LMFDB - Embedded newform 7168.2.a.bj.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7168,2,Mod(1,7168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7168, 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("7168.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7168 = 2^{10} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7168.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:	$57.2367681689$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 24x^{10} + 221x^{8} - 968x^{6} + 2008x^{4} - 1640x^{2} + 196 $ x^12 - 24x^10 + 221x^8 - 968x^6 + 2008x^4 - 1640*x^2 + 196
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 112)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.66376$ of defining polynomial
Character	$\chi$	$=$	7168.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.96750 q^{3} +3.06146 q^{5} +1.00000 q^{7} +0.871066 q^{9} +O(q^{10})$	$q-1.96750 q^{3} +3.06146 q^{5} +1.00000 q^{7} +0.871066 q^{9} +4.37788 q^{11} +2.48067 q^{13} -6.02343 q^{15} +5.20470 q^{17} -1.20437 q^{19} -1.96750 q^{21} +6.15500 q^{23} +4.37253 q^{25} +4.18868 q^{27} +8.82605 q^{29} -2.78247 q^{31} -8.61348 q^{33} +3.06146 q^{35} +5.81528 q^{37} -4.88072 q^{39} +6.32956 q^{41} -4.32660 q^{43} +2.66673 q^{45} -3.60383 q^{47} +1.00000 q^{49} -10.2403 q^{51} -7.47260 q^{53} +13.4027 q^{55} +2.36961 q^{57} -10.0912 q^{59} -1.47067 q^{61} +0.871066 q^{63} +7.59446 q^{65} +1.36711 q^{67} -12.1100 q^{69} +10.0597 q^{71} +15.1717 q^{73} -8.60296 q^{75} +4.37788 q^{77} +6.61348 q^{79} -10.8544 q^{81} -10.4896 q^{83} +15.9340 q^{85} -17.3653 q^{87} -3.26144 q^{89} +2.48067 q^{91} +5.47451 q^{93} -3.68714 q^{95} -7.66352 q^{97} +3.81342 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q + 12 * q^7 + 12 * q^9	$ 12 q + 12 q^{7} + 12 q^{9} + 24 q^{15} + 8 q^{17} + 16 q^{23} + 20 q^{25} - 8 q^{31} + 16 q^{39} + 32 q^{41} - 16 q^{47} + 12 q^{49} + 24 q^{55} + 64 q^{57} + 12 q^{63} + 32 q^{65} + 8 q^{71} - 24 q^{79} + 44 q^{81} - 32 q^{87} + 24 q^{89} + 48 q^{97}+O(q^{100}) $ 12 * q + 12 * q^7 + 12 * q^9 + 24 * q^15 + 8 * q^17 + 16 * q^23 + 20 * q^25 - 8 * q^31 + 16 * q^39 + 32 * q^41 - 16 * q^47 + 12 * q^49 + 24 * q^55 + 64 * q^57 + 12 * q^63 + 32 * q^65 + 8 * q^71 - 24 * q^79 + 44 * q^81 - 32 * q^87 + 24 * q^89 + 48 * 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
$3$	−1.96750	−1.13594	−0.567969	−	0.823050i	$-0.692271\pi$
$3$	−1.96750	−1.13594	−0.567969	+	0.823050i	$0.692271\pi$
$4$	0	0
$5$	3.06146	1.36913	0.684563	−	0.728954i	$-0.259993\pi$
$5$	3.06146	1.36913	0.684563	+	0.728954i	$0.259993\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0.871066	0.290355
$10$	0	0
$11$	4.37788	1.31998	0.659990	−	0.751275i	$-0.270561\pi$
$11$	4.37788	1.31998	0.659990	+	0.751275i	$0.270561\pi$
$12$	0	0
$13$	2.48067	0.688013	0.344007	−	0.938967i	$-0.388216\pi$
$13$	2.48067	0.688013	0.344007	+	0.938967i	$0.388216\pi$
$14$	0	0
$15$	−6.02343	−1.55524
$16$	0	0
$17$	5.20470	1.26233	0.631163	−	0.775651i	$-0.282578\pi$
$17$	5.20470	1.26233	0.631163	+	0.775651i	$0.282578\pi$
$18$	0	0
$19$	−1.20437	−0.276302	−0.138151	−	0.990411i	$-0.544116\pi$
$19$	−1.20437	−0.276302	−0.138151	+	0.990411i	$0.544116\pi$
$20$	0	0
$21$	−1.96750	−0.429344
$22$	0	0
$23$	6.15500	1.28341	0.641703	−	0.766954i	$-0.278228\pi$
$23$	6.15500	1.28341	0.641703	+	0.766954i	$0.278228\pi$
$24$	0	0
$25$	4.37253	0.874505
$26$	0	0
$27$	4.18868	0.806112
$28$	0	0
$29$	8.82605	1.63896	0.819478	−	0.573111i	$-0.194264\pi$
$29$	8.82605	1.63896	0.819478	+	0.573111i	$0.194264\pi$
$30$	0	0
$31$	−2.78247	−0.499746	−0.249873	−	0.968279i	$-0.580389\pi$
$31$	−2.78247	−0.499746	−0.249873	+	0.968279i	$0.580389\pi$
$32$	0	0
$33$	−8.61348	−1.49942
$34$	0	0
$35$	3.06146	0.517481
$36$	0	0
$37$	5.81528	0.956026	0.478013	−	0.878353i	$-0.341357\pi$
$37$	5.81528	0.956026	0.478013	+	0.878353i	$0.341357\pi$
$38$	0	0
$39$	−4.88072	−0.781541
$40$	0	0
$41$	6.32956	0.988511	0.494255	−	0.869317i	$-0.335441\pi$
$41$	6.32956	0.988511	0.494255	+	0.869317i	$0.335441\pi$
$42$	0	0
$43$	−4.32660	−0.659800	−0.329900	−	0.944016i	$-0.607015\pi$
$43$	−4.32660	−0.659800	−0.329900	+	0.944016i	$0.607015\pi$
$44$	0	0
$45$	2.66673	0.397533
$46$	0	0
$47$	−3.60383	−0.525673	−0.262836	−	0.964840i	$-0.584658\pi$
$47$	−3.60383	−0.525673	−0.262836	+	0.964840i	$0.584658\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−10.2403	−1.43392
$52$	0	0
$53$	−7.47260	−1.02644	−0.513221	−	0.858257i	$-0.671548\pi$
$53$	−7.47260	−1.02644	−0.513221	+	0.858257i	$0.671548\pi$
$54$	0	0
$55$	13.4027	1.80722
$56$	0	0
$57$	2.36961	0.313862
$58$	0	0
$59$	−10.0912	−1.31376	−0.656881	−	0.753995i	$-0.728124\pi$
$59$	−10.0912	−1.31376	−0.656881	+	0.753995i	$0.728124\pi$
$60$	0	0
$61$	−1.47067	−0.188300	−0.0941501	−	0.995558i	$-0.530013\pi$
$61$	−1.47067	−0.188300	−0.0941501	+	0.995558i	$0.530013\pi$
$62$	0	0
$63$	0.871066	0.109744
$64$	0	0
$65$	7.59446	0.941977
$66$	0	0
$67$	1.36711	0.167019	0.0835096	−	0.996507i	$-0.473387\pi$
$67$	1.36711	0.167019	0.0835096	+	0.996507i	$0.473387\pi$
$68$	0	0
$69$	−12.1100	−1.45787
$70$	0	0
$71$	10.0597	1.19386	0.596932	−	0.802291i	$-0.296386\pi$
$71$	10.0597	1.19386	0.596932	+	0.802291i	$0.296386\pi$
$72$	0	0
$73$	15.1717	1.77571	0.887857	−	0.460119i	$-0.152193\pi$
$73$	15.1717	1.77571	0.887857	+	0.460119i	$0.152193\pi$
$74$	0	0
$75$	−8.60296	−0.993384
$76$	0	0
$77$	4.37788	0.498905
$78$	0	0
$79$	6.61348	0.744075	0.372038	−	0.928218i	$-0.378659\pi$
$79$	6.61348	0.744075	0.372038	+	0.928218i	$0.378659\pi$
$80$	0	0
$81$	−10.8544	−1.20605
$82$	0	0
$83$	−10.4896	−1.15139	−0.575694	−	0.817665i	$-0.695268\pi$
$83$	−10.4896	−1.15139	−0.575694	+	0.817665i	$0.695268\pi$
$84$	0	0
$85$	15.9340	1.72828
$86$	0	0
$87$	−17.3653	−1.86175
$88$	0	0
$89$	−3.26144	−0.345712	−0.172856	−	0.984947i	$-0.555300\pi$
$89$	−3.26144	−0.345712	−0.172856	+	0.984947i	$0.555300\pi$
$90$	0	0
$91$	2.48067	0.260045
$92$	0	0
$93$	5.47451	0.567681
$94$	0	0
$95$	−3.68714	−0.378292
$96$	0	0
$97$	−7.66352	−0.778112	−0.389056	−	0.921214i	$-0.627199\pi$
$97$	−7.66352	−0.778112	−0.389056	+	0.921214i	$0.627199\pi$
$98$	0	0
$99$	3.81342	0.383263
$100$	0	0
$101$	14.0467	1.39770	0.698850	−	0.715269i	$-0.253696\pi$
$101$	14.0467	1.39770	0.698850	+	0.715269i	$0.253696\pi$
$102$	0	0
$103$	1.61199	0.158834	0.0794169	−	0.996841i	$-0.474694\pi$
$103$	1.61199	0.158834	0.0794169	+	0.996841i	$0.474694\pi$
$104$	0	0
$105$	−6.02343	−0.587826
$106$	0	0
$107$	−11.2058	−1.08331	−0.541655	−	0.840601i	$-0.682202\pi$
$107$	−11.2058	−1.08331	−0.541655	+	0.840601i	$0.682202\pi$
$108$	0	0
$109$	8.04658	0.770723	0.385361	−	0.922766i	$-0.374077\pi$
$109$	8.04658	0.770723	0.385361	+	0.922766i	$0.374077\pi$
$110$	0	0
$111$	−11.4416	−1.08599
$112$	0	0
$113$	−15.2609	−1.43563	−0.717813	−	0.696235i	$-0.754857\pi$
$113$	−15.2609	−1.43563	−0.717813	+	0.696235i	$0.754857\pi$
$114$	0	0
$115$	18.8433	1.75714
$116$	0	0
$117$	2.16083	0.199768
$118$	0	0
$119$	5.20470	0.477114
$120$	0	0
$121$	8.16581	0.742346
$122$	0	0
$123$	−12.4534	−1.12289
$124$	0	0
$125$	−1.92098	−0.171818
$126$	0	0
$127$	−1.80529	−0.160193	−0.0800966	−	0.996787i	$-0.525523\pi$
$127$	−1.80529	−0.160193	−0.0800966	+	0.996787i	$0.525523\pi$
$128$	0	0
$129$	8.51260	0.749492
$130$	0	0
$131$	−13.1320	−1.14735	−0.573674	−	0.819084i	$-0.694482\pi$
$131$	−13.1320	−1.14735	−0.573674	+	0.819084i	$0.694482\pi$
$132$	0	0
$133$	−1.20437	−0.104432
$134$	0	0
$135$	12.8235	1.10367
$136$	0	0
$137$	7.93652	0.678063	0.339031	−	0.940775i	$-0.389901\pi$
$137$	7.93652	0.678063	0.339031	+	0.940775i	$0.389901\pi$
$138$	0	0
$139$	2.92622	0.248199	0.124099	−	0.992270i	$-0.460396\pi$
$139$	2.92622	0.248199	0.124099	+	0.992270i	$0.460396\pi$
$140$	0	0
$141$	7.09055	0.597132
$142$	0	0
$143$	10.8601	0.908164
$144$	0	0
$145$	27.0206	2.24394
$146$	0	0
$147$	−1.96750	−0.162277
$148$	0	0
$149$	−12.9471	−1.06067	−0.530335	−	0.847788i	$-0.677934\pi$
$149$	−12.9471	−1.06067	−0.530335	+	0.847788i	$0.677934\pi$
$150$	0	0
$151$	2.80295	0.228101	0.114051	−	0.993475i	$-0.463617\pi$
$151$	2.80295	0.228101	0.114051	+	0.993475i	$0.463617\pi$
$152$	0	0
$153$	4.53364	0.366523
$154$	0	0
$155$	−8.51841	−0.684215
$156$	0	0
$157$	−1.35594	−0.108216	−0.0541081	−	0.998535i	$-0.517232\pi$
$157$	−1.35594	−0.108216	−0.0541081	+	0.998535i	$0.517232\pi$
$158$	0	0
$159$	14.7024	1.16597
$160$	0	0
$161$	6.15500	0.485082
$162$	0	0
$163$	−18.2724	−1.43121	−0.715603	−	0.698507i	$-0.753848\pi$
$163$	−18.2724	−1.43121	−0.715603	+	0.698507i	$0.753848\pi$
$164$	0	0
$165$	−26.3698	−2.05289
$166$	0	0
$167$	1.96111	0.151755	0.0758775	−	0.997117i	$-0.475824\pi$
$167$	1.96111	0.151755	0.0758775	+	0.997117i	$0.475824\pi$
$168$	0	0
$169$	−6.84629	−0.526637
$170$	0	0
$171$	−1.04909	−0.0802258
$172$	0	0
$173$	−5.85540	−0.445178	−0.222589	−	0.974912i	$-0.571451\pi$
$173$	−5.85540	−0.445178	−0.222589	+	0.974912i	$0.571451\pi$
$174$	0	0
$175$	4.37253	0.330532
$176$	0	0
$177$	19.8544	1.49235
$178$	0	0
$179$	−13.4566	−1.00579	−0.502895	−	0.864347i	$-0.667732\pi$
$179$	−13.4566	−1.00579	−0.502895	+	0.864347i	$0.667732\pi$
$180$	0	0
$181$	−10.7540	−0.799340	−0.399670	−	0.916659i	$-0.630875\pi$
$181$	−10.7540	−0.799340	−0.399670	+	0.916659i	$0.630875\pi$
$182$	0	0
$183$	2.89355	0.213897
$184$	0	0
$185$	17.8032	1.30892
$186$	0	0
$187$	22.7855	1.66624
$188$	0	0
$189$	4.18868	0.304682
$190$	0	0
$191$	−20.4878	−1.48245	−0.741223	−	0.671259i	$-0.765754\pi$
$191$	−20.4878	−1.48245	−0.741223	+	0.671259i	$0.765754\pi$
$192$	0	0
$193$	12.7155	0.915284	0.457642	−	0.889137i	$-0.348694\pi$
$193$	12.7155	0.915284	0.457642	+	0.889137i	$0.348694\pi$
$194$	0	0
$195$	−14.9421	−1.07003
$196$	0	0
$197$	−18.1922	−1.29614	−0.648069	−	0.761581i	$-0.724423\pi$
$197$	−18.1922	−1.29614	−0.648069	+	0.761581i	$0.724423\pi$
$198$	0	0
$199$	1.46847	0.104097	0.0520487	−	0.998645i	$-0.483425\pi$
$199$	1.46847	0.104097	0.0520487	+	0.998645i	$0.483425\pi$
$200$	0	0
$201$	−2.68979	−0.189723
$202$	0	0
$203$	8.82605	0.619467
$204$	0	0
$205$	19.3777	1.35340
$206$	0	0
$207$	5.36141	0.372644
$208$	0	0
$209$	−5.27260	−0.364713
$210$	0	0
$211$	0.756275	0.0520641	0.0260320	−	0.999661i	$-0.491713\pi$
$211$	0.756275	0.0520641	0.0260320	+	0.999661i	$0.491713\pi$
$212$	0	0
$213$	−19.7925	−1.35616
$214$	0	0
$215$	−13.2457	−0.903350
$216$	0	0
$217$	−2.78247	−0.188886
$218$	0	0
$219$	−29.8504	−2.01710
$220$	0	0
$221$	12.9111	0.868497
$222$	0	0
$223$	−7.83775	−0.524855	−0.262427	−	0.964952i	$-0.584523\pi$
$223$	−7.83775	−0.524855	−0.262427	+	0.964952i	$0.584523\pi$
$224$	0	0
$225$	3.80876	0.253917
$226$	0	0
$227$	−0.350122	−0.0232384	−0.0116192	−	0.999932i	$-0.503699\pi$
$227$	−0.350122	−0.0232384	−0.0116192	+	0.999932i	$0.503699\pi$
$228$	0	0
$229$	−22.0659	−1.45816	−0.729079	−	0.684430i	$-0.760051\pi$
$229$	−22.0659	−1.45816	−0.729079	+	0.684430i	$0.760051\pi$
$230$	0	0
$231$	−8.61348	−0.566726
$232$	0	0
$233$	9.51493	0.623344	0.311672	−	0.950190i	$-0.399111\pi$
$233$	9.51493	0.623344	0.311672	+	0.950190i	$0.399111\pi$
$234$	0	0
$235$	−11.0330	−0.719712
$236$	0	0
$237$	−13.0120	−0.845223
$238$	0	0
$239$	−18.8469	−1.21910	−0.609552	−	0.792746i	$-0.708651\pi$
$239$	−18.8469	−1.21910	−0.609552	+	0.792746i	$0.708651\pi$
$240$	0	0
$241$	6.39828	0.412150	0.206075	−	0.978536i	$-0.433931\pi$
$241$	6.39828	0.412150	0.206075	+	0.978536i	$0.433931\pi$
$242$	0	0
$243$	8.79010	0.563885
$244$	0	0
$245$	3.06146	0.195589
$246$	0	0
$247$	−2.98765	−0.190100
$248$	0	0
$249$	20.6384	1.30791
$250$	0	0
$251$	−4.14589	−0.261686	−0.130843	−	0.991403i	$-0.541768\pi$
$251$	−4.14589	−0.261686	−0.130843	+	0.991403i	$0.541768\pi$
$252$	0	0
$253$	26.9458	1.69407
$254$	0	0
$255$	−31.3501	−1.96322
$256$	0	0
$257$	28.9676	1.80695	0.903475	−	0.428640i	$-0.141007\pi$
$257$	28.9676	1.80695	0.903475	+	0.428640i	$0.141007\pi$
$258$	0	0
$259$	5.81528	0.361344
$260$	0	0
$261$	7.68807	0.475880
$262$	0	0
$263$	−0.344446	−0.0212395	−0.0106197	−	0.999944i	$-0.503380\pi$
$263$	−0.344446	−0.0212395	−0.0106197	+	0.999944i	$0.503380\pi$
$264$	0	0
$265$	−22.8771	−1.40533
$266$	0	0
$267$	6.41690	0.392708
$268$	0	0
$269$	−13.9706	−0.851805	−0.425903	−	0.904769i	$-0.640043\pi$
$269$	−13.9706	−0.851805	−0.425903	+	0.904769i	$0.640043\pi$
$270$	0	0
$271$	12.4969	0.759130	0.379565	−	0.925165i	$-0.376074\pi$
$271$	12.4969	0.759130	0.379565	+	0.925165i	$0.376074\pi$
$272$	0	0
$273$	−4.88072	−0.295395
$274$	0	0
$275$	19.1424	1.15433
$276$	0	0
$277$	8.48451	0.509785	0.254893	−	0.966969i	$-0.417960\pi$
$277$	8.48451	0.509785	0.254893	+	0.966969i	$0.417960\pi$
$278$	0	0
$279$	−2.42372	−0.145104
$280$	0	0
$281$	9.56494	0.570596	0.285298	−	0.958439i	$-0.407907\pi$
$281$	9.56494	0.570596	0.285298	+	0.958439i	$0.407907\pi$
$282$	0	0
$283$	16.5290	0.982550	0.491275	−	0.871005i	$-0.336531\pi$
$283$	16.5290	0.982550	0.491275	+	0.871005i	$0.336531\pi$
$284$	0	0
$285$	7.25445	0.429717
$286$	0	0
$287$	6.32956	0.373622
$288$	0	0
$289$	10.0889	0.593465
$290$	0	0
$291$	15.0780	0.883887
$292$	0	0
$293$	−21.6252	−1.26336	−0.631678	−	0.775231i	$-0.717634\pi$
$293$	−21.6252	−1.26336	−0.631678	+	0.775231i	$0.717634\pi$
$294$	0	0
$295$	−30.8938	−1.79870
$296$	0	0
$297$	18.3375	1.06405
$298$	0	0
$299$	15.2685	0.883000
$300$	0	0
$301$	−4.32660	−0.249381
$302$	0	0
$303$	−27.6369	−1.58770
$304$	0	0
$305$	−4.50240	−0.257807
$306$	0	0
$307$	−5.88679	−0.335977	−0.167988	−	0.985789i	$-0.553727\pi$
$307$	−5.88679	−0.335977	−0.167988	+	0.985789i	$0.553727\pi$
$308$	0	0
$309$	−3.17159	−0.180425
$310$	0	0
$311$	0.802623	0.0455126	0.0227563	−	0.999741i	$-0.492756\pi$
$311$	0.802623	0.0455126	0.0227563	+	0.999741i	$0.492756\pi$
$312$	0	0
$313$	−17.7285	−1.00207	−0.501036	−	0.865427i	$-0.667047\pi$
$313$	−17.7285	−1.00207	−0.501036	+	0.865427i	$0.667047\pi$
$314$	0	0
$315$	2.66673	0.150253
$316$	0	0
$317$	−14.6641	−0.823616	−0.411808	−	0.911271i	$-0.635103\pi$
$317$	−14.6641	−0.823616	−0.411808	+	0.911271i	$0.635103\pi$
$318$	0	0
$319$	38.6394	2.16339
$320$	0	0
$321$	22.0475	1.23057
$322$	0	0
$323$	−6.26840	−0.348783
$324$	0	0
$325$	10.8468	0.601671
$326$	0	0
$327$	−15.8317	−0.875493
$328$	0	0
$329$	−3.60383	−0.198686
$330$	0	0
$331$	0.315457	0.0173391	0.00866955	−	0.999962i	$-0.497240\pi$
$331$	0.315457	0.0173391	0.00866955	+	0.999962i	$0.497240\pi$
$332$	0	0
$333$	5.06550	0.277587
$334$	0	0
$335$	4.18535	0.228670
$336$	0	0
$337$	26.7633	1.45789	0.728944	−	0.684573i	$-0.240011\pi$
$337$	26.7633	1.45789	0.728944	+	0.684573i	$0.240011\pi$
$338$	0	0
$339$	30.0259	1.63078
$340$	0	0
$341$	−12.1813	−0.659655
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−37.0742	−1.99601
$346$	0	0
$347$	34.8005	1.86819	0.934095	−	0.357025i	$-0.116209\pi$
$347$	34.8005	1.86819	0.934095	+	0.357025i	$0.116209\pi$
$348$	0	0
$349$	−19.4905	−1.04330	−0.521650	−	0.853159i	$-0.674683\pi$
$349$	−19.4905	−1.04330	−0.521650	+	0.853159i	$0.674683\pi$
$350$	0	0
$351$	10.3907	0.554616
$352$	0	0
$353$	−26.0443	−1.38620	−0.693099	−	0.720842i	$-0.743755\pi$
$353$	−26.0443	−1.38620	−0.693099	+	0.720842i	$0.743755\pi$
$354$	0	0
$355$	30.7973	1.63455
$356$	0	0
$357$	−10.2403	−0.541972
$358$	0	0
$359$	21.0382	1.11035	0.555176	−	0.831733i	$-0.312651\pi$
$359$	21.0382	1.11035	0.555176	+	0.831733i	$0.312651\pi$
$360$	0	0
$361$	−17.5495	−0.923657
$362$	0	0
$363$	−16.0662	−0.843259
$364$	0	0
$365$	46.4476	2.43118
$366$	0	0
$367$	−7.41340	−0.386977	−0.193488	−	0.981103i	$-0.561980\pi$
$367$	−7.41340	−0.386977	−0.193488	+	0.981103i	$0.561980\pi$
$368$	0	0
$369$	5.51346	0.287019
$370$	0	0
$371$	−7.47260	−0.387958
$372$	0	0
$373$	−5.30747	−0.274810	−0.137405	−	0.990515i	$-0.543876\pi$
$373$	−5.30747	−0.274810	−0.137405	+	0.990515i	$0.543876\pi$
$374$	0	0
$375$	3.77954	0.195175
$376$	0	0
$377$	21.8945	1.12762
$378$	0	0
$379$	26.6419	1.36850	0.684251	−	0.729247i	$-0.260129\pi$
$379$	26.6419	1.36850	0.684251	+	0.729247i	$0.260129\pi$
$380$	0	0
$381$	3.55190	0.181969
$382$	0	0
$383$	−4.94620	−0.252739	−0.126369	−	0.991983i	$-0.540332\pi$
$383$	−4.94620	−0.252739	−0.126369	+	0.991983i	$0.540332\pi$
$384$	0	0
$385$	13.4027	0.683064
$386$	0	0
$387$	−3.76876	−0.191577
$388$	0	0
$389$	19.8367	1.00576	0.502881	−	0.864356i	$-0.332273\pi$
$389$	19.8367	1.00576	0.502881	+	0.864356i	$0.332273\pi$
$390$	0	0
$391$	32.0349	1.62007
$392$	0	0
$393$	25.8372	1.30332
$394$	0	0
$395$	20.2469	1.01873
$396$	0	0
$397$	−0.0903561	−0.00453484	−0.00226742	−	0.999997i	$-0.500722\pi$
$397$	−0.0903561	−0.00453484	−0.00226742	+	0.999997i	$0.500722\pi$
$398$	0	0
$399$	2.36961	0.118629
$400$	0	0
$401$	25.4103	1.26893	0.634466	−	0.772951i	$-0.281220\pi$
$401$	25.4103	1.26893	0.634466	+	0.772951i	$0.281220\pi$
$402$	0	0
$403$	−6.90238	−0.343832
$404$	0	0
$405$	−33.2304	−1.65123
$406$	0	0
$407$	25.4586	1.26194
$408$	0	0
$409$	−22.4054	−1.10788	−0.553939	−	0.832557i	$-0.686876\pi$
$409$	−22.4054	−1.10788	−0.553939	+	0.832557i	$0.686876\pi$
$410$	0	0
$411$	−15.6151	−0.770237
$412$	0	0
$413$	−10.0912	−0.496555
$414$	0	0
$415$	−32.1136	−1.57639
$416$	0	0
$417$	−5.75735	−0.281939
$418$	0	0
$419$	34.3312	1.67719	0.838594	−	0.544757i	$-0.183378\pi$
$419$	34.3312	1.67719	0.838594	+	0.544757i	$0.183378\pi$
$420$	0	0
$421$	22.6656	1.10465	0.552327	−	0.833628i	$-0.313740\pi$
$421$	22.6656	1.10465	0.552327	+	0.833628i	$0.313740\pi$
$422$	0	0
$423$	−3.13918	−0.152632
$424$	0	0
$425$	22.7577	1.10391
$426$	0	0
$427$	−1.47067	−0.0711708
$428$	0	0
$429$	−21.3672	−1.03162
$430$	0	0
$431$	31.6615	1.52508	0.762540	−	0.646941i	$-0.223952\pi$
$431$	31.6615	1.52508	0.762540	+	0.646941i	$0.223952\pi$
$432$	0	0
$433$	11.6823	0.561413	0.280707	−	0.959794i	$-0.409431\pi$
$433$	11.6823	0.561413	0.280707	+	0.959794i	$0.409431\pi$
$434$	0	0
$435$	−53.1631	−2.54897
$436$	0	0
$437$	−7.41291	−0.354608
$438$	0	0
$439$	−13.1217	−0.626265	−0.313133	−	0.949709i	$-0.601378\pi$
$439$	−13.1217	−0.626265	−0.313133	+	0.949709i	$0.601378\pi$
$440$	0	0
$441$	0.871066	0.0414794
$442$	0	0
$443$	−22.9385	−1.08984	−0.544920	−	0.838488i	$-0.683440\pi$
$443$	−22.9385	−1.08984	−0.544920	+	0.838488i	$0.683440\pi$
$444$	0	0
$445$	−9.98477	−0.473324
$446$	0	0
$447$	25.4735	1.20485
$448$	0	0
$449$	−15.6396	−0.738077	−0.369038	−	0.929414i	$-0.620313\pi$
$449$	−15.6396	−0.738077	−0.369038	+	0.929414i	$0.620313\pi$
$450$	0	0
$451$	27.7100	1.30481
$452$	0	0
$453$	−5.51482	−0.259109
$454$	0	0
$455$	7.59446	0.356034
$456$	0	0
$457$	−12.2305	−0.572117	−0.286058	−	0.958212i	$-0.592345\pi$
$457$	−12.2305	−0.572117	−0.286058	+	0.958212i	$0.592345\pi$
$458$	0	0
$459$	21.8008	1.01758
$460$	0	0
$461$	−19.0434	−0.886942	−0.443471	−	0.896289i	$-0.646253\pi$
$461$	−19.0434	−0.886942	−0.443471	+	0.896289i	$0.646253\pi$
$462$	0	0
$463$	24.7807	1.15166	0.575829	−	0.817570i	$-0.304679\pi$
$463$	24.7807	1.15166	0.575829	+	0.817570i	$0.304679\pi$
$464$	0	0
$465$	16.7600	0.777226
$466$	0	0
$467$	32.7650	1.51618	0.758091	−	0.652148i	$-0.226132\pi$
$467$	32.7650	1.51618	0.758091	+	0.652148i	$0.226132\pi$
$468$	0	0
$469$	1.36711	0.0631273
$470$	0	0
$471$	2.66782	0.122927
$472$	0	0
$473$	−18.9413	−0.870923
$474$	0	0
$475$	−5.26615	−0.241628
$476$	0	0
$477$	−6.50913	−0.298033
$478$	0	0
$479$	35.3648	1.61586	0.807930	−	0.589278i	$-0.200588\pi$
$479$	35.3648	1.61586	0.807930	+	0.589278i	$0.200588\pi$
$480$	0	0
$481$	14.4258	0.657759
$482$	0	0
$483$	−12.1100	−0.551023
$484$	0	0
$485$	−23.4615	−1.06533
$486$	0	0
$487$	−16.7258	−0.757921	−0.378960	−	0.925413i	$-0.623718\pi$
$487$	−16.7258	−0.757921	−0.378960	+	0.925413i	$0.623718\pi$
$488$	0	0
$489$	35.9510	1.62576
$490$	0	0
$491$	−8.38973	−0.378623	−0.189312	−	0.981917i	$-0.560626\pi$
$491$	−8.38973	−0.378623	−0.189312	+	0.981917i	$0.560626\pi$
$492$	0	0
$493$	45.9369	2.06890
$494$	0	0
$495$	11.6746	0.524736
$496$	0	0
$497$	10.0597	0.451239
$498$	0	0
$499$	−7.85843	−0.351792	−0.175896	−	0.984409i	$-0.556282\pi$
$499$	−7.85843	−0.351792	−0.175896	+	0.984409i	$0.556282\pi$
$500$	0	0
$501$	−3.85848	−0.172384
$502$	0	0
$503$	−6.99765	−0.312010	−0.156005	−	0.987756i	$-0.549862\pi$
$503$	−6.99765	−0.312010	−0.156005	+	0.987756i	$0.549862\pi$
$504$	0	0
$505$	43.0034	1.91363
$506$	0	0
$507$	13.4701	0.598228
$508$	0	0
$509$	−5.05914	−0.224243	−0.112121	−	0.993695i	$-0.535765\pi$
$509$	−5.05914	−0.224243	−0.112121	+	0.993695i	$0.535765\pi$
$510$	0	0
$511$	15.1717	0.671157
$512$	0	0
$513$	−5.04473	−0.222730
$514$	0	0
$515$	4.93503	0.217463
$516$	0	0
$517$	−15.7771	−0.693877
$518$	0	0
$519$	11.5205	0.505695
$520$	0	0
$521$	17.3647	0.760761	0.380380	−	0.924830i	$-0.375793\pi$
$521$	17.3647	0.760761	0.380380	+	0.924830i	$0.375793\pi$
$522$	0	0
$523$	36.7033	1.60492	0.802461	−	0.596705i	$-0.203524\pi$
$523$	36.7033	1.60492	0.802461	+	0.596705i	$0.203524\pi$
$524$	0	0
$525$	−8.60296	−0.375464
$526$	0	0
$527$	−14.4819	−0.630842
$528$	0	0
$529$	14.8840	0.647129
$530$	0	0
$531$	−8.79010	−0.381458
$532$	0	0
$533$	15.7015	0.680109
$534$	0	0
$535$	−34.3062	−1.48319
$536$	0	0
$537$	26.4758	1.14252
$538$	0	0
$539$	4.37788	0.188569
$540$	0	0
$541$	16.0208	0.688787	0.344393	−	0.938825i	$-0.388085\pi$
$541$	16.0208	0.688787	0.344393	+	0.938825i	$0.388085\pi$
$542$	0	0
$543$	21.1586	0.908001
$544$	0	0
$545$	24.6343	1.05522
$546$	0	0
$547$	15.3332	0.655599	0.327799	−	0.944747i	$-0.393693\pi$
$547$	15.3332	0.655599	0.327799	+	0.944747i	$0.393693\pi$
$548$	0	0
$549$	−1.28105	−0.0546740
$550$	0	0
$551$	−10.6299	−0.452847
$552$	0	0
$553$	6.61348	0.281234
$554$	0	0
$555$	−35.0279	−1.48685
$556$	0	0
$557$	−22.3144	−0.945492	−0.472746	−	0.881199i	$-0.656737\pi$
$557$	−22.3144	−0.945492	−0.472746	+	0.881199i	$0.656737\pi$
$558$	0	0
$559$	−10.7329	−0.453952
$560$	0	0
$561$	−44.8306	−1.89275
$562$	0	0
$563$	20.8827	0.880099	0.440050	−	0.897974i	$-0.354961\pi$
$563$	20.8827	0.880099	0.440050	+	0.897974i	$0.354961\pi$
$564$	0	0
$565$	−46.7207	−1.96555
$566$	0	0
$567$	−10.8544	−0.455844
$568$	0	0
$569$	17.6170	0.738542	0.369271	−	0.929322i	$-0.379607\pi$
$569$	17.6170	0.738542	0.369271	+	0.929322i	$0.379607\pi$
$570$	0	0
$571$	−8.58708	−0.359358	−0.179679	−	0.983725i	$-0.557506\pi$
$571$	−8.58708	−0.359358	−0.179679	+	0.983725i	$0.557506\pi$
$572$	0	0
$573$	40.3098	1.68397
$574$	0	0
$575$	26.9129	1.12234
$576$	0	0
$577$	−35.3028	−1.46968	−0.734838	−	0.678242i	$-0.762742\pi$
$577$	−35.3028	−1.46968	−0.734838	+	0.678242i	$0.762742\pi$
$578$	0	0
$579$	−25.0178	−1.03971
$580$	0	0
$581$	−10.4896	−0.435184
$582$	0	0
$583$	−32.7141	−1.35488
$584$	0	0
$585$	6.61528	0.273508
$586$	0	0
$587$	3.37768	0.139412	0.0697059	−	0.997568i	$-0.477794\pi$
$587$	3.37768	0.139412	0.0697059	+	0.997568i	$0.477794\pi$
$588$	0	0
$589$	3.35113	0.138081
$590$	0	0
$591$	35.7932	1.47233
$592$	0	0
$593$	−11.5641	−0.474882	−0.237441	−	0.971402i	$-0.576309\pi$
$593$	−11.5641	−0.474882	−0.237441	+	0.971402i	$0.576309\pi$
$594$	0	0
$595$	15.9340	0.653229
$596$	0	0
$597$	−2.88923	−0.118248
$598$	0	0
$599$	12.2713	0.501393	0.250697	−	0.968066i	$-0.419340\pi$
$599$	12.2713	0.501393	0.250697	+	0.968066i	$0.419340\pi$
$600$	0	0
$601$	−14.1538	−0.577347	−0.288674	−	0.957428i	$-0.593214\pi$
$601$	−14.1538	−0.577347	−0.288674	+	0.957428i	$0.593214\pi$
$602$	0	0
$603$	1.19084	0.0484949
$604$	0	0
$605$	24.9993	1.01637
$606$	0	0
$607$	−25.6349	−1.04049	−0.520244	−	0.854018i	$-0.674159\pi$
$607$	−25.6349	−1.04049	−0.520244	+	0.854018i	$0.674159\pi$
$608$	0	0
$609$	−17.3653	−0.703676
$610$	0	0
$611$	−8.93991	−0.361670
$612$	0	0
$613$	9.62828	0.388883	0.194441	−	0.980914i	$-0.437711\pi$
$613$	9.62828	0.388883	0.194441	+	0.980914i	$0.437711\pi$
$614$	0	0
$615$	−38.1256	−1.53737
$616$	0	0
$617$	23.1951	0.933800	0.466900	−	0.884310i	$-0.345371\pi$
$617$	23.1951	0.933800	0.466900	+	0.884310i	$0.345371\pi$
$618$	0	0
$619$	−33.5706	−1.34932	−0.674658	−	0.738131i	$-0.735709\pi$
$619$	−33.5706	−1.34932	−0.674658	+	0.738131i	$0.735709\pi$
$620$	0	0
$621$	25.7813	1.03457
$622$	0	0
$623$	−3.26144	−0.130667
$624$	0	0
$625$	−27.7436	−1.10975
$626$	0	0
$627$	10.3738	0.414292
$628$	0	0
$629$	30.2668	1.20682
$630$	0	0
$631$	30.2574	1.20453	0.602265	−	0.798296i	$-0.294265\pi$
$631$	30.2574	1.20453	0.602265	+	0.798296i	$0.294265\pi$
$632$	0	0
$633$	−1.48797	−0.0591416
$634$	0	0
$635$	−5.52680	−0.219325
$636$	0	0
$637$	2.48067	0.0982876
$638$	0	0
$639$	8.76265	0.346645
$640$	0	0
$641$	−6.69390	−0.264393	−0.132197	−	0.991224i	$-0.542203\pi$
$641$	−6.69390	−0.264393	−0.132197	+	0.991224i	$0.542203\pi$
$642$	0	0
$643$	−35.0004	−1.38028	−0.690141	−	0.723674i	$-0.742452\pi$
$643$	−35.0004	−1.38028	−0.690141	+	0.723674i	$0.742452\pi$
$644$	0	0
$645$	26.0610	1.02615
$646$	0	0
$647$	−43.6311	−1.71532	−0.857659	−	0.514219i	$-0.828082\pi$
$647$	−43.6311	−1.71532	−0.857659	+	0.514219i	$0.828082\pi$
$648$	0	0
$649$	−44.1780	−1.73414
$650$	0	0
$651$	5.47451	0.214563
$652$	0	0
$653$	−25.2975	−0.989968	−0.494984	−	0.868902i	$-0.664826\pi$
$653$	−25.2975	−0.989968	−0.494984	+	0.868902i	$0.664826\pi$
$654$	0	0
$655$	−40.2031	−1.57086
$656$	0	0
$657$	13.2156	0.515588
$658$	0	0
$659$	−22.6779	−0.883406	−0.441703	−	0.897161i	$-0.645626\pi$
$659$	−22.6779	−0.883406	−0.441703	+	0.897161i	$0.645626\pi$
$660$	0	0
$661$	−10.9014	−0.424016	−0.212008	−	0.977268i	$-0.568000\pi$
$661$	−10.9014	−0.424016	−0.212008	+	0.977268i	$0.568000\pi$
$662$	0	0
$663$	−25.4027	−0.986559
$664$	0	0
$665$	−3.68714	−0.142981
$666$	0	0
$667$	54.3243	2.10344
$668$	0	0
$669$	15.4208	0.596202
$670$	0	0
$671$	−6.43842	−0.248553
$672$	0	0
$673$	1.82580	0.0703795	0.0351897	−	0.999381i	$-0.488796\pi$
$673$	1.82580	0.0703795	0.0351897	+	0.999381i	$0.488796\pi$
$674$	0	0
$675$	18.3151	0.704949
$676$	0	0
$677$	27.0659	1.04023	0.520114	−	0.854097i	$-0.325889\pi$
$677$	27.0659	1.04023	0.520114	+	0.854097i	$0.325889\pi$
$678$	0	0
$679$	−7.66352	−0.294099
$680$	0	0
$681$	0.688866	0.0263974
$682$	0	0
$683$	20.0208	0.766073	0.383037	−	0.923733i	$-0.374878\pi$
$683$	20.0208	0.766073	0.383037	+	0.923733i	$0.374878\pi$
$684$	0	0
$685$	24.2973	0.928353
$686$	0	0
$687$	43.4148	1.65638
$688$	0	0
$689$	−18.5370	−0.706205
$690$	0	0
$691$	−15.3000	−0.582040	−0.291020	−	0.956717i	$-0.593995\pi$
$691$	−15.3000	−0.582040	−0.291020	+	0.956717i	$0.593995\pi$
$692$	0	0
$693$	3.81342	0.144860
$694$	0	0
$695$	8.95851	0.339815
$696$	0	0
$697$	32.9434	1.24782
$698$	0	0
$699$	−18.7206	−0.708080
$700$	0	0
$701$	−3.73365	−0.141018	−0.0705090	−	0.997511i	$-0.522462\pi$
$701$	−3.73365	−0.141018	−0.0705090	+	0.997511i	$0.522462\pi$
$702$	0	0
$703$	−7.00377	−0.264152
$704$	0	0
$705$	21.7074	0.817548
$706$	0	0
$707$	14.0467	0.528281
$708$	0	0
$709$	−28.6695	−1.07670	−0.538352	−	0.842720i	$-0.680953\pi$
$709$	−28.6695	−1.07670	−0.538352	+	0.842720i	$0.680953\pi$
$710$	0	0
$711$	5.76078	0.216046
$712$	0	0
$713$	−17.1261	−0.641377
$714$	0	0
$715$	33.2476	1.24339
$716$	0	0
$717$	37.0813	1.38483
$718$	0	0
$719$	−20.1357	−0.750936	−0.375468	−	0.926835i	$-0.622518\pi$
$719$	−20.1357	−0.750936	−0.375468	+	0.926835i	$0.622518\pi$
$720$	0	0
$721$	1.61199	0.0600335
$722$	0	0
$723$	−12.5886	−0.468177
$724$	0	0
$725$	38.5921	1.43328
$726$	0	0
$727$	30.0313	1.11380	0.556900	−	0.830580i	$-0.311991\pi$
$727$	30.0313	1.11380	0.556900	+	0.830580i	$0.311991\pi$
$728$	0	0
$729$	15.2688	0.565511
$730$	0	0
$731$	−22.5187	−0.832883
$732$	0	0
$733$	12.8023	0.472862	0.236431	−	0.971648i	$-0.424022\pi$
$733$	12.8023	0.472862	0.236431	+	0.971648i	$0.424022\pi$
$734$	0	0
$735$	−6.02343	−0.222177
$736$	0	0
$737$	5.98504	0.220462
$738$	0	0
$739$	−28.5681	−1.05090	−0.525448	−	0.850826i	$-0.676102\pi$
$739$	−28.5681	−1.05090	−0.525448	+	0.850826i	$0.676102\pi$
$740$	0	0
$741$	5.87821	0.215941
$742$	0	0
$743$	32.8469	1.20504	0.602518	−	0.798106i	$-0.294164\pi$
$743$	32.8469	1.20504	0.602518	+	0.798106i	$0.294164\pi$
$744$	0	0
$745$	−39.6371	−1.45219
$746$	0	0
$747$	−9.13717	−0.334312
$748$	0	0
$749$	−11.2058	−0.409452
$750$	0	0
$751$	−17.6012	−0.642277	−0.321138	−	0.947032i	$-0.604065\pi$
$751$	−17.6012	−0.642277	−0.321138	+	0.947032i	$0.604065\pi$
$752$	0	0
$753$	8.15705	0.297259
$754$	0	0
$755$	8.58113	0.312299
$756$	0	0
$757$	15.3104	0.556465	0.278233	−	0.960514i	$-0.410251\pi$
$757$	15.3104	0.556465	0.278233	+	0.960514i	$0.410251\pi$
$758$	0	0
$759$	−53.0160	−1.92436
$760$	0	0
$761$	1.30937	0.0474648	0.0237324	−	0.999718i	$-0.492445\pi$
$761$	1.30937	0.0474648	0.0237324	+	0.999718i	$0.492445\pi$
$762$	0	0
$763$	8.04658	0.291306
$764$	0	0
$765$	13.8795	0.501816
$766$	0	0
$767$	−25.0329	−0.903885
$768$	0	0
$769$	22.6761	0.817720	0.408860	−	0.912597i	$-0.365926\pi$
$769$	22.6761	0.817720	0.408860	+	0.912597i	$0.365926\pi$
$770$	0	0
$771$	−56.9939	−2.05258
$772$	0	0
$773$	−3.92043	−0.141008	−0.0705041	−	0.997511i	$-0.522461\pi$
$773$	−3.92043	−0.141008	−0.0705041	+	0.997511i	$0.522461\pi$
$774$	0	0
$775$	−12.1664	−0.437031
$776$	0	0
$777$	−11.4416	−0.410464
$778$	0	0
$779$	−7.62314	−0.273127
$780$	0	0
$781$	44.0401	1.57588
$782$	0	0
$783$	36.9695	1.32118
$784$	0	0
$785$	−4.15117	−0.148161
$786$	0	0
$787$	31.0794	1.10786	0.553931	−	0.832563i	$-0.313127\pi$
$787$	31.0794	1.10786	0.553931	+	0.832563i	$0.313127\pi$
$788$	0	0
$789$	0.677699	0.0241267
$790$	0	0
$791$	−15.2609	−0.542616
$792$	0	0
$793$	−3.64825	−0.129553
$794$	0	0
$795$	45.0107	1.59636
$796$	0	0
$797$	−39.3132	−1.39255	−0.696273	−	0.717777i	$-0.745160\pi$
$797$	−39.3132	−1.39255	−0.696273	+	0.717777i	$0.745160\pi$
$798$	0	0
$799$	−18.7569	−0.663570
$800$	0	0
$801$	−2.84093	−0.100379
$802$	0	0
$803$	66.4199	2.34391
$804$	0	0
$805$	18.8433	0.664138
$806$	0	0
$807$	27.4873	0.967598
$808$	0	0
$809$	49.3996	1.73680	0.868398	−	0.495868i	$-0.165150\pi$
$809$	49.3996	1.73680	0.868398	+	0.495868i	$0.165150\pi$
$810$	0	0
$811$	18.5867	0.652668	0.326334	−	0.945254i	$-0.394187\pi$
$811$	18.5867	0.652668	0.326334	+	0.945254i	$0.394187\pi$
$812$	0	0
$813$	−24.5876	−0.862325
$814$	0	0
$815$	−55.9402	−1.95950
$816$	0	0
$817$	5.21084	0.182304
$818$	0	0
$819$	2.16083	0.0755054
$820$	0	0
$821$	−1.07524	−0.0375261	−0.0187631	−	0.999824i	$-0.505973\pi$
$821$	−1.07524	−0.0375261	−0.0187631	+	0.999824i	$0.505973\pi$
$822$	0	0
$823$	−20.0833	−0.700058	−0.350029	−	0.936739i	$-0.613828\pi$
$823$	−20.0833	−0.700058	−0.350029	+	0.936739i	$0.613828\pi$
$824$	0	0
$825$	−37.6627	−1.31125
$826$	0	0
$827$	9.89624	0.344126	0.172063	−	0.985086i	$-0.444957\pi$
$827$	9.89624	0.344126	0.172063	+	0.985086i	$0.444957\pi$
$828$	0	0
$829$	46.2758	1.60722	0.803612	−	0.595153i	$-0.202909\pi$
$829$	46.2758	1.60722	0.803612	+	0.595153i	$0.202909\pi$
$830$	0	0
$831$	−16.6933	−0.579084
$832$	0	0
$833$	5.20470	0.180332
$834$	0	0
$835$	6.00385	0.207772
$836$	0	0
$837$	−11.6549	−0.402852
$838$	0	0
$839$	11.1872	0.386225	0.193113	−	0.981177i	$-0.438142\pi$
$839$	11.1872	0.386225	0.193113	+	0.981177i	$0.438142\pi$
$840$	0	0
$841$	48.8991	1.68618
$842$	0	0
$843$	−18.8190	−0.648162
$844$	0	0
$845$	−20.9596	−0.721033
$846$	0	0
$847$	8.16581	0.280580
$848$	0	0
$849$	−32.5209	−1.11612
$850$	0	0
$851$	35.7930	1.22697
$852$	0	0
$853$	35.9732	1.23170	0.615850	−	0.787863i	$-0.288813\pi$
$853$	35.9732	1.23170	0.615850	+	0.787863i	$0.288813\pi$
$854$	0	0
$855$	−3.21174	−0.109839
$856$	0	0
$857$	32.9414	1.12526	0.562628	−	0.826710i	$-0.309790\pi$
$857$	32.9414	1.12526	0.562628	+	0.826710i	$0.309790\pi$
$858$	0	0
$859$	−42.3121	−1.44367	−0.721835	−	0.692065i	$-0.756701\pi$
$859$	−42.3121	−1.44367	−0.721835	+	0.692065i	$0.756701\pi$
$860$	0	0
$861$	−12.4534	−0.424411
$862$	0	0
$863$	21.8142	0.742562	0.371281	−	0.928520i	$-0.378919\pi$
$863$	21.8142	0.742562	0.371281	+	0.928520i	$0.378919\pi$
$864$	0	0
$865$	−17.9261	−0.609505
$866$	0	0
$867$	−19.8499	−0.674139
$868$	0	0
$869$	28.9530	0.982164
$870$	0	0
$871$	3.39135	0.114911
$872$	0	0
$873$	−6.67543	−0.225929
$874$	0	0
$875$	−1.92098	−0.0649411
$876$	0	0
$877$	13.9938	0.472537	0.236268	−	0.971688i	$-0.424076\pi$
$877$	13.9938	0.472537	0.236268	+	0.971688i	$0.424076\pi$
$878$	0	0
$879$	42.5476	1.43510
$880$	0	0
$881$	12.3319	0.415471	0.207735	−	0.978185i	$-0.433391\pi$
$881$	12.3319	0.415471	0.207735	+	0.978185i	$0.433391\pi$
$882$	0	0
$883$	34.4372	1.15890	0.579452	−	0.815006i	$-0.303267\pi$
$883$	34.4372	1.15890	0.579452	+	0.815006i	$0.303267\pi$
$884$	0	0
$885$	60.7835	2.04322
$886$	0	0
$887$	−4.26921	−0.143346	−0.0716731	−	0.997428i	$-0.522834\pi$
$887$	−4.26921	−0.143346	−0.0716731	+	0.997428i	$0.522834\pi$
$888$	0	0
$889$	−1.80529	−0.0605473
$890$	0	0
$891$	−47.5194	−1.59196
$892$	0	0
$893$	4.34036	0.145244
$894$	0	0
$895$	−41.1967	−1.37705
$896$	0	0
$897$	−30.0408	−1.00303
$898$	0	0
$899$	−24.5582	−0.819062
$900$	0	0
$901$	−38.8927	−1.29570
$902$	0	0
$903$	8.51260	0.283282
$904$	0	0
$905$	−32.9230	−1.09440
$906$	0	0
$907$	33.6720	1.11806	0.559030	−	0.829148i	$-0.311174\pi$
$907$	33.6720	1.11806	0.559030	+	0.829148i	$0.311174\pi$
$908$	0	0
$909$	12.2356	0.405830
$910$	0	0
$911$	−4.22749	−0.140063	−0.0700315	−	0.997545i	$-0.522310\pi$
$911$	−4.22749	−0.140063	−0.0700315	+	0.997545i	$0.522310\pi$
$912$	0	0
$913$	−45.9224	−1.51981
$914$	0	0
$915$	8.85849	0.292853
$916$	0	0
$917$	−13.1320	−0.433657
$918$	0	0
$919$	−21.7824	−0.718536	−0.359268	−	0.933234i	$-0.616974\pi$
$919$	−21.7824	−0.718536	−0.359268	+	0.933234i	$0.616974\pi$
$920$	0	0
$921$	11.5823	0.381649
$922$	0	0
$923$	24.9547	0.821395
$924$	0	0
$925$	25.4275	0.836050
$926$	0	0
$927$	1.40415	0.0461182
$928$	0	0
$929$	−31.3367	−1.02812	−0.514062	−	0.857753i	$-0.671860\pi$
$929$	−31.3367	−1.02812	−0.514062	+	0.857753i	$0.671860\pi$
$930$	0	0
$931$	−1.20437	−0.0394717
$932$	0	0
$933$	−1.57916	−0.0516995
$934$	0	0
$935$	69.7570	2.28130
$936$	0	0
$937$	−53.9341	−1.76195	−0.880975	−	0.473162i	$-0.843113\pi$
$937$	−53.9341	−1.76195	−0.880975	+	0.473162i	$0.843113\pi$
$938$	0	0
$939$	34.8808	1.13829
$940$	0	0
$941$	−49.5836	−1.61638	−0.808190	−	0.588922i	$-0.799552\pi$
$941$	−49.5836	−1.61638	−0.808190	+	0.588922i	$0.799552\pi$
$942$	0	0
$943$	38.9584	1.26866
$944$	0	0
$945$	12.8235	0.417148
$946$	0	0
$947$	0.0908970	0.00295375	0.00147688	−	0.999999i	$-0.499530\pi$
$947$	0.0908970	0.00295375	0.00147688	+	0.999999i	$0.499530\pi$
$948$	0	0
$949$	37.6360	1.22172
$950$	0	0
$951$	28.8516	0.935577
$952$	0	0
$953$	45.4195	1.47128	0.735641	−	0.677371i	$-0.236881\pi$
$953$	45.4195	1.47128	0.735641	+	0.677371i	$0.236881\pi$
$954$	0	0
$955$	−62.7226	−2.02966
$956$	0	0
$957$	−76.0230	−2.45748
$958$	0	0
$959$	7.93652	0.256284
$960$	0	0
$961$	−23.2579	−0.750254
$962$	0	0
$963$	−9.76103	−0.314545
$964$	0	0
$965$	38.9281	1.25314
$966$	0	0
$967$	60.1289	1.93362	0.966808	−	0.255506i	$-0.0822420\pi$
$967$	60.1289	1.93362	0.966808	+	0.255506i	$0.0822420\pi$
$968$	0	0
$969$	12.3331	0.396196
$970$	0	0
$971$	−37.6975	−1.20977	−0.604885	−	0.796313i	$-0.706781\pi$
$971$	−37.6975	−1.20977	−0.604885	+	0.796313i	$0.706781\pi$
$972$	0	0
$973$	2.92622	0.0938104
$974$	0	0
$975$	−21.3411	−0.683462
$976$	0	0
$977$	24.6888	0.789864	0.394932	−	0.918710i	$-0.370768\pi$
$977$	24.6888	0.789864	0.394932	+	0.918710i	$0.370768\pi$
$978$	0	0
$979$	−14.2782	−0.456333
$980$	0	0
$981$	7.00911	0.223784
$982$	0	0
$983$	5.11704	0.163208	0.0816041	−	0.996665i	$-0.473996\pi$
$983$	5.11704	0.163208	0.0816041	+	0.996665i	$0.473996\pi$
$984$	0	0
$985$	−55.6946	−1.77458
$986$	0	0
$987$	7.09055	0.225695
$988$	0	0
$989$	−26.6302	−0.846791
$990$	0	0
$991$	−5.43929	−0.172785	−0.0863924	−	0.996261i	$-0.527534\pi$
$991$	−5.43929	−0.172785	−0.0863924	+	0.996261i	$0.527534\pi$
$992$	0	0
$993$	−0.620663	−0.0196961
$994$	0	0
$995$	4.49567	0.142522
$996$	0	0
$997$	−12.0919	−0.382954	−0.191477	−	0.981497i	$-0.561328\pi$
$997$	−12.0919	−0.382954	−0.191477	+	0.981497i	$0.561328\pi$
$998$	0	0
$999$	24.3584	0.770665

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7168.2.a.bj.1.3		12
4.3	odd	2		7168.2.a.bi.1.10		12
8.3	odd	2		7168.2.a.bi.1.3		12
8.5	even	2	inner	7168.2.a.bj.1.10		12
32.3	odd	8		896.2.m.h.673.5		12
32.5	even	8		112.2.m.d.85.1	yes	12
32.11	odd	8		896.2.m.h.225.5		12
32.13	even	8		112.2.m.d.29.1	✓	12
32.19	odd	8		448.2.m.d.337.2		12
32.21	even	8		896.2.m.g.225.2		12
32.27	odd	8		448.2.m.d.113.2		12
32.29	even	8		896.2.m.g.673.2		12
224.5	odd	24		784.2.x.m.165.4		24
224.13	odd	8		784.2.m.h.589.1		12
224.37	even	24		784.2.x.l.165.4		24
224.45	odd	24		784.2.x.m.765.4		24
224.69	odd	8		784.2.m.h.197.1		12
224.101	odd	24		784.2.x.m.373.6		24
224.109	even	24		784.2.x.l.765.4		24
224.165	even	24		784.2.x.l.373.6		24
224.173	odd	24		784.2.x.m.557.6		24
224.205	even	24		784.2.x.l.557.6		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.m.d.29.1	✓	12	32.13	even	8
112.2.m.d.85.1	yes	12	32.5	even	8
448.2.m.d.113.2		12	32.27	odd	8
448.2.m.d.337.2		12	32.19	odd	8
784.2.m.h.197.1		12	224.69	odd	8
784.2.m.h.589.1		12	224.13	odd	8
784.2.x.l.165.4		24	224.37	even	24
784.2.x.l.373.6		24	224.165	even	24
784.2.x.l.557.6		24	224.205	even	24
784.2.x.l.765.4		24	224.109	even	24
784.2.x.m.165.4		24	224.5	odd	24
784.2.x.m.373.6		24	224.101	odd	24
784.2.x.m.557.6		24	224.173	odd	24
784.2.x.m.765.4		24	224.45	odd	24
896.2.m.g.225.2		12	32.21	even	8
896.2.m.g.673.2		12	32.29	even	8
896.2.m.h.225.5		12	32.11	odd	8
896.2.m.h.673.5		12	32.3	odd	8
7168.2.a.bi.1.3		12	8.3	odd	2
7168.2.a.bi.1.10		12	4.3	odd	2
7168.2.a.bj.1.3		12	1.1	even	1	trivial
7168.2.a.bj.1.10		12	8.5	even	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:	\( N \)	\(=\)	\( 7168 = 2^{10} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7168.a (trivial)

\(f(q)\)	\(=\)	\(q-1.96750 q^{3} +3.06146 q^{5} +1.00000 q^{7} +0.871066 q^{9} +O(q^{10})\)	\(q-1.96750 q^{3} +3.06146 q^{5} +1.00000 q^{7} +0.871066 q^{9} +4.37788 q^{11} +2.48067 q^{13} -6.02343 q^{15} +5.20470 q^{17} -1.20437 q^{19} -1.96750 q^{21} +6.15500 q^{23} +4.37253 q^{25} +4.18868 q^{27} +8.82605 q^{29} -2.78247 q^{31} -8.61348 q^{33} +3.06146 q^{35} +5.81528 q^{37} -4.88072 q^{39} +6.32956 q^{41} -4.32660 q^{43} +2.66673 q^{45} -3.60383 q^{47} +1.00000 q^{49} -10.2403 q^{51} -7.47260 q^{53} +13.4027 q^{55} +2.36961 q^{57} -10.0912 q^{59} -1.47067 q^{61} +0.871066 q^{63} +7.59446 q^{65} +1.36711 q^{67} -12.1100 q^{69} +10.0597 q^{71} +15.1717 q^{73} -8.60296 q^{75} +4.37788 q^{77} +6.61348 q^{79} -10.8544 q^{81} -10.4896 q^{83} +15.9340 q^{85} -17.3653 q^{87} -3.26144 q^{89} +2.48067 q^{91} +5.47451 q^{93} -3.68714 q^{95} -7.66352 q^{97} +3.81342 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q + 12 * q^7 + 12 * q^9	\( 12 q + 12 q^{7} + 12 q^{9} + 24 q^{15} + 8 q^{17} + 16 q^{23} + 20 q^{25} - 8 q^{31} + 16 q^{39} + 32 q^{41} - 16 q^{47} + 12 q^{49} + 24 q^{55} + 64 q^{57} + 12 q^{63} + 32 q^{65} + 8 q^{71} - 24 q^{79} + 44 q^{81} - 32 q^{87} + 24 q^{89} + 48 q^{97}+O(q^{100}) \) 12 * q + 12 * q^7 + 12 * q^9 + 24 * q^15 + 8 * q^17 + 16 * q^23 + 20 * q^25 - 8 * q^31 + 16 * q^39 + 32 * q^41 - 16 * q^47 + 12 * q^49 + 24 * q^55 + 64 * q^57 + 12 * q^63 + 32 * q^65 + 8 * q^71 - 24 * q^79 + 44 * q^81 - 32 * q^87 + 24 * q^89 + 48 * q^97

Introduction

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