LMFDB - Embedded newform 1470.2.a.l.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:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 210)
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)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -5.00000 q^{11} -1.00000 q^{12} -5.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} -7.00000 q^{19} +1.00000 q^{20} -5.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -5.00000 q^{26} -1.00000 q^{27} -1.00000 q^{30} -2.00000 q^{31} +1.00000 q^{32} +5.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} +1.00000 q^{37} -7.00000 q^{38} +5.00000 q^{39} +1.00000 q^{40} +5.00000 q^{41} +12.0000 q^{43} -5.00000 q^{44} +1.00000 q^{45} +1.00000 q^{46} -11.0000 q^{47} -1.00000 q^{48} +1.00000 q^{50} +4.00000 q^{51} -5.00000 q^{52} -9.00000 q^{53} -1.00000 q^{54} -5.00000 q^{55} +7.00000 q^{57} +4.00000 q^{59} -1.00000 q^{60} +4.00000 q^{61} -2.00000 q^{62} +1.00000 q^{64} -5.00000 q^{65} +5.00000 q^{66} -12.0000 q^{67} -4.00000 q^{68} -1.00000 q^{69} +2.00000 q^{71} +1.00000 q^{72} +10.0000 q^{73} +1.00000 q^{74} -1.00000 q^{75} -7.00000 q^{76} +5.00000 q^{78} -12.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +5.00000 q^{82} -12.0000 q^{83} -4.00000 q^{85} +12.0000 q^{86} -5.00000 q^{88} +14.0000 q^{89} +1.00000 q^{90} +1.00000 q^{92} +2.00000 q^{93} -11.0000 q^{94} -7.00000 q^{95} -1.00000 q^{96} -8.00000 q^{97} -5.00000 q^{99} +O(q^{100})\)

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
$3$	−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
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	−1.00000	−0.288675
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	1.00000	0.235702
$19$	−7.00000	−1.60591	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000	−1.60591	−0.802955	+	0.596040i	$0.796740\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−5.00000	−1.06600
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	−5.00000	−0.980581
$27$	−1.00000	−0.192450
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	−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
$33$	5.00000	0.870388
$34$	−4.00000	−0.685994
$35$	0	0
$36$	1.00000	0.166667
$37$	1.00000	0.164399	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000	0.164399	0.0821995	+	0.996616i	$0.473806\pi$
$38$	−7.00000	−1.13555
$39$	5.00000	0.800641
$40$	1.00000	0.158114
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	0	0
$43$	12.0000	1.82998	0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000	1.82998	0.914991	+	0.403473i	$0.132197\pi$
$44$	−5.00000	−0.753778
$45$	1.00000	0.149071
$46$	1.00000	0.147442
$47$	−11.0000	−1.60451	−0.802257	−	0.596978i	$-0.796368\pi$
$47$	−11.0000	−1.60451	−0.802257	+	0.596978i	$0.796368\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	1.00000	0.141421
$51$	4.00000	0.560112
$52$	−5.00000	−0.693375
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	−1.00000	−0.136083
$55$	−5.00000	−0.674200
$56$	0	0
$57$	7.00000	0.927173
$58$	0	0
$59$	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
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	−2.00000	−0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	−5.00000	−0.620174
$66$	5.00000	0.615457
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	−4.00000	−0.485071
$69$	−1.00000	−0.120386
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	1.00000	0.117851
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	1.00000	0.116248
$75$	−1.00000	−0.115470
$76$	−7.00000	−0.802955
$77$	0	0
$78$	5.00000	0.566139
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	5.00000	0.552158
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	12.0000	1.29399
$87$	0	0
$88$	−5.00000	−0.533002
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	1.00000	0.104257
$93$	2.00000	0.207390
$94$	−11.0000	−1.13456
$95$	−7.00000	−0.718185
$96$	−1.00000	−0.102062
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	0	0
$99$	−5.00000	−0.502519
$100$	1.00000	0.100000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	4.00000	0.396059
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	−5.00000	−0.490290
$105$	0	0
$106$	−9.00000	−0.874157
$107$	−2.00000	−0.193347	−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000	−0.193347	−0.0966736	+	0.995316i	$0.530820\pi$
$108$	−1.00000	−0.0962250
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	−5.00000	−0.476731
$111$	−1.00000	−0.0949158
$112$	0	0
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	7.00000	0.655610
$115$	1.00000	0.0932505
$116$	0	0
$117$	−5.00000	−0.462250
$118$	4.00000	0.368230
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	14.0000	1.27273
$122$	4.00000	0.362143
$123$	−5.00000	−0.450835
$124$	−2.00000	−0.179605
$125$	1.00000	0.0894427
$126$	0	0
$127$	9.00000	0.798621	0.399310	−	0.916816i	$-0.369250\pi$
$127$	9.00000	0.798621	0.399310	+	0.916816i	$0.369250\pi$
$128$	1.00000	0.0883883
$129$	−12.0000	−1.05654
$130$	−5.00000	−0.438529
$131$	−9.00000	−0.786334	−0.393167	−	0.919467i	$-0.628621\pi$
$131$	−9.00000	−0.786334	−0.393167	+	0.919467i	$0.628621\pi$
$132$	5.00000	0.435194
$133$	0	0
$134$	−12.0000	−1.03664
$135$	−1.00000	−0.0860663
$136$	−4.00000	−0.342997
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	−1.00000	−0.0851257
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	11.0000	0.926367
$142$	2.00000	0.167836
$143$	25.0000	2.09061
$144$	1.00000	0.0833333
$145$	0	0
$146$	10.0000	0.827606
$147$	0	0
$148$	1.00000	0.0821995
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	−1.00000	−0.0816497
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	−7.00000	−0.567775
$153$	−4.00000	−0.323381
$154$	0	0
$155$	−2.00000	−0.160644
$156$	5.00000	0.400320
$157$	11.0000	0.877896	0.438948	−	0.898513i	$-0.355351\pi$
$157$	11.0000	0.877896	0.438948	+	0.898513i	$0.355351\pi$
$158$	−12.0000	−0.954669
$159$	9.00000	0.713746
$160$	1.00000	0.0790569
$161$	0	0
$162$	1.00000	0.0785674
$163$	−24.0000	−1.87983	−0.939913	−	0.341415i	$-0.889094\pi$
$163$	−24.0000	−1.87983	−0.939913	+	0.341415i	$0.889094\pi$
$164$	5.00000	0.390434
$165$	5.00000	0.389249
$166$	−12.0000	−0.931381
$167$	11.0000	0.851206	0.425603	−	0.904910i	$-0.360062\pi$
$167$	11.0000	0.851206	0.425603	+	0.904910i	$0.360062\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	−4.00000	−0.306786
$171$	−7.00000	−0.535303
$172$	12.0000	0.914991
$173$	−13.0000	−0.988372	−0.494186	−	0.869356i	$-0.664534\pi$
$173$	−13.0000	−0.988372	−0.494186	+	0.869356i	$0.664534\pi$
$174$	0	0
$175$	0	0
$176$	−5.00000	−0.376889
$177$	−4.00000	−0.300658
$178$	14.0000	1.04934
$179$	−23.0000	−1.71910	−0.859550	−	0.511051i	$-0.829256\pi$
$179$	−23.0000	−1.71910	−0.859550	+	0.511051i	$0.829256\pi$
$180$	1.00000	0.0745356
$181$	20.0000	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$182$	0	0
$183$	−4.00000	−0.295689
$184$	1.00000	0.0737210
$185$	1.00000	0.0735215
$186$	2.00000	0.146647
$187$	20.0000	1.46254
$188$	−11.0000	−0.802257
$189$	0	0
$190$	−7.00000	−0.507833
$191$	14.0000	1.01300	0.506502	−	0.862239i	$-0.330938\pi$
$191$	14.0000	1.01300	0.506502	+	0.862239i	$0.330938\pi$
$192$	−1.00000	−0.0721688
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	−8.00000	−0.574367
$195$	5.00000	0.358057
$196$	0	0
$197$	−3.00000	−0.213741	−0.106871	−	0.994273i	$-0.534083\pi$
$197$	−3.00000	−0.213741	−0.106871	+	0.994273i	$0.534083\pi$
$198$	−5.00000	−0.355335
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	1.00000	0.0707107
$201$	12.0000	0.846415
$202$	0	0
$203$	0	0
$204$	4.00000	0.280056
$205$	5.00000	0.349215
$206$	8.00000	0.557386
$207$	1.00000	0.0695048
$208$	−5.00000	−0.346688
$209$	35.0000	2.42100
$210$	0	0
$211$	17.0000	1.17033	0.585164	−	0.810915i	$-0.301030\pi$
$211$	17.0000	1.17033	0.585164	+	0.810915i	$0.301030\pi$
$212$	−9.00000	−0.618123
$213$	−2.00000	−0.137038
$214$	−2.00000	−0.136717
$215$	12.0000	0.818393
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−2.00000	−0.135457
$219$	−10.0000	−0.675737
$220$	−5.00000	−0.337100
$221$	20.0000	1.34535
$222$	−1.00000	−0.0671156
$223$	−12.0000	−0.803579	−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000	−0.803579	−0.401790	+	0.915732i	$0.631612\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	−14.0000	−0.931266
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	7.00000	0.463586
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	1.00000	0.0659380
$231$	0	0
$232$	0	0
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	−5.00000	−0.326860
$235$	−11.0000	−0.717561
$236$	4.00000	0.260378
$237$	12.0000	0.779484
$238$	0	0
$239$	−22.0000	−1.42306	−0.711531	−	0.702655i	$-0.751998\pi$
$239$	−22.0000	−1.42306	−0.711531	+	0.702655i	$0.751998\pi$
$240$	−1.00000	−0.0645497
$241$	15.0000	0.966235	0.483117	−	0.875556i	$-0.339504\pi$
$241$	15.0000	0.966235	0.483117	+	0.875556i	$0.339504\pi$
$242$	14.0000	0.899954
$243$	−1.00000	−0.0641500
$244$	4.00000	0.256074
$245$	0	0
$246$	−5.00000	−0.318788
$247$	35.0000	2.22700
$248$	−2.00000	−0.127000
$249$	12.0000	0.760469
$250$	1.00000	0.0632456
$251$	1.00000	0.0631194	0.0315597	−	0.999502i	$-0.489953\pi$
$251$	1.00000	0.0631194	0.0315597	+	0.999502i	$0.489953\pi$
$252$	0	0
$253$	−5.00000	−0.314347
$254$	9.00000	0.564710
$255$	4.00000	0.250490
$256$	1.00000	0.0625000
$257$	−16.0000	−0.998053	−0.499026	−	0.866587i	$-0.666309\pi$
$257$	−16.0000	−0.998053	−0.499026	+	0.866587i	$0.666309\pi$
$258$	−12.0000	−0.747087
$259$	0	0
$260$	−5.00000	−0.310087
$261$	0	0
$262$	−9.00000	−0.556022
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	5.00000	0.307729
$265$	−9.00000	−0.552866
$266$	0	0
$267$	−14.0000	−0.856786
$268$	−12.0000	−0.733017
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	−1.00000	−0.0608581
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	−2.00000	−0.120824
$275$	−5.00000	−0.301511
$276$	−1.00000	−0.0601929
$277$	14.0000	0.841178	0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000	0.841178	0.420589	+	0.907251i	$0.361823\pi$
$278$	−4.00000	−0.239904
$279$	−2.00000	−0.119737
$280$	0	0
$281$	7.00000	0.417585	0.208792	−	0.977960i	$-0.433047\pi$
$281$	7.00000	0.417585	0.208792	+	0.977960i	$0.433047\pi$
$282$	11.0000	0.655040
$283$	−10.0000	−0.594438	−0.297219	−	0.954809i	$-0.596059\pi$
$283$	−10.0000	−0.594438	−0.297219	+	0.954809i	$0.596059\pi$
$284$	2.00000	0.118678
$285$	7.00000	0.414644
$286$	25.0000	1.47828
$287$	0	0
$288$	1.00000	0.0589256
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	8.00000	0.468968
$292$	10.0000	0.585206
$293$	9.00000	0.525786	0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000	0.525786	0.262893	+	0.964825i	$0.415323\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	1.00000	0.0581238
$297$	5.00000	0.290129
$298$	12.0000	0.695141
$299$	−5.00000	−0.289157
$300$	−1.00000	−0.0577350
$301$	0	0
$302$	14.0000	0.805609
$303$	0	0
$304$	−7.00000	−0.401478
$305$	4.00000	0.229039
$306$	−4.00000	−0.228665
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	−2.00000	−0.113592
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	5.00000	0.283069
$313$	16.0000	0.904373	0.452187	−	0.891923i	$-0.350644\pi$
$313$	16.0000	0.904373	0.452187	+	0.891923i	$0.350644\pi$
$314$	11.0000	0.620766
$315$	0	0
$316$	−12.0000	−0.675053
$317$	−22.0000	−1.23564	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	−22.0000	−1.23564	−0.617822	+	0.786318i	$0.711985\pi$
$318$	9.00000	0.504695
$319$	0	0
$320$	1.00000	0.0559017
$321$	2.00000	0.111629
$322$	0	0
$323$	28.0000	1.55796
$324$	1.00000	0.0555556
$325$	−5.00000	−0.277350
$326$	−24.0000	−1.32924
$327$	2.00000	0.110600
$328$	5.00000	0.276079
$329$	0	0
$330$	5.00000	0.275241
$331$	31.0000	1.70391	0.851957	−	0.523612i	$-0.175416\pi$
$331$	31.0000	1.70391	0.851957	+	0.523612i	$0.175416\pi$
$332$	−12.0000	−0.658586
$333$	1.00000	0.0547997
$334$	11.0000	0.601893
$335$	−12.0000	−0.655630
$336$	0	0
$337$	−16.0000	−0.871576	−0.435788	−	0.900049i	$-0.643530\pi$
$337$	−16.0000	−0.871576	−0.435788	+	0.900049i	$0.643530\pi$
$338$	12.0000	0.652714
$339$	14.0000	0.760376
$340$	−4.00000	−0.216930
$341$	10.0000	0.541530
$342$	−7.00000	−0.378517
$343$	0	0
$344$	12.0000	0.646997
$345$	−1.00000	−0.0538382
$346$	−13.0000	−0.698884
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	0	0
$349$	−4.00000	−0.214115	−0.107058	−	0.994253i	$-0.534143\pi$
$349$	−4.00000	−0.214115	−0.107058	+	0.994253i	$0.534143\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	−5.00000	−0.266501
$353$	24.0000	1.27739	0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000	1.27739	0.638696	+	0.769460i	$0.279474\pi$
$354$	−4.00000	−0.212598
$355$	2.00000	0.106149
$356$	14.0000	0.741999
$357$	0	0
$358$	−23.0000	−1.21559
$359$	−20.0000	−1.05556	−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000	−1.05556	−0.527780	+	0.849381i	$0.676975\pi$
$360$	1.00000	0.0527046
$361$	30.0000	1.57895
$362$	20.0000	1.05118
$363$	−14.0000	−0.734809
$364$	0	0
$365$	10.0000	0.523424
$366$	−4.00000	−0.209083
$367$	7.00000	0.365397	0.182699	−	0.983169i	$-0.441517\pi$
$367$	7.00000	0.365397	0.182699	+	0.983169i	$0.441517\pi$
$368$	1.00000	0.0521286
$369$	5.00000	0.260290
$370$	1.00000	0.0519875
$371$	0	0
$372$	2.00000	0.103695
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	20.0000	1.03418
$375$	−1.00000	−0.0516398
$376$	−11.0000	−0.567282
$377$	0	0
$378$	0	0
$379$	1.00000	0.0513665	0.0256833	−	0.999670i	$-0.491824\pi$
$379$	1.00000	0.0513665	0.0256833	+	0.999670i	$0.491824\pi$
$380$	−7.00000	−0.359092
$381$	−9.00000	−0.461084
$382$	14.0000	0.716302
$383$	21.0000	1.07305	0.536525	−	0.843884i	$-0.319737\pi$
$383$	21.0000	1.07305	0.536525	+	0.843884i	$0.319737\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	10.0000	0.508987
$387$	12.0000	0.609994
$388$	−8.00000	−0.406138
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	5.00000	0.253185
$391$	−4.00000	−0.202289
$392$	0	0
$393$	9.00000	0.453990
$394$	−3.00000	−0.151138
$395$	−12.0000	−0.603786
$396$	−5.00000	−0.251259
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	−4.00000	−0.200502
$399$	0	0
$400$	1.00000	0.0500000
$401$	−21.0000	−1.04869	−0.524345	−	0.851506i	$-0.675690\pi$
$401$	−21.0000	−1.04869	−0.524345	+	0.851506i	$0.675690\pi$
$402$	12.0000	0.598506
$403$	10.0000	0.498135
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−5.00000	−0.247841
$408$	4.00000	0.198030
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	5.00000	0.246932
$411$	2.00000	0.0986527
$412$	8.00000	0.394132
$413$	0	0
$414$	1.00000	0.0491473
$415$	−12.0000	−0.589057
$416$	−5.00000	−0.245145
$417$	4.00000	0.195881
$418$	35.0000	1.71191
$419$	15.0000	0.732798	0.366399	−	0.930458i	$-0.380591\pi$
$419$	15.0000	0.732798	0.366399	+	0.930458i	$0.380591\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	17.0000	0.827547
$423$	−11.0000	−0.534838
$424$	−9.00000	−0.437079
$425$	−4.00000	−0.194029
$426$	−2.00000	−0.0969003
$427$	0	0
$428$	−2.00000	−0.0966736
$429$	−25.0000	−1.20701
$430$	12.0000	0.578691
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	−1.00000	−0.0481125
$433$	24.0000	1.15337	0.576683	−	0.816968i	$-0.304347\pi$
$433$	24.0000	1.15337	0.576683	+	0.816968i	$0.304347\pi$
$434$	0	0
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	−7.00000	−0.334855
$438$	−10.0000	−0.477818
$439$	−40.0000	−1.90910	−0.954548	−	0.298057i	$-0.903661\pi$
$439$	−40.0000	−1.90910	−0.954548	+	0.298057i	$0.903661\pi$
$440$	−5.00000	−0.238366
$441$	0	0
$442$	20.0000	0.951303
$443$	−28.0000	−1.33032	−0.665160	−	0.746701i	$-0.731637\pi$
$443$	−28.0000	−1.33032	−0.665160	+	0.746701i	$0.731637\pi$
$444$	−1.00000	−0.0474579
$445$	14.0000	0.663664
$446$	−12.0000	−0.568216
$447$	−12.0000	−0.567581
$448$	0	0
$449$	−29.0000	−1.36859	−0.684297	−	0.729203i	$-0.739891\pi$
$449$	−29.0000	−1.36859	−0.684297	+	0.729203i	$0.739891\pi$
$450$	1.00000	0.0471405
$451$	−25.0000	−1.17720
$452$	−14.0000	−0.658505
$453$	−14.0000	−0.657777
$454$	−12.0000	−0.563188
$455$	0	0
$456$	7.00000	0.327805
$457$	−14.0000	−0.654892	−0.327446	−	0.944870i	$-0.606188\pi$
$457$	−14.0000	−0.654892	−0.327446	+	0.944870i	$0.606188\pi$
$458$	10.0000	0.467269
$459$	4.00000	0.186704
$460$	1.00000	0.0466252
$461$	−4.00000	−0.186299	−0.0931493	−	0.995652i	$-0.529693\pi$
$461$	−4.00000	−0.186299	−0.0931493	+	0.995652i	$0.529693\pi$
$462$	0	0
$463$	−19.0000	−0.883005	−0.441502	−	0.897260i	$-0.645554\pi$
$463$	−19.0000	−0.883005	−0.441502	+	0.897260i	$0.645554\pi$
$464$	0	0
$465$	2.00000	0.0927478
$466$	−14.0000	−0.648537
$467$	−20.0000	−0.925490	−0.462745	−	0.886492i	$-0.653135\pi$
$467$	−20.0000	−0.925490	−0.462745	+	0.886492i	$0.653135\pi$
$468$	−5.00000	−0.231125
$469$	0	0
$470$	−11.0000	−0.507392
$471$	−11.0000	−0.506853
$472$	4.00000	0.184115
$473$	−60.0000	−2.75880
$474$	12.0000	0.551178
$475$	−7.00000	−0.321182
$476$	0	0
$477$	−9.00000	−0.412082
$478$	−22.0000	−1.00626
$479$	18.0000	0.822441	0.411220	−	0.911536i	$-0.365103\pi$
$479$	18.0000	0.822441	0.411220	+	0.911536i	$0.365103\pi$
$480$	−1.00000	−0.0456435
$481$	−5.00000	−0.227980
$482$	15.0000	0.683231
$483$	0	0
$484$	14.0000	0.636364
$485$	−8.00000	−0.363261
$486$	−1.00000	−0.0453609
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	4.00000	0.181071
$489$	24.0000	1.08532
$490$	0	0
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	−5.00000	−0.225417
$493$	0	0
$494$	35.0000	1.57472
$495$	−5.00000	−0.224733
$496$	−2.00000	−0.0898027
$497$	0	0
$498$	12.0000	0.537733
$499$	40.0000	1.79065	0.895323	−	0.445418i	$-0.146945\pi$
$499$	40.0000	1.79065	0.895323	+	0.445418i	$0.146945\pi$
$500$	1.00000	0.0447214
$501$	−11.0000	−0.491444
$502$	1.00000	0.0446322
$503$	−20.0000	−0.891756	−0.445878	−	0.895094i	$-0.647108\pi$
$503$	−20.0000	−0.891756	−0.445878	+	0.895094i	$0.647108\pi$
$504$	0	0
$505$	0	0
$506$	−5.00000	−0.222277
$507$	−12.0000	−0.532939
$508$	9.00000	0.399310
$509$	−10.0000	−0.443242	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000	−0.443242	−0.221621	+	0.975133i	$0.571135\pi$
$510$	4.00000	0.177123
$511$	0	0
$512$	1.00000	0.0441942
$513$	7.00000	0.309058
$514$	−16.0000	−0.705730
$515$	8.00000	0.352522
$516$	−12.0000	−0.528271
$517$	55.0000	2.41890
$518$	0	0
$519$	13.0000	0.570637
$520$	−5.00000	−0.219265
$521$	−33.0000	−1.44576	−0.722878	−	0.690976i	$-0.757181\pi$
$521$	−33.0000	−1.44576	−0.722878	+	0.690976i	$0.757181\pi$
$522$	0	0
$523$	−22.0000	−0.961993	−0.480996	−	0.876723i	$-0.659725\pi$
$523$	−22.0000	−0.961993	−0.480996	+	0.876723i	$0.659725\pi$
$524$	−9.00000	−0.393167
$525$	0	0
$526$	0	0
$527$	8.00000	0.348485
$528$	5.00000	0.217597
$529$	−22.0000	−0.956522
$530$	−9.00000	−0.390935
$531$	4.00000	0.173585
$532$	0	0
$533$	−25.0000	−1.08287
$534$	−14.0000	−0.605839
$535$	−2.00000	−0.0864675
$536$	−12.0000	−0.518321
$537$	23.0000	0.992523
$538$	−24.0000	−1.03471
$539$	0	0
$540$	−1.00000	−0.0430331
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	0	0
$543$	−20.0000	−0.858282
$544$	−4.00000	−0.171499
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−2.00000	−0.0854358
$549$	4.00000	0.170716
$550$	−5.00000	−0.213201
$551$	0	0
$552$	−1.00000	−0.0425628
$553$	0	0
$554$	14.0000	0.594803
$555$	−1.00000	−0.0424476
$556$	−4.00000	−0.169638
$557$	37.0000	1.56774	0.783870	−	0.620925i	$-0.213243\pi$
$557$	37.0000	1.56774	0.783870	+	0.620925i	$0.213243\pi$
$558$	−2.00000	−0.0846668
$559$	−60.0000	−2.53773
$560$	0	0
$561$	−20.0000	−0.844401
$562$	7.00000	0.295277
$563$	−18.0000	−0.758610	−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000	−0.758610	−0.379305	+	0.925272i	$0.623837\pi$
$564$	11.0000	0.463184
$565$	−14.0000	−0.588984
$566$	−10.0000	−0.420331
$567$	0	0
$568$	2.00000	0.0839181
$569$	39.0000	1.63497	0.817483	−	0.575953i	$-0.195369\pi$
$569$	39.0000	1.63497	0.817483	+	0.575953i	$0.195369\pi$
$570$	7.00000	0.293198
$571$	40.0000	1.67395	0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000	1.67395	0.836974	+	0.547243i	$0.184323\pi$
$572$	25.0000	1.04530
$573$	−14.0000	−0.584858
$574$	0	0
$575$	1.00000	0.0417029
$576$	1.00000	0.0416667
$577$	26.0000	1.08239	0.541197	−	0.840896i	$-0.317971\pi$
$577$	26.0000	1.08239	0.541197	+	0.840896i	$0.317971\pi$
$578$	−1.00000	−0.0415945
$579$	−10.0000	−0.415586
$580$	0	0
$581$	0	0
$582$	8.00000	0.331611
$583$	45.0000	1.86371
$584$	10.0000	0.413803
$585$	−5.00000	−0.206725
$586$	9.00000	0.371787
$587$	−30.0000	−1.23823	−0.619116	−	0.785299i	$-0.712509\pi$
$587$	−30.0000	−1.23823	−0.619116	+	0.785299i	$0.712509\pi$
$588$	0	0
$589$	14.0000	0.576860
$590$	4.00000	0.164677
$591$	3.00000	0.123404
$592$	1.00000	0.0410997
$593$	36.0000	1.47834	0.739171	−	0.673517i	$-0.235217\pi$
$593$	36.0000	1.47834	0.739171	+	0.673517i	$0.235217\pi$
$594$	5.00000	0.205152
$595$	0	0
$596$	12.0000	0.491539
$597$	4.00000	0.163709
$598$	−5.00000	−0.204465
$599$	−10.0000	−0.408589	−0.204294	−	0.978909i	$-0.565490\pi$
$599$	−10.0000	−0.408589	−0.204294	+	0.978909i	$0.565490\pi$
$600$	−1.00000	−0.0408248
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	−12.0000	−0.488678
$604$	14.0000	0.569652
$605$	14.0000	0.569181
$606$	0	0
$607$	−27.0000	−1.09590	−0.547948	−	0.836512i	$-0.684591\pi$
$607$	−27.0000	−1.09590	−0.547948	+	0.836512i	$0.684591\pi$
$608$	−7.00000	−0.283887
$609$	0	0
$610$	4.00000	0.161955
$611$	55.0000	2.22506
$612$	−4.00000	−0.161690
$613$	−43.0000	−1.73675	−0.868377	−	0.495905i	$-0.834836\pi$
$613$	−43.0000	−1.73675	−0.868377	+	0.495905i	$0.834836\pi$
$614$	8.00000	0.322854
$615$	−5.00000	−0.201619
$616$	0	0
$617$	−8.00000	−0.322068	−0.161034	−	0.986949i	$-0.551483\pi$
$617$	−8.00000	−0.322068	−0.161034	+	0.986949i	$0.551483\pi$
$618$	−8.00000	−0.321807
$619$	25.0000	1.00483	0.502417	−	0.864625i	$-0.332444\pi$
$619$	25.0000	1.00483	0.502417	+	0.864625i	$0.332444\pi$
$620$	−2.00000	−0.0803219
$621$	−1.00000	−0.0401286
$622$	8.00000	0.320771
$623$	0	0
$624$	5.00000	0.200160
$625$	1.00000	0.0400000
$626$	16.0000	0.639489
$627$	−35.0000	−1.39777
$628$	11.0000	0.438948
$629$	−4.00000	−0.159490
$630$	0	0
$631$	6.00000	0.238856	0.119428	−	0.992843i	$-0.461894\pi$
$631$	6.00000	0.238856	0.119428	+	0.992843i	$0.461894\pi$
$632$	−12.0000	−0.477334
$633$	−17.0000	−0.675689
$634$	−22.0000	−0.873732
$635$	9.00000	0.357154
$636$	9.00000	0.356873
$637$	0	0
$638$	0	0
$639$	2.00000	0.0791188
$640$	1.00000	0.0395285
$641$	21.0000	0.829450	0.414725	−	0.909947i	$-0.363878\pi$
$641$	21.0000	0.829450	0.414725	+	0.909947i	$0.363878\pi$
$642$	2.00000	0.0789337
$643$	34.0000	1.34083	0.670415	−	0.741987i	$-0.266116\pi$
$643$	34.0000	1.34083	0.670415	+	0.741987i	$0.266116\pi$
$644$	0	0
$645$	−12.0000	−0.472500
$646$	28.0000	1.10165
$647$	−23.0000	−0.904223	−0.452112	−	0.891961i	$-0.649329\pi$
$647$	−23.0000	−0.904223	−0.452112	+	0.891961i	$0.649329\pi$
$648$	1.00000	0.0392837
$649$	−20.0000	−0.785069
$650$	−5.00000	−0.196116
$651$	0	0
$652$	−24.0000	−0.939913
$653$	−19.0000	−0.743527	−0.371764	−	0.928327i	$-0.621247\pi$
$653$	−19.0000	−0.743527	−0.371764	+	0.928327i	$0.621247\pi$
$654$	2.00000	0.0782062
$655$	−9.00000	−0.351659
$656$	5.00000	0.195217
$657$	10.0000	0.390137
$658$	0	0
$659$	−20.0000	−0.779089	−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000	−0.779089	−0.389545	+	0.921008i	$0.627368\pi$
$660$	5.00000	0.194625
$661$	−40.0000	−1.55582	−0.777910	−	0.628376i	$-0.783720\pi$
$661$	−40.0000	−1.55582	−0.777910	+	0.628376i	$0.783720\pi$
$662$	31.0000	1.20485
$663$	−20.0000	−0.776736
$664$	−12.0000	−0.465690
$665$	0	0
$666$	1.00000	0.0387492
$667$	0	0
$668$	11.0000	0.425603
$669$	12.0000	0.463947
$670$	−12.0000	−0.463600
$671$	−20.0000	−0.772091
$672$	0	0
$673$	−4.00000	−0.154189	−0.0770943	−	0.997024i	$-0.524564\pi$
$673$	−4.00000	−0.154189	−0.0770943	+	0.997024i	$0.524564\pi$
$674$	−16.0000	−0.616297
$675$	−1.00000	−0.0384900
$676$	12.0000	0.461538
$677$	33.0000	1.26829	0.634147	−	0.773213i	$-0.281352\pi$
$677$	33.0000	1.26829	0.634147	+	0.773213i	$0.281352\pi$
$678$	14.0000	0.537667
$679$	0	0
$680$	−4.00000	−0.153393
$681$	12.0000	0.459841
$682$	10.0000	0.382920
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	−7.00000	−0.267652
$685$	−2.00000	−0.0764161
$686$	0	0
$687$	−10.0000	−0.381524
$688$	12.0000	0.457496
$689$	45.0000	1.71436
$690$	−1.00000	−0.0380693
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	−13.0000	−0.494186
$693$	0	0
$694$	−18.0000	−0.683271
$695$	−4.00000	−0.151729
$696$	0	0
$697$	−20.0000	−0.757554
$698$	−4.00000	−0.151402
$699$	14.0000	0.529529
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	5.00000	0.188713
$703$	−7.00000	−0.264010
$704$	−5.00000	−0.188445
$705$	11.0000	0.414284
$706$	24.0000	0.903252
$707$	0	0
$708$	−4.00000	−0.150329
$709$	28.0000	1.05156	0.525781	−	0.850620i	$-0.323773\pi$
$709$	28.0000	1.05156	0.525781	+	0.850620i	$0.323773\pi$
$710$	2.00000	0.0750587
$711$	−12.0000	−0.450035
$712$	14.0000	0.524672
$713$	−2.00000	−0.0749006
$714$	0	0
$715$	25.0000	0.934947
$716$	−23.0000	−0.859550
$717$	22.0000	0.821605
$718$	−20.0000	−0.746393
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	1.00000	0.0372678
$721$	0	0
$722$	30.0000	1.11648
$723$	−15.0000	−0.557856
$724$	20.0000	0.743294
$725$	0	0
$726$	−14.0000	−0.519589
$727$	−3.00000	−0.111264	−0.0556319	−	0.998451i	$-0.517717\pi$
$727$	−3.00000	−0.111264	−0.0556319	+	0.998451i	$0.517717\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	10.0000	0.370117
$731$	−48.0000	−1.77534
$732$	−4.00000	−0.147844
$733$	9.00000	0.332423	0.166211	−	0.986090i	$-0.446847\pi$
$733$	9.00000	0.332423	0.166211	+	0.986090i	$0.446847\pi$
$734$	7.00000	0.258375
$735$	0	0
$736$	1.00000	0.0368605
$737$	60.0000	2.21013
$738$	5.00000	0.184053
$739$	−15.0000	−0.551784	−0.275892	−	0.961189i	$-0.588973\pi$
$739$	−15.0000	−0.551784	−0.275892	+	0.961189i	$0.588973\pi$
$740$	1.00000	0.0367607
$741$	−35.0000	−1.28576
$742$	0	0
$743$	15.0000	0.550297	0.275148	−	0.961402i	$-0.411273\pi$
$743$	15.0000	0.550297	0.275148	+	0.961402i	$0.411273\pi$
$744$	2.00000	0.0733236
$745$	12.0000	0.439646
$746$	−22.0000	−0.805477
$747$	−12.0000	−0.439057
$748$	20.0000	0.731272
$749$	0	0
$750$	−1.00000	−0.0365148
$751$	−50.0000	−1.82453	−0.912263	−	0.409605i	$-0.865667\pi$
$751$	−50.0000	−1.82453	−0.912263	+	0.409605i	$0.865667\pi$
$752$	−11.0000	−0.401129
$753$	−1.00000	−0.0364420
$754$	0	0
$755$	14.0000	0.509512
$756$	0	0
$757$	−34.0000	−1.23575	−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000	−1.23575	−0.617876	+	0.786276i	$0.712006\pi$
$758$	1.00000	0.0363216
$759$	5.00000	0.181489
$760$	−7.00000	−0.253917
$761$	−37.0000	−1.34125	−0.670624	−	0.741797i	$-0.733974\pi$
$761$	−37.0000	−1.34125	−0.670624	+	0.741797i	$0.733974\pi$
$762$	−9.00000	−0.326036
$763$	0	0
$764$	14.0000	0.506502
$765$	−4.00000	−0.144620
$766$	21.0000	0.758761
$767$	−20.0000	−0.722158
$768$	−1.00000	−0.0360844
$769$	−5.00000	−0.180305	−0.0901523	−	0.995928i	$-0.528735\pi$
$769$	−5.00000	−0.180305	−0.0901523	+	0.995928i	$0.528735\pi$
$770$	0	0
$771$	16.0000	0.576226
$772$	10.0000	0.359908
$773$	−5.00000	−0.179838	−0.0899188	−	0.995949i	$-0.528661\pi$
$773$	−5.00000	−0.179838	−0.0899188	+	0.995949i	$0.528661\pi$
$774$	12.0000	0.431331
$775$	−2.00000	−0.0718421
$776$	−8.00000	−0.287183
$777$	0	0
$778$	18.0000	0.645331
$779$	−35.0000	−1.25401
$780$	5.00000	0.179029
$781$	−10.0000	−0.357828
$782$	−4.00000	−0.143040
$783$	0	0
$784$	0	0
$785$	11.0000	0.392607
$786$	9.00000	0.321019
$787$	2.00000	0.0712923	0.0356462	−	0.999364i	$-0.488651\pi$
$787$	2.00000	0.0712923	0.0356462	+	0.999364i	$0.488651\pi$
$788$	−3.00000	−0.106871
$789$	0	0
$790$	−12.0000	−0.426941
$791$	0	0
$792$	−5.00000	−0.177667
$793$	−20.0000	−0.710221
$794$	−14.0000	−0.496841
$795$	9.00000	0.319197
$796$	−4.00000	−0.141776
$797$	−54.0000	−1.91278	−0.956389	−	0.292096i	$-0.905647\pi$
$797$	−54.0000	−1.91278	−0.956389	+	0.292096i	$0.905647\pi$
$798$	0	0
$799$	44.0000	1.55661
$800$	1.00000	0.0353553
$801$	14.0000	0.494666
$802$	−21.0000	−0.741536
$803$	−50.0000	−1.76446
$804$	12.0000	0.423207
$805$	0	0
$806$	10.0000	0.352235
$807$	24.0000	0.844840
$808$	0	0
$809$	25.0000	0.878953	0.439477	−	0.898254i	$-0.355164\pi$
$809$	25.0000	0.878953	0.439477	+	0.898254i	$0.355164\pi$
$810$	1.00000	0.0351364
$811$	21.0000	0.737410	0.368705	−	0.929547i	$-0.379801\pi$
$811$	21.0000	0.737410	0.368705	+	0.929547i	$0.379801\pi$
$812$	0	0
$813$	0	0
$814$	−5.00000	−0.175250
$815$	−24.0000	−0.840683
$816$	4.00000	0.140028
$817$	−84.0000	−2.93879
$818$	10.0000	0.349642
$819$	0	0
$820$	5.00000	0.174608
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	2.00000	0.0697580
$823$	40.0000	1.39431	0.697156	−	0.716919i	$-0.254448\pi$
$823$	40.0000	1.39431	0.697156	+	0.716919i	$0.254448\pi$
$824$	8.00000	0.278693
$825$	5.00000	0.174078
$826$	0	0
$827$	−6.00000	−0.208640	−0.104320	−	0.994544i	$-0.533267\pi$
$827$	−6.00000	−0.208640	−0.104320	+	0.994544i	$0.533267\pi$
$828$	1.00000	0.0347524
$829$	−52.0000	−1.80603	−0.903017	−	0.429604i	$-0.858653\pi$
$829$	−52.0000	−1.80603	−0.903017	+	0.429604i	$0.858653\pi$
$830$	−12.0000	−0.416526
$831$	−14.0000	−0.485655
$832$	−5.00000	−0.173344
$833$	0	0
$834$	4.00000	0.138509
$835$	11.0000	0.380671
$836$	35.0000	1.21050
$837$	2.00000	0.0691301
$838$	15.0000	0.518166
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	10.0000	0.344623
$843$	−7.00000	−0.241093
$844$	17.0000	0.585164
$845$	12.0000	0.412813
$846$	−11.0000	−0.378188
$847$	0	0
$848$	−9.00000	−0.309061
$849$	10.0000	0.343199
$850$	−4.00000	−0.137199
$851$	1.00000	0.0342796
$852$	−2.00000	−0.0685189
$853$	−11.0000	−0.376633	−0.188316	−	0.982108i	$-0.560303\pi$
$853$	−11.0000	−0.376633	−0.188316	+	0.982108i	$0.560303\pi$
$854$	0	0
$855$	−7.00000	−0.239395
$856$	−2.00000	−0.0683586
$857$	2.00000	0.0683187	0.0341593	−	0.999416i	$-0.489125\pi$
$857$	2.00000	0.0683187	0.0341593	+	0.999416i	$0.489125\pi$
$858$	−25.0000	−0.853486
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	12.0000	0.409197
$861$	0	0
$862$	12.0000	0.408722
$863$	45.0000	1.53182	0.765909	−	0.642949i	$-0.222289\pi$
$863$	45.0000	1.53182	0.765909	+	0.642949i	$0.222289\pi$
$864$	−1.00000	−0.0340207
$865$	−13.0000	−0.442013
$866$	24.0000	0.815553
$867$	1.00000	0.0339618
$868$	0	0
$869$	60.0000	2.03536
$870$	0	0
$871$	60.0000	2.03302
$872$	−2.00000	−0.0677285
$873$	−8.00000	−0.270759
$874$	−7.00000	−0.236779
$875$	0	0
$876$	−10.0000	−0.337869
$877$	−37.0000	−1.24940	−0.624701	−	0.780864i	$-0.714779\pi$
$877$	−37.0000	−1.24940	−0.624701	+	0.780864i	$0.714779\pi$
$878$	−40.0000	−1.34993
$879$	−9.00000	−0.303562
$880$	−5.00000	−0.168550
$881$	3.00000	0.101073	0.0505363	−	0.998722i	$-0.483907\pi$
$881$	3.00000	0.101073	0.0505363	+	0.998722i	$0.483907\pi$
$882$	0	0
$883$	−50.0000	−1.68263	−0.841317	−	0.540542i	$-0.818219\pi$
$883$	−50.0000	−1.68263	−0.841317	+	0.540542i	$0.818219\pi$
$884$	20.0000	0.672673
$885$	−4.00000	−0.134459
$886$	−28.0000	−0.940678
$887$	−44.0000	−1.47738	−0.738688	−	0.674048i	$-0.764554\pi$
$887$	−44.0000	−1.47738	−0.738688	+	0.674048i	$0.764554\pi$
$888$	−1.00000	−0.0335578
$889$	0	0
$890$	14.0000	0.469281
$891$	−5.00000	−0.167506
$892$	−12.0000	−0.401790
$893$	77.0000	2.57671
$894$	−12.0000	−0.401340
$895$	−23.0000	−0.768805
$896$	0	0
$897$	5.00000	0.166945
$898$	−29.0000	−0.967743
$899$	0	0
$900$	1.00000	0.0333333
$901$	36.0000	1.19933
$902$	−25.0000	−0.832409
$903$	0	0
$904$	−14.0000	−0.465633
$905$	20.0000	0.664822
$906$	−14.0000	−0.465119
$907$	−18.0000	−0.597680	−0.298840	−	0.954303i	$-0.596600\pi$
$907$	−18.0000	−0.597680	−0.298840	+	0.954303i	$0.596600\pi$
$908$	−12.0000	−0.398234
$909$	0	0
$910$	0	0
$911$	30.0000	0.993944	0.496972	−	0.867766i	$-0.334445\pi$
$911$	30.0000	0.993944	0.496972	+	0.867766i	$0.334445\pi$
$912$	7.00000	0.231793
$913$	60.0000	1.98571
$914$	−14.0000	−0.463079
$915$	−4.00000	−0.132236
$916$	10.0000	0.330409
$917$	0	0
$918$	4.00000	0.132020
$919$	28.0000	0.923635	0.461817	−	0.886975i	$-0.347198\pi$
$919$	28.0000	0.923635	0.461817	+	0.886975i	$0.347198\pi$
$920$	1.00000	0.0329690
$921$	−8.00000	−0.263609
$922$	−4.00000	−0.131733
$923$	−10.0000	−0.329154
$924$	0	0
$925$	1.00000	0.0328798
$926$	−19.0000	−0.624379
$927$	8.00000	0.262754
$928$	0	0
$929$	19.0000	0.623370	0.311685	−	0.950186i	$-0.399107\pi$
$929$	19.0000	0.623370	0.311685	+	0.950186i	$0.399107\pi$
$930$	2.00000	0.0655826
$931$	0	0
$932$	−14.0000	−0.458585
$933$	−8.00000	−0.261908
$934$	−20.0000	−0.654420
$935$	20.0000	0.654070
$936$	−5.00000	−0.163430
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	0	0
$939$	−16.0000	−0.522140
$940$	−11.0000	−0.358780
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	−11.0000	−0.358399
$943$	5.00000	0.162822
$944$	4.00000	0.130189
$945$	0	0
$946$	−60.0000	−1.95077
$947$	−42.0000	−1.36482	−0.682408	−	0.730971i	$-0.739067\pi$
$947$	−42.0000	−1.36482	−0.682408	+	0.730971i	$0.739067\pi$
$948$	12.0000	0.389742
$949$	−50.0000	−1.62307
$950$	−7.00000	−0.227110
$951$	22.0000	0.713399
$952$	0	0
$953$	12.0000	0.388718	0.194359	−	0.980930i	$-0.437737\pi$
$953$	12.0000	0.388718	0.194359	+	0.980930i	$0.437737\pi$
$954$	−9.00000	−0.291386
$955$	14.0000	0.453029
$956$	−22.0000	−0.711531
$957$	0	0
$958$	18.0000	0.581554
$959$	0	0
$960$	−1.00000	−0.0322749
$961$	−27.0000	−0.870968
$962$	−5.00000	−0.161206
$963$	−2.00000	−0.0644491
$964$	15.0000	0.483117
$965$	10.0000	0.321911
$966$	0	0
$967$	4.00000	0.128631	0.0643157	−	0.997930i	$-0.479514\pi$
$967$	4.00000	0.128631	0.0643157	+	0.997930i	$0.479514\pi$
$968$	14.0000	0.449977
$969$	−28.0000	−0.899490
$970$	−8.00000	−0.256865
$971$	15.0000	0.481373	0.240686	−	0.970603i	$-0.422627\pi$
$971$	15.0000	0.481373	0.240686	+	0.970603i	$0.422627\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	0	0
$975$	5.00000	0.160128
$976$	4.00000	0.128037
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	24.0000	0.767435
$979$	−70.0000	−2.23721
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	−36.0000	−1.14881
$983$	−25.0000	−0.797376	−0.398688	−	0.917087i	$-0.630534\pi$
$983$	−25.0000	−0.797376	−0.398688	+	0.917087i	$0.630534\pi$
$984$	−5.00000	−0.159394
$985$	−3.00000	−0.0955879
$986$	0	0
$987$	0	0
$988$	35.0000	1.11350
$989$	12.0000	0.381578
$990$	−5.00000	−0.158910
$991$	−38.0000	−1.20711	−0.603555	−	0.797321i	$-0.706250\pi$
$991$	−38.0000	−1.20711	−0.603555	+	0.797321i	$0.706250\pi$
$992$	−2.00000	−0.0635001
$993$	−31.0000	−0.983755
$994$	0	0
$995$	−4.00000	−0.126809
$996$	12.0000	0.380235
$997$	−22.0000	−0.696747	−0.348373	−	0.937356i	$-0.613266\pi$
$997$	−22.0000	−0.696747	−0.348373	+	0.937356i	$0.613266\pi$
$998$	40.0000	1.26618
$999$	−1.00000	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1470.2.a.l.1.1		1
3.2	odd	2		4410.2.a.j.1.1		1
5.4	even	2		7350.2.a.u.1.1		1
7.2	even	3		210.2.i.b.151.1	yes	2
7.3	odd	6		1470.2.i.e.961.1		2
7.4	even	3		210.2.i.b.121.1	✓	2
7.5	odd	6		1470.2.i.e.361.1		2
7.6	odd	2		1470.2.a.o.1.1		1
21.2	odd	6		630.2.k.g.361.1		2
21.11	odd	6		630.2.k.g.541.1		2
21.20	even	2		4410.2.a.u.1.1		1
28.11	odd	6		1680.2.bg.d.961.1		2
28.23	odd	6		1680.2.bg.d.1201.1		2
35.2	odd	12		1050.2.o.g.949.2		4
35.4	even	6		1050.2.i.p.751.1		2
35.9	even	6		1050.2.i.p.151.1		2
35.18	odd	12		1050.2.o.g.499.2		4
35.23	odd	12		1050.2.o.g.949.1		4
35.32	odd	12		1050.2.o.g.499.1		4
35.34	odd	2		7350.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.i.b.121.1	✓	2	7.4	even	3
210.2.i.b.151.1	yes	2	7.2	even	3
630.2.k.g.361.1		2	21.2	odd	6
630.2.k.g.541.1		2	21.11	odd	6
1050.2.i.p.151.1		2	35.9	even	6
1050.2.i.p.751.1		2	35.4	even	6
1050.2.o.g.499.1		4	35.32	odd	12
1050.2.o.g.499.2		4	35.18	odd	12
1050.2.o.g.949.1		4	35.23	odd	12
1050.2.o.g.949.2		4	35.2	odd	12
1470.2.a.l.1.1		1	1.1	even	1	trivial
1470.2.a.o.1.1		1	7.6	odd	2
1470.2.i.e.361.1		2	7.5	odd	6
1470.2.i.e.961.1		2	7.3	odd	6
1680.2.bg.d.961.1		2	28.11	odd	6
1680.2.bg.d.1201.1		2	28.23	odd	6
4410.2.a.j.1.1		1	3.2	odd	2
4410.2.a.u.1.1		1	21.20	even	2
7350.2.a.a.1.1		1	35.34	odd	2
7350.2.a.u.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

Related objects

Downloads

Learn more