LMFDB - Embedded newform 5577.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5577,2,Mod(1,5577)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5577, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5577.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5577 = 3 \cdot 11 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5577.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$44.5325692073$
Analytic rank:	$1$
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	$\chi$	$=$	5577.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} +4.00000 q^{14} -2.00000 q^{15} -1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} -2.00000 q^{20} +4.00000 q^{21} +1.00000 q^{22} +8.00000 q^{23} -3.00000 q^{24} -1.00000 q^{25} -1.00000 q^{27} +4.00000 q^{28} -6.00000 q^{29} +2.00000 q^{30} +8.00000 q^{31} -5.00000 q^{32} +1.00000 q^{33} +2.00000 q^{34} -8.00000 q^{35} -1.00000 q^{36} -6.00000 q^{37} +6.00000 q^{40} +2.00000 q^{41} -4.00000 q^{42} +1.00000 q^{44} +2.00000 q^{45} -8.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} +9.00000 q^{49} +1.00000 q^{50} +2.00000 q^{51} +6.00000 q^{53} +1.00000 q^{54} -2.00000 q^{55} -12.0000 q^{56} +6.00000 q^{58} +4.00000 q^{59} +2.00000 q^{60} +6.00000 q^{61} -8.00000 q^{62} -4.00000 q^{63} +7.00000 q^{64} -1.00000 q^{66} +4.00000 q^{67} +2.00000 q^{68} -8.00000 q^{69} +8.00000 q^{70} +3.00000 q^{72} +14.0000 q^{73} +6.00000 q^{74} +1.00000 q^{75} +4.00000 q^{77} -4.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -12.0000 q^{83} -4.00000 q^{84} -4.00000 q^{85} +6.00000 q^{87} -3.00000 q^{88} +6.00000 q^{89} -2.00000 q^{90} -8.00000 q^{92} -8.00000 q^{93} +8.00000 q^{94} +5.00000 q^{96} -2.00000 q^{97} -9.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107	−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	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	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
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	4.00000	1.06904
$15$	−2.00000	−0.516398
$16$	−1.00000	−0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	−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
$22$	1.00000	0.213201
$23$	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
$26$	0	0
$27$	−1.00000	−0.192450
$28$	4.00000	0.755929
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	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
$33$	1.00000	0.174078
$34$	2.00000	0.342997
$35$	−8.00000	−1.35225
$36$	−1.00000	−0.166667
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	0	0
$40$	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	$0.5\pi$
$44$	1.00000	0.150756
$45$	2.00000	0.298142
$46$	−8.00000	−1.17954
$47$	−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
$49$	9.00000	1.28571
$50$	1.00000	0.141421
$51$	2.00000	0.280056
$52$	0	0
$53$	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
$55$	−2.00000	−0.269680
$56$	−12.0000	−1.60357
$57$	0	0
$58$	6.00000	0.787839
$59$	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
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	−8.00000	−1.01600
$63$	−4.00000	−0.503953
$64$	7.00000	0.875000
$65$	0	0
$66$	−1.00000	−0.123091
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	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
$77$	4.00000	0.455842
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	−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
$88$	−3.00000	−0.319801
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	−2.00000	−0.210819
$91$	0	0
$92$	−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	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	−9.00000	−0.909137
$99$	−1.00000	−0.100504
$100$	1.00000	0.100000
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	−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	$0.371045\pi$
$104$	0	0
$105$	8.00000	0.780720
$106$	−6.00000	−0.582772
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	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	$0.469465\pi$
$110$	2.00000	0.190693
$111$	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
$117$	0	0
$118$	−4.00000	−0.368230
$119$	8.00000	0.733359
$120$	−6.00000	−0.547723
$121$	1.00000	0.0909091
$122$	−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	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	3.00000	0.265165
$129$	0	0
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	−1.00000	−0.0870388
$133$	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	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	8.00000	0.681005
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	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
$154$	−4.00000	−0.322329
$155$	16.0000	1.28515
$156$	0	0
$157$	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	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−2.00000	−0.156174
$165$	2.00000	0.155700
$166$	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
$169$	0	0
$170$	4.00000	0.306786
$171$	0	0
$172$	0	0
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	−6.00000	−0.454859
$175$	4.00000	0.302372
$176$	1.00000	0.0753778
$177$	−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	$0.195291\pi$
$182$	0	0
$183$	−6.00000	−0.443533
$184$	24.0000	1.76930
$185$	−12.0000	−0.882258
$186$	8.00000	0.586588
$187$	2.00000	0.146254
$188$	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
$195$	0	0
$196$	−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	$0.333800\pi$
$198$	1.00000	0.0710669
$199$	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
$208$	0	0
$209$	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
$220$	2.00000	0.134840
$221$	0	0
$222$	−6.00000	−0.402694
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	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	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	−16.0000	−1.05501
$231$	−4.00000	−0.263181
$232$	−18.0000	−1.18176
$233$	30.0000	1.96537	0.982683	−	0.185296i	$-0.0593245\pi$
$233$	30.0000	1.96537	0.982683	+	0.185296i	$0.0593245\pi$
$234$	0	0
$235$	−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	$0.604381\pi$
$242$	−1.00000	−0.0642824
$243$	−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
$253$	−8.00000	−0.502956
$254$	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
$260$	0	0
$261$	−6.00000	−0.371391
$262$	12.0000	0.741362
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	3.00000	0.184637
$265$	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
$273$	0	0
$274$	2.00000	0.120824
$275$	1.00000	0.0603023
$276$	8.00000	0.481543
$277$	−26.0000	−1.56219	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000	−1.56219	−0.781094	+	0.624413i	$0.785338\pi$
$278$	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	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	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
$297$	1.00000	0.0580259
$298$	−22.0000	−1.27443
$299$	0	0
$300$	−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	$0.866372\pi$
$308$	−4.00000	−0.227921
$309$	−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	$0.738219\pi$
$312$	0	0
$313$	−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	$-0.711985\pi$
$317$	−22.0000	−1.23564	−0.617822	+	0.786318i	$0.711985\pi$
$318$	6.00000	0.336463
$319$	6.00000	0.335936
$320$	14.0000	0.782624
$321$	12.0000	0.669775
$322$	32.0000	1.78329
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	4.00000	0.221540
$327$	−2.00000	−0.110600
$328$	6.00000	0.331295
$329$	32.0000	1.76422
$330$	−2.00000	−0.110096
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	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	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	4.00000	0.216930
$341$	−8.00000	−0.433224
$342$	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	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	−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
$351$	0	0
$352$	5.00000	0.266501
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	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
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	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	$-0.516489\pi$
$373$	−2.00000	−0.103556	−0.0517780	+	0.998659i	$0.516489\pi$
$374$	−2.00000	−0.103418
$375$	12.0000	0.619677
$376$	−24.0000	−1.23771
$377$	0	0
$378$	−4.00000	−0.205738
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	0	0
$381$	4.00000	0.204926
$382$	−8.00000	−0.409316
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	−3.00000	−0.153093
$385$	8.00000	0.407718
$386$	−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	$0.650832\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	27.0000	1.36371
$393$	12.0000	0.605320
$394$	−14.0000	−0.705310
$395$	−8.00000	−0.402524
$396$	1.00000	0.0502519
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	0	0
$399$	0	0
$400$	1.00000	0.0500000
$401$	−26.0000	−1.29838	−0.649189	−	0.760627i	$-0.724892\pi$
$401$	−26.0000	−1.29838	−0.649189	+	0.760627i	$0.724892\pi$
$402$	4.00000	0.199502
$403$	0	0
$404$	−2.00000	−0.0995037
$405$	2.00000	0.0993808
$406$	−24.0000	−1.19110
$407$	6.00000	0.297409
$408$	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
$416$	0	0
$417$	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	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	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	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	−6.00000	−0.286039
$441$	9.00000	0.428571
$442$	0	0
$443$	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	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	1.00000	0.0471405
$451$	−2.00000	−0.0941763
$452$	6.00000	0.282216
$453$	20.0000	0.939682
$454$	12.0000	0.563188
$455$	0	0
$456$	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	$0.253798\pi$
$462$	4.00000	0.186097
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	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	$0.589555\pi$
$468$	0	0
$469$	−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
$481$	0	0
$482$	10.0000	0.455488
$483$	32.0000	1.45605
$484$	−1.00000	−0.0454545
$485$	−4.00000	−0.181631
$486$	1.00000	0.0453609
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	18.0000	0.814822
$489$	4.00000	0.180886
$490$	−18.0000	−0.813157
$491$	4.00000	0.180517	0.0902587	−	0.995918i	$-0.471231\pi$
$491$	4.00000	0.180517	0.0902587	+	0.995918i	$0.471231\pi$
$492$	2.00000	0.0901670
$493$	12.0000	0.540453
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	−8.00000	−0.359211
$497$	0	0
$498$	−12.0000	−0.537733
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	12.0000	0.536656
$501$	0	0
$502$	−4.00000	−0.178529
$503$	−32.0000	−1.42681	−0.713405	−	0.700752i	$-0.752848\pi$
$503$	−32.0000	−1.42681	−0.713405	+	0.700752i	$0.752848\pi$
$504$	−12.0000	−0.534522
$505$	4.00000	0.177998
$506$	8.00000	0.355643
$507$	0	0
$508$	4.00000	0.177471
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	−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
$517$	8.00000	0.351840
$518$	−24.0000	−1.05450
$519$	6.00000	0.263371
$520$	0	0
$521$	−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	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	12.0000	0.524222
$525$	−4.00000	−0.174574
$526$	16.0000	0.697633
$527$	−16.0000	−0.696971
$528$	−1.00000	−0.0435194
$529$	41.0000	1.78261
$530$	−12.0000	−0.521247
$531$	4.00000	0.173585
$532$	0	0
$533$	0	0
$534$	6.00000	0.259645
$535$	−24.0000	−1.03761
$536$	12.0000	0.518321
$537$	−12.0000	−0.517838
$538$	2.00000	0.0862261
$539$	−9.00000	−0.387657
$540$	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
$546$	0	0
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	2.00000	0.0854358
$549$	6.00000	0.256074
$550$	−1.00000	−0.0426401
$551$	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
$561$	−2.00000	−0.0844401
$562$	−18.0000	−0.759284
$563$	−44.0000	−1.85438	−0.927189	−	0.374593i	$-0.877783\pi$
$563$	−44.0000	−1.85438	−0.927189	+	0.374593i	$0.877783\pi$
$564$	−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	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	0	0
$571$	−16.0000	−0.669579	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000	−0.669579	−0.334790	+	0.942293i	$0.608665\pi$
$572$	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	$-0.285320\pi$
$577$	30.0000	1.24892	0.624458	+	0.781058i	$0.285320\pi$
$578$	13.0000	0.540729
$579$	−14.0000	−0.581820
$580$	12.0000	0.498273
$581$	48.0000	1.99138
$582$	−2.00000	−0.0829027
$583$	−6.00000	−0.248495
$584$	42.0000	1.73797
$585$	0	0
$586$	−6.00000	−0.247858
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	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	$0.784901\pi$
$594$	−1.00000	−0.0410305
$595$	16.0000	0.655936
$596$	−22.0000	−0.901155
$597$	0	0
$598$	0	0
$599$	−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	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	20.0000	0.813788
$605$	2.00000	0.0813116
$606$	2.00000	0.0812444
$607$	−4.00000	−0.162355	−0.0811775	−	0.996700i	$-0.525868\pi$
$607$	−4.00000	−0.162355	−0.0811775	+	0.996700i	$0.525868\pi$
$608$	0	0
$609$	−24.0000	−0.972529
$610$	−12.0000	−0.485866
$611$	0	0
$612$	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
$616$	12.0000	0.483494
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	8.00000	0.321807
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	−16.0000	−0.642575
$621$	−8.00000	−0.321029
$622$	24.0000	0.962312
$623$	−24.0000	−0.961540
$624$	0	0
$625$	−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	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	−12.0000	−0.477334
$633$	0	0
$634$	22.0000	0.873732
$635$	−8.00000	−0.317470
$636$	6.00000	0.237915
$637$	0	0
$638$	−6.00000	−0.237542
$639$	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	$-0.629034\pi$
$643$	−20.0000	−0.788723	−0.394362	+	0.918955i	$0.629034\pi$
$644$	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
$649$	−4.00000	−0.157014
$650$	0	0
$651$	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	$-0.475176\pi$
$659$	4.00000	0.155818	0.0779089	+	0.996960i	$0.475176\pi$
$660$	−2.00000	−0.0778499
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	−20.0000	−0.777322
$663$	0	0
$664$	−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
$671$	−6.00000	−0.231627
$672$	−20.0000	−0.771517
$673$	−46.0000	−1.77317	−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000	−1.77317	−0.886585	+	0.462566i	$0.846929\pi$
$674$	22.0000	0.847408
$675$	1.00000	0.0384900
$676$	0	0
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	−6.00000	−0.230429
$679$	8.00000	0.307012
$680$	−12.0000	−0.460179
$681$	12.0000	0.459841
$682$	8.00000	0.306336
$683$	−20.0000	−0.765279	−0.382639	−	0.923898i	$-0.624985\pi$
$683$	−20.0000	−0.765279	−0.382639	+	0.923898i	$0.624985\pi$
$684$	0	0
$685$	−4.00000	−0.152832
$686$	8.00000	0.305441
$687$	6.00000	0.228914
$688$	0	0
$689$	0	0
$690$	16.0000	0.609110
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	6.00000	0.228086
$693$	4.00000	0.151947
$694$	−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	$-0.106802\pi$
$701$	50.0000	1.88847	0.944237	+	0.329267i	$0.106802\pi$
$702$	0	0
$703$	0	0
$704$	−7.00000	−0.263822
$705$	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	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	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
$726$	1.00000	0.0371135
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	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
$737$	−4.00000	−0.147342
$738$	−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
$748$	−2.00000	−0.0731272
$749$	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
$754$	0	0
$755$	−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
$759$	8.00000	0.290382
$760$	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
$767$	0	0
$768$	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	$0.511481\pi$
$770$	−8.00000	−0.288300
$771$	14.0000	0.504198
$772$	−14.0000	−0.503871
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	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
$780$	0	0
$781$	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
$792$	−3.00000	−0.106600
$793$	0	0
$794$	−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
$803$	−14.0000	−0.494049
$804$	4.00000	0.141069
$805$	−64.0000	−2.25570
$806$	0	0
$807$	2.00000	0.0704033
$808$	6.00000	0.211079
$809$	54.0000	1.89854	0.949269	−	0.314464i	$-0.101825\pi$
$809$	54.0000	1.89854	0.949269	+	0.314464i	$0.101825\pi$
$810$	−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
$814$	−6.00000	−0.210300
$815$	−8.00000	−0.280228
$816$	−2.00000	−0.0700140
$817$	0	0
$818$	18.0000	0.629355
$819$	0	0
$820$	−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
$825$	−1.00000	−0.0348155
$826$	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
$832$	0	0
$833$	−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	$-0.0824164\pi$
$839$	56.0000	1.93333	0.966667	+	0.256036i	$0.0824164\pi$
$840$	24.0000	0.828079
$841$	7.00000	0.241379
$842$	−26.0000	−0.896019
$843$	−18.0000	−0.619953
$844$	0	0
$845$	0	0
$846$	8.00000	0.275046
$847$	−4.00000	−0.137442
$848$	−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	$-0.554634\pi$
$857$	−10.0000	−0.341593	−0.170797	+	0.985306i	$0.554634\pi$
$858$	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	$-0.804348\pi$
$863$	−48.0000	−1.63394	−0.816970	+	0.576681i	$0.804348\pi$
$864$	5.00000	0.170103
$865$	−12.0000	−0.408012
$866$	−34.0000	−1.15537
$867$	13.0000	0.441503
$868$	32.0000	1.08615
$869$	4.00000	0.135691
$870$	−12.0000	−0.406838
$871$	0	0
$872$	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
$880$	2.00000	0.0674200
$881$	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	$0.609252\pi$
$884$	0	0
$885$	−8.00000	−0.268917
$886$	−28.0000	−0.940678
$887$	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
$890$	−12.0000	−0.402241
$891$	−1.00000	−0.0335013
$892$	16.0000	0.535720
$893$	0	0
$894$	22.0000	0.735790
$895$	24.0000	0.802232
$896$	−12.0000	−0.400892
$897$	0	0
$898$	2.00000	0.0667409
$899$	−48.0000	−1.60089
$900$	1.00000	0.0333333
$901$	−12.0000	−0.399778
$902$	2.00000	0.0665927
$903$	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
$910$	0	0
$911$	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
$913$	12.0000	0.397142
$914$	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	$-0.607005\pi$
$919$	−20.0000	−0.659739	−0.329870	+	0.944027i	$0.607005\pi$
$920$	48.0000	1.58251
$921$	32.0000	1.05444
$922$	−30.0000	−0.987997
$923$	0	0
$924$	4.00000	0.131590
$925$	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	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	16.0000	0.524661
$931$	0	0
$932$	−30.0000	−0.982683
$933$	24.0000	0.785725
$934$	12.0000	0.392652
$935$	4.00000	0.130814
$936$	0	0
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	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	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	−4.00000	−0.129914
$949$	0	0
$950$	0	0
$951$	22.0000	0.713399
$952$	24.0000	0.777844
$953$	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
$955$	16.0000	0.517748
$956$	24.0000	0.776215
$957$	−6.00000	−0.193952
$958$	8.00000	0.258468
$959$	8.00000	0.258333
$960$	−14.0000	−0.451848
$961$	33.0000	1.06452
$962$	0	0
$963$	−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	$0.520486\pi$
$968$	3.00000	0.0964237
$969$	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
$975$	0	0
$976$	−6.00000	−0.192055
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	−4.00000	−0.127906
$979$	−6.00000	−0.191761
$980$	−18.0000	−0.574989
$981$	2.00000	0.0638551
$982$	−4.00000	−0.127645
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	−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
$990$	2.00000	0.0635642
$991$	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	$-0.428842\pi$
$997$	14.0000	0.443384	0.221692	+	0.975117i	$0.428842\pi$
$998$	−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	5577.2.a.a.1.1		1
13.12	even	2		33.2.a.a.1.1	✓	1
39.38	odd	2		99.2.a.b.1.1		1
52.51	odd	2		528.2.a.g.1.1		1
65.12	odd	4		825.2.c.a.199.2		2
65.38	odd	4		825.2.c.a.199.1		2
65.64	even	2		825.2.a.a.1.1		1
91.90	odd	2		1617.2.a.j.1.1		1
104.51	odd	2		2112.2.a.j.1.1		1
104.77	even	2		2112.2.a.bb.1.1		1
117.25	even	6		891.2.e.e.595.1		2
117.38	odd	6		891.2.e.g.595.1		2
117.77	odd	6		891.2.e.g.298.1		2
117.103	even	6		891.2.e.e.298.1		2
143.25	even	10		363.2.e.e.130.1		4
143.38	even	10		363.2.e.e.124.1		4
143.51	odd	10		363.2.e.g.148.1		4
143.64	even	10		363.2.e.e.202.1		4
143.90	odd	10		363.2.e.g.202.1		4
143.103	even	10		363.2.e.e.148.1		4
143.116	odd	10		363.2.e.g.124.1		4
143.129	odd	10		363.2.e.g.130.1		4
143.142	odd	2		363.2.a.b.1.1		1
156.155	even	2		1584.2.a.o.1.1		1
195.38	even	4		2475.2.c.d.199.2		2
195.77	even	4		2475.2.c.d.199.1		2
195.194	odd	2		2475.2.a.g.1.1		1
221.220	even	2		9537.2.a.m.1.1		1
273.272	even	2		4851.2.a.b.1.1		1
312.77	odd	2		6336.2.a.x.1.1		1
312.155	even	2		6336.2.a.n.1.1		1
429.428	even	2		1089.2.a.j.1.1		1
572.571	even	2		5808.2.a.t.1.1		1
715.714	odd	2		9075.2.a.q.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.2.a.a.1.1	✓	1	13.12	even	2
99.2.a.b.1.1		1	39.38	odd	2
363.2.a.b.1.1		1	143.142	odd	2
363.2.e.e.124.1		4	143.38	even	10
363.2.e.e.130.1		4	143.25	even	10
363.2.e.e.148.1		4	143.103	even	10
363.2.e.e.202.1		4	143.64	even	10
363.2.e.g.124.1		4	143.116	odd	10
363.2.e.g.130.1		4	143.129	odd	10
363.2.e.g.148.1		4	143.51	odd	10
363.2.e.g.202.1		4	143.90	odd	10
528.2.a.g.1.1		1	52.51	odd	2
825.2.a.a.1.1		1	65.64	even	2
825.2.c.a.199.1		2	65.38	odd	4
825.2.c.a.199.2		2	65.12	odd	4
891.2.e.e.298.1		2	117.103	even	6
891.2.e.e.595.1		2	117.25	even	6
891.2.e.g.298.1		2	117.77	odd	6
891.2.e.g.595.1		2	117.38	odd	6
1089.2.a.j.1.1		1	429.428	even	2
1584.2.a.o.1.1		1	156.155	even	2
1617.2.a.j.1.1		1	91.90	odd	2
2112.2.a.j.1.1		1	104.51	odd	2
2112.2.a.bb.1.1		1	104.77	even	2
2475.2.a.g.1.1		1	195.194	odd	2
2475.2.c.d.199.1		2	195.77	even	4
2475.2.c.d.199.2		2	195.38	even	4
4851.2.a.b.1.1		1	273.272	even	2
5577.2.a.a.1.1		1	1.1	even	1	trivial
5808.2.a.t.1.1		1	572.571	even	2
6336.2.a.n.1.1		1	312.155	even	2
6336.2.a.x.1.1		1	312.77	odd	2
9075.2.a.q.1.1		1	715.714	odd	2
9537.2.a.m.1.1		1	221.220	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5577 = 3 \cdot 11 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5577.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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