LMFDB - Embedded newform 8470.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8470,2,Mod(1,8470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8470, 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("8470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8470.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:	$67.6332905120$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8470.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -2.00000 q^{12} +4.00000 q^{13} +1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} +2.00000 q^{21} +2.00000 q^{24} +1.00000 q^{25} -4.00000 q^{26} +4.00000 q^{27} -1.00000 q^{28} +6.00000 q^{29} -2.00000 q^{30} -10.0000 q^{31} -1.00000 q^{32} +1.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} -4.00000 q^{38} -8.00000 q^{39} +1.00000 q^{40} +12.0000 q^{41} -2.00000 q^{42} +4.00000 q^{43} -1.00000 q^{45} +6.00000 q^{47} -2.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +4.00000 q^{52} -6.00000 q^{53} -4.00000 q^{54} +1.00000 q^{56} -8.00000 q^{57} -6.00000 q^{58} -6.00000 q^{59} +2.00000 q^{60} +4.00000 q^{61} +10.0000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} -4.00000 q^{67} -1.00000 q^{70} +12.0000 q^{71} -1.00000 q^{72} +4.00000 q^{73} -2.00000 q^{74} -2.00000 q^{75} +4.00000 q^{76} +8.00000 q^{78} -8.00000 q^{79} -1.00000 q^{80} -11.0000 q^{81} -12.0000 q^{82} -12.0000 q^{83} +2.00000 q^{84} -4.00000 q^{86} -12.0000 q^{87} +18.0000 q^{89} +1.00000 q^{90} -4.00000 q^{91} +20.0000 q^{93} -6.00000 q^{94} -4.00000 q^{95} +2.00000 q^{96} -10.0000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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
$11$	0	0
$11$	0	0
$12$	−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
$15$	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
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	−1.00000	−0.223607
$21$	2.00000	0.436436
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	2.00000	0.408248
$25$	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	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	−2.00000	−0.365148
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0	0
$35$	1.00000	0.169031
$36$	1.00000	0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−4.00000	−0.648886
$39$	−8.00000	−1.28103
$40$	1.00000	0.158114
$41$	12.0000	1.87409	0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000	1.87409	0.937043	+	0.349215i	$0.113552\pi$
$42$	−2.00000	−0.308607
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	−2.00000	−0.288675
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	0	0
$52$	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
$55$	0	0
$56$	1.00000	0.133631
$57$	−8.00000	−1.05963
$58$	−6.00000	−0.787839
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	2.00000	0.258199
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	10.0000	1.27000
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	0	0
$70$	−1.00000	−0.119523
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	−1.00000	−0.117851
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	−2.00000	−0.232495
$75$	−2.00000	−0.230940
$76$	4.00000	0.458831
$77$	0	0
$78$	8.00000	0.905822
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	−1.00000	−0.111803
$81$	−11.0000	−1.22222
$82$	−12.0000	−1.32518
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	−4.00000	−0.431331
$87$	−12.0000	−1.28654
$88$	0	0
$89$	18.0000	1.90800	0.953998	−	0.299813i	$-0.0969242\pi$
$89$	18.0000	1.90800	0.953998	+	0.299813i	$0.0969242\pi$
$90$	1.00000	0.105409
$91$	−4.00000	−0.419314
$92$	0	0
$93$	20.0000	2.07390
$94$	−6.00000	−0.618853
$95$	−4.00000	−0.410391
$96$	2.00000	0.204124
$97$	−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
$99$	0	0
$100$	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	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	−4.00000	−0.392232
$105$	−2.00000	−0.195180
$106$	6.00000	0.582772
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	4.00000	0.384900
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	−1.00000	−0.0944911
$113$	−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
$117$	4.00000	0.369800
$118$	6.00000	0.552345
$119$	0	0
$120$	−2.00000	−0.182574
$121$	0	0
$122$	−4.00000	−0.362143
$123$	−24.0000	−2.16401
$124$	−10.0000	−0.898027
$125$	−1.00000	−0.0894427
$126$	1.00000	0.0890871
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	−1.00000	−0.0883883
$129$	−8.00000	−0.704361
$130$	4.00000	0.350823
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−4.00000	−0.346844
$134$	4.00000	0.345547
$135$	−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	$0.417494\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	1.00000	0.0845154
$141$	−12.0000	−1.01058
$142$	−12.0000	−1.00702
$143$	0	0
$144$	1.00000	0.0833333
$145$	−6.00000	−0.498273
$146$	−4.00000	−0.331042
$147$	−2.00000	−0.164957
$148$	2.00000	0.164399
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	2.00000	0.163299
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	−4.00000	−0.324443
$153$	0	0
$154$	0	0
$155$	10.0000	0.803219
$156$	−8.00000	−0.640513
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	8.00000	0.636446
$159$	12.0000	0.951662
$160$	1.00000	0.0790569
$161$	0	0
$162$	11.0000	0.864242
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	12.0000	0.937043
$165$	0	0
$166$	12.0000	0.931381
$167$	−24.0000	−1.85718	−0.928588	−	0.371113i	$-0.878976\pi$
$167$	−24.0000	−1.85718	−0.928588	+	0.371113i	$0.878976\pi$
$168$	−2.00000	−0.154303
$169$	3.00000	0.230769
$170$	0	0
$171$	4.00000	0.305888
$172$	4.00000	0.304997
$173$	−12.0000	−0.912343	−0.456172	−	0.889892i	$-0.650780\pi$
$173$	−12.0000	−0.912343	−0.456172	+	0.889892i	$0.650780\pi$
$174$	12.0000	0.909718
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	12.0000	0.901975
$178$	−18.0000	−1.34916
$179$	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
$181$	−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
$183$	−8.00000	−0.591377
$184$	0	0
$185$	−2.00000	−0.147043
$186$	−20.0000	−1.46647
$187$	0	0
$188$	6.00000	0.437595
$189$	−4.00000	−0.290957
$190$	4.00000	0.290191
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−2.00000	−0.144338
$193$	−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
$195$	8.00000	0.572892
$196$	1.00000	0.0714286
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	−1.00000	−0.0707107
$201$	8.00000	0.564276
$202$	0	0
$203$	−6.00000	−0.421117
$204$	0	0
$205$	−12.0000	−0.838116
$206$	−14.0000	−0.975426
$207$	0	0
$208$	4.00000	0.277350
$209$	0	0
$210$	2.00000	0.138013
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	−6.00000	−0.412082
$213$	−24.0000	−1.64445
$214$	12.0000	0.820303
$215$	−4.00000	−0.272798
$216$	−4.00000	−0.272166
$217$	10.0000	0.678844
$218$	2.00000	0.135457
$219$	−8.00000	−0.540590
$220$	0	0
$221$	0	0
$222$	4.00000	0.268462
$223$	−10.0000	−0.669650	−0.334825	−	0.942280i	$-0.608677\pi$
$223$	−10.0000	−0.669650	−0.334825	+	0.942280i	$0.608677\pi$
$224$	1.00000	0.0668153
$225$	1.00000	0.0666667
$226$	6.00000	0.399114
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	−8.00000	−0.529813
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	0	0
$232$	−6.00000	−0.393919
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	−4.00000	−0.261488
$235$	−6.00000	−0.391397
$236$	−6.00000	−0.390567
$237$	16.0000	1.03931
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	2.00000	0.129099
$241$	4.00000	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$242$	0	0
$243$	10.0000	0.641500
$244$	4.00000	0.256074
$245$	−1.00000	−0.0638877
$246$	24.0000	1.53018
$247$	16.0000	1.01806
$248$	10.0000	0.635001
$249$	24.0000	1.52094
$250$	1.00000	0.0632456
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	−1.00000	−0.0629941
$253$	0	0
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	8.00000	0.498058
$259$	−2.00000	−0.124274
$260$	−4.00000	−0.248069
$261$	6.00000	0.371391
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	4.00000	0.245256
$267$	−36.0000	−2.20316
$268$	−4.00000	−0.244339
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	4.00000	0.243432
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	0	0
$273$	8.00000	0.484182
$274$	−6.00000	−0.362473
$275$	0	0
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	−4.00000	−0.239904
$279$	−10.0000	−0.598684
$280$	−1.00000	−0.0597614
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	12.0000	0.714590
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	12.0000	0.712069
$285$	8.00000	0.473879
$286$	0	0
$287$	−12.0000	−0.708338
$288$	−1.00000	−0.0589256
$289$	−17.0000	−1.00000
$290$	6.00000	0.352332
$291$	20.0000	1.17242
$292$	4.00000	0.234082
$293$	−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
$295$	6.00000	0.349334
$296$	−2.00000	−0.116248
$297$	0	0
$298$	−18.0000	−1.04271
$299$	0	0
$300$	−2.00000	−0.115470
$301$	−4.00000	−0.230556
$302$	−16.0000	−0.920697
$303$	0	0
$304$	4.00000	0.229416
$305$	−4.00000	−0.229039
$306$	0	0
$307$	−32.0000	−1.82634	−0.913168	−	0.407583i	$-0.866372\pi$
$307$	−32.0000	−1.82634	−0.913168	+	0.407583i	$0.866372\pi$
$308$	0	0
$309$	−28.0000	−1.59286
$310$	−10.0000	−0.567962
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	8.00000	0.452911
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	22.0000	1.24153
$315$	1.00000	0.0563436
$316$	−8.00000	−0.450035
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	−12.0000	−0.672927
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	24.0000	1.33955
$322$	0	0
$323$	0	0
$324$	−11.0000	−0.611111
$325$	4.00000	0.221880
$326$	4.00000	0.221540
$327$	4.00000	0.221201
$328$	−12.0000	−0.662589
$329$	−6.00000	−0.330791
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	−12.0000	−0.658586
$333$	2.00000	0.109599
$334$	24.0000	1.31322
$335$	4.00000	0.218543
$336$	2.00000	0.109109
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	−3.00000	−0.163178
$339$	12.0000	0.651751
$340$	0	0
$341$	0	0
$342$	−4.00000	−0.216295
$343$	−1.00000	−0.0539949
$344$	−4.00000	−0.215666
$345$	0	0
$346$	12.0000	0.645124
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	−12.0000	−0.643268
$349$	−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
$351$	16.0000	0.854017
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	−12.0000	−0.637793
$355$	−12.0000	−0.636894
$356$	18.0000	0.953998
$357$	0	0
$358$	−12.0000	−0.634220
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	1.00000	0.0527046
$361$	−3.00000	−0.157895
$362$	10.0000	0.525588
$363$	0	0
$364$	−4.00000	−0.209657
$365$	−4.00000	−0.209370
$366$	8.00000	0.418167
$367$	−34.0000	−1.77479	−0.887393	−	0.461014i	$-0.847486\pi$
$367$	−34.0000	−1.77479	−0.887393	+	0.461014i	$0.847486\pi$
$368$	0	0
$369$	12.0000	0.624695
$370$	2.00000	0.103975
$371$	6.00000	0.311504
$372$	20.0000	1.03695
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	0	0
$375$	2.00000	0.103280
$376$	−6.00000	−0.309426
$377$	24.0000	1.23606
$378$	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	$0.634797\pi$
$380$	−4.00000	−0.205196
$381$	16.0000	0.819705
$382$	0	0
$383$	−6.00000	−0.306586	−0.153293	−	0.988181i	$-0.548988\pi$
$383$	−6.00000	−0.306586	−0.153293	+	0.988181i	$0.548988\pi$
$384$	2.00000	0.102062
$385$	0	0
$386$	14.0000	0.712581
$387$	4.00000	0.203331
$388$	−10.0000	−0.507673
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	−8.00000	−0.405096
$391$	0	0
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−6.00000	−0.302276
$395$	8.00000	0.402524
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	−2.00000	−0.100251
$399$	8.00000	0.400501
$400$	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	$0.547867\pi$
$402$	−8.00000	−0.399004
$403$	−40.0000	−1.99254
$404$	0	0
$405$	11.0000	0.546594
$406$	6.00000	0.297775
$407$	0	0
$408$	0	0
$409$	−32.0000	−1.58230	−0.791149	−	0.611623i	$-0.790517\pi$
$409$	−32.0000	−1.58230	−0.791149	+	0.611623i	$0.790517\pi$
$410$	12.0000	0.592638
$411$	−12.0000	−0.591916
$412$	14.0000	0.689730
$413$	6.00000	0.295241
$414$	0	0
$415$	12.0000	0.589057
$416$	−4.00000	−0.196116
$417$	−8.00000	−0.391762
$418$	0	0
$419$	−6.00000	−0.293119	−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000	−0.293119	−0.146560	+	0.989202i	$0.546820\pi$
$420$	−2.00000	−0.0975900
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	−4.00000	−0.194717
$423$	6.00000	0.291730
$424$	6.00000	0.291386
$425$	0	0
$426$	24.0000	1.16280
$427$	−4.00000	−0.193574
$428$	−12.0000	−0.580042
$429$	0	0
$430$	4.00000	0.192897
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	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	$0.390791\pi$
$434$	−10.0000	−0.480015
$435$	12.0000	0.575356
$436$	−2.00000	−0.0957826
$437$	0	0
$438$	8.00000	0.382255
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	−4.00000	−0.189832
$445$	−18.0000	−0.853282
$446$	10.0000	0.473514
$447$	−36.0000	−1.70274
$448$	−1.00000	−0.0472456
$449$	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
$451$	0	0
$452$	−6.00000	−0.282216
$453$	−32.0000	−1.50349
$454$	−24.0000	−1.12638
$455$	4.00000	0.187523
$456$	8.00000	0.374634
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	10.0000	0.467269
$459$	0	0
$460$	0	0
$461$	36.0000	1.67669	0.838344	−	0.545142i	$-0.183524\pi$
$461$	36.0000	1.67669	0.838344	+	0.545142i	$0.183524\pi$
$462$	0	0
$463$	−40.0000	−1.85896	−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000	−1.85896	−0.929479	+	0.368875i	$0.879743\pi$
$464$	6.00000	0.278543
$465$	−20.0000	−0.927478
$466$	−6.00000	−0.277945
$467$	42.0000	1.94353	0.971764	−	0.235954i	$-0.0758216\pi$
$467$	42.0000	1.94353	0.971764	+	0.235954i	$0.0758216\pi$
$468$	4.00000	0.184900
$469$	4.00000	0.184703
$470$	6.00000	0.276759
$471$	44.0000	2.02741
$472$	6.00000	0.276172
$473$	0	0
$474$	−16.0000	−0.734904
$475$	4.00000	0.183533
$476$	0	0
$477$	−6.00000	−0.274721
$478$	−24.0000	−1.09773
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	−2.00000	−0.0912871
$481$	8.00000	0.364769
$482$	−4.00000	−0.182195
$483$	0	0
$484$	0	0
$485$	10.0000	0.454077
$486$	−10.0000	−0.453609
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	−4.00000	−0.181071
$489$	8.00000	0.361773
$490$	1.00000	0.0451754
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	−24.0000	−1.08200
$493$	0	0
$494$	−16.0000	−0.719874
$495$	0	0
$496$	−10.0000	−0.449013
$497$	−12.0000	−0.538274
$498$	−24.0000	−1.07547
$499$	−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
$501$	48.0000	2.14448
$502$	18.0000	0.803379
$503$	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
$506$	0	0
$507$	−6.00000	−0.266469
$508$	−8.00000	−0.354943
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	−4.00000	−0.176950
$512$	−1.00000	−0.0441942
$513$	16.0000	0.706417
$514$	−18.0000	−0.793946
$515$	−14.0000	−0.616914
$516$	−8.00000	−0.352180
$517$	0	0
$518$	2.00000	0.0878750
$519$	24.0000	1.05348
$520$	4.00000	0.175412
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	−6.00000	−0.262613
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	0	0
$525$	2.00000	0.0872872
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−23.0000	−1.00000
$530$	−6.00000	−0.260623
$531$	−6.00000	−0.260378
$532$	−4.00000	−0.173422
$533$	48.0000	2.07911
$534$	36.0000	1.55787
$535$	12.0000	0.518805
$536$	4.00000	0.172774
$537$	−24.0000	−1.03568
$538$	−6.00000	−0.258678
$539$	0	0
$540$	−4.00000	−0.172133
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	20.0000	0.859074
$543$	20.0000	0.858282
$544$	0	0
$545$	2.00000	0.0856706
$546$	−8.00000	−0.342368
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	6.00000	0.256307
$549$	4.00000	0.170716
$550$	0	0
$551$	24.0000	1.02243
$552$	0	0
$553$	8.00000	0.340195
$554$	−22.0000	−0.934690
$555$	4.00000	0.169791
$556$	4.00000	0.169638
$557$	6.00000	0.254228	0.127114	−	0.991888i	$-0.459429\pi$
$557$	6.00000	0.254228	0.127114	+	0.991888i	$0.459429\pi$
$558$	10.0000	0.423334
$559$	16.0000	0.676728
$560$	1.00000	0.0422577
$561$	0	0
$562$	−18.0000	−0.759284
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	−12.0000	−0.505291
$565$	6.00000	0.252422
$566$	−16.0000	−0.672530
$567$	11.0000	0.461957
$568$	−12.0000	−0.503509
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	−8.00000	−0.335083
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	0	0
$574$	12.0000	0.500870
$575$	0	0
$576$	1.00000	0.0416667
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	17.0000	0.707107
$579$	28.0000	1.16364
$580$	−6.00000	−0.249136
$581$	12.0000	0.497844
$582$	−20.0000	−0.829027
$583$	0	0
$584$	−4.00000	−0.165521
$585$	−4.00000	−0.165380
$586$	24.0000	0.991431
$587$	18.0000	0.742940	0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000	0.742940	0.371470	+	0.928445i	$0.378854\pi$
$588$	−2.00000	−0.0824786
$589$	−40.0000	−1.64817
$590$	−6.00000	−0.247016
$591$	−12.0000	−0.493614
$592$	2.00000	0.0821995
$593$	12.0000	0.492781	0.246390	−	0.969171i	$-0.420755\pi$
$593$	12.0000	0.492781	0.246390	+	0.969171i	$0.420755\pi$
$594$	0	0
$595$	0	0
$596$	18.0000	0.737309
$597$	−4.00000	−0.163709
$598$	0	0
$599$	36.0000	1.47092	0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000	1.47092	0.735460	+	0.677568i	$0.236966\pi$
$600$	2.00000	0.0816497
$601$	4.00000	0.163163	0.0815817	−	0.996667i	$-0.474003\pi$
$601$	4.00000	0.163163	0.0815817	+	0.996667i	$0.474003\pi$
$602$	4.00000	0.163028
$603$	−4.00000	−0.162893
$604$	16.0000	0.651031
$605$	0	0
$606$	0	0
$607$	40.0000	1.62355	0.811775	−	0.583970i	$-0.198502\pi$
$607$	40.0000	1.62355	0.811775	+	0.583970i	$0.198502\pi$
$608$	−4.00000	−0.162221
$609$	12.0000	0.486265
$610$	4.00000	0.161955
$611$	24.0000	0.970936
$612$	0	0
$613$	10.0000	0.403896	0.201948	−	0.979396i	$-0.435273\pi$
$613$	10.0000	0.403896	0.201948	+	0.979396i	$0.435273\pi$
$614$	32.0000	1.29141
$615$	24.0000	0.967773
$616$	0	0
$617$	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
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	10.0000	0.401610
$621$	0	0
$622$	−18.0000	−0.721734
$623$	−18.0000	−0.721155
$624$	−8.00000	−0.320256
$625$	1.00000	0.0400000
$626$	−2.00000	−0.0799361
$627$	0	0
$628$	−22.0000	−0.877896
$629$	0	0
$630$	−1.00000	−0.0398410
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	8.00000	0.318223
$633$	−8.00000	−0.317971
$634$	−6.00000	−0.238290
$635$	8.00000	0.317470
$636$	12.0000	0.475831
$637$	4.00000	0.158486
$638$	0	0
$639$	12.0000	0.474713
$640$	1.00000	0.0395285
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	−24.0000	−0.947204
$643$	2.00000	0.0788723	0.0394362	−	0.999222i	$-0.487444\pi$
$643$	2.00000	0.0788723	0.0394362	+	0.999222i	$0.487444\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	0	0
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	11.0000	0.432121
$649$	0	0
$650$	−4.00000	−0.156893
$651$	−20.0000	−0.783862
$652$	−4.00000	−0.156652
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	−4.00000	−0.156412
$655$	0	0
$656$	12.0000	0.468521
$657$	4.00000	0.156055
$658$	6.00000	0.233904
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	−20.0000	−0.777322
$663$	0	0
$664$	12.0000	0.465690
$665$	4.00000	0.155113
$666$	−2.00000	−0.0774984
$667$	0	0
$668$	−24.0000	−0.928588
$669$	20.0000	0.773245
$670$	−4.00000	−0.154533
$671$	0	0
$672$	−2.00000	−0.0771517
$673$	−2.00000	−0.0770943	−0.0385472	−	0.999257i	$-0.512273\pi$
$673$	−2.00000	−0.0770943	−0.0385472	+	0.999257i	$0.512273\pi$
$674$	−22.0000	−0.847408
$675$	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	$0.574068\pi$
$678$	−12.0000	−0.460857
$679$	10.0000	0.383765
$680$	0	0
$681$	−48.0000	−1.83936
$682$	0	0
$683$	−24.0000	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$684$	4.00000	0.152944
$685$	−6.00000	−0.229248
$686$	1.00000	0.0381802
$687$	20.0000	0.763048
$688$	4.00000	0.152499
$689$	−24.0000	−0.914327
$690$	0	0
$691$	2.00000	0.0760836	0.0380418	−	0.999276i	$-0.487888\pi$
$691$	2.00000	0.0760836	0.0380418	+	0.999276i	$0.487888\pi$
$692$	−12.0000	−0.456172
$693$	0	0
$694$	12.0000	0.455514
$695$	−4.00000	−0.151729
$696$	12.0000	0.454859
$697$	0	0
$698$	20.0000	0.757011
$699$	−12.0000	−0.453882
$700$	−1.00000	−0.0377964
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	−16.0000	−0.603881
$703$	8.00000	0.301726
$704$	0	0
$705$	12.0000	0.451946
$706$	6.00000	0.225813
$707$	0	0
$708$	12.0000	0.450988
$709$	38.0000	1.42712	0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000	1.42712	0.713560	+	0.700594i	$0.247082\pi$
$710$	12.0000	0.450352
$711$	−8.00000	−0.300023
$712$	−18.0000	−0.674579
$713$	0	0
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	−48.0000	−1.79259
$718$	−24.0000	−0.895672
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	−1.00000	−0.0372678
$721$	−14.0000	−0.521387
$722$	3.00000	0.111648
$723$	−8.00000	−0.297523
$724$	−10.0000	−0.371647
$725$	6.00000	0.222834
$726$	0	0
$727$	2.00000	0.0741759	0.0370879	−	0.999312i	$-0.488192\pi$
$727$	2.00000	0.0741759	0.0370879	+	0.999312i	$0.488192\pi$
$728$	4.00000	0.148250
$729$	13.0000	0.481481
$730$	4.00000	0.148047
$731$	0	0
$732$	−8.00000	−0.295689
$733$	40.0000	1.47743	0.738717	−	0.674016i	$-0.235432\pi$
$733$	40.0000	1.47743	0.738717	+	0.674016i	$0.235432\pi$
$734$	34.0000	1.25496
$735$	2.00000	0.0737711
$736$	0	0
$737$	0	0
$738$	−12.0000	−0.441726
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	−2.00000	−0.0735215
$741$	−32.0000	−1.17555
$742$	−6.00000	−0.220267
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	−20.0000	−0.733236
$745$	−18.0000	−0.659469
$746$	−22.0000	−0.805477
$747$	−12.0000	−0.439057
$748$	0	0
$749$	12.0000	0.438470
$750$	−2.00000	−0.0730297
$751$	−28.0000	−1.02173	−0.510867	−	0.859660i	$-0.670676\pi$
$751$	−28.0000	−1.02173	−0.510867	+	0.859660i	$0.670676\pi$
$752$	6.00000	0.218797
$753$	36.0000	1.31191
$754$	−24.0000	−0.874028
$755$	−16.0000	−0.582300
$756$	−4.00000	−0.145479
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	16.0000	0.581146
$759$	0	0
$760$	4.00000	0.145095
$761$	−24.0000	−0.869999	−0.435000	−	0.900431i	$-0.643252\pi$
$761$	−24.0000	−0.869999	−0.435000	+	0.900431i	$0.643252\pi$
$762$	−16.0000	−0.579619
$763$	2.00000	0.0724049
$764$	0	0
$765$	0	0
$766$	6.00000	0.216789
$767$	−24.0000	−0.866590
$768$	−2.00000	−0.0721688
$769$	16.0000	0.576975	0.288487	−	0.957484i	$-0.406848\pi$
$769$	16.0000	0.576975	0.288487	+	0.957484i	$0.406848\pi$
$770$	0	0
$771$	−36.0000	−1.29651
$772$	−14.0000	−0.503871
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	−4.00000	−0.143777
$775$	−10.0000	−0.359211
$776$	10.0000	0.358979
$777$	4.00000	0.143499
$778$	−30.0000	−1.07555
$779$	48.0000	1.71978
$780$	8.00000	0.286446
$781$	0	0
$782$	0	0
$783$	24.0000	0.857690
$784$	1.00000	0.0357143
$785$	22.0000	0.785214
$786$	0	0
$787$	−44.0000	−1.56843	−0.784215	−	0.620489i	$-0.786934\pi$
$787$	−44.0000	−1.56843	−0.784215	+	0.620489i	$0.786934\pi$
$788$	6.00000	0.213741
$789$	0	0
$790$	−8.00000	−0.284627
$791$	6.00000	0.213335
$792$	0	0
$793$	16.0000	0.568177
$794$	−2.00000	−0.0709773
$795$	−12.0000	−0.425596
$796$	2.00000	0.0708881
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	−8.00000	−0.283197
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	18.0000	0.635999
$802$	6.00000	0.211867
$803$	0	0
$804$	8.00000	0.282138
$805$	0	0
$806$	40.0000	1.40894
$807$	−12.0000	−0.422420
$808$	0	0
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	−11.0000	−0.386501
$811$	16.0000	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$812$	−6.00000	−0.210559
$813$	40.0000	1.40286
$814$	0	0
$815$	4.00000	0.140114
$816$	0	0
$817$	16.0000	0.559769
$818$	32.0000	1.11885
$819$	−4.00000	−0.139771
$820$	−12.0000	−0.419058
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	12.0000	0.418548
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	−14.0000	−0.487713
$825$	0	0
$826$	−6.00000	−0.208767
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	26.0000	0.903017	0.451509	−	0.892267i	$-0.350886\pi$
$829$	26.0000	0.903017	0.451509	+	0.892267i	$0.350886\pi$
$830$	−12.0000	−0.416526
$831$	−44.0000	−1.52634
$832$	4.00000	0.138675
$833$	0	0
$834$	8.00000	0.277017
$835$	24.0000	0.830554
$836$	0	0
$837$	−40.0000	−1.38260
$838$	6.00000	0.207267
$839$	30.0000	1.03572	0.517858	−	0.855467i	$-0.326730\pi$
$839$	30.0000	1.03572	0.517858	+	0.855467i	$0.326730\pi$
$840$	2.00000	0.0690066
$841$	7.00000	0.241379
$842$	−26.0000	−0.896019
$843$	−36.0000	−1.23991
$844$	4.00000	0.137686
$845$	−3.00000	−0.103203
$846$	−6.00000	−0.206284
$847$	0	0
$848$	−6.00000	−0.206041
$849$	−32.0000	−1.09824
$850$	0	0
$851$	0	0
$852$	−24.0000	−0.822226
$853$	4.00000	0.136957	0.0684787	−	0.997653i	$-0.478185\pi$
$853$	4.00000	0.136957	0.0684787	+	0.997653i	$0.478185\pi$
$854$	4.00000	0.136877
$855$	−4.00000	−0.136797
$856$	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
$859$	−10.0000	−0.341196	−0.170598	−	0.985341i	$-0.554570\pi$
$859$	−10.0000	−0.341196	−0.170598	+	0.985341i	$0.554570\pi$
$860$	−4.00000	−0.136399
$861$	24.0000	0.817918
$862$	24.0000	0.817443
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	−4.00000	−0.136083
$865$	12.0000	0.408012
$866$	−14.0000	−0.475739
$867$	34.0000	1.15470
$868$	10.0000	0.339422
$869$	0	0
$870$	−12.0000	−0.406838
$871$	−16.0000	−0.542139
$872$	2.00000	0.0677285
$873$	−10.0000	−0.338449
$874$	0	0
$875$	1.00000	0.0338062
$876$	−8.00000	−0.270295
$877$	34.0000	1.14810	0.574049	−	0.818821i	$-0.305372\pi$
$877$	34.0000	1.14810	0.574049	+	0.818821i	$0.305372\pi$
$878$	−28.0000	−0.944954
$879$	48.0000	1.61900
$880$	0	0
$881$	6.00000	0.202145	0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000	0.202145	0.101073	+	0.994879i	$0.467773\pi$
$882$	−1.00000	−0.0336718
$883$	−16.0000	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	0	0
$885$	−12.0000	−0.403376
$886$	24.0000	0.806296
$887$	−12.0000	−0.402921	−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000	−0.402921	−0.201460	+	0.979497i	$0.564569\pi$
$888$	4.00000	0.134231
$889$	8.00000	0.268311
$890$	18.0000	0.603361
$891$	0	0
$892$	−10.0000	−0.334825
$893$	24.0000	0.803129
$894$	36.0000	1.20402
$895$	−12.0000	−0.401116
$896$	1.00000	0.0334077
$897$	0	0
$898$	−6.00000	−0.200223
$899$	−60.0000	−2.00111
$900$	1.00000	0.0333333
$901$	0	0
$902$	0	0
$903$	8.00000	0.266223
$904$	6.00000	0.199557
$905$	10.0000	0.332411
$906$	32.0000	1.06313
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	24.0000	0.796468
$909$	0	0
$910$	−4.00000	−0.132599
$911$	36.0000	1.19273	0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000	1.19273	0.596367	+	0.802712i	$0.296610\pi$
$912$	−8.00000	−0.264906
$913$	0	0
$914$	−10.0000	−0.330771
$915$	8.00000	0.264472
$916$	−10.0000	−0.330409
$917$	0	0
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	64.0000	2.10887
$922$	−36.0000	−1.18560
$923$	48.0000	1.57994
$924$	0	0
$925$	2.00000	0.0657596
$926$	40.0000	1.31448
$927$	14.0000	0.459820
$928$	−6.00000	−0.196960
$929$	−42.0000	−1.37798	−0.688988	−	0.724773i	$-0.741945\pi$
$929$	−42.0000	−1.37798	−0.688988	+	0.724773i	$0.741945\pi$
$930$	20.0000	0.655826
$931$	4.00000	0.131095
$932$	6.00000	0.196537
$933$	−36.0000	−1.17859
$934$	−42.0000	−1.37428
$935$	0	0
$936$	−4.00000	−0.130744
$937$	52.0000	1.69877	0.849383	−	0.527777i	$-0.176974\pi$
$937$	52.0000	1.69877	0.849383	+	0.527777i	$0.176974\pi$
$938$	−4.00000	−0.130605
$939$	−4.00000	−0.130535
$940$	−6.00000	−0.195698
$941$	36.0000	1.17357	0.586783	−	0.809744i	$-0.300394\pi$
$941$	36.0000	1.17357	0.586783	+	0.809744i	$0.300394\pi$
$942$	−44.0000	−1.43360
$943$	0	0
$944$	−6.00000	−0.195283
$945$	4.00000	0.130120
$946$	0	0
$947$	48.0000	1.55979	0.779895	−	0.625910i	$-0.215272\pi$
$947$	48.0000	1.55979	0.779895	+	0.625910i	$0.215272\pi$
$948$	16.0000	0.519656
$949$	16.0000	0.519382
$950$	−4.00000	−0.129777
$951$	−12.0000	−0.389127
$952$	0	0
$953$	18.0000	0.583077	0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000	0.583077	0.291539	+	0.956559i	$0.405833\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	24.0000	0.776215
$957$	0	0
$958$	12.0000	0.387702
$959$	−6.00000	−0.193750
$960$	2.00000	0.0645497
$961$	69.0000	2.22581
$962$	−8.00000	−0.257930
$963$	−12.0000	−0.386695
$964$	4.00000	0.128831
$965$	14.0000	0.450676
$966$	0	0
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	0	0
$969$	0	0
$970$	−10.0000	−0.321081
$971$	30.0000	0.962746	0.481373	−	0.876516i	$-0.340138\pi$
$971$	30.0000	0.962746	0.481373	+	0.876516i	$0.340138\pi$
$972$	10.0000	0.320750
$973$	−4.00000	−0.128234
$974$	−32.0000	−1.02535
$975$	−8.00000	−0.256205
$976$	4.00000	0.128037
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	−8.00000	−0.255812
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	−2.00000	−0.0638551
$982$	−12.0000	−0.382935
$983$	−42.0000	−1.33959	−0.669796	−	0.742545i	$-0.733618\pi$
$983$	−42.0000	−1.33959	−0.669796	+	0.742545i	$0.733618\pi$
$984$	24.0000	0.765092
$985$	−6.00000	−0.191176
$986$	0	0
$987$	12.0000	0.381964
$988$	16.0000	0.509028
$989$	0	0
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	10.0000	0.317500
$993$	−40.0000	−1.26936
$994$	12.0000	0.380617
$995$	−2.00000	−0.0634043
$996$	24.0000	0.760469
$997$	52.0000	1.64686	0.823428	−	0.567420i	$-0.192059\pi$
$997$	52.0000	1.64686	0.823428	+	0.567420i	$0.192059\pi$
$998$	40.0000	1.26618
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.c.1.1		1
11.10	odd	2		770.2.a.f.1.1	✓	1
33.32	even	2		6930.2.a.o.1.1		1
44.43	even	2		6160.2.a.j.1.1		1
55.32	even	4		3850.2.c.b.1849.2		2
55.43	even	4		3850.2.c.b.1849.1		2
55.54	odd	2		3850.2.a.k.1.1		1
77.76	even	2		5390.2.a.bj.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.f.1.1	✓	1	11.10	odd	2
3850.2.a.k.1.1		1	55.54	odd	2
3850.2.c.b.1849.1		2	55.43	even	4
3850.2.c.b.1849.2		2	55.32	even	4
5390.2.a.bj.1.1		1	77.76	even	2
6160.2.a.j.1.1		1	44.43	even	2
6930.2.a.o.1.1		1	33.32	even	2
8470.2.a.c.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8470.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more