LMFDB - Embedded newform 77.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [77,2,Mod(1,77)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(77, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("77.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 77 = 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	77.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$0.614848095564$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	77.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +1.00000 q^{11} -2.00000 q^{12} +4.00000 q^{13} -1.00000 q^{14} -4.00000 q^{15} -1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} +2.00000 q^{20} -2.00000 q^{21} +1.00000 q^{22} -4.00000 q^{23} -6.00000 q^{24} -1.00000 q^{25} +4.00000 q^{26} -4.00000 q^{27} +1.00000 q^{28} -6.00000 q^{29} -4.00000 q^{30} +10.0000 q^{31} +5.00000 q^{32} +2.00000 q^{33} +4.00000 q^{34} +2.00000 q^{35} -1.00000 q^{36} -6.00000 q^{37} +8.00000 q^{39} +6.00000 q^{40} +4.00000 q^{41} -2.00000 q^{42} +12.0000 q^{43} -1.00000 q^{44} -2.00000 q^{45} -4.00000 q^{46} -10.0000 q^{47} -2.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +8.00000 q^{51} -4.00000 q^{52} -6.00000 q^{53} -4.00000 q^{54} -2.00000 q^{55} +3.00000 q^{56} -6.00000 q^{58} +2.00000 q^{59} +4.00000 q^{60} +10.0000 q^{62} -1.00000 q^{63} +7.00000 q^{64} -8.00000 q^{65} +2.00000 q^{66} +8.00000 q^{67} -4.00000 q^{68} -8.00000 q^{69} +2.00000 q^{70} -12.0000 q^{71} -3.00000 q^{72} -8.00000 q^{73} -6.00000 q^{74} -2.00000 q^{75} -1.00000 q^{77} +8.00000 q^{78} +8.00000 q^{79} +2.00000 q^{80} -11.0000 q^{81} +4.00000 q^{82} +2.00000 q^{84} -8.00000 q^{85} +12.0000 q^{86} -12.0000 q^{87} -3.00000 q^{88} -6.00000 q^{89} -2.00000 q^{90} -4.00000 q^{91} +4.00000 q^{92} +20.0000 q^{93} -10.0000 q^{94} +10.0000 q^{96} -10.0000 q^{97} +1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	−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	$0.647584\pi$
$6$	2.00000	0.816497
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−2.00000	−0.577350
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	−1.00000	−0.267261
$15$	−4.00000	−1.03280
$16$	−1.00000	−0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	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
$21$	−2.00000	−0.436436
$22$	1.00000	0.213201
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	−6.00000	−1.22474
$25$	−1.00000	−0.200000
$26$	4.00000	0.784465
$27$	−4.00000	−0.769800
$28$	1.00000	0.188982
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	−4.00000	−0.730297
$31$	10.0000	1.79605	0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000	1.79605	0.898027	+	0.439941i	$0.145001\pi$
$32$	5.00000	0.883883
$33$	2.00000	0.348155
$34$	4.00000	0.685994
$35$	2.00000	0.338062
$36$	−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
$39$	8.00000	1.28103
$40$	6.00000	0.948683
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	−2.00000	−0.308607
$43$	12.0000	1.82998	0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000	1.82998	0.914991	+	0.403473i	$0.132197\pi$
$44$	−1.00000	−0.150756
$45$	−2.00000	−0.298142
$46$	−4.00000	−0.589768
$47$	−10.0000	−1.45865	−0.729325	−	0.684167i	$-0.760166\pi$
$47$	−10.0000	−1.45865	−0.729325	+	0.684167i	$0.760166\pi$
$48$	−2.00000	−0.288675
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	8.00000	1.12022
$52$	−4.00000	−0.554700
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−4.00000	−0.544331
$55$	−2.00000	−0.269680
$56$	3.00000	0.400892
$57$	0	0
$58$	−6.00000	−0.787839
$59$	2.00000	0.260378	0.130189	−	0.991489i	$-0.458442\pi$
$59$	2.00000	0.260378	0.130189	+	0.991489i	$0.458442\pi$
$60$	4.00000	0.516398
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	10.0000	1.27000
$63$	−1.00000	−0.125988
$64$	7.00000	0.875000
$65$	−8.00000	−0.992278
$66$	2.00000	0.246183
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	−4.00000	−0.485071
$69$	−8.00000	−0.963087
$70$	2.00000	0.239046
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	−3.00000	−0.353553
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	−6.00000	−0.697486
$75$	−2.00000	−0.230940
$76$	0	0
$77$	−1.00000	−0.113961
$78$	8.00000	0.905822
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	2.00000	0.223607
$81$	−11.0000	−1.22222
$82$	4.00000	0.441726
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	2.00000	0.218218
$85$	−8.00000	−0.867722
$86$	12.0000	1.29399
$87$	−12.0000	−1.28654
$88$	−3.00000	−0.319801
$89$	−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
$91$	−4.00000	−0.419314
$92$	4.00000	0.417029
$93$	20.0000	2.07390
$94$	−10.0000	−1.03142
$95$	0	0
$96$	10.0000	1.02062
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	1.00000	0.101015
$99$	1.00000	0.100504
$100$	1.00000	0.100000
$101$	−4.00000	−0.398015	−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000	−0.398015	−0.199007	+	0.979998i	$0.563772\pi$
$102$	8.00000	0.792118
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	−12.0000	−1.17670
$105$	4.00000	0.390360
$106$	−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	$0.303036\pi$
$108$	4.00000	0.384900
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	−2.00000	−0.190693
$111$	−12.0000	−1.13899
$112$	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	$0.178617\pi$
$114$	0	0
$115$	8.00000	0.746004
$116$	6.00000	0.557086
$117$	4.00000	0.369800
$118$	2.00000	0.184115
$119$	−4.00000	−0.366679
$120$	12.0000	1.09545
$121$	1.00000	0.0909091
$122$	0	0
$123$	8.00000	0.721336
$124$	−10.0000	−0.898027
$125$	12.0000	1.07331
$126$	−1.00000	−0.0890871
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	−3.00000	−0.265165
$129$	24.0000	2.11308
$130$	−8.00000	−0.701646
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	−2.00000	−0.174078
$133$	0	0
$134$	8.00000	0.691095
$135$	8.00000	0.688530
$136$	−12.0000	−1.02899
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	−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	$0.610182\pi$
$140$	−2.00000	−0.169031
$141$	−20.0000	−1.68430
$142$	−12.0000	−1.00702
$143$	4.00000	0.334497
$144$	−1.00000	−0.0833333
$145$	12.0000	0.996546
$146$	−8.00000	−0.662085
$147$	2.00000	0.164957
$148$	6.00000	0.493197
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	−2.00000	−0.163299
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	4.00000	0.323381
$154$	−1.00000	−0.0805823
$155$	−20.0000	−1.60644
$156$	−8.00000	−0.640513
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	8.00000	0.636446
$159$	−12.0000	−0.951662
$160$	−10.0000	−0.790569
$161$	4.00000	0.315244
$162$	−11.0000	−0.864242
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	−4.00000	−0.312348
$165$	−4.00000	−0.311400
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	6.00000	0.462910
$169$	3.00000	0.230769
$170$	−8.00000	−0.613572
$171$	0	0
$172$	−12.0000	−0.914991
$173$	12.0000	0.912343	0.456172	−	0.889892i	$-0.349220\pi$
$173$	12.0000	0.912343	0.456172	+	0.889892i	$0.349220\pi$
$174$	−12.0000	−0.909718
$175$	1.00000	0.0755929
$176$	−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
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	−4.00000	−0.296500
$183$	0	0
$184$	12.0000	0.884652
$185$	12.0000	0.882258
$186$	20.0000	1.46647
$187$	4.00000	0.292509
$188$	10.0000	0.729325
$189$	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	$0.406534\pi$
$192$	14.0000	1.01036
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	−10.0000	−0.717958
$195$	−16.0000	−1.14578
$196$	−1.00000	−0.0714286
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	1.00000	0.0710669
$199$	−18.0000	−1.27599	−0.637993	−	0.770042i	$-0.720235\pi$
$199$	−18.0000	−1.27599	−0.637993	+	0.770042i	$0.720235\pi$
$200$	3.00000	0.212132
$201$	16.0000	1.12855
$202$	−4.00000	−0.281439
$203$	6.00000	0.421117
$204$	−8.00000	−0.560112
$205$	−8.00000	−0.558744
$206$	14.0000	0.975426
$207$	−4.00000	−0.278019
$208$	−4.00000	−0.277350
$209$	0	0
$210$	4.00000	0.276026
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	6.00000	0.412082
$213$	−24.0000	−1.64445
$214$	12.0000	0.820303
$215$	−24.0000	−1.63679
$216$	12.0000	0.816497
$217$	−10.0000	−0.678844
$218$	−14.0000	−0.948200
$219$	−16.0000	−1.08118
$220$	2.00000	0.134840
$221$	16.0000	1.07628
$222$	−12.0000	−0.805387
$223$	22.0000	1.47323	0.736614	−	0.676313i	$-0.236423\pi$
$223$	22.0000	1.47323	0.736614	+	0.676313i	$0.236423\pi$
$224$	−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	$0.369623\pi$
$228$	0	0
$229$	18.0000	1.18947	0.594737	−	0.803921i	$-0.297256\pi$
$229$	18.0000	1.18947	0.594737	+	0.803921i	$0.297256\pi$
$230$	8.00000	0.527504
$231$	−2.00000	−0.131590
$232$	18.0000	1.18176
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	4.00000	0.261488
$235$	20.0000	1.30466
$236$	−2.00000	−0.130189
$237$	16.0000	1.03931
$238$	−4.00000	−0.259281
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	4.00000	0.258199
$241$	−20.0000	−1.28831	−0.644157	−	0.764894i	$-0.722792\pi$
$241$	−20.0000	−1.28831	−0.644157	+	0.764894i	$0.722792\pi$
$242$	1.00000	0.0642824
$243$	−10.0000	−0.641500
$244$	0	0
$245$	−2.00000	−0.127775
$246$	8.00000	0.510061
$247$	0	0
$248$	−30.0000	−1.90500
$249$	0	0
$250$	12.0000	0.758947
$251$	−2.00000	−0.126239	−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000	−0.126239	−0.0631194	+	0.998006i	$0.520105\pi$
$252$	1.00000	0.0629941
$253$	−4.00000	−0.251478
$254$	8.00000	0.501965
$255$	−16.0000	−1.00196
$256$	−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	$0.643834\pi$
$258$	24.0000	1.49417
$259$	6.00000	0.372822
$260$	8.00000	0.496139
$261$	−6.00000	−0.371391
$262$	12.0000	0.741362
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	−6.00000	−0.369274
$265$	12.0000	0.737154
$266$	0	0
$267$	−12.0000	−0.734388
$268$	−8.00000	−0.488678
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	8.00000	0.486864
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	−4.00000	−0.242536
$273$	−8.00000	−0.484182
$274$	−10.0000	−0.604122
$275$	−1.00000	−0.0603023
$276$	8.00000	0.481543
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	−8.00000	−0.479808
$279$	10.0000	0.598684
$280$	−6.00000	−0.358569
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	−20.0000	−1.19098
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	4.00000	0.236525
$287$	−4.00000	−0.236113
$288$	5.00000	0.294628
$289$	−1.00000	−0.0588235
$290$	12.0000	0.704664
$291$	−20.0000	−1.17242
$292$	8.00000	0.468165
$293$	−24.0000	−1.40209	−0.701047	−	0.713115i	$-0.747284\pi$
$293$	−24.0000	−1.40209	−0.701047	+	0.713115i	$0.747284\pi$
$294$	2.00000	0.116642
$295$	−4.00000	−0.232889
$296$	18.0000	1.04623
$297$	−4.00000	−0.232104
$298$	−10.0000	−0.579284
$299$	−16.0000	−0.925304
$300$	2.00000	0.115470
$301$	−12.0000	−0.691669
$302$	−16.0000	−0.920697
$303$	−8.00000	−0.459588
$304$	0	0
$305$	0	0
$306$	4.00000	0.228665
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	1.00000	0.0569803
$309$	28.0000	1.59286
$310$	−20.0000	−1.13592
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	−24.0000	−1.35873
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	14.0000	0.790066
$315$	2.00000	0.112687
$316$	−8.00000	−0.450035
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	−12.0000	−0.672927
$319$	−6.00000	−0.335936
$320$	−14.0000	−0.782624
$321$	24.0000	1.33955
$322$	4.00000	0.222911
$323$	0	0
$324$	11.0000	0.611111
$325$	−4.00000	−0.221880
$326$	−8.00000	−0.443079
$327$	−28.0000	−1.54840
$328$	−12.0000	−0.662589
$329$	10.0000	0.551318
$330$	−4.00000	−0.220193
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	0	0
$333$	−6.00000	−0.328798
$334$	0	0
$335$	−16.0000	−0.874173
$336$	2.00000	0.109109
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	3.00000	0.163178
$339$	36.0000	1.95525
$340$	8.00000	0.433861
$341$	10.0000	0.541530
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−36.0000	−1.94099
$345$	16.0000	0.861411
$346$	12.0000	0.645124
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	12.0000	0.643268
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	1.00000	0.0534522
$351$	−16.0000	−0.854017
$352$	5.00000	0.266501
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	4.00000	0.212598
$355$	24.0000	1.27379
$356$	6.00000	0.317999
$357$	−8.00000	−0.423405
$358$	12.0000	0.634220
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	6.00000	0.316228
$361$	−19.0000	−1.00000
$362$	10.0000	0.525588
$363$	2.00000	0.104973
$364$	4.00000	0.209657
$365$	16.0000	0.837478
$366$	0	0
$367$	22.0000	1.14839	0.574195	−	0.818718i	$-0.305315\pi$
$367$	22.0000	1.14839	0.574195	+	0.818718i	$0.305315\pi$
$368$	4.00000	0.208514
$369$	4.00000	0.208232
$370$	12.0000	0.623850
$371$	6.00000	0.311504
$372$	−20.0000	−1.03695
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	4.00000	0.206835
$375$	24.0000	1.23935
$376$	30.0000	1.54713
$377$	−24.0000	−1.23606
$378$	4.00000	0.205738
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	8.00000	0.409316
$383$	2.00000	0.102195	0.0510976	−	0.998694i	$-0.483728\pi$
$383$	2.00000	0.102195	0.0510976	+	0.998694i	$0.483728\pi$
$384$	−6.00000	−0.306186
$385$	2.00000	0.101929
$386$	−14.0000	−0.712581
$387$	12.0000	0.609994
$388$	10.0000	0.507673
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	−16.0000	−0.810191
$391$	−16.0000	−0.809155
$392$	−3.00000	−0.151523
$393$	24.0000	1.21064
$394$	22.0000	1.10834
$395$	−16.0000	−0.805047
$396$	−1.00000	−0.0502519
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	−18.0000	−0.902258
$399$	0	0
$400$	1.00000	0.0500000
$401$	−22.0000	−1.09863	−0.549314	−	0.835616i	$-0.685111\pi$
$401$	−22.0000	−1.09863	−0.549314	+	0.835616i	$0.685111\pi$
$402$	16.0000	0.798007
$403$	40.0000	1.99254
$404$	4.00000	0.199007
$405$	22.0000	1.09319
$406$	6.00000	0.297775
$407$	−6.00000	−0.297409
$408$	−24.0000	−1.18818
$409$	24.0000	1.18672	0.593362	−	0.804936i	$-0.297800\pi$
$409$	24.0000	1.18672	0.593362	+	0.804936i	$0.297800\pi$
$410$	−8.00000	−0.395092
$411$	−20.0000	−0.986527
$412$	−14.0000	−0.689730
$413$	−2.00000	−0.0984136
$414$	−4.00000	−0.196589
$415$	0	0
$416$	20.0000	0.980581
$417$	−16.0000	−0.783523
$418$	0	0
$419$	2.00000	0.0977064	0.0488532	−	0.998806i	$-0.484443\pi$
$419$	2.00000	0.0977064	0.0488532	+	0.998806i	$0.484443\pi$
$420$	−4.00000	−0.195180
$421$	−14.0000	−0.682318	−0.341159	−	0.940006i	$-0.610819\pi$
$421$	−14.0000	−0.682318	−0.341159	+	0.940006i	$0.610819\pi$
$422$	−12.0000	−0.584151
$423$	−10.0000	−0.486217
$424$	18.0000	0.874157
$425$	−4.00000	−0.194029
$426$	−24.0000	−1.16280
$427$	0	0
$428$	−12.0000	−0.580042
$429$	8.00000	0.386244
$430$	−24.0000	−1.15738
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	4.00000	0.192450
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	−10.0000	−0.480015
$435$	24.0000	1.15071
$436$	14.0000	0.670478
$437$	0	0
$438$	−16.0000	−0.764510
$439$	−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
$441$	1.00000	0.0476190
$442$	16.0000	0.761042
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	12.0000	0.569495
$445$	12.0000	0.568855
$446$	22.0000	1.04173
$447$	−20.0000	−0.945968
$448$	−7.00000	−0.330719
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	−1.00000	−0.0471405
$451$	4.00000	0.188353
$452$	−18.0000	−0.846649
$453$	−32.0000	−1.50349
$454$	12.0000	0.563188
$455$	8.00000	0.375046
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	18.0000	0.841085
$459$	−16.0000	−0.746816
$460$	−8.00000	−0.373002
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	−2.00000	−0.0930484
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	6.00000	0.278543
$465$	−40.0000	−1.85496
$466$	−18.0000	−0.833834
$467$	30.0000	1.38823	0.694117	−	0.719862i	$-0.255795\pi$
$467$	30.0000	1.38823	0.694117	+	0.719862i	$0.255795\pi$
$468$	−4.00000	−0.184900
$469$	−8.00000	−0.369406
$470$	20.0000	0.922531
$471$	28.0000	1.29017
$472$	−6.00000	−0.276172
$473$	12.0000	0.551761
$474$	16.0000	0.734904
$475$	0	0
$476$	4.00000	0.183340
$477$	−6.00000	−0.274721
$478$	0	0
$479$	−4.00000	−0.182765	−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000	−0.182765	−0.0913823	+	0.995816i	$0.529129\pi$
$480$	−20.0000	−0.912871
$481$	−24.0000	−1.09431
$482$	−20.0000	−0.910975
$483$	8.00000	0.364013
$484$	−1.00000	−0.0454545
$485$	20.0000	0.908153
$486$	−10.0000	−0.453609
$487$	−28.0000	−1.26880	−0.634401	−	0.773004i	$-0.718753\pi$
$487$	−28.0000	−1.26880	−0.634401	+	0.773004i	$0.718753\pi$
$488$	0	0
$489$	−16.0000	−0.723545
$490$	−2.00000	−0.0903508
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	−8.00000	−0.360668
$493$	−24.0000	−1.08091
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	−10.0000	−0.449013
$497$	12.0000	0.538274
$498$	0	0
$499$	−16.0000	−0.716258	−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000	−0.716258	−0.358129	+	0.933672i	$0.616585\pi$
$500$	−12.0000	−0.536656
$501$	0	0
$502$	−2.00000	−0.0892644
$503$	4.00000	0.178351	0.0891756	−	0.996016i	$-0.471577\pi$
$503$	4.00000	0.178351	0.0891756	+	0.996016i	$0.471577\pi$
$504$	3.00000	0.133631
$505$	8.00000	0.355995
$506$	−4.00000	−0.177822
$507$	6.00000	0.266469
$508$	−8.00000	−0.354943
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	−16.0000	−0.708492
$511$	8.00000	0.353899
$512$	−11.0000	−0.486136
$513$	0	0
$514$	−14.0000	−0.617514
$515$	−28.0000	−1.23383
$516$	−24.0000	−1.05654
$517$	−10.0000	−0.439799
$518$	6.00000	0.263625
$519$	24.0000	1.05348
$520$	24.0000	1.05247
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	−6.00000	−0.262613
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	−12.0000	−0.524222
$525$	2.00000	0.0872872
$526$	8.00000	0.348817
$527$	40.0000	1.74243
$528$	−2.00000	−0.0870388
$529$	−7.00000	−0.304348
$530$	12.0000	0.521247
$531$	2.00000	0.0867926
$532$	0	0
$533$	16.0000	0.693037
$534$	−12.0000	−0.519291
$535$	−24.0000	−1.03761
$536$	−24.0000	−1.03664
$537$	24.0000	1.03568
$538$	10.0000	0.431131
$539$	1.00000	0.0430730
$540$	−8.00000	−0.344265
$541$	−26.0000	−1.11783	−0.558914	−	0.829226i	$-0.688782\pi$
$541$	−26.0000	−1.11783	−0.558914	+	0.829226i	$0.688782\pi$
$542$	−4.00000	−0.171815
$543$	20.0000	0.858282
$544$	20.0000	0.857493
$545$	28.0000	1.19939
$546$	−8.00000	−0.342368
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	10.0000	0.427179
$549$	0	0
$550$	−1.00000	−0.0426401
$551$	0	0
$552$	24.0000	1.02151
$553$	−8.00000	−0.340195
$554$	22.0000	0.934690
$555$	24.0000	1.01874
$556$	8.00000	0.339276
$557$	22.0000	0.932170	0.466085	−	0.884740i	$-0.345664\pi$
$557$	22.0000	0.932170	0.466085	+	0.884740i	$0.345664\pi$
$558$	10.0000	0.423334
$559$	48.0000	2.03018
$560$	−2.00000	−0.0845154
$561$	8.00000	0.337760
$562$	6.00000	0.253095
$563$	−32.0000	−1.34864	−0.674320	−	0.738440i	$-0.735563\pi$
$563$	−32.0000	−1.34864	−0.674320	+	0.738440i	$0.735563\pi$
$564$	20.0000	0.842152
$565$	−36.0000	−1.51453
$566$	4.00000	0.168133
$567$	11.0000	0.461957
$568$	36.0000	1.51053
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	−4.00000	−0.167248
$573$	16.0000	0.668410
$574$	−4.00000	−0.166957
$575$	4.00000	0.166812
$576$	7.00000	0.291667
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	−1.00000	−0.0415945
$579$	−28.0000	−1.16364
$580$	−12.0000	−0.498273
$581$	0	0
$582$	−20.0000	−0.829027
$583$	−6.00000	−0.248495
$584$	24.0000	0.993127
$585$	−8.00000	−0.330759
$586$	−24.0000	−0.991431
$587$	−2.00000	−0.0825488	−0.0412744	−	0.999148i	$-0.513142\pi$
$587$	−2.00000	−0.0825488	−0.0412744	+	0.999148i	$0.513142\pi$
$588$	−2.00000	−0.0824786
$589$	0	0
$590$	−4.00000	−0.164677
$591$	44.0000	1.80992
$592$	6.00000	0.246598
$593$	32.0000	1.31408	0.657041	−	0.753855i	$-0.271808\pi$
$593$	32.0000	1.31408	0.657041	+	0.753855i	$0.271808\pi$
$594$	−4.00000	−0.164122
$595$	8.00000	0.327968
$596$	10.0000	0.409616
$597$	−36.0000	−1.47338
$598$	−16.0000	−0.654289
$599$	−20.0000	−0.817178	−0.408589	−	0.912719i	$-0.633979\pi$
$599$	−20.0000	−0.817178	−0.408589	+	0.912719i	$0.633979\pi$
$600$	6.00000	0.244949
$601$	−28.0000	−1.14214	−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000	−1.14214	−0.571072	+	0.820900i	$0.693472\pi$
$602$	−12.0000	−0.489083
$603$	8.00000	0.325785
$604$	16.0000	0.651031
$605$	−2.00000	−0.0813116
$606$	−8.00000	−0.324978
$607$	−40.0000	−1.62355	−0.811775	−	0.583970i	$-0.801498\pi$
$607$	−40.0000	−1.62355	−0.811775	+	0.583970i	$0.801498\pi$
$608$	0	0
$609$	12.0000	0.486265
$610$	0	0
$611$	−40.0000	−1.61823
$612$	−4.00000	−0.161690
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	20.0000	0.807134
$615$	−16.0000	−0.645182
$616$	3.00000	0.120873
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	28.0000	1.12633
$619$	14.0000	0.562708	0.281354	−	0.959604i	$-0.409217\pi$
$619$	14.0000	0.562708	0.281354	+	0.959604i	$0.409217\pi$
$620$	20.0000	0.803219
$621$	16.0000	0.642058
$622$	−18.0000	−0.721734
$623$	6.00000	0.240385
$624$	−8.00000	−0.320256
$625$	−19.0000	−0.760000
$626$	2.00000	0.0799361
$627$	0	0
$628$	−14.0000	−0.558661
$629$	−24.0000	−0.956943
$630$	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	$0.550904\pi$
$632$	−24.0000	−0.954669
$633$	−24.0000	−0.953914
$634$	−2.00000	−0.0794301
$635$	−16.0000	−0.634941
$636$	12.0000	0.475831
$637$	4.00000	0.158486
$638$	−6.00000	−0.237542
$639$	−12.0000	−0.474713
$640$	6.00000	0.237171
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	24.0000	0.947204
$643$	14.0000	0.552106	0.276053	−	0.961142i	$-0.410973\pi$
$643$	14.0000	0.552106	0.276053	+	0.961142i	$0.410973\pi$
$644$	−4.00000	−0.157622
$645$	−48.0000	−1.89000
$646$	0	0
$647$	−22.0000	−0.864909	−0.432455	−	0.901656i	$-0.642352\pi$
$647$	−22.0000	−0.864909	−0.432455	+	0.901656i	$0.642352\pi$
$648$	33.0000	1.29636
$649$	2.00000	0.0785069
$650$	−4.00000	−0.156893
$651$	−20.0000	−0.783862
$652$	8.00000	0.313304
$653$	−26.0000	−1.01746	−0.508729	−	0.860927i	$-0.669885\pi$
$653$	−26.0000	−1.01746	−0.508729	+	0.860927i	$0.669885\pi$
$654$	−28.0000	−1.09489
$655$	−24.0000	−0.937758
$656$	−4.00000	−0.156174
$657$	−8.00000	−0.312110
$658$	10.0000	0.389841
$659$	4.00000	0.155818	0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000	0.155818	0.0779089	+	0.996960i	$0.475176\pi$
$660$	4.00000	0.155700
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	−20.0000	−0.777322
$663$	32.0000	1.24278
$664$	0	0
$665$	0	0
$666$	−6.00000	−0.232495
$667$	24.0000	0.929284
$668$	0	0
$669$	44.0000	1.70114
$670$	−16.0000	−0.618134
$671$	0	0
$672$	−10.0000	−0.385758
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	14.0000	0.539260
$675$	4.00000	0.153960
$676$	−3.00000	−0.115385
$677$	−12.0000	−0.461197	−0.230599	−	0.973049i	$-0.574068\pi$
$677$	−12.0000	−0.461197	−0.230599	+	0.973049i	$0.574068\pi$
$678$	36.0000	1.38257
$679$	10.0000	0.383765
$680$	24.0000	0.920358
$681$	24.0000	0.919682
$682$	10.0000	0.382920
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	0	0
$685$	20.0000	0.764161
$686$	−1.00000	−0.0381802
$687$	36.0000	1.37349
$688$	−12.0000	−0.457496
$689$	−24.0000	−0.914327
$690$	16.0000	0.609110
$691$	−46.0000	−1.74992	−0.874961	−	0.484193i	$-0.839113\pi$
$691$	−46.0000	−1.74992	−0.874961	+	0.484193i	$0.839113\pi$
$692$	−12.0000	−0.456172
$693$	−1.00000	−0.0379869
$694$	4.00000	0.151838
$695$	16.0000	0.606915
$696$	36.0000	1.36458
$697$	16.0000	0.606043
$698$	0	0
$699$	−36.0000	−1.36165
$700$	−1.00000	−0.0377964
$701$	−22.0000	−0.830929	−0.415464	−	0.909610i	$-0.636381\pi$
$701$	−22.0000	−0.830929	−0.415464	+	0.909610i	$0.636381\pi$
$702$	−16.0000	−0.603881
$703$	0	0
$704$	7.00000	0.263822
$705$	40.0000	1.50649
$706$	−30.0000	−1.12906
$707$	4.00000	0.150435
$708$	−4.00000	−0.150329
$709$	−34.0000	−1.27690	−0.638448	−	0.769665i	$-0.720423\pi$
$709$	−34.0000	−1.27690	−0.638448	+	0.769665i	$0.720423\pi$
$710$	24.0000	0.900704
$711$	8.00000	0.300023
$712$	18.0000	0.674579
$713$	−40.0000	−1.49801
$714$	−8.00000	−0.299392
$715$	−8.00000	−0.299183
$716$	−12.0000	−0.448461
$717$	0	0
$718$	16.0000	0.597115
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	2.00000	0.0745356
$721$	−14.0000	−0.521387
$722$	−19.0000	−0.707107
$723$	−40.0000	−1.48762
$724$	−10.0000	−0.371647
$725$	6.00000	0.222834
$726$	2.00000	0.0742270
$727$	18.0000	0.667583	0.333792	−	0.942647i	$-0.391672\pi$
$727$	18.0000	0.667583	0.333792	+	0.942647i	$0.391672\pi$
$728$	12.0000	0.444750
$729$	13.0000	0.481481
$730$	16.0000	0.592187
$731$	48.0000	1.77534
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	22.0000	0.812035
$735$	−4.00000	−0.147542
$736$	−20.0000	−0.737210
$737$	8.00000	0.294684
$738$	4.00000	0.147242
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	−12.0000	−0.441129
$741$	0	0
$742$	6.00000	0.220267
$743$	−8.00000	−0.293492	−0.146746	−	0.989174i	$-0.546880\pi$
$743$	−8.00000	−0.293492	−0.146746	+	0.989174i	$0.546880\pi$
$744$	−60.0000	−2.19971
$745$	20.0000	0.732743
$746$	−26.0000	−0.951928
$747$	0	0
$748$	−4.00000	−0.146254
$749$	−12.0000	−0.438470
$750$	24.0000	0.876356
$751$	−20.0000	−0.729810	−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000	−0.729810	−0.364905	+	0.931045i	$0.618899\pi$
$752$	10.0000	0.364662
$753$	−4.00000	−0.145768
$754$	−24.0000	−0.874028
$755$	32.0000	1.16460
$756$	−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	$0.558169\pi$
$758$	−8.00000	−0.290573
$759$	−8.00000	−0.290382
$760$	0	0
$761$	48.0000	1.74000	0.869999	−	0.493053i	$-0.164119\pi$
$761$	48.0000	1.74000	0.869999	+	0.493053i	$0.164119\pi$
$762$	16.0000	0.579619
$763$	14.0000	0.506834
$764$	−8.00000	−0.289430
$765$	−8.00000	−0.289241
$766$	2.00000	0.0722629
$767$	8.00000	0.288863
$768$	−34.0000	−1.22687
$769$	32.0000	1.15395	0.576975	−	0.816762i	$-0.304233\pi$
$769$	32.0000	1.15395	0.576975	+	0.816762i	$0.304233\pi$
$770$	2.00000	0.0720750
$771$	−28.0000	−1.00840
$772$	14.0000	0.503871
$773$	−30.0000	−1.07903	−0.539513	−	0.841978i	$-0.681391\pi$
$773$	−30.0000	−1.07903	−0.539513	+	0.841978i	$0.681391\pi$
$774$	12.0000	0.431331
$775$	−10.0000	−0.359211
$776$	30.0000	1.07694
$777$	12.0000	0.430498
$778$	6.00000	0.215110
$779$	0	0
$780$	16.0000	0.572892
$781$	−12.0000	−0.429394
$782$	−16.0000	−0.572159
$783$	24.0000	0.857690
$784$	−1.00000	−0.0357143
$785$	−28.0000	−0.999363
$786$	24.0000	0.856052
$787$	16.0000	0.570338	0.285169	−	0.958477i	$-0.407950\pi$
$787$	16.0000	0.570338	0.285169	+	0.958477i	$0.407950\pi$
$788$	−22.0000	−0.783718
$789$	16.0000	0.569615
$790$	−16.0000	−0.569254
$791$	−18.0000	−0.640006
$792$	−3.00000	−0.106600
$793$	0	0
$794$	22.0000	0.780751
$795$	24.0000	0.851192
$796$	18.0000	0.637993
$797$	14.0000	0.495905	0.247953	−	0.968772i	$-0.420242\pi$
$797$	14.0000	0.495905	0.247953	+	0.968772i	$0.420242\pi$
$798$	0	0
$799$	−40.0000	−1.41510
$800$	−5.00000	−0.176777
$801$	−6.00000	−0.212000
$802$	−22.0000	−0.776847
$803$	−8.00000	−0.282314
$804$	−16.0000	−0.564276
$805$	−8.00000	−0.281963
$806$	40.0000	1.40894
$807$	20.0000	0.704033
$808$	12.0000	0.422159
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	22.0000	0.773001
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	−6.00000	−0.210559
$813$	−8.00000	−0.280572
$814$	−6.00000	−0.210300
$815$	16.0000	0.560456
$816$	−8.00000	−0.280056
$817$	0	0
$818$	24.0000	0.839140
$819$	−4.00000	−0.139771
$820$	8.00000	0.279372
$821$	46.0000	1.60541	0.802706	−	0.596376i	$-0.203393\pi$
$821$	46.0000	1.60541	0.802706	+	0.596376i	$0.203393\pi$
$822$	−20.0000	−0.697580
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	−42.0000	−1.46314
$825$	−2.00000	−0.0696311
$826$	−2.00000	−0.0695889
$827$	−28.0000	−0.973655	−0.486828	−	0.873498i	$-0.661846\pi$
$827$	−28.0000	−0.973655	−0.486828	+	0.873498i	$0.661846\pi$
$828$	4.00000	0.139010
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	44.0000	1.52634
$832$	28.0000	0.970725
$833$	4.00000	0.138592
$834$	−16.0000	−0.554035
$835$	0	0
$836$	0	0
$837$	−40.0000	−1.38260
$838$	2.00000	0.0690889
$839$	34.0000	1.17381	0.586905	−	0.809656i	$-0.300346\pi$
$839$	34.0000	1.17381	0.586905	+	0.809656i	$0.300346\pi$
$840$	−12.0000	−0.414039
$841$	7.00000	0.241379
$842$	−14.0000	−0.482472
$843$	12.0000	0.413302
$844$	12.0000	0.413057
$845$	−6.00000	−0.206406
$846$	−10.0000	−0.343807
$847$	−1.00000	−0.0343604
$848$	6.00000	0.206041
$849$	8.00000	0.274559
$850$	−4.00000	−0.137199
$851$	24.0000	0.822709
$852$	24.0000	0.822226
$853$	44.0000	1.50653	0.753266	−	0.657716i	$-0.228477\pi$
$853$	44.0000	1.50653	0.753266	+	0.657716i	$0.228477\pi$
$854$	0	0
$855$	0	0
$856$	−36.0000	−1.23045
$857$	56.0000	1.91292	0.956462	−	0.291858i	$-0.0942733\pi$
$857$	56.0000	1.91292	0.956462	+	0.291858i	$0.0942733\pi$
$858$	8.00000	0.273115
$859$	6.00000	0.204717	0.102359	−	0.994748i	$-0.467361\pi$
$859$	6.00000	0.204717	0.102359	+	0.994748i	$0.467361\pi$
$860$	24.0000	0.818393
$861$	−8.00000	−0.272639
$862$	0	0
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	−20.0000	−0.680414
$865$	−24.0000	−0.816024
$866$	−26.0000	−0.883516
$867$	−2.00000	−0.0679236
$868$	10.0000	0.339422
$869$	8.00000	0.271381
$870$	24.0000	0.813676
$871$	32.0000	1.08428
$872$	42.0000	1.42230
$873$	−10.0000	−0.338449
$874$	0	0
$875$	−12.0000	−0.405674
$876$	16.0000	0.540590
$877$	42.0000	1.41824	0.709120	−	0.705088i	$-0.249093\pi$
$877$	42.0000	1.41824	0.709120	+	0.705088i	$0.249093\pi$
$878$	−20.0000	−0.674967
$879$	−48.0000	−1.61900
$880$	2.00000	0.0674200
$881$	−34.0000	−1.14549	−0.572745	−	0.819734i	$-0.694121\pi$
$881$	−34.0000	−1.14549	−0.572745	+	0.819734i	$0.694121\pi$
$882$	1.00000	0.0336718
$883$	28.0000	0.942275	0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000	0.942275	0.471138	+	0.882060i	$0.343844\pi$
$884$	−16.0000	−0.538138
$885$	−8.00000	−0.268917
$886$	4.00000	0.134383
$887$	−28.0000	−0.940148	−0.470074	−	0.882627i	$-0.655773\pi$
$887$	−28.0000	−0.940148	−0.470074	+	0.882627i	$0.655773\pi$
$888$	36.0000	1.20808
$889$	−8.00000	−0.268311
$890$	12.0000	0.402241
$891$	−11.0000	−0.368514
$892$	−22.0000	−0.736614
$893$	0	0
$894$	−20.0000	−0.668900
$895$	−24.0000	−0.802232
$896$	3.00000	0.100223
$897$	−32.0000	−1.06845
$898$	−10.0000	−0.333704
$899$	−60.0000	−2.00111
$900$	1.00000	0.0333333
$901$	−24.0000	−0.799556
$902$	4.00000	0.133185
$903$	−24.0000	−0.798670
$904$	−54.0000	−1.79601
$905$	−20.0000	−0.664822
$906$	−32.0000	−1.06313
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	−12.0000	−0.398234
$909$	−4.00000	−0.132672
$910$	8.00000	0.265197
$911$	36.0000	1.19273	0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000	1.19273	0.596367	+	0.802712i	$0.296610\pi$
$912$	0	0
$913$	0	0
$914$	18.0000	0.595387
$915$	0	0
$916$	−18.0000	−0.594737
$917$	−12.0000	−0.396275
$918$	−16.0000	−0.528079
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	−24.0000	−0.791257
$921$	40.0000	1.31804
$922$	0	0
$923$	−48.0000	−1.57994
$924$	2.00000	0.0657952
$925$	6.00000	0.197279
$926$	4.00000	0.131448
$927$	14.0000	0.459820
$928$	−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	$0.468619\pi$
$930$	−40.0000	−1.31165
$931$	0	0
$932$	18.0000	0.589610
$933$	−36.0000	−1.17859
$934$	30.0000	0.981630
$935$	−8.00000	−0.261628
$936$	−12.0000	−0.392232
$937$	−16.0000	−0.522697	−0.261349	−	0.965244i	$-0.584167\pi$
$937$	−16.0000	−0.522697	−0.261349	+	0.965244i	$0.584167\pi$
$938$	−8.00000	−0.261209
$939$	4.00000	0.130535
$940$	−20.0000	−0.652328
$941$	−24.0000	−0.782378	−0.391189	−	0.920310i	$-0.627936\pi$
$941$	−24.0000	−0.782378	−0.391189	+	0.920310i	$0.627936\pi$
$942$	28.0000	0.912289
$943$	−16.0000	−0.521032
$944$	−2.00000	−0.0650945
$945$	−8.00000	−0.260240
$946$	12.0000	0.390154
$947$	−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
$949$	−32.0000	−1.03876
$950$	0	0
$951$	−4.00000	−0.129709
$952$	12.0000	0.388922
$953$	34.0000	1.10137	0.550684	−	0.834714i	$-0.314367\pi$
$953$	34.0000	1.10137	0.550684	+	0.834714i	$0.314367\pi$
$954$	−6.00000	−0.194257
$955$	−16.0000	−0.517748
$956$	0	0
$957$	−12.0000	−0.387905
$958$	−4.00000	−0.129234
$959$	10.0000	0.322917
$960$	−28.0000	−0.903696
$961$	69.0000	2.22581
$962$	−24.0000	−0.773791
$963$	12.0000	0.386695
$964$	20.0000	0.644157
$965$	28.0000	0.901352
$966$	8.00000	0.257396
$967$	40.0000	1.28631	0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000	1.28631	0.643157	+	0.765735i	$0.277624\pi$
$968$	−3.00000	−0.0964237
$969$	0	0
$970$	20.0000	0.642161
$971$	14.0000	0.449281	0.224641	−	0.974442i	$-0.427879\pi$
$971$	14.0000	0.449281	0.224641	+	0.974442i	$0.427879\pi$
$972$	10.0000	0.320750
$973$	8.00000	0.256468
$974$	−28.0000	−0.897178
$975$	−8.00000	−0.256205
$976$	0	0
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	−16.0000	−0.511624
$979$	−6.00000	−0.191761
$980$	2.00000	0.0638877
$981$	−14.0000	−0.446986
$982$	28.0000	0.893516
$983$	54.0000	1.72233	0.861166	−	0.508323i	$-0.169735\pi$
$983$	54.0000	1.72233	0.861166	+	0.508323i	$0.169735\pi$
$984$	−24.0000	−0.765092
$985$	−44.0000	−1.40196
$986$	−24.0000	−0.764316
$987$	20.0000	0.636607
$988$	0	0
$989$	−48.0000	−1.52631
$990$	−2.00000	−0.0635642
$991$	52.0000	1.65183	0.825917	−	0.563791i	$-0.190658\pi$
$991$	52.0000	1.65183	0.825917	+	0.563791i	$0.190658\pi$
$992$	50.0000	1.58750
$993$	−40.0000	−1.26936
$994$	12.0000	0.380617
$995$	36.0000	1.14128
$996$	0	0
$997$	20.0000	0.633406	0.316703	−	0.948525i	$-0.397424\pi$
$997$	20.0000	0.633406	0.316703	+	0.948525i	$0.397424\pi$
$998$	−16.0000	−0.506471
$999$	24.0000	0.759326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	77.2.a.c.1.1	✓	1
3.2	odd	2		693.2.a.a.1.1		1
4.3	odd	2		1232.2.a.a.1.1		1
5.2	odd	4		1925.2.b.d.1849.2		2
5.3	odd	4		1925.2.b.d.1849.1		2
5.4	even	2		1925.2.a.c.1.1		1
7.2	even	3		539.2.e.a.67.1		2
7.3	odd	6		539.2.e.b.177.1		2
7.4	even	3		539.2.e.a.177.1		2
7.5	odd	6		539.2.e.b.67.1		2
7.6	odd	2		539.2.a.d.1.1		1
8.3	odd	2		4928.2.a.bi.1.1		1
8.5	even	2		4928.2.a.g.1.1		1
11.2	odd	10		847.2.f.k.323.1		4
11.3	even	5		847.2.f.e.372.1		4
11.4	even	5		847.2.f.e.148.1		4
11.5	even	5		847.2.f.e.729.1		4
11.6	odd	10		847.2.f.k.729.1		4
11.7	odd	10		847.2.f.k.148.1		4
11.8	odd	10		847.2.f.k.372.1		4
11.9	even	5		847.2.f.e.323.1		4
11.10	odd	2		847.2.a.a.1.1		1
21.20	even	2		4851.2.a.a.1.1		1
28.27	even	2		8624.2.a.bc.1.1		1
33.32	even	2		7623.2.a.n.1.1		1
77.76	even	2		5929.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.a.c.1.1	✓	1	1.1	even	1	trivial
539.2.a.d.1.1		1	7.6	odd	2
539.2.e.a.67.1		2	7.2	even	3
539.2.e.a.177.1		2	7.4	even	3
539.2.e.b.67.1		2	7.5	odd	6
539.2.e.b.177.1		2	7.3	odd	6
693.2.a.a.1.1		1	3.2	odd	2
847.2.a.a.1.1		1	11.10	odd	2
847.2.f.e.148.1		4	11.4	even	5
847.2.f.e.323.1		4	11.9	even	5
847.2.f.e.372.1		4	11.3	even	5
847.2.f.e.729.1		4	11.5	even	5
847.2.f.k.148.1		4	11.7	odd	10
847.2.f.k.323.1		4	11.2	odd	10
847.2.f.k.372.1		4	11.8	odd	10
847.2.f.k.729.1		4	11.6	odd	10
1232.2.a.a.1.1		1	4.3	odd	2
1925.2.a.c.1.1		1	5.4	even	2
1925.2.b.d.1849.1		2	5.3	odd	4
1925.2.b.d.1849.2		2	5.2	odd	4
4851.2.a.a.1.1		1	21.20	even	2
4928.2.a.g.1.1		1	8.5	even	2
4928.2.a.bi.1.1		1	8.3	odd	2
5929.2.a.b.1.1		1	77.76	even	2
7623.2.a.n.1.1		1	33.32	even	2
8624.2.a.bc.1.1		1	28.27	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:	\( N \)	\(=\)	\( 77 = 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	77.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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