LMFDB - Embedded newform 78.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [78,2,Mod(1,78)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(78, 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("78.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 78 = 2 \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	78.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:	$0.622833135766$
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	$\chi$	$=$	78.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} -4.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} -8.00000 q^{19} +2.00000 q^{20} -4.00000 q^{21} +4.00000 q^{22} +1.00000 q^{24} -1.00000 q^{25} -1.00000 q^{26} -1.00000 q^{27} +4.00000 q^{28} +6.00000 q^{29} +2.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} +4.00000 q^{33} -2.00000 q^{34} +8.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} +8.00000 q^{38} -1.00000 q^{39} -2.00000 q^{40} -10.0000 q^{41} +4.00000 q^{42} +4.00000 q^{43} -4.00000 q^{44} +2.00000 q^{45} +8.00000 q^{47} -1.00000 q^{48} +9.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} +1.00000 q^{52} -10.0000 q^{53} +1.00000 q^{54} -8.00000 q^{55} -4.00000 q^{56} +8.00000 q^{57} -6.00000 q^{58} +4.00000 q^{59} -2.00000 q^{60} -2.00000 q^{61} +4.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} -4.00000 q^{66} -16.0000 q^{67} +2.00000 q^{68} -8.00000 q^{70} -8.00000 q^{71} -1.00000 q^{72} +2.00000 q^{73} +2.00000 q^{74} +1.00000 q^{75} -8.00000 q^{76} -16.0000 q^{77} +1.00000 q^{78} +8.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} +12.0000 q^{83} -4.00000 q^{84} +4.00000 q^{85} -4.00000 q^{86} -6.00000 q^{87} +4.00000 q^{88} +14.0000 q^{89} -2.00000 q^{90} +4.00000 q^{91} +4.00000 q^{93} -8.00000 q^{94} -16.0000 q^{95} +1.00000 q^{96} +10.0000 q^{97} -9.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−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	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	−1.00000	−0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−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
$19$	−8.00000	−1.83533	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	2.00000	0.447214
$21$	−4.00000	−0.872872
$22$	4.00000	0.852803
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	1.00000	0.204124
$25$	−1.00000	−0.200000
$26$	−1.00000	−0.196116
$27$	−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
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000	−0.176777
$33$	4.00000	0.696311
$34$	−2.00000	−0.342997
$35$	8.00000	1.35225
$36$	1.00000	0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	8.00000	1.29777
$39$	−1.00000	−0.160128
$40$	−2.00000	−0.316228
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	4.00000	0.617213
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−4.00000	−0.603023
$45$	2.00000	0.298142
$46$	0	0
$47$	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
$52$	1.00000	0.138675
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	1.00000	0.136083
$55$	−8.00000	−1.07872
$56$	−4.00000	−0.534522
$57$	8.00000	1.05963
$58$	−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
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	4.00000	0.508001
$63$	4.00000	0.503953
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	−4.00000	−0.492366
$67$	−16.0000	−1.95471	−0.977356	−	0.211604i	$-0.932131\pi$
$67$	−16.0000	−1.95471	−0.977356	+	0.211604i	$0.932131\pi$
$68$	2.00000	0.242536
$69$	0	0
$70$	−8.00000	−0.956183
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	−1.00000	−0.117851
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	2.00000	0.232495
$75$	1.00000	0.115470
$76$	−8.00000	−0.917663
$77$	−16.0000	−1.82337
$78$	1.00000	0.113228
$79$	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
$81$	1.00000	0.111111
$82$	10.0000	1.10432
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	−4.00000	−0.436436
$85$	4.00000	0.433861
$86$	−4.00000	−0.431331
$87$	−6.00000	−0.643268
$88$	4.00000	0.426401
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	−2.00000	−0.210819
$91$	4.00000	0.419314
$92$	0	0
$93$	4.00000	0.414781
$94$	−8.00000	−0.825137
$95$	−16.0000	−1.64157
$96$	1.00000	0.102062
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	−9.00000	−0.909137
$99$	−4.00000	−0.402015
$100$	−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
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	−1.00000	−0.0980581
$105$	−8.00000	−0.780720
$106$	10.0000	0.971286
$107$	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	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	8.00000	0.762770
$111$	2.00000	0.189832
$112$	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	$0.591070\pi$
$114$	−8.00000	−0.749269
$115$	0	0
$116$	6.00000	0.557086
$117$	1.00000	0.0924500
$118$	−4.00000	−0.368230
$119$	8.00000	0.733359
$120$	2.00000	0.182574
$121$	5.00000	0.454545
$122$	2.00000	0.181071
$123$	10.0000	0.901670
$124$	−4.00000	−0.359211
$125$	−12.0000	−1.07331
$126$	−4.00000	−0.356348
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−1.00000	−0.0883883
$129$	−4.00000	−0.352180
$130$	−2.00000	−0.175412
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	4.00000	0.348155
$133$	−32.0000	−2.77475
$134$	16.0000	1.38219
$135$	−2.00000	−0.172133
$136$	−2.00000	−0.171499
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	8.00000	0.676123
$141$	−8.00000	−0.673722
$142$	8.00000	0.671345
$143$	−4.00000	−0.334497
$144$	1.00000	0.0833333
$145$	12.0000	0.996546
$146$	−2.00000	−0.165521
$147$	−9.00000	−0.742307
$148$	−2.00000	−0.164399
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	−1.00000	−0.0816497
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	8.00000	0.648886
$153$	2.00000	0.161690
$154$	16.0000	1.28932
$155$	−8.00000	−0.642575
$156$	−1.00000	−0.0800641
$157$	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
$159$	10.0000	0.793052
$160$	−2.00000	−0.158114
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	−10.0000	−0.780869
$165$	8.00000	0.622799
$166$	−12.0000	−0.931381
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	4.00000	0.308607
$169$	1.00000	0.0769231
$170$	−4.00000	−0.306786
$171$	−8.00000	−0.611775
$172$	4.00000	0.304997
$173$	−10.0000	−0.760286	−0.380143	−	0.924928i	$-0.624125\pi$
$173$	−10.0000	−0.760286	−0.380143	+	0.924928i	$0.624125\pi$
$174$	6.00000	0.454859
$175$	−4.00000	−0.302372
$176$	−4.00000	−0.301511
$177$	−4.00000	−0.300658
$178$	−14.0000	−1.04934
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	2.00000	0.149071
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	−4.00000	−0.296500
$183$	2.00000	0.147844
$184$	0	0
$185$	−4.00000	−0.294086
$186$	−4.00000	−0.293294
$187$	−8.00000	−0.585018
$188$	8.00000	0.583460
$189$	−4.00000	−0.290957
$190$	16.0000	1.16076
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	−1.00000	−0.0721688
$193$	−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
$195$	−2.00000	−0.143223
$196$	9.00000	0.642857
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	4.00000	0.284268
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	1.00000	0.0707107
$201$	16.0000	1.12855
$202$	2.00000	0.140720
$203$	24.0000	1.68447
$204$	−2.00000	−0.140028
$205$	−20.0000	−1.39686
$206$	−16.0000	−1.11477
$207$	0	0
$208$	1.00000	0.0693375
$209$	32.0000	2.21349
$210$	8.00000	0.552052
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	−10.0000	−0.686803
$213$	8.00000	0.548151
$214$	−12.0000	−0.820303
$215$	8.00000	0.545595
$216$	1.00000	0.0680414
$217$	−16.0000	−1.08615
$218$	2.00000	0.135457
$219$	−2.00000	−0.135147
$220$	−8.00000	−0.539360
$221$	2.00000	0.134535
$222$	−2.00000	−0.134231
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	−4.00000	−0.267261
$225$	−1.00000	−0.0666667
$226$	6.00000	0.399114
$227$	20.0000	1.32745	0.663723	−	0.747978i	$-0.268975\pi$
$227$	20.0000	1.32745	0.663723	+	0.747978i	$0.268975\pi$
$228$	8.00000	0.529813
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	0	0
$231$	16.0000	1.05272
$232$	−6.00000	−0.393919
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	−1.00000	−0.0653720
$235$	16.0000	1.04372
$236$	4.00000	0.260378
$237$	−8.00000	−0.519656
$238$	−8.00000	−0.518563
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	−2.00000	−0.129099
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	−5.00000	−0.321412
$243$	−1.00000	−0.0641500
$244$	−2.00000	−0.128037
$245$	18.0000	1.14998
$246$	−10.0000	−0.637577
$247$	−8.00000	−0.509028
$248$	4.00000	0.254000
$249$	−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
$254$	0	0
$255$	−4.00000	−0.250490
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	4.00000	0.249029
$259$	−8.00000	−0.497096
$260$	2.00000	0.124035
$261$	6.00000	0.371391
$262$	−4.00000	−0.247121
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	−4.00000	−0.246183
$265$	−20.0000	−1.22859
$266$	32.0000	1.96205
$267$	−14.0000	−0.856786
$268$	−16.0000	−0.977356
$269$	−26.0000	−1.58525	−0.792624	−	0.609711i	$-0.791286\pi$
$269$	−26.0000	−1.58525	−0.792624	+	0.609711i	$0.791286\pi$
$270$	2.00000	0.121716
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	2.00000	0.121268
$273$	−4.00000	−0.242091
$274$	10.0000	0.604122
$275$	4.00000	0.241209
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	−12.0000	−0.719712
$279$	−4.00000	−0.239474
$280$	−8.00000	−0.478091
$281$	−26.0000	−1.55103	−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−26.0000	−1.55103	−0.775515	+	0.631329i	$0.782510\pi$
$282$	8.00000	0.476393
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−8.00000	−0.474713
$285$	16.0000	0.947758
$286$	4.00000	0.236525
$287$	−40.0000	−2.36113
$288$	−1.00000	−0.0589256
$289$	−13.0000	−0.764706
$290$	−12.0000	−0.704664
$291$	−10.0000	−0.586210
$292$	2.00000	0.117041
$293$	26.0000	1.51894	0.759468	−	0.650545i	$-0.225459\pi$
$293$	26.0000	1.51894	0.759468	+	0.650545i	$0.225459\pi$
$294$	9.00000	0.524891
$295$	8.00000	0.465778
$296$	2.00000	0.116248
$297$	4.00000	0.232104
$298$	6.00000	0.347571
$299$	0	0
$300$	1.00000	0.0577350
$301$	16.0000	0.922225
$302$	−12.0000	−0.690522
$303$	2.00000	0.114897
$304$	−8.00000	−0.458831
$305$	−4.00000	−0.229039
$306$	−2.00000	−0.114332
$307$	−8.00000	−0.456584	−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000	−0.456584	−0.228292	+	0.973593i	$0.573314\pi$
$308$	−16.0000	−0.911685
$309$	−16.0000	−0.910208
$310$	8.00000	0.454369
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	1.00000	0.0566139
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	−14.0000	−0.790066
$315$	8.00000	0.450749
$316$	8.00000	0.450035
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	−10.0000	−0.560772
$319$	−24.0000	−1.34374
$320$	2.00000	0.111803
$321$	−12.0000	−0.669775
$322$	0	0
$323$	−16.0000	−0.890264
$324$	1.00000	0.0555556
$325$	−1.00000	−0.0554700
$326$	16.0000	0.886158
$327$	2.00000	0.110600
$328$	10.0000	0.552158
$329$	32.0000	1.76422
$330$	−8.00000	−0.440386
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	12.0000	0.658586
$333$	−2.00000	−0.109599
$334$	0	0
$335$	−32.0000	−1.74835
$336$	−4.00000	−0.218218
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	−1.00000	−0.0543928
$339$	6.00000	0.325875
$340$	4.00000	0.216930
$341$	16.0000	0.866449
$342$	8.00000	0.432590
$343$	8.00000	0.431959
$344$	−4.00000	−0.215666
$345$	0	0
$346$	10.0000	0.537603
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	−6.00000	−0.321634
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	4.00000	0.213809
$351$	−1.00000	−0.0533761
$352$	4.00000	0.213201
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	4.00000	0.212598
$355$	−16.0000	−0.849192
$356$	14.0000	0.741999
$357$	−8.00000	−0.423405
$358$	12.0000	0.634220
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	−2.00000	−0.105409
$361$	45.0000	2.36842
$362$	10.0000	0.525588
$363$	−5.00000	−0.262432
$364$	4.00000	0.209657
$365$	4.00000	0.209370
$366$	−2.00000	−0.104542
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	0	0
$369$	−10.0000	−0.520579
$370$	4.00000	0.207950
$371$	−40.0000	−2.07670
$372$	4.00000	0.207390
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	8.00000	0.413670
$375$	12.0000	0.619677
$376$	−8.00000	−0.412568
$377$	6.00000	0.309016
$378$	4.00000	0.205738
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	−16.0000	−0.820783
$381$	0	0
$382$	8.00000	0.409316
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	1.00000	0.0510310
$385$	−32.0000	−1.63087
$386$	14.0000	0.712581
$387$	4.00000	0.203331
$388$	10.0000	0.507673
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	2.00000	0.101274
$391$	0	0
$392$	−9.00000	−0.454569
$393$	−4.00000	−0.201773
$394$	−18.0000	−0.906827
$395$	16.0000	0.805047
$396$	−4.00000	−0.201008
$397$	6.00000	0.301131	0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000	0.301131	0.150566	+	0.988600i	$0.451890\pi$
$398$	8.00000	0.401004
$399$	32.0000	1.60200
$400$	−1.00000	−0.0500000
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	−16.0000	−0.798007
$403$	−4.00000	−0.199254
$404$	−2.00000	−0.0995037
$405$	2.00000	0.0993808
$406$	−24.0000	−1.19110
$407$	8.00000	0.396545
$408$	2.00000	0.0990148
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	20.0000	0.987730
$411$	10.0000	0.493264
$412$	16.0000	0.788263
$413$	16.0000	0.787309
$414$	0	0
$415$	24.0000	1.17811
$416$	−1.00000	−0.0490290
$417$	−12.0000	−0.587643
$418$	−32.0000	−1.56517
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	−8.00000	−0.390360
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	−12.0000	−0.584151
$423$	8.00000	0.388973
$424$	10.0000	0.485643
$425$	−2.00000	−0.0970143
$426$	−8.00000	−0.387601
$427$	−8.00000	−0.387147
$428$	12.0000	0.580042
$429$	4.00000	0.193122
$430$	−8.00000	−0.385794
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	−1.00000	−0.0481125
$433$	−30.0000	−1.44171	−0.720854	−	0.693087i	$-0.756250\pi$
$433$	−30.0000	−1.44171	−0.720854	+	0.693087i	$0.756250\pi$
$434$	16.0000	0.768025
$435$	−12.0000	−0.575356
$436$	−2.00000	−0.0957826
$437$	0	0
$438$	2.00000	0.0955637
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	8.00000	0.381385
$441$	9.00000	0.428571
$442$	−2.00000	−0.0951303
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	2.00000	0.0949158
$445$	28.0000	1.32733
$446$	4.00000	0.189405
$447$	6.00000	0.283790
$448$	4.00000	0.188982
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	1.00000	0.0471405
$451$	40.0000	1.88353
$452$	−6.00000	−0.282216
$453$	−12.0000	−0.563809
$454$	−20.0000	−0.938647
$455$	8.00000	0.375046
$456$	−8.00000	−0.374634
$457$	−30.0000	−1.40334	−0.701670	−	0.712502i	$-0.747562\pi$
$457$	−30.0000	−1.40334	−0.701670	+	0.712502i	$0.747562\pi$
$458$	−22.0000	−1.02799
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	−16.0000	−0.744387
$463$	−20.0000	−0.929479	−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000	−0.929479	−0.464739	+	0.885448i	$0.653852\pi$
$464$	6.00000	0.278543
$465$	8.00000	0.370991
$466$	−18.0000	−0.833834
$467$	−4.00000	−0.185098	−0.0925490	−	0.995708i	$-0.529501\pi$
$467$	−4.00000	−0.185098	−0.0925490	+	0.995708i	$0.529501\pi$
$468$	1.00000	0.0462250
$469$	−64.0000	−2.95525
$470$	−16.0000	−0.738025
$471$	−14.0000	−0.645086
$472$	−4.00000	−0.184115
$473$	−16.0000	−0.735681
$474$	8.00000	0.367452
$475$	8.00000	0.367065
$476$	8.00000	0.366679
$477$	−10.0000	−0.457869
$478$	0	0
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	2.00000	0.0912871
$481$	−2.00000	−0.0911922
$482$	−10.0000	−0.455488
$483$	0	0
$484$	5.00000	0.227273
$485$	20.0000	0.908153
$486$	1.00000	0.0453609
$487$	4.00000	0.181257	0.0906287	−	0.995885i	$-0.471112\pi$
$487$	4.00000	0.181257	0.0906287	+	0.995885i	$0.471112\pi$
$488$	2.00000	0.0905357
$489$	16.0000	0.723545
$490$	−18.0000	−0.813157
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	10.0000	0.450835
$493$	12.0000	0.540453
$494$	8.00000	0.359937
$495$	−8.00000	−0.359573
$496$	−4.00000	−0.179605
$497$	−32.0000	−1.43540
$498$	12.0000	0.537733
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	−12.0000	−0.536656
$501$	0	0
$502$	−4.00000	−0.178529
$503$	40.0000	1.78351	0.891756	−	0.452517i	$-0.149474\pi$
$503$	40.0000	1.78351	0.891756	+	0.452517i	$0.149474\pi$
$504$	−4.00000	−0.178174
$505$	−4.00000	−0.177998
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	42.0000	1.86162	0.930809	−	0.365507i	$-0.119104\pi$
$509$	42.0000	1.86162	0.930809	+	0.365507i	$0.119104\pi$
$510$	4.00000	0.177123
$511$	8.00000	0.353899
$512$	−1.00000	−0.0441942
$513$	8.00000	0.353209
$514$	6.00000	0.264649
$515$	32.0000	1.41009
$516$	−4.00000	−0.176090
$517$	−32.0000	−1.40736
$518$	8.00000	0.351500
$519$	10.0000	0.438951
$520$	−2.00000	−0.0877058
$521$	−14.0000	−0.613351	−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000	−0.613351	−0.306676	+	0.951814i	$0.599217\pi$
$522$	−6.00000	−0.262613
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	4.00000	0.174741
$525$	4.00000	0.174574
$526$	−8.00000	−0.348817
$527$	−8.00000	−0.348485
$528$	4.00000	0.174078
$529$	−23.0000	−1.00000
$530$	20.0000	0.868744
$531$	4.00000	0.173585
$532$	−32.0000	−1.38738
$533$	−10.0000	−0.433148
$534$	14.0000	0.605839
$535$	24.0000	1.03761
$536$	16.0000	0.691095
$537$	12.0000	0.517838
$538$	26.0000	1.12094
$539$	−36.0000	−1.55063
$540$	−2.00000	−0.0860663
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	4.00000	0.171815
$543$	10.0000	0.429141
$544$	−2.00000	−0.0857493
$545$	−4.00000	−0.171341
$546$	4.00000	0.171184
$547$	−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
$549$	−2.00000	−0.0853579
$550$	−4.00000	−0.170561
$551$	−48.0000	−2.04487
$552$	0	0
$553$	32.0000	1.36078
$554$	−22.0000	−0.934690
$555$	4.00000	0.169791
$556$	12.0000	0.508913
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	4.00000	0.169334
$559$	4.00000	0.169182
$560$	8.00000	0.338062
$561$	8.00000	0.337760
$562$	26.0000	1.09674
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	−8.00000	−0.336861
$565$	−12.0000	−0.504844
$566$	4.00000	0.168133
$567$	4.00000	0.167984
$568$	8.00000	0.335673
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	−16.0000	−0.670166
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	−4.00000	−0.167248
$573$	8.00000	0.334205
$574$	40.0000	1.66957
$575$	0	0
$576$	1.00000	0.0416667
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	13.0000	0.540729
$579$	14.0000	0.581820
$580$	12.0000	0.498273
$581$	48.0000	1.99138
$582$	10.0000	0.414513
$583$	40.0000	1.65663
$584$	−2.00000	−0.0827606
$585$	2.00000	0.0826898
$586$	−26.0000	−1.07405
$587$	4.00000	0.165098	0.0825488	−	0.996587i	$-0.473694\pi$
$587$	4.00000	0.165098	0.0825488	+	0.996587i	$0.473694\pi$
$588$	−9.00000	−0.371154
$589$	32.0000	1.31854
$590$	−8.00000	−0.329355
$591$	−18.0000	−0.740421
$592$	−2.00000	−0.0821995
$593$	−42.0000	−1.72473	−0.862367	−	0.506284i	$-0.831019\pi$
$593$	−42.0000	−1.72473	−0.862367	+	0.506284i	$0.831019\pi$
$594$	−4.00000	−0.164122
$595$	16.0000	0.655936
$596$	−6.00000	−0.245770
$597$	8.00000	0.327418
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	−1.00000	−0.0408248
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	−16.0000	−0.652111
$603$	−16.0000	−0.651570
$604$	12.0000	0.488273
$605$	10.0000	0.406558
$606$	−2.00000	−0.0812444
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	8.00000	0.324443
$609$	−24.0000	−0.972529
$610$	4.00000	0.161955
$611$	8.00000	0.323645
$612$	2.00000	0.0808452
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	8.00000	0.322854
$615$	20.0000	0.806478
$616$	16.0000	0.644658
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	16.0000	0.643614
$619$	32.0000	1.28619	0.643094	−	0.765787i	$-0.277650\pi$
$619$	32.0000	1.28619	0.643094	+	0.765787i	$0.277650\pi$
$620$	−8.00000	−0.321288
$621$	0	0
$622$	0	0
$623$	56.0000	2.24359
$624$	−1.00000	−0.0400320
$625$	−19.0000	−0.760000
$626$	6.00000	0.239808
$627$	−32.0000	−1.27796
$628$	14.0000	0.558661
$629$	−4.00000	−0.159490
$630$	−8.00000	−0.318728
$631$	−36.0000	−1.43314	−0.716569	−	0.697517i	$-0.754288\pi$
$631$	−36.0000	−1.43314	−0.716569	+	0.697517i	$0.754288\pi$
$632$	−8.00000	−0.318223
$633$	−12.0000	−0.476957
$634$	6.00000	0.238290
$635$	0	0
$636$	10.0000	0.396526
$637$	9.00000	0.356593
$638$	24.0000	0.950169
$639$	−8.00000	−0.316475
$640$	−2.00000	−0.0790569
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	12.0000	0.473602
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	0	0
$645$	−8.00000	−0.315000
$646$	16.0000	0.629512
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	−1.00000	−0.0392837
$649$	−16.0000	−0.628055
$650$	1.00000	0.0392232
$651$	16.0000	0.627089
$652$	−16.0000	−0.626608
$653$	−10.0000	−0.391330	−0.195665	−	0.980671i	$-0.562687\pi$
$653$	−10.0000	−0.391330	−0.195665	+	0.980671i	$0.562687\pi$
$654$	−2.00000	−0.0782062
$655$	8.00000	0.312586
$656$	−10.0000	−0.390434
$657$	2.00000	0.0780274
$658$	−32.0000	−1.24749
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	8.00000	0.311400
$661$	−2.00000	−0.0777910	−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000	−0.0777910	−0.0388955	+	0.999243i	$0.512384\pi$
$662$	−8.00000	−0.310929
$663$	−2.00000	−0.0776736
$664$	−12.0000	−0.465690
$665$	−64.0000	−2.48181
$666$	2.00000	0.0774984
$667$	0	0
$668$	0	0
$669$	4.00000	0.154649
$670$	32.0000	1.23627
$671$	8.00000	0.308837
$672$	4.00000	0.154303
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	−18.0000	−0.693334
$675$	1.00000	0.0384900
$676$	1.00000	0.0384615
$677$	38.0000	1.46046	0.730229	−	0.683202i	$-0.239413\pi$
$677$	38.0000	1.46046	0.730229	+	0.683202i	$0.239413\pi$
$678$	−6.00000	−0.230429
$679$	40.0000	1.53506
$680$	−4.00000	−0.153393
$681$	−20.0000	−0.766402
$682$	−16.0000	−0.612672
$683$	44.0000	1.68361	0.841807	−	0.539779i	$-0.181492\pi$
$683$	44.0000	1.68361	0.841807	+	0.539779i	$0.181492\pi$
$684$	−8.00000	−0.305888
$685$	−20.0000	−0.764161
$686$	−8.00000	−0.305441
$687$	−22.0000	−0.839352
$688$	4.00000	0.152499
$689$	−10.0000	−0.380970
$690$	0	0
$691$	−32.0000	−1.21734	−0.608669	−	0.793424i	$-0.708296\pi$
$691$	−32.0000	−1.21734	−0.608669	+	0.793424i	$0.708296\pi$
$692$	−10.0000	−0.380143
$693$	−16.0000	−0.607790
$694$	12.0000	0.455514
$695$	24.0000	0.910372
$696$	6.00000	0.227429
$697$	−20.0000	−0.757554
$698$	−6.00000	−0.227103
$699$	−18.0000	−0.680823
$700$	−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	$0.893198\pi$
$702$	1.00000	0.0377426
$703$	16.0000	0.603451
$704$	−4.00000	−0.150756
$705$	−16.0000	−0.602595
$706$	−14.0000	−0.526897
$707$	−8.00000	−0.300871
$708$	−4.00000	−0.150329
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	16.0000	0.600469
$711$	8.00000	0.300023
$712$	−14.0000	−0.524672
$713$	0	0
$714$	8.00000	0.299392
$715$	−8.00000	−0.299183
$716$	−12.0000	−0.448461
$717$	0	0
$718$	0	0
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	2.00000	0.0745356
$721$	64.0000	2.38348
$722$	−45.0000	−1.67473
$723$	−10.0000	−0.371904
$724$	−10.0000	−0.371647
$725$	−6.00000	−0.222834
$726$	5.00000	0.185567
$727$	−40.0000	−1.48352	−0.741759	−	0.670667i	$-0.766008\pi$
$727$	−40.0000	−1.48352	−0.741759	+	0.670667i	$0.766008\pi$
$728$	−4.00000	−0.148250
$729$	1.00000	0.0370370
$730$	−4.00000	−0.148047
$731$	8.00000	0.295891
$732$	2.00000	0.0739221
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	−16.0000	−0.590571
$735$	−18.0000	−0.663940
$736$	0	0
$737$	64.0000	2.35747
$738$	10.0000	0.368105
$739$	40.0000	1.47142	0.735712	−	0.677295i	$-0.236848\pi$
$739$	40.0000	1.47142	0.735712	+	0.677295i	$0.236848\pi$
$740$	−4.00000	−0.147043
$741$	8.00000	0.293887
$742$	40.0000	1.46845
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	−4.00000	−0.146647
$745$	−12.0000	−0.439646
$746$	−6.00000	−0.219676
$747$	12.0000	0.439057
$748$	−8.00000	−0.292509
$749$	48.0000	1.75388
$750$	−12.0000	−0.438178
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	8.00000	0.291730
$753$	−4.00000	−0.145768
$754$	−6.00000	−0.218507
$755$	24.0000	0.873449
$756$	−4.00000	−0.145479
$757$	54.0000	1.96266	0.981332	−	0.192323i	$-0.0616021\pi$
$757$	54.0000	1.96266	0.981332	+	0.192323i	$0.0616021\pi$
$758$	0	0
$759$	0	0
$760$	16.0000	0.580381
$761$	−26.0000	−0.942499	−0.471250	−	0.882000i	$-0.656197\pi$
$761$	−26.0000	−0.942499	−0.471250	+	0.882000i	$0.656197\pi$
$762$	0	0
$763$	−8.00000	−0.289619
$764$	−8.00000	−0.289430
$765$	4.00000	0.144620
$766$	24.0000	0.867155
$767$	4.00000	0.144432
$768$	−1.00000	−0.0360844
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	32.0000	1.15320
$771$	6.00000	0.216085
$772$	−14.0000	−0.503871
$773$	−54.0000	−1.94225	−0.971123	−	0.238581i	$-0.923318\pi$
$773$	−54.0000	−1.94225	−0.971123	+	0.238581i	$0.923318\pi$
$774$	−4.00000	−0.143777
$775$	4.00000	0.143684
$776$	−10.0000	−0.358979
$777$	8.00000	0.286998
$778$	26.0000	0.932145
$779$	80.0000	2.86630
$780$	−2.00000	−0.0716115
$781$	32.0000	1.14505
$782$	0	0
$783$	−6.00000	−0.214423
$784$	9.00000	0.321429
$785$	28.0000	0.999363
$786$	4.00000	0.142675
$787$	40.0000	1.42585	0.712923	−	0.701242i	$-0.247371\pi$
$787$	40.0000	1.42585	0.712923	+	0.701242i	$0.247371\pi$
$788$	18.0000	0.641223
$789$	−8.00000	−0.284808
$790$	−16.0000	−0.569254
$791$	−24.0000	−0.853342
$792$	4.00000	0.142134
$793$	−2.00000	−0.0710221
$794$	−6.00000	−0.212932
$795$	20.0000	0.709327
$796$	−8.00000	−0.283552
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	−32.0000	−1.13279
$799$	16.0000	0.566039
$800$	1.00000	0.0353553
$801$	14.0000	0.494666
$802$	−6.00000	−0.211867
$803$	−8.00000	−0.282314
$804$	16.0000	0.564276
$805$	0	0
$806$	4.00000	0.140894
$807$	26.0000	0.915243
$808$	2.00000	0.0703598
$809$	2.00000	0.0703163	0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000	0.0703163	0.0351581	+	0.999382i	$0.488807\pi$
$810$	−2.00000	−0.0702728
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	24.0000	0.842235
$813$	4.00000	0.140286
$814$	−8.00000	−0.280400
$815$	−32.0000	−1.12091
$816$	−2.00000	−0.0700140
$817$	−32.0000	−1.11954
$818$	−2.00000	−0.0699284
$819$	4.00000	0.139771
$820$	−20.0000	−0.698430
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	−10.0000	−0.348790
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	−16.0000	−0.557386
$825$	−4.00000	−0.139262
$826$	−16.0000	−0.556711
$827$	28.0000	0.973655	0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000	0.973655	0.486828	+	0.873498i	$0.338154\pi$
$828$	0	0
$829$	−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
$831$	−22.0000	−0.763172
$832$	1.00000	0.0346688
$833$	18.0000	0.623663
$834$	12.0000	0.415526
$835$	0	0
$836$	32.0000	1.10674
$837$	4.00000	0.138260
$838$	−4.00000	−0.138178
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	8.00000	0.276026
$841$	7.00000	0.241379
$842$	−22.0000	−0.758170
$843$	26.0000	0.895488
$844$	12.0000	0.413057
$845$	2.00000	0.0688021
$846$	−8.00000	−0.275046
$847$	20.0000	0.687208
$848$	−10.0000	−0.343401
$849$	4.00000	0.137280
$850$	2.00000	0.0685994
$851$	0	0
$852$	8.00000	0.274075
$853$	−2.00000	−0.0684787	−0.0342393	−	0.999414i	$-0.510901\pi$
$853$	−2.00000	−0.0684787	−0.0342393	+	0.999414i	$0.510901\pi$
$854$	8.00000	0.273754
$855$	−16.0000	−0.547188
$856$	−12.0000	−0.410152
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	−4.00000	−0.136558
$859$	−44.0000	−1.50126	−0.750630	−	0.660722i	$-0.770250\pi$
$859$	−44.0000	−1.50126	−0.750630	+	0.660722i	$0.770250\pi$
$860$	8.00000	0.272798
$861$	40.0000	1.36320
$862$	8.00000	0.272481
$863$	40.0000	1.36162	0.680808	−	0.732462i	$-0.261629\pi$
$863$	40.0000	1.36162	0.680808	+	0.732462i	$0.261629\pi$
$864$	1.00000	0.0340207
$865$	−20.0000	−0.680020
$866$	30.0000	1.01944
$867$	13.0000	0.441503
$868$	−16.0000	−0.543075
$869$	−32.0000	−1.08553
$870$	12.0000	0.406838
$871$	−16.0000	−0.542139
$872$	2.00000	0.0677285
$873$	10.0000	0.338449
$874$	0	0
$875$	−48.0000	−1.62270
$876$	−2.00000	−0.0675737
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	−16.0000	−0.539974
$879$	−26.0000	−0.876958
$880$	−8.00000	−0.269680
$881$	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
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	2.00000	0.0672673
$885$	−8.00000	−0.268917
$886$	4.00000	0.134383
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	−2.00000	−0.0671156
$889$	0	0
$890$	−28.0000	−0.938562
$891$	−4.00000	−0.134005
$892$	−4.00000	−0.133930
$893$	−64.0000	−2.14168
$894$	−6.00000	−0.200670
$895$	−24.0000	−0.802232
$896$	−4.00000	−0.133631
$897$	0	0
$898$	−6.00000	−0.200223
$899$	−24.0000	−0.800445
$900$	−1.00000	−0.0333333
$901$	−20.0000	−0.666297
$902$	−40.0000	−1.33185
$903$	−16.0000	−0.532447
$904$	6.00000	0.199557
$905$	−20.0000	−0.664822
$906$	12.0000	0.398673
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	20.0000	0.663723
$909$	−2.00000	−0.0663358
$910$	−8.00000	−0.265197
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	8.00000	0.264906
$913$	−48.0000	−1.58857
$914$	30.0000	0.992312
$915$	4.00000	0.132236
$916$	22.0000	0.726900
$917$	16.0000	0.528367
$918$	2.00000	0.0660098
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	8.00000	0.263609
$922$	6.00000	0.197599
$923$	−8.00000	−0.263323
$924$	16.0000	0.526361
$925$	2.00000	0.0657596
$926$	20.0000	0.657241
$927$	16.0000	0.525509
$928$	−6.00000	−0.196960
$929$	46.0000	1.50921	0.754606	−	0.656179i	$-0.227828\pi$
$929$	46.0000	1.50921	0.754606	+	0.656179i	$0.227828\pi$
$930$	−8.00000	−0.262330
$931$	−72.0000	−2.35970
$932$	18.0000	0.589610
$933$	0	0
$934$	4.00000	0.130884
$935$	−16.0000	−0.523256
$936$	−1.00000	−0.0326860
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	64.0000	2.08967
$939$	6.00000	0.195803
$940$	16.0000	0.521862
$941$	−46.0000	−1.49956	−0.749779	−	0.661689i	$-0.769840\pi$
$941$	−46.0000	−1.49956	−0.749779	+	0.661689i	$0.769840\pi$
$942$	14.0000	0.456145
$943$	0	0
$944$	4.00000	0.130189
$945$	−8.00000	−0.260240
$946$	16.0000	0.520205
$947$	−4.00000	−0.129983	−0.0649913	−	0.997886i	$-0.520702\pi$
$947$	−4.00000	−0.129983	−0.0649913	+	0.997886i	$0.520702\pi$
$948$	−8.00000	−0.259828
$949$	2.00000	0.0649227
$950$	−8.00000	−0.259554
$951$	6.00000	0.194563
$952$	−8.00000	−0.259281
$953$	−30.0000	−0.971795	−0.485898	−	0.874016i	$-0.661507\pi$
$953$	−30.0000	−0.971795	−0.485898	+	0.874016i	$0.661507\pi$
$954$	10.0000	0.323762
$955$	−16.0000	−0.517748
$956$	0	0
$957$	24.0000	0.775810
$958$	16.0000	0.516937
$959$	−40.0000	−1.29167
$960$	−2.00000	−0.0645497
$961$	−15.0000	−0.483871
$962$	2.00000	0.0644826
$963$	12.0000	0.386695
$964$	10.0000	0.322078
$965$	−28.0000	−0.901352
$966$	0	0
$967$	4.00000	0.128631	0.0643157	−	0.997930i	$-0.479514\pi$
$967$	4.00000	0.128631	0.0643157	+	0.997930i	$0.479514\pi$
$968$	−5.00000	−0.160706
$969$	16.0000	0.513994
$970$	−20.0000	−0.642161
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	−1.00000	−0.0320750
$973$	48.0000	1.53881
$974$	−4.00000	−0.128168
$975$	1.00000	0.0320256
$976$	−2.00000	−0.0640184
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	−16.0000	−0.511624
$979$	−56.0000	−1.78977
$980$	18.0000	0.574989
$981$	−2.00000	−0.0638551
$982$	−36.0000	−1.14881
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	−10.0000	−0.318788
$985$	36.0000	1.14706
$986$	−12.0000	−0.382158
$987$	−32.0000	−1.01857
$988$	−8.00000	−0.254514
$989$	0	0
$990$	8.00000	0.254257
$991$	−48.0000	−1.52477	−0.762385	−	0.647124i	$-0.775972\pi$
$991$	−48.0000	−1.52477	−0.762385	+	0.647124i	$0.775972\pi$
$992$	4.00000	0.127000
$993$	−8.00000	−0.253872
$994$	32.0000	1.01498
$995$	−16.0000	−0.507234
$996$	−12.0000	−0.380235
$997$	−26.0000	−0.823428	−0.411714	−	0.911313i	$-0.635070\pi$
$997$	−26.0000	−0.823428	−0.411714	+	0.911313i	$0.635070\pi$
$998$	0	0
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	78.2.a.a.1.1	✓	1
3.2	odd	2		234.2.a.c.1.1		1
4.3	odd	2		624.2.a.h.1.1		1
5.2	odd	4		1950.2.e.i.1249.1		2
5.3	odd	4		1950.2.e.i.1249.2		2
5.4	even	2		1950.2.a.w.1.1		1
7.6	odd	2		3822.2.a.j.1.1		1
8.3	odd	2		2496.2.a.b.1.1		1
8.5	even	2		2496.2.a.t.1.1		1
9.2	odd	6		2106.2.e.j.1405.1		2
9.4	even	3		2106.2.e.q.703.1		2
9.5	odd	6		2106.2.e.j.703.1		2
9.7	even	3		2106.2.e.q.1405.1		2
11.10	odd	2		9438.2.a.t.1.1		1
12.11	even	2		1872.2.a.c.1.1		1
13.2	odd	12		1014.2.i.d.823.1		4
13.3	even	3		1014.2.e.f.529.1		2
13.4	even	6		1014.2.e.c.991.1		2
13.5	odd	4		1014.2.b.b.337.2		2
13.6	odd	12		1014.2.i.d.361.2		4
13.7	odd	12		1014.2.i.d.361.1		4
13.8	odd	4		1014.2.b.b.337.1		2
13.9	even	3		1014.2.e.f.991.1		2
13.10	even	6		1014.2.e.c.529.1		2
13.11	odd	12		1014.2.i.d.823.2		4
13.12	even	2		1014.2.a.d.1.1		1
15.2	even	4		5850.2.e.bb.5149.2		2
15.8	even	4		5850.2.e.bb.5149.1		2
15.14	odd	2		5850.2.a.d.1.1		1
24.5	odd	2		7488.2.a.bz.1.1		1
24.11	even	2		7488.2.a.bk.1.1		1
39.5	even	4		3042.2.b.g.1351.1		2
39.8	even	4		3042.2.b.g.1351.2		2
39.38	odd	2		3042.2.a.f.1.1		1
52.51	odd	2		8112.2.a.v.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.2.a.a.1.1	✓	1	1.1	even	1	trivial
234.2.a.c.1.1		1	3.2	odd	2
624.2.a.h.1.1		1	4.3	odd	2
1014.2.a.d.1.1		1	13.12	even	2
1014.2.b.b.337.1		2	13.8	odd	4
1014.2.b.b.337.2		2	13.5	odd	4
1014.2.e.c.529.1		2	13.10	even	6
1014.2.e.c.991.1		2	13.4	even	6
1014.2.e.f.529.1		2	13.3	even	3
1014.2.e.f.991.1		2	13.9	even	3
1014.2.i.d.361.1		4	13.7	odd	12
1014.2.i.d.361.2		4	13.6	odd	12
1014.2.i.d.823.1		4	13.2	odd	12
1014.2.i.d.823.2		4	13.11	odd	12
1872.2.a.c.1.1		1	12.11	even	2
1950.2.a.w.1.1		1	5.4	even	2
1950.2.e.i.1249.1		2	5.2	odd	4
1950.2.e.i.1249.2		2	5.3	odd	4
2106.2.e.j.703.1		2	9.5	odd	6
2106.2.e.j.1405.1		2	9.2	odd	6
2106.2.e.q.703.1		2	9.4	even	3
2106.2.e.q.1405.1		2	9.7	even	3
2496.2.a.b.1.1		1	8.3	odd	2
2496.2.a.t.1.1		1	8.5	even	2
3042.2.a.f.1.1		1	39.38	odd	2
3042.2.b.g.1351.1		2	39.5	even	4
3042.2.b.g.1351.2		2	39.8	even	4
3822.2.a.j.1.1		1	7.6	odd	2
5850.2.a.d.1.1		1	15.14	odd	2
5850.2.e.bb.5149.1		2	15.8	even	4
5850.2.e.bb.5149.2		2	15.2	even	4
7488.2.a.bk.1.1		1	24.11	even	2
7488.2.a.bz.1.1		1	24.5	odd	2
8112.2.a.v.1.1		1	52.51	odd	2
9438.2.a.t.1.1		1	11.10	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 78 = 2 \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	78.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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