LMFDB - Embedded newform 1470.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1470,2,Mod(1,1470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1470, 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("1470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1470.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:	$11.7380090971$
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	$\chi$	$=$	1470.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +6.00000 q^{11} -1.00000 q^{12} -6.00000 q^{13} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} -6.00000 q^{22} +1.00000 q^{24} +1.00000 q^{25} +6.00000 q^{26} -1.00000 q^{27} -8.00000 q^{29} -1.00000 q^{30} -2.00000 q^{31} -1.00000 q^{32} -6.00000 q^{33} +1.00000 q^{36} +4.00000 q^{37} -4.00000 q^{38} +6.00000 q^{39} +1.00000 q^{40} +10.0000 q^{41} -6.00000 q^{43} +6.00000 q^{44} -1.00000 q^{45} +2.00000 q^{47} -1.00000 q^{48} -1.00000 q^{50} -6.00000 q^{52} +10.0000 q^{53} +1.00000 q^{54} -6.00000 q^{55} -4.00000 q^{57} +8.00000 q^{58} +4.00000 q^{59} +1.00000 q^{60} -14.0000 q^{61} +2.00000 q^{62} +1.00000 q^{64} +6.00000 q^{65} +6.00000 q^{66} +14.0000 q^{67} +8.00000 q^{71} -1.00000 q^{72} -6.00000 q^{73} -4.00000 q^{74} -1.00000 q^{75} +4.00000 q^{76} -6.00000 q^{78} -8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} +8.00000 q^{83} +6.00000 q^{86} +8.00000 q^{87} -6.00000 q^{88} +18.0000 q^{89} +1.00000 q^{90} +2.00000 q^{93} -2.00000 q^{94} -4.00000 q^{95} +1.00000 q^{96} -2.00000 q^{97} +6.00000 q^{99} +O(q^{100})\)

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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	6.00000	1.80907	0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$12$	−1.00000	−0.288675
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	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
$21$	0	0
$22$	−6.00000	−1.27920
$23$	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
$26$	6.00000	1.17670
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	−1.00000	−0.182574
$31$	−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
$33$	−6.00000	−1.04447
$34$	0	0
$35$	0	0
$36$	1.00000	0.166667
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	−4.00000	−0.648886
$39$	6.00000	0.960769
$40$	1.00000	0.158114
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	6.00000	0.904534
$45$	−1.00000	−0.149071
$46$	0	0
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−6.00000	−0.832050
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	1.00000	0.136083
$55$	−6.00000	−0.809040
$56$	0	0
$57$	−4.00000	−0.529813
$58$	8.00000	1.05045
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	1.00000	0.129099
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	2.00000	0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	6.00000	0.744208
$66$	6.00000	0.738549
$67$	14.0000	1.71037	0.855186	−	0.518321i	$-0.173443\pi$
$67$	14.0000	1.71037	0.855186	+	0.518321i	$0.173443\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	−1.00000	−0.117851
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	−4.00000	−0.464991
$75$	−1.00000	−0.115470
$76$	4.00000	0.458831
$77$	0	0
$78$	−6.00000	−0.679366
$79$	−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
$81$	1.00000	0.111111
$82$	−10.0000	−1.10432
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	0	0
$85$	0	0
$86$	6.00000	0.646997
$87$	8.00000	0.857690
$88$	−6.00000	−0.639602
$89$	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
$91$	0	0
$92$	0	0
$93$	2.00000	0.207390
$94$	−2.00000	−0.206284
$95$	−4.00000	−0.410391
$96$	1.00000	0.102062
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	6.00000	0.603023
$100$	1.00000	0.100000
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	0	0
$103$	12.0000	1.18240	0.591198	−	0.806527i	$-0.298655\pi$
$103$	12.0000	1.18240	0.591198	+	0.806527i	$0.298655\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	−10.0000	−0.971286
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	−1.00000	−0.0962250
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	6.00000	0.572078
$111$	−4.00000	−0.379663
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	−8.00000	−0.742781
$117$	−6.00000	−0.554700
$118$	−4.00000	−0.368230
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	25.0000	2.27273
$122$	14.0000	1.26750
$123$	−10.0000	−0.901670
$124$	−2.00000	−0.179605
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	−1.00000	−0.0883883
$129$	6.00000	0.528271
$130$	−6.00000	−0.526235
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	−6.00000	−0.522233
$133$	0	0
$134$	−14.0000	−1.20942
$135$	1.00000	0.0860663
$136$	0	0
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	−2.00000	−0.168430
$142$	−8.00000	−0.671345
$143$	−36.0000	−3.01047
$144$	1.00000	0.0833333
$145$	8.00000	0.664364
$146$	6.00000	0.496564
$147$	0	0
$148$	4.00000	0.328798
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	1.00000	0.0816497
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	−4.00000	−0.324443
$153$	0	0
$154$	0	0
$155$	2.00000	0.160644
$156$	6.00000	0.480384
$157$	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
$160$	1.00000	0.0790569
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	10.0000	0.780869
$165$	6.00000	0.467099
$166$	−8.00000	−0.620920
$167$	18.0000	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	0	0
$171$	4.00000	0.305888
$172$	−6.00000	−0.457496
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	−8.00000	−0.606478
$175$	0	0
$176$	6.00000	0.452267
$177$	−4.00000	−0.300658
$178$	−18.0000	−1.34916
$179$	2.00000	0.149487	0.0747435	−	0.997203i	$-0.476186\pi$
$179$	2.00000	0.149487	0.0747435	+	0.997203i	$0.476186\pi$
$180$	−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	$0.621207\pi$
$182$	0	0
$183$	14.0000	1.03491
$184$	0	0
$185$	−4.00000	−0.294086
$186$	−2.00000	−0.146647
$187$	0	0
$188$	2.00000	0.145865
$189$	0	0
$190$	4.00000	0.290191
$191$	−20.0000	−1.44715	−0.723575	−	0.690246i	$-0.757502\pi$
$191$	−20.0000	−1.44715	−0.723575	+	0.690246i	$0.757502\pi$
$192$	−1.00000	−0.0721688
$193$	22.0000	1.58359	0.791797	−	0.610784i	$-0.209146\pi$
$193$	22.0000	1.58359	0.791797	+	0.610784i	$0.209146\pi$
$194$	2.00000	0.143592
$195$	−6.00000	−0.429669
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	−6.00000	−0.426401
$199$	18.0000	1.27599	0.637993	−	0.770042i	$-0.279765\pi$
$199$	18.0000	1.27599	0.637993	+	0.770042i	$0.279765\pi$
$200$	−1.00000	−0.0707107
$201$	−14.0000	−0.987484
$202$	−14.0000	−0.985037
$203$	0	0
$204$	0	0
$205$	−10.0000	−0.698430
$206$	−12.0000	−0.836080
$207$	0	0
$208$	−6.00000	−0.416025
$209$	24.0000	1.66011
$210$	0	0
$211$	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
$214$	−4.00000	−0.273434
$215$	6.00000	0.409197
$216$	1.00000	0.0680414
$217$	0	0
$218$	−6.00000	−0.406371
$219$	6.00000	0.405442
$220$	−6.00000	−0.404520
$221$	0	0
$222$	4.00000	0.268462
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	−6.00000	−0.399114
$227$	−20.0000	−1.32745	−0.663723	−	0.747978i	$-0.731025\pi$
$227$	−20.0000	−1.32745	−0.663723	+	0.747978i	$0.731025\pi$
$228$	−4.00000	−0.264906
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	0	0
$232$	8.00000	0.525226
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	6.00000	0.392232
$235$	−2.00000	−0.130466
$236$	4.00000	0.260378
$237$	8.00000	0.519656
$238$	0	0
$239$	−4.00000	−0.258738	−0.129369	−	0.991596i	$-0.541295\pi$
$239$	−4.00000	−0.258738	−0.129369	+	0.991596i	$0.541295\pi$
$240$	1.00000	0.0645497
$241$	28.0000	1.80364	0.901819	−	0.432113i	$-0.142232\pi$
$241$	28.0000	1.80364	0.901819	+	0.432113i	$0.142232\pi$
$242$	−25.0000	−1.60706
$243$	−1.00000	−0.0641500
$244$	−14.0000	−0.896258
$245$	0	0
$246$	10.0000	0.637577
$247$	−24.0000	−1.52708
$248$	2.00000	0.127000
$249$	−8.00000	−0.506979
$250$	1.00000	0.0632456
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	0	0
$253$	0	0
$254$	−4.00000	−0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	8.00000	0.499026	0.249513	−	0.968371i	$-0.419729\pi$
$257$	8.00000	0.499026	0.249513	+	0.968371i	$0.419729\pi$
$258$	−6.00000	−0.373544
$259$	0	0
$260$	6.00000	0.372104
$261$	−8.00000	−0.495188
$262$	12.0000	0.741362
$263$	−32.0000	−1.97320	−0.986602	−	0.163144i	$-0.947836\pi$
$263$	−32.0000	−1.97320	−0.986602	+	0.163144i	$0.947836\pi$
$264$	6.00000	0.369274
$265$	−10.0000	−0.614295
$266$	0	0
$267$	−18.0000	−1.10158
$268$	14.0000	0.855186
$269$	−22.0000	−1.34136	−0.670682	−	0.741745i	$-0.733998\pi$
$269$	−22.0000	−1.34136	−0.670682	+	0.741745i	$0.733998\pi$
$270$	−1.00000	−0.0608581
$271$	18.0000	1.09342	0.546711	−	0.837321i	$-0.315880\pi$
$271$	18.0000	1.09342	0.546711	+	0.837321i	$0.315880\pi$
$272$	0	0
$273$	0	0
$274$	−10.0000	−0.604122
$275$	6.00000	0.361814
$276$	0	0
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	8.00000	0.479808
$279$	−2.00000	−0.119737
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	2.00000	0.119098
$283$	−28.0000	−1.66443	−0.832214	−	0.554455i	$-0.812927\pi$
$283$	−28.0000	−1.66443	−0.832214	+	0.554455i	$0.812927\pi$
$284$	8.00000	0.474713
$285$	4.00000	0.236940
$286$	36.0000	2.12872
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−17.0000	−1.00000
$290$	−8.00000	−0.469776
$291$	2.00000	0.117242
$292$	−6.00000	−0.351123
$293$	−22.0000	−1.28525	−0.642627	−	0.766179i	$-0.722155\pi$
$293$	−22.0000	−1.28525	−0.642627	+	0.766179i	$0.722155\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	−4.00000	−0.232495
$297$	−6.00000	−0.348155
$298$	0	0
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	0	0
$302$	8.00000	0.460348
$303$	−14.0000	−0.804279
$304$	4.00000	0.229416
$305$	14.0000	0.801638
$306$	0	0
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	0	0
$309$	−12.0000	−0.682656
$310$	−2.00000	−0.113592
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	−6.00000	−0.339683
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	−8.00000	−0.450035
$317$	10.0000	0.561656	0.280828	−	0.959758i	$-0.409391\pi$
$317$	10.0000	0.561656	0.280828	+	0.959758i	$0.409391\pi$
$318$	10.0000	0.560772
$319$	−48.0000	−2.68748
$320$	−1.00000	−0.0559017
$321$	−4.00000	−0.223258
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	−6.00000	−0.332820
$326$	−2.00000	−0.110770
$327$	−6.00000	−0.331801
$328$	−10.0000	−0.552158
$329$	0	0
$330$	−6.00000	−0.330289
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	8.00000	0.439057
$333$	4.00000	0.219199
$334$	−18.0000	−0.984916
$335$	−14.0000	−0.764902
$336$	0	0
$337$	−26.0000	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$337$	−26.0000	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$338$	−23.0000	−1.25104
$339$	−6.00000	−0.325875
$340$	0	0
$341$	−12.0000	−0.649836
$342$	−4.00000	−0.216295
$343$	0	0
$344$	6.00000	0.323498
$345$	0	0
$346$	−6.00000	−0.322562
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	8.00000	0.428845
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	−6.00000	−0.319801
$353$	−4.00000	−0.212899	−0.106449	−	0.994318i	$-0.533948\pi$
$353$	−4.00000	−0.212899	−0.106449	+	0.994318i	$0.533948\pi$
$354$	4.00000	0.212598
$355$	−8.00000	−0.424596
$356$	18.0000	0.953998
$357$	0	0
$358$	−2.00000	−0.105703
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	1.00000	0.0527046
$361$	−3.00000	−0.157895
$362$	10.0000	0.525588
$363$	−25.0000	−1.31216
$364$	0	0
$365$	6.00000	0.314054
$366$	−14.0000	−0.731792
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	0	0
$369$	10.0000	0.520579
$370$	4.00000	0.207950
$371$	0	0
$372$	2.00000	0.103695
$373$	−8.00000	−0.414224	−0.207112	−	0.978317i	$-0.566407\pi$
$373$	−8.00000	−0.414224	−0.207112	+	0.978317i	$0.566407\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	−2.00000	−0.103142
$377$	48.0000	2.47213
$378$	0	0
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	−4.00000	−0.205196
$381$	−4.00000	−0.204926
$382$	20.0000	1.02329
$383$	14.0000	0.715367	0.357683	−	0.933843i	$-0.383567\pi$
$383$	14.0000	0.715367	0.357683	+	0.933843i	$0.383567\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−22.0000	−1.11977
$387$	−6.00000	−0.304997
$388$	−2.00000	−0.101535
$389$	24.0000	1.21685	0.608424	−	0.793612i	$-0.291802\pi$
$389$	24.0000	1.21685	0.608424	+	0.793612i	$0.291802\pi$
$390$	6.00000	0.303822
$391$	0	0
$392$	0	0
$393$	12.0000	0.605320
$394$	−6.00000	−0.302276
$395$	8.00000	0.402524
$396$	6.00000	0.301511
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	−18.0000	−0.902258
$399$	0	0
$400$	1.00000	0.0500000
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	14.0000	0.698257
$403$	12.0000	0.597763
$404$	14.0000	0.696526
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	24.0000	1.18964
$408$	0	0
$409$	16.0000	0.791149	0.395575	−	0.918434i	$-0.370545\pi$
$409$	16.0000	0.791149	0.395575	+	0.918434i	$0.370545\pi$
$410$	10.0000	0.493865
$411$	−10.0000	−0.493264
$412$	12.0000	0.591198
$413$	0	0
$414$	0	0
$415$	−8.00000	−0.392705
$416$	6.00000	0.294174
$417$	8.00000	0.391762
$418$	−24.0000	−1.17388
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−34.0000	−1.65706	−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000	−1.65706	−0.828529	+	0.559946i	$0.810822\pi$
$422$	−12.0000	−0.584151
$423$	2.00000	0.0972433
$424$	−10.0000	−0.485643
$425$	0	0
$426$	8.00000	0.387601
$427$	0	0
$428$	4.00000	0.193347
$429$	36.0000	1.73810
$430$	−6.00000	−0.289346
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	−1.00000	−0.0481125
$433$	−18.0000	−0.865025	−0.432512	−	0.901628i	$-0.642373\pi$
$433$	−18.0000	−0.865025	−0.432512	+	0.901628i	$0.642373\pi$
$434$	0	0
$435$	−8.00000	−0.383571
$436$	6.00000	0.287348
$437$	0	0
$438$	−6.00000	−0.286691
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	6.00000	0.286039
$441$	0	0
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	−4.00000	−0.189832
$445$	−18.0000	−0.853282
$446$	16.0000	0.757622
$447$	0	0
$448$	0	0
$449$	−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
$451$	60.0000	2.82529
$452$	6.00000	0.282216
$453$	8.00000	0.375873
$454$	20.0000	0.938647
$455$	0	0
$456$	4.00000	0.187317
$457$	38.0000	1.77757	0.888783	−	0.458329i	$-0.151552\pi$
$457$	38.0000	1.77757	0.888783	+	0.458329i	$0.151552\pi$
$458$	−14.0000	−0.654177
$459$	0	0
$460$	0	0
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	−8.00000	−0.371391
$465$	−2.00000	−0.0927478
$466$	10.0000	0.463241
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	−6.00000	−0.277350
$469$	0	0
$470$	2.00000	0.0922531
$471$	−14.0000	−0.645086
$472$	−4.00000	−0.184115
$473$	−36.0000	−1.65528
$474$	−8.00000	−0.367452
$475$	4.00000	0.183533
$476$	0	0
$477$	10.0000	0.457869
$478$	4.00000	0.182956
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	−1.00000	−0.0456435
$481$	−24.0000	−1.09431
$482$	−28.0000	−1.27537
$483$	0	0
$484$	25.0000	1.13636
$485$	2.00000	0.0908153
$486$	1.00000	0.0453609
$487$	12.0000	0.543772	0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000	0.543772	0.271886	+	0.962329i	$0.412353\pi$
$488$	14.0000	0.633750
$489$	−2.00000	−0.0904431
$490$	0	0
$491$	−22.0000	−0.992846	−0.496423	−	0.868081i	$-0.665354\pi$
$491$	−22.0000	−0.992846	−0.496423	+	0.868081i	$0.665354\pi$
$492$	−10.0000	−0.450835
$493$	0	0
$494$	24.0000	1.07981
$495$	−6.00000	−0.269680
$496$	−2.00000	−0.0898027
$497$	0	0
$498$	8.00000	0.358489
$499$	−32.0000	−1.43252	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000	−1.43252	−0.716258	+	0.697835i	$0.754147\pi$
$500$	−1.00000	−0.0447214
$501$	−18.0000	−0.804181
$502$	−20.0000	−0.892644
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	0	0
$505$	−14.0000	−0.622992
$506$	0	0
$507$	−23.0000	−1.02147
$508$	4.00000	0.177471
$509$	−34.0000	−1.50702	−0.753512	−	0.657434i	$-0.771642\pi$
$509$	−34.0000	−1.50702	−0.753512	+	0.657434i	$0.771642\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−4.00000	−0.176604
$514$	−8.00000	−0.352865
$515$	−12.0000	−0.528783
$516$	6.00000	0.264135
$517$	12.0000	0.527759
$518$	0	0
$519$	−6.00000	−0.263371
$520$	−6.00000	−0.263117
$521$	2.00000	0.0876216	0.0438108	−	0.999040i	$-0.486050\pi$
$521$	2.00000	0.0876216	0.0438108	+	0.999040i	$0.486050\pi$
$522$	8.00000	0.350150
$523$	44.0000	1.92399	0.961993	−	0.273075i	$-0.0880406\pi$
$523$	44.0000	1.92399	0.961993	+	0.273075i	$0.0880406\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	32.0000	1.39527
$527$	0	0
$528$	−6.00000	−0.261116
$529$	−23.0000	−1.00000
$530$	10.0000	0.434372
$531$	4.00000	0.173585
$532$	0	0
$533$	−60.0000	−2.59889
$534$	18.0000	0.778936
$535$	−4.00000	−0.172935
$536$	−14.0000	−0.604708
$537$	−2.00000	−0.0863064
$538$	22.0000	0.948487
$539$	0	0
$540$	1.00000	0.0430331
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	−18.0000	−0.773166
$543$	10.0000	0.429141
$544$	0	0
$545$	−6.00000	−0.257012
$546$	0	0
$547$	30.0000	1.28271	0.641354	−	0.767245i	$-0.278373\pi$
$547$	30.0000	1.28271	0.641354	+	0.767245i	$0.278373\pi$
$548$	10.0000	0.427179
$549$	−14.0000	−0.597505
$550$	−6.00000	−0.255841
$551$	−32.0000	−1.36325
$552$	0	0
$553$	0	0
$554$	8.00000	0.339887
$555$	4.00000	0.169791
$556$	−8.00000	−0.339276
$557$	2.00000	0.0847427	0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000	0.0847427	0.0423714	+	0.999102i	$0.486509\pi$
$558$	2.00000	0.0846668
$559$	36.0000	1.52264
$560$	0	0
$561$	0	0
$562$	−6.00000	−0.253095
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	−2.00000	−0.0842152
$565$	−6.00000	−0.252422
$566$	28.0000	1.17693
$567$	0	0
$568$	−8.00000	−0.335673
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	−4.00000	−0.167542
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	−36.0000	−1.50524
$573$	20.0000	0.835512
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	14.0000	0.582828	0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000	0.582828	0.291414	+	0.956597i	$0.405874\pi$
$578$	17.0000	0.707107
$579$	−22.0000	−0.914289
$580$	8.00000	0.332182
$581$	0	0
$582$	−2.00000	−0.0829027
$583$	60.0000	2.48495
$584$	6.00000	0.248282
$585$	6.00000	0.248069
$586$	22.0000	0.908812
$587$	−32.0000	−1.32078	−0.660391	−	0.750922i	$-0.729609\pi$
$587$	−32.0000	−1.32078	−0.660391	+	0.750922i	$0.729609\pi$
$588$	0	0
$589$	−8.00000	−0.329634
$590$	4.00000	0.164677
$591$	−6.00000	−0.246807
$592$	4.00000	0.164399
$593$	16.0000	0.657041	0.328521	−	0.944497i	$-0.393450\pi$
$593$	16.0000	0.657041	0.328521	+	0.944497i	$0.393450\pi$
$594$	6.00000	0.246183
$595$	0	0
$596$	0	0
$597$	−18.0000	−0.736691
$598$	0	0
$599$	44.0000	1.79779	0.898896	−	0.438163i	$-0.144371\pi$
$599$	44.0000	1.79779	0.898896	+	0.438163i	$0.144371\pi$
$600$	1.00000	0.0408248
$601$	−12.0000	−0.489490	−0.244745	−	0.969587i	$-0.578704\pi$
$601$	−12.0000	−0.489490	−0.244745	+	0.969587i	$0.578704\pi$
$602$	0	0
$603$	14.0000	0.570124
$604$	−8.00000	−0.325515
$605$	−25.0000	−1.01639
$606$	14.0000	0.568711
$607$	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
$609$	0	0
$610$	−14.0000	−0.566843
$611$	−12.0000	−0.485468
$612$	0	0
$613$	36.0000	1.45403	0.727013	−	0.686624i	$-0.240908\pi$
$613$	36.0000	1.45403	0.727013	+	0.686624i	$0.240908\pi$
$614$	−20.0000	−0.807134
$615$	10.0000	0.403239
$616$	0	0
$617$	42.0000	1.69086	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000	1.69086	0.845428	+	0.534089i	$0.179345\pi$
$618$	12.0000	0.482711
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	2.00000	0.0803219
$621$	0	0
$622$	−12.0000	−0.481156
$623$	0	0
$624$	6.00000	0.240192
$625$	1.00000	0.0400000
$626$	26.0000	1.03917
$627$	−24.0000	−0.958468
$628$	14.0000	0.558661
$629$	0	0
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	8.00000	0.318223
$633$	−12.0000	−0.476957
$634$	−10.0000	−0.397151
$635$	−4.00000	−0.158735
$636$	−10.0000	−0.396526
$637$	0	0
$638$	48.0000	1.90034
$639$	8.00000	0.316475
$640$	1.00000	0.0395285
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	4.00000	0.157867
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	0	0
$645$	−6.00000	−0.236250
$646$	0	0
$647$	46.0000	1.80845	0.904223	−	0.427060i	$-0.140451\pi$
$647$	46.0000	1.80845	0.904223	+	0.427060i	$0.140451\pi$
$648$	−1.00000	−0.0392837
$649$	24.0000	0.942082
$650$	6.00000	0.235339
$651$	0	0
$652$	2.00000	0.0783260
$653$	14.0000	0.547862	0.273931	−	0.961749i	$-0.411676\pi$
$653$	14.0000	0.547862	0.273931	+	0.961749i	$0.411676\pi$
$654$	6.00000	0.234619
$655$	12.0000	0.468879
$656$	10.0000	0.390434
$657$	−6.00000	−0.234082
$658$	0	0
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	6.00000	0.233550
$661$	18.0000	0.700119	0.350059	−	0.936727i	$-0.386161\pi$
$661$	18.0000	0.700119	0.350059	+	0.936727i	$0.386161\pi$
$662$	20.0000	0.777322
$663$	0	0
$664$	−8.00000	−0.310460
$665$	0	0
$666$	−4.00000	−0.154997
$667$	0	0
$668$	18.0000	0.696441
$669$	16.0000	0.618596
$670$	14.0000	0.540867
$671$	−84.0000	−3.24278
$672$	0	0
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	26.0000	1.00148
$675$	−1.00000	−0.0384900
$676$	23.0000	0.884615
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	6.00000	0.230429
$679$	0	0
$680$	0	0
$681$	20.0000	0.766402
$682$	12.0000	0.459504
$683$	−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
$685$	−10.0000	−0.382080
$686$	0	0
$687$	−14.0000	−0.534133
$688$	−6.00000	−0.228748
$689$	−60.0000	−2.28582
$690$	0	0
$691$	−40.0000	−1.52167	−0.760836	−	0.648944i	$-0.775211\pi$
$691$	−40.0000	−1.52167	−0.760836	+	0.648944i	$0.775211\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	28.0000	1.06287
$695$	8.00000	0.303457
$696$	−8.00000	−0.303239
$697$	0	0
$698$	2.00000	0.0757011
$699$	10.0000	0.378235
$700$	0	0
$701$	24.0000	0.906467	0.453234	−	0.891392i	$-0.350270\pi$
$701$	24.0000	0.906467	0.453234	+	0.891392i	$0.350270\pi$
$702$	−6.00000	−0.226455
$703$	16.0000	0.603451
$704$	6.00000	0.226134
$705$	2.00000	0.0753244
$706$	4.00000	0.150542
$707$	0	0
$708$	−4.00000	−0.150329
$709$	2.00000	0.0751116	0.0375558	−	0.999295i	$-0.488043\pi$
$709$	2.00000	0.0751116	0.0375558	+	0.999295i	$0.488043\pi$
$710$	8.00000	0.300235
$711$	−8.00000	−0.300023
$712$	−18.0000	−0.674579
$713$	0	0
$714$	0	0
$715$	36.0000	1.34632
$716$	2.00000	0.0747435
$717$	4.00000	0.149383
$718$	24.0000	0.895672
$719$	16.0000	0.596699	0.298350	−	0.954457i	$-0.403564\pi$
$719$	16.0000	0.596699	0.298350	+	0.954457i	$0.403564\pi$
$720$	−1.00000	−0.0372678
$721$	0	0
$722$	3.00000	0.111648
$723$	−28.0000	−1.04133
$724$	−10.0000	−0.371647
$725$	−8.00000	−0.297113
$726$	25.0000	0.927837
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−6.00000	−0.222070
$731$	0	0
$732$	14.0000	0.517455
$733$	46.0000	1.69905	0.849524	−	0.527549i	$-0.176889\pi$
$733$	46.0000	1.69905	0.849524	+	0.527549i	$0.176889\pi$
$734$	−4.00000	−0.147643
$735$	0	0
$736$	0	0
$737$	84.0000	3.09418
$738$	−10.0000	−0.368105
$739$	−40.0000	−1.47142	−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000	−1.47142	−0.735712	+	0.677295i	$0.763152\pi$
$740$	−4.00000	−0.147043
$741$	24.0000	0.881662
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	−2.00000	−0.0733236
$745$	0	0
$746$	8.00000	0.292901
$747$	8.00000	0.292705
$748$	0	0
$749$	0	0
$750$	−1.00000	−0.0365148
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	2.00000	0.0729325
$753$	−20.0000	−0.728841
$754$	−48.0000	−1.74806
$755$	8.00000	0.291150
$756$	0	0
$757$	−40.0000	−1.45382	−0.726912	−	0.686730i	$-0.759045\pi$
$757$	−40.0000	−1.45382	−0.726912	+	0.686730i	$0.759045\pi$
$758$	−16.0000	−0.581146
$759$	0	0
$760$	4.00000	0.145095
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	4.00000	0.144905
$763$	0	0
$764$	−20.0000	−0.723575
$765$	0	0
$766$	−14.0000	−0.505841
$767$	−24.0000	−0.866590
$768$	−1.00000	−0.0360844
$769$	−40.0000	−1.44244	−0.721218	−	0.692708i	$-0.756418\pi$
$769$	−40.0000	−1.44244	−0.721218	+	0.692708i	$0.756418\pi$
$770$	0	0
$771$	−8.00000	−0.288113
$772$	22.0000	0.791797
$773$	26.0000	0.935155	0.467578	−	0.883952i	$-0.345127\pi$
$773$	26.0000	0.935155	0.467578	+	0.883952i	$0.345127\pi$
$774$	6.00000	0.215666
$775$	−2.00000	−0.0718421
$776$	2.00000	0.0717958
$777$	0	0
$778$	−24.0000	−0.860442
$779$	40.0000	1.43315
$780$	−6.00000	−0.214834
$781$	48.0000	1.71758
$782$	0	0
$783$	8.00000	0.285897
$784$	0	0
$785$	−14.0000	−0.499681
$786$	−12.0000	−0.428026
$787$	−20.0000	−0.712923	−0.356462	−	0.934310i	$-0.616017\pi$
$787$	−20.0000	−0.712923	−0.356462	+	0.934310i	$0.616017\pi$
$788$	6.00000	0.213741
$789$	32.0000	1.13923
$790$	−8.00000	−0.284627
$791$	0	0
$792$	−6.00000	−0.213201
$793$	84.0000	2.98293
$794$	−14.0000	−0.496841
$795$	10.0000	0.354663
$796$	18.0000	0.637993
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	18.0000	0.635999
$802$	−2.00000	−0.0706225
$803$	−36.0000	−1.27041
$804$	−14.0000	−0.493742
$805$	0	0
$806$	−12.0000	−0.422682
$807$	22.0000	0.774437
$808$	−14.0000	−0.492518
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	1.00000	0.0351364
$811$	16.0000	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$812$	0	0
$813$	−18.0000	−0.631288
$814$	−24.0000	−0.841200
$815$	−2.00000	−0.0700569
$816$	0	0
$817$	−24.0000	−0.839654
$818$	−16.0000	−0.559427
$819$	0	0
$820$	−10.0000	−0.349215
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	10.0000	0.348790
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	−12.0000	−0.418040
$825$	−6.00000	−0.208893
$826$	0	0
$827$	−24.0000	−0.834562	−0.417281	−	0.908778i	$-0.637017\pi$
$827$	−24.0000	−0.834562	−0.417281	+	0.908778i	$0.637017\pi$
$828$	0	0
$829$	−22.0000	−0.764092	−0.382046	−	0.924143i	$-0.624780\pi$
$829$	−22.0000	−0.764092	−0.382046	+	0.924143i	$0.624780\pi$
$830$	8.00000	0.277684
$831$	8.00000	0.277517
$832$	−6.00000	−0.208013
$833$	0	0
$834$	−8.00000	−0.277017
$835$	−18.0000	−0.622916
$836$	24.0000	0.830057
$837$	2.00000	0.0691301
$838$	12.0000	0.414533
$839$	44.0000	1.51905	0.759524	−	0.650479i	$-0.225432\pi$
$839$	44.0000	1.51905	0.759524	+	0.650479i	$0.225432\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	34.0000	1.17172
$843$	−6.00000	−0.206651
$844$	12.0000	0.413057
$845$	−23.0000	−0.791224
$846$	−2.00000	−0.0687614
$847$	0	0
$848$	10.0000	0.343401
$849$	28.0000	0.960958
$850$	0	0
$851$	0	0
$852$	−8.00000	−0.274075
$853$	22.0000	0.753266	0.376633	−	0.926363i	$-0.377082\pi$
$853$	22.0000	0.753266	0.376633	+	0.926363i	$0.377082\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	−4.00000	−0.136717
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	−36.0000	−1.22902
$859$	−16.0000	−0.545913	−0.272956	−	0.962026i	$-0.588002\pi$
$859$	−16.0000	−0.545913	−0.272956	+	0.962026i	$0.588002\pi$
$860$	6.00000	0.204598
$861$	0	0
$862$	0	0
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	1.00000	0.0340207
$865$	−6.00000	−0.204006
$866$	18.0000	0.611665
$867$	17.0000	0.577350
$868$	0	0
$869$	−48.0000	−1.62829
$870$	8.00000	0.271225
$871$	−84.0000	−2.84623
$872$	−6.00000	−0.203186
$873$	−2.00000	−0.0676897
$874$	0	0
$875$	0	0
$876$	6.00000	0.202721
$877$	−32.0000	−1.08056	−0.540282	−	0.841484i	$-0.681682\pi$
$877$	−32.0000	−1.08056	−0.540282	+	0.841484i	$0.681682\pi$
$878$	10.0000	0.337484
$879$	22.0000	0.742042
$880$	−6.00000	−0.202260
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	14.0000	0.471138	0.235569	−	0.971858i	$-0.424305\pi$
$883$	14.0000	0.471138	0.235569	+	0.971858i	$0.424305\pi$
$884$	0	0
$885$	4.00000	0.134459
$886$	12.0000	0.403148
$887$	−26.0000	−0.872995	−0.436497	−	0.899706i	$-0.643781\pi$
$887$	−26.0000	−0.872995	−0.436497	+	0.899706i	$0.643781\pi$
$888$	4.00000	0.134231
$889$	0	0
$890$	18.0000	0.603361
$891$	6.00000	0.201008
$892$	−16.0000	−0.535720
$893$	8.00000	0.267710
$894$	0	0
$895$	−2.00000	−0.0668526
$896$	0	0
$897$	0	0
$898$	10.0000	0.333704
$899$	16.0000	0.533630
$900$	1.00000	0.0333333
$901$	0	0
$902$	−60.0000	−1.99778
$903$	0	0
$904$	−6.00000	−0.199557
$905$	10.0000	0.332411
$906$	−8.00000	−0.265782
$907$	−10.0000	−0.332045	−0.166022	−	0.986122i	$-0.553092\pi$
$907$	−10.0000	−0.332045	−0.166022	+	0.986122i	$0.553092\pi$
$908$	−20.0000	−0.663723
$909$	14.0000	0.464351
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	−4.00000	−0.132453
$913$	48.0000	1.58857
$914$	−38.0000	−1.25693
$915$	−14.0000	−0.462826
$916$	14.0000	0.462573
$917$	0	0
$918$	0	0
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	−2.00000	−0.0658665
$923$	−48.0000	−1.57994
$924$	0	0
$925$	4.00000	0.131519
$926$	−8.00000	−0.262896
$927$	12.0000	0.394132
$928$	8.00000	0.262613
$929$	−42.0000	−1.37798	−0.688988	−	0.724773i	$-0.741945\pi$
$929$	−42.0000	−1.37798	−0.688988	+	0.724773i	$0.741945\pi$
$930$	2.00000	0.0655826
$931$	0	0
$932$	−10.0000	−0.327561
$933$	−12.0000	−0.392862
$934$	0	0
$935$	0	0
$936$	6.00000	0.196116
$937$	−30.0000	−0.980057	−0.490029	−	0.871706i	$-0.663014\pi$
$937$	−30.0000	−0.980057	−0.490029	+	0.871706i	$0.663014\pi$
$938$	0	0
$939$	26.0000	0.848478
$940$	−2.00000	−0.0652328
$941$	−2.00000	−0.0651981	−0.0325991	−	0.999469i	$-0.510378\pi$
$941$	−2.00000	−0.0651981	−0.0325991	+	0.999469i	$0.510378\pi$
$942$	14.0000	0.456145
$943$	0	0
$944$	4.00000	0.130189
$945$	0	0
$946$	36.0000	1.17046
$947$	−48.0000	−1.55979	−0.779895	−	0.625910i	$-0.784728\pi$
$947$	−48.0000	−1.55979	−0.779895	+	0.625910i	$0.784728\pi$
$948$	8.00000	0.259828
$949$	36.0000	1.16861
$950$	−4.00000	−0.129777
$951$	−10.0000	−0.324272
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	−10.0000	−0.323762
$955$	20.0000	0.647185
$956$	−4.00000	−0.129369
$957$	48.0000	1.55162
$958$	−36.0000	−1.16311
$959$	0	0
$960$	1.00000	0.0322749
$961$	−27.0000	−0.870968
$962$	24.0000	0.773791
$963$	4.00000	0.128898
$964$	28.0000	0.901819
$965$	−22.0000	−0.708205
$966$	0	0
$967$	−48.0000	−1.54358	−0.771788	−	0.635880i	$-0.780637\pi$
$967$	−48.0000	−1.54358	−0.771788	+	0.635880i	$0.780637\pi$
$968$	−25.0000	−0.803530
$969$	0	0
$970$	−2.00000	−0.0642161
$971$	4.00000	0.128366	0.0641831	−	0.997938i	$-0.479556\pi$
$971$	4.00000	0.128366	0.0641831	+	0.997938i	$0.479556\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−12.0000	−0.384505
$975$	6.00000	0.192154
$976$	−14.0000	−0.448129
$977$	54.0000	1.72761	0.863807	−	0.503824i	$-0.168074\pi$
$977$	54.0000	1.72761	0.863807	+	0.503824i	$0.168074\pi$
$978$	2.00000	0.0639529
$979$	108.000	3.45169
$980$	0	0
$981$	6.00000	0.191565
$982$	22.0000	0.702048
$983$	42.0000	1.33959	0.669796	−	0.742545i	$-0.266382\pi$
$983$	42.0000	1.33959	0.669796	+	0.742545i	$0.266382\pi$
$984$	10.0000	0.318788
$985$	−6.00000	−0.191176
$986$	0	0
$987$	0	0
$988$	−24.0000	−0.763542
$989$	0	0
$990$	6.00000	0.190693
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	2.00000	0.0635001
$993$	20.0000	0.634681
$994$	0	0
$995$	−18.0000	−0.570638
$996$	−8.00000	−0.253490
$997$	−42.0000	−1.33015	−0.665077	−	0.746775i	$-0.731601\pi$
$997$	−42.0000	−1.33015	−0.665077	+	0.746775i	$0.731601\pi$
$998$	32.0000	1.01294
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1470.2.a.c.1.1	✓	1
3.2	odd	2		4410.2.a.be.1.1		1
5.4	even	2		7350.2.a.da.1.1		1
7.2	even	3		1470.2.i.r.361.1		2
7.3	odd	6		1470.2.i.k.961.1		2
7.4	even	3		1470.2.i.r.961.1		2
7.5	odd	6		1470.2.i.k.361.1		2
7.6	odd	2		1470.2.a.i.1.1	yes	1
21.20	even	2		4410.2.a.v.1.1		1
35.34	odd	2		7350.2.a.cf.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1470.2.a.c.1.1	✓	1	1.1	even	1	trivial
1470.2.a.i.1.1	yes	1	7.6	odd	2
1470.2.i.k.361.1		2	7.5	odd	6
1470.2.i.k.961.1		2	7.3	odd	6
1470.2.i.r.361.1		2	7.2	even	3
1470.2.i.r.961.1		2	7.4	even	3
4410.2.a.v.1.1		1	21.20	even	2
4410.2.a.be.1.1		1	3.2	odd	2
7350.2.a.cf.1.1		1	35.34	odd	2
7350.2.a.da.1.1		1	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1470.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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