LMFDB - Embedded newform 5610.2.a.bl.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)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} -4.00000 q^{13} -2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} -6.00000 q^{19} +1.00000 q^{20} -2.00000 q^{21} -1.00000 q^{22} -2.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -4.00000 q^{26} +1.00000 q^{27} -2.00000 q^{28} -6.00000 q^{29} +1.00000 q^{30} -4.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} +1.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} -6.00000 q^{38} -4.00000 q^{39} +1.00000 q^{40} -6.00000 q^{41} -2.00000 q^{42} +6.00000 q^{43} -1.00000 q^{44} +1.00000 q^{45} -2.00000 q^{46} -12.0000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} +1.00000 q^{51} -4.00000 q^{52} +1.00000 q^{54} -1.00000 q^{55} -2.00000 q^{56} -6.00000 q^{57} -6.00000 q^{58} +6.00000 q^{59} +1.00000 q^{60} +2.00000 q^{61} -4.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} -1.00000 q^{66} +1.00000 q^{68} -2.00000 q^{69} -2.00000 q^{70} +6.00000 q^{71} +1.00000 q^{72} -4.00000 q^{73} -2.00000 q^{74} +1.00000 q^{75} -6.00000 q^{76} +2.00000 q^{77} -4.00000 q^{78} +10.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -4.00000 q^{83} -2.00000 q^{84} +1.00000 q^{85} +6.00000 q^{86} -6.00000 q^{87} -1.00000 q^{88} +6.00000 q^{89} +1.00000 q^{90} +8.00000 q^{91} -2.00000 q^{92} -4.00000 q^{93} -12.0000 q^{94} -6.00000 q^{95} +1.00000 q^{96} +14.0000 q^{97} -3.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−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
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	1.00000	0.223607
$21$	−2.00000	−0.436436
$22$	−1.00000	−0.213201
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	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	$-0.688081\pi$
$29$	−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	$0.552568\pi$
$38$	−6.00000	−0.973329
$39$	−4.00000	−0.640513
$40$	1.00000	0.158114
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	−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
$46$	−2.00000	−0.294884
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	1.00000	0.144338
$49$	−3.00000	−0.428571
$50$	1.00000	0.141421
$51$	1.00000	0.140028
$52$	−4.00000	−0.554700
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	1.00000	0.136083
$55$	−1.00000	−0.134840
$56$	−2.00000	−0.267261
$57$	−6.00000	−0.794719
$58$	−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	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−4.00000	−0.508001
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	−1.00000	−0.123091
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	1.00000	0.121268
$69$	−2.00000	−0.240772
$70$	−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
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	−2.00000	−0.232495
$75$	1.00000	0.115470
$76$	−6.00000	−0.688247
$77$	2.00000	0.227921
$78$	−4.00000	−0.452911
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	−2.00000	−0.218218
$85$	1.00000	0.108465
$86$	6.00000	0.646997
$87$	−6.00000	−0.643268
$88$	−1.00000	−0.106600
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	1.00000	0.105409
$91$	8.00000	0.838628
$92$	−2.00000	−0.208514
$93$	−4.00000	−0.414781
$94$	−12.0000	−1.23771
$95$	−6.00000	−0.615587
$96$	1.00000	0.102062
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	−3.00000	−0.303046
$99$	−1.00000	−0.100504
$100$	1.00000	0.100000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	1.00000	0.0990148
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	−4.00000	−0.392232
$105$	−2.00000	−0.195180
$106$	0	0
$107$	−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
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	−1.00000	−0.0953463
$111$	−2.00000	−0.189832
$112$	−2.00000	−0.188982
$113$	−4.00000	−0.376288	−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000	−0.376288	−0.188144	+	0.982141i	$0.560247\pi$
$114$	−6.00000	−0.561951
$115$	−2.00000	−0.186501
$116$	−6.00000	−0.557086
$117$	−4.00000	−0.369800
$118$	6.00000	0.552345
$119$	−2.00000	−0.183340
$120$	1.00000	0.0912871
$121$	1.00000	0.0909091
$122$	2.00000	0.181071
$123$	−6.00000	−0.541002
$124$	−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
$129$	6.00000	0.528271
$130$	−4.00000	−0.350823
$131$	20.0000	1.74741	0.873704	−	0.486458i	$-0.161711\pi$
$131$	20.0000	1.74741	0.873704	+	0.486458i	$0.161711\pi$
$132$	−1.00000	−0.0870388
$133$	12.0000	1.04053
$134$	0	0
$135$	1.00000	0.0860663
$136$	1.00000	0.0857493
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	−2.00000	−0.170251
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	−2.00000	−0.169031
$141$	−12.0000	−1.01058
$142$	6.00000	0.503509
$143$	4.00000	0.334497
$144$	1.00000	0.0833333
$145$	−6.00000	−0.498273
$146$	−4.00000	−0.331042
$147$	−3.00000	−0.247436
$148$	−2.00000	−0.164399
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	1.00000	0.0816497
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	−6.00000	−0.486664
$153$	1.00000	0.0808452
$154$	2.00000	0.161165
$155$	−4.00000	−0.321288
$156$	−4.00000	−0.320256
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	10.0000	0.795557
$159$	0	0
$160$	1.00000	0.0790569
$161$	4.00000	0.315244
$162$	1.00000	0.0785674
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−6.00000	−0.468521
$165$	−1.00000	−0.0778499
$166$	−4.00000	−0.310460
$167$	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
$169$	3.00000	0.230769
$170$	1.00000	0.0766965
$171$	−6.00000	−0.458831
$172$	6.00000	0.457496
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	−6.00000	−0.454859
$175$	−2.00000	−0.151186
$176$	−1.00000	−0.0753778
$177$	6.00000	0.450988
$178$	6.00000	0.449719
$179$	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
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	8.00000	0.592999
$183$	2.00000	0.147844
$184$	−2.00000	−0.147442
$185$	−2.00000	−0.147043
$186$	−4.00000	−0.293294
$187$	−1.00000	−0.0731272
$188$	−12.0000	−0.875190
$189$	−2.00000	−0.145479
$190$	−6.00000	−0.435286
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	1.00000	0.0721688
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	14.0000	1.00514
$195$	−4.00000	−0.286446
$196$	−3.00000	−0.214286
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	−1.00000	−0.0710669
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−10.0000	−0.703598
$203$	12.0000	0.842235
$204$	1.00000	0.0700140
$205$	−6.00000	−0.419058
$206$	0	0
$207$	−2.00000	−0.139010
$208$	−4.00000	−0.277350
$209$	6.00000	0.415029
$210$	−2.00000	−0.138013
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	0	0
$213$	6.00000	0.411113
$214$	−20.0000	−1.36717
$215$	6.00000	0.409197
$216$	1.00000	0.0680414
$217$	8.00000	0.543075
$218$	−10.0000	−0.677285
$219$	−4.00000	−0.270295
$220$	−1.00000	−0.0674200
$221$	−4.00000	−0.269069
$222$	−2.00000	−0.134231
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	−2.00000	−0.133631
$225$	1.00000	0.0666667
$226$	−4.00000	−0.266076
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	−6.00000	−0.397360
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	−2.00000	−0.131876
$231$	2.00000	0.131590
$232$	−6.00000	−0.393919
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	−4.00000	−0.261488
$235$	−12.0000	−0.782794
$236$	6.00000	0.390567
$237$	10.0000	0.649570
$238$	−2.00000	−0.129641
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	1.00000	0.0645497
$241$	−12.0000	−0.772988	−0.386494	−	0.922292i	$-0.626314\pi$
$241$	−12.0000	−0.772988	−0.386494	+	0.922292i	$0.626314\pi$
$242$	1.00000	0.0642824
$243$	1.00000	0.0641500
$244$	2.00000	0.128037
$245$	−3.00000	−0.191663
$246$	−6.00000	−0.382546
$247$	24.0000	1.52708
$248$	−4.00000	−0.254000
$249$	−4.00000	−0.253490
$250$	1.00000	0.0632456
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	−2.00000	−0.125988
$253$	2.00000	0.125739
$254$	12.0000	0.752947
$255$	1.00000	0.0626224
$256$	1.00000	0.0625000
$257$	26.0000	1.62184	0.810918	−	0.585160i	$-0.198968\pi$
$257$	26.0000	1.62184	0.810918	+	0.585160i	$0.198968\pi$
$258$	6.00000	0.373544
$259$	4.00000	0.248548
$260$	−4.00000	−0.248069
$261$	−6.00000	−0.371391
$262$	20.0000	1.23560
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	−1.00000	−0.0615457
$265$	0	0
$266$	12.0000	0.735767
$267$	6.00000	0.367194
$268$	0	0
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	1.00000	0.0608581
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	1.00000	0.0606339
$273$	8.00000	0.484182
$274$	−14.0000	−0.845771
$275$	−1.00000	−0.0603023
$276$	−2.00000	−0.120386
$277$	14.0000	0.841178	0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000	0.841178	0.420589	+	0.907251i	$0.361823\pi$
$278$	8.00000	0.479808
$279$	−4.00000	−0.239474
$280$	−2.00000	−0.119523
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	−12.0000	−0.714590
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	6.00000	0.356034
$285$	−6.00000	−0.355409
$286$	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
$292$	−4.00000	−0.234082
$293$	−26.0000	−1.51894	−0.759468	−	0.650545i	$-0.774541\pi$
$293$	−26.0000	−1.51894	−0.759468	+	0.650545i	$0.774541\pi$
$294$	−3.00000	−0.174964
$295$	6.00000	0.349334
$296$	−2.00000	−0.116248
$297$	−1.00000	−0.0580259
$298$	−14.0000	−0.810998
$299$	8.00000	0.462652
$300$	1.00000	0.0577350
$301$	−12.0000	−0.691669
$302$	−8.00000	−0.460348
$303$	−10.0000	−0.574485
$304$	−6.00000	−0.344124
$305$	2.00000	0.114520
$306$	1.00000	0.0571662
$307$	18.0000	1.02731	0.513657	−	0.857996i	$-0.328290\pi$
$307$	18.0000	1.02731	0.513657	+	0.857996i	$0.328290\pi$
$308$	2.00000	0.113961
$309$	0	0
$310$	−4.00000	−0.227185
$311$	−2.00000	−0.113410	−0.0567048	−	0.998391i	$-0.518059\pi$
$311$	−2.00000	−0.113410	−0.0567048	+	0.998391i	$0.518059\pi$
$312$	−4.00000	−0.226455
$313$	−34.0000	−1.92179	−0.960897	−	0.276907i	$-0.910691\pi$
$313$	−34.0000	−1.92179	−0.960897	+	0.276907i	$0.910691\pi$
$314$	−10.0000	−0.564333
$315$	−2.00000	−0.112687
$316$	10.0000	0.562544
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	1.00000	0.0559017
$321$	−20.0000	−1.11629
$322$	4.00000	0.222911
$323$	−6.00000	−0.333849
$324$	1.00000	0.0555556
$325$	−4.00000	−0.221880
$326$	4.00000	0.221540
$327$	−10.0000	−0.553001
$328$	−6.00000	−0.331295
$329$	24.0000	1.32316
$330$	−1.00000	−0.0550482
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	−4.00000	−0.219529
$333$	−2.00000	−0.109599
$334$	12.0000	0.656611
$335$	0	0
$336$	−2.00000	−0.109109
$337$	−20.0000	−1.08947	−0.544735	−	0.838608i	$-0.683370\pi$
$337$	−20.0000	−1.08947	−0.544735	+	0.838608i	$0.683370\pi$
$338$	3.00000	0.163178
$339$	−4.00000	−0.217250
$340$	1.00000	0.0542326
$341$	4.00000	0.216612
$342$	−6.00000	−0.324443
$343$	20.0000	1.07990
$344$	6.00000	0.323498
$345$	−2.00000	−0.107676
$346$	14.0000	0.752645
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	−6.00000	−0.321634
$349$	−24.0000	−1.28469	−0.642345	−	0.766415i	$-0.722038\pi$
$349$	−24.0000	−1.28469	−0.642345	+	0.766415i	$0.722038\pi$
$350$	−2.00000	−0.106904
$351$	−4.00000	−0.213504
$352$	−1.00000	−0.0533002
$353$	26.0000	1.38384	0.691920	−	0.721974i	$-0.256765\pi$
$353$	26.0000	1.38384	0.691920	+	0.721974i	$0.256765\pi$
$354$	6.00000	0.318896
$355$	6.00000	0.318447
$356$	6.00000	0.317999
$357$	−2.00000	−0.105851
$358$	2.00000	0.105703
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	1.00000	0.0527046
$361$	17.0000	0.894737
$362$	−2.00000	−0.105118
$363$	1.00000	0.0524864
$364$	8.00000	0.419314
$365$	−4.00000	−0.209370
$366$	2.00000	0.104542
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	−2.00000	−0.104257
$369$	−6.00000	−0.312348
$370$	−2.00000	−0.103975
$371$	0	0
$372$	−4.00000	−0.207390
$373$	−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
$376$	−12.0000	−0.618853
$377$	24.0000	1.23606
$378$	−2.00000	−0.102869
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	−6.00000	−0.307794
$381$	12.0000	0.614779
$382$	16.0000	0.818631
$383$	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
$386$	0	0
$387$	6.00000	0.304997
$388$	14.0000	0.710742
$389$	16.0000	0.811232	0.405616	−	0.914044i	$-0.367057\pi$
$389$	16.0000	0.811232	0.405616	+	0.914044i	$0.367057\pi$
$390$	−4.00000	−0.202548
$391$	−2.00000	−0.101144
$392$	−3.00000	−0.151523
$393$	20.0000	1.00887
$394$	−6.00000	−0.302276
$395$	10.0000	0.503155
$396$	−1.00000	−0.0502519
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	−20.0000	−1.00251
$399$	12.0000	0.600751
$400$	1.00000	0.0500000
$401$	32.0000	1.59800	0.799002	−	0.601329i	$-0.205362\pi$
$401$	32.0000	1.59800	0.799002	+	0.601329i	$0.205362\pi$
$402$	0	0
$403$	16.0000	0.797017
$404$	−10.0000	−0.497519
$405$	1.00000	0.0496904
$406$	12.0000	0.595550
$407$	2.00000	0.0991363
$408$	1.00000	0.0495074
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	−6.00000	−0.296319
$411$	−14.0000	−0.690569
$412$	0	0
$413$	−12.0000	−0.590481
$414$	−2.00000	−0.0982946
$415$	−4.00000	−0.196352
$416$	−4.00000	−0.196116
$417$	8.00000	0.391762
$418$	6.00000	0.293470
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	−2.00000	−0.0975900
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	−8.00000	−0.389434
$423$	−12.0000	−0.583460
$424$	0	0
$425$	1.00000	0.0485071
$426$	6.00000	0.290701
$427$	−4.00000	−0.193574
$428$	−20.0000	−0.966736
$429$	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	$0.390791\pi$
$434$	8.00000	0.384012
$435$	−6.00000	−0.287678
$436$	−10.0000	−0.478913
$437$	12.0000	0.574038
$438$	−4.00000	−0.191127
$439$	−30.0000	−1.43182	−0.715911	−	0.698192i	$-0.753988\pi$
$439$	−30.0000	−1.43182	−0.715911	+	0.698192i	$0.753988\pi$
$440$	−1.00000	−0.0476731
$441$	−3.00000	−0.142857
$442$	−4.00000	−0.190261
$443$	18.0000	0.855206	0.427603	−	0.903967i	$-0.359358\pi$
$443$	18.0000	0.855206	0.427603	+	0.903967i	$0.359358\pi$
$444$	−2.00000	−0.0949158
$445$	6.00000	0.284427
$446$	0	0
$447$	−14.0000	−0.662177
$448$	−2.00000	−0.0944911
$449$	−12.0000	−0.566315	−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000	−0.566315	−0.283158	+	0.959073i	$0.591382\pi$
$450$	1.00000	0.0471405
$451$	6.00000	0.282529
$452$	−4.00000	−0.188144
$453$	−8.00000	−0.375873
$454$	12.0000	0.563188
$455$	8.00000	0.375046
$456$	−6.00000	−0.280976
$457$	−18.0000	−0.842004	−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000	−0.842004	−0.421002	+	0.907060i	$0.638322\pi$
$458$	10.0000	0.467269
$459$	1.00000	0.0466760
$460$	−2.00000	−0.0932505
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	2.00000	0.0930484
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	−6.00000	−0.278543
$465$	−4.00000	−0.185496
$466$	−10.0000	−0.463241
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	−4.00000	−0.184900
$469$	0	0
$470$	−12.0000	−0.553519
$471$	−10.0000	−0.460776
$472$	6.00000	0.276172
$473$	−6.00000	−0.275880
$474$	10.0000	0.459315
$475$	−6.00000	−0.275299
$476$	−2.00000	−0.0916698
$477$	0	0
$478$	−24.0000	−1.09773
$479$	32.0000	1.46212	0.731059	−	0.682315i	$-0.239027\pi$
$479$	32.0000	1.46212	0.731059	+	0.682315i	$0.239027\pi$
$480$	1.00000	0.0456435
$481$	8.00000	0.364769
$482$	−12.0000	−0.546585
$483$	4.00000	0.182006
$484$	1.00000	0.0454545
$485$	14.0000	0.635707
$486$	1.00000	0.0453609
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	2.00000	0.0905357
$489$	4.00000	0.180886
$490$	−3.00000	−0.135526
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	−6.00000	−0.270501
$493$	−6.00000	−0.270226
$494$	24.0000	1.07981
$495$	−1.00000	−0.0449467
$496$	−4.00000	−0.179605
$497$	−12.0000	−0.538274
$498$	−4.00000	−0.179244
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	1.00000	0.0447214
$501$	12.0000	0.536120
$502$	18.0000	0.803379
$503$	40.0000	1.78351	0.891756	−	0.452517i	$-0.149474\pi$
$503$	40.0000	1.78351	0.891756	+	0.452517i	$0.149474\pi$
$504$	−2.00000	−0.0890871
$505$	−10.0000	−0.444994
$506$	2.00000	0.0889108
$507$	3.00000	0.133235
$508$	12.0000	0.532414
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	1.00000	0.0442807
$511$	8.00000	0.353899
$512$	1.00000	0.0441942
$513$	−6.00000	−0.264906
$514$	26.0000	1.14681
$515$	0	0
$516$	6.00000	0.264135
$517$	12.0000	0.527759
$518$	4.00000	0.175750
$519$	14.0000	0.614532
$520$	−4.00000	−0.175412
$521$	24.0000	1.05146	0.525730	−	0.850652i	$-0.323792\pi$
$521$	24.0000	1.05146	0.525730	+	0.850652i	$0.323792\pi$
$522$	−6.00000	−0.262613
$523$	26.0000	1.13690	0.568450	−	0.822718i	$-0.307543\pi$
$523$	26.0000	1.13690	0.568450	+	0.822718i	$0.307543\pi$
$524$	20.0000	0.873704
$525$	−2.00000	−0.0872872
$526$	0	0
$527$	−4.00000	−0.174243
$528$	−1.00000	−0.0435194
$529$	−19.0000	−0.826087
$530$	0	0
$531$	6.00000	0.260378
$532$	12.0000	0.520266
$533$	24.0000	1.03956
$534$	6.00000	0.259645
$535$	−20.0000	−0.864675
$536$	0	0
$537$	2.00000	0.0863064
$538$	6.00000	0.258678
$539$	3.00000	0.129219
$540$	1.00000	0.0430331
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	−20.0000	−0.859074
$543$	−2.00000	−0.0858282
$544$	1.00000	0.0428746
$545$	−10.0000	−0.428353
$546$	8.00000	0.342368
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	−14.0000	−0.598050
$549$	2.00000	0.0853579
$550$	−1.00000	−0.0426401
$551$	36.0000	1.53365
$552$	−2.00000	−0.0851257
$553$	−20.0000	−0.850487
$554$	14.0000	0.594803
$555$	−2.00000	−0.0848953
$556$	8.00000	0.339276
$557$	−10.0000	−0.423714	−0.211857	−	0.977301i	$-0.567951\pi$
$557$	−10.0000	−0.423714	−0.211857	+	0.977301i	$0.567951\pi$
$558$	−4.00000	−0.169334
$559$	−24.0000	−1.01509
$560$	−2.00000	−0.0845154
$561$	−1.00000	−0.0422200
$562$	−10.0000	−0.421825
$563$	28.0000	1.18006	0.590030	−	0.807382i	$-0.299116\pi$
$563$	28.0000	1.18006	0.590030	+	0.807382i	$0.299116\pi$
$564$	−12.0000	−0.505291
$565$	−4.00000	−0.168281
$566$	16.0000	0.672530
$567$	−2.00000	−0.0839921
$568$	6.00000	0.251754
$569$	−38.0000	−1.59304	−0.796521	−	0.604610i	$-0.793329\pi$
$569$	−38.0000	−1.59304	−0.796521	+	0.604610i	$0.793329\pi$
$570$	−6.00000	−0.251312
$571$	−16.0000	−0.669579	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000	−0.669579	−0.334790	+	0.942293i	$0.608665\pi$
$572$	4.00000	0.167248
$573$	16.0000	0.668410
$574$	12.0000	0.500870
$575$	−2.00000	−0.0834058
$576$	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
$579$	0	0
$580$	−6.00000	−0.249136
$581$	8.00000	0.331896
$582$	14.0000	0.580319
$583$	0	0
$584$	−4.00000	−0.165521
$585$	−4.00000	−0.165380
$586$	−26.0000	−1.07405
$587$	18.0000	0.742940	0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000	0.742940	0.371470	+	0.928445i	$0.378854\pi$
$588$	−3.00000	−0.123718
$589$	24.0000	0.988903
$590$	6.00000	0.247016
$591$	−6.00000	−0.246807
$592$	−2.00000	−0.0821995
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	−1.00000	−0.0410305
$595$	−2.00000	−0.0819920
$596$	−14.0000	−0.573462
$597$	−20.0000	−0.818546
$598$	8.00000	0.327144
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	1.00000	0.0408248
$601$	−16.0000	−0.652654	−0.326327	−	0.945257i	$-0.605811\pi$
$601$	−16.0000	−0.652654	−0.326327	+	0.945257i	$0.605811\pi$
$602$	−12.0000	−0.489083
$603$	0	0
$604$	−8.00000	−0.325515
$605$	1.00000	0.0406558
$606$	−10.0000	−0.406222
$607$	46.0000	1.86708	0.933541	−	0.358470i	$-0.116702\pi$
$607$	46.0000	1.86708	0.933541	+	0.358470i	$0.116702\pi$
$608$	−6.00000	−0.243332
$609$	12.0000	0.486265
$610$	2.00000	0.0809776
$611$	48.0000	1.94187
$612$	1.00000	0.0404226
$613$	4.00000	0.161558	0.0807792	−	0.996732i	$-0.474259\pi$
$613$	4.00000	0.161558	0.0807792	+	0.996732i	$0.474259\pi$
$614$	18.0000	0.726421
$615$	−6.00000	−0.241943
$616$	2.00000	0.0805823
$617$	−8.00000	−0.322068	−0.161034	−	0.986949i	$-0.551483\pi$
$617$	−8.00000	−0.322068	−0.161034	+	0.986949i	$0.551483\pi$
$618$	0	0
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	−4.00000	−0.160644
$621$	−2.00000	−0.0802572
$622$	−2.00000	−0.0801927
$623$	−12.0000	−0.480770
$624$	−4.00000	−0.160128
$625$	1.00000	0.0400000
$626$	−34.0000	−1.35891
$627$	6.00000	0.239617
$628$	−10.0000	−0.399043
$629$	−2.00000	−0.0797452
$630$	−2.00000	−0.0796819
$631$	−32.0000	−1.27390	−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000	−1.27390	−0.636950	+	0.770905i	$0.719804\pi$
$632$	10.0000	0.397779
$633$	−8.00000	−0.317971
$634$	2.00000	0.0794301
$635$	12.0000	0.476205
$636$	0	0
$637$	12.0000	0.475457
$638$	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	$0.5\pi$
$642$	−20.0000	−0.789337
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	4.00000	0.157622
$645$	6.00000	0.236250
$646$	−6.00000	−0.236067
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	1.00000	0.0392837
$649$	−6.00000	−0.235521
$650$	−4.00000	−0.156893
$651$	8.00000	0.313545
$652$	4.00000	0.156652
$653$	22.0000	0.860927	0.430463	−	0.902608i	$-0.358350\pi$
$653$	22.0000	0.860927	0.430463	+	0.902608i	$0.358350\pi$
$654$	−10.0000	−0.391031
$655$	20.0000	0.781465
$656$	−6.00000	−0.234261
$657$	−4.00000	−0.156055
$658$	24.0000	0.935617
$659$	28.0000	1.09073	0.545363	−	0.838200i	$-0.316392\pi$
$659$	28.0000	1.09073	0.545363	+	0.838200i	$0.316392\pi$
$660$	−1.00000	−0.0389249
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	20.0000	0.777322
$663$	−4.00000	−0.155347
$664$	−4.00000	−0.155230
$665$	12.0000	0.465340
$666$	−2.00000	−0.0774984
$667$	12.0000	0.464642
$668$	12.0000	0.464294
$669$	0	0
$670$	0	0
$671$	−2.00000	−0.0772091
$672$	−2.00000	−0.0771517
$673$	32.0000	1.23351	0.616755	−	0.787155i	$-0.288447\pi$
$673$	32.0000	1.23351	0.616755	+	0.787155i	$0.288447\pi$
$674$	−20.0000	−0.770371
$675$	1.00000	0.0384900
$676$	3.00000	0.115385
$677$	−38.0000	−1.46046	−0.730229	−	0.683202i	$-0.760587\pi$
$677$	−38.0000	−1.46046	−0.730229	+	0.683202i	$0.760587\pi$
$678$	−4.00000	−0.153619
$679$	−28.0000	−1.07454
$680$	1.00000	0.0383482
$681$	12.0000	0.459841
$682$	4.00000	0.153168
$683$	44.0000	1.68361	0.841807	−	0.539779i	$-0.181492\pi$
$683$	44.0000	1.68361	0.841807	+	0.539779i	$0.181492\pi$
$684$	−6.00000	−0.229416
$685$	−14.0000	−0.534913
$686$	20.0000	0.763604
$687$	10.0000	0.381524
$688$	6.00000	0.228748
$689$	0	0
$690$	−2.00000	−0.0761387
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	14.0000	0.532200
$693$	2.00000	0.0759737
$694$	−28.0000	−1.06287
$695$	8.00000	0.303457
$696$	−6.00000	−0.227429
$697$	−6.00000	−0.227266
$698$	−24.0000	−0.908413
$699$	−10.0000	−0.378235
$700$	−2.00000	−0.0755929
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	−4.00000	−0.150970
$703$	12.0000	0.452589
$704$	−1.00000	−0.0376889
$705$	−12.0000	−0.451946
$706$	26.0000	0.978523
$707$	20.0000	0.752177
$708$	6.00000	0.225494
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	6.00000	0.225176
$711$	10.0000	0.375029
$712$	6.00000	0.224860
$713$	8.00000	0.299602
$714$	−2.00000	−0.0748481
$715$	4.00000	0.149592
$716$	2.00000	0.0747435
$717$	−24.0000	−0.896296
$718$	−16.0000	−0.597115
$719$	−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
$721$	0	0
$722$	17.0000	0.632674
$723$	−12.0000	−0.446285
$724$	−2.00000	−0.0743294
$725$	−6.00000	−0.222834
$726$	1.00000	0.0371135
$727$	−40.0000	−1.48352	−0.741759	−	0.670667i	$-0.766008\pi$
$727$	−40.0000	−1.48352	−0.741759	+	0.670667i	$0.766008\pi$
$728$	8.00000	0.296500
$729$	1.00000	0.0370370
$730$	−4.00000	−0.148047
$731$	6.00000	0.221918
$732$	2.00000	0.0739221
$733$	−4.00000	−0.147743	−0.0738717	−	0.997268i	$-0.523536\pi$
$733$	−4.00000	−0.147743	−0.0738717	+	0.997268i	$0.523536\pi$
$734$	32.0000	1.18114
$735$	−3.00000	−0.110657
$736$	−2.00000	−0.0737210
$737$	0	0
$738$	−6.00000	−0.220863
$739$	−30.0000	−1.10357	−0.551784	−	0.833987i	$-0.686053\pi$
$739$	−30.0000	−1.10357	−0.551784	+	0.833987i	$0.686053\pi$
$740$	−2.00000	−0.0735215
$741$	24.0000	0.881662
$742$	0	0
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	−4.00000	−0.146647
$745$	−14.0000	−0.512920
$746$	−12.0000	−0.439351
$747$	−4.00000	−0.146352
$748$	−1.00000	−0.0365636
$749$	40.0000	1.46157
$750$	1.00000	0.0365148
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	−12.0000	−0.437595
$753$	18.0000	0.655956
$754$	24.0000	0.874028
$755$	−8.00000	−0.291150
$756$	−2.00000	−0.0727393
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	−28.0000	−1.01701
$759$	2.00000	0.0725954
$760$	−6.00000	−0.217643
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	12.0000	0.434714
$763$	20.0000	0.724049
$764$	16.0000	0.578860
$765$	1.00000	0.0361551
$766$	0	0
$767$	−24.0000	−0.866590
$768$	1.00000	0.0360844
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	2.00000	0.0720750
$771$	26.0000	0.936367
$772$	0	0
$773$	−20.0000	−0.719350	−0.359675	−	0.933078i	$-0.617112\pi$
$773$	−20.0000	−0.719350	−0.359675	+	0.933078i	$0.617112\pi$
$774$	6.00000	0.215666
$775$	−4.00000	−0.143684
$776$	14.0000	0.502571
$777$	4.00000	0.143499
$778$	16.0000	0.573628
$779$	36.0000	1.28983
$780$	−4.00000	−0.143223
$781$	−6.00000	−0.214697
$782$	−2.00000	−0.0715199
$783$	−6.00000	−0.214423
$784$	−3.00000	−0.107143
$785$	−10.0000	−0.356915
$786$	20.0000	0.713376
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	−6.00000	−0.213741
$789$	0	0
$790$	10.0000	0.355784
$791$	8.00000	0.284447
$792$	−1.00000	−0.0355335
$793$	−8.00000	−0.284088
$794$	−14.0000	−0.496841
$795$	0	0
$796$	−20.0000	−0.708881
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	12.0000	0.424795
$799$	−12.0000	−0.424529
$800$	1.00000	0.0353553
$801$	6.00000	0.212000
$802$	32.0000	1.12996
$803$	4.00000	0.141157
$804$	0	0
$805$	4.00000	0.140981
$806$	16.0000	0.563576
$807$	6.00000	0.211210
$808$	−10.0000	−0.351799
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	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	$0.522373\pi$
$812$	12.0000	0.421117
$813$	−20.0000	−0.701431
$814$	2.00000	0.0701000
$815$	4.00000	0.140114
$816$	1.00000	0.0350070
$817$	−36.0000	−1.25948
$818$	14.0000	0.489499
$819$	8.00000	0.279543
$820$	−6.00000	−0.209529
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	−14.0000	−0.488306
$823$	36.0000	1.25488	0.627441	−	0.778664i	$-0.284103\pi$
$823$	36.0000	1.25488	0.627441	+	0.778664i	$0.284103\pi$
$824$	0	0
$825$	−1.00000	−0.0348155
$826$	−12.0000	−0.417533
$827$	−28.0000	−0.973655	−0.486828	−	0.873498i	$-0.661846\pi$
$827$	−28.0000	−0.973655	−0.486828	+	0.873498i	$0.661846\pi$
$828$	−2.00000	−0.0695048
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	−4.00000	−0.138842
$831$	14.0000	0.485655
$832$	−4.00000	−0.138675
$833$	−3.00000	−0.103944
$834$	8.00000	0.277017
$835$	12.0000	0.415277
$836$	6.00000	0.207514
$837$	−4.00000	−0.138260
$838$	−12.0000	−0.414533
$839$	−10.0000	−0.345238	−0.172619	−	0.984989i	$-0.555223\pi$
$839$	−10.0000	−0.345238	−0.172619	+	0.984989i	$0.555223\pi$
$840$	−2.00000	−0.0690066
$841$	7.00000	0.241379
$842$	6.00000	0.206774
$843$	−10.0000	−0.344418
$844$	−8.00000	−0.275371
$845$	3.00000	0.103203
$846$	−12.0000	−0.412568
$847$	−2.00000	−0.0687208
$848$	0	0
$849$	16.0000	0.549119
$850$	1.00000	0.0342997
$851$	4.00000	0.137118
$852$	6.00000	0.205557
$853$	−34.0000	−1.16414	−0.582069	−	0.813139i	$-0.697757\pi$
$853$	−34.0000	−1.16414	−0.582069	+	0.813139i	$0.697757\pi$
$854$	−4.00000	−0.136877
$855$	−6.00000	−0.205196
$856$	−20.0000	−0.683586
$857$	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
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	6.00000	0.204598
$861$	12.0000	0.408959
$862$	−12.0000	−0.408722
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	1.00000	0.0340207
$865$	14.0000	0.476014
$866$	14.0000	0.475739
$867$	1.00000	0.0339618
$868$	8.00000	0.271538
$869$	−10.0000	−0.339227
$870$	−6.00000	−0.203419
$871$	0	0
$872$	−10.0000	−0.338643
$873$	14.0000	0.473828
$874$	12.0000	0.405906
$875$	−2.00000	−0.0676123
$876$	−4.00000	−0.135147
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	−30.0000	−1.01245
$879$	−26.0000	−0.876958
$880$	−1.00000	−0.0337100
$881$	−4.00000	−0.134763	−0.0673817	−	0.997727i	$-0.521465\pi$
$881$	−4.00000	−0.134763	−0.0673817	+	0.997727i	$0.521465\pi$
$882$	−3.00000	−0.101015
$883$	12.0000	0.403832	0.201916	−	0.979403i	$-0.435283\pi$
$883$	12.0000	0.403832	0.201916	+	0.979403i	$0.435283\pi$
$884$	−4.00000	−0.134535
$885$	6.00000	0.201688
$886$	18.0000	0.604722
$887$	−12.0000	−0.402921	−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000	−0.402921	−0.201460	+	0.979497i	$0.564569\pi$
$888$	−2.00000	−0.0671156
$889$	−24.0000	−0.804934
$890$	6.00000	0.201120
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	72.0000	2.40939
$894$	−14.0000	−0.468230
$895$	2.00000	0.0668526
$896$	−2.00000	−0.0668153
$897$	8.00000	0.267112
$898$	−12.0000	−0.400445
$899$	24.0000	0.800445
$900$	1.00000	0.0333333
$901$	0	0
$902$	6.00000	0.199778
$903$	−12.0000	−0.399335
$904$	−4.00000	−0.133038
$905$	−2.00000	−0.0664822
$906$	−8.00000	−0.265782
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	12.0000	0.398234
$909$	−10.0000	−0.331679
$910$	8.00000	0.265197
$911$	22.0000	0.728893	0.364446	−	0.931224i	$-0.381258\pi$
$911$	22.0000	0.728893	0.364446	+	0.931224i	$0.381258\pi$
$912$	−6.00000	−0.198680
$913$	4.00000	0.132381
$914$	−18.0000	−0.595387
$915$	2.00000	0.0661180
$916$	10.0000	0.330409
$917$	−40.0000	−1.32092
$918$	1.00000	0.0330049
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	−2.00000	−0.0659380
$921$	18.0000	0.593120
$922$	−6.00000	−0.197599
$923$	−24.0000	−0.789970
$924$	2.00000	0.0657952
$925$	−2.00000	−0.0657596
$926$	8.00000	0.262896
$927$	0	0
$928$	−6.00000	−0.196960
$929$	4.00000	0.131236	0.0656179	−	0.997845i	$-0.479098\pi$
$929$	4.00000	0.131236	0.0656179	+	0.997845i	$0.479098\pi$
$930$	−4.00000	−0.131165
$931$	18.0000	0.589926
$932$	−10.0000	−0.327561
$933$	−2.00000	−0.0654771
$934$	−6.00000	−0.196326
$935$	−1.00000	−0.0327035
$936$	−4.00000	−0.130744
$937$	−10.0000	−0.326686	−0.163343	−	0.986569i	$-0.552228\pi$
$937$	−10.0000	−0.326686	−0.163343	+	0.986569i	$0.552228\pi$
$938$	0	0
$939$	−34.0000	−1.10955
$940$	−12.0000	−0.391397
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	−10.0000	−0.325818
$943$	12.0000	0.390774
$944$	6.00000	0.195283
$945$	−2.00000	−0.0650600
$946$	−6.00000	−0.195077
$947$	8.00000	0.259965	0.129983	−	0.991516i	$-0.458508\pi$
$947$	8.00000	0.259965	0.129983	+	0.991516i	$0.458508\pi$
$948$	10.0000	0.324785
$949$	16.0000	0.519382
$950$	−6.00000	−0.194666
$951$	2.00000	0.0648544
$952$	−2.00000	−0.0648204
$953$	14.0000	0.453504	0.226752	−	0.973952i	$-0.427189\pi$
$953$	14.0000	0.453504	0.226752	+	0.973952i	$0.427189\pi$
$954$	0	0
$955$	16.0000	0.517748
$956$	−24.0000	−0.776215
$957$	6.00000	0.193952
$958$	32.0000	1.03387
$959$	28.0000	0.904167
$960$	1.00000	0.0322749
$961$	−15.0000	−0.483871
$962$	8.00000	0.257930
$963$	−20.0000	−0.644491
$964$	−12.0000	−0.386494
$965$	0	0
$966$	4.00000	0.128698
$967$	−12.0000	−0.385894	−0.192947	−	0.981209i	$-0.561805\pi$
$967$	−12.0000	−0.385894	−0.192947	+	0.981209i	$0.561805\pi$
$968$	1.00000	0.0321412
$969$	−6.00000	−0.192748
$970$	14.0000	0.449513
$971$	−42.0000	−1.34784	−0.673922	−	0.738802i	$-0.735392\pi$
$971$	−42.0000	−1.34784	−0.673922	+	0.738802i	$0.735392\pi$
$972$	1.00000	0.0320750
$973$	−16.0000	−0.512936
$974$	−32.0000	−1.02535
$975$	−4.00000	−0.128103
$976$	2.00000	0.0640184
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	4.00000	0.127906
$979$	−6.00000	−0.191761
$980$	−3.00000	−0.0958315
$981$	−10.0000	−0.319275
$982$	−20.0000	−0.638226
$983$	6.00000	0.191370	0.0956851	−	0.995412i	$-0.469496\pi$
$983$	6.00000	0.191370	0.0956851	+	0.995412i	$0.469496\pi$
$984$	−6.00000	−0.191273
$985$	−6.00000	−0.191176
$986$	−6.00000	−0.191079
$987$	24.0000	0.763928
$988$	24.0000	0.763542
$989$	−12.0000	−0.381578
$990$	−1.00000	−0.0317821
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	−4.00000	−0.127000
$993$	20.0000	0.634681
$994$	−12.0000	−0.380617
$995$	−20.0000	−0.634043
$996$	−4.00000	−0.126745
$997$	−22.0000	−0.696747	−0.348373	−	0.937356i	$-0.613266\pi$
$997$	−22.0000	−0.696747	−0.348373	+	0.937356i	$0.613266\pi$
$998$	−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.bl.1.1	✓	1

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