LMFDB - Embedded newform 5610.2.a.bg.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)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} -4.00000 q^{13} -1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} -1.00000 q^{21} -1.00000 q^{22} +9.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -4.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +9.00000 q^{29} -1.00000 q^{30} -1.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} +1.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} -4.00000 q^{38} -4.00000 q^{39} -1.00000 q^{40} +12.0000 q^{41} -1.00000 q^{42} -1.00000 q^{43} -1.00000 q^{44} -1.00000 q^{45} +9.00000 q^{46} +1.00000 q^{48} -6.00000 q^{49} +1.00000 q^{50} +1.00000 q^{51} -4.00000 q^{52} +1.00000 q^{54} +1.00000 q^{55} -1.00000 q^{56} -4.00000 q^{57} +9.00000 q^{58} +6.00000 q^{59} -1.00000 q^{60} +2.00000 q^{61} -1.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} -1.00000 q^{66} +8.00000 q^{67} +1.00000 q^{68} +9.00000 q^{69} +1.00000 q^{70} -6.00000 q^{71} +1.00000 q^{72} +2.00000 q^{73} +2.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} +1.00000 q^{77} -4.00000 q^{78} -4.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +12.0000 q^{82} +12.0000 q^{83} -1.00000 q^{84} -1.00000 q^{85} -1.00000 q^{86} +9.00000 q^{87} -1.00000 q^{88} -1.00000 q^{90} +4.00000 q^{91} +9.00000 q^{92} -1.00000 q^{93} +4.00000 q^{95} +1.00000 q^{96} +17.0000 q^{97} -6.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	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	$0.687167\pi$
$14$	−1.00000	−0.267261
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	1.00000	0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	−1.00000	−0.223607
$21$	−1.00000	−0.218218
$22$	−1.00000	−0.213201
$23$	9.00000	1.87663	0.938315	−	0.345782i	$-0.112386\pi$
$23$	9.00000	1.87663	0.938315	+	0.345782i	$0.112386\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	−4.00000	−0.784465
$27$	1.00000	0.192450
$28$	−1.00000	−0.188982
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	−1.00000	−0.182574
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	1.00000	0.171499
$35$	1.00000	0.169031
$36$	1.00000	0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−4.00000	−0.648886
$39$	−4.00000	−0.640513
$40$	−1.00000	−0.158114
$41$	12.0000	1.87409	0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000	1.87409	0.937043	+	0.349215i	$0.113552\pi$
$42$	−1.00000	−0.154303
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	−1.00000	−0.150756
$45$	−1.00000	−0.149071
$46$	9.00000	1.32698
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	1.00000	0.144338
$49$	−6.00000	−0.857143
$50$	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
$56$	−1.00000	−0.133631
$57$	−4.00000	−0.529813
$58$	9.00000	1.18176
$59$	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	$0.459132\pi$
$62$	−1.00000	−0.127000
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	4.00000	0.496139
$66$	−1.00000	−0.123091
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	1.00000	0.121268
$69$	9.00000	1.08347
$70$	1.00000	0.119523
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	1.00000	0.117851
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	2.00000	0.232495
$75$	1.00000	0.115470
$76$	−4.00000	−0.458831
$77$	1.00000	0.113961
$78$	−4.00000	−0.452911
$79$	−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
$82$	12.0000	1.32518
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	−1.00000	−0.109109
$85$	−1.00000	−0.108465
$86$	−1.00000	−0.107833
$87$	9.00000	0.964901
$88$	−1.00000	−0.106600
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	−1.00000	−0.105409
$91$	4.00000	0.419314
$92$	9.00000	0.938315
$93$	−1.00000	−0.103695
$94$	0	0
$95$	4.00000	0.410391
$96$	1.00000	0.102062
$97$	17.0000	1.72609	0.863044	−	0.505128i	$-0.168555\pi$
$97$	17.0000	1.72609	0.863044	+	0.505128i	$0.168555\pi$
$98$	−6.00000	−0.606092
$99$	−1.00000	−0.100504
$100$	1.00000	0.100000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	1.00000	0.0990148
$103$	−1.00000	−0.0985329	−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000	−0.0985329	−0.0492665	+	0.998786i	$0.515688\pi$
$104$	−4.00000	−0.392232
$105$	1.00000	0.0975900
$106$	0	0
$107$	3.00000	0.290021	0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000	0.290021	0.145010	+	0.989430i	$0.453678\pi$
$108$	1.00000	0.0962250
$109$	20.0000	1.91565	0.957826	−	0.287348i	$-0.0927736\pi$
$109$	20.0000	1.91565	0.957826	+	0.287348i	$0.0927736\pi$
$110$	1.00000	0.0953463
$111$	2.00000	0.189832
$112$	−1.00000	−0.0944911
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	−4.00000	−0.374634
$115$	−9.00000	−0.839254
$116$	9.00000	0.835629
$117$	−4.00000	−0.369800
$118$	6.00000	0.552345
$119$	−1.00000	−0.0916698
$120$	−1.00000	−0.0912871
$121$	1.00000	0.0909091
$122$	2.00000	0.181071
$123$	12.0000	1.08200
$124$	−1.00000	−0.0898027
$125$	−1.00000	−0.0894427
$126$	−1.00000	−0.0890871
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	1.00000	0.0883883
$129$	−1.00000	−0.0880451
$130$	4.00000	0.350823
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	−1.00000	−0.0870388
$133$	4.00000	0.346844
$134$	8.00000	0.691095
$135$	−1.00000	−0.0860663
$136$	1.00000	0.0857493
$137$	−21.0000	−1.79415	−0.897076	−	0.441877i	$-0.854313\pi$
$137$	−21.0000	−1.79415	−0.897076	+	0.441877i	$0.854313\pi$
$138$	9.00000	0.766131
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	1.00000	0.0845154
$141$	0	0
$142$	−6.00000	−0.503509
$143$	4.00000	0.334497
$144$	1.00000	0.0833333
$145$	−9.00000	−0.747409
$146$	2.00000	0.165521
$147$	−6.00000	−0.494872
$148$	2.00000	0.164399
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	1.00000	0.0816497
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	−4.00000	−0.324443
$153$	1.00000	0.0808452
$154$	1.00000	0.0805823
$155$	1.00000	0.0803219
$156$	−4.00000	−0.320256
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	−4.00000	−0.318223
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	−9.00000	−0.709299
$162$	1.00000	0.0785674
$163$	−1.00000	−0.0783260	−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000	−0.0783260	−0.0391630	+	0.999233i	$0.512469\pi$
$164$	12.0000	0.937043
$165$	1.00000	0.0778499
$166$	12.0000	0.931381
$167$	−18.0000	−1.39288	−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000	−1.39288	−0.696441	+	0.717614i	$0.745234\pi$
$168$	−1.00000	−0.0771517
$169$	3.00000	0.230769
$170$	−1.00000	−0.0766965
$171$	−4.00000	−0.305888
$172$	−1.00000	−0.0762493
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	9.00000	0.682288
$175$	−1.00000	−0.0755929
$176$	−1.00000	−0.0753778
$177$	6.00000	0.450988
$178$	0	0
$179$	−18.0000	−1.34538	−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000	−1.34538	−0.672692	+	0.739923i	$0.734862\pi$
$180$	−1.00000	−0.0745356
$181$	11.0000	0.817624	0.408812	−	0.912619i	$-0.365943\pi$
$181$	11.0000	0.817624	0.408812	+	0.912619i	$0.365943\pi$
$182$	4.00000	0.296500
$183$	2.00000	0.147844
$184$	9.00000	0.663489
$185$	−2.00000	−0.147043
$186$	−1.00000	−0.0733236
$187$	−1.00000	−0.0731272
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	4.00000	0.290191
$191$	−9.00000	−0.651217	−0.325609	−	0.945505i	$-0.605569\pi$
$191$	−9.00000	−0.651217	−0.325609	+	0.945505i	$0.605569\pi$
$192$	1.00000	0.0721688
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	17.0000	1.22053
$195$	4.00000	0.286446
$196$	−6.00000	−0.428571
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	−1.00000	−0.0710669
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	1.00000	0.0707107
$201$	8.00000	0.564276
$202$	6.00000	0.422159
$203$	−9.00000	−0.631676
$204$	1.00000	0.0700140
$205$	−12.0000	−0.838116
$206$	−1.00000	−0.0696733
$207$	9.00000	0.625543
$208$	−4.00000	−0.277350
$209$	4.00000	0.276686
$210$	1.00000	0.0690066
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	0	0
$213$	−6.00000	−0.411113
$214$	3.00000	0.205076
$215$	1.00000	0.0681994
$216$	1.00000	0.0680414
$217$	1.00000	0.0678844
$218$	20.0000	1.35457
$219$	2.00000	0.135147
$220$	1.00000	0.0674200
$221$	−4.00000	−0.269069
$222$	2.00000	0.134231
$223$	23.0000	1.54019	0.770097	−	0.637927i	$-0.220208\pi$
$223$	23.0000	1.54019	0.770097	+	0.637927i	$0.220208\pi$
$224$	−1.00000	−0.0668153
$225$	1.00000	0.0666667
$226$	−6.00000	−0.399114
$227$	3.00000	0.199117	0.0995585	−	0.995032i	$-0.468257\pi$
$227$	3.00000	0.199117	0.0995585	+	0.995032i	$0.468257\pi$
$228$	−4.00000	−0.264906
$229$	−28.0000	−1.85029	−0.925146	−	0.379611i	$-0.876058\pi$
$229$	−28.0000	−1.85029	−0.925146	+	0.379611i	$0.876058\pi$
$230$	−9.00000	−0.593442
$231$	1.00000	0.0657952
$232$	9.00000	0.590879
$233$	−3.00000	−0.196537	−0.0982683	−	0.995160i	$-0.531330\pi$
$233$	−3.00000	−0.196537	−0.0982683	+	0.995160i	$0.531330\pi$
$234$	−4.00000	−0.261488
$235$	0	0
$236$	6.00000	0.390567
$237$	−4.00000	−0.259828
$238$	−1.00000	−0.0648204
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	−1.00000	−0.0645497
$241$	−25.0000	−1.61039	−0.805196	−	0.593009i	$-0.797940\pi$
$241$	−25.0000	−1.61039	−0.805196	+	0.593009i	$0.797940\pi$
$242$	1.00000	0.0642824
$243$	1.00000	0.0641500
$244$	2.00000	0.128037
$245$	6.00000	0.383326
$246$	12.0000	0.765092
$247$	16.0000	1.01806
$248$	−1.00000	−0.0635001
$249$	12.0000	0.760469
$250$	−1.00000	−0.0632456
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	−1.00000	−0.0629941
$253$	−9.00000	−0.565825
$254$	2.00000	0.125491
$255$	−1.00000	−0.0626224
$256$	1.00000	0.0625000
$257$	−3.00000	−0.187135	−0.0935674	−	0.995613i	$-0.529827\pi$
$257$	−3.00000	−0.187135	−0.0935674	+	0.995613i	$0.529827\pi$
$258$	−1.00000	−0.0622573
$259$	−2.00000	−0.124274
$260$	4.00000	0.248069
$261$	9.00000	0.557086
$262$	0	0
$263$	−3.00000	−0.184988	−0.0924940	−	0.995713i	$-0.529484\pi$
$263$	−3.00000	−0.184988	−0.0924940	+	0.995713i	$0.529484\pi$
$264$	−1.00000	−0.0615457
$265$	0	0
$266$	4.00000	0.245256
$267$	0	0
$268$	8.00000	0.488678
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	−1.00000	−0.0608581
$271$	11.0000	0.668202	0.334101	−	0.942537i	$-0.391567\pi$
$271$	11.0000	0.668202	0.334101	+	0.942537i	$0.391567\pi$
$272$	1.00000	0.0606339
$273$	4.00000	0.242091
$274$	−21.0000	−1.26866
$275$	−1.00000	−0.0603023
$276$	9.00000	0.541736
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	5.00000	0.299880
$279$	−1.00000	−0.0598684
$280$	1.00000	0.0597614
$281$	15.0000	0.894825	0.447412	−	0.894328i	$-0.352346\pi$
$281$	15.0000	0.894825	0.447412	+	0.894328i	$0.352346\pi$
$282$	0	0
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	−6.00000	−0.356034
$285$	4.00000	0.236940
$286$	4.00000	0.236525
$287$	−12.0000	−0.708338
$288$	1.00000	0.0589256
$289$	1.00000	0.0588235
$290$	−9.00000	−0.528498
$291$	17.0000	0.996558
$292$	2.00000	0.117041
$293$	−9.00000	−0.525786	−0.262893	−	0.964825i	$-0.584677\pi$
$293$	−9.00000	−0.525786	−0.262893	+	0.964825i	$0.584677\pi$
$294$	−6.00000	−0.349927
$295$	−6.00000	−0.349334
$296$	2.00000	0.116248
$297$	−1.00000	−0.0580259
$298$	12.0000	0.695141
$299$	−36.0000	−2.08193
$300$	1.00000	0.0577350
$301$	1.00000	0.0576390
$302$	20.0000	1.15087
$303$	6.00000	0.344691
$304$	−4.00000	−0.229416
$305$	−2.00000	−0.114520
$306$	1.00000	0.0571662
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	1.00000	0.0569803
$309$	−1.00000	−0.0568880
$310$	1.00000	0.0567962
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	−4.00000	−0.226455
$313$	−1.00000	−0.0565233	−0.0282617	−	0.999601i	$-0.508997\pi$
$313$	−1.00000	−0.0565233	−0.0282617	+	0.999601i	$0.508997\pi$
$314$	2.00000	0.112867
$315$	1.00000	0.0563436
$316$	−4.00000	−0.225018
$317$	3.00000	0.168497	0.0842484	−	0.996445i	$-0.473151\pi$
$317$	3.00000	0.168497	0.0842484	+	0.996445i	$0.473151\pi$
$318$	0	0
$319$	−9.00000	−0.503903
$320$	−1.00000	−0.0559017
$321$	3.00000	0.167444
$322$	−9.00000	−0.501550
$323$	−4.00000	−0.222566
$324$	1.00000	0.0555556
$325$	−4.00000	−0.221880
$326$	−1.00000	−0.0553849
$327$	20.0000	1.10600
$328$	12.0000	0.662589
$329$	0	0
$330$	1.00000	0.0550482
$331$	−7.00000	−0.384755	−0.192377	−	0.981321i	$-0.561620\pi$
$331$	−7.00000	−0.384755	−0.192377	+	0.981321i	$0.561620\pi$
$332$	12.0000	0.658586
$333$	2.00000	0.109599
$334$	−18.0000	−0.984916
$335$	−8.00000	−0.437087
$336$	−1.00000	−0.0545545
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	3.00000	0.163178
$339$	−6.00000	−0.325875
$340$	−1.00000	−0.0542326
$341$	1.00000	0.0541530
$342$	−4.00000	−0.216295
$343$	13.0000	0.701934
$344$	−1.00000	−0.0539164
$345$	−9.00000	−0.484544
$346$	−6.00000	−0.322562
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	9.00000	0.482451
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	−1.00000	−0.0534522
$351$	−4.00000	−0.213504
$352$	−1.00000	−0.0533002
$353$	3.00000	0.159674	0.0798369	−	0.996808i	$-0.474560\pi$
$353$	3.00000	0.159674	0.0798369	+	0.996808i	$0.474560\pi$
$354$	6.00000	0.318896
$355$	6.00000	0.318447
$356$	0	0
$357$	−1.00000	−0.0529256
$358$	−18.0000	−0.951330
$359$	18.0000	0.950004	0.475002	−	0.879985i	$-0.342447\pi$
$359$	18.0000	0.950004	0.475002	+	0.879985i	$0.342447\pi$
$360$	−1.00000	−0.0527046
$361$	−3.00000	−0.157895
$362$	11.0000	0.578147
$363$	1.00000	0.0524864
$364$	4.00000	0.209657
$365$	−2.00000	−0.104685
$366$	2.00000	0.104542
$367$	−22.0000	−1.14839	−0.574195	−	0.818718i	$-0.694685\pi$
$367$	−22.0000	−1.14839	−0.574195	+	0.818718i	$0.694685\pi$
$368$	9.00000	0.469157
$369$	12.0000	0.624695
$370$	−2.00000	−0.103975
$371$	0	0
$372$	−1.00000	−0.0518476
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	−1.00000	−0.0517088
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−36.0000	−1.85409
$378$	−1.00000	−0.0514344
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	4.00000	0.205196
$381$	2.00000	0.102463
$382$	−9.00000	−0.460480
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	1.00000	0.0510310
$385$	−1.00000	−0.0509647
$386$	2.00000	0.101797
$387$	−1.00000	−0.0508329
$388$	17.0000	0.863044
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	4.00000	0.202548
$391$	9.00000	0.455150
$392$	−6.00000	−0.303046
$393$	0	0
$394$	−12.0000	−0.604551
$395$	4.00000	0.201262
$396$	−1.00000	−0.0502519
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	−16.0000	−0.802008
$399$	4.00000	0.200250
$400$	1.00000	0.0500000
$401$	15.0000	0.749064	0.374532	−	0.927214i	$-0.377803\pi$
$401$	15.0000	0.749064	0.374532	+	0.927214i	$0.377803\pi$
$402$	8.00000	0.399004
$403$	4.00000	0.199254
$404$	6.00000	0.298511
$405$	−1.00000	−0.0496904
$406$	−9.00000	−0.446663
$407$	−2.00000	−0.0991363
$408$	1.00000	0.0495074
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	−12.0000	−0.592638
$411$	−21.0000	−1.03585
$412$	−1.00000	−0.0492665
$413$	−6.00000	−0.295241
$414$	9.00000	0.442326
$415$	−12.0000	−0.589057
$416$	−4.00000	−0.196116
$417$	5.00000	0.244851
$418$	4.00000	0.195646
$419$	−39.0000	−1.90527	−0.952637	−	0.304109i	$-0.901641\pi$
$419$	−39.0000	−1.90527	−0.952637	+	0.304109i	$0.901641\pi$
$420$	1.00000	0.0487950
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	−13.0000	−0.632830
$423$	0	0
$424$	0	0
$425$	1.00000	0.0485071
$426$	−6.00000	−0.290701
$427$	−2.00000	−0.0967868
$428$	3.00000	0.145010
$429$	4.00000	0.193122
$430$	1.00000	0.0482243
$431$	27.0000	1.30054	0.650272	−	0.759701i	$-0.274655\pi$
$431$	27.0000	1.30054	0.650272	+	0.759701i	$0.274655\pi$
$432$	1.00000	0.0481125
$433$	8.00000	0.384455	0.192228	−	0.981350i	$-0.438429\pi$
$433$	8.00000	0.384455	0.192228	+	0.981350i	$0.438429\pi$
$434$	1.00000	0.0480015
$435$	−9.00000	−0.431517
$436$	20.0000	0.957826
$437$	−36.0000	−1.72211
$438$	2.00000	0.0955637
$439$	26.0000	1.24091	0.620456	−	0.784241i	$-0.286947\pi$
$439$	26.0000	1.24091	0.620456	+	0.784241i	$0.286947\pi$
$440$	1.00000	0.0476731
$441$	−6.00000	−0.285714
$442$	−4.00000	−0.190261
$443$	3.00000	0.142534	0.0712672	−	0.997457i	$-0.477296\pi$
$443$	3.00000	0.142534	0.0712672	+	0.997457i	$0.477296\pi$
$444$	2.00000	0.0949158
$445$	0	0
$446$	23.0000	1.08908
$447$	12.0000	0.567581
$448$	−1.00000	−0.0472456
$449$	−3.00000	−0.141579	−0.0707894	−	0.997491i	$-0.522552\pi$
$449$	−3.00000	−0.141579	−0.0707894	+	0.997491i	$0.522552\pi$
$450$	1.00000	0.0471405
$451$	−12.0000	−0.565058
$452$	−6.00000	−0.282216
$453$	20.0000	0.939682
$454$	3.00000	0.140797
$455$	−4.00000	−0.187523
$456$	−4.00000	−0.187317
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	−28.0000	−1.30835
$459$	1.00000	0.0466760
$460$	−9.00000	−0.419627
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	1.00000	0.0465242
$463$	32.0000	1.48717	0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000	1.48717	0.743583	+	0.668644i	$0.233125\pi$
$464$	9.00000	0.417815
$465$	1.00000	0.0463739
$466$	−3.00000	−0.138972
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	−4.00000	−0.184900
$469$	−8.00000	−0.369406
$470$	0	0
$471$	2.00000	0.0921551
$472$	6.00000	0.276172
$473$	1.00000	0.0459800
$474$	−4.00000	−0.183726
$475$	−4.00000	−0.183533
$476$	−1.00000	−0.0458349
$477$	0	0
$478$	6.00000	0.274434
$479$	−3.00000	−0.137073	−0.0685367	−	0.997649i	$-0.521833\pi$
$479$	−3.00000	−0.137073	−0.0685367	+	0.997649i	$0.521833\pi$
$480$	−1.00000	−0.0456435
$481$	−8.00000	−0.364769
$482$	−25.0000	−1.13872
$483$	−9.00000	−0.409514
$484$	1.00000	0.0454545
$485$	−17.0000	−0.771930
$486$	1.00000	0.0453609
$487$	−34.0000	−1.54069	−0.770344	−	0.637629i	$-0.779915\pi$
$487$	−34.0000	−1.54069	−0.770344	+	0.637629i	$0.779915\pi$
$488$	2.00000	0.0905357
$489$	−1.00000	−0.0452216
$490$	6.00000	0.271052
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	12.0000	0.541002
$493$	9.00000	0.405340
$494$	16.0000	0.719874
$495$	1.00000	0.0449467
$496$	−1.00000	−0.0449013
$497$	6.00000	0.269137
$498$	12.0000	0.537733
$499$	−40.0000	−1.79065	−0.895323	−	0.445418i	$-0.853055\pi$
$499$	−40.0000	−1.79065	−0.895323	+	0.445418i	$0.853055\pi$
$500$	−1.00000	−0.0447214
$501$	−18.0000	−0.804181
$502$	−12.0000	−0.535586
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	−1.00000	−0.0445435
$505$	−6.00000	−0.266996
$506$	−9.00000	−0.400099
$507$	3.00000	0.133235
$508$	2.00000	0.0887357
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	−1.00000	−0.0442807
$511$	−2.00000	−0.0884748
$512$	1.00000	0.0441942
$513$	−4.00000	−0.176604
$514$	−3.00000	−0.132324
$515$	1.00000	0.0440653
$516$	−1.00000	−0.0440225
$517$	0	0
$518$	−2.00000	−0.0878750
$519$	−6.00000	−0.263371
$520$	4.00000	0.175412
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	9.00000	0.393919
$523$	−25.0000	−1.09317	−0.546587	−	0.837402i	$-0.684073\pi$
$523$	−25.0000	−1.09317	−0.546587	+	0.837402i	$0.684073\pi$
$524$	0	0
$525$	−1.00000	−0.0436436
$526$	−3.00000	−0.130806
$527$	−1.00000	−0.0435607
$528$	−1.00000	−0.0435194
$529$	58.0000	2.52174
$530$	0	0
$531$	6.00000	0.260378
$532$	4.00000	0.173422
$533$	−48.0000	−2.07911
$534$	0	0
$535$	−3.00000	−0.129701
$536$	8.00000	0.345547
$537$	−18.0000	−0.776757
$538$	−18.0000	−0.776035
$539$	6.00000	0.258438
$540$	−1.00000	−0.0430331
$541$	−28.0000	−1.20381	−0.601907	−	0.798566i	$-0.705592\pi$
$541$	−28.0000	−1.20381	−0.601907	+	0.798566i	$0.705592\pi$
$542$	11.0000	0.472490
$543$	11.0000	0.472055
$544$	1.00000	0.0428746
$545$	−20.0000	−0.856706
$546$	4.00000	0.171184
$547$	−10.0000	−0.427569	−0.213785	−	0.976881i	$-0.568579\pi$
$547$	−10.0000	−0.427569	−0.213785	+	0.976881i	$0.568579\pi$
$548$	−21.0000	−0.897076
$549$	2.00000	0.0853579
$550$	−1.00000	−0.0426401
$551$	−36.0000	−1.53365
$552$	9.00000	0.383065
$553$	4.00000	0.170097
$554$	26.0000	1.10463
$555$	−2.00000	−0.0848953
$556$	5.00000	0.212047
$557$	33.0000	1.39825	0.699127	−	0.714997i	$-0.253572\pi$
$557$	33.0000	1.39825	0.699127	+	0.714997i	$0.253572\pi$
$558$	−1.00000	−0.0423334
$559$	4.00000	0.169182
$560$	1.00000	0.0422577
$561$	−1.00000	−0.0422200
$562$	15.0000	0.632737
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	14.0000	0.588464
$567$	−1.00000	−0.0419961
$568$	−6.00000	−0.251754
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	4.00000	0.167542
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	4.00000	0.167248
$573$	−9.00000	−0.375980
$574$	−12.0000	−0.500870
$575$	9.00000	0.375326
$576$	1.00000	0.0416667
$577$	26.0000	1.08239	0.541197	−	0.840896i	$-0.317971\pi$
$577$	26.0000	1.08239	0.541197	+	0.840896i	$0.317971\pi$
$578$	1.00000	0.0415945
$579$	2.00000	0.0831172
$580$	−9.00000	−0.373705
$581$	−12.0000	−0.497844
$582$	17.0000	0.704673
$583$	0	0
$584$	2.00000	0.0827606
$585$	4.00000	0.165380
$586$	−9.00000	−0.371787
$587$	−27.0000	−1.11441	−0.557205	−	0.830375i	$-0.688126\pi$
$587$	−27.0000	−1.11441	−0.557205	+	0.830375i	$0.688126\pi$
$588$	−6.00000	−0.247436
$589$	4.00000	0.164817
$590$	−6.00000	−0.247016
$591$	−12.0000	−0.493614
$592$	2.00000	0.0821995
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	−1.00000	−0.0410305
$595$	1.00000	0.0409960
$596$	12.0000	0.491539
$597$	−16.0000	−0.654836
$598$	−36.0000	−1.47215
$599$	9.00000	0.367730	0.183865	−	0.982952i	$-0.441139\pi$
$599$	9.00000	0.367730	0.183865	+	0.982952i	$0.441139\pi$
$600$	1.00000	0.0408248
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	1.00000	0.0407570
$603$	8.00000	0.325785
$604$	20.0000	0.813788
$605$	−1.00000	−0.0406558
$606$	6.00000	0.243733
$607$	−40.0000	−1.62355	−0.811775	−	0.583970i	$-0.801498\pi$
$607$	−40.0000	−1.62355	−0.811775	+	0.583970i	$0.801498\pi$
$608$	−4.00000	−0.162221
$609$	−9.00000	−0.364698
$610$	−2.00000	−0.0809776
$611$	0	0
$612$	1.00000	0.0404226
$613$	−46.0000	−1.85792	−0.928961	−	0.370177i	$-0.879297\pi$
$613$	−46.0000	−1.85792	−0.928961	+	0.370177i	$0.879297\pi$
$614$	20.0000	0.807134
$615$	−12.0000	−0.483887
$616$	1.00000	0.0402911
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	−1.00000	−0.0402259
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	1.00000	0.0401610
$621$	9.00000	0.361158
$622$	0	0
$623$	0	0
$624$	−4.00000	−0.160128
$625$	1.00000	0.0400000
$626$	−1.00000	−0.0399680
$627$	4.00000	0.159745
$628$	2.00000	0.0798087
$629$	2.00000	0.0797452
$630$	1.00000	0.0398410
$631$	26.0000	1.03504	0.517522	−	0.855670i	$-0.326855\pi$
$631$	26.0000	1.03504	0.517522	+	0.855670i	$0.326855\pi$
$632$	−4.00000	−0.159111
$633$	−13.0000	−0.516704
$634$	3.00000	0.119145
$635$	−2.00000	−0.0793676
$636$	0	0
$637$	24.0000	0.950915
$638$	−9.00000	−0.356313
$639$	−6.00000	−0.237356
$640$	−1.00000	−0.0395285
$641$	33.0000	1.30342	0.651711	−	0.758468i	$-0.274052\pi$
$641$	33.0000	1.30342	0.651711	+	0.758468i	$0.274052\pi$
$642$	3.00000	0.118401
$643$	−7.00000	−0.276053	−0.138027	−	0.990429i	$-0.544076\pi$
$643$	−7.00000	−0.276053	−0.138027	+	0.990429i	$0.544076\pi$
$644$	−9.00000	−0.354650
$645$	1.00000	0.0393750
$646$	−4.00000	−0.157378
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	1.00000	0.0392837
$649$	−6.00000	−0.235521
$650$	−4.00000	−0.156893
$651$	1.00000	0.0391931
$652$	−1.00000	−0.0391630
$653$	3.00000	0.117399	0.0586995	−	0.998276i	$-0.481305\pi$
$653$	3.00000	0.117399	0.0586995	+	0.998276i	$0.481305\pi$
$654$	20.0000	0.782062
$655$	0	0
$656$	12.0000	0.468521
$657$	2.00000	0.0780274
$658$	0	0
$659$	−15.0000	−0.584317	−0.292159	−	0.956370i	$-0.594373\pi$
$659$	−15.0000	−0.584317	−0.292159	+	0.956370i	$0.594373\pi$
$660$	1.00000	0.0389249
$661$	−16.0000	−0.622328	−0.311164	−	0.950356i	$-0.600719\pi$
$661$	−16.0000	−0.622328	−0.311164	+	0.950356i	$0.600719\pi$
$662$	−7.00000	−0.272063
$663$	−4.00000	−0.155347
$664$	12.0000	0.465690
$665$	−4.00000	−0.155113
$666$	2.00000	0.0774984
$667$	81.0000	3.13633
$668$	−18.0000	−0.696441
$669$	23.0000	0.889231
$670$	−8.00000	−0.309067
$671$	−2.00000	−0.0772091
$672$	−1.00000	−0.0385758
$673$	−16.0000	−0.616755	−0.308377	−	0.951264i	$-0.599786\pi$
$673$	−16.0000	−0.616755	−0.308377	+	0.951264i	$0.599786\pi$
$674$	14.0000	0.539260
$675$	1.00000	0.0384900
$676$	3.00000	0.115385
$677$	−42.0000	−1.61419	−0.807096	−	0.590421i	$-0.798962\pi$
$677$	−42.0000	−1.61419	−0.807096	+	0.590421i	$0.798962\pi$
$678$	−6.00000	−0.230429
$679$	−17.0000	−0.652400
$680$	−1.00000	−0.0383482
$681$	3.00000	0.114960
$682$	1.00000	0.0382920
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	−4.00000	−0.152944
$685$	21.0000	0.802369
$686$	13.0000	0.496342
$687$	−28.0000	−1.06827
$688$	−1.00000	−0.0381246
$689$	0	0
$690$	−9.00000	−0.342624
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	−6.00000	−0.228086
$693$	1.00000	0.0379869
$694$	12.0000	0.455514
$695$	−5.00000	−0.189661
$696$	9.00000	0.341144
$697$	12.0000	0.454532
$698$	−22.0000	−0.832712
$699$	−3.00000	−0.113470
$700$	−1.00000	−0.0377964
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	−4.00000	−0.150970
$703$	−8.00000	−0.301726
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	3.00000	0.112906
$707$	−6.00000	−0.225653
$708$	6.00000	0.225494
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	6.00000	0.225176
$711$	−4.00000	−0.150012
$712$	0	0
$713$	−9.00000	−0.337053
$714$	−1.00000	−0.0374241
$715$	−4.00000	−0.149592
$716$	−18.0000	−0.672692
$717$	6.00000	0.224074
$718$	18.0000	0.671754
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	−1.00000	−0.0372678
$721$	1.00000	0.0372419
$722$	−3.00000	−0.111648
$723$	−25.0000	−0.929760
$724$	11.0000	0.408812
$725$	9.00000	0.334252
$726$	1.00000	0.0371135
$727$	−49.0000	−1.81731	−0.908655	−	0.417548i	$-0.862889\pi$
$727$	−49.0000	−1.81731	−0.908655	+	0.417548i	$0.862889\pi$
$728$	4.00000	0.148250
$729$	1.00000	0.0370370
$730$	−2.00000	−0.0740233
$731$	−1.00000	−0.0369863
$732$	2.00000	0.0739221
$733$	32.0000	1.18195	0.590973	−	0.806691i	$-0.298744\pi$
$733$	32.0000	1.18195	0.590973	+	0.806691i	$0.298744\pi$
$734$	−22.0000	−0.812035
$735$	6.00000	0.221313
$736$	9.00000	0.331744
$737$	−8.00000	−0.294684
$738$	12.0000	0.441726
$739$	−22.0000	−0.809283	−0.404642	−	0.914475i	$-0.632604\pi$
$739$	−22.0000	−0.809283	−0.404642	+	0.914475i	$0.632604\pi$
$740$	−2.00000	−0.0735215
$741$	16.0000	0.587775
$742$	0	0
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	−1.00000	−0.0366618
$745$	−12.0000	−0.439646
$746$	14.0000	0.512576
$747$	12.0000	0.439057
$748$	−1.00000	−0.0365636
$749$	−3.00000	−0.109618
$750$	−1.00000	−0.0365148
$751$	−31.0000	−1.13121	−0.565603	−	0.824678i	$-0.691357\pi$
$751$	−31.0000	−1.13121	−0.565603	+	0.824678i	$0.691357\pi$
$752$	0	0
$753$	−12.0000	−0.437304
$754$	−36.0000	−1.31104
$755$	−20.0000	−0.727875
$756$	−1.00000	−0.0363696
$757$	47.0000	1.70824	0.854122	−	0.520073i	$-0.174095\pi$
$757$	47.0000	1.70824	0.854122	+	0.520073i	$0.174095\pi$
$758$	−16.0000	−0.581146
$759$	−9.00000	−0.326679
$760$	4.00000	0.145095
$761$	9.00000	0.326250	0.163125	−	0.986605i	$-0.447843\pi$
$761$	9.00000	0.326250	0.163125	+	0.986605i	$0.447843\pi$
$762$	2.00000	0.0724524
$763$	−20.0000	−0.724049
$764$	−9.00000	−0.325609
$765$	−1.00000	−0.0361551
$766$	0	0
$767$	−24.0000	−0.866590
$768$	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	$0.557708\pi$
$770$	−1.00000	−0.0360375
$771$	−3.00000	−0.108042
$772$	2.00000	0.0719816
$773$	24.0000	0.863220	0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000	0.863220	0.431610	+	0.902060i	$0.357946\pi$
$774$	−1.00000	−0.0359443
$775$	−1.00000	−0.0359211
$776$	17.0000	0.610264
$777$	−2.00000	−0.0717496
$778$	30.0000	1.07555
$779$	−48.0000	−1.71978
$780$	4.00000	0.143223
$781$	6.00000	0.214697
$782$	9.00000	0.321839
$783$	9.00000	0.321634
$784$	−6.00000	−0.214286
$785$	−2.00000	−0.0713831
$786$	0	0
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	−12.0000	−0.427482
$789$	−3.00000	−0.106803
$790$	4.00000	0.142314
$791$	6.00000	0.213335
$792$	−1.00000	−0.0355335
$793$	−8.00000	−0.284088
$794$	2.00000	0.0709773
$795$	0	0
$796$	−16.0000	−0.567105
$797$	−24.0000	−0.850124	−0.425062	−	0.905164i	$-0.639748\pi$
$797$	−24.0000	−0.850124	−0.425062	+	0.905164i	$0.639748\pi$
$798$	4.00000	0.141598
$799$	0	0
$800$	1.00000	0.0353553
$801$	0	0
$802$	15.0000	0.529668
$803$	−2.00000	−0.0705785
$804$	8.00000	0.282138
$805$	9.00000	0.317208
$806$	4.00000	0.140894
$807$	−18.0000	−0.633630
$808$	6.00000	0.211079
$809$	−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
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	−9.00000	−0.315838
$813$	11.0000	0.385787
$814$	−2.00000	−0.0701000
$815$	1.00000	0.0350285
$816$	1.00000	0.0350070
$817$	4.00000	0.139942
$818$	32.0000	1.11885
$819$	4.00000	0.139771
$820$	−12.0000	−0.419058
$821$	27.0000	0.942306	0.471153	−	0.882051i	$-0.343838\pi$
$821$	27.0000	0.942306	0.471153	+	0.882051i	$0.343838\pi$
$822$	−21.0000	−0.732459
$823$	14.0000	0.488009	0.244005	−	0.969774i	$-0.421539\pi$
$823$	14.0000	0.488009	0.244005	+	0.969774i	$0.421539\pi$
$824$	−1.00000	−0.0348367
$825$	−1.00000	−0.0348155
$826$	−6.00000	−0.208767
$827$	−27.0000	−0.938882	−0.469441	−	0.882964i	$-0.655545\pi$
$827$	−27.0000	−0.938882	−0.469441	+	0.882964i	$0.655545\pi$
$828$	9.00000	0.312772
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	−12.0000	−0.416526
$831$	26.0000	0.901930
$832$	−4.00000	−0.138675
$833$	−6.00000	−0.207888
$834$	5.00000	0.173136
$835$	18.0000	0.622916
$836$	4.00000	0.138343
$837$	−1.00000	−0.0345651
$838$	−39.0000	−1.34723
$839$	−18.0000	−0.621429	−0.310715	−	0.950503i	$-0.600568\pi$
$839$	−18.0000	−0.621429	−0.310715	+	0.950503i	$0.600568\pi$
$840$	1.00000	0.0345033
$841$	52.0000	1.79310
$842$	8.00000	0.275698
$843$	15.0000	0.516627
$844$	−13.0000	−0.447478
$845$	−3.00000	−0.103203
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	14.0000	0.480479
$850$	1.00000	0.0342997
$851$	18.0000	0.617032
$852$	−6.00000	−0.205557
$853$	−13.0000	−0.445112	−0.222556	−	0.974920i	$-0.571440\pi$
$853$	−13.0000	−0.445112	−0.222556	+	0.974920i	$0.571440\pi$
$854$	−2.00000	−0.0684386
$855$	4.00000	0.136797
$856$	3.00000	0.102538
$857$	−45.0000	−1.53717	−0.768585	−	0.639747i	$-0.779039\pi$
$857$	−45.0000	−1.53717	−0.768585	+	0.639747i	$0.779039\pi$
$858$	4.00000	0.136558
$859$	−1.00000	−0.0341196	−0.0170598	−	0.999854i	$-0.505431\pi$
$859$	−1.00000	−0.0341196	−0.0170598	+	0.999854i	$0.505431\pi$
$860$	1.00000	0.0340997
$861$	−12.0000	−0.408959
$862$	27.0000	0.919624
$863$	36.0000	1.22545	0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000	1.22545	0.612727	+	0.790295i	$0.290072\pi$
$864$	1.00000	0.0340207
$865$	6.00000	0.204006
$866$	8.00000	0.271851
$867$	1.00000	0.0339618
$868$	1.00000	0.0339422
$869$	4.00000	0.135691
$870$	−9.00000	−0.305129
$871$	−32.0000	−1.08428
$872$	20.0000	0.677285
$873$	17.0000	0.575363
$874$	−36.0000	−1.21772
$875$	1.00000	0.0338062
$876$	2.00000	0.0675737
$877$	−37.0000	−1.24940	−0.624701	−	0.780864i	$-0.714779\pi$
$877$	−37.0000	−1.24940	−0.624701	+	0.780864i	$0.714779\pi$
$878$	26.0000	0.877457
$879$	−9.00000	−0.303562
$880$	1.00000	0.0337100
$881$	−9.00000	−0.303218	−0.151609	−	0.988441i	$-0.548445\pi$
$881$	−9.00000	−0.303218	−0.151609	+	0.988441i	$0.548445\pi$
$882$	−6.00000	−0.202031
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	−4.00000	−0.134535
$885$	−6.00000	−0.201688
$886$	3.00000	0.100787
$887$	36.0000	1.20876	0.604381	−	0.796696i	$-0.293421\pi$
$887$	36.0000	1.20876	0.604381	+	0.796696i	$0.293421\pi$
$888$	2.00000	0.0671156
$889$	−2.00000	−0.0670778
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	23.0000	0.770097
$893$	0	0
$894$	12.0000	0.401340
$895$	18.0000	0.601674
$896$	−1.00000	−0.0334077
$897$	−36.0000	−1.20201
$898$	−3.00000	−0.100111
$899$	−9.00000	−0.300167
$900$	1.00000	0.0333333
$901$	0	0
$902$	−12.0000	−0.399556
$903$	1.00000	0.0332779
$904$	−6.00000	−0.199557
$905$	−11.0000	−0.365652
$906$	20.0000	0.664455
$907$	−55.0000	−1.82625	−0.913123	−	0.407685i	$-0.866336\pi$
$907$	−55.0000	−1.82625	−0.913123	+	0.407685i	$0.866336\pi$
$908$	3.00000	0.0995585
$909$	6.00000	0.199007
$910$	−4.00000	−0.132599
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	−4.00000	−0.132453
$913$	−12.0000	−0.397142
$914$	−10.0000	−0.330771
$915$	−2.00000	−0.0661180
$916$	−28.0000	−0.925146
$917$	0	0
$918$	1.00000	0.0330049
$919$	29.0000	0.956622	0.478311	−	0.878191i	$-0.341249\pi$
$919$	29.0000	0.956622	0.478311	+	0.878191i	$0.341249\pi$
$920$	−9.00000	−0.296721
$921$	20.0000	0.659022
$922$	−18.0000	−0.592798
$923$	24.0000	0.789970
$924$	1.00000	0.0328976
$925$	2.00000	0.0657596
$926$	32.0000	1.05159
$927$	−1.00000	−0.0328443
$928$	9.00000	0.295439
$929$	−27.0000	−0.885841	−0.442921	−	0.896561i	$-0.646058\pi$
$929$	−27.0000	−0.885841	−0.442921	+	0.896561i	$0.646058\pi$
$930$	1.00000	0.0327913
$931$	24.0000	0.786568
$932$	−3.00000	−0.0982683
$933$	0	0
$934$	12.0000	0.392652
$935$	1.00000	0.0327035
$936$	−4.00000	−0.130744
$937$	−34.0000	−1.11073	−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000	−1.11073	−0.555366	+	0.831606i	$0.687422\pi$
$938$	−8.00000	−0.261209
$939$	−1.00000	−0.0326338
$940$	0	0
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	2.00000	0.0651635
$943$	108.000	3.51696
$944$	6.00000	0.195283
$945$	1.00000	0.0325300
$946$	1.00000	0.0325128
$947$	−6.00000	−0.194974	−0.0974869	−	0.995237i	$-0.531080\pi$
$947$	−6.00000	−0.194974	−0.0974869	+	0.995237i	$0.531080\pi$
$948$	−4.00000	−0.129914
$949$	−8.00000	−0.259691
$950$	−4.00000	−0.129777
$951$	3.00000	0.0972817
$952$	−1.00000	−0.0324102
$953$	18.0000	0.583077	0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000	0.583077	0.291539	+	0.956559i	$0.405833\pi$
$954$	0	0
$955$	9.00000	0.291233
$956$	6.00000	0.194054
$957$	−9.00000	−0.290929
$958$	−3.00000	−0.0969256
$959$	21.0000	0.678125
$960$	−1.00000	−0.0322749
$961$	−30.0000	−0.967742
$962$	−8.00000	−0.257930
$963$	3.00000	0.0966736
$964$	−25.0000	−0.805196
$965$	−2.00000	−0.0643823
$966$	−9.00000	−0.289570
$967$	−22.0000	−0.707472	−0.353736	−	0.935345i	$-0.615089\pi$
$967$	−22.0000	−0.707472	−0.353736	+	0.935345i	$0.615089\pi$
$968$	1.00000	0.0321412
$969$	−4.00000	−0.128499
$970$	−17.0000	−0.545837
$971$	24.0000	0.770197	0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000	0.770197	0.385098	+	0.922876i	$0.374168\pi$
$972$	1.00000	0.0320750
$973$	−5.00000	−0.160293
$974$	−34.0000	−1.08943
$975$	−4.00000	−0.128103
$976$	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	$0.340676\pi$
$978$	−1.00000	−0.0319765
$979$	0	0
$980$	6.00000	0.191663
$981$	20.0000	0.638551
$982$	24.0000	0.765871
$983$	33.0000	1.05254	0.526268	−	0.850319i	$-0.323591\pi$
$983$	33.0000	1.05254	0.526268	+	0.850319i	$0.323591\pi$
$984$	12.0000	0.382546
$985$	12.0000	0.382352
$986$	9.00000	0.286618
$987$	0	0
$988$	16.0000	0.509028
$989$	−9.00000	−0.286183
$990$	1.00000	0.0317821
$991$	29.0000	0.921215	0.460608	−	0.887604i	$-0.347632\pi$
$991$	29.0000	0.921215	0.460608	+	0.887604i	$0.347632\pi$
$992$	−1.00000	−0.0317500
$993$	−7.00000	−0.222138
$994$	6.00000	0.190308
$995$	16.0000	0.507234
$996$	12.0000	0.380235
$997$	−7.00000	−0.221692	−0.110846	−	0.993838i	$-0.535356\pi$
$997$	−7.00000	−0.221692	−0.110846	+	0.993838i	$0.535356\pi$
$998$	−40.0000	−1.26618
$999$	2.00000	0.0632772

Twists

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

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