LMFDB - Embedded newform 2541.2.a.h.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2541,2,Mod(1,2541)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2541, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2541.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2541 = 3 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2541.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,-1,-1,-2,-1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$20.2899871536$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2541.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +1.00000 q^{12} -6.00000 q^{13} -1.00000 q^{14} +2.00000 q^{15} -1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} +2.00000 q^{20} +1.00000 q^{21} +3.00000 q^{24} -1.00000 q^{25} -6.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} +2.00000 q^{29} +2.00000 q^{30} +8.00000 q^{31} +5.00000 q^{32} -2.00000 q^{34} +2.00000 q^{35} -1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} +6.00000 q^{39} +6.00000 q^{40} -10.0000 q^{41} +1.00000 q^{42} +4.00000 q^{43} -2.00000 q^{45} -8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +2.00000 q^{51} +6.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} +3.00000 q^{56} +4.00000 q^{57} +2.00000 q^{58} +4.00000 q^{59} -2.00000 q^{60} +10.0000 q^{61} +8.00000 q^{62} -1.00000 q^{63} +7.00000 q^{64} +12.0000 q^{65} -12.0000 q^{67} +2.00000 q^{68} +2.00000 q^{70} -3.00000 q^{72} -2.00000 q^{73} +6.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} +6.00000 q^{78} -16.0000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} -4.00000 q^{83} -1.00000 q^{84} +4.00000 q^{85} +4.00000 q^{86} -2.00000 q^{87} +18.0000 q^{89} -2.00000 q^{90} +6.00000 q^{91} -8.00000 q^{93} -8.00000 q^{94} +8.00000 q^{95} -5.00000 q^{96} +2.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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 $

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.00000	−0.500000
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	−1.00000	−0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	0	0
$11$	0	0
$12$	1.00000	0.288675
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	−1.00000	−0.267261
$15$	2.00000	0.516398
$16$	−1.00000	−0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	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	$0.651732\pi$
$20$	2.00000	0.447214
$21$	1.00000	0.218218
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	3.00000	0.612372
$25$	−1.00000	−0.200000
$26$	−6.00000	−1.17670
$27$	−1.00000	−0.192450
$28$	1.00000	0.188982
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	2.00000	0.365148
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	5.00000	0.883883
$33$	0	0
$34$	−2.00000	−0.342997
$35$	2.00000	0.338062
$36$	−1.00000	−0.166667
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	−4.00000	−0.648886
$39$	6.00000	0.960769
$40$	6.00000	0.948683
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	1.00000	0.154303
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	2.00000	0.280056
$52$	6.00000	0.832050
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	3.00000	0.400892
$57$	4.00000	0.529813
$58$	2.00000	0.262613
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	−2.00000	−0.258199
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	8.00000	1.01600
$63$	−1.00000	−0.125988
$64$	7.00000	0.875000
$65$	12.0000	1.48842
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	2.00000	0.242536
$69$	0	0
$70$	2.00000	0.239046
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−3.00000	−0.353553
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	6.00000	0.697486
$75$	1.00000	0.115470
$76$	4.00000	0.458831
$77$	0	0
$78$	6.00000	0.679366
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	2.00000	0.223607
$81$	1.00000	0.111111
$82$	−10.0000	−1.10432
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	−1.00000	−0.109109
$85$	4.00000	0.433861
$86$	4.00000	0.431331
$87$	−2.00000	−0.214423
$88$	0	0
$89$	18.0000	1.90800	0.953998	−	0.299813i	$-0.0969242\pi$
$89$	18.0000	1.90800	0.953998	+	0.299813i	$0.0969242\pi$
$90$	−2.00000	−0.210819
$91$	6.00000	0.628971
$92$	0	0
$93$	−8.00000	−0.829561
$94$	−8.00000	−0.825137
$95$	8.00000	0.820783
$96$	−5.00000	−0.510310
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	2.00000	0.198030
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	18.0000	1.76505
$105$	−2.00000	−0.195180
$106$	6.00000	0.582772
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	1.00000	0.0962250
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	1.00000	0.0944911
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	−2.00000	−0.185695
$117$	−6.00000	−0.554700
$118$	4.00000	0.368230
$119$	2.00000	0.183340
$120$	−6.00000	−0.547723
$121$	0	0
$122$	10.0000	0.905357
$123$	10.0000	0.901670
$124$	−8.00000	−0.718421
$125$	12.0000	1.07331
$126$	−1.00000	−0.0890871
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	−3.00000	−0.265165
$129$	−4.00000	−0.352180
$130$	12.0000	1.05247
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	−12.0000	−1.03664
$135$	2.00000	0.172133
$136$	6.00000	0.514496
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	−2.00000	−0.169031
$141$	8.00000	0.673722
$142$	0	0
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	−4.00000	−0.332182
$146$	−2.00000	−0.165521
$147$	−1.00000	−0.0824786
$148$	−6.00000	−0.493197
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	1.00000	0.0816497
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	12.0000	0.973329
$153$	−2.00000	−0.161690
$154$	0	0
$155$	−16.0000	−1.28515
$156$	−6.00000	−0.480384
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	−16.0000	−1.27289
$159$	−6.00000	−0.475831
$160$	−10.0000	−0.790569
$161$	0	0
$162$	1.00000	0.0785674
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	−4.00000	−0.310460
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	−3.00000	−0.231455
$169$	23.0000	1.76923
$170$	4.00000	0.306786
$171$	−4.00000	−0.305888
$172$	−4.00000	−0.304997
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	−2.00000	−0.151620
$175$	1.00000	0.0755929
$176$	0	0
$177$	−4.00000	−0.300658
$178$	18.0000	1.34916
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	2.00000	0.149071
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	6.00000	0.444750
$183$	−10.0000	−0.739221
$184$	0	0
$185$	−12.0000	−0.882258
$186$	−8.00000	−0.586588
$187$	0	0
$188$	8.00000	0.583460
$189$	1.00000	0.0727393
$190$	8.00000	0.580381
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	−7.00000	−0.505181
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	2.00000	0.143592
$195$	−12.0000	−0.859338
$196$	−1.00000	−0.0714286
$197$	26.0000	1.85242	0.926212	−	0.377004i	$-0.123046\pi$
$197$	26.0000	1.85242	0.926212	+	0.377004i	$0.123046\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	3.00000	0.212132
$201$	12.0000	0.846415
$202$	−6.00000	−0.422159
$203$	−2.00000	−0.140372
$204$	−2.00000	−0.140028
$205$	20.0000	1.39686
$206$	0	0
$207$	0	0
$208$	6.00000	0.416025
$209$	0	0
$210$	−2.00000	−0.138013
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	12.0000	0.820303
$215$	−8.00000	−0.545595
$216$	3.00000	0.204124
$217$	−8.00000	−0.543075
$218$	2.00000	0.135457
$219$	2.00000	0.135147
$220$	0	0
$221$	12.0000	0.807207
$222$	−6.00000	−0.402694
$223$	−24.0000	−1.60716	−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000	−1.60716	−0.803579	+	0.595198i	$0.797074\pi$
$224$	−5.00000	−0.334077
$225$	−1.00000	−0.0666667
$226$	18.0000	1.19734
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	−4.00000	−0.264906
$229$	−26.0000	−1.71813	−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000	−1.71813	−0.859064	+	0.511868i	$0.828954\pi$
$230$	0	0
$231$	0	0
$232$	−6.00000	−0.393919
$233$	−26.0000	−1.70332	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−26.0000	−1.70332	−0.851658	+	0.524097i	$0.824403\pi$
$234$	−6.00000	−0.392232
$235$	16.0000	1.04372
$236$	−4.00000	−0.260378
$237$	16.0000	1.03931
$238$	2.00000	0.129641
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	−2.00000	−0.129099
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	−10.0000	−0.640184
$245$	−2.00000	−0.127775
$246$	10.0000	0.637577
$247$	24.0000	1.52708
$248$	−24.0000	−1.52400
$249$	4.00000	0.253490
$250$	12.0000	0.758947
$251$	−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
$253$	0	0
$254$	16.0000	1.00393
$255$	−4.00000	−0.250490
$256$	−17.0000	−1.06250
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	−4.00000	−0.249029
$259$	−6.00000	−0.372822
$260$	−12.0000	−0.744208
$261$	2.00000	0.123797
$262$	−4.00000	−0.247121
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	4.00000	0.245256
$267$	−18.0000	−1.10158
$268$	12.0000	0.733017
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	2.00000	0.121716
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	2.00000	0.121268
$273$	−6.00000	−0.363137
$274$	−6.00000	−0.362473
$275$	0	0
$276$	0	0
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	20.0000	1.19952
$279$	8.00000	0.478947
$280$	−6.00000	−0.358569
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	8.00000	0.476393
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	−8.00000	−0.473879
$286$	0	0
$287$	10.0000	0.590281
$288$	5.00000	0.294628
$289$	−13.0000	−0.764706
$290$	−4.00000	−0.234888
$291$	−2.00000	−0.117242
$292$	2.00000	0.117041
$293$	10.0000	0.584206	0.292103	−	0.956387i	$-0.405645\pi$
$293$	10.0000	0.584206	0.292103	+	0.956387i	$0.405645\pi$
$294$	−1.00000	−0.0583212
$295$	−8.00000	−0.465778
$296$	−18.0000	−1.04623
$297$	0	0
$298$	10.0000	0.579284
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	−4.00000	−0.230556
$302$	−8.00000	−0.460348
$303$	6.00000	0.344691
$304$	4.00000	0.229416
$305$	−20.0000	−1.14520
$306$	−2.00000	−0.114332
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	0	0
$310$	−16.0000	−0.908739
$311$	32.0000	1.81455	0.907277	−	0.420534i	$-0.138157\pi$
$311$	32.0000	1.81455	0.907277	+	0.420534i	$0.138157\pi$
$312$	−18.0000	−1.01905
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	−2.00000	−0.112867
$315$	2.00000	0.112687
$316$	16.0000	0.900070
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	−6.00000	−0.336463
$319$	0	0
$320$	−14.0000	−0.782624
$321$	−12.0000	−0.669775
$322$	0	0
$323$	8.00000	0.445132
$324$	−1.00000	−0.0555556
$325$	6.00000	0.332820
$326$	−12.0000	−0.664619
$327$	−2.00000	−0.110600
$328$	30.0000	1.65647
$329$	8.00000	0.441054
$330$	0	0
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	4.00000	0.219529
$333$	6.00000	0.328798
$334$	8.00000	0.437741
$335$	24.0000	1.31126
$336$	−1.00000	−0.0545545
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	23.0000	1.25104
$339$	−18.0000	−0.977626
$340$	−4.00000	−0.216930
$341$	0	0
$342$	−4.00000	−0.216295
$343$	−1.00000	−0.0539949
$344$	−12.0000	−0.646997
$345$	0	0
$346$	−14.0000	−0.752645
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	2.00000	0.107211
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	1.00000	0.0534522
$351$	6.00000	0.320256
$352$	0	0
$353$	−22.0000	−1.17094	−0.585471	−	0.810693i	$-0.699090\pi$
$353$	−22.0000	−1.17094	−0.585471	+	0.810693i	$0.699090\pi$
$354$	−4.00000	−0.212598
$355$	0	0
$356$	−18.0000	−0.953998
$357$	−2.00000	−0.105851
$358$	12.0000	0.634220
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	6.00000	0.316228
$361$	−3.00000	−0.157895
$362$	−10.0000	−0.525588
$363$	0	0
$364$	−6.00000	−0.314485
$365$	4.00000	0.209370
$366$	−10.0000	−0.522708
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	0	0
$369$	−10.0000	−0.520579
$370$	−12.0000	−0.623850
$371$	−6.00000	−0.311504
$372$	8.00000	0.414781
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	−12.0000	−0.619677
$376$	24.0000	1.23771
$377$	−12.0000	−0.618031
$378$	1.00000	0.0514344
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	−8.00000	−0.410391
$381$	−16.0000	−0.819705
$382$	−24.0000	−1.22795
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	3.00000	0.153093
$385$	0	0
$386$	−2.00000	−0.101797
$387$	4.00000	0.203331
$388$	−2.00000	−0.101535
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	−12.0000	−0.607644
$391$	0	0
$392$	−3.00000	−0.151523
$393$	4.00000	0.201773
$394$	26.0000	1.30986
$395$	32.0000	1.61009
$396$	0	0
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	16.0000	0.802008
$399$	−4.00000	−0.200250
$400$	1.00000	0.0500000
$401$	34.0000	1.69788	0.848939	−	0.528490i	$-0.177242\pi$
$401$	34.0000	1.69788	0.848939	+	0.528490i	$0.177242\pi$
$402$	12.0000	0.598506
$403$	−48.0000	−2.39105
$404$	6.00000	0.298511
$405$	−2.00000	−0.0993808
$406$	−2.00000	−0.0992583
$407$	0	0
$408$	−6.00000	−0.297044
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	20.0000	0.987730
$411$	6.00000	0.295958
$412$	0	0
$413$	−4.00000	−0.196827
$414$	0	0
$415$	8.00000	0.392705
$416$	−30.0000	−1.47087
$417$	−20.0000	−0.979404
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	2.00000	0.0975900
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	12.0000	0.584151
$423$	−8.00000	−0.388973
$424$	−18.0000	−0.874157
$425$	2.00000	0.0970143
$426$	0	0
$427$	−10.0000	−0.483934
$428$	−12.0000	−0.580042
$429$	0	0
$430$	−8.00000	−0.385794
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	1.00000	0.0481125
$433$	−30.0000	−1.44171	−0.720854	−	0.693087i	$-0.756250\pi$
$433$	−30.0000	−1.44171	−0.720854	+	0.693087i	$0.756250\pi$
$434$	−8.00000	−0.384012
$435$	4.00000	0.191785
$436$	−2.00000	−0.0957826
$437$	0	0
$438$	2.00000	0.0955637
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	12.0000	0.570782
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	6.00000	0.284747
$445$	−36.0000	−1.70656
$446$	−24.0000	−1.13643
$447$	−10.0000	−0.472984
$448$	−7.00000	−0.330719
$449$	34.0000	1.60456	0.802280	−	0.596948i	$-0.203620\pi$
$449$	34.0000	1.60456	0.802280	+	0.596948i	$0.203620\pi$
$450$	−1.00000	−0.0471405
$451$	0	0
$452$	−18.0000	−0.846649
$453$	8.00000	0.375873
$454$	12.0000	0.563188
$455$	−12.0000	−0.562569
$456$	−12.0000	−0.561951
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	−26.0000	−1.21490
$459$	2.00000	0.0933520
$460$	0	0
$461$	34.0000	1.58354	0.791769	−	0.610821i	$-0.209160\pi$
$461$	34.0000	1.58354	0.791769	+	0.610821i	$0.209160\pi$
$462$	0	0
$463$	−32.0000	−1.48717	−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000	−1.48717	−0.743583	+	0.668644i	$0.766875\pi$
$464$	−2.00000	−0.0928477
$465$	16.0000	0.741982
$466$	−26.0000	−1.20443
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	6.00000	0.277350
$469$	12.0000	0.554109
$470$	16.0000	0.738025
$471$	2.00000	0.0921551
$472$	−12.0000	−0.552345
$473$	0	0
$474$	16.0000	0.734904
$475$	4.00000	0.183533
$476$	−2.00000	−0.0916698
$477$	6.00000	0.274721
$478$	24.0000	1.09773
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	10.0000	0.456435
$481$	−36.0000	−1.64146
$482$	−26.0000	−1.18427
$483$	0	0
$484$	0	0
$485$	−4.00000	−0.181631
$486$	−1.00000	−0.0453609
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	−30.0000	−1.35804
$489$	12.0000	0.542659
$490$	−2.00000	−0.0903508
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	−10.0000	−0.450835
$493$	−4.00000	−0.180151
$494$	24.0000	1.07981
$495$	0	0
$496$	−8.00000	−0.359211
$497$	0	0
$498$	4.00000	0.179244
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	−12.0000	−0.536656
$501$	−8.00000	−0.357414
$502$	−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	$0.320291\pi$
$504$	3.00000	0.133631
$505$	12.0000	0.533993
$506$	0	0
$507$	−23.0000	−1.02147
$508$	−16.0000	−0.709885
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	−4.00000	−0.177123
$511$	2.00000	0.0884748
$512$	−11.0000	−0.486136
$513$	4.00000	0.176604
$514$	−6.00000	−0.264649
$515$	0	0
$516$	4.00000	0.176090
$517$	0	0
$518$	−6.00000	−0.263625
$519$	14.0000	0.614532
$520$	−36.0000	−1.57870
$521$	2.00000	0.0876216	0.0438108	−	0.999040i	$-0.486050\pi$
$521$	2.00000	0.0876216	0.0438108	+	0.999040i	$0.486050\pi$
$522$	2.00000	0.0875376
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	4.00000	0.174741
$525$	−1.00000	−0.0436436
$526$	0	0
$527$	−16.0000	−0.696971
$528$	0	0
$529$	−23.0000	−1.00000
$530$	−12.0000	−0.521247
$531$	4.00000	0.173585
$532$	−4.00000	−0.173422
$533$	60.0000	2.59889
$534$	−18.0000	−0.778936
$535$	−24.0000	−1.03761
$536$	36.0000	1.55496
$537$	−12.0000	−0.517838
$538$	−10.0000	−0.431131
$539$	0	0
$540$	−2.00000	−0.0860663
$541$	34.0000	1.46177	0.730887	−	0.682498i	$-0.239107\pi$
$541$	34.0000	1.46177	0.730887	+	0.682498i	$0.239107\pi$
$542$	−16.0000	−0.687259
$543$	10.0000	0.429141
$544$	−10.0000	−0.428746
$545$	−4.00000	−0.171341
$546$	−6.00000	−0.256776
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	6.00000	0.256307
$549$	10.0000	0.426790
$550$	0	0
$551$	−8.00000	−0.340811
$552$	0	0
$553$	16.0000	0.680389
$554$	−22.0000	−0.934690
$555$	12.0000	0.509372
$556$	−20.0000	−0.848189
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	8.00000	0.338667
$559$	−24.0000	−1.01509
$560$	−2.00000	−0.0845154
$561$	0	0
$562$	−10.0000	−0.421825
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	−8.00000	−0.336861
$565$	−36.0000	−1.51453
$566$	4.00000	0.168133
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	22.0000	0.922288	0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000	0.922288	0.461144	+	0.887325i	$0.347439\pi$
$570$	−8.00000	−0.335083
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	24.0000	1.00261
$574$	10.0000	0.417392
$575$	0	0
$576$	7.00000	0.291667
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	−13.0000	−0.540729
$579$	2.00000	0.0831172
$580$	4.00000	0.166091
$581$	4.00000	0.165948
$582$	−2.00000	−0.0829027
$583$	0	0
$584$	6.00000	0.248282
$585$	12.0000	0.496139
$586$	10.0000	0.413096
$587$	36.0000	1.48588	0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000	1.48588	0.742940	+	0.669359i	$0.233431\pi$
$588$	1.00000	0.0412393
$589$	−32.0000	−1.31854
$590$	−8.00000	−0.329355
$591$	−26.0000	−1.06950
$592$	−6.00000	−0.246598
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	0	0
$595$	−4.00000	−0.163984
$596$	−10.0000	−0.409616
$597$	−16.0000	−0.654836
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	−3.00000	−0.122474
$601$	14.0000	0.571072	0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000	0.571072	0.285536	+	0.958368i	$0.407828\pi$
$602$	−4.00000	−0.163028
$603$	−12.0000	−0.488678
$604$	8.00000	0.325515
$605$	0	0
$606$	6.00000	0.243733
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	−20.0000	−0.811107
$609$	2.00000	0.0810441
$610$	−20.0000	−0.809776
$611$	48.0000	1.94187
$612$	2.00000	0.0808452
$613$	10.0000	0.403896	0.201948	−	0.979396i	$-0.435273\pi$
$613$	10.0000	0.403896	0.201948	+	0.979396i	$0.435273\pi$
$614$	−20.0000	−0.807134
$615$	−20.0000	−0.806478
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	16.0000	0.642575
$621$	0	0
$622$	32.0000	1.28308
$623$	−18.0000	−0.721155
$624$	−6.00000	−0.240192
$625$	−19.0000	−0.760000
$626$	−6.00000	−0.239808
$627$	0	0
$628$	2.00000	0.0798087
$629$	−12.0000	−0.478471
$630$	2.00000	0.0796819
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	48.0000	1.90934
$633$	−12.0000	−0.476957
$634$	−18.0000	−0.714871
$635$	−32.0000	−1.26988
$636$	6.00000	0.237915
$637$	−6.00000	−0.237729
$638$	0	0
$639$	0	0
$640$	6.00000	0.237171
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	−12.0000	−0.473602
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	8.00000	0.314756
$647$	32.0000	1.25805	0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000	1.25805	0.629025	+	0.777385i	$0.283454\pi$
$648$	−3.00000	−0.117851
$649$	0	0
$650$	6.00000	0.235339
$651$	8.00000	0.313545
$652$	12.0000	0.469956
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	−2.00000	−0.0782062
$655$	8.00000	0.312586
$656$	10.0000	0.390434
$657$	−2.00000	−0.0780274
$658$	8.00000	0.311872
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	28.0000	1.08825
$663$	−12.0000	−0.466041
$664$	12.0000	0.465690
$665$	−8.00000	−0.310227
$666$	6.00000	0.232495
$667$	0	0
$668$	−8.00000	−0.309529
$669$	24.0000	0.927894
$670$	24.0000	0.927201
$671$	0	0
$672$	5.00000	0.192879
$673$	−18.0000	−0.693849	−0.346925	−	0.937893i	$-0.612774\pi$
$673$	−18.0000	−0.693849	−0.346925	+	0.937893i	$0.612774\pi$
$674$	−2.00000	−0.0770371
$675$	1.00000	0.0384900
$676$	−23.0000	−0.884615
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	−18.0000	−0.691286
$679$	−2.00000	−0.0767530
$680$	−12.0000	−0.460179
$681$	−12.0000	−0.459841
$682$	0	0
$683$	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
$685$	12.0000	0.458496
$686$	−1.00000	−0.0381802
$687$	26.0000	0.991962
$688$	−4.00000	−0.152499
$689$	−36.0000	−1.37149
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	14.0000	0.532200
$693$	0	0
$694$	−4.00000	−0.151838
$695$	−40.0000	−1.51729
$696$	6.00000	0.227429
$697$	20.0000	0.757554
$698$	10.0000	0.378506
$699$	26.0000	0.983410
$700$	−1.00000	−0.0377964
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	6.00000	0.226455
$703$	−24.0000	−0.905177
$704$	0	0
$705$	−16.0000	−0.602595
$706$	−22.0000	−0.827981
$707$	6.00000	0.225653
$708$	4.00000	0.150329
$709$	38.0000	1.42712	0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000	1.42712	0.713560	+	0.700594i	$0.247082\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	−54.0000	−2.02374
$713$	0	0
$714$	−2.00000	−0.0748481
$715$	0	0
$716$	−12.0000	−0.448461
$717$	−24.0000	−0.896296
$718$	0	0
$719$	−40.0000	−1.49175	−0.745874	−	0.666087i	$-0.767968\pi$
$719$	−40.0000	−1.49175	−0.745874	+	0.666087i	$0.767968\pi$
$720$	2.00000	0.0745356
$721$	0	0
$722$	−3.00000	−0.111648
$723$	26.0000	0.966950
$724$	10.0000	0.371647
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	−18.0000	−0.667124
$729$	1.00000	0.0370370
$730$	4.00000	0.148047
$731$	−8.00000	−0.295891
$732$	10.0000	0.369611
$733$	−6.00000	−0.221615	−0.110808	−	0.993842i	$-0.535344\pi$
$733$	−6.00000	−0.221615	−0.110808	+	0.993842i	$0.535344\pi$
$734$	−8.00000	−0.295285
$735$	2.00000	0.0737711
$736$	0	0
$737$	0	0
$738$	−10.0000	−0.368105
$739$	44.0000	1.61857	0.809283	−	0.587419i	$-0.199856\pi$
$739$	44.0000	1.61857	0.809283	+	0.587419i	$0.199856\pi$
$740$	12.0000	0.441129
$741$	−24.0000	−0.881662
$742$	−6.00000	−0.220267
$743$	48.0000	1.76095	0.880475	−	0.474093i	$-0.157224\pi$
$743$	48.0000	1.76095	0.880475	+	0.474093i	$0.157224\pi$
$744$	24.0000	0.879883
$745$	−20.0000	−0.732743
$746$	−22.0000	−0.805477
$747$	−4.00000	−0.146352
$748$	0	0
$749$	−12.0000	−0.438470
$750$	−12.0000	−0.438178
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	8.00000	0.291730
$753$	12.0000	0.437304
$754$	−12.0000	−0.437014
$755$	16.0000	0.582300
$756$	−1.00000	−0.0363696
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	28.0000	1.01701
$759$	0	0
$760$	−24.0000	−0.870572
$761$	−10.0000	−0.362500	−0.181250	−	0.983437i	$-0.558014\pi$
$761$	−10.0000	−0.362500	−0.181250	+	0.983437i	$0.558014\pi$
$762$	−16.0000	−0.579619
$763$	−2.00000	−0.0724049
$764$	24.0000	0.868290
$765$	4.00000	0.144620
$766$	8.00000	0.289052
$767$	−24.0000	−0.866590
$768$	17.0000	0.613435
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	2.00000	0.0719816
$773$	30.0000	1.07903	0.539513	−	0.841978i	$-0.318609\pi$
$773$	30.0000	1.07903	0.539513	+	0.841978i	$0.318609\pi$
$774$	4.00000	0.143777
$775$	−8.00000	−0.287368
$776$	−6.00000	−0.215387
$777$	6.00000	0.215249
$778$	6.00000	0.215110
$779$	40.0000	1.43315
$780$	12.0000	0.429669
$781$	0	0
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	−1.00000	−0.0357143
$785$	4.00000	0.142766
$786$	4.00000	0.142675
$787$	12.0000	0.427754	0.213877	−	0.976861i	$-0.431391\pi$
$787$	12.0000	0.427754	0.213877	+	0.976861i	$0.431391\pi$
$788$	−26.0000	−0.926212
$789$	0	0
$790$	32.0000	1.13851
$791$	−18.0000	−0.640006
$792$	0	0
$793$	−60.0000	−2.13066
$794$	14.0000	0.496841
$795$	12.0000	0.425596
$796$	−16.0000	−0.567105
$797$	6.00000	0.212531	0.106265	−	0.994338i	$-0.466111\pi$
$797$	6.00000	0.212531	0.106265	+	0.994338i	$0.466111\pi$
$798$	−4.00000	−0.141598
$799$	16.0000	0.566039
$800$	−5.00000	−0.176777
$801$	18.0000	0.635999
$802$	34.0000	1.20058
$803$	0	0
$804$	−12.0000	−0.423207
$805$	0	0
$806$	−48.0000	−1.69073
$807$	10.0000	0.352017
$808$	18.0000	0.633238
$809$	38.0000	1.33601	0.668004	−	0.744157i	$-0.267149\pi$
$809$	38.0000	1.33601	0.668004	+	0.744157i	$0.267149\pi$
$810$	−2.00000	−0.0702728
$811$	4.00000	0.140459	0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000	0.140459	0.0702295	+	0.997531i	$0.477627\pi$
$812$	2.00000	0.0701862
$813$	16.0000	0.561144
$814$	0	0
$815$	24.0000	0.840683
$816$	−2.00000	−0.0700140
$817$	−16.0000	−0.559769
$818$	14.0000	0.489499
$819$	6.00000	0.209657
$820$	−20.0000	−0.698430
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	6.00000	0.209274
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	0	0
$825$	0	0
$826$	−4.00000	−0.139178
$827$	44.0000	1.53003	0.765015	−	0.644013i	$-0.222732\pi$
$827$	44.0000	1.53003	0.765015	+	0.644013i	$0.222732\pi$
$828$	0	0
$829$	−50.0000	−1.73657	−0.868286	−	0.496064i	$-0.834778\pi$
$829$	−50.0000	−1.73657	−0.868286	+	0.496064i	$0.834778\pi$
$830$	8.00000	0.277684
$831$	22.0000	0.763172
$832$	−42.0000	−1.45609
$833$	−2.00000	−0.0692959
$834$	−20.0000	−0.692543
$835$	−16.0000	−0.553703
$836$	0	0
$837$	−8.00000	−0.276520
$838$	12.0000	0.414533
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	6.00000	0.207020
$841$	−25.0000	−0.862069
$842$	6.00000	0.206774
$843$	10.0000	0.344418
$844$	−12.0000	−0.413057
$845$	−46.0000	−1.58245
$846$	−8.00000	−0.275046
$847$	0	0
$848$	−6.00000	−0.206041
$849$	−4.00000	−0.137280
$850$	2.00000	0.0685994
$851$	0	0
$852$	0	0
$853$	34.0000	1.16414	0.582069	−	0.813139i	$-0.302243\pi$
$853$	34.0000	1.16414	0.582069	+	0.813139i	$0.302243\pi$
$854$	−10.0000	−0.342193
$855$	8.00000	0.273594
$856$	−36.0000	−1.23045
$857$	−10.0000	−0.341593	−0.170797	−	0.985306i	$-0.554634\pi$
$857$	−10.0000	−0.341593	−0.170797	+	0.985306i	$0.554634\pi$
$858$	0	0
$859$	52.0000	1.77422	0.887109	−	0.461561i	$-0.152710\pi$
$859$	52.0000	1.77422	0.887109	+	0.461561i	$0.152710\pi$
$860$	8.00000	0.272798
$861$	−10.0000	−0.340799
$862$	24.0000	0.817443
$863$	−40.0000	−1.36162	−0.680808	−	0.732462i	$-0.738371\pi$
$863$	−40.0000	−1.36162	−0.680808	+	0.732462i	$0.738371\pi$
$864$	−5.00000	−0.170103
$865$	28.0000	0.952029
$866$	−30.0000	−1.01944
$867$	13.0000	0.441503
$868$	8.00000	0.271538
$869$	0	0
$870$	4.00000	0.135613
$871$	72.0000	2.43963
$872$	−6.00000	−0.203186
$873$	2.00000	0.0676897
$874$	0	0
$875$	−12.0000	−0.405674
$876$	−2.00000	−0.0675737
$877$	34.0000	1.14810	0.574049	−	0.818821i	$-0.305372\pi$
$877$	34.0000	1.14810	0.574049	+	0.818821i	$0.305372\pi$
$878$	−8.00000	−0.269987
$879$	−10.0000	−0.337292
$880$	0	0
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	1.00000	0.0336718
$883$	36.0000	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$884$	−12.0000	−0.403604
$885$	8.00000	0.268917
$886$	−12.0000	−0.403148
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	18.0000	0.604040
$889$	−16.0000	−0.536623
$890$	−36.0000	−1.20672
$891$	0	0
$892$	24.0000	0.803579
$893$	32.0000	1.07084
$894$	−10.0000	−0.334450
$895$	−24.0000	−0.802232
$896$	3.00000	0.100223
$897$	0	0
$898$	34.0000	1.13459
$899$	16.0000	0.533630
$900$	1.00000	0.0333333
$901$	−12.0000	−0.399778
$902$	0	0
$903$	4.00000	0.133112
$904$	−54.0000	−1.79601
$905$	20.0000	0.664822
$906$	8.00000	0.265782
$907$	−20.0000	−0.664089	−0.332045	−	0.943264i	$-0.607738\pi$
$907$	−20.0000	−0.664089	−0.332045	+	0.943264i	$0.607738\pi$
$908$	−12.0000	−0.398234
$909$	−6.00000	−0.199007
$910$	−12.0000	−0.397796
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	−4.00000	−0.132453
$913$	0	0
$914$	22.0000	0.727695
$915$	20.0000	0.661180
$916$	26.0000	0.859064
$917$	4.00000	0.132092
$918$	2.00000	0.0660098
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	34.0000	1.11973
$923$	0	0
$924$	0	0
$925$	−6.00000	−0.197279
$926$	−32.0000	−1.05159
$927$	0	0
$928$	10.0000	0.328266
$929$	−54.0000	−1.77168	−0.885841	−	0.463988i	$-0.846418\pi$
$929$	−54.0000	−1.77168	−0.885841	+	0.463988i	$0.846418\pi$
$930$	16.0000	0.524661
$931$	−4.00000	−0.131095
$932$	26.0000	0.851658
$933$	−32.0000	−1.04763
$934$	12.0000	0.392652
$935$	0	0
$936$	18.0000	0.588348
$937$	−50.0000	−1.63343	−0.816714	−	0.577042i	$-0.804207\pi$
$937$	−50.0000	−1.63343	−0.816714	+	0.577042i	$0.804207\pi$
$938$	12.0000	0.391814
$939$	6.00000	0.195803
$940$	−16.0000	−0.521862
$941$	−14.0000	−0.456387	−0.228193	−	0.973616i	$-0.573282\pi$
$941$	−14.0000	−0.456387	−0.228193	+	0.973616i	$0.573282\pi$
$942$	2.00000	0.0651635
$943$	0	0
$944$	−4.00000	−0.130189
$945$	−2.00000	−0.0650600
$946$	0	0
$947$	−36.0000	−1.16984	−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000	−1.16984	−0.584921	+	0.811090i	$0.698875\pi$
$948$	−16.0000	−0.519656
$949$	12.0000	0.389536
$950$	4.00000	0.129777
$951$	18.0000	0.583690
$952$	−6.00000	−0.194461
$953$	22.0000	0.712650	0.356325	−	0.934362i	$-0.384030\pi$
$953$	22.0000	0.712650	0.356325	+	0.934362i	$0.384030\pi$
$954$	6.00000	0.194257
$955$	48.0000	1.55324
$956$	−24.0000	−0.776215
$957$	0	0
$958$	0	0
$959$	6.00000	0.193750
$960$	14.0000	0.451848
$961$	33.0000	1.06452
$962$	−36.0000	−1.16069
$963$	12.0000	0.386695
$964$	26.0000	0.837404
$965$	4.00000	0.128765
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	0	0
$969$	−8.00000	−0.256997
$970$	−4.00000	−0.128432
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	1.00000	0.0320750
$973$	−20.0000	−0.641171
$974$	8.00000	0.256337
$975$	−6.00000	−0.192154
$976$	−10.0000	−0.320092
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	12.0000	0.383718
$979$	0	0
$980$	2.00000	0.0638877
$981$	2.00000	0.0638551
$982$	−20.0000	−0.638226
$983$	−16.0000	−0.510321	−0.255160	−	0.966899i	$-0.582128\pi$
$983$	−16.0000	−0.510321	−0.255160	+	0.966899i	$0.582128\pi$
$984$	−30.0000	−0.956365
$985$	−52.0000	−1.65686
$986$	−4.00000	−0.127386
$987$	−8.00000	−0.254643
$988$	−24.0000	−0.763542
$989$	0	0
$990$	0	0
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	40.0000	1.27000
$993$	−28.0000	−0.888553
$994$	0	0
$995$	−32.0000	−1.01447
$996$	−4.00000	−0.126745
$997$	−46.0000	−1.45683	−0.728417	−	0.685134i	$-0.759744\pi$
$997$	−46.0000	−1.45683	−0.728417	+	0.685134i	$0.759744\pi$
$998$	20.0000	0.633089
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2541.2.a.h.1.1		1
3.2	odd	2		7623.2.a.f.1.1		1
11.10	odd	2		231.2.a.a.1.1	✓	1
33.32	even	2		693.2.a.d.1.1		1
44.43	even	2		3696.2.a.t.1.1		1
55.54	odd	2		5775.2.a.t.1.1		1
77.76	even	2		1617.2.a.e.1.1		1
231.230	odd	2		4851.2.a.p.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.a.a.1.1	✓	1	11.10	odd	2
693.2.a.d.1.1		1	33.32	even	2
1617.2.a.e.1.1		1	77.76	even	2
2541.2.a.h.1.1		1	1.1	even	1	trivial
3696.2.a.t.1.1		1	44.43	even	2
4851.2.a.p.1.1		1	231.230	odd	2
5775.2.a.t.1.1		1	55.54	odd	2
7623.2.a.f.1.1		1	3.2	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:	\( N \)	\(=\)	\( 2541 = 3 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2541.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more