LMFDB - Embedded newform 384.9.g.a.127.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,9,Mod(127,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("384.127");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	384.g (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$156.433386263$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.6
Character	$\chi$	$=$	384.127
Dual form			384.9.g.a.127.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+46.7654i q^{3} -679.203 q^{5} +2831.65i q^{7} -2187.00 q^{9} +O(q^{10})$	$q+46.7654i q^{3} -679.203 q^{5} +2831.65i q^{7} -2187.00 q^{9} +19758.3i q^{11} -9310.42 q^{13} -31763.2i q^{15} -11898.2 q^{17} -186421. i q^{19} -132423. q^{21} -161519. i q^{23} +70691.7 q^{25} -102276. i q^{27} -364415. q^{29} -1.63024e6i q^{31} -924005. q^{33} -1.92327e6i q^{35} +1.25222e6 q^{37} -435405. i q^{39} -4.90663e6 q^{41} +6.24745e6i q^{43} +1.48542e6 q^{45} +1.78822e6i q^{47} -2.25346e6 q^{49} -556422. i q^{51} -6.12766e6 q^{53} -1.34199e7i q^{55} +8.71806e6 q^{57} +1.94288e6i q^{59} -2.08497e7 q^{61} -6.19282e6i q^{63} +6.32366e6 q^{65} +4.51436e6i q^{67} +7.55348e6 q^{69} -3.25315e6i q^{71} +2.10836e7 q^{73} +3.30592e6i q^{75} -5.59487e7 q^{77} +2.98235e6i q^{79} +4.78297e6 q^{81} -3.64290e7i q^{83} +8.08127e6 q^{85} -1.70420e7i q^{87} +8.65018e7 q^{89} -2.63639e7i q^{91} +7.62388e7 q^{93} +1.26618e8i q^{95} -6.39157e7 q^{97} -4.32114e7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 1344 q^{5} - 69984 q^{9}+O(q^{10}) $ 32 * q - 1344 * q^5 - 69984 * q^9	$ 32 q - 1344 q^{5} - 69984 q^{9} - 114240 q^{13} - 154560 q^{17} + 1791712 q^{25} - 275520 q^{29} + 2421440 q^{37} - 4374720 q^{41} + 2939328 q^{45} - 14219104 q^{49} - 6224448 q^{53} + 3100032 q^{57} - 13005632 q^{61} + 75175296 q^{65} - 85710400 q^{73} + 154517760 q^{77} + 153055008 q^{81} - 384830848 q^{85} - 182669760 q^{89} + 149817600 q^{93} - 149408192 q^{97}+O(q^{100}) $ 32 * q - 1344 * q^5 - 69984 * q^9 - 114240 * q^13 - 154560 * q^17 + 1791712 * q^25 - 275520 * q^29 + 2421440 * q^37 - 4374720 * q^41 + 2939328 * q^45 - 14219104 * q^49 - 6224448 * q^53 + 3100032 * q^57 - 13005632 * q^61 + 75175296 * q^65 - 85710400 * q^73 + 154517760 * q^77 + 153055008 * q^81 - 384830848 * q^85 - 182669760 * q^89 + 149817600 * q^93 - 149408192 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	46.7654i	0.577350i
$3$	46.7654i	0.577350i
$4$	0	0
$5$	−679.203	−1.08672	−0.543362	−	0.839498i	$-0.682849\pi$
$5$	−679.203	−1.08672	−0.543362	+	0.839498i	$0.682849\pi$
$6$	0	0
$7$	2831.65i	1.17936i	0.807636	+	0.589682i	$0.200747\pi$
$7$	2831.65i	1.17936i	−0.807636	+	0.589682i	$0.799253\pi$
$8$	0	0
$9$	−2187.00	−0.333333
$10$	0	0
$11$	19758.3i	1.34952i	0.738038	+	0.674759i	$0.235753\pi$
$11$	19758.3i	1.34952i	−0.738038	+	0.674759i	$0.764247\pi$
$12$	0	0
$13$	−9310.42	−0.325984	−0.162992	−	0.986627i	$-0.552114\pi$
$13$	−9310.42	−0.325984	−0.162992	+	0.986627i	$0.552114\pi$
$14$	0	0
$15$	− 31763.2i	− 0.627421i
$16$	0	0
$17$	−11898.2	−0.142457	−0.0712286	−	0.997460i	$-0.522692\pi$
$17$	−11898.2	−0.142457	−0.0712286	+	0.997460i	$0.522692\pi$
$18$	0	0
$19$	− 186421.i	− 1.43048i	−0.698880	−	0.715239i	$-0.746318\pi$
$19$	− 186421.i	− 1.43048i	0.698880	−	0.715239i	$-0.253682\pi$
$20$	0	0
$21$	−132423.	−0.680906
$22$	0	0
$23$	− 161519.i	− 0.577180i	−0.957453	−	0.288590i	$-0.906814\pi$
$23$	− 161519.i	− 0.577180i	0.957453	−	0.288590i	$-0.0931865\pi$
$24$	0	0
$25$	70691.7	0.180971
$26$	0	0
$27$	− 102276.i	− 0.192450i
$28$	0	0
$29$	−364415.	−0.515234	−0.257617	−	0.966247i	$-0.582937\pi$
$29$	−364415.	−0.515234	−0.257617	+	0.966247i	$0.582937\pi$
$30$	0	0
$31$	− 1.63024e6i	− 1.76524i	−0.470083	−	0.882622i	$-0.655776\pi$
$31$	− 1.63024e6i	− 1.76524i	0.470083	−	0.882622i	$-0.344224\pi$
$32$	0	0
$33$	−924005.	−0.779145
$34$	0	0
$35$	− 1.92327e6i	− 1.28164i
$36$	0	0
$37$	1.25222e6	0.668150	0.334075	−	0.942547i	$-0.391576\pi$
$37$	1.25222e6	0.668150	0.334075	+	0.942547i	$0.391576\pi$
$38$	0	0
$39$	− 435405.i	− 0.188207i
$40$	0	0
$41$	−4.90663e6	−1.73639	−0.868197	−	0.496220i	$-0.834721\pi$
$41$	−4.90663e6	−1.73639	−0.868197	+	0.496220i	$0.834721\pi$
$42$	0	0
$43$	6.24745e6i	1.82738i	0.406412	+	0.913690i	$0.366780\pi$
$43$	6.24745e6i	1.82738i	−0.406412	+	0.913690i	$0.633220\pi$
$44$	0	0
$45$	1.48542e6	0.362242
$46$	0	0
$47$	1.78822e6i	0.366462i	0.983070	+	0.183231i	$0.0586557\pi$
$47$	1.78822e6i	0.366462i	−0.983070	+	0.183231i	$0.941344\pi$
$48$	0	0
$49$	−2.25346e6	−0.390899
$50$	0	0
$51$	− 556422.i	− 0.0822477i
$52$	0	0
$53$	−6.12766e6	−0.776589	−0.388294	−	0.921535i	$-0.626936\pi$
$53$	−6.12766e6	−0.776589	−0.388294	+	0.921535i	$0.626936\pi$
$54$	0	0
$55$	− 1.34199e7i	− 1.46656i
$56$	0	0
$57$	8.71806e6	0.825887
$58$	0	0
$59$	1.94288e6i	0.160338i	0.996781	+	0.0801691i	$0.0255460\pi$
$59$	1.94288e6i	0.160338i	−0.996781	+	0.0801691i	$0.974454\pi$
$60$	0	0
$61$	−2.08497e7	−1.50585	−0.752923	−	0.658108i	$-0.771357\pi$
$61$	−2.08497e7	−1.50585	−0.752923	+	0.658108i	$0.771357\pi$
$62$	0	0
$63$	− 6.19282e6i	− 0.393121i
$64$	0	0
$65$	6.32366e6	0.354255
$66$	0	0
$67$	4.51436e6i	0.224025i	0.993707	+	0.112013i	$0.0357297\pi$
$67$	4.51436e6i	0.224025i	−0.993707	+	0.112013i	$0.964270\pi$
$68$	0	0
$69$	7.55348e6	0.333235
$70$	0	0
$71$	− 3.25315e6i	− 0.128018i	−0.997949	−	0.0640089i	$-0.979611\pi$
$71$	− 3.25315e6i	− 0.128018i	0.997949	−	0.0640089i	$-0.0203886\pi$
$72$	0	0
$73$	2.10836e7	0.742426	0.371213	−	0.928548i	$-0.378942\pi$
$73$	2.10836e7	0.742426	0.371213	+	0.928548i	$0.378942\pi$
$74$	0	0
$75$	3.30592e6i	0.104484i
$76$	0	0
$77$	−5.59487e7	−1.59157
$78$	0	0
$79$	2.98235e6i	0.0765685i	0.999267	+	0.0382842i	$0.0121892\pi$
$79$	2.98235e6i	0.0765685i	−0.999267	+	0.0382842i	$0.987811\pi$
$80$	0	0
$81$	4.78297e6	0.111111
$82$	0	0
$83$	− 3.64290e7i	− 0.767600i	−0.923416	−	0.383800i	$-0.874615\pi$
$83$	− 3.64290e7i	− 0.767600i	0.923416	−	0.383800i	$-0.125385\pi$
$84$	0	0
$85$	8.08127e6	0.154812
$86$	0	0
$87$	− 1.70420e7i	− 0.297470i
$88$	0	0
$89$	8.65018e7	1.37869	0.689343	−	0.724435i	$-0.257899\pi$
$89$	8.65018e7	1.37869	0.689343	+	0.724435i	$0.257899\pi$
$90$	0	0
$91$	− 2.63639e7i	− 0.384453i
$92$	0	0
$93$	7.62388e7	1.01916
$94$	0	0
$95$	1.26618e8i	1.55454i
$96$	0	0
$97$	−6.39157e7	−0.721972	−0.360986	−	0.932571i	$-0.617560\pi$
$97$	−6.39157e7	−0.721972	−0.360986	+	0.932571i	$0.617560\pi$
$98$	0	0
$99$	− 4.32114e7i	− 0.449840i
$100$	0	0
$101$	1.09765e8	1.05482	0.527409	−	0.849612i	$-0.323164\pi$
$101$	1.09765e8	1.05482	0.527409	+	0.849612i	$0.323164\pi$
$102$	0	0
$103$	− 1.48445e8i	− 1.31892i	−0.751741	−	0.659458i	$-0.770786\pi$
$103$	− 1.48445e8i	− 1.31892i	0.751741	−	0.659458i	$-0.229214\pi$
$104$	0	0
$105$	8.99423e7	0.739958
$106$	0	0
$107$	− 4.73296e7i	− 0.361075i	−0.983568	−	0.180537i	$-0.942216\pi$
$107$	− 4.73296e7i	− 0.361075i	0.983568	−	0.180537i	$-0.0577837\pi$
$108$	0	0
$109$	−7.68647e7	−0.544529	−0.272264	−	0.962223i	$-0.587773\pi$
$109$	−7.68647e7	−0.544529	−0.272264	+	0.962223i	$0.587773\pi$
$110$	0	0
$111$	5.85605e7i	0.385756i
$112$	0	0
$113$	1.32023e8	0.809724	0.404862	−	0.914378i	$-0.367320\pi$
$113$	1.32023e8	0.809724	0.404862	+	0.914378i	$0.367320\pi$
$114$	0	0
$115$	1.09704e8i	0.627236i
$116$	0	0
$117$	2.03619e7	0.108661
$118$	0	0
$119$	− 3.36915e7i	− 0.168009i
$120$	0	0
$121$	−1.76032e8	−0.821202
$122$	0	0
$123$	− 2.29461e8i	− 1.00251i
$124$	0	0
$125$	2.17300e8	0.890059
$126$	0	0
$127$	− 4.30030e8i	− 1.65304i	−0.562905	−	0.826522i	$-0.690316\pi$
$127$	− 4.30030e8i	− 1.65304i	0.562905	−	0.826522i	$-0.309684\pi$
$128$	0	0
$129$	−2.92164e8	−1.05504
$130$	0	0
$131$	3.60441e8i	1.22391i	0.790894	+	0.611954i	$0.209616\pi$
$131$	3.60441e8i	1.22391i	−0.790894	+	0.611954i	$0.790384\pi$
$132$	0	0
$133$	5.27880e8	1.68705
$134$	0	0
$135$	6.94661e7i	0.209140i
$136$	0	0
$137$	5.15425e8	1.46313	0.731565	−	0.681772i	$-0.238790\pi$
$137$	5.15425e8	1.46313	0.731565	+	0.681772i	$0.238790\pi$
$138$	0	0
$139$	− 1.37680e8i	− 0.368817i	−0.982850	−	0.184408i	$-0.940963\pi$
$139$	− 1.37680e8i	− 0.368817i	0.982850	−	0.184408i	$-0.0590369\pi$
$140$	0	0
$141$	−8.36267e7	−0.211577
$142$	0	0
$143$	− 1.83958e8i	− 0.439921i
$144$	0	0
$145$	2.47512e8	0.559918
$146$	0	0
$147$	− 1.05384e8i	− 0.225686i
$148$	0	0
$149$	−8.72878e8	−1.77096	−0.885480	−	0.464678i	$-0.846170\pi$
$149$	−8.72878e8	−1.77096	−0.885480	+	0.464678i	$0.846170\pi$
$150$	0	0
$151$	1.65985e8i	0.319273i	0.987176	+	0.159636i	$0.0510321\pi$
$151$	1.65985e8i	0.319273i	−0.987176	+	0.159636i	$0.948968\pi$
$152$	0	0
$153$	2.60213e7	0.0474857
$154$	0	0
$155$	1.10726e9i	1.91834i
$156$	0	0
$157$	−7.10228e8	−1.16896	−0.584479	−	0.811409i	$-0.698701\pi$
$157$	−7.10228e8	−1.16896	−0.584479	+	0.811409i	$0.698701\pi$
$158$	0	0
$159$	− 2.86562e8i	− 0.448364i
$160$	0	0
$161$	4.57365e8	0.680706
$162$	0	0
$163$	9.92019e8i	1.40530i	0.711535	+	0.702651i	$0.248000\pi$
$163$	9.92019e8i	1.40530i	−0.711535	+	0.702651i	$0.752000\pi$
$164$	0	0
$165$	6.27587e8	0.846716
$166$	0	0
$167$	1.81406e8i	0.233231i	0.993177	+	0.116616i	$0.0372045\pi$
$167$	1.81406e8i	0.233231i	−0.993177	+	0.116616i	$0.962795\pi$
$168$	0	0
$169$	−7.29047e8	−0.893735
$170$	0	0
$171$	4.07703e8i	0.476826i
$172$	0	0
$173$	1.35248e9	1.50989	0.754946	−	0.655787i	$-0.227663\pi$
$173$	1.35248e9	1.50989	0.754946	+	0.655787i	$0.227663\pi$
$174$	0	0
$175$	2.00174e8i	0.213430i
$176$	0	0
$177$	−9.08593e7	−0.0925713
$178$	0	0
$179$	1.97272e9i	1.92156i	0.277320	+	0.960778i	$0.410554\pi$
$179$	1.97272e9i	1.92156i	−0.277320	+	0.960778i	$0.589446\pi$
$180$	0	0
$181$	9.23522e8	0.860464	0.430232	−	0.902718i	$-0.358432\pi$
$181$	9.23522e8	0.860464	0.430232	+	0.902718i	$0.358432\pi$
$182$	0	0
$183$	− 9.75045e8i	− 0.869401i
$184$	0	0
$185$	−8.50512e8	−0.726095
$186$	0	0
$187$	− 2.35088e8i	− 0.192249i
$188$	0	0
$189$	2.89610e8	0.226969
$190$	0	0
$191$	9.74384e8i	0.732144i	0.930586	+	0.366072i	$0.119298\pi$
$191$	9.74384e8i	0.732144i	−0.930586	+	0.366072i	$0.880702\pi$
$192$	0	0
$193$	5.81973e8	0.419444	0.209722	−	0.977761i	$-0.432744\pi$
$193$	5.81973e8	0.419444	0.209722	+	0.977761i	$0.432744\pi$
$194$	0	0
$195$	2.95729e8i	0.204529i
$196$	0	0
$197$	−4.74424e7	−0.0314993	−0.0157497	−	0.999876i	$-0.505013\pi$
$197$	−4.74424e7	−0.0314993	−0.0157497	+	0.999876i	$0.505013\pi$
$198$	0	0
$199$	− 1.54532e9i	− 0.985383i	−0.870204	−	0.492691i	$-0.836013\pi$
$199$	− 1.54532e9i	− 0.985383i	0.870204	−	0.492691i	$-0.163987\pi$
$200$	0	0
$201$	−2.11116e8	−0.129341
$202$	0	0
$203$	− 1.03190e9i	− 0.607648i
$204$	0	0
$205$	3.33260e9	1.88698
$206$	0	0
$207$	3.53241e8i	0.192393i
$208$	0	0
$209$	3.68337e9	1.93046
$210$	0	0
$211$	1.32819e8i	0.0670085i	0.999439	+	0.0335043i	$0.0106667\pi$
$211$	1.32819e8i	0.0670085i	−0.999439	+	0.0335043i	$0.989333\pi$
$212$	0	0
$213$	1.52135e8	0.0739111
$214$	0	0
$215$	− 4.24329e9i	− 1.98586i
$216$	0	0
$217$	4.61628e9	2.08187
$218$	0	0
$219$	9.85981e8i	0.428640i
$220$	0	0
$221$	1.10777e8	0.0464387
$222$	0	0
$223$	− 1.51125e9i	− 0.611105i	−0.952175	−	0.305553i	$-0.901159\pi$
$223$	− 1.51125e9i	− 0.611105i	0.952175	−	0.305553i	$-0.0988412\pi$
$224$	0	0
$225$	−1.54603e8	−0.0603236
$226$	0	0
$227$	− 2.53571e9i	− 0.954985i	−0.878636	−	0.477492i	$-0.841546\pi$
$227$	− 2.53571e9i	− 0.954985i	0.878636	−	0.477492i	$-0.158454\pi$
$228$	0	0
$229$	3.73429e9	1.35789	0.678947	−	0.734188i	$-0.262437\pi$
$229$	3.73429e9	1.35789	0.678947	+	0.734188i	$0.262437\pi$
$230$	0	0
$231$	− 2.61646e9i	− 0.918896i
$232$	0	0
$233$	−3.29885e9	−1.11928	−0.559640	−	0.828736i	$-0.689061\pi$
$233$	−3.29885e9	−1.11928	−0.559640	+	0.828736i	$0.689061\pi$
$234$	0	0
$235$	− 1.21456e9i	− 0.398244i
$236$	0	0
$237$	−1.39471e8	−0.0442068
$238$	0	0
$239$	3.50003e8i	0.107270i	0.998561	+	0.0536352i	$0.0170808\pi$
$239$	3.50003e8i	0.107270i	−0.998561	+	0.0536352i	$0.982919\pi$
$240$	0	0
$241$	6.21113e9	1.84121	0.920603	−	0.390499i	$-0.127697\pi$
$241$	6.21113e9	1.84121	0.920603	+	0.390499i	$0.127697\pi$
$242$	0	0
$243$	2.23677e8i	0.0641500i
$244$	0	0
$245$	1.53055e9	0.424800
$246$	0	0
$247$	1.73566e9i	0.466312i
$248$	0	0
$249$	1.70362e9	0.443174
$250$	0	0
$251$	6.60672e9i	1.66453i	0.554380	+	0.832264i	$0.312955\pi$
$251$	6.60672e9i	1.66453i	−0.554380	+	0.832264i	$0.687045\pi$
$252$	0	0
$253$	3.19134e9	0.778916
$254$	0	0
$255$	3.77924e8i	0.0893806i
$256$	0	0
$257$	2.35786e9	0.540487	0.270244	−	0.962792i	$-0.412896\pi$
$257$	2.35786e9	0.540487	0.270244	+	0.962792i	$0.412896\pi$
$258$	0	0
$259$	3.54585e9i	0.787992i
$260$	0	0
$261$	7.96976e8	0.171745
$262$	0	0
$263$	1.19679e9i	0.250148i	0.992147	+	0.125074i	$0.0399168\pi$
$263$	1.19679e9i	0.250148i	−0.992147	+	0.125074i	$0.960083\pi$
$264$	0	0
$265$	4.16192e9	0.843938
$266$	0	0
$267$	4.04529e9i	0.795984i
$268$	0	0
$269$	2.03090e9	0.387864	0.193932	−	0.981015i	$-0.437876\pi$
$269$	2.03090e9	0.387864	0.193932	+	0.981015i	$0.437876\pi$
$270$	0	0
$271$	− 8.07813e9i	− 1.49773i	−0.662723	−	0.748865i	$-0.730599\pi$
$271$	− 8.07813e9i	− 1.49773i	0.662723	−	0.748865i	$-0.269401\pi$
$272$	0	0
$273$	1.23292e9	0.221964
$274$	0	0
$275$	1.39675e9i	0.244224i
$276$	0	0
$277$	−3.24262e9	−0.550779	−0.275389	−	0.961333i	$-0.588807\pi$
$277$	−3.24262e9	−0.550779	−0.275389	+	0.961333i	$0.588807\pi$
$278$	0	0
$279$	3.56534e9i	0.588415i
$280$	0	0
$281$	1.05010e9	0.168424	0.0842122	−	0.996448i	$-0.473163\pi$
$281$	1.05010e9	0.168424	0.0842122	+	0.996448i	$0.473163\pi$
$282$	0	0
$283$	3.51472e9i	0.547955i	0.961736	+	0.273978i	$0.0883394\pi$
$283$	3.51472e9i	0.547955i	−0.961736	+	0.273978i	$0.911661\pi$
$284$	0	0
$285$	−5.92133e9	−0.897512
$286$	0	0
$287$	− 1.38939e10i	− 2.04784i
$288$	0	0
$289$	−6.83419e9	−0.979706
$290$	0	0
$291$	− 2.98904e9i	− 0.416831i
$292$	0	0
$293$	9.46780e9	1.28463	0.642316	−	0.766440i	$-0.277974\pi$
$293$	9.46780e9	1.28463	0.642316	+	0.766440i	$0.277974\pi$
$294$	0	0
$295$	− 1.31961e9i	− 0.174243i
$296$	0	0
$297$	2.02080e9	0.259715
$298$	0	0
$299$	1.50381e9i	0.188151i
$300$	0	0
$301$	−1.76906e10	−2.15515
$302$	0	0
$303$	5.13319e9i	0.608999i
$304$	0	0
$305$	1.41612e10	1.63644
$306$	0	0
$307$	1.13587e10i	1.27872i	0.768908	+	0.639360i	$0.220801\pi$
$307$	1.13587e10i	1.27872i	−0.768908	+	0.639360i	$0.779199\pi$
$308$	0	0
$309$	6.94209e9	0.761477
$310$	0	0
$311$	− 1.57767e10i	− 1.68646i	−0.537554	−	0.843229i	$-0.680652\pi$
$311$	− 1.57767e10i	− 1.68646i	0.537554	−	0.843229i	$-0.319348\pi$
$312$	0	0
$313$	4.86560e9	0.506943	0.253471	−	0.967343i	$-0.418428\pi$
$313$	4.86560e9	0.506943	0.253471	+	0.967343i	$0.418428\pi$
$314$	0	0
$315$	4.20619e9i	0.427215i
$316$	0	0
$317$	5.58450e9	0.553028	0.276514	−	0.961010i	$-0.410821\pi$
$317$	5.58450e9	0.553028	0.276514	+	0.961010i	$0.410821\pi$
$318$	0	0
$319$	− 7.20023e9i	− 0.695318i
$320$	0	0
$321$	2.21338e9	0.208467
$322$	0	0
$323$	2.21807e9i	0.203782i
$324$	0	0
$325$	−6.58170e8	−0.0589935
$326$	0	0
$327$	− 3.59460e9i	− 0.314384i
$328$	0	0
$329$	−5.06362e9	−0.432192
$330$	0	0
$331$	− 9.43318e9i	− 0.785862i	−0.919568	−	0.392931i	$-0.871461\pi$
$331$	− 9.43318e9i	− 0.785862i	0.919568	−	0.392931i	$-0.128539\pi$
$332$	0	0
$333$	−2.73861e9	−0.222717
$334$	0	0
$335$	− 3.06617e9i	− 0.243454i
$336$	0	0
$337$	5.02290e9	0.389435	0.194717	−	0.980859i	$-0.437621\pi$
$337$	5.02290e9	0.389435	0.194717	+	0.980859i	$0.437621\pi$
$338$	0	0
$339$	6.17412e9i	0.467494i
$340$	0	0
$341$	3.22108e10	2.38223
$342$	0	0
$343$	9.94291e9i	0.718351i
$344$	0	0
$345$	−5.13035e9	−0.362135
$346$	0	0
$347$	2.08241e9i	0.143631i	0.997418	+	0.0718156i	$0.0228793\pi$
$347$	2.08241e9i	0.143631i	−0.997418	+	0.0718156i	$0.977121\pi$
$348$	0	0
$349$	2.94552e9	0.198545	0.0992727	−	0.995060i	$-0.468348\pi$
$349$	2.94552e9	0.198545	0.0992727	+	0.995060i	$0.468348\pi$
$350$	0	0
$351$	9.52231e8i	0.0627356i
$352$	0	0
$353$	−2.15734e10	−1.38937	−0.694686	−	0.719313i	$-0.744457\pi$
$353$	−2.15734e10	−1.38937	−0.694686	+	0.719313i	$0.744457\pi$
$354$	0	0
$355$	2.20955e9i	0.139120i
$356$	0	0
$357$	1.57559e9	0.0970000
$358$	0	0
$359$	1.14762e10i	0.690911i	0.938435	+	0.345455i	$0.112276\pi$
$359$	1.14762e10i	0.690911i	−0.938435	+	0.345455i	$0.887724\pi$
$360$	0	0
$361$	−1.77693e10	−1.04627
$362$	0	0
$363$	− 8.23219e9i	− 0.474121i
$364$	0	0
$365$	−1.43200e10	−0.806812
$366$	0	0
$367$	− 2.96940e10i	− 1.63683i	−0.574624	−	0.818417i	$-0.694852\pi$
$367$	− 2.96940e10i	− 1.63683i	0.574624	−	0.818417i	$-0.305148\pi$
$368$	0	0
$369$	1.07308e10	0.578798
$370$	0	0
$371$	− 1.73514e10i	− 0.915881i
$372$	0	0
$373$	2.28770e10	1.18185	0.590927	−	0.806725i	$-0.298762\pi$
$373$	2.28770e10	1.18185	0.590927	+	0.806725i	$0.298762\pi$
$374$	0	0
$375$	1.01621e10i	0.513876i
$376$	0	0
$377$	3.39286e9	0.167958
$378$	0	0
$379$	1.93611e10i	0.938367i	0.883101	+	0.469183i	$0.155452\pi$
$379$	1.93611e10i	0.938367i	−0.883101	+	0.469183i	$0.844548\pi$
$380$	0	0
$381$	2.01105e10	0.954385
$382$	0	0
$383$	3.48862e10i	1.62128i	0.585542	+	0.810642i	$0.300882\pi$
$383$	3.48862e10i	1.62128i	−0.585542	+	0.810642i	$0.699118\pi$
$384$	0	0
$385$	3.80005e10	1.72960
$386$	0	0
$387$	− 1.36632e10i	− 0.609127i
$388$	0	0
$389$	−1.35659e9	−0.0592448	−0.0296224	−	0.999561i	$-0.509430\pi$
$389$	−1.35659e9	−0.0592448	−0.0296224	+	0.999561i	$0.509430\pi$
$390$	0	0
$391$	1.92178e9i	0.0822235i
$392$	0	0
$393$	−1.68561e10	−0.706623
$394$	0	0
$395$	− 2.02562e9i	− 0.0832089i
$396$	0	0
$397$	−2.17380e10	−0.875099	−0.437550	−	0.899194i	$-0.644154\pi$
$397$	−2.17380e10	−0.875099	−0.437550	+	0.899194i	$0.644154\pi$
$398$	0	0
$399$	2.46865e10i	0.974021i
$400$	0	0
$401$	−1.16187e10	−0.449346	−0.224673	−	0.974434i	$-0.572131\pi$
$401$	−1.16187e10	−0.449346	−0.224673	+	0.974434i	$0.572131\pi$
$402$	0	0
$403$	1.51782e10i	0.575441i
$404$	0	0
$405$	−3.24861e9	−0.120747
$406$	0	0
$407$	2.47417e10i	0.901681i
$408$	0	0
$409$	−3.41316e10	−1.21973	−0.609864	−	0.792506i	$-0.708776\pi$
$409$	−3.41316e10	−1.21973	−0.609864	+	0.792506i	$0.708776\pi$
$410$	0	0
$411$	2.41040e10i	0.844738i
$412$	0	0
$413$	−5.50155e9	−0.189097
$414$	0	0
$415$	2.47427e10i	0.834170i
$416$	0	0
$417$	6.43864e9	0.212936
$418$	0	0
$419$	− 1.35221e10i	− 0.438719i	−0.975644	−	0.219360i	$-0.929603\pi$
$419$	− 1.35221e10i	− 0.438719i	0.975644	−	0.219360i	$-0.0703968\pi$
$420$	0	0
$421$	1.31223e9	0.0417716	0.0208858	−	0.999782i	$-0.493351\pi$
$421$	1.31223e9	0.0417716	0.0208858	+	0.999782i	$0.493351\pi$
$422$	0	0
$423$	− 3.91083e9i	− 0.122154i
$424$	0	0
$425$	−8.41102e8	−0.0257806
$426$	0	0
$427$	− 5.90392e10i	− 1.77594i
$428$	0	0
$429$	8.60287e9	0.253989
$430$	0	0
$431$	2.66857e10i	0.773338i	0.922219	+	0.386669i	$0.126374\pi$
$431$	2.66857e10i	0.773338i	−0.922219	+	0.386669i	$0.873626\pi$
$432$	0	0
$433$	5.82024e10	1.65573	0.827864	−	0.560929i	$-0.189556\pi$
$433$	5.82024e10	1.65573	0.827864	+	0.560929i	$0.189556\pi$
$434$	0	0
$435$	1.15750e10i	0.323269i
$436$	0	0
$437$	−3.01105e10	−0.825644
$438$	0	0
$439$	− 1.30602e9i	− 0.0351634i	−0.999845	−	0.0175817i	$-0.994403\pi$
$439$	− 1.30602e9i	− 0.0351634i	0.999845	−	0.0175817i	$-0.00559672\pi$
$440$	0	0
$441$	4.92831e9	0.130300
$442$	0	0
$443$	5.14164e10i	1.33502i	0.744602	+	0.667508i	$0.232639\pi$
$443$	5.14164e10i	1.33502i	−0.744602	+	0.667508i	$0.767361\pi$
$444$	0	0
$445$	−5.87523e10	−1.49825
$446$	0	0
$447$	− 4.08205e10i	− 1.02246i
$448$	0	0
$449$	3.55586e10	0.874902	0.437451	−	0.899242i	$-0.355881\pi$
$449$	3.55586e10	0.874902	0.437451	+	0.899242i	$0.355881\pi$
$450$	0	0
$451$	− 9.69468e10i	− 2.34330i
$452$	0	0
$453$	−7.76236e9	−0.184332
$454$	0	0
$455$	1.79064e10i	0.417795i
$456$	0	0
$457$	−2.81024e10	−0.644285	−0.322142	−	0.946691i	$-0.604403\pi$
$457$	−2.81024e10	−0.644285	−0.322142	+	0.946691i	$0.604403\pi$
$458$	0	0
$459$	1.21690e9i	0.0274159i
$460$	0	0
$461$	−8.86381e10	−1.96253	−0.981266	−	0.192660i	$-0.938289\pi$
$461$	−8.86381e10	−1.96253	−0.981266	+	0.192660i	$0.938289\pi$
$462$	0	0
$463$	− 6.11431e8i	− 0.0133053i	−0.999978	−	0.00665263i	$-0.997882\pi$
$463$	− 6.11431e8i	− 0.0133053i	0.999978	−	0.00665263i	$-0.00211761\pi$
$464$	0	0
$465$	−5.17816e10	−1.10755
$466$	0	0
$467$	7.76748e9i	0.163310i	0.996661	+	0.0816550i	$0.0260206\pi$
$467$	7.76748e9i	0.163310i	−0.996661	+	0.0816550i	$0.973979\pi$
$468$	0	0
$469$	−1.27831e10	−0.264207
$470$	0	0
$471$	− 3.32141e10i	− 0.674899i
$472$	0	0
$473$	−1.23439e11	−2.46608
$474$	0	0
$475$	− 1.31784e10i	− 0.258875i
$476$	0	0
$477$	1.34012e10	0.258863
$478$	0	0
$479$	− 7.96633e9i	− 0.151327i	−0.997133	−	0.0756635i	$-0.975893\pi$
$479$	− 7.96633e9i	− 0.151327i	0.997133	−	0.0756635i	$-0.0241075\pi$
$480$	0	0
$481$	−1.16587e10	−0.217806
$482$	0	0
$483$	2.13888e10i	0.393006i
$484$	0	0
$485$	4.34117e10	0.784585
$486$	0	0
$487$	− 7.09691e10i	− 1.26169i	−0.775908	−	0.630846i	$-0.782708\pi$
$487$	− 7.09691e10i	− 1.26169i	0.775908	−	0.630846i	$-0.217292\pi$
$488$	0	0
$489$	−4.63921e10	−0.811351
$490$	0	0
$491$	− 3.62383e10i	− 0.623508i	−0.950163	−	0.311754i	$-0.899083\pi$
$491$	− 3.62383e10i	− 0.623508i	0.950163	−	0.311754i	$-0.100917\pi$
$492$	0	0
$493$	4.33587e9	0.0733988
$494$	0	0
$495$	2.93493e10i	0.488852i
$496$	0	0
$497$	9.21179e9	0.150980
$498$	0	0
$499$	− 7.87064e10i	− 1.26943i	−0.772748	−	0.634713i	$-0.781118\pi$
$499$	− 7.87064e10i	− 1.26943i	0.772748	−	0.634713i	$-0.218882\pi$
$500$	0	0
$501$	−8.48353e9	−0.134656
$502$	0	0
$503$	1.00940e10i	0.157686i	0.996887	+	0.0788430i	$0.0251226\pi$
$503$	1.00940e10i	0.157686i	−0.996887	+	0.0788430i	$0.974877\pi$
$504$	0	0
$505$	−7.45525e10	−1.14630
$506$	0	0
$507$	− 3.40941e10i	− 0.515998i
$508$	0	0
$509$	−9.21767e10	−1.37325	−0.686626	−	0.727011i	$-0.740909\pi$
$509$	−9.21767e10	−1.37325	−0.686626	+	0.727011i	$0.740909\pi$
$510$	0	0
$511$	5.97014e10i	0.875590i
$512$	0	0
$513$	−1.90664e10	−0.275296
$514$	0	0
$515$	1.00824e11i	1.43330i
$516$	0	0
$517$	−3.53322e10	−0.494548
$518$	0	0
$519$	6.32492e10i	0.871737i
$520$	0	0
$521$	−1.66354e10	−0.225778	−0.112889	−	0.993608i	$-0.536010\pi$
$521$	−1.66354e10	−0.225778	−0.112889	+	0.993608i	$0.536010\pi$
$522$	0	0
$523$	− 1.05141e11i	− 1.40528i	−0.711544	−	0.702641i	$-0.752004\pi$
$523$	− 1.05141e11i	− 1.40528i	0.711544	−	0.702641i	$-0.247996\pi$
$524$	0	0
$525$	−9.36123e9	−0.123224
$526$	0	0
$527$	1.93969e10i	0.251472i
$528$	0	0
$529$	5.22227e10	0.666863
$530$	0	0
$531$	− 4.24907e9i	− 0.0534461i
$532$	0	0
$533$	4.56828e10	0.566036
$534$	0	0
$535$	3.21464e10i	0.392389i
$536$	0	0
$537$	−9.22549e10	−1.10941
$538$	0	0
$539$	− 4.45245e10i	− 0.527526i
$540$	0	0
$541$	−8.71349e10	−1.01719	−0.508596	−	0.861005i	$-0.669835\pi$
$541$	−8.71349e10	−1.01719	−0.508596	+	0.861005i	$0.669835\pi$
$542$	0	0
$543$	4.31888e10i	0.496789i
$544$	0	0
$545$	5.22067e10	0.591753
$546$	0	0
$547$	− 2.14425e10i	− 0.239511i	−0.992803	−	0.119756i	$-0.961789\pi$
$547$	− 2.14425e10i	− 0.239511i	0.992803	−	0.119756i	$-0.0382111\pi$
$548$	0	0
$549$	4.55983e10	0.501949
$550$	0	0
$551$	6.79347e10i	0.737031i
$552$	0	0
$553$	−8.44498e9	−0.0903021
$554$	0	0
$555$	− 3.97745e10i	− 0.419211i
$556$	0	0
$557$	−6.77095e10	−0.703443	−0.351721	−	0.936105i	$-0.614404\pi$
$557$	−6.77095e10	−0.703443	−0.351721	+	0.936105i	$0.614404\pi$
$558$	0	0
$559$	− 5.81664e10i	− 0.595696i
$560$	0	0
$561$	1.09940e10	0.110995
$562$	0	0
$563$	1.59348e10i	0.158604i	0.996851	+	0.0793019i	$0.0252691\pi$
$563$	1.59348e10i	0.158604i	−0.996851	+	0.0793019i	$0.974731\pi$
$564$	0	0
$565$	−8.96707e10	−0.879947
$566$	0	0
$567$	1.35437e10i	0.131040i
$568$	0	0
$569$	2.52251e10	0.240649	0.120324	−	0.992735i	$-0.461607\pi$
$569$	2.52251e10	0.240649	0.120324	+	0.992735i	$0.461607\pi$
$570$	0	0
$571$	− 1.90350e11i	− 1.79064i	−0.445422	−	0.895321i	$-0.646946\pi$
$571$	− 1.90350e11i	− 1.79064i	0.445422	−	0.895321i	$-0.353054\pi$
$572$	0	0
$573$	−4.55674e10	−0.422704
$574$	0	0
$575$	− 1.14180e10i	− 0.104453i
$576$	0	0
$577$	1.86718e11	1.68454	0.842271	−	0.539054i	$-0.181218\pi$
$577$	1.86718e11	1.68454	0.842271	+	0.539054i	$0.181218\pi$
$578$	0	0
$579$	2.72162e10i	0.242166i
$580$	0	0
$581$	1.03154e11	0.905280
$582$	0	0
$583$	− 1.21072e11i	− 1.04802i
$584$	0	0
$585$	−1.38299e10	−0.118085
$586$	0	0
$587$	− 1.54829e11i	− 1.30407i	−0.758190	−	0.652033i	$-0.773916\pi$
$587$	− 1.54829e11i	− 1.30407i	0.758190	−	0.652033i	$-0.226084\pi$
$588$	0	0
$589$	−3.03912e11	−2.52514
$590$	0	0
$591$	− 2.21866e9i	− 0.0181861i
$592$	0	0
$593$	−1.92874e11	−1.55975	−0.779876	−	0.625934i	$-0.784718\pi$
$593$	−1.92874e11	−1.55975	−0.779876	+	0.625934i	$0.784718\pi$
$594$	0	0
$595$	2.28833e10i	0.182579i
$596$	0	0
$597$	7.22673e10	0.568911
$598$	0	0
$599$	1.61172e11i	1.25193i	0.779850	+	0.625967i	$0.215295\pi$
$599$	1.61172e11i	1.25193i	−0.779850	+	0.625967i	$0.784705\pi$
$600$	0	0
$601$	2.22491e11	1.70535	0.852677	−	0.522439i	$-0.174978\pi$
$601$	2.22491e11	1.70535	0.852677	+	0.522439i	$0.174978\pi$
$602$	0	0
$603$	− 9.87291e9i	− 0.0746751i
$604$	0	0
$605$	1.19561e11	0.892420
$606$	0	0
$607$	2.00435e11i	1.47645i	0.674555	+	0.738224i	$0.264335\pi$
$607$	2.00435e11i	1.47645i	−0.674555	+	0.738224i	$0.735665\pi$
$608$	0	0
$609$	4.82571e10	0.350826
$610$	0	0
$611$	− 1.66491e10i	− 0.119461i
$612$	0	0
$613$	2.43639e11	1.72546	0.862729	−	0.505666i	$-0.168753\pi$
$613$	2.43639e11	1.72546	0.862729	+	0.505666i	$0.168753\pi$
$614$	0	0
$615$	1.55850e11i	1.08945i
$616$	0	0
$617$	−1.38038e11	−0.952483	−0.476241	−	0.879315i	$-0.658001\pi$
$617$	−1.38038e11	−0.952483	−0.476241	+	0.879315i	$0.658001\pi$
$618$	0	0
$619$	2.61228e11i	1.77934i	0.456609	+	0.889668i	$0.349064\pi$
$619$	2.61228e11i	1.77934i	−0.456609	+	0.889668i	$0.650936\pi$
$620$	0	0
$621$	−1.65195e10	−0.111078
$622$	0	0
$623$	2.44943e11i	1.62597i
$624$	0	0
$625$	−1.75205e11	−1.14822
$626$	0	0
$627$	1.72254e11i	1.11455i
$628$	0	0
$629$	−1.48991e10	−0.0951827
$630$	0	0
$631$	− 1.13779e11i	− 0.717702i	−0.933395	−	0.358851i	$-0.883169\pi$
$631$	− 1.13779e11i	− 0.717702i	0.933395	−	0.358851i	$-0.116831\pi$
$632$	0	0
$633$	−6.21132e9	−0.0386874
$634$	0	0
$635$	2.92078e11i	1.79640i
$636$	0	0
$637$	2.09806e10	0.127427
$638$	0	0
$639$	7.11464e9i	0.0426726i
$640$	0	0
$641$	1.60385e11	0.950018	0.475009	−	0.879981i	$-0.342445\pi$
$641$	1.60385e11	0.950018	0.475009	+	0.879981i	$0.342445\pi$
$642$	0	0
$643$	− 2.50437e11i	− 1.46506i	−0.680737	−	0.732528i	$-0.738340\pi$
$643$	− 2.50437e11i	− 1.46506i	0.680737	−	0.732528i	$-0.261660\pi$
$644$	0	0
$645$	1.98439e11	1.14654
$646$	0	0
$647$	7.56500e10i	0.431710i	0.976425	+	0.215855i	$0.0692538\pi$
$647$	7.56500e10i	0.431710i	−0.976425	+	0.215855i	$0.930746\pi$
$648$	0	0
$649$	−3.83879e10	−0.216379
$650$	0	0
$651$	2.15882e11i	1.20197i
$652$	0	0
$653$	8.15262e10	0.448378	0.224189	−	0.974546i	$-0.428027\pi$
$653$	8.15262e10	0.448378	0.224189	+	0.974546i	$0.428027\pi$
$654$	0	0
$655$	− 2.44812e11i	− 1.33005i
$656$	0	0
$657$	−4.61098e10	−0.247475
$658$	0	0
$659$	− 2.95585e11i	− 1.56726i	−0.621230	−	0.783628i	$-0.713367\pi$
$659$	− 2.95585e11i	− 1.56726i	0.621230	−	0.783628i	$-0.286633\pi$
$660$	0	0
$661$	−2.61473e11	−1.36968	−0.684842	−	0.728691i	$-0.740129\pi$
$661$	−2.61473e11	−1.36968	−0.684842	+	0.728691i	$0.740129\pi$
$662$	0	0
$663$	5.18052e9i	0.0268114i
$664$	0	0
$665$	−3.58538e11	−1.83336
$666$	0	0
$667$	5.88599e10i	0.297383i
$668$	0	0
$669$	7.06740e10	0.352822
$670$	0	0
$671$	− 4.11955e11i	− 2.03217i
$672$	0	0
$673$	−1.37858e11	−0.672006	−0.336003	−	0.941861i	$-0.609075\pi$
$673$	−1.37858e11	−0.672006	−0.336003	+	0.941861i	$0.609075\pi$
$674$	0	0
$675$	− 7.23006e9i	− 0.0348278i
$676$	0	0
$677$	3.01911e11	1.43722	0.718611	−	0.695412i	$-0.244778\pi$
$677$	3.01911e11	1.43722	0.718611	+	0.695412i	$0.244778\pi$
$678$	0	0
$679$	− 1.80987e11i	− 0.851468i
$680$	0	0
$681$	1.18583e11	0.551361
$682$	0	0
$683$	− 1.68192e10i	− 0.0772900i	−0.999253	−	0.0386450i	$-0.987696\pi$
$683$	− 1.68192e10i	− 0.0772900i	0.999253	−	0.0386450i	$-0.0123042\pi$
$684$	0	0
$685$	−3.50078e11	−1.59002
$686$	0	0
$687$	1.74635e11i	0.783980i
$688$	0	0
$689$	5.70511e10	0.253155
$690$	0	0
$691$	− 3.35023e10i	− 0.146948i	−0.997297	−	0.0734738i	$-0.976591\pi$
$691$	− 3.35023e10i	− 0.146948i	0.997297	−	0.0734738i	$-0.0234085\pi$
$692$	0	0
$693$	1.22360e11	0.530525
$694$	0	0
$695$	9.35124e10i	0.400802i
$696$	0	0
$697$	5.83799e10	0.247362
$698$	0	0
$699$	− 1.54272e11i	− 0.646217i
$700$	0	0
$701$	2.67420e11	1.10744	0.553722	−	0.832702i	$-0.313207\pi$
$701$	2.67420e11	1.10744	0.553722	+	0.832702i	$0.313207\pi$
$702$	0	0
$703$	− 2.33440e11i	− 0.955773i
$704$	0	0
$705$	5.67995e10	0.229926
$706$	0	0
$707$	3.10816e11i	1.24401i
$708$	0	0
$709$	−3.07071e11	−1.21522	−0.607609	−	0.794236i	$-0.707871\pi$
$709$	−3.07071e11	−1.21522	−0.607609	+	0.794236i	$0.707871\pi$
$710$	0	0
$711$	− 6.52240e9i	− 0.0255228i
$712$	0	0
$713$	−2.63314e11	−1.01886
$714$	0	0
$715$	1.24945e11i	0.478073i
$716$	0	0
$717$	−1.63680e10	−0.0619326
$718$	0	0
$719$	6.98430e10i	0.261341i	0.991426	+	0.130671i	$0.0417130\pi$
$719$	6.98430e10i	0.261341i	−0.991426	+	0.130671i	$0.958287\pi$
$720$	0	0
$721$	4.20345e11	1.55548
$722$	0	0
$723$	2.90466e11i	1.06302i
$724$	0	0
$725$	−2.57611e10	−0.0932423
$726$	0	0
$727$	− 5.06395e11i	− 1.81281i	−0.422413	−	0.906404i	$-0.638817\pi$
$727$	− 5.06395e11i	− 1.81281i	0.422413	−	0.906404i	$-0.361183\pi$
$728$	0	0
$729$	−1.04604e10	−0.0370370
$730$	0	0
$731$	− 7.43332e10i	− 0.260323i
$732$	0	0
$733$	2.03361e11	0.704451	0.352226	−	0.935915i	$-0.385425\pi$
$733$	2.03361e11	0.704451	0.352226	+	0.935915i	$0.385425\pi$
$734$	0	0
$735$	7.15770e10i	0.245258i
$736$	0	0
$737$	−8.91962e10	−0.302326
$738$	0	0
$739$	9.52283e9i	0.0319292i	0.999873	+	0.0159646i	$0.00508191\pi$
$739$	9.52283e9i	0.0319292i	−0.999873	+	0.0159646i	$0.994918\pi$
$740$	0	0
$741$	−8.11688e10	−0.269226
$742$	0	0
$743$	− 5.11665e11i	− 1.67892i	−0.543419	−	0.839461i	$-0.682871\pi$
$743$	− 5.11665e11i	− 1.67892i	0.543419	−	0.839461i	$-0.317129\pi$
$744$	0	0
$745$	5.92861e11	1.92455
$746$	0	0
$747$	7.96703e10i	0.255867i
$748$	0	0
$749$	1.34021e11	0.425839
$750$	0	0
$751$	− 9.01221e10i	− 0.283316i	−0.989916	−	0.141658i	$-0.954757\pi$
$751$	− 9.01221e10i	− 0.283316i	0.989916	−	0.141658i	$-0.0452434\pi$
$752$	0	0
$753$	−3.08966e11	−0.961015
$754$	0	0
$755$	− 1.12738e11i	− 0.346961i
$756$	0	0
$757$	3.52773e11	1.07427	0.537133	−	0.843498i	$-0.319507\pi$
$757$	3.52773e11	1.07427	0.537133	+	0.843498i	$0.319507\pi$
$758$	0	0
$759$	1.49244e11i	0.449707i
$760$	0	0
$761$	−1.29273e11	−0.385450	−0.192725	−	0.981253i	$-0.561732\pi$
$761$	−1.29273e11	−0.385450	−0.192725	+	0.981253i	$0.561732\pi$
$762$	0	0
$763$	− 2.17654e11i	− 0.642197i
$764$	0	0
$765$	−1.76737e10	−0.0516039
$766$	0	0
$767$	− 1.80890e10i	− 0.0522676i
$768$	0	0
$769$	6.01912e11	1.72118	0.860592	−	0.509295i	$-0.170094\pi$
$769$	6.01912e11	1.72118	0.860592	+	0.509295i	$0.170094\pi$
$770$	0	0
$771$	1.10266e11i	0.312051i
$772$	0	0
$773$	−2.92681e11	−0.819740	−0.409870	−	0.912144i	$-0.634426\pi$
$773$	−2.92681e11	−0.819740	−0.409870	+	0.912144i	$0.634426\pi$
$774$	0	0
$775$	− 1.15245e11i	− 0.319458i
$776$	0	0
$777$	−1.65823e11	−0.454947
$778$	0	0
$779$	9.14701e11i	2.48387i
$780$	0	0
$781$	6.42767e10	0.172763
$782$	0	0
$783$	3.72709e10i	0.0991568i
$784$	0	0
$785$	4.82389e11	1.27034
$786$	0	0
$787$	3.50970e10i	0.0914894i	0.998953	+	0.0457447i	$0.0145661\pi$
$787$	3.50970e10i	0.0914894i	−0.998953	+	0.0457447i	$0.985434\pi$
$788$	0	0
$789$	−5.59685e10	−0.144423
$790$	0	0
$791$	3.73844e11i	0.954959i
$792$	0	0
$793$	1.94120e11	0.490881
$794$	0	0
$795$	1.94634e11i	0.487248i
$796$	0	0
$797$	3.20201e11	0.793579	0.396789	−	0.917910i	$-0.370124\pi$
$797$	3.20201e11	0.793579	0.396789	+	0.917910i	$0.370124\pi$
$798$	0	0
$799$	− 2.12765e10i	− 0.0522052i
$800$	0	0
$801$	−1.89179e11	−0.459562
$802$	0	0
$803$	4.16576e11i	1.00192i
$804$	0	0
$805$	−3.10644e11	−0.739740
$806$	0	0
$807$	9.49759e10i	0.223934i
$808$	0	0
$809$	8.49727e11	1.98374	0.991872	−	0.127241i	$-0.0406122\pi$
$809$	8.49727e11	1.98374	0.991872	+	0.127241i	$0.0406122\pi$
$810$	0	0
$811$	− 5.10908e11i	− 1.18103i	−0.807028	−	0.590513i	$-0.798925\pi$
$811$	− 5.10908e11i	− 1.18103i	0.807028	−	0.590513i	$-0.201075\pi$
$812$	0	0
$813$	3.77777e11	0.864715
$814$	0	0
$815$	− 6.73782e11i	− 1.52718i
$816$	0	0
$817$	1.16466e12	2.61403
$818$	0	0
$819$	5.76578e10i	0.128151i
$820$	0	0
$821$	7.41595e11	1.63228	0.816139	−	0.577856i	$-0.196110\pi$
$821$	7.41595e11	1.63228	0.816139	+	0.577856i	$0.196110\pi$
$822$	0	0
$823$	2.86895e11i	0.625351i	0.949860	+	0.312676i	$0.101225\pi$
$823$	2.86895e11i	0.625351i	−0.949860	+	0.312676i	$0.898775\pi$
$824$	0	0
$825$	−6.53195e10	−0.141003
$826$	0	0
$827$	2.22522e11i	0.475720i	0.971299	+	0.237860i	$0.0764460\pi$
$827$	2.22522e11i	0.475720i	−0.971299	+	0.237860i	$0.923554\pi$
$828$	0	0
$829$	−2.27339e11	−0.481344	−0.240672	−	0.970607i	$-0.577368\pi$
$829$	−2.27339e11	−0.481344	−0.240672	+	0.970607i	$0.577368\pi$
$830$	0	0
$831$	− 1.51642e11i	− 0.317992i
$832$	0	0
$833$	2.68120e10	0.0556864
$834$	0	0
$835$	− 1.23212e11i	− 0.253458i
$836$	0	0
$837$	−1.66734e11	−0.339722
$838$	0	0
$839$	5.82182e11i	1.17493i	0.809251	+	0.587463i	$0.199873\pi$
$839$	5.82182e11i	1.17493i	−0.809251	+	0.587463i	$0.800127\pi$
$840$	0	0
$841$	−3.67448e11	−0.734534
$842$	0	0
$843$	4.91083e10i	0.0972399i
$844$	0	0
$845$	4.95171e11	0.971244
$846$	0	0
$847$	− 4.98461e11i	− 0.968495i
$848$	0	0
$849$	−1.64367e11	−0.316362
$850$	0	0
$851$	− 2.02257e11i	− 0.385643i
$852$	0	0
$853$	−6.27190e10	−0.118469	−0.0592343	−	0.998244i	$-0.518866\pi$
$853$	−6.27190e10	−0.118469	−0.0592343	+	0.998244i	$0.518866\pi$
$854$	0	0
$855$	− 2.76913e11i	− 0.518179i
$856$	0	0
$857$	7.61729e11	1.41214	0.706069	−	0.708143i	$-0.250467\pi$
$857$	7.61729e11	1.41214	0.706069	+	0.708143i	$0.250467\pi$
$858$	0	0
$859$	− 8.94336e11i	− 1.64259i	−0.570506	−	0.821293i	$-0.693253\pi$
$859$	− 8.94336e11i	− 1.64259i	0.570506	−	0.821293i	$-0.306747\pi$
$860$	0	0
$861$	6.49753e11	1.18232
$862$	0	0
$863$	− 9.62950e10i	− 0.173604i	−0.996226	−	0.0868022i	$-0.972335\pi$
$863$	− 9.62950e10i	− 0.173604i	0.996226	−	0.0868022i	$-0.0276648\pi$
$864$	0	0
$865$	−9.18608e11	−1.64084
$866$	0	0
$867$	− 3.19603e11i	− 0.565633i
$868$	0	0
$869$	−5.89262e10	−0.103331
$870$	0	0
$871$	− 4.20306e10i	− 0.0730286i
$872$	0	0
$873$	1.39784e11	0.240657
$874$	0	0
$875$	6.15317e11i	1.04970i
$876$	0	0
$877$	2.86258e11	0.483905	0.241952	−	0.970288i	$-0.422212\pi$
$877$	2.86258e11	0.483905	0.241952	+	0.970288i	$0.422212\pi$
$878$	0	0
$879$	4.42765e11i	0.741683i
$880$	0	0
$881$	−7.76381e11	−1.28876	−0.644379	−	0.764706i	$-0.722884\pi$
$881$	−7.76381e11	−1.28876	−0.644379	+	0.764706i	$0.722884\pi$
$882$	0	0
$883$	− 5.96574e11i	− 0.981345i	−0.871344	−	0.490672i	$-0.836751\pi$
$883$	− 5.96574e11i	− 0.981345i	0.871344	−	0.490672i	$-0.163249\pi$
$884$	0	0
$885$	6.17119e10	0.100600
$886$	0	0
$887$	− 4.47553e10i	− 0.0723020i	−0.999346	−	0.0361510i	$-0.988490\pi$
$887$	− 4.47553e10i	− 0.0723020i	0.999346	−	0.0361510i	$-0.0115097\pi$
$888$	0	0
$889$	1.21770e12	1.94954
$890$	0	0
$891$	9.45034e10i	0.149947i
$892$	0	0
$893$	3.33362e11	0.524216
$894$	0	0
$895$	− 1.33988e12i	− 2.08820i
$896$	0	0
$897$	−7.03261e10	−0.108629
$898$	0	0
$899$	5.94084e11i	0.909514i
$900$	0	0
$901$	7.29079e10	0.110631
$902$	0	0
$903$	− 8.27308e11i	− 1.24427i
$904$	0	0
$905$	−6.27259e11	−0.935088
$906$	0	0
$907$	1.55076e10i	0.0229148i	0.999934	+	0.0114574i	$0.00364708\pi$
$907$	1.55076e10i	0.0229148i	−0.999934	+	0.0114574i	$0.996353\pi$
$908$	0	0
$909$	−2.40056e11	−0.351606
$910$	0	0
$911$	− 2.63676e11i	− 0.382822i	−0.981510	−	0.191411i	$-0.938694\pi$
$911$	− 2.63676e11i	− 0.382822i	0.981510	−	0.191411i	$-0.0613064\pi$
$912$	0	0
$913$	7.19776e11	1.03589
$914$	0	0
$915$	6.62253e11i	0.944800i
$916$	0	0
$917$	−1.02064e12	−1.44343
$918$	0	0
$919$	− 5.84929e11i	− 0.820051i	−0.912074	−	0.410025i	$-0.865520\pi$
$919$	− 5.84929e11i	− 0.820051i	0.912074	−	0.410025i	$-0.134480\pi$
$920$	0	0
$921$	−5.31194e11	−0.738269
$922$	0	0
$923$	3.02882e10i	0.0417317i
$924$	0	0
$925$	8.85216e10	0.120916
$926$	0	0
$927$	3.24650e11i	0.439639i
$928$	0	0
$929$	−7.20851e11	−0.967794	−0.483897	−	0.875125i	$-0.660779\pi$
$929$	−7.20851e11	−0.967794	−0.483897	+	0.875125i	$0.660779\pi$
$930$	0	0
$931$	4.20092e11i	0.559173i
$932$	0	0
$933$	7.37805e11	0.973677
$934$	0	0
$935$	1.59672e11i	0.208921i
$936$	0	0
$937$	4.55505e11	0.590929	0.295464	−	0.955354i	$-0.404526\pi$
$937$	4.55505e11	0.590929	0.295464	+	0.955354i	$0.404526\pi$
$938$	0	0
$939$	2.27541e11i	0.292683i
$940$	0	0
$941$	−1.18851e12	−1.51580	−0.757901	−	0.652369i	$-0.773775\pi$
$941$	−1.18851e12	−1.51580	−0.757901	+	0.652369i	$0.773775\pi$
$942$	0	0
$943$	7.92513e11i	1.00221i
$944$	0	0
$945$	−1.96704e11	−0.246653
$946$	0	0
$947$	− 8.54579e11i	− 1.06256i	−0.847197	−	0.531279i	$-0.821712\pi$
$947$	− 8.54579e11i	− 1.06256i	0.847197	−	0.531279i	$-0.178288\pi$
$948$	0	0
$949$	−1.96297e11	−0.242019
$950$	0	0
$951$	2.61161e11i	0.319291i
$952$	0	0
$953$	1.13431e12	1.37518	0.687590	−	0.726099i	$-0.258669\pi$
$953$	1.13431e12	1.37518	0.687590	+	0.726099i	$0.258669\pi$
$954$	0	0
$955$	− 6.61804e11i	− 0.795639i
$956$	0	0
$957$	3.36721e11	0.401442
$958$	0	0
$959$	1.45950e12i	1.72556i
$960$	0	0
$961$	−1.80479e12	−2.11609
$962$	0	0
$963$	1.03510e11i	0.120358i
$964$	0	0
$965$	−3.95278e11	−0.455820
$966$	0	0
$967$	− 1.29630e12i	− 1.48252i	−0.671220	−	0.741258i	$-0.734229\pi$
$967$	− 1.29630e12i	− 1.48252i	0.671220	−	0.741258i	$-0.265771\pi$
$968$	0	0
$969$	−1.03729e11	−0.117653
$970$	0	0
$971$	7.65514e11i	0.861145i	0.902556	+	0.430573i	$0.141688\pi$
$971$	7.65514e11i	0.861145i	−0.902556	+	0.430573i	$0.858312\pi$
$972$	0	0
$973$	3.89861e11	0.434969
$974$	0	0
$975$	− 3.07795e10i	− 0.0340599i
$976$	0	0
$977$	−1.61885e12	−1.77676	−0.888380	−	0.459109i	$-0.848169\pi$
$977$	−1.61885e12	−1.77676	−0.888380	+	0.459109i	$0.848169\pi$
$978$	0	0
$979$	1.70913e12i	1.86056i
$980$	0	0
$981$	1.68103e11	0.181510
$982$	0	0
$983$	6.49521e11i	0.695631i	0.937563	+	0.347815i	$0.113076\pi$
$983$	6.49521e11i	0.695631i	−0.937563	+	0.347815i	$0.886924\pi$
$984$	0	0
$985$	3.22230e10	0.0342311
$986$	0	0
$987$	− 2.36802e11i	− 0.249526i
$988$	0	0
$989$	1.00908e12	1.05473
$990$	0	0
$991$	− 4.90041e11i	− 0.508087i	−0.967193	−	0.254043i	$-0.918239\pi$
$991$	− 4.90041e11i	− 0.508087i	0.967193	−	0.254043i	$-0.0817606\pi$
$992$	0	0
$993$	4.41146e11	0.453718
$994$	0	0
$995$	1.04958e12i	1.07084i
$996$	0	0
$997$	−8.17541e11	−0.827426	−0.413713	−	0.910407i	$-0.635768\pi$
$997$	−8.17541e11	−0.827426	−0.413713	+	0.910407i	$0.635768\pi$
$998$	0	0
$999$	− 1.28072e11i	− 0.128585i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.9.g.a.127.6	yes	32
4.3	odd	2	inner	384.9.g.a.127.5	✓	32
8.3	odd	2		384.9.g.b.127.28	yes	32
8.5	even	2		384.9.g.b.127.27	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.9.g.a.127.5	✓	32	4.3	odd	2	inner
384.9.g.a.127.6	yes	32	1.1	even	1	trivial
384.9.g.b.127.27	yes	32	8.5	even	2
384.9.g.b.127.28	yes	32	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	384.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+46.7654i q^{3} -679.203 q^{5} +2831.65i q^{7} -2187.00 q^{9} +O(q^{10})\)	\(q+46.7654i q^{3} -679.203 q^{5} +2831.65i q^{7} -2187.00 q^{9} +19758.3i q^{11} -9310.42 q^{13} -31763.2i q^{15} -11898.2 q^{17} -186421. i q^{19} -132423. q^{21} -161519. i q^{23} +70691.7 q^{25} -102276. i q^{27} -364415. q^{29} -1.63024e6i q^{31} -924005. q^{33} -1.92327e6i q^{35} +1.25222e6 q^{37} -435405. i q^{39} -4.90663e6 q^{41} +6.24745e6i q^{43} +1.48542e6 q^{45} +1.78822e6i q^{47} -2.25346e6 q^{49} -556422. i q^{51} -6.12766e6 q^{53} -1.34199e7i q^{55} +8.71806e6 q^{57} +1.94288e6i q^{59} -2.08497e7 q^{61} -6.19282e6i q^{63} +6.32366e6 q^{65} +4.51436e6i q^{67} +7.55348e6 q^{69} -3.25315e6i q^{71} +2.10836e7 q^{73} +3.30592e6i q^{75} -5.59487e7 q^{77} +2.98235e6i q^{79} +4.78297e6 q^{81} -3.64290e7i q^{83} +8.08127e6 q^{85} -1.70420e7i q^{87} +8.65018e7 q^{89} -2.63639e7i q^{91} +7.62388e7 q^{93} +1.26618e8i q^{95} -6.39157e7 q^{97} -4.32114e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 1344 q^{5} - 69984 q^{9}+O(q^{10}) \) 32 * q - 1344 * q^5 - 69984 * q^9	\( 32 q - 1344 q^{5} - 69984 q^{9} - 114240 q^{13} - 154560 q^{17} + 1791712 q^{25} - 275520 q^{29} + 2421440 q^{37} - 4374720 q^{41} + 2939328 q^{45} - 14219104 q^{49} - 6224448 q^{53} + 3100032 q^{57} - 13005632 q^{61} + 75175296 q^{65} - 85710400 q^{73} + 154517760 q^{77} + 153055008 q^{81} - 384830848 q^{85} - 182669760 q^{89} + 149817600 q^{93} - 149408192 q^{97}+O(q^{100}) \) 32 * q - 1344 * q^5 - 69984 * q^9 - 114240 * q^13 - 154560 * q^17 + 1791712 * q^25 - 275520 * q^29 + 2421440 * q^37 - 4374720 * q^41 + 2939328 * q^45 - 14219104 * q^49 - 6224448 * q^53 + 3100032 * q^57 - 13005632 * q^61 + 75175296 * q^65 - 85710400 * q^73 + 154517760 * q^77 + 153055008 * q^81 - 384830848 * q^85 - 182669760 * q^89 + 149817600 * q^93 - 149408192 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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