LMFDB - Embedded newform 1470.2.a.o.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:	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	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	1.00000	0.235702
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$20$	−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	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	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	$-0.627675\pi$
$41$	−5.00000	−0.780869	−0.390434	+	0.920631i	$0.627675\pi$
$42$	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	$-0.203632\pi$
$47$	11.0000	1.60451	0.802257	+	0.596978i	$0.203632\pi$
$48$	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	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	−1.00000	−0.129099
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	2.00000	0.254000
$63$	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	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	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	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	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	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	−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	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	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	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	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	$-0.371379\pi$
$131$	9.00000	0.786334	0.393167	+	0.919467i	$0.371379\pi$
$132$	−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	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	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	$-0.644649\pi$
$157$	−11.0000	−0.877896	−0.438948	+	0.898513i	$0.644649\pi$
$158$	−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	$-0.639938\pi$
$167$	−11.0000	−0.851206	−0.425603	+	0.904910i	$0.639938\pi$
$168$	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	$-0.335466\pi$
$173$	13.0000	0.988372	0.494186	+	0.869356i	$0.335466\pi$
$174$	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	$-0.766738\pi$
$181$	−20.0000	−1.48659	−0.743294	+	0.668965i	$0.766738\pi$
$182$	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	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	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	$-0.368388\pi$
$223$	12.0000	0.803579	0.401790	+	0.915732i	$0.368388\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	−14.0000	−0.931266
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	7.00000	0.463586
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	−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	$-0.660496\pi$
$241$	−15.0000	−0.966235	−0.483117	+	0.875556i	$0.660496\pi$
$242$	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	$-0.510047\pi$
$251$	−1.00000	−0.0631194	−0.0315597	+	0.999502i	$0.510047\pi$
$252$	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	$-0.333691\pi$
$257$	16.0000	0.998053	0.499026	+	0.866587i	$0.333691\pi$
$258$	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	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	−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	$-0.403941\pi$
$283$	10.0000	0.594438	0.297219	+	0.954809i	$0.403941\pi$
$284$	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	$-0.584677\pi$
$293$	−9.00000	−0.525786	−0.262893	+	0.964825i	$0.584677\pi$
$294$	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	$-0.573314\pi$
$307$	−8.00000	−0.456584	−0.228292	+	0.973593i	$0.573314\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	−2.00000	−0.113592
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	5.00000	0.283069
$313$	−16.0000	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$314$	−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	$-0.465857\pi$
$349$	4.00000	0.214115	0.107058	+	0.994253i	$0.465857\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	−5.00000	−0.266501
$353$	−24.0000	−1.27739	−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000	−1.27739	−0.638696	+	0.769460i	$0.720526\pi$
$354$	−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	$-0.558483\pi$
$367$	−7.00000	−0.365397	−0.182699	+	0.983169i	$0.558483\pi$
$368$	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	$-0.680263\pi$
$383$	−21.0000	−1.07305	−0.536525	+	0.843884i	$0.680263\pi$
$384$	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	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	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	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	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	$-0.619409\pi$
$419$	−15.0000	−0.732798	−0.366399	+	0.930458i	$0.619409\pi$
$420$	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	$-0.695653\pi$
$433$	−24.0000	−1.15337	−0.576683	+	0.816968i	$0.695653\pi$
$434$	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	$-0.0963387\pi$
$439$	40.0000	1.90910	0.954548	+	0.298057i	$0.0963387\pi$
$440$	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	$-0.470307\pi$
$461$	4.00000	0.186299	0.0931493	+	0.995652i	$0.470307\pi$
$462$	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	$-0.346865\pi$
$467$	20.0000	0.925490	0.462745	+	0.886492i	$0.346865\pi$
$468$	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	$-0.634897\pi$
$479$	−18.0000	−0.822441	−0.411220	+	0.911536i	$0.634897\pi$
$480$	−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	$-0.352892\pi$
$503$	20.0000	0.891756	0.445878	+	0.895094i	$0.352892\pi$
$504$	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	$-0.428865\pi$
$509$	10.0000	0.443242	0.221621	+	0.975133i	$0.428865\pi$
$510$	−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	$-0.242819\pi$
$521$	33.0000	1.44576	0.722878	+	0.690976i	$0.242819\pi$
$522$	0	0
$523$	22.0000	0.961993	0.480996	−	0.876723i	$-0.340275\pi$
$523$	22.0000	0.961993	0.480996	+	0.876723i	$0.340275\pi$
$524$	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	$-0.376163\pi$
$563$	18.0000	0.758610	0.379305	+	0.925272i	$0.376163\pi$
$564$	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	$-0.682029\pi$
$577$	−26.0000	−1.08239	−0.541197	+	0.840896i	$0.682029\pi$
$578$	−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	$-0.287491\pi$
$587$	30.0000	1.23823	0.619116	+	0.785299i	$0.287491\pi$
$588$	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	$-0.764783\pi$
$593$	−36.0000	−1.47834	−0.739171	+	0.673517i	$0.764783\pi$
$594$	−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	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	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	$-0.315409\pi$
$607$	27.0000	1.09590	0.547948	+	0.836512i	$0.315409\pi$
$608$	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	$-0.667556\pi$
$619$	−25.0000	−1.00483	−0.502417	+	0.864625i	$0.667556\pi$
$620$	−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	$-0.733884\pi$
$643$	−34.0000	−1.34083	−0.670415	+	0.741987i	$0.733884\pi$
$644$	0	0
$645$	−12.0000	−0.472500
$646$	28.0000	1.10165
$647$	23.0000	0.904223	0.452112	−	0.891961i	$-0.350671\pi$
$647$	23.0000	0.904223	0.452112	+	0.891961i	$0.350671\pi$
$648$	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	$-0.216280\pi$
$661$	40.0000	1.55582	0.777910	+	0.628376i	$0.216280\pi$
$662$	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	$-0.718648\pi$
$677$	−33.0000	−1.26829	−0.634147	+	0.773213i	$0.718648\pi$
$678$	−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	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	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	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	−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	$-0.482283\pi$
$727$	3.00000	0.111264	0.0556319	+	0.998451i	$0.482283\pi$
$728$	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	$-0.553153\pi$
$733$	−9.00000	−0.332423	−0.166211	+	0.986090i	$0.553153\pi$
$734$	−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	$-0.266026\pi$
$761$	37.0000	1.34125	0.670624	+	0.741797i	$0.266026\pi$
$762$	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	$-0.471265\pi$
$769$	5.00000	0.180305	0.0901523	+	0.995928i	$0.471265\pi$
$770$	0	0
$771$	16.0000	0.576226
$772$	10.0000	0.359908
$773$	5.00000	0.179838	0.0899188	−	0.995949i	$-0.471339\pi$
$773$	5.00000	0.179838	0.0899188	+	0.995949i	$0.471339\pi$
$774$	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	$-0.511349\pi$
$787$	−2.00000	−0.0712923	−0.0356462	+	0.999364i	$0.511349\pi$
$788$	−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	$-0.0943526\pi$
$797$	54.0000	1.91278	0.956389	+	0.292096i	$0.0943526\pi$
$798$	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	$-0.620199\pi$
$811$	−21.0000	−0.737410	−0.368705	+	0.929547i	$0.620199\pi$
$812$	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	$-0.141347\pi$
$829$	52.0000	1.80603	0.903017	+	0.429604i	$0.141347\pi$
$830$	−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	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	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	$-0.439697\pi$
$853$	11.0000	0.376633	0.188316	+	0.982108i	$0.439697\pi$
$854$	0	0
$855$	−7.00000	−0.239395
$856$	−2.00000	−0.0683586
$857$	−2.00000	−0.0683187	−0.0341593	−	0.999416i	$-0.510875\pi$
$857$	−2.00000	−0.0683187	−0.0341593	+	0.999416i	$0.510875\pi$
$858$	−25.0000	−0.853486
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	−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	$-0.516093\pi$
$881$	−3.00000	−0.101073	−0.0505363	+	0.998722i	$0.516093\pi$
$882$	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	$-0.235446\pi$
$887$	44.0000	1.47738	0.738688	+	0.674048i	$0.235446\pi$
$888$	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	$-0.600893\pi$
$929$	−19.0000	−0.623370	−0.311685	+	0.950186i	$0.600893\pi$
$930$	−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	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	−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	$-0.577373\pi$
$971$	−15.0000	−0.481373	−0.240686	+	0.970603i	$0.577373\pi$
$972$	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	$-0.369466\pi$
$983$	25.0000	0.797376	0.398688	+	0.917087i	$0.369466\pi$
$984$	−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	$-0.386734\pi$
$997$	22.0000	0.696747	0.348373	+	0.937356i	$0.386734\pi$
$998$	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.o.1.1		1
3.2	odd	2		4410.2.a.u.1.1		1
5.4	even	2		7350.2.a.a.1.1		1
7.2	even	3		1470.2.i.e.361.1		2
7.3	odd	6		210.2.i.b.121.1	✓	2
7.4	even	3		1470.2.i.e.961.1		2
7.5	odd	6		210.2.i.b.151.1	yes	2
7.6	odd	2		1470.2.a.l.1.1		1
21.5	even	6		630.2.k.g.361.1		2
21.17	even	6		630.2.k.g.541.1		2
21.20	even	2		4410.2.a.j.1.1		1
28.3	even	6		1680.2.bg.d.961.1		2
28.19	even	6		1680.2.bg.d.1201.1		2
35.3	even	12		1050.2.o.g.499.2		4
35.12	even	12		1050.2.o.g.949.2		4
35.17	even	12		1050.2.o.g.499.1		4
35.19	odd	6		1050.2.i.p.151.1		2
35.24	odd	6		1050.2.i.p.751.1		2
35.33	even	12		1050.2.o.g.949.1		4
35.34	odd	2		7350.2.a.u.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.i.b.121.1	✓	2	7.3	odd	6
210.2.i.b.151.1	yes	2	7.5	odd	6
630.2.k.g.361.1		2	21.5	even	6
630.2.k.g.541.1		2	21.17	even	6
1050.2.i.p.151.1		2	35.19	odd	6
1050.2.i.p.751.1		2	35.24	odd	6
1050.2.o.g.499.1		4	35.17	even	12
1050.2.o.g.499.2		4	35.3	even	12
1050.2.o.g.949.1		4	35.33	even	12
1050.2.o.g.949.2		4	35.12	even	12
1470.2.a.l.1.1		1	7.6	odd	2
1470.2.a.o.1.1		1	1.1	even	1	trivial
1470.2.i.e.361.1		2	7.2	even	3
1470.2.i.e.961.1		2	7.4	even	3
1680.2.bg.d.961.1		2	28.3	even	6
1680.2.bg.d.1201.1		2	28.19	even	6
4410.2.a.j.1.1		1	21.20	even	2
4410.2.a.u.1.1		1	3.2	odd	2
7350.2.a.a.1.1		1	5.4	even	2
7350.2.a.u.1.1		1	35.34	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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