LMFDB - Embedded newform 450.8.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,8,Mod(1,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("450.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	450.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:	$140.573261468$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 150)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+8.00000 q^{2} +64.0000 q^{4} -1427.00 q^{7} +512.000 q^{8} +O(q^{10})$

$q+8.00000 q^{2} +64.0000 q^{4} -1427.00 q^{7} +512.000 q^{8} -5070.00 q^{11} +11899.0 q^{13} -11416.0 q^{14} +4096.00 q^{16} +19242.0 q^{17} -43711.0 q^{19} -40560.0 q^{22} -3150.00 q^{23} +95192.0 q^{26} -91328.0 q^{28} -140106. q^{29} +147563. q^{31} +32768.0 q^{32} +153936. q^{34} -561674. q^{37} -349688. q^{38} +270336. q^{41} -180683. q^{43} -324480. q^{44} -25200.0 q^{46} +97470.0 q^{47} +1.21279e6 q^{49} +761536. q^{52} +2.13013e6 q^{53} -730624. q^{56} -1.12085e6 q^{58} +935070. q^{59} +135875. q^{61} +1.18050e6 q^{62} +262144. q^{64} +1.44343e6 q^{67} +1.23149e6 q^{68} +2.68584e6 q^{71} -3.28047e6 q^{73} -4.49339e6 q^{74} -2.79750e6 q^{76} +7.23489e6 q^{77} +5.67267e6 q^{79} +2.16269e6 q^{82} +4.22791e6 q^{83} -1.44546e6 q^{86} -2.59584e6 q^{88} +1.18908e7 q^{89} -1.69799e7 q^{91} -201600. q^{92} +779760. q^{94} -3.80853e6 q^{97} +9.70229e6 q^{98} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	8.00000	0.707107
$2$	8.00000	0.707107
$3$	0	0
$3$	0	0
$4$	64.0000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1427.00	−1.57246	−0.786232	−	0.617931i	$-0.787971\pi$
$7$	−1427.00	−1.57246	−0.786232	+	0.617931i	$0.787971\pi$
$8$	512.000	0.353553
$9$	0	0
$10$	0	0
$11$	−5070.00	−1.14851	−0.574253	−	0.818678i	$-0.694708\pi$
$11$	−5070.00	−1.14851	−0.574253	+	0.818678i	$0.694708\pi$
$12$	0	0
$13$	11899.0	1.50213	0.751067	−	0.660226i	$-0.229539\pi$
$13$	11899.0	1.50213	0.751067	+	0.660226i	$0.229539\pi$
$14$	−11416.0	−1.11190
$15$	0	0
$16$	4096.00	0.250000
$17$	19242.0	0.949902	0.474951	−	0.880012i	$-0.342466\pi$
$17$	19242.0	0.949902	0.474951	+	0.880012i	$0.342466\pi$
$18$	0	0
$19$	−43711.0	−1.46202	−0.731010	−	0.682367i	$-0.760951\pi$
$19$	−43711.0	−1.46202	−0.731010	+	0.682367i	$0.760951\pi$
$20$	0	0
$21$	0	0
$22$	−40560.0	−0.812117
$23$	−3150.00	−0.0539838	−0.0269919	−	0.999636i	$-0.508593\pi$
$23$	−3150.00	−0.0539838	−0.0269919	+	0.999636i	$0.508593\pi$
$24$	0	0
$25$	0	0
$26$	95192.0	1.06217
$27$	0	0
$28$	−91328.0	−0.786232
$29$	−140106.	−1.06675	−0.533376	−	0.845878i	$-0.679077\pi$
$29$	−140106.	−1.06675	−0.533376	+	0.845878i	$0.679077\pi$
$30$	0	0
$31$	147563.	0.889634	0.444817	−	0.895621i	$-0.353269\pi$
$31$	147563.	0.889634	0.444817	+	0.895621i	$0.353269\pi$
$32$	32768.0	0.176777
$33$	0	0
$34$	153936.	0.671682
$35$	0	0
$36$	0	0
$37$	−561674.	−1.82296	−0.911482	−	0.411339i	$-0.865061\pi$
$37$	−561674.	−1.82296	−0.911482	+	0.411339i	$0.865061\pi$
$38$	−349688.	−1.03380
$39$	0	0
$40$	0	0
$41$	270336.	0.612577	0.306288	−	0.951939i	$-0.400913\pi$
$41$	270336.	0.612577	0.306288	+	0.951939i	$0.400913\pi$
$42$	0	0
$43$	−180683.	−0.346559	−0.173280	−	0.984873i	$-0.555436\pi$
$43$	−180683.	−0.346559	−0.173280	+	0.984873i	$0.555436\pi$
$44$	−324480.	−0.574253
$45$	0	0
$46$	−25200.0	−0.0381723
$47$	97470.0	0.136939	0.0684697	−	0.997653i	$-0.478188\pi$
$47$	97470.0	0.136939	0.0684697	+	0.997653i	$0.478188\pi$
$48$	0	0
$49$	1.21279e6	1.47264
$50$	0	0
$51$	0	0
$52$	761536.	0.751067
$53$	2.13013e6	1.96535	0.982677	−	0.185324i	$-0.0593335\pi$
$53$	2.13013e6	1.96535	0.982677	+	0.185324i	$0.0593335\pi$
$54$	0	0
$55$	0	0
$56$	−730624.	−0.555950
$57$	0	0
$58$	−1.12085e6	−0.754308
$59$	935070.	0.592737	0.296369	−	0.955074i	$-0.404224\pi$
$59$	935070.	0.592737	0.296369	+	0.955074i	$0.404224\pi$
$60$	0	0
$61$	135875.	0.0766452	0.0383226	−	0.999265i	$-0.487799\pi$
$61$	135875.	0.0766452	0.0383226	+	0.999265i	$0.487799\pi$
$62$	1.18050e6	0.629066
$63$	0	0
$64$	262144.	0.125000
$65$	0	0
$66$	0	0
$67$	1.44343e6	0.586320	0.293160	−	0.956063i	$-0.405293\pi$
$67$	1.44343e6	0.586320	0.293160	+	0.956063i	$0.405293\pi$
$68$	1.23149e6	0.474951
$69$	0	0
$70$	0	0
$71$	2.68584e6	0.890586	0.445293	−	0.895385i	$-0.353100\pi$
$71$	2.68584e6	0.890586	0.445293	+	0.895385i	$0.353100\pi$
$72$	0	0
$73$	−3.28047e6	−0.986974	−0.493487	−	0.869753i	$-0.664278\pi$
$73$	−3.28047e6	−0.986974	−0.493487	+	0.869753i	$0.664278\pi$
$74$	−4.49339e6	−1.28903
$75$	0	0
$76$	−2.79750e6	−0.731010
$77$	7.23489e6	1.80599
$78$	0	0
$79$	5.67267e6	1.29447	0.647236	−	0.762289i	$-0.275925\pi$
$79$	5.67267e6	1.29447	0.647236	+	0.762289i	$0.275925\pi$
$80$	0	0
$81$	0	0
$82$	2.16269e6	0.433157
$83$	4.22791e6	0.811619	0.405809	−	0.913958i	$-0.366990\pi$
$83$	4.22791e6	0.811619	0.405809	+	0.913958i	$0.366990\pi$
$84$	0	0
$85$	0	0
$86$	−1.44546e6	−0.245055
$87$	0	0
$88$	−2.59584e6	−0.406058
$89$	1.18908e7	1.78792	0.893959	−	0.448148i	$-0.147916\pi$
$89$	1.18908e7	1.78792	0.893959	+	0.448148i	$0.147916\pi$
$90$	0	0
$91$	−1.69799e7	−2.36205
$92$	−201600.	−0.0269919
$93$	0	0
$94$	779760.	0.0968308
$95$	0	0
$96$	0	0
$97$	−3.80853e6	−0.423698	−0.211849	−	0.977302i	$-0.567948\pi$
$97$	−3.80853e6	−0.423698	−0.211849	+	0.977302i	$0.567948\pi$
$98$	9.70229e6	1.04132
$99$	0	0
$100$	0	0
$101$	1.20593e7	1.16465	0.582327	−	0.812955i	$-0.302142\pi$
$101$	1.20593e7	1.16465	0.582327	+	0.812955i	$0.302142\pi$
$102$	0	0
$103$	−3.61951e6	−0.326377	−0.163188	−	0.986595i	$-0.552178\pi$
$103$	−3.61951e6	−0.326377	−0.163188	+	0.986595i	$0.552178\pi$
$104$	6.09229e6	0.531085
$105$	0	0
$106$	1.70411e7	1.38972
$107$	−2.56928e6	−0.202754	−0.101377	−	0.994848i	$-0.532325\pi$
$107$	−2.56928e6	−0.202754	−0.101377	+	0.994848i	$0.532325\pi$
$108$	0	0
$109$	6.71514e6	0.496664	0.248332	−	0.968675i	$-0.420118\pi$
$109$	6.71514e6	0.496664	0.248332	+	0.968675i	$0.420118\pi$
$110$	0	0
$111$	0	0
$112$	−5.84499e6	−0.393116
$113$	1.37565e7	0.896876	0.448438	−	0.893814i	$-0.351980\pi$
$113$	1.37565e7	0.896876	0.448438	+	0.893814i	$0.351980\pi$
$114$	0	0
$115$	0	0
$116$	−8.96678e6	−0.533376
$117$	0	0
$118$	7.48056e6	0.419128
$119$	−2.74583e7	−1.49369
$120$	0	0
$121$	6.21773e6	0.319068
$122$	1.08700e6	0.0541964
$123$	0	0
$124$	9.44403e6	0.444817
$125$	0	0
$126$	0	0
$127$	−4.01769e7	−1.74046	−0.870229	−	0.492647i	$-0.836029\pi$
$127$	−4.01769e7	−1.74046	−0.870229	+	0.492647i	$0.836029\pi$
$128$	2.09715e6	0.0883883
$129$	0	0
$130$	0	0
$131$	−3.44456e7	−1.33870	−0.669351	−	0.742946i	$-0.733428\pi$
$131$	−3.44456e7	−1.33870	−0.669351	+	0.742946i	$0.733428\pi$
$132$	0	0
$133$	6.23756e7	2.29897
$134$	1.15475e7	0.414591
$135$	0	0
$136$	9.85190e6	0.335841
$137$	2.78217e7	0.924405	0.462203	−	0.886774i	$-0.347059\pi$
$137$	2.78217e7	0.924405	0.462203	+	0.886774i	$0.347059\pi$
$138$	0	0
$139$	−8.75852e6	−0.276617	−0.138308	−	0.990389i	$-0.544167\pi$
$139$	−8.75852e6	−0.276617	−0.138308	+	0.990389i	$0.544167\pi$
$140$	0	0
$141$	0	0
$142$	2.14867e7	0.629739
$143$	−6.03279e7	−1.72521
$144$	0	0
$145$	0	0
$146$	−2.62437e7	−0.697896
$147$	0	0
$148$	−3.59471e7	−0.911482
$149$	7.68194e7	1.90247	0.951237	−	0.308460i	$-0.0998137\pi$
$149$	7.68194e7	1.90247	0.951237	+	0.308460i	$0.0998137\pi$
$150$	0	0
$151$	−1.42992e7	−0.337981	−0.168991	−	0.985618i	$-0.554051\pi$
$151$	−1.42992e7	−0.337981	−0.168991	+	0.985618i	$0.554051\pi$
$152$	−2.23800e7	−0.516902
$153$	0	0
$154$	5.78791e7	1.27702
$155$	0	0
$156$	0	0
$157$	6.38587e7	1.31696	0.658478	−	0.752600i	$-0.271200\pi$
$157$	6.38587e7	1.31696	0.658478	+	0.752600i	$0.271200\pi$
$158$	4.53814e7	0.915330
$159$	0	0
$160$	0	0
$161$	4.49505e6	0.0848875
$162$	0	0
$163$	8.81015e7	1.59341	0.796703	−	0.604371i	$-0.206575\pi$
$163$	8.81015e7	1.59341	0.796703	+	0.604371i	$0.206575\pi$
$164$	1.73015e7	0.306288
$165$	0	0
$166$	3.38232e7	0.573901
$167$	6.22383e7	1.03407	0.517035	−	0.855964i	$-0.327036\pi$
$167$	6.22383e7	1.03407	0.517035	+	0.855964i	$0.327036\pi$
$168$	0	0
$169$	7.88377e7	1.25641
$170$	0	0
$171$	0	0
$172$	−1.15637e7	−0.173280
$173$	−8.69882e7	−1.27732	−0.638659	−	0.769490i	$-0.720510\pi$
$173$	−8.69882e7	−1.27732	−0.638659	+	0.769490i	$0.720510\pi$
$174$	0	0
$175$	0	0
$176$	−2.07667e7	−0.287127
$177$	0	0
$178$	9.51268e7	1.26425
$179$	1.03821e8	1.35300	0.676500	−	0.736443i	$-0.263496\pi$
$179$	1.03821e8	1.35300	0.676500	+	0.736443i	$0.263496\pi$
$180$	0	0
$181$	−1.21159e7	−0.151873	−0.0759365	−	0.997113i	$-0.524195\pi$
$181$	−1.21159e7	−0.151873	−0.0759365	+	0.997113i	$0.524195\pi$
$182$	−1.35839e8	−1.67022
$183$	0	0
$184$	−1.61280e6	−0.0190861
$185$	0	0
$186$	0	0
$187$	−9.75569e7	−1.09097
$188$	6.23808e6	0.0684697
$189$	0	0
$190$	0	0
$191$	8.64043e7	0.897260	0.448630	−	0.893718i	$-0.351912\pi$
$191$	8.64043e7	0.897260	0.448630	+	0.893718i	$0.351912\pi$
$192$	0	0
$193$	8.89367e7	0.890493	0.445247	−	0.895408i	$-0.353116\pi$
$193$	8.89367e7	0.890493	0.445247	+	0.895408i	$0.353116\pi$
$194$	−3.04682e7	−0.299600
$195$	0	0
$196$	7.76183e7	0.736322
$197$	−1.32518e8	−1.23493	−0.617465	−	0.786599i	$-0.711840\pi$
$197$	−1.32518e8	−1.23493	−0.617465	+	0.786599i	$0.711840\pi$
$198$	0	0
$199$	−3.94534e7	−0.354894	−0.177447	−	0.984130i	$-0.556784\pi$
$199$	−3.94534e7	−0.354894	−0.177447	+	0.984130i	$0.556784\pi$
$200$	0	0
$201$	0	0
$202$	9.64743e7	0.823535
$203$	1.99931e8	1.67743
$204$	0	0
$205$	0	0
$206$	−2.89561e7	−0.230783
$207$	0	0
$208$	4.87383e7	0.375534
$209$	2.21615e8	1.67914
$210$	0	0
$211$	1.93707e8	1.41957	0.709786	−	0.704417i	$-0.248792\pi$
$211$	1.93707e8	1.41957	0.709786	+	0.704417i	$0.248792\pi$
$212$	1.36328e8	0.982677
$213$	0	0
$214$	−2.05543e7	−0.143369
$215$	0	0
$216$	0	0
$217$	−2.10572e8	−1.39892
$218$	5.37212e7	0.351194
$219$	0	0
$220$	0	0
$221$	2.28961e8	1.42688
$222$	0	0
$223$	−1.36236e8	−0.822671	−0.411335	−	0.911484i	$-0.634938\pi$
$223$	−1.36236e8	−0.822671	−0.411335	+	0.911484i	$0.634938\pi$
$224$	−4.67599e7	−0.277975
$225$	0	0
$226$	1.10052e8	0.634187
$227$	−2.86433e8	−1.62530	−0.812648	−	0.582755i	$-0.801975\pi$
$227$	−2.86433e8	−1.62530	−0.812648	+	0.582755i	$0.801975\pi$
$228$	0	0
$229$	2.84427e8	1.56512	0.782558	−	0.622578i	$-0.213915\pi$
$229$	2.84427e8	1.56512	0.782558	+	0.622578i	$0.213915\pi$
$230$	0	0
$231$	0	0
$232$	−7.17343e7	−0.377154
$233$	−3.24032e8	−1.67820	−0.839098	−	0.543981i	$-0.816917\pi$
$233$	−3.24032e8	−1.67820	−0.839098	+	0.543981i	$0.816917\pi$
$234$	0	0
$235$	0	0
$236$	5.98445e7	0.296369
$237$	0	0
$238$	−2.19667e8	−1.05620
$239$	−2.53375e8	−1.20053	−0.600263	−	0.799802i	$-0.704938\pi$
$239$	−2.53375e8	−1.20053	−0.600263	+	0.799802i	$0.704938\pi$
$240$	0	0
$241$	1.07795e8	0.496067	0.248034	−	0.968751i	$-0.420216\pi$
$241$	1.07795e8	0.496067	0.248034	+	0.968751i	$0.420216\pi$
$242$	4.97418e7	0.225615
$243$	0	0
$244$	8.69600e6	0.0383226
$245$	0	0
$246$	0	0
$247$	−5.20117e8	−2.19615
$248$	7.55523e7	0.314533
$249$	0	0
$250$	0	0
$251$	−2.09909e8	−0.837863	−0.418932	−	0.908018i	$-0.637595\pi$
$251$	−2.09909e8	−0.837863	−0.418932	+	0.908018i	$0.637595\pi$
$252$	0	0
$253$	1.59705e7	0.0620007
$254$	−3.21415e8	−1.23069
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	−1.88661e8	−0.693293	−0.346646	−	0.937996i	$-0.612680\pi$
$257$	−1.88661e8	−0.693293	−0.346646	+	0.937996i	$0.612680\pi$
$258$	0	0
$259$	8.01509e8	2.86655
$260$	0	0
$261$	0	0
$262$	−2.75565e8	−0.946605
$263$	4.38730e8	1.48714	0.743571	−	0.668657i	$-0.233131\pi$
$263$	4.38730e8	1.48714	0.743571	+	0.668657i	$0.233131\pi$
$264$	0	0
$265$	0	0
$266$	4.99005e8	1.62562
$267$	0	0
$268$	9.23797e7	0.293160
$269$	1.19910e8	0.375598	0.187799	−	0.982207i	$-0.439865\pi$
$269$	1.19910e8	0.375598	0.187799	+	0.982207i	$0.439865\pi$
$270$	0	0
$271$	−3.99299e8	−1.21872	−0.609362	−	0.792892i	$-0.708574\pi$
$271$	−3.99299e8	−1.21872	−0.609362	+	0.792892i	$0.708574\pi$
$272$	7.88152e7	0.237476
$273$	0	0
$274$	2.22574e8	0.653653
$275$	0	0
$276$	0	0
$277$	9.45520e7	0.267295	0.133648	−	0.991029i	$-0.457331\pi$
$277$	9.45520e7	0.267295	0.133648	+	0.991029i	$0.457331\pi$
$278$	−7.00681e7	−0.195598
$279$	0	0
$280$	0	0
$281$	2.37829e8	0.639430	0.319715	−	0.947514i	$-0.396413\pi$
$281$	2.37829e8	0.639430	0.319715	+	0.947514i	$0.396413\pi$
$282$	0	0
$283$	5.34471e8	1.40175	0.700877	−	0.713282i	$-0.252792\pi$
$283$	5.34471e8	1.40175	0.700877	+	0.713282i	$0.252792\pi$
$284$	1.71894e8	0.445293
$285$	0	0
$286$	−4.82623e8	−1.21991
$287$	−3.85769e8	−0.963255
$288$	0	0
$289$	−4.00841e7	−0.0976854
$290$	0	0
$291$	0	0
$292$	−2.09950e8	−0.493487
$293$	6.18845e8	1.43729	0.718647	−	0.695375i	$-0.244762\pi$
$293$	6.18845e8	1.43729	0.718647	+	0.695375i	$0.244762\pi$
$294$	0	0
$295$	0	0
$296$	−2.87577e8	−0.644515
$297$	0	0
$298$	6.14555e8	1.34525
$299$	−3.74818e7	−0.0810909
$300$	0	0
$301$	2.57835e8	0.544952
$302$	−1.14394e8	−0.238989
$303$	0	0
$304$	−1.79040e8	−0.365505
$305$	0	0
$306$	0	0
$307$	1.74921e8	0.345030	0.172515	−	0.985007i	$-0.444811\pi$
$307$	1.74921e8	0.345030	0.172515	+	0.985007i	$0.444811\pi$
$308$	4.63033e8	0.902993
$309$	0	0
$310$	0	0
$311$	4.03443e7	0.0760537	0.0380269	−	0.999277i	$-0.487893\pi$
$311$	4.03443e7	0.0760537	0.0380269	+	0.999277i	$0.487893\pi$
$312$	0	0
$313$	−8.33163e8	−1.53577	−0.767883	−	0.640590i	$-0.778690\pi$
$313$	−8.33163e8	−1.53577	−0.767883	+	0.640590i	$0.778690\pi$
$314$	5.10870e8	0.931229
$315$	0	0
$316$	3.63051e8	0.647236
$317$	1.06497e9	1.87771	0.938857	−	0.344307i	$-0.111886\pi$
$317$	1.06497e9	1.87771	0.938857	+	0.344307i	$0.111886\pi$
$318$	0	0
$319$	7.10337e8	1.22517
$320$	0	0
$321$	0	0
$322$	3.59604e7	0.0600246
$323$	−8.41087e8	−1.38878
$324$	0	0
$325$	0	0
$326$	7.04812e8	1.12671
$327$	0	0
$328$	1.38412e8	0.216579
$329$	−1.39090e8	−0.215332
$330$	0	0
$331$	1.92701e8	0.292070	0.146035	−	0.989279i	$-0.453349\pi$
$331$	1.92701e8	0.292070	0.146035	+	0.989279i	$0.453349\pi$
$332$	2.70586e8	0.405809
$333$	0	0
$334$	4.97906e8	0.731198
$335$	0	0
$336$	0	0
$337$	−2.05520e8	−0.292516	−0.146258	−	0.989247i	$-0.546723\pi$
$337$	−2.05520e8	−0.292516	−0.146258	+	0.989247i	$0.546723\pi$
$338$	6.30701e8	0.888414
$339$	0	0
$340$	0	0
$341$	−7.48144e8	−1.02175
$342$	0	0
$343$	−5.55450e8	−0.743217
$344$	−9.25097e7	−0.122527
$345$	0	0
$346$	−6.95905e8	−0.903200
$347$	−1.24873e9	−1.60441	−0.802204	−	0.597050i	$-0.796340\pi$
$347$	−1.24873e9	−1.60441	−0.802204	+	0.597050i	$0.796340\pi$
$348$	0	0
$349$	−9.75467e8	−1.22835	−0.614177	−	0.789168i	$-0.710512\pi$
$349$	−9.75467e8	−1.22835	−0.614177	+	0.789168i	$0.710512\pi$
$350$	0	0
$351$	0	0
$352$	−1.66134e8	−0.203029
$353$	−8.12177e8	−0.982742	−0.491371	−	0.870950i	$-0.663504\pi$
$353$	−8.12177e8	−0.982742	−0.491371	+	0.870950i	$0.663504\pi$
$354$	0	0
$355$	0	0
$356$	7.61014e8	0.893959
$357$	0	0
$358$	8.30565e8	0.956716
$359$	8.19316e8	0.934589	0.467294	−	0.884102i	$-0.345229\pi$
$359$	8.19316e8	0.934589	0.467294	+	0.884102i	$0.345229\pi$
$360$	0	0
$361$	1.01678e9	1.13750
$362$	−9.69271e7	−0.107390
$363$	0	0
$364$	−1.08671e9	−1.18103
$365$	0	0
$366$	0	0
$367$	4.44464e8	0.469360	0.234680	−	0.972073i	$-0.424596\pi$
$367$	4.44464e8	0.469360	0.234680	+	0.972073i	$0.424596\pi$
$368$	−1.29024e7	−0.0134959
$369$	0	0
$370$	0	0
$371$	−3.03970e9	−3.09045
$372$	0	0
$373$	−5.23822e8	−0.522640	−0.261320	−	0.965252i	$-0.584158\pi$
$373$	−5.23822e8	−0.522640	−0.261320	+	0.965252i	$0.584158\pi$
$374$	−7.80456e8	−0.771432
$375$	0	0
$376$	4.99046e7	0.0484154
$377$	−1.66712e9	−1.60241
$378$	0	0
$379$	−6.01805e8	−0.567830	−0.283915	−	0.958849i	$-0.591633\pi$
$379$	−6.01805e8	−0.567830	−0.283915	+	0.958849i	$0.591633\pi$
$380$	0	0
$381$	0	0
$382$	6.91234e8	0.634459
$383$	−4.67737e8	−0.425408	−0.212704	−	0.977117i	$-0.568227\pi$
$383$	−4.67737e8	−0.425408	−0.212704	+	0.977117i	$0.568227\pi$
$384$	0	0
$385$	0	0
$386$	7.11494e8	0.629674
$387$	0	0
$388$	−2.43746e8	−0.211849
$389$	1.11784e9	0.962841	0.481421	−	0.876490i	$-0.340121\pi$
$389$	1.11784e9	0.962841	0.481421	+	0.876490i	$0.340121\pi$
$390$	0	0
$391$	−6.06123e7	−0.0512793
$392$	6.20946e8	0.520658
$393$	0	0
$394$	−1.06014e9	−0.873227
$395$	0	0
$396$	0	0
$397$	1.01062e9	0.810626	0.405313	−	0.914178i	$-0.367162\pi$
$397$	1.01062e9	0.810626	0.405313	+	0.914178i	$0.367162\pi$
$398$	−3.15627e8	−0.250948
$399$	0	0
$400$	0	0
$401$	4.30852e8	0.333675	0.166837	−	0.985984i	$-0.446645\pi$
$401$	4.30852e8	0.333675	0.166837	+	0.985984i	$0.446645\pi$
$402$	0	0
$403$	1.75585e9	1.33635
$404$	7.71795e8	0.582327
$405$	0	0
$406$	1.59945e9	1.18612
$407$	2.84769e9	2.09369
$408$	0	0
$409$	7.35181e8	0.531328	0.265664	−	0.964066i	$-0.414409\pi$
$409$	7.35181e8	0.531328	0.265664	+	0.964066i	$0.414409\pi$
$410$	0	0
$411$	0	0
$412$	−2.31649e8	−0.163188
$413$	−1.33434e9	−0.932058
$414$	0	0
$415$	0	0
$416$	3.89906e8	0.265542
$417$	0	0
$418$	1.77292e9	1.18733
$419$	7.97408e8	0.529580	0.264790	−	0.964306i	$-0.414697\pi$
$419$	7.97408e8	0.529580	0.264790	+	0.964306i	$0.414697\pi$
$420$	0	0
$421$	−2.16840e8	−0.141629	−0.0708144	−	0.997490i	$-0.522560\pi$
$421$	−2.16840e8	−0.141629	−0.0708144	+	0.997490i	$0.522560\pi$
$422$	1.54966e9	1.00379
$423$	0	0
$424$	1.09063e9	0.694858
$425$	0	0
$426$	0	0
$427$	−1.93894e8	−0.120522
$428$	−1.64434e8	−0.101377
$429$	0	0
$430$	0	0
$431$	−1.05196e9	−0.632890	−0.316445	−	0.948611i	$-0.602489\pi$
$431$	−1.05196e9	−0.632890	−0.316445	+	0.948611i	$0.602489\pi$
$432$	0	0
$433$	−8.42421e8	−0.498680	−0.249340	−	0.968416i	$-0.580214\pi$
$433$	−8.42421e8	−0.498680	−0.249340	+	0.968416i	$0.580214\pi$
$434$	−1.68458e9	−0.989185
$435$	0	0
$436$	4.29769e8	0.248332
$437$	1.37690e8	0.0789253
$438$	0	0
$439$	8.78820e8	0.495763	0.247882	−	0.968790i	$-0.420266\pi$
$439$	8.78820e8	0.495763	0.247882	+	0.968790i	$0.420266\pi$
$440$	0	0
$441$	0	0
$442$	1.83168e9	1.00896
$443$	−2.09919e9	−1.14720	−0.573599	−	0.819136i	$-0.694453\pi$
$443$	−2.09919e9	−1.14720	−0.573599	+	0.819136i	$0.694453\pi$
$444$	0	0
$445$	0	0
$446$	−1.08989e9	−0.581716
$447$	0	0
$448$	−3.74079e8	−0.196558
$449$	1.30937e9	0.682656	0.341328	−	0.939944i	$-0.389123\pi$
$449$	1.30937e9	0.682656	0.341328	+	0.939944i	$0.389123\pi$
$450$	0	0
$451$	−1.37060e9	−0.703548
$452$	8.80414e8	0.448438
$453$	0	0
$454$	−2.29146e9	−1.14926
$455$	0	0
$456$	0	0
$457$	2.54163e9	1.24568	0.622839	−	0.782350i	$-0.285979\pi$
$457$	2.54163e9	1.24568	0.622839	+	0.782350i	$0.285979\pi$
$458$	2.27541e9	1.10670
$459$	0	0
$460$	0	0
$461$	−5.86141e8	−0.278644	−0.139322	−	0.990247i	$-0.544492\pi$
$461$	−5.86141e8	−0.278644	−0.139322	+	0.990247i	$0.544492\pi$
$462$	0	0
$463$	−1.57279e9	−0.736439	−0.368220	−	0.929739i	$-0.620033\pi$
$463$	−1.57279e9	−0.736439	−0.368220	+	0.929739i	$0.620033\pi$
$464$	−5.73874e8	−0.266688
$465$	0	0
$466$	−2.59226e9	−1.18666
$467$	1.58046e9	0.718083	0.359042	−	0.933322i	$-0.383104\pi$
$467$	1.58046e9	0.718083	0.359042	+	0.933322i	$0.383104\pi$
$468$	0	0
$469$	−2.05978e9	−0.921968
$470$	0	0
$471$	0	0
$472$	4.78756e8	0.209564
$473$	9.16063e8	0.398026
$474$	0	0
$475$	0	0
$476$	−1.75733e9	−0.746844
$477$	0	0
$478$	−2.02700e9	−0.848901
$479$	−3.70114e9	−1.53873	−0.769364	−	0.638811i	$-0.779427\pi$
$479$	−3.70114e9	−1.53873	−0.769364	+	0.638811i	$0.779427\pi$
$480$	0	0
$481$	−6.68336e9	−2.73834
$482$	8.62363e8	0.350773
$483$	0	0
$484$	3.97935e8	0.159534
$485$	0	0
$486$	0	0
$487$	9.78261e8	0.383798	0.191899	−	0.981415i	$-0.438535\pi$
$487$	9.78261e8	0.383798	0.191899	+	0.981415i	$0.438535\pi$
$488$	6.95680e7	0.0270982
$489$	0	0
$490$	0	0
$491$	8.54921e8	0.325942	0.162971	−	0.986631i	$-0.447892\pi$
$491$	8.54921e8	0.325942	0.162971	+	0.986631i	$0.447892\pi$
$492$	0	0
$493$	−2.69592e9	−1.01331
$494$	−4.16094e9	−1.55291
$495$	0	0
$496$	6.04418e8	0.222409
$497$	−3.83269e9	−1.40041
$498$	0	0
$499$	−2.00214e9	−0.721345	−0.360672	−	0.932693i	$-0.617453\pi$
$499$	−2.00214e9	−0.721345	−0.360672	+	0.932693i	$0.617453\pi$
$500$	0	0
$501$	0	0
$502$	−1.67927e9	−0.592459
$503$	−3.51483e9	−1.23145	−0.615724	−	0.787962i	$-0.711136\pi$
$503$	−3.51483e9	−1.23145	−0.615724	+	0.787962i	$0.711136\pi$
$504$	0	0
$505$	0	0
$506$	1.27764e8	0.0438411
$507$	0	0
$508$	−2.57132e9	−0.870229
$509$	3.70703e9	1.24599	0.622995	−	0.782226i	$-0.285916\pi$
$509$	3.70703e9	1.24599	0.622995	+	0.782226i	$0.285916\pi$
$510$	0	0
$511$	4.68122e9	1.55198
$512$	1.34218e8	0.0441942
$513$	0	0
$514$	−1.50929e9	−0.490232
$515$	0	0
$516$	0	0
$517$	−4.94173e8	−0.157276
$518$	6.41207e9	2.02696
$519$	0	0
$520$	0	0
$521$	1.31752e9	0.408156	0.204078	−	0.978955i	$-0.434580\pi$
$521$	1.31752e9	0.408156	0.204078	+	0.978955i	$0.434580\pi$
$522$	0	0
$523$	4.45839e8	0.136277	0.0681385	−	0.997676i	$-0.478294\pi$
$523$	4.45839e8	0.136277	0.0681385	+	0.997676i	$0.478294\pi$
$524$	−2.20452e9	−0.669351
$525$	0	0
$526$	3.50984e9	1.05157
$527$	2.83941e9	0.845066
$528$	0	0
$529$	−3.39490e9	−0.997086
$530$	0	0
$531$	0	0
$532$	3.99204e9	1.14949
$533$	3.21673e9	0.920172
$534$	0	0
$535$	0	0
$536$	7.39038e8	0.207295
$537$	0	0
$538$	9.59282e8	0.265588
$539$	−6.14883e9	−1.69134
$540$	0	0
$541$	−7.31894e9	−1.98727	−0.993637	−	0.112631i	$-0.964072\pi$
$541$	−7.31894e9	−1.98727	−0.993637	+	0.112631i	$0.964072\pi$
$542$	−3.19439e9	−0.861768
$543$	0	0
$544$	6.30522e8	0.167921
$545$	0	0
$546$	0	0
$547$	−5.21813e9	−1.36320	−0.681600	−	0.731725i	$-0.738715\pi$
$547$	−5.21813e9	−1.36320	−0.681600	+	0.731725i	$0.738715\pi$
$548$	1.78059e9	0.462203
$549$	0	0
$550$	0	0
$551$	6.12417e9	1.55961
$552$	0	0
$553$	−8.09490e9	−2.03551
$554$	7.56416e8	0.189006
$555$	0	0
$556$	−5.60545e8	−0.138308
$557$	5.22742e7	0.0128172	0.00640862	−	0.999979i	$-0.497960\pi$
$557$	5.22742e7	0.0128172	0.00640862	+	0.999979i	$0.497960\pi$
$558$	0	0
$559$	−2.14995e9	−0.520579
$560$	0	0
$561$	0	0
$562$	1.90263e9	0.452145
$563$	−7.63499e9	−1.80314	−0.901569	−	0.432636i	$-0.857584\pi$
$563$	−7.63499e9	−1.80314	−0.901569	+	0.432636i	$0.857584\pi$
$564$	0	0
$565$	0	0
$566$	4.27577e9	0.991190
$567$	0	0
$568$	1.37515e9	0.314870
$569$	2.52617e9	0.574871	0.287435	−	0.957800i	$-0.407197\pi$
$569$	2.52617e9	0.574871	0.287435	+	0.957800i	$0.407197\pi$
$570$	0	0
$571$	−4.75335e9	−1.06850	−0.534248	−	0.845328i	$-0.679405\pi$
$571$	−4.75335e9	−1.06850	−0.534248	+	0.845328i	$0.679405\pi$
$572$	−3.86099e9	−0.862606
$573$	0	0
$574$	−3.08616e9	−0.681124
$575$	0	0
$576$	0	0
$577$	−7.07337e9	−1.53289	−0.766445	−	0.642309i	$-0.777976\pi$
$577$	−7.07337e9	−1.53289	−0.766445	+	0.642309i	$0.777976\pi$
$578$	−3.20673e8	−0.0690740
$579$	0	0
$580$	0	0
$581$	−6.03322e9	−1.27624
$582$	0	0
$583$	−1.07998e10	−2.25722
$584$	−1.67960e9	−0.348948
$585$	0	0
$586$	4.95076e9	1.01632
$587$	−6.82995e8	−0.139375	−0.0696874	−	0.997569i	$-0.522200\pi$
$587$	−6.82995e8	−0.139375	−0.0696874	+	0.997569i	$0.522200\pi$
$588$	0	0
$589$	−6.45013e9	−1.30066
$590$	0	0
$591$	0	0
$592$	−2.30062e9	−0.455741
$593$	4.11466e9	0.810294	0.405147	−	0.914251i	$-0.367220\pi$
$593$	4.11466e9	0.810294	0.405147	+	0.914251i	$0.367220\pi$
$594$	0	0
$595$	0	0
$596$	4.91644e9	0.951237
$597$	0	0
$598$	−2.99855e8	−0.0573399
$599$	2.71835e8	0.0516787	0.0258394	−	0.999666i	$-0.491774\pi$
$599$	2.71835e8	0.0516787	0.0258394	+	0.999666i	$0.491774\pi$
$600$	0	0
$601$	−1.63646e9	−0.307500	−0.153750	−	0.988110i	$-0.549135\pi$
$601$	−1.63646e9	−0.307500	−0.153750	+	0.988110i	$0.549135\pi$
$602$	2.06268e9	0.385340
$603$	0	0
$604$	−9.15149e8	−0.168991
$605$	0	0
$606$	0	0
$607$	7.25840e8	0.131729	0.0658644	−	0.997829i	$-0.479020\pi$
$607$	7.25840e8	0.131729	0.0658644	+	0.997829i	$0.479020\pi$
$608$	−1.43232e9	−0.258451
$609$	0	0
$610$	0	0
$611$	1.15980e9	0.205701
$612$	0	0
$613$	9.79507e9	1.71750	0.858749	−	0.512397i	$-0.171242\pi$
$613$	9.79507e9	1.71750	0.858749	+	0.512397i	$0.171242\pi$
$614$	1.39937e9	0.243973
$615$	0	0
$616$	3.70426e9	0.638512
$617$	−7.54736e8	−0.129359	−0.0646796	−	0.997906i	$-0.520603\pi$
$617$	−7.54736e8	−0.129359	−0.0646796	+	0.997906i	$0.520603\pi$
$618$	0	0
$619$	9.48066e9	1.60665	0.803325	−	0.595541i	$-0.203062\pi$
$619$	9.48066e9	1.60665	0.803325	+	0.595541i	$0.203062\pi$
$620$	0	0
$621$	0	0
$622$	3.22754e8	0.0537781
$623$	−1.69682e10	−2.81144
$624$	0	0
$625$	0	0
$626$	−6.66531e9	−1.08595
$627$	0	0
$628$	4.08696e9	0.658478
$629$	−1.08077e10	−1.73164
$630$	0	0
$631$	2.11281e8	0.0334779	0.0167389	−	0.999860i	$-0.494672\pi$
$631$	2.11281e8	0.0334779	0.0167389	+	0.999860i	$0.494672\pi$
$632$	2.90441e9	0.457665
$633$	0	0
$634$	8.51975e9	1.32774
$635$	0	0
$636$	0	0
$637$	1.44309e10	2.21211
$638$	5.68270e9	0.866328
$639$	0	0
$640$	0	0
$641$	−7.52664e9	−1.12875	−0.564375	−	0.825518i	$-0.690883\pi$
$641$	−7.52664e9	−1.12875	−0.564375	+	0.825518i	$0.690883\pi$
$642$	0	0
$643$	3.52375e9	0.522717	0.261358	−	0.965242i	$-0.415830\pi$
$643$	3.52375e9	0.522717	0.261358	+	0.965242i	$0.415830\pi$
$644$	2.87683e8	0.0424438
$645$	0	0
$646$	−6.72870e9	−0.982013
$647$	3.90185e9	0.566377	0.283189	−	0.959064i	$-0.408608\pi$
$647$	3.90185e9	0.566377	0.283189	+	0.959064i	$0.408608\pi$
$648$	0	0
$649$	−4.74080e9	−0.680763
$650$	0	0
$651$	0	0
$652$	5.63850e9	0.796703
$653$	−8.20768e9	−1.15352	−0.576759	−	0.816915i	$-0.695683\pi$
$653$	−8.20768e9	−1.15352	−0.576759	+	0.816915i	$0.695683\pi$
$654$	0	0
$655$	0	0
$656$	1.10730e9	0.153144
$657$	0	0
$658$	−1.11272e9	−0.152263
$659$	−4.49137e9	−0.611336	−0.305668	−	0.952138i	$-0.598880\pi$
$659$	−4.49137e9	−0.611336	−0.305668	+	0.952138i	$0.598880\pi$
$660$	0	0
$661$	9.78484e9	1.31780	0.658898	−	0.752232i	$-0.271023\pi$
$661$	9.78484e9	1.31780	0.658898	+	0.752232i	$0.271023\pi$
$662$	1.54161e9	0.206525
$663$	0	0
$664$	2.16469e9	0.286951
$665$	0	0
$666$	0	0
$667$	4.41334e8	0.0575873
$668$	3.98325e9	0.517035
$669$	0	0
$670$	0	0
$671$	−6.88886e8	−0.0880276
$672$	0	0
$673$	1.99688e9	0.252522	0.126261	−	0.991997i	$-0.459702\pi$
$673$	1.99688e9	0.252522	0.126261	+	0.991997i	$0.459702\pi$
$674$	−1.64416e9	−0.206840
$675$	0	0
$676$	5.04561e9	0.628204
$677$	−1.76852e9	−0.219053	−0.109526	−	0.993984i	$-0.534933\pi$
$677$	−1.76852e9	−0.219053	−0.109526	+	0.993984i	$0.534933\pi$
$678$	0	0
$679$	5.43477e9	0.666250
$680$	0	0
$681$	0	0
$682$	−5.98516e9	−0.722487
$683$	1.33982e10	1.60906	0.804531	−	0.593910i	$-0.202417\pi$
$683$	1.33982e10	1.60906	0.804531	+	0.593910i	$0.202417\pi$
$684$	0	0
$685$	0	0
$686$	−4.44360e9	−0.525533
$687$	0	0
$688$	−7.40078e8	−0.0866399
$689$	2.53464e10	2.95223
$690$	0	0
$691$	1.36908e10	1.57854	0.789270	−	0.614046i	$-0.210459\pi$
$691$	1.36908e10	1.57854	0.789270	+	0.614046i	$0.210459\pi$
$692$	−5.56724e9	−0.638659
$693$	0	0
$694$	−9.98983e9	−1.13449
$695$	0	0
$696$	0	0
$697$	5.20181e9	0.581888
$698$	−7.80374e9	−0.868578
$699$	0	0
$700$	0	0
$701$	8.98742e9	0.985422	0.492711	−	0.870193i	$-0.336006\pi$
$701$	8.98742e9	0.985422	0.492711	+	0.870193i	$0.336006\pi$
$702$	0	0
$703$	2.45513e10	2.66521
$704$	−1.32907e9	−0.143563
$705$	0	0
$706$	−6.49742e9	−0.694903
$707$	−1.72086e10	−1.83138
$708$	0	0
$709$	−4.36499e9	−0.459962	−0.229981	−	0.973195i	$-0.573866\pi$
$709$	−4.36499e9	−0.459962	−0.229981	+	0.973195i	$0.573866\pi$
$710$	0	0
$711$	0	0
$712$	6.08811e9	0.632125
$713$	−4.64823e8	−0.0480258
$714$	0	0
$715$	0	0
$716$	6.64452e9	0.676500
$717$	0	0
$718$	6.55452e9	0.660854
$719$	1.46045e10	1.46534	0.732668	−	0.680587i	$-0.238275\pi$
$719$	1.46045e10	1.46534	0.732668	+	0.680587i	$0.238275\pi$
$720$	0	0
$721$	5.16504e9	0.513216
$722$	8.13424e9	0.804335
$723$	0	0
$724$	−7.75417e8	−0.0759365
$725$	0	0
$726$	0	0
$727$	−1.42461e10	−1.37507	−0.687536	−	0.726150i	$-0.741308\pi$
$727$	−1.42461e10	−1.37507	−0.687536	+	0.726150i	$0.741308\pi$
$728$	−8.69369e9	−0.835112
$729$	0	0
$730$	0	0
$731$	−3.47670e9	−0.329198
$732$	0	0
$733$	−8.78749e9	−0.824140	−0.412070	−	0.911152i	$-0.635194\pi$
$733$	−8.78749e9	−0.824140	−0.412070	+	0.911152i	$0.635194\pi$
$734$	3.55571e9	0.331887
$735$	0	0
$736$	−1.03219e8	−0.00954307
$737$	−7.31821e9	−0.673393
$738$	0	0
$739$	1.59660e10	1.45526	0.727632	−	0.685968i	$-0.240621\pi$
$739$	1.59660e10	1.45526	0.727632	+	0.685968i	$0.240621\pi$
$740$	0	0
$741$	0	0
$742$	−2.43176e10	−2.18528
$743$	4.01209e9	0.358847	0.179424	−	0.983772i	$-0.442577\pi$
$743$	4.01209e9	0.358847	0.179424	+	0.983772i	$0.442577\pi$
$744$	0	0
$745$	0	0
$746$	−4.19057e9	−0.369562
$747$	0	0
$748$	−6.24364e9	−0.545485
$749$	3.66637e9	0.318823
$750$	0	0
$751$	7.41663e9	0.638950	0.319475	−	0.947595i	$-0.396493\pi$
$751$	7.41663e9	0.638950	0.319475	+	0.947595i	$0.396493\pi$
$752$	3.99237e8	0.0342349
$753$	0	0
$754$	−1.33370e10	−1.13307
$755$	0	0
$756$	0	0
$757$	−1.51143e8	−0.0126634	−0.00633171	−	0.999980i	$-0.502015\pi$
$757$	−1.51143e8	−0.0126634	−0.00633171	+	0.999980i	$0.502015\pi$
$758$	−4.81444e9	−0.401516
$759$	0	0
$760$	0	0
$761$	−3.85268e9	−0.316896	−0.158448	−	0.987367i	$-0.550649\pi$
$761$	−3.85268e9	−0.316896	−0.158448	+	0.987367i	$0.550649\pi$
$762$	0	0
$763$	−9.58251e9	−0.780986
$764$	5.52987e9	0.448630
$765$	0	0
$766$	−3.74189e9	−0.300809
$767$	1.11264e10	0.890371
$768$	0	0
$769$	1.56192e10	1.23856	0.619280	−	0.785170i	$-0.287424\pi$
$769$	1.56192e10	1.23856	0.619280	+	0.785170i	$0.287424\pi$
$770$	0	0
$771$	0	0
$772$	5.69195e9	0.445247
$773$	1.44828e10	1.12778	0.563889	−	0.825850i	$-0.309304\pi$
$773$	1.44828e10	1.12778	0.563889	+	0.825850i	$0.309304\pi$
$774$	0	0
$775$	0	0
$776$	−1.94997e9	−0.149800
$777$	0	0
$778$	8.94269e9	0.680831
$779$	−1.18167e10	−0.895599
$780$	0	0
$781$	−1.36172e10	−1.02284
$782$	−4.84898e8	−0.0362599
$783$	0	0
$784$	4.96757e9	0.368161
$785$	0	0
$786$	0	0
$787$	−1.37729e10	−1.00720	−0.503599	−	0.863938i	$-0.667991\pi$
$787$	−1.37729e10	−1.00720	−0.503599	+	0.863938i	$0.667991\pi$
$788$	−8.48113e9	−0.617465
$789$	0	0
$790$	0	0
$791$	−1.96305e10	−1.41031
$792$	0	0
$793$	1.61678e9	0.115131
$794$	8.08495e9	0.573199
$795$	0	0
$796$	−2.52502e9	−0.177447
$797$	−6.88529e9	−0.481746	−0.240873	−	0.970557i	$-0.577434\pi$
$797$	−6.88529e9	−0.481746	−0.240873	+	0.970557i	$0.577434\pi$
$798$	0	0
$799$	1.87552e9	0.130079
$800$	0	0
$801$	0	0
$802$	3.44682e9	0.235944
$803$	1.66320e10	1.13355
$804$	0	0
$805$	0	0
$806$	1.40468e10	0.944942
$807$	0	0
$808$	6.17436e9	0.411767
$809$	1.28256e10	0.851644	0.425822	−	0.904807i	$-0.359985\pi$
$809$	1.28256e10	0.851644	0.425822	+	0.904807i	$0.359985\pi$
$810$	0	0
$811$	−7.36094e9	−0.484574	−0.242287	−	0.970205i	$-0.577898\pi$
$811$	−7.36094e9	−0.484574	−0.242287	+	0.970205i	$0.577898\pi$
$812$	1.27956e10	0.838715
$813$	0	0
$814$	2.27815e10	1.48046
$815$	0	0
$816$	0	0
$817$	7.89783e9	0.506677
$818$	5.88144e9	0.375705
$819$	0	0
$820$	0	0
$821$	1.09919e10	0.693219	0.346610	−	0.938009i	$-0.387333\pi$
$821$	1.09919e10	0.693219	0.346610	+	0.938009i	$0.387333\pi$
$822$	0	0
$823$	1.15788e10	0.724041	0.362020	−	0.932170i	$-0.382087\pi$
$823$	1.15788e10	0.724041	0.362020	+	0.932170i	$0.382087\pi$
$824$	−1.85319e9	−0.115392
$825$	0	0
$826$	−1.06748e10	−0.659064
$827$	4.97823e9	0.306059	0.153030	−	0.988222i	$-0.451097\pi$
$827$	4.97823e9	0.306059	0.153030	+	0.988222i	$0.451097\pi$
$828$	0	0
$829$	−1.15589e10	−0.704655	−0.352328	−	0.935877i	$-0.614610\pi$
$829$	−1.15589e10	−0.704655	−0.352328	+	0.935877i	$0.614610\pi$
$830$	0	0
$831$	0	0
$832$	3.11925e9	0.187767
$833$	2.33364e10	1.39887
$834$	0	0
$835$	0	0
$836$	1.41833e10	0.839570
$837$	0	0
$838$	6.37926e9	0.374469
$839$	−4.50868e9	−0.263562	−0.131781	−	0.991279i	$-0.542070\pi$
$839$	−4.50868e9	−0.263562	−0.131781	+	0.991279i	$0.542070\pi$
$840$	0	0
$841$	2.37981e9	0.137961
$842$	−1.73472e9	−0.100147
$843$	0	0
$844$	1.23973e10	0.709786
$845$	0	0
$846$	0	0
$847$	−8.87270e9	−0.501723
$848$	8.72502e9	0.491339
$849$	0	0
$850$	0	0
$851$	1.76927e9	0.0984105
$852$	0	0
$853$	9.35143e9	0.515889	0.257945	−	0.966160i	$-0.416955\pi$
$853$	9.35143e9	0.515889	0.257945	+	0.966160i	$0.416955\pi$
$854$	−1.55115e9	−0.0852219
$855$	0	0
$856$	−1.31547e9	−0.0716843
$857$	1.85745e9	0.100806	0.0504029	−	0.998729i	$-0.483949\pi$
$857$	1.85745e9	0.100806	0.0504029	+	0.998729i	$0.483949\pi$
$858$	0	0
$859$	3.34304e10	1.79956	0.899779	−	0.436346i	$-0.143728\pi$
$859$	3.34304e10	1.79956	0.899779	+	0.436346i	$0.143728\pi$
$860$	0	0
$861$	0	0
$862$	−8.41567e9	−0.447521
$863$	1.34259e10	0.711061	0.355531	−	0.934665i	$-0.384300\pi$
$863$	1.34259e10	0.711061	0.355531	+	0.934665i	$0.384300\pi$
$864$	0	0
$865$	0	0
$866$	−6.73937e9	−0.352620
$867$	0	0
$868$	−1.34766e10	−0.699459
$869$	−2.87604e10	−1.48671
$870$	0	0
$871$	1.71754e10	0.880732
$872$	3.43815e9	0.175597
$873$	0	0
$874$	1.10152e9	0.0558086
$875$	0	0
$876$	0	0
$877$	−3.08992e9	−0.154685	−0.0773426	−	0.997005i	$-0.524644\pi$
$877$	−3.08992e9	−0.154685	−0.0773426	+	0.997005i	$0.524644\pi$
$878$	7.03056e9	0.350558
$879$	0	0
$880$	0	0
$881$	−8.72396e9	−0.429831	−0.214916	−	0.976633i	$-0.568948\pi$
$881$	−8.72396e9	−0.429831	−0.214916	+	0.976633i	$0.568948\pi$
$882$	0	0
$883$	3.90723e9	0.190988	0.0954942	−	0.995430i	$-0.469557\pi$
$883$	3.90723e9	0.190988	0.0954942	+	0.995430i	$0.469557\pi$
$884$	1.46535e10	0.713440
$885$	0	0
$886$	−1.67935e10	−0.811192
$887$	1.53854e10	0.740246	0.370123	−	0.928983i	$-0.379316\pi$
$887$	1.53854e10	0.740246	0.370123	+	0.928983i	$0.379316\pi$
$888$	0	0
$889$	5.73325e10	2.73681
$890$	0	0
$891$	0	0
$892$	−8.71913e9	−0.411335
$893$	−4.26051e9	−0.200208
$894$	0	0
$895$	0	0
$896$	−2.99264e9	−0.138988
$897$	0	0
$898$	1.04750e10	0.482711
$899$	−2.06745e10	−0.949020
$900$	0	0
$901$	4.09880e10	1.86690
$902$	−1.09648e10	−0.497484
$903$	0	0
$904$	7.04332e9	0.317094
$905$	0	0
$906$	0	0
$907$	2.63986e10	1.17478	0.587388	−	0.809305i	$-0.300156\pi$
$907$	2.63986e10	1.17478	0.587388	+	0.809305i	$0.300156\pi$
$908$	−1.83317e10	−0.812648
$909$	0	0
$910$	0	0
$911$	2.12340e10	0.930503	0.465251	−	0.885179i	$-0.345964\pi$
$911$	2.12340e10	0.930503	0.465251	+	0.885179i	$0.345964\pi$
$912$	0	0
$913$	−2.14355e10	−0.932149
$914$	2.03331e10	0.880828
$915$	0	0
$916$	1.82033e10	0.782558
$917$	4.91538e10	2.10506
$918$	0	0
$919$	9.50036e9	0.403771	0.201886	−	0.979409i	$-0.435293\pi$
$919$	9.50036e9	0.403771	0.201886	+	0.979409i	$0.435293\pi$
$920$	0	0
$921$	0	0
$922$	−4.68913e9	−0.197031
$923$	3.19588e10	1.33778
$924$	0	0
$925$	0	0
$926$	−1.25823e10	−0.520741
$927$	0	0
$928$	−4.59099e9	−0.188577
$929$	4.29492e10	1.75752	0.878760	−	0.477264i	$-0.158371\pi$
$929$	4.29492e10	1.75752	0.878760	+	0.477264i	$0.158371\pi$
$930$	0	0
$931$	−5.30121e10	−2.15303
$932$	−2.07381e10	−0.839098
$933$	0	0
$934$	1.26437e10	0.507761
$935$	0	0
$936$	0	0
$937$	−1.52405e10	−0.605216	−0.302608	−	0.953115i	$-0.597857\pi$
$937$	−1.52405e10	−0.605216	−0.302608	+	0.953115i	$0.597857\pi$
$938$	−1.64782e10	−0.651930
$939$	0	0
$940$	0	0
$941$	2.27448e10	0.889853	0.444926	−	0.895567i	$-0.353230\pi$
$941$	2.27448e10	0.889853	0.444926	+	0.895567i	$0.353230\pi$
$942$	0	0
$943$	−8.51558e8	−0.0330692
$944$	3.83005e9	0.148184
$945$	0	0
$946$	7.32850e9	0.281447
$947$	−8.80667e9	−0.336967	−0.168483	−	0.985705i	$-0.553887\pi$
$947$	−8.80667e9	−0.336967	−0.168483	+	0.985705i	$0.553887\pi$
$948$	0	0
$949$	−3.90343e10	−1.48257
$950$	0	0
$951$	0	0
$952$	−1.40587e10	−0.528098
$953$	−1.37186e9	−0.0513434	−0.0256717	−	0.999670i	$-0.508172\pi$
$953$	−1.37186e9	−0.0513434	−0.0256717	+	0.999670i	$0.508172\pi$
$954$	0	0
$955$	0	0
$956$	−1.62160e10	−0.600263
$957$	0	0
$958$	−2.96092e10	−1.08804
$959$	−3.97016e10	−1.45359
$960$	0	0
$961$	−5.73778e9	−0.208551
$962$	−5.34669e10	−1.93630
$963$	0	0
$964$	6.89891e9	0.248034
$965$	0	0
$966$	0	0
$967$	−2.51907e10	−0.895874	−0.447937	−	0.894065i	$-0.647841\pi$
$967$	−2.51907e10	−0.895874	−0.447937	+	0.894065i	$0.647841\pi$
$968$	3.18348e9	0.112808
$969$	0	0
$970$	0	0
$971$	5.11915e9	0.179445	0.0897223	−	0.995967i	$-0.471402\pi$
$971$	5.11915e9	0.179445	0.0897223	+	0.995967i	$0.471402\pi$
$972$	0	0
$973$	1.24984e10	0.434970
$974$	7.82608e9	0.271386
$975$	0	0
$976$	5.56544e8	0.0191613
$977$	5.06512e8	0.0173764	0.00868818	−	0.999962i	$-0.497234\pi$
$977$	5.06512e8	0.0173764	0.00868818	+	0.999962i	$0.497234\pi$
$978$	0	0
$979$	−6.02866e10	−2.05344
$980$	0	0
$981$	0	0
$982$	6.83937e9	0.230476
$983$	−1.00756e10	−0.338323	−0.169162	−	0.985588i	$-0.554106\pi$
$983$	−1.00756e10	−0.338323	−0.169162	+	0.985588i	$0.554106\pi$
$984$	0	0
$985$	0	0
$986$	−2.15674e10	−0.716519
$987$	0	0
$988$	−3.32875e10	−1.09807
$989$	5.69151e8	0.0187086
$990$	0	0
$991$	4.56127e10	1.48877	0.744386	−	0.667749i	$-0.232742\pi$
$991$	4.56127e10	1.48877	0.744386	+	0.667749i	$0.232742\pi$
$992$	4.83534e9	0.157267
$993$	0	0
$994$	−3.06615e10	−0.990243
$995$	0	0
$996$	0	0
$997$	−3.05555e10	−0.976463	−0.488232	−	0.872714i	$-0.662358\pi$
$997$	−3.05555e10	−0.976463	−0.488232	+	0.872714i	$0.662358\pi$
$998$	−1.60171e10	−0.510068
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.8.a.o.1.1		1
3.2	odd	2		150.8.a.f.1.1	✓	1
5.2	odd	4		450.8.c.d.199.2		2
5.3	odd	4		450.8.c.d.199.1		2
5.4	even	2		450.8.a.m.1.1		1
15.2	even	4		150.8.c.d.49.1		2
15.8	even	4		150.8.c.d.49.2		2
15.14	odd	2		150.8.a.l.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.8.a.f.1.1	✓	1	3.2	odd	2
150.8.a.l.1.1	yes	1	15.14	odd	2
150.8.c.d.49.1		2	15.2	even	4
150.8.c.d.49.2		2	15.8	even	4
450.8.a.m.1.1		1	5.4	even	2
450.8.a.o.1.1		1	1.1	even	1	trivial
450.8.c.d.199.1		2	5.3	odd	4
450.8.c.d.199.2		2	5.2	odd	4

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:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	450.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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