LMFDB - Embedded newform 7406.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7406,2,Mod(1,7406)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7406, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7406.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7406 = 2 \cdot 7 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7406.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:	$59.1372077370$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 322)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	7406.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +2.00000 q^{11} -2.00000 q^{12} -4.00000 q^{13} +1.00000 q^{14} -4.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +2.00000 q^{20} -2.00000 q^{21} +2.00000 q^{22} -2.00000 q^{24} -1.00000 q^{25} -4.00000 q^{26} +4.00000 q^{27} +1.00000 q^{28} -2.00000 q^{29} -4.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} -4.00000 q^{33} +6.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} +8.00000 q^{39} +2.00000 q^{40} +6.00000 q^{41} -2.00000 q^{42} -6.00000 q^{43} +2.00000 q^{44} +2.00000 q^{45} -2.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -12.0000 q^{51} -4.00000 q^{52} +12.0000 q^{53} +4.00000 q^{54} +4.00000 q^{55} +1.00000 q^{56} -2.00000 q^{58} -10.0000 q^{59} -4.00000 q^{60} -2.00000 q^{61} +4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -8.00000 q^{65} -4.00000 q^{66} +2.00000 q^{67} +6.00000 q^{68} +2.00000 q^{70} +8.00000 q^{71} +1.00000 q^{72} +2.00000 q^{73} +2.00000 q^{75} +2.00000 q^{77} +8.00000 q^{78} -8.00000 q^{79} +2.00000 q^{80} -11.0000 q^{81} +6.00000 q^{82} +16.0000 q^{83} -2.00000 q^{84} +12.0000 q^{85} -6.00000 q^{86} +4.00000 q^{87} +2.00000 q^{88} -6.00000 q^{89} +2.00000 q^{90} -4.00000 q^{91} -8.00000 q^{93} -2.00000 q^{96} +2.00000 q^{97} +1.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	1.00000	0.500000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	−2.00000	−0.816497
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	−2.00000	−0.577350
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	1.00000	0.267261
$15$	−4.00000	−1.03280
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	1.00000	0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	2.00000	0.447214
$21$	−2.00000	−0.436436
$22$	2.00000	0.426401
$23$	0	0
$23$	0	0
$24$	−2.00000	−0.408248
$25$	−1.00000	−0.200000
$26$	−4.00000	−0.784465
$27$	4.00000	0.769800
$28$	1.00000	0.188982
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	−4.00000	−0.730297
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000	0.176777
$33$	−4.00000	−0.696311
$34$	6.00000	1.02899
$35$	2.00000	0.338062
$36$	1.00000	0.166667
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	8.00000	1.28103
$40$	2.00000	0.316228
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	−2.00000	−0.308607
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	2.00000	0.301511
$45$	2.00000	0.298142
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−2.00000	−0.288675
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	−12.0000	−1.68034
$52$	−4.00000	−0.554700
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	4.00000	0.544331
$55$	4.00000	0.539360
$56$	1.00000	0.133631
$57$	0	0
$58$	−2.00000	−0.262613
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	−4.00000	−0.516398
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	4.00000	0.508001
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	−8.00000	−0.992278
$66$	−4.00000	−0.492366
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	6.00000	0.727607
$69$	0	0
$70$	2.00000	0.239046
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	1.00000	0.117851
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	2.00000	0.230940
$76$	0	0
$77$	2.00000	0.227921
$78$	8.00000	0.905822
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	2.00000	0.223607
$81$	−11.0000	−1.22222
$82$	6.00000	0.662589
$83$	16.0000	1.75623	0.878114	−	0.478451i	$-0.158802\pi$
$83$	16.0000	1.75623	0.878114	+	0.478451i	$0.158802\pi$
$84$	−2.00000	−0.218218
$85$	12.0000	1.30158
$86$	−6.00000	−0.646997
$87$	4.00000	0.428845
$88$	2.00000	0.213201
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	2.00000	0.210819
$91$	−4.00000	−0.419314
$92$	0	0
$93$	−8.00000	−0.829561
$94$	0	0
$95$	0	0
$96$	−2.00000	−0.204124
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	1.00000	0.101015
$99$	2.00000	0.201008
$100$	−1.00000	−0.100000
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	−12.0000	−1.18818
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	−4.00000	−0.392232
$105$	−4.00000	−0.390360
$106$	12.0000	1.16554
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	4.00000	0.384900
$109$	20.0000	1.91565	0.957826	−	0.287348i	$-0.0927736\pi$
$109$	20.0000	1.91565	0.957826	+	0.287348i	$0.0927736\pi$
$110$	4.00000	0.381385
$111$	0	0
$112$	1.00000	0.0944911
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0	0
$116$	−2.00000	−0.185695
$117$	−4.00000	−0.369800
$118$	−10.0000	−0.920575
$119$	6.00000	0.550019
$120$	−4.00000	−0.365148
$121$	−7.00000	−0.636364
$122$	−2.00000	−0.181071
$123$	−12.0000	−1.08200
$124$	4.00000	0.359211
$125$	−12.0000	−1.07331
$126$	1.00000	0.0890871
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	1.00000	0.0883883
$129$	12.0000	1.05654
$130$	−8.00000	−0.701646
$131$	22.0000	1.92215	0.961074	−	0.276289i	$-0.0891049\pi$
$131$	22.0000	1.92215	0.961074	+	0.276289i	$0.0891049\pi$
$132$	−4.00000	−0.348155
$133$	0	0
$134$	2.00000	0.172774
$135$	8.00000	0.688530
$136$	6.00000	0.514496
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	2.00000	0.169031
$141$	0	0
$142$	8.00000	0.671345
$143$	−8.00000	−0.668994
$144$	1.00000	0.0833333
$145$	−4.00000	−0.332182
$146$	2.00000	0.165521
$147$	−2.00000	−0.164957
$148$	0	0
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	2.00000	0.163299
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	2.00000	0.161165
$155$	8.00000	0.642575
$156$	8.00000	0.640513
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	−8.00000	−0.636446
$159$	−24.0000	−1.90332
$160$	2.00000	0.158114
$161$	0	0
$162$	−11.0000	−0.864242
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	6.00000	0.468521
$165$	−8.00000	−0.622799
$166$	16.0000	1.24184
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	−2.00000	−0.154303
$169$	3.00000	0.230769
$170$	12.0000	0.920358
$171$	0	0
$172$	−6.00000	−0.457496
$173$	−24.0000	−1.82469	−0.912343	−	0.409426i	$-0.865729\pi$
$173$	−24.0000	−1.82469	−0.912343	+	0.409426i	$0.865729\pi$
$174$	4.00000	0.303239
$175$	−1.00000	−0.0755929
$176$	2.00000	0.150756
$177$	20.0000	1.50329
$178$	−6.00000	−0.449719
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	2.00000	0.149071
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	−4.00000	−0.296500
$183$	4.00000	0.295689
$184$	0	0
$185$	0	0
$186$	−8.00000	−0.586588
$187$	12.0000	0.877527
$188$	0	0
$189$	4.00000	0.290957
$190$	0	0
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	−2.00000	−0.144338
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	2.00000	0.143592
$195$	16.0000	1.14578
$196$	1.00000	0.0714286
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	2.00000	0.142134
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	−1.00000	−0.0707107
$201$	−4.00000	−0.282138
$202$	−8.00000	−0.562878
$203$	−2.00000	−0.140372
$204$	−12.0000	−0.840168
$205$	12.0000	0.838116
$206$	0	0
$207$	0	0
$208$	−4.00000	−0.277350
$209$	0	0
$210$	−4.00000	−0.276026
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	12.0000	0.824163
$213$	−16.0000	−1.09630
$214$	18.0000	1.23045
$215$	−12.0000	−0.818393
$216$	4.00000	0.272166
$217$	4.00000	0.271538
$218$	20.0000	1.35457
$219$	−4.00000	−0.270295
$220$	4.00000	0.269680
$221$	−24.0000	−1.61441
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	1.00000	0.0668153
$225$	−1.00000	−0.0666667
$226$	6.00000	0.399114
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	−2.00000	−0.131306
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	−4.00000	−0.261488
$235$	0	0
$236$	−10.0000	−0.650945
$237$	16.0000	1.03931
$238$	6.00000	0.388922
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	−4.00000	−0.258199
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	−7.00000	−0.449977
$243$	10.0000	0.641500
$244$	−2.00000	−0.128037
$245$	2.00000	0.127775
$246$	−12.0000	−0.765092
$247$	0	0
$248$	4.00000	0.254000
$249$	−32.0000	−2.02792
$250$	−12.0000	−0.758947
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	1.00000	0.0629941
$253$	0	0
$254$	16.0000	1.00393
$255$	−24.0000	−1.50294
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	12.0000	0.747087
$259$	0	0
$260$	−8.00000	−0.496139
$261$	−2.00000	−0.123797
$262$	22.0000	1.35916
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	−4.00000	−0.246183
$265$	24.0000	1.47431
$266$	0	0
$267$	12.0000	0.734388
$268$	2.00000	0.122169
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	8.00000	0.486864
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	6.00000	0.363803
$273$	8.00000	0.484182
$274$	2.00000	0.120824
$275$	−2.00000	−0.120605
$276$	0	0
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	−14.0000	−0.839664
$279$	4.00000	0.239474
$280$	2.00000	0.119523
$281$	22.0000	1.31241	0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000	1.31241	0.656205	+	0.754583i	$0.272161\pi$
$282$	0	0
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	−8.00000	−0.473050
$287$	6.00000	0.354169
$288$	1.00000	0.0589256
$289$	19.0000	1.11765
$290$	−4.00000	−0.234888
$291$	−4.00000	−0.234484
$292$	2.00000	0.117041
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	−2.00000	−0.116642
$295$	−20.0000	−1.16445
$296$	0	0
$297$	8.00000	0.464207
$298$	12.0000	0.695141
$299$	0	0
$300$	2.00000	0.115470
$301$	−6.00000	−0.345834
$302$	0	0
$303$	16.0000	0.919176
$304$	0	0
$305$	−4.00000	−0.229039
$306$	6.00000	0.342997
$307$	10.0000	0.570730	0.285365	−	0.958419i	$-0.407885\pi$
$307$	10.0000	0.570730	0.285365	+	0.958419i	$0.407885\pi$
$308$	2.00000	0.113961
$309$	0	0
$310$	8.00000	0.454369
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	8.00000	0.452911
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	−18.0000	−1.01580
$315$	2.00000	0.112687
$316$	−8.00000	−0.450035
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	−24.0000	−1.34585
$319$	−4.00000	−0.223957
$320$	2.00000	0.111803
$321$	−36.0000	−2.00932
$322$	0	0
$323$	0	0
$324$	−11.0000	−0.611111
$325$	4.00000	0.221880
$326$	−4.00000	−0.221540
$327$	−40.0000	−2.21201
$328$	6.00000	0.331295
$329$	0	0
$330$	−8.00000	−0.440386
$331$	16.0000	0.879440	0.439720	−	0.898135i	$-0.355078\pi$
$331$	16.0000	0.879440	0.439720	+	0.898135i	$0.355078\pi$
$332$	16.0000	0.878114
$333$	0	0
$334$	−16.0000	−0.875481
$335$	4.00000	0.218543
$336$	−2.00000	−0.109109
$337$	−26.0000	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$337$	−26.0000	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$338$	3.00000	0.163178
$339$	−12.0000	−0.651751
$340$	12.0000	0.650791
$341$	8.00000	0.433224
$342$	0	0
$343$	1.00000	0.0539949
$344$	−6.00000	−0.323498
$345$	0	0
$346$	−24.0000	−1.29025
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	4.00000	0.214423
$349$	24.0000	1.28469	0.642345	−	0.766415i	$-0.277962\pi$
$349$	24.0000	1.28469	0.642345	+	0.766415i	$0.277962\pi$
$350$	−1.00000	−0.0534522
$351$	−16.0000	−0.854017
$352$	2.00000	0.106600
$353$	10.0000	0.532246	0.266123	−	0.963939i	$-0.414257\pi$
$353$	10.0000	0.532246	0.266123	+	0.963939i	$0.414257\pi$
$354$	20.0000	1.06299
$355$	16.0000	0.849192
$356$	−6.00000	−0.317999
$357$	−12.0000	−0.635107
$358$	16.0000	0.845626
$359$	4.00000	0.211112	0.105556	−	0.994413i	$-0.466338\pi$
$359$	4.00000	0.211112	0.105556	+	0.994413i	$0.466338\pi$
$360$	2.00000	0.105409
$361$	−19.0000	−1.00000
$362$	2.00000	0.105118
$363$	14.0000	0.734809
$364$	−4.00000	−0.209657
$365$	4.00000	0.209370
$366$	4.00000	0.209083
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	12.0000	0.623009
$372$	−8.00000	−0.414781
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	12.0000	0.620505
$375$	24.0000	1.23935
$376$	0	0
$377$	8.00000	0.412021
$378$	4.00000	0.205738
$379$	−2.00000	−0.102733	−0.0513665	−	0.998680i	$-0.516358\pi$
$379$	−2.00000	−0.102733	−0.0513665	+	0.998680i	$0.516358\pi$
$380$	0	0
$381$	−32.0000	−1.63941
$382$	16.0000	0.818631
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	−2.00000	−0.102062
$385$	4.00000	0.203859
$386$	−18.0000	−0.916176
$387$	−6.00000	−0.304997
$388$	2.00000	0.101535
$389$	28.0000	1.41966	0.709828	−	0.704375i	$-0.248773\pi$
$389$	28.0000	1.41966	0.709828	+	0.704375i	$0.248773\pi$
$390$	16.0000	0.810191
$391$	0	0
$392$	1.00000	0.0505076
$393$	−44.0000	−2.21951
$394$	18.0000	0.906827
$395$	−16.0000	−0.805047
$396$	2.00000	0.100504
$397$	−20.0000	−1.00377	−0.501886	−	0.864934i	$-0.667360\pi$
$397$	−20.0000	−1.00377	−0.501886	+	0.864934i	$0.667360\pi$
$398$	−24.0000	−1.20301
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	−4.00000	−0.199502
$403$	−16.0000	−0.797017
$404$	−8.00000	−0.398015
$405$	−22.0000	−1.09319
$406$	−2.00000	−0.0992583
$407$	0	0
$408$	−12.0000	−0.594089
$409$	−34.0000	−1.68119	−0.840596	−	0.541663i	$-0.817795\pi$
$409$	−34.0000	−1.68119	−0.840596	+	0.541663i	$0.817795\pi$
$410$	12.0000	0.592638
$411$	−4.00000	−0.197305
$412$	0	0
$413$	−10.0000	−0.492068
$414$	0	0
$415$	32.0000	1.57082
$416$	−4.00000	−0.196116
$417$	28.0000	1.37117
$418$	0	0
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	−4.00000	−0.195180
$421$	20.0000	0.974740	0.487370	−	0.873195i	$-0.337956\pi$
$421$	20.0000	0.974740	0.487370	+	0.873195i	$0.337956\pi$
$422$	4.00000	0.194717
$423$	0	0
$424$	12.0000	0.582772
$425$	−6.00000	−0.291043
$426$	−16.0000	−0.775203
$427$	−2.00000	−0.0967868
$428$	18.0000	0.870063
$429$	16.0000	0.772487
$430$	−12.0000	−0.578691
$431$	−28.0000	−1.34871	−0.674356	−	0.738406i	$-0.735579\pi$
$431$	−28.0000	−1.34871	−0.674356	+	0.738406i	$0.735579\pi$
$432$	4.00000	0.192450
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	4.00000	0.192006
$435$	8.00000	0.383571
$436$	20.0000	0.957826
$437$	0	0
$438$	−4.00000	−0.191127
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	4.00000	0.190693
$441$	1.00000	0.0476190
$442$	−24.0000	−1.14156
$443$	−16.0000	−0.760183	−0.380091	−	0.924949i	$-0.624107\pi$
$443$	−16.0000	−0.760183	−0.380091	+	0.924949i	$0.624107\pi$
$444$	0	0
$445$	−12.0000	−0.568855
$446$	−16.0000	−0.757622
$447$	−24.0000	−1.13516
$448$	1.00000	0.0472456
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	−1.00000	−0.0471405
$451$	12.0000	0.565058
$452$	6.00000	0.282216
$453$	0	0
$454$	12.0000	0.563188
$455$	−8.00000	−0.375046
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	−6.00000	−0.280362
$459$	24.0000	1.12022
$460$	0	0
$461$	8.00000	0.372597	0.186299	−	0.982493i	$-0.440351\pi$
$461$	8.00000	0.372597	0.186299	+	0.982493i	$0.440351\pi$
$462$	−4.00000	−0.186097
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	−2.00000	−0.0928477
$465$	−16.0000	−0.741982
$466$	10.0000	0.463241
$467$	40.0000	1.85098	0.925490	−	0.378773i	$-0.123654\pi$
$467$	40.0000	1.85098	0.925490	+	0.378773i	$0.123654\pi$
$468$	−4.00000	−0.184900
$469$	2.00000	0.0923514
$470$	0	0
$471$	36.0000	1.65879
$472$	−10.0000	−0.460287
$473$	−12.0000	−0.551761
$474$	16.0000	0.734904
$475$	0	0
$476$	6.00000	0.275010
$477$	12.0000	0.549442
$478$	−16.0000	−0.731823
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	−4.00000	−0.182574
$481$	0	0
$482$	−14.0000	−0.637683
$483$	0	0
$484$	−7.00000	−0.318182
$485$	4.00000	0.181631
$486$	10.0000	0.453609
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	−2.00000	−0.0905357
$489$	8.00000	0.361773
$490$	2.00000	0.0903508
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	−12.0000	−0.541002
$493$	−12.0000	−0.540453
$494$	0	0
$495$	4.00000	0.179787
$496$	4.00000	0.179605
$497$	8.00000	0.358849
$498$	−32.0000	−1.43395
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	−12.0000	−0.536656
$501$	32.0000	1.42965
$502$	12.0000	0.535586
$503$	32.0000	1.42681	0.713405	−	0.700752i	$-0.247152\pi$
$503$	32.0000	1.42681	0.713405	+	0.700752i	$0.247152\pi$
$504$	1.00000	0.0445435
$505$	−16.0000	−0.711991
$506$	0	0
$507$	−6.00000	−0.266469
$508$	16.0000	0.709885
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	−24.0000	−1.06274
$511$	2.00000	0.0884748
$512$	1.00000	0.0441942
$513$	0	0
$514$	−6.00000	−0.264649
$515$	0	0
$516$	12.0000	0.528271
$517$	0	0
$518$	0	0
$519$	48.0000	2.10697
$520$	−8.00000	−0.350823
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	−2.00000	−0.0875376
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	22.0000	0.961074
$525$	2.00000	0.0872872
$526$	12.0000	0.523225
$527$	24.0000	1.04546
$528$	−4.00000	−0.174078
$529$	0	0
$530$	24.0000	1.04249
$531$	−10.0000	−0.433963
$532$	0	0
$533$	−24.0000	−1.03956
$534$	12.0000	0.519291
$535$	36.0000	1.55642
$536$	2.00000	0.0863868
$537$	−32.0000	−1.38090
$538$	24.0000	1.03471
$539$	2.00000	0.0861461
$540$	8.00000	0.344265
$541$	−6.00000	−0.257960	−0.128980	−	0.991647i	$-0.541170\pi$
$541$	−6.00000	−0.257960	−0.128980	+	0.991647i	$0.541170\pi$
$542$	20.0000	0.859074
$543$	−4.00000	−0.171656
$544$	6.00000	0.257248
$545$	40.0000	1.71341
$546$	8.00000	0.342368
$547$	−40.0000	−1.71028	−0.855138	−	0.518400i	$-0.826528\pi$
$547$	−40.0000	−1.71028	−0.855138	+	0.518400i	$0.826528\pi$
$548$	2.00000	0.0854358
$549$	−2.00000	−0.0853579
$550$	−2.00000	−0.0852803
$551$	0	0
$552$	0	0
$553$	−8.00000	−0.340195
$554$	26.0000	1.10463
$555$	0	0
$556$	−14.0000	−0.593732
$557$	−32.0000	−1.35588	−0.677942	−	0.735116i	$-0.737128\pi$
$557$	−32.0000	−1.35588	−0.677942	+	0.735116i	$0.737128\pi$
$558$	4.00000	0.169334
$559$	24.0000	1.01509
$560$	2.00000	0.0845154
$561$	−24.0000	−1.01328
$562$	22.0000	0.928014
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	0	0
$565$	12.0000	0.504844
$566$	12.0000	0.504398
$567$	−11.0000	−0.461957
$568$	8.00000	0.335673
$569$	42.0000	1.76073	0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000	1.76073	0.880366	+	0.474295i	$0.157297\pi$
$570$	0	0
$571$	38.0000	1.59025	0.795125	−	0.606445i	$-0.207405\pi$
$571$	38.0000	1.59025	0.795125	+	0.606445i	$0.207405\pi$
$572$	−8.00000	−0.334497
$573$	−32.0000	−1.33682
$574$	6.00000	0.250435
$575$	0	0
$576$	1.00000	0.0416667
$577$	−38.0000	−1.58196	−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000	−1.58196	−0.790980	+	0.611842i	$0.790429\pi$
$578$	19.0000	0.790296
$579$	36.0000	1.49611
$580$	−4.00000	−0.166091
$581$	16.0000	0.663792
$582$	−4.00000	−0.165805
$583$	24.0000	0.993978
$584$	2.00000	0.0827606
$585$	−8.00000	−0.330759
$586$	−14.0000	−0.578335
$587$	18.0000	0.742940	0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000	0.742940	0.371470	+	0.928445i	$0.378854\pi$
$588$	−2.00000	−0.0824786
$589$	0	0
$590$	−20.0000	−0.823387
$591$	−36.0000	−1.48084
$592$	0	0
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	8.00000	0.328244
$595$	12.0000	0.491952
$596$	12.0000	0.491539
$597$	48.0000	1.96451
$598$	0	0
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	2.00000	0.0816497
$601$	−46.0000	−1.87638	−0.938190	−	0.346122i	$-0.887498\pi$
$601$	−46.0000	−1.87638	−0.938190	+	0.346122i	$0.887498\pi$
$602$	−6.00000	−0.244542
$603$	2.00000	0.0814463
$604$	0	0
$605$	−14.0000	−0.569181
$606$	16.0000	0.649956
$607$	−40.0000	−1.62355	−0.811775	−	0.583970i	$-0.801498\pi$
$607$	−40.0000	−1.62355	−0.811775	+	0.583970i	$0.801498\pi$
$608$	0	0
$609$	4.00000	0.162088
$610$	−4.00000	−0.161955
$611$	0	0
$612$	6.00000	0.242536
$613$	−44.0000	−1.77714	−0.888572	−	0.458738i	$-0.848302\pi$
$613$	−44.0000	−1.77714	−0.888572	+	0.458738i	$0.848302\pi$
$614$	10.0000	0.403567
$615$	−24.0000	−0.967773
$616$	2.00000	0.0805823
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	28.0000	1.12542	0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000	1.12542	0.562708	+	0.826656i	$0.309760\pi$
$620$	8.00000	0.321288
$621$	0	0
$622$	24.0000	0.962312
$623$	−6.00000	−0.240385
$624$	8.00000	0.320256
$625$	−19.0000	−0.760000
$626$	10.0000	0.399680
$627$	0	0
$628$	−18.0000	−0.718278
$629$	0	0
$630$	2.00000	0.0796819
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	−8.00000	−0.318223
$633$	−8.00000	−0.317971
$634$	2.00000	0.0794301
$635$	32.0000	1.26988
$636$	−24.0000	−0.951662
$637$	−4.00000	−0.158486
$638$	−4.00000	−0.158362
$639$	8.00000	0.316475
$640$	2.00000	0.0790569
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	−36.0000	−1.42081
$643$	−24.0000	−0.946468	−0.473234	−	0.880937i	$-0.656913\pi$
$643$	−24.0000	−0.946468	−0.473234	+	0.880937i	$0.656913\pi$
$644$	0	0
$645$	24.0000	0.944999
$646$	0	0
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	−11.0000	−0.432121
$649$	−20.0000	−0.785069
$650$	4.00000	0.156893
$651$	−8.00000	−0.313545
$652$	−4.00000	−0.156652
$653$	26.0000	1.01746	0.508729	−	0.860927i	$-0.330115\pi$
$653$	26.0000	1.01746	0.508729	+	0.860927i	$0.330115\pi$
$654$	−40.0000	−1.56412
$655$	44.0000	1.71922
$656$	6.00000	0.234261
$657$	2.00000	0.0780274
$658$	0	0
$659$	6.00000	0.233727	0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000	0.233727	0.116863	+	0.993148i	$0.462716\pi$
$660$	−8.00000	−0.311400
$661$	30.0000	1.16686	0.583432	−	0.812162i	$-0.301709\pi$
$661$	30.0000	1.16686	0.583432	+	0.812162i	$0.301709\pi$
$662$	16.0000	0.621858
$663$	48.0000	1.86417
$664$	16.0000	0.620920
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−16.0000	−0.619059
$669$	32.0000	1.23719
$670$	4.00000	0.154533
$671$	−4.00000	−0.154418
$672$	−2.00000	−0.0771517
$673$	−50.0000	−1.92736	−0.963679	−	0.267063i	$-0.913947\pi$
$673$	−50.0000	−1.92736	−0.963679	+	0.267063i	$0.913947\pi$
$674$	−26.0000	−1.00148
$675$	−4.00000	−0.153960
$676$	3.00000	0.115385
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	−12.0000	−0.460857
$679$	2.00000	0.0767530
$680$	12.0000	0.460179
$681$	−24.0000	−0.919682
$682$	8.00000	0.306336
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	0	0
$685$	4.00000	0.152832
$686$	1.00000	0.0381802
$687$	12.0000	0.457829
$688$	−6.00000	−0.228748
$689$	−48.0000	−1.82865
$690$	0	0
$691$	46.0000	1.74992	0.874961	−	0.484193i	$-0.160887\pi$
$691$	46.0000	1.74992	0.874961	+	0.484193i	$0.160887\pi$
$692$	−24.0000	−0.912343
$693$	2.00000	0.0759737
$694$	12.0000	0.455514
$695$	−28.0000	−1.06210
$696$	4.00000	0.151620
$697$	36.0000	1.36360
$698$	24.0000	0.908413
$699$	−20.0000	−0.756469
$700$	−1.00000	−0.0377964
$701$	−20.0000	−0.755390	−0.377695	−	0.925930i	$-0.623283\pi$
$701$	−20.0000	−0.755390	−0.377695	+	0.925930i	$0.623283\pi$
$702$	−16.0000	−0.603881
$703$	0	0
$704$	2.00000	0.0753778
$705$	0	0
$706$	10.0000	0.376355
$707$	−8.00000	−0.300871
$708$	20.0000	0.751646
$709$	24.0000	0.901339	0.450669	−	0.892691i	$-0.351185\pi$
$709$	24.0000	0.901339	0.450669	+	0.892691i	$0.351185\pi$
$710$	16.0000	0.600469
$711$	−8.00000	−0.300023
$712$	−6.00000	−0.224860
$713$	0	0
$714$	−12.0000	−0.449089
$715$	−16.0000	−0.598366
$716$	16.0000	0.597948
$717$	32.0000	1.19506
$718$	4.00000	0.149279
$719$	−20.0000	−0.745874	−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000	−0.745874	−0.372937	+	0.927857i	$0.621649\pi$
$720$	2.00000	0.0745356
$721$	0	0
$722$	−19.0000	−0.707107
$723$	28.0000	1.04133
$724$	2.00000	0.0743294
$725$	2.00000	0.0742781
$726$	14.0000	0.519589
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	−4.00000	−0.148250
$729$	13.0000	0.481481
$730$	4.00000	0.148047
$731$	−36.0000	−1.33151
$732$	4.00000	0.147844
$733$	−38.0000	−1.40356	−0.701781	−	0.712393i	$-0.747612\pi$
$733$	−38.0000	−1.40356	−0.701781	+	0.712393i	$0.747612\pi$
$734$	0	0
$735$	−4.00000	−0.147542
$736$	0	0
$737$	4.00000	0.147342
$738$	6.00000	0.220863
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	0	0
$742$	12.0000	0.440534
$743$	−48.0000	−1.76095	−0.880475	−	0.474093i	$-0.842776\pi$
$743$	−48.0000	−1.76095	−0.880475	+	0.474093i	$0.842776\pi$
$744$	−8.00000	−0.293294
$745$	24.0000	0.879292
$746$	−4.00000	−0.146450
$747$	16.0000	0.585409
$748$	12.0000	0.438763
$749$	18.0000	0.657706
$750$	24.0000	0.876356
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	−24.0000	−0.874609
$754$	8.00000	0.291343
$755$	0	0
$756$	4.00000	0.145479
$757$	−44.0000	−1.59921	−0.799604	−	0.600528i	$-0.794957\pi$
$757$	−44.0000	−1.59921	−0.799604	+	0.600528i	$0.794957\pi$
$758$	−2.00000	−0.0726433
$759$	0	0
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	−32.0000	−1.15924
$763$	20.0000	0.724049
$764$	16.0000	0.578860
$765$	12.0000	0.433861
$766$	24.0000	0.867155
$767$	40.0000	1.44432
$768$	−2.00000	−0.0721688
$769$	−26.0000	−0.937584	−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000	−0.937584	−0.468792	+	0.883309i	$0.655311\pi$
$770$	4.00000	0.144150
$771$	12.0000	0.432169
$772$	−18.0000	−0.647834
$773$	−10.0000	−0.359675	−0.179838	−	0.983696i	$-0.557557\pi$
$773$	−10.0000	−0.359675	−0.179838	+	0.983696i	$0.557557\pi$
$774$	−6.00000	−0.215666
$775$	−4.00000	−0.143684
$776$	2.00000	0.0717958
$777$	0	0
$778$	28.0000	1.00385
$779$	0	0
$780$	16.0000	0.572892
$781$	16.0000	0.572525
$782$	0	0
$783$	−8.00000	−0.285897
$784$	1.00000	0.0357143
$785$	−36.0000	−1.28490
$786$	−44.0000	−1.56943
$787$	−20.0000	−0.712923	−0.356462	−	0.934310i	$-0.616017\pi$
$787$	−20.0000	−0.712923	−0.356462	+	0.934310i	$0.616017\pi$
$788$	18.0000	0.641223
$789$	−24.0000	−0.854423
$790$	−16.0000	−0.569254
$791$	6.00000	0.213335
$792$	2.00000	0.0710669
$793$	8.00000	0.284088
$794$	−20.0000	−0.709773
$795$	−48.0000	−1.70238
$796$	−24.0000	−0.850657
$797$	−14.0000	−0.495905	−0.247953	−	0.968772i	$-0.579758\pi$
$797$	−14.0000	−0.495905	−0.247953	+	0.968772i	$0.579758\pi$
$798$	0	0
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	−6.00000	−0.212000
$802$	−6.00000	−0.211867
$803$	4.00000	0.141157
$804$	−4.00000	−0.141069
$805$	0	0
$806$	−16.0000	−0.563576
$807$	−48.0000	−1.68968
$808$	−8.00000	−0.281439
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	−22.0000	−0.773001
$811$	−2.00000	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$812$	−2.00000	−0.0701862
$813$	−40.0000	−1.40286
$814$	0	0
$815$	−8.00000	−0.280228
$816$	−12.0000	−0.420084
$817$	0	0
$818$	−34.0000	−1.18878
$819$	−4.00000	−0.139771
$820$	12.0000	0.419058
$821$	54.0000	1.88461	0.942306	−	0.334751i	$-0.108652\pi$
$821$	54.0000	1.88461	0.942306	+	0.334751i	$0.108652\pi$
$822$	−4.00000	−0.139516
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	0	0
$825$	4.00000	0.139262
$826$	−10.0000	−0.347945
$827$	−14.0000	−0.486828	−0.243414	−	0.969923i	$-0.578267\pi$
$827$	−14.0000	−0.486828	−0.243414	+	0.969923i	$0.578267\pi$
$828$	0	0
$829$	12.0000	0.416777	0.208389	−	0.978046i	$-0.433178\pi$
$829$	12.0000	0.416777	0.208389	+	0.978046i	$0.433178\pi$
$830$	32.0000	1.11074
$831$	−52.0000	−1.80386
$832$	−4.00000	−0.138675
$833$	6.00000	0.207888
$834$	28.0000	0.969561
$835$	−32.0000	−1.10741
$836$	0	0
$837$	16.0000	0.553041
$838$	−4.00000	−0.138178
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	−4.00000	−0.138013
$841$	−25.0000	−0.862069
$842$	20.0000	0.689246
$843$	−44.0000	−1.51544
$844$	4.00000	0.137686
$845$	6.00000	0.206406
$846$	0	0
$847$	−7.00000	−0.240523
$848$	12.0000	0.412082
$849$	−24.0000	−0.823678
$850$	−6.00000	−0.205798
$851$	0	0
$852$	−16.0000	−0.548151
$853$	−44.0000	−1.50653	−0.753266	−	0.657716i	$-0.771523\pi$
$853$	−44.0000	−1.50653	−0.753266	+	0.657716i	$0.771523\pi$
$854$	−2.00000	−0.0684386
$855$	0	0
$856$	18.0000	0.615227
$857$	38.0000	1.29806	0.649028	−	0.760765i	$-0.275176\pi$
$857$	38.0000	1.29806	0.649028	+	0.760765i	$0.275176\pi$
$858$	16.0000	0.546231
$859$	−42.0000	−1.43302	−0.716511	−	0.697576i	$-0.754262\pi$
$859$	−42.0000	−1.43302	−0.716511	+	0.697576i	$0.754262\pi$
$860$	−12.0000	−0.409197
$861$	−12.0000	−0.408959
$862$	−28.0000	−0.953684
$863$	−8.00000	−0.272323	−0.136162	−	0.990687i	$-0.543477\pi$
$863$	−8.00000	−0.272323	−0.136162	+	0.990687i	$0.543477\pi$
$864$	4.00000	0.136083
$865$	−48.0000	−1.63205
$866$	14.0000	0.475739
$867$	−38.0000	−1.29055
$868$	4.00000	0.135769
$869$	−16.0000	−0.542763
$870$	8.00000	0.271225
$871$	−8.00000	−0.271070
$872$	20.0000	0.677285
$873$	2.00000	0.0676897
$874$	0	0
$875$	−12.0000	−0.405674
$876$	−4.00000	−0.135147
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	20.0000	0.674967
$879$	28.0000	0.944417
$880$	4.00000	0.134840
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	1.00000	0.0336718
$883$	12.0000	0.403832	0.201916	−	0.979403i	$-0.435283\pi$
$883$	12.0000	0.403832	0.201916	+	0.979403i	$0.435283\pi$
$884$	−24.0000	−0.807207
$885$	40.0000	1.34459
$886$	−16.0000	−0.537531
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	−12.0000	−0.402241
$891$	−22.0000	−0.737028
$892$	−16.0000	−0.535720
$893$	0	0
$894$	−24.0000	−0.802680
$895$	32.0000	1.06964
$896$	1.00000	0.0334077
$897$	0	0
$898$	30.0000	1.00111
$899$	−8.00000	−0.266815
$900$	−1.00000	−0.0333333
$901$	72.0000	2.39867
$902$	12.0000	0.399556
$903$	12.0000	0.399335
$904$	6.00000	0.199557
$905$	4.00000	0.132964
$906$	0	0
$907$	46.0000	1.52740	0.763702	−	0.645568i	$-0.223379\pi$
$907$	46.0000	1.52740	0.763702	+	0.645568i	$0.223379\pi$
$908$	12.0000	0.398234
$909$	−8.00000	−0.265343
$910$	−8.00000	−0.265197
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	0	0
$913$	32.0000	1.05905
$914$	18.0000	0.595387
$915$	8.00000	0.264472
$916$	−6.00000	−0.198246
$917$	22.0000	0.726504
$918$	24.0000	0.792118
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	8.00000	0.263466
$923$	−32.0000	−1.05329
$924$	−4.00000	−0.131590
$925$	0	0
$926$	16.0000	0.525793
$927$	0	0
$928$	−2.00000	−0.0656532
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	−16.0000	−0.524661
$931$	0	0
$932$	10.0000	0.327561
$933$	−48.0000	−1.57145
$934$	40.0000	1.30884
$935$	24.0000	0.784884
$936$	−4.00000	−0.130744
$937$	−30.0000	−0.980057	−0.490029	−	0.871706i	$-0.663014\pi$
$937$	−30.0000	−0.980057	−0.490029	+	0.871706i	$0.663014\pi$
$938$	2.00000	0.0653023
$939$	−20.0000	−0.652675
$940$	0	0
$941$	10.0000	0.325991	0.162995	−	0.986627i	$-0.447884\pi$
$941$	10.0000	0.325991	0.162995	+	0.986627i	$0.447884\pi$
$942$	36.0000	1.17294
$943$	0	0
$944$	−10.0000	−0.325472
$945$	8.00000	0.260240
$946$	−12.0000	−0.390154
$947$	60.0000	1.94974	0.974869	−	0.222779i	$-0.0715128\pi$
$947$	60.0000	1.94974	0.974869	+	0.222779i	$0.0715128\pi$
$948$	16.0000	0.519656
$949$	−8.00000	−0.259691
$950$	0	0
$951$	−4.00000	−0.129709
$952$	6.00000	0.194461
$953$	26.0000	0.842223	0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000	0.842223	0.421111	+	0.907009i	$0.361640\pi$
$954$	12.0000	0.388514
$955$	32.0000	1.03550
$956$	−16.0000	−0.517477
$957$	8.00000	0.258603
$958$	8.00000	0.258468
$959$	2.00000	0.0645834
$960$	−4.00000	−0.129099
$961$	−15.0000	−0.483871
$962$	0	0
$963$	18.0000	0.580042
$964$	−14.0000	−0.450910
$965$	−36.0000	−1.15888
$966$	0	0
$967$	24.0000	0.771788	0.385894	−	0.922543i	$-0.373893\pi$
$967$	24.0000	0.771788	0.385894	+	0.922543i	$0.373893\pi$
$968$	−7.00000	−0.224989
$969$	0	0
$970$	4.00000	0.128432
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	10.0000	0.320750
$973$	−14.0000	−0.448819
$974$	8.00000	0.256337
$975$	−8.00000	−0.256205
$976$	−2.00000	−0.0640184
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	8.00000	0.255812
$979$	−12.0000	−0.383522
$980$	2.00000	0.0638877
$981$	20.0000	0.638551
$982$	0	0
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	−12.0000	−0.382546
$985$	36.0000	1.14706
$986$	−12.0000	−0.382158
$987$	0	0
$988$	0	0
$989$	0	0
$990$	4.00000	0.127128
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	4.00000	0.127000
$993$	−32.0000	−1.01549
$994$	8.00000	0.253745
$995$	−48.0000	−1.52170
$996$	−32.0000	−1.01396
$997$	−36.0000	−1.14013	−0.570066	−	0.821599i	$-0.693082\pi$
$997$	−36.0000	−1.14013	−0.570066	+	0.821599i	$0.693082\pi$
$998$	−24.0000	−0.759707
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7406.2.a.f.1.1		1
23.22	odd	2		322.2.a.c.1.1	✓	1
69.68	even	2		2898.2.a.g.1.1		1
92.91	even	2		2576.2.a.m.1.1		1
115.114	odd	2		8050.2.a.m.1.1		1
161.160	even	2		2254.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.a.c.1.1	✓	1	23.22	odd	2
2254.2.a.g.1.1		1	161.160	even	2
2576.2.a.m.1.1		1	92.91	even	2
2898.2.a.g.1.1		1	69.68	even	2
7406.2.a.f.1.1		1	1.1	even	1	trivial
8050.2.a.m.1.1		1	115.114	odd	2

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:	\( N \)	\(=\)	\( 7406 = 2 \cdot 7 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7406.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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