i	− 0.551391i
$263$	− 6.24892e10i	− 0.805386i	−0.915335	−	0.402693i	$-0.868074\pi$
$263$	− 6.24892e10i	− 0.805386i	0.915335	−	0.402693i	$-0.131926\pi$
$264$	2.69683e10	0.341693
$265$	0	0
$266$	−1.22393e9	−0.0149896
$267$	− 3.43878e10i	− 0.414098i
$268$	7.58321e10i	0.897938i
$269$	−1.27214e11	−1.48132	−0.740659	−	0.671881i	$-0.765487\pi$
$269$	−1.27214e11	−1.48132	−0.740659	+	0.671881i	$0.765487\pi$
$270$	0	0
$271$	1.54116e10	0.173574	0.0867871	−	0.996227i	$-0.472340\pi$
$271$	1.54116e10	0.173574	0.0867871	+	0.996227i	$0.472340\pi$
$272$	4.37946e10i	0.485133i
$273$	− 2.03551e10i	− 0.221790i
$274$	9.33596e7	0.00100065
$275$	0	0
$276$	−3.58085e10	−0.371445
$277$	9.20867e10i	0.939806i	0.882718	+	0.469903i	$0.155711\pi$
$277$	9.20867e10i	0.939806i	−0.882718	+	0.469903i	$0.844289\pi$
$278$	− 7.12818e9i	− 0.0715776i
$279$	−5.87070e10	−0.580057
$280$	0	0
$281$	−7.78782e10	−0.745139	−0.372570	−	0.928004i	$-0.621523\pi$
$281$	−7.78782e10	−0.745139	−0.372570	+	0.928004i	$0.621523\pi$
$282$	3.88985e10i	0.366279i
$283$	2.92463e10i	0.271039i	0.990775	+	0.135520i	$0.0432704\pi$
$283$	2.92463e10i	0.271039i	−0.990775	+	0.135520i	$0.956730\pi$
$284$	9.52334e10	0.868674
$285$	0	0
$286$	−4.72034e10	−0.417182
$287$	− 4.50572e10i	− 0.392009i
$288$	− 3.40203e10i	− 0.291388i
$289$	5.42796e10	0.457717
$290$	0	0
$291$	−9.71231e10	−0.793970
$292$	− 6.47877e10i	− 0.521519i
$293$	− 2.27403e11i	− 1.80257i	−0.433227	−	0.901285i	$-0.642625\pi$
$293$	− 2.27403e11i	− 1.80257i	0.433227	−	0.901285i	$-0.357375\pi$
$294$	2.33155e10	0.182005
$295$	0	0
$296$	1.17065e11	0.886368
$297$	2.35885e10i	0.175912i
$298$	− 3.89673e10i	− 0.286238i
$299$	1.33771e11	0.967927
$300$	0	0
$301$	2.34754e10	0.164840
$302$	2.49592e10i	0.172663i
$303$	1.43439e11i	0.977632i
$304$	−1.47546e10	−0.0990823
$305$	0	0
$306$	1.29548e10	0.0844668
$307$	6.02579e10i	0.387160i	0.981084	+	0.193580i	$0.0620100\pi$
$307$	6.02579e10i	0.387160i	−0.981084	+	0.193580i	$0.937990\pi$
$308$	− 3.68613e10i	− 0.233395i
$309$	2.45691e10	0.153312
$310$	0	0
$311$	1.36816e11	0.829308	0.414654	−	0.909979i	$-0.363903\pi$
$311$	1.36816e11	0.829308	0.414654	+	0.909979i	$0.363903\pi$
$312$	8.29866e10i	0.495807i
$313$	2.25535e11i	1.32820i	0.747643	+	0.664101i	$0.231185\pi$
$313$	2.25535e11i	1.32820i	−0.747643	+	0.664101i	$0.768815\pi$
$314$	−3.60839e9	−0.0209474
$315$	0	0
$316$	2.05809e11	1.16111
$317$	1.28386e11i	0.714089i	0.934087	+	0.357045i	$0.116216\pi$
$317$	1.28386e11i	0.714089i	−0.934087	+	0.357045i	$0.883784\pi$
$318$	3.59755e9i	0.0197281i
$319$	−1.14800e11	−0.620701
$320$	0	0
$321$	1.58285e10	0.0832086
$322$	− 1.40308e10i	− 0.0727328i
$323$	− 2.16657e10i	− 0.110755i
$324$	1.94302e10	0.0979545
$325$	0	0
$326$	9.96239e10	0.488522
$327$	1.87510e11i	0.906902i
$328$	1.83696e11i	0.876329i
$329$	1.13477e11	0.533981
$330$	0	0
$331$	1.15018e10	0.0526673	0.0263337	−	0.999653i	$-0.491617\pi$
$331$	1.15018e10	0.0526673	0.0263337	+	0.999653i	$0.491617\pi$
$332$	− 1.60286e11i	− 0.724058i
$333$	1.02394e11i	0.456324i
$334$	−1.44590e11	−0.635740
$335$	0	0
$336$	−2.57372e10	−0.110163
$337$	2.79631e11i	1.18100i	0.807036	+	0.590502i	$0.201070\pi$
$337$	2.79631e11i	1.18100i	−0.807036	+	0.590502i	$0.798930\pi$
$338$	− 6.26843e10i	− 0.261236i
$339$	1.06875e11	0.439519
$340$	0	0
$341$	3.97159e11	1.59063
$342$	4.36454e9i	0.0172513i
$343$	− 1.42263e11i	− 0.554969i
$344$	−9.57078e10	−0.368497
$345$	0	0
$346$	−3.82107e10	−0.143332
$347$	− 9.07088e10i	− 0.335867i	−0.985798	−	0.167933i	$-0.946291\pi$
$347$	− 9.07088e10i	− 0.335867i	0.985798	−	0.167933i	$-0.0537093\pi$
$348$	9.45621e10i	0.345629i
$349$	3.98825e11	1.43902	0.719511	−	0.694481i	$-0.244366\pi$
$349$	3.98825e11	1.43902	0.719511	+	0.694481i	$0.244366\pi$
$350$	0	0
$351$	−7.25862e10	−0.255254
$352$	2.30151e11i	0.799044i
$353$	− 4.56192e11i	− 1.56373i	−0.623448	−	0.781865i	$-0.714269\pi$
$353$	− 4.56192e11i	− 1.56373i	0.623448	−	0.781865i	$-0.285731\pi$
$354$	−5.26684e10	−0.178252
$355$	0	0
$356$	1.91627e11	0.632311
$357$	− 3.77926e10i	− 0.123140i
$358$	1.00389e11i	0.323006i
$359$	−1.89176e11	−0.601094	−0.300547	−	0.953767i	$-0.597169\pi$
$359$	−1.89176e11	−0.601094	−0.300547	+	0.953767i	$0.597169\pi$
$360$	0	0
$361$	−3.15388e11	−0.977380
$362$	2.07296e11i	0.634458i
$363$	3.14152e10i	0.0949643i
$364$	1.13429e11	0.338664
$365$	0	0
$366$	9.37339e10	0.273044
$367$	3.17920e10i	0.0914787i	0.998953	+	0.0457394i	$0.0145644\pi$
$367$	3.17920e10i	0.0914787i	−0.998953	+	0.0457394i	$0.985436\pi$
$368$	− 1.69142e11i	− 0.480769i
$369$	−1.60674e11	−0.451156
$370$	0	0
$371$	1.04950e10	0.0287607
$372$	− 3.27146e11i	− 0.885723i
$373$	− 4.46388e11i	− 1.19405i	−0.802222	−	0.597025i	$-0.796349\pi$
$373$	− 4.46388e11i	− 1.19405i	0.802222	−	0.597025i	$-0.203651\pi$
$374$	−8.76409e10	−0.231625
$375$	0	0
$376$	−4.62639e11	−1.19371
$377$	− 3.53260e11i	− 0.900655i
$378$	7.61330e9i	0.0191805i
$379$	−3.53467e11	−0.879978	−0.439989	−	0.898003i	$-0.645018\pi$
$379$	−3.53467e11	−0.879978	−0.439989	+	0.898003i	$0.645018\pi$
$380$	0	0
$381$	−2.58357e11	−0.628143
$382$	− 2.12411e11i	− 0.510377i
$383$	3.14974e11i	0.747963i	0.927436	+	0.373981i	$0.122008\pi$
$383$	3.14974e11i	0.747963i	−0.927436	+	0.373981i	$0.877992\pi$
$384$	2.45345e11	0.575820
$385$	0	0
$386$	7.51718e10	0.172350
$387$	− 8.37131e10i	− 0.189712i
$388$	− 5.41220e11i	− 1.21236i
$389$	4.20729e11	0.931600	0.465800	−	0.884890i	$-0.345767\pi$
$389$	4.20729e11	0.931600	0.465800	+	0.884890i	$0.345767\pi$
$390$	0	0
$391$	2.48369e11	0.537405
$392$	2.77303e11i	0.593155i
$393$	4.37493e11i	0.925134i
$394$	9.42843e10	0.197109
$395$	0	0
$396$	−1.31447e11	−0.268611
$397$	− 5.63093e11i	− 1.13769i	−0.822446	−	0.568844i	$-0.807391\pi$
$397$	− 5.63093e11i	− 1.13769i	0.822446	−	0.568844i	$-0.192609\pi$
$398$	− 1.47737e11i	− 0.295132i
$399$	1.27325e10	0.0251498
$400$	0	0
$401$	2.75897e11	0.532841	0.266420	−	0.963857i	$-0.414159\pi$
$401$	2.75897e11	0.532841	0.266420	+	0.963857i	$0.414159\pi$
$402$	1.05957e11i	0.202355i
$403$	1.22213e12i	2.30805i
$404$	−7.99317e11	−1.49280
$405$	0	0
$406$	−3.70521e10	−0.0676778
$407$	− 6.92703e11i	− 1.25133i
$408$	1.54079e11i	0.275278i
$409$	6.54552e10	0.115662	0.0578308	−	0.998326i	$-0.481582\pi$
$409$	6.54552e10	0.115662	0.0578308	+	0.998326i	$0.481582\pi$
$410$	0	0
$411$	−9.71214e8	−0.00167891
$412$	1.36912e11i	0.234101i
$413$	1.53647e11i	0.259866i
$414$	−5.00337e10	−0.0837069
$415$	0	0
$416$	−7.08218e11	−1.15944
$417$	7.41541e10i	0.120094i
$418$	− 2.95266e10i	− 0.0473064i
$419$	−2.70250e10	−0.0428354	−0.0214177	−	0.999771i	$-0.506818\pi$
$419$	−2.70250e10	−0.0428354	−0.0214177	+	0.999771i	$0.506818\pi$
$420$	0	0
$421$	−4.29698e11	−0.666644	−0.333322	−	0.942813i	$-0.608170\pi$
$421$	−4.29698e11	−0.666644	−0.333322	+	0.942813i	$0.608170\pi$
$422$	− 6.94478e10i	− 0.106599i
$423$	− 4.04659e11i	− 0.614550i
$424$	−4.27875e10	−0.0642940
$425$	0	0
$426$	1.33066e11	0.195760
$427$	− 2.73446e11i	− 0.398058i
$428$	8.82048e10i	0.127056i
$429$	4.91054e11	0.699957
$430$	0	0
$431$	−9.44700e11	−1.31870	−0.659350	−	0.751836i	$-0.729169\pi$
$431$	−9.44700e11	−1.31870	−0.659350	+	0.751836i	$0.729169\pi$
$432$	9.17789e10i	0.126784i
$433$	2.10762e11i	0.288136i	0.989568	+	0.144068i	$0.0460184\pi$
$433$	2.10762e11i	0.288136i	−0.989568	+	0.144068i	$0.953982\pi$
$434$	1.28185e11	0.173434
$435$	0	0
$436$	−1.04491e12	−1.38480
$437$	8.36765e10i	0.109758i
$438$	− 9.05253e10i	− 0.117527i
$439$	−7.69079e11	−0.988282	−0.494141	−	0.869382i	$-0.664517\pi$
$439$	−7.69079e11	−0.988282	−0.494141	+	0.869382i	$0.664517\pi$
$440$	0	0
$441$	−2.42550e11	−0.305371
$442$	− 2.69688e11i	− 0.336094i
$443$	− 6.49894e11i	− 0.801726i	−0.916138	−	0.400863i	$-0.868710\pi$
$443$	− 6.49894e11i	− 0.801726i	0.916138	−	0.400863i	$-0.131290\pi$
$444$	−5.70590e11	−0.696788
$445$	0	0
$446$	3.44741e11	0.412559
$447$	4.05374e11i	0.480255i
$448$	− 8.84023e10i	− 0.103684i
$449$	4.51858e11	0.524679	0.262339	−	0.964976i	$-0.415506\pi$
$449$	4.51858e11	0.524679	0.262339	+	0.964976i	$0.415506\pi$
$450$	0	0
$451$	1.08698e12	1.23716
$452$	5.95564e11i	0.671128i
$453$	− 2.59649e11i	− 0.289697i
$454$	−4.20493e11	−0.464523
$455$	0	0
$456$	−5.19097e10	−0.0562220
$457$	6.18434e11i	0.663240i	0.943413	+	0.331620i	$0.107595\pi$
$457$	6.18434e11i	0.663240i	−0.943413	+	0.331620i	$0.892405\pi$
$458$	− 4.51798e10i	− 0.0479788i
$459$	−1.34768e11	−0.141720
$460$	0	0
$461$	−2.76450e11	−0.285078	−0.142539	−	0.989789i	$-0.545527\pi$
$461$	−2.76450e11	−0.285078	−0.142539	+	0.989789i	$0.545527\pi$
$462$	− 5.15048e10i	− 0.0525967i
$463$	6.09969e11i	0.616870i	0.951245	+	0.308435i	$0.0998052\pi$
$463$	6.09969e11i	0.616870i	−0.951245	+	0.308435i	$0.900195\pi$
$464$	−4.46666e11	−0.447355
$465$	0	0
$466$	−4.21910e11	−0.414461
$467$	− 2.83260e11i	− 0.275587i	−0.990461	−	0.137794i	$-0.955999\pi$
$467$	− 2.83260e11i	− 0.275587i	0.990461	−	0.137794i	$-0.0440011\pi$
$468$	− 4.04488e11i	− 0.389762i
$469$	3.09104e11	0.295004
$470$	0	0
$471$	3.75378e10	0.0351459
$472$	− 6.26412e11i	− 0.580926i
$473$	5.66328e11i	0.520227i
$474$	2.87568e11	0.261661
$475$	0	0
$476$	2.10600e11	0.188030
$477$	− 3.74251e10i	− 0.0331002i
$478$	− 6.16485e11i	− 0.540129i
$479$	1.39149e12	1.20773	0.603865	−	0.797087i	$-0.293627\pi$
$479$	1.39149e12	1.20773	0.603865	+	0.797087i	$0.293627\pi$
$480$	0	0
$481$	2.13158e12	1.81572
$482$	− 4.78793e11i	− 0.404051i
$483$	1.45961e11i	0.122033i
$484$	−1.75062e11	−0.145007
$485$	0	0
$486$	2.71490e10	0.0220745
$487$	1.41532e11i	0.114018i	0.998374	+	0.0570092i	$0.0181564\pi$
$487$	1.41532e11i	0.114018i	−0.998374	+	0.0570092i	$0.981844\pi$
$488$	1.11483e12i	0.889851i
$489$	−1.03638e12	−0.819652
$490$	0	0
$491$	−4.66246e11	−0.362034	−0.181017	−	0.983480i	$-0.557939\pi$
$491$	−4.66246e11	−0.362034	−0.181017	+	0.983480i	$0.557939\pi$
$492$	− 8.95359e11i	− 0.688897i
$493$	− 6.55887e11i	− 0.500055i
$494$	9.08589e10	0.0686430
$495$	0	0
$496$	1.54528e12	1.14641
$497$	− 3.88188e11i	− 0.285389i
$498$	− 2.23961e11i	− 0.163170i
$499$	2.21254e12	1.59749	0.798746	−	0.601668i	$-0.205497\pi$
$499$	2.21254e12	1.59749	0.798746	+	0.601668i	$0.205497\pi$
$500$	0	0
$501$	1.50416e12	1.06666
$502$	2.25271e11i	0.158321i
$503$	3.39574e11i	0.236526i	0.992982	+	0.118263i	$0.0377326\pi$
$503$	3.39574e11i	0.236526i	−0.992982	+	0.118263i	$0.962267\pi$
$504$	−9.05488e10	−0.0625094
$505$	0	0
$506$	3.38484e11	0.229541
$507$	6.52101e11i	0.438308i
$508$	− 1.43970e12i	− 0.959149i
$509$	5.66691e11	0.374211	0.187106	−	0.982340i	$-0.440089\pi$
$509$	5.66691e11	0.374211	0.187106	+	0.982340i	$0.440089\pi$
$510$	0	0
$511$	−2.64086e11	−0.171337
$512$	1.55874e12i	1.00244i
$513$	− 4.54040e10i	− 0.0289445i
$514$	9.52947e11	0.602191
$515$	0	0
$516$	4.66493e11	0.289682
$517$	2.73756e12i	1.68522i
$518$	− 2.23574e11i	− 0.136438i
$519$	3.97504e11	0.240485
$520$	0	0
$521$	4.42970e11	0.263393	0.131697	−	0.991290i	$-0.457958\pi$
$521$	4.42970e11	0.263393	0.131697	+	0.991290i	$0.457958\pi$
$522$	1.32128e11i	0.0778892i
$523$	− 1.44683e11i	− 0.0845591i	−0.999106	−	0.0422796i	$-0.986538\pi$
$523$	− 1.44683e11i	− 0.0845591i	0.999106	−	0.0422796i	$-0.0134620\pi$
$524$	−2.43794e12	−1.41264
$525$	0	0
$526$	4.86557e11	0.277139
$527$	2.26910e12i	1.28146i
$528$	− 6.20894e11i	− 0.347668i
$529$	8.41911e11	0.467429
$530$	0	0
$531$	5.47906e11	0.299075
$532$	7.09520e10i	0.0384028i
$533$	3.34484e12i	1.79516i
$534$	2.67752e11	0.142494
$535$	0	0
$536$	−1.26020e12	−0.659475
$537$	− 1.04434e12i	− 0.541946i
$538$	− 9.90519e11i	− 0.509733i
$539$	1.64088e12	0.837388
$540$	0	0
$541$	−3.10308e11	−0.155742	−0.0778710	−	0.996963i	$-0.524812\pi$
$541$	−3.10308e11	−0.155742	−0.0778710	+	0.996963i	$0.524812\pi$
$542$	1.19999e11i	0.0597281i
$543$	− 2.15649e12i	− 1.06451i
$544$	−1.31493e12	−0.643733
$545$	0	0
$546$	1.58490e11	0.0763194
$547$	− 1.68502e12i	− 0.804750i	−0.915475	−	0.402375i	$-0.868185\pi$
$547$	− 1.68502e12i	− 0.804750i	0.915475	−	0.402375i	$-0.131815\pi$
$548$	− 5.41211e9i	− 0.00256362i
$549$	−9.75108e11	−0.458118
$550$	0	0
$551$	2.20971e11	0.102130
$552$	− 5.95077e11i	− 0.272802i
$553$	− 8.38911e11i	− 0.381463i
$554$	−7.17011e11	−0.323394
$555$	0	0
$556$	−4.13225e11	−0.183379
$557$	2.31328e12i	1.01831i	0.860674	+	0.509156i	$0.170042\pi$
$557$	2.31328e12i	1.01831i	−0.860674	+	0.509156i	$0.829958\pi$
$558$	− 4.57108e11i	− 0.199602i
$559$	−1.74270e12	−0.754865
$560$	0	0
$561$	9.11723e11	0.388624
$562$	− 6.06380e11i	− 0.256408i
$563$	7.03648e11i	0.295167i	0.989050	+	0.147583i	$0.0471495\pi$
$563$	7.03648e11i	0.295167i	−0.989050	+	0.147583i	$0.952850\pi$
$564$	2.25497e12	0.938392
$565$	0	0
$566$	−2.27720e11	−0.0932666
$567$	− 7.92007e10i	− 0.0321814i
$568$	1.58262e12i	0.637983i
$569$	−3.21997e12	−1.28779	−0.643896	−	0.765113i	$-0.722683\pi$
$569$	−3.21997e12	−1.28779	−0.643896	+	0.765113i	$0.722683\pi$
$570$	0	0
$571$	−1.21116e12	−0.476801	−0.238401	−	0.971167i	$-0.576623\pi$
$571$	−1.21116e12	−0.476801	−0.238401	+	0.971167i	$0.576623\pi$
$572$	2.73641e12i	1.06880i
$573$	2.20969e12i	0.856320i
$574$	3.50827e11	0.134893
$575$	0	0
$576$	−3.15242e11	−0.119328
$577$	− 7.30673e11i	− 0.274430i	−0.990541	−	0.137215i	$-0.956185\pi$
$577$	− 7.30673e11i	− 0.274430i	0.990541	−	0.137215i	$-0.0438151\pi$
$578$	4.22635e11i	0.157504i
$579$	−7.82008e11	−0.289173
$580$	0	0
$581$	−6.53352e11	−0.237878
$582$	− 7.56226e11i	− 0.273211i
$583$	2.53185e11i	0.0907673i
$584$	1.07666e12	0.383021
$585$	0	0
$586$	1.77062e12	0.620278
$587$	− 4.55331e12i	− 1.58291i	−0.611230	−	0.791453i	$-0.709325\pi$
$587$	− 4.55331e12i	− 1.58291i	0.611230	−	0.791453i	$-0.290675\pi$
$588$	− 1.35162e12i	− 0.466289i
$589$	−7.64467e11	−0.261722
$590$	0	0
$591$	−9.80833e11	−0.330713
$592$	− 2.69520e12i	− 0.901867i
$593$	3.00074e12i	0.996510i	0.867030	+	0.498255i	$0.166026\pi$
$593$	3.00074e12i	0.996510i	−0.867030	+	0.498255i	$0.833974\pi$
$594$	−1.83666e11	−0.0605327
$595$	0	0
$596$	−2.25896e12	−0.733330
$597$	1.53690e12i	0.495178i
$598$	1.04158e12i	0.333071i
$599$	4.03514e12	1.28067	0.640336	−	0.768095i	$-0.278795\pi$
$599$	4.03514e12	1.28067	0.640336	+	0.768095i	$0.278795\pi$
$600$	0	0
$601$	2.04760e12	0.640192	0.320096	−	0.947385i	$-0.396285\pi$
$601$	2.04760e12	0.640192	0.320096	+	0.947385i	$0.396285\pi$
$602$	1.82785e11i	0.0567227i
$603$	− 1.10227e12i	− 0.339514i
$604$	1.44690e12	0.442355
$605$	0	0
$606$	−1.11685e12	−0.336410
$607$	− 3.15792e12i	− 0.944175i	−0.881552	−	0.472088i	$-0.843501\pi$
$607$	− 3.15792e12i	− 0.944175i	0.881552	−	0.472088i	$-0.156499\pi$
$608$	− 4.43004e11i	− 0.131474i
$609$	3.85451e11	0.113551
$610$	0	0
$611$	−8.42399e12	−2.44530
$612$	− 7.51000e11i	− 0.216401i
$613$	− 2.89302e12i	− 0.827520i	−0.910386	−	0.413760i	$-0.864215\pi$
$613$	− 2.89302e12i	− 0.827520i	0.910386	−	0.413760i	$-0.135785\pi$
$614$	−4.69183e11	−0.133225
$615$	0	0
$616$	6.12573e11	0.171413
$617$	− 6.03603e12i	− 1.67675i	−0.545094	−	0.838375i	$-0.683506\pi$
$617$	− 6.03603e12i	− 1.67675i	0.545094	−	0.838375i	$-0.316494\pi$
$618$	1.91301e11i	0.0527558i
$619$	−4.05606e12	−1.11044	−0.555222	−	0.831702i	$-0.687367\pi$
$619$	−4.05606e12	−1.11044	−0.555222	+	0.831702i	$0.687367\pi$
$620$	0	0
$621$	5.20498e11	0.140445
$622$	1.06529e12i	0.285371i
$623$	− 7.81102e11i	− 0.207736i
$624$	1.91061e12	0.504477
$625$	0	0
$626$	−1.75607e12	−0.457044
$627$	3.07163e11i	0.0793716i
$628$	2.09180e11i	0.0536664i
$629$	3.95764e12	1.00811
$630$	0	0
$631$	−5.34498e12	−1.34219	−0.671095	−	0.741372i	$-0.734176\pi$
$631$	−5.34498e12	−1.34219	−0.671095	+	0.741372i	$0.734176\pi$
$632$	3.42020e12i	0.852755i
$633$	7.22461e11i	0.178854i
$634$	−9.99651e11	−0.245723
$635$	0	0
$636$	2.08552e11	0.0505426
$637$	5.04929e12i	1.21507i
$638$	− 8.93859e11i	− 0.213588i
$639$	−1.38428e12	−0.328450
$640$	0	0
$641$	4.81739e12	1.12707	0.563534	−	0.826093i	$-0.309441\pi$
$641$	4.81739e12	1.12707	0.563534	+	0.826093i	$0.309441\pi$
$642$	1.23245e11i	0.0286327i
$643$	− 5.85505e11i	− 0.135077i	−0.997717	−	0.0675385i	$-0.978485\pi$
$643$	− 5.85505e11i	− 0.135077i	0.997717	−	0.0675385i	$-0.0215145\pi$
$644$	−8.13372e11	−0.186339
$645$	0	0
$646$	1.68695e11	0.0381114
$647$	− 4.54903e12i	− 1.02059i	−0.860001	−	0.510293i	$-0.829537\pi$
$647$	− 4.54903e12i	− 1.02059i	0.860001	−	0.510293i	$-0.170463\pi$
$648$	3.22897e11i	0.0719410i
$649$	−3.70664e12	−0.820124
$650$	0	0
$651$	−1.33350e12	−0.290991
$652$	− 5.77526e12i	− 1.25158i
$653$	− 7.04350e12i	− 1.51593i	−0.652295	−	0.757965i	$-0.726194\pi$
$653$	− 7.04350e12i	− 1.51593i	0.652295	−	0.757965i	$-0.273806\pi$
$654$	−1.46001e12	−0.312072
$655$	0	0
$656$	4.22925e12	0.891653
$657$	9.41729e11i	0.197189i
$658$	8.83561e11i	0.183747i
$659$	−5.23559e12	−1.08139	−0.540694	−	0.841219i	$-0.681838\pi$
$659$	−5.23559e12	−1.08139	−0.540694	+	0.841219i	$0.681838\pi$
$660$	0	0
$661$	−5.79017e12	−1.17974	−0.589868	−	0.807500i	$-0.700820\pi$
$661$	−5.79017e12	−1.17974	−0.589868	+	0.807500i	$0.700820\pi$
$662$	8.95563e10i	0.0181232i
$663$	2.80555e12i	0.563906i
$664$	2.66368e12	0.531772
$665$	0	0
$666$	−7.97263e11	−0.157025
$667$	2.53314e12i	0.495557i
$668$	8.38197e12i	1.62874i
$669$	−3.58632e12	−0.692199
$670$	0	0
$671$	6.59671e12	1.25625
$672$	− 7.72755e11i	− 0.146177i
$673$	− 5.92328e10i	− 0.0111300i	−0.999985	−	0.00556499i	$-0.998229\pi$
$673$	− 5.92328e10i	− 0.0111300i	0.999985	−	0.00556499i	$-0.00177140\pi$
$674$	−2.17728e12	−0.406392
$675$	0	0
$676$	−3.63385e12	−0.669278
$677$	− 4.57177e12i	− 0.836440i	−0.908346	−	0.418220i	$-0.862654\pi$
$677$	− 4.57177e12i	− 0.836440i	0.908346	−	0.418220i	$-0.137346\pi$
$678$	8.32157e11i	0.151242i
$679$	−2.20611e12	−0.398302
$680$	0	0
$681$	4.37436e12	0.779386
$682$	3.09238e12i	0.547348i
$683$	− 9.71286e12i	− 1.70787i	−0.520382	−	0.853934i	$-0.674210\pi$
$683$	− 9.71286e12i	− 1.70787i	0.520382	−	0.853934i	$-0.325790\pi$
$684$	2.53015e11	0.0441971
$685$	0	0
$686$	1.10770e12	0.190969
$687$	4.70003e11i	0.0804998i
$688$	2.20349e12i	0.374941i
$689$	−7.79098e11	−0.131706
$690$	0	0
$691$	2.23726e12	0.373306	0.186653	−	0.982426i	$-0.440236\pi$
$691$	2.23726e12	0.373306	0.186653	+	0.982426i	$0.440236\pi$
$692$	2.21510e12i	0.367211i
$693$	5.35801e11i	0.0882478i
$694$	7.06282e11	0.115574
$695$	0	0
$696$	−1.57146e12	−0.253842
$697$	6.21025e12i	0.996693i
$698$	3.10535e12i	0.495178i
$699$	4.38911e12	0.695391
$700$	0	0
$701$	−2.29477e10	−0.00358929	−0.00179464	−	0.999998i	$-0.500571\pi$
$701$	−2.29477e10	−0.00358929	−0.00179464	+	0.999998i	$0.500571\pi$
$702$	− 5.65175e11i	− 0.0878347i
$703$	1.33334e12i	0.205894i
$704$	2.13265e12	0.327222
$705$	0	0
$706$	3.55203e12	0.538091
$707$	3.25815e12i	0.490438i
$708$	3.05322e12i	0.456676i
$709$	9.57637e12	1.42329	0.711644	−	0.702540i	$-0.247951\pi$
$709$	9.57637e12	1.42329	0.711644	+	0.702540i	$0.247951\pi$
$710$	0	0
$711$	−2.99156e12	−0.439020
$712$	3.18451e12i	0.464390i
$713$	− 8.76362e12i	− 1.26993i
$714$	2.94263e11	0.0423735
$715$	0	0
$716$	5.81959e12	0.827529
$717$	6.41326e12i	0.906239i
$718$	− 1.47298e12i	− 0.206841i
$719$	5.46390e12	0.762469	0.381235	−	0.924478i	$-0.375499\pi$
$719$	5.46390e12	0.762469	0.381235	+	0.924478i	$0.375499\pi$
$720$	0	0
$721$	5.58075e11	0.0769102
$722$	− 2.45570e12i	− 0.336324i
$723$	4.98085e12i	0.677925i
$724$	1.20171e13	1.62546
$725$	0	0
$726$	−2.44607e11	−0.0326779
$727$	− 6.59842e12i	− 0.876062i	−0.898960	−	0.438031i	$-0.855676\pi$
$727$	− 6.59842e12i	− 0.876062i	0.898960	−	0.438031i	$-0.144324\pi$
$728$	1.88500e12i	0.248726i
$729$	−2.82430e11	−0.0370370
$730$	0	0
$731$	−3.23561e12	−0.419111
$732$	− 5.43381e12i	− 0.699527i
$733$	6.91041e11i	0.0884170i	0.999022	+	0.0442085i	$0.0140766\pi$
$733$	6.91041e11i	0.0884170i	−0.999022	+	0.0442085i	$0.985923\pi$
$734$	−2.47541e11	−0.0314785
$735$	0	0
$736$	5.07846e12	0.637943
$737$	7.45694e12i	0.931016i
$738$	− 1.25105e12i	− 0.155246i
$739$	−7.60189e12	−0.937608	−0.468804	−	0.883302i	$-0.655315\pi$
$739$	−7.60189e12	−0.937608	−0.468804	+	0.883302i	$0.655315\pi$
$740$	0	0
$741$	−9.45200e11	−0.115171
$742$	8.17167e10i	0.00989677i
$743$	− 2.23002e12i	− 0.268448i	−0.990951	−	0.134224i	$-0.957146\pi$
$743$	− 2.23002e12i	− 0.268448i	0.990951	−	0.134224i	$-0.0428541\pi$
$744$	5.43661e12	0.650504
$745$	0	0
$746$	3.47569e12	0.410882
$747$	2.32985e12i	0.273770i
$748$	5.08060e12i	0.593414i
$749$	3.59538e11	0.0417423
$750$	0	0
$751$	1.65708e12	0.190092	0.0950459	−	0.995473i	$-0.469700\pi$
$751$	1.65708e12	0.190092	0.0950459	+	0.995473i	$0.469700\pi$
$752$	1.06514e13i	1.21458i
$753$	− 2.34348e12i	− 0.265635i
$754$	2.75058e12	0.309922
$755$	0	0
$756$	4.41347e11	0.0491397
$757$	1.28420e13i	1.42135i	0.703521	+	0.710675i	$0.251610\pi$
$757$	1.28420e13i	1.42135i	−0.703521	+	0.710675i	$0.748390\pi$
$758$	− 2.75218e12i	− 0.302807i
$759$	−3.52122e12	−0.385129
$760$	0	0
$761$	1.61852e13	1.74939	0.874694	−	0.484676i	$-0.161063\pi$
$761$	1.61852e13	1.74939	0.874694	+	0.484676i	$0.161063\pi$
$762$	− 2.01164e12i	− 0.216149i
$763$	4.25921e12i	0.454955i
$764$	−1.23136e13	−1.30757
$765$	0	0
$766$	−2.45247e12	−0.257380
$767$	− 1.14060e13i	− 1.19002i
$768$	− 8.23228e10i	− 0.00853875i
$769$	−5.30462e12	−0.546998	−0.273499	−	0.961872i	$-0.588181\pi$
$769$	−5.30462e12	−0.546998	−0.273499	+	0.961872i	$0.588181\pi$
$770$	0	0
$771$	−9.91345e12	−1.01037
$772$	− 4.35775e12i	− 0.441555i
$773$	− 1.26188e13i	− 1.27119i	−0.772022	−	0.635596i	$-0.780755\pi$
$773$	− 1.26188e13i	− 1.27119i	0.772022	−	0.635596i	$-0.219245\pi$
$774$	6.51812e11	0.0652812
$775$	0	0
$776$	8.99418e12	0.890397
$777$	2.32582e12i	0.228919i
$778$	3.27591e12i	0.320571i
$779$	−2.09226e12	−0.203562
$780$	0	0
$781$	9.36478e12	0.900675
$782$	1.93387e12i	0.184925i
$783$	− 1.37452e12i	− 0.130684i
$784$	6.38438e12	0.603527
$785$	0	0
$786$	−3.40644e12	−0.318346
$787$	− 1.22887e13i	− 1.14188i	−0.820991	−	0.570941i	$-0.806579\pi$
$787$	− 1.22887e13i	− 1.14188i	0.820991	−	0.570941i	$-0.193421\pi$
$788$	− 5.46571e12i	− 0.504985i
$789$	−5.06162e12	−0.464990
$790$	0	0
$791$	2.42762e12	0.220489
$792$	− 2.18443e12i	− 0.197277i
$793$	2.02993e13i	1.82286i
$794$	4.38439e12	0.391487
$795$	0	0
$796$	−8.56441e12	−0.756116
$797$	3.56495e12i	0.312962i	0.987681	+	0.156481i	$0.0500150\pi$
$797$	3.56495e12i	0.312962i	−0.987681	+	0.156481i	$0.949985\pi$
$798$	9.91385e10i	0.00865425i
$799$	−1.56405e13	−1.35766
$800$	0	0
$801$	−2.78541e12	−0.239080
$802$	2.14821e12i	0.183354i
$803$	− 6.37090e12i	− 0.540730i
$804$	6.14240e12	0.518425
$805$	0	0
$806$	−9.51585e12	−0.794218
$807$	1.03043e13i	0.855239i
$808$	− 1.32833e13i	− 1.09636i
$809$	3.11624e12	0.255778	0.127889	−	0.991789i	$-0.459180\pi$
$809$	3.11624e12	0.255778	0.127889	+	0.991789i	$0.459180\pi$
$810$	0	0
$811$	−2.30256e12	−0.186904	−0.0934518	−	0.995624i	$-0.529790\pi$
$811$	−2.30256e12	−0.186904	−0.0934518	+	0.995624i	$0.529790\pi$
$812$	2.14793e12i	0.173388i
$813$	− 1.24834e12i	− 0.100213i
$814$	5.39357e12	0.430593
$815$	0	0
$816$	3.54737e12	0.280092
$817$	− 1.09009e12i	− 0.0855980i
$818$	5.09651e11i	0.0398000i
$819$	−1.64876e12	−0.128050
$820$	0	0
$821$	−6.28720e12	−0.482962	−0.241481	−	0.970406i	$-0.577633\pi$
$821$	−6.28720e12	−0.482962	−0.241481	+	0.970406i	$0.577633\pi$
$822$	− 7.56212e9i	0 0.000577724i
$823$	3.76056e12i	0.285728i	0.989742	+	0.142864i	$0.0456312\pi$
$823$	3.76056e12i	0.285728i	−0.989742	+	0.142864i	$0.954369\pi$
$824$	−2.27524e12	−0.171932
$825$	0	0
$826$	−1.19634e12	−0.0894219
$827$	− 1.75054e13i	− 1.30136i	−0.759353	−	0.650679i	$-0.774484\pi$
$827$	− 1.75054e13i	− 1.30136i	0.759353	−	0.650679i	$-0.225516\pi$
$828$	2.90049e12i	0.214454i
$829$	2.42570e12	0.178378	0.0891891	−	0.996015i	$-0.471572\pi$
$829$	2.42570e12	0.178378	0.0891891	+	0.996015i	$0.471572\pi$
$830$	0	0
$831$	7.45902e12	0.542597
$832$	6.56257e12i	0.474809i
$833$	9.37486e12i	0.674625i
$834$	−5.77383e11	−0.0413254
$835$	0	0
$836$	−1.71167e12	−0.121197
$837$	4.75526e12i	0.334896i
$838$	− 2.10424e11i	− 0.0147400i
$839$	−9.69596e11	−0.0675557	−0.0337778	−	0.999429i	$-0.510754\pi$
$839$	−9.69596e11	−0.0675557	−0.0337778	+	0.999429i	$0.510754\pi$
$840$	0	0
$841$	−7.81769e12	−0.538885
$842$	− 3.34574e12i	− 0.229397i
$843$	6.30813e12i	0.430206i
$844$	−4.02593e12	−0.273102
$845$	0	0
$846$	3.15078e12	0.211471
$847$	7.13582e11i	0.0476397i
$848$	9.85101e11i	0.0654183i
$849$	2.36895e12	0.156485
$850$	0	0
$851$	−1.52850e13	−0.999042
$852$	− 7.71391e12i	− 0.501529i
$853$	− 2.11898e13i	− 1.37043i	−0.728340	−	0.685215i	$-0.759708\pi$
$853$	− 2.11898e13i	− 1.37043i	0.728340	−	0.685215i	$-0.240292\pi$
$854$	2.12912e12	0.136975
$855$	0	0
$856$	−1.46582e12	−0.0933142
$857$	− 8.34904e12i	− 0.528717i	−0.964425	−	0.264358i	$-0.914840\pi$
$857$	− 8.34904e12i	− 0.528717i	0.964425	−	0.264358i	$-0.0851602\pi$
$858$	3.82347e12i	0.240860i
$859$	8.24621e12	0.516755	0.258378	−	0.966044i	$-0.416812\pi$
$859$	8.24621e12	0.516755	0.258378	+	0.966044i	$0.416812\pi$
$860$	0	0
$861$	−3.64964e12	−0.226326
$862$	− 7.35568e12i	− 0.453775i
$863$	4.35663e12i	0.267364i	0.991024	+	0.133682i	$0.0426800\pi$
$863$	4.35663e12i	0.267364i	−0.991024	+	0.133682i	$0.957320\pi$
$864$	−2.75564e12	−0.168233
$865$	0	0
$866$	−1.64105e12	−0.0991497
$867$	− 4.39665e12i	− 0.264263i
$868$	− 7.43096e12i	− 0.444331i
$869$	2.02382e13	1.20388
$870$	0	0
$871$	−2.29464e13	−1.35093
$872$	− 1.73646e13i	− 1.01705i
$873$	7.86697e12i	0.458399i
$874$	−6.51527e11	−0.0377686
$875$	0	0
$876$	−5.24780e12	−0.301099
$877$	1.21873e12i	0.0695680i	0.999395	+	0.0347840i	$0.0110743\pi$
$877$	1.21873e12i	0.0695680i	−0.999395	+	0.0347840i	$0.988926\pi$
$878$	− 5.98825e12i	− 0.340075i
$879$	−1.84197e13	−1.04071
$880$	0	0
$881$	3.98577e12	0.222905	0.111453	−	0.993770i	$-0.464450\pi$
$881$	3.98577e12	0.222905	0.111453	+	0.993770i	$0.464450\pi$
$882$	− 1.88856e12i	− 0.105080i
$883$	2.19963e13i	1.21766i	0.793300	+	0.608831i	$0.208361\pi$
$883$	2.19963e13i	1.21766i	−0.793300	+	0.608831i	$0.791639\pi$
$884$	−1.56340e13	−0.861061
$885$	0	0
$886$	5.06025e12	0.275880
$887$	− 1.39860e13i	− 0.758640i	−0.925266	−	0.379320i	$-0.876158\pi$
$887$	− 1.39860e13i	− 0.758640i	0.925266	−	0.379320i	$-0.123842\pi$
$888$	− 9.48226e12i	− 0.511745i
$889$	−5.86847e12	−0.315113
$890$	0	0
$891$	1.91067e12	0.101563
$892$	− 1.99848e13i	− 1.05696i
$893$	− 5.26936e12i	− 0.277285i
$894$	−3.15635e12	−0.165259
$895$	0	0
$896$	5.57290e12	0.288865
$897$	− 1.08355e13i	− 0.558833i
$898$	3.51829e12i	0.180546i
$899$	−2.31428e13	−1.18167
$900$	0	0
$901$	−1.44653e12	−0.0731248
$902$	8.46349e12i	0.425716i
$903$	− 1.90150e12i	− 0.0951705i
$904$	−9.89727e12	−0.492899
$905$	0	0
$906$	2.02169e12	0.0996869
$907$	8.00749e12i	0.392884i	0.980515	+	0.196442i	$0.0629387\pi$
$907$	8.00749e12i	0.392884i	−0.980515	+	0.196442i	$0.937061\pi$
$908$	2.43762e13i	1.19009i
$909$	1.16186e13	0.564436
$910$	0	0
$911$	−4.09248e13	−1.96858	−0.984291	−	0.176555i	$-0.943505\pi$
$911$	−4.09248e13	−1.96858	−0.984291	+	0.176555i	$0.943505\pi$
$912$	1.19512e12i	0.0572052i
$913$	− 1.57617e13i	− 0.750731i
$914$	−4.81529e12	−0.228226
$915$	0	0
$916$	−2.61910e12	−0.122920
$917$	9.93745e12i	0.464102i
$918$	− 1.04934e12i	− 0.0487669i
$919$	−8.12730e12	−0.375860	−0.187930	−	0.982182i	$-0.560178\pi$
$919$	−8.12730e12	−0.375860	−0.187930	+	0.982182i	$0.560178\pi$
$920$	0	0
$921$	4.88089e12	0.223527
$922$	− 2.15252e12i	− 0.0980973i
$923$	2.88172e13i	1.30691i
$924$	−2.98576e12	−0.134751
$925$	0	0
$926$	−4.74938e12	−0.212269
$927$	− 1.99010e12i	− 0.0885147i
$928$	− 1.34111e13i	− 0.593605i
$929$	4.43436e13	1.95326	0.976631	−	0.214923i	$-0.0689500\pi$
$929$	4.43436e13	1.95326	0.976631	+	0.214923i	$0.0689500\pi$
$930$	0	0
$931$	−3.15843e12	−0.137783
$932$	2.44584e13i	1.06183i
$933$	− 1.10821e13i	− 0.478801i
$934$	2.20554e12	0.0948317
$935$	0	0
$936$	6.72192e12	0.286254
$937$	3.56882e13i	1.51250i	0.654281	+	0.756251i	$0.272971\pi$
$937$	3.56882e13i	1.51250i	−0.654281	+	0.756251i	$0.727029\pi$
$938$	2.40677e12i	0.101513i
$939$	1.82683e13	0.766838
$940$	0	0
$941$	3.76351e13	1.56473	0.782366	−	0.622818i	$-0.214012\pi$
$941$	3.76351e13	1.56473	0.782366	+	0.622818i	$0.214012\pi$
$942$	2.92279e11i	0.0120940i
$943$	− 2.39850e13i	− 0.987727i
$944$	−1.44219e13	−0.591085
$945$	0	0
$946$	−4.40958e12	−0.179014
$947$	1.01014e13i	0.408139i	0.978956	+	0.204070i	$0.0654169\pi$
$947$	1.01014e13i	0.408139i	−0.978956	+	0.204070i	$0.934583\pi$
$948$	− 1.66705e13i	− 0.670365i
$949$	1.96045e13	0.784616
$950$	0	0
$951$	1.03993e13	0.412280
$952$	3.49982e12i	0.138096i
$953$	1.49469e13i	0.586993i	0.955960	+	0.293497i	$0.0948190\pi$
$953$	1.49469e13i	0.586993i	−0.955960	+	0.293497i	$0.905181\pi$
$954$	2.91402e11	0.0113900
$955$	0	0
$956$	−3.57380e13	−1.38379
$957$	9.29876e12i	0.358362i
$958$	1.08345e13i	0.415589i
$959$	−2.20607e10	−0.000842238 0
$960$	0	0
$961$	5.36247e13	2.02819
$962$	1.65970e13i	0.624803i
$963$	− 1.28211e12i	− 0.0480405i
$964$	−2.77559e13	−1.03516
$965$	0	0
$966$	−1.13649e12	−0.0419923
$967$	− 2.42942e13i	− 0.893475i	−0.894665	−	0.446738i	$-0.852586\pi$
$967$	− 2.42942e13i	− 0.893475i	0.894665	−	0.446738i	$-0.147414\pi$
$968$	− 2.90924e12i	− 0.106498i
$969$	−1.75492e12	−0.0639441
$970$	0	0
$971$	−1.07032e13	−0.386392	−0.193196	−	0.981160i	$-0.561885\pi$
$971$	−1.07032e13	−0.386392	−0.193196	+	0.981160i	$0.561885\pi$
$972$	− 1.57384e12i	− 0.0565540i
$973$	1.68437e12i	0.0602464i
$974$	−1.10201e12	−0.0392346
$975$	0	0
$976$	2.56667e13	0.905412
$977$	1.53038e13i	0.537372i	0.963228	+	0.268686i	$0.0865894\pi$
$977$	1.53038e13i	0.537372i	−0.963228	+	0.268686i	$0.913411\pi$
$978$	− 8.06954e12i	− 0.282048i
$979$	1.88436e13	0.655604
$980$	0	0
$981$	1.51883e13	0.523600
$982$	− 3.63032e12i	− 0.124578i
$983$	3.27251e13i	1.11787i	0.829212	+	0.558934i	$0.188790\pi$
$983$	3.27251e13i	1.11787i	−0.829212	+	0.558934i	$0.811210\pi$
$984$	1.48794e13	0.505949
$985$	0	0
$986$	5.10691e12	0.172073
$987$	− 9.19163e12i	− 0.308294i
$988$	− 5.26714e12i	− 0.175861i
$989$	1.24965e13	0.415340
$990$	0	0
$991$	3.05488e13	1.00615	0.503075	−	0.864243i	$-0.332202\pi$
$991$	3.05488e13	1.00615	0.503075	+	0.864243i	$0.332202\pi$
$992$	4.63967e13i	1.52119i
$993$	− 9.31649e11i	− 0.0304075i
$994$	3.02253e12	0.0982046
$995$	0	0
$996$	−1.29831e13	−0.418035
$997$	− 4.87857e13i	− 1.56374i	−0.623441	−	0.781870i	$-0.714266\pi$
$997$	− 4.87857e13i	− 1.56374i	0.623441	−	0.781870i	$-0.285734\pi$
$998$	1.72274e13i	0.549709i
$999$	8.29388e12	0.263459

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.10.b.f.49.3		4
3.2	odd	2		225.10.b.i.199.2		4
5.2	odd	4		15.10.a.d.1.1	✓	2
5.3	odd	4		75.10.a.f.1.2		2
5.4	even	2	inner	75.10.b.f.49.2		4
15.2	even	4		45.10.a.d.1.2		2
15.8	even	4		225.10.a.k.1.1		2
15.14	odd	2		225.10.b.i.199.3		4
20.7	even	4		240.10.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.10.a.d.1.1	✓	2	5.2	odd	4
45.10.a.d.1.2		2	15.2	even	4
75.10.a.f.1.2		2	5.3	odd	4
75.10.b.f.49.2		4	5.4	even	2	inner
75.10.b.f.49.3		4	1.1	even	1	trivial
225.10.a.k.1.1		2	15.8	even	4
225.10.b.i.199.2		4	3.2	odd	2
225.10.b.i.199.3		4	15.14	odd	2
240.10.a.r.1.2		2	20.7	even	4

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:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	75.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+7.78626i q^{2} -81.0000i q^{3} +451.374 q^{4} +630.687 q^{6} -1839.88i q^{7} +7501.08i q^{8} -6561.00 q^{9} +O(q^{10})\)	\(q+7.78626i q^{2} -81.0000i q^{3} +451.374 q^{4} +630.687 q^{6} -1839.88i q^{7} +7501.08i q^{8} -6561.00 q^{9} +44385.9 q^{11} -36561.3i q^{12} +136584. i q^{13} +14325.8 q^{14} +172698. q^{16} +253591. i q^{17} -51085.7i q^{18} -85435.7 q^{19} -149030. q^{21} +345600. i q^{22} -979409. i q^{23} +607588. q^{24} -1.06348e6 q^{26} +531441. i q^{27} -830473. i q^{28} -2.58640e6 q^{29} +8.94787e6 q^{31} +5.18523e6i q^{32} -3.59526e6i q^{33} -1.97452e6 q^{34} -2.96147e6 q^{36} -1.56064e7i q^{37} -665225. i q^{38} +1.10633e7 q^{39} +2.44893e7 q^{41} -1.16039e6i q^{42} +1.27592e7i q^{43} +2.00346e7 q^{44} +7.62593e6 q^{46} +6.16764e7i q^{47} -1.39886e7i q^{48} +3.69685e7 q^{49} +2.05408e7 q^{51} +6.16504e7i q^{52} +5.70418e6i q^{53} -4.13794e6 q^{54} +1.38011e7 q^{56} +6.92029e6i q^{57} -2.01384e7i q^{58} -8.35095e7 q^{59} +1.48622e8 q^{61} +6.96704e7i q^{62} +1.20714e7i q^{63} +4.80479e7 q^{64} +2.79936e7 q^{66} +1.68003e8i q^{67} +1.14464e8i q^{68} -7.93321e7 q^{69} +2.10986e8 q^{71} -4.92146e7i q^{72} -1.43534e8i q^{73} +1.21515e8 q^{74} -3.85635e7 q^{76} -8.16646e7i q^{77} +8.61416e7i q^{78} +4.55960e8 q^{79} +4.30467e7 q^{81} +1.90680e8i q^{82} -3.55106e8i q^{83} -6.72683e7 q^{84} -9.93465e7 q^{86} +2.09498e8i q^{87} +3.32942e8i q^{88} +4.24540e8 q^{89} +2.51297e8 q^{91} -4.42080e8i q^{92} -7.24777e8i q^{93} -4.80228e8 q^{94} +4.20003e8 q^{96} -1.19905e9i q^{97} +2.87846e8i q^{98} -2.91216e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 1082 q^{4} - 5022 q^{6} - 26244 q^{9}+O(q^{10}) \) 4 * q - 1082 * q^4 - 5022 * q^6 - 26244 * q^9	\( 4 q - 1082 q^{4} - 5022 q^{6} - 26244 q^{9} - 43024 q^{11} - 923328 q^{14} + 774530 q^{16} + 191792 q^{19} - 2286144 q^{21} + 4233546 q^{24} - 10838332 q^{26} + 5356424 q^{29} + 21564864 q^{31} - 11445244 q^{34} + 7099002 q^{36} + 3934008 q^{39} + 52120744 q^{41} + 170859944 q^{44} + 34190784 q^{46} - 146565860 q^{49} + 25426872 q^{51} + 32949342 q^{54} + 484908480 q^{56} + 70989328 q^{59} + 682994680 q^{61} + 410240942 q^{64} + 470048184 q^{66} - 119112768 q^{69} + 420572128 q^{71} - 250480468 q^{74} - 437024728 q^{76} + 49510080 q^{79} + 172186884 q^{81} + 1838386368 q^{84} - 1746292744 q^{86} + 855278232 q^{89} - 2253718656 q^{91} - 3295087792 q^{94} + 1037208726 q^{96} + 282280464 q^{99}+O(q^{100}) \) 4 * q - 1082 * q^4 - 5022 * q^6 - 26244 * q^9 - 43024 * q^11 - 923328 * q^14 + 774530 * q^16 + 191792 * q^19 - 2286144 * q^21 + 4233546 * q^24 - 10838332 * q^26 + 5356424 * q^29 + 21564864 * q^31 - 11445244 * q^34 + 7099002 * q^36 + 3934008 * q^39 + 52120744 * q^41 + 170859944 * q^44 + 34190784 * q^46 - 146565860 * q^49 + 25426872 * q^51 + 32949342 * q^54 + 484908480 * q^56 + 70989328 * q^59 + 682994680 * q^61 + 410240942 * q^64 + 470048184 * q^66 - 119112768 * q^69 + 420572128 * q^71 - 250480468 * q^74 - 437024728 * q^76 + 49510080 * q^79 + 172186884 * q^81 + 1838386368 * q^84 - 1746292744 * q^86 + 855278232 * q^89 - 2253718656 * q^91 - 3295087792 * q^94 + 1037208726 * q^96 + 282280464 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more