LMFDB - Embedded newform 8464.2.a.ch.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8464,2,Mod(1,8464)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8464, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8464.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8464 = 2^{4} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8464.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:	$67.5853802708$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 30 x^{13} + 28 x^{12} + 354 x^{11} - 302 x^{10} - 2111 x^{9} + 1596 x^{8} + 6777 x^{7} + \cdots - 419 $ x^15 - x^14 - 30x^13 + 28x^12 + 354x^11 - 302x^10 - 2111x^9 + 1596x^8 + 6777x^7 - 4292x^6 - 11506x^5 + 5297x^4 + 9480x^3 - 1825x^2 - 3136*x - 419
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 184)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.54853$ of defining polynomial
Character	$\chi$	$=$	8464.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.54853 q^{3} -0.818556 q^{5} +2.13290 q^{7} +3.49500 q^{9} +O(q^{10})$	$q-2.54853 q^{3} -0.818556 q^{5} +2.13290 q^{7} +3.49500 q^{9} -0.530031 q^{11} +1.02097 q^{13} +2.08611 q^{15} -7.45195 q^{17} -0.396151 q^{19} -5.43576 q^{21} -4.32997 q^{25} -1.26151 q^{27} -8.23107 q^{29} -7.63291 q^{31} +1.35080 q^{33} -1.74590 q^{35} -1.74623 q^{37} -2.60198 q^{39} +8.78714 q^{41} +10.6135 q^{43} -2.86085 q^{45} +4.39680 q^{47} -2.45073 q^{49} +18.9915 q^{51} +5.42746 q^{53} +0.433860 q^{55} +1.00960 q^{57} +7.80983 q^{59} -7.82506 q^{61} +7.45448 q^{63} -0.835724 q^{65} +11.6988 q^{67} -2.48859 q^{71} -5.05495 q^{73} +11.0350 q^{75} -1.13050 q^{77} -17.2072 q^{79} -7.26999 q^{81} -9.63725 q^{83} +6.09984 q^{85} +20.9771 q^{87} +0.481240 q^{89} +2.17764 q^{91} +19.4527 q^{93} +0.324272 q^{95} -7.70108 q^{97} -1.85246 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{3} + 10 q^{7} + 16 q^{9}+O(q^{10}) $ 15 * q - q^3 + 10 * q^7 + 16 * q^9	$ 15 q - q^{3} + 10 q^{7} + 16 q^{9} + 23 q^{11} + 10 q^{15} + 29 q^{19} - q^{21} + 23 q^{25} - q^{27} - 2 q^{29} - 20 q^{31} - 18 q^{33} + 18 q^{35} - 24 q^{37} + 19 q^{39} + 9 q^{41} + 48 q^{43} - 4 q^{45} + 36 q^{47} + 25 q^{49} + 35 q^{51} + 5 q^{53} + 10 q^{55} - 23 q^{57} + 22 q^{59} - 12 q^{61} + 35 q^{63} + 26 q^{65} + 58 q^{67} - 2 q^{71} + 5 q^{73} + 17 q^{75} + 26 q^{77} + 26 q^{79} - 21 q^{81} + 68 q^{83} - 72 q^{85} - 19 q^{87} + 6 q^{89} + 71 q^{91} - 55 q^{93} + 12 q^{95} - 40 q^{97} + 63 q^{99}+O(q^{100}) $ 15 * q - q^3 + 10 * q^7 + 16 * q^9 + 23 * q^11 + 10 * q^15 + 29 * q^19 - q^21 + 23 * q^25 - q^27 - 2 * q^29 - 20 * q^31 - 18 * q^33 + 18 * q^35 - 24 * q^37 + 19 * q^39 + 9 * q^41 + 48 * q^43 - 4 * q^45 + 36 * q^47 + 25 * q^49 + 35 * q^51 + 5 * q^53 + 10 * q^55 - 23 * q^57 + 22 * q^59 - 12 * q^61 + 35 * q^63 + 26 * q^65 + 58 * q^67 - 2 * q^71 + 5 * q^73 + 17 * q^75 + 26 * q^77 + 26 * q^79 - 21 * q^81 + 68 * q^83 - 72 * q^85 - 19 * q^87 + 6 * q^89 + 71 * q^91 - 55 * q^93 + 12 * q^95 - 40 * q^97 + 63 * q^99

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
$3$	−2.54853	−1.47139	−0.735697	−	0.677311i	$-0.763145\pi$
$3$	−2.54853	−1.47139	−0.735697	+	0.677311i	$0.763145\pi$
$4$	0	0
$5$	−0.818556	−0.366069	−0.183035	−	0.983106i	$-0.558592\pi$
$5$	−0.818556	−0.366069	−0.183035	+	0.983106i	$0.558592\pi$
$6$	0	0
$7$	2.13290	0.806161	0.403080	−	0.915165i	$-0.367940\pi$
$7$	2.13290	0.806161	0.403080	+	0.915165i	$0.367940\pi$
$8$	0	0
$9$	3.49500	1.16500
$10$	0	0
$11$	−0.530031	−0.159810	−0.0799052	−	0.996802i	$-0.525462\pi$
$11$	−0.530031	−0.159810	−0.0799052	+	0.996802i	$0.525462\pi$
$12$	0	0
$13$	1.02097	0.283167	0.141584	−	0.989926i	$-0.454781\pi$
$13$	1.02097	0.283167	0.141584	+	0.989926i	$0.454781\pi$
$14$	0	0
$15$	2.08611	0.538632
$16$	0	0
$17$	−7.45195	−1.80736	−0.903682	−	0.428205i	$-0.859146\pi$
$17$	−7.45195	−1.80736	−0.903682	+	0.428205i	$0.859146\pi$
$18$	0	0
$19$	−0.396151	−0.0908832	−0.0454416	−	0.998967i	$-0.514469\pi$
$19$	−0.396151	−0.0908832	−0.0454416	+	0.998967i	$0.514469\pi$
$20$	0	0
$21$	−5.43576	−1.18618
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−4.32997	−0.865993
$26$	0	0
$27$	−1.26151	−0.242778
$28$	0	0
$29$	−8.23107	−1.52847	−0.764235	−	0.644938i	$-0.776883\pi$
$29$	−8.23107	−1.52847	−0.764235	+	0.644938i	$0.776883\pi$
$30$	0	0
$31$	−7.63291	−1.37091	−0.685456	−	0.728114i	$-0.740397\pi$
$31$	−7.63291	−1.37091	−0.685456	+	0.728114i	$0.740397\pi$
$32$	0	0
$33$	1.35080	0.235144
$34$	0	0
$35$	−1.74590	−0.295111
$36$	0	0
$37$	−1.74623	−0.287079	−0.143539	−	0.989645i	$-0.545848\pi$
$37$	−1.74623	−0.287079	−0.143539	+	0.989645i	$0.545848\pi$
$38$	0	0
$39$	−2.60198	−0.416650
$40$	0	0
$41$	8.78714	1.37232	0.686161	−	0.727450i	$-0.259295\pi$
$41$	8.78714	1.37232	0.686161	+	0.727450i	$0.259295\pi$
$42$	0	0
$43$	10.6135	1.61854	0.809272	−	0.587434i	$-0.199862\pi$
$43$	10.6135	1.61854	0.809272	+	0.587434i	$0.199862\pi$
$44$	0	0
$45$	−2.86085	−0.426470
$46$	0	0
$47$	4.39680	0.641339	0.320669	−	0.947191i	$-0.396092\pi$
$47$	4.39680	0.641339	0.320669	+	0.947191i	$0.396092\pi$
$48$	0	0
$49$	−2.45073	−0.350105
$50$	0	0
$51$	18.9915	2.65934
$52$	0	0
$53$	5.42746	0.745518	0.372759	−	0.927928i	$-0.378412\pi$
$53$	5.42746	0.745518	0.372759	+	0.927928i	$0.378412\pi$
$54$	0	0
$55$	0.433860	0.0585017
$56$	0	0
$57$	1.00960	0.133725
$58$	0	0
$59$	7.80983	1.01675	0.508376	−	0.861135i	$-0.330246\pi$
$59$	7.80983	1.01675	0.508376	+	0.861135i	$0.330246\pi$
$60$	0	0
$61$	−7.82506	−1.00190	−0.500948	−	0.865477i	$-0.667015\pi$
$61$	−7.82506	−1.00190	−0.500948	+	0.865477i	$0.667015\pi$
$62$	0	0
$63$	7.45448	0.939176
$64$	0	0
$65$	−0.835724	−0.103659
$66$	0	0
$67$	11.6988	1.42923	0.714617	−	0.699516i	$-0.246601\pi$
$67$	11.6988	1.42923	0.714617	+	0.699516i	$0.246601\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−2.48859	−0.295341	−0.147671	−	0.989037i	$-0.547178\pi$
$71$	−2.48859	−0.295341	−0.147671	+	0.989037i	$0.547178\pi$
$72$	0	0
$73$	−5.05495	−0.591637	−0.295819	−	0.955244i	$-0.595592\pi$
$73$	−5.05495	−0.591637	−0.295819	+	0.955244i	$0.595592\pi$
$74$	0	0
$75$	11.0350	1.27422
$76$	0	0
$77$	−1.13050	−0.128833
$78$	0	0
$79$	−17.2072	−1.93596	−0.967981	−	0.251024i	$-0.919233\pi$
$79$	−17.2072	−1.93596	−0.967981	+	0.251024i	$0.919233\pi$
$80$	0	0
$81$	−7.26999	−0.807777
$82$	0	0
$83$	−9.63725	−1.05783	−0.528913	−	0.848676i	$-0.677400\pi$
$83$	−9.63725	−1.05783	−0.528913	+	0.848676i	$0.677400\pi$
$84$	0	0
$85$	6.09984	0.661620
$86$	0	0
$87$	20.9771	2.24898
$88$	0	0
$89$	0.481240	0.0510113	0.0255056	−	0.999675i	$-0.491880\pi$
$89$	0.481240	0.0510113	0.0255056	+	0.999675i	$0.491880\pi$
$90$	0	0
$91$	2.17764	0.228278
$92$	0	0
$93$	19.4527	2.01715
$94$	0	0
$95$	0.324272	0.0332696
$96$	0	0
$97$	−7.70108	−0.781926	−0.390963	−	0.920406i	$-0.627858\pi$
$97$	−7.70108	−0.781926	−0.390963	+	0.920406i	$0.627858\pi$
$98$	0	0
$99$	−1.85246	−0.186179
$100$	0	0
$101$	17.9173	1.78284	0.891418	−	0.453181i	$-0.149711\pi$
$101$	17.9173	1.78284	0.891418	+	0.453181i	$0.149711\pi$
$102$	0	0
$103$	2.29906	0.226533	0.113267	−	0.993565i	$-0.463869\pi$
$103$	2.29906	0.226533	0.113267	+	0.993565i	$0.463869\pi$
$104$	0	0
$105$	4.44947	0.434224
$106$	0	0
$107$	2.85445	0.275950	0.137975	−	0.990436i	$-0.455941\pi$
$107$	2.85445	0.275950	0.137975	+	0.990436i	$0.455941\pi$
$108$	0	0
$109$	6.11372	0.585588	0.292794	−	0.956176i	$-0.405415\pi$
$109$	6.11372	0.585588	0.292794	+	0.956176i	$0.405415\pi$
$110$	0	0
$111$	4.45032	0.422406
$112$	0	0
$113$	−10.7666	−1.01283	−0.506416	−	0.862289i	$-0.669030\pi$
$113$	−10.7666	−1.01283	−0.506416	+	0.862289i	$0.669030\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.56830	0.329889
$118$	0	0
$119$	−15.8943	−1.45703
$120$	0	0
$121$	−10.7191	−0.974461
$122$	0	0
$123$	−22.3943	−2.01922
$124$	0	0
$125$	7.63710	0.683083
$126$	0	0
$127$	12.8989	1.14459	0.572294	−	0.820048i	$-0.306054\pi$
$127$	12.8989	1.14459	0.572294	+	0.820048i	$0.306054\pi$
$128$	0	0
$129$	−27.0488	−2.38152
$130$	0	0
$131$	2.92070	0.255182	0.127591	−	0.991827i	$-0.459275\pi$
$131$	2.92070	0.255182	0.127591	+	0.991827i	$0.459275\pi$
$132$	0	0
$133$	−0.844950	−0.0732665
$134$	0	0
$135$	1.03262	0.0888737
$136$	0	0
$137$	9.20821	0.786711	0.393355	−	0.919386i	$-0.371314\pi$
$137$	9.20821	0.786711	0.393355	+	0.919386i	$0.371314\pi$
$138$	0	0
$139$	−4.47037	−0.379172	−0.189586	−	0.981864i	$-0.560715\pi$
$139$	−4.47037	−0.379172	−0.189586	+	0.981864i	$0.560715\pi$
$140$	0	0
$141$	−11.2054	−0.943662
$142$	0	0
$143$	−0.541148	−0.0452531
$144$	0	0
$145$	6.73759	0.559526
$146$	0	0
$147$	6.24577	0.515142
$148$	0	0
$149$	−14.3954	−1.17932	−0.589658	−	0.807653i	$-0.700737\pi$
$149$	−14.3954	−1.17932	−0.589658	+	0.807653i	$0.700737\pi$
$150$	0	0
$151$	−2.81738	−0.229275	−0.114637	−	0.993407i	$-0.536571\pi$
$151$	−2.81738	−0.229275	−0.114637	+	0.993407i	$0.536571\pi$
$152$	0	0
$153$	−26.0445	−2.10558
$154$	0	0
$155$	6.24797	0.501849
$156$	0	0
$157$	−8.38103	−0.668879	−0.334440	−	0.942417i	$-0.608547\pi$
$157$	−8.38103	−0.668879	−0.334440	+	0.942417i	$0.608547\pi$
$158$	0	0
$159$	−13.8320	−1.09695
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−12.0459	−0.943511	−0.471755	−	0.881730i	$-0.656379\pi$
$163$	−12.0459	−0.943511	−0.471755	+	0.881730i	$0.656379\pi$
$164$	0	0
$165$	−1.10571	−0.0860790
$166$	0	0
$167$	−0.823955	−0.0637596	−0.0318798	−	0.999492i	$-0.510149\pi$
$167$	−0.823955	−0.0637596	−0.0318798	+	0.999492i	$0.510149\pi$
$168$	0	0
$169$	−11.9576	−0.919816
$170$	0	0
$171$	−1.38455	−0.105879
$172$	0	0
$173$	−8.32249	−0.632747	−0.316374	−	0.948635i	$-0.602465\pi$
$173$	−8.32249	−0.632747	−0.316374	+	0.948635i	$0.602465\pi$
$174$	0	0
$175$	−9.23539	−0.698130
$176$	0	0
$177$	−19.9036	−1.49604
$178$	0	0
$179$	7.58472	0.566909	0.283454	−	0.958986i	$-0.408520\pi$
$179$	7.58472	0.566909	0.283454	+	0.958986i	$0.408520\pi$
$180$	0	0
$181$	0.280275	0.0208327	0.0104163	−	0.999946i	$-0.496684\pi$
$181$	0.280275	0.0208327	0.0104163	+	0.999946i	$0.496684\pi$
$182$	0	0
$183$	19.9424	1.47418
$184$	0	0
$185$	1.42939	0.105091
$186$	0	0
$187$	3.94977	0.288836
$188$	0	0
$189$	−2.69068	−0.195718
$190$	0	0
$191$	−3.08436	−0.223177	−0.111588	−	0.993755i	$-0.535594\pi$
$191$	−3.08436	−0.223177	−0.111588	+	0.993755i	$0.535594\pi$
$192$	0	0
$193$	−22.6876	−1.63309	−0.816546	−	0.577280i	$-0.804114\pi$
$193$	−22.6876	−1.63309	−0.816546	+	0.577280i	$0.804114\pi$
$194$	0	0
$195$	2.12987	0.152523
$196$	0	0
$197$	16.0393	1.14275	0.571376	−	0.820688i	$-0.306410\pi$
$197$	16.0393	1.14275	0.571376	+	0.820688i	$0.306410\pi$
$198$	0	0
$199$	9.59589	0.680235	0.340117	−	0.940383i	$-0.389533\pi$
$199$	9.59589	0.680235	0.340117	+	0.940383i	$0.389533\pi$
$200$	0	0
$201$	−29.8147	−2.10296
$202$	0	0
$203$	−17.5560	−1.23219
$204$	0	0
$205$	−7.19277	−0.502365
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.209972	0.0145241
$210$	0	0
$211$	−23.2965	−1.60380	−0.801898	−	0.597460i	$-0.796176\pi$
$211$	−23.2965	−1.60380	−0.801898	+	0.597460i	$0.796176\pi$
$212$	0	0
$213$	6.34224	0.434563
$214$	0	0
$215$	−8.68775	−0.592500
$216$	0	0
$217$	−16.2802	−1.10518
$218$	0	0
$219$	12.8827	0.870531
$220$	0	0
$221$	−7.60824	−0.511786
$222$	0	0
$223$	15.9710	1.06950	0.534750	−	0.845011i	$-0.320406\pi$
$223$	15.9710	1.06950	0.534750	+	0.845011i	$0.320406\pi$
$224$	0	0
$225$	−15.1332	−1.00888
$226$	0	0
$227$	−2.55666	−0.169691	−0.0848456	−	0.996394i	$-0.527040\pi$
$227$	−2.55666	−0.169691	−0.0848456	+	0.996394i	$0.527040\pi$
$228$	0	0
$229$	18.7176	1.23690	0.618448	−	0.785826i	$-0.287762\pi$
$229$	18.7176	1.23690	0.618448	+	0.785826i	$0.287762\pi$
$230$	0	0
$231$	2.88112	0.189564
$232$	0	0
$233$	13.1266	0.859950	0.429975	−	0.902841i	$-0.358522\pi$
$233$	13.1266	0.859950	0.429975	+	0.902841i	$0.358522\pi$
$234$	0	0
$235$	−3.59903	−0.234775
$236$	0	0
$237$	43.8530	2.84856
$238$	0	0
$239$	28.0256	1.81282	0.906412	−	0.422394i	$-0.138810\pi$
$239$	28.0256	1.81282	0.906412	+	0.422394i	$0.138810\pi$
$240$	0	0
$241$	15.7613	1.01527	0.507636	−	0.861571i	$-0.330519\pi$
$241$	15.7613	1.01527	0.507636	+	0.861571i	$0.330519\pi$
$242$	0	0
$243$	22.3123	1.43134
$244$	0	0
$245$	2.00606	0.128163
$246$	0	0
$247$	−0.404459	−0.0257351
$248$	0	0
$249$	24.5608	1.55648
$250$	0	0
$251$	15.5921	0.984162	0.492081	−	0.870549i	$-0.336236\pi$
$251$	15.5921	0.984162	0.492081	+	0.870549i	$0.336236\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−15.5456	−0.973504
$256$	0	0
$257$	3.79714	0.236859	0.118430	−	0.992962i	$-0.462214\pi$
$257$	3.79714	0.236859	0.118430	+	0.992962i	$0.462214\pi$
$258$	0	0
$259$	−3.72454	−0.231432
$260$	0	0
$261$	−28.7675	−1.78067
$262$	0	0
$263$	15.1719	0.935537	0.467768	−	0.883851i	$-0.345058\pi$
$263$	15.1719	0.935537	0.467768	+	0.883851i	$0.345058\pi$
$264$	0	0
$265$	−4.44268	−0.272911
$266$	0	0
$267$	−1.22645	−0.0750577
$268$	0	0
$269$	−0.339546	−0.0207025	−0.0103512	−	0.999946i	$-0.503295\pi$
$269$	−0.339546	−0.0207025	−0.0103512	+	0.999946i	$0.503295\pi$
$270$	0	0
$271$	−13.1984	−0.801743	−0.400872	−	0.916134i	$-0.631293\pi$
$271$	−13.1984	−0.801743	−0.400872	+	0.916134i	$0.631293\pi$
$272$	0	0
$273$	−5.54976	−0.335887
$274$	0	0
$275$	2.29502	0.138395
$276$	0	0
$277$	−11.4417	−0.687466	−0.343733	−	0.939067i	$-0.611691\pi$
$277$	−11.4417	−0.687466	−0.343733	+	0.939067i	$0.611691\pi$
$278$	0	0
$279$	−26.6770	−1.59711
$280$	0	0
$281$	29.6709	1.77002	0.885008	−	0.465576i	$-0.154153\pi$
$281$	29.6709	1.77002	0.885008	+	0.465576i	$0.154153\pi$
$282$	0	0
$283$	22.4261	1.33309	0.666545	−	0.745464i	$-0.267772\pi$
$283$	22.4261	1.33309	0.666545	+	0.745464i	$0.267772\pi$
$284$	0	0
$285$	−0.826415	−0.0489526
$286$	0	0
$287$	18.7421	1.10631
$288$	0	0
$289$	38.5316	2.26656
$290$	0	0
$291$	19.6264	1.15052
$292$	0	0
$293$	29.0133	1.69497	0.847486	−	0.530817i	$-0.178115\pi$
$293$	29.0133	1.69497	0.847486	+	0.530817i	$0.178115\pi$
$294$	0	0
$295$	−6.39278	−0.372202
$296$	0	0
$297$	0.668641	0.0387985
$298$	0	0
$299$	0	0
$300$	0	0
$301$	22.6376	1.30481
$302$	0	0
$303$	−45.6627	−2.62325
$304$	0	0
$305$	6.40525	0.366763
$306$	0	0
$307$	−0.572611	−0.0326807	−0.0163403	−	0.999866i	$-0.505202\pi$
$307$	−0.572611	−0.0326807	−0.0163403	+	0.999866i	$0.505202\pi$
$308$	0	0
$309$	−5.85923	−0.333320
$310$	0	0
$311$	21.6057	1.22515	0.612575	−	0.790413i	$-0.290134\pi$
$311$	21.6057	1.22515	0.612575	+	0.790413i	$0.290134\pi$
$312$	0	0
$313$	−4.97168	−0.281016	−0.140508	−	0.990080i	$-0.544874\pi$
$313$	−4.97168	−0.281016	−0.140508	+	0.990080i	$0.544874\pi$
$314$	0	0
$315$	−6.10191	−0.343804
$316$	0	0
$317$	−28.7590	−1.61527	−0.807634	−	0.589685i	$-0.799252\pi$
$317$	−28.7590	−1.61527	−0.807634	+	0.589685i	$0.799252\pi$
$318$	0	0
$319$	4.36272	0.244266
$320$	0	0
$321$	−7.27464	−0.406031
$322$	0	0
$323$	2.95209	0.164259
$324$	0	0
$325$	−4.42078	−0.245221
$326$	0	0
$327$	−15.5810	−0.861631
$328$	0	0
$329$	9.37794	0.517022
$330$	0	0
$331$	−15.3807	−0.845398	−0.422699	−	0.906270i	$-0.638917\pi$
$331$	−15.3807	−0.845398	−0.422699	+	0.906270i	$0.638917\pi$
$332$	0	0
$333$	−6.10307	−0.334446
$334$	0	0
$335$	−9.57611	−0.523199
$336$	0	0
$337$	−19.3841	−1.05592	−0.527959	−	0.849270i	$-0.677042\pi$
$337$	−19.3841	−1.05592	−0.527959	+	0.849270i	$0.677042\pi$
$338$	0	0
$339$	27.4389	1.49027
$340$	0	0
$341$	4.04568	0.219086
$342$	0	0
$343$	−20.1575	−1.08840
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10.1351	−0.544082	−0.272041	−	0.962286i	$-0.587699\pi$
$347$	−10.1351	−0.544082	−0.272041	+	0.962286i	$0.587699\pi$
$348$	0	0
$349$	16.6359	0.890500	0.445250	−	0.895406i	$-0.353115\pi$
$349$	16.6359	0.890500	0.445250	+	0.895406i	$0.353115\pi$
$350$	0	0
$351$	−1.28797	−0.0687468
$352$	0	0
$353$	19.8423	1.05610	0.528049	−	0.849214i	$-0.322924\pi$
$353$	19.8423	1.05610	0.528049	+	0.849214i	$0.322924\pi$
$354$	0	0
$355$	2.03705	0.108115
$356$	0	0
$357$	40.5070	2.14386
$358$	0	0
$359$	8.91916	0.470735	0.235368	−	0.971906i	$-0.424371\pi$
$359$	8.91916	0.470735	0.235368	+	0.971906i	$0.424371\pi$
$360$	0	0
$361$	−18.8431	−0.991740
$362$	0	0
$363$	27.3178	1.43382
$364$	0	0
$365$	4.13776	0.216580
$366$	0	0
$367$	−32.8280	−1.71361	−0.856804	−	0.515642i	$-0.827554\pi$
$367$	−32.8280	−1.71361	−0.856804	+	0.515642i	$0.827554\pi$
$368$	0	0
$369$	30.7110	1.59875
$370$	0	0
$371$	11.5762	0.601008
$372$	0	0
$373$	8.99485	0.465736	0.232868	−	0.972508i	$-0.425189\pi$
$373$	8.99485	0.465736	0.232868	+	0.972508i	$0.425189\pi$
$374$	0	0
$375$	−19.4634	−1.00508
$376$	0	0
$377$	−8.40370	−0.432813
$378$	0	0
$379$	30.9479	1.58969	0.794843	−	0.606815i	$-0.207553\pi$
$379$	30.9479	1.58969	0.794843	+	0.606815i	$0.207553\pi$
$380$	0	0
$381$	−32.8731	−1.68414
$382$	0	0
$383$	−24.1424	−1.23362	−0.616810	−	0.787112i	$-0.711575\pi$
$383$	−24.1424	−1.23362	−0.616810	+	0.787112i	$0.711575\pi$
$384$	0	0
$385$	0.925381	0.0471618
$386$	0	0
$387$	37.0942	1.88560
$388$	0	0
$389$	26.0987	1.32326	0.661628	−	0.749832i	$-0.269866\pi$
$389$	26.0987	1.32326	0.661628	+	0.749832i	$0.269866\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−7.44348	−0.375474
$394$	0	0
$395$	14.0851	0.708696
$396$	0	0
$397$	−17.6837	−0.887517	−0.443759	−	0.896146i	$-0.646355\pi$
$397$	−17.6837	−0.887517	−0.443759	+	0.896146i	$0.646355\pi$
$398$	0	0
$399$	2.15338	0.107804
$400$	0	0
$401$	2.53664	0.126674	0.0633370	−	0.997992i	$-0.479826\pi$
$401$	2.53664	0.126674	0.0633370	+	0.997992i	$0.479826\pi$
$402$	0	0
$403$	−7.79300	−0.388197
$404$	0	0
$405$	5.95089	0.295702
$406$	0	0
$407$	0.925557	0.0458782
$408$	0	0
$409$	32.3535	1.59978	0.799888	−	0.600149i	$-0.204892\pi$
$409$	32.3535	1.59978	0.799888	+	0.600149i	$0.204892\pi$
$410$	0	0
$411$	−23.4674	−1.15756
$412$	0	0
$413$	16.6576	0.819666
$414$	0	0
$415$	7.88863	0.387238
$416$	0	0
$417$	11.3929	0.557911
$418$	0	0
$419$	9.52838	0.465492	0.232746	−	0.972538i	$-0.425229\pi$
$419$	9.52838	0.465492	0.232746	+	0.972538i	$0.425229\pi$
$420$	0	0
$421$	21.0664	1.02671	0.513356	−	0.858176i	$-0.328402\pi$
$421$	21.0664	1.02671	0.513356	+	0.858176i	$0.328402\pi$
$422$	0	0
$423$	15.3668	0.747159
$424$	0	0
$425$	32.2667	1.56516
$426$	0	0
$427$	−16.6901	−0.807689
$428$	0	0
$429$	1.37913	0.0665851
$430$	0	0
$431$	−14.0573	−0.677117	−0.338559	−	0.940945i	$-0.609939\pi$
$431$	−14.0573	−0.677117	−0.338559	+	0.940945i	$0.609939\pi$
$432$	0	0
$433$	21.4859	1.03255	0.516274	−	0.856423i	$-0.327319\pi$
$433$	21.4859	1.03255	0.516274	+	0.856423i	$0.327319\pi$
$434$	0	0
$435$	−17.1709	−0.823283
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−12.5859	−0.600692	−0.300346	−	0.953830i	$-0.597102\pi$
$439$	−12.5859	−0.600692	−0.300346	+	0.953830i	$0.597102\pi$
$440$	0	0
$441$	−8.56531	−0.407872
$442$	0	0
$443$	10.3754	0.492952	0.246476	−	0.969149i	$-0.420727\pi$
$443$	10.3754	0.492952	0.246476	+	0.969149i	$0.420727\pi$
$444$	0	0
$445$	−0.393922	−0.0186737
$446$	0	0
$447$	36.6870	1.73524
$448$	0	0
$449$	−9.49764	−0.448221	−0.224111	−	0.974564i	$-0.571948\pi$
$449$	−9.49764	−0.448221	−0.224111	+	0.974564i	$0.571948\pi$
$450$	0	0
$451$	−4.65746	−0.219311
$452$	0	0
$453$	7.18016	0.337354
$454$	0	0
$455$	−1.78252	−0.0835657
$456$	0	0
$457$	−26.1737	−1.22435	−0.612176	−	0.790721i	$-0.709706\pi$
$457$	−26.1737	−1.22435	−0.612176	+	0.790721i	$0.709706\pi$
$458$	0	0
$459$	9.40073	0.438788
$460$	0	0
$461$	28.4685	1.32591	0.662955	−	0.748660i	$-0.269302\pi$
$461$	28.4685	1.32591	0.662955	+	0.748660i	$0.269302\pi$
$462$	0	0
$463$	4.52760	0.210415	0.105208	−	0.994450i	$-0.466449\pi$
$463$	4.52760	0.210415	0.105208	+	0.994450i	$0.466449\pi$
$464$	0	0
$465$	−15.9231	−0.738417
$466$	0	0
$467$	21.5533	0.997368	0.498684	−	0.866784i	$-0.333817\pi$
$467$	21.5533	0.997368	0.498684	+	0.866784i	$0.333817\pi$
$468$	0	0
$469$	24.9523	1.15219
$470$	0	0
$471$	21.3593	0.984185
$472$	0	0
$473$	−5.62549	−0.258660
$474$	0	0
$475$	1.71532	0.0787042
$476$	0	0
$477$	18.9689	0.868528
$478$	0	0
$479$	2.33643	0.106754	0.0533772	−	0.998574i	$-0.483001\pi$
$479$	2.33643	0.106754	0.0533772	+	0.998574i	$0.483001\pi$
$480$	0	0
$481$	−1.78286	−0.0812912
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6.30377	0.286239
$486$	0	0
$487$	24.4371	1.10735	0.553676	−	0.832732i	$-0.313225\pi$
$487$	24.4371	1.10735	0.553676	+	0.832732i	$0.313225\pi$
$488$	0	0
$489$	30.6994	1.38828
$490$	0	0
$491$	−3.72246	−0.167992	−0.0839961	−	0.996466i	$-0.526768\pi$
$491$	−3.72246	−0.167992	−0.0839961	+	0.996466i	$0.526768\pi$
$492$	0	0
$493$	61.3375	2.76250
$494$	0	0
$495$	1.51634	0.0681544
$496$	0	0
$497$	−5.30792	−0.238093
$498$	0	0
$499$	−18.2965	−0.819062	−0.409531	−	0.912296i	$-0.634308\pi$
$499$	−18.2965	−0.819062	−0.409531	+	0.912296i	$0.634308\pi$
$500$	0	0
$501$	2.09987	0.0938154
$502$	0	0
$503$	−16.2566	−0.724847	−0.362424	−	0.932013i	$-0.618051\pi$
$503$	−16.2566	−0.724847	−0.362424	+	0.932013i	$0.618051\pi$
$504$	0	0
$505$	−14.6663	−0.652642
$506$	0	0
$507$	30.4743	1.35341
$508$	0	0
$509$	−34.1710	−1.51460	−0.757301	−	0.653066i	$-0.773482\pi$
$509$	−34.1710	−1.51460	−0.757301	+	0.653066i	$0.773482\pi$
$510$	0	0
$511$	−10.7817	−0.476955
$512$	0	0
$513$	0.499749	0.0220645
$514$	0	0
$515$	−1.88191	−0.0829270
$516$	0	0
$517$	−2.33044	−0.102493
$518$	0	0
$519$	21.2101	0.931020
$520$	0	0
$521$	25.0495	1.09744	0.548718	−	0.836007i	$-0.315116\pi$
$521$	25.0495	1.09744	0.548718	+	0.836007i	$0.315116\pi$
$522$	0	0
$523$	−25.2632	−1.10468	−0.552342	−	0.833618i	$-0.686266\pi$
$523$	−25.2632	−1.10468	−0.552342	+	0.833618i	$0.686266\pi$
$524$	0	0
$525$	23.5366	1.02722
$526$	0	0
$527$	56.8801	2.47774
$528$	0	0
$529$	0	0
$530$	0	0
$531$	27.2953	1.18452
$532$	0	0
$533$	8.97144	0.388596
$534$	0	0
$535$	−2.33653	−0.101017
$536$	0	0
$537$	−19.3299	−0.834146
$538$	0	0
$539$	1.29897	0.0559504
$540$	0	0
$541$	−22.1335	−0.951594	−0.475797	−	0.879555i	$-0.657840\pi$
$541$	−22.1335	−0.951594	−0.475797	+	0.879555i	$0.657840\pi$
$542$	0	0
$543$	−0.714288	−0.0306530
$544$	0	0
$545$	−5.00442	−0.214366
$546$	0	0
$547$	−22.0901	−0.944505	−0.472253	−	0.881463i	$-0.656559\pi$
$547$	−22.0901	−0.944505	−0.472253	+	0.881463i	$0.656559\pi$
$548$	0	0
$549$	−27.3485	−1.16721
$550$	0	0
$551$	3.26074	0.138912
$552$	0	0
$553$	−36.7013	−1.56070
$554$	0	0
$555$	−3.64284	−0.154630
$556$	0	0
$557$	−8.24762	−0.349463	−0.174731	−	0.984616i	$-0.555906\pi$
$557$	−8.24762	−0.349463	−0.174731	+	0.984616i	$0.555906\pi$
$558$	0	0
$559$	10.8361	0.458318
$560$	0	0
$561$	−10.0661	−0.424991
$562$	0	0
$563$	−20.1471	−0.849100	−0.424550	−	0.905404i	$-0.639568\pi$
$563$	−20.1471	−0.849100	−0.424550	+	0.905404i	$0.639568\pi$
$564$	0	0
$565$	8.81303	0.370767
$566$	0	0
$567$	−15.5062	−0.651198
$568$	0	0
$569$	5.14985	0.215893	0.107947	−	0.994157i	$-0.465572\pi$
$569$	5.14985	0.215893	0.107947	+	0.994157i	$0.465572\pi$
$570$	0	0
$571$	38.7843	1.62307	0.811536	−	0.584302i	$-0.198632\pi$
$571$	38.7843	1.62307	0.811536	+	0.584302i	$0.198632\pi$
$572$	0	0
$573$	7.86059	0.328381
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−39.6230	−1.64953	−0.824763	−	0.565479i	$-0.808691\pi$
$577$	−39.6230	−1.64953	−0.824763	+	0.565479i	$0.808691\pi$
$578$	0	0
$579$	57.8201	2.40292
$580$	0	0
$581$	−20.5553	−0.852777
$582$	0	0
$583$	−2.87672	−0.119142
$584$	0	0
$585$	−2.92085	−0.120762
$586$	0	0
$587$	30.3759	1.25375	0.626875	−	0.779120i	$-0.284334\pi$
$587$	30.3759	1.25375	0.626875	+	0.779120i	$0.284334\pi$
$588$	0	0
$589$	3.02378	0.124593
$590$	0	0
$591$	−40.8766	−1.68144
$592$	0	0
$593$	−5.28670	−0.217099	−0.108549	−	0.994091i	$-0.534621\pi$
$593$	−5.28670	−0.217099	−0.108549	+	0.994091i	$0.534621\pi$
$594$	0	0
$595$	13.0104	0.533372
$596$	0	0
$597$	−24.4554	−1.00089
$598$	0	0
$599$	−23.0512	−0.941846	−0.470923	−	0.882174i	$-0.656079\pi$
$599$	−23.0512	−0.941846	−0.470923	+	0.882174i	$0.656079\pi$
$600$	0	0
$601$	24.7873	1.01110	0.505548	−	0.862798i	$-0.331290\pi$
$601$	24.7873	1.01110	0.505548	+	0.862798i	$0.331290\pi$
$602$	0	0
$603$	40.8872	1.66506
$604$	0	0
$605$	8.77416	0.356720
$606$	0	0
$607$	−5.81069	−0.235848	−0.117924	−	0.993023i	$-0.537624\pi$
$607$	−5.81069	−0.235848	−0.117924	+	0.993023i	$0.537624\pi$
$608$	0	0
$609$	44.7421	1.81304
$610$	0	0
$611$	4.48901	0.181606
$612$	0	0
$613$	30.3801	1.22704	0.613520	−	0.789679i	$-0.289753\pi$
$613$	30.3801	1.22704	0.613520	+	0.789679i	$0.289753\pi$
$614$	0	0
$615$	18.3310	0.739176
$616$	0	0
$617$	−13.9443	−0.561375	−0.280688	−	0.959799i	$-0.590562\pi$
$617$	−13.9443	−0.561375	−0.280688	+	0.959799i	$0.590562\pi$
$618$	0	0
$619$	32.4314	1.30353	0.651765	−	0.758421i	$-0.274029\pi$
$619$	32.4314	1.30353	0.651765	+	0.758421i	$0.274029\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.02644	0.0411233
$624$	0	0
$625$	15.3984	0.615937
$626$	0	0
$627$	−0.535120	−0.0213706
$628$	0	0
$629$	13.0128	0.518855
$630$	0	0
$631$	48.5083	1.93109	0.965543	−	0.260244i	$-0.0838031\pi$
$631$	48.5083	1.93109	0.965543	+	0.260244i	$0.0838031\pi$
$632$	0	0
$633$	59.3718	2.35982
$634$	0	0
$635$	−10.5584	−0.418999
$636$	0	0
$637$	−2.50213	−0.0991382
$638$	0	0
$639$	−8.69761	−0.344072
$640$	0	0
$641$	41.2073	1.62759	0.813794	−	0.581153i	$-0.197398\pi$
$641$	41.2073	1.62759	0.813794	+	0.581153i	$0.197398\pi$
$642$	0	0
$643$	23.8211	0.939413	0.469706	−	0.882823i	$-0.344360\pi$
$643$	23.8211	0.939413	0.469706	+	0.882823i	$0.344360\pi$
$644$	0	0
$645$	22.1410	0.871800
$646$	0	0
$647$	10.3218	0.405793	0.202896	−	0.979200i	$-0.434965\pi$
$647$	10.3218	0.405793	0.202896	+	0.979200i	$0.434965\pi$
$648$	0	0
$649$	−4.13945	−0.162488
$650$	0	0
$651$	41.4907	1.62615
$652$	0	0
$653$	5.56981	0.217964	0.108982	−	0.994044i	$-0.465241\pi$
$653$	5.56981	0.217964	0.108982	+	0.994044i	$0.465241\pi$
$654$	0	0
$655$	−2.39075	−0.0934145
$656$	0	0
$657$	−17.6670	−0.689257
$658$	0	0
$659$	0.206535	0.00804548	0.00402274	−	0.999992i	$-0.498720\pi$
$659$	0.206535	0.00804548	0.00402274	+	0.999992i	$0.498720\pi$
$660$	0	0
$661$	15.1079	0.587627	0.293814	−	0.955863i	$-0.405075\pi$
$661$	15.1079	0.587627	0.293814	+	0.955863i	$0.405075\pi$
$662$	0	0
$663$	19.3898	0.753038
$664$	0	0
$665$	0.691639	0.0268206
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−40.7026	−1.57365
$670$	0	0
$671$	4.14753	0.160113
$672$	0	0
$673$	2.53537	0.0977313	0.0488656	−	0.998805i	$-0.484439\pi$
$673$	2.53537	0.0977313	0.0488656	+	0.998805i	$0.484439\pi$
$674$	0	0
$675$	5.46231	0.210244
$676$	0	0
$677$	36.9435	1.41985	0.709927	−	0.704275i	$-0.248728\pi$
$677$	36.9435	1.41985	0.709927	+	0.704275i	$0.248728\pi$
$678$	0	0
$679$	−16.4256	−0.630358
$680$	0	0
$681$	6.51571	0.249683
$682$	0	0
$683$	−7.33956	−0.280840	−0.140420	−	0.990092i	$-0.544845\pi$
$683$	−7.33956	−0.280840	−0.140420	+	0.990092i	$0.544845\pi$
$684$	0	0
$685$	−7.53744	−0.287991
$686$	0	0
$687$	−47.7024	−1.81996
$688$	0	0
$689$	5.54129	0.211106
$690$	0	0
$691$	20.8536	0.793307	0.396654	−	0.917968i	$-0.370171\pi$
$691$	20.8536	0.793307	0.396654	+	0.917968i	$0.370171\pi$
$692$	0	0
$693$	−3.95111	−0.150090
$694$	0	0
$695$	3.65925	0.138803
$696$	0	0
$697$	−65.4813	−2.48028
$698$	0	0
$699$	−33.4534	−1.26533
$700$	0	0
$701$	0.776098	0.0293128	0.0146564	−	0.999893i	$-0.495335\pi$
$701$	0.776098	0.0293128	0.0146564	+	0.999893i	$0.495335\pi$
$702$	0	0
$703$	0.691771	0.0260906
$704$	0	0
$705$	9.17222	0.345446
$706$	0	0
$707$	38.2158	1.43725
$708$	0	0
$709$	−5.91255	−0.222050	−0.111025	−	0.993818i	$-0.535413\pi$
$709$	−5.91255	−0.222050	−0.111025	+	0.993818i	$0.535413\pi$
$710$	0	0
$711$	−60.1391	−2.25539
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0.442960	0.0165658
$716$	0	0
$717$	−71.4240	−2.66738
$718$	0	0
$719$	−26.3306	−0.981964	−0.490982	−	0.871170i	$-0.663362\pi$
$719$	−26.3306	−0.981964	−0.490982	+	0.871170i	$0.663362\pi$
$720$	0	0
$721$	4.90367	0.182622
$722$	0	0
$723$	−40.1681	−1.49387
$724$	0	0
$725$	35.6402	1.32365
$726$	0	0
$727$	8.75822	0.324825	0.162412	−	0.986723i	$-0.448073\pi$
$727$	8.75822	0.324825	0.162412	+	0.986723i	$0.448073\pi$
$728$	0	0
$729$	−35.0536	−1.29828
$730$	0	0
$731$	−79.0913	−2.92530
$732$	0	0
$733$	−43.9598	−1.62369	−0.811847	−	0.583870i	$-0.801538\pi$
$733$	−43.9598	−1.62369	−0.811847	+	0.583870i	$0.801538\pi$
$734$	0	0
$735$	−5.11251	−0.188578
$736$	0	0
$737$	−6.20072	−0.228406
$738$	0	0
$739$	−7.89060	−0.290261	−0.145130	−	0.989413i	$-0.546360\pi$
$739$	−7.89060	−0.290261	−0.145130	+	0.989413i	$0.546360\pi$
$740$	0	0
$741$	1.03078	0.0378665
$742$	0	0
$743$	33.6204	1.23341	0.616707	−	0.787193i	$-0.288466\pi$
$743$	33.6204	1.23341	0.616707	+	0.787193i	$0.288466\pi$
$744$	0	0
$745$	11.7834	0.431711
$746$	0	0
$747$	−33.6822	−1.23237
$748$	0	0
$749$	6.08825	0.222460
$750$	0	0
$751$	40.9180	1.49312	0.746559	−	0.665319i	$-0.231705\pi$
$751$	40.9180	1.49312	0.746559	+	0.665319i	$0.231705\pi$
$752$	0	0
$753$	−39.7368	−1.44809
$754$	0	0
$755$	2.30618	0.0839305
$756$	0	0
$757$	17.9173	0.651215	0.325608	−	0.945505i	$-0.394431\pi$
$757$	17.9173	0.651215	0.325608	+	0.945505i	$0.394431\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−25.4111	−0.921153	−0.460576	−	0.887620i	$-0.652357\pi$
$761$	−25.4111	−0.921153	−0.460576	+	0.887620i	$0.652357\pi$
$762$	0	0
$763$	13.0400	0.472078
$764$	0	0
$765$	21.3189	0.770787
$766$	0	0
$767$	7.97363	0.287911
$768$	0	0
$769$	28.2796	1.01979	0.509895	−	0.860237i	$-0.329684\pi$
$769$	28.2796	1.01979	0.509895	+	0.860237i	$0.329684\pi$
$770$	0	0
$771$	−9.67713	−0.348513
$772$	0	0
$773$	40.2087	1.44621	0.723103	−	0.690740i	$-0.242715\pi$
$773$	40.2087	1.44621	0.723103	+	0.690740i	$0.242715\pi$
$774$	0	0
$775$	33.0503	1.18720
$776$	0	0
$777$	9.49209	0.340527
$778$	0	0
$779$	−3.48103	−0.124721
$780$	0	0
$781$	1.31903	0.0471986
$782$	0	0
$783$	10.3836	0.371079
$784$	0	0
$785$	6.86034	0.244856
$786$	0	0
$787$	39.6192	1.41227	0.706135	−	0.708077i	$-0.250437\pi$
$787$	39.6192	1.41227	0.706135	+	0.708077i	$0.250437\pi$
$788$	0	0
$789$	−38.6659	−1.37654
$790$	0	0
$791$	−22.9640	−0.816506
$792$	0	0
$793$	−7.98918	−0.283704
$794$	0	0
$795$	11.3223	0.401560
$796$	0	0
$797$	6.09971	0.216063	0.108031	−	0.994147i	$-0.465545\pi$
$797$	6.09971	0.216063	0.108031	+	0.994147i	$0.465545\pi$
$798$	0	0
$799$	−32.7647	−1.15913
$800$	0	0
$801$	1.68193	0.0594281
$802$	0	0
$803$	2.67928	0.0945498
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0.865342	0.0304615
$808$	0	0
$809$	−23.2520	−0.817498	−0.408749	−	0.912647i	$-0.634035\pi$
$809$	−23.2520	−0.817498	−0.408749	+	0.912647i	$0.634035\pi$
$810$	0	0
$811$	51.5055	1.80860	0.904302	−	0.426894i	$-0.140392\pi$
$811$	51.5055	1.80860	0.904302	+	0.426894i	$0.140392\pi$
$812$	0	0
$813$	33.6364	1.17968
$814$	0	0
$815$	9.86027	0.345390
$816$	0	0
$817$	−4.20455	−0.147098
$818$	0	0
$819$	7.61083	0.265944
$820$	0	0
$821$	43.9279	1.53309	0.766546	−	0.642189i	$-0.221974\pi$
$821$	43.9279	1.53309	0.766546	+	0.642189i	$0.221974\pi$
$822$	0	0
$823$	12.0492	0.420008	0.210004	−	0.977701i	$-0.432652\pi$
$823$	12.0492	0.420008	0.210004	+	0.977701i	$0.432652\pi$
$824$	0	0
$825$	−5.84892	−0.203633
$826$	0	0
$827$	−30.8000	−1.07102	−0.535510	−	0.844529i	$-0.679881\pi$
$827$	−30.8000	−1.07102	−0.535510	+	0.844529i	$0.679881\pi$
$828$	0	0
$829$	−28.3389	−0.984252	−0.492126	−	0.870524i	$-0.663780\pi$
$829$	−28.3389	−0.984252	−0.492126	+	0.870524i	$0.663780\pi$
$830$	0	0
$831$	29.1595	1.01153
$832$	0	0
$833$	18.2627	0.632767
$834$	0	0
$835$	0.674454	0.0233404
$836$	0	0
$837$	9.62902	0.332828
$838$	0	0
$839$	4.45378	0.153762	0.0768809	−	0.997040i	$-0.475504\pi$
$839$	4.45378	0.153762	0.0768809	+	0.997040i	$0.475504\pi$
$840$	0	0
$841$	38.7504	1.33622
$842$	0	0
$843$	−75.6171	−2.60439
$844$	0	0
$845$	9.78798	0.336717
$846$	0	0
$847$	−22.8627	−0.785572
$848$	0	0
$849$	−57.1534	−1.96150
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−4.80183	−0.164412	−0.0822058	−	0.996615i	$-0.526196\pi$
$853$	−4.80183	−0.164412	−0.0822058	+	0.996615i	$0.526196\pi$
$854$	0	0
$855$	1.13333	0.0387590
$856$	0	0
$857$	8.17365	0.279207	0.139603	−	0.990208i	$-0.455417\pi$
$857$	8.17365	0.279207	0.139603	+	0.990208i	$0.455417\pi$
$858$	0	0
$859$	4.31226	0.147133	0.0735663	−	0.997290i	$-0.476562\pi$
$859$	4.31226	0.147133	0.0735663	+	0.997290i	$0.476562\pi$
$860$	0	0
$861$	−47.7648	−1.62782
$862$	0	0
$863$	7.65832	0.260692	0.130346	−	0.991469i	$-0.458391\pi$
$863$	7.65832	0.260692	0.130346	+	0.991469i	$0.458391\pi$
$864$	0	0
$865$	6.81242	0.231629
$866$	0	0
$867$	−98.1988	−3.33500
$868$	0	0
$869$	9.12036	0.309387
$870$	0	0
$871$	11.9441	0.404712
$872$	0	0
$873$	−26.9152	−0.910943
$874$	0	0
$875$	16.2892	0.550675
$876$	0	0
$877$	9.22949	0.311658	0.155829	−	0.987784i	$-0.450195\pi$
$877$	9.22949	0.311658	0.155829	+	0.987784i	$0.450195\pi$
$878$	0	0
$879$	−73.9411	−2.49397
$880$	0	0
$881$	10.7320	0.361572	0.180786	−	0.983522i	$-0.442136\pi$
$881$	10.7320	0.361572	0.180786	+	0.983522i	$0.442136\pi$
$882$	0	0
$883$	−29.1306	−0.980323	−0.490161	−	0.871632i	$-0.663062\pi$
$883$	−29.1306	−0.980323	−0.490161	+	0.871632i	$0.663062\pi$
$884$	0	0
$885$	16.2922	0.547656
$886$	0	0
$887$	21.8390	0.733283	0.366641	−	0.930362i	$-0.380508\pi$
$887$	21.8390	0.733283	0.366641	+	0.930362i	$0.380508\pi$
$888$	0	0
$889$	27.5120	0.922722
$890$	0	0
$891$	3.85332	0.129091
$892$	0	0
$893$	−1.74179	−0.0582869
$894$	0	0
$895$	−6.20852	−0.207528
$896$	0	0
$897$	0	0
$898$	0	0
$899$	62.8270	2.09540
$900$	0	0
$901$	−40.4451	−1.34742
$902$	0	0
$903$	−57.6924	−1.91988
$904$	0	0
$905$	−0.229421	−0.00762620
$906$	0	0
$907$	−2.44976	−0.0813430	−0.0406715	−	0.999173i	$-0.512950\pi$
$907$	−2.44976	−0.0813430	−0.0406715	+	0.999173i	$0.512950\pi$
$908$	0	0
$909$	62.6209	2.07700
$910$	0	0
$911$	−29.8002	−0.987325	−0.493663	−	0.869654i	$-0.664342\pi$
$911$	−29.8002	−0.987325	−0.493663	+	0.869654i	$0.664342\pi$
$912$	0	0
$913$	5.10804	0.169052
$914$	0	0
$915$	−16.3240	−0.539653
$916$	0	0
$917$	6.22956	0.205718
$918$	0	0
$919$	23.8700	0.787398	0.393699	−	0.919239i	$-0.371195\pi$
$919$	23.8700	0.787398	0.393699	+	0.919239i	$0.371195\pi$
$920$	0	0
$921$	1.45932	0.0480861
$922$	0	0
$923$	−2.54078	−0.0836309
$924$	0	0
$925$	7.56112	0.248608
$926$	0	0
$927$	8.03522	0.263911
$928$	0	0
$929$	12.8380	0.421200	0.210600	−	0.977572i	$-0.432458\pi$
$929$	12.8380	0.421200	0.210600	+	0.977572i	$0.432458\pi$
$930$	0	0
$931$	0.970860	0.0318187
$932$	0	0
$933$	−55.0629	−1.80268
$934$	0	0
$935$	−3.23310	−0.105734
$936$	0	0
$937$	34.7153	1.13410	0.567050	−	0.823683i	$-0.308085\pi$
$937$	34.7153	1.13410	0.567050	+	0.823683i	$0.308085\pi$
$938$	0	0
$939$	12.6705	0.413485
$940$	0	0
$941$	7.07162	0.230528	0.115264	−	0.993335i	$-0.463229\pi$
$941$	7.07162	0.230528	0.115264	+	0.993335i	$0.463229\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	2.20247	0.0716465
$946$	0	0
$947$	4.66427	0.151568	0.0757841	−	0.997124i	$-0.475854\pi$
$947$	4.66427	0.151568	0.0757841	+	0.997124i	$0.475854\pi$
$948$	0	0
$949$	−5.16097	−0.167532
$950$	0	0
$951$	73.2932	2.37669
$952$	0	0
$953$	26.1745	0.847874	0.423937	−	0.905692i	$-0.360648\pi$
$953$	26.1745	0.847874	0.423937	+	0.905692i	$0.360648\pi$
$954$	0	0
$955$	2.52472	0.0816982
$956$	0	0
$957$	−11.1185	−0.359411
$958$	0	0
$959$	19.6402	0.634215
$960$	0	0
$961$	27.2614	0.879399
$962$	0	0
$963$	9.97629	0.321481
$964$	0	0
$965$	18.5711	0.597825
$966$	0	0
$967$	8.44710	0.271640	0.135820	−	0.990734i	$-0.456633\pi$
$967$	8.44710	0.271640	0.135820	+	0.990734i	$0.456633\pi$
$968$	0	0
$969$	−7.52350	−0.241690
$970$	0	0
$971$	−23.7113	−0.760933	−0.380466	−	0.924795i	$-0.624237\pi$
$971$	−23.7113	−0.760933	−0.380466	+	0.924795i	$0.624237\pi$
$972$	0	0
$973$	−9.53486	−0.305674
$974$	0	0
$975$	11.2665	0.360816
$976$	0	0
$977$	−30.6795	−0.981523	−0.490762	−	0.871294i	$-0.663281\pi$
$977$	−30.6795	−0.981523	−0.490762	+	0.871294i	$0.663281\pi$
$978$	0	0
$979$	−0.255072	−0.00815214
$980$	0	0
$981$	21.3674	0.682210
$982$	0	0
$983$	10.5504	0.336505	0.168252	−	0.985744i	$-0.446188\pi$
$983$	10.5504	0.336505	0.168252	+	0.985744i	$0.446188\pi$
$984$	0	0
$985$	−13.1291	−0.418327
$986$	0	0
$987$	−23.8999	−0.760743
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−43.8761	−1.39377	−0.696885	−	0.717183i	$-0.745431\pi$
$991$	−43.8761	−1.39377	−0.696885	+	0.717183i	$0.745431\pi$
$992$	0	0
$993$	39.1981	1.24391
$994$	0	0
$995$	−7.85477	−0.249013
$996$	0	0
$997$	−54.8115	−1.73590	−0.867949	−	0.496653i	$-0.834562\pi$
$997$	−54.8115	−1.73590	−0.867949	+	0.496653i	$0.834562\pi$
$998$	0	0
$999$	2.20289	0.0696965

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8464.2.a.ch.1.2		15
4.3	odd	2		4232.2.a.ba.1.14		15
23.7	odd	22		368.2.m.e.49.1		30
23.10	odd	22		368.2.m.e.353.1		30
23.22	odd	2		8464.2.a.cg.1.2		15
92.7	even	22		184.2.i.b.49.3	✓	30
92.79	even	22		184.2.i.b.169.3	yes	30
92.91	even	2		4232.2.a.bb.1.14		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
184.2.i.b.49.3	✓	30	92.7	even	22
184.2.i.b.169.3	yes	30	92.79	even	22
368.2.m.e.49.1		30	23.7	odd	22
368.2.m.e.353.1		30	23.10	odd	22
4232.2.a.ba.1.14		15	4.3	odd	2
4232.2.a.bb.1.14		15	92.91	even	2
8464.2.a.cg.1.2		15	23.22	odd	2
8464.2.a.ch.1.2		15	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8464 = 2^{4} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8464.a (trivial)

\(f(q)\)	\(=\)	\(q-2.54853 q^{3} -0.818556 q^{5} +2.13290 q^{7} +3.49500 q^{9} +O(q^{10})\)	\(q-2.54853 q^{3} -0.818556 q^{5} +2.13290 q^{7} +3.49500 q^{9} -0.530031 q^{11} +1.02097 q^{13} +2.08611 q^{15} -7.45195 q^{17} -0.396151 q^{19} -5.43576 q^{21} -4.32997 q^{25} -1.26151 q^{27} -8.23107 q^{29} -7.63291 q^{31} +1.35080 q^{33} -1.74590 q^{35} -1.74623 q^{37} -2.60198 q^{39} +8.78714 q^{41} +10.6135 q^{43} -2.86085 q^{45} +4.39680 q^{47} -2.45073 q^{49} +18.9915 q^{51} +5.42746 q^{53} +0.433860 q^{55} +1.00960 q^{57} +7.80983 q^{59} -7.82506 q^{61} +7.45448 q^{63} -0.835724 q^{65} +11.6988 q^{67} -2.48859 q^{71} -5.05495 q^{73} +11.0350 q^{75} -1.13050 q^{77} -17.2072 q^{79} -7.26999 q^{81} -9.63725 q^{83} +6.09984 q^{85} +20.9771 q^{87} +0.481240 q^{89} +2.17764 q^{91} +19.4527 q^{93} +0.324272 q^{95} -7.70108 q^{97} -1.85246 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + 10 q^{7} + 16 q^{9}+O(q^{10}) \) 15 * q - q^3 + 10 * q^7 + 16 * q^9	\( 15 q - q^{3} + 10 q^{7} + 16 q^{9} + 23 q^{11} + 10 q^{15} + 29 q^{19} - q^{21} + 23 q^{25} - q^{27} - 2 q^{29} - 20 q^{31} - 18 q^{33} + 18 q^{35} - 24 q^{37} + 19 q^{39} + 9 q^{41} + 48 q^{43} - 4 q^{45} + 36 q^{47} + 25 q^{49} + 35 q^{51} + 5 q^{53} + 10 q^{55} - 23 q^{57} + 22 q^{59} - 12 q^{61} + 35 q^{63} + 26 q^{65} + 58 q^{67} - 2 q^{71} + 5 q^{73} + 17 q^{75} + 26 q^{77} + 26 q^{79} - 21 q^{81} + 68 q^{83} - 72 q^{85} - 19 q^{87} + 6 q^{89} + 71 q^{91} - 55 q^{93} + 12 q^{95} - 40 q^{97} + 63 q^{99}+O(q^{100}) \) 15 * q - q^3 + 10 * q^7 + 16 * q^9 + 23 * q^11 + 10 * q^15 + 29 * q^19 - q^21 + 23 * q^25 - q^27 - 2 * q^29 - 20 * q^31 - 18 * q^33 + 18 * q^35 - 24 * q^37 + 19 * q^39 + 9 * q^41 + 48 * q^43 - 4 * q^45 + 36 * q^47 + 25 * q^49 + 35 * q^51 + 5 * q^53 + 10 * q^55 - 23 * q^57 + 22 * q^59 - 12 * q^61 + 35 * q^63 + 26 * q^65 + 58 * q^67 - 2 * q^71 + 5 * q^73 + 17 * q^75 + 26 * q^77 + 26 * q^79 - 21 * q^81 + 68 * q^83 - 72 * q^85 - 19 * q^87 + 6 * q^89 + 71 * q^91 - 55 * q^93 + 12 * q^95 - 40 * q^97 + 63 * q^99

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