LMFDB - Embedded newform 8712.2.a.cc.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8712,2,Mod(1,8712)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8712, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8712.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8712 = 2^{3} \cdot 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8712.a (trivial)

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:	yes
Analytic conductor:	$69.5656702409$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.13625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 11x^{2} + 12x + 31 $ x^4 - 2x^3 - 11x^2 + 12*x + 31
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 264)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.50348$ of defining polynomial
Character	$\chi$	$=$	8712.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.50348 q^{5} -3.66875 q^{7} +O(q^{10})$	$q+1.50348 q^{5} -3.66875 q^{7} -4.97992 q^{13} +7.05072 q^{17} -4.17223 q^{19} +1.12151 q^{23} -2.73955 q^{25} -7.20357 q^{29} +5.20472 q^{31} -5.51589 q^{35} -4.35758 q^{37} +1.49222 q^{41} +9.12151 q^{43} -3.27982 q^{47} +6.45972 q^{49} -0.732588 q^{53} +13.7264 q^{59} -10.5429 q^{61} -7.48721 q^{65} +12.5655 q^{67} +1.77900 q^{71} -14.9118 q^{73} +5.31813 q^{79} +6.01937 q^{83} +10.6006 q^{85} -5.96435 q^{89} +18.2701 q^{91} -6.27286 q^{95} +3.11025 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{5} + 5 q^{7}+O(q^{10}) $ 4 * q - 2 * q^5 + 5 * q^7	$ 4 q - 2 q^{5} + 5 q^{7} - 2 q^{13} + 13 q^{17} + 11 q^{19} - 8 q^{23} + 6 q^{25} - 11 q^{29} + 2 q^{31} - 5 q^{35} + 4 q^{37} + 6 q^{41} + 24 q^{43} - 5 q^{47} + 17 q^{49} - 2 q^{53} + 4 q^{59} - 27 q^{61} + 21 q^{65} + 19 q^{67} - 17 q^{71} - 15 q^{73} + 7 q^{79} - q^{83} - 4 q^{85} - 6 q^{89} + 50 q^{91} - 33 q^{95} + 8 q^{97}+O(q^{100}) $ 4 * q - 2 * q^5 + 5 * q^7 - 2 * q^13 + 13 * q^17 + 11 * q^19 - 8 * q^23 + 6 * q^25 - 11 * q^29 + 2 * q^31 - 5 * q^35 + 4 * q^37 + 6 * q^41 + 24 * q^43 - 5 * q^47 + 17 * q^49 - 2 * q^53 + 4 * q^59 - 27 * q^61 + 21 * q^65 + 19 * q^67 - 17 * q^71 - 15 * q^73 + 7 * q^79 - q^83 - 4 * q^85 - 6 * q^89 + 50 * q^91 - 33 * q^95 + 8 * q^97

Display 100 coefficients 10 coefficients

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
$5$	1.50348	0.672377	0.336188	−	0.941795i	$-0.390862\pi$
$5$	1.50348	0.672377	0.336188	+	0.941795i	$0.390862\pi$
$6$	0	0
$7$	−3.66875	−1.38666	−0.693329	−	0.720622i	$-0.743857\pi$
$7$	−3.66875	−1.38666	−0.693329	+	0.720622i	$0.743857\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−4.97992	−1.38118	−0.690590	−	0.723246i	$-0.742649\pi$
$13$	−4.97992	−1.38118	−0.690590	+	0.723246i	$0.742649\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.05072	1.71005	0.855025	−	0.518587i	$-0.173542\pi$
$17$	7.05072	1.71005	0.855025	+	0.518587i	$0.173542\pi$
$18$	0	0
$19$	−4.17223	−0.957175	−0.478588	−	0.878040i	$-0.658851\pi$
$19$	−4.17223	−0.957175	−0.478588	+	0.878040i	$0.658851\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.12151	0.233852	0.116926	−	0.993141i	$-0.462696\pi$
$23$	1.12151	0.233852	0.116926	+	0.993141i	$0.462696\pi$
$24$	0	0
$25$	−2.73955	−0.547910
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.20357	−1.33767	−0.668835	−	0.743411i	$-0.733207\pi$
$29$	−7.20357	−1.33767	−0.668835	+	0.743411i	$0.733207\pi$
$30$	0	0
$31$	5.20472	0.934796	0.467398	−	0.884047i	$-0.345192\pi$
$31$	5.20472	0.934796	0.467398	+	0.884047i	$0.345192\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−5.51589	−0.932356
$36$	0	0
$37$	−4.35758	−0.716382	−0.358191	−	0.933648i	$-0.616606\pi$
$37$	−4.35758	−0.716382	−0.358191	+	0.933648i	$0.616606\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.49222	0.233045	0.116523	−	0.993188i	$-0.462825\pi$
$41$	1.49222	0.233045	0.116523	+	0.993188i	$0.462825\pi$
$42$	0	0
$43$	9.12151	1.39102	0.695509	−	0.718517i	$-0.255179\pi$
$43$	9.12151	1.39102	0.695509	+	0.718517i	$0.255179\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.27982	−0.478411	−0.239206	−	0.970969i	$-0.576887\pi$
$47$	−3.27982	−0.478411	−0.239206	+	0.970969i	$0.576887\pi$
$48$	0	0
$49$	6.45972	0.922818
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−0.732588	−0.100629	−0.0503144	−	0.998733i	$-0.516022\pi$
$53$	−0.732588	−0.100629	−0.0503144	+	0.998733i	$0.516022\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	13.7264	1.78703	0.893514	−	0.449035i	$-0.148232\pi$
$59$	13.7264	1.78703	0.893514	+	0.449035i	$0.148232\pi$
$60$	0	0
$61$	−10.5429	−1.34988	−0.674942	−	0.737871i	$-0.735831\pi$
$61$	−10.5429	−1.34988	−0.674942	+	0.737871i	$0.735831\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−7.48721	−0.928674
$66$	0	0
$67$	12.5655	1.53511	0.767557	−	0.640980i	$-0.221472\pi$
$67$	12.5655	1.53511	0.767557	+	0.640980i	$0.221472\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.77900	0.211129	0.105564	−	0.994412i	$-0.466335\pi$
$71$	1.77900	0.211129	0.105564	+	0.994412i	$0.466335\pi$
$72$	0	0
$73$	−14.9118	−1.74529	−0.872646	−	0.488354i	$-0.837598\pi$
$73$	−14.9118	−1.74529	−0.872646	+	0.488354i	$0.837598\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5.31813	0.598336	0.299168	−	0.954200i	$-0.403291\pi$
$79$	5.31813	0.598336	0.299168	+	0.954200i	$0.403291\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.01937	0.660712	0.330356	−	0.943856i	$-0.392831\pi$
$83$	6.01937	0.660712	0.330356	+	0.943856i	$0.392831\pi$
$84$	0	0
$85$	10.6006	1.14980
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−5.96435	−0.632220	−0.316110	−	0.948722i	$-0.602377\pi$
$89$	−5.96435	−0.632220	−0.316110	+	0.948722i	$0.602377\pi$
$90$	0	0
$91$	18.2701	1.91522
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.27286	−0.643582
$96$	0	0
$97$	3.11025	0.315798	0.157899	−	0.987455i	$-0.449528\pi$
$97$	3.11025	0.315798	0.157899	+	0.987455i	$0.449528\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.45276	0.642074	0.321037	−	0.947067i	$-0.395969\pi$
$101$	6.45276	0.642074	0.321037	+	0.947067i	$0.395969\pi$
$102$	0	0
$103$	2.85295	0.281110	0.140555	−	0.990073i	$-0.455111\pi$
$103$	2.85295	0.281110	0.140555	+	0.990073i	$0.455111\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.96751	−0.286880	−0.143440	−	0.989659i	$-0.545816\pi$
$107$	−2.96751	−0.286880	−0.143440	+	0.989659i	$0.545816\pi$
$108$	0	0
$109$	4.70124	0.450298	0.225149	−	0.974324i	$-0.427713\pi$
$109$	4.70124	0.450298	0.225149	+	0.974324i	$0.427713\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0.485259	0.0456493	0.0228246	−	0.999739i	$-0.492734\pi$
$113$	0.485259	0.0456493	0.0228246	+	0.999739i	$0.492734\pi$
$114$	0	0
$115$	1.68617	0.157237
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−25.8673	−2.37125
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.6363	−1.04078
$126$	0	0
$127$	−1.38627	−0.123011	−0.0615057	−	0.998107i	$-0.519590\pi$
$127$	−1.38627	−0.123011	−0.0615057	+	0.998107i	$0.519590\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0.503480	0.0439892	0.0219946	−	0.999758i	$-0.492998\pi$
$131$	0.503480	0.0439892	0.0219946	+	0.999758i	$0.492998\pi$
$132$	0	0
$133$	15.3069	1.32727
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.22295	0.787969	0.393985	−	0.919117i	$-0.371096\pi$
$137$	9.22295	0.787969	0.393985	+	0.919117i	$0.371096\pi$
$138$	0	0
$139$	16.0577	1.36199	0.680997	−	0.732286i	$-0.261547\pi$
$139$	16.0577	1.36199	0.680997	+	0.732286i	$0.261547\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−10.8304	−0.899418
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−13.2105	−1.08225	−0.541125	−	0.840942i	$-0.682001\pi$
$149$	−13.2105	−1.08225	−0.541125	+	0.840942i	$0.682001\pi$
$150$	0	0
$151$	16.5411	1.34609	0.673047	−	0.739600i	$-0.264985\pi$
$151$	16.5411	1.34609	0.673047	+	0.739600i	$0.264985\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	7.82520	0.628535
$156$	0	0
$157$	11.3150	0.903033	0.451517	−	0.892263i	$-0.350883\pi$
$157$	11.3150	0.903033	0.451517	+	0.892263i	$0.350883\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.11455	−0.324272
$162$	0	0
$163$	15.8936	1.24488	0.622440	−	0.782668i	$-0.286142\pi$
$163$	15.8936	1.24488	0.622440	+	0.782668i	$0.286142\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.84169	−0.452044	−0.226022	−	0.974122i	$-0.572572\pi$
$167$	−5.84169	−0.452044	−0.226022	+	0.974122i	$0.572572\pi$
$168$	0	0
$169$	11.7996	0.907660
$170$	0	0
$171$	0	0
$172$	0	0
$173$	15.9154	1.21002	0.605012	−	0.796217i	$-0.293168\pi$
$173$	15.9154	1.21002	0.605012	+	0.796217i	$0.293168\pi$
$174$	0	0
$175$	10.0507	0.759763
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.56116	0.116686	0.0583431	−	0.998297i	$-0.481418\pi$
$179$	1.56116	0.116686	0.0583431	+	0.998297i	$0.481418\pi$
$180$	0	0
$181$	24.1758	1.79697	0.898487	−	0.439000i	$-0.144667\pi$
$181$	24.1758	1.79697	0.898487	+	0.439000i	$0.144667\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.55154	−0.481679
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−10.4590	−0.756788	−0.378394	−	0.925645i	$-0.623523\pi$
$191$	−10.4590	−0.756788	−0.378394	+	0.925645i	$0.623523\pi$
$192$	0	0
$193$	−20.5963	−1.48255	−0.741277	−	0.671199i	$-0.765780\pi$
$193$	−20.5963	−1.48255	−0.741277	+	0.671199i	$0.765780\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−15.1986	−1.08285	−0.541426	−	0.840748i	$-0.682115\pi$
$197$	−15.1986	−1.08285	−0.541426	+	0.840748i	$0.682115\pi$
$198$	0	0
$199$	−26.6069	−1.88611	−0.943055	−	0.332637i	$-0.892062\pi$
$199$	−26.6069	−1.88611	−0.943055	+	0.332637i	$0.892062\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	26.4281	1.85489
$204$	0	0
$205$	2.24352	0.156694
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−0.199272	−0.0137185	−0.00685923	−	0.999976i	$-0.502183\pi$
$211$	−0.199272	−0.0137185	−0.00685923	+	0.999976i	$0.502183\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	13.7140	0.935288
$216$	0	0
$217$	−19.0948	−1.29624
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−35.1120	−2.36189
$222$	0	0
$223$	6.76938	0.453311	0.226656	−	0.973975i	$-0.427221\pi$
$223$	6.76938	0.453311	0.226656	+	0.973975i	$0.427221\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3.06650	−0.203531	−0.101765	−	0.994808i	$-0.532449\pi$
$227$	−3.06650	−0.203531	−0.101765	+	0.994808i	$0.532449\pi$
$228$	0	0
$229$	16.8004	1.11020	0.555100	−	0.831784i	$-0.312680\pi$
$229$	16.8004	1.11020	0.555100	+	0.831784i	$0.312680\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	25.5097	1.67120	0.835599	−	0.549340i	$-0.185121\pi$
$233$	25.5097	1.67120	0.835599	+	0.549340i	$0.185121\pi$
$234$	0	0
$235$	−4.93115	−0.321673
$236$	0	0
$237$	0	0
$238$	0	0
$239$	21.5368	1.39310	0.696549	−	0.717509i	$-0.254718\pi$
$239$	21.5368	1.39310	0.696549	+	0.717509i	$0.254718\pi$
$240$	0	0
$241$	7.53167	0.485158	0.242579	−	0.970132i	$-0.422007\pi$
$241$	7.53167	0.485158	0.242579	+	0.970132i	$0.422007\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	9.71207	0.620481
$246$	0	0
$247$	20.7774	1.32203
$248$	0	0
$249$	0	0
$250$	0	0
$251$	29.6143	1.86924	0.934619	−	0.355650i	$-0.115740\pi$
$251$	29.6143	1.86924	0.934619	+	0.355650i	$0.115740\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7.32939	0.457195	0.228597	−	0.973521i	$-0.426586\pi$
$257$	7.32939	0.457195	0.228597	+	0.973521i	$0.426586\pi$
$258$	0	0
$259$	15.9869	0.993376
$260$	0	0
$261$	0	0
$262$	0	0
$263$	15.5747	0.960378	0.480189	−	0.877165i	$-0.340568\pi$
$263$	15.5747	0.960378	0.480189	+	0.877165i	$0.340568\pi$
$264$	0	0
$265$	−1.10143	−0.0676604
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−19.1471	−1.16742	−0.583711	−	0.811962i	$-0.698400\pi$
$269$	−19.1471	−1.16742	−0.583711	+	0.811962i	$0.698400\pi$
$270$	0	0
$271$	7.09168	0.430789	0.215394	−	0.976527i	$-0.430896\pi$
$271$	7.09168	0.430789	0.215394	+	0.976527i	$0.430896\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0.569977	0.0342466	0.0171233	−	0.999853i	$-0.494549\pi$
$277$	0.569977	0.0342466	0.0171233	+	0.999853i	$0.494549\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	26.4373	1.57712	0.788558	−	0.614960i	$-0.210828\pi$
$281$	26.4373	1.57712	0.788558	+	0.614960i	$0.210828\pi$
$282$	0	0
$283$	8.61373	0.512033	0.256017	−	0.966672i	$-0.417590\pi$
$283$	8.61373	0.512033	0.256017	+	0.966672i	$0.417590\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.47458	−0.323154
$288$	0	0
$289$	32.7126	1.92427
$290$	0	0
$291$	0	0
$292$	0	0
$293$	23.8855	1.39541	0.697704	−	0.716386i	$-0.254205\pi$
$293$	23.8855	1.39541	0.697704	+	0.716386i	$0.254205\pi$
$294$	0	0
$295$	20.6374	1.20156
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−5.58505	−0.322992
$300$	0	0
$301$	−33.4646	−1.92886
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−15.8511	−0.907631
$306$	0	0
$307$	11.3425	0.647351	0.323676	−	0.946168i	$-0.395081\pi$
$307$	11.3425	0.647351	0.323676	+	0.946168i	$0.395081\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−26.7561	−1.51720	−0.758600	−	0.651556i	$-0.774116\pi$
$311$	−26.7561	−1.51720	−0.758600	+	0.651556i	$0.774116\pi$
$312$	0	0
$313$	−16.9613	−0.958712	−0.479356	−	0.877621i	$-0.659130\pi$
$313$	−16.9613	−0.958712	−0.479356	+	0.877621i	$0.659130\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	25.4908	1.43171	0.715853	−	0.698251i	$-0.246038\pi$
$317$	25.4908	1.43171	0.715853	+	0.698251i	$0.246038\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−29.4172	−1.63682
$324$	0	0
$325$	13.6427	0.756762
$326$	0	0
$327$	0	0
$328$	0	0
$329$	12.0329	0.663393
$330$	0	0
$331$	−8.27007	−0.454564	−0.227282	−	0.973829i	$-0.572984\pi$
$331$	−8.27007	−0.454564	−0.227282	+	0.973829i	$0.572984\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	18.8919	1.03218
$336$	0	0
$337$	−24.5596	−1.33785	−0.668925	−	0.743330i	$-0.733245\pi$
$337$	−24.5596	−1.33785	−0.668925	+	0.743330i	$0.733245\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.98214	0.107025
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−14.5837	−0.782893	−0.391446	−	0.920201i	$-0.628025\pi$
$347$	−14.5837	−0.782893	−0.391446	+	0.920201i	$0.628025\pi$
$348$	0	0
$349$	27.5697	1.47577	0.737886	−	0.674925i	$-0.235824\pi$
$349$	27.5697	1.47577	0.737886	+	0.674925i	$0.235824\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	12.4334	0.661763	0.330881	−	0.943672i	$-0.392654\pi$
$353$	12.4334	0.661763	0.330881	+	0.943672i	$0.392654\pi$
$354$	0	0
$355$	2.67469	0.141958
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−32.8693	−1.73477	−0.867387	−	0.497635i	$-0.834202\pi$
$359$	−32.8693	−1.73477	−0.867387	+	0.497635i	$0.834202\pi$
$360$	0	0
$361$	−1.59250	−0.0838158
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−22.4196	−1.17349
$366$	0	0
$367$	−14.3602	−0.749598	−0.374799	−	0.927106i	$-0.622288\pi$
$367$	−14.3602	−0.749598	−0.374799	+	0.927106i	$0.622288\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.68768	0.139537
$372$	0	0
$373$	−11.4836	−0.594599	−0.297300	−	0.954784i	$-0.596086\pi$
$373$	−11.4836	−0.594599	−0.297300	+	0.954784i	$0.596086\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	35.8732	1.84756
$378$	0	0
$379$	−7.82046	−0.401710	−0.200855	−	0.979621i	$-0.564372\pi$
$379$	−7.82046	−0.401710	−0.200855	+	0.979621i	$0.564372\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7.93421	0.405419	0.202710	−	0.979239i	$-0.435025\pi$
$383$	7.93421	0.405419	0.202710	+	0.979239i	$0.435025\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	32.2380	1.63453	0.817266	−	0.576260i	$-0.195489\pi$
$389$	32.2380	1.63453	0.817266	+	0.576260i	$0.195489\pi$
$390$	0	0
$391$	7.90748	0.399898
$392$	0	0
$393$	0	0
$394$	0	0
$395$	7.99570	0.402307
$396$	0	0
$397$	−19.2617	−0.966716	−0.483358	−	0.875423i	$-0.660583\pi$
$397$	−19.2617	−0.966716	−0.483358	+	0.875423i	$0.660583\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−20.8130	−1.03935	−0.519676	−	0.854363i	$-0.673947\pi$
$401$	−20.8130	−1.03935	−0.519676	+	0.854363i	$0.673947\pi$
$402$	0	0
$403$	−25.9191	−1.29112
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−14.8893	−0.736226	−0.368113	−	0.929781i	$-0.619996\pi$
$409$	−14.8893	−0.736226	−0.368113	+	0.929781i	$0.619996\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−50.3588	−2.47800
$414$	0	0
$415$	9.05000	0.444247
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−10.7133	−0.523379	−0.261690	−	0.965152i	$-0.584280\pi$
$419$	−10.7133	−0.523379	−0.261690	+	0.965152i	$0.584280\pi$
$420$	0	0
$421$	12.4415	0.606362	0.303181	−	0.952933i	$-0.401951\pi$
$421$	12.4415	0.606362	0.303181	+	0.952933i	$0.401951\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−19.3158	−0.936953
$426$	0	0
$427$	38.6794	1.87183
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−19.5477	−0.941578	−0.470789	−	0.882246i	$-0.656031\pi$
$431$	−19.5477	−0.941578	−0.470789	+	0.882246i	$0.656031\pi$
$432$	0	0
$433$	38.4970	1.85005	0.925025	−	0.379906i	$-0.124044\pi$
$433$	38.4970	1.85005	0.925025	+	0.379906i	$0.124044\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.67921	−0.223837
$438$	0	0
$439$	−7.52715	−0.359251	−0.179626	−	0.983735i	$-0.557489\pi$
$439$	−7.52715	−0.359251	−0.179626	+	0.983735i	$0.557489\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	19.8708	0.944091	0.472045	−	0.881574i	$-0.343516\pi$
$443$	19.8708	0.944091	0.472045	+	0.881574i	$0.343516\pi$
$444$	0	0
$445$	−8.96729	−0.425090
$446$	0	0
$447$	0	0
$448$	0	0
$449$	8.34676	0.393908	0.196954	−	0.980413i	$-0.436895\pi$
$449$	8.34676	0.393908	0.196954	+	0.980413i	$0.436895\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	27.4687	1.28775
$456$	0	0
$457$	10.9068	0.510197	0.255098	−	0.966915i	$-0.417892\pi$
$457$	10.9068	0.510197	0.255098	+	0.966915i	$0.417892\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−12.9763	−0.604368	−0.302184	−	0.953250i	$-0.597716\pi$
$461$	−12.9763	−0.604368	−0.302184	+	0.953250i	$0.597716\pi$
$462$	0	0
$463$	20.4036	0.948238	0.474119	−	0.880461i	$-0.342767\pi$
$463$	20.4036	0.948238	0.474119	+	0.880461i	$0.342767\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−24.8298	−1.14899	−0.574493	−	0.818509i	$-0.694801\pi$
$467$	−24.8298	−1.14899	−0.574493	+	0.818509i	$0.694801\pi$
$468$	0	0
$469$	−46.0995	−2.12868
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	11.4300	0.524445
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−5.68697	−0.259844	−0.129922	−	0.991524i	$-0.541473\pi$
$479$	−5.68697	−0.259844	−0.129922	+	0.991524i	$0.541473\pi$
$480$	0	0
$481$	21.7004	0.989453
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.67620	0.212335
$486$	0	0
$487$	−21.3368	−0.966862	−0.483431	−	0.875382i	$-0.660610\pi$
$487$	−21.3368	−0.966862	−0.483431	+	0.875382i	$0.660610\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	19.0669	0.860479	0.430239	−	0.902715i	$-0.358429\pi$
$491$	19.0669	0.860479	0.430239	+	0.902715i	$0.358429\pi$
$492$	0	0
$493$	−50.7904	−2.28748
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6.52671	−0.292763
$498$	0	0
$499$	−1.34029	−0.0599997	−0.0299999	−	0.999550i	$-0.509551\pi$
$499$	−1.34029	−0.0599997	−0.0299999	+	0.999550i	$0.509551\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	25.7469	1.14800	0.574000	−	0.818855i	$-0.305391\pi$
$503$	25.7469	1.14800	0.574000	+	0.818855i	$0.305391\pi$
$504$	0	0
$505$	9.70160	0.431716
$506$	0	0
$507$	0	0
$508$	0	0
$509$	24.1955	1.07245	0.536223	−	0.844077i	$-0.319851\pi$
$509$	24.1955	1.07245	0.536223	+	0.844077i	$0.319851\pi$
$510$	0	0
$511$	54.7076	2.42012
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.28936	0.189012
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−12.7865	−0.560185	−0.280092	−	0.959973i	$-0.590365\pi$
$521$	−12.7865	−0.560185	−0.280092	+	0.959973i	$0.590365\pi$
$522$	0	0
$523$	−14.0855	−0.615917	−0.307958	−	0.951400i	$-0.599646\pi$
$523$	−14.0855	−0.615917	−0.307958	+	0.951400i	$0.599646\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	36.6970	1.59855
$528$	0	0
$529$	−21.7422	−0.945313
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−7.43113	−0.321878
$534$	0	0
$535$	−4.46159	−0.192891
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	6.45011	0.277312	0.138656	−	0.990341i	$-0.455722\pi$
$541$	6.45011	0.277312	0.138656	+	0.990341i	$0.455722\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7.06823	0.302770
$546$	0	0
$547$	8.79908	0.376222	0.188111	−	0.982148i	$-0.439764\pi$
$547$	8.79908	0.376222	0.188111	+	0.982148i	$0.439764\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	30.0550	1.28038
$552$	0	0
$553$	−19.5109	−0.829687
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−15.9781	−0.677012	−0.338506	−	0.940964i	$-0.609922\pi$
$557$	−15.9781	−0.677012	−0.338506	+	0.940964i	$0.609922\pi$
$558$	0	0
$559$	−45.4244	−1.92125
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0.957395	0.0403494	0.0201747	−	0.999796i	$-0.493578\pi$
$563$	0.957395	0.0403494	0.0201747	+	0.999796i	$0.493578\pi$
$564$	0	0
$565$	0.729577	0.0306935
$566$	0	0
$567$	0	0
$568$	0	0
$569$	20.1014	0.842696	0.421348	−	0.906899i	$-0.361557\pi$
$569$	20.1014	0.842696	0.421348	+	0.906899i	$0.361557\pi$
$570$	0	0
$571$	−3.31936	−0.138911	−0.0694555	−	0.997585i	$-0.522126\pi$
$571$	−3.31936	−0.138911	−0.0694555	+	0.997585i	$0.522126\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3.07244	−0.128130
$576$	0	0
$577$	43.8322	1.82476	0.912380	−	0.409344i	$-0.134242\pi$
$577$	43.8322	1.82476	0.912380	+	0.409344i	$0.134242\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−22.0836	−0.916181
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−34.3376	−1.41726	−0.708632	−	0.705578i	$-0.750687\pi$
$587$	−34.3376	−1.41726	−0.708632	+	0.705578i	$0.750687\pi$
$588$	0	0
$589$	−21.7153	−0.894763
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−16.3665	−0.672091	−0.336046	−	0.941846i	$-0.609090\pi$
$593$	−16.3665	−0.672091	−0.336046	+	0.941846i	$0.609090\pi$
$594$	0	0
$595$	−38.8910	−1.59437
$596$	0	0
$597$	0	0
$598$	0	0
$599$	23.6031	0.964394	0.482197	−	0.876063i	$-0.339839\pi$
$599$	23.6031	0.964394	0.482197	+	0.876063i	$0.339839\pi$
$600$	0	0
$601$	−17.3382	−0.707240	−0.353620	−	0.935389i	$-0.615049\pi$
$601$	−17.3382	−0.707240	−0.353620	+	0.935389i	$0.615049\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−27.4230	−1.11307	−0.556533	−	0.830826i	$-0.687869\pi$
$607$	−27.4230	−1.11307	−0.556533	+	0.830826i	$0.687869\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	16.3333	0.660773
$612$	0	0
$613$	46.5225	1.87903	0.939513	−	0.342513i	$-0.111278\pi$
$613$	46.5225	1.87903	0.939513	+	0.342513i	$0.111278\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	38.9398	1.56766	0.783828	−	0.620978i	$-0.213265\pi$
$617$	38.9398	1.56766	0.783828	+	0.620978i	$0.213265\pi$
$618$	0	0
$619$	−9.68976	−0.389465	−0.194732	−	0.980856i	$-0.562384\pi$
$619$	−9.68976	−0.389465	−0.194732	+	0.980856i	$0.562384\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	21.8817	0.876673
$624$	0	0
$625$	−3.79714	−0.151885
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−30.7241	−1.22505
$630$	0	0
$631$	11.8697	0.472524	0.236262	−	0.971689i	$-0.424078\pi$
$631$	11.8697	0.472524	0.236262	+	0.971689i	$0.424078\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−2.08423	−0.0827100
$636$	0	0
$637$	−32.1689	−1.27458
$638$	0	0
$639$	0	0
$640$	0	0
$641$	37.0429	1.46311	0.731553	−	0.681784i	$-0.238796\pi$
$641$	37.0429	1.46311	0.731553	+	0.681784i	$0.238796\pi$
$642$	0	0
$643$	9.85339	0.388580	0.194290	−	0.980944i	$-0.437760\pi$
$643$	9.85339	0.388580	0.194290	+	0.980944i	$0.437760\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	33.3325	1.31044	0.655218	−	0.755440i	$-0.272577\pi$
$647$	33.3325	1.31044	0.655218	+	0.755440i	$0.272577\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−3.19856	−0.125169	−0.0625847	−	0.998040i	$-0.519934\pi$
$653$	−3.19856	−0.125169	−0.0625847	+	0.998040i	$0.519934\pi$
$654$	0	0
$655$	0.756972	0.0295773
$656$	0	0
$657$	0	0
$658$	0	0
$659$	42.6114	1.65990	0.829952	−	0.557835i	$-0.188368\pi$
$659$	42.6114	1.65990	0.829952	+	0.557835i	$0.188368\pi$
$660$	0	0
$661$	16.4734	0.640742	0.320371	−	0.947292i	$-0.396192\pi$
$661$	16.4734	0.640742	0.320371	+	0.947292i	$0.396192\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	23.0136	0.892428
$666$	0	0
$667$	−8.07891	−0.312817
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−6.27100	−0.241729	−0.120865	−	0.992669i	$-0.538567\pi$
$673$	−6.27100	−0.241729	−0.120865	+	0.992669i	$0.538567\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−19.5804	−0.752537	−0.376269	−	0.926511i	$-0.622793\pi$
$677$	−19.5804	−0.752537	−0.376269	+	0.926511i	$0.622793\pi$
$678$	0	0
$679$	−11.4107	−0.437904
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−10.5225	−0.402632	−0.201316	−	0.979526i	$-0.564522\pi$
$683$	−10.5225	−0.402632	−0.201316	+	0.979526i	$0.564522\pi$
$684$	0	0
$685$	13.8665	0.529812
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3.64823	0.138986
$690$	0	0
$691$	−0.723767	−0.0275334	−0.0137667	−	0.999905i	$-0.504382\pi$
$691$	−0.723767	−0.0275334	−0.0137667	+	0.999905i	$0.504382\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	24.1424	0.915773
$696$	0	0
$697$	10.5212	0.398519
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−6.82081	−0.257618	−0.128809	−	0.991669i	$-0.541115\pi$
$701$	−6.82081	−0.257618	−0.128809	+	0.991669i	$0.541115\pi$
$702$	0	0
$703$	18.1808	0.685703
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−23.6736	−0.890336
$708$	0	0
$709$	−6.78286	−0.254736	−0.127368	−	0.991856i	$-0.540653\pi$
$709$	−6.78286	−0.254736	−0.127368	+	0.991856i	$0.540653\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5.83717	0.218604
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−19.2684	−0.718591	−0.359296	−	0.933224i	$-0.616983\pi$
$719$	−19.2684	−0.718591	−0.359296	+	0.933224i	$0.616983\pi$
$720$	0	0
$721$	−10.4668	−0.389803
$722$	0	0
$723$	0	0
$724$	0	0
$725$	19.7345	0.732922
$726$	0	0
$727$	−31.5105	−1.16866	−0.584330	−	0.811516i	$-0.698643\pi$
$727$	−31.5105	−1.16866	−0.584330	+	0.811516i	$0.698643\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	64.3132	2.37871
$732$	0	0
$733$	20.7066	0.764814	0.382407	−	0.923994i	$-0.375095\pi$
$733$	20.7066	0.764814	0.382407	+	0.923994i	$0.375095\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−8.00310	−0.294399	−0.147199	−	0.989107i	$-0.547026\pi$
$739$	−8.00310	−0.294399	−0.147199	+	0.989107i	$0.547026\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−25.3158	−0.928746	−0.464373	−	0.885640i	$-0.653720\pi$
$743$	−25.3158	−0.928746	−0.464373	+	0.885640i	$0.653720\pi$
$744$	0	0
$745$	−19.8618	−0.727679
$746$	0	0
$747$	0	0
$748$	0	0
$749$	10.8870	0.397804
$750$	0	0
$751$	14.0058	0.511079	0.255540	−	0.966799i	$-0.417747\pi$
$751$	14.0058	0.511079	0.255540	+	0.966799i	$0.417747\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	24.8692	0.905082
$756$	0	0
$757$	−4.70346	−0.170950	−0.0854751	−	0.996340i	$-0.527241\pi$
$757$	−4.70346	−0.170950	−0.0854751	+	0.996340i	$0.527241\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	25.0574	0.908330	0.454165	−	0.890918i	$-0.349938\pi$
$761$	25.0574	0.908330	0.454165	+	0.890918i	$0.349938\pi$
$762$	0	0
$763$	−17.2477	−0.624408
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−68.3565	−2.46821
$768$	0	0
$769$	−0.757190	−0.0273050	−0.0136525	−	0.999907i	$-0.504346\pi$
$769$	−0.757190	−0.0273050	−0.0136525	+	0.999907i	$0.504346\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−9.68511	−0.348349	−0.174175	−	0.984715i	$-0.555726\pi$
$773$	−9.68511	−0.348349	−0.174175	+	0.984715i	$0.555726\pi$
$774$	0	0
$775$	−14.2586	−0.512184
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6.22588	−0.223065
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	17.0118	0.607179
$786$	0	0
$787$	30.3693	1.08255	0.541274	−	0.840846i	$-0.317942\pi$
$787$	30.3693	1.08255	0.541274	+	0.840846i	$0.317942\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.78029	−0.0632999
$792$	0	0
$793$	52.5029	1.86443
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4.53633	0.160685	0.0803426	−	0.996767i	$-0.474399\pi$
$797$	4.53633	0.160685	0.0803426	+	0.996767i	$0.474399\pi$
$798$	0	0
$799$	−23.1251	−0.818107
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−6.18615	−0.218033
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−39.4974	−1.38865	−0.694327	−	0.719659i	$-0.744298\pi$
$809$	−39.4974	−1.38865	−0.694327	+	0.719659i	$0.744298\pi$
$810$	0	0
$811$	−0.567676	−0.0199338	−0.00996690	−	0.999950i	$-0.503173\pi$
$811$	−0.567676	−0.0199338	−0.00996690	+	0.999950i	$0.503173\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	23.8956	0.837028
$816$	0	0
$817$	−38.0571	−1.33145
$818$	0	0
$819$	0	0
$820$	0	0
$821$	6.28018	0.219180	0.109590	−	0.993977i	$-0.465046\pi$
$821$	6.28018	0.219180	0.109590	+	0.993977i	$0.465046\pi$
$822$	0	0
$823$	−32.6370	−1.13765	−0.568827	−	0.822457i	$-0.692603\pi$
$823$	−32.6370	−1.13765	−0.568827	+	0.822457i	$0.692603\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	14.3181	0.497890	0.248945	−	0.968518i	$-0.419916\pi$
$827$	14.3181	0.497890	0.248945	+	0.968518i	$0.419916\pi$
$828$	0	0
$829$	−5.56302	−0.193212	−0.0966058	−	0.995323i	$-0.530799\pi$
$829$	−5.56302	−0.193212	−0.0966058	+	0.995323i	$0.530799\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	45.5457	1.57806
$834$	0	0
$835$	−8.78286	−0.303944
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−23.6951	−0.818045	−0.409023	−	0.912524i	$-0.634130\pi$
$839$	−23.6951	−0.818045	−0.409023	+	0.912524i	$0.634130\pi$
$840$	0	0
$841$	22.8915	0.789361
$842$	0	0
$843$	0	0
$844$	0	0
$845$	17.7404	0.610289
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−4.88709	−0.167527
$852$	0	0
$853$	44.6217	1.52782	0.763909	−	0.645324i	$-0.223278\pi$
$853$	44.6217	1.52782	0.763909	+	0.645324i	$0.223278\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−22.2787	−0.761025	−0.380512	−	0.924776i	$-0.624252\pi$
$857$	−22.2787	−0.761025	−0.380512	+	0.924776i	$0.624252\pi$
$858$	0	0
$859$	9.11149	0.310880	0.155440	−	0.987845i	$-0.450320\pi$
$859$	9.11149	0.310880	0.155440	+	0.987845i	$0.450320\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	13.6843	0.465818	0.232909	−	0.972499i	$-0.425176\pi$
$863$	13.6843	0.465818	0.232909	+	0.972499i	$0.425176\pi$
$864$	0	0
$865$	23.9284	0.813591
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−62.5749	−2.12027
$872$	0	0
$873$	0	0
$874$	0	0
$875$	42.6905	1.44320
$876$	0	0
$877$	1.37545	0.0464455	0.0232227	−	0.999730i	$-0.492607\pi$
$877$	1.37545	0.0464455	0.0232227	+	0.999730i	$0.492607\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	26.2582	0.884663	0.442331	−	0.896852i	$-0.354152\pi$
$881$	26.2582	0.884663	0.442331	+	0.896852i	$0.354152\pi$
$882$	0	0
$883$	−29.7695	−1.00182	−0.500911	−	0.865499i	$-0.667002\pi$
$883$	−29.7695	−1.00182	−0.500911	+	0.865499i	$0.667002\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0.918651	0.0308453	0.0154226	−	0.999881i	$-0.495091\pi$
$887$	0.918651	0.0308453	0.0154226	+	0.999881i	$0.495091\pi$
$888$	0	0
$889$	5.08587	0.170575
$890$	0	0
$891$	0	0
$892$	0	0
$893$	13.6842	0.457924
$894$	0	0
$895$	2.34717	0.0784571
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−37.4926	−1.25045
$900$	0	0
$901$	−5.16527	−0.172080
$902$	0	0
$903$	0	0
$904$	0	0
$905$	36.3479	1.20824
$906$	0	0
$907$	53.2931	1.76957	0.884784	−	0.466000i	$-0.154305\pi$
$907$	53.2931	1.76957	0.884784	+	0.466000i	$0.154305\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−16.1695	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$911$	−16.1695	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.84714	−0.0609980
$918$	0	0
$919$	34.1595	1.12682	0.563410	−	0.826178i	$-0.309489\pi$
$919$	34.1595	1.12682	0.563410	+	0.826178i	$0.309489\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−8.85928	−0.291607
$924$	0	0
$925$	11.9378	0.392513
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−10.5217	−0.345206	−0.172603	−	0.984991i	$-0.555218\pi$
$929$	−10.5217	−0.345206	−0.172603	+	0.984991i	$0.555218\pi$
$930$	0	0
$931$	−26.9515	−0.883298
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	32.2075	1.05217	0.526086	−	0.850431i	$-0.323659\pi$
$937$	32.2075	1.05217	0.526086	+	0.850431i	$0.323659\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−38.3116	−1.24892	−0.624461	−	0.781056i	$-0.714681\pi$
$941$	−38.3116	−1.24892	−0.624461	+	0.781056i	$0.714681\pi$
$942$	0	0
$943$	1.67354	0.0544981
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−30.2459	−0.982860	−0.491430	−	0.870917i	$-0.663526\pi$
$947$	−30.2459	−0.982860	−0.491430	+	0.870917i	$0.663526\pi$
$948$	0	0
$949$	74.2594	2.41056
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−56.6526	−1.83516	−0.917580	−	0.397552i	$-0.869860\pi$
$953$	−56.6526	−1.83516	−0.917580	+	0.397552i	$0.869860\pi$
$954$	0	0
$955$	−15.7249	−0.508846
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−33.8367	−1.09264
$960$	0	0
$961$	−3.91085	−0.126156
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−30.9661	−0.996835
$966$	0	0
$967$	36.7258	1.18102	0.590511	−	0.807030i	$-0.298926\pi$
$967$	36.7258	1.18102	0.590511	+	0.807030i	$0.298926\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−35.3623	−1.13483	−0.567415	−	0.823432i	$-0.692057\pi$
$971$	−35.3623	−1.13483	−0.567415	+	0.823432i	$0.692057\pi$
$972$	0	0
$973$	−58.9116	−1.88862
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4.75985	−0.152281	−0.0761405	−	0.997097i	$-0.524260\pi$
$977$	−4.75985	−0.152281	−0.0761405	+	0.997097i	$0.524260\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	53.3113	1.70036	0.850182	−	0.526488i	$-0.176492\pi$
$983$	53.3113	1.70036	0.850182	+	0.526488i	$0.176492\pi$
$984$	0	0
$985$	−22.8507	−0.728085
$986$	0	0
$987$	0	0
$988$	0	0
$989$	10.2299	0.325292
$990$	0	0
$991$	48.5492	1.54222	0.771108	−	0.636705i	$-0.219703\pi$
$991$	48.5492	1.54222	0.771108	+	0.636705i	$0.219703\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−40.0029	−1.26818
$996$	0	0
$997$	14.1722	0.448839	0.224420	−	0.974493i	$-0.427951\pi$
$997$	14.1722	0.448839	0.224420	+	0.974493i	$0.427951\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8712.2.a.cc.1.3		4
3.2	odd	2		2904.2.a.be.1.2		4
11.7	odd	10		792.2.r.f.577.2		8
11.8	odd	10		792.2.r.f.361.2		8
11.10	odd	2		8712.2.a.bz.1.3		4
12.11	even	2		5808.2.a.cl.1.2		4
33.8	even	10		264.2.q.e.97.1	yes	8
33.29	even	10		264.2.q.e.49.1	✓	8
33.32	even	2		2904.2.a.bb.1.2		4
132.95	odd	10		528.2.y.k.49.1		8
132.107	odd	10		528.2.y.k.97.1		8
132.131	odd	2		5808.2.a.co.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
264.2.q.e.49.1	✓	8	33.29	even	10
264.2.q.e.97.1	yes	8	33.8	even	10
528.2.y.k.49.1		8	132.95	odd	10
528.2.y.k.97.1		8	132.107	odd	10
792.2.r.f.361.2		8	11.8	odd	10
792.2.r.f.577.2		8	11.7	odd	10
2904.2.a.bb.1.2		4	33.32	even	2
2904.2.a.be.1.2		4	3.2	odd	2
5808.2.a.cl.1.2		4	12.11	even	2
5808.2.a.co.1.2		4	132.131	odd	2
8712.2.a.bz.1.3		4	11.10	odd	2
8712.2.a.cc.1.3		4	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8712 = 2^{3} \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8712.a (trivial)

\(f(q)\)	\(=\)	\(q+1.50348 q^{5} -3.66875 q^{7} +O(q^{10})\)	\(q+1.50348 q^{5} -3.66875 q^{7} -4.97992 q^{13} +7.05072 q^{17} -4.17223 q^{19} +1.12151 q^{23} -2.73955 q^{25} -7.20357 q^{29} +5.20472 q^{31} -5.51589 q^{35} -4.35758 q^{37} +1.49222 q^{41} +9.12151 q^{43} -3.27982 q^{47} +6.45972 q^{49} -0.732588 q^{53} +13.7264 q^{59} -10.5429 q^{61} -7.48721 q^{65} +12.5655 q^{67} +1.77900 q^{71} -14.9118 q^{73} +5.31813 q^{79} +6.01937 q^{83} +10.6006 q^{85} -5.96435 q^{89} +18.2701 q^{91} -6.27286 q^{95} +3.11025 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{5} + 5 q^{7}+O(q^{10}) \) 4 * q - 2 * q^5 + 5 * q^7	\( 4 q - 2 q^{5} + 5 q^{7} - 2 q^{13} + 13 q^{17} + 11 q^{19} - 8 q^{23} + 6 q^{25} - 11 q^{29} + 2 q^{31} - 5 q^{35} + 4 q^{37} + 6 q^{41} + 24 q^{43} - 5 q^{47} + 17 q^{49} - 2 q^{53} + 4 q^{59} - 27 q^{61} + 21 q^{65} + 19 q^{67} - 17 q^{71} - 15 q^{73} + 7 q^{79} - q^{83} - 4 q^{85} - 6 q^{89} + 50 q^{91} - 33 q^{95} + 8 q^{97}+O(q^{100}) \) 4 * q - 2 * q^5 + 5 * q^7 - 2 * q^13 + 13 * q^17 + 11 * q^19 - 8 * q^23 + 6 * q^25 - 11 * q^29 + 2 * q^31 - 5 * q^35 + 4 * q^37 + 6 * q^41 + 24 * q^43 - 5 * q^47 + 17 * q^49 - 2 * q^53 + 4 * q^59 - 27 * q^61 + 21 * q^65 + 19 * q^67 - 17 * q^71 - 15 * q^73 + 7 * q^79 - q^83 - 4 * q^85 - 6 * q^89 + 50 * q^91 - 33 * q^95 + 8 * q^97

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