LMFDB - Embedded newform 150.8.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [150,8,Mod(1,150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(150, 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("150.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 150 = 2 \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	150.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:	$46.8577538226$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	150.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -216.000 q^{6} +1427.00 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})$

\(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -216.000 q^{6} +1427.00 q^{7} +512.000 q^{8} +729.000 q^{9} +5070.00 q^{11} -1728.00 q^{12} -11899.0 q^{13} +11416.0 q^{14} +4096.00 q^{16} +19242.0 q^{17} +5832.00 q^{18} -43711.0 q^{19} -38529.0 q^{21} +40560.0 q^{22} -3150.00 q^{23} -13824.0 q^{24} -95192.0 q^{26} -19683.0 q^{27} +91328.0 q^{28} +140106. q^{29} +147563. q^{31} +32768.0 q^{32} -136890. q^{33} +153936. q^{34} +46656.0 q^{36} +561674. q^{37} -349688. q^{38} +321273. q^{39} -270336. q^{41} -308232. q^{42} +180683. q^{43} +324480. q^{44} -25200.0 q^{46} +97470.0 q^{47} -110592. q^{48} +1.21279e6 q^{49} -519534. q^{51} -761536. q^{52} +2.13013e6 q^{53} -157464. q^{54} +730624. q^{56} +1.18020e6 q^{57} +1.12085e6 q^{58} -935070. q^{59} +135875. q^{61} +1.18050e6 q^{62} +1.04028e6 q^{63} +262144. q^{64} -1.09512e6 q^{66} -1.44343e6 q^{67} +1.23149e6 q^{68} +85050.0 q^{69} -2.68584e6 q^{71} +373248. q^{72} +3.28047e6 q^{73} +4.49339e6 q^{74} -2.79750e6 q^{76} +7.23489e6 q^{77} +2.57018e6 q^{78} +5.67267e6 q^{79} +531441. q^{81} -2.16269e6 q^{82} +4.22791e6 q^{83} -2.46586e6 q^{84} +1.44546e6 q^{86} -3.78286e6 q^{87} +2.59584e6 q^{88} -1.18908e7 q^{89} -1.69799e7 q^{91} -201600. q^{92} -3.98420e6 q^{93} +779760. q^{94} -884736. q^{96} +3.80853e6 q^{97} +9.70229e6 q^{98} +3.69603e6 q^{99} +O(q^{100})\)

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 $

$2$	8.00000	0.707107
$2$	8.00000	0.707107
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	64.0000	0.500000
$5$	0	0
$5$	0	0
$6$	−216.000	−0.408248
$7$	1427.00	1.57246	0.786232	−	0.617931i	$-0.212029\pi$
$7$	1427.00	1.57246	0.786232	+	0.617931i	$0.212029\pi$
$8$	512.000	0.353553
$9$	729.000	0.333333
$10$	0	0
$11$	5070.00	1.14851	0.574253	−	0.818678i	$-0.305292\pi$
$11$	5070.00	1.14851	0.574253	+	0.818678i	$0.305292\pi$
$12$	−1728.00	−0.288675
$13$	−11899.0	−1.50213	−0.751067	−	0.660226i	$-0.770461\pi$
$13$	−11899.0	−1.50213	−0.751067	+	0.660226i	$0.770461\pi$
$14$	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	$0.342466\pi$
$18$	5832.00	0.235702
$19$	−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
$21$	−38529.0	−0.907863
$22$	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	$0.508593\pi$
$24$	−13824.0	−0.204124
$25$	0	0
$26$	−95192.0	−1.06217
$27$	−19683.0	−0.192450
$28$	91328.0	0.786232
$29$	140106.	1.06675	0.533376	−	0.845878i	$-0.320923\pi$
$29$	140106.	1.06675	0.533376	+	0.845878i	$0.320923\pi$
$30$	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
$33$	−136890.	−0.663091
$34$	153936.	0.671682
$35$	0	0
$36$	46656.0	0.166667
$37$	561674.	1.82296	0.911482	−	0.411339i	$-0.134939\pi$
$37$	561674.	1.82296	0.911482	+	0.411339i	$0.134939\pi$
$38$	−349688.	−1.03380
$39$	321273.	0.867258
$40$	0	0
$41$	−270336.	−0.612577	−0.306288	−	0.951939i	$-0.599087\pi$
$41$	−270336.	−0.612577	−0.306288	+	0.951939i	$0.599087\pi$
$42$	−308232.	−0.641956
$43$	180683.	0.346559	0.173280	−	0.984873i	$-0.444564\pi$
$43$	180683.	0.346559	0.173280	+	0.984873i	$0.444564\pi$
$44$	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	$0.478188\pi$
$48$	−110592.	−0.144338
$49$	1.21279e6	1.47264
$50$	0	0
$51$	−519534.	−0.548426
$52$	−761536.	−0.751067
$53$	2.13013e6	1.96535	0.982677	−	0.185324i	$-0.0593335\pi$
$53$	2.13013e6	1.96535	0.982677	+	0.185324i	$0.0593335\pi$
$54$	−157464.	−0.136083
$55$	0	0
$56$	730624.	0.555950
$57$	1.18020e6	0.844097
$58$	1.12085e6	0.754308
$59$	−935070.	−0.592737	−0.296369	−	0.955074i	$-0.595776\pi$
$59$	−935070.	−0.592737	−0.296369	+	0.955074i	$0.595776\pi$
$60$	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
$63$	1.04028e6	0.524155
$64$	262144.	0.125000
$65$	0	0
$66$	−1.09512e6	−0.468876
$67$	−1.44343e6	−0.586320	−0.293160	−	0.956063i	$-0.594707\pi$
$67$	−1.44343e6	−0.586320	−0.293160	+	0.956063i	$0.594707\pi$
$68$	1.23149e6	0.474951
$69$	85050.0	0.0311675
$70$	0	0
$71$	−2.68584e6	−0.890586	−0.445293	−	0.895385i	$-0.646900\pi$
$71$	−2.68584e6	−0.890586	−0.445293	+	0.895385i	$0.646900\pi$
$72$	373248.	0.117851
$73$	3.28047e6	0.986974	0.493487	−	0.869753i	$-0.335722\pi$
$73$	3.28047e6	0.986974	0.493487	+	0.869753i	$0.335722\pi$
$74$	4.49339e6	1.28903
$75$	0	0
$76$	−2.79750e6	−0.731010
$77$	7.23489e6	1.80599
$78$	2.57018e6	0.613244
$79$	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
$81$	531441.	0.111111
$82$	−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	$0.366990\pi$
$84$	−2.46586e6	−0.453931
$85$	0	0
$86$	1.44546e6	0.245055
$87$	−3.78286e6	−0.615890
$88$	2.59584e6	0.406058
$89$	−1.18908e7	−1.78792	−0.893959	−	0.448148i	$-0.852084\pi$
$89$	−1.18908e7	−1.78792	−0.893959	+	0.448148i	$0.852084\pi$
$90$	0	0
$91$	−1.69799e7	−2.36205
$92$	−201600.	−0.0269919
$93$	−3.98420e6	−0.513631
$94$	779760.	0.0968308
$95$	0	0
$96$	−884736.	−0.102062
$97$	3.80853e6	0.423698	0.211849	−	0.977302i	$-0.432052\pi$
$97$	3.80853e6	0.423698	0.211849	+	0.977302i	$0.432052\pi$
$98$	9.70229e6	1.04132
$99$	3.69603e6	0.382836
$100$	0	0
$101$	−1.20593e7	−1.16465	−0.582327	−	0.812955i	$-0.697858\pi$
$101$	−1.20593e7	−1.16465	−0.582327	+	0.812955i	$0.697858\pi$
$102$	−4.15627e6	−0.387796
$103$	3.61951e6	0.326377	0.163188	−	0.986595i	$-0.447822\pi$
$103$	3.61951e6	0.326377	0.163188	+	0.986595i	$0.447822\pi$
$104$	−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	$0.532325\pi$
$108$	−1.25971e6	−0.0962250
$109$	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
$111$	−1.51652e7	−1.05249
$112$	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	$0.351980\pi$
$114$	9.44158e6	0.596867
$115$	0	0
$116$	8.96678e6	0.533376
$117$	−8.67437e6	−0.500711
$118$	−7.48056e6	−0.419128
$119$	2.74583e7	1.49369
$120$	0	0
$121$	6.21773e6	0.319068
$122$	1.08700e6	0.0541964
$123$	7.29907e6	0.353671
$124$	9.44403e6	0.444817
$125$	0	0
$126$	8.32226e6	0.370633
$127$	4.01769e7	1.74046	0.870229	−	0.492647i	$-0.163971\pi$
$127$	4.01769e7	1.74046	0.870229	+	0.492647i	$0.163971\pi$
$128$	2.09715e6	0.0883883
$129$	−4.87844e6	−0.200086
$130$	0	0
$131$	3.44456e7	1.33870	0.669351	−	0.742946i	$-0.266572\pi$
$131$	3.44456e7	1.33870	0.669351	+	0.742946i	$0.266572\pi$
$132$	−8.76096e6	−0.331545
$133$	−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	$0.347059\pi$
$138$	680400.	0.0220388
$139$	−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
$141$	−2.63169e6	−0.0790620
$142$	−2.14867e7	−0.629739
$143$	−6.03279e7	−1.72521
$144$	2.98598e6	0.0833333
$145$	0	0
$146$	2.62437e7	0.697896
$147$	−3.27452e7	−0.850232
$148$	3.59471e7	0.911482
$149$	−7.68194e7	−1.90247	−0.951237	−	0.308460i	$-0.900186\pi$
$149$	−7.68194e7	−1.90247	−0.951237	+	0.308460i	$0.900186\pi$
$150$	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
$153$	1.40274e7	0.316634
$154$	5.78791e7	1.27702
$155$	0	0
$156$	2.05615e7	0.433629
$157$	−6.38587e7	−1.31696	−0.658478	−	0.752600i	$-0.728800\pi$
$157$	−6.38587e7	−1.31696	−0.658478	+	0.752600i	$0.728800\pi$
$158$	4.53814e7	0.915330
$159$	−5.75136e7	−1.13470
$160$	0	0
$161$	−4.49505e6	−0.0848875
$162$	4.25153e6	0.0785674
$163$	−8.81015e7	−1.59341	−0.796703	−	0.604371i	$-0.793425\pi$
$163$	−8.81015e7	−1.59341	−0.796703	+	0.604371i	$0.793425\pi$
$164$	−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	$0.327036\pi$
$168$	−1.97268e7	−0.320978
$169$	7.88377e7	1.25641
$170$	0	0
$171$	−3.18653e7	−0.487340
$172$	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	$0.720510\pi$
$174$	−3.02629e7	−0.435500
$175$	0	0
$176$	2.07667e7	0.287127
$177$	2.52469e7	0.342217
$178$	−9.51268e7	−1.26425
$179$	−1.03821e8	−1.35300	−0.676500	−	0.736443i	$-0.736504\pi$
$179$	−1.03821e8	−1.35300	−0.676500	+	0.736443i	$0.736504\pi$
$180$	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
$183$	−3.66862e6	−0.0442511
$184$	−1.61280e6	−0.0190861
$185$	0	0
$186$	−3.18736e7	−0.363192
$187$	9.75569e7	1.09097
$188$	6.23808e6	0.0684697
$189$	−2.80876e7	−0.302621
$190$	0	0
$191$	−8.64043e7	−0.897260	−0.448630	−	0.893718i	$-0.648088\pi$
$191$	−8.64043e7	−0.897260	−0.448630	+	0.893718i	$0.648088\pi$
$192$	−7.07789e6	−0.0721688
$193$	−8.89367e7	−0.890493	−0.445247	−	0.895408i	$-0.646884\pi$
$193$	−8.89367e7	−0.890493	−0.445247	+	0.895408i	$0.646884\pi$
$194$	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	$0.711840\pi$
$198$	2.95682e7	0.270706
$199$	−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
$201$	3.89727e7	0.338512
$202$	−9.64743e7	−0.823535
$203$	1.99931e8	1.67743
$204$	−3.32502e7	−0.274213
$205$	0	0
$206$	2.89561e7	0.230783
$207$	−2.29635e6	−0.0179946
$208$	−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
$213$	7.25177e7	0.514180
$214$	−2.05543e7	−0.143369
$215$	0	0
$216$	−1.00777e7	−0.0680414
$217$	2.10572e8	1.39892
$218$	5.37212e7	0.351194
$219$	−8.85726e7	−0.569829
$220$	0	0
$221$	−2.28961e8	−1.42688
$222$	−1.21322e8	−0.744222
$223$	1.36236e8	0.822671	0.411335	−	0.911484i	$-0.365062\pi$
$223$	1.36236e8	0.822671	0.411335	+	0.911484i	$0.365062\pi$
$224$	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	$0.801975\pi$
$228$	7.55326e7	0.422049
$229$	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
$231$	−1.95342e8	−1.04269
$232$	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	$0.816917\pi$
$234$	−6.93950e7	−0.354056
$235$	0	0
$236$	−5.98445e7	−0.296369
$237$	−1.53162e8	−0.747364
$238$	2.19667e8	1.05620
$239$	2.53375e8	1.20053	0.600263	−	0.799802i	$-0.295062\pi$
$239$	2.53375e8	1.20053	0.600263	+	0.799802i	$0.295062\pi$
$240$	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
$243$	−1.43489e7	−0.0641500
$244$	8.69600e6	0.0383226
$245$	0	0
$246$	5.83926e7	0.250083
$247$	5.20117e8	2.19615
$248$	7.55523e7	0.314533
$249$	−1.14153e8	−0.468588
$250$	0	0
$251$	2.09909e8	0.837863	0.418932	−	0.908018i	$-0.362405\pi$
$251$	2.09909e8	0.837863	0.418932	+	0.908018i	$0.362405\pi$
$252$	6.65781e7	0.262077
$253$	−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	$0.612680\pi$
$258$	−3.90275e7	−0.141482
$259$	8.01509e8	2.86655
$260$	0	0
$261$	1.02137e8	0.355584
$262$	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	$0.233131\pi$
$264$	−7.00877e7	−0.234438
$265$	0	0
$266$	−4.99005e8	−1.62562
$267$	3.21053e8	1.03226
$268$	−9.23797e7	−0.293160
$269$	−1.19910e8	−0.375598	−0.187799	−	0.982207i	$-0.560135\pi$
$269$	−1.19910e8	−0.375598	−0.187799	+	0.982207i	$0.560135\pi$
$270$	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
$273$	4.58457e8	1.36373
$274$	2.22574e8	0.653653
$275$	0	0
$276$	5.44320e6	0.0155838
$277$	−9.45520e7	−0.267295	−0.133648	−	0.991029i	$-0.542669\pi$
$277$	−9.45520e7	−0.267295	−0.133648	+	0.991029i	$0.542669\pi$
$278$	−7.00681e7	−0.195598
$279$	1.07573e8	0.296545
$280$	0	0
$281$	−2.37829e8	−0.639430	−0.319715	−	0.947514i	$-0.603587\pi$
$281$	−2.37829e8	−0.639430	−0.319715	+	0.947514i	$0.603587\pi$
$282$	−2.10535e7	−0.0559053
$283$	−5.34471e8	−1.40175	−0.700877	−	0.713282i	$-0.747208\pi$
$283$	−5.34471e8	−1.40175	−0.700877	+	0.713282i	$0.747208\pi$
$284$	−1.71894e8	−0.445293
$285$	0	0
$286$	−4.82623e8	−1.21991
$287$	−3.85769e8	−0.963255
$288$	2.38879e7	0.0589256
$289$	−4.00841e7	−0.0976854
$290$	0	0
$291$	−1.02830e8	−0.244622
$292$	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	$0.244762\pi$
$294$	−2.61962e8	−0.601205
$295$	0	0
$296$	2.87577e8	0.644515
$297$	−9.97928e7	−0.221030
$298$	−6.14555e8	−1.34525
$299$	3.74818e7	0.0810909
$300$	0	0
$301$	2.57835e8	0.544952
$302$	−1.14394e8	−0.238989
$303$	3.25601e8	0.672413
$304$	−1.79040e8	−0.365505
$305$	0	0
$306$	1.12219e8	0.223894
$307$	−1.74921e8	−0.345030	−0.172515	−	0.985007i	$-0.555189\pi$
$307$	−1.74921e8	−0.345030	−0.172515	+	0.985007i	$0.555189\pi$
$308$	4.63033e8	0.902993
$309$	−9.77267e7	−0.188434
$310$	0	0
$311$	−4.03443e7	−0.0760537	−0.0380269	−	0.999277i	$-0.512107\pi$
$311$	−4.03443e7	−0.0760537	−0.0380269	+	0.999277i	$0.512107\pi$
$312$	1.64492e8	0.306622
$313$	8.33163e8	1.53577	0.767883	−	0.640590i	$-0.221310\pi$
$313$	8.33163e8	1.53577	0.767883	+	0.640590i	$0.221310\pi$
$314$	−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	$0.111886\pi$
$318$	−4.60109e8	−0.802353
$319$	7.10337e8	1.22517
$320$	0	0
$321$	6.93707e7	0.117060
$322$	−3.59604e7	−0.0600246
$323$	−8.41087e8	−1.38878
$324$	3.40122e7	0.0555556
$325$	0	0
$326$	−7.04812e8	−1.12671
$327$	−1.81309e8	−0.286749
$328$	−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
$333$	4.09460e8	0.607655
$334$	4.97906e8	0.731198
$335$	0	0
$336$	−1.57815e8	−0.226966
$337$	2.05520e8	0.292516	0.146258	−	0.989247i	$-0.453277\pi$
$337$	2.05520e8	0.292516	0.146258	+	0.989247i	$0.453277\pi$
$338$	6.30701e8	0.888414
$339$	−3.71425e8	−0.517812
$340$	0	0
$341$	7.48144e8	1.02175
$342$	−2.54923e8	−0.344601
$343$	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	$0.796340\pi$
$348$	−2.42103e8	−0.307945
$349$	−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
$351$	2.34208e8	0.289086
$352$	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	$0.663504\pi$
$354$	2.01975e8	0.241984
$355$	0	0
$356$	−7.61014e8	−0.893959
$357$	−7.41375e8	−0.862381
$358$	−8.30565e8	−0.956716
$359$	−8.19316e8	−0.934589	−0.467294	−	0.884102i	$-0.654771\pi$
$359$	−8.19316e8	−0.934589	−0.467294	+	0.884102i	$0.654771\pi$
$360$	0	0
$361$	1.01678e9	1.13750
$362$	−9.69271e7	−0.107390
$363$	−1.67879e8	−0.184214
$364$	−1.08671e9	−1.18103
$365$	0	0
$366$	−2.93490e7	−0.0312903
$367$	−4.44464e8	−0.469360	−0.234680	−	0.972073i	$-0.575404\pi$
$367$	−4.44464e8	−0.469360	−0.234680	+	0.972073i	$0.575404\pi$
$368$	−1.29024e7	−0.0134959
$369$	−1.97075e8	−0.204192
$370$	0	0
$371$	3.03970e9	3.09045
$372$	−2.54989e8	−0.256815
$373$	5.23822e8	0.522640	0.261320	−	0.965252i	$-0.415842\pi$
$373$	5.23822e8	0.522640	0.261320	+	0.965252i	$0.415842\pi$
$374$	7.80456e8	0.771432
$375$	0	0
$376$	4.99046e7	0.0484154
$377$	−1.66712e9	−1.60241
$378$	−2.24701e8	−0.213985
$379$	−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
$381$	−1.08478e9	−1.00485
$382$	−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	$0.568227\pi$
$384$	−5.66231e7	−0.0510310
$385$	0	0
$386$	−7.11494e8	−0.629674
$387$	1.31718e8	0.115520
$388$	2.43746e8	0.211849
$389$	−1.11784e9	−0.962841	−0.481421	−	0.876490i	$-0.659879\pi$
$389$	−1.11784e9	−0.962841	−0.481421	+	0.876490i	$0.659879\pi$
$390$	0	0
$391$	−6.06123e7	−0.0512793
$392$	6.20946e8	0.520658
$393$	−9.30031e8	−0.772900
$394$	−1.06014e9	−0.873227
$395$	0	0
$396$	2.36546e8	0.191418
$397$	−1.01062e9	−0.810626	−0.405313	−	0.914178i	$-0.632838\pi$
$397$	−1.01062e9	−0.810626	−0.405313	+	0.914178i	$0.632838\pi$
$398$	−3.15627e8	−0.250948
$399$	1.68414e9	1.32731
$400$	0	0
$401$	−4.30852e8	−0.333675	−0.166837	−	0.985984i	$-0.553355\pi$
$401$	−4.30852e8	−0.333675	−0.166837	+	0.985984i	$0.553355\pi$
$402$	3.11782e8	0.239364
$403$	−1.75585e9	−1.33635
$404$	−7.71795e8	−0.582327
$405$	0	0
$406$	1.59945e9	1.18612
$407$	2.84769e9	2.09369
$408$	−2.66001e8	−0.193898
$409$	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
$411$	−7.51187e8	−0.533706
$412$	2.31649e8	0.163188
$413$	−1.33434e9	−0.932058
$414$	−1.83708e7	−0.0127241
$415$	0	0
$416$	−3.89906e8	−0.265542
$417$	2.36480e8	0.159705
$418$	−1.77292e9	−1.18733
$419$	−7.97408e8	−0.529580	−0.264790	−	0.964306i	$-0.585303\pi$
$419$	−7.97408e8	−0.529580	−0.264790	+	0.964306i	$0.585303\pi$
$420$	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
$423$	7.10556e7	0.0456465
$424$	1.09063e9	0.694858
$425$	0	0
$426$	5.80141e8	0.363580
$427$	1.93894e8	0.120522
$428$	−1.64434e8	−0.101377
$429$	1.62885e9	0.996051
$430$	0	0
$431$	1.05196e9	0.632890	0.316445	−	0.948611i	$-0.397511\pi$
$431$	1.05196e9	0.632890	0.316445	+	0.948611i	$0.397511\pi$
$432$	−8.06216e7	−0.0481125
$433$	8.42421e8	0.498680	0.249340	−	0.968416i	$-0.419786\pi$
$433$	8.42421e8	0.498680	0.249340	+	0.968416i	$0.419786\pi$
$434$	1.68458e9	0.989185
$435$	0	0
$436$	4.29769e8	0.248332
$437$	1.37690e8	0.0789253
$438$	−7.08581e8	−0.402930
$439$	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
$441$	8.84121e8	0.490881
$442$	−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	$0.694453\pi$
$444$	−9.70573e8	−0.526245
$445$	0	0
$446$	1.08989e9	0.581716
$447$	2.07412e9	1.09839
$448$	3.74079e8	0.196558
$449$	−1.30937e9	−0.682656	−0.341328	−	0.939944i	$-0.610877\pi$
$449$	−1.30937e9	−0.682656	−0.341328	+	0.939944i	$0.610877\pi$
$450$	0	0
$451$	−1.37060e9	−0.703548
$452$	8.80414e8	0.448438
$453$	3.86079e8	0.195134
$454$	−2.29146e9	−1.14926
$455$	0	0
$456$	6.04261e8	0.298433
$457$	−2.54163e9	−1.24568	−0.622839	−	0.782350i	$-0.714021\pi$
$457$	−2.54163e9	−1.24568	−0.622839	+	0.782350i	$0.714021\pi$
$458$	2.27541e9	1.10670
$459$	−3.78740e8	−0.182809
$460$	0	0
$461$	5.86141e8	0.278644	0.139322	−	0.990247i	$-0.455508\pi$
$461$	5.86141e8	0.278644	0.139322	+	0.990247i	$0.455508\pi$
$462$	−1.56274e9	−0.737291
$463$	1.57279e9	0.736439	0.368220	−	0.929739i	$-0.379967\pi$
$463$	1.57279e9	0.736439	0.368220	+	0.929739i	$0.379967\pi$
$464$	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	$0.383104\pi$
$468$	−5.55160e8	−0.250356
$469$	−2.05978e9	−0.921968
$470$	0	0
$471$	1.72419e9	0.760345
$472$	−4.78756e8	−0.209564
$473$	9.16063e8	0.398026
$474$	−1.22530e9	−0.528466
$475$	0	0
$476$	1.75733e9	0.746844
$477$	1.55287e9	0.655118
$478$	2.02700e9	0.848901
$479$	3.70114e9	1.53873	0.769364	−	0.638811i	$-0.220573\pi$
$479$	3.70114e9	1.53873	0.769364	+	0.638811i	$0.220573\pi$
$480$	0	0
$481$	−6.68336e9	−2.73834
$482$	8.62363e8	0.350773
$483$	1.21366e8	0.0490098
$484$	3.97935e8	0.159534
$485$	0	0
$486$	−1.14791e8	−0.0453609
$487$	−9.78261e8	−0.383798	−0.191899	−	0.981415i	$-0.561465\pi$
$487$	−9.78261e8	−0.383798	−0.191899	+	0.981415i	$0.561465\pi$
$488$	6.95680e7	0.0270982
$489$	2.37874e9	0.919954
$490$	0	0
$491$	−8.54921e8	−0.325942	−0.162971	−	0.986631i	$-0.552108\pi$
$491$	−8.54921e8	−0.325942	−0.162971	+	0.986631i	$0.552108\pi$
$492$	4.67141e8	0.176836
$493$	2.69592e9	1.01331
$494$	4.16094e9	1.55291
$495$	0	0
$496$	6.04418e8	0.222409
$497$	−3.83269e9	−1.40041
$498$	−9.13228e8	−0.331342
$499$	−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
$501$	−1.68043e9	−0.597021
$502$	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	$0.711136\pi$
$504$	5.32625e8	0.185317
$505$	0	0
$506$	−1.27764e8	−0.0438411
$507$	−2.12862e9	−0.725387
$508$	2.57132e9	0.870229
$509$	−3.70703e9	−1.24599	−0.622995	−	0.782226i	$-0.714084\pi$
$509$	−3.70703e9	−1.24599	−0.622995	+	0.782226i	$0.714084\pi$
$510$	0	0
$511$	4.68122e9	1.55198
$512$	1.34218e8	0.0441942
$513$	8.60364e8	0.281366
$514$	−1.50929e9	−0.490232
$515$	0	0
$516$	−3.12220e8	−0.100043
$517$	4.94173e8	0.157276
$518$	6.41207e9	2.02696
$519$	2.34868e9	0.737460
$520$	0	0
$521$	−1.31752e9	−0.408156	−0.204078	−	0.978955i	$-0.565420\pi$
$521$	−1.31752e9	−0.408156	−0.204078	+	0.978955i	$0.565420\pi$
$522$	8.17098e8	0.251436
$523$	−4.45839e8	−0.136277	−0.0681385	−	0.997676i	$-0.521706\pi$
$523$	−4.45839e8	−0.136277	−0.0681385	+	0.997676i	$0.521706\pi$
$524$	2.20452e9	0.669351
$525$	0	0
$526$	3.50984e9	1.05157
$527$	2.83941e9	0.845066
$528$	−5.60701e8	−0.165773
$529$	−3.39490e9	−0.997086
$530$	0	0
$531$	−6.81666e8	−0.197579
$532$	−3.99204e9	−1.14949
$533$	3.21673e9	0.920172
$534$	2.56842e9	0.729915
$535$	0	0
$536$	−7.39038e8	−0.207295
$537$	2.80316e9	0.781155
$538$	−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
$543$	3.27129e8	0.0876839
$544$	6.30522e8	0.167921
$545$	0	0
$546$	3.66765e9	0.964304
$547$	5.21813e9	1.36320	0.681600	−	0.731725i	$-0.261285\pi$
$547$	5.21813e9	1.36320	0.681600	+	0.731725i	$0.261285\pi$
$548$	1.78059e9	0.462203
$549$	9.90529e7	0.0255484
$550$	0	0
$551$	−6.12417e9	−1.55961
$552$	4.35456e7	0.0110194
$553$	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	$0.497960\pi$
$558$	8.60587e8	0.209689
$559$	−2.14995e9	−0.520579
$560$	0	0
$561$	−2.63404e9	−0.629871
$562$	−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	$0.857584\pi$
$564$	−1.68428e8	−0.0395310
$565$	0	0
$566$	−4.27577e9	−0.991190
$567$	7.58366e8	0.174718
$568$	−1.37515e9	−0.314870
$569$	−2.52617e9	−0.574871	−0.287435	−	0.957800i	$-0.592803\pi$
$569$	−2.52617e9	−0.574871	−0.287435	+	0.957800i	$0.592803\pi$
$570$	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
$573$	2.33291e9	0.518033
$574$	−3.08616e9	−0.681124
$575$	0	0
$576$	1.91103e8	0.0416667
$577$	7.07337e9	1.53289	0.766445	−	0.642309i	$-0.222024\pi$
$577$	7.07337e9	1.53289	0.766445	+	0.642309i	$0.222024\pi$
$578$	−3.20673e8	−0.0690740
$579$	2.40129e9	0.514127
$580$	0	0
$581$	6.03322e9	1.27624
$582$	−8.22642e8	−0.172974
$583$	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	$0.522200\pi$
$588$	−2.09569e9	−0.425116
$589$	−6.45013e9	−1.30066
$590$	0	0
$591$	3.57798e9	0.712987
$592$	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	$0.367220\pi$
$594$	−7.98342e8	−0.156292
$595$	0	0
$596$	−4.91644e9	−0.951237
$597$	1.06524e9	0.204898
$598$	2.99855e8	0.0573399
$599$	−2.71835e8	−0.0516787	−0.0258394	−	0.999666i	$-0.508226\pi$
$599$	−2.71835e8	−0.0516787	−0.0258394	+	0.999666i	$0.508226\pi$
$600$	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
$603$	−1.05226e9	−0.195440
$604$	−9.15149e8	−0.168991
$605$	0	0
$606$	2.60481e9	0.475468
$607$	−7.25840e8	−0.131729	−0.0658644	−	0.997829i	$-0.520980\pi$
$607$	−7.25840e8	−0.131729	−0.0658644	+	0.997829i	$0.520980\pi$
$608$	−1.43232e9	−0.258451
$609$	−5.39814e9	−0.968465
$610$	0	0
$611$	−1.15980e9	−0.205701
$612$	8.97755e8	0.158317
$613$	−9.79507e9	−1.71750	−0.858749	−	0.512397i	$-0.828758\pi$
$613$	−9.79507e9	−1.71750	−0.858749	+	0.512397i	$0.828758\pi$
$614$	−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	$0.520603\pi$
$618$	−7.81814e8	−0.133243
$619$	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
$621$	6.20014e7	0.0103892
$622$	−3.22754e8	−0.0537781
$623$	−1.69682e10	−2.81144
$624$	1.31593e9	0.216814
$625$	0	0
$626$	6.66531e9	1.08595
$627$	5.98360e9	0.969451
$628$	−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
$633$	−5.23010e9	−0.819590
$634$	8.51975e9	1.32774
$635$	0	0
$636$	−3.68087e9	−0.567349
$637$	−1.44309e10	−2.21211
$638$	5.68270e9	0.866328
$639$	−1.95798e9	−0.296862
$640$	0	0
$641$	7.52664e9	1.12875	0.564375	−	0.825518i	$-0.309117\pi$
$641$	7.52664e9	1.12875	0.564375	+	0.825518i	$0.309117\pi$
$642$	5.54965e8	0.0827739
$643$	−3.52375e9	−0.522717	−0.261358	−	0.965242i	$-0.584170\pi$
$643$	−3.52375e9	−0.522717	−0.261358	+	0.965242i	$0.584170\pi$
$644$	−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	$0.408608\pi$
$648$	2.72098e8	0.0392837
$649$	−4.74080e9	−0.680763
$650$	0	0
$651$	−5.68545e9	−0.807666
$652$	−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	$0.695683\pi$
$654$	−1.45047e9	−0.202762
$655$	0	0
$656$	−1.10730e9	−0.153144
$657$	2.39146e9	0.328991
$658$	1.11272e9	0.152263
$659$	4.49137e9	0.611336	0.305668	−	0.952138i	$-0.401120\pi$
$659$	4.49137e9	0.611336	0.305668	+	0.952138i	$0.401120\pi$
$660$	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
$663$	6.18194e9	0.823810
$664$	2.16469e9	0.286951
$665$	0	0
$666$	3.27568e9	0.429677
$667$	−4.41334e8	−0.0575873
$668$	3.98325e9	0.517035
$669$	−3.67838e9	−0.474969
$670$	0	0
$671$	6.88886e8	0.0880276
$672$	−1.26252e9	−0.160489
$673$	−1.99688e9	−0.252522	−0.126261	−	0.991997i	$-0.540298\pi$
$673$	−1.99688e9	−0.252522	−0.126261	+	0.991997i	$0.540298\pi$
$674$	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	$0.534933\pi$
$678$	−2.97140e9	−0.366148
$679$	5.43477e9	0.666250
$680$	0	0
$681$	7.73369e9	0.938365
$682$	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	$0.202417\pi$
$684$	−2.03938e9	−0.243670
$685$	0	0
$686$	4.44360e9	0.525533
$687$	−7.67952e9	−0.903620
$688$	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
$693$	5.27423e9	0.601995
$694$	−9.98983e9	−1.13449
$695$	0	0
$696$	−1.93683e9	−0.217750
$697$	−5.20181e9	−0.581888
$698$	−7.80374e9	−0.868578
$699$	8.74887e9	0.968907
$700$	0	0
$701$	−8.98742e9	−0.985422	−0.492711	−	0.870193i	$-0.663994\pi$
$701$	−8.98742e9	−0.985422	−0.492711	+	0.870193i	$0.663994\pi$
$702$	1.87366e9	0.204415
$703$	−2.45513e10	−2.66521
$704$	1.32907e9	0.143563
$705$	0	0
$706$	−6.49742e9	−0.694903
$707$	−1.72086e10	−1.83138
$708$	1.61580e9	0.171108
$709$	−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
$711$	4.13538e9	0.431491
$712$	−6.08811e9	−0.632125
$713$	−4.64823e8	−0.0480258
$714$	−5.93100e9	−0.609795
$715$	0	0
$716$	−6.64452e9	−0.676500
$717$	−6.84114e9	−0.693125
$718$	−6.55452e9	−0.660854
$719$	−1.46045e10	−1.46534	−0.732668	−	0.680587i	$-0.761725\pi$
$719$	−1.46045e10	−1.46534	−0.732668	+	0.680587i	$0.761725\pi$
$720$	0	0
$721$	5.16504e9	0.513216
$722$	8.13424e9	0.804335
$723$	−2.91048e9	−0.286405
$724$	−7.75417e8	−0.0759365
$725$	0	0
$726$	−1.34303e9	−0.130259
$727$	1.42461e10	1.37507	0.687536	−	0.726150i	$-0.258692\pi$
$727$	1.42461e10	1.37507	0.687536	+	0.726150i	$0.258692\pi$
$728$	−8.69369e9	−0.835112
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	3.47670e9	0.329198
$732$	−2.34792e8	−0.0221256
$733$	8.78749e9	0.824140	0.412070	−	0.911152i	$-0.364806\pi$
$733$	8.78749e9	0.824140	0.412070	+	0.911152i	$0.364806\pi$
$734$	−3.55571e9	−0.331887
$735$	0	0
$736$	−1.03219e8	−0.00954307
$737$	−7.31821e9	−0.673393
$738$	−1.57660e9	−0.144386
$739$	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
$741$	−1.40432e10	−1.26795
$742$	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	$0.442577\pi$
$744$	−2.03991e9	−0.181596
$745$	0	0
$746$	4.19057e9	0.369562
$747$	3.08214e9	0.270540
$748$	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
$753$	−5.66755e9	−0.483741
$754$	−1.33370e10	−1.13307
$755$	0	0
$756$	−1.79761e9	−0.151310
$757$	1.51143e8	0.0126634	0.00633171	−	0.999980i	$-0.497985\pi$
$757$	1.51143e8	0.0126634	0.00633171	+	0.999980i	$0.497985\pi$
$758$	−4.81444e9	−0.401516
$759$	4.31204e8	0.0357961
$760$	0	0
$761$	3.85268e9	0.316896	0.158448	−	0.987367i	$-0.449351\pi$
$761$	3.85268e9	0.316896	0.158448	+	0.987367i	$0.449351\pi$
$762$	−8.67821e9	−0.710539
$763$	9.58251e9	0.780986
$764$	−5.52987e9	−0.448630
$765$	0	0
$766$	−3.74189e9	−0.300809
$767$	1.11264e10	0.890371
$768$	−4.52985e8	−0.0360844
$769$	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
$771$	5.09385e9	0.400273
$772$	−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	$0.309304\pi$
$774$	1.05374e9	0.0816848
$775$	0	0
$776$	1.94997e9	0.149800
$777$	−2.16407e10	−1.65500
$778$	−8.94269e9	−0.680831
$779$	1.18167e10	0.895599
$780$	0	0
$781$	−1.36172e10	−1.02284
$782$	−4.84898e8	−0.0362599
$783$	−2.75771e9	−0.205297
$784$	4.96757e9	0.368161
$785$	0	0
$786$	−7.44025e9	−0.546523
$787$	1.37729e10	1.00720	0.503599	−	0.863938i	$-0.332009\pi$
$787$	1.37729e10	1.00720	0.503599	+	0.863938i	$0.332009\pi$
$788$	−8.48113e9	−0.617465
$789$	−1.18457e10	−0.858601
$790$	0	0
$791$	1.96305e10	1.41031
$792$	1.89237e9	0.135353
$793$	−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	$0.577434\pi$
$798$	1.34731e10	0.938552
$799$	1.87552e9	0.130079
$800$	0	0
$801$	−8.66843e9	−0.595973
$802$	−3.44682e9	−0.235944
$803$	1.66320e10	1.13355
$804$	2.49425e9	0.169256
$805$	0	0
$806$	−1.40468e10	−0.944942
$807$	3.23758e9	0.216852
$808$	−6.17436e9	−0.411767
$809$	−1.28256e10	−0.851644	−0.425822	−	0.904807i	$-0.640015\pi$
$809$	−1.28256e10	−0.851644	−0.425822	+	0.904807i	$0.640015\pi$
$810$	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
$813$	1.07811e10	0.703631
$814$	2.27815e10	1.48046
$815$	0	0
$816$	−2.12801e9	−0.137107
$817$	−7.89783e9	−0.506677
$818$	5.88144e9	0.375705
$819$	−1.23783e10	−0.787351
$820$	0	0
$821$	−1.09919e10	−0.693219	−0.346610	−	0.938009i	$-0.612667\pi$
$821$	−1.09919e10	−0.693219	−0.346610	+	0.938009i	$0.612667\pi$
$822$	−6.00950e9	−0.377387
$823$	−1.15788e10	−0.724041	−0.362020	−	0.932170i	$-0.617913\pi$
$823$	−1.15788e10	−0.724041	−0.362020	+	0.932170i	$0.617913\pi$
$824$	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	$0.451097\pi$
$828$	−1.46966e8	−0.00899729
$829$	−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
$831$	2.55290e9	0.154323
$832$	−3.11925e9	−0.187767
$833$	2.33364e10	1.39887
$834$	1.89184e9	0.112928
$835$	0	0
$836$	−1.41833e10	−0.839570
$837$	−2.90448e9	−0.171210
$838$	−6.37926e9	−0.374469
$839$	4.50868e9	0.263562	0.131781	−	0.991279i	$-0.457930\pi$
$839$	4.50868e9	0.263562	0.131781	+	0.991279i	$0.457930\pi$
$840$	0	0
$841$	2.37981e9	0.137961
$842$	−1.73472e9	−0.100147
$843$	6.42139e9	0.369175
$844$	1.23973e10	0.709786
$845$	0	0
$846$	5.68445e8	0.0322769
$847$	8.87270e9	0.501723
$848$	8.72502e9	0.491339
$849$	1.44307e10	0.809303
$850$	0	0
$851$	−1.76927e9	−0.0984105
$852$	4.64113e9	0.257090
$853$	−9.35143e9	−0.515889	−0.257945	−	0.966160i	$-0.583045\pi$
$853$	−9.35143e9	−0.515889	−0.257945	+	0.966160i	$0.583045\pi$
$854$	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	$0.483949\pi$
$858$	1.30308e10	0.704315
$859$	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
$861$	1.04158e10	0.556135
$862$	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	$0.384300\pi$
$864$	−6.44973e8	−0.0340207
$865$	0	0
$866$	6.73937e9	0.352620
$867$	1.08227e9	0.0563987
$868$	1.34766e10	0.699459
$869$	2.87604e10	1.48671
$870$	0	0
$871$	1.71754e10	0.880732
$872$	3.43815e9	0.175597
$873$	2.77642e9	0.141233
$874$	1.10152e9	0.0558086
$875$	0	0
$876$	−5.66865e9	−0.284915
$877$	3.08992e9	0.154685	0.0773426	−	0.997005i	$-0.475356\pi$
$877$	3.08992e9	0.154685	0.0773426	+	0.997005i	$0.475356\pi$
$878$	7.03056e9	0.350558
$879$	−1.67088e10	−0.829822
$880$	0	0
$881$	8.72396e9	0.429831	0.214916	−	0.976633i	$-0.431052\pi$
$881$	8.72396e9	0.429831	0.214916	+	0.976633i	$0.431052\pi$
$882$	7.07297e9	0.347106
$883$	−3.90723e9	−0.190988	−0.0954942	−	0.995430i	$-0.530443\pi$
$883$	−3.90723e9	−0.190988	−0.0954942	+	0.995430i	$0.530443\pi$
$884$	−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	$0.379316\pi$
$888$	−7.76458e9	−0.372111
$889$	5.73325e10	2.73681
$890$	0	0
$891$	2.69441e9	0.127612
$892$	8.71913e9	0.411335
$893$	−4.26051e9	−0.200208
$894$	1.65930e10	0.776682
$895$	0	0
$896$	2.99264e9	0.138988
$897$	−1.01201e9	−0.0468178
$898$	−1.04750e10	−0.482711
$899$	2.06745e10	0.949020
$900$	0	0
$901$	4.09880e10	1.86690
$902$	−1.09648e10	−0.497484
$903$	−6.96154e9	−0.314628
$904$	7.04332e9	0.317094
$905$	0	0
$906$	3.08863e9	0.137980
$907$	−2.63986e10	−1.17478	−0.587388	−	0.809305i	$-0.699844\pi$
$907$	−2.63986e10	−1.17478	−0.587388	+	0.809305i	$0.699844\pi$
$908$	−1.83317e10	−0.812648
$909$	−8.79122e9	−0.388218
$910$	0	0
$911$	−2.12340e10	−0.930503	−0.465251	−	0.885179i	$-0.654036\pi$
$911$	−2.12340e10	−0.930503	−0.465251	+	0.885179i	$0.654036\pi$
$912$	4.83409e9	0.211024
$913$	2.14355e10	0.932149
$914$	−2.03331e10	−0.880828
$915$	0	0
$916$	1.82033e10	0.782558
$917$	4.91538e10	2.10506
$918$	−3.02992e9	−0.129265
$919$	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
$921$	4.72286e9	0.199203
$922$	4.68913e9	0.197031
$923$	3.19588e10	1.33778
$924$	−1.25019e10	−0.521343
$925$	0	0
$926$	1.25823e10	0.520741
$927$	2.63862e9	0.108792
$928$	4.59099e9	0.188577
$929$	−4.29492e10	−1.75752	−0.878760	−	0.477264i	$-0.841629\pi$
$929$	−4.29492e10	−1.75752	−0.878760	+	0.477264i	$0.841629\pi$
$930$	0	0
$931$	−5.30121e10	−2.15303
$932$	−2.07381e10	−0.839098
$933$	1.08930e9	0.0439096
$934$	1.26437e10	0.507761
$935$	0	0
$936$	−4.44128e9	−0.177028
$937$	1.52405e10	0.605216	0.302608	−	0.953115i	$-0.402143\pi$
$937$	1.52405e10	0.605216	0.302608	+	0.953115i	$0.402143\pi$
$938$	−1.64782e10	−0.651930
$939$	−2.24954e10	−0.886675
$940$	0	0
$941$	−2.27448e10	−0.889853	−0.444926	−	0.895567i	$-0.646770\pi$
$941$	−2.27448e10	−0.889853	−0.444926	+	0.895567i	$0.646770\pi$
$942$	1.37935e10	0.537645
$943$	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	$0.553887\pi$
$948$	−9.80238e9	−0.373682
$949$	−3.90343e10	−1.48257
$950$	0	0
$951$	−2.87542e10	−1.08410
$952$	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	$0.508172\pi$
$954$	1.24229e10	0.463239
$955$	0	0
$956$	1.62160e10	0.600263
$957$	−1.91791e10	−0.707354
$958$	2.96092e10	1.08804
$959$	3.97016e10	1.45359
$960$	0	0
$961$	−5.73778e9	−0.208551
$962$	−5.34669e10	−1.93630
$963$	−1.87301e9	−0.0675846
$964$	6.89891e9	0.248034
$965$	0	0
$966$	9.70931e8	0.0346552
$967$	2.51907e10	0.895874	0.447937	−	0.894065i	$-0.352159\pi$
$967$	2.51907e10	0.895874	0.447937	+	0.894065i	$0.352159\pi$
$968$	3.18348e9	0.112808
$969$	2.27094e10	0.801810
$970$	0	0
$971$	−5.11915e9	−0.179445	−0.0897223	−	0.995967i	$-0.528598\pi$
$971$	−5.11915e9	−0.179445	−0.0897223	+	0.995967i	$0.528598\pi$
$972$	−9.18330e8	−0.0320750
$973$	−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	$0.497234\pi$
$978$	1.90299e10	0.650506
$979$	−6.02866e10	−2.05344
$980$	0	0
$981$	4.89534e9	0.165555
$982$	−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	$0.554106\pi$
$984$	3.73712e9	0.125042
$985$	0	0
$986$	2.15674e10	0.716519
$987$	−3.75542e9	−0.124322
$988$	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
$993$	−5.20294e9	−0.168627
$994$	−3.06615e10	−0.990243
$995$	0	0
$996$	−7.30582e9	−0.234294
$997$	3.05555e10	0.976463	0.488232	−	0.872714i	$-0.337642\pi$
$997$	3.05555e10	0.976463	0.488232	+	0.872714i	$0.337642\pi$
$998$	−1.60171e10	−0.510068
$999$	−1.10554e10	−0.350830

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.8.a.f.1.1	✓	1	5.4	even	2
150.8.a.l.1.1	yes	1	1.1	even	1	trivial
150.8.c.d.49.1		2	5.3	odd	4
150.8.c.d.49.2		2	5.2	odd	4
450.8.a.m.1.1		1	3.2	odd	2
450.8.a.o.1.1		1	15.14	odd	2
450.8.c.d.199.1		2	15.2	even	4
450.8.c.d.199.2		2	15.8	even	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:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	150.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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