LMFDB - Embedded newform 1155.2.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1155,2,Mod(1,1155)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1155.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1155, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1155 = 3 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1155.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-2,1,2,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$9.22272143346$
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	$\chi$	$=$	1155.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{10} +1.00000 q^{11} +2.00000 q^{12} -6.00000 q^{13} -2.00000 q^{14} +1.00000 q^{15} -4.00000 q^{16} -7.00000 q^{17} -2.00000 q^{18} -5.00000 q^{19} +2.00000 q^{20} +1.00000 q^{21} -2.00000 q^{22} -1.00000 q^{23} +1.00000 q^{25} +12.0000 q^{26} +1.00000 q^{27} +2.00000 q^{28} -5.00000 q^{29} -2.00000 q^{30} -8.00000 q^{31} +8.00000 q^{32} +1.00000 q^{33} +14.0000 q^{34} +1.00000 q^{35} +2.00000 q^{36} -2.00000 q^{37} +10.0000 q^{38} -6.00000 q^{39} +12.0000 q^{41} -2.00000 q^{42} -11.0000 q^{43} +2.00000 q^{44} +1.00000 q^{45} +2.00000 q^{46} +8.00000 q^{47} -4.00000 q^{48} +1.00000 q^{49} -2.00000 q^{50} -7.00000 q^{51} -12.0000 q^{52} -11.0000 q^{53} -2.00000 q^{54} +1.00000 q^{55} -5.00000 q^{57} +10.0000 q^{58} -5.00000 q^{59} +2.00000 q^{60} +7.00000 q^{61} +16.0000 q^{62} +1.00000 q^{63} -8.00000 q^{64} -6.00000 q^{65} -2.00000 q^{66} -2.00000 q^{67} -14.0000 q^{68} -1.00000 q^{69} -2.00000 q^{70} +12.0000 q^{71} +4.00000 q^{73} +4.00000 q^{74} +1.00000 q^{75} -10.0000 q^{76} +1.00000 q^{77} +12.0000 q^{78} -10.0000 q^{79} -4.00000 q^{80} +1.00000 q^{81} -24.0000 q^{82} -1.00000 q^{83} +2.00000 q^{84} -7.00000 q^{85} +22.0000 q^{86} -5.00000 q^{87} +15.0000 q^{89} -2.00000 q^{90} -6.00000 q^{91} -2.00000 q^{92} -8.00000 q^{93} -16.0000 q^{94} -5.00000 q^{95} +8.00000 q^{96} +3.00000 q^{97} -2.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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 $

$2$	−2.00000	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−2.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.00000	1.00000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−2.00000	−0.816497
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	2.00000	0.577350
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	−2.00000	−0.534522
$15$	1.00000	0.258199
$16$	−4.00000	−1.00000
$17$	−7.00000	−1.69775	−0.848875	−	0.528594i	$-0.822719\pi$
$17$	−7.00000	−1.69775	−0.848875	+	0.528594i	$0.822719\pi$
$18$	−2.00000	−0.471405
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	2.00000	0.447214
$21$	1.00000	0.218218
$22$	−2.00000	−0.426401
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	12.0000	2.35339
$27$	1.00000	0.192450
$28$	2.00000	0.377964
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	−2.00000	−0.365148
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	8.00000	1.41421
$33$	1.00000	0.174078
$34$	14.0000	2.40098
$35$	1.00000	0.169031
$36$	2.00000	0.333333
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	10.0000	1.62221
$39$	−6.00000	−0.960769
$40$	0	0
$41$	12.0000	1.87409	0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000	1.87409	0.937043	+	0.349215i	$0.113552\pi$
$42$	−2.00000	−0.308607
$43$	−11.0000	−1.67748	−0.838742	−	0.544529i	$-0.816708\pi$
$43$	−11.0000	−1.67748	−0.838742	+	0.544529i	$0.816708\pi$
$44$	2.00000	0.301511
$45$	1.00000	0.149071
$46$	2.00000	0.294884
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	−4.00000	−0.577350
$49$	1.00000	0.142857
$50$	−2.00000	−0.282843
$51$	−7.00000	−0.980196
$52$	−12.0000	−1.66410
$53$	−11.0000	−1.51097	−0.755483	−	0.655168i	$-0.772598\pi$
$53$	−11.0000	−1.51097	−0.755483	+	0.655168i	$0.772598\pi$
$54$	−2.00000	−0.272166
$55$	1.00000	0.134840
$56$	0	0
$57$	−5.00000	−0.662266
$58$	10.0000	1.31306
$59$	−5.00000	−0.650945	−0.325472	−	0.945552i	$-0.605523\pi$
$59$	−5.00000	−0.650945	−0.325472	+	0.945552i	$0.605523\pi$
$60$	2.00000	0.258199
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	16.0000	2.03200
$63$	1.00000	0.125988
$64$	−8.00000	−1.00000
$65$	−6.00000	−0.744208
$66$	−2.00000	−0.246183
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	−14.0000	−1.69775
$69$	−1.00000	−0.120386
$70$	−2.00000	−0.239046
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	4.00000	0.464991
$75$	1.00000	0.115470
$76$	−10.0000	−1.14708
$77$	1.00000	0.113961
$78$	12.0000	1.35873
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	−24.0000	−2.65036
$83$	−1.00000	−0.109764	−0.0548821	−	0.998493i	$-0.517478\pi$
$83$	−1.00000	−0.109764	−0.0548821	+	0.998493i	$0.517478\pi$
$84$	2.00000	0.218218
$85$	−7.00000	−0.759257
$86$	22.0000	2.37232
$87$	−5.00000	−0.536056
$88$	0	0
$89$	15.0000	1.59000	0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000	1.59000	0.794998	+	0.606612i	$0.207472\pi$
$90$	−2.00000	−0.210819
$91$	−6.00000	−0.628971
$92$	−2.00000	−0.208514
$93$	−8.00000	−0.829561
$94$	−16.0000	−1.65027
$95$	−5.00000	−0.512989
$96$	8.00000	0.816497
$97$	3.00000	0.304604	0.152302	−	0.988334i	$-0.451331\pi$
$97$	3.00000	0.304604	0.152302	+	0.988334i	$0.451331\pi$
$98$	−2.00000	−0.202031
$99$	1.00000	0.100504
$100$	2.00000	0.200000
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	14.0000	1.38621
$103$	−1.00000	−0.0985329	−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000	−0.0985329	−0.0492665	+	0.998786i	$0.515688\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	22.0000	2.13683
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	2.00000	0.192450
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	−2.00000	−0.190693
$111$	−2.00000	−0.189832
$112$	−4.00000	−0.377964
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	10.0000	0.936586
$115$	−1.00000	−0.0932505
$116$	−10.0000	−0.928477
$117$	−6.00000	−0.554700
$118$	10.0000	0.920575
$119$	−7.00000	−0.641689
$120$	0	0
$121$	1.00000	0.0909091
$122$	−14.0000	−1.26750
$123$	12.0000	1.08200
$124$	−16.0000	−1.43684
$125$	1.00000	0.0894427
$126$	−2.00000	−0.178174
$127$	−7.00000	−0.621150	−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000	−0.621150	−0.310575	+	0.950549i	$0.600522\pi$
$128$	0	0
$129$	−11.0000	−0.968496
$130$	12.0000	1.05247
$131$	2.00000	0.174741	0.0873704	−	0.996176i	$-0.472154\pi$
$131$	2.00000	0.174741	0.0873704	+	0.996176i	$0.472154\pi$
$132$	2.00000	0.174078
$133$	−5.00000	−0.433555
$134$	4.00000	0.345547
$135$	1.00000	0.0860663
$136$	0	0
$137$	−22.0000	−1.87959	−0.939793	−	0.341743i	$-0.888983\pi$
$137$	−22.0000	−1.87959	−0.939793	+	0.341743i	$0.888983\pi$
$138$	2.00000	0.170251
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	2.00000	0.169031
$141$	8.00000	0.673722
$142$	−24.0000	−2.01404
$143$	−6.00000	−0.501745
$144$	−4.00000	−0.333333
$145$	−5.00000	−0.415227
$146$	−8.00000	−0.662085
$147$	1.00000	0.0824786
$148$	−4.00000	−0.328798
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	−2.00000	−0.163299
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	0	0
$153$	−7.00000	−0.565916
$154$	−2.00000	−0.161165
$155$	−8.00000	−0.642575
$156$	−12.0000	−0.960769
$157$	−17.0000	−1.35675	−0.678374	−	0.734717i	$-0.737315\pi$
$157$	−17.0000	−1.35675	−0.678374	+	0.734717i	$0.737315\pi$
$158$	20.0000	1.59111
$159$	−11.0000	−0.872357
$160$	8.00000	0.632456
$161$	−1.00000	−0.0788110
$162$	−2.00000	−0.157135
$163$	−6.00000	−0.469956	−0.234978	−	0.972001i	$-0.575502\pi$
$163$	−6.00000	−0.469956	−0.234978	+	0.972001i	$0.575502\pi$
$164$	24.0000	1.87409
$165$	1.00000	0.0778499
$166$	2.00000	0.155230
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	14.0000	1.07375
$171$	−5.00000	−0.382360
$172$	−22.0000	−1.67748
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	10.0000	0.758098
$175$	1.00000	0.0755929
$176$	−4.00000	−0.301511
$177$	−5.00000	−0.375823
$178$	−30.0000	−2.24860
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	2.00000	0.149071
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	12.0000	0.889499
$183$	7.00000	0.517455
$184$	0	0
$185$	−2.00000	−0.147043
$186$	16.0000	1.17318
$187$	−7.00000	−0.511891
$188$	16.0000	1.16692
$189$	1.00000	0.0727393
$190$	10.0000	0.725476
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	−8.00000	−0.577350
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	−6.00000	−0.430775
$195$	−6.00000	−0.429669
$196$	2.00000	0.142857
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	−2.00000	−0.142134
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	−4.00000	−0.281439
$203$	−5.00000	−0.350931
$204$	−14.0000	−0.980196
$205$	12.0000	0.838116
$206$	2.00000	0.139347
$207$	−1.00000	−0.0695048
$208$	24.0000	1.66410
$209$	−5.00000	−0.345857
$210$	−2.00000	−0.138013
$211$	22.0000	1.51454	0.757271	−	0.653101i	$-0.226532\pi$
$211$	22.0000	1.51454	0.757271	+	0.653101i	$0.226532\pi$
$212$	−22.0000	−1.51097
$213$	12.0000	0.822226
$214$	24.0000	1.64061
$215$	−11.0000	−0.750194
$216$	0	0
$217$	−8.00000	−0.543075
$218$	0	0
$219$	4.00000	0.270295
$220$	2.00000	0.134840
$221$	42.0000	2.82523
$222$	4.00000	0.268462
$223$	9.00000	0.602685	0.301342	−	0.953516i	$-0.402565\pi$
$223$	9.00000	0.602685	0.301342	+	0.953516i	$0.402565\pi$
$224$	8.00000	0.534522
$225$	1.00000	0.0666667
$226$	−18.0000	−1.19734
$227$	23.0000	1.52656	0.763282	−	0.646066i	$-0.223587\pi$
$227$	23.0000	1.52656	0.763282	+	0.646066i	$0.223587\pi$
$228$	−10.0000	−0.662266
$229$	−20.0000	−1.32164	−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000	−1.32164	−0.660819	+	0.750546i	$0.729791\pi$
$230$	2.00000	0.131876
$231$	1.00000	0.0657952
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	12.0000	0.784465
$235$	8.00000	0.521862
$236$	−10.0000	−0.650945
$237$	−10.0000	−0.649570
$238$	14.0000	0.907485
$239$	5.00000	0.323423	0.161712	−	0.986838i	$-0.448299\pi$
$239$	5.00000	0.323423	0.161712	+	0.986838i	$0.448299\pi$
$240$	−4.00000	−0.258199
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	−2.00000	−0.128565
$243$	1.00000	0.0641500
$244$	14.0000	0.896258
$245$	1.00000	0.0638877
$246$	−24.0000	−1.53018
$247$	30.0000	1.90885
$248$	0	0
$249$	−1.00000	−0.0633724
$250$	−2.00000	−0.126491
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	2.00000	0.125988
$253$	−1.00000	−0.0628695
$254$	14.0000	0.878438
$255$	−7.00000	−0.438357
$256$	16.0000	1.00000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	22.0000	1.36966
$259$	−2.00000	−0.124274
$260$	−12.0000	−0.744208
$261$	−5.00000	−0.309492
$262$	−4.00000	−0.247121
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	0	0
$265$	−11.0000	−0.675725
$266$	10.0000	0.613139
$267$	15.0000	0.917985
$268$	−4.00000	−0.244339
$269$	−25.0000	−1.52428	−0.762138	−	0.647414i	$-0.775850\pi$
$269$	−25.0000	−1.52428	−0.762138	+	0.647414i	$0.775850\pi$
$270$	−2.00000	−0.121716
$271$	−3.00000	−0.182237	−0.0911185	−	0.995840i	$-0.529044\pi$
$271$	−3.00000	−0.182237	−0.0911185	+	0.995840i	$0.529044\pi$
$272$	28.0000	1.69775
$273$	−6.00000	−0.363137
$274$	44.0000	2.65814
$275$	1.00000	0.0603023
$276$	−2.00000	−0.120386
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	0	0
$279$	−8.00000	−0.478947
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	−16.0000	−0.952786
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	24.0000	1.42414
$285$	−5.00000	−0.296174
$286$	12.0000	0.709575
$287$	12.0000	0.708338
$288$	8.00000	0.471405
$289$	32.0000	1.88235
$290$	10.0000	0.587220
$291$	3.00000	0.175863
$292$	8.00000	0.468165
$293$	29.0000	1.69420	0.847099	−	0.531435i	$-0.178347\pi$
$293$	29.0000	1.69420	0.847099	+	0.531435i	$0.178347\pi$
$294$	−2.00000	−0.116642
$295$	−5.00000	−0.291111
$296$	0	0
$297$	1.00000	0.0580259
$298$	20.0000	1.15857
$299$	6.00000	0.346989
$300$	2.00000	0.115470
$301$	−11.0000	−0.634029
$302$	−4.00000	−0.230174
$303$	2.00000	0.114897
$304$	20.0000	1.14708
$305$	7.00000	0.400819
$306$	14.0000	0.800327
$307$	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
$309$	−1.00000	−0.0568880
$310$	16.0000	0.908739
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	−31.0000	−1.75222	−0.876112	−	0.482108i	$-0.839871\pi$
$313$	−31.0000	−1.75222	−0.876112	+	0.482108i	$0.839871\pi$
$314$	34.0000	1.91873
$315$	1.00000	0.0563436
$316$	−20.0000	−1.12509
$317$	−22.0000	−1.23564	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	−22.0000	−1.23564	−0.617822	+	0.786318i	$0.711985\pi$
$318$	22.0000	1.23370
$319$	−5.00000	−0.279946
$320$	−8.00000	−0.447214
$321$	−12.0000	−0.669775
$322$	2.00000	0.111456
$323$	35.0000	1.94745
$324$	2.00000	0.111111
$325$	−6.00000	−0.332820
$326$	12.0000	0.664619
$327$	0	0
$328$	0	0
$329$	8.00000	0.441054
$330$	−2.00000	−0.110096
$331$	−3.00000	−0.164895	−0.0824475	−	0.996595i	$-0.526274\pi$
$331$	−3.00000	−0.164895	−0.0824475	+	0.996595i	$0.526274\pi$
$332$	−2.00000	−0.109764
$333$	−2.00000	−0.109599
$334$	−16.0000	−0.875481
$335$	−2.00000	−0.109272
$336$	−4.00000	−0.218218
$337$	13.0000	0.708155	0.354078	−	0.935216i	$-0.384795\pi$
$337$	13.0000	0.708155	0.354078	+	0.935216i	$0.384795\pi$
$338$	−46.0000	−2.50207
$339$	9.00000	0.488813
$340$	−14.0000	−0.759257
$341$	−8.00000	−0.433224
$342$	10.0000	0.540738
$343$	1.00000	0.0539949
$344$	0	0
$345$	−1.00000	−0.0538382
$346$	12.0000	0.645124
$347$	−32.0000	−1.71785	−0.858925	−	0.512101i	$-0.828867\pi$
$347$	−32.0000	−1.71785	−0.858925	+	0.512101i	$0.828867\pi$
$348$	−10.0000	−0.536056
$349$	−25.0000	−1.33822	−0.669110	−	0.743164i	$-0.733324\pi$
$349$	−25.0000	−1.33822	−0.669110	+	0.743164i	$0.733324\pi$
$350$	−2.00000	−0.106904
$351$	−6.00000	−0.320256
$352$	8.00000	0.426401
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	10.0000	0.531494
$355$	12.0000	0.636894
$356$	30.0000	1.59000
$357$	−7.00000	−0.370479
$358$	0	0
$359$	−15.0000	−0.791670	−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000	−0.791670	−0.395835	+	0.918322i	$0.629545\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	−44.0000	−2.31259
$363$	1.00000	0.0524864
$364$	−12.0000	−0.628971
$365$	4.00000	0.209370
$366$	−14.0000	−0.731792
$367$	−17.0000	−0.887393	−0.443696	−	0.896177i	$-0.646333\pi$
$367$	−17.0000	−0.887393	−0.443696	+	0.896177i	$0.646333\pi$
$368$	4.00000	0.208514
$369$	12.0000	0.624695
$370$	4.00000	0.207950
$371$	−11.0000	−0.571092
$372$	−16.0000	−0.829561
$373$	29.0000	1.50156	0.750782	−	0.660551i	$-0.229677\pi$
$373$	29.0000	1.50156	0.750782	+	0.660551i	$0.229677\pi$
$374$	14.0000	0.723923
$375$	1.00000	0.0516398
$376$	0	0
$377$	30.0000	1.54508
$378$	−2.00000	−0.102869
$379$	5.00000	0.256833	0.128416	−	0.991720i	$-0.459011\pi$
$379$	5.00000	0.256833	0.128416	+	0.991720i	$0.459011\pi$
$380$	−10.0000	−0.512989
$381$	−7.00000	−0.358621
$382$	16.0000	0.818631
$383$	34.0000	1.73732	0.868659	−	0.495410i	$-0.164982\pi$
$383$	34.0000	1.73732	0.868659	+	0.495410i	$0.164982\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	12.0000	0.610784
$387$	−11.0000	−0.559161
$388$	6.00000	0.304604
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	12.0000	0.607644
$391$	7.00000	0.354005
$392$	0	0
$393$	2.00000	0.100887
$394$	24.0000	1.20910
$395$	−10.0000	−0.503155
$396$	2.00000	0.100504
$397$	38.0000	1.90717	0.953583	−	0.301131i	$-0.0973643\pi$
$397$	38.0000	1.90717	0.953583	+	0.301131i	$0.0973643\pi$
$398$	−40.0000	−2.00502
$399$	−5.00000	−0.250313
$400$	−4.00000	−0.200000
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	4.00000	0.199502
$403$	48.0000	2.39105
$404$	4.00000	0.199007
$405$	1.00000	0.0496904
$406$	10.0000	0.496292
$407$	−2.00000	−0.0991363
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	−24.0000	−1.18528
$411$	−22.0000	−1.08518
$412$	−2.00000	−0.0985329
$413$	−5.00000	−0.246034
$414$	2.00000	0.0982946
$415$	−1.00000	−0.0490881
$416$	−48.0000	−2.35339
$417$	0	0
$418$	10.0000	0.489116
$419$	−15.0000	−0.732798	−0.366399	−	0.930458i	$-0.619409\pi$
$419$	−15.0000	−0.732798	−0.366399	+	0.930458i	$0.619409\pi$
$420$	2.00000	0.0975900
$421$	−23.0000	−1.12095	−0.560476	−	0.828171i	$-0.689382\pi$
$421$	−23.0000	−1.12095	−0.560476	+	0.828171i	$0.689382\pi$
$422$	−44.0000	−2.14189
$423$	8.00000	0.388973
$424$	0	0
$425$	−7.00000	−0.339550
$426$	−24.0000	−1.16280
$427$	7.00000	0.338754
$428$	−24.0000	−1.16008
$429$	−6.00000	−0.289683
$430$	22.0000	1.06093
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	−4.00000	−0.192450
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	16.0000	0.768025
$435$	−5.00000	−0.239732
$436$	0	0
$437$	5.00000	0.239182
$438$	−8.00000	−0.382255
$439$	−5.00000	−0.238637	−0.119318	−	0.992856i	$-0.538071\pi$
$439$	−5.00000	−0.238637	−0.119318	+	0.992856i	$0.538071\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	−84.0000	−3.99547
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	−4.00000	−0.189832
$445$	15.0000	0.711068
$446$	−18.0000	−0.852325
$447$	−10.0000	−0.472984
$448$	−8.00000	−0.377964
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	−2.00000	−0.0942809
$451$	12.0000	0.565058
$452$	18.0000	0.846649
$453$	2.00000	0.0939682
$454$	−46.0000	−2.15889
$455$	−6.00000	−0.281284
$456$	0	0
$457$	33.0000	1.54367	0.771837	−	0.635820i	$-0.219338\pi$
$457$	33.0000	1.54367	0.771837	+	0.635820i	$0.219338\pi$
$458$	40.0000	1.86908
$459$	−7.00000	−0.326732
$460$	−2.00000	−0.0932505
$461$	−38.0000	−1.76984	−0.884918	−	0.465746i	$-0.845786\pi$
$461$	−38.0000	−1.76984	−0.884918	+	0.465746i	$0.845786\pi$
$462$	−2.00000	−0.0930484
$463$	−6.00000	−0.278844	−0.139422	−	0.990233i	$-0.544524\pi$
$463$	−6.00000	−0.278844	−0.139422	+	0.990233i	$0.544524\pi$
$464$	20.0000	0.928477
$465$	−8.00000	−0.370991
$466$	12.0000	0.555889
$467$	28.0000	1.29569	0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000	1.29569	0.647843	+	0.761774i	$0.275671\pi$
$468$	−12.0000	−0.554700
$469$	−2.00000	−0.0923514
$470$	−16.0000	−0.738025
$471$	−17.0000	−0.783319
$472$	0	0
$473$	−11.0000	−0.505781
$474$	20.0000	0.918630
$475$	−5.00000	−0.229416
$476$	−14.0000	−0.641689
$477$	−11.0000	−0.503655
$478$	−10.0000	−0.457389
$479$	−10.0000	−0.456912	−0.228456	−	0.973554i	$-0.573368\pi$
$479$	−10.0000	−0.456912	−0.228456	+	0.973554i	$0.573368\pi$
$480$	8.00000	0.365148
$481$	12.0000	0.547153
$482$	36.0000	1.63976
$483$	−1.00000	−0.0455016
$484$	2.00000	0.0909091
$485$	3.00000	0.136223
$486$	−2.00000	−0.0907218
$487$	−22.0000	−0.996915	−0.498458	−	0.866914i	$-0.666100\pi$
$487$	−22.0000	−0.996915	−0.498458	+	0.866914i	$0.666100\pi$
$488$	0	0
$489$	−6.00000	−0.271329
$490$	−2.00000	−0.0903508
$491$	27.0000	1.21849	0.609246	−	0.792981i	$-0.291472\pi$
$491$	27.0000	1.21849	0.609246	+	0.792981i	$0.291472\pi$
$492$	24.0000	1.08200
$493$	35.0000	1.57632
$494$	−60.0000	−2.69953
$495$	1.00000	0.0449467
$496$	32.0000	1.43684
$497$	12.0000	0.538274
$498$	2.00000	0.0896221
$499$	−25.0000	−1.11915	−0.559577	−	0.828778i	$-0.689036\pi$
$499$	−25.0000	−1.11915	−0.559577	+	0.828778i	$0.689036\pi$
$500$	2.00000	0.0894427
$501$	8.00000	0.357414
$502$	−24.0000	−1.07117
$503$	−11.0000	−0.490466	−0.245233	−	0.969464i	$-0.578864\pi$
$503$	−11.0000	−0.490466	−0.245233	+	0.969464i	$0.578864\pi$
$504$	0	0
$505$	2.00000	0.0889988
$506$	2.00000	0.0889108
$507$	23.0000	1.02147
$508$	−14.0000	−0.621150
$509$	−35.0000	−1.55135	−0.775674	−	0.631134i	$-0.782590\pi$
$509$	−35.0000	−1.55135	−0.775674	+	0.631134i	$0.782590\pi$
$510$	14.0000	0.619930
$511$	4.00000	0.176950
$512$	−32.0000	−1.41421
$513$	−5.00000	−0.220755
$514$	−36.0000	−1.58789
$515$	−1.00000	−0.0440653
$516$	−22.0000	−0.968496
$517$	8.00000	0.351840
$518$	4.00000	0.175750
$519$	−6.00000	−0.263371
$520$	0	0
$521$	−23.0000	−1.00765	−0.503824	−	0.863806i	$-0.668074\pi$
$521$	−23.0000	−1.00765	−0.503824	+	0.863806i	$0.668074\pi$
$522$	10.0000	0.437688
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	4.00000	0.174741
$525$	1.00000	0.0436436
$526$	−8.00000	−0.348817
$527$	56.0000	2.43940
$528$	−4.00000	−0.174078
$529$	−22.0000	−0.956522
$530$	22.0000	0.955619
$531$	−5.00000	−0.216982
$532$	−10.0000	−0.433555
$533$	−72.0000	−3.11867
$534$	−30.0000	−1.29823
$535$	−12.0000	−0.518805
$536$	0	0
$537$	0	0
$538$	50.0000	2.15565
$539$	1.00000	0.0430730
$540$	2.00000	0.0860663
$541$	12.0000	0.515920	0.257960	−	0.966156i	$-0.416950\pi$
$541$	12.0000	0.515920	0.257960	+	0.966156i	$0.416950\pi$
$542$	6.00000	0.257722
$543$	22.0000	0.944110
$544$	−56.0000	−2.40098
$545$	0	0
$546$	12.0000	0.513553
$547$	3.00000	0.128271	0.0641354	−	0.997941i	$-0.479571\pi$
$547$	3.00000	0.128271	0.0641354	+	0.997941i	$0.479571\pi$
$548$	−44.0000	−1.87959
$549$	7.00000	0.298753
$550$	−2.00000	−0.0852803
$551$	25.0000	1.06504
$552$	0	0
$553$	−10.0000	−0.425243
$554$	44.0000	1.86938
$555$	−2.00000	−0.0848953
$556$	0	0
$557$	38.0000	1.61011	0.805056	−	0.593199i	$-0.202135\pi$
$557$	38.0000	1.61011	0.805056	+	0.593199i	$0.202135\pi$
$558$	16.0000	0.677334
$559$	66.0000	2.79150
$560$	−4.00000	−0.169031
$561$	−7.00000	−0.295540
$562$	36.0000	1.51857
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	16.0000	0.673722
$565$	9.00000	0.378633
$566$	−28.0000	−1.17693
$567$	1.00000	0.0419961
$568$	0	0
$569$	15.0000	0.628833	0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000	0.628833	0.314416	+	0.949285i	$0.398191\pi$
$570$	10.0000	0.418854
$571$	−18.0000	−0.753277	−0.376638	−	0.926360i	$-0.622920\pi$
$571$	−18.0000	−0.753277	−0.376638	+	0.926360i	$0.622920\pi$
$572$	−12.0000	−0.501745
$573$	−8.00000	−0.334205
$574$	−24.0000	−1.00174
$575$	−1.00000	−0.0417029
$576$	−8.00000	−0.333333
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	−64.0000	−2.66205
$579$	−6.00000	−0.249351
$580$	−10.0000	−0.415227
$581$	−1.00000	−0.0414870
$582$	−6.00000	−0.248708
$583$	−11.0000	−0.455573
$584$	0	0
$585$	−6.00000	−0.248069
$586$	−58.0000	−2.39596
$587$	−42.0000	−1.73353	−0.866763	−	0.498721i	$-0.833803\pi$
$587$	−42.0000	−1.73353	−0.866763	+	0.498721i	$0.833803\pi$
$588$	2.00000	0.0824786
$589$	40.0000	1.64817
$590$	10.0000	0.411693
$591$	−12.0000	−0.493614
$592$	8.00000	0.328798
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	−2.00000	−0.0820610
$595$	−7.00000	−0.286972
$596$	−20.0000	−0.819232
$597$	20.0000	0.818546
$598$	−12.0000	−0.490716
$599$	−10.0000	−0.408589	−0.204294	−	0.978909i	$-0.565490\pi$
$599$	−10.0000	−0.408589	−0.204294	+	0.978909i	$0.565490\pi$
$600$	0	0
$601$	−3.00000	−0.122373	−0.0611863	−	0.998126i	$-0.519488\pi$
$601$	−3.00000	−0.122373	−0.0611863	+	0.998126i	$0.519488\pi$
$602$	22.0000	0.896653
$603$	−2.00000	−0.0814463
$604$	4.00000	0.162758
$605$	1.00000	0.0406558
$606$	−4.00000	−0.162489
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	−40.0000	−1.62221
$609$	−5.00000	−0.202610
$610$	−14.0000	−0.566843
$611$	−48.0000	−1.94187
$612$	−14.0000	−0.565916
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	−36.0000	−1.45284
$615$	12.0000	0.483887
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	2.00000	0.0804518
$619$	30.0000	1.20580	0.602901	−	0.797816i	$-0.294011\pi$
$619$	30.0000	1.20580	0.602901	+	0.797816i	$0.294011\pi$
$620$	−16.0000	−0.642575
$621$	−1.00000	−0.0401286
$622$	−24.0000	−0.962312
$623$	15.0000	0.600962
$624$	24.0000	0.960769
$625$	1.00000	0.0400000
$626$	62.0000	2.47802
$627$	−5.00000	−0.199681
$628$	−34.0000	−1.35675
$629$	14.0000	0.558217
$630$	−2.00000	−0.0796819
$631$	−3.00000	−0.119428	−0.0597141	−	0.998216i	$-0.519019\pi$
$631$	−3.00000	−0.119428	−0.0597141	+	0.998216i	$0.519019\pi$
$632$	0	0
$633$	22.0000	0.874421
$634$	44.0000	1.74746
$635$	−7.00000	−0.277787
$636$	−22.0000	−0.872357
$637$	−6.00000	−0.237729
$638$	10.0000	0.395904
$639$	12.0000	0.474713
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	24.0000	0.947204
$643$	19.0000	0.749287	0.374643	−	0.927169i	$-0.377765\pi$
$643$	19.0000	0.749287	0.374643	+	0.927169i	$0.377765\pi$
$644$	−2.00000	−0.0788110
$645$	−11.0000	−0.433125
$646$	−70.0000	−2.75411
$647$	48.0000	1.88707	0.943537	−	0.331266i	$-0.107476\pi$
$647$	48.0000	1.88707	0.943537	+	0.331266i	$0.107476\pi$
$648$	0	0
$649$	−5.00000	−0.196267
$650$	12.0000	0.470679
$651$	−8.00000	−0.313545
$652$	−12.0000	−0.469956
$653$	−1.00000	−0.0391330	−0.0195665	−	0.999809i	$-0.506229\pi$
$653$	−1.00000	−0.0391330	−0.0195665	+	0.999809i	$0.506229\pi$
$654$	0	0
$655$	2.00000	0.0781465
$656$	−48.0000	−1.87409
$657$	4.00000	0.156055
$658$	−16.0000	−0.623745
$659$	15.0000	0.584317	0.292159	−	0.956370i	$-0.405627\pi$
$659$	15.0000	0.584317	0.292159	+	0.956370i	$0.405627\pi$
$660$	2.00000	0.0778499
$661$	−28.0000	−1.08907	−0.544537	−	0.838737i	$-0.683295\pi$
$661$	−28.0000	−1.08907	−0.544537	+	0.838737i	$0.683295\pi$
$662$	6.00000	0.233197
$663$	42.0000	1.63114
$664$	0	0
$665$	−5.00000	−0.193892
$666$	4.00000	0.154997
$667$	5.00000	0.193601
$668$	16.0000	0.619059
$669$	9.00000	0.347960
$670$	4.00000	0.154533
$671$	7.00000	0.270232
$672$	8.00000	0.308607
$673$	−1.00000	−0.0385472	−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000	−0.0385472	−0.0192736	+	0.999814i	$0.506135\pi$
$674$	−26.0000	−1.00148
$675$	1.00000	0.0384900
$676$	46.0000	1.76923
$677$	13.0000	0.499631	0.249815	−	0.968294i	$-0.419630\pi$
$677$	13.0000	0.499631	0.249815	+	0.968294i	$0.419630\pi$
$678$	−18.0000	−0.691286
$679$	3.00000	0.115129
$680$	0	0
$681$	23.0000	0.881362
$682$	16.0000	0.612672
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	−10.0000	−0.382360
$685$	−22.0000	−0.840577
$686$	−2.00000	−0.0763604
$687$	−20.0000	−0.763048
$688$	44.0000	1.67748
$689$	66.0000	2.51440
$690$	2.00000	0.0761387
$691$	42.0000	1.59776	0.798878	−	0.601494i	$-0.205427\pi$
$691$	42.0000	1.59776	0.798878	+	0.601494i	$0.205427\pi$
$692$	−12.0000	−0.456172
$693$	1.00000	0.0379869
$694$	64.0000	2.42941
$695$	0	0
$696$	0	0
$697$	−84.0000	−3.18173
$698$	50.0000	1.89253
$699$	−6.00000	−0.226941
$700$	2.00000	0.0755929
$701$	−23.0000	−0.868698	−0.434349	−	0.900745i	$-0.643022\pi$
$701$	−23.0000	−0.868698	−0.434349	+	0.900745i	$0.643022\pi$
$702$	12.0000	0.452911
$703$	10.0000	0.377157
$704$	−8.00000	−0.301511
$705$	8.00000	0.301297
$706$	12.0000	0.451626
$707$	2.00000	0.0752177
$708$	−10.0000	−0.375823
$709$	−35.0000	−1.31445	−0.657226	−	0.753693i	$-0.728270\pi$
$709$	−35.0000	−1.31445	−0.657226	+	0.753693i	$0.728270\pi$
$710$	−24.0000	−0.900704
$711$	−10.0000	−0.375029
$712$	0	0
$713$	8.00000	0.299602
$714$	14.0000	0.523937
$715$	−6.00000	−0.224387
$716$	0	0
$717$	5.00000	0.186728
$718$	30.0000	1.11959
$719$	−25.0000	−0.932343	−0.466171	−	0.884694i	$-0.654367\pi$
$719$	−25.0000	−0.932343	−0.466171	+	0.884694i	$0.654367\pi$
$720$	−4.00000	−0.149071
$721$	−1.00000	−0.0372419
$722$	−12.0000	−0.446594
$723$	−18.0000	−0.669427
$724$	44.0000	1.63525
$725$	−5.00000	−0.185695
$726$	−2.00000	−0.0742270
$727$	−47.0000	−1.74313	−0.871567	−	0.490277i	$-0.836896\pi$
$727$	−47.0000	−1.74313	−0.871567	+	0.490277i	$0.836896\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−8.00000	−0.296093
$731$	77.0000	2.84795
$732$	14.0000	0.517455
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	34.0000	1.25496
$735$	1.00000	0.0368856
$736$	−8.00000	−0.294884
$737$	−2.00000	−0.0736709
$738$	−24.0000	−0.883452
$739$	−10.0000	−0.367856	−0.183928	−	0.982940i	$-0.558881\pi$
$739$	−10.0000	−0.367856	−0.183928	+	0.982940i	$0.558881\pi$
$740$	−4.00000	−0.147043
$741$	30.0000	1.10208
$742$	22.0000	0.807645
$743$	−6.00000	−0.220119	−0.110059	−	0.993925i	$-0.535104\pi$
$743$	−6.00000	−0.220119	−0.110059	+	0.993925i	$0.535104\pi$
$744$	0	0
$745$	−10.0000	−0.366372
$746$	−58.0000	−2.12353
$747$	−1.00000	−0.0365881
$748$	−14.0000	−0.511891
$749$	−12.0000	−0.438470
$750$	−2.00000	−0.0730297
$751$	17.0000	0.620339	0.310169	−	0.950681i	$-0.399614\pi$
$751$	17.0000	0.620339	0.310169	+	0.950681i	$0.399614\pi$
$752$	−32.0000	−1.16692
$753$	12.0000	0.437304
$754$	−60.0000	−2.18507
$755$	2.00000	0.0727875
$756$	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	$0.630921\pi$
$758$	−10.0000	−0.363216
$759$	−1.00000	−0.0362977
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	14.0000	0.507166
$763$	0	0
$764$	−16.0000	−0.578860
$765$	−7.00000	−0.253086
$766$	−68.0000	−2.45694
$767$	30.0000	1.08324
$768$	16.0000	0.577350
$769$	35.0000	1.26213	0.631066	−	0.775729i	$-0.282618\pi$
$769$	35.0000	1.26213	0.631066	+	0.775729i	$0.282618\pi$
$770$	−2.00000	−0.0720750
$771$	18.0000	0.648254
$772$	−12.0000	−0.431889
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	22.0000	0.790774
$775$	−8.00000	−0.287368
$776$	0	0
$777$	−2.00000	−0.0717496
$778$	−60.0000	−2.15110
$779$	−60.0000	−2.14972
$780$	−12.0000	−0.429669
$781$	12.0000	0.429394
$782$	−14.0000	−0.500639
$783$	−5.00000	−0.178685
$784$	−4.00000	−0.142857
$785$	−17.0000	−0.606756
$786$	−4.00000	−0.142675
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	−24.0000	−0.854965
$789$	4.00000	0.142404
$790$	20.0000	0.711568
$791$	9.00000	0.320003
$792$	0	0
$793$	−42.0000	−1.49146
$794$	−76.0000	−2.69714
$795$	−11.0000	−0.390130
$796$	40.0000	1.41776
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	10.0000	0.353996
$799$	−56.0000	−1.98114
$800$	8.00000	0.282843
$801$	15.0000	0.529999
$802$	−24.0000	−0.847469
$803$	4.00000	0.141157
$804$	−4.00000	−0.141069
$805$	−1.00000	−0.0352454
$806$	−96.0000	−3.38146
$807$	−25.0000	−0.880042
$808$	0	0
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	−2.00000	−0.0702728
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	−10.0000	−0.350931
$813$	−3.00000	−0.105215
$814$	4.00000	0.140200
$815$	−6.00000	−0.210171
$816$	28.0000	0.980196
$817$	55.0000	1.92421
$818$	20.0000	0.699284
$819$	−6.00000	−0.209657
$820$	24.0000	0.838116
$821$	−3.00000	−0.104701	−0.0523504	−	0.998629i	$-0.516671\pi$
$821$	−3.00000	−0.104701	−0.0523504	+	0.998629i	$0.516671\pi$
$822$	44.0000	1.53468
$823$	14.0000	0.488009	0.244005	−	0.969774i	$-0.421539\pi$
$823$	14.0000	0.488009	0.244005	+	0.969774i	$0.421539\pi$
$824$	0	0
$825$	1.00000	0.0348155
$826$	10.0000	0.347945
$827$	48.0000	1.66912	0.834562	−	0.550914i	$-0.185721\pi$
$827$	48.0000	1.66912	0.834562	+	0.550914i	$0.185721\pi$
$828$	−2.00000	−0.0695048
$829$	−30.0000	−1.04194	−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000	−1.04194	−0.520972	+	0.853574i	$0.674430\pi$
$830$	2.00000	0.0694210
$831$	−22.0000	−0.763172
$832$	48.0000	1.66410
$833$	−7.00000	−0.242536
$834$	0	0
$835$	8.00000	0.276851
$836$	−10.0000	−0.345857
$837$	−8.00000	−0.276520
$838$	30.0000	1.03633
$839$	15.0000	0.517858	0.258929	−	0.965896i	$-0.416631\pi$
$839$	15.0000	0.517858	0.258929	+	0.965896i	$0.416631\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	46.0000	1.58526
$843$	−18.0000	−0.619953
$844$	44.0000	1.51454
$845$	23.0000	0.791224
$846$	−16.0000	−0.550091
$847$	1.00000	0.0343604
$848$	44.0000	1.51097
$849$	14.0000	0.480479
$850$	14.0000	0.480196
$851$	2.00000	0.0685591
$852$	24.0000	0.822226
$853$	−36.0000	−1.23262	−0.616308	−	0.787505i	$-0.711372\pi$
$853$	−36.0000	−1.23262	−0.616308	+	0.787505i	$0.711372\pi$
$854$	−14.0000	−0.479070
$855$	−5.00000	−0.170996
$856$	0	0
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	12.0000	0.409673
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	−22.0000	−0.750194
$861$	12.0000	0.408959
$862$	16.0000	0.544962
$863$	39.0000	1.32758	0.663788	−	0.747921i	$-0.268948\pi$
$863$	39.0000	1.32758	0.663788	+	0.747921i	$0.268948\pi$
$864$	8.00000	0.272166
$865$	−6.00000	−0.204006
$866$	52.0000	1.76703
$867$	32.0000	1.08678
$868$	−16.0000	−0.543075
$869$	−10.0000	−0.339227
$870$	10.0000	0.339032
$871$	12.0000	0.406604
$872$	0	0
$873$	3.00000	0.101535
$874$	−10.0000	−0.338255
$875$	1.00000	0.0338062
$876$	8.00000	0.270295
$877$	23.0000	0.776655	0.388327	−	0.921521i	$-0.373053\pi$
$877$	23.0000	0.776655	0.388327	+	0.921521i	$0.373053\pi$
$878$	10.0000	0.337484
$879$	29.0000	0.978146
$880$	−4.00000	−0.134840
$881$	−33.0000	−1.11180	−0.555899	−	0.831250i	$-0.687626\pi$
$881$	−33.0000	−1.11180	−0.555899	+	0.831250i	$0.687626\pi$
$882$	−2.00000	−0.0673435
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	84.0000	2.82523
$885$	−5.00000	−0.168073
$886$	72.0000	2.41889
$887$	43.0000	1.44380	0.721899	−	0.691998i	$-0.243269\pi$
$887$	43.0000	1.44380	0.721899	+	0.691998i	$0.243269\pi$
$888$	0	0
$889$	−7.00000	−0.234772
$890$	−30.0000	−1.00560
$891$	1.00000	0.0335013
$892$	18.0000	0.602685
$893$	−40.0000	−1.33855
$894$	20.0000	0.668900
$895$	0	0
$896$	0	0
$897$	6.00000	0.200334
$898$	0	0
$899$	40.0000	1.33407
$900$	2.00000	0.0666667
$901$	77.0000	2.56524
$902$	−24.0000	−0.799113
$903$	−11.0000	−0.366057
$904$	0	0
$905$	22.0000	0.731305
$906$	−4.00000	−0.132891
$907$	−2.00000	−0.0664089	−0.0332045	−	0.999449i	$-0.510571\pi$
$907$	−2.00000	−0.0664089	−0.0332045	+	0.999449i	$0.510571\pi$
$908$	46.0000	1.52656
$909$	2.00000	0.0663358
$910$	12.0000	0.397796
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	20.0000	0.662266
$913$	−1.00000	−0.0330952
$914$	−66.0000	−2.18309
$915$	7.00000	0.231413
$916$	−40.0000	−1.32164
$917$	2.00000	0.0660458
$918$	14.0000	0.462069
$919$	30.0000	0.989609	0.494804	−	0.869004i	$-0.335240\pi$
$919$	30.0000	0.989609	0.494804	+	0.869004i	$0.335240\pi$
$920$	0	0
$921$	18.0000	0.593120
$922$	76.0000	2.50293
$923$	−72.0000	−2.36991
$924$	2.00000	0.0657952
$925$	−2.00000	−0.0657596
$926$	12.0000	0.394344
$927$	−1.00000	−0.0328443
$928$	−40.0000	−1.31306
$929$	−50.0000	−1.64045	−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000	−1.64045	−0.820223	+	0.572043i	$0.806151\pi$
$930$	16.0000	0.524661
$931$	−5.00000	−0.163868
$932$	−12.0000	−0.393073
$933$	12.0000	0.392862
$934$	−56.0000	−1.83238
$935$	−7.00000	−0.228924
$936$	0	0
$937$	28.0000	0.914720	0.457360	−	0.889282i	$-0.348795\pi$
$937$	28.0000	0.914720	0.457360	+	0.889282i	$0.348795\pi$
$938$	4.00000	0.130605
$939$	−31.0000	−1.01165
$940$	16.0000	0.521862
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	34.0000	1.10778
$943$	−12.0000	−0.390774
$944$	20.0000	0.650945
$945$	1.00000	0.0325300
$946$	22.0000	0.715282
$947$	23.0000	0.747400	0.373700	−	0.927550i	$-0.378089\pi$
$947$	23.0000	0.747400	0.373700	+	0.927550i	$0.378089\pi$
$948$	−20.0000	−0.649570
$949$	−24.0000	−0.779073
$950$	10.0000	0.324443
$951$	−22.0000	−0.713399
$952$	0	0
$953$	34.0000	1.10137	0.550684	−	0.834714i	$-0.314367\pi$
$953$	34.0000	1.10137	0.550684	+	0.834714i	$0.314367\pi$
$954$	22.0000	0.712276
$955$	−8.00000	−0.258874
$956$	10.0000	0.323423
$957$	−5.00000	−0.161627
$958$	20.0000	0.646171
$959$	−22.0000	−0.710417
$960$	−8.00000	−0.258199
$961$	33.0000	1.06452
$962$	−24.0000	−0.773791
$963$	−12.0000	−0.386695
$964$	−36.0000	−1.15948
$965$	−6.00000	−0.193147
$966$	2.00000	0.0643489
$967$	−47.0000	−1.51142	−0.755709	−	0.654907i	$-0.772708\pi$
$967$	−47.0000	−1.51142	−0.755709	+	0.654907i	$0.772708\pi$
$968$	0	0
$969$	35.0000	1.12436
$970$	−6.00000	−0.192648
$971$	−13.0000	−0.417190	−0.208595	−	0.978002i	$-0.566889\pi$
$971$	−13.0000	−0.417190	−0.208595	+	0.978002i	$0.566889\pi$
$972$	2.00000	0.0641500
$973$	0	0
$974$	44.0000	1.40985
$975$	−6.00000	−0.192154
$976$	−28.0000	−0.896258
$977$	−7.00000	−0.223950	−0.111975	−	0.993711i	$-0.535718\pi$
$977$	−7.00000	−0.223950	−0.111975	+	0.993711i	$0.535718\pi$
$978$	12.0000	0.383718
$979$	15.0000	0.479402
$980$	2.00000	0.0638877
$981$	0	0
$982$	−54.0000	−1.72321
$983$	14.0000	0.446531	0.223265	−	0.974758i	$-0.428328\pi$
$983$	14.0000	0.446531	0.223265	+	0.974758i	$0.428328\pi$
$984$	0	0
$985$	−12.0000	−0.382352
$986$	−70.0000	−2.22925
$987$	8.00000	0.254643
$988$	60.0000	1.90885
$989$	11.0000	0.349780
$990$	−2.00000	−0.0635642
$991$	−33.0000	−1.04828	−0.524140	−	0.851632i	$-0.675613\pi$
$991$	−33.0000	−1.04828	−0.524140	+	0.851632i	$0.675613\pi$
$992$	−64.0000	−2.03200
$993$	−3.00000	−0.0952021
$994$	−24.0000	−0.761234
$995$	20.0000	0.634043
$996$	−2.00000	−0.0633724
$997$	−22.0000	−0.696747	−0.348373	−	0.937356i	$-0.613266\pi$
$997$	−22.0000	−0.696747	−0.348373	+	0.937356i	$0.613266\pi$
$998$	50.0000	1.58272
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1155.2.a.c.1.1	✓	1
3.2	odd	2		3465.2.a.r.1.1		1
5.4	even	2		5775.2.a.y.1.1		1
7.6	odd	2		8085.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.2.a.c.1.1	✓	1	1.1	even	1	trivial
3465.2.a.r.1.1		1	3.2	odd	2
5775.2.a.y.1.1		1	5.4	even	2
8085.2.a.b.1.1		1	7.6	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1155.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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