LMFDB - Embedded newform 110.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [110,2,Mod(1,110)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(110, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("110.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 110 = 2 \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	110.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:	$0.878354422234$
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	$\chi$	$=$	110.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} -2.00000 q^{18} -1.00000 q^{19} -1.00000 q^{20} -1.00000 q^{21} -1.00000 q^{22} +6.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -5.00000 q^{27} -1.00000 q^{28} -9.00000 q^{29} -1.00000 q^{30} +5.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} -3.00000 q^{34} +1.00000 q^{35} -2.00000 q^{36} +5.00000 q^{37} -1.00000 q^{38} +2.00000 q^{39} -1.00000 q^{40} -6.00000 q^{41} -1.00000 q^{42} +8.00000 q^{43} -1.00000 q^{44} +2.00000 q^{45} +6.00000 q^{46} +6.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} +1.00000 q^{50} -3.00000 q^{51} +2.00000 q^{52} +9.00000 q^{53} -5.00000 q^{54} +1.00000 q^{55} -1.00000 q^{56} -1.00000 q^{57} -9.00000 q^{58} +6.00000 q^{59} -1.00000 q^{60} +5.00000 q^{61} +5.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} -1.00000 q^{66} +8.00000 q^{67} -3.00000 q^{68} +6.00000 q^{69} +1.00000 q^{70} -9.00000 q^{71} -2.00000 q^{72} -10.0000 q^{73} +5.00000 q^{74} +1.00000 q^{75} -1.00000 q^{76} +1.00000 q^{77} +2.00000 q^{78} +14.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -6.00000 q^{83} -1.00000 q^{84} +3.00000 q^{85} +8.00000 q^{86} -9.00000 q^{87} -1.00000 q^{88} -15.0000 q^{89} +2.00000 q^{90} -2.00000 q^{91} +6.00000 q^{92} +5.00000 q^{93} +6.00000 q^{94} +1.00000 q^{95} +1.00000 q^{96} +8.00000 q^{97} -6.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	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
$9$	−2.00000	−0.666667
$10$	−1.00000	−0.316228
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.00000	0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−1.00000	−0.267261
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	−2.00000	−0.471405
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	−1.00000	−0.223607
$21$	−1.00000	−0.218218
$22$	−1.00000	−0.213201
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	−5.00000	−0.962250
$28$	−1.00000	−0.188982
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	−1.00000	−0.182574
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	−3.00000	−0.514496
$35$	1.00000	0.169031
$36$	−2.00000	−0.333333
$37$	5.00000	0.821995	0.410997	−	0.911636i	$-0.365181\pi$
$37$	5.00000	0.821995	0.410997	+	0.911636i	$0.365181\pi$
$38$	−1.00000	−0.162221
$39$	2.00000	0.320256
$40$	−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	$0.655213\pi$
$42$	−1.00000	−0.154303
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	−1.00000	−0.150756
$45$	2.00000	0.298142
$46$	6.00000	0.884652
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	1.00000	0.144338
$49$	−6.00000	−0.857143
$50$	1.00000	0.141421
$51$	−3.00000	−0.420084
$52$	2.00000	0.277350
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	−5.00000	−0.680414
$55$	1.00000	0.134840
$56$	−1.00000	−0.133631
$57$	−1.00000	−0.132453
$58$	−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
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	5.00000	0.635001
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	−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	$0.337479\pi$
$68$	−3.00000	−0.363803
$69$	6.00000	0.722315
$70$	1.00000	0.119523
$71$	−9.00000	−1.06810	−0.534052	−	0.845452i	$-0.679331\pi$
$71$	−9.00000	−1.06810	−0.534052	+	0.845452i	$0.679331\pi$
$72$	−2.00000	−0.235702
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	5.00000	0.581238
$75$	1.00000	0.115470
$76$	−1.00000	−0.114708
$77$	1.00000	0.113961
$78$	2.00000	0.226455
$79$	14.0000	1.57512	0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000	1.57512	0.787562	+	0.616236i	$0.211343\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	−1.00000	−0.109109
$85$	3.00000	0.325396
$86$	8.00000	0.862662
$87$	−9.00000	−0.964901
$88$	−1.00000	−0.106600
$89$	−15.0000	−1.59000	−0.794998	−	0.606612i	$-0.792528\pi$
$89$	−15.0000	−1.59000	−0.794998	+	0.606612i	$0.792528\pi$
$90$	2.00000	0.210819
$91$	−2.00000	−0.209657
$92$	6.00000	0.625543
$93$	5.00000	0.518476
$94$	6.00000	0.618853
$95$	1.00000	0.102598
$96$	1.00000	0.102062
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	−6.00000	−0.606092
$99$	2.00000	0.201008
$100$	1.00000	0.100000
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	−3.00000	−0.297044
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	2.00000	0.196116
$105$	1.00000	0.0975900
$106$	9.00000	0.874157
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	−5.00000	−0.481125
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	1.00000	0.0953463
$111$	5.00000	0.474579
$112$	−1.00000	−0.0944911
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	−1.00000	−0.0936586
$115$	−6.00000	−0.559503
$116$	−9.00000	−0.835629
$117$	−4.00000	−0.369800
$118$	6.00000	0.552345
$119$	3.00000	0.275010
$120$	−1.00000	−0.0912871
$121$	1.00000	0.0909091
$122$	5.00000	0.452679
$123$	−6.00000	−0.541002
$124$	5.00000	0.449013
$125$	−1.00000	−0.0894427
$126$	2.00000	0.178174
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	1.00000	0.0883883
$129$	8.00000	0.704361
$130$	−2.00000	−0.175412
$131$	3.00000	0.262111	0.131056	−	0.991375i	$-0.458163\pi$
$131$	3.00000	0.262111	0.131056	+	0.991375i	$0.458163\pi$
$132$	−1.00000	−0.0870388
$133$	1.00000	0.0867110
$134$	8.00000	0.691095
$135$	5.00000	0.430331
$136$	−3.00000	−0.257248
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	6.00000	0.510754
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	1.00000	0.0845154
$141$	6.00000	0.505291
$142$	−9.00000	−0.755263
$143$	−2.00000	−0.167248
$144$	−2.00000	−0.166667
$145$	9.00000	0.747409
$146$	−10.0000	−0.827606
$147$	−6.00000	−0.494872
$148$	5.00000	0.410997
$149$	21.0000	1.72039	0.860194	−	0.509968i	$-0.170343\pi$
$149$	21.0000	1.72039	0.860194	+	0.509968i	$0.170343\pi$
$150$	1.00000	0.0816497
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	−1.00000	−0.0811107
$153$	6.00000	0.485071
$154$	1.00000	0.0805823
$155$	−5.00000	−0.401610
$156$	2.00000	0.160128
$157$	5.00000	0.399043	0.199522	−	0.979893i	$-0.436061\pi$
$157$	5.00000	0.399043	0.199522	+	0.979893i	$0.436061\pi$
$158$	14.0000	1.11378
$159$	9.00000	0.713746
$160$	−1.00000	−0.0790569
$161$	−6.00000	−0.472866
$162$	1.00000	0.0785674
$163$	5.00000	0.391630	0.195815	−	0.980641i	$-0.437265\pi$
$163$	5.00000	0.391630	0.195815	+	0.980641i	$0.437265\pi$
$164$	−6.00000	−0.468521
$165$	1.00000	0.0778499
$166$	−6.00000	−0.465690
$167$	−9.00000	−0.696441	−0.348220	−	0.937413i	$-0.613214\pi$
$167$	−9.00000	−0.696441	−0.348220	+	0.937413i	$0.613214\pi$
$168$	−1.00000	−0.0771517
$169$	−9.00000	−0.692308
$170$	3.00000	0.230089
$171$	2.00000	0.152944
$172$	8.00000	0.609994
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	−9.00000	−0.682288
$175$	−1.00000	−0.0755929
$176$	−1.00000	−0.0753778
$177$	6.00000	0.450988
$178$	−15.0000	−1.12430
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	2.00000	0.149071
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	−2.00000	−0.148250
$183$	5.00000	0.369611
$184$	6.00000	0.442326
$185$	−5.00000	−0.367607
$186$	5.00000	0.366618
$187$	3.00000	0.219382
$188$	6.00000	0.437595
$189$	5.00000	0.363696
$190$	1.00000	0.0725476
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	1.00000	0.0721688
$193$	−7.00000	−0.503871	−0.251936	−	0.967744i	$-0.581067\pi$
$193$	−7.00000	−0.503871	−0.251936	+	0.967744i	$0.581067\pi$
$194$	8.00000	0.574367
$195$	−2.00000	−0.143223
$196$	−6.00000	−0.428571
$197$	12.0000	0.854965	0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000	0.854965	0.427482	+	0.904024i	$0.359401\pi$
$198$	2.00000	0.142134
$199$	−25.0000	−1.77220	−0.886102	−	0.463491i	$-0.846597\pi$
$199$	−25.0000	−1.77220	−0.886102	+	0.463491i	$0.846597\pi$
$200$	1.00000	0.0707107
$201$	8.00000	0.564276
$202$	−18.0000	−1.26648
$203$	9.00000	0.631676
$204$	−3.00000	−0.210042
$205$	6.00000	0.419058
$206$	−16.0000	−1.11477
$207$	−12.0000	−0.834058
$208$	2.00000	0.138675
$209$	1.00000	0.0691714
$210$	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	$0.647678\pi$
$212$	9.00000	0.618123
$213$	−9.00000	−0.616670
$214$	−12.0000	−0.820303
$215$	−8.00000	−0.545595
$216$	−5.00000	−0.340207
$217$	−5.00000	−0.339422
$218$	2.00000	0.135457
$219$	−10.0000	−0.675737
$220$	1.00000	0.0674200
$221$	−6.00000	−0.403604
$222$	5.00000	0.335578
$223$	−10.0000	−0.669650	−0.334825	−	0.942280i	$-0.608677\pi$
$223$	−10.0000	−0.669650	−0.334825	+	0.942280i	$0.608677\pi$
$224$	−1.00000	−0.0668153
$225$	−2.00000	−0.133333
$226$	−12.0000	−0.798228
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	−1.00000	−0.0662266
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	−6.00000	−0.395628
$231$	1.00000	0.0657952
$232$	−9.00000	−0.590879
$233$	−27.0000	−1.76883	−0.884414	−	0.466702i	$-0.845442\pi$
$233$	−27.0000	−1.76883	−0.884414	+	0.466702i	$0.845442\pi$
$234$	−4.00000	−0.261488
$235$	−6.00000	−0.391397
$236$	6.00000	0.390567
$237$	14.0000	0.909398
$238$	3.00000	0.194461
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	−1.00000	−0.0645497
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	1.00000	0.0642824
$243$	16.0000	1.02640
$244$	5.00000	0.320092
$245$	6.00000	0.383326
$246$	−6.00000	−0.382546
$247$	−2.00000	−0.127257
$248$	5.00000	0.317500
$249$	−6.00000	−0.380235
$250$	−1.00000	−0.0632456
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	2.00000	0.125988
$253$	−6.00000	−0.377217
$254$	−16.0000	−1.00393
$255$	3.00000	0.187867
$256$	1.00000	0.0625000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	8.00000	0.498058
$259$	−5.00000	−0.310685
$260$	−2.00000	−0.124035
$261$	18.0000	1.11417
$262$	3.00000	0.185341
$263$	21.0000	1.29492	0.647458	−	0.762101i	$-0.275832\pi$
$263$	21.0000	1.29492	0.647458	+	0.762101i	$0.275832\pi$
$264$	−1.00000	−0.0615457
$265$	−9.00000	−0.552866
$266$	1.00000	0.0613139
$267$	−15.0000	−0.917985
$268$	8.00000	0.488678
$269$	−12.0000	−0.731653	−0.365826	−	0.930683i	$-0.619214\pi$
$269$	−12.0000	−0.731653	−0.365826	+	0.930683i	$0.619214\pi$
$270$	5.00000	0.304290
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	−3.00000	−0.181902
$273$	−2.00000	−0.121046
$274$	12.0000	0.724947
$275$	−1.00000	−0.0603023
$276$	6.00000	0.361158
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	−4.00000	−0.239904
$279$	−10.0000	−0.598684
$280$	1.00000	0.0597614
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	6.00000	0.357295
$283$	−22.0000	−1.30776	−0.653882	−	0.756596i	$-0.726861\pi$
$283$	−22.0000	−1.30776	−0.653882	+	0.756596i	$0.726861\pi$
$284$	−9.00000	−0.534052
$285$	1.00000	0.0592349
$286$	−2.00000	−0.118262
$287$	6.00000	0.354169
$288$	−2.00000	−0.117851
$289$	−8.00000	−0.470588
$290$	9.00000	0.528498
$291$	8.00000	0.468968
$292$	−10.0000	−0.585206
$293$	30.0000	1.75262	0.876309	−	0.481749i	$-0.159998\pi$
$293$	30.0000	1.75262	0.876309	+	0.481749i	$0.159998\pi$
$294$	−6.00000	−0.349927
$295$	−6.00000	−0.349334
$296$	5.00000	0.290619
$297$	5.00000	0.290129
$298$	21.0000	1.21650
$299$	12.0000	0.693978
$300$	1.00000	0.0577350
$301$	−8.00000	−0.461112
$302$	2.00000	0.115087
$303$	−18.0000	−1.03407
$304$	−1.00000	−0.0573539
$305$	−5.00000	−0.286299
$306$	6.00000	0.342997
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	1.00000	0.0569803
$309$	−16.0000	−0.910208
$310$	−5.00000	−0.283981
$311$	−3.00000	−0.170114	−0.0850572	−	0.996376i	$-0.527107\pi$
$311$	−3.00000	−0.170114	−0.0850572	+	0.996376i	$0.527107\pi$
$312$	2.00000	0.113228
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	5.00000	0.282166
$315$	−2.00000	−0.112687
$316$	14.0000	0.787562
$317$	−9.00000	−0.505490	−0.252745	−	0.967533i	$-0.581333\pi$
$317$	−9.00000	−0.505490	−0.252745	+	0.967533i	$0.581333\pi$
$318$	9.00000	0.504695
$319$	9.00000	0.503903
$320$	−1.00000	−0.0559017
$321$	−12.0000	−0.669775
$322$	−6.00000	−0.334367
$323$	3.00000	0.166924
$324$	1.00000	0.0555556
$325$	2.00000	0.110940
$326$	5.00000	0.276924
$327$	2.00000	0.110600
$328$	−6.00000	−0.331295
$329$	−6.00000	−0.330791
$330$	1.00000	0.0550482
$331$	32.0000	1.75888	0.879440	−	0.476011i	$-0.157918\pi$
$331$	32.0000	1.75888	0.879440	+	0.476011i	$0.157918\pi$
$332$	−6.00000	−0.329293
$333$	−10.0000	−0.547997
$334$	−9.00000	−0.492458
$335$	−8.00000	−0.437087
$336$	−1.00000	−0.0545545
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	−9.00000	−0.489535
$339$	−12.0000	−0.651751
$340$	3.00000	0.162698
$341$	−5.00000	−0.270765
$342$	2.00000	0.108148
$343$	13.0000	0.701934
$344$	8.00000	0.431331
$345$	−6.00000	−0.323029
$346$	6.00000	0.322562
$347$	18.0000	0.966291	0.483145	−	0.875540i	$-0.339494\pi$
$347$	18.0000	0.966291	0.483145	+	0.875540i	$0.339494\pi$
$348$	−9.00000	−0.482451
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	−1.00000	−0.0534522
$351$	−10.0000	−0.533761
$352$	−1.00000	−0.0533002
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	6.00000	0.318896
$355$	9.00000	0.477670
$356$	−15.0000	−0.794998
$357$	3.00000	0.158777
$358$	24.0000	1.26844
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	2.00000	0.105409
$361$	−18.0000	−0.947368
$362$	14.0000	0.735824
$363$	1.00000	0.0524864
$364$	−2.00000	−0.104828
$365$	10.0000	0.523424
$366$	5.00000	0.261354
$367$	−4.00000	−0.208798	−0.104399	−	0.994535i	$-0.533292\pi$
$367$	−4.00000	−0.208798	−0.104399	+	0.994535i	$0.533292\pi$
$368$	6.00000	0.312772
$369$	12.0000	0.624695
$370$	−5.00000	−0.259938
$371$	−9.00000	−0.467257
$372$	5.00000	0.259238
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	3.00000	0.155126
$375$	−1.00000	−0.0516398
$376$	6.00000	0.309426
$377$	−18.0000	−0.927047
$378$	5.00000	0.257172
$379$	−10.0000	−0.513665	−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000	−0.513665	−0.256833	+	0.966456i	$0.582679\pi$
$380$	1.00000	0.0512989
$381$	−16.0000	−0.819705
$382$	12.0000	0.613973
$383$	30.0000	1.53293	0.766464	−	0.642287i	$-0.222014\pi$
$383$	30.0000	1.53293	0.766464	+	0.642287i	$0.222014\pi$
$384$	1.00000	0.0510310
$385$	−1.00000	−0.0509647
$386$	−7.00000	−0.356291
$387$	−16.0000	−0.813326
$388$	8.00000	0.406138
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	−2.00000	−0.101274
$391$	−18.0000	−0.910299
$392$	−6.00000	−0.303046
$393$	3.00000	0.151330
$394$	12.0000	0.604551
$395$	−14.0000	−0.704416
$396$	2.00000	0.100504
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	−25.0000	−1.25314
$399$	1.00000	0.0500626
$400$	1.00000	0.0500000
$401$	3.00000	0.149813	0.0749064	−	0.997191i	$-0.476134\pi$
$401$	3.00000	0.149813	0.0749064	+	0.997191i	$0.476134\pi$
$402$	8.00000	0.399004
$403$	10.0000	0.498135
$404$	−18.0000	−0.895533
$405$	−1.00000	−0.0496904
$406$	9.00000	0.446663
$407$	−5.00000	−0.247841
$408$	−3.00000	−0.148522
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	6.00000	0.296319
$411$	12.0000	0.591916
$412$	−16.0000	−0.788263
$413$	−6.00000	−0.295241
$414$	−12.0000	−0.589768
$415$	6.00000	0.294528
$416$	2.00000	0.0980581
$417$	−4.00000	−0.195881
$418$	1.00000	0.0489116
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	1.00000	0.0487950
$421$	−28.0000	−1.36464	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000	−1.36464	−0.682318	+	0.731055i	$0.739028\pi$
$422$	−13.0000	−0.632830
$423$	−12.0000	−0.583460
$424$	9.00000	0.437079
$425$	−3.00000	−0.145521
$426$	−9.00000	−0.436051
$427$	−5.00000	−0.241967
$428$	−12.0000	−0.580042
$429$	−2.00000	−0.0965609
$430$	−8.00000	−0.385794
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	−5.00000	−0.240563
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	−5.00000	−0.240008
$435$	9.00000	0.431517
$436$	2.00000	0.0957826
$437$	−6.00000	−0.287019
$438$	−10.0000	−0.477818
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	1.00000	0.0476731
$441$	12.0000	0.571429
$442$	−6.00000	−0.285391
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	5.00000	0.237289
$445$	15.0000	0.711068
$446$	−10.0000	−0.473514
$447$	21.0000	0.993266
$448$	−1.00000	−0.0472456
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	−2.00000	−0.0942809
$451$	6.00000	0.282529
$452$	−12.0000	−0.564433
$453$	2.00000	0.0939682
$454$	18.0000	0.844782
$455$	2.00000	0.0937614
$456$	−1.00000	−0.0468293
$457$	−1.00000	−0.0467780	−0.0233890	−	0.999726i	$-0.507446\pi$
$457$	−1.00000	−0.0467780	−0.0233890	+	0.999726i	$0.507446\pi$
$458$	−22.0000	−1.02799
$459$	15.0000	0.700140
$460$	−6.00000	−0.279751
$461$	−15.0000	−0.698620	−0.349310	−	0.937007i	$-0.613584\pi$
$461$	−15.0000	−0.698620	−0.349310	+	0.937007i	$0.613584\pi$
$462$	1.00000	0.0465242
$463$	14.0000	0.650635	0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000	0.650635	0.325318	+	0.945605i	$0.394529\pi$
$464$	−9.00000	−0.417815
$465$	−5.00000	−0.231869
$466$	−27.0000	−1.25075
$467$	−15.0000	−0.694117	−0.347059	−	0.937843i	$-0.612820\pi$
$467$	−15.0000	−0.694117	−0.347059	+	0.937843i	$0.612820\pi$
$468$	−4.00000	−0.184900
$469$	−8.00000	−0.369406
$470$	−6.00000	−0.276759
$471$	5.00000	0.230388
$472$	6.00000	0.276172
$473$	−8.00000	−0.367840
$474$	14.0000	0.643041
$475$	−1.00000	−0.0458831
$476$	3.00000	0.137505
$477$	−18.0000	−0.824163
$478$	−6.00000	−0.274434
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	−1.00000	−0.0456435
$481$	10.0000	0.455961
$482$	14.0000	0.637683
$483$	−6.00000	−0.273009
$484$	1.00000	0.0454545
$485$	−8.00000	−0.363261
$486$	16.0000	0.725775
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	5.00000	0.226339
$489$	5.00000	0.226108
$490$	6.00000	0.271052
$491$	−9.00000	−0.406164	−0.203082	−	0.979162i	$-0.565096\pi$
$491$	−9.00000	−0.406164	−0.203082	+	0.979162i	$0.565096\pi$
$492$	−6.00000	−0.270501
$493$	27.0000	1.21602
$494$	−2.00000	−0.0899843
$495$	−2.00000	−0.0898933
$496$	5.00000	0.224507
$497$	9.00000	0.403705
$498$	−6.00000	−0.268866
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	−1.00000	−0.0447214
$501$	−9.00000	−0.402090
$502$	−18.0000	−0.803379
$503$	−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
$505$	18.0000	0.800989
$506$	−6.00000	−0.266733
$507$	−9.00000	−0.399704
$508$	−16.0000	−0.709885
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	3.00000	0.132842
$511$	10.0000	0.442374
$512$	1.00000	0.0441942
$513$	5.00000	0.220755
$514$	18.0000	0.793946
$515$	16.0000	0.705044
$516$	8.00000	0.352180
$517$	−6.00000	−0.263880
$518$	−5.00000	−0.219687
$519$	6.00000	0.263371
$520$	−2.00000	−0.0877058
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	18.0000	0.787839
$523$	−28.0000	−1.22435	−0.612177	−	0.790721i	$-0.709706\pi$
$523$	−28.0000	−1.22435	−0.612177	+	0.790721i	$0.709706\pi$
$524$	3.00000	0.131056
$525$	−1.00000	−0.0436436
$526$	21.0000	0.915644
$527$	−15.0000	−0.653410
$528$	−1.00000	−0.0435194
$529$	13.0000	0.565217
$530$	−9.00000	−0.390935
$531$	−12.0000	−0.520756
$532$	1.00000	0.0433555
$533$	−12.0000	−0.519778
$534$	−15.0000	−0.649113
$535$	12.0000	0.518805
$536$	8.00000	0.345547
$537$	24.0000	1.03568
$538$	−12.0000	−0.517357
$539$	6.00000	0.258438
$540$	5.00000	0.215166
$541$	−13.0000	−0.558914	−0.279457	−	0.960158i	$-0.590154\pi$
$541$	−13.0000	−0.558914	−0.279457	+	0.960158i	$0.590154\pi$
$542$	20.0000	0.859074
$543$	14.0000	0.600798
$544$	−3.00000	−0.128624
$545$	−2.00000	−0.0856706
$546$	−2.00000	−0.0855921
$547$	44.0000	1.88130	0.940652	−	0.339372i	$-0.110215\pi$
$547$	44.0000	1.88130	0.940652	+	0.339372i	$0.110215\pi$
$548$	12.0000	0.512615
$549$	−10.0000	−0.426790
$550$	−1.00000	−0.0426401
$551$	9.00000	0.383413
$552$	6.00000	0.255377
$553$	−14.0000	−0.595341
$554$	8.00000	0.339887
$555$	−5.00000	−0.212238
$556$	−4.00000	−0.169638
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	−10.0000	−0.423334
$559$	16.0000	0.676728
$560$	1.00000	0.0422577
$561$	3.00000	0.126660
$562$	18.0000	0.759284
$563$	6.00000	0.252870	0.126435	−	0.991975i	$-0.459647\pi$
$563$	6.00000	0.252870	0.126435	+	0.991975i	$0.459647\pi$
$564$	6.00000	0.252646
$565$	12.0000	0.504844
$566$	−22.0000	−0.924729
$567$	−1.00000	−0.0419961
$568$	−9.00000	−0.377632
$569$	−12.0000	−0.503066	−0.251533	−	0.967849i	$-0.580935\pi$
$569$	−12.0000	−0.503066	−0.251533	+	0.967849i	$0.580935\pi$
$570$	1.00000	0.0418854
$571$	−31.0000	−1.29731	−0.648655	−	0.761083i	$-0.724668\pi$
$571$	−31.0000	−1.29731	−0.648655	+	0.761083i	$0.724668\pi$
$572$	−2.00000	−0.0836242
$573$	12.0000	0.501307
$574$	6.00000	0.250435
$575$	6.00000	0.250217
$576$	−2.00000	−0.0833333
$577$	26.0000	1.08239	0.541197	−	0.840896i	$-0.317971\pi$
$577$	26.0000	1.08239	0.541197	+	0.840896i	$0.317971\pi$
$578$	−8.00000	−0.332756
$579$	−7.00000	−0.290910
$580$	9.00000	0.373705
$581$	6.00000	0.248922
$582$	8.00000	0.331611
$583$	−9.00000	−0.372742
$584$	−10.0000	−0.413803
$585$	4.00000	0.165380
$586$	30.0000	1.23929
$587$	27.0000	1.11441	0.557205	−	0.830375i	$-0.311874\pi$
$587$	27.0000	1.11441	0.557205	+	0.830375i	$0.311874\pi$
$588$	−6.00000	−0.247436
$589$	−5.00000	−0.206021
$590$	−6.00000	−0.247016
$591$	12.0000	0.493614
$592$	5.00000	0.205499
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	5.00000	0.205152
$595$	−3.00000	−0.122988
$596$	21.0000	0.860194
$597$	−25.0000	−1.02318
$598$	12.0000	0.490716
$599$	−3.00000	−0.122577	−0.0612883	−	0.998120i	$-0.519521\pi$
$599$	−3.00000	−0.122577	−0.0612883	+	0.998120i	$0.519521\pi$
$600$	1.00000	0.0408248
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	−8.00000	−0.326056
$603$	−16.0000	−0.651570
$604$	2.00000	0.0813788
$605$	−1.00000	−0.0406558
$606$	−18.0000	−0.731200
$607$	5.00000	0.202944	0.101472	−	0.994838i	$-0.467645\pi$
$607$	5.00000	0.202944	0.101472	+	0.994838i	$0.467645\pi$
$608$	−1.00000	−0.0405554
$609$	9.00000	0.364698
$610$	−5.00000	−0.202444
$611$	12.0000	0.485468
$612$	6.00000	0.242536
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	2.00000	0.0807134
$615$	6.00000	0.241943
$616$	1.00000	0.0402911
$617$	−36.0000	−1.44931	−0.724653	−	0.689114i	$-0.758000\pi$
$617$	−36.0000	−1.44931	−0.724653	+	0.689114i	$0.758000\pi$
$618$	−16.0000	−0.643614
$619$	−16.0000	−0.643094	−0.321547	−	0.946894i	$-0.604203\pi$
$619$	−16.0000	−0.643094	−0.321547	+	0.946894i	$0.604203\pi$
$620$	−5.00000	−0.200805
$621$	−30.0000	−1.20386
$622$	−3.00000	−0.120289
$623$	15.0000	0.600962
$624$	2.00000	0.0800641
$625$	1.00000	0.0400000
$626$	14.0000	0.559553
$627$	1.00000	0.0399362
$628$	5.00000	0.199522
$629$	−15.0000	−0.598089
$630$	−2.00000	−0.0796819
$631$	−1.00000	−0.0398094	−0.0199047	−	0.999802i	$-0.506336\pi$
$631$	−1.00000	−0.0398094	−0.0199047	+	0.999802i	$0.506336\pi$
$632$	14.0000	0.556890
$633$	−13.0000	−0.516704
$634$	−9.00000	−0.357436
$635$	16.0000	0.634941
$636$	9.00000	0.356873
$637$	−12.0000	−0.475457
$638$	9.00000	0.356313
$639$	18.0000	0.712069
$640$	−1.00000	−0.0395285
$641$	−9.00000	−0.355479	−0.177739	−	0.984078i	$-0.556878\pi$
$641$	−9.00000	−0.355479	−0.177739	+	0.984078i	$0.556878\pi$
$642$	−12.0000	−0.473602
$643$	5.00000	0.197181	0.0985904	−	0.995128i	$-0.468567\pi$
$643$	5.00000	0.197181	0.0985904	+	0.995128i	$0.468567\pi$
$644$	−6.00000	−0.236433
$645$	−8.00000	−0.315000
$646$	3.00000	0.118033
$647$	−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
$650$	2.00000	0.0784465
$651$	−5.00000	−0.195965
$652$	5.00000	0.195815
$653$	−9.00000	−0.352197	−0.176099	−	0.984373i	$-0.556348\pi$
$653$	−9.00000	−0.352197	−0.176099	+	0.984373i	$0.556348\pi$
$654$	2.00000	0.0782062
$655$	−3.00000	−0.117220
$656$	−6.00000	−0.234261
$657$	20.0000	0.780274
$658$	−6.00000	−0.233904
$659$	−27.0000	−1.05177	−0.525885	−	0.850555i	$-0.676266\pi$
$659$	−27.0000	−1.05177	−0.525885	+	0.850555i	$0.676266\pi$
$660$	1.00000	0.0389249
$661$	−34.0000	−1.32245	−0.661223	−	0.750189i	$-0.729962\pi$
$661$	−34.0000	−1.32245	−0.661223	+	0.750189i	$0.729962\pi$
$662$	32.0000	1.24372
$663$	−6.00000	−0.233021
$664$	−6.00000	−0.232845
$665$	−1.00000	−0.0387783
$666$	−10.0000	−0.387492
$667$	−54.0000	−2.09089
$668$	−9.00000	−0.348220
$669$	−10.0000	−0.386622
$670$	−8.00000	−0.309067
$671$	−5.00000	−0.193023
$672$	−1.00000	−0.0385758
$673$	41.0000	1.58043	0.790217	−	0.612827i	$-0.209968\pi$
$673$	41.0000	1.58043	0.790217	+	0.612827i	$0.209968\pi$
$674$	−13.0000	−0.500741
$675$	−5.00000	−0.192450
$676$	−9.00000	−0.346154
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	−12.0000	−0.460857
$679$	−8.00000	−0.307012
$680$	3.00000	0.115045
$681$	18.0000	0.689761
$682$	−5.00000	−0.191460
$683$	−9.00000	−0.344375	−0.172188	−	0.985064i	$-0.555084\pi$
$683$	−9.00000	−0.344375	−0.172188	+	0.985064i	$0.555084\pi$
$684$	2.00000	0.0764719
$685$	−12.0000	−0.458496
$686$	13.0000	0.496342
$687$	−22.0000	−0.839352
$688$	8.00000	0.304997
$689$	18.0000	0.685745
$690$	−6.00000	−0.228416
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	6.00000	0.228086
$693$	−2.00000	−0.0759737
$694$	18.0000	0.683271
$695$	4.00000	0.151729
$696$	−9.00000	−0.341144
$697$	18.0000	0.681799
$698$	2.00000	0.0757011
$699$	−27.0000	−1.02123
$700$	−1.00000	−0.0377964
$701$	−3.00000	−0.113308	−0.0566542	−	0.998394i	$-0.518043\pi$
$701$	−3.00000	−0.113308	−0.0566542	+	0.998394i	$0.518043\pi$
$702$	−10.0000	−0.377426
$703$	−5.00000	−0.188579
$704$	−1.00000	−0.0376889
$705$	−6.00000	−0.225973
$706$	−30.0000	−1.12906
$707$	18.0000	0.676960
$708$	6.00000	0.225494
$709$	−34.0000	−1.27690	−0.638448	−	0.769665i	$-0.720423\pi$
$709$	−34.0000	−1.27690	−0.638448	+	0.769665i	$0.720423\pi$
$710$	9.00000	0.337764
$711$	−28.0000	−1.05008
$712$	−15.0000	−0.562149
$713$	30.0000	1.12351
$714$	3.00000	0.112272
$715$	2.00000	0.0747958
$716$	24.0000	0.896922
$717$	−6.00000	−0.224074
$718$	24.0000	0.895672
$719$	−15.0000	−0.559406	−0.279703	−	0.960087i	$-0.590236\pi$
$719$	−15.0000	−0.559406	−0.279703	+	0.960087i	$0.590236\pi$
$720$	2.00000	0.0745356
$721$	16.0000	0.595871
$722$	−18.0000	−0.669891
$723$	14.0000	0.520666
$724$	14.0000	0.520306
$725$	−9.00000	−0.334252
$726$	1.00000	0.0371135
$727$	26.0000	0.964287	0.482143	−	0.876092i	$-0.339858\pi$
$727$	26.0000	0.964287	0.482143	+	0.876092i	$0.339858\pi$
$728$	−2.00000	−0.0741249
$729$	13.0000	0.481481
$730$	10.0000	0.370117
$731$	−24.0000	−0.887672
$732$	5.00000	0.184805
$733$	−4.00000	−0.147743	−0.0738717	−	0.997268i	$-0.523536\pi$
$733$	−4.00000	−0.147743	−0.0738717	+	0.997268i	$0.523536\pi$
$734$	−4.00000	−0.147643
$735$	6.00000	0.221313
$736$	6.00000	0.221163
$737$	−8.00000	−0.294684
$738$	12.0000	0.441726
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	−5.00000	−0.183804
$741$	−2.00000	−0.0734718
$742$	−9.00000	−0.330400
$743$	−9.00000	−0.330178	−0.165089	−	0.986279i	$-0.552791\pi$
$743$	−9.00000	−0.330178	−0.165089	+	0.986279i	$0.552791\pi$
$744$	5.00000	0.183309
$745$	−21.0000	−0.769380
$746$	14.0000	0.512576
$747$	12.0000	0.439057
$748$	3.00000	0.109691
$749$	12.0000	0.438470
$750$	−1.00000	−0.0365148
$751$	−7.00000	−0.255434	−0.127717	−	0.991811i	$-0.540765\pi$
$751$	−7.00000	−0.255434	−0.127717	+	0.991811i	$0.540765\pi$
$752$	6.00000	0.218797
$753$	−18.0000	−0.655956
$754$	−18.0000	−0.655521
$755$	−2.00000	−0.0727875
$756$	5.00000	0.181848
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	−10.0000	−0.363216
$759$	−6.00000	−0.217786
$760$	1.00000	0.0362738
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	−16.0000	−0.579619
$763$	−2.00000	−0.0724049
$764$	12.0000	0.434145
$765$	−6.00000	−0.216930
$766$	30.0000	1.08394
$767$	12.0000	0.433295
$768$	1.00000	0.0360844
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	−1.00000	−0.0360375
$771$	18.0000	0.648254
$772$	−7.00000	−0.251936
$773$	45.0000	1.61854	0.809269	−	0.587439i	$-0.199864\pi$
$773$	45.0000	1.61854	0.809269	+	0.587439i	$0.199864\pi$
$774$	−16.0000	−0.575108
$775$	5.00000	0.179605
$776$	8.00000	0.287183
$777$	−5.00000	−0.179374
$778$	6.00000	0.215110
$779$	6.00000	0.214972
$780$	−2.00000	−0.0716115
$781$	9.00000	0.322045
$782$	−18.0000	−0.643679
$783$	45.0000	1.60817
$784$	−6.00000	−0.214286
$785$	−5.00000	−0.178458
$786$	3.00000	0.107006
$787$	44.0000	1.56843	0.784215	−	0.620489i	$-0.213066\pi$
$787$	44.0000	1.56843	0.784215	+	0.620489i	$0.213066\pi$
$788$	12.0000	0.427482
$789$	21.0000	0.747620
$790$	−14.0000	−0.498098
$791$	12.0000	0.426671
$792$	2.00000	0.0710669
$793$	10.0000	0.355110
$794$	2.00000	0.0709773
$795$	−9.00000	−0.319197
$796$	−25.0000	−0.886102
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	1.00000	0.0353996
$799$	−18.0000	−0.636794
$800$	1.00000	0.0353553
$801$	30.0000	1.06000
$802$	3.00000	0.105934
$803$	10.0000	0.352892
$804$	8.00000	0.282138
$805$	6.00000	0.211472
$806$	10.0000	0.352235
$807$	−12.0000	−0.422420
$808$	−18.0000	−0.633238
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	−1.00000	−0.0351364
$811$	−43.0000	−1.50993	−0.754967	−	0.655763i	$-0.772347\pi$
$811$	−43.0000	−1.50993	−0.754967	+	0.655763i	$0.772347\pi$
$812$	9.00000	0.315838
$813$	20.0000	0.701431
$814$	−5.00000	−0.175250
$815$	−5.00000	−0.175142
$816$	−3.00000	−0.105021
$817$	−8.00000	−0.279885
$818$	32.0000	1.11885
$819$	4.00000	0.139771
$820$	6.00000	0.209529
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	12.0000	0.418548
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	−16.0000	−0.557386
$825$	−1.00000	−0.0348155
$826$	−6.00000	−0.208767
$827$	−48.0000	−1.66912	−0.834562	−	0.550914i	$-0.814279\pi$
$827$	−48.0000	−1.66912	−0.834562	+	0.550914i	$0.814279\pi$
$828$	−12.0000	−0.417029
$829$	−4.00000	−0.138926	−0.0694629	−	0.997585i	$-0.522129\pi$
$829$	−4.00000	−0.138926	−0.0694629	+	0.997585i	$0.522129\pi$
$830$	6.00000	0.208263
$831$	8.00000	0.277517
$832$	2.00000	0.0693375
$833$	18.0000	0.623663
$834$	−4.00000	−0.138509
$835$	9.00000	0.311458
$836$	1.00000	0.0345857
$837$	−25.0000	−0.864126
$838$	12.0000	0.414533
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	1.00000	0.0345033
$841$	52.0000	1.79310
$842$	−28.0000	−0.964944
$843$	18.0000	0.619953
$844$	−13.0000	−0.447478
$845$	9.00000	0.309609
$846$	−12.0000	−0.412568
$847$	−1.00000	−0.0343604
$848$	9.00000	0.309061
$849$	−22.0000	−0.755038
$850$	−3.00000	−0.102899
$851$	30.0000	1.02839
$852$	−9.00000	−0.308335
$853$	38.0000	1.30110	0.650548	−	0.759465i	$-0.274539\pi$
$853$	38.0000	1.30110	0.650548	+	0.759465i	$0.274539\pi$
$854$	−5.00000	−0.171096
$855$	−2.00000	−0.0683986
$856$	−12.0000	−0.410152
$857$	−3.00000	−0.102478	−0.0512390	−	0.998686i	$-0.516317\pi$
$857$	−3.00000	−0.102478	−0.0512390	+	0.998686i	$0.516317\pi$
$858$	−2.00000	−0.0682789
$859$	26.0000	0.887109	0.443554	−	0.896248i	$-0.353717\pi$
$859$	26.0000	0.887109	0.443554	+	0.896248i	$0.353717\pi$
$860$	−8.00000	−0.272798
$861$	6.00000	0.204479
$862$	0	0
$863$	−6.00000	−0.204242	−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000	−0.204242	−0.102121	+	0.994772i	$0.532563\pi$
$864$	−5.00000	−0.170103
$865$	−6.00000	−0.204006
$866$	−16.0000	−0.543702
$867$	−8.00000	−0.271694
$868$	−5.00000	−0.169711
$869$	−14.0000	−0.474917
$870$	9.00000	0.305129
$871$	16.0000	0.542139
$872$	2.00000	0.0677285
$873$	−16.0000	−0.541518
$874$	−6.00000	−0.202953
$875$	1.00000	0.0338062
$876$	−10.0000	−0.337869
$877$	−34.0000	−1.14810	−0.574049	−	0.818821i	$-0.694628\pi$
$877$	−34.0000	−1.14810	−0.574049	+	0.818821i	$0.694628\pi$
$878$	−28.0000	−0.944954
$879$	30.0000	1.01187
$880$	1.00000	0.0337100
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	12.0000	0.404061
$883$	−25.0000	−0.841317	−0.420658	−	0.907219i	$-0.638201\pi$
$883$	−25.0000	−0.841317	−0.420658	+	0.907219i	$0.638201\pi$
$884$	−6.00000	−0.201802
$885$	−6.00000	−0.201688
$886$	12.0000	0.403148
$887$	48.0000	1.61168	0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000	1.61168	0.805841	+	0.592132i	$0.201714\pi$
$888$	5.00000	0.167789
$889$	16.0000	0.536623
$890$	15.0000	0.502801
$891$	−1.00000	−0.0335013
$892$	−10.0000	−0.334825
$893$	−6.00000	−0.200782
$894$	21.0000	0.702345
$895$	−24.0000	−0.802232
$896$	−1.00000	−0.0334077
$897$	12.0000	0.400668
$898$	−6.00000	−0.200223
$899$	−45.0000	−1.50083
$900$	−2.00000	−0.0666667
$901$	−27.0000	−0.899500
$902$	6.00000	0.199778
$903$	−8.00000	−0.266223
$904$	−12.0000	−0.399114
$905$	−14.0000	−0.465376
$906$	2.00000	0.0664455
$907$	−7.00000	−0.232431	−0.116216	−	0.993224i	$-0.537076\pi$
$907$	−7.00000	−0.232431	−0.116216	+	0.993224i	$0.537076\pi$
$908$	18.0000	0.597351
$909$	36.0000	1.19404
$910$	2.00000	0.0662994
$911$	3.00000	0.0993944	0.0496972	−	0.998764i	$-0.484174\pi$
$911$	3.00000	0.0993944	0.0496972	+	0.998764i	$0.484174\pi$
$912$	−1.00000	−0.0331133
$913$	6.00000	0.198571
$914$	−1.00000	−0.0330771
$915$	−5.00000	−0.165295
$916$	−22.0000	−0.726900
$917$	−3.00000	−0.0990687
$918$	15.0000	0.495074
$919$	−52.0000	−1.71532	−0.857661	−	0.514216i	$-0.828083\pi$
$919$	−52.0000	−1.71532	−0.857661	+	0.514216i	$0.828083\pi$
$920$	−6.00000	−0.197814
$921$	2.00000	0.0659022
$922$	−15.0000	−0.493999
$923$	−18.0000	−0.592477
$924$	1.00000	0.0328976
$925$	5.00000	0.164399
$926$	14.0000	0.460069
$927$	32.0000	1.05102
$928$	−9.00000	−0.295439
$929$	−21.0000	−0.688988	−0.344494	−	0.938789i	$-0.611949\pi$
$929$	−21.0000	−0.688988	−0.344494	+	0.938789i	$0.611949\pi$
$930$	−5.00000	−0.163956
$931$	6.00000	0.196642
$932$	−27.0000	−0.884414
$933$	−3.00000	−0.0982156
$934$	−15.0000	−0.490815
$935$	−3.00000	−0.0981105
$936$	−4.00000	−0.130744
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	−8.00000	−0.261209
$939$	14.0000	0.456873
$940$	−6.00000	−0.195698
$941$	27.0000	0.880175	0.440087	−	0.897955i	$-0.354947\pi$
$941$	27.0000	0.880175	0.440087	+	0.897955i	$0.354947\pi$
$942$	5.00000	0.162909
$943$	−36.0000	−1.17232
$944$	6.00000	0.195283
$945$	−5.00000	−0.162650
$946$	−8.00000	−0.260102
$947$	3.00000	0.0974869	0.0487435	−	0.998811i	$-0.484478\pi$
$947$	3.00000	0.0974869	0.0487435	+	0.998811i	$0.484478\pi$
$948$	14.0000	0.454699
$949$	−20.0000	−0.649227
$950$	−1.00000	−0.0324443
$951$	−9.00000	−0.291845
$952$	3.00000	0.0972306
$953$	−21.0000	−0.680257	−0.340128	−	0.940379i	$-0.610471\pi$
$953$	−21.0000	−0.680257	−0.340128	+	0.940379i	$0.610471\pi$
$954$	−18.0000	−0.582772
$955$	−12.0000	−0.388311
$956$	−6.00000	−0.194054
$957$	9.00000	0.290929
$958$	0	0
$959$	−12.0000	−0.387500
$960$	−1.00000	−0.0322749
$961$	−6.00000	−0.193548
$962$	10.0000	0.322413
$963$	24.0000	0.773389
$964$	14.0000	0.450910
$965$	7.00000	0.225338
$966$	−6.00000	−0.193047
$967$	17.0000	0.546683	0.273342	−	0.961917i	$-0.411871\pi$
$967$	17.0000	0.546683	0.273342	+	0.961917i	$0.411871\pi$
$968$	1.00000	0.0321412
$969$	3.00000	0.0963739
$970$	−8.00000	−0.256865
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	16.0000	0.513200
$973$	4.00000	0.128234
$974$	8.00000	0.256337
$975$	2.00000	0.0640513
$976$	5.00000	0.160046
$977$	−48.0000	−1.53566	−0.767828	−	0.640656i	$-0.778662\pi$
$977$	−48.0000	−1.53566	−0.767828	+	0.640656i	$0.778662\pi$
$978$	5.00000	0.159882
$979$	15.0000	0.479402
$980$	6.00000	0.191663
$981$	−4.00000	−0.127710
$982$	−9.00000	−0.287202
$983$	−42.0000	−1.33959	−0.669796	−	0.742545i	$-0.733618\pi$
$983$	−42.0000	−1.33959	−0.669796	+	0.742545i	$0.733618\pi$
$984$	−6.00000	−0.191273
$985$	−12.0000	−0.382352
$986$	27.0000	0.859855
$987$	−6.00000	−0.190982
$988$	−2.00000	−0.0636285
$989$	48.0000	1.52631
$990$	−2.00000	−0.0635642
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	5.00000	0.158750
$993$	32.0000	1.01549
$994$	9.00000	0.285463
$995$	25.0000	0.792553
$996$	−6.00000	−0.190117
$997$	−28.0000	−0.886769	−0.443384	−	0.896332i	$-0.646222\pi$
$997$	−28.0000	−0.886769	−0.443384	+	0.896332i	$0.646222\pi$
$998$	32.0000	1.01294
$999$	−25.0000	−0.790965

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	110.2.a.c.1.1	✓	1
3.2	odd	2		990.2.a.f.1.1		1
4.3	odd	2		880.2.a.d.1.1		1
5.2	odd	4		550.2.b.c.199.2		2
5.3	odd	4		550.2.b.c.199.1		2
5.4	even	2		550.2.a.d.1.1		1
7.6	odd	2		5390.2.a.x.1.1		1
8.3	odd	2		3520.2.a.ba.1.1		1
8.5	even	2		3520.2.a.k.1.1		1
11.10	odd	2		1210.2.a.e.1.1		1
12.11	even	2		7920.2.a.bc.1.1		1
15.2	even	4		4950.2.c.s.199.1		2
15.8	even	4		4950.2.c.s.199.2		2
15.14	odd	2		4950.2.a.bm.1.1		1
20.3	even	4		4400.2.b.j.4049.1		2
20.7	even	4		4400.2.b.j.4049.2		2
20.19	odd	2		4400.2.a.t.1.1		1
44.43	even	2		9680.2.a.g.1.1		1
55.54	odd	2		6050.2.a.bc.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.2.a.c.1.1	✓	1	1.1	even	1	trivial
550.2.a.d.1.1		1	5.4	even	2
550.2.b.c.199.1		2	5.3	odd	4
550.2.b.c.199.2		2	5.2	odd	4
880.2.a.d.1.1		1	4.3	odd	2
990.2.a.f.1.1		1	3.2	odd	2
1210.2.a.e.1.1		1	11.10	odd	2
3520.2.a.k.1.1		1	8.5	even	2
3520.2.a.ba.1.1		1	8.3	odd	2
4400.2.a.t.1.1		1	20.19	odd	2
4400.2.b.j.4049.1		2	20.3	even	4
4400.2.b.j.4049.2		2	20.7	even	4
4950.2.a.bm.1.1		1	15.14	odd	2
4950.2.c.s.199.1		2	15.2	even	4
4950.2.c.s.199.2		2	15.8	even	4
5390.2.a.x.1.1		1	7.6	odd	2
6050.2.a.bc.1.1		1	55.54	odd	2
7920.2.a.bc.1.1		1	12.11	even	2
9680.2.a.g.1.1		1	44.43	even	2

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:	\( N \)	\(=\)	\( 110 = 2 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	110.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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