LMFDB - Embedded newform 8379.2.a.bl.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8379,2,Mod(1,8379)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8379.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8379, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8379 = 3^{2} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8379.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,2,0,2,2,0,0,6,0,2,-2,0,-4,0,0,6,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$66.9066518536$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 133)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	8379.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{5} +4.41421 q^{8} +2.41421 q^{10} +0.414214 q^{11} +2.24264 q^{13} +3.00000 q^{16} -4.00000 q^{17} +1.00000 q^{19} +3.82843 q^{20} +1.00000 q^{22} +5.58579 q^{23} -4.00000 q^{25} +5.41421 q^{26} +6.58579 q^{29} +6.24264 q^{31} -1.58579 q^{32} -9.65685 q^{34} +9.07107 q^{37} +2.41421 q^{38} +4.41421 q^{40} -3.17157 q^{41} +8.07107 q^{43} +1.58579 q^{44} +13.4853 q^{46} -4.41421 q^{47} -9.65685 q^{50} +8.58579 q^{52} +4.24264 q^{53} +0.414214 q^{55} +15.8995 q^{58} +6.82843 q^{59} -11.8284 q^{61} +15.0711 q^{62} -9.82843 q^{64} +2.24264 q^{65} +1.17157 q^{67} -15.3137 q^{68} +14.2426 q^{71} -15.1421 q^{73} +21.8995 q^{74} +3.82843 q^{76} +1.41421 q^{79} +3.00000 q^{80} -7.65685 q^{82} +9.24264 q^{83} -4.00000 q^{85} +19.4853 q^{86} +1.82843 q^{88} -11.4142 q^{89} +21.3848 q^{92} -10.6569 q^{94} +1.00000 q^{95} +3.17157 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 6 q^{8} + 2 q^{10} - 2 q^{11} - 4 q^{13} + 6 q^{16} - 8 q^{17} + 2 q^{19} + 2 q^{20} + 2 q^{22} + 14 q^{23} - 8 q^{25} + 8 q^{26} + 16 q^{29} + 4 q^{31} - 6 q^{32} - 8 q^{34}+ \cdots + 12 q^{97}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 6 * q^8 + 2 * q^10 - 2 * q^11 - 4 * q^13 + 6 * q^16 - 8 * q^17 + 2 * q^19 + 2 * q^20 + 2 * q^22 + 14 * q^23 - 8 * q^25 + 8 * q^26 + 16 * q^29 + 4 * q^31 - 6 * q^32 - 8 * q^34 + 4 * q^37 + 2 * q^38 + 6 * q^40 - 12 * q^41 + 2 * q^43 + 6 * q^44 + 10 * q^46 - 6 * q^47 - 8 * q^50 + 20 * q^52 - 2 * q^55 + 12 * q^58 + 8 * q^59 - 18 * q^61 + 16 * q^62 - 14 * q^64 - 4 * q^65 + 8 * q^67 - 8 * q^68 + 20 * q^71 - 2 * q^73 + 24 * q^74 + 2 * q^76 + 6 * q^80 - 4 * q^82 + 10 * q^83 - 8 * q^85 + 22 * q^86 - 2 * q^88 - 20 * q^89 + 6 * q^92 - 10 * q^94 + 2 * q^95 + 12 * q^97

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 $

$2$	2.41421	1.70711	0.853553	−	0.521005i	$-0.174443\pi$
$2$	2.41421	1.70711	0.853553	+	0.521005i	$0.174443\pi$
$3$	0	0
$3$	0	0
$4$	3.82843	1.91421
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	4.41421	1.56066
$9$	0	0
$10$	2.41421	0.763441
$11$	0.414214	0.124890	0.0624450	−	0.998048i	$-0.480110\pi$
$11$	0.414214	0.124890	0.0624450	+	0.998048i	$0.480110\pi$
$12$	0	0
$13$	2.24264	0.621997	0.310998	−	0.950410i	$-0.399337\pi$
$13$	2.24264	0.621997	0.310998	+	0.950410i	$0.399337\pi$
$14$	0	0
$15$	0	0
$16$	3.00000	0.750000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	3.82843	0.856062
$21$	0	0
$22$	1.00000	0.213201
$23$	5.58579	1.16472	0.582358	−	0.812932i	$-0.302130\pi$
$23$	5.58579	1.16472	0.582358	+	0.812932i	$0.302130\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	5.41421	1.06181
$27$	0	0
$28$	0	0
$29$	6.58579	1.22295	0.611475	−	0.791264i	$-0.290577\pi$
$29$	6.58579	1.22295	0.611475	+	0.791264i	$0.290577\pi$
$30$	0	0
$31$	6.24264	1.12121	0.560606	−	0.828083i	$-0.310568\pi$
$31$	6.24264	1.12121	0.560606	+	0.828083i	$0.310568\pi$
$32$	−1.58579	−0.280330
$33$	0	0
$34$	−9.65685	−1.65614
$35$	0	0
$36$	0	0
$37$	9.07107	1.49127	0.745637	−	0.666352i	$-0.232145\pi$
$37$	9.07107	1.49127	0.745637	+	0.666352i	$0.232145\pi$
$38$	2.41421	0.391637
$39$	0	0
$40$	4.41421	0.697948
$41$	−3.17157	−0.495316	−0.247658	−	0.968847i	$-0.579661\pi$
$41$	−3.17157	−0.495316	−0.247658	+	0.968847i	$0.579661\pi$
$42$	0	0
$43$	8.07107	1.23083	0.615413	−	0.788205i	$-0.288989\pi$
$43$	8.07107	1.23083	0.615413	+	0.788205i	$0.288989\pi$
$44$	1.58579	0.239066
$45$	0	0
$46$	13.4853	1.98830
$47$	−4.41421	−0.643879	−0.321940	−	0.946760i	$-0.604335\pi$
$47$	−4.41421	−0.643879	−0.321940	+	0.946760i	$0.604335\pi$
$48$	0	0
$49$	0	0
$50$	−9.65685	−1.36569
$51$	0	0
$52$	8.58579	1.19063
$53$	4.24264	0.582772	0.291386	−	0.956606i	$-0.405884\pi$
$53$	4.24264	0.582772	0.291386	+	0.956606i	$0.405884\pi$
$54$	0	0
$55$	0.414214	0.0558525
$56$	0	0
$57$	0	0
$58$	15.8995	2.08771
$59$	6.82843	0.888985	0.444493	−	0.895782i	$-0.353384\pi$
$59$	6.82843	0.888985	0.444493	+	0.895782i	$0.353384\pi$
$60$	0	0
$61$	−11.8284	−1.51447	−0.757237	−	0.653140i	$-0.773451\pi$
$61$	−11.8284	−1.51447	−0.757237	+	0.653140i	$0.773451\pi$
$62$	15.0711	1.91403
$63$	0	0
$64$	−9.82843	−1.22855
$65$	2.24264	0.278165
$66$	0	0
$67$	1.17157	0.143130	0.0715652	−	0.997436i	$-0.477201\pi$
$67$	1.17157	0.143130	0.0715652	+	0.997436i	$0.477201\pi$
$68$	−15.3137	−1.85706
$69$	0	0
$70$	0	0
$71$	14.2426	1.69029	0.845145	−	0.534537i	$-0.179514\pi$
$71$	14.2426	1.69029	0.845145	+	0.534537i	$0.179514\pi$
$72$	0	0
$73$	−15.1421	−1.77225	−0.886126	−	0.463444i	$-0.846614\pi$
$73$	−15.1421	−1.77225	−0.886126	+	0.463444i	$0.846614\pi$
$74$	21.8995	2.54576
$75$	0	0
$76$	3.82843	0.439151
$77$	0	0
$78$	0	0
$79$	1.41421	0.159111	0.0795557	−	0.996830i	$-0.474650\pi$
$79$	1.41421	0.159111	0.0795557	+	0.996830i	$0.474650\pi$
$80$	3.00000	0.335410
$81$	0	0
$82$	−7.65685	−0.845558
$83$	9.24264	1.01451	0.507256	−	0.861796i	$-0.330660\pi$
$83$	9.24264	1.01451	0.507256	+	0.861796i	$0.330660\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	19.4853	2.10115
$87$	0	0
$88$	1.82843	0.194911
$89$	−11.4142	−1.20990	−0.604952	−	0.796262i	$-0.706808\pi$
$89$	−11.4142	−1.20990	−0.604952	+	0.796262i	$0.706808\pi$
$90$	0	0
$91$	0	0
$92$	21.3848	2.22952
$93$	0	0
$94$	−10.6569	−1.09917
$95$	1.00000	0.102598
$96$	0	0
$97$	3.17157	0.322024	0.161012	−	0.986952i	$-0.448524\pi$
$97$	3.17157	0.322024	0.161012	+	0.986952i	$0.448524\pi$
$98$	0	0
$99$	0	0
$100$	−15.3137	−1.53137
$101$	18.3137	1.82228	0.911141	−	0.412095i	$-0.135203\pi$
$101$	18.3137	1.82228	0.911141	+	0.412095i	$0.135203\pi$
$102$	0	0
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	9.89949	0.970725
$105$	0	0
$106$	10.2426	0.994853
$107$	13.8995	1.34371	0.671857	−	0.740680i	$-0.265497\pi$
$107$	13.8995	1.34371	0.671857	+	0.740680i	$0.265497\pi$
$108$	0	0
$109$	−9.65685	−0.924959	−0.462479	−	0.886630i	$-0.653040\pi$
$109$	−9.65685	−0.924959	−0.462479	+	0.886630i	$0.653040\pi$
$110$	1.00000	0.0953463
$111$	0	0
$112$	0	0
$113$	0.242641	0.0228257	0.0114129	−	0.999935i	$-0.496367\pi$
$113$	0.242641	0.0228257	0.0114129	+	0.999935i	$0.496367\pi$
$114$	0	0
$115$	5.58579	0.520877
$116$	25.2132	2.34099
$117$	0	0
$118$	16.4853	1.51759
$119$	0	0
$120$	0	0
$121$	−10.8284	−0.984402
$122$	−28.5563	−2.58537
$123$	0	0
$124$	23.8995	2.14624
$125$	−9.00000	−0.804984
$126$	0	0
$127$	4.24264	0.376473	0.188237	−	0.982124i	$-0.439723\pi$
$127$	4.24264	0.376473	0.188237	+	0.982124i	$0.439723\pi$
$128$	−20.5563	−1.81694
$129$	0	0
$130$	5.41421	0.474858
$131$	3.51472	0.307082	0.153541	−	0.988142i	$-0.450932\pi$
$131$	3.51472	0.307082	0.153541	+	0.988142i	$0.450932\pi$
$132$	0	0
$133$	0	0
$134$	2.82843	0.244339
$135$	0	0
$136$	−17.6569	−1.51406
$137$	19.8284	1.69406	0.847028	−	0.531548i	$-0.178389\pi$
$137$	19.8284	1.69406	0.847028	+	0.531548i	$0.178389\pi$
$138$	0	0
$139$	6.07107	0.514941	0.257471	−	0.966286i	$-0.417111\pi$
$139$	6.07107	0.514941	0.257471	+	0.966286i	$0.417111\pi$
$140$	0	0
$141$	0	0
$142$	34.3848	2.88551
$143$	0.928932	0.0776812
$144$	0	0
$145$	6.58579	0.546920
$146$	−36.5563	−3.02542
$147$	0	0
$148$	34.7279	2.85462
$149$	0.171573	0.0140558	0.00702790	−	0.999975i	$-0.497763\pi$
$149$	0.171573	0.0140558	0.00702790	+	0.999975i	$0.497763\pi$
$150$	0	0
$151$	−17.3137	−1.40897	−0.704485	−	0.709719i	$-0.748822\pi$
$151$	−17.3137	−1.40897	−0.704485	+	0.709719i	$0.748822\pi$
$152$	4.41421	0.358040
$153$	0	0
$154$	0	0
$155$	6.24264	0.501421
$156$	0	0
$157$	−6.17157	−0.492545	−0.246273	−	0.969201i	$-0.579206\pi$
$157$	−6.17157	−0.492545	−0.246273	+	0.969201i	$0.579206\pi$
$158$	3.41421	0.271620
$159$	0	0
$160$	−1.58579	−0.125367
$161$	0	0
$162$	0	0
$163$	3.92893	0.307738	0.153869	−	0.988091i	$-0.450827\pi$
$163$	3.92893	0.307738	0.153869	+	0.988091i	$0.450827\pi$
$164$	−12.1421	−0.948141
$165$	0	0
$166$	22.3137	1.73188
$167$	13.0711	1.01147	0.505735	−	0.862689i	$-0.331221\pi$
$167$	13.0711	1.01147	0.505735	+	0.862689i	$0.331221\pi$
$168$	0	0
$169$	−7.97056	−0.613120
$170$	−9.65685	−0.740647
$171$	0	0
$172$	30.8995	2.35606
$173$	−7.31371	−0.556051	−0.278025	−	0.960574i	$-0.589680\pi$
$173$	−7.31371	−0.556051	−0.278025	+	0.960574i	$0.589680\pi$
$174$	0	0
$175$	0	0
$176$	1.24264	0.0936676
$177$	0	0
$178$	−27.5563	−2.06544
$179$	−17.7990	−1.33036	−0.665179	−	0.746684i	$-0.731645\pi$
$179$	−17.7990	−1.33036	−0.665179	+	0.746684i	$0.731645\pi$
$180$	0	0
$181$	−9.89949	−0.735824	−0.367912	−	0.929861i	$-0.619927\pi$
$181$	−9.89949	−0.735824	−0.367912	+	0.929861i	$0.619927\pi$
$182$	0	0
$183$	0	0
$184$	24.6569	1.81773
$185$	9.07107	0.666918
$186$	0	0
$187$	−1.65685	−0.121161
$188$	−16.8995	−1.23252
$189$	0	0
$190$	2.41421	0.175145
$191$	−8.41421	−0.608831	−0.304416	−	0.952539i	$-0.598461\pi$
$191$	−8.41421	−0.608831	−0.304416	+	0.952539i	$0.598461\pi$
$192$	0	0
$193$	7.89949	0.568618	0.284309	−	0.958733i	$-0.408236\pi$
$193$	7.89949	0.568618	0.284309	+	0.958733i	$0.408236\pi$
$194$	7.65685	0.549730
$195$	0	0
$196$	0	0
$197$	−4.51472	−0.321660	−0.160830	−	0.986982i	$-0.551417\pi$
$197$	−4.51472	−0.321660	−0.160830	+	0.986982i	$0.551417\pi$
$198$	0	0
$199$	−16.0711	−1.13925	−0.569624	−	0.821905i	$-0.692911\pi$
$199$	−16.0711	−1.13925	−0.569624	+	0.821905i	$0.692911\pi$
$200$	−17.6569	−1.24853
$201$	0	0
$202$	44.2132	3.11083
$203$	0	0
$204$	0	0
$205$	−3.17157	−0.221512
$206$	14.4853	1.00924
$207$	0	0
$208$	6.72792	0.466497
$209$	0.414214	0.0286518
$210$	0	0
$211$	−1.17157	−0.0806544	−0.0403272	−	0.999187i	$-0.512840\pi$
$211$	−1.17157	−0.0806544	−0.0403272	+	0.999187i	$0.512840\pi$
$212$	16.2426	1.11555
$213$	0	0
$214$	33.5563	2.29386
$215$	8.07107	0.550442
$216$	0	0
$217$	0	0
$218$	−23.3137	−1.57900
$219$	0	0
$220$	1.58579	0.106914
$221$	−8.97056	−0.603425
$222$	0	0
$223$	−2.24264	−0.150178	−0.0750892	−	0.997177i	$-0.523924\pi$
$223$	−2.24264	−0.150178	−0.0750892	+	0.997177i	$0.523924\pi$
$224$	0	0
$225$	0	0
$226$	0.585786	0.0389659
$227$	−20.2426	−1.34355	−0.671776	−	0.740755i	$-0.734468\pi$
$227$	−20.2426	−1.34355	−0.671776	+	0.740755i	$0.734468\pi$
$228$	0	0
$229$	−12.4853	−0.825051	−0.412525	−	0.910946i	$-0.635353\pi$
$229$	−12.4853	−0.825051	−0.412525	+	0.910946i	$0.635353\pi$
$230$	13.4853	0.889193
$231$	0	0
$232$	29.0711	1.90861
$233$	−0.485281	−0.0317918	−0.0158959	−	0.999874i	$-0.505060\pi$
$233$	−0.485281	−0.0317918	−0.0158959	+	0.999874i	$0.505060\pi$
$234$	0	0
$235$	−4.41421	−0.287952
$236$	26.1421	1.70171
$237$	0	0
$238$	0	0
$239$	−2.00000	−0.129369	−0.0646846	−	0.997906i	$-0.520604\pi$
$239$	−2.00000	−0.129369	−0.0646846	+	0.997906i	$0.520604\pi$
$240$	0	0
$241$	26.0416	1.67749	0.838744	−	0.544525i	$-0.183290\pi$
$241$	26.0416	1.67749	0.838744	+	0.544525i	$0.183290\pi$
$242$	−26.1421	−1.68048
$243$	0	0
$244$	−45.2843	−2.89903
$245$	0	0
$246$	0	0
$247$	2.24264	0.142696
$248$	27.5563	1.74983
$249$	0	0
$250$	−21.7279	−1.37419
$251$	−13.7279	−0.866499	−0.433249	−	0.901274i	$-0.642633\pi$
$251$	−13.7279	−0.866499	−0.433249	+	0.901274i	$0.642633\pi$
$252$	0	0
$253$	2.31371	0.145462
$254$	10.2426	0.642680
$255$	0	0
$256$	−29.9706	−1.87316
$257$	9.31371	0.580973	0.290487	−	0.956879i	$-0.406183\pi$
$257$	9.31371	0.580973	0.290487	+	0.956879i	$0.406183\pi$
$258$	0	0
$259$	0	0
$260$	8.58579	0.532468
$261$	0	0
$262$	8.48528	0.524222
$263$	−16.4853	−1.01653	−0.508263	−	0.861202i	$-0.669712\pi$
$263$	−16.4853	−1.01653	−0.508263	+	0.861202i	$0.669712\pi$
$264$	0	0
$265$	4.24264	0.260623
$266$	0	0
$267$	0	0
$268$	4.48528	0.273982
$269$	−4.00000	−0.243884	−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000	−0.243884	−0.121942	+	0.992537i	$0.538912\pi$
$270$	0	0
$271$	15.2426	0.925924	0.462962	−	0.886378i	$-0.346787\pi$
$271$	15.2426	0.925924	0.462962	+	0.886378i	$0.346787\pi$
$272$	−12.0000	−0.727607
$273$	0	0
$274$	47.8701	2.89194
$275$	−1.65685	−0.0999121
$276$	0	0
$277$	−22.4558	−1.34924	−0.674620	−	0.738165i	$-0.735693\pi$
$277$	−22.4558	−1.34924	−0.674620	+	0.738165i	$0.735693\pi$
$278$	14.6569	0.879060
$279$	0	0
$280$	0	0
$281$	28.2426	1.68481	0.842407	−	0.538841i	$-0.181138\pi$
$281$	28.2426	1.68481	0.842407	+	0.538841i	$0.181138\pi$
$282$	0	0
$283$	−30.0711	−1.78754	−0.893770	−	0.448526i	$-0.851949\pi$
$283$	−30.0711	−1.78754	−0.893770	+	0.448526i	$0.851949\pi$
$284$	54.5269	3.23558
$285$	0	0
$286$	2.24264	0.132610
$287$	0	0
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	15.8995	0.933650
$291$	0	0
$292$	−57.9706	−3.39247
$293$	3.17157	0.185285	0.0926426	−	0.995699i	$-0.470469\pi$
$293$	3.17157	0.185285	0.0926426	+	0.995699i	$0.470469\pi$
$294$	0	0
$295$	6.82843	0.397566
$296$	40.0416	2.32737
$297$	0	0
$298$	0.414214	0.0239947
$299$	12.5269	0.724450
$300$	0	0
$301$	0	0
$302$	−41.7990	−2.40526
$303$	0	0
$304$	3.00000	0.172062
$305$	−11.8284	−0.677294
$306$	0	0
$307$	−29.3137	−1.67302	−0.836511	−	0.547950i	$-0.815408\pi$
$307$	−29.3137	−1.67302	−0.836511	+	0.547950i	$0.815408\pi$
$308$	0	0
$309$	0	0
$310$	15.0711	0.855979
$311$	−23.7990	−1.34952	−0.674758	−	0.738039i	$-0.735752\pi$
$311$	−23.7990	−1.34952	−0.674758	+	0.738039i	$0.735752\pi$
$312$	0	0
$313$	6.65685	0.376268	0.188134	−	0.982143i	$-0.439756\pi$
$313$	6.65685	0.376268	0.188134	+	0.982143i	$0.439756\pi$
$314$	−14.8995	−0.840827
$315$	0	0
$316$	5.41421	0.304573
$317$	20.2426	1.13694	0.568470	−	0.822704i	$-0.307536\pi$
$317$	20.2426	1.13694	0.568470	+	0.822704i	$0.307536\pi$
$318$	0	0
$319$	2.72792	0.152734
$320$	−9.82843	−0.549426
$321$	0	0
$322$	0	0
$323$	−4.00000	−0.222566
$324$	0	0
$325$	−8.97056	−0.497597
$326$	9.48528	0.525341
$327$	0	0
$328$	−14.0000	−0.773021
$329$	0	0
$330$	0	0
$331$	−31.4558	−1.72897	−0.864485	−	0.502659i	$-0.832355\pi$
$331$	−31.4558	−1.72897	−0.864485	+	0.502659i	$0.832355\pi$
$332$	35.3848	1.94199
$333$	0	0
$334$	31.5563	1.72669
$335$	1.17157	0.0640099
$336$	0	0
$337$	19.5563	1.06530	0.532651	−	0.846335i	$-0.321196\pi$
$337$	19.5563	1.06530	0.532651	+	0.846335i	$0.321196\pi$
$338$	−19.2426	−1.04666
$339$	0	0
$340$	−15.3137	−0.830502
$341$	2.58579	0.140028
$342$	0	0
$343$	0	0
$344$	35.6274	1.92090
$345$	0	0
$346$	−17.6569	−0.949238
$347$	−27.0416	−1.45167	−0.725835	−	0.687868i	$-0.758547\pi$
$347$	−27.0416	−1.45167	−0.725835	+	0.687868i	$0.758547\pi$
$348$	0	0
$349$	−4.48528	−0.240092	−0.120046	−	0.992768i	$-0.538304\pi$
$349$	−4.48528	−0.240092	−0.120046	+	0.992768i	$0.538304\pi$
$350$	0	0
$351$	0	0
$352$	−0.656854	−0.0350104
$353$	0.485281	0.0258289	0.0129145	−	0.999917i	$-0.495889\pi$
$353$	0.485281	0.0258289	0.0129145	+	0.999917i	$0.495889\pi$
$354$	0	0
$355$	14.2426	0.755921
$356$	−43.6985	−2.31602
$357$	0	0
$358$	−42.9706	−2.27106
$359$	−5.24264	−0.276696	−0.138348	−	0.990384i	$-0.544179\pi$
$359$	−5.24264	−0.276696	−0.138348	+	0.990384i	$0.544179\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−23.8995	−1.25613
$363$	0	0
$364$	0	0
$365$	−15.1421	−0.792576
$366$	0	0
$367$	24.3431	1.27070	0.635351	−	0.772224i	$-0.280855\pi$
$367$	24.3431	1.27070	0.635351	+	0.772224i	$0.280855\pi$
$368$	16.7574	0.873538
$369$	0	0
$370$	21.8995	1.13850
$371$	0	0
$372$	0	0
$373$	11.3137	0.585802	0.292901	−	0.956143i	$-0.405379\pi$
$373$	11.3137	0.585802	0.292901	+	0.956143i	$0.405379\pi$
$374$	−4.00000	−0.206835
$375$	0	0
$376$	−19.4853	−1.00488
$377$	14.7696	0.760671
$378$	0	0
$379$	4.24264	0.217930	0.108965	−	0.994046i	$-0.465246\pi$
$379$	4.24264	0.217930	0.108965	+	0.994046i	$0.465246\pi$
$380$	3.82843	0.196394
$381$	0	0
$382$	−20.3137	−1.03934
$383$	19.7574	1.00955	0.504777	−	0.863250i	$-0.331575\pi$
$383$	19.7574	1.00955	0.504777	+	0.863250i	$0.331575\pi$
$384$	0	0
$385$	0	0
$386$	19.0711	0.970692
$387$	0	0
$388$	12.1421	0.616424
$389$	11.5147	0.583819	0.291910	−	0.956446i	$-0.405709\pi$
$389$	11.5147	0.583819	0.291910	+	0.956446i	$0.405709\pi$
$390$	0	0
$391$	−22.3431	−1.12994
$392$	0	0
$393$	0	0
$394$	−10.8995	−0.549109
$395$	1.41421	0.0711568
$396$	0	0
$397$	8.62742	0.432998	0.216499	−	0.976283i	$-0.430536\pi$
$397$	8.62742	0.432998	0.216499	+	0.976283i	$0.430536\pi$
$398$	−38.7990	−1.94482
$399$	0	0
$400$	−12.0000	−0.600000
$401$	−0.485281	−0.0242338	−0.0121169	−	0.999927i	$-0.503857\pi$
$401$	−0.485281	−0.0242338	−0.0121169	+	0.999927i	$0.503857\pi$
$402$	0	0
$403$	14.0000	0.697390
$404$	70.1127	3.48824
$405$	0	0
$406$	0	0
$407$	3.75736	0.186245
$408$	0	0
$409$	2.68629	0.132829	0.0664143	−	0.997792i	$-0.478844\pi$
$409$	2.68629	0.132829	0.0664143	+	0.997792i	$0.478844\pi$
$410$	−7.65685	−0.378145
$411$	0	0
$412$	22.9706	1.13168
$413$	0	0
$414$	0	0
$415$	9.24264	0.453703
$416$	−3.55635	−0.174364
$417$	0	0
$418$	1.00000	0.0489116
$419$	6.07107	0.296591	0.148296	−	0.988943i	$-0.452621\pi$
$419$	6.07107	0.296591	0.148296	+	0.988943i	$0.452621\pi$
$420$	0	0
$421$	−37.2132	−1.81366	−0.906830	−	0.421496i	$-0.861505\pi$
$421$	−37.2132	−1.81366	−0.906830	+	0.421496i	$0.861505\pi$
$422$	−2.82843	−0.137686
$423$	0	0
$424$	18.7279	0.909508
$425$	16.0000	0.776114
$426$	0	0
$427$	0	0
$428$	53.2132	2.57216
$429$	0	0
$430$	19.4853	0.939664
$431$	10.0416	0.483688	0.241844	−	0.970315i	$-0.422248\pi$
$431$	10.0416	0.483688	0.241844	+	0.970315i	$0.422248\pi$
$432$	0	0
$433$	−28.0000	−1.34559	−0.672797	−	0.739827i	$-0.734907\pi$
$433$	−28.0000	−1.34559	−0.672797	+	0.739827i	$0.734907\pi$
$434$	0	0
$435$	0	0
$436$	−36.9706	−1.77057
$437$	5.58579	0.267204
$438$	0	0
$439$	24.4853	1.16862	0.584309	−	0.811531i	$-0.301365\pi$
$439$	24.4853	1.16862	0.584309	+	0.811531i	$0.301365\pi$
$440$	1.82843	0.0871668
$441$	0	0
$442$	−21.6569	−1.03011
$443$	−17.6569	−0.838902	−0.419451	−	0.907778i	$-0.637777\pi$
$443$	−17.6569	−0.838902	−0.419451	+	0.907778i	$0.637777\pi$
$444$	0	0
$445$	−11.4142	−0.541086
$446$	−5.41421	−0.256370
$447$	0	0
$448$	0	0
$449$	−9.65685	−0.455735	−0.227868	−	0.973692i	$-0.573175\pi$
$449$	−9.65685	−0.455735	−0.227868	+	0.973692i	$0.573175\pi$
$450$	0	0
$451$	−1.31371	−0.0618601
$452$	0.928932	0.0436933
$453$	0	0
$454$	−48.8701	−2.29359
$455$	0	0
$456$	0	0
$457$	−7.82843	−0.366198	−0.183099	−	0.983094i	$-0.558613\pi$
$457$	−7.82843	−0.366198	−0.183099	+	0.983094i	$0.558613\pi$
$458$	−30.1421	−1.40845
$459$	0	0
$460$	21.3848	0.997070
$461$	29.9706	1.39587	0.697934	−	0.716162i	$-0.254103\pi$
$461$	29.9706	1.39587	0.697934	+	0.716162i	$0.254103\pi$
$462$	0	0
$463$	−4.07107	−0.189199	−0.0945993	−	0.995515i	$-0.530157\pi$
$463$	−4.07107	−0.189199	−0.0945993	+	0.995515i	$0.530157\pi$
$464$	19.7574	0.917212
$465$	0	0
$466$	−1.17157	−0.0542721
$467$	19.1005	0.883866	0.441933	−	0.897048i	$-0.354293\pi$
$467$	19.1005	0.883866	0.441933	+	0.897048i	$0.354293\pi$
$468$	0	0
$469$	0	0
$470$	−10.6569	−0.491564
$471$	0	0
$472$	30.1421	1.38740
$473$	3.34315	0.153718
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	0	0
$478$	−4.82843	−0.220847
$479$	−25.3848	−1.15986	−0.579930	−	0.814666i	$-0.696920\pi$
$479$	−25.3848	−1.15986	−0.579930	+	0.814666i	$0.696920\pi$
$480$	0	0
$481$	20.3431	0.927568
$482$	62.8701	2.86365
$483$	0	0
$484$	−41.4558	−1.88436
$485$	3.17157	0.144014
$486$	0	0
$487$	15.1716	0.687490	0.343745	−	0.939063i	$-0.388304\pi$
$487$	15.1716	0.687490	0.343745	+	0.939063i	$0.388304\pi$
$488$	−52.2132	−2.36358
$489$	0	0
$490$	0	0
$491$	1.44365	0.0651510	0.0325755	−	0.999469i	$-0.489629\pi$
$491$	1.44365	0.0651510	0.0325755	+	0.999469i	$0.489629\pi$
$492$	0	0
$493$	−26.3431	−1.18644
$494$	5.41421	0.243597
$495$	0	0
$496$	18.7279	0.840909
$497$	0	0
$498$	0	0
$499$	34.5563	1.54695	0.773477	−	0.633824i	$-0.218516\pi$
$499$	34.5563	1.54695	0.773477	+	0.633824i	$0.218516\pi$
$500$	−34.4558	−1.54091
$501$	0	0
$502$	−33.1421	−1.47921
$503$	−29.0416	−1.29490	−0.647451	−	0.762107i	$-0.724165\pi$
$503$	−29.0416	−1.29490	−0.647451	+	0.762107i	$0.724165\pi$
$504$	0	0
$505$	18.3137	0.814949
$506$	5.58579	0.248318
$507$	0	0
$508$	16.2426	0.720651
$509$	−14.9706	−0.663559	−0.331779	−	0.943357i	$-0.607649\pi$
$509$	−14.9706	−0.663559	−0.331779	+	0.943357i	$0.607649\pi$
$510$	0	0
$511$	0	0
$512$	−31.2426	−1.38074
$513$	0	0
$514$	22.4853	0.991783
$515$	6.00000	0.264392
$516$	0	0
$517$	−1.82843	−0.0804141
$518$	0	0
$519$	0	0
$520$	9.89949	0.434122
$521$	41.5563	1.82062	0.910308	−	0.413931i	$-0.135844\pi$
$521$	41.5563	1.82062	0.910308	+	0.413931i	$0.135844\pi$
$522$	0	0
$523$	12.3431	0.539728	0.269864	−	0.962898i	$-0.413021\pi$
$523$	12.3431	0.539728	0.269864	+	0.962898i	$0.413021\pi$
$524$	13.4558	0.587821
$525$	0	0
$526$	−39.7990	−1.73532
$527$	−24.9706	−1.08773
$528$	0	0
$529$	8.20101	0.356566
$530$	10.2426	0.444912
$531$	0	0
$532$	0	0
$533$	−7.11270	−0.308085
$534$	0	0
$535$	13.8995	0.600928
$536$	5.17157	0.223378
$537$	0	0
$538$	−9.65685	−0.416337
$539$	0	0
$540$	0	0
$541$	−23.1421	−0.994958	−0.497479	−	0.867476i	$-0.665741\pi$
$541$	−23.1421	−0.994958	−0.497479	+	0.867476i	$0.665741\pi$
$542$	36.7990	1.58065
$543$	0	0
$544$	6.34315	0.271960
$545$	−9.65685	−0.413654
$546$	0	0
$547$	−1.55635	−0.0665447	−0.0332723	−	0.999446i	$-0.510593\pi$
$547$	−1.55635	−0.0665447	−0.0332723	+	0.999446i	$0.510593\pi$
$548$	75.9117	3.24279
$549$	0	0
$550$	−4.00000	−0.170561
$551$	6.58579	0.280564
$552$	0	0
$553$	0	0
$554$	−54.2132	−2.30330
$555$	0	0
$556$	23.2426	0.985708
$557$	−15.8284	−0.670672	−0.335336	−	0.942099i	$-0.608850\pi$
$557$	−15.8284	−0.670672	−0.335336	+	0.942099i	$0.608850\pi$
$558$	0	0
$559$	18.1005	0.765570
$560$	0	0
$561$	0	0
$562$	68.1838	2.87616
$563$	5.89949	0.248634	0.124317	−	0.992243i	$-0.460326\pi$
$563$	5.89949	0.248634	0.124317	+	0.992243i	$0.460326\pi$
$564$	0	0
$565$	0.242641	0.0102080
$566$	−72.5980	−3.05152
$567$	0	0
$568$	62.8701	2.63797
$569$	9.02944	0.378534	0.189267	−	0.981926i	$-0.439389\pi$
$569$	9.02944	0.378534	0.189267	+	0.981926i	$0.439389\pi$
$570$	0	0
$571$	0.899495	0.0376427	0.0188213	−	0.999823i	$-0.494009\pi$
$571$	0.899495	0.0376427	0.0188213	+	0.999823i	$0.494009\pi$
$572$	3.55635	0.148698
$573$	0	0
$574$	0	0
$575$	−22.3431	−0.931774
$576$	0	0
$577$	5.97056	0.248558	0.124279	−	0.992247i	$-0.460338\pi$
$577$	5.97056	0.248558	0.124279	+	0.992247i	$0.460338\pi$
$578$	−2.41421	−0.100418
$579$	0	0
$580$	25.2132	1.04692
$581$	0	0
$582$	0	0
$583$	1.75736	0.0727824
$584$	−66.8406	−2.76588
$585$	0	0
$586$	7.65685	0.316302
$587$	14.0000	0.577842	0.288921	−	0.957353i	$-0.406704\pi$
$587$	14.0000	0.577842	0.288921	+	0.957353i	$0.406704\pi$
$588$	0	0
$589$	6.24264	0.257224
$590$	16.4853	0.678688
$591$	0	0
$592$	27.2132	1.11846
$593$	11.8284	0.485735	0.242868	−	0.970059i	$-0.421912\pi$
$593$	11.8284	0.485735	0.242868	+	0.970059i	$0.421912\pi$
$594$	0	0
$595$	0	0
$596$	0.656854	0.0269058
$597$	0	0
$598$	30.2426	1.23671
$599$	−27.6985	−1.13173	−0.565865	−	0.824498i	$-0.691458\pi$
$599$	−27.6985	−1.13173	−0.565865	+	0.824498i	$0.691458\pi$
$600$	0	0
$601$	34.7279	1.41658	0.708291	−	0.705921i	$-0.249467\pi$
$601$	34.7279	1.41658	0.708291	+	0.705921i	$0.249467\pi$
$602$	0	0
$603$	0	0
$604$	−66.2843	−2.69707
$605$	−10.8284	−0.440238
$606$	0	0
$607$	32.5269	1.32023	0.660113	−	0.751166i	$-0.270508\pi$
$607$	32.5269	1.32023	0.660113	+	0.751166i	$0.270508\pi$
$608$	−1.58579	−0.0643121
$609$	0	0
$610$	−28.5563	−1.15621
$611$	−9.89949	−0.400491
$612$	0	0
$613$	−31.3137	−1.26475	−0.632374	−	0.774663i	$-0.717920\pi$
$613$	−31.3137	−1.26475	−0.632374	+	0.774663i	$0.717920\pi$
$614$	−70.7696	−2.85603
$615$	0	0
$616$	0	0
$617$	−8.45584	−0.340419	−0.170210	−	0.985408i	$-0.554445\pi$
$617$	−8.45584	−0.340419	−0.170210	+	0.985408i	$0.554445\pi$
$618$	0	0
$619$	−34.5563	−1.38894	−0.694468	−	0.719523i	$-0.744360\pi$
$619$	−34.5563	−1.38894	−0.694468	+	0.719523i	$0.744360\pi$
$620$	23.8995	0.959827
$621$	0	0
$622$	−57.4558	−2.30377
$623$	0	0
$624$	0	0
$625$	11.0000	0.440000
$626$	16.0711	0.642329
$627$	0	0
$628$	−23.6274	−0.942837
$629$	−36.2843	−1.44675
$630$	0	0
$631$	−36.5563	−1.45529	−0.727643	−	0.685956i	$-0.759384\pi$
$631$	−36.5563	−1.45529	−0.727643	+	0.685956i	$0.759384\pi$
$632$	6.24264	0.248319
$633$	0	0
$634$	48.8701	1.94088
$635$	4.24264	0.168364
$636$	0	0
$637$	0	0
$638$	6.58579	0.260734
$639$	0	0
$640$	−20.5563	−0.812561
$641$	35.7990	1.41398	0.706988	−	0.707226i	$-0.250054\pi$
$641$	35.7990	1.41398	0.706988	+	0.707226i	$0.250054\pi$
$642$	0	0
$643$	−1.85786	−0.0732670	−0.0366335	−	0.999329i	$-0.511663\pi$
$643$	−1.85786	−0.0732670	−0.0366335	+	0.999329i	$0.511663\pi$
$644$	0	0
$645$	0	0
$646$	−9.65685	−0.379944
$647$	−36.2132	−1.42369	−0.711844	−	0.702338i	$-0.752140\pi$
$647$	−36.2132	−1.42369	−0.711844	+	0.702338i	$0.752140\pi$
$648$	0	0
$649$	2.82843	0.111025
$650$	−21.6569	−0.849452
$651$	0	0
$652$	15.0416	0.589076
$653$	30.1421	1.17955	0.589776	−	0.807567i	$-0.299216\pi$
$653$	30.1421	1.17955	0.589776	+	0.807567i	$0.299216\pi$
$654$	0	0
$655$	3.51472	0.137331
$656$	−9.51472	−0.371487
$657$	0	0
$658$	0	0
$659$	−10.0416	−0.391166	−0.195583	−	0.980687i	$-0.562660\pi$
$659$	−10.0416	−0.391166	−0.195583	+	0.980687i	$0.562660\pi$
$660$	0	0
$661$	−2.72792	−0.106104	−0.0530519	−	0.998592i	$-0.516895\pi$
$661$	−2.72792	−0.106104	−0.0530519	+	0.998592i	$0.516895\pi$
$662$	−75.9411	−2.95154
$663$	0	0
$664$	40.7990	1.58331
$665$	0	0
$666$	0	0
$667$	36.7868	1.42439
$668$	50.0416	1.93617
$669$	0	0
$670$	2.82843	0.109272
$671$	−4.89949	−0.189143
$672$	0	0
$673$	−3.79899	−0.146440	−0.0732201	−	0.997316i	$-0.523328\pi$
$673$	−3.79899	−0.146440	−0.0732201	+	0.997316i	$0.523328\pi$
$674$	47.2132	1.81858
$675$	0	0
$676$	−30.5147	−1.17364
$677$	23.3137	0.896019	0.448009	−	0.894029i	$-0.352133\pi$
$677$	23.3137	0.896019	0.448009	+	0.894029i	$0.352133\pi$
$678$	0	0
$679$	0	0
$680$	−17.6569	−0.677109
$681$	0	0
$682$	6.24264	0.239043
$683$	6.48528	0.248152	0.124076	−	0.992273i	$-0.460403\pi$
$683$	6.48528	0.248152	0.124076	+	0.992273i	$0.460403\pi$
$684$	0	0
$685$	19.8284	0.757605
$686$	0	0
$687$	0	0
$688$	24.2132	0.923120
$689$	9.51472	0.362482
$690$	0	0
$691$	−30.9706	−1.17818	−0.589088	−	0.808069i	$-0.700513\pi$
$691$	−30.9706	−1.17818	−0.589088	+	0.808069i	$0.700513\pi$
$692$	−28.0000	−1.06440
$693$	0	0
$694$	−65.2843	−2.47816
$695$	6.07107	0.230289
$696$	0	0
$697$	12.6863	0.480528
$698$	−10.8284	−0.409862
$699$	0	0
$700$	0	0
$701$	1.82843	0.0690587	0.0345294	−	0.999404i	$-0.489007\pi$
$701$	1.82843	0.0690587	0.0345294	+	0.999404i	$0.489007\pi$
$702$	0	0
$703$	9.07107	0.342122
$704$	−4.07107	−0.153434
$705$	0	0
$706$	1.17157	0.0440927
$707$	0	0
$708$	0	0
$709$	−42.9411	−1.61269	−0.806344	−	0.591447i	$-0.798557\pi$
$709$	−42.9411	−1.61269	−0.806344	+	0.591447i	$0.798557\pi$
$710$	34.3848	1.29044
$711$	0	0
$712$	−50.3848	−1.88825
$713$	34.8701	1.30589
$714$	0	0
$715$	0.928932	0.0347401
$716$	−68.1421	−2.54659
$717$	0	0
$718$	−12.6569	−0.472350
$719$	52.0833	1.94238	0.971189	−	0.238311i	$-0.0765937\pi$
$719$	52.0833	1.94238	0.971189	+	0.238311i	$0.0765937\pi$
$720$	0	0
$721$	0	0
$722$	2.41421	0.0898477
$723$	0	0
$724$	−37.8995	−1.40852
$725$	−26.3431	−0.978360
$726$	0	0
$727$	−20.0711	−0.744395	−0.372197	−	0.928154i	$-0.621396\pi$
$727$	−20.0711	−0.744395	−0.372197	+	0.928154i	$0.621396\pi$
$728$	0	0
$729$	0	0
$730$	−36.5563	−1.35301
$731$	−32.2843	−1.19408
$732$	0	0
$733$	−21.4558	−0.792490	−0.396245	−	0.918145i	$-0.629687\pi$
$733$	−21.4558	−0.792490	−0.396245	+	0.918145i	$0.629687\pi$
$734$	58.7696	2.16922
$735$	0	0
$736$	−8.85786	−0.326505
$737$	0.485281	0.0178756
$738$	0	0
$739$	−8.34315	−0.306908	−0.153454	−	0.988156i	$-0.549040\pi$
$739$	−8.34315	−0.306908	−0.153454	+	0.988156i	$0.549040\pi$
$740$	34.7279	1.27662
$741$	0	0
$742$	0	0
$743$	2.87006	0.105292	0.0526461	−	0.998613i	$-0.483234\pi$
$743$	2.87006	0.105292	0.0526461	+	0.998613i	$0.483234\pi$
$744$	0	0
$745$	0.171573	0.00628594
$746$	27.3137	1.00003
$747$	0	0
$748$	−6.34315	−0.231928
$749$	0	0
$750$	0	0
$751$	9.45584	0.345049	0.172524	−	0.985005i	$-0.444808\pi$
$751$	9.45584	0.345049	0.172524	+	0.985005i	$0.444808\pi$
$752$	−13.2426	−0.482909
$753$	0	0
$754$	35.6569	1.29855
$755$	−17.3137	−0.630110
$756$	0	0
$757$	−5.00000	−0.181728	−0.0908640	−	0.995863i	$-0.528963\pi$
$757$	−5.00000	−0.181728	−0.0908640	+	0.995863i	$0.528963\pi$
$758$	10.2426	0.372029
$759$	0	0
$760$	4.41421	0.160120
$761$	−34.1127	−1.23658	−0.618292	−	0.785948i	$-0.712175\pi$
$761$	−34.1127	−1.23658	−0.618292	+	0.785948i	$0.712175\pi$
$762$	0	0
$763$	0	0
$764$	−32.2132	−1.16543
$765$	0	0
$766$	47.6985	1.72342
$767$	15.3137	0.552946
$768$	0	0
$769$	43.4264	1.56600	0.782998	−	0.622024i	$-0.213689\pi$
$769$	43.4264	1.56600	0.782998	+	0.622024i	$0.213689\pi$
$770$	0	0
$771$	0	0
$772$	30.2426	1.08846
$773$	−34.2426	−1.23162	−0.615811	−	0.787894i	$-0.711172\pi$
$773$	−34.2426	−1.23162	−0.615811	+	0.787894i	$0.711172\pi$
$774$	0	0
$775$	−24.9706	−0.896969
$776$	14.0000	0.502571
$777$	0	0
$778$	27.7990	0.996642
$779$	−3.17157	−0.113633
$780$	0	0
$781$	5.89949	0.211101
$782$	−53.9411	−1.92893
$783$	0	0
$784$	0	0
$785$	−6.17157	−0.220273
$786$	0	0
$787$	−41.0122	−1.46193	−0.730963	−	0.682417i	$-0.760929\pi$
$787$	−41.0122	−1.46193	−0.730963	+	0.682417i	$0.760929\pi$
$788$	−17.2843	−0.615727
$789$	0	0
$790$	3.41421	0.121472
$791$	0	0
$792$	0	0
$793$	−26.5269	−0.941998
$794$	20.8284	0.739173
$795$	0	0
$796$	−61.5269	−2.18076
$797$	−30.2426	−1.07125	−0.535625	−	0.844456i	$-0.679924\pi$
$797$	−30.2426	−1.07125	−0.535625	+	0.844456i	$0.679924\pi$
$798$	0	0
$799$	17.6569	0.624655
$800$	6.34315	0.224264
$801$	0	0
$802$	−1.17157	−0.0413697
$803$	−6.27208	−0.221337
$804$	0	0
$805$	0	0
$806$	33.7990	1.19052
$807$	0	0
$808$	80.8406	2.84396
$809$	−31.1421	−1.09490	−0.547450	−	0.836839i	$-0.684401\pi$
$809$	−31.1421	−1.09490	−0.547450	+	0.836839i	$0.684401\pi$
$810$	0	0
$811$	0.544156	0.0191079	0.00955395	−	0.999954i	$-0.496959\pi$
$811$	0.544156	0.0191079	0.00955395	+	0.999954i	$0.496959\pi$
$812$	0	0
$813$	0	0
$814$	9.07107	0.317941
$815$	3.92893	0.137624
$816$	0	0
$817$	8.07107	0.282371
$818$	6.48528	0.226753
$819$	0	0
$820$	−12.1421	−0.424022
$821$	−10.1127	−0.352936	−0.176468	−	0.984306i	$-0.556467\pi$
$821$	−10.1127	−0.352936	−0.176468	+	0.984306i	$0.556467\pi$
$822$	0	0
$823$	−6.21320	−0.216579	−0.108289	−	0.994119i	$-0.534537\pi$
$823$	−6.21320	−0.216579	−0.108289	+	0.994119i	$0.534537\pi$
$824$	26.4853	0.922658
$825$	0	0
$826$	0	0
$827$	−12.2843	−0.427166	−0.213583	−	0.976925i	$-0.568513\pi$
$827$	−12.2843	−0.427166	−0.213583	+	0.976925i	$0.568513\pi$
$828$	0	0
$829$	28.2843	0.982353	0.491177	−	0.871060i	$-0.336567\pi$
$829$	28.2843	0.982353	0.491177	+	0.871060i	$0.336567\pi$
$830$	22.3137	0.774520
$831$	0	0
$832$	−22.0416	−0.764156
$833$	0	0
$834$	0	0
$835$	13.0711	0.452343
$836$	1.58579	0.0548456
$837$	0	0
$838$	14.6569	0.506313
$839$	−19.7990	−0.683537	−0.341769	−	0.939784i	$-0.611026\pi$
$839$	−19.7990	−0.683537	−0.341769	+	0.939784i	$0.611026\pi$
$840$	0	0
$841$	14.3726	0.495606
$842$	−89.8406	−3.09611
$843$	0	0
$844$	−4.48528	−0.154390
$845$	−7.97056	−0.274196
$846$	0	0
$847$	0	0
$848$	12.7279	0.437079
$849$	0	0
$850$	38.6274	1.32491
$851$	50.6690	1.73691
$852$	0	0
$853$	−43.9706	−1.50552	−0.752762	−	0.658293i	$-0.771279\pi$
$853$	−43.9706	−1.50552	−0.752762	+	0.658293i	$0.771279\pi$
$854$	0	0
$855$	0	0
$856$	61.3553	2.09708
$857$	−12.9706	−0.443066	−0.221533	−	0.975153i	$-0.571106\pi$
$857$	−12.9706	−0.443066	−0.221533	+	0.975153i	$0.571106\pi$
$858$	0	0
$859$	12.0711	0.411860	0.205930	−	0.978567i	$-0.433978\pi$
$859$	12.0711	0.411860	0.205930	+	0.978567i	$0.433978\pi$
$860$	30.8995	1.05366
$861$	0	0
$862$	24.2426	0.825708
$863$	−4.58579	−0.156102	−0.0780510	−	0.996949i	$-0.524870\pi$
$863$	−4.58579	−0.156102	−0.0780510	+	0.996949i	$0.524870\pi$
$864$	0	0
$865$	−7.31371	−0.248674
$866$	−67.5980	−2.29707
$867$	0	0
$868$	0	0
$869$	0.585786	0.0198714
$870$	0	0
$871$	2.62742	0.0890266
$872$	−42.6274	−1.44355
$873$	0	0
$874$	13.4853	0.456146
$875$	0	0
$876$	0	0
$877$	−27.8995	−0.942099	−0.471050	−	0.882107i	$-0.656125\pi$
$877$	−27.8995	−0.942099	−0.471050	+	0.882107i	$0.656125\pi$
$878$	59.1127	1.99496
$879$	0	0
$880$	1.24264	0.0418894
$881$	−52.2843	−1.76150	−0.880751	−	0.473580i	$-0.842962\pi$
$881$	−52.2843	−1.76150	−0.880751	+	0.473580i	$0.842962\pi$
$882$	0	0
$883$	−5.65685	−0.190368	−0.0951842	−	0.995460i	$-0.530344\pi$
$883$	−5.65685	−0.190368	−0.0951842	+	0.995460i	$0.530344\pi$
$884$	−34.3431	−1.15508
$885$	0	0
$886$	−42.6274	−1.43210
$887$	−21.3137	−0.715644	−0.357822	−	0.933790i	$-0.616481\pi$
$887$	−21.3137	−0.715644	−0.357822	+	0.933790i	$0.616481\pi$
$888$	0	0
$889$	0	0
$890$	−27.5563	−0.923691
$891$	0	0
$892$	−8.58579	−0.287473
$893$	−4.41421	−0.147716
$894$	0	0
$895$	−17.7990	−0.594955
$896$	0	0
$897$	0	0
$898$	−23.3137	−0.777989
$899$	41.1127	1.37119
$900$	0	0
$901$	−16.9706	−0.565371
$902$	−3.17157	−0.105602
$903$	0	0
$904$	1.07107	0.0356232
$905$	−9.89949	−0.329070
$906$	0	0
$907$	31.4142	1.04309	0.521546	−	0.853223i	$-0.325356\pi$
$907$	31.4142	1.04309	0.521546	+	0.853223i	$0.325356\pi$
$908$	−77.4975	−2.57184
$909$	0	0
$910$	0	0
$911$	−9.27208	−0.307198	−0.153599	−	0.988133i	$-0.549086\pi$
$911$	−9.27208	−0.307198	−0.153599	+	0.988133i	$0.549086\pi$
$912$	0	0
$913$	3.82843	0.126702
$914$	−18.8995	−0.625140
$915$	0	0
$916$	−47.7990	−1.57932
$917$	0	0
$918$	0	0
$919$	−19.0416	−0.628125	−0.314063	−	0.949402i	$-0.601690\pi$
$919$	−19.0416	−0.628125	−0.314063	+	0.949402i	$0.601690\pi$
$920$	24.6569	0.812912
$921$	0	0
$922$	72.3553	2.38290
$923$	31.9411	1.05135
$924$	0	0
$925$	−36.2843	−1.19302
$926$	−9.82843	−0.322982
$927$	0	0
$928$	−10.4437	−0.342830
$929$	15.0000	0.492134	0.246067	−	0.969253i	$-0.420862\pi$
$929$	15.0000	0.492134	0.246067	+	0.969253i	$0.420862\pi$
$930$	0	0
$931$	0	0
$932$	−1.85786	−0.0608564
$933$	0	0
$934$	46.1127	1.50885
$935$	−1.65685	−0.0541849
$936$	0	0
$937$	33.6863	1.10048	0.550242	−	0.835006i	$-0.314536\pi$
$937$	33.6863	1.10048	0.550242	+	0.835006i	$0.314536\pi$
$938$	0	0
$939$	0	0
$940$	−16.8995	−0.551201
$941$	4.04163	0.131753	0.0658767	−	0.997828i	$-0.479016\pi$
$941$	4.04163	0.131753	0.0658767	+	0.997828i	$0.479016\pi$
$942$	0	0
$943$	−17.7157	−0.576904
$944$	20.4853	0.666739
$945$	0	0
$946$	8.07107	0.262413
$947$	−3.65685	−0.118832	−0.0594159	−	0.998233i	$-0.518924\pi$
$947$	−3.65685	−0.118832	−0.0594159	+	0.998233i	$0.518924\pi$
$948$	0	0
$949$	−33.9584	−1.10234
$950$	−9.65685	−0.313310
$951$	0	0
$952$	0	0
$953$	3.41421	0.110597	0.0552986	−	0.998470i	$-0.482389\pi$
$953$	3.41421	0.110597	0.0552986	+	0.998470i	$0.482389\pi$
$954$	0	0
$955$	−8.41421	−0.272278
$956$	−7.65685	−0.247640
$957$	0	0
$958$	−61.2843	−1.98000
$959$	0	0
$960$	0	0
$961$	7.97056	0.257115
$962$	49.1127	1.58346
$963$	0	0
$964$	99.6985	3.21107
$965$	7.89949	0.254294
$966$	0	0
$967$	−36.2843	−1.16682	−0.583412	−	0.812177i	$-0.698283\pi$
$967$	−36.2843	−1.16682	−0.583412	+	0.812177i	$0.698283\pi$
$968$	−47.7990	−1.53632
$969$	0	0
$970$	7.65685	0.245847
$971$	26.8701	0.862301	0.431151	−	0.902280i	$-0.358108\pi$
$971$	26.8701	0.862301	0.431151	+	0.902280i	$0.358108\pi$
$972$	0	0
$973$	0	0
$974$	36.6274	1.17362
$975$	0	0
$976$	−35.4853	−1.13586
$977$	44.7696	1.43230	0.716152	−	0.697944i	$-0.245902\pi$
$977$	44.7696	1.43230	0.716152	+	0.697944i	$0.245902\pi$
$978$	0	0
$979$	−4.72792	−0.151105
$980$	0	0
$981$	0	0
$982$	3.48528	0.111220
$983$	−46.0000	−1.46717	−0.733586	−	0.679597i	$-0.762155\pi$
$983$	−46.0000	−1.46717	−0.733586	+	0.679597i	$0.762155\pi$
$984$	0	0
$985$	−4.51472	−0.143851
$986$	−63.5980	−2.02537
$987$	0	0
$988$	8.58579	0.273150
$989$	45.0833	1.43356
$990$	0	0
$991$	3.79899	0.120679	0.0603394	−	0.998178i	$-0.480782\pi$
$991$	3.79899	0.120679	0.0603394	+	0.998178i	$0.480782\pi$
$992$	−9.89949	−0.314309
$993$	0	0
$994$	0	0
$995$	−16.0711	−0.509487
$996$	0	0
$997$	28.2843	0.895772	0.447886	−	0.894091i	$-0.352177\pi$
$997$	28.2843	0.895772	0.447886	+	0.894091i	$0.352177\pi$
$998$	83.4264	2.64082
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8379.2.a.bl.1.2		2
3.2	odd	2		931.2.a.e.1.1		2
7.3	odd	6		1197.2.j.e.856.1		4
7.5	odd	6		1197.2.j.e.172.1		4
7.6	odd	2		8379.2.a.bi.1.2		2
21.2	odd	6		931.2.f.i.704.2		4
21.5	even	6		133.2.f.c.39.2	✓	4
21.11	odd	6		931.2.f.i.324.2		4
21.17	even	6		133.2.f.c.58.2	yes	4
21.20	even	2		931.2.a.f.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
133.2.f.c.39.2	✓	4	21.5	even	6
133.2.f.c.58.2	yes	4	21.17	even	6
931.2.a.e.1.1		2	3.2	odd	2
931.2.a.f.1.1		2	21.20	even	2
931.2.f.i.324.2		4	21.11	odd	6
931.2.f.i.704.2		4	21.2	odd	6
1197.2.j.e.172.1		4	7.5	odd	6
1197.2.j.e.856.1		4	7.3	odd	6
8379.2.a.bi.1.2		2	7.6	odd	2
8379.2.a.bl.1.2		2	1.1	even	1	trivial

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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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8379 = 3^{2} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8379.a (trivial)

\(f(q)\)	\(=\)	\(q+2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{5} +4.41421 q^{8} +2.41421 q^{10} +0.414214 q^{11} +2.24264 q^{13} +3.00000 q^{16} -4.00000 q^{17} +1.00000 q^{19} +3.82843 q^{20} +1.00000 q^{22} +5.58579 q^{23} -4.00000 q^{25} +5.41421 q^{26} +6.58579 q^{29} +6.24264 q^{31} -1.58579 q^{32} -9.65685 q^{34} +9.07107 q^{37} +2.41421 q^{38} +4.41421 q^{40} -3.17157 q^{41} +8.07107 q^{43} +1.58579 q^{44} +13.4853 q^{46} -4.41421 q^{47} -9.65685 q^{50} +8.58579 q^{52} +4.24264 q^{53} +0.414214 q^{55} +15.8995 q^{58} +6.82843 q^{59} -11.8284 q^{61} +15.0711 q^{62} -9.82843 q^{64} +2.24264 q^{65} +1.17157 q^{67} -15.3137 q^{68} +14.2426 q^{71} -15.1421 q^{73} +21.8995 q^{74} +3.82843 q^{76} +1.41421 q^{79} +3.00000 q^{80} -7.65685 q^{82} +9.24264 q^{83} -4.00000 q^{85} +19.4853 q^{86} +1.82843 q^{88} -11.4142 q^{89} +21.3848 q^{92} -10.6569 q^{94} +1.00000 q^{95} +3.17157 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 6 q^{8} + 2 q^{10} - 2 q^{11} - 4 q^{13} + 6 q^{16} - 8 q^{17} + 2 q^{19} + 2 q^{20} + 2 q^{22} + 14 q^{23} - 8 q^{25} + 8 q^{26} + 16 q^{29} + 4 q^{31} - 6 q^{32} - 8 q^{34}+ \cdots + 12 q^{97}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 6 * q^8 + 2 * q^10 - 2 * q^11 - 4 * q^13 + 6 * q^16 - 8 * q^17 + 2 * q^19 + 2 * q^20 + 2 * q^22 + 14 * q^23 - 8 * q^25 + 8 * q^26 + 16 * q^29 + 4 * q^31 - 6 * q^32 - 8 * q^34 + 4 * q^37 + 2 * q^38 + 6 * q^40 - 12 * q^41 + 2 * q^43 + 6 * q^44 + 10 * q^46 - 6 * q^47 - 8 * q^50 + 20 * q^52 - 2 * q^55 + 12 * q^58 + 8 * q^59 - 18 * q^61 + 16 * q^62 - 14 * q^64 - 4 * q^65 + 8 * q^67 - 8 * q^68 + 20 * q^71 - 2 * q^73 + 24 * q^74 + 2 * q^76 + 6 * q^80 - 4 * q^82 + 10 * q^83 - 8 * q^85 + 22 * q^86 - 2 * q^88 - 20 * q^89 + 6 * q^92 - 10 * q^94 + 2 * q^95 + 12 * q^97

Introduction

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