LMFDB - Embedded newform 5610.2.a.b.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:	$1$
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)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} -1.00000 q^{20} +2.00000 q^{21} -1.00000 q^{22} -4.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} -2.00000 q^{28} +2.00000 q^{29} -1.00000 q^{30} +4.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} +1.00000 q^{40} -2.00000 q^{42} -6.00000 q^{43} +1.00000 q^{44} -1.00000 q^{45} +4.00000 q^{46} -1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} -1.00000 q^{51} -6.00000 q^{53} +1.00000 q^{54} -1.00000 q^{55} +2.00000 q^{56} -2.00000 q^{58} +14.0000 q^{59} +1.00000 q^{60} -2.00000 q^{61} -4.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +1.00000 q^{66} +14.0000 q^{67} +1.00000 q^{68} +4.00000 q^{69} -2.00000 q^{70} -2.00000 q^{71} -1.00000 q^{72} +16.0000 q^{73} +2.00000 q^{74} -1.00000 q^{75} -2.00000 q^{77} +12.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{84} -1.00000 q^{85} +6.00000 q^{86} -2.00000 q^{87} -1.00000 q^{88} -18.0000 q^{89} +1.00000 q^{90} -4.00000 q^{92} -4.00000 q^{93} +1.00000 q^{96} +3.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−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
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	−1.00000	−0.182574
$31$	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
$33$	−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	$0.552568\pi$
$38$	0	0
$39$	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
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	1.00000	0.150756
$45$	−1.00000	−0.149071
$46$	4.00000	0.589768
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−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
$57$	0	0
$58$	−2.00000	−0.262613
$59$	14.0000	1.82264	0.911322	−	0.411693i	$-0.135063\pi$
$59$	14.0000	1.82264	0.911322	+	0.411693i	$0.135063\pi$
$60$	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
$67$	14.0000	1.71037	0.855186	−	0.518321i	$-0.173443\pi$
$67$	14.0000	1.71037	0.855186	+	0.518321i	$0.173443\pi$
$68$	1.00000	0.121268
$69$	4.00000	0.481543
$70$	−2.00000	−0.239046
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	−1.00000	−0.117851
$73$	16.0000	1.87266	0.936329	−	0.351123i	$-0.114200\pi$
$73$	16.0000	1.87266	0.936329	+	0.351123i	$0.114200\pi$
$74$	2.00000	0.232495
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	2.00000	0.218218
$85$	−1.00000	−0.108465
$86$	6.00000	0.646997
$87$	−2.00000	−0.214423
$88$	−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
$94$	0	0
$95$	0	0
$96$	1.00000	0.102062
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	3.00000	0.303046
$99$	1.00000	0.100504
$100$	1.00000	0.100000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	1.00000	0.0990148
$103$	−12.0000	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	6.00000	0.582772
$107$	−20.0000	−1.93347	−0.966736	−	0.255774i	$-0.917670\pi$
$107$	−20.0000	−1.93347	−0.966736	+	0.255774i	$0.917670\pi$
$108$	−1.00000	−0.0962250
$109$	18.0000	1.72409	0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000	1.72409	0.862044	+	0.506834i	$0.169184\pi$
$110$	1.00000	0.0953463
$111$	2.00000	0.189832
$112$	−2.00000	−0.188982
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	2.00000	0.185695
$117$	0	0
$118$	−14.0000	−1.28880
$119$	−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	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	−1.00000	−0.0883883
$129$	6.00000	0.528271
$130$	0	0
$131$	−16.0000	−1.39793	−0.698963	−	0.715158i	$-0.746355\pi$
$131$	−16.0000	−1.39793	−0.698963	+	0.715158i	$0.746355\pi$
$132$	−1.00000	−0.0870388
$133$	0	0
$134$	−14.0000	−1.20942
$135$	1.00000	0.0860663
$136$	−1.00000	−0.0857493
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	−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
$141$	0	0
$142$	2.00000	0.167836
$143$	0	0
$144$	1.00000	0.0833333
$145$	−2.00000	−0.166091
$146$	−16.0000	−1.32417
$147$	3.00000	0.247436
$148$	−2.00000	−0.164399
$149$	16.0000	1.31077	0.655386	−	0.755295i	$-0.272506\pi$
$149$	16.0000	1.31077	0.655386	+	0.755295i	$0.272506\pi$
$150$	1.00000	0.0816497
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	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	$0.448974\pi$
$158$	−12.0000	−0.954669
$159$	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	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	0	0
$165$	1.00000	0.0778499
$166$	0	0
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	−2.00000	−0.154303
$169$	−13.0000	−1.00000
$170$	1.00000	0.0766965
$171$	0	0
$172$	−6.00000	−0.457496
$173$	22.0000	1.67263	0.836315	−	0.548250i	$-0.184706\pi$
$173$	22.0000	1.67263	0.836315	+	0.548250i	$0.184706\pi$
$174$	2.00000	0.151620
$175$	−2.00000	−0.151186
$176$	1.00000	0.0753778
$177$	−14.0000	−1.05230
$178$	18.0000	1.34916
$179$	−14.0000	−1.04641	−0.523205	−	0.852207i	$-0.675264\pi$
$179$	−14.0000	−1.04641	−0.523205	+	0.852207i	$0.675264\pi$
$180$	−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	$0.378793\pi$
$182$	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
$188$	0	0
$189$	2.00000	0.145479
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	−1.00000	−0.0721688
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	0	0
$195$	0	0
$196$	−3.00000	−0.214286
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	−1.00000	−0.0710669
$199$	12.0000	0.850657	0.425329	−	0.905039i	$-0.360158\pi$
$199$	12.0000	0.850657	0.425329	+	0.905039i	$0.360158\pi$
$200$	−1.00000	−0.0707107
$201$	−14.0000	−0.987484
$202$	0	0
$203$	−4.00000	−0.280745
$204$	−1.00000	−0.0700140
$205$	0	0
$206$	12.0000	0.836080
$207$	−4.00000	−0.278019
$208$	0	0
$209$	0	0
$210$	2.00000	0.138013
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	−6.00000	−0.412082
$213$	2.00000	0.137038
$214$	20.0000	1.36717
$215$	6.00000	0.409197
$216$	1.00000	0.0680414
$217$	−8.00000	−0.543075
$218$	−18.0000	−1.21911
$219$	−16.0000	−1.08118
$220$	−1.00000	−0.0674200
$221$	0	0
$222$	−2.00000	−0.134231
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	2.00000	0.133631
$225$	1.00000	0.0666667
$226$	−10.0000	−0.665190
$227$	−28.0000	−1.85843	−0.929213	−	0.369546i	$-0.879513\pi$
$227$	−28.0000	−1.85843	−0.929213	+	0.369546i	$0.879513\pi$
$228$	0	0
$229$	−18.0000	−1.18947	−0.594737	−	0.803921i	$-0.702744\pi$
$229$	−18.0000	−1.18947	−0.594737	+	0.803921i	$0.702744\pi$
$230$	−4.00000	−0.263752
$231$	2.00000	0.131590
$232$	−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	$0.393773\pi$
$234$	0	0
$235$	0	0
$236$	14.0000	0.911322
$237$	−12.0000	−0.779484
$238$	2.00000	0.129641
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	1.00000	0.0645497
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	−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
$249$	0	0
$250$	1.00000	0.0632456
$251$	−10.0000	−0.631194	−0.315597	−	0.948893i	$-0.602205\pi$
$251$	−10.0000	−0.631194	−0.315597	+	0.948893i	$0.602205\pi$
$252$	−2.00000	−0.125988
$253$	−4.00000	−0.251478
$254$	12.0000	0.752947
$255$	1.00000	0.0626224
$256$	1.00000	0.0625000
$257$	−26.0000	−1.62184	−0.810918	−	0.585160i	$-0.801032\pi$
$257$	−26.0000	−1.62184	−0.810918	+	0.585160i	$0.801032\pi$
$258$	−6.00000	−0.373544
$259$	4.00000	0.248548
$260$	0	0
$261$	2.00000	0.123797
$262$	16.0000	0.988483
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	1.00000	0.0615457
$265$	6.00000	0.368577
$266$	0	0
$267$	18.0000	1.10158
$268$	14.0000	0.855186
$269$	−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
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	−10.0000	−0.604122
$275$	1.00000	0.0603023
$276$	4.00000	0.240772
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	4.00000	0.239904
$279$	4.00000	0.239474
$280$	−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	$0.782510\pi$
$282$	0	0
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	−2.00000	−0.118678
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	1.00000	0.0588235
$290$	2.00000	0.117444
$291$	0	0
$292$	16.0000	0.936329
$293$	−30.0000	−1.75262	−0.876309	−	0.481749i	$-0.840002\pi$
$293$	−30.0000	−1.75262	−0.876309	+	0.481749i	$0.840002\pi$
$294$	−3.00000	−0.174964
$295$	−14.0000	−0.815112
$296$	2.00000	0.116248
$297$	−1.00000	−0.0580259
$298$	−16.0000	−0.926855
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	12.0000	0.691669
$302$	−16.0000	−0.920697
$303$	0	0
$304$	0	0
$305$	2.00000	0.114520
$306$	−1.00000	−0.0571662
$307$	34.0000	1.94048	0.970241	−	0.242140i	$-0.0778494\pi$
$307$	34.0000	1.94048	0.970241	+	0.242140i	$0.0778494\pi$
$308$	−2.00000	−0.113961
$309$	12.0000	0.682656
$310$	4.00000	0.227185
$311$	26.0000	1.47432	0.737162	−	0.675716i	$-0.236165\pi$
$311$	26.0000	1.47432	0.737162	+	0.675716i	$0.236165\pi$
$312$	0	0
$313$	16.0000	0.904373	0.452187	−	0.891923i	$-0.350644\pi$
$313$	16.0000	0.904373	0.452187	+	0.891923i	$0.350644\pi$
$314$	−4.00000	−0.225733
$315$	2.00000	0.112687
$316$	12.0000	0.675053
$317$	−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
$319$	2.00000	0.111979
$320$	−1.00000	−0.0559017
$321$	20.0000	1.11629
$322$	−8.00000	−0.445823
$323$	0	0
$324$	1.00000	0.0555556
$325$	0	0
$326$	8.00000	0.443079
$327$	−18.0000	−0.995402
$328$	0	0
$329$	0	0
$330$	−1.00000	−0.0550482
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	0	0
$333$	−2.00000	−0.109599
$334$	12.0000	0.656611
$335$	−14.0000	−0.764902
$336$	2.00000	0.109109
$337$	16.0000	0.871576	0.435788	−	0.900049i	$-0.356470\pi$
$337$	16.0000	0.871576	0.435788	+	0.900049i	$0.356470\pi$
$338$	13.0000	0.707107
$339$	−10.0000	−0.543125
$340$	−1.00000	−0.0542326
$341$	4.00000	0.216612
$342$	0	0
$343$	20.0000	1.07990
$344$	6.00000	0.323498
$345$	−4.00000	−0.215353
$346$	−22.0000	−1.18273
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	−2.00000	−0.107211
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	2.00000	0.106904
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	−34.0000	−1.80964	−0.904819	−	0.425797i	$-0.859994\pi$
$353$	−34.0000	−1.80964	−0.904819	+	0.425797i	$0.859994\pi$
$354$	14.0000	0.744092
$355$	2.00000	0.106149
$356$	−18.0000	−0.953998
$357$	2.00000	0.105851
$358$	14.0000	0.739923
$359$	36.0000	1.90001	0.950004	−	0.312239i	$-0.101079\pi$
$359$	36.0000	1.90001	0.950004	+	0.312239i	$0.101079\pi$
$360$	1.00000	0.0527046
$361$	−19.0000	−1.00000
$362$	−10.0000	−0.525588
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−16.0000	−0.837478
$366$	−2.00000	−0.104542
$367$	−26.0000	−1.35719	−0.678594	−	0.734513i	$-0.737411\pi$
$367$	−26.0000	−1.35719	−0.678594	+	0.734513i	$0.737411\pi$
$368$	−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
$376$	0	0
$377$	0	0
$378$	−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	$0.532759\pi$
$380$	0	0
$381$	12.0000	0.614779
$382$	4.00000	0.204658
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	1.00000	0.0510310
$385$	2.00000	0.101929
$386$	4.00000	0.203595
$387$	−6.00000	−0.304997
$388$	0	0
$389$	−36.0000	−1.82527	−0.912636	−	0.408773i	$-0.865957\pi$
$389$	−36.0000	−1.82527	−0.912636	+	0.408773i	$0.865957\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	3.00000	0.151523
$393$	16.0000	0.807093
$394$	−6.00000	−0.302276
$395$	−12.0000	−0.603786
$396$	1.00000	0.0502519
$397$	−26.0000	−1.30490	−0.652451	−	0.757831i	$-0.726259\pi$
$397$	−26.0000	−1.30490	−0.652451	+	0.757831i	$0.726259\pi$
$398$	−12.0000	−0.601506
$399$	0	0
$400$	1.00000	0.0500000
$401$	−8.00000	−0.399501	−0.199750	−	0.979847i	$-0.564013\pi$
$401$	−8.00000	−0.399501	−0.199750	+	0.979847i	$0.564013\pi$
$402$	14.0000	0.698257
$403$	0	0
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	4.00000	0.198517
$407$	−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
$411$	−10.0000	−0.493264
$412$	−12.0000	−0.591198
$413$	−28.0000	−1.37779
$414$	4.00000	0.196589
$415$	0	0
$416$	0	0
$417$	4.00000	0.195881
$418$	0	0
$419$	24.0000	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$420$	−2.00000	−0.0975900
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	20.0000	0.973585
$423$	0	0
$424$	6.00000	0.291386
$425$	1.00000	0.0485071
$426$	−2.00000	−0.0969003
$427$	4.00000	0.193574
$428$	−20.0000	−0.966736
$429$	0	0
$430$	−6.00000	−0.289346
$431$	−26.0000	−1.25238	−0.626188	−	0.779672i	$-0.715386\pi$
$431$	−26.0000	−1.25238	−0.626188	+	0.779672i	$0.715386\pi$
$432$	−1.00000	−0.0481125
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	8.00000	0.384012
$435$	2.00000	0.0958927
$436$	18.0000	0.862044
$437$	0	0
$438$	16.0000	0.764510
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	1.00000	0.0476731
$441$	−3.00000	−0.142857
$442$	0	0
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	2.00000	0.0949158
$445$	18.0000	0.853282
$446$	4.00000	0.189405
$447$	−16.0000	−0.756774
$448$	−2.00000	−0.0944911
$449$	32.0000	1.51017	0.755087	−	0.655625i	$-0.227595\pi$
$449$	32.0000	1.51017	0.755087	+	0.655625i	$0.227595\pi$
$450$	−1.00000	−0.0471405
$451$	0	0
$452$	10.0000	0.470360
$453$	−16.0000	−0.751746
$454$	28.0000	1.31411
$455$	0	0
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	18.0000	0.841085
$459$	−1.00000	−0.0466760
$460$	4.00000	0.186501
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	−2.00000	−0.0930484
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	2.00000	0.0928477
$465$	4.00000	0.185496
$466$	−10.0000	−0.463241
$467$	−8.00000	−0.370196	−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000	−0.370196	−0.185098	+	0.982720i	$0.559260\pi$
$468$	0	0
$469$	−28.0000	−1.29292
$470$	0	0
$471$	−4.00000	−0.184310
$472$	−14.0000	−0.644402
$473$	−6.00000	−0.275880
$474$	12.0000	0.551178
$475$	0	0
$476$	−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
$482$	10.0000	0.455488
$483$	−8.00000	−0.364013
$484$	1.00000	0.0454545
$485$	0	0
$486$	1.00000	0.0453609
$487$	−38.0000	−1.72194	−0.860972	−	0.508652i	$-0.830144\pi$
$487$	−38.0000	−1.72194	−0.860972	+	0.508652i	$0.830144\pi$
$488$	2.00000	0.0905357
$489$	8.00000	0.361773
$490$	−3.00000	−0.135526
$491$	−42.0000	−1.89543	−0.947717	−	0.319113i	$-0.896615\pi$
$491$	−42.0000	−1.89543	−0.947717	+	0.319113i	$0.896615\pi$
$492$	0	0
$493$	2.00000	0.0900755
$494$	0	0
$495$	−1.00000	−0.0449467
$496$	4.00000	0.179605
$497$	4.00000	0.179425
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	−1.00000	−0.0447214
$501$	12.0000	0.536120
$502$	10.0000	0.446322
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	2.00000	0.0890871
$505$	0	0
$506$	4.00000	0.177822
$507$	13.0000	0.577350
$508$	−12.0000	−0.532414
$509$	12.0000	0.531891	0.265945	−	0.963988i	$-0.414316\pi$
$509$	12.0000	0.531891	0.265945	+	0.963988i	$0.414316\pi$
$510$	−1.00000	−0.0442807
$511$	−32.0000	−1.41560
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	26.0000	1.14681
$515$	12.0000	0.528783
$516$	6.00000	0.264135
$517$	0	0
$518$	−4.00000	−0.175750
$519$	−22.0000	−0.965693
$520$	0	0
$521$	−16.0000	−0.700973	−0.350486	−	0.936568i	$-0.613984\pi$
$521$	−16.0000	−0.700973	−0.350486	+	0.936568i	$0.613984\pi$
$522$	−2.00000	−0.0875376
$523$	−18.0000	−0.787085	−0.393543	−	0.919306i	$-0.628751\pi$
$523$	−18.0000	−0.787085	−0.393543	+	0.919306i	$0.628751\pi$
$524$	−16.0000	−0.698963
$525$	2.00000	0.0872872
$526$	16.0000	0.697633
$527$	4.00000	0.174243
$528$	−1.00000	−0.0435194
$529$	−7.00000	−0.304348
$530$	−6.00000	−0.260623
$531$	14.0000	0.607548
$532$	0	0
$533$	0	0
$534$	−18.0000	−0.778936
$535$	20.0000	0.864675
$536$	−14.0000	−0.604708
$537$	14.0000	0.604145
$538$	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	$0.760893\pi$
$542$	−8.00000	−0.343629
$543$	−10.0000	−0.429141
$544$	−1.00000	−0.0428746
$545$	−18.0000	−0.771035
$546$	0	0
$547$	44.0000	1.88130	0.940652	−	0.339372i	$-0.110215\pi$
$547$	44.0000	1.88130	0.940652	+	0.339372i	$0.110215\pi$
$548$	10.0000	0.427179
$549$	−2.00000	−0.0853579
$550$	−1.00000	−0.0426401
$551$	0	0
$552$	−4.00000	−0.170251
$553$	−24.0000	−1.02058
$554$	−10.0000	−0.424859
$555$	−2.00000	−0.0848953
$556$	−4.00000	−0.169638
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	−4.00000	−0.169334
$559$	0	0
$560$	2.00000	0.0845154
$561$	−1.00000	−0.0422200
$562$	26.0000	1.09674
$563$	−8.00000	−0.337160	−0.168580	−	0.985688i	$-0.553918\pi$
$563$	−8.00000	−0.337160	−0.168580	+	0.985688i	$0.553918\pi$
$564$	0	0
$565$	−10.0000	−0.420703
$566$	20.0000	0.840663
$567$	−2.00000	−0.0839921
$568$	2.00000	0.0839181
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	0	0
$573$	4.00000	0.167102
$574$	0	0
$575$	−4.00000	−0.166812
$576$	1.00000	0.0416667
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	−1.00000	−0.0415945
$579$	4.00000	0.166234
$580$	−2.00000	−0.0830455
$581$	0	0
$582$	0	0
$583$	−6.00000	−0.248495
$584$	−16.0000	−0.662085
$585$	0	0
$586$	30.0000	1.23929
$587$	−36.0000	−1.48588	−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000	−1.48588	−0.742940	+	0.669359i	$0.766569\pi$
$588$	3.00000	0.123718
$589$	0	0
$590$	14.0000	0.576371
$591$	−6.00000	−0.246807
$592$	−2.00000	−0.0821995
$593$	−2.00000	−0.0821302	−0.0410651	−	0.999156i	$-0.513075\pi$
$593$	−2.00000	−0.0821302	−0.0410651	+	0.999156i	$0.513075\pi$
$594$	1.00000	0.0410305
$595$	2.00000	0.0819920
$596$	16.0000	0.655386
$597$	−12.0000	−0.491127
$598$	0	0
$599$	4.00000	0.163436	0.0817178	−	0.996656i	$-0.473959\pi$
$599$	4.00000	0.163436	0.0817178	+	0.996656i	$0.473959\pi$
$600$	1.00000	0.0408248
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	−12.0000	−0.489083
$603$	14.0000	0.570124
$604$	16.0000	0.651031
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−18.0000	−0.730597	−0.365299	−	0.930890i	$-0.619033\pi$
$607$	−18.0000	−0.730597	−0.365299	+	0.930890i	$0.619033\pi$
$608$	0	0
$609$	4.00000	0.162088
$610$	−2.00000	−0.0809776
$611$	0	0
$612$	1.00000	0.0404226
$613$	44.0000	1.77714	0.888572	−	0.458738i	$-0.151698\pi$
$613$	44.0000	1.77714	0.888572	+	0.458738i	$0.151698\pi$
$614$	−34.0000	−1.37213
$615$	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	$0.353970\pi$
$618$	−12.0000	−0.482711
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	−4.00000	−0.160644
$621$	4.00000	0.160514
$622$	−26.0000	−1.04251
$623$	36.0000	1.44231
$624$	0	0
$625$	1.00000	0.0400000
$626$	−16.0000	−0.639489
$627$	0	0
$628$	4.00000	0.159617
$629$	−2.00000	−0.0797452
$630$	−2.00000	−0.0796819
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	−12.0000	−0.477334
$633$	20.0000	0.794929
$634$	18.0000	0.714871
$635$	12.0000	0.476205
$636$	6.00000	0.237915
$637$	0	0
$638$	−2.00000	−0.0791808
$639$	−2.00000	−0.0791188
$640$	1.00000	0.0395285
$641$	48.0000	1.89589	0.947943	−	0.318440i	$-0.103159\pi$
$641$	48.0000	1.89589	0.947943	+	0.318440i	$0.103159\pi$
$642$	−20.0000	−0.789337
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	8.00000	0.315244
$645$	−6.00000	−0.236250
$646$	0	0
$647$	−48.0000	−1.88707	−0.943537	−	0.331266i	$-0.892524\pi$
$647$	−48.0000	−1.88707	−0.943537	+	0.331266i	$0.892524\pi$
$648$	−1.00000	−0.0392837
$649$	14.0000	0.549548
$650$	0	0
$651$	8.00000	0.313545
$652$	−8.00000	−0.313304
$653$	−14.0000	−0.547862	−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	18.0000	0.703856
$655$	16.0000	0.625172
$656$	0	0
$657$	16.0000	0.624219
$658$	0	0
$659$	−42.0000	−1.63609	−0.818044	−	0.575156i	$-0.804941\pi$
$659$	−42.0000	−1.63609	−0.818044	+	0.575156i	$0.804941\pi$
$660$	1.00000	0.0389249
$661$	−38.0000	−1.47803	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000	−1.47803	−0.739014	+	0.673690i	$0.764708\pi$
$662$	8.00000	0.310929
$663$	0	0
$664$	0	0
$665$	0	0
$666$	2.00000	0.0774984
$667$	−8.00000	−0.309761
$668$	−12.0000	−0.464294
$669$	4.00000	0.154649
$670$	14.0000	0.540867
$671$	−2.00000	−0.0772091
$672$	−2.00000	−0.0771517
$673$	−20.0000	−0.770943	−0.385472	−	0.922720i	$-0.625961\pi$
$673$	−20.0000	−0.770943	−0.385472	+	0.922720i	$0.625961\pi$
$674$	−16.0000	−0.616297
$675$	−1.00000	−0.0384900
$676$	−13.0000	−0.500000
$677$	38.0000	1.46046	0.730229	−	0.683202i	$-0.239413\pi$
$677$	38.0000	1.46046	0.730229	+	0.683202i	$0.239413\pi$
$678$	10.0000	0.384048
$679$	0	0
$680$	1.00000	0.0383482
$681$	28.0000	1.07296
$682$	−4.00000	−0.153168
$683$	−44.0000	−1.68361	−0.841807	−	0.539779i	$-0.818508\pi$
$683$	−44.0000	−1.68361	−0.841807	+	0.539779i	$0.818508\pi$
$684$	0	0
$685$	−10.0000	−0.382080
$686$	−20.0000	−0.763604
$687$	18.0000	0.686743
$688$	−6.00000	−0.228748
$689$	0	0
$690$	4.00000	0.152277
$691$	12.0000	0.456502	0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000	0.456502	0.228251	+	0.973602i	$0.426699\pi$
$692$	22.0000	0.836315
$693$	−2.00000	−0.0759737
$694$	4.00000	0.151838
$695$	4.00000	0.151729
$696$	2.00000	0.0758098
$697$	0	0
$698$	−30.0000	−1.13552
$699$	−10.0000	−0.378235
$700$	−2.00000	−0.0755929
$701$	−12.0000	−0.453234	−0.226617	−	0.973984i	$-0.572767\pi$
$701$	−12.0000	−0.453234	−0.226617	+	0.973984i	$0.572767\pi$
$702$	0	0
$703$	0	0
$704$	1.00000	0.0376889
$705$	0	0
$706$	34.0000	1.27961
$707$	0	0
$708$	−14.0000	−0.526152
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	−2.00000	−0.0750587
$711$	12.0000	0.450035
$712$	18.0000	0.674579
$713$	−16.0000	−0.599205
$714$	−2.00000	−0.0748481
$715$	0	0
$716$	−14.0000	−0.523205
$717$	16.0000	0.597531
$718$	−36.0000	−1.34351
$719$	−34.0000	−1.26799	−0.633993	−	0.773339i	$-0.718585\pi$
$719$	−34.0000	−1.26799	−0.633993	+	0.773339i	$0.718585\pi$
$720$	−1.00000	−0.0372678
$721$	24.0000	0.893807
$722$	19.0000	0.707107
$723$	10.0000	0.371904
$724$	10.0000	0.371647
$725$	2.00000	0.0742781
$726$	1.00000	0.0371135
$727$	12.0000	0.445055	0.222528	−	0.974926i	$-0.428569\pi$
$727$	12.0000	0.445055	0.222528	+	0.974926i	$0.428569\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	16.0000	0.592187
$731$	−6.00000	−0.221918
$732$	2.00000	0.0739221
$733$	−36.0000	−1.32969	−0.664845	−	0.746981i	$-0.731502\pi$
$733$	−36.0000	−1.32969	−0.664845	+	0.746981i	$0.731502\pi$
$734$	26.0000	0.959678
$735$	−3.00000	−0.110657
$736$	4.00000	0.147442
$737$	14.0000	0.515697
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	2.00000	0.0735215
$741$	0	0
$742$	−12.0000	−0.440534
$743$	36.0000	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$744$	4.00000	0.146647
$745$	−16.0000	−0.586195
$746$	4.00000	0.146450
$747$	0	0
$748$	1.00000	0.0365636
$749$	40.0000	1.46157
$750$	−1.00000	−0.0365148
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	10.0000	0.364420
$754$	0	0
$755$	−16.0000	−0.582300
$756$	2.00000	0.0727393
$757$	4.00000	0.145382	0.0726912	−	0.997354i	$-0.476841\pi$
$757$	4.00000	0.145382	0.0726912	+	0.997354i	$0.476841\pi$
$758$	4.00000	0.145287
$759$	4.00000	0.145191
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	−12.0000	−0.434714
$763$	−36.0000	−1.30329
$764$	−4.00000	−0.144715
$765$	−1.00000	−0.0361551
$766$	0	0
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	−2.00000	−0.0720750
$771$	26.0000	0.936367
$772$	−4.00000	−0.143963
$773$	42.0000	1.51064	0.755318	−	0.655359i	$-0.227483\pi$
$773$	42.0000	1.51064	0.755318	+	0.655359i	$0.227483\pi$
$774$	6.00000	0.215666
$775$	4.00000	0.143684
$776$	0	0
$777$	−4.00000	−0.143499
$778$	36.0000	1.29066
$779$	0	0
$780$	0	0
$781$	−2.00000	−0.0715656
$782$	4.00000	0.143040
$783$	−2.00000	−0.0714742
$784$	−3.00000	−0.107143
$785$	−4.00000	−0.142766
$786$	−16.0000	−0.570701
$787$	−16.0000	−0.570338	−0.285169	−	0.958477i	$-0.592050\pi$
$787$	−16.0000	−0.570338	−0.285169	+	0.958477i	$0.592050\pi$
$788$	6.00000	0.213741
$789$	16.0000	0.569615
$790$	12.0000	0.426941
$791$	−20.0000	−0.711118
$792$	−1.00000	−0.0355335
$793$	0	0
$794$	26.0000	0.922705
$795$	−6.00000	−0.212798
$796$	12.0000	0.425329
$797$	−50.0000	−1.77109	−0.885545	−	0.464553i	$-0.846215\pi$
$797$	−50.0000	−1.77109	−0.885545	+	0.464553i	$0.846215\pi$
$798$	0	0
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	−18.0000	−0.635999
$802$	8.00000	0.282490
$803$	16.0000	0.564628
$804$	−14.0000	−0.493742
$805$	−8.00000	−0.281963
$806$	0	0
$807$	6.00000	0.211210
$808$	0	0
$809$	32.0000	1.12506	0.562530	−	0.826777i	$-0.309828\pi$
$809$	32.0000	1.12506	0.562530	+	0.826777i	$0.309828\pi$
$810$	1.00000	0.0351364
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	−4.00000	−0.140372
$813$	−8.00000	−0.280572
$814$	2.00000	0.0701000
$815$	8.00000	0.280228
$816$	−1.00000	−0.0350070
$817$	0	0
$818$	−6.00000	−0.209785
$819$	0	0
$820$	0	0
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	10.0000	0.348790
$823$	−10.0000	−0.348578	−0.174289	−	0.984695i	$-0.555763\pi$
$823$	−10.0000	−0.348578	−0.174289	+	0.984695i	$0.555763\pi$
$824$	12.0000	0.418040
$825$	−1.00000	−0.0348155
$826$	28.0000	0.974245
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	−4.00000	−0.139010
$829$	18.0000	0.625166	0.312583	−	0.949890i	$-0.398806\pi$
$829$	18.0000	0.625166	0.312583	+	0.949890i	$0.398806\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	0	0
$833$	−3.00000	−0.103944
$834$	−4.00000	−0.138509
$835$	12.0000	0.415277
$836$	0	0
$837$	−4.00000	−0.138260
$838$	−24.0000	−0.829066
$839$	34.0000	1.17381	0.586905	−	0.809656i	$-0.300346\pi$
$839$	34.0000	1.17381	0.586905	+	0.809656i	$0.300346\pi$
$840$	2.00000	0.0690066
$841$	−25.0000	−0.862069
$842$	−14.0000	−0.482472
$843$	26.0000	0.895488
$844$	−20.0000	−0.688428
$845$	13.0000	0.447214
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	−6.00000	−0.206041
$849$	20.0000	0.686398
$850$	−1.00000	−0.0342997
$851$	8.00000	0.274236
$852$	2.00000	0.0685189
$853$	−14.0000	−0.479351	−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000	−0.479351	−0.239675	+	0.970853i	$0.577041\pi$
$854$	−4.00000	−0.136877
$855$	0	0
$856$	20.0000	0.683586
$857$	−54.0000	−1.84460	−0.922302	−	0.386469i	$-0.873695\pi$
$857$	−54.0000	−1.84460	−0.922302	+	0.386469i	$0.873695\pi$
$858$	0	0
$859$	−32.0000	−1.09183	−0.545913	−	0.837842i	$-0.683817\pi$
$859$	−32.0000	−1.09183	−0.545913	+	0.837842i	$0.683817\pi$
$860$	6.00000	0.204598
$861$	0	0
$862$	26.0000	0.885564
$863$	16.0000	0.544646	0.272323	−	0.962206i	$-0.412208\pi$
$863$	16.0000	0.544646	0.272323	+	0.962206i	$0.412208\pi$
$864$	1.00000	0.0340207
$865$	−22.0000	−0.748022
$866$	2.00000	0.0679628
$867$	−1.00000	−0.0339618
$868$	−8.00000	−0.271538
$869$	12.0000	0.407072
$870$	−2.00000	−0.0678064
$871$	0	0
$872$	−18.0000	−0.609557
$873$	0	0
$874$	0	0
$875$	2.00000	0.0676123
$876$	−16.0000	−0.540590
$877$	50.0000	1.68838	0.844190	−	0.536044i	$-0.180082\pi$
$877$	50.0000	1.68838	0.844190	+	0.536044i	$0.180082\pi$
$878$	−20.0000	−0.674967
$879$	30.0000	1.01187
$880$	−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
$883$	26.0000	0.874970	0.437485	−	0.899226i	$-0.355869\pi$
$883$	26.0000	0.874970	0.437485	+	0.899226i	$0.355869\pi$
$884$	0	0
$885$	14.0000	0.470605
$886$	4.00000	0.134383
$887$	32.0000	1.07445	0.537227	−	0.843437i	$-0.319472\pi$
$887$	32.0000	1.07445	0.537227	+	0.843437i	$0.319472\pi$
$888$	−2.00000	−0.0671156
$889$	24.0000	0.804934
$890$	−18.0000	−0.603361
$891$	1.00000	0.0335013
$892$	−4.00000	−0.133930
$893$	0	0
$894$	16.0000	0.535120
$895$	14.0000	0.467968
$896$	2.00000	0.0668153
$897$	0	0
$898$	−32.0000	−1.06785
$899$	8.00000	0.266815
$900$	1.00000	0.0333333
$901$	−6.00000	−0.199889
$902$	0	0
$903$	−12.0000	−0.399335
$904$	−10.0000	−0.332595
$905$	−10.0000	−0.332411
$906$	16.0000	0.531564
$907$	48.0000	1.59381	0.796907	−	0.604102i	$-0.206468\pi$
$907$	48.0000	1.59381	0.796907	+	0.604102i	$0.206468\pi$
$908$	−28.0000	−0.929213
$909$	0	0
$910$	0	0
$911$	−18.0000	−0.596367	−0.298183	−	0.954509i	$-0.596381\pi$
$911$	−18.0000	−0.596367	−0.298183	+	0.954509i	$0.596381\pi$
$912$	0	0
$913$	0	0
$914$	10.0000	0.330771
$915$	−2.00000	−0.0661180
$916$	−18.0000	−0.594737
$917$	32.0000	1.05673
$918$	1.00000	0.0330049
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	−4.00000	−0.131876
$921$	−34.0000	−1.12034
$922$	12.0000	0.395199
$923$	0	0
$924$	2.00000	0.0657952
$925$	−2.00000	−0.0657596
$926$	−4.00000	−0.131448
$927$	−12.0000	−0.394132
$928$	−2.00000	−0.0656532
$929$	28.0000	0.918650	0.459325	−	0.888268i	$-0.348091\pi$
$929$	28.0000	0.918650	0.459325	+	0.888268i	$0.348091\pi$
$930$	−4.00000	−0.131165
$931$	0	0
$932$	10.0000	0.327561
$933$	−26.0000	−0.851202
$934$	8.00000	0.261768
$935$	−1.00000	−0.0327035
$936$	0	0
$937$	46.0000	1.50275	0.751377	−	0.659873i	$-0.229390\pi$
$937$	46.0000	1.50275	0.751377	+	0.659873i	$0.229390\pi$
$938$	28.0000	0.914232
$939$	−16.0000	−0.522140
$940$	0	0
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	4.00000	0.130327
$943$	0	0
$944$	14.0000	0.455661
$945$	−2.00000	−0.0650600
$946$	6.00000	0.195077
$947$	−36.0000	−1.16984	−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000	−1.16984	−0.584921	+	0.811090i	$0.698875\pi$
$948$	−12.0000	−0.389742
$949$	0	0
$950$	0	0
$951$	18.0000	0.583690
$952$	2.00000	0.0648204
$953$	10.0000	0.323932	0.161966	−	0.986796i	$-0.448217\pi$
$953$	10.0000	0.323932	0.161966	+	0.986796i	$0.448217\pi$
$954$	6.00000	0.194257
$955$	4.00000	0.129437
$956$	−16.0000	−0.517477
$957$	−2.00000	−0.0646508
$958$	6.00000	0.193851
$959$	−20.0000	−0.645834
$960$	1.00000	0.0322749
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−20.0000	−0.644491
$964$	−10.0000	−0.322078
$965$	4.00000	0.128765
$966$	8.00000	0.257396
$967$	−56.0000	−1.80084	−0.900419	−	0.435023i	$-0.856740\pi$
$967$	−56.0000	−1.80084	−0.900419	+	0.435023i	$0.856740\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	0	0
$971$	−30.0000	−0.962746	−0.481373	−	0.876516i	$-0.659862\pi$
$971$	−30.0000	−0.962746	−0.481373	+	0.876516i	$0.659862\pi$
$972$	−1.00000	−0.0320750
$973$	8.00000	0.256468
$974$	38.0000	1.21760
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	46.0000	1.47167	0.735835	−	0.677161i	$-0.236790\pi$
$977$	46.0000	1.47167	0.735835	+	0.677161i	$0.236790\pi$
$978$	−8.00000	−0.255812
$979$	−18.0000	−0.575282
$980$	3.00000	0.0958315
$981$	18.0000	0.574696
$982$	42.0000	1.34027
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	−2.00000	−0.0636930
$987$	0	0
$988$	0	0
$989$	24.0000	0.763156
$990$	1.00000	0.0317821
$991$	−4.00000	−0.127064	−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000	−0.127064	−0.0635321	+	0.997980i	$0.520237\pi$
$992$	−4.00000	−0.127000
$993$	8.00000	0.253872
$994$	−4.00000	−0.126872
$995$	−12.0000	−0.380426
$996$	0	0
$997$	46.0000	1.45683	0.728417	−	0.685134i	$-0.240256\pi$
$997$	46.0000	1.45683	0.728417	+	0.685134i	$0.240256\pi$
$998$	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.b.1.1	✓	1

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

Downloads

Learn more