LMFDB - Embedded newform 770.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [770,2,Mod(1,770)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(770, 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("770.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 770 = 2 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	770.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:	$6.14848095564$
Analytic rank:	$1$
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	$\chi$	$=$	770.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{11} -2.00000 q^{12} +1.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +1.00000 q^{20} +2.00000 q^{21} -1.00000 q^{22} -4.00000 q^{23} +2.00000 q^{24} +1.00000 q^{25} +4.00000 q^{27} -1.00000 q^{28} +2.00000 q^{29} +2.00000 q^{30} -2.00000 q^{31} -1.00000 q^{32} -2.00000 q^{33} -1.00000 q^{35} +1.00000 q^{36} -6.00000 q^{37} -1.00000 q^{40} +8.00000 q^{41} -2.00000 q^{42} -12.0000 q^{43} +1.00000 q^{44} +1.00000 q^{45} +4.00000 q^{46} -6.00000 q^{47} -2.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -6.00000 q^{53} -4.00000 q^{54} +1.00000 q^{55} +1.00000 q^{56} -2.00000 q^{58} -10.0000 q^{59} -2.00000 q^{60} -4.00000 q^{61} +2.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{66} -8.00000 q^{67} +8.00000 q^{69} +1.00000 q^{70} -4.00000 q^{71} -1.00000 q^{72} -4.00000 q^{73} +6.00000 q^{74} -2.00000 q^{75} -1.00000 q^{77} -16.0000 q^{79} +1.00000 q^{80} -11.0000 q^{81} -8.00000 q^{82} +2.00000 q^{84} +12.0000 q^{86} -4.00000 q^{87} -1.00000 q^{88} -6.00000 q^{89} -1.00000 q^{90} -4.00000 q^{92} +4.00000 q^{93} +6.00000 q^{94} +2.00000 q^{96} +14.0000 q^{97} -1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
$2$	−1.00000	−0.707107
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	2.00000	0.816497
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−2.00000	−0.577350
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	1.00000	0.267261
$15$	−2.00000	−0.516398
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−1.00000	−0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	1.00000	0.223607
$21$	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	$0.636929\pi$
$24$	2.00000	0.408248
$25$	1.00000	0.200000
$26$	0	0
$27$	4.00000	0.769800
$28$	−1.00000	−0.188982
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	2.00000	0.365148
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	−1.00000	−0.176777
$33$	−2.00000	−0.348155
$34$	0	0
$35$	−1.00000	−0.169031
$36$	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
$40$	−1.00000	−0.158114
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	−2.00000	−0.308607
$43$	−12.0000	−1.82998	−0.914991	−	0.403473i	$-0.867803\pi$
$43$	−12.0000	−1.82998	−0.914991	+	0.403473i	$0.867803\pi$
$44$	1.00000	0.150756
$45$	1.00000	0.149071
$46$	4.00000	0.589768
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	−2.00000	−0.288675
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	0	0
$52$	0	0
$53$	−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
$55$	1.00000	0.134840
$56$	1.00000	0.133631
$57$	0	0
$58$	−2.00000	−0.262613
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	−2.00000	−0.258199
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	2.00000	0.254000
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	2.00000	0.246183
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	8.00000	0.963087
$70$	1.00000	0.119523
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	−1.00000	−0.117851
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	6.00000	0.697486
$75$	−2.00000	−0.230940
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	1.00000	0.111803
$81$	−11.0000	−1.22222
$82$	−8.00000	−0.883452
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	12.0000	1.29399
$87$	−4.00000	−0.428845
$88$	−1.00000	−0.106600
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	−4.00000	−0.417029
$93$	4.00000	0.414781
$94$	6.00000	0.618853
$95$	0	0
$96$	2.00000	0.204124
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	−1.00000	−0.101015
$99$	1.00000	0.100504
$100$	1.00000	0.100000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	6.00000	0.582772
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	4.00000	0.384900
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	−1.00000	−0.0953463
$111$	12.0000	1.13899
$112$	−1.00000	−0.0944911
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	2.00000	0.185695
$117$	0	0
$118$	10.0000	0.920575
$119$	0	0
$120$	2.00000	0.182574
$121$	1.00000	0.0909091
$122$	4.00000	0.362143
$123$	−16.0000	−1.44267
$124$	−2.00000	−0.179605
$125$	1.00000	0.0894427
$126$	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	$0.384500\pi$
$128$	−1.00000	−0.0883883
$129$	24.0000	2.11308
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	−2.00000	−0.174078
$133$	0	0
$134$	8.00000	0.691095
$135$	4.00000	0.344265
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	−8.00000	−0.681005
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	−1.00000	−0.0845154
$141$	12.0000	1.01058
$142$	4.00000	0.335673
$143$	0	0
$144$	1.00000	0.0833333
$145$	2.00000	0.166091
$146$	4.00000	0.331042
$147$	−2.00000	−0.164957
$148$	−6.00000	−0.493197
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	2.00000	0.163299
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	0	0
$154$	1.00000	0.0805823
$155$	−2.00000	−0.160644
$156$	0	0
$157$	6.00000	0.478852	0.239426	−	0.970915i	$-0.423041\pi$
$157$	6.00000	0.478852	0.239426	+	0.970915i	$0.423041\pi$
$158$	16.0000	1.27289
$159$	12.0000	0.951662
$160$	−1.00000	−0.0790569
$161$	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	$0.601436\pi$
$164$	8.00000	0.624695
$165$	−2.00000	−0.155700
$166$	0	0
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	−2.00000	−0.154303
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	−12.0000	−0.914991
$173$	24.0000	1.82469	0.912343	−	0.409426i	$-0.134271\pi$
$173$	24.0000	1.82469	0.912343	+	0.409426i	$0.134271\pi$
$174$	4.00000	0.303239
$175$	−1.00000	−0.0755929
$176$	1.00000	0.0753778
$177$	20.0000	1.50329
$178$	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	$0.351972\pi$
$180$	1.00000	0.0745356
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	8.00000	0.591377
$184$	4.00000	0.294884
$185$	−6.00000	−0.441129
$186$	−4.00000	−0.293294
$187$	0	0
$188$	−6.00000	−0.437595
$189$	−4.00000	−0.290957
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	−2.00000	−0.144338
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	1.00000	0.0714286
$197$	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
$199$	−6.00000	−0.425329	−0.212664	−	0.977125i	$-0.568214\pi$
$199$	−6.00000	−0.425329	−0.212664	+	0.977125i	$0.568214\pi$
$200$	−1.00000	−0.0707107
$201$	16.0000	1.12855
$202$	0	0
$203$	−2.00000	−0.140372
$204$	0	0
$205$	8.00000	0.558744
$206$	14.0000	0.975426
$207$	−4.00000	−0.278019
$208$	0	0
$209$	0	0
$210$	−2.00000	−0.138013
$211$	−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
$213$	8.00000	0.548151
$214$	−12.0000	−0.820303
$215$	−12.0000	−0.818393
$216$	−4.00000	−0.272166
$217$	2.00000	0.135769
$218$	6.00000	0.406371
$219$	8.00000	0.540590
$220$	1.00000	0.0674200
$221$	0	0
$222$	−12.0000	−0.805387
$223$	−6.00000	−0.401790	−0.200895	−	0.979613i	$-0.564385\pi$
$223$	−6.00000	−0.401790	−0.200895	+	0.979613i	$0.564385\pi$
$224$	1.00000	0.0668153
$225$	1.00000	0.0666667
$226$	−2.00000	−0.133038
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	4.00000	0.263752
$231$	2.00000	0.131590
$232$	−2.00000	−0.131306
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	−6.00000	−0.391397
$236$	−10.0000	−0.650945
$237$	32.0000	2.07862
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	−2.00000	−0.129099
$241$	−16.0000	−1.03065	−0.515325	−	0.856995i	$-0.672329\pi$
$241$	−16.0000	−1.03065	−0.515325	+	0.856995i	$0.672329\pi$
$242$	−1.00000	−0.0642824
$243$	10.0000	0.641500
$244$	−4.00000	−0.256074
$245$	1.00000	0.0638877
$246$	16.0000	1.02012
$247$	0	0
$248$	2.00000	0.127000
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	10.0000	0.631194	0.315597	−	0.948893i	$-0.397795\pi$
$251$	10.0000	0.631194	0.315597	+	0.948893i	$0.397795\pi$
$252$	−1.00000	−0.0629941
$253$	−4.00000	−0.251478
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	−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
$260$	0	0
$261$	2.00000	0.123797
$262$	−4.00000	−0.247121
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	2.00000	0.123091
$265$	−6.00000	−0.368577
$266$	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	$0.401392\pi$
$270$	−4.00000	−0.243432
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	0	0
$273$	0	0
$274$	−6.00000	−0.362473
$275$	1.00000	0.0603023
$276$	8.00000	0.481543
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	−8.00000	−0.479808
$279$	−2.00000	−0.119737
$280$	1.00000	0.0597614
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	−12.0000	−0.714590
$283$	28.0000	1.66443	0.832214	−	0.554455i	$-0.187073\pi$
$283$	28.0000	1.66443	0.832214	+	0.554455i	$0.187073\pi$
$284$	−4.00000	−0.237356
$285$	0	0
$286$	0	0
$287$	−8.00000	−0.472225
$288$	−1.00000	−0.0589256
$289$	−17.0000	−1.00000
$290$	−2.00000	−0.117444
$291$	−28.0000	−1.64139
$292$	−4.00000	−0.234082
$293$	−4.00000	−0.233682	−0.116841	−	0.993151i	$-0.537277\pi$
$293$	−4.00000	−0.233682	−0.116841	+	0.993151i	$0.537277\pi$
$294$	2.00000	0.116642
$295$	−10.0000	−0.582223
$296$	6.00000	0.348743
$297$	4.00000	0.232104
$298$	−6.00000	−0.347571
$299$	0	0
$300$	−2.00000	−0.115470
$301$	12.0000	0.691669
$302$	8.00000	0.460348
$303$	0	0
$304$	0	0
$305$	−4.00000	−0.229039
$306$	0	0
$307$	28.0000	1.59804	0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000	1.59804	0.799022	+	0.601302i	$0.205351\pi$
$308$	−1.00000	−0.0569803
$309$	28.0000	1.59286
$310$	2.00000	0.113592
$311$	−6.00000	−0.340229	−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000	−0.340229	−0.170114	+	0.985424i	$0.554414\pi$
$312$	0	0
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	−6.00000	−0.338600
$315$	−1.00000	−0.0563436
$316$	−16.0000	−0.900070
$317$	−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
$319$	2.00000	0.111979
$320$	1.00000	0.0559017
$321$	−24.0000	−1.33955
$322$	−4.00000	−0.222911
$323$	0	0
$324$	−11.0000	−0.611111
$325$	0	0
$326$	8.00000	0.443079
$327$	12.0000	0.663602
$328$	−8.00000	−0.441726
$329$	6.00000	0.330791
$330$	2.00000	0.110096
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	0	0
$333$	−6.00000	−0.328798
$334$	−16.0000	−0.875481
$335$	−8.00000	−0.437087
$336$	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	$0.375472\pi$
$338$	13.0000	0.707107
$339$	−4.00000	−0.217250
$340$	0	0
$341$	−2.00000	−0.108306
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	12.0000	0.646997
$345$	8.00000	0.430706
$346$	−24.0000	−1.29025
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	−4.00000	−0.214423
$349$	−20.0000	−1.07058	−0.535288	−	0.844670i	$-0.679797\pi$
$349$	−20.0000	−1.07058	−0.535288	+	0.844670i	$0.679797\pi$
$350$	1.00000	0.0534522
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	−20.0000	−1.06299
$355$	−4.00000	−0.212298
$356$	−6.00000	−0.317999
$357$	0	0
$358$	−12.0000	−0.634220
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	−1.00000	−0.0527046
$361$	−19.0000	−1.00000
$362$	6.00000	0.315353
$363$	−2.00000	−0.104973
$364$	0	0
$365$	−4.00000	−0.209370
$366$	−8.00000	−0.418167
$367$	−22.0000	−1.14839	−0.574195	−	0.818718i	$-0.694685\pi$
$367$	−22.0000	−1.14839	−0.574195	+	0.818718i	$0.694685\pi$
$368$	−4.00000	−0.208514
$369$	8.00000	0.416463
$370$	6.00000	0.311925
$371$	6.00000	0.311504
$372$	4.00000	0.207390
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	0	0
$375$	−2.00000	−0.103280
$376$	6.00000	0.309426
$377$	0	0
$378$	4.00000	0.205738
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	8.00000	0.409316
$383$	−2.00000	−0.102195	−0.0510976	−	0.998694i	$-0.516272\pi$
$383$	−2.00000	−0.102195	−0.0510976	+	0.998694i	$0.516272\pi$
$384$	2.00000	0.102062
$385$	−1.00000	−0.0509647
$386$	22.0000	1.11977
$387$	−12.0000	−0.609994
$388$	14.0000	0.710742
$389$	22.0000	1.11544	0.557722	−	0.830028i	$-0.311675\pi$
$389$	22.0000	1.11544	0.557722	+	0.830028i	$0.311675\pi$
$390$	0	0
$391$	0	0
$392$	−1.00000	−0.0505076
$393$	−8.00000	−0.403547
$394$	−14.0000	−0.705310
$395$	−16.0000	−0.805047
$396$	1.00000	0.0502519
$397$	−34.0000	−1.70641	−0.853206	−	0.521575i	$-0.825345\pi$
$397$	−34.0000	−1.70641	−0.853206	+	0.521575i	$0.825345\pi$
$398$	6.00000	0.300753
$399$	0	0
$400$	1.00000	0.0500000
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	−16.0000	−0.798007
$403$	0	0
$404$	0	0
$405$	−11.0000	−0.546594
$406$	2.00000	0.0992583
$407$	−6.00000	−0.297409
$408$	0	0
$409$	4.00000	0.197787	0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000	0.197787	0.0988936	+	0.995098i	$0.468470\pi$
$410$	−8.00000	−0.395092
$411$	−12.0000	−0.591916
$412$	−14.0000	−0.689730
$413$	10.0000	0.492068
$414$	4.00000	0.196589
$415$	0	0
$416$	0	0
$417$	−16.0000	−0.783523
$418$	0	0
$419$	−26.0000	−1.27018	−0.635092	−	0.772437i	$-0.719038\pi$
$419$	−26.0000	−1.27018	−0.635092	+	0.772437i	$0.719038\pi$
$420$	2.00000	0.0975900
$421$	18.0000	0.877266	0.438633	−	0.898666i	$-0.355463\pi$
$421$	18.0000	0.877266	0.438633	+	0.898666i	$0.355463\pi$
$422$	12.0000	0.584151
$423$	−6.00000	−0.291730
$424$	6.00000	0.291386
$425$	0	0
$426$	−8.00000	−0.387601
$427$	4.00000	0.193574
$428$	12.0000	0.580042
$429$	0	0
$430$	12.0000	0.578691
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	4.00000	0.192450
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	−2.00000	−0.0960031
$435$	−4.00000	−0.191785
$436$	−6.00000	−0.287348
$437$	0	0
$438$	−8.00000	−0.382255
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	−1.00000	−0.0476731
$441$	1.00000	0.0476190
$442$	0	0
$443$	−20.0000	−0.950229	−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000	−0.950229	−0.475114	+	0.879924i	$0.657593\pi$
$444$	12.0000	0.569495
$445$	−6.00000	−0.284427
$446$	6.00000	0.284108
$447$	−12.0000	−0.567581
$448$	−1.00000	−0.0472456
$449$	−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
$451$	8.00000	0.376705
$452$	2.00000	0.0940721
$453$	16.0000	0.751746
$454$	−4.00000	−0.187729
$455$	0	0
$456$	0	0
$457$	26.0000	1.21623	0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000	1.21623	0.608114	+	0.793849i	$0.291926\pi$
$458$	−2.00000	−0.0934539
$459$	0	0
$460$	−4.00000	−0.186501
$461$	−20.0000	−0.931493	−0.465746	−	0.884918i	$-0.654214\pi$
$461$	−20.0000	−0.931493	−0.465746	+	0.884918i	$0.654214\pi$
$462$	−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	$0.470371\pi$
$464$	2.00000	0.0928477
$465$	4.00000	0.185496
$466$	18.0000	0.833834
$467$	−30.0000	−1.38823	−0.694117	−	0.719862i	$-0.744205\pi$
$467$	−30.0000	−1.38823	−0.694117	+	0.719862i	$0.744205\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	6.00000	0.276759
$471$	−12.0000	−0.552931
$472$	10.0000	0.460287
$473$	−12.0000	−0.551761
$474$	−32.0000	−1.46981
$475$	0	0
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	20.0000	0.913823	0.456912	−	0.889512i	$-0.348956\pi$
$479$	20.0000	0.913823	0.456912	+	0.889512i	$0.348956\pi$
$480$	2.00000	0.0912871
$481$	0	0
$482$	16.0000	0.728780
$483$	−8.00000	−0.364013
$484$	1.00000	0.0454545
$485$	14.0000	0.635707
$486$	−10.0000	−0.453609
$487$	20.0000	0.906287	0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000	0.906287	0.453143	+	0.891438i	$0.350303\pi$
$488$	4.00000	0.181071
$489$	16.0000	0.723545
$490$	−1.00000	−0.0451754
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	−16.0000	−0.721336
$493$	0	0
$494$	0	0
$495$	1.00000	0.0449467
$496$	−2.00000	−0.0898027
$497$	4.00000	0.179425
$498$	0	0
$499$	−40.0000	−1.79065	−0.895323	−	0.445418i	$-0.853055\pi$
$499$	−40.0000	−1.79065	−0.895323	+	0.445418i	$0.853055\pi$
$500$	1.00000	0.0447214
$501$	−32.0000	−1.42965
$502$	−10.0000	−0.446322
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	1.00000	0.0445435
$505$	0	0
$506$	4.00000	0.177822
$507$	26.0000	1.15470
$508$	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	$0.369386\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	14.0000	0.617514
$515$	−14.0000	−0.616914
$516$	24.0000	1.05654
$517$	−6.00000	−0.263880
$518$	−6.00000	−0.263625
$519$	−48.0000	−2.10697
$520$	0	0
$521$	−34.0000	−1.48957	−0.744784	−	0.667306i	$-0.767447\pi$
$521$	−34.0000	−1.48957	−0.744784	+	0.667306i	$0.767447\pi$
$522$	−2.00000	−0.0875376
$523$	28.0000	1.22435	0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000	1.22435	0.612177	+	0.790721i	$0.290294\pi$
$524$	4.00000	0.174741
$525$	2.00000	0.0872872
$526$	24.0000	1.04645
$527$	0	0
$528$	−2.00000	−0.0870388
$529$	−7.00000	−0.304348
$530$	6.00000	0.260623
$531$	−10.0000	−0.433963
$532$	0	0
$533$	0	0
$534$	−12.0000	−0.519291
$535$	12.0000	0.518805
$536$	8.00000	0.345547
$537$	−24.0000	−1.03568
$538$	−10.0000	−0.431131
$539$	1.00000	0.0430730
$540$	4.00000	0.172133
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	−20.0000	−0.859074
$543$	12.0000	0.514969
$544$	0	0
$545$	−6.00000	−0.257012
$546$	0	0
$547$	−44.0000	−1.88130	−0.940652	−	0.339372i	$-0.889785\pi$
$547$	−44.0000	−1.88130	−0.940652	+	0.339372i	$0.889785\pi$
$548$	6.00000	0.256307
$549$	−4.00000	−0.170716
$550$	−1.00000	−0.0426401
$551$	0	0
$552$	−8.00000	−0.340503
$553$	16.0000	0.680389
$554$	2.00000	0.0849719
$555$	12.0000	0.509372
$556$	8.00000	0.339276
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	2.00000	0.0846668
$559$	0	0
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	2.00000	0.0843649
$563$	32.0000	1.34864	0.674320	−	0.738440i	$-0.264437\pi$
$563$	32.0000	1.34864	0.674320	+	0.738440i	$0.264437\pi$
$564$	12.0000	0.505291
$565$	2.00000	0.0841406
$566$	−28.0000	−1.17693
$567$	11.0000	0.461957
$568$	4.00000	0.167836
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	16.0000	0.668410
$574$	8.00000	0.333914
$575$	−4.00000	−0.166812
$576$	1.00000	0.0416667
$577$	−26.0000	−1.08239	−0.541197	−	0.840896i	$-0.682029\pi$
$577$	−26.0000	−1.08239	−0.541197	+	0.840896i	$0.682029\pi$
$578$	17.0000	0.707107
$579$	44.0000	1.82858
$580$	2.00000	0.0830455
$581$	0	0
$582$	28.0000	1.16064
$583$	−6.00000	−0.248495
$584$	4.00000	0.165521
$585$	0	0
$586$	4.00000	0.165238
$587$	2.00000	0.0825488	0.0412744	−	0.999148i	$-0.486858\pi$
$587$	2.00000	0.0825488	0.0412744	+	0.999148i	$0.486858\pi$
$588$	−2.00000	−0.0824786
$589$	0	0
$590$	10.0000	0.411693
$591$	−28.0000	−1.15177
$592$	−6.00000	−0.246598
$593$	−12.0000	−0.492781	−0.246390	−	0.969171i	$-0.579245\pi$
$593$	−12.0000	−0.492781	−0.246390	+	0.969171i	$0.579245\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	6.00000	0.245770
$597$	12.0000	0.491127
$598$	0	0
$599$	20.0000	0.817178	0.408589	−	0.912719i	$-0.366021\pi$
$599$	20.0000	0.817178	0.408589	+	0.912719i	$0.366021\pi$
$600$	2.00000	0.0816497
$601$	−8.00000	−0.326327	−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000	−0.326327	−0.163163	+	0.986599i	$0.552170\pi$
$602$	−12.0000	−0.489083
$603$	−8.00000	−0.325785
$604$	−8.00000	−0.325515
$605$	1.00000	0.0406558
$606$	0	0
$607$	24.0000	0.974130	0.487065	−	0.873366i	$-0.338067\pi$
$607$	24.0000	0.974130	0.487065	+	0.873366i	$0.338067\pi$
$608$	0	0
$609$	4.00000	0.162088
$610$	4.00000	0.161955
$611$	0	0
$612$	0	0
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	−28.0000	−1.12999
$615$	−16.0000	−0.645182
$616$	1.00000	0.0402911
$617$	−42.0000	−1.69086	−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000	−1.69086	−0.845428	+	0.534089i	$0.820655\pi$
$618$	−28.0000	−1.12633
$619$	−22.0000	−0.884255	−0.442127	−	0.896952i	$-0.645776\pi$
$619$	−22.0000	−0.884255	−0.442127	+	0.896952i	$0.645776\pi$
$620$	−2.00000	−0.0803219
$621$	−16.0000	−0.642058
$622$	6.00000	0.240578
$623$	6.00000	0.240385
$624$	0	0
$625$	1.00000	0.0400000
$626$	−2.00000	−0.0799361
$627$	0	0
$628$	6.00000	0.239426
$629$	0	0
$630$	1.00000	0.0398410
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	16.0000	0.636446
$633$	24.0000	0.953914
$634$	2.00000	0.0794301
$635$	8.00000	0.317470
$636$	12.0000	0.475831
$637$	0	0
$638$	−2.00000	−0.0791808
$639$	−4.00000	−0.158238
$640$	−1.00000	−0.0395285
$641$	46.0000	1.81689	0.908445	−	0.418004i	$-0.137270\pi$
$641$	46.0000	1.81689	0.908445	+	0.418004i	$0.137270\pi$
$642$	24.0000	0.947204
$643$	34.0000	1.34083	0.670415	−	0.741987i	$-0.266116\pi$
$643$	34.0000	1.34083	0.670415	+	0.741987i	$0.266116\pi$
$644$	4.00000	0.157622
$645$	24.0000	0.944999
$646$	0	0
$647$	−42.0000	−1.65119	−0.825595	−	0.564263i	$-0.809160\pi$
$647$	−42.0000	−1.65119	−0.825595	+	0.564263i	$0.809160\pi$
$648$	11.0000	0.432121
$649$	−10.0000	−0.392534
$650$	0	0
$651$	−4.00000	−0.156772
$652$	−8.00000	−0.313304
$653$	22.0000	0.860927	0.430463	−	0.902608i	$-0.358350\pi$
$653$	22.0000	0.860927	0.430463	+	0.902608i	$0.358350\pi$
$654$	−12.0000	−0.469237
$655$	4.00000	0.156293
$656$	8.00000	0.312348
$657$	−4.00000	−0.156055
$658$	−6.00000	−0.233904
$659$	−20.0000	−0.779089	−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000	−0.779089	−0.389545	+	0.921008i	$0.627368\pi$
$660$	−2.00000	−0.0778499
$661$	46.0000	1.78919	0.894596	−	0.446875i	$-0.147463\pi$
$661$	46.0000	1.78919	0.894596	+	0.446875i	$0.147463\pi$
$662$	−28.0000	−1.08825
$663$	0	0
$664$	0	0
$665$	0	0
$666$	6.00000	0.232495
$667$	−8.00000	−0.309761
$668$	16.0000	0.619059
$669$	12.0000	0.463947
$670$	8.00000	0.309067
$671$	−4.00000	−0.154418
$672$	−2.00000	−0.0771517
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	−14.0000	−0.539260
$675$	4.00000	0.153960
$676$	−13.0000	−0.500000
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	4.00000	0.153619
$679$	−14.0000	−0.537271
$680$	0	0
$681$	−8.00000	−0.306561
$682$	2.00000	0.0765840
$683$	20.0000	0.765279	0.382639	−	0.923898i	$-0.375015\pi$
$683$	20.0000	0.765279	0.382639	+	0.923898i	$0.375015\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	1.00000	0.0381802
$687$	−4.00000	−0.152610
$688$	−12.0000	−0.457496
$689$	0	0
$690$	−8.00000	−0.304555
$691$	6.00000	0.228251	0.114125	−	0.993466i	$-0.463593\pi$
$691$	6.00000	0.228251	0.114125	+	0.993466i	$0.463593\pi$
$692$	24.0000	0.912343
$693$	−1.00000	−0.0379869
$694$	−28.0000	−1.06287
$695$	8.00000	0.303457
$696$	4.00000	0.151620
$697$	0	0
$698$	20.0000	0.757011
$699$	36.0000	1.36165
$700$	−1.00000	−0.0377964
$701$	−14.0000	−0.528773	−0.264386	−	0.964417i	$-0.585169\pi$
$701$	−14.0000	−0.528773	−0.264386	+	0.964417i	$0.585169\pi$
$702$	0	0
$703$	0	0
$704$	1.00000	0.0376889
$705$	12.0000	0.451946
$706$	−18.0000	−0.677439
$707$	0	0
$708$	20.0000	0.751646
$709$	30.0000	1.12667	0.563337	−	0.826227i	$-0.309517\pi$
$709$	30.0000	1.12667	0.563337	+	0.826227i	$0.309517\pi$
$710$	4.00000	0.150117
$711$	−16.0000	−0.600047
$712$	6.00000	0.224860
$713$	8.00000	0.299602
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	0	0
$718$	8.00000	0.298557
$719$	−34.0000	−1.26799	−0.633993	−	0.773339i	$-0.718585\pi$
$719$	−34.0000	−1.26799	−0.633993	+	0.773339i	$0.718585\pi$
$720$	1.00000	0.0372678
$721$	14.0000	0.521387
$722$	19.0000	0.707107
$723$	32.0000	1.19009
$724$	−6.00000	−0.222988
$725$	2.00000	0.0742781
$726$	2.00000	0.0742270
$727$	14.0000	0.519231	0.259616	−	0.965712i	$-0.416404\pi$
$727$	14.0000	0.519231	0.259616	+	0.965712i	$0.416404\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	4.00000	0.148047
$731$	0	0
$732$	8.00000	0.295689
$733$	4.00000	0.147743	0.0738717	−	0.997268i	$-0.476464\pi$
$733$	4.00000	0.147743	0.0738717	+	0.997268i	$0.476464\pi$
$734$	22.0000	0.812035
$735$	−2.00000	−0.0737711
$736$	4.00000	0.147442
$737$	−8.00000	−0.294684
$738$	−8.00000	−0.294484
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	−6.00000	−0.220564
$741$	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	$0.546880\pi$
$744$	−4.00000	−0.146647
$745$	6.00000	0.219823
$746$	−14.0000	−0.512576
$747$	0	0
$748$	0	0
$749$	−12.0000	−0.438470
$750$	2.00000	0.0730297
$751$	52.0000	1.89751	0.948753	−	0.316017i	$-0.102346\pi$
$751$	52.0000	1.89751	0.948753	+	0.316017i	$0.102346\pi$
$752$	−6.00000	−0.218797
$753$	−20.0000	−0.728841
$754$	0	0
$755$	−8.00000	−0.291150
$756$	−4.00000	−0.145479
$757$	−26.0000	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$758$	16.0000	0.581146
$759$	8.00000	0.290382
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	16.0000	0.579619
$763$	6.00000	0.217215
$764$	−8.00000	−0.289430
$765$	0	0
$766$	2.00000	0.0722629
$767$	0	0
$768$	−2.00000	−0.0721688
$769$	12.0000	0.432731	0.216366	−	0.976312i	$-0.430580\pi$
$769$	12.0000	0.432731	0.216366	+	0.976312i	$0.430580\pi$
$770$	1.00000	0.0360375
$771$	28.0000	1.00840
$772$	−22.0000	−0.791797
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	12.0000	0.431331
$775$	−2.00000	−0.0718421
$776$	−14.0000	−0.502571
$777$	−12.0000	−0.430498
$778$	−22.0000	−0.788738
$779$	0	0
$780$	0	0
$781$	−4.00000	−0.143131
$782$	0	0
$783$	8.00000	0.285897
$784$	1.00000	0.0357143
$785$	6.00000	0.214149
$786$	8.00000	0.285351
$787$	48.0000	1.71102	0.855508	−	0.517790i	$-0.173245\pi$
$787$	48.0000	1.71102	0.855508	+	0.517790i	$0.173245\pi$
$788$	14.0000	0.498729
$789$	48.0000	1.70885
$790$	16.0000	0.569254
$791$	−2.00000	−0.0711118
$792$	−1.00000	−0.0355335
$793$	0	0
$794$	34.0000	1.20661
$795$	12.0000	0.425596
$796$	−6.00000	−0.212664
$797$	6.00000	0.212531	0.106265	−	0.994338i	$-0.466111\pi$
$797$	6.00000	0.212531	0.106265	+	0.994338i	$0.466111\pi$
$798$	0	0
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	−6.00000	−0.212000
$802$	6.00000	0.211867
$803$	−4.00000	−0.141157
$804$	16.0000	0.564276
$805$	4.00000	0.140981
$806$	0	0
$807$	−20.0000	−0.704033
$808$	0	0
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	11.0000	0.386501
$811$	−44.0000	−1.54505	−0.772524	−	0.634985i	$-0.781006\pi$
$811$	−44.0000	−1.54505	−0.772524	+	0.634985i	$0.781006\pi$
$812$	−2.00000	−0.0701862
$813$	−40.0000	−1.40286
$814$	6.00000	0.210300
$815$	−8.00000	−0.280228
$816$	0	0
$817$	0	0
$818$	−4.00000	−0.139857
$819$	0	0
$820$	8.00000	0.279372
$821$	−34.0000	−1.18661	−0.593304	−	0.804978i	$-0.702177\pi$
$821$	−34.0000	−1.18661	−0.593304	+	0.804978i	$0.702177\pi$
$822$	12.0000	0.418548
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	14.0000	0.487713
$825$	−2.00000	−0.0696311
$826$	−10.0000	−0.347945
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	−4.00000	−0.139010
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	0	0
$831$	4.00000	0.138758
$832$	0	0
$833$	0	0
$834$	16.0000	0.554035
$835$	16.0000	0.553703
$836$	0	0
$837$	−8.00000	−0.276520
$838$	26.0000	0.898155
$839$	−10.0000	−0.345238	−0.172619	−	0.984989i	$-0.555223\pi$
$839$	−10.0000	−0.345238	−0.172619	+	0.984989i	$0.555223\pi$
$840$	−2.00000	−0.0690066
$841$	−25.0000	−0.862069
$842$	−18.0000	−0.620321
$843$	4.00000	0.137767
$844$	−12.0000	−0.413057
$845$	−13.0000	−0.447214
$846$	6.00000	0.206284
$847$	−1.00000	−0.0343604
$848$	−6.00000	−0.206041
$849$	−56.0000	−1.92192
$850$	0	0
$851$	24.0000	0.822709
$852$	8.00000	0.274075
$853$	−8.00000	−0.273915	−0.136957	−	0.990577i	$-0.543732\pi$
$853$	−8.00000	−0.273915	−0.136957	+	0.990577i	$0.543732\pi$
$854$	−4.00000	−0.136877
$855$	0	0
$856$	−12.0000	−0.410152
$857$	12.0000	0.409912	0.204956	−	0.978771i	$-0.434295\pi$
$857$	12.0000	0.409912	0.204956	+	0.978771i	$0.434295\pi$
$858$	0	0
$859$	2.00000	0.0682391	0.0341196	−	0.999418i	$-0.489137\pi$
$859$	2.00000	0.0682391	0.0341196	+	0.999418i	$0.489137\pi$
$860$	−12.0000	−0.409197
$861$	16.0000	0.545279
$862$	−8.00000	−0.272481
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	−4.00000	−0.136083
$865$	24.0000	0.816024
$866$	−14.0000	−0.475739
$867$	34.0000	1.15470
$868$	2.00000	0.0678844
$869$	−16.0000	−0.542763
$870$	4.00000	0.135613
$871$	0	0
$872$	6.00000	0.203186
$873$	14.0000	0.473828
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	8.00000	0.270295
$877$	10.0000	0.337676	0.168838	−	0.985644i	$-0.445999\pi$
$877$	10.0000	0.337676	0.168838	+	0.985644i	$0.445999\pi$
$878$	−4.00000	−0.134993
$879$	8.00000	0.269833
$880$	1.00000	0.0337100
$881$	−26.0000	−0.875962	−0.437981	−	0.898984i	$-0.644306\pi$
$881$	−26.0000	−0.875962	−0.437981	+	0.898984i	$0.644306\pi$
$882$	−1.00000	−0.0336718
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	0	0
$885$	20.0000	0.672293
$886$	20.0000	0.671913
$887$	−52.0000	−1.74599	−0.872995	−	0.487730i	$-0.837825\pi$
$887$	−52.0000	−1.74599	−0.872995	+	0.487730i	$0.837825\pi$
$888$	−12.0000	−0.402694
$889$	−8.00000	−0.268311
$890$	6.00000	0.201120
$891$	−11.0000	−0.368514
$892$	−6.00000	−0.200895
$893$	0	0
$894$	12.0000	0.401340
$895$	12.0000	0.401116
$896$	1.00000	0.0334077
$897$	0	0
$898$	10.0000	0.333704
$899$	−4.00000	−0.133407
$900$	1.00000	0.0333333
$901$	0	0
$902$	−8.00000	−0.266371
$903$	−24.0000	−0.798670
$904$	−2.00000	−0.0665190
$905$	−6.00000	−0.199447
$906$	−16.0000	−0.531564
$907$	48.0000	1.59381	0.796907	−	0.604102i	$-0.206468\pi$
$907$	48.0000	1.59381	0.796907	+	0.604102i	$0.206468\pi$
$908$	4.00000	0.132745
$909$	0	0
$910$	0	0
$911$	28.0000	0.927681	0.463841	−	0.885919i	$-0.346471\pi$
$911$	28.0000	0.927681	0.463841	+	0.885919i	$0.346471\pi$
$912$	0	0
$913$	0	0
$914$	−26.0000	−0.860004
$915$	8.00000	0.264472
$916$	2.00000	0.0660819
$917$	−4.00000	−0.132092
$918$	0	0
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	4.00000	0.131876
$921$	−56.0000	−1.84526
$922$	20.0000	0.658665
$923$	0	0
$924$	2.00000	0.0657952
$925$	−6.00000	−0.197279
$926$	−4.00000	−0.131448
$927$	−14.0000	−0.459820
$928$	−2.00000	−0.0656532
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	−4.00000	−0.131165
$931$	0	0
$932$	−18.0000	−0.589610
$933$	12.0000	0.392862
$934$	30.0000	0.981630
$935$	0	0
$936$	0	0
$937$	36.0000	1.17607	0.588034	−	0.808836i	$-0.299902\pi$
$937$	36.0000	1.17607	0.588034	+	0.808836i	$0.299902\pi$
$938$	−8.00000	−0.261209
$939$	−4.00000	−0.130535
$940$	−6.00000	−0.195698
$941$	−60.0000	−1.95594	−0.977972	−	0.208736i	$-0.933065\pi$
$941$	−60.0000	−1.95594	−0.977972	+	0.208736i	$0.933065\pi$
$942$	12.0000	0.390981
$943$	−32.0000	−1.04206
$944$	−10.0000	−0.325472
$945$	−4.00000	−0.130120
$946$	12.0000	0.390154
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	32.0000	1.03931
$949$	0	0
$950$	0	0
$951$	4.00000	0.129709
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	6.00000	0.194257
$955$	−8.00000	−0.258874
$956$	0	0
$957$	−4.00000	−0.129302
$958$	−20.0000	−0.646171
$959$	−6.00000	−0.193750
$960$	−2.00000	−0.0645497
$961$	−27.0000	−0.870968
$962$	0	0
$963$	12.0000	0.386695
$964$	−16.0000	−0.515325
$965$	−22.0000	−0.708205
$966$	8.00000	0.257396
$967$	−48.0000	−1.54358	−0.771788	−	0.635880i	$-0.780637\pi$
$967$	−48.0000	−1.54358	−0.771788	+	0.635880i	$0.780637\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	−14.0000	−0.449513
$971$	26.0000	0.834380	0.417190	−	0.908819i	$-0.363015\pi$
$971$	26.0000	0.834380	0.417190	+	0.908819i	$0.363015\pi$
$972$	10.0000	0.320750
$973$	−8.00000	−0.256468
$974$	−20.0000	−0.640841
$975$	0	0
$976$	−4.00000	−0.128037
$977$	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
$980$	1.00000	0.0319438
$981$	−6.00000	−0.191565
$982$	−36.0000	−1.14881
$983$	−22.0000	−0.701691	−0.350846	−	0.936433i	$-0.614106\pi$
$983$	−22.0000	−0.701691	−0.350846	+	0.936433i	$0.614106\pi$
$984$	16.0000	0.510061
$985$	14.0000	0.446077
$986$	0	0
$987$	−12.0000	−0.381964
$988$	0	0
$989$	48.0000	1.52631
$990$	−1.00000	−0.0317821
$991$	−4.00000	−0.127064	−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000	−0.127064	−0.0635321	+	0.997980i	$0.520237\pi$
$992$	2.00000	0.0635001
$993$	−56.0000	−1.77711
$994$	−4.00000	−0.126872
$995$	−6.00000	−0.190213
$996$	0	0
$997$	−48.0000	−1.52018	−0.760088	−	0.649821i	$-0.774844\pi$
$997$	−48.0000	−1.52018	−0.760088	+	0.649821i	$0.774844\pi$
$998$	40.0000	1.26618
$999$	−24.0000	−0.759326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	770.2.a.b.1.1	✓	1
3.2	odd	2		6930.2.a.s.1.1		1
4.3	odd	2		6160.2.a.p.1.1		1
5.2	odd	4		3850.2.c.p.1849.1		2
5.3	odd	4		3850.2.c.p.1849.2		2
5.4	even	2		3850.2.a.bb.1.1		1
7.6	odd	2		5390.2.a.q.1.1		1
11.10	odd	2		8470.2.a.v.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.b.1.1	✓	1	1.1	even	1	trivial
3850.2.a.bb.1.1		1	5.4	even	2
3850.2.c.p.1849.1		2	5.2	odd	4
3850.2.c.p.1849.2		2	5.3	odd	4
5390.2.a.q.1.1		1	7.6	odd	2
6160.2.a.p.1.1		1	4.3	odd	2
6930.2.a.s.1.1		1	3.2	odd	2
8470.2.a.v.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:	\( N \)	\(=\)	\( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	770.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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