LMFDB - Embedded newform 363.2.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [363,2,Mod(1,363)] 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("363.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(363, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 363 = 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	363.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$2.89856959337$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	363.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +1.00000 q^{12} +2.00000 q^{13} +4.00000 q^{14} +2.00000 q^{15} -1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} +2.00000 q^{20} +4.00000 q^{21} +8.00000 q^{23} -3.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} +4.00000 q^{28} +6.00000 q^{29} -2.00000 q^{30} -8.00000 q^{31} -5.00000 q^{32} -2.00000 q^{34} +8.00000 q^{35} -1.00000 q^{36} +6.00000 q^{37} -2.00000 q^{39} -6.00000 q^{40} +2.00000 q^{41} -4.00000 q^{42} -2.00000 q^{45} -8.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} +9.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} -2.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} -12.0000 q^{56} -6.00000 q^{58} -4.00000 q^{59} -2.00000 q^{60} -6.00000 q^{61} +8.00000 q^{62} -4.00000 q^{63} +7.00000 q^{64} -4.00000 q^{65} -4.00000 q^{67} -2.00000 q^{68} -8.00000 q^{69} -8.00000 q^{70} +3.00000 q^{72} +14.0000 q^{73} -6.00000 q^{74} +1.00000 q^{75} +2.00000 q^{78} +4.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -12.0000 q^{83} -4.00000 q^{84} -4.00000 q^{85} -6.00000 q^{87} -6.00000 q^{89} +2.00000 q^{90} -8.00000 q^{91} -8.00000 q^{92} +8.00000 q^{93} -8.00000 q^{94} +5.00000 q^{96} +2.00000 q^{97} -9.00000 q^{98} +O(q^{100})\)

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	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	−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
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	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
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	4.00000	1.06904
$15$	2.00000	0.516398
$16$	−1.00000	−0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	−1.00000	−0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	2.00000	0.447214
$21$	4.00000	0.872872
$22$	0	0
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	−3.00000	−0.612372
$25$	−1.00000	−0.200000
$26$	−2.00000	−0.392232
$27$	−1.00000	−0.192450
$28$	4.00000	0.755929
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	−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	$0.755135\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	−2.00000	−0.342997
$35$	8.00000	1.35225
$36$	−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	$0.335828\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	−6.00000	−0.948683
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	−4.00000	−0.617213
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	−8.00000	−1.17954
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	1.00000	0.144338
$49$	9.00000	1.28571
$50$	1.00000	0.141421
$51$	−2.00000	−0.280056
$52$	−2.00000	−0.277350
$53$	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
$56$	−12.0000	−1.60357
$57$	0	0
$58$	−6.00000	−0.787839
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	−2.00000	−0.258199
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	8.00000	1.01600
$63$	−4.00000	−0.503953
$64$	7.00000	0.875000
$65$	−4.00000	−0.496139
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−2.00000	−0.242536
$69$	−8.00000	−0.963087
$70$	−8.00000	−0.956183
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	3.00000	0.353553
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	−6.00000	−0.697486
$75$	1.00000	0.115470
$76$	0	0
$77$	0	0
$78$	2.00000	0.226455
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	2.00000	0.223607
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	−4.00000	−0.436436
$85$	−4.00000	−0.433861
$86$	0	0
$87$	−6.00000	−0.643268
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	2.00000	0.210819
$91$	−8.00000	−0.838628
$92$	−8.00000	−0.834058
$93$	8.00000	0.829561
$94$	−8.00000	−0.825137
$95$	0	0
$96$	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	$0.467625\pi$
$98$	−9.00000	−0.909137
$99$	0	0
$100$	1.00000	0.100000
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	2.00000	0.198030
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	6.00000	0.588348
$105$	−8.00000	−0.780720
$106$	−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
$112$	4.00000	0.377964
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	−16.0000	−1.49201
$116$	−6.00000	−0.557086
$117$	2.00000	0.184900
$118$	4.00000	0.368230
$119$	−8.00000	−0.733359
$120$	6.00000	0.547723
$121$	0	0
$122$	6.00000	0.543214
$123$	−2.00000	−0.180334
$124$	8.00000	0.718421
$125$	12.0000	1.07331
$126$	4.00000	0.356348
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	3.00000	0.265165
$129$	0	0
$130$	4.00000	0.350823
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	0	0
$134$	4.00000	0.345547
$135$	2.00000	0.172133
$136$	6.00000	0.514496
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	8.00000	0.681005
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	−8.00000	−0.676123
$141$	−8.00000	−0.673722
$142$	0	0
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	−12.0000	−0.996546
$146$	−14.0000	−1.15865
$147$	−9.00000	−0.742307
$148$	−6.00000	−0.493197
$149$	22.0000	1.80231	0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000	1.80231	0.901155	+	0.433497i	$0.142720\pi$
$150$	−1.00000	−0.0816497
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	16.0000	1.28515
$156$	2.00000	0.160128
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	−4.00000	−0.318223
$159$	−6.00000	−0.475831
$160$	10.0000	0.790569
$161$	−32.0000	−2.52195
$162$	−1.00000	−0.0785674
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	12.0000	0.931381
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	12.0000	0.925820
$169$	−9.00000	−0.692308
$170$	4.00000	0.306786
$171$	0	0
$172$	0	0
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	6.00000	0.454859
$175$	4.00000	0.302372
$176$	0	0
$177$	4.00000	0.300658
$178$	6.00000	0.449719
$179$	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
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	8.00000	0.592999
$183$	6.00000	0.443533
$184$	24.0000	1.76930
$185$	−12.0000	−0.882258
$186$	−8.00000	−0.586588
$187$	0	0
$188$	−8.00000	−0.583460
$189$	4.00000	0.290957
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	−7.00000	−0.505181
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	−2.00000	−0.143592
$195$	4.00000	0.286446
$196$	−9.00000	−0.642857
$197$	14.0000	0.997459	0.498729	−	0.866758i	$-0.333800\pi$
$197$	14.0000	0.997459	0.498729	+	0.866758i	$0.333800\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	−3.00000	−0.212132
$201$	4.00000	0.282138
$202$	2.00000	0.140720
$203$	−24.0000	−1.68447
$204$	2.00000	0.140028
$205$	−4.00000	−0.279372
$206$	−8.00000	−0.557386
$207$	8.00000	0.556038
$208$	−2.00000	−0.138675
$209$	0	0
$210$	8.00000	0.552052
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	−12.0000	−0.820303
$215$	0	0
$216$	−3.00000	−0.204124
$217$	32.0000	2.17230
$218$	−2.00000	−0.135457
$219$	−14.0000	−0.946032
$220$	0	0
$221$	4.00000	0.269069
$222$	6.00000	0.402694
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	20.0000	1.33631
$225$	−1.00000	−0.0666667
$226$	6.00000	0.399114
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	16.0000	1.05501
$231$	0	0
$232$	18.0000	1.18176
$233$	−30.0000	−1.96537	−0.982683	−	0.185296i	$-0.940675\pi$
$233$	−30.0000	−1.96537	−0.982683	+	0.185296i	$0.940675\pi$
$234$	−2.00000	−0.130744
$235$	−16.0000	−1.04372
$236$	4.00000	0.260378
$237$	−4.00000	−0.259828
$238$	8.00000	0.518563
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	−2.00000	−0.129099
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	6.00000	0.384111
$245$	−18.0000	−1.14998
$246$	2.00000	0.127515
$247$	0	0
$248$	−24.0000	−1.52400
$249$	12.0000	0.760469
$250$	−12.0000	−0.758947
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	4.00000	0.251976
$253$	0	0
$254$	−4.00000	−0.250982
$255$	4.00000	0.250490
$256$	−17.0000	−1.06250
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	0	0
$259$	−24.0000	−1.49129
$260$	4.00000	0.248069
$261$	6.00000	0.371391
$262$	−12.0000	−0.741362
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	0	0
$267$	6.00000	0.367194
$268$	4.00000	0.244339
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	−2.00000	−0.121716
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	−2.00000	−0.121268
$273$	8.00000	0.484182
$274$	−2.00000	−0.120824
$275$	0	0
$276$	8.00000	0.481543
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	−8.00000	−0.479808
$279$	−8.00000	−0.478947
$280$	24.0000	1.43427
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	8.00000	0.476393
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.00000	−0.472225
$288$	−5.00000	−0.294628
$289$	−13.0000	−0.764706
$290$	12.0000	0.704664
$291$	−2.00000	−0.117242
$292$	−14.0000	−0.819288
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	9.00000	0.524891
$295$	8.00000	0.465778
$296$	18.0000	1.04623
$297$	0	0
$298$	−22.0000	−1.27443
$299$	16.0000	0.925304
$300$	−1.00000	−0.0577350
$301$	0	0
$302$	20.0000	1.15087
$303$	2.00000	0.114897
$304$	0	0
$305$	12.0000	0.687118
$306$	−2.00000	−0.114332
$307$	−32.0000	−1.82634	−0.913168	−	0.407583i	$-0.866372\pi$
$307$	−32.0000	−1.82634	−0.913168	+	0.407583i	$0.866372\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	−16.0000	−0.908739
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	−6.00000	−0.339683
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	−14.0000	−0.790066
$315$	8.00000	0.450749
$316$	−4.00000	−0.225018
$317$	22.0000	1.23564	0.617822	−	0.786318i	$-0.288015\pi$
$317$	22.0000	1.23564	0.617822	+	0.786318i	$0.288015\pi$
$318$	6.00000	0.336463
$319$	0	0
$320$	−14.0000	−0.782624
$321$	−12.0000	−0.669775
$322$	32.0000	1.78329
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	−2.00000	−0.110940
$326$	−4.00000	−0.221540
$327$	−2.00000	−0.110600
$328$	6.00000	0.331295
$329$	−32.0000	−1.76422
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	12.0000	0.658586
$333$	6.00000	0.328798
$334$	0	0
$335$	8.00000	0.437087
$336$	−4.00000	−0.218218
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	9.00000	0.489535
$339$	6.00000	0.325875
$340$	4.00000	0.216930
$341$	0	0
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	16.0000	0.861411
$346$	−6.00000	−0.322562
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	6.00000	0.321634
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	−4.00000	−0.213809
$351$	−2.00000	−0.106752
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	−4.00000	−0.212598
$355$	0	0
$356$	6.00000	0.317999
$357$	8.00000	0.423405
$358$	−12.0000	−0.634220
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	−6.00000	−0.316228
$361$	−19.0000	−1.00000
$362$	−22.0000	−1.15629
$363$	0	0
$364$	8.00000	0.419314
$365$	−28.0000	−1.46559
$366$	−6.00000	−0.313625
$367$	−32.0000	−1.67039	−0.835193	−	0.549957i	$-0.814644\pi$
$367$	−32.0000	−1.67039	−0.835193	+	0.549957i	$0.814644\pi$
$368$	−8.00000	−0.417029
$369$	2.00000	0.104116
$370$	12.0000	0.623850
$371$	−24.0000	−1.24602
$372$	−8.00000	−0.414781
$373$	2.00000	0.103556	0.0517780	−	0.998659i	$-0.483511\pi$
$373$	2.00000	0.103556	0.0517780	+	0.998659i	$0.483511\pi$
$374$	0	0
$375$	−12.0000	−0.619677
$376$	24.0000	1.23771
$377$	12.0000	0.618031
$378$	−4.00000	−0.205738
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	−8.00000	−0.409316
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	−14.0000	−0.712581
$387$	0	0
$388$	−2.00000	−0.101535
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	−4.00000	−0.202548
$391$	16.0000	0.809155
$392$	27.0000	1.36371
$393$	−12.0000	−0.605320
$394$	−14.0000	−0.705310
$395$	−8.00000	−0.402524
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	0	0
$399$	0	0
$400$	1.00000	0.0500000
$401$	26.0000	1.29838	0.649189	−	0.760627i	$-0.275108\pi$
$401$	26.0000	1.29838	0.649189	+	0.760627i	$0.275108\pi$
$402$	−4.00000	−0.199502
$403$	−16.0000	−0.797017
$404$	2.00000	0.0995037
$405$	−2.00000	−0.0993808
$406$	24.0000	1.19110
$407$	0	0
$408$	−6.00000	−0.297044
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	4.00000	0.197546
$411$	−2.00000	−0.0986527
$412$	−8.00000	−0.394132
$413$	16.0000	0.787309
$414$	−8.00000	−0.393179
$415$	24.0000	1.17811
$416$	−10.0000	−0.490290
$417$	−8.00000	−0.391762
$418$	0	0
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	8.00000	0.390360
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	0	0
$423$	8.00000	0.388973
$424$	18.0000	0.874157
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	24.0000	1.16144
$428$	−12.0000	−0.580042
$429$	0	0
$430$	0	0
$431$	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
$433$	34.0000	1.63394	0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000	1.63394	0.816968	+	0.576683i	$0.195653\pi$
$434$	−32.0000	−1.53605
$435$	12.0000	0.575356
$436$	−2.00000	−0.0957826
$437$	0	0
$438$	14.0000	0.668946
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	−4.00000	−0.190261
$443$	28.0000	1.33032	0.665160	−	0.746701i	$-0.268363\pi$
$443$	28.0000	1.33032	0.665160	+	0.746701i	$0.268363\pi$
$444$	6.00000	0.284747
$445$	12.0000	0.568855
$446$	−16.0000	−0.757622
$447$	−22.0000	−1.04056
$448$	−28.0000	−1.32288
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	1.00000	0.0471405
$451$	0	0
$452$	6.00000	0.282216
$453$	20.0000	0.939682
$454$	12.0000	0.563188
$455$	16.0000	0.750092
$456$	0	0
$457$	−18.0000	−0.842004	−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000	−0.842004	−0.421002	+	0.907060i	$0.638322\pi$
$458$	−6.00000	−0.280362
$459$	−2.00000	−0.0933520
$460$	16.0000	0.746004
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	−6.00000	−0.278543
$465$	−16.0000	−0.741982
$466$	30.0000	1.38972
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	−2.00000	−0.0924500
$469$	16.0000	0.738811
$470$	16.0000	0.738025
$471$	−14.0000	−0.645086
$472$	−12.0000	−0.552345
$473$	0	0
$474$	4.00000	0.183726
$475$	0	0
$476$	8.00000	0.366679
$477$	6.00000	0.274721
$478$	24.0000	1.09773
$479$	−8.00000	−0.365529	−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000	−0.365529	−0.182765	+	0.983157i	$0.558505\pi$
$480$	−10.0000	−0.456435
$481$	12.0000	0.547153
$482$	10.0000	0.455488
$483$	32.0000	1.45605
$484$	0	0
$485$	−4.00000	−0.181631
$486$	1.00000	0.0453609
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	−18.0000	−0.814822
$489$	−4.00000	−0.180886
$490$	18.0000	0.813157
$491$	−4.00000	−0.180517	−0.0902587	−	0.995918i	$-0.528769\pi$
$491$	−4.00000	−0.180517	−0.0902587	+	0.995918i	$0.528769\pi$
$492$	2.00000	0.0901670
$493$	12.0000	0.540453
$494$	0	0
$495$	0	0
$496$	8.00000	0.359211
$497$	0	0
$498$	−12.0000	−0.537733
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	−12.0000	−0.536656
$501$	0	0
$502$	−4.00000	−0.178529
$503$	32.0000	1.42681	0.713405	−	0.700752i	$-0.247152\pi$
$503$	32.0000	1.42681	0.713405	+	0.700752i	$0.247152\pi$
$504$	−12.0000	−0.534522
$505$	4.00000	0.177998
$506$	0	0
$507$	9.00000	0.399704
$508$	−4.00000	−0.177471
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	−4.00000	−0.177123
$511$	−56.0000	−2.47729
$512$	11.0000	0.486136
$513$	0	0
$514$	14.0000	0.617514
$515$	−16.0000	−0.705044
$516$	0	0
$517$	0	0
$518$	24.0000	1.05450
$519$	−6.00000	−0.263371
$520$	−12.0000	−0.526235
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	−6.00000	−0.262613
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	−12.0000	−0.524222
$525$	−4.00000	−0.174574
$526$	−16.0000	−0.697633
$527$	−16.0000	−0.696971
$528$	0	0
$529$	41.0000	1.78261
$530$	12.0000	0.521247
$531$	−4.00000	−0.173585
$532$	0	0
$533$	4.00000	0.173259
$534$	−6.00000	−0.259645
$535$	−24.0000	−1.03761
$536$	−12.0000	−0.518321
$537$	−12.0000	−0.517838
$538$	2.00000	0.0862261
$539$	0	0
$540$	−2.00000	−0.0860663
$541$	−46.0000	−1.97769	−0.988847	−	0.148933i	$-0.952416\pi$
$541$	−46.0000	−1.97769	−0.988847	+	0.148933i	$0.952416\pi$
$542$	20.0000	0.859074
$543$	−22.0000	−0.944110
$544$	−10.0000	−0.428746
$545$	−4.00000	−0.171341
$546$	−8.00000	−0.342368
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	−2.00000	−0.0854358
$549$	−6.00000	−0.256074
$550$	0	0
$551$	0	0
$552$	−24.0000	−1.02151
$553$	−16.0000	−0.680389
$554$	−26.0000	−1.10463
$555$	12.0000	0.509372
$556$	−8.00000	−0.339276
$557$	14.0000	0.593199	0.296600	−	0.955002i	$-0.404147\pi$
$557$	14.0000	0.593199	0.296600	+	0.955002i	$0.404147\pi$
$558$	8.00000	0.338667
$559$	0	0
$560$	−8.00000	−0.338062
$561$	0	0
$562$	−18.0000	−0.759284
$563$	44.0000	1.85438	0.927189	−	0.374593i	$-0.122217\pi$
$563$	44.0000	1.85438	0.927189	+	0.374593i	$0.122217\pi$
$564$	8.00000	0.336861
$565$	12.0000	0.504844
$566$	16.0000	0.672530
$567$	−4.00000	−0.167984
$568$	0	0
$569$	42.0000	1.76073	0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000	1.76073	0.880366	+	0.474295i	$0.157297\pi$
$570$	0	0
$571$	16.0000	0.669579	0.334790	−	0.942293i	$-0.391335\pi$
$571$	16.0000	0.669579	0.334790	+	0.942293i	$0.391335\pi$
$572$	0	0
$573$	−8.00000	−0.334205
$574$	8.00000	0.333914
$575$	−8.00000	−0.333623
$576$	7.00000	0.291667
$577$	−30.0000	−1.24892	−0.624458	−	0.781058i	$-0.714680\pi$
$577$	−30.0000	−1.24892	−0.624458	+	0.781058i	$0.714680\pi$
$578$	13.0000	0.540729
$579$	−14.0000	−0.581820
$580$	12.0000	0.498273
$581$	48.0000	1.99138
$582$	2.00000	0.0829027
$583$	0	0
$584$	42.0000	1.73797
$585$	−4.00000	−0.165380
$586$	−6.00000	−0.247858
$587$	28.0000	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$588$	9.00000	0.371154
$589$	0	0
$590$	−8.00000	−0.329355
$591$	−14.0000	−0.575883
$592$	−6.00000	−0.246598
$593$	−38.0000	−1.56047	−0.780236	−	0.625485i	$-0.784901\pi$
$593$	−38.0000	−1.56047	−0.780236	+	0.625485i	$0.784901\pi$
$594$	0	0
$595$	16.0000	0.655936
$596$	−22.0000	−0.901155
$597$	0	0
$598$	−16.0000	−0.654289
$599$	−8.00000	−0.326871	−0.163436	−	0.986554i	$-0.552258\pi$
$599$	−8.00000	−0.326871	−0.163436	+	0.986554i	$0.552258\pi$
$600$	3.00000	0.122474
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	20.0000	0.813788
$605$	0	0
$606$	−2.00000	−0.0812444
$607$	4.00000	0.162355	0.0811775	−	0.996700i	$-0.474132\pi$
$607$	4.00000	0.162355	0.0811775	+	0.996700i	$0.474132\pi$
$608$	0	0
$609$	24.0000	0.972529
$610$	−12.0000	−0.485866
$611$	16.0000	0.647291
$612$	−2.00000	−0.0808452
$613$	−14.0000	−0.565455	−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000	−0.565455	−0.282727	+	0.959200i	$0.591239\pi$
$614$	32.0000	1.29141
$615$	4.00000	0.161296
$616$	0	0
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	8.00000	0.321807
$619$	44.0000	1.76851	0.884255	−	0.467005i	$-0.154667\pi$
$619$	44.0000	1.76851	0.884255	+	0.467005i	$0.154667\pi$
$620$	−16.0000	−0.642575
$621$	−8.00000	−0.321029
$622$	24.0000	0.962312
$623$	24.0000	0.961540
$624$	2.00000	0.0800641
$625$	−19.0000	−0.760000
$626$	22.0000	0.879297
$627$	0	0
$628$	−14.0000	−0.558661
$629$	12.0000	0.478471
$630$	−8.00000	−0.318728
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	12.0000	0.477334
$633$	0	0
$634$	−22.0000	−0.873732
$635$	−8.00000	−0.317470
$636$	6.00000	0.237915
$637$	18.0000	0.713186
$638$	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
$643$	20.0000	0.788723	0.394362	−	0.918955i	$-0.370966\pi$
$643$	20.0000	0.788723	0.394362	+	0.918955i	$0.370966\pi$
$644$	32.0000	1.26098
$645$	0	0
$646$	0	0
$647$	8.00000	0.314512	0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000	0.314512	0.157256	+	0.987558i	$0.449735\pi$
$648$	3.00000	0.117851
$649$	0	0
$650$	2.00000	0.0784465
$651$	−32.0000	−1.25418
$652$	−4.00000	−0.156652
$653$	−2.00000	−0.0782660	−0.0391330	−	0.999234i	$-0.512460\pi$
$653$	−2.00000	−0.0782660	−0.0391330	+	0.999234i	$0.512460\pi$
$654$	2.00000	0.0782062
$655$	−24.0000	−0.937758
$656$	−2.00000	−0.0780869
$657$	14.0000	0.546192
$658$	32.0000	1.24749
$659$	−4.00000	−0.155818	−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000	−0.155818	−0.0779089	+	0.996960i	$0.524824\pi$
$660$	0	0
$661$	−26.0000	−1.01128	−0.505641	−	0.862744i	$-0.668744\pi$
$661$	−26.0000	−1.01128	−0.505641	+	0.862744i	$0.668744\pi$
$662$	20.0000	0.777322
$663$	−4.00000	−0.155347
$664$	−36.0000	−1.39707
$665$	0	0
$666$	−6.00000	−0.232495
$667$	48.0000	1.85857
$668$	0	0
$669$	−16.0000	−0.618596
$670$	−8.00000	−0.309067
$671$	0	0
$672$	−20.0000	−0.771517
$673$	46.0000	1.77317	0.886585	−	0.462566i	$-0.153071\pi$
$673$	46.0000	1.77317	0.886585	+	0.462566i	$0.153071\pi$
$674$	−22.0000	−0.847408
$675$	1.00000	0.0384900
$676$	9.00000	0.346154
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	−6.00000	−0.230429
$679$	−8.00000	−0.307012
$680$	−12.0000	−0.460179
$681$	12.0000	0.459841
$682$	0	0
$683$	20.0000	0.765279	0.382639	−	0.923898i	$-0.375015\pi$
$683$	20.0000	0.765279	0.382639	+	0.923898i	$0.375015\pi$
$684$	0	0
$685$	−4.00000	−0.152832
$686$	8.00000	0.305441
$687$	−6.00000	−0.228914
$688$	0	0
$689$	12.0000	0.457164
$690$	−16.0000	−0.609110
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	4.00000	0.151838
$695$	−16.0000	−0.606915
$696$	−18.0000	−0.682288
$697$	4.00000	0.151511
$698$	6.00000	0.227103
$699$	30.0000	1.13470
$700$	−4.00000	−0.151186
$701$	−50.0000	−1.88847	−0.944237	−	0.329267i	$-0.893198\pi$
$701$	−50.0000	−1.88847	−0.944237	+	0.329267i	$0.893198\pi$
$702$	2.00000	0.0754851
$703$	0	0
$704$	0	0
$705$	16.0000	0.602595
$706$	−18.0000	−0.677439
$707$	8.00000	0.300871
$708$	−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
$711$	4.00000	0.150012
$712$	−18.0000	−0.674579
$713$	−64.0000	−2.39682
$714$	−8.00000	−0.299392
$715$	0	0
$716$	−12.0000	−0.448461
$717$	24.0000	0.896296
$718$	−8.00000	−0.298557
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	2.00000	0.0745356
$721$	−32.0000	−1.19174
$722$	19.0000	0.707107
$723$	10.0000	0.371904
$724$	−22.0000	−0.817624
$725$	−6.00000	−0.222834
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	−24.0000	−0.889499
$729$	1.00000	0.0370370
$730$	28.0000	1.03633
$731$	0	0
$732$	−6.00000	−0.221766
$733$	−30.0000	−1.10808	−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000	−1.10808	−0.554038	+	0.832492i	$0.686914\pi$
$734$	32.0000	1.18114
$735$	18.0000	0.663940
$736$	−40.0000	−1.47442
$737$	0	0
$738$	−2.00000	−0.0736210
$739$	−8.00000	−0.294285	−0.147142	−	0.989115i	$-0.547008\pi$
$739$	−8.00000	−0.294285	−0.147142	+	0.989115i	$0.547008\pi$
$740$	12.0000	0.441129
$741$	0	0
$742$	24.0000	0.881068
$743$	−40.0000	−1.46746	−0.733729	−	0.679442i	$-0.762222\pi$
$743$	−40.0000	−1.46746	−0.733729	+	0.679442i	$0.762222\pi$
$744$	24.0000	0.879883
$745$	−44.0000	−1.61204
$746$	−2.00000	−0.0732252
$747$	−12.0000	−0.439057
$748$	0	0
$749$	−48.0000	−1.75388
$750$	12.0000	0.438178
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	−8.00000	−0.291730
$753$	−4.00000	−0.145768
$754$	−12.0000	−0.437014
$755$	40.0000	1.45575
$756$	−4.00000	−0.145479
$757$	−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
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	4.00000	0.144905
$763$	−8.00000	−0.289619
$764$	−8.00000	−0.289430
$765$	−4.00000	−0.144620
$766$	16.0000	0.578103
$767$	−8.00000	−0.288863
$768$	17.0000	0.613435
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	14.0000	0.504198
$772$	−14.0000	−0.503871
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	6.00000	0.215387
$777$	24.0000	0.860995
$778$	18.0000	0.645331
$779$	0	0
$780$	−4.00000	−0.143223
$781$	0	0
$782$	−16.0000	−0.572159
$783$	−6.00000	−0.214423
$784$	−9.00000	−0.321429
$785$	−28.0000	−0.999363
$786$	12.0000	0.428026
$787$	8.00000	0.285169	0.142585	−	0.989783i	$-0.454459\pi$
$787$	8.00000	0.285169	0.142585	+	0.989783i	$0.454459\pi$
$788$	−14.0000	−0.498729
$789$	−16.0000	−0.569615
$790$	8.00000	0.284627
$791$	24.0000	0.853342
$792$	0	0
$793$	−12.0000	−0.426132
$794$	2.00000	0.0709773
$795$	12.0000	0.425596
$796$	0	0
$797$	−10.0000	−0.354218	−0.177109	−	0.984191i	$-0.556675\pi$
$797$	−10.0000	−0.354218	−0.177109	+	0.984191i	$0.556675\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	5.00000	0.176777
$801$	−6.00000	−0.212000
$802$	−26.0000	−0.918092
$803$	0	0
$804$	−4.00000	−0.141069
$805$	64.0000	2.25570
$806$	16.0000	0.563576
$807$	2.00000	0.0704033
$808$	−6.00000	−0.211079
$809$	−54.0000	−1.89854	−0.949269	−	0.314464i	$-0.898175\pi$
$809$	−54.0000	−1.89854	−0.949269	+	0.314464i	$0.898175\pi$
$810$	2.00000	0.0702728
$811$	56.0000	1.96643	0.983213	−	0.182462i	$-0.0584065\pi$
$811$	56.0000	1.96643	0.983213	+	0.182462i	$0.0584065\pi$
$812$	24.0000	0.842235
$813$	20.0000	0.701431
$814$	0	0
$815$	−8.00000	−0.280228
$816$	2.00000	0.0700140
$817$	0	0
$818$	18.0000	0.629355
$819$	−8.00000	−0.279543
$820$	4.00000	0.139686
$821$	14.0000	0.488603	0.244302	−	0.969699i	$-0.421441\pi$
$821$	14.0000	0.488603	0.244302	+	0.969699i	$0.421441\pi$
$822$	2.00000	0.0697580
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	24.0000	0.836080
$825$	0	0
$826$	−16.0000	−0.556711
$827$	−20.0000	−0.695468	−0.347734	−	0.937593i	$-0.613049\pi$
$827$	−20.0000	−0.695468	−0.347734	+	0.937593i	$0.613049\pi$
$828$	−8.00000	−0.278019
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	−24.0000	−0.833052
$831$	−26.0000	−0.901930
$832$	14.0000	0.485363
$833$	18.0000	0.623663
$834$	8.00000	0.277017
$835$	0	0
$836$	0	0
$837$	8.00000	0.276520
$838$	4.00000	0.138178
$839$	−56.0000	−1.93333	−0.966667	−	0.256036i	$-0.917584\pi$
$839$	−56.0000	−1.93333	−0.966667	+	0.256036i	$0.917584\pi$
$840$	−24.0000	−0.828079
$841$	7.00000	0.241379
$842$	26.0000	0.896019
$843$	−18.0000	−0.619953
$844$	0	0
$845$	18.0000	0.619219
$846$	−8.00000	−0.275046
$847$	0	0
$848$	−6.00000	−0.206041
$849$	16.0000	0.549119
$850$	2.00000	0.0685994
$851$	48.0000	1.64542
$852$	0	0
$853$	34.0000	1.16414	0.582069	−	0.813139i	$-0.302243\pi$
$853$	34.0000	1.16414	0.582069	+	0.813139i	$0.302243\pi$
$854$	−24.0000	−0.821263
$855$	0	0
$856$	36.0000	1.23045
$857$	10.0000	0.341593	0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000	0.341593	0.170797	+	0.985306i	$0.445366\pi$
$858$	0	0
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	0	0
$861$	8.00000	0.272639
$862$	−24.0000	−0.817443
$863$	48.0000	1.63394	0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000	1.63394	0.816970	+	0.576681i	$0.195652\pi$
$864$	5.00000	0.170103
$865$	−12.0000	−0.408012
$866$	−34.0000	−1.15537
$867$	13.0000	0.441503
$868$	−32.0000	−1.08615
$869$	0	0
$870$	−12.0000	−0.406838
$871$	−8.00000	−0.271070
$872$	6.00000	0.203186
$873$	2.00000	0.0676897
$874$	0	0
$875$	−48.0000	−1.62270
$876$	14.0000	0.473016
$877$	−6.00000	−0.202606	−0.101303	−	0.994856i	$-0.532301\pi$
$877$	−6.00000	−0.202606	−0.101303	+	0.994856i	$0.532301\pi$
$878$	−20.0000	−0.674967
$879$	−6.00000	−0.202375
$880$	0	0
$881$	26.0000	0.875962	0.437981	−	0.898984i	$-0.355694\pi$
$881$	26.0000	0.875962	0.437981	+	0.898984i	$0.355694\pi$
$882$	−9.00000	−0.303046
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	−4.00000	−0.134535
$885$	−8.00000	−0.268917
$886$	−28.0000	−0.940678
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	−18.0000	−0.604040
$889$	−16.0000	−0.536623
$890$	−12.0000	−0.402241
$891$	0	0
$892$	−16.0000	−0.535720
$893$	0	0
$894$	22.0000	0.735790
$895$	−24.0000	−0.802232
$896$	−12.0000	−0.400892
$897$	−16.0000	−0.534224
$898$	−2.00000	−0.0667409
$899$	−48.0000	−1.60089
$900$	1.00000	0.0333333
$901$	12.0000	0.399778
$902$	0	0
$903$	0	0
$904$	−18.0000	−0.598671
$905$	−44.0000	−1.46261
$906$	−20.0000	−0.664455
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	12.0000	0.398234
$909$	−2.00000	−0.0663358
$910$	−16.0000	−0.530395
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	0	0
$913$	0	0
$914$	18.0000	0.595387
$915$	−12.0000	−0.396708
$916$	−6.00000	−0.198246
$917$	−48.0000	−1.58510
$918$	2.00000	0.0660098
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	−48.0000	−1.58251
$921$	32.0000	1.05444
$922$	−30.0000	−0.987997
$923$	0	0
$924$	0	0
$925$	−6.00000	−0.197279
$926$	−16.0000	−0.525793
$927$	8.00000	0.262754
$928$	−30.0000	−0.984798
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	16.0000	0.524661
$931$	0	0
$932$	30.0000	0.982683
$933$	24.0000	0.785725
$934$	12.0000	0.392652
$935$	0	0
$936$	6.00000	0.196116
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	−16.0000	−0.522419
$939$	22.0000	0.717943
$940$	16.0000	0.521862
$941$	54.0000	1.76035	0.880175	−	0.474650i	$-0.157425\pi$
$941$	54.0000	1.76035	0.880175	+	0.474650i	$0.157425\pi$
$942$	14.0000	0.456145
$943$	16.0000	0.521032
$944$	4.00000	0.130189
$945$	−8.00000	−0.260240
$946$	0	0
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	4.00000	0.129914
$949$	28.0000	0.908918
$950$	0	0
$951$	−22.0000	−0.713399
$952$	−24.0000	−0.777844
$953$	−22.0000	−0.712650	−0.356325	−	0.934362i	$-0.615970\pi$
$953$	−22.0000	−0.712650	−0.356325	+	0.934362i	$0.615970\pi$
$954$	−6.00000	−0.194257
$955$	−16.0000	−0.517748
$956$	24.0000	0.776215
$957$	0	0
$958$	8.00000	0.258468
$959$	−8.00000	−0.258333
$960$	14.0000	0.451848
$961$	33.0000	1.06452
$962$	−12.0000	−0.386896
$963$	12.0000	0.386695
$964$	10.0000	0.322078
$965$	−28.0000	−0.901352
$966$	−32.0000	−1.02958
$967$	−4.00000	−0.128631	−0.0643157	−	0.997930i	$-0.520486\pi$
$967$	−4.00000	−0.128631	−0.0643157	+	0.997930i	$0.520486\pi$
$968$	0	0
$969$	0	0
$970$	4.00000	0.128432
$971$	−52.0000	−1.66876	−0.834380	−	0.551190i	$-0.814174\pi$
$971$	−52.0000	−1.66876	−0.834380	+	0.551190i	$0.814174\pi$
$972$	1.00000	0.0320750
$973$	−32.0000	−1.02587
$974$	16.0000	0.512673
$975$	2.00000	0.0640513
$976$	6.00000	0.192055
$977$	−6.00000	−0.191957	−0.0959785	−	0.995383i	$-0.530598\pi$
$977$	−6.00000	−0.191957	−0.0959785	+	0.995383i	$0.530598\pi$
$978$	4.00000	0.127906
$979$	0	0
$980$	18.0000	0.574989
$981$	2.00000	0.0638551
$982$	4.00000	0.127645
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	−6.00000	−0.191273
$985$	−28.0000	−0.892154
$986$	−12.0000	−0.382158
$987$	32.0000	1.01857
$988$	0	0
$989$	0	0
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	40.0000	1.27000
$993$	20.0000	0.634681
$994$	0	0
$995$	0	0
$996$	−12.0000	−0.380235
$997$	−14.0000	−0.443384	−0.221692	−	0.975117i	$-0.571158\pi$
$997$	−14.0000	−0.443384	−0.221692	+	0.975117i	$0.571158\pi$
$998$	4.00000	0.126618
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.2.a.b.1.1		1
3.2	odd	2		1089.2.a.j.1.1		1
4.3	odd	2		5808.2.a.t.1.1		1
5.4	even	2		9075.2.a.q.1.1		1
11.2	odd	10		363.2.e.e.202.1		4
11.3	even	5		363.2.e.g.130.1		4
11.4	even	5		363.2.e.g.148.1		4
11.5	even	5		363.2.e.g.124.1		4
11.6	odd	10		363.2.e.e.124.1		4
11.7	odd	10		363.2.e.e.148.1		4
11.8	odd	10		363.2.e.e.130.1		4
11.9	even	5		363.2.e.g.202.1		4
11.10	odd	2		33.2.a.a.1.1	✓	1
33.32	even	2		99.2.a.b.1.1		1
44.43	even	2		528.2.a.g.1.1		1
55.32	even	4		825.2.c.a.199.2		2
55.43	even	4		825.2.c.a.199.1		2
55.54	odd	2		825.2.a.a.1.1		1
77.76	even	2		1617.2.a.j.1.1		1
88.21	odd	2		2112.2.a.bb.1.1		1
88.43	even	2		2112.2.a.j.1.1		1
99.32	even	6		891.2.e.g.298.1		2
99.43	odd	6		891.2.e.e.595.1		2
99.65	even	6		891.2.e.g.595.1		2
99.76	odd	6		891.2.e.e.298.1		2
132.131	odd	2		1584.2.a.o.1.1		1
143.142	odd	2		5577.2.a.a.1.1		1
165.32	odd	4		2475.2.c.d.199.1		2
165.98	odd	4		2475.2.c.d.199.2		2
165.164	even	2		2475.2.a.g.1.1		1
187.186	odd	2		9537.2.a.m.1.1		1
231.230	odd	2		4851.2.a.b.1.1		1
264.131	odd	2		6336.2.a.n.1.1		1
264.197	even	2		6336.2.a.x.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.2.a.a.1.1	✓	1	11.10	odd	2
99.2.a.b.1.1		1	33.32	even	2
363.2.a.b.1.1		1	1.1	even	1	trivial
363.2.e.e.124.1		4	11.6	odd	10
363.2.e.e.130.1		4	11.8	odd	10
363.2.e.e.148.1		4	11.7	odd	10
363.2.e.e.202.1		4	11.2	odd	10
363.2.e.g.124.1		4	11.5	even	5
363.2.e.g.130.1		4	11.3	even	5
363.2.e.g.148.1		4	11.4	even	5
363.2.e.g.202.1		4	11.9	even	5
528.2.a.g.1.1		1	44.43	even	2
825.2.a.a.1.1		1	55.54	odd	2
825.2.c.a.199.1		2	55.43	even	4
825.2.c.a.199.2		2	55.32	even	4
891.2.e.e.298.1		2	99.76	odd	6
891.2.e.e.595.1		2	99.43	odd	6
891.2.e.g.298.1		2	99.32	even	6
891.2.e.g.595.1		2	99.65	even	6
1089.2.a.j.1.1		1	3.2	odd	2
1584.2.a.o.1.1		1	132.131	odd	2
1617.2.a.j.1.1		1	77.76	even	2
2112.2.a.j.1.1		1	88.43	even	2
2112.2.a.bb.1.1		1	88.21	odd	2
2475.2.a.g.1.1		1	165.164	even	2
2475.2.c.d.199.1		2	165.32	odd	4
2475.2.c.d.199.2		2	165.98	odd	4
4851.2.a.b.1.1		1	231.230	odd	2
5577.2.a.a.1.1		1	143.142	odd	2
5808.2.a.t.1.1		1	4.3	odd	2
6336.2.a.n.1.1		1	264.131	odd	2
6336.2.a.x.1.1		1	264.197	even	2
9075.2.a.q.1.1		1	5.4	even	2
9537.2.a.m.1.1		1	187.186	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:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	363.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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