LMFDB - Embedded newform 882.6.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,6,Mod(1,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("882.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	882.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:	$141.458529075$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	882.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} +16.0000 q^{4} +86.0000 q^{5} +64.0000 q^{8} +344.000 q^{10} -34.0000 q^{11} +3.00000 q^{13} +256.000 q^{16} -1904.00 q^{17} +1489.00 q^{19} +1376.00 q^{20} -136.000 q^{22} +224.000 q^{23} +4271.00 q^{25} +12.0000 q^{26} +6508.00 q^{29} -1731.00 q^{31} +1024.00 q^{32} -7616.00 q^{34} -7633.00 q^{37} +5956.00 q^{38} +5504.00 q^{40} +15414.0 q^{41} +18491.0 q^{43} -544.000 q^{44} +896.000 q^{46} +18462.0 q^{47} +17084.0 q^{50} +48.0000 q^{52} +19956.0 q^{53} -2924.00 q^{55} +26032.0 q^{58} -31828.0 q^{59} +57654.0 q^{61} -6924.00 q^{62} +4096.00 q^{64} +258.000 q^{65} -60563.0 q^{67} -30464.0 q^{68} +44834.0 q^{71} -20821.0 q^{73} -30532.0 q^{74} +23824.0 q^{76} -30531.0 q^{79} +22016.0 q^{80} +61656.0 q^{82} +110602. q^{83} -163744. q^{85} +73964.0 q^{86} -2176.00 q^{88} -58992.0 q^{89} +3584.00 q^{92} +73848.0 q^{94} +128054. q^{95} +119846. q^{97} +O(q^{100})$

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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	86.0000	1.53841	0.769207	−	0.638999i	$-0.220651\pi$
$5$	86.0000	1.53841	0.769207	+	0.638999i	$0.220651\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	64.0000	0.353553
$9$	0	0
$10$	344.000	1.08782
$11$	−34.0000	−0.0847222	−0.0423611	−	0.999102i	$-0.513488\pi$
$11$	−34.0000	−0.0847222	−0.0423611	+	0.999102i	$0.513488\pi$
$12$	0	0
$13$	3.00000	0.00492337	0.00246169	−	0.999997i	$-0.499216\pi$
$13$	3.00000	0.00492337	0.00246169	+	0.999997i	$0.499216\pi$
$14$	0	0
$15$	0	0
$16$	256.000	0.250000
$17$	−1904.00	−1.59788	−0.798941	−	0.601410i	$-0.794606\pi$
$17$	−1904.00	−1.59788	−0.798941	+	0.601410i	$0.794606\pi$
$18$	0	0
$19$	1489.00	0.946260	0.473130	−	0.880992i	$-0.343124\pi$
$19$	1489.00	0.946260	0.473130	+	0.880992i	$0.343124\pi$
$20$	1376.00	0.769207
$21$	0	0
$22$	−136.000	−0.0599076
$23$	224.000	0.0882934	0.0441467	−	0.999025i	$-0.485943\pi$
$23$	224.000	0.0882934	0.0441467	+	0.999025i	$0.485943\pi$
$24$	0	0
$25$	4271.00	1.36672
$26$	12.0000	0.00348135
$27$	0	0
$28$	0	0
$29$	6508.00	1.43699	0.718493	−	0.695534i	$-0.244832\pi$
$29$	6508.00	1.43699	0.718493	+	0.695534i	$0.244832\pi$
$30$	0	0
$31$	−1731.00	−0.323514	−0.161757	−	0.986831i	$-0.551716\pi$
$31$	−1731.00	−0.323514	−0.161757	+	0.986831i	$0.551716\pi$
$32$	1024.00	0.176777
$33$	0	0
$34$	−7616.00	−1.12987
$35$	0	0
$36$	0	0
$37$	−7633.00	−0.916623	−0.458312	−	0.888792i	$-0.651546\pi$
$37$	−7633.00	−0.916623	−0.458312	+	0.888792i	$0.651546\pi$
$38$	5956.00	0.669107
$39$	0	0
$40$	5504.00	0.543912
$41$	15414.0	1.43204	0.716021	−	0.698079i	$-0.245961\pi$
$41$	15414.0	1.43204	0.716021	+	0.698079i	$0.245961\pi$
$42$	0	0
$43$	18491.0	1.52507	0.762534	−	0.646948i	$-0.223955\pi$
$43$	18491.0	1.52507	0.762534	+	0.646948i	$0.223955\pi$
$44$	−544.000	−0.0423611
$45$	0	0
$46$	896.000	0.0624329
$47$	18462.0	1.21909	0.609543	−	0.792753i	$-0.291353\pi$
$47$	18462.0	1.21909	0.609543	+	0.792753i	$0.291353\pi$
$48$	0	0
$49$	0	0
$50$	17084.0	0.966417
$51$	0	0
$52$	48.0000	0.00246169
$53$	19956.0	0.975852	0.487926	−	0.872885i	$-0.337754\pi$
$53$	19956.0	0.975852	0.487926	+	0.872885i	$0.337754\pi$
$54$	0	0
$55$	−2924.00	−0.130338
$56$	0	0
$57$	0	0
$58$	26032.0	1.01610
$59$	−31828.0	−1.19036	−0.595181	−	0.803591i	$-0.702920\pi$
$59$	−31828.0	−1.19036	−0.595181	+	0.803591i	$0.702920\pi$
$60$	0	0
$61$	57654.0	1.98383	0.991916	−	0.126897i	$-0.0405017\pi$
$61$	57654.0	1.98383	0.991916	+	0.126897i	$0.0405017\pi$
$62$	−6924.00	−0.228759
$63$	0	0
$64$	4096.00	0.125000
$65$	258.000	0.00757419
$66$	0	0
$67$	−60563.0	−1.64824	−0.824120	−	0.566415i	$-0.808330\pi$
$67$	−60563.0	−1.64824	−0.824120	+	0.566415i	$0.808330\pi$
$68$	−30464.0	−0.798941
$69$	0	0
$70$	0	0
$71$	44834.0	1.05551	0.527754	−	0.849397i	$-0.323034\pi$
$71$	44834.0	1.05551	0.527754	+	0.849397i	$0.323034\pi$
$72$	0	0
$73$	−20821.0	−0.457293	−0.228646	−	0.973510i	$-0.573430\pi$
$73$	−20821.0	−0.457293	−0.228646	+	0.973510i	$0.573430\pi$
$74$	−30532.0	−0.648151
$75$	0	0
$76$	23824.0	0.473130
$77$	0	0
$78$	0	0
$79$	−30531.0	−0.550394	−0.275197	−	0.961388i	$-0.588743\pi$
$79$	−30531.0	−0.550394	−0.275197	+	0.961388i	$0.588743\pi$
$80$	22016.0	0.384604
$81$	0	0
$82$	61656.0	1.01261
$83$	110602.	1.76225	0.881125	−	0.472883i	$-0.156787\pi$
$83$	110602.	1.76225	0.881125	+	0.472883i	$0.156787\pi$
$84$	0	0
$85$	−163744.	−2.45820
$86$	73964.0	1.07839
$87$	0	0
$88$	−2176.00	−0.0299538
$89$	−58992.0	−0.789438	−0.394719	−	0.918802i	$-0.629158\pi$
$89$	−58992.0	−0.789438	−0.394719	+	0.918802i	$0.629158\pi$
$90$	0	0
$91$	0	0
$92$	3584.00	0.0441467
$93$	0	0
$94$	73848.0	0.862023
$95$	128054.	1.45574
$96$	0	0
$97$	119846.	1.29328	0.646642	−	0.762793i	$-0.276173\pi$
$97$	119846.	1.29328	0.646642	+	0.762793i	$0.276173\pi$
$98$	0	0
$99$	0	0
$100$	68336.0	0.683360
$101$	100010.	0.975529	0.487764	−	0.872975i	$-0.337813\pi$
$101$	100010.	0.975529	0.487764	+	0.872975i	$0.337813\pi$
$102$	0	0
$103$	−121691.	−1.13023	−0.565113	−	0.825013i	$-0.691167\pi$
$103$	−121691.	−1.13023	−0.565113	+	0.825013i	$0.691167\pi$
$104$	192.000	0.00174068
$105$	0	0
$106$	79824.0	0.690031
$107$	48648.0	0.410776	0.205388	−	0.978681i	$-0.434154\pi$
$107$	48648.0	0.410776	0.205388	+	0.978681i	$0.434154\pi$
$108$	0	0
$109$	−152075.	−1.22600	−0.613002	−	0.790082i	$-0.710038\pi$
$109$	−152075.	−1.22600	−0.613002	+	0.790082i	$0.710038\pi$
$110$	−11696.0	−0.0921628
$111$	0	0
$112$	0	0
$113$	60886.0	0.448561	0.224280	−	0.974525i	$-0.427997\pi$
$113$	60886.0	0.448561	0.224280	+	0.974525i	$0.427997\pi$
$114$	0	0
$115$	19264.0	0.135832
$116$	104128.	0.718493
$117$	0	0
$118$	−127312.	−0.841714
$119$	0	0
$120$	0	0
$121$	−159895.	−0.992822
$122$	230616.	1.40278
$123$	0	0
$124$	−27696.0	−0.161757
$125$	98556.0	0.564167
$126$	0	0
$127$	−151965.	−0.836054	−0.418027	−	0.908435i	$-0.637278\pi$
$127$	−151965.	−0.836054	−0.418027	+	0.908435i	$0.637278\pi$
$128$	16384.0	0.0883883
$129$	0	0
$130$	1032.00	0.00535576
$131$	235502.	1.19899	0.599496	−	0.800378i	$-0.295368\pi$
$131$	235502.	1.19899	0.599496	+	0.800378i	$0.295368\pi$
$132$	0	0
$133$	0	0
$134$	−242252.	−1.16548
$135$	0	0
$136$	−121856.	−0.564937
$137$	−325508.	−1.48170	−0.740850	−	0.671671i	$-0.765577\pi$
$137$	−325508.	−1.48170	−0.740850	+	0.671671i	$0.765577\pi$
$138$	0	0
$139$	−3211.00	−0.0140962	−0.00704812	−	0.999975i	$-0.502244\pi$
$139$	−3211.00	−0.0140962	−0.00704812	+	0.999975i	$0.502244\pi$
$140$	0	0
$141$	0	0
$142$	179336.	0.746357
$143$	−102.000	−0.000417119 0
$144$	0	0
$145$	559688.	2.21068
$146$	−83284.0	−0.323355
$147$	0	0
$148$	−122128.	−0.458312
$149$	151884.	0.560462	0.280231	−	0.959933i	$-0.409589\pi$
$149$	151884.	0.560462	0.280231	+	0.959933i	$0.409589\pi$
$150$	0	0
$151$	76648.0	0.273564	0.136782	−	0.990601i	$-0.456324\pi$
$151$	76648.0	0.273564	0.136782	+	0.990601i	$0.456324\pi$
$152$	95296.0	0.334554
$153$	0	0
$154$	0	0
$155$	−148866.	−0.497698
$156$	0	0
$157$	389710.	1.26181	0.630903	−	0.775862i	$-0.282685\pi$
$157$	389710.	1.26181	0.630903	+	0.775862i	$0.282685\pi$
$158$	−122124.	−0.389187
$159$	0	0
$160$	88064.0	0.271956
$161$	0	0
$162$	0	0
$163$	−112372.	−0.331275	−0.165638	−	0.986187i	$-0.552968\pi$
$163$	−112372.	−0.331275	−0.165638	+	0.986187i	$0.552968\pi$
$164$	246624.	0.716021
$165$	0	0
$166$	442408.	1.24610
$167$	52550.0	0.145808	0.0729040	−	0.997339i	$-0.476773\pi$
$167$	52550.0	0.145808	0.0729040	+	0.997339i	$0.476773\pi$
$168$	0	0
$169$	−371284.	−0.999976
$170$	−654976.	−1.73821
$171$	0	0
$172$	295856.	0.762534
$173$	−135256.	−0.343591	−0.171795	−	0.985133i	$-0.554957\pi$
$173$	−135256.	−0.343591	−0.171795	+	0.985133i	$0.554957\pi$
$174$	0	0
$175$	0	0
$176$	−8704.00	−0.0211805
$177$	0	0
$178$	−235968.	−0.558217
$179$	−250638.	−0.584675	−0.292337	−	0.956315i	$-0.594433\pi$
$179$	−250638.	−0.584675	−0.292337	+	0.956315i	$0.594433\pi$
$180$	0	0
$181$	−199233.	−0.452027	−0.226014	−	0.974124i	$-0.572569\pi$
$181$	−199233.	−0.452027	−0.226014	+	0.974124i	$0.572569\pi$
$182$	0	0
$183$	0	0
$184$	14336.0	0.0312164
$185$	−656438.	−1.41015
$186$	0	0
$187$	64736.0	0.135376
$188$	295392.	0.609543
$189$	0	0
$190$	512216.	1.02936
$191$	238770.	0.473583	0.236792	−	0.971560i	$-0.423904\pi$
$191$	238770.	0.473583	0.236792	+	0.971560i	$0.423904\pi$
$192$	0	0
$193$	85691.0	0.165593	0.0827965	−	0.996566i	$-0.473615\pi$
$193$	85691.0	0.165593	0.0827965	+	0.996566i	$0.473615\pi$
$194$	479384.	0.914491
$195$	0	0
$196$	0	0
$197$	71408.0	0.131094	0.0655468	−	0.997849i	$-0.479121\pi$
$197$	71408.0	0.131094	0.0655468	+	0.997849i	$0.479121\pi$
$198$	0	0
$199$	711352.	1.27336	0.636681	−	0.771127i	$-0.280307\pi$
$199$	711352.	1.27336	0.636681	+	0.771127i	$0.280307\pi$
$200$	273344.	0.483208
$201$	0	0
$202$	400040.	0.689803
$203$	0	0
$204$	0	0
$205$	1.32560e6	2.20307
$206$	−486764.	−0.799191
$207$	0	0
$208$	768.000	0.00123084
$209$	−50626.0	−0.0801693
$210$	0	0
$211$	−260260.	−0.402440	−0.201220	−	0.979546i	$-0.564491\pi$
$211$	−260260.	−0.402440	−0.201220	+	0.979546i	$0.564491\pi$
$212$	319296.	0.487926
$213$	0	0
$214$	194592.	0.290463
$215$	1.59023e6	2.34619
$216$	0	0
$217$	0	0
$218$	−608300.	−0.866915
$219$	0	0
$220$	−46784.0	−0.0651689
$221$	−5712.00	−0.00786697
$222$	0	0
$223$	−105656.	−0.142276	−0.0711381	−	0.997466i	$-0.522663\pi$
$223$	−105656.	−0.142276	−0.0711381	+	0.997466i	$0.522663\pi$
$224$	0	0
$225$	0	0
$226$	243544.	0.317180
$227$	654750.	0.843356	0.421678	−	0.906746i	$-0.361441\pi$
$227$	654750.	0.843356	0.421678	+	0.906746i	$0.361441\pi$
$228$	0	0
$229$	−557713.	−0.702784	−0.351392	−	0.936228i	$-0.614292\pi$
$229$	−557713.	−0.702784	−0.351392	+	0.936228i	$0.614292\pi$
$230$	77056.0	0.0960477
$231$	0	0
$232$	416512.	0.508051
$233$	1.24759e6	1.50551	0.752755	−	0.658301i	$-0.228725\pi$
$233$	1.24759e6	1.50551	0.752755	+	0.658301i	$0.228725\pi$
$234$	0	0
$235$	1.58773e6	1.87546
$236$	−509248.	−0.595181
$237$	0	0
$238$	0	0
$239$	496926.	0.562726	0.281363	−	0.959601i	$-0.409214\pi$
$239$	496926.	0.562726	0.281363	+	0.959601i	$0.409214\pi$
$240$	0	0
$241$	277618.	0.307897	0.153948	−	0.988079i	$-0.450801\pi$
$241$	277618.	0.307897	0.153948	+	0.988079i	$0.450801\pi$
$242$	−639580.	−0.702031
$243$	0	0
$244$	922464.	0.991916
$245$	0	0
$246$	0	0
$247$	4467.00	0.00465879
$248$	−110784.	−0.114379
$249$	0	0
$250$	394224.	0.398927
$251$	−308328.	−0.308908	−0.154454	−	0.988000i	$-0.549362\pi$
$251$	−308328.	−0.308908	−0.154454	+	0.988000i	$0.549362\pi$
$252$	0	0
$253$	−7616.00	−0.00748041
$254$	−607860.	−0.591179
$255$	0	0
$256$	65536.0	0.0625000
$257$	408762.	0.386045	0.193022	−	0.981194i	$-0.438171\pi$
$257$	408762.	0.386045	0.193022	+	0.981194i	$0.438171\pi$
$258$	0	0
$259$	0	0
$260$	4128.00	0.00378710
$261$	0	0
$262$	942008.	0.847816
$263$	−1.08812e6	−0.970039	−0.485019	−	0.874503i	$-0.661187\pi$
$263$	−1.08812e6	−0.970039	−0.485019	+	0.874503i	$0.661187\pi$
$264$	0	0
$265$	1.71622e6	1.50126
$266$	0	0
$267$	0	0
$268$	−969008.	−0.824120
$269$	−668290.	−0.563098	−0.281549	−	0.959547i	$-0.590848\pi$
$269$	−668290.	−0.563098	−0.281549	+	0.959547i	$0.590848\pi$
$270$	0	0
$271$	830664.	0.687072	0.343536	−	0.939140i	$-0.388375\pi$
$271$	830664.	0.687072	0.343536	+	0.939140i	$0.388375\pi$
$272$	−487424.	−0.399470
$273$	0	0
$274$	−1.30203e6	−1.04772
$275$	−145214.	−0.115792
$276$	0	0
$277$	925073.	0.724397	0.362198	−	0.932101i	$-0.382026\pi$
$277$	925073.	0.724397	0.362198	+	0.932101i	$0.382026\pi$
$278$	−12844.0	−0.00996755
$279$	0	0
$280$	0	0
$281$	−1.33635e6	−1.00961	−0.504805	−	0.863233i	$-0.668436\pi$
$281$	−1.33635e6	−1.00961	−0.504805	+	0.863233i	$0.668436\pi$
$282$	0	0
$283$	992957.	0.736995	0.368497	−	0.929629i	$-0.379872\pi$
$283$	992957.	0.736995	0.368497	+	0.929629i	$0.379872\pi$
$284$	717344.	0.527754
$285$	0	0
$286$	−408.000	−0.000294948 0
$287$	0	0
$288$	0	0
$289$	2.20536e6	1.55323
$290$	2.23875e6	1.56319
$291$	0	0
$292$	−333136.	−0.228646
$293$	563544.	0.383494	0.191747	−	0.981444i	$-0.438585\pi$
$293$	563544.	0.383494	0.191747	+	0.981444i	$0.438585\pi$
$294$	0	0
$295$	−2.73721e6	−1.83127
$296$	−488512.	−0.324075
$297$	0	0
$298$	607536.	0.396307
$299$	672.000	0.000434702 0
$300$	0	0
$301$	0	0
$302$	306592.	0.193439
$303$	0	0
$304$	381184.	0.236565
$305$	4.95824e6	3.05196
$306$	0	0
$307$	−2.82703e6	−1.71193	−0.855963	−	0.517037i	$-0.827035\pi$
$307$	−2.82703e6	−1.71193	−0.855963	+	0.517037i	$0.827035\pi$
$308$	0	0
$309$	0	0
$310$	−595464.	−0.351926
$311$	−1.12731e6	−0.660912	−0.330456	−	0.943821i	$-0.607203\pi$
$311$	−1.12731e6	−0.660912	−0.330456	+	0.943821i	$0.607203\pi$
$312$	0	0
$313$	2.36013e6	1.36168	0.680840	−	0.732432i	$-0.261615\pi$
$313$	2.36013e6	1.36168	0.680840	+	0.732432i	$0.261615\pi$
$314$	1.55884e6	0.892231
$315$	0	0
$316$	−488496.	−0.275197
$317$	2.22420e6	1.24316	0.621578	−	0.783352i	$-0.286492\pi$
$317$	2.22420e6	1.24316	0.621578	+	0.783352i	$0.286492\pi$
$318$	0	0
$319$	−221272.	−0.121745
$320$	352256.	0.192302
$321$	0	0
$322$	0	0
$323$	−2.83506e6	−1.51201
$324$	0	0
$325$	12813.0	0.00672887
$326$	−449488.	−0.234247
$327$	0	0
$328$	986496.	0.506303
$329$	0	0
$330$	0	0
$331$	−3.70304e6	−1.85775	−0.928877	−	0.370388i	$-0.879225\pi$
$331$	−3.70304e6	−1.85775	−0.928877	+	0.370388i	$0.879225\pi$
$332$	1.76963e6	0.881125
$333$	0	0
$334$	210200.	0.103102
$335$	−5.20842e6	−2.53568
$336$	0	0
$337$	1.21432e6	0.582452	0.291226	−	0.956654i	$-0.405937\pi$
$337$	1.21432e6	0.582452	0.291226	+	0.956654i	$0.405937\pi$
$338$	−1.48514e6	−0.707090
$339$	0	0
$340$	−2.61990e6	−1.22910
$341$	58854.0	0.0274088
$342$	0	0
$343$	0	0
$344$	1.18342e6	0.539193
$345$	0	0
$346$	−541024.	−0.242955
$347$	977904.	0.435986	0.217993	−	0.975950i	$-0.430049\pi$
$347$	977904.	0.435986	0.217993	+	0.975950i	$0.430049\pi$
$348$	0	0
$349$	−511282.	−0.224697	−0.112348	−	0.993669i	$-0.535837\pi$
$349$	−511282.	−0.224697	−0.112348	+	0.993669i	$0.535837\pi$
$350$	0	0
$351$	0	0
$352$	−34816.0	−0.0149769
$353$	3.02752e6	1.29315	0.646577	−	0.762848i	$-0.276200\pi$
$353$	3.02752e6	1.29315	0.646577	+	0.762848i	$0.276200\pi$
$354$	0	0
$355$	3.85572e6	1.62381
$356$	−943872.	−0.394719
$357$	0	0
$358$	−1.00255e6	−0.413427
$359$	−4.59456e6	−1.88151	−0.940757	−	0.339082i	$-0.889884\pi$
$359$	−4.59456e6	−1.88151	−0.940757	+	0.339082i	$0.889884\pi$
$360$	0	0
$361$	−258978.	−0.104591
$362$	−796932.	−0.319632
$363$	0	0
$364$	0	0
$365$	−1.79061e6	−0.703506
$366$	0	0
$367$	1.11273e6	0.431246	0.215623	−	0.976477i	$-0.430822\pi$
$367$	1.11273e6	0.431246	0.215623	+	0.976477i	$0.430822\pi$
$368$	57344.0	0.0220734
$369$	0	0
$370$	−2.62575e6	−0.997125
$371$	0	0
$372$	0	0
$373$	−2.45895e6	−0.915119	−0.457559	−	0.889179i	$-0.651276\pi$
$373$	−2.45895e6	−0.915119	−0.457559	+	0.889179i	$0.651276\pi$
$374$	258944.	0.0957253
$375$	0	0
$376$	1.18157e6	0.431012
$377$	19524.0	0.00707482
$378$	0	0
$379$	−4.10130e6	−1.46664	−0.733320	−	0.679884i	$-0.762030\pi$
$379$	−4.10130e6	−1.46664	−0.733320	+	0.679884i	$0.762030\pi$
$380$	2.04886e6	0.727871
$381$	0	0
$382$	955080.	0.334874
$383$	−2.59413e6	−0.903639	−0.451820	−	0.892109i	$-0.649225\pi$
$383$	−2.59413e6	−0.903639	−0.451820	+	0.892109i	$0.649225\pi$
$384$	0	0
$385$	0	0
$386$	342764.	0.117092
$387$	0	0
$388$	1.91754e6	0.646642
$389$	−2.23407e6	−0.748552	−0.374276	−	0.927317i	$-0.622109\pi$
$389$	−2.23407e6	−0.748552	−0.374276	+	0.927317i	$0.622109\pi$
$390$	0	0
$391$	−426496.	−0.141082
$392$	0	0
$393$	0	0
$394$	285632.	0.0926971
$395$	−2.62567e6	−0.846733
$396$	0	0
$397$	2.06400e6	0.657253	0.328627	−	0.944460i	$-0.393414\pi$
$397$	2.06400e6	0.657253	0.328627	+	0.944460i	$0.393414\pi$
$398$	2.84541e6	0.900403
$399$	0	0
$400$	1.09338e6	0.341680
$401$	−1.15283e6	−0.358017	−0.179008	−	0.983848i	$-0.557289\pi$
$401$	−1.15283e6	−0.358017	−0.179008	+	0.983848i	$0.557289\pi$
$402$	0	0
$403$	−5193.00	−0.00159278
$404$	1.60016e6	0.487764
$405$	0	0
$406$	0	0
$407$	259522.	0.0776583
$408$	0	0
$409$	−5.93412e6	−1.75408	−0.877038	−	0.480421i	$-0.840484\pi$
$409$	−5.93412e6	−1.75408	−0.877038	+	0.480421i	$0.840484\pi$
$410$	5.30242e6	1.55781
$411$	0	0
$412$	−1.94706e6	−0.565113
$413$	0	0
$414$	0	0
$415$	9.51177e6	2.71107
$416$	3072.00	0.000870338 0
$417$	0	0
$418$	−202504.	−0.0566882
$419$	771666.	0.214731	0.107365	−	0.994220i	$-0.465759\pi$
$419$	771666.	0.214731	0.107365	+	0.994220i	$0.465759\pi$
$420$	0	0
$421$	−2.87542e6	−0.790671	−0.395336	−	0.918537i	$-0.629372\pi$
$421$	−2.87542e6	−0.790671	−0.395336	+	0.918537i	$0.629372\pi$
$422$	−1.04104e6	−0.284568
$423$	0	0
$424$	1.27718e6	0.345016
$425$	−8.13198e6	−2.18386
$426$	0	0
$427$	0	0
$428$	778368.	0.205388
$429$	0	0
$430$	6.36090e6	1.65901
$431$	137862.	0.0357480	0.0178740	−	0.999840i	$-0.494310\pi$
$431$	137862.	0.0357480	0.0178740	+	0.999840i	$0.494310\pi$
$432$	0	0
$433$	−1.56526e6	−0.401204	−0.200602	−	0.979673i	$-0.564290\pi$
$433$	−1.56526e6	−0.401204	−0.200602	+	0.979673i	$0.564290\pi$
$434$	0	0
$435$	0	0
$436$	−2.43320e6	−0.613002
$437$	333536.	0.0835486
$438$	0	0
$439$	−4.88158e6	−1.20892	−0.604462	−	0.796634i	$-0.706612\pi$
$439$	−4.88158e6	−1.20892	−0.604462	+	0.796634i	$0.706612\pi$
$440$	−187136.	−0.0460814
$441$	0	0
$442$	−22848.0	−0.00556279
$443$	1.30152e6	0.315094	0.157547	−	0.987511i	$-0.449641\pi$
$443$	1.30152e6	0.315094	0.157547	+	0.987511i	$0.449641\pi$
$444$	0	0
$445$	−5.07331e6	−1.21448
$446$	−422624.	−0.100604
$447$	0	0
$448$	0	0
$449$	3.13141e6	0.733034	0.366517	−	0.930411i	$-0.380550\pi$
$449$	3.13141e6	0.733034	0.366517	+	0.930411i	$0.380550\pi$
$450$	0	0
$451$	−524076.	−0.121326
$452$	974176.	0.224280
$453$	0	0
$454$	2.61900e6	0.596343
$455$	0	0
$456$	0	0
$457$	6.49268e6	1.45423	0.727116	−	0.686514i	$-0.240860\pi$
$457$	6.49268e6	1.45423	0.727116	+	0.686514i	$0.240860\pi$
$458$	−2.23085e6	−0.496944
$459$	0	0
$460$	308224.	0.0679160
$461$	5.34717e6	1.17185	0.585925	−	0.810365i	$-0.300731\pi$
$461$	5.34717e6	1.17185	0.585925	+	0.810365i	$0.300731\pi$
$462$	0	0
$463$	3.37285e6	0.731215	0.365607	−	0.930769i	$-0.380861\pi$
$463$	3.37285e6	0.731215	0.365607	+	0.930769i	$0.380861\pi$
$464$	1.66605e6	0.359247
$465$	0	0
$466$	4.99038e6	1.06456
$467$	2.23452e6	0.474125	0.237062	−	0.971494i	$-0.423815\pi$
$467$	2.23452e6	0.474125	0.237062	+	0.971494i	$0.423815\pi$
$468$	0	0
$469$	0	0
$470$	6.35093e6	1.32615
$471$	0	0
$472$	−2.03699e6	−0.420857
$473$	−628694.	−0.129207
$474$	0	0
$475$	6.35952e6	1.29327
$476$	0	0
$477$	0	0
$478$	1.98770e6	0.397907
$479$	−2.52136e6	−0.502108	−0.251054	−	0.967973i	$-0.580777\pi$
$479$	−2.52136e6	−0.502108	−0.251054	+	0.967973i	$0.580777\pi$
$480$	0	0
$481$	−22899.0	−0.00451288
$482$	1.11047e6	0.217716
$483$	0	0
$484$	−2.55832e6	−0.496411
$485$	1.03068e7	1.98961
$486$	0	0
$487$	1.91672e6	0.366215	0.183107	−	0.983093i	$-0.441384\pi$
$487$	1.91672e6	0.366215	0.183107	+	0.983093i	$0.441384\pi$
$488$	3.68986e6	0.701390
$489$	0	0
$490$	0	0
$491$	−5.82875e6	−1.09112	−0.545559	−	0.838073i	$-0.683683\pi$
$491$	−5.82875e6	−1.09112	−0.545559	+	0.838073i	$0.683683\pi$
$492$	0	0
$493$	−1.23912e7	−2.29613
$494$	17868.0	0.00329427
$495$	0	0
$496$	−443136.	−0.0808785
$497$	0	0
$498$	0	0
$499$	−1.00049e7	−1.79870	−0.899352	−	0.437225i	$-0.855961\pi$
$499$	−1.00049e7	−1.79870	−0.899352	+	0.437225i	$0.855961\pi$
$500$	1.57690e6	0.282084
$501$	0	0
$502$	−1.23331e6	−0.218431
$503$	1.13666e6	0.200313	0.100157	−	0.994972i	$-0.468066\pi$
$503$	1.13666e6	0.200313	0.100157	+	0.994972i	$0.468066\pi$
$504$	0	0
$505$	8.60086e6	1.50077
$506$	−30464.0	−0.00528945
$507$	0	0
$508$	−2.43144e6	−0.418027
$509$	4.01937e6	0.687644	0.343822	−	0.939035i	$-0.388278\pi$
$509$	4.01937e6	0.687644	0.343822	+	0.939035i	$0.388278\pi$
$510$	0	0
$511$	0	0
$512$	262144.	0.0441942
$513$	0	0
$514$	1.63505e6	0.272975
$515$	−1.04654e7	−1.73876
$516$	0	0
$517$	−627708.	−0.103284
$518$	0	0
$519$	0	0
$520$	16512.0	0.00267788
$521$	−5.52028e6	−0.890977	−0.445488	−	0.895288i	$-0.646970\pi$
$521$	−5.52028e6	−0.890977	−0.445488	+	0.895288i	$0.646970\pi$
$522$	0	0
$523$	−8.94472e6	−1.42992	−0.714961	−	0.699164i	$-0.753556\pi$
$523$	−8.94472e6	−1.42992	−0.714961	+	0.699164i	$0.753556\pi$
$524$	3.76803e6	0.599496
$525$	0	0
$526$	−4.35250e6	−0.685921
$527$	3.29582e6	0.516937
$528$	0	0
$529$	−6.38617e6	−0.992204
$530$	6.86486e6	1.06155
$531$	0	0
$532$	0	0
$533$	46242.0	0.00705048
$534$	0	0
$535$	4.18373e6	0.631945
$536$	−3.87603e6	−0.582741
$537$	0	0
$538$	−2.67316e6	−0.398171
$539$	0	0
$540$	0	0
$541$	849057.	0.124722	0.0623611	−	0.998054i	$-0.480137\pi$
$541$	849057.	0.124722	0.0623611	+	0.998054i	$0.480137\pi$
$542$	3.32266e6	0.485833
$543$	0	0
$544$	−1.94970e6	−0.282468
$545$	−1.30784e7	−1.88610
$546$	0	0
$547$	8.61340e6	1.23085	0.615426	−	0.788194i	$-0.288984\pi$
$547$	8.61340e6	1.23085	0.615426	+	0.788194i	$0.288984\pi$
$548$	−5.20813e6	−0.740850
$549$	0	0
$550$	−580856.	−0.0818770
$551$	9.69041e6	1.35976
$552$	0	0
$553$	0	0
$554$	3.70029e6	0.512226
$555$	0	0
$556$	−51376.0	−0.00704812
$557$	−7.79879e6	−1.06510	−0.532549	−	0.846399i	$-0.678766\pi$
$557$	−7.79879e6	−1.06510	−0.532549	+	0.846399i	$0.678766\pi$
$558$	0	0
$559$	55473.0	0.00750848
$560$	0	0
$561$	0	0
$562$	−5.34539e6	−0.713902
$563$	1.07271e6	0.142631	0.0713153	−	0.997454i	$-0.477280\pi$
$563$	1.07271e6	0.142631	0.0713153	+	0.997454i	$0.477280\pi$
$564$	0	0
$565$	5.23620e6	0.690073
$566$	3.97183e6	0.521134
$567$	0	0
$568$	2.86938e6	0.373179
$569$	1.01524e6	0.131459	0.0657293	−	0.997837i	$-0.479063\pi$
$569$	1.01524e6	0.131459	0.0657293	+	0.997837i	$0.479063\pi$
$570$	0	0
$571$	−6.78093e6	−0.870361	−0.435180	−	0.900343i	$-0.643315\pi$
$571$	−6.78093e6	−0.870361	−0.435180	+	0.900343i	$0.643315\pi$
$572$	−1632.00	−0.000208560 0
$573$	0	0
$574$	0	0
$575$	956704.	0.120672
$576$	0	0
$577$	3.30537e6	0.413314	0.206657	−	0.978413i	$-0.433742\pi$
$577$	3.30537e6	0.413314	0.206657	+	0.978413i	$0.433742\pi$
$578$	8.82144e6	1.09830
$579$	0	0
$580$	8.95501e6	1.10534
$581$	0	0
$582$	0	0
$583$	−678504.	−0.0826763
$584$	−1.33254e6	−0.161677
$585$	0	0
$586$	2.25418e6	0.271171
$587$	−1.19833e7	−1.43543	−0.717715	−	0.696337i	$-0.754812\pi$
$587$	−1.19833e7	−1.43543	−0.717715	+	0.696337i	$0.754812\pi$
$588$	0	0
$589$	−2.57746e6	−0.306128
$590$	−1.09488e7	−1.29490
$591$	0	0
$592$	−1.95405e6	−0.229156
$593$	−5.31020e6	−0.620117	−0.310059	−	0.950717i	$-0.600349\pi$
$593$	−5.31020e6	−0.620117	−0.310059	+	0.950717i	$0.600349\pi$
$594$	0	0
$595$	0	0
$596$	2.43014e6	0.280231
$597$	0	0
$598$	2688.00	0.000307380 0
$599$	−2.42236e6	−0.275849	−0.137924	−	0.990443i	$-0.544043\pi$
$599$	−2.42236e6	−0.275849	−0.137924	+	0.990443i	$0.544043\pi$
$600$	0	0
$601$	7.10659e6	0.802556	0.401278	−	0.915956i	$-0.368566\pi$
$601$	7.10659e6	0.802556	0.401278	+	0.915956i	$0.368566\pi$
$602$	0	0
$603$	0	0
$604$	1.22637e6	0.136782
$605$	−1.37510e7	−1.52737
$606$	0	0
$607$	−1.79194e7	−1.97402	−0.987011	−	0.160655i	$-0.948640\pi$
$607$	−1.79194e7	−1.97402	−0.987011	+	0.160655i	$0.948640\pi$
$608$	1.52474e6	0.167277
$609$	0	0
$610$	1.98330e7	2.15806
$611$	55386.0	0.00600201
$612$	0	0
$613$	1.39790e7	1.50254	0.751269	−	0.659997i	$-0.229442\pi$
$613$	1.39790e7	1.50254	0.751269	+	0.659997i	$0.229442\pi$
$614$	−1.13081e7	−1.21051
$615$	0	0
$616$	0	0
$617$	5.25594e6	0.555824	0.277912	−	0.960606i	$-0.410358\pi$
$617$	5.25594e6	0.555824	0.277912	+	0.960606i	$0.410358\pi$
$618$	0	0
$619$	−1.44301e6	−0.151371	−0.0756857	−	0.997132i	$-0.524115\pi$
$619$	−1.44301e6	−0.151371	−0.0756857	+	0.997132i	$0.524115\pi$
$620$	−2.38186e6	−0.248849
$621$	0	0
$622$	−4.50926e6	−0.467336
$623$	0	0
$624$	0	0
$625$	−4.87106e6	−0.498796
$626$	9.44052e6	0.962853
$627$	0	0
$628$	6.23536e6	0.630903
$629$	1.45332e7	1.46466
$630$	0	0
$631$	1.51723e7	1.51697	0.758487	−	0.651688i	$-0.225939\pi$
$631$	1.51723e7	1.51697	0.758487	+	0.651688i	$0.225939\pi$
$632$	−1.95398e6	−0.194593
$633$	0	0
$634$	8.89680e6	0.879044
$635$	−1.30690e7	−1.28620
$636$	0	0
$637$	0	0
$638$	−885088.	−0.0860864
$639$	0	0
$640$	1.40902e6	0.135978
$641$	144848.	0.0139241	0.00696205	−	0.999976i	$-0.497784\pi$
$641$	144848.	0.0139241	0.00696205	+	0.999976i	$0.497784\pi$
$642$	0	0
$643$	27469.0	0.00262009	0.00131004	−	0.999999i	$-0.499583\pi$
$643$	27469.0	0.00262009	0.00131004	+	0.999999i	$0.499583\pi$
$644$	0	0
$645$	0	0
$646$	−1.13402e7	−1.06915
$647$	8.55783e6	0.803717	0.401858	−	0.915702i	$-0.368364\pi$
$647$	8.55783e6	0.803717	0.401858	+	0.915702i	$0.368364\pi$
$648$	0	0
$649$	1.08215e6	0.100850
$650$	51252.0	0.00475803
$651$	0	0
$652$	−1.79795e6	−0.165638
$653$	−1.25030e7	−1.14745	−0.573723	−	0.819049i	$-0.694502\pi$
$653$	−1.25030e7	−1.14745	−0.573723	+	0.819049i	$0.694502\pi$
$654$	0	0
$655$	2.02532e7	1.84455
$656$	3.94598e6	0.358010
$657$	0	0
$658$	0	0
$659$	−1.80471e7	−1.61880	−0.809400	−	0.587258i	$-0.800207\pi$
$659$	−1.80471e7	−1.61880	−0.809400	+	0.587258i	$0.800207\pi$
$660$	0	0
$661$	−3.34144e6	−0.297461	−0.148731	−	0.988878i	$-0.547519\pi$
$661$	−3.34144e6	−0.297461	−0.148731	+	0.988878i	$0.547519\pi$
$662$	−1.48122e7	−1.31363
$663$	0	0
$664$	7.07853e6	0.623050
$665$	0	0
$666$	0	0
$667$	1.45779e6	0.126876
$668$	840800.	0.0729040
$669$	0	0
$670$	−2.08337e7	−1.79299
$671$	−1.96024e6	−0.168075
$672$	0	0
$673$	−8.47066e6	−0.720907	−0.360454	−	0.932777i	$-0.617378\pi$
$673$	−8.47066e6	−0.720907	−0.360454	+	0.932777i	$0.617378\pi$
$674$	4.85730e6	0.411856
$675$	0	0
$676$	−5.94054e6	−0.499988
$677$	1.46553e7	1.22892	0.614458	−	0.788949i	$-0.289375\pi$
$677$	1.46553e7	1.22892	0.614458	+	0.788949i	$0.289375\pi$
$678$	0	0
$679$	0	0
$680$	−1.04796e7	−0.869107
$681$	0	0
$682$	235416.	0.0193809
$683$	−1.97616e7	−1.62095	−0.810477	−	0.585771i	$-0.800792\pi$
$683$	−1.97616e7	−1.62095	−0.810477	+	0.585771i	$0.800792\pi$
$684$	0	0
$685$	−2.79937e7	−2.27947
$686$	0	0
$687$	0	0
$688$	4.73370e6	0.381267
$689$	59868.0	0.00480448
$690$	0	0
$691$	−1.35832e7	−1.08220	−0.541101	−	0.840958i	$-0.681992\pi$
$691$	−1.35832e7	−1.08220	−0.541101	+	0.840958i	$0.681992\pi$
$692$	−2.16410e6	−0.171795
$693$	0	0
$694$	3.91162e6	0.308289
$695$	−276146.	−0.0216859
$696$	0	0
$697$	−2.93483e7	−2.28823
$698$	−2.04513e6	−0.158885
$699$	0	0
$700$	0	0
$701$	1.29915e7	0.998538	0.499269	−	0.866447i	$-0.333602\pi$
$701$	1.29915e7	0.998538	0.499269	+	0.866447i	$0.333602\pi$
$702$	0	0
$703$	−1.13655e7	−0.867365
$704$	−139264.	−0.0105903
$705$	0	0
$706$	1.21101e7	0.914399
$707$	0	0
$708$	0	0
$709$	1.90873e6	0.142603	0.0713017	−	0.997455i	$-0.477285\pi$
$709$	1.90873e6	0.142603	0.0713017	+	0.997455i	$0.477285\pi$
$710$	1.54229e7	1.14821
$711$	0	0
$712$	−3.77549e6	−0.279109
$713$	−387744.	−0.0285641
$714$	0	0
$715$	−8772.00	−0.000641702 0
$716$	−4.01021e6	−0.292337
$717$	0	0
$718$	−1.83782e7	−1.33043
$719$	−1.11200e7	−0.802198	−0.401099	−	0.916035i	$-0.631372\pi$
$719$	−1.11200e7	−0.802198	−0.401099	+	0.916035i	$0.631372\pi$
$720$	0	0
$721$	0	0
$722$	−1.03591e6	−0.0739571
$723$	0	0
$724$	−3.18773e6	−0.226014
$725$	2.77957e7	1.96396
$726$	0	0
$727$	−8.37406e6	−0.587624	−0.293812	−	0.955863i	$-0.594924\pi$
$727$	−8.37406e6	−0.587624	−0.293812	+	0.955863i	$0.594924\pi$
$728$	0	0
$729$	0	0
$730$	−7.16242e6	−0.497454
$731$	−3.52069e7	−2.43688
$732$	0	0
$733$	−4.64448e6	−0.319284	−0.159642	−	0.987175i	$-0.551034\pi$
$733$	−4.64448e6	−0.319284	−0.159642	+	0.987175i	$0.551034\pi$
$734$	4.45092e6	0.304937
$735$	0	0
$736$	229376.	0.0156082
$737$	2.05914e6	0.139642
$738$	0	0
$739$	−1.10623e7	−0.745136	−0.372568	−	0.928005i	$-0.621523\pi$
$739$	−1.10623e7	−0.745136	−0.372568	+	0.928005i	$0.621523\pi$
$740$	−1.05030e7	−0.705074
$741$	0	0
$742$	0	0
$743$	−1.97245e6	−0.131079	−0.0655395	−	0.997850i	$-0.520877\pi$
$743$	−1.97245e6	−0.131079	−0.0655395	+	0.997850i	$0.520877\pi$
$744$	0	0
$745$	1.30620e7	0.862223
$746$	−9.83580e6	−0.647087
$747$	0	0
$748$	1.03578e6	0.0676880
$749$	0	0
$750$	0	0
$751$	1.50246e7	0.972085	0.486042	−	0.873935i	$-0.338440\pi$
$751$	1.50246e7	0.972085	0.486042	+	0.873935i	$0.338440\pi$
$752$	4.72627e6	0.304771
$753$	0	0
$754$	78096.0	0.00500265
$755$	6.59173e6	0.420854
$756$	0	0
$757$	−4.72426e6	−0.299636	−0.149818	−	0.988714i	$-0.547869\pi$
$757$	−4.72426e6	−0.299636	−0.149818	+	0.988714i	$0.547869\pi$
$758$	−1.64052e7	−1.03707
$759$	0	0
$760$	8.19546e6	0.514682
$761$	−8.57835e6	−0.536960	−0.268480	−	0.963285i	$-0.586521\pi$
$761$	−8.57835e6	−0.536960	−0.268480	+	0.963285i	$0.586521\pi$
$762$	0	0
$763$	0	0
$764$	3.82032e6	0.236792
$765$	0	0
$766$	−1.03765e7	−0.638970
$767$	−95484.0	−0.00586060
$768$	0	0
$769$	−1.76168e7	−1.07426	−0.537131	−	0.843499i	$-0.680492\pi$
$769$	−1.76168e7	−1.07426	−0.537131	+	0.843499i	$0.680492\pi$
$770$	0	0
$771$	0	0
$772$	1.37106e6	0.0827965
$773$	−1.25420e6	−0.0754951	−0.0377475	−	0.999287i	$-0.512018\pi$
$773$	−1.25420e6	−0.0754951	−0.0377475	+	0.999287i	$0.512018\pi$
$774$	0	0
$775$	−7.39310e6	−0.442153
$776$	7.67014e6	0.457245
$777$	0	0
$778$	−8.93626e6	−0.529306
$779$	2.29514e7	1.35508
$780$	0	0
$781$	−1.52436e6	−0.0894250
$782$	−1.70598e6	−0.0997604
$783$	0	0
$784$	0	0
$785$	3.35151e7	1.94118
$786$	0	0
$787$	−7.09121e6	−0.408116	−0.204058	−	0.978959i	$-0.565413\pi$
$787$	−7.09121e6	−0.408116	−0.204058	+	0.978959i	$0.565413\pi$
$788$	1.14253e6	0.0655468
$789$	0	0
$790$	−1.05027e7	−0.598731
$791$	0	0
$792$	0	0
$793$	172962.	0.00976715
$794$	8.25599e6	0.464748
$795$	0	0
$796$	1.13816e7	0.636681
$797$	−2.71630e6	−0.151472	−0.0757358	−	0.997128i	$-0.524131\pi$
$797$	−2.71630e6	−0.151472	−0.0757358	+	0.997128i	$0.524131\pi$
$798$	0	0
$799$	−3.51516e7	−1.94795
$800$	4.37350e6	0.241604
$801$	0	0
$802$	−4.61131e6	−0.253156
$803$	707914.	0.0387429
$804$	0	0
$805$	0	0
$806$	−20772.0	−0.00112627
$807$	0	0
$808$	6.40064e6	0.344901
$809$	2.18739e7	1.17505	0.587524	−	0.809207i	$-0.300103\pi$
$809$	2.18739e7	1.17505	0.587524	+	0.809207i	$0.300103\pi$
$810$	0	0
$811$	1.72352e7	0.920164	0.460082	−	0.887876i	$-0.347820\pi$
$811$	1.72352e7	0.920164	0.460082	+	0.887876i	$0.347820\pi$
$812$	0	0
$813$	0	0
$814$	1.03809e6	0.0549127
$815$	−9.66399e6	−0.509639
$816$	0	0
$817$	2.75331e7	1.44311
$818$	−2.37365e7	−1.24032
$819$	0	0
$820$	2.12097e7	1.10154
$821$	2.26191e7	1.17116	0.585582	−	0.810613i	$-0.300866\pi$
$821$	2.26191e7	1.17116	0.585582	+	0.810613i	$0.300866\pi$
$822$	0	0
$823$	1.62633e6	0.0836967	0.0418484	−	0.999124i	$-0.486675\pi$
$823$	1.62633e6	0.0836967	0.0418484	+	0.999124i	$0.486675\pi$
$824$	−7.78822e6	−0.399595
$825$	0	0
$826$	0	0
$827$	2.28304e7	1.16078	0.580391	−	0.814338i	$-0.302900\pi$
$827$	2.28304e7	1.16078	0.580391	+	0.814338i	$0.302900\pi$
$828$	0	0
$829$	−2.58593e7	−1.30686	−0.653432	−	0.756985i	$-0.726672\pi$
$829$	−2.58593e7	−1.30686	−0.653432	+	0.756985i	$0.726672\pi$
$830$	3.80471e7	1.91702
$831$	0	0
$832$	12288.0	0.000615422 0
$833$	0	0
$834$	0	0
$835$	4.51930e6	0.224313
$836$	−810016.	−0.0400846
$837$	0	0
$838$	3.08666e6	0.151838
$839$	−1.23061e7	−0.603554	−0.301777	−	0.953379i	$-0.597580\pi$
$839$	−1.23061e7	−0.603554	−0.301777	+	0.953379i	$0.597580\pi$
$840$	0	0
$841$	2.18429e7	1.06493
$842$	−1.15017e7	−0.559089
$843$	0	0
$844$	−4.16416e6	−0.201220
$845$	−3.19304e7	−1.53838
$846$	0	0
$847$	0	0
$848$	5.10874e6	0.243963
$849$	0	0
$850$	−3.25279e7	−1.54422
$851$	−1.70979e6	−0.0809318
$852$	0	0
$853$	1.91416e7	0.900753	0.450377	−	0.892839i	$-0.351290\pi$
$853$	1.91416e7	0.900753	0.450377	+	0.892839i	$0.351290\pi$
$854$	0	0
$855$	0	0
$856$	3.11347e6	0.145231
$857$	−4.75507e6	−0.221159	−0.110580	−	0.993867i	$-0.535271\pi$
$857$	−4.75507e6	−0.221159	−0.110580	+	0.993867i	$0.535271\pi$
$858$	0	0
$859$	9.19876e6	0.425350	0.212675	−	0.977123i	$-0.431782\pi$
$859$	9.19876e6	0.425350	0.212675	+	0.977123i	$0.431782\pi$
$860$	2.54436e7	1.17309
$861$	0	0
$862$	551448.	0.0252776
$863$	1.73532e7	0.793144	0.396572	−	0.918004i	$-0.370200\pi$
$863$	1.73532e7	0.793144	0.396572	+	0.918004i	$0.370200\pi$
$864$	0	0
$865$	−1.16320e7	−0.528585
$866$	−6.26102e6	−0.283694
$867$	0	0
$868$	0	0
$869$	1.03805e6	0.0466305
$870$	0	0
$871$	−181689.	−0.00811490
$872$	−9.73280e6	−0.433458
$873$	0	0
$874$	1.33414e6	0.0590778
$875$	0	0
$876$	0	0
$877$	−3.55622e7	−1.56131	−0.780655	−	0.624962i	$-0.785115\pi$
$877$	−3.55622e7	−1.56131	−0.780655	+	0.624962i	$0.785115\pi$
$878$	−1.95263e7	−0.854838
$879$	0	0
$880$	−748544.	−0.0325845
$881$	−2.63056e7	−1.14185	−0.570923	−	0.821003i	$-0.693415\pi$
$881$	−2.63056e7	−1.14185	−0.570923	+	0.821003i	$0.693415\pi$
$882$	0	0
$883$	1.20394e7	0.519642	0.259821	−	0.965657i	$-0.416336\pi$
$883$	1.20394e7	0.519642	0.259821	+	0.965657i	$0.416336\pi$
$884$	−91392.0	−0.00393349
$885$	0	0
$886$	5.20606e6	0.222805
$887$	−133590.	−0.00570118	−0.00285059	−	0.999996i	$-0.500907\pi$
$887$	−133590.	−0.00570118	−0.00285059	+	0.999996i	$0.500907\pi$
$888$	0	0
$889$	0	0
$890$	−2.02932e7	−0.858769
$891$	0	0
$892$	−1.69050e6	−0.0711381
$893$	2.74899e7	1.15357
$894$	0	0
$895$	−2.15549e7	−0.899472
$896$	0	0
$897$	0	0
$898$	1.25256e7	0.518333
$899$	−1.12653e7	−0.464885
$900$	0	0
$901$	−3.79962e7	−1.55930
$902$	−2.09630e6	−0.0857902
$903$	0	0
$904$	3.89670e6	0.158590
$905$	−1.71340e7	−0.695406
$906$	0	0
$907$	−1.27476e7	−0.514529	−0.257264	−	0.966341i	$-0.582821\pi$
$907$	−1.27476e7	−0.514529	−0.257264	+	0.966341i	$0.582821\pi$
$908$	1.04760e7	0.421678
$909$	0	0
$910$	0	0
$911$	−2.43253e7	−0.971095	−0.485548	−	0.874210i	$-0.661380\pi$
$911$	−2.43253e7	−0.971095	−0.485548	+	0.874210i	$0.661380\pi$
$912$	0	0
$913$	−3.76047e6	−0.149302
$914$	2.59707e7	1.02830
$915$	0	0
$916$	−8.92341e6	−0.351392
$917$	0	0
$918$	0	0
$919$	−3.84805e7	−1.50297	−0.751487	−	0.659748i	$-0.770663\pi$
$919$	−3.84805e7	−1.50297	−0.751487	+	0.659748i	$0.770663\pi$
$920$	1.23290e6	0.0480238
$921$	0	0
$922$	2.13887e7	0.828623
$923$	134502.	0.00519666
$924$	0	0
$925$	−3.26005e7	−1.25277
$926$	1.34914e7	0.517047
$927$	0	0
$928$	6.66419e6	0.254026
$929$	3.96678e7	1.50799	0.753996	−	0.656879i	$-0.228124\pi$
$929$	3.96678e7	1.50799	0.753996	+	0.656879i	$0.228124\pi$
$930$	0	0
$931$	0	0
$932$	1.99615e7	0.752755
$933$	0	0
$934$	8.93809e6	0.335257
$935$	5.56730e6	0.208265
$936$	0	0
$937$	−1.48428e7	−0.552288	−0.276144	−	0.961116i	$-0.589057\pi$
$937$	−1.48428e7	−0.552288	−0.276144	+	0.961116i	$0.589057\pi$
$938$	0	0
$939$	0	0
$940$	2.54037e7	0.937729
$941$	4.67063e7	1.71950	0.859748	−	0.510718i	$-0.170620\pi$
$941$	4.67063e7	1.71950	0.859748	+	0.510718i	$0.170620\pi$
$942$	0	0
$943$	3.45274e6	0.126440
$944$	−8.14797e6	−0.297591
$945$	0	0
$946$	−2.51478e6	−0.0913632
$947$	−4.02379e7	−1.45801	−0.729006	−	0.684508i	$-0.760017\pi$
$947$	−4.02379e7	−1.45801	−0.729006	+	0.684508i	$0.760017\pi$
$948$	0	0
$949$	−62463.0	−0.00225142
$950$	2.54381e7	0.914482
$951$	0	0
$952$	0	0
$953$	−2.45579e7	−0.875908	−0.437954	−	0.898997i	$-0.644297\pi$
$953$	−2.45579e7	−0.875908	−0.437954	+	0.898997i	$0.644297\pi$
$954$	0	0
$955$	2.05342e7	0.728567
$956$	7.95082e6	0.281363
$957$	0	0
$958$	−1.00855e7	−0.355044
$959$	0	0
$960$	0	0
$961$	−2.56328e7	−0.895339
$962$	−91596.0	−0.00319109
$963$	0	0
$964$	4.44189e6	0.153948
$965$	7.36943e6	0.254751
$966$	0	0
$967$	5.34313e7	1.83751	0.918754	−	0.394830i	$-0.129197\pi$
$967$	5.34313e7	1.83751	0.918754	+	0.394830i	$0.129197\pi$
$968$	−1.02333e7	−0.351016
$969$	0	0
$970$	4.12270e7	1.40687
$971$	−2.81485e6	−0.0958093	−0.0479046	−	0.998852i	$-0.515254\pi$
$971$	−2.81485e6	−0.0958093	−0.0479046	+	0.998852i	$0.515254\pi$
$972$	0	0
$973$	0	0
$974$	7.66687e6	0.258953
$975$	0	0
$976$	1.47594e7	0.495958
$977$	−2.54713e7	−0.853718	−0.426859	−	0.904318i	$-0.640380\pi$
$977$	−2.54713e7	−0.853718	−0.426859	+	0.904318i	$0.640380\pi$
$978$	0	0
$979$	2.00573e6	0.0668829
$980$	0	0
$981$	0	0
$982$	−2.33150e7	−0.771537
$983$	−2.90135e6	−0.0957670	−0.0478835	−	0.998853i	$-0.515248\pi$
$983$	−2.90135e6	−0.0957670	−0.0478835	+	0.998853i	$0.515248\pi$
$984$	0	0
$985$	6.14109e6	0.201676
$986$	−4.95649e7	−1.62361
$987$	0	0
$988$	71472.0	0.00232940
$989$	4.14198e6	0.134654
$990$	0	0
$991$	2.73784e6	0.0885570	0.0442785	−	0.999019i	$-0.485901\pi$
$991$	2.73784e6	0.0885570	0.0442785	+	0.999019i	$0.485901\pi$
$992$	−1.77254e6	−0.0571897
$993$	0	0
$994$	0	0
$995$	6.11763e7	1.95896
$996$	0	0
$997$	2.74079e7	0.873249	0.436624	−	0.899644i	$-0.356174\pi$
$997$	2.74079e7	0.873249	0.436624	+	0.899644i	$0.356174\pi$
$998$	−4.00194e7	−1.27188
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.6.a.y.1.1		1
3.2	odd	2		294.6.a.e.1.1		1
7.3	odd	6		126.6.g.b.37.1		2
7.5	odd	6		126.6.g.b.109.1		2
7.6	odd	2		882.6.a.m.1.1		1
21.2	odd	6		294.6.e.m.67.1		2
21.5	even	6		42.6.e.b.25.1	✓	2
21.11	odd	6		294.6.e.m.79.1		2
21.17	even	6		42.6.e.b.37.1	yes	2
21.20	even	2		294.6.a.d.1.1		1
84.47	odd	6		336.6.q.a.193.1		2
84.59	odd	6		336.6.q.a.289.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.6.e.b.25.1	✓	2	21.5	even	6
42.6.e.b.37.1	yes	2	21.17	even	6
126.6.g.b.37.1		2	7.3	odd	6
126.6.g.b.109.1		2	7.5	odd	6
294.6.a.d.1.1		1	21.20	even	2
294.6.a.e.1.1		1	3.2	odd	2
294.6.e.m.67.1		2	21.2	odd	6
294.6.e.m.79.1		2	21.11	odd	6
336.6.q.a.193.1		2	84.47	odd	6
336.6.q.a.289.1		2	84.59	odd	6
882.6.a.m.1.1		1	7.6	odd	2
882.6.a.y.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	882.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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