LMFDB - Embedded newform 864.4.f.b.431.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,4,Mod(431,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("864.431");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	864.f (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:	$50.9776502450$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			431.12
Character	$\chi$	$=$	864.431
Dual form			864.4.f.b.431.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.898373 q^{5} +20.7150i q^{7} +O(q^{10})$	$q-0.898373 q^{5} +20.7150i q^{7} +0.351756i q^{11} -44.6629i q^{13} -58.0727i q^{17} +42.4793 q^{19} -196.511 q^{23} -124.193 q^{25} +71.7647 q^{29} +96.1796i q^{31} -18.6098i q^{35} +18.9148i q^{37} -322.069i q^{41} +264.193 q^{43} +370.645 q^{47} -86.1123 q^{49} +512.874 q^{53} -0.316008i q^{55} -240.841i q^{59} -526.724i q^{61} +40.1240i q^{65} +156.199 q^{67} +395.199 q^{71} -723.326 q^{73} -7.28663 q^{77} +972.700i q^{79} -1222.96i q^{83} +52.1710i q^{85} -1231.35i q^{89} +925.194 q^{91} -38.1623 q^{95} -499.594 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 48 q^{19} + 600 q^{25} + 432 q^{43} - 816 q^{49} - 1632 q^{67} - 216 q^{73} - 3600 q^{91} + 2280 q^{97}+O(q^{100}) $ 24 * q + 48 * q^19 + 600 * q^25 + 432 * q^43 - 816 * q^49 - 1632 * q^67 - 216 * q^73 - 3600 * q^91 + 2280 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$.

$n$	$325$	$353$	$703$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−0.898373	−0.0803530	−0.0401765	−	0.999193i	$-0.512792\pi$
$5$	−0.898373	−0.0803530	−0.0401765	+	0.999193i	$0.512792\pi$
$6$	0	0
$7$	20.7150i	1.11851i	0.828997	+	0.559253i	$0.188912\pi$
$7$	20.7150i	1.11851i	−0.828997	+	0.559253i	$0.811088\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.351756i	0.00964166i	0.999988	+	0.00482083i	$0.00153452\pi$
$11$	0.351756i	0.00964166i	−0.999988	+	0.00482083i	$0.998465\pi$
$12$	0	0
$13$	− 44.6629i	− 0.952867i	−0.879211	−	0.476433i	$-0.841929\pi$
$13$	− 44.6629i	− 0.952867i	0.879211	−	0.476433i	$-0.158071\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 58.0727i	− 0.828512i	−0.910160	−	0.414256i	$-0.864042\pi$
$17$	− 58.0727i	− 0.828512i	0.910160	−	0.414256i	$-0.135958\pi$
$18$	0	0
$19$	42.4793	0.512917	0.256459	−	0.966555i	$-0.417444\pi$
$19$	42.4793	0.512917	0.256459	+	0.966555i	$0.417444\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−196.511	−1.78154	−0.890770	−	0.454455i	$-0.849834\pi$
$23$	−196.511	−1.78154	−0.890770	+	0.454455i	$0.849834\pi$
$24$	0	0
$25$	−124.193	−0.993543
$26$	0	0
$27$	0	0
$28$	0	0
$29$	71.7647	0.459530	0.229765	−	0.973246i	$-0.426204\pi$
$29$	71.7647	0.459530	0.229765	+	0.973246i	$0.426204\pi$
$30$	0	0
$31$	96.1796i	0.557238i	0.960402	+	0.278619i	$0.0898766\pi$
$31$	96.1796i	0.557238i	−0.960402	+	0.278619i	$0.910123\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 18.6098i	− 0.0898753i
$36$	0	0
$37$	18.9148i	0.0840423i	0.999117	+	0.0420212i	$0.0133797\pi$
$37$	18.9148i	0.0840423i	−0.999117	+	0.0420212i	$0.986620\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 322.069i	− 1.22680i	−0.789772	−	0.613400i	$-0.789801\pi$
$41$	− 322.069i	− 1.22680i	0.789772	−	0.613400i	$-0.210199\pi$
$42$	0	0
$43$	264.193	0.936954	0.468477	−	0.883476i	$-0.344803\pi$
$43$	264.193	0.936954	0.468477	+	0.883476i	$0.344803\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	370.645	1.15030	0.575151	−	0.818048i	$-0.304943\pi$
$47$	370.645	1.15030	0.575151	+	0.818048i	$0.304943\pi$
$48$	0	0
$49$	−86.1123	−0.251056
$50$	0	0
$51$	0	0
$52$	0	0
$53$	512.874	1.32922	0.664610	−	0.747190i	$-0.268598\pi$
$53$	512.874	1.32922	0.664610	+	0.747190i	$0.268598\pi$
$54$	0	0
$55$	− 0.316008i	0 0.000774736i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 240.841i	− 0.531438i	−0.964050	−	0.265719i	$-0.914391\pi$
$59$	− 240.841i	− 0.531438i	0.964050	−	0.265719i	$-0.0856094\pi$
$60$	0	0
$61$	− 526.724i	− 1.10558i	−0.833322	−	0.552788i	$-0.813564\pi$
$61$	− 526.724i	− 1.10558i	0.833322	−	0.552788i	$-0.186436\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	40.1240i	0.0765657i
$66$	0	0
$67$	156.199	0.284818	0.142409	−	0.989808i	$-0.454515\pi$
$67$	156.199	0.284818	0.142409	+	0.989808i	$0.454515\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	395.199	0.660585	0.330293	−	0.943879i	$-0.392853\pi$
$71$	395.199	0.660585	0.330293	+	0.943879i	$0.392853\pi$
$72$	0	0
$73$	−723.326	−1.15971	−0.579856	−	0.814719i	$-0.696891\pi$
$73$	−723.326	−1.15971	−0.579856	+	0.814719i	$0.696891\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−7.28663	−0.0107843
$78$	0	0
$79$	972.700i	1.38528i	0.721282	+	0.692641i	$0.243553\pi$
$79$	972.700i	1.38528i	−0.721282	+	0.692641i	$0.756447\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 1222.96i	− 1.61732i	−0.588279	−	0.808658i	$-0.700194\pi$
$83$	− 1222.96i	− 1.61732i	0.588279	−	0.808658i	$-0.299806\pi$
$84$	0	0
$85$	52.1710i	0.0665734i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 1231.35i	− 1.46655i	−0.679931	−	0.733276i	$-0.737990\pi$
$89$	− 1231.35i	− 1.46655i	0.679931	−	0.733276i	$-0.262010\pi$
$90$	0	0
$91$	925.194	1.06579
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−38.1623	−0.0412144
$96$	0	0
$97$	−499.594	−0.522949	−0.261474	−	0.965210i	$-0.584209\pi$
$97$	−499.594	−0.522949	−0.261474	+	0.965210i	$0.584209\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1546.93	−1.52401	−0.762007	−	0.647568i	$-0.775786\pi$
$101$	−1546.93	−1.52401	−0.762007	+	0.647568i	$0.775786\pi$
$102$	0	0
$103$	− 799.583i	− 0.764906i	−0.923975	−	0.382453i	$-0.875079\pi$
$103$	− 799.583i	− 0.764906i	0.923975	−	0.382453i	$-0.124921\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 2113.60i	− 1.90962i	−0.297209	−	0.954812i	$-0.596056\pi$
$107$	− 2113.60i	− 1.90962i	0.297209	−	0.954812i	$-0.403944\pi$
$108$	0	0
$109$	− 1197.70i	− 1.05247i	−0.850339	−	0.526235i	$-0.823603\pi$
$109$	− 1197.70i	− 1.05247i	0.850339	−	0.526235i	$-0.176397\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1094.53i	− 0.911192i	−0.890187	−	0.455596i	$-0.849426\pi$
$113$	− 1094.53i	− 0.911192i	0.890187	−	0.455596i	$-0.150574\pi$
$114$	0	0
$115$	176.540	0.143152
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1202.98	0.926696
$120$	0	0
$121$	1330.88	0.999907
$122$	0	0
$123$	0	0
$124$	0	0
$125$	223.868	0.160187
$126$	0	0
$127$	1729.91i	1.20870i	0.796718	+	0.604351i	$0.206567\pi$
$127$	1729.91i	1.20870i	−0.796718	+	0.604351i	$0.793433\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 614.718i	− 0.409986i	−0.978763	−	0.204993i	$-0.934283\pi$
$131$	− 614.718i	− 0.409986i	0.978763	−	0.204993i	$-0.0657171\pi$
$132$	0	0
$133$	879.961i	0.573701i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 99.5146i	− 0.0620592i	−0.999518	−	0.0310296i	$-0.990121\pi$
$137$	− 99.5146i	− 0.0620592i	0.999518	−	0.0310296i	$-0.00987861\pi$
$138$	0	0
$139$	−2788.14	−1.70134	−0.850671	−	0.525698i	$-0.823804\pi$
$139$	−2788.14	−1.70134	−0.850671	+	0.525698i	$0.823804\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	15.7104	0.00918722
$144$	0	0
$145$	−64.4715	−0.0369246
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3125.06	1.71822	0.859111	−	0.511789i	$-0.171017\pi$
$149$	3125.06	1.71822	0.859111	+	0.511789i	$0.171017\pi$
$150$	0	0
$151$	− 2896.76i	− 1.56116i	−0.625056	−	0.780580i	$-0.714924\pi$
$151$	− 2896.76i	− 1.56116i	0.625056	−	0.780580i	$-0.285076\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 86.4052i	− 0.0447757i
$156$	0	0
$157$	3385.64i	1.72104i	0.509416	+	0.860521i	$0.329862\pi$
$157$	3385.64i	1.72104i	−0.509416	+	0.860521i	$0.670138\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 4070.73i	− 1.99266i
$162$	0	0
$163$	336.736	0.161811	0.0809054	−	0.996722i	$-0.474219\pi$
$163$	336.736	0.161811	0.0809054	+	0.996722i	$0.474219\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	769.709	0.356658	0.178329	−	0.983971i	$-0.442931\pi$
$167$	769.709	0.356658	0.178329	+	0.983971i	$0.442931\pi$
$168$	0	0
$169$	202.224	0.0920454
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1997.45	0.877823	0.438911	−	0.898530i	$-0.355364\pi$
$173$	1997.45	0.877823	0.438911	+	0.898530i	$0.355364\pi$
$174$	0	0
$175$	− 2572.66i	− 1.11128i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 2606.09i	− 1.08820i	−0.839019	−	0.544102i	$-0.816870\pi$
$179$	− 2606.09i	− 1.08820i	0.839019	−	0.544102i	$-0.183130\pi$
$180$	0	0
$181$	− 458.311i	− 0.188210i	−0.995562	−	0.0941050i	$-0.970001\pi$
$181$	− 458.311i	− 0.188210i	0.995562	−	0.0941050i	$-0.0299989\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 16.9925i	− 0.00675305i
$186$	0	0
$187$	20.4274	0.00798823
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1432.72	−0.542762	−0.271381	−	0.962472i	$-0.587480\pi$
$191$	−1432.72	−0.542762	−0.271381	+	0.962472i	$0.587480\pi$
$192$	0	0
$193$	4839.58	1.80498	0.902489	−	0.430713i	$-0.141738\pi$
$193$	4839.58	1.80498	0.902489	+	0.430713i	$0.141738\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1209.90	0.437572	0.218786	−	0.975773i	$-0.429790\pi$
$197$	1209.90	0.437572	0.218786	+	0.975773i	$0.429790\pi$
$198$	0	0
$199$	53.0166i	0.0188857i	0.999955	+	0.00944283i	$0.00300579\pi$
$199$	53.0166i	0.0188857i	−0.999955	+	0.00944283i	$0.996994\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1486.61i	0.513987i
$204$	0	0
$205$	289.339i	0.0985770i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	14.9423i	0.00494538i
$210$	0	0
$211$	−3491.01	−1.13901	−0.569505	−	0.821988i	$-0.692865\pi$
$211$	−3491.01	−1.13901	−0.569505	+	0.821988i	$0.692865\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−237.344	−0.0752870
$216$	0	0
$217$	−1992.36	−0.623274
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2593.70	−0.789461
$222$	0	0
$223$	4119.13i	1.23694i	0.785808	+	0.618470i	$0.212247\pi$
$223$	4119.13i	1.23694i	−0.785808	+	0.618470i	$0.787753\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3739.71i	1.09345i	0.837312	+	0.546725i	$0.184126\pi$
$227$	3739.71i	1.09345i	−0.837312	+	0.546725i	$0.815874\pi$
$228$	0	0
$229$	− 175.001i	− 0.0504995i	−0.999681	−	0.0252498i	$-0.991962\pi$
$229$	− 175.001i	− 0.0504995i	0.999681	−	0.0252498i	$-0.00803810\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 3450.11i	− 0.970062i	−0.874497	−	0.485031i	$-0.838808\pi$
$233$	− 3450.11i	− 0.970062i	0.874497	−	0.485031i	$-0.161192\pi$
$234$	0	0
$235$	−332.978	−0.0924301
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1133.29	0.306722	0.153361	−	0.988170i	$-0.450990\pi$
$239$	1133.29	0.306722	0.153361	+	0.988170i	$0.450990\pi$
$240$	0	0
$241$	1350.31	0.360918	0.180459	−	0.983583i	$-0.442242\pi$
$241$	1350.31	0.360918	0.180459	+	0.983583i	$0.442242\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	77.3610	0.0201731
$246$	0	0
$247$	− 1897.25i	− 0.488742i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 5787.89i	− 1.45549i	−0.685846	−	0.727746i	$-0.740568\pi$
$251$	− 5787.89i	− 1.45549i	0.685846	−	0.727746i	$-0.259432\pi$
$252$	0	0
$253$	− 69.1239i	− 0.0171770i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4252.21i	1.03208i	0.856563	+	0.516042i	$0.172595\pi$
$257$	4252.21i	1.03208i	−0.856563	+	0.516042i	$0.827405\pi$
$258$	0	0
$259$	−391.820	−0.0940019
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6519.24	−1.52849	−0.764246	−	0.644924i	$-0.776889\pi$
$263$	−6519.24	−1.52849	−0.764246	+	0.644924i	$0.776889\pi$
$264$	0	0
$265$	−460.752	−0.106807
$266$	0	0
$267$	0	0
$268$	0	0
$269$	938.205	0.212652	0.106326	−	0.994331i	$-0.466091\pi$
$269$	938.205	0.212652	0.106326	+	0.994331i	$0.466091\pi$
$270$	0	0
$271$	− 3123.60i	− 0.700167i	−0.936719	−	0.350083i	$-0.886153\pi$
$271$	− 3123.60i	− 0.700167i	0.936719	−	0.350083i	$-0.113847\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 43.6856i	− 0.00957941i
$276$	0	0
$277$	− 4328.77i	− 0.938954i	−0.882945	−	0.469477i	$-0.844442\pi$
$277$	− 4328.77i	− 0.938954i	0.882945	−	0.469477i	$-0.155558\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4644.87i	0.986084i	0.870006	+	0.493042i	$0.164115\pi$
$281$	4644.87i	0.986084i	−0.870006	+	0.493042i	$0.835885\pi$
$282$	0	0
$283$	864.160	0.181516	0.0907580	−	0.995873i	$-0.471071\pi$
$283$	864.160	0.181516	0.0907580	+	0.995873i	$0.471071\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6671.68	1.37218
$288$	0	0
$289$	1540.56	0.313568
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−8487.85	−1.69237	−0.846187	−	0.532886i	$-0.821108\pi$
$293$	−8487.85	−1.69237	−0.846187	+	0.532886i	$0.821108\pi$
$294$	0	0
$295$	216.365i	0.0427027i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8776.76i	1.69757i
$300$	0	0
$301$	5472.76i	1.04799i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	473.195i	0.0888362i
$306$	0	0
$307$	4082.00	0.758867	0.379433	−	0.925219i	$-0.376119\pi$
$307$	4082.00	0.758867	0.379433	+	0.925219i	$0.376119\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1366.89	−0.249226	−0.124613	−	0.992205i	$-0.539769\pi$
$311$	−1366.89	−0.249226	−0.124613	+	0.992205i	$0.539769\pi$
$312$	0	0
$313$	−5039.16	−0.910001	−0.455001	−	0.890491i	$-0.650361\pi$
$313$	−5039.16	−0.910001	−0.455001	+	0.890491i	$0.650361\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4699.69	−0.832685	−0.416342	−	0.909208i	$-0.636688\pi$
$317$	−4699.69	−0.832685	−0.416342	+	0.909208i	$0.636688\pi$
$318$	0	0
$319$	25.2436i	0.00443064i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 2466.89i	− 0.424958i
$324$	0	0
$325$	5546.82i	0.946714i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	7677.92i	1.28662i
$330$	0	0
$331$	3220.43	0.534776	0.267388	−	0.963589i	$-0.413839\pi$
$331$	3220.43	0.534776	0.267388	+	0.963589i	$0.413839\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−140.325	−0.0228860
$336$	0	0
$337$	−1198.67	−0.193756	−0.0968778	−	0.995296i	$-0.530886\pi$
$337$	−1198.67	−0.193756	−0.0968778	+	0.995296i	$0.530886\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−33.8317	−0.00537270
$342$	0	0
$343$	5321.44i	0.837698i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4230.19i	0.654434i	0.944949	+	0.327217i	$0.106111\pi$
$347$	4230.19i	0.654434i	−0.944949	+	0.327217i	$0.893889\pi$
$348$	0	0
$349$	− 3597.47i	− 0.551771i	−0.961190	−	0.275886i	$-0.911029\pi$
$349$	− 3597.47i	− 0.551771i	0.961190	−	0.275886i	$-0.0889711\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10954.3i	1.65167i	0.563911	+	0.825836i	$0.309296\pi$
$353$	10954.3i	1.65167i	−0.563911	+	0.825836i	$0.690704\pi$
$354$	0	0
$355$	−355.037	−0.0530800
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8651.30	1.27186	0.635931	−	0.771746i	$-0.280616\pi$
$359$	8651.30	1.27186	0.635931	+	0.771746i	$0.280616\pi$
$360$	0	0
$361$	−5054.51	−0.736916
$362$	0	0
$363$	0	0
$364$	0	0
$365$	649.817	0.0931862
$366$	0	0
$367$	− 1016.25i	− 0.144544i	−0.997385	−	0.0722720i	$-0.976975\pi$
$367$	− 1016.25i	− 0.144544i	0.997385	−	0.0722720i	$-0.0230250\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	10624.2i	1.48674i
$372$	0	0
$373$	− 13256.0i	− 1.84013i	−0.391767	−	0.920065i	$-0.628136\pi$
$373$	− 13256.0i	− 1.84013i	0.391767	−	0.920065i	$-0.371864\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 3205.22i	− 0.437871i
$378$	0	0
$379$	−121.292	−0.0164389	−0.00821945	−	0.999966i	$-0.502616\pi$
$379$	−121.292	−0.0164389	−0.00821945	+	0.999966i	$0.502616\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−8469.57	−1.12996	−0.564980	−	0.825105i	$-0.691116\pi$
$383$	−8469.57	−1.12996	−0.564980	+	0.825105i	$0.691116\pi$
$384$	0	0
$385$	6.54611	0.000866547 0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	10022.4	1.30631	0.653156	−	0.757223i	$-0.273445\pi$
$389$	10022.4	1.30631	0.653156	+	0.757223i	$0.273445\pi$
$390$	0	0
$391$	11411.9i	1.47603i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 873.848i	− 0.111312i
$396$	0	0
$397$	− 14528.4i	− 1.83668i	−0.395793	−	0.918340i	$-0.629530\pi$
$397$	− 14528.4i	− 1.83668i	0.395793	−	0.918340i	$-0.370470\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7080.47i	0.881750i	0.897569	+	0.440875i	$0.145332\pi$
$401$	7080.47i	0.881750i	−0.897569	+	0.440875i	$0.854668\pi$
$402$	0	0
$403$	4295.66	0.530973
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6.65337	−0.000810308 0
$408$	0	0
$409$	−6672.53	−0.806688	−0.403344	−	0.915048i	$-0.632152\pi$
$409$	−6672.53	−0.806688	−0.403344	+	0.915048i	$0.632152\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4989.03	0.594417
$414$	0	0
$415$	1098.67i	0.129956i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	13972.2i	1.62909i	0.580101	+	0.814545i	$0.303013\pi$
$419$	13972.2i	1.62909i	−0.580101	+	0.814545i	$0.696987\pi$
$420$	0	0
$421$	13047.1i	1.51039i	0.655499	+	0.755196i	$0.272459\pi$
$421$	13047.1i	1.51039i	−0.655499	+	0.755196i	$0.727541\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	7212.22i	0.823162i
$426$	0	0
$427$	10911.1	1.23659
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−13232.4	−1.47884	−0.739422	−	0.673242i	$-0.764901\pi$
$431$	−13232.4	−1.47884	−0.739422	+	0.673242i	$0.764901\pi$
$432$	0	0
$433$	1480.79	0.164347	0.0821736	−	0.996618i	$-0.473814\pi$
$433$	1480.79	0.164347	0.0821736	+	0.996618i	$0.473814\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−8347.66	−0.913782
$438$	0	0
$439$	5195.44i	0.564840i	0.959291	+	0.282420i	$0.0911371\pi$
$439$	5195.44i	0.564840i	−0.959291	+	0.282420i	$0.908863\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	11273.5i	1.20908i	0.796576	+	0.604539i	$0.206643\pi$
$443$	11273.5i	1.20908i	−0.796576	+	0.604539i	$0.793357\pi$
$444$	0	0
$445$	1106.21i	0.117842i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 8863.37i	− 0.931600i	−0.884890	−	0.465800i	$-0.845767\pi$
$449$	− 8863.37i	− 0.931600i	0.884890	−	0.465800i	$-0.154233\pi$
$450$	0	0
$451$	113.290	0.0118284
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−831.169	−0.0856392
$456$	0	0
$457$	−5663.24	−0.579683	−0.289842	−	0.957075i	$-0.593603\pi$
$457$	−5663.24	−0.579683	−0.289842	+	0.957075i	$0.593603\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−14366.0	−1.45139	−0.725695	−	0.688017i	$-0.758482\pi$
$461$	−14366.0	−1.45139	−0.725695	+	0.688017i	$0.758482\pi$
$462$	0	0
$463$	− 7300.37i	− 0.732779i	−0.930462	−	0.366390i	$-0.880594\pi$
$463$	− 7300.37i	− 0.732779i	0.930462	−	0.366390i	$-0.119406\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 9456.68i	− 0.937051i	−0.883450	−	0.468526i	$-0.844785\pi$
$467$	− 9456.68i	− 0.937051i	0.883450	−	0.468526i	$-0.155215\pi$
$468$	0	0
$469$	3235.68i	0.318571i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	92.9312i	0.00903379i
$474$	0	0
$475$	−5275.63	−0.509606
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−10398.2	−0.991869	−0.495935	−	0.868360i	$-0.665174\pi$
$479$	−10398.2	−0.991869	−0.495935	+	0.868360i	$0.665174\pi$
$480$	0	0
$481$	844.788	0.0800811
$482$	0	0
$483$	0	0
$484$	0	0
$485$	448.822	0.0420205
$486$	0	0
$487$	1299.05i	0.120874i	0.998172	+	0.0604369i	$0.0192494\pi$
$487$	1299.05i	0.120874i	−0.998172	+	0.0604369i	$0.980751\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	3740.15i	0.343769i	0.985117	+	0.171885i	$0.0549856\pi$
$491$	3740.15i	0.343769i	−0.985117	+	0.171885i	$0.945014\pi$
$492$	0	0
$493$	− 4167.57i	− 0.380726i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8186.57i	0.738869i
$498$	0	0
$499$	4346.25	0.389910	0.194955	−	0.980812i	$-0.437544\pi$
$499$	4346.25	0.389910	0.194955	+	0.980812i	$0.437544\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−11191.4	−0.992052	−0.496026	−	0.868308i	$-0.665208\pi$
$503$	−11191.4	−0.992052	−0.496026	+	0.868308i	$0.665208\pi$
$504$	0	0
$505$	1389.72	0.122459
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14092.9	1.22722	0.613610	−	0.789609i	$-0.289717\pi$
$509$	14092.9	1.22722	0.613610	+	0.789609i	$0.289717\pi$
$510$	0	0
$511$	− 14983.7i	− 1.29714i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	718.325i	0.0614624i
$516$	0	0
$517$	130.376i	0.0110908i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 9859.21i	− 0.829060i	−0.910036	−	0.414530i	$-0.863946\pi$
$521$	− 9859.21i	− 0.829060i	0.910036	−	0.414530i	$-0.136054\pi$
$522$	0	0
$523$	−13061.8	−1.09207	−0.546037	−	0.837761i	$-0.683864\pi$
$523$	−13061.8	−1.09207	−0.546037	+	0.837761i	$0.683864\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5585.41	0.461678
$528$	0	0
$529$	26449.6	2.17388
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−14384.6	−1.16898
$534$	0	0
$535$	1898.81i	0.153444i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 30.2905i	− 0.00242060i
$540$	0	0
$541$	10232.9i	0.813212i	0.913604	+	0.406606i	$0.133288\pi$
$541$	10232.9i	0.813212i	−0.913604	+	0.406606i	$0.866712\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1075.99i	0.0845691i
$546$	0	0
$547$	−14051.5	−1.09835	−0.549176	−	0.835706i	$-0.685059\pi$
$547$	−14051.5	−1.09835	−0.549176	+	0.835706i	$0.685059\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3048.52	0.235701
$552$	0	0
$553$	−20149.5	−1.54945
$554$	0	0
$555$	0	0
$556$	0	0
$557$	390.845	0.0297318	0.0148659	−	0.999889i	$-0.495268\pi$
$557$	390.845	0.0297318	0.0148659	+	0.999889i	$0.495268\pi$
$558$	0	0
$559$	− 11799.6i	− 0.892792i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	7548.18i	0.565041i	0.959261	+	0.282520i	$0.0911705\pi$
$563$	7548.18i	0.565041i	−0.959261	+	0.282520i	$0.908830\pi$
$564$	0	0
$565$	983.296i	0.0732170i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	13968.1i	1.02912i	0.857453	+	0.514562i	$0.172045\pi$
$569$	13968.1i	1.02912i	−0.857453	+	0.514562i	$0.827955\pi$
$570$	0	0
$571$	−1253.22	−0.0918488	−0.0459244	−	0.998945i	$-0.514623\pi$
$571$	−1253.22	−0.0918488	−0.0459244	+	0.998945i	$0.514623\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	24405.3	1.77004
$576$	0	0
$577$	23968.2	1.72930	0.864651	−	0.502373i	$-0.167539\pi$
$577$	23968.2	1.72930	0.864651	+	0.502373i	$0.167539\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	25333.6	1.80898
$582$	0	0
$583$	180.406i	0.0128159i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 18003.7i	− 1.26592i	−0.774186	−	0.632958i	$-0.781841\pi$
$587$	− 18003.7i	− 1.26592i	0.774186	−	0.632958i	$-0.218159\pi$
$588$	0	0
$589$	4085.65i	0.285817i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 12257.6i	− 0.848835i	−0.905467	−	0.424417i	$-0.860479\pi$
$593$	− 12257.6i	− 0.848835i	0.905467	−	0.424417i	$-0.139521\pi$
$594$	0	0
$595$	−1080.72	−0.0744627
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1985.47	0.135433	0.0677164	−	0.997705i	$-0.478429\pi$
$599$	1985.47	0.135433	0.0677164	+	0.997705i	$0.478429\pi$
$600$	0	0
$601$	−8652.96	−0.587291	−0.293645	−	0.955914i	$-0.594868\pi$
$601$	−8652.96	−0.587291	−0.293645	+	0.955914i	$0.594868\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1195.62	−0.0803455
$606$	0	0
$607$	16866.0i	1.12779i	0.825846	+	0.563896i	$0.190698\pi$
$607$	16866.0i	1.12779i	−0.825846	+	0.563896i	$0.809302\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 16554.1i	− 1.09608i
$612$	0	0
$613$	20971.3i	1.38176i	0.722968	+	0.690882i	$0.242778\pi$
$613$	20971.3i	1.38176i	−0.722968	+	0.690882i	$0.757222\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7835.37i	0.511248i	0.966776	+	0.255624i	$0.0822809\pi$
$617$	7835.37i	0.511248i	−0.966776	+	0.255624i	$0.917719\pi$
$618$	0	0
$619$	−2340.76	−0.151992	−0.0759960	−	0.997108i	$-0.524214\pi$
$619$	−2340.76	−0.151992	−0.0759960	+	0.997108i	$0.524214\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	25507.5	1.64035
$624$	0	0
$625$	15323.0	0.980672
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1098.43	0.0696301
$630$	0	0
$631$	− 18345.7i	− 1.15742i	−0.815534	−	0.578709i	$-0.803557\pi$
$631$	− 18345.7i	− 1.15742i	0.815534	−	0.578709i	$-0.196443\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 1554.11i	− 0.0971228i
$636$	0	0
$637$	3846.03i	0.239223i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 10560.3i	− 0.650713i	−0.945591	−	0.325357i	$-0.894516\pi$
$641$	− 10560.3i	− 0.650713i	0.945591	−	0.325357i	$-0.105484\pi$
$642$	0	0
$643$	10020.1	0.614548	0.307274	−	0.951621i	$-0.400583\pi$
$643$	10020.1	0.614548	0.307274	+	0.951621i	$0.400583\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3986.01	0.242204	0.121102	−	0.992640i	$-0.461357\pi$
$647$	3986.01	0.242204	0.121102	+	0.992640i	$0.461357\pi$
$648$	0	0
$649$	84.7173	0.00512395
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−16847.0	−1.00961	−0.504804	−	0.863234i	$-0.668435\pi$
$653$	−16847.0	−1.00961	−0.504804	+	0.863234i	$0.668435\pi$
$654$	0	0
$655$	552.246i	0.0329436i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 2518.82i	− 0.148891i	−0.997225	−	0.0744456i	$-0.976281\pi$
$659$	− 2518.82i	− 0.148891i	0.997225	−	0.0744456i	$-0.0237187\pi$
$660$	0	0
$661$	− 30098.8i	− 1.77111i	−0.464530	−	0.885557i	$-0.653777\pi$
$661$	− 30098.8i	− 1.77111i	0.464530	−	0.885557i	$-0.346223\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 790.533i	− 0.0460986i
$666$	0	0
$667$	−14102.6	−0.818671
$668$	0	0
$669$	0	0
$670$	0	0
$671$	185.278	0.0106596
$672$	0	0
$673$	4759.03	0.272581	0.136291	−	0.990669i	$-0.456482\pi$
$673$	4759.03	0.272581	0.136291	+	0.990669i	$0.456482\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−19148.9	−1.08708	−0.543540	−	0.839383i	$-0.682916\pi$
$677$	−19148.9	−1.08708	−0.543540	+	0.839383i	$0.682916\pi$
$678$	0	0
$679$	− 10349.1i	− 0.584922i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 9631.82i	− 0.539607i	−0.962915	−	0.269803i	$-0.913041\pi$
$683$	− 9631.82i	− 0.539607i	0.962915	−	0.269803i	$-0.0869587\pi$
$684$	0	0
$685$	89.4013i	0.00498664i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 22906.5i	− 1.26657i
$690$	0	0
$691$	−8176.16	−0.450124	−0.225062	−	0.974344i	$-0.572258\pi$
$691$	−8176.16	−0.450124	−0.225062	+	0.974344i	$0.572258\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2504.79	0.136708
$696$	0	0
$697$	−18703.4	−1.01642
$698$	0	0
$699$	0	0
$700$	0	0
$701$	6120.23	0.329754	0.164877	−	0.986314i	$-0.447277\pi$
$701$	6120.23	0.329754	0.164877	+	0.986314i	$0.447277\pi$
$702$	0	0
$703$	803.486i	0.0431068i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 32044.7i	− 1.70462i
$708$	0	0
$709$	10678.5i	0.565639i	0.959173	+	0.282819i	$0.0912697\pi$
$709$	10678.5i	0.565639i	−0.959173	+	0.282819i	$0.908730\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 18900.4i	− 0.992741i
$714$	0	0
$715$	−14.1138	−0.000738220 0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−17698.3	−0.917992	−0.458996	−	0.888438i	$-0.651791\pi$
$719$	−17698.3	−0.917992	−0.458996	+	0.888438i	$0.651791\pi$
$720$	0	0
$721$	16563.4	0.855552
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−8912.67	−0.456563
$726$	0	0
$727$	9538.33i	0.486599i	0.969951	+	0.243299i	$0.0782297\pi$
$727$	9538.33i	0.486599i	−0.969951	+	0.243299i	$0.921770\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 15342.4i	− 0.776277i
$732$	0	0
$733$	10543.9i	0.531307i	0.964069	+	0.265653i	$0.0855877\pi$
$733$	10543.9i	0.531307i	−0.964069	+	0.265653i	$0.914412\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	54.9440i	0.00274612i
$738$	0	0
$739$	−20329.2	−1.01194	−0.505969	−	0.862552i	$-0.668865\pi$
$739$	−20329.2	−1.01194	−0.505969	+	0.862552i	$0.668865\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−19114.3	−0.943791	−0.471895	−	0.881655i	$-0.656430\pi$
$743$	−19114.3	−0.943791	−0.471895	+	0.881655i	$0.656430\pi$
$744$	0	0
$745$	−2807.47	−0.138064
$746$	0	0
$747$	0	0
$748$	0	0
$749$	43783.4	2.13593
$750$	0	0
$751$	24797.9i	1.20491i	0.798152	+	0.602456i	$0.205811\pi$
$751$	24797.9i	1.20491i	−0.798152	+	0.602456i	$0.794189\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2602.38i	0.125444i
$756$	0	0
$757$	− 7338.12i	− 0.352323i	−0.984361	−	0.176161i	$-0.943632\pi$
$757$	− 7338.12i	− 0.352323i	0.984361	−	0.176161i	$-0.0563681\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 34547.3i	− 1.64565i	−0.568296	−	0.822824i	$-0.692397\pi$
$761$	− 34547.3i	− 1.64565i	0.568296	−	0.822824i	$-0.307603\pi$
$762$	0	0
$763$	24810.5	1.17719
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−10756.7	−0.506390
$768$	0	0
$769$	13988.7	0.655977	0.327988	−	0.944682i	$-0.393629\pi$
$769$	13988.7	0.655977	0.327988	+	0.944682i	$0.393629\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	11871.5	0.552380	0.276190	−	0.961103i	$-0.410928\pi$
$773$	11871.5	0.552380	0.276190	+	0.961103i	$0.410928\pi$
$774$	0	0
$775$	− 11944.8i	− 0.553640i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 13681.3i	− 0.629247i
$780$	0	0
$781$	139.014i	0.00636914i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 3041.57i	− 0.138291i
$786$	0	0
$787$	1362.56	0.0617155	0.0308577	−	0.999524i	$-0.490176\pi$
$787$	1362.56	0.0617155	0.0308577	+	0.999524i	$0.490176\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	22673.2	1.01917
$792$	0	0
$793$	−23525.0	−1.05347
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−16568.7	−0.736380	−0.368190	−	0.929751i	$-0.620022\pi$
$797$	−16568.7	−0.736380	−0.368190	+	0.929751i	$0.620022\pi$
$798$	0	0
$799$	− 21524.4i	− 0.953038i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 254.434i	− 0.0111815i
$804$	0	0
$805$	3657.04i	0.160116i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 17086.8i	− 0.742571i	−0.928519	−	0.371286i	$-0.878917\pi$
$809$	− 17086.8i	− 0.742571i	0.928519	−	0.371286i	$-0.121083\pi$
$810$	0	0
$811$	15277.5	0.661486	0.330743	−	0.943721i	$-0.392701\pi$
$811$	15277.5	0.661486	0.330743	+	0.943721i	$0.392701\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−302.514	−0.0130020
$816$	0	0
$817$	11222.7	0.480580
$818$	0	0
$819$	0	0
$820$	0	0
$821$	27304.1	1.16068	0.580342	−	0.814373i	$-0.302919\pi$
$821$	27304.1	1.16068	0.580342	+	0.814373i	$0.302919\pi$
$822$	0	0
$823$	− 13760.9i	− 0.582835i	−0.956596	−	0.291418i	$-0.905873\pi$
$823$	− 13760.9i	− 0.582835i	0.956596	−	0.291418i	$-0.0941270\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7095.81i	0.298362i	0.988810	+	0.149181i	$0.0476637\pi$
$827$	7095.81i	0.298362i	−0.988810	+	0.149181i	$0.952336\pi$
$828$	0	0
$829$	16873.6i	0.706930i	0.935448	+	0.353465i	$0.114997\pi$
$829$	16873.6i	0.706930i	−0.935448	+	0.353465i	$0.885003\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	5000.77i	0.208003i
$834$	0	0
$835$	−691.487	−0.0286585
$836$	0	0
$837$	0	0
$838$	0	0
$839$	3582.07	0.147398	0.0736989	−	0.997281i	$-0.476520\pi$
$839$	3582.07	0.147398	0.0736989	+	0.997281i	$0.476520\pi$
$840$	0	0
$841$	−19238.8	−0.788832
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−181.672	−0.00739612
$846$	0	0
$847$	27569.1i	1.11840i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 3716.96i	− 0.149725i
$852$	0	0
$853$	− 32087.2i	− 1.28798i	−0.765035	−	0.643988i	$-0.777279\pi$
$853$	− 32087.2i	− 1.28798i	0.765035	−	0.643988i	$-0.222721\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 34615.5i	− 1.37975i	−0.723931	−	0.689873i	$-0.757666\pi$
$857$	− 34615.5i	− 1.37975i	0.723931	−	0.689873i	$-0.242334\pi$
$858$	0	0
$859$	9163.98	0.363994	0.181997	−	0.983299i	$-0.441744\pi$
$859$	9163.98	0.363994	0.181997	+	0.983299i	$0.441744\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−9624.28	−0.379622	−0.189811	−	0.981821i	$-0.560788\pi$
$863$	−9624.28	−0.379622	−0.189811	+	0.981821i	$0.560788\pi$
$864$	0	0
$865$	−1794.46	−0.0705357
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−342.153	−0.0133564
$870$	0	0
$871$	− 6976.32i	− 0.271393i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4637.44i	0.179170i
$876$	0	0
$877$	− 6459.97i	− 0.248732i	−0.992236	−	0.124366i	$-0.960310\pi$
$877$	− 6459.97i	− 0.248732i	0.992236	−	0.124366i	$-0.0396896\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	16282.5i	0.622670i	0.950300	+	0.311335i	$0.100776\pi$
$881$	16282.5i	0.622670i	−0.950300	+	0.311335i	$0.899224\pi$
$882$	0	0
$883$	44992.8	1.71476	0.857378	−	0.514687i	$-0.172092\pi$
$883$	44992.8	1.71476	0.857378	+	0.514687i	$0.172092\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	28759.6	1.08867	0.544336	−	0.838867i	$-0.316782\pi$
$887$	28759.6	1.08867	0.544336	+	0.838867i	$0.316782\pi$
$888$	0	0
$889$	−35835.2	−1.35194
$890$	0	0
$891$	0	0
$892$	0	0
$893$	15744.8	0.590009
$894$	0	0
$895$	2341.24i	0.0874404i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6902.31i	0.256068i
$900$	0	0
$901$	− 29784.0i	− 1.10127i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	411.735i	0.0151232i
$906$	0	0
$907$	−10771.9	−0.394351	−0.197175	−	0.980368i	$-0.563177\pi$
$907$	−10771.9	−0.394351	−0.197175	+	0.980368i	$0.563177\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−6368.46	−0.231610	−0.115805	−	0.993272i	$-0.536945\pi$
$911$	−6368.46	−0.231610	−0.115805	+	0.993272i	$0.536945\pi$
$912$	0	0
$913$	430.183	0.0155936
$914$	0	0
$915$	0	0
$916$	0	0
$917$	12733.9	0.458572
$918$	0	0
$919$	21188.6i	0.760553i	0.924873	+	0.380277i	$0.124171\pi$
$919$	21188.6i	0.760553i	−0.924873	+	0.380277i	$0.875829\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 17650.8i	− 0.629449i
$924$	0	0
$925$	− 2349.08i	− 0.0834997i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 39628.4i	− 1.39953i	−0.714371	−	0.699767i	$-0.753287\pi$
$929$	− 39628.4i	− 1.39953i	0.714371	−	0.699767i	$-0.246713\pi$
$930$	0	0
$931$	−3657.99	−0.128771
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−18.3514	−0.000641878 0
$936$	0	0
$937$	−44882.9	−1.56485	−0.782423	−	0.622747i	$-0.786017\pi$
$937$	−44882.9	−1.56485	−0.782423	+	0.622747i	$0.786017\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	19290.8	0.668290	0.334145	−	0.942522i	$-0.391553\pi$
$941$	19290.8	0.668290	0.334145	+	0.942522i	$0.391553\pi$
$942$	0	0
$943$	63290.2i	2.18559i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 25216.9i	− 0.865301i	−0.901562	−	0.432651i	$-0.857578\pi$
$947$	− 25216.9i	− 0.865301i	0.901562	−	0.432651i	$-0.142422\pi$
$948$	0	0
$949$	32305.8i	1.10505i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	43956.3i	1.49411i	0.664764	+	0.747054i	$0.268532\pi$
$953$	43956.3i	1.49411i	−0.664764	+	0.747054i	$0.731468\pi$
$954$	0	0
$955$	1287.11	0.0436126
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2061.45	0.0694136
$960$	0	0
$961$	20540.5	0.689486
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−4347.75	−0.145035
$966$	0	0
$967$	− 12899.3i	− 0.428969i	−0.976727	−	0.214485i	$-0.931193\pi$
$967$	− 12899.3i	− 0.428969i	0.976727	−	0.214485i	$-0.0688072\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	15206.8i	0.502584i	0.967911	+	0.251292i	$0.0808554\pi$
$971$	15206.8i	0.502584i	−0.967911	+	0.251292i	$0.919145\pi$
$972$	0	0
$973$	− 57756.3i	− 1.90296i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	7912.42i	0.259100i	0.991573	+	0.129550i	$0.0413533\pi$
$977$	7912.42i	0.259100i	−0.991573	+	0.129550i	$0.958647\pi$
$978$	0	0
$979$	433.135	0.0141400
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−28573.8	−0.927123	−0.463562	−	0.886065i	$-0.653429\pi$
$983$	−28573.8	−0.927123	−0.463562	+	0.886065i	$0.653429\pi$
$984$	0	0
$985$	−1086.94	−0.0351602
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−51916.8	−1.66922
$990$	0	0
$991$	27399.6i	0.878282i	0.898418	+	0.439141i	$0.144717\pi$
$991$	27399.6i	0.878282i	−0.898418	+	0.439141i	$0.855283\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 47.6287i	− 0.00151752i
$996$	0	0
$997$	484.325i	0.0153849i	0.999970	+	0.00769244i	$0.00244860\pi$
$997$	484.325i	0.0153849i	−0.999970	+	0.00769244i	$0.997551\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.4.f.b.431.12		24
3.2	odd	2	inner	864.4.f.b.431.14		24
4.3	odd	2		216.4.f.b.107.16	yes	24
8.3	odd	2	inner	864.4.f.b.431.13		24
8.5	even	2		216.4.f.b.107.10	yes	24
12.11	even	2		216.4.f.b.107.9	✓	24
24.5	odd	2		216.4.f.b.107.15	yes	24
24.11	even	2	inner	864.4.f.b.431.11		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.4.f.b.107.9	✓	24	12.11	even	2
216.4.f.b.107.10	yes	24	8.5	even	2
216.4.f.b.107.15	yes	24	24.5	odd	2
216.4.f.b.107.16	yes	24	4.3	odd	2
864.4.f.b.431.11		24	24.11	even	2	inner
864.4.f.b.431.12		24	1.1	even	1	trivial
864.4.f.b.431.13		24	8.3	odd	2	inner
864.4.f.b.431.14		24	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	864.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.898373 q^{5} +20.7150i q^{7} +O(q^{10})\)	\(q-0.898373 q^{5} +20.7150i q^{7} +0.351756i q^{11} -44.6629i q^{13} -58.0727i q^{17} +42.4793 q^{19} -196.511 q^{23} -124.193 q^{25} +71.7647 q^{29} +96.1796i q^{31} -18.6098i q^{35} +18.9148i q^{37} -322.069i q^{41} +264.193 q^{43} +370.645 q^{47} -86.1123 q^{49} +512.874 q^{53} -0.316008i q^{55} -240.841i q^{59} -526.724i q^{61} +40.1240i q^{65} +156.199 q^{67} +395.199 q^{71} -723.326 q^{73} -7.28663 q^{77} +972.700i q^{79} -1222.96i q^{83} +52.1710i q^{85} -1231.35i q^{89} +925.194 q^{91} -38.1623 q^{95} -499.594 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 48 q^{19} + 600 q^{25} + 432 q^{43} - 816 q^{49} - 1632 q^{67} - 216 q^{73} - 3600 q^{91} + 2280 q^{97}+O(q^{100}) \) 24 * q + 48 * q^19 + 600 * q^25 + 432 * q^43 - 816 * q^49 - 1632 * q^67 - 216 * q^73 - 3600 * q^91 + 2280 * q^97

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