LMFDB - Embedded newform 5610.2.a.x.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5610,2,Mod(1,5610)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5610, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5610.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5610.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:	$44.7960755339$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5610.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +4.00000 q^{13} -2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} -2.00000 q^{19} -1.00000 q^{20} +2.00000 q^{21} +1.00000 q^{22} -6.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +4.00000 q^{26} -1.00000 q^{27} -2.00000 q^{28} +6.00000 q^{29} +1.00000 q^{30} -8.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} +1.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} -2.00000 q^{38} -4.00000 q^{39} -1.00000 q^{40} +6.00000 q^{41} +2.00000 q^{42} +6.00000 q^{43} +1.00000 q^{44} -1.00000 q^{45} -6.00000 q^{46} -4.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} -1.00000 q^{51} +4.00000 q^{52} +4.00000 q^{53} -1.00000 q^{54} -1.00000 q^{55} -2.00000 q^{56} +2.00000 q^{57} +6.00000 q^{58} -2.00000 q^{59} +1.00000 q^{60} +14.0000 q^{61} -8.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} -1.00000 q^{66} +12.0000 q^{67} +1.00000 q^{68} +6.00000 q^{69} +2.00000 q^{70} -10.0000 q^{71} +1.00000 q^{72} +2.00000 q^{74} -1.00000 q^{75} -2.00000 q^{76} -2.00000 q^{77} -4.00000 q^{78} -10.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +12.0000 q^{83} +2.00000 q^{84} -1.00000 q^{85} +6.00000 q^{86} -6.00000 q^{87} +1.00000 q^{88} -2.00000 q^{89} -1.00000 q^{90} -8.00000 q^{91} -6.00000 q^{92} +8.00000 q^{93} -4.00000 q^{94} +2.00000 q^{95} -1.00000 q^{96} -14.0000 q^{97} -3.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.00000	−0.408248
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−1.00000	−0.288675
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	−2.00000	−0.534522
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	1.00000	0.235702
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	−1.00000	−0.223607
$21$	2.00000	0.436436
$22$	1.00000	0.213201
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	4.00000	0.784465
$27$	−1.00000	−0.192450
$28$	−2.00000	−0.377964
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	1.00000	0.182574
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	1.00000	0.171499
$35$	2.00000	0.338062
$36$	1.00000	0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−2.00000	−0.324443
$39$	−4.00000	−0.640513
$40$	−1.00000	−0.158114
$41$	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	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	1.00000	0.150756
$45$	−1.00000	−0.149071
$46$	−6.00000	−0.884652
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	−1.00000	−0.144338
$49$	−3.00000	−0.428571
$50$	1.00000	0.141421
$51$	−1.00000	−0.140028
$52$	4.00000	0.554700
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	−1.00000	−0.136083
$55$	−1.00000	−0.134840
$56$	−2.00000	−0.267261
$57$	2.00000	0.264906
$58$	6.00000	0.787839
$59$	−2.00000	−0.260378	−0.130189	−	0.991489i	$-0.541558\pi$
$59$	−2.00000	−0.260378	−0.130189	+	0.991489i	$0.541558\pi$
$60$	1.00000	0.129099
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	−8.00000	−1.01600
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	−1.00000	−0.123091
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	1.00000	0.121268
$69$	6.00000	0.722315
$70$	2.00000	0.239046
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	1.00000	0.117851
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	2.00000	0.232495
$75$	−1.00000	−0.115470
$76$	−2.00000	−0.229416
$77$	−2.00000	−0.227921
$78$	−4.00000	−0.452911
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	2.00000	0.218218
$85$	−1.00000	−0.108465
$86$	6.00000	0.646997
$87$	−6.00000	−0.643268
$88$	1.00000	0.106600
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	−1.00000	−0.105409
$91$	−8.00000	−0.838628
$92$	−6.00000	−0.625543
$93$	8.00000	0.829561
$94$	−4.00000	−0.412568
$95$	2.00000	0.205196
$96$	−1.00000	−0.102062
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	−3.00000	−0.303046
$99$	1.00000	0.100504
$100$	1.00000	0.100000
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	−1.00000	−0.0990148
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	4.00000	0.392232
$105$	−2.00000	−0.195180
$106$	4.00000	0.388514
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	−1.00000	−0.0962250
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	−1.00000	−0.0953463
$111$	−2.00000	−0.189832
$112$	−2.00000	−0.188982
$113$	4.00000	0.376288	0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000	0.376288	0.188144	+	0.982141i	$0.439753\pi$
$114$	2.00000	0.187317
$115$	6.00000	0.559503
$116$	6.00000	0.557086
$117$	4.00000	0.369800
$118$	−2.00000	−0.184115
$119$	−2.00000	−0.183340
$120$	1.00000	0.0912871
$121$	1.00000	0.0909091
$122$	14.0000	1.26750
$123$	−6.00000	−0.541002
$124$	−8.00000	−0.718421
$125$	−1.00000	−0.0894427
$126$	−2.00000	−0.178174
$127$	20.0000	1.77471	0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000	1.77471	0.887357	+	0.461084i	$0.152539\pi$
$128$	1.00000	0.0883883
$129$	−6.00000	−0.528271
$130$	−4.00000	−0.350823
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	−1.00000	−0.0870388
$133$	4.00000	0.346844
$134$	12.0000	1.03664
$135$	1.00000	0.0860663
$136$	1.00000	0.0857493
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	6.00000	0.510754
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	2.00000	0.169031
$141$	4.00000	0.336861
$142$	−10.0000	−0.839181
$143$	4.00000	0.334497
$144$	1.00000	0.0833333
$145$	−6.00000	−0.498273
$146$	0	0
$147$	3.00000	0.247436
$148$	2.00000	0.164399
$149$	22.0000	1.80231	0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000	1.80231	0.901155	+	0.433497i	$0.142720\pi$
$150$	−1.00000	−0.0816497
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	−2.00000	−0.162221
$153$	1.00000	0.0808452
$154$	−2.00000	−0.161165
$155$	8.00000	0.642575
$156$	−4.00000	−0.320256
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	−10.0000	−0.795557
$159$	−4.00000	−0.317221
$160$	−1.00000	−0.0790569
$161$	12.0000	0.945732
$162$	1.00000	0.0785674
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	6.00000	0.468521
$165$	1.00000	0.0778499
$166$	12.0000	0.931381
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	2.00000	0.154303
$169$	3.00000	0.230769
$170$	−1.00000	−0.0766965
$171$	−2.00000	−0.152944
$172$	6.00000	0.457496
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	−6.00000	−0.454859
$175$	−2.00000	−0.151186
$176$	1.00000	0.0753778
$177$	2.00000	0.150329
$178$	−2.00000	−0.149906
$179$	2.00000	0.149487	0.0747435	−	0.997203i	$-0.476186\pi$
$179$	2.00000	0.149487	0.0747435	+	0.997203i	$0.476186\pi$
$180$	−1.00000	−0.0745356
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	−8.00000	−0.592999
$183$	−14.0000	−1.03491
$184$	−6.00000	−0.442326
$185$	−2.00000	−0.147043
$186$	8.00000	0.586588
$187$	1.00000	0.0731272
$188$	−4.00000	−0.291730
$189$	2.00000	0.145479
$190$	2.00000	0.145095
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−1.00000	−0.0721688
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	−14.0000	−1.00514
$195$	4.00000	0.286446
$196$	−3.00000	−0.214286
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	1.00000	0.0710669
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	1.00000	0.0707107
$201$	−12.0000	−0.846415
$202$	10.0000	0.703598
$203$	−12.0000	−0.842235
$204$	−1.00000	−0.0700140
$205$	−6.00000	−0.419058
$206$	−8.00000	−0.557386
$207$	−6.00000	−0.417029
$208$	4.00000	0.277350
$209$	−2.00000	−0.138343
$210$	−2.00000	−0.138013
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	4.00000	0.274721
$213$	10.0000	0.685189
$214$	12.0000	0.820303
$215$	−6.00000	−0.409197
$216$	−1.00000	−0.0680414
$217$	16.0000	1.08615
$218$	10.0000	0.677285
$219$	0	0
$220$	−1.00000	−0.0674200
$221$	4.00000	0.269069
$222$	−2.00000	−0.134231
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	−2.00000	−0.133631
$225$	1.00000	0.0666667
$226$	4.00000	0.266076
$227$	20.0000	1.32745	0.663723	−	0.747978i	$-0.268975\pi$
$227$	20.0000	1.32745	0.663723	+	0.747978i	$0.268975\pi$
$228$	2.00000	0.132453
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	6.00000	0.395628
$231$	2.00000	0.131590
$232$	6.00000	0.393919
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	4.00000	0.261488
$235$	4.00000	0.260931
$236$	−2.00000	−0.130189
$237$	10.0000	0.649570
$238$	−2.00000	−0.129641
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	1.00000	0.0645497
$241$	28.0000	1.80364	0.901819	−	0.432113i	$-0.142232\pi$
$241$	28.0000	1.80364	0.901819	+	0.432113i	$0.142232\pi$
$242$	1.00000	0.0642824
$243$	−1.00000	−0.0641500
$244$	14.0000	0.896258
$245$	3.00000	0.191663
$246$	−6.00000	−0.382546
$247$	−8.00000	−0.509028
$248$	−8.00000	−0.508001
$249$	−12.0000	−0.760469
$250$	−1.00000	−0.0632456
$251$	−6.00000	−0.378717	−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000	−0.378717	−0.189358	+	0.981908i	$0.560641\pi$
$252$	−2.00000	−0.125988
$253$	−6.00000	−0.377217
$254$	20.0000	1.25491
$255$	1.00000	0.0626224
$256$	1.00000	0.0625000
$257$	10.0000	0.623783	0.311891	−	0.950118i	$-0.399037\pi$
$257$	10.0000	0.623783	0.311891	+	0.950118i	$0.399037\pi$
$258$	−6.00000	−0.373544
$259$	−4.00000	−0.248548
$260$	−4.00000	−0.248069
$261$	6.00000	0.371391
$262$	−12.0000	−0.741362
$263$	32.0000	1.97320	0.986602	−	0.163144i	$-0.0521635\pi$
$263$	32.0000	1.97320	0.986602	+	0.163144i	$0.0521635\pi$
$264$	−1.00000	−0.0615457
$265$	−4.00000	−0.245718
$266$	4.00000	0.245256
$267$	2.00000	0.122398
$268$	12.0000	0.733017
$269$	26.0000	1.58525	0.792624	−	0.609711i	$-0.208714\pi$
$269$	26.0000	1.58525	0.792624	+	0.609711i	$0.208714\pi$
$270$	1.00000	0.0608581
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	1.00000	0.0606339
$273$	8.00000	0.484182
$274$	2.00000	0.120824
$275$	1.00000	0.0603023
$276$	6.00000	0.361158
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	−16.0000	−0.959616
$279$	−8.00000	−0.478947
$280$	2.00000	0.119523
$281$	−26.0000	−1.55103	−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−26.0000	−1.55103	−0.775515	+	0.631329i	$0.782510\pi$
$282$	4.00000	0.238197
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	−10.0000	−0.593391
$285$	−2.00000	−0.118470
$286$	4.00000	0.236525
$287$	−12.0000	−0.708338
$288$	1.00000	0.0589256
$289$	1.00000	0.0588235
$290$	−6.00000	−0.352332
$291$	14.0000	0.820695
$292$	0	0
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	3.00000	0.174964
$295$	2.00000	0.116445
$296$	2.00000	0.116248
$297$	−1.00000	−0.0580259
$298$	22.0000	1.27443
$299$	−24.0000	−1.38796
$300$	−1.00000	−0.0577350
$301$	−12.0000	−0.691669
$302$	0	0
$303$	−10.0000	−0.574485
$304$	−2.00000	−0.114708
$305$	−14.0000	−0.801638
$306$	1.00000	0.0571662
$307$	−22.0000	−1.25561	−0.627803	−	0.778372i	$-0.716046\pi$
$307$	−22.0000	−1.25561	−0.627803	+	0.778372i	$0.716046\pi$
$308$	−2.00000	−0.113961
$309$	8.00000	0.455104
$310$	8.00000	0.454369
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	−4.00000	−0.226455
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	2.00000	0.112867
$315$	2.00000	0.112687
$316$	−10.0000	−0.562544
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	−4.00000	−0.224309
$319$	6.00000	0.335936
$320$	−1.00000	−0.0559017
$321$	−12.0000	−0.669775
$322$	12.0000	0.668734
$323$	−2.00000	−0.111283
$324$	1.00000	0.0555556
$325$	4.00000	0.221880
$326$	12.0000	0.664619
$327$	−10.0000	−0.553001
$328$	6.00000	0.331295
$329$	8.00000	0.441054
$330$	1.00000	0.0550482
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	12.0000	0.658586
$333$	2.00000	0.109599
$334$	16.0000	0.875481
$335$	−12.0000	−0.655630
$336$	2.00000	0.109109
$337$	−32.0000	−1.74315	−0.871576	−	0.490261i	$-0.836901\pi$
$337$	−32.0000	−1.74315	−0.871576	+	0.490261i	$0.836901\pi$
$338$	3.00000	0.163178
$339$	−4.00000	−0.217250
$340$	−1.00000	−0.0542326
$341$	−8.00000	−0.433224
$342$	−2.00000	−0.108148
$343$	20.0000	1.07990
$344$	6.00000	0.323498
$345$	−6.00000	−0.323029
$346$	18.0000	0.967686
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	−6.00000	−0.321634
$349$	−12.0000	−0.642345	−0.321173	−	0.947021i	$-0.604077\pi$
$349$	−12.0000	−0.642345	−0.321173	+	0.947021i	$0.604077\pi$
$350$	−2.00000	−0.106904
$351$	−4.00000	−0.213504
$352$	1.00000	0.0533002
$353$	34.0000	1.80964	0.904819	−	0.425797i	$-0.140006\pi$
$353$	34.0000	1.80964	0.904819	+	0.425797i	$0.140006\pi$
$354$	2.00000	0.106299
$355$	10.0000	0.530745
$356$	−2.00000	−0.106000
$357$	2.00000	0.105851
$358$	2.00000	0.105703
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	−1.00000	−0.0527046
$361$	−15.0000	−0.789474
$362$	10.0000	0.525588
$363$	−1.00000	−0.0524864
$364$	−8.00000	−0.419314
$365$	0	0
$366$	−14.0000	−0.731792
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	−6.00000	−0.312772
$369$	6.00000	0.312348
$370$	−2.00000	−0.103975
$371$	−8.00000	−0.415339
$372$	8.00000	0.414781
$373$	−12.0000	−0.621336	−0.310668	−	0.950518i	$-0.600553\pi$
$373$	−12.0000	−0.621336	−0.310668	+	0.950518i	$0.600553\pi$
$374$	1.00000	0.0517088
$375$	1.00000	0.0516398
$376$	−4.00000	−0.206284
$377$	24.0000	1.23606
$378$	2.00000	0.102869
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	2.00000	0.102598
$381$	−20.0000	−1.02463
$382$	0	0
$383$	32.0000	1.63512	0.817562	−	0.575841i	$-0.195325\pi$
$383$	32.0000	1.63512	0.817562	+	0.575841i	$0.195325\pi$
$384$	−1.00000	−0.0510310
$385$	2.00000	0.101929
$386$	4.00000	0.203595
$387$	6.00000	0.304997
$388$	−14.0000	−0.710742
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	4.00000	0.202548
$391$	−6.00000	−0.303433
$392$	−3.00000	−0.151523
$393$	12.0000	0.605320
$394$	−10.0000	−0.503793
$395$	10.0000	0.503155
$396$	1.00000	0.0502519
$397$	−34.0000	−1.70641	−0.853206	−	0.521575i	$-0.825345\pi$
$397$	−34.0000	−1.70641	−0.853206	+	0.521575i	$0.825345\pi$
$398$	16.0000	0.802008
$399$	−4.00000	−0.200250
$400$	1.00000	0.0500000
$401$	−20.0000	−0.998752	−0.499376	−	0.866385i	$-0.666437\pi$
$401$	−20.0000	−0.998752	−0.499376	+	0.866385i	$0.666437\pi$
$402$	−12.0000	−0.598506
$403$	−32.0000	−1.59403
$404$	10.0000	0.497519
$405$	−1.00000	−0.0496904
$406$	−12.0000	−0.595550
$407$	2.00000	0.0991363
$408$	−1.00000	−0.0495074
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	−6.00000	−0.296319
$411$	−2.00000	−0.0986527
$412$	−8.00000	−0.394132
$413$	4.00000	0.196827
$414$	−6.00000	−0.294884
$415$	−12.0000	−0.589057
$416$	4.00000	0.196116
$417$	16.0000	0.783523
$418$	−2.00000	−0.0978232
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	−2.00000	−0.0975900
$421$	−18.0000	−0.877266	−0.438633	−	0.898666i	$-0.644537\pi$
$421$	−18.0000	−0.877266	−0.438633	+	0.898666i	$0.644537\pi$
$422$	16.0000	0.778868
$423$	−4.00000	−0.194487
$424$	4.00000	0.194257
$425$	1.00000	0.0485071
$426$	10.0000	0.484502
$427$	−28.0000	−1.35501
$428$	12.0000	0.580042
$429$	−4.00000	−0.193122
$430$	−6.00000	−0.289346
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	−1.00000	−0.0481125
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	16.0000	0.768025
$435$	6.00000	0.287678
$436$	10.0000	0.478913
$437$	12.0000	0.574038
$438$	0	0
$439$	−26.0000	−1.24091	−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000	−1.24091	−0.620456	+	0.784241i	$0.713053\pi$
$440$	−1.00000	−0.0476731
$441$	−3.00000	−0.142857
$442$	4.00000	0.190261
$443$	30.0000	1.42534	0.712672	−	0.701498i	$-0.247485\pi$
$443$	30.0000	1.42534	0.712672	+	0.701498i	$0.247485\pi$
$444$	−2.00000	−0.0949158
$445$	2.00000	0.0948091
$446$	8.00000	0.378811
$447$	−22.0000	−1.04056
$448$	−2.00000	−0.0944911
$449$	−16.0000	−0.755087	−0.377543	−	0.925992i	$-0.623231\pi$
$449$	−16.0000	−0.755087	−0.377543	+	0.925992i	$0.623231\pi$
$450$	1.00000	0.0471405
$451$	6.00000	0.282529
$452$	4.00000	0.188144
$453$	0	0
$454$	20.0000	0.938647
$455$	8.00000	0.375046
$456$	2.00000	0.0936586
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	−14.0000	−0.654177
$459$	−1.00000	−0.0466760
$460$	6.00000	0.279751
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	2.00000	0.0930484
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	6.00000	0.278543
$465$	−8.00000	−0.370991
$466$	−18.0000	−0.833834
$467$	−18.0000	−0.832941	−0.416470	−	0.909149i	$-0.636733\pi$
$467$	−18.0000	−0.832941	−0.416470	+	0.909149i	$0.636733\pi$
$468$	4.00000	0.184900
$469$	−24.0000	−1.10822
$470$	4.00000	0.184506
$471$	−2.00000	−0.0921551
$472$	−2.00000	−0.0920575
$473$	6.00000	0.275880
$474$	10.0000	0.459315
$475$	−2.00000	−0.0917663
$476$	−2.00000	−0.0916698
$477$	4.00000	0.183147
$478$	16.0000	0.731823
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	1.00000	0.0456435
$481$	8.00000	0.364769
$482$	28.0000	1.27537
$483$	−12.0000	−0.546019
$484$	1.00000	0.0454545
$485$	14.0000	0.635707
$486$	−1.00000	−0.0453609
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	14.0000	0.633750
$489$	−12.0000	−0.542659
$490$	3.00000	0.135526
$491$	−40.0000	−1.80517	−0.902587	−	0.430507i	$-0.858335\pi$
$491$	−40.0000	−1.80517	−0.902587	+	0.430507i	$0.858335\pi$
$492$	−6.00000	−0.270501
$493$	6.00000	0.270226
$494$	−8.00000	−0.359937
$495$	−1.00000	−0.0449467
$496$	−8.00000	−0.359211
$497$	20.0000	0.897123
$498$	−12.0000	−0.537733
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	−1.00000	−0.0447214
$501$	−16.0000	−0.714827
$502$	−6.00000	−0.267793
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	−2.00000	−0.0890871
$505$	−10.0000	−0.444994
$506$	−6.00000	−0.266733
$507$	−3.00000	−0.133235
$508$	20.0000	0.887357
$509$	−32.0000	−1.41838	−0.709188	−	0.705020i	$-0.750938\pi$
$509$	−32.0000	−1.41838	−0.709188	+	0.705020i	$0.750938\pi$
$510$	1.00000	0.0442807
$511$	0	0
$512$	1.00000	0.0441942
$513$	2.00000	0.0883022
$514$	10.0000	0.441081
$515$	8.00000	0.352522
$516$	−6.00000	−0.264135
$517$	−4.00000	−0.175920
$518$	−4.00000	−0.175750
$519$	−18.0000	−0.790112
$520$	−4.00000	−0.175412
$521$	−28.0000	−1.22670	−0.613351	−	0.789810i	$-0.710179\pi$
$521$	−28.0000	−1.22670	−0.613351	+	0.789810i	$0.710179\pi$
$522$	6.00000	0.262613
$523$	−30.0000	−1.31181	−0.655904	−	0.754844i	$-0.727712\pi$
$523$	−30.0000	−1.31181	−0.655904	+	0.754844i	$0.727712\pi$
$524$	−12.0000	−0.524222
$525$	2.00000	0.0872872
$526$	32.0000	1.39527
$527$	−8.00000	−0.348485
$528$	−1.00000	−0.0435194
$529$	13.0000	0.565217
$530$	−4.00000	−0.173749
$531$	−2.00000	−0.0867926
$532$	4.00000	0.173422
$533$	24.0000	1.03956
$534$	2.00000	0.0865485
$535$	−12.0000	−0.518805
$536$	12.0000	0.518321
$537$	−2.00000	−0.0863064
$538$	26.0000	1.12094
$539$	−3.00000	−0.129219
$540$	1.00000	0.0430331
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	12.0000	0.515444
$543$	−10.0000	−0.429141
$544$	1.00000	0.0428746
$545$	−10.0000	−0.428353
$546$	8.00000	0.342368
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	2.00000	0.0854358
$549$	14.0000	0.597505
$550$	1.00000	0.0426401
$551$	−12.0000	−0.511217
$552$	6.00000	0.255377
$553$	20.0000	0.850487
$554$	18.0000	0.764747
$555$	2.00000	0.0848953
$556$	−16.0000	−0.678551
$557$	−42.0000	−1.77960	−0.889799	−	0.456354i	$-0.849155\pi$
$557$	−42.0000	−1.77960	−0.889799	+	0.456354i	$0.849155\pi$
$558$	−8.00000	−0.338667
$559$	24.0000	1.01509
$560$	2.00000	0.0845154
$561$	−1.00000	−0.0422200
$562$	−26.0000	−1.09674
$563$	44.0000	1.85438	0.927189	−	0.374593i	$-0.122217\pi$
$563$	44.0000	1.85438	0.927189	+	0.374593i	$0.122217\pi$
$564$	4.00000	0.168430
$565$	−4.00000	−0.168281
$566$	0	0
$567$	−2.00000	−0.0839921
$568$	−10.0000	−0.419591
$569$	−22.0000	−0.922288	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000	−0.922288	−0.461144	+	0.887325i	$0.652561\pi$
$570$	−2.00000	−0.0837708
$571$	24.0000	1.00437	0.502184	−	0.864761i	$-0.332530\pi$
$571$	24.0000	1.00437	0.502184	+	0.864761i	$0.332530\pi$
$572$	4.00000	0.167248
$573$	0	0
$574$	−12.0000	−0.500870
$575$	−6.00000	−0.250217
$576$	1.00000	0.0416667
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	1.00000	0.0415945
$579$	−4.00000	−0.166234
$580$	−6.00000	−0.249136
$581$	−24.0000	−0.995688
$582$	14.0000	0.580319
$583$	4.00000	0.165663
$584$	0	0
$585$	−4.00000	−0.165380
$586$	6.00000	0.247858
$587$	−18.0000	−0.742940	−0.371470	−	0.928445i	$-0.621146\pi$
$587$	−18.0000	−0.742940	−0.371470	+	0.928445i	$0.621146\pi$
$588$	3.00000	0.123718
$589$	16.0000	0.659269
$590$	2.00000	0.0823387
$591$	10.0000	0.411345
$592$	2.00000	0.0821995
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	−1.00000	−0.0410305
$595$	2.00000	0.0819920
$596$	22.0000	0.901155
$597$	−16.0000	−0.654836
$598$	−24.0000	−0.981433
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	−1.00000	−0.0408248
$601$	−40.0000	−1.63163	−0.815817	−	0.578310i	$-0.803712\pi$
$601$	−40.0000	−1.63163	−0.815817	+	0.578310i	$0.803712\pi$
$602$	−12.0000	−0.489083
$603$	12.0000	0.488678
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	−10.0000	−0.406222
$607$	−10.0000	−0.405887	−0.202944	−	0.979190i	$-0.565051\pi$
$607$	−10.0000	−0.405887	−0.202944	+	0.979190i	$0.565051\pi$
$608$	−2.00000	−0.0811107
$609$	12.0000	0.486265
$610$	−14.0000	−0.566843
$611$	−16.0000	−0.647291
$612$	1.00000	0.0404226
$613$	−20.0000	−0.807792	−0.403896	−	0.914805i	$-0.632344\pi$
$613$	−20.0000	−0.807792	−0.403896	+	0.914805i	$0.632344\pi$
$614$	−22.0000	−0.887848
$615$	6.00000	0.241943
$616$	−2.00000	−0.0805823
$617$	8.00000	0.322068	0.161034	−	0.986949i	$-0.448517\pi$
$617$	8.00000	0.322068	0.161034	+	0.986949i	$0.448517\pi$
$618$	8.00000	0.321807
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	8.00000	0.321288
$621$	6.00000	0.240772
$622$	−18.0000	−0.721734
$623$	4.00000	0.160257
$624$	−4.00000	−0.160128
$625$	1.00000	0.0400000
$626$	−14.0000	−0.559553
$627$	2.00000	0.0798723
$628$	2.00000	0.0798087
$629$	2.00000	0.0797452
$630$	2.00000	0.0796819
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	−10.0000	−0.397779
$633$	−16.0000	−0.635943
$634$	−2.00000	−0.0794301
$635$	−20.0000	−0.793676
$636$	−4.00000	−0.158610
$637$	−12.0000	−0.475457
$638$	6.00000	0.237542
$639$	−10.0000	−0.395594
$640$	−1.00000	−0.0395285
$641$	12.0000	0.473972	0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000	0.473972	0.236986	+	0.971513i	$0.423841\pi$
$642$	−12.0000	−0.473602
$643$	36.0000	1.41970	0.709851	−	0.704352i	$-0.248762\pi$
$643$	36.0000	1.41970	0.709851	+	0.704352i	$0.248762\pi$
$644$	12.0000	0.472866
$645$	6.00000	0.236250
$646$	−2.00000	−0.0786889
$647$	−4.00000	−0.157256	−0.0786281	−	0.996904i	$-0.525054\pi$
$647$	−4.00000	−0.157256	−0.0786281	+	0.996904i	$0.525054\pi$
$648$	1.00000	0.0392837
$649$	−2.00000	−0.0785069
$650$	4.00000	0.156893
$651$	−16.0000	−0.627089
$652$	12.0000	0.469956
$653$	34.0000	1.33052	0.665261	−	0.746611i	$-0.268320\pi$
$653$	34.0000	1.33052	0.665261	+	0.746611i	$0.268320\pi$
$654$	−10.0000	−0.391031
$655$	12.0000	0.468879
$656$	6.00000	0.234261
$657$	0	0
$658$	8.00000	0.311872
$659$	−16.0000	−0.623272	−0.311636	−	0.950202i	$-0.600877\pi$
$659$	−16.0000	−0.623272	−0.311636	+	0.950202i	$0.600877\pi$
$660$	1.00000	0.0389249
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	12.0000	0.466393
$663$	−4.00000	−0.155347
$664$	12.0000	0.465690
$665$	−4.00000	−0.155113
$666$	2.00000	0.0774984
$667$	−36.0000	−1.39393
$668$	16.0000	0.619059
$669$	−8.00000	−0.309298
$670$	−12.0000	−0.463600
$671$	14.0000	0.540464
$672$	2.00000	0.0771517
$673$	36.0000	1.38770	0.693849	−	0.720121i	$-0.255914\pi$
$673$	36.0000	1.38770	0.693849	+	0.720121i	$0.255914\pi$
$674$	−32.0000	−1.23259
$675$	−1.00000	−0.0384900
$676$	3.00000	0.115385
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	−4.00000	−0.153619
$679$	28.0000	1.07454
$680$	−1.00000	−0.0383482
$681$	−20.0000	−0.766402
$682$	−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	$0.573736\pi$
$684$	−2.00000	−0.0764719
$685$	−2.00000	−0.0764161
$686$	20.0000	0.763604
$687$	14.0000	0.534133
$688$	6.00000	0.228748
$689$	16.0000	0.609551
$690$	−6.00000	−0.228416
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	18.0000	0.684257
$693$	−2.00000	−0.0759737
$694$	−4.00000	−0.151838
$695$	16.0000	0.606915
$696$	−6.00000	−0.227429
$697$	6.00000	0.227266
$698$	−12.0000	−0.454207
$699$	18.0000	0.680823
$700$	−2.00000	−0.0755929
$701$	38.0000	1.43524	0.717620	−	0.696435i	$-0.245231\pi$
$701$	38.0000	1.43524	0.717620	+	0.696435i	$0.245231\pi$
$702$	−4.00000	−0.150970
$703$	−4.00000	−0.150863
$704$	1.00000	0.0376889
$705$	−4.00000	−0.150649
$706$	34.0000	1.27961
$707$	−20.0000	−0.752177
$708$	2.00000	0.0751646
$709$	−34.0000	−1.27690	−0.638448	−	0.769665i	$-0.720423\pi$
$709$	−34.0000	−1.27690	−0.638448	+	0.769665i	$0.720423\pi$
$710$	10.0000	0.375293
$711$	−10.0000	−0.375029
$712$	−2.00000	−0.0749532
$713$	48.0000	1.79761
$714$	2.00000	0.0748481
$715$	−4.00000	−0.149592
$716$	2.00000	0.0747435
$717$	−16.0000	−0.597531
$718$	24.0000	0.895672
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	−1.00000	−0.0372678
$721$	16.0000	0.595871
$722$	−15.0000	−0.558242
$723$	−28.0000	−1.04133
$724$	10.0000	0.371647
$725$	6.00000	0.222834
$726$	−1.00000	−0.0371135
$727$	−24.0000	−0.890111	−0.445055	−	0.895503i	$-0.646816\pi$
$727$	−24.0000	−0.890111	−0.445055	+	0.895503i	$0.646816\pi$
$728$	−8.00000	−0.296500
$729$	1.00000	0.0370370
$730$	0	0
$731$	6.00000	0.221918
$732$	−14.0000	−0.517455
$733$	−4.00000	−0.147743	−0.0738717	−	0.997268i	$-0.523536\pi$
$733$	−4.00000	−0.147743	−0.0738717	+	0.997268i	$0.523536\pi$
$734$	−8.00000	−0.295285
$735$	−3.00000	−0.110657
$736$	−6.00000	−0.221163
$737$	12.0000	0.442026
$738$	6.00000	0.220863
$739$	14.0000	0.514998	0.257499	−	0.966279i	$-0.417102\pi$
$739$	14.0000	0.514998	0.257499	+	0.966279i	$0.417102\pi$
$740$	−2.00000	−0.0735215
$741$	8.00000	0.293887
$742$	−8.00000	−0.293689
$743$	28.0000	1.02722	0.513610	−	0.858024i	$-0.328308\pi$
$743$	28.0000	1.02722	0.513610	+	0.858024i	$0.328308\pi$
$744$	8.00000	0.293294
$745$	−22.0000	−0.806018
$746$	−12.0000	−0.439351
$747$	12.0000	0.439057
$748$	1.00000	0.0365636
$749$	−24.0000	−0.876941
$750$	1.00000	0.0365148
$751$	−20.0000	−0.729810	−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000	−0.729810	−0.364905	+	0.931045i	$0.618899\pi$
$752$	−4.00000	−0.145865
$753$	6.00000	0.218652
$754$	24.0000	0.874028
$755$	0	0
$756$	2.00000	0.0727393
$757$	−18.0000	−0.654221	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000	−0.654221	−0.327111	+	0.944986i	$0.606075\pi$
$758$	−4.00000	−0.145287
$759$	6.00000	0.217786
$760$	2.00000	0.0725476
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	−20.0000	−0.724524
$763$	−20.0000	−0.724049
$764$	0	0
$765$	−1.00000	−0.0361551
$766$	32.0000	1.15621
$767$	−8.00000	−0.288863
$768$	−1.00000	−0.0360844
$769$	−38.0000	−1.37032	−0.685158	−	0.728395i	$-0.740267\pi$
$769$	−38.0000	−1.37032	−0.685158	+	0.728395i	$0.740267\pi$
$770$	2.00000	0.0720750
$771$	−10.0000	−0.360141
$772$	4.00000	0.143963
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	6.00000	0.215666
$775$	−8.00000	−0.287368
$776$	−14.0000	−0.502571
$777$	4.00000	0.143499
$778$	0	0
$779$	−12.0000	−0.429945
$780$	4.00000	0.143223
$781$	−10.0000	−0.357828
$782$	−6.00000	−0.214560
$783$	−6.00000	−0.214423
$784$	−3.00000	−0.107143
$785$	−2.00000	−0.0713831
$786$	12.0000	0.428026
$787$	12.0000	0.427754	0.213877	−	0.976861i	$-0.431391\pi$
$787$	12.0000	0.427754	0.213877	+	0.976861i	$0.431391\pi$
$788$	−10.0000	−0.356235
$789$	−32.0000	−1.13923
$790$	10.0000	0.355784
$791$	−8.00000	−0.284447
$792$	1.00000	0.0355335
$793$	56.0000	1.98862
$794$	−34.0000	−1.20661
$795$	4.00000	0.141865
$796$	16.0000	0.567105
$797$	12.0000	0.425062	0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000	0.425062	0.212531	+	0.977154i	$0.431829\pi$
$798$	−4.00000	−0.141598
$799$	−4.00000	−0.141510
$800$	1.00000	0.0353553
$801$	−2.00000	−0.0706665
$802$	−20.0000	−0.706225
$803$	0	0
$804$	−12.0000	−0.423207
$805$	−12.0000	−0.422944
$806$	−32.0000	−1.12715
$807$	−26.0000	−0.915243
$808$	10.0000	0.351799
$809$	34.0000	1.19538	0.597688	−	0.801729i	$-0.296086\pi$
$809$	34.0000	1.19538	0.597688	+	0.801729i	$0.296086\pi$
$810$	−1.00000	−0.0351364
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	−12.0000	−0.421117
$813$	−12.0000	−0.420858
$814$	2.00000	0.0701000
$815$	−12.0000	−0.420342
$816$	−1.00000	−0.0350070
$817$	−12.0000	−0.419827
$818$	6.00000	0.209785
$819$	−8.00000	−0.279543
$820$	−6.00000	−0.209529
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	−2.00000	−0.0697580
$823$	−20.0000	−0.697156	−0.348578	−	0.937280i	$-0.613335\pi$
$823$	−20.0000	−0.697156	−0.348578	+	0.937280i	$0.613335\pi$
$824$	−8.00000	−0.278693
$825$	−1.00000	−0.0348155
$826$	4.00000	0.139178
$827$	28.0000	0.973655	0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000	0.973655	0.486828	+	0.873498i	$0.338154\pi$
$828$	−6.00000	−0.208514
$829$	−42.0000	−1.45872	−0.729360	−	0.684130i	$-0.760182\pi$
$829$	−42.0000	−1.45872	−0.729360	+	0.684130i	$0.760182\pi$
$830$	−12.0000	−0.416526
$831$	−18.0000	−0.624413
$832$	4.00000	0.138675
$833$	−3.00000	−0.103944
$834$	16.0000	0.554035
$835$	−16.0000	−0.553703
$836$	−2.00000	−0.0691714
$837$	8.00000	0.276520
$838$	−20.0000	−0.690889
$839$	−18.0000	−0.621429	−0.310715	−	0.950503i	$-0.600568\pi$
$839$	−18.0000	−0.621429	−0.310715	+	0.950503i	$0.600568\pi$
$840$	−2.00000	−0.0690066
$841$	7.00000	0.241379
$842$	−18.0000	−0.620321
$843$	26.0000	0.895488
$844$	16.0000	0.550743
$845$	−3.00000	−0.103203
$846$	−4.00000	−0.137523
$847$	−2.00000	−0.0687208
$848$	4.00000	0.137361
$849$	0	0
$850$	1.00000	0.0342997
$851$	−12.0000	−0.411355
$852$	10.0000	0.342594
$853$	−14.0000	−0.479351	−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000	−0.479351	−0.239675	+	0.970853i	$0.577041\pi$
$854$	−28.0000	−0.958140
$855$	2.00000	0.0683986
$856$	12.0000	0.410152
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	−4.00000	−0.136558
$859$	−44.0000	−1.50126	−0.750630	−	0.660722i	$-0.770250\pi$
$859$	−44.0000	−1.50126	−0.750630	+	0.660722i	$0.770250\pi$
$860$	−6.00000	−0.204598
$861$	12.0000	0.408959
$862$	−12.0000	−0.408722
$863$	−8.00000	−0.272323	−0.136162	−	0.990687i	$-0.543477\pi$
$863$	−8.00000	−0.272323	−0.136162	+	0.990687i	$0.543477\pi$
$864$	−1.00000	−0.0340207
$865$	−18.0000	−0.612018
$866$	14.0000	0.475739
$867$	−1.00000	−0.0339618
$868$	16.0000	0.543075
$869$	−10.0000	−0.339227
$870$	6.00000	0.203419
$871$	48.0000	1.62642
$872$	10.0000	0.338643
$873$	−14.0000	−0.473828
$874$	12.0000	0.405906
$875$	2.00000	0.0676123
$876$	0	0
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	−26.0000	−0.877457
$879$	−6.00000	−0.202375
$880$	−1.00000	−0.0337100
$881$	32.0000	1.07811	0.539054	−	0.842271i	$-0.318782\pi$
$881$	32.0000	1.07811	0.539054	+	0.842271i	$0.318782\pi$
$882$	−3.00000	−0.101015
$883$	56.0000	1.88455	0.942275	−	0.334840i	$-0.108682\pi$
$883$	56.0000	1.88455	0.942275	+	0.334840i	$0.108682\pi$
$884$	4.00000	0.134535
$885$	−2.00000	−0.0672293
$886$	30.0000	1.00787
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	−2.00000	−0.0671156
$889$	−40.0000	−1.34156
$890$	2.00000	0.0670402
$891$	1.00000	0.0335013
$892$	8.00000	0.267860
$893$	8.00000	0.267710
$894$	−22.0000	−0.735790
$895$	−2.00000	−0.0668526
$896$	−2.00000	−0.0668153
$897$	24.0000	0.801337
$898$	−16.0000	−0.533927
$899$	−48.0000	−1.60089
$900$	1.00000	0.0333333
$901$	4.00000	0.133259
$902$	6.00000	0.199778
$903$	12.0000	0.399335
$904$	4.00000	0.133038
$905$	−10.0000	−0.332411
$906$	0	0
$907$	36.0000	1.19536	0.597680	−	0.801735i	$-0.296089\pi$
$907$	36.0000	1.19536	0.597680	+	0.801735i	$0.296089\pi$
$908$	20.0000	0.663723
$909$	10.0000	0.331679
$910$	8.00000	0.265197
$911$	−42.0000	−1.39152	−0.695761	−	0.718273i	$-0.744933\pi$
$911$	−42.0000	−1.39152	−0.695761	+	0.718273i	$0.744933\pi$
$912$	2.00000	0.0662266
$913$	12.0000	0.397142
$914$	−2.00000	−0.0661541
$915$	14.0000	0.462826
$916$	−14.0000	−0.462573
$917$	24.0000	0.792550
$918$	−1.00000	−0.0330049
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	6.00000	0.197814
$921$	22.0000	0.724925
$922$	30.0000	0.987997
$923$	−40.0000	−1.31662
$924$	2.00000	0.0657952
$925$	2.00000	0.0657596
$926$	16.0000	0.525793
$927$	−8.00000	−0.262754
$928$	6.00000	0.196960
$929$	40.0000	1.31236	0.656179	−	0.754606i	$-0.272172\pi$
$929$	40.0000	1.31236	0.656179	+	0.754606i	$0.272172\pi$
$930$	−8.00000	−0.262330
$931$	6.00000	0.196642
$932$	−18.0000	−0.589610
$933$	18.0000	0.589294
$934$	−18.0000	−0.588978
$935$	−1.00000	−0.0327035
$936$	4.00000	0.130744
$937$	−42.0000	−1.37208	−0.686040	−	0.727564i	$-0.740653\pi$
$937$	−42.0000	−1.37208	−0.686040	+	0.727564i	$0.740653\pi$
$938$	−24.0000	−0.783628
$939$	14.0000	0.456873
$940$	4.00000	0.130466
$941$	−14.0000	−0.456387	−0.228193	−	0.973616i	$-0.573282\pi$
$941$	−14.0000	−0.456387	−0.228193	+	0.973616i	$0.573282\pi$
$942$	−2.00000	−0.0651635
$943$	−36.0000	−1.17232
$944$	−2.00000	−0.0650945
$945$	−2.00000	−0.0650600
$946$	6.00000	0.195077
$947$	−8.00000	−0.259965	−0.129983	−	0.991516i	$-0.541492\pi$
$947$	−8.00000	−0.259965	−0.129983	+	0.991516i	$0.541492\pi$
$948$	10.0000	0.324785
$949$	0	0
$950$	−2.00000	−0.0648886
$951$	2.00000	0.0648544
$952$	−2.00000	−0.0648204
$953$	−50.0000	−1.61966	−0.809829	−	0.586665i	$-0.800440\pi$
$953$	−50.0000	−1.61966	−0.809829	+	0.586665i	$0.800440\pi$
$954$	4.00000	0.129505
$955$	0	0
$956$	16.0000	0.517477
$957$	−6.00000	−0.193952
$958$	0	0
$959$	−4.00000	−0.129167
$960$	1.00000	0.0322749
$961$	33.0000	1.06452
$962$	8.00000	0.257930
$963$	12.0000	0.386695
$964$	28.0000	0.901819
$965$	−4.00000	−0.128765
$966$	−12.0000	−0.386094
$967$	−4.00000	−0.128631	−0.0643157	−	0.997930i	$-0.520486\pi$
$967$	−4.00000	−0.128631	−0.0643157	+	0.997930i	$0.520486\pi$
$968$	1.00000	0.0321412
$969$	2.00000	0.0642493
$970$	14.0000	0.449513
$971$	54.0000	1.73294	0.866471	−	0.499227i	$-0.166383\pi$
$971$	54.0000	1.73294	0.866471	+	0.499227i	$0.166383\pi$
$972$	−1.00000	−0.0320750
$973$	32.0000	1.02587
$974$	16.0000	0.512673
$975$	−4.00000	−0.128103
$976$	14.0000	0.448129
$977$	−14.0000	−0.447900	−0.223950	−	0.974601i	$-0.571895\pi$
$977$	−14.0000	−0.447900	−0.223950	+	0.974601i	$0.571895\pi$
$978$	−12.0000	−0.383718
$979$	−2.00000	−0.0639203
$980$	3.00000	0.0958315
$981$	10.0000	0.319275
$982$	−40.0000	−1.27645
$983$	−30.0000	−0.956851	−0.478426	−	0.878128i	$-0.658792\pi$
$983$	−30.0000	−0.956851	−0.478426	+	0.878128i	$0.658792\pi$
$984$	−6.00000	−0.191273
$985$	10.0000	0.318626
$986$	6.00000	0.191079
$987$	−8.00000	−0.254643
$988$	−8.00000	−0.254514
$989$	−36.0000	−1.14473
$990$	−1.00000	−0.0317821
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	−8.00000	−0.254000
$993$	−12.0000	−0.380808
$994$	20.0000	0.634361
$995$	−16.0000	−0.507234
$996$	−12.0000	−0.380235
$997$	22.0000	0.696747	0.348373	−	0.937356i	$-0.386734\pi$
$997$	22.0000	0.696747	0.348373	+	0.937356i	$0.386734\pi$
$998$	20.0000	0.633089
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5610.2.a.x.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5610.2.a.x.1.1	✓	1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5610.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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