LMFDB - Embedded newform 5610.2.a.a.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)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} +2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} -8.00000 q^{19} -1.00000 q^{20} +2.00000 q^{21} +1.00000 q^{22} -4.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} -2.00000 q^{28} -6.00000 q^{29} -1.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} -1.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} +8.00000 q^{38} +1.00000 q^{40} -2.00000 q^{42} +10.0000 q^{43} -1.00000 q^{44} -1.00000 q^{45} +4.00000 q^{46} +8.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} -1.00000 q^{51} -6.00000 q^{53} +1.00000 q^{54} +1.00000 q^{55} +2.00000 q^{56} +8.00000 q^{57} +6.00000 q^{58} +6.00000 q^{59} +1.00000 q^{60} -2.00000 q^{61} +4.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -1.00000 q^{66} -10.0000 q^{67} +1.00000 q^{68} +4.00000 q^{69} -2.00000 q^{70} +6.00000 q^{71} -1.00000 q^{72} -8.00000 q^{73} +2.00000 q^{74} -1.00000 q^{75} -8.00000 q^{76} +2.00000 q^{77} -4.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +16.0000 q^{83} +2.00000 q^{84} -1.00000 q^{85} -10.0000 q^{86} +6.00000 q^{87} +1.00000 q^{88} -18.0000 q^{89} +1.00000 q^{90} -4.00000 q^{92} +4.00000 q^{93} -8.00000 q^{94} +8.00000 q^{95} +1.00000 q^{96} -8.00000 q^{97} +3.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	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
$19$	−8.00000	−1.83533	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	−1.00000	−0.223607
$21$	2.00000	0.436436
$22$	1.00000	0.213201
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	−2.00000	−0.377964
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	−1.00000	−0.182574
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−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	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	8.00000	1.29777
$39$	0	0
$40$	1.00000	0.158114
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	−2.00000	−0.308607
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	−1.00000	−0.150756
$45$	−1.00000	−0.149071
$46$	4.00000	0.589768
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	−1.00000	−0.144338
$49$	−3.00000	−0.428571
$50$	−1.00000	−0.141421
$51$	−1.00000	−0.140028
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	1.00000	0.136083
$55$	1.00000	0.134840
$56$	2.00000	0.267261
$57$	8.00000	1.05963
$58$	6.00000	0.787839
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	1.00000	0.129099
$61$	−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
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.00000	−0.123091
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	1.00000	0.121268
$69$	4.00000	0.481543
$70$	−2.00000	−0.239046
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	−1.00000	−0.117851
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	2.00000	0.232495
$75$	−1.00000	−0.115470
$76$	−8.00000	−0.917663
$77$	2.00000	0.227921
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	0	0
$83$	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
$85$	−1.00000	−0.108465
$86$	−10.0000	−1.07833
$87$	6.00000	0.643268
$88$	1.00000	0.106600
$89$	−18.0000	−1.90800	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000	−1.90800	−0.953998	+	0.299813i	$0.903076\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	−4.00000	−0.417029
$93$	4.00000	0.414781
$94$	−8.00000	−0.825137
$95$	8.00000	0.820783
$96$	1.00000	0.102062
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	3.00000	0.303046
$99$	−1.00000	−0.100504
$100$	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	$0.630301\pi$
$102$	1.00000	0.0990148
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	6.00000	0.582772
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	−1.00000	−0.0962250
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	−1.00000	−0.0953463
$111$	2.00000	0.189832
$112$	−2.00000	−0.188982
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	−8.00000	−0.749269
$115$	4.00000	0.373002
$116$	−6.00000	−0.557086
$117$	0	0
$118$	−6.00000	−0.552345
$119$	−2.00000	−0.183340
$120$	−1.00000	−0.0912871
$121$	1.00000	0.0909091
$122$	2.00000	0.181071
$123$	0	0
$124$	−4.00000	−0.359211
$125$	−1.00000	−0.0894427
$126$	2.00000	0.178174
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	−1.00000	−0.0883883
$129$	−10.0000	−0.880451
$130$	0	0
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	1.00000	0.0870388
$133$	16.0000	1.38738
$134$	10.0000	0.863868
$135$	1.00000	0.0860663
$136$	−1.00000	−0.0857493
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	−4.00000	−0.340503
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	2.00000	0.169031
$141$	−8.00000	−0.673722
$142$	−6.00000	−0.503509
$143$	0	0
$144$	1.00000	0.0833333
$145$	6.00000	0.498273
$146$	8.00000	0.662085
$147$	3.00000	0.247436
$148$	−2.00000	−0.164399
$149$	−8.00000	−0.655386	−0.327693	−	0.944784i	$-0.606271\pi$
$149$	−8.00000	−0.655386	−0.327693	+	0.944784i	$0.606271\pi$
$150$	1.00000	0.0816497
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	8.00000	0.648886
$153$	1.00000	0.0808452
$154$	−2.00000	−0.161165
$155$	4.00000	0.321288
$156$	0	0
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	4.00000	0.318223
$159$	6.00000	0.475831
$160$	1.00000	0.0790569
$161$	8.00000	0.630488
$162$	−1.00000	−0.0785674
$163$	8.00000	0.626608	0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000	0.626608	0.313304	+	0.949653i	$0.398564\pi$
$164$	0	0
$165$	−1.00000	−0.0778499
$166$	−16.0000	−1.24184
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	−2.00000	−0.154303
$169$	−13.0000	−1.00000
$170$	1.00000	0.0766965
$171$	−8.00000	−0.611775
$172$	10.0000	0.762493
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	−6.00000	−0.454859
$175$	−2.00000	−0.151186
$176$	−1.00000	−0.0753778
$177$	−6.00000	−0.450988
$178$	18.0000	1.34916
$179$	26.0000	1.94333	0.971666	−	0.236360i	$-0.0759544\pi$
$179$	26.0000	1.94333	0.971666	+	0.236360i	$0.0759544\pi$
$180$	−1.00000	−0.0745356
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	4.00000	0.294884
$185$	2.00000	0.147043
$186$	−4.00000	−0.293294
$187$	−1.00000	−0.0731272
$188$	8.00000	0.583460
$189$	2.00000	0.145479
$190$	−8.00000	−0.580381
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	−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	$0.454015\pi$
$194$	8.00000	0.574367
$195$	0	0
$196$	−3.00000	−0.214286
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	1.00000	0.0710669
$199$	−12.0000	−0.850657	−0.425329	−	0.905039i	$-0.639842\pi$
$199$	−12.0000	−0.850657	−0.425329	+	0.905039i	$0.639842\pi$
$200$	−1.00000	−0.0707107
$201$	10.0000	0.705346
$202$	8.00000	0.562878
$203$	12.0000	0.842235
$204$	−1.00000	−0.0700140
$205$	0	0
$206$	4.00000	0.278693
$207$	−4.00000	−0.278019
$208$	0	0
$209$	8.00000	0.553372
$210$	2.00000	0.138013
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−6.00000	−0.412082
$213$	−6.00000	−0.411113
$214$	−4.00000	−0.273434
$215$	−10.0000	−0.681994
$216$	1.00000	0.0680414
$217$	8.00000	0.543075
$218$	−2.00000	−0.135457
$219$	8.00000	0.540590
$220$	1.00000	0.0674200
$221$	0	0
$222$	−2.00000	−0.134231
$223$	−12.0000	−0.803579	−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000	−0.803579	−0.401790	+	0.915732i	$0.631612\pi$
$224$	2.00000	0.133631
$225$	1.00000	0.0666667
$226$	6.00000	0.399114
$227$	28.0000	1.85843	0.929213	−	0.369546i	$-0.120487\pi$
$227$	28.0000	1.85843	0.929213	+	0.369546i	$0.120487\pi$
$228$	8.00000	0.529813
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	−4.00000	−0.263752
$231$	−2.00000	−0.131590
$232$	6.00000	0.393919
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	6.00000	0.390567
$237$	4.00000	0.259828
$238$	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
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	−1.00000	−0.0642824
$243$	−1.00000	−0.0641500
$244$	−2.00000	−0.128037
$245$	3.00000	0.191663
$246$	0	0
$247$	0	0
$248$	4.00000	0.254000
$249$	−16.0000	−1.01396
$250$	1.00000	0.0632456
$251$	−2.00000	−0.126239	−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000	−0.126239	−0.0631194	+	0.998006i	$0.520105\pi$
$252$	−2.00000	−0.125988
$253$	4.00000	0.251478
$254$	−12.0000	−0.752947
$255$	1.00000	0.0626224
$256$	1.00000	0.0625000
$257$	22.0000	1.37232	0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000	1.37232	0.686161	+	0.727450i	$0.259294\pi$
$258$	10.0000	0.622573
$259$	4.00000	0.248548
$260$	0	0
$261$	−6.00000	−0.371391
$262$	8.00000	0.494242
$263$	−32.0000	−1.97320	−0.986602	−	0.163144i	$-0.947836\pi$
$263$	−32.0000	−1.97320	−0.986602	+	0.163144i	$0.947836\pi$
$264$	−1.00000	−0.0615457
$265$	6.00000	0.368577
$266$	−16.0000	−0.981023
$267$	18.0000	1.10158
$268$	−10.0000	−0.610847
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	−1.00000	−0.0608581
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	6.00000	0.362473
$275$	−1.00000	−0.0603023
$276$	4.00000	0.240772
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	4.00000	0.239904
$279$	−4.00000	−0.239474
$280$	−2.00000	−0.119523
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	8.00000	0.476393
$283$	28.0000	1.66443	0.832214	−	0.554455i	$-0.187073\pi$
$283$	28.0000	1.66443	0.832214	+	0.554455i	$0.187073\pi$
$284$	6.00000	0.356034
$285$	−8.00000	−0.473879
$286$	0	0
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	1.00000	0.0588235
$290$	−6.00000	−0.352332
$291$	8.00000	0.468968
$292$	−8.00000	−0.468165
$293$	18.0000	1.05157	0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000	1.05157	0.525786	+	0.850617i	$0.323771\pi$
$294$	−3.00000	−0.174964
$295$	−6.00000	−0.349334
$296$	2.00000	0.116248
$297$	1.00000	0.0580259
$298$	8.00000	0.463428
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	−20.0000	−1.15278
$302$	−8.00000	−0.460348
$303$	8.00000	0.459588
$304$	−8.00000	−0.458831
$305$	2.00000	0.114520
$306$	−1.00000	−0.0571662
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	2.00000	0.113961
$309$	4.00000	0.227552
$310$	−4.00000	−0.227185
$311$	2.00000	0.113410	0.0567048	−	0.998391i	$-0.481941\pi$
$311$	2.00000	0.113410	0.0567048	+	0.998391i	$0.481941\pi$
$312$	0	0
$313$	−8.00000	−0.452187	−0.226093	−	0.974106i	$-0.572595\pi$
$313$	−8.00000	−0.452187	−0.226093	+	0.974106i	$0.572595\pi$
$314$	−4.00000	−0.225733
$315$	2.00000	0.112687
$316$	−4.00000	−0.225018
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	−6.00000	−0.336463
$319$	6.00000	0.335936
$320$	−1.00000	−0.0559017
$321$	−4.00000	−0.223258
$322$	−8.00000	−0.445823
$323$	−8.00000	−0.445132
$324$	1.00000	0.0555556
$325$	0	0
$326$	−8.00000	−0.443079
$327$	−2.00000	−0.110600
$328$	0	0
$329$	−16.0000	−0.882109
$330$	1.00000	0.0550482
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	16.0000	0.878114
$333$	−2.00000	−0.109599
$334$	−12.0000	−0.656611
$335$	10.0000	0.546358
$336$	2.00000	0.109109
$337$	−8.00000	−0.435788	−0.217894	−	0.975972i	$-0.569919\pi$
$337$	−8.00000	−0.435788	−0.217894	+	0.975972i	$0.569919\pi$
$338$	13.0000	0.707107
$339$	6.00000	0.325875
$340$	−1.00000	−0.0542326
$341$	4.00000	0.216612
$342$	8.00000	0.432590
$343$	20.0000	1.07990
$344$	−10.0000	−0.539164
$345$	−4.00000	−0.215353
$346$	−6.00000	−0.322562
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	6.00000	0.321634
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	2.00000	0.106904
$351$	0	0
$352$	1.00000	0.0533002
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	6.00000	0.318896
$355$	−6.00000	−0.318447
$356$	−18.0000	−0.953998
$357$	2.00000	0.105851
$358$	−26.0000	−1.37414
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	1.00000	0.0527046
$361$	45.0000	2.36842
$362$	22.0000	1.15629
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	8.00000	0.418739
$366$	−2.00000	−0.104542
$367$	−18.0000	−0.939592	−0.469796	−	0.882775i	$-0.655673\pi$
$367$	−18.0000	−0.939592	−0.469796	+	0.882775i	$0.655673\pi$
$368$	−4.00000	−0.208514
$369$	0	0
$370$	−2.00000	−0.103975
$371$	12.0000	0.623009
$372$	4.00000	0.207390
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	1.00000	0.0517088
$375$	1.00000	0.0516398
$376$	−8.00000	−0.412568
$377$	0	0
$378$	−2.00000	−0.102869
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	8.00000	0.410391
$381$	−12.0000	−0.614779
$382$	−12.0000	−0.613973
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	1.00000	0.0510310
$385$	−2.00000	−0.101929
$386$	−4.00000	−0.203595
$387$	10.0000	0.508329
$388$	−8.00000	−0.406138
$389$	−28.0000	−1.41966	−0.709828	−	0.704375i	$-0.751227\pi$
$389$	−28.0000	−1.41966	−0.709828	+	0.704375i	$0.751227\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	3.00000	0.151523
$393$	8.00000	0.403547
$394$	−22.0000	−1.10834
$395$	4.00000	0.201262
$396$	−1.00000	−0.0502519
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	12.0000	0.601506
$399$	−16.0000	−0.801002
$400$	1.00000	0.0500000
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	−10.0000	−0.498755
$403$	0	0
$404$	−8.00000	−0.398015
$405$	−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	$0.452607\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	−4.00000	−0.197066
$413$	−12.0000	−0.590481
$414$	4.00000	0.196589
$415$	−16.0000	−0.785409
$416$	0	0
$417$	4.00000	0.195881
$418$	−8.00000	−0.391293
$419$	16.0000	0.781651	0.390826	−	0.920465i	$-0.372190\pi$
$419$	16.0000	0.781651	0.390826	+	0.920465i	$0.372190\pi$
$420$	−2.00000	−0.0975900
$421$	30.0000	1.46211	0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000	1.46211	0.731055	+	0.682318i	$0.239028\pi$
$422$	4.00000	0.194717
$423$	8.00000	0.388973
$424$	6.00000	0.291386
$425$	1.00000	0.0485071
$426$	6.00000	0.290701
$427$	4.00000	0.193574
$428$	4.00000	0.193347
$429$	0	0
$430$	10.0000	0.482243
$431$	6.00000	0.289010	0.144505	−	0.989504i	$-0.453841\pi$
$431$	6.00000	0.289010	0.144505	+	0.989504i	$0.453841\pi$
$432$	−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	$0.390791\pi$
$434$	−8.00000	−0.384012
$435$	−6.00000	−0.287678
$436$	2.00000	0.0957826
$437$	32.0000	1.53077
$438$	−8.00000	−0.382255
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	−1.00000	−0.0476731
$441$	−3.00000	−0.142857
$442$	0	0
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	2.00000	0.0949158
$445$	18.0000	0.853282
$446$	12.0000	0.568216
$447$	8.00000	0.378387
$448$	−2.00000	−0.0944911
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	−1.00000	−0.0471405
$451$	0	0
$452$	−6.00000	−0.282216
$453$	−8.00000	−0.375873
$454$	−28.0000	−1.31411
$455$	0	0
$456$	−8.00000	−0.374634
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	2.00000	0.0934539
$459$	−1.00000	−0.0466760
$460$	4.00000	0.186501
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	2.00000	0.0930484
$463$	−20.0000	−0.929479	−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000	−0.929479	−0.464739	+	0.885448i	$0.653852\pi$
$464$	−6.00000	−0.278543
$465$	−4.00000	−0.185496
$466$	6.00000	0.277945
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	20.0000	0.923514
$470$	8.00000	0.369012
$471$	−4.00000	−0.184310
$472$	−6.00000	−0.276172
$473$	−10.0000	−0.459800
$474$	−4.00000	−0.183726
$475$	−8.00000	−0.367065
$476$	−2.00000	−0.0916698
$477$	−6.00000	−0.274721
$478$	−16.0000	−0.731823
$479$	−6.00000	−0.274147	−0.137073	−	0.990561i	$-0.543770\pi$
$479$	−6.00000	−0.274147	−0.137073	+	0.990561i	$0.543770\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	26.0000	1.18427
$483$	−8.00000	−0.364013
$484$	1.00000	0.0454545
$485$	8.00000	0.363261
$486$	1.00000	0.0453609
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	2.00000	0.0905357
$489$	−8.00000	−0.361773
$490$	−3.00000	−0.135526
$491$	22.0000	0.992846	0.496423	−	0.868081i	$-0.334646\pi$
$491$	22.0000	0.992846	0.496423	+	0.868081i	$0.334646\pi$
$492$	0	0
$493$	−6.00000	−0.270226
$494$	0	0
$495$	1.00000	0.0449467
$496$	−4.00000	−0.179605
$497$	−12.0000	−0.538274
$498$	16.0000	0.716977
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	−1.00000	−0.0447214
$501$	−12.0000	−0.536120
$502$	2.00000	0.0892644
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	2.00000	0.0890871
$505$	8.00000	0.355995
$506$	−4.00000	−0.177822
$507$	13.0000	0.577350
$508$	12.0000	0.532414
$509$	36.0000	1.59567	0.797836	−	0.602875i	$-0.205978\pi$
$509$	36.0000	1.59567	0.797836	+	0.602875i	$0.205978\pi$
$510$	−1.00000	−0.0442807
$511$	16.0000	0.707798
$512$	−1.00000	−0.0441942
$513$	8.00000	0.353209
$514$	−22.0000	−0.970378
$515$	4.00000	0.176261
$516$	−10.0000	−0.440225
$517$	−8.00000	−0.351840
$518$	−4.00000	−0.175750
$519$	−6.00000	−0.263371
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	6.00000	0.262613
$523$	−2.00000	−0.0874539	−0.0437269	−	0.999044i	$-0.513923\pi$
$523$	−2.00000	−0.0874539	−0.0437269	+	0.999044i	$0.513923\pi$
$524$	−8.00000	−0.349482
$525$	2.00000	0.0872872
$526$	32.0000	1.39527
$527$	−4.00000	−0.174243
$528$	1.00000	0.0435194
$529$	−7.00000	−0.304348
$530$	−6.00000	−0.260623
$531$	6.00000	0.260378
$532$	16.0000	0.693688
$533$	0	0
$534$	−18.0000	−0.778936
$535$	−4.00000	−0.172935
$536$	10.0000	0.431934
$537$	−26.0000	−1.12198
$538$	6.00000	0.258678
$539$	3.00000	0.129219
$540$	1.00000	0.0430331
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	0	0
$543$	22.0000	0.944110
$544$	−1.00000	−0.0428746
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	−6.00000	−0.256307
$549$	−2.00000	−0.0853579
$550$	1.00000	0.0426401
$551$	48.0000	2.04487
$552$	−4.00000	−0.170251
$553$	8.00000	0.340195
$554$	−2.00000	−0.0849719
$555$	−2.00000	−0.0848953
$556$	−4.00000	−0.169638
$557$	14.0000	0.593199	0.296600	−	0.955002i	$-0.404147\pi$
$557$	14.0000	0.593199	0.296600	+	0.955002i	$0.404147\pi$
$558$	4.00000	0.169334
$559$	0	0
$560$	2.00000	0.0845154
$561$	1.00000	0.0422200
$562$	−30.0000	−1.26547
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	−8.00000	−0.336861
$565$	6.00000	0.252422
$566$	−28.0000	−1.17693
$567$	−2.00000	−0.0839921
$568$	−6.00000	−0.251754
$569$	−14.0000	−0.586911	−0.293455	−	0.955973i	$-0.594805\pi$
$569$	−14.0000	−0.586911	−0.293455	+	0.955973i	$0.594805\pi$
$570$	8.00000	0.335083
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	−12.0000	−0.501307
$574$	0	0
$575$	−4.00000	−0.166812
$576$	1.00000	0.0416667
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	−1.00000	−0.0415945
$579$	−4.00000	−0.166234
$580$	6.00000	0.249136
$581$	−32.0000	−1.32758
$582$	−8.00000	−0.331611
$583$	6.00000	0.248495
$584$	8.00000	0.331042
$585$	0	0
$586$	−18.0000	−0.743573
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	3.00000	0.123718
$589$	32.0000	1.31854
$590$	6.00000	0.247016
$591$	−22.0000	−0.904959
$592$	−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
$596$	−8.00000	−0.327693
$597$	12.0000	0.491127
$598$	0	0
$599$	−28.0000	−1.14405	−0.572024	−	0.820237i	$-0.693842\pi$
$599$	−28.0000	−1.14405	−0.572024	+	0.820237i	$0.693842\pi$
$600$	1.00000	0.0408248
$601$	42.0000	1.71322	0.856608	−	0.515968i	$-0.172568\pi$
$601$	42.0000	1.71322	0.856608	+	0.515968i	$0.172568\pi$
$602$	20.0000	0.815139
$603$	−10.0000	−0.407231
$604$	8.00000	0.325515
$605$	−1.00000	−0.0406558
$606$	−8.00000	−0.324978
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	8.00000	0.324443
$609$	−12.0000	−0.486265
$610$	−2.00000	−0.0809776
$611$	0	0
$612$	1.00000	0.0404226
$613$	12.0000	0.484675	0.242338	−	0.970192i	$-0.422086\pi$
$613$	12.0000	0.484675	0.242338	+	0.970192i	$0.422086\pi$
$614$	−2.00000	−0.0807134
$615$	0	0
$616$	−2.00000	−0.0805823
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	−4.00000	−0.160904
$619$	−36.0000	−1.44696	−0.723481	−	0.690344i	$-0.757459\pi$
$619$	−36.0000	−1.44696	−0.723481	+	0.690344i	$0.757459\pi$
$620$	4.00000	0.160644
$621$	4.00000	0.160514
$622$	−2.00000	−0.0801927
$623$	36.0000	1.44231
$624$	0	0
$625$	1.00000	0.0400000
$626$	8.00000	0.319744
$627$	−8.00000	−0.319489
$628$	4.00000	0.159617
$629$	−2.00000	−0.0797452
$630$	−2.00000	−0.0796819
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	4.00000	0.159111
$633$	4.00000	0.158986
$634$	18.0000	0.714871
$635$	−12.0000	−0.476205
$636$	6.00000	0.237915
$637$	0	0
$638$	−6.00000	−0.237542
$639$	6.00000	0.237356
$640$	1.00000	0.0395285
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	4.00000	0.157867
$643$	32.0000	1.26196	0.630978	−	0.775800i	$-0.282654\pi$
$643$	32.0000	1.26196	0.630978	+	0.775800i	$0.282654\pi$
$644$	8.00000	0.315244
$645$	10.0000	0.393750
$646$	8.00000	0.314756
$647$	−8.00000	−0.314512	−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000	−0.314512	−0.157256	+	0.987558i	$0.550265\pi$
$648$	−1.00000	−0.0392837
$649$	−6.00000	−0.235521
$650$	0	0
$651$	−8.00000	−0.313545
$652$	8.00000	0.313304
$653$	50.0000	1.95665	0.978326	−	0.207072i	$-0.0663936\pi$
$653$	50.0000	1.95665	0.978326	+	0.207072i	$0.0663936\pi$
$654$	2.00000	0.0782062
$655$	8.00000	0.312586
$656$	0	0
$657$	−8.00000	−0.312110
$658$	16.0000	0.623745
$659$	6.00000	0.233727	0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000	0.233727	0.116863	+	0.993148i	$0.462716\pi$
$660$	−1.00000	−0.0389249
$661$	42.0000	1.63361	0.816805	−	0.576913i	$-0.195743\pi$
$661$	42.0000	1.63361	0.816805	+	0.576913i	$0.195743\pi$
$662$	−8.00000	−0.310929
$663$	0	0
$664$	−16.0000	−0.620920
$665$	−16.0000	−0.620453
$666$	2.00000	0.0774984
$667$	24.0000	0.929284
$668$	12.0000	0.464294
$669$	12.0000	0.463947
$670$	−10.0000	−0.386334
$671$	2.00000	0.0772091
$672$	−2.00000	−0.0771517
$673$	4.00000	0.154189	0.0770943	−	0.997024i	$-0.475436\pi$
$673$	4.00000	0.154189	0.0770943	+	0.997024i	$0.475436\pi$
$674$	8.00000	0.308148
$675$	−1.00000	−0.0384900
$676$	−13.0000	−0.500000
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	−6.00000	−0.230429
$679$	16.0000	0.614024
$680$	1.00000	0.0383482
$681$	−28.0000	−1.07296
$682$	−4.00000	−0.153168
$683$	4.00000	0.153056	0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000	0.153056	0.0765279	+	0.997067i	$0.475617\pi$
$684$	−8.00000	−0.305888
$685$	6.00000	0.229248
$686$	−20.0000	−0.763604
$687$	2.00000	0.0763048
$688$	10.0000	0.381246
$689$	0	0
$690$	4.00000	0.152277
$691$	36.0000	1.36950	0.684752	−	0.728776i	$-0.259910\pi$
$691$	36.0000	1.36950	0.684752	+	0.728776i	$0.259910\pi$
$692$	6.00000	0.228086
$693$	2.00000	0.0759737
$694$	−4.00000	−0.151838
$695$	4.00000	0.151729
$696$	−6.00000	−0.227429
$697$	0	0
$698$	−14.0000	−0.529908
$699$	6.00000	0.226941
$700$	−2.00000	−0.0755929
$701$	12.0000	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$702$	0	0
$703$	16.0000	0.603451
$704$	−1.00000	−0.0376889
$705$	8.00000	0.301297
$706$	−14.0000	−0.526897
$707$	16.0000	0.601742
$708$	−6.00000	−0.225494
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	6.00000	0.225176
$711$	−4.00000	−0.150012
$712$	18.0000	0.674579
$713$	16.0000	0.599205
$714$	−2.00000	−0.0748481
$715$	0	0
$716$	26.0000	0.971666
$717$	−16.0000	−0.597531
$718$	12.0000	0.447836
$719$	−26.0000	−0.969636	−0.484818	−	0.874615i	$-0.661114\pi$
$719$	−26.0000	−0.969636	−0.484818	+	0.874615i	$0.661114\pi$
$720$	−1.00000	−0.0372678
$721$	8.00000	0.297936
$722$	−45.0000	−1.67473
$723$	26.0000	0.966950
$724$	−22.0000	−0.817624
$725$	−6.00000	−0.222834
$726$	1.00000	0.0371135
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−8.00000	−0.296093
$731$	10.0000	0.369863
$732$	2.00000	0.0739221
$733$	12.0000	0.443230	0.221615	−	0.975134i	$-0.428867\pi$
$733$	12.0000	0.443230	0.221615	+	0.975134i	$0.428867\pi$
$734$	18.0000	0.664392
$735$	−3.00000	−0.110657
$736$	4.00000	0.147442
$737$	10.0000	0.368355
$738$	0	0
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	2.00000	0.0735215
$741$	0	0
$742$	−12.0000	−0.440534
$743$	−4.00000	−0.146746	−0.0733729	−	0.997305i	$-0.523376\pi$
$743$	−4.00000	−0.146746	−0.0733729	+	0.997305i	$0.523376\pi$
$744$	−4.00000	−0.146647
$745$	8.00000	0.293097
$746$	4.00000	0.146450
$747$	16.0000	0.585409
$748$	−1.00000	−0.0365636
$749$	−8.00000	−0.292314
$750$	−1.00000	−0.0365148
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	8.00000	0.291730
$753$	2.00000	0.0728841
$754$	0	0
$755$	−8.00000	−0.291150
$756$	2.00000	0.0727393
$757$	36.0000	1.30844	0.654221	−	0.756303i	$-0.272997\pi$
$757$	36.0000	1.30844	0.654221	+	0.756303i	$0.272997\pi$
$758$	−20.0000	−0.726433
$759$	−4.00000	−0.145191
$760$	−8.00000	−0.290191
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	12.0000	0.434714
$763$	−4.00000	−0.144810
$764$	12.0000	0.434145
$765$	−1.00000	−0.0361551
$766$	−8.00000	−0.289052
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	2.00000	0.0720750
$771$	−22.0000	−0.792311
$772$	4.00000	0.143963
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	−10.0000	−0.359443
$775$	−4.00000	−0.143684
$776$	8.00000	0.287183
$777$	−4.00000	−0.143499
$778$	28.0000	1.00385
$779$	0	0
$780$	0	0
$781$	−6.00000	−0.214697
$782$	4.00000	0.143040
$783$	6.00000	0.214423
$784$	−3.00000	−0.107143
$785$	−4.00000	−0.142766
$786$	−8.00000	−0.285351
$787$	−16.0000	−0.570338	−0.285169	−	0.958477i	$-0.592050\pi$
$787$	−16.0000	−0.570338	−0.285169	+	0.958477i	$0.592050\pi$
$788$	22.0000	0.783718
$789$	32.0000	1.13923
$790$	−4.00000	−0.142314
$791$	12.0000	0.426671
$792$	1.00000	0.0355335
$793$	0	0
$794$	−22.0000	−0.780751
$795$	−6.00000	−0.212798
$796$	−12.0000	−0.425329
$797$	−34.0000	−1.20434	−0.602171	−	0.798367i	$-0.705697\pi$
$797$	−34.0000	−1.20434	−0.602171	+	0.798367i	$0.705697\pi$
$798$	16.0000	0.566394
$799$	8.00000	0.283020
$800$	−1.00000	−0.0353553
$801$	−18.0000	−0.635999
$802$	−24.0000	−0.847469
$803$	8.00000	0.282314
$804$	10.0000	0.352673
$805$	−8.00000	−0.281963
$806$	0	0
$807$	6.00000	0.211210
$808$	8.00000	0.281439
$809$	−16.0000	−0.562530	−0.281265	−	0.959630i	$-0.590754\pi$
$809$	−16.0000	−0.562530	−0.281265	+	0.959630i	$0.590754\pi$
$810$	1.00000	0.0351364
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	12.0000	0.421117
$813$	0	0
$814$	−2.00000	−0.0701000
$815$	−8.00000	−0.280228
$816$	−1.00000	−0.0350070
$817$	−80.0000	−2.79885
$818$	−6.00000	−0.209785
$819$	0	0
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	−6.00000	−0.209274
$823$	−34.0000	−1.18517	−0.592583	−	0.805510i	$-0.701892\pi$
$823$	−34.0000	−1.18517	−0.592583	+	0.805510i	$0.701892\pi$
$824$	4.00000	0.139347
$825$	1.00000	0.0348155
$826$	12.0000	0.417533
$827$	−44.0000	−1.53003	−0.765015	−	0.644013i	$-0.777268\pi$
$827$	−44.0000	−1.53003	−0.765015	+	0.644013i	$0.777268\pi$
$828$	−4.00000	−0.139010
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	16.0000	0.555368
$831$	−2.00000	−0.0693792
$832$	0	0
$833$	−3.00000	−0.103944
$834$	−4.00000	−0.138509
$835$	−12.0000	−0.415277
$836$	8.00000	0.276686
$837$	4.00000	0.138260
$838$	−16.0000	−0.552711
$839$	42.0000	1.45000	0.725001	−	0.688748i	$-0.241839\pi$
$839$	42.0000	1.45000	0.725001	+	0.688748i	$0.241839\pi$
$840$	2.00000	0.0690066
$841$	7.00000	0.241379
$842$	−30.0000	−1.03387
$843$	−30.0000	−1.03325
$844$	−4.00000	−0.137686
$845$	13.0000	0.447214
$846$	−8.00000	−0.275046
$847$	−2.00000	−0.0687208
$848$	−6.00000	−0.206041
$849$	−28.0000	−0.960958
$850$	−1.00000	−0.0342997
$851$	8.00000	0.274236
$852$	−6.00000	−0.205557
$853$	−38.0000	−1.30110	−0.650548	−	0.759465i	$-0.725461\pi$
$853$	−38.0000	−1.30110	−0.650548	+	0.759465i	$0.725461\pi$
$854$	−4.00000	−0.136877
$855$	8.00000	0.273594
$856$	−4.00000	−0.136717
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	0	0
$859$	−48.0000	−1.63774	−0.818869	−	0.573980i	$-0.805399\pi$
$859$	−48.0000	−1.63774	−0.818869	+	0.573980i	$0.805399\pi$
$860$	−10.0000	−0.340997
$861$	0	0
$862$	−6.00000	−0.204361
$863$	40.0000	1.36162	0.680808	−	0.732462i	$-0.261629\pi$
$863$	40.0000	1.36162	0.680808	+	0.732462i	$0.261629\pi$
$864$	1.00000	0.0340207
$865$	−6.00000	−0.204006
$866$	−14.0000	−0.475739
$867$	−1.00000	−0.0339618
$868$	8.00000	0.271538
$869$	4.00000	0.135691
$870$	6.00000	0.203419
$871$	0	0
$872$	−2.00000	−0.0677285
$873$	−8.00000	−0.270759
$874$	−32.0000	−1.08242
$875$	2.00000	0.0676123
$876$	8.00000	0.270295
$877$	42.0000	1.41824	0.709120	−	0.705088i	$-0.249093\pi$
$877$	42.0000	1.41824	0.709120	+	0.705088i	$0.249093\pi$
$878$	−4.00000	−0.134993
$879$	−18.0000	−0.607125
$880$	1.00000	0.0337100
$881$	16.0000	0.539054	0.269527	−	0.962993i	$-0.413133\pi$
$881$	16.0000	0.539054	0.269527	+	0.962993i	$0.413133\pi$
$882$	3.00000	0.101015
$883$	2.00000	0.0673054	0.0336527	−	0.999434i	$-0.489286\pi$
$883$	2.00000	0.0673054	0.0336527	+	0.999434i	$0.489286\pi$
$884$	0	0
$885$	6.00000	0.201688
$886$	−4.00000	−0.134383
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	−2.00000	−0.0671156
$889$	−24.0000	−0.804934
$890$	−18.0000	−0.603361
$891$	−1.00000	−0.0335013
$892$	−12.0000	−0.401790
$893$	−64.0000	−2.14168
$894$	−8.00000	−0.267560
$895$	−26.0000	−0.869084
$896$	2.00000	0.0668153
$897$	0	0
$898$	0	0
$899$	24.0000	0.800445
$900$	1.00000	0.0333333
$901$	−6.00000	−0.199889
$902$	0	0
$903$	20.0000	0.665558
$904$	6.00000	0.199557
$905$	22.0000	0.731305
$906$	8.00000	0.265782
$907$	−32.0000	−1.06254	−0.531271	−	0.847202i	$-0.678286\pi$
$907$	−32.0000	−1.06254	−0.531271	+	0.847202i	$0.678286\pi$
$908$	28.0000	0.929213
$909$	−8.00000	−0.265343
$910$	0	0
$911$	6.00000	0.198789	0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000	0.198789	0.0993944	+	0.995048i	$0.468309\pi$
$912$	8.00000	0.264906
$913$	−16.0000	−0.529523
$914$	2.00000	0.0661541
$915$	−2.00000	−0.0661180
$916$	−2.00000	−0.0660819
$917$	16.0000	0.528367
$918$	1.00000	0.0330049
$919$	−56.0000	−1.84727	−0.923635	−	0.383274i	$-0.874797\pi$
$919$	−56.0000	−1.84727	−0.923635	+	0.383274i	$0.874797\pi$
$920$	−4.00000	−0.131876
$921$	−2.00000	−0.0659022
$922$	−12.0000	−0.395199
$923$	0	0
$924$	−2.00000	−0.0657952
$925$	−2.00000	−0.0657596
$926$	20.0000	0.657241
$927$	−4.00000	−0.131377
$928$	6.00000	0.196960
$929$	−20.0000	−0.656179	−0.328089	−	0.944647i	$-0.606405\pi$
$929$	−20.0000	−0.656179	−0.328089	+	0.944647i	$0.606405\pi$
$930$	4.00000	0.131165
$931$	24.0000	0.786568
$932$	−6.00000	−0.196537
$933$	−2.00000	−0.0654771
$934$	0	0
$935$	1.00000	0.0327035
$936$	0	0
$937$	6.00000	0.196011	0.0980057	−	0.995186i	$-0.468754\pi$
$937$	6.00000	0.196011	0.0980057	+	0.995186i	$0.468754\pi$
$938$	−20.0000	−0.653023
$939$	8.00000	0.261070
$940$	−8.00000	−0.260931
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	4.00000	0.130327
$943$	0	0
$944$	6.00000	0.195283
$945$	−2.00000	−0.0650600
$946$	10.0000	0.325128
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	4.00000	0.129914
$949$	0	0
$950$	8.00000	0.259554
$951$	18.0000	0.583690
$952$	2.00000	0.0648204
$953$	−38.0000	−1.23094	−0.615470	−	0.788160i	$-0.711034\pi$
$953$	−38.0000	−1.23094	−0.615470	+	0.788160i	$0.711034\pi$
$954$	6.00000	0.194257
$955$	−12.0000	−0.388311
$956$	16.0000	0.517477
$957$	−6.00000	−0.193952
$958$	6.00000	0.193851
$959$	12.0000	0.387500
$960$	1.00000	0.0322749
$961$	−15.0000	−0.483871
$962$	0	0
$963$	4.00000	0.128898
$964$	−26.0000	−0.837404
$965$	−4.00000	−0.128765
$966$	8.00000	0.257396
$967$	48.0000	1.54358	0.771788	−	0.635880i	$-0.219363\pi$
$967$	48.0000	1.54358	0.771788	+	0.635880i	$0.219363\pi$
$968$	−1.00000	−0.0321412
$969$	8.00000	0.256997
$970$	−8.00000	−0.256865
$971$	58.0000	1.86131	0.930654	−	0.365900i	$-0.119239\pi$
$971$	58.0000	1.86131	0.930654	+	0.365900i	$0.119239\pi$
$972$	−1.00000	−0.0320750
$973$	8.00000	0.256468
$974$	−2.00000	−0.0640841
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	8.00000	0.255812
$979$	18.0000	0.575282
$980$	3.00000	0.0958315
$981$	2.00000	0.0638551
$982$	−22.0000	−0.702048
$983$	−16.0000	−0.510321	−0.255160	−	0.966899i	$-0.582128\pi$
$983$	−16.0000	−0.510321	−0.255160	+	0.966899i	$0.582128\pi$
$984$	0	0
$985$	−22.0000	−0.700978
$986$	6.00000	0.191079
$987$	16.0000	0.509286
$988$	0	0
$989$	−40.0000	−1.27193
$990$	−1.00000	−0.0317821
$991$	36.0000	1.14358	0.571789	−	0.820401i	$-0.306250\pi$
$991$	36.0000	1.14358	0.571789	+	0.820401i	$0.306250\pi$
$992$	4.00000	0.127000
$993$	−8.00000	−0.253872
$994$	12.0000	0.380617
$995$	12.0000	0.380426
$996$	−16.0000	−0.506979
$997$	−42.0000	−1.33015	−0.665077	−	0.746775i	$-0.731601\pi$
$997$	−42.0000	−1.33015	−0.665077	+	0.746775i	$0.731601\pi$
$998$	−4.00000	−0.126618
$999$	2.00000	0.0632772

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5610.2.a.a.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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