LMFDB - Embedded newform 150.8.a.c.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)

$f(q)$

$=$

$q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +216.000 q^{6} +391.000 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})$

\(q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +216.000 q^{6} +391.000 q^{7} -512.000 q^{8} +729.000 q^{9} -4398.00 q^{11} -1728.00 q^{12} +13447.0 q^{13} -3128.00 q^{14} +4096.00 q^{16} +7686.00 q^{17} -5832.00 q^{18} -13705.0 q^{19} -10557.0 q^{21} +35184.0 q^{22} -35478.0 q^{23} +13824.0 q^{24} -107576. q^{26} -19683.0 q^{27} +25024.0 q^{28} -157470. q^{29} -99343.0 q^{31} -32768.0 q^{32} +118746. q^{33} -61488.0 q^{34} +46656.0 q^{36} +161926. q^{37} +109640. q^{38} -363069. q^{39} +521952. q^{41} +84456.0 q^{42} -340973. q^{43} -281472. q^{44} +283824. q^{46} +50886.0 q^{47} -110592. q^{48} -670662. q^{49} -207522. q^{51} +860608. q^{52} +891132. q^{53} +157464. q^{54} -200192. q^{56} +370035. q^{57} +1.25976e6 q^{58} -1.34421e6 q^{59} +3.39413e6 q^{61} +794744. q^{62} +285039. q^{63} +262144. q^{64} -949968. q^{66} +2.24895e6 q^{67} +491904. q^{68} +957906. q^{69} +2.73187e6 q^{71} -373248. q^{72} +5.02862e6 q^{73} -1.29541e6 q^{74} -877120. q^{76} -1.71962e6 q^{77} +2.90455e6 q^{78} +1.57148e6 q^{79} +531441. q^{81} -4.17562e6 q^{82} +7.79296e6 q^{83} -675648. q^{84} +2.72778e6 q^{86} +4.25169e6 q^{87} +2.25178e6 q^{88} -5.80224e6 q^{89} +5.25778e6 q^{91} -2.27059e6 q^{92} +2.68226e6 q^{93} -407088. q^{94} +884736. q^{96} +2.49831e6 q^{97} +5.36530e6 q^{98} -3.20614e6 q^{99} +O(q^{100})\)

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
$7$	391.000	0.430857	0.215429	−	0.976520i	$-0.430885\pi$
$7$	391.000	0.430857	0.215429	+	0.976520i	$0.430885\pi$
$8$	−512.000	−0.353553
$9$	729.000	0.333333
$10$	0	0
$11$	−4398.00	−0.996279	−0.498139	−	0.867097i	$-0.665983\pi$
$11$	−4398.00	−0.996279	−0.498139	+	0.867097i	$0.665983\pi$
$12$	−1728.00	−0.288675
$13$	13447.0	1.69755	0.848777	−	0.528751i	$-0.177339\pi$
$13$	13447.0	1.69755	0.848777	+	0.528751i	$0.177339\pi$
$14$	−3128.00	−0.304662
$15$	0	0
$16$	4096.00	0.250000
$17$	7686.00	0.379428	0.189714	−	0.981839i	$-0.439244\pi$
$17$	7686.00	0.379428	0.189714	+	0.981839i	$0.439244\pi$
$18$	−5832.00	−0.235702
$19$	−13705.0	−0.458397	−0.229198	−	0.973380i	$-0.573610\pi$
$19$	−13705.0	−0.458397	−0.229198	+	0.973380i	$0.573610\pi$
$20$	0	0
$21$	−10557.0	−0.248756
$22$	35184.0	0.704475
$23$	−35478.0	−0.608011	−0.304006	−	0.952670i	$-0.598324\pi$
$23$	−35478.0	−0.608011	−0.304006	+	0.952670i	$0.598324\pi$
$24$	13824.0	0.204124
$25$	0	0
$26$	−107576.	−1.20035
$27$	−19683.0	−0.192450
$28$	25024.0	0.215429
$29$	−157470.	−1.19896	−0.599480	−	0.800390i	$-0.704626\pi$
$29$	−157470.	−1.19896	−0.599480	+	0.800390i	$0.704626\pi$
$30$	0	0
$31$	−99343.0	−0.598923	−0.299462	−	0.954108i	$-0.596807\pi$
$31$	−99343.0	−0.598923	−0.299462	+	0.954108i	$0.596807\pi$
$32$	−32768.0	−0.176777
$33$	118746.	0.575202
$34$	−61488.0	−0.268296
$35$	0	0
$36$	46656.0	0.166667
$37$	161926.	0.525546	0.262773	−	0.964858i	$-0.415363\pi$
$37$	161926.	0.525546	0.262773	+	0.964858i	$0.415363\pi$
$38$	109640.	0.324135
$39$	−363069.	−0.980083
$40$	0	0
$41$	521952.	1.18273	0.591367	−	0.806403i	$-0.298588\pi$
$41$	521952.	1.18273	0.591367	+	0.806403i	$0.298588\pi$
$42$	84456.0	0.175897
$43$	−340973.	−0.654004	−0.327002	−	0.945024i	$-0.606038\pi$
$43$	−340973.	−0.654004	−0.327002	+	0.945024i	$0.606038\pi$
$44$	−281472.	−0.498139
$45$	0	0
$46$	283824.	0.429929
$47$	50886.0	0.0714917	0.0357459	−	0.999361i	$-0.488619\pi$
$47$	50886.0	0.0714917	0.0357459	+	0.999361i	$0.488619\pi$
$48$	−110592.	−0.144338
$49$	−670662.	−0.814362
$50$	0	0
$51$	−207522.	−0.219063
$52$	860608.	0.848777
$53$	891132.	0.822198	0.411099	−	0.911591i	$-0.365145\pi$
$53$	891132.	0.822198	0.411099	+	0.911591i	$0.365145\pi$
$54$	157464.	0.136083
$55$	0	0
$56$	−200192.	−0.152331
$57$	370035.	0.264655
$58$	1.25976e6	0.847793
$59$	−1.34421e6	−0.852089	−0.426045	−	0.904702i	$-0.640093\pi$
$59$	−1.34421e6	−0.852089	−0.426045	+	0.904702i	$0.640093\pi$
$60$	0	0
$61$	3.39413e6	1.91458	0.957290	−	0.289128i	$-0.0933655\pi$
$61$	3.39413e6	1.91458	0.957290	+	0.289128i	$0.0933655\pi$
$62$	794744.	0.423503
$63$	285039.	0.143619
$64$	262144.	0.125000
$65$	0	0
$66$	−949968.	−0.406729
$67$	2.24895e6	0.913520	0.456760	−	0.889590i	$-0.349010\pi$
$67$	2.24895e6	0.913520	0.456760	+	0.889590i	$0.349010\pi$
$68$	491904.	0.189714
$69$	957906.	0.351036
$70$	0	0
$71$	2.73187e6	0.905850	0.452925	−	0.891549i	$-0.350381\pi$
$71$	2.73187e6	0.905850	0.452925	+	0.891549i	$0.350381\pi$
$72$	−373248.	−0.117851
$73$	5.02862e6	1.51293	0.756465	−	0.654034i	$-0.226925\pi$
$73$	5.02862e6	1.51293	0.756465	+	0.654034i	$0.226925\pi$
$74$	−1.29541e6	−0.371617
$75$	0	0
$76$	−877120.	−0.229198
$77$	−1.71962e6	−0.429254
$78$	2.90455e6	0.693024
$79$	1.57148e6	0.358603	0.179302	−	0.983794i	$-0.442616\pi$
$79$	1.57148e6	0.358603	0.179302	+	0.983794i	$0.442616\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	−4.17562e6	−0.836319
$83$	7.79296e6	1.49599	0.747996	−	0.663703i	$-0.231016\pi$
$83$	7.79296e6	1.49599	0.747996	+	0.663703i	$0.231016\pi$
$84$	−675648.	−0.124378
$85$	0	0
$86$	2.72778e6	0.462451
$87$	4.25169e6	0.692220
$88$	2.25178e6	0.352238
$89$	−5.80224e6	−0.872430	−0.436215	−	0.899842i	$-0.643681\pi$
$89$	−5.80224e6	−0.872430	−0.436215	+	0.899842i	$0.643681\pi$
$90$	0	0
$91$	5.25778e6	0.731404
$92$	−2.27059e6	−0.304006
$93$	2.68226e6	0.345789
$94$	−407088.	−0.0505523
$95$	0	0
$96$	884736.	0.102062
$97$	2.49831e6	0.277936	0.138968	−	0.990297i	$-0.455621\pi$
$97$	2.49831e6	0.277936	0.138968	+	0.990297i	$0.455621\pi$
$98$	5.36530e6	0.575841
$99$	−3.20614e6	−0.332093
$100$	0	0
$101$	1.14049e7	1.10146	0.550729	−	0.834684i	$-0.314350\pi$
$101$	1.14049e7	1.10146	0.550729	+	0.834684i	$0.314350\pi$
$102$	1.66018e6	0.154901
$103$	1.16323e7	1.04890	0.524449	−	0.851442i	$-0.324271\pi$
$103$	1.16323e7	1.04890	0.524449	+	0.851442i	$0.324271\pi$
$104$	−6.88486e6	−0.600176
$105$	0	0
$106$	−7.12906e6	−0.581382
$107$	−5.78912e6	−0.456846	−0.228423	−	0.973562i	$-0.573357\pi$
$107$	−5.78912e6	−0.456846	−0.228423	+	0.973562i	$0.573357\pi$
$108$	−1.25971e6	−0.0962250
$109$	5.27676e6	0.390279	0.195139	−	0.980776i	$-0.437484\pi$
$109$	5.27676e6	0.390279	0.195139	+	0.980776i	$0.437484\pi$
$110$	0	0
$111$	−4.37200e6	−0.303424
$112$	1.60154e6	0.107714
$113$	−2.25596e7	−1.47081	−0.735406	−	0.677626i	$-0.763009\pi$
$113$	−2.25596e7	−1.47081	−0.735406	+	0.677626i	$0.763009\pi$
$114$	−2.96028e6	−0.187140
$115$	0	0
$116$	−1.00781e7	−0.599480
$117$	9.80286e6	0.565851
$118$	1.07537e7	0.602518
$119$	3.00523e6	0.163479
$120$	0	0
$121$	−144767.	−0.00742884
$122$	−2.71530e7	−1.35381
$123$	−1.40927e7	−0.682852
$124$	−6.35795e6	−0.299462
$125$	0	0
$126$	−2.28031e6	−0.101554
$127$	2.64183e7	1.14443	0.572217	−	0.820102i	$-0.306083\pi$
$127$	2.64183e7	1.14443	0.572217	+	0.820102i	$0.306083\pi$
$128$	−2.09715e6	−0.0883883
$129$	9.20627e6	0.377589
$130$	0	0
$131$	3.30870e7	1.28590	0.642951	−	0.765907i	$-0.277710\pi$
$131$	3.30870e7	1.28590	0.642951	+	0.765907i	$0.277710\pi$
$132$	7.59974e6	0.287601
$133$	−5.35866e6	−0.197504
$134$	−1.79916e7	−0.645956
$135$	0	0
$136$	−3.93523e6	−0.134148
$137$	5.31303e7	1.76531	0.882654	−	0.470024i	$-0.155755\pi$
$137$	5.31303e7	1.76531	0.882654	+	0.470024i	$0.155755\pi$
$138$	−7.66325e6	−0.248220
$139$	4.77106e7	1.50683	0.753413	−	0.657548i	$-0.228406\pi$
$139$	4.77106e7	1.50683	0.753413	+	0.657548i	$0.228406\pi$
$140$	0	0
$141$	−1.37392e6	−0.0412758
$142$	−2.18550e7	−0.640532
$143$	−5.91399e7	−1.69124
$144$	2.98598e6	0.0833333
$145$	0	0
$146$	−4.02290e7	−1.06980
$147$	1.81079e7	0.470172
$148$	1.03633e7	0.262773
$149$	−6.29027e7	−1.55782	−0.778910	−	0.627135i	$-0.784227\pi$
$149$	−6.29027e7	−1.55782	−0.778910	+	0.627135i	$0.784227\pi$
$150$	0	0
$151$	−2.39117e7	−0.565185	−0.282593	−	0.959240i	$-0.591194\pi$
$151$	−2.39117e7	−0.565185	−0.282593	+	0.959240i	$0.591194\pi$
$152$	7.01696e6	0.162068
$153$	5.60309e6	0.126476
$154$	1.37569e7	0.303528
$155$	0	0
$156$	−2.32364e7	−0.490042
$157$	−8.35247e7	−1.72253	−0.861264	−	0.508158i	$-0.830327\pi$
$157$	−8.35247e7	−1.72253	−0.861264	+	0.508158i	$0.830327\pi$
$158$	−1.25718e7	−0.253571
$159$	−2.40606e7	−0.474696
$160$	0	0
$161$	−1.38719e7	−0.261966
$162$	−4.25153e6	−0.0785674
$163$	−9.82251e7	−1.77650	−0.888251	−	0.459358i	$-0.848079\pi$
$163$	−9.82251e7	−1.77650	−0.888251	+	0.459358i	$0.848079\pi$
$164$	3.34049e7	0.591367
$165$	0	0
$166$	−6.23437e7	−1.05783
$167$	−8.13980e7	−1.35240	−0.676201	−	0.736717i	$-0.736375\pi$
$167$	−8.13980e7	−1.35240	−0.676201	+	0.736717i	$0.736375\pi$
$168$	5.40518e6	0.0879484
$169$	1.18073e8	1.88169
$170$	0	0
$171$	−9.99094e6	−0.152799
$172$	−2.18223e7	−0.327002
$173$	−3.34611e7	−0.491337	−0.245668	−	0.969354i	$-0.579007\pi$
$173$	−3.34611e7	−0.491337	−0.245668	+	0.969354i	$0.579007\pi$
$174$	−3.40135e7	−0.489474
$175$	0	0
$176$	−1.80142e7	−0.249070
$177$	3.62937e7	0.491954
$178$	4.64179e7	0.616901
$179$	5.88533e7	0.766982	0.383491	−	0.923545i	$-0.374722\pi$
$179$	5.88533e7	0.766982	0.383491	+	0.923545i	$0.374722\pi$
$180$	0	0
$181$	1.48383e8	1.85999	0.929994	−	0.367574i	$-0.119812\pi$
$181$	1.48383e8	1.85999	0.929994	+	0.367574i	$0.119812\pi$
$182$	−4.20622e7	−0.517181
$183$	−9.16414e7	−1.10538
$184$	1.81647e7	0.214964
$185$	0	0
$186$	−2.14581e7	−0.244509
$187$	−3.38030e7	−0.378016
$188$	3.25670e6	0.0357459
$189$	−7.69605e6	−0.0829186
$190$	0	0
$191$	−8.22340e7	−0.853954	−0.426977	−	0.904262i	$-0.640421\pi$
$191$	−8.22340e7	−0.853954	−0.426977	+	0.904262i	$0.640421\pi$
$192$	−7.07789e6	−0.0721688
$193$	9.32235e7	0.933415	0.466707	−	0.884412i	$-0.345440\pi$
$193$	9.32235e7	0.933415	0.466707	+	0.884412i	$0.345440\pi$
$194$	−1.99865e7	−0.196531
$195$	0	0
$196$	−4.29224e7	−0.407181
$197$	3.49597e7	0.325788	0.162894	−	0.986644i	$-0.447917\pi$
$197$	3.49597e7	0.325788	0.162894	+	0.986644i	$0.447917\pi$
$198$	2.56491e7	0.234825
$199$	1.53518e8	1.38093	0.690467	−	0.723364i	$-0.257405\pi$
$199$	1.53518e8	1.38093	0.690467	+	0.723364i	$0.257405\pi$
$200$	0	0
$201$	−6.07217e7	−0.527421
$202$	−9.12395e7	−0.778848
$203$	−6.15708e7	−0.516581
$204$	−1.32814e7	−0.109531
$205$	0	0
$206$	−9.30580e7	−0.741683
$207$	−2.58635e7	−0.202670
$208$	5.50789e7	0.424389
$209$	6.02746e7	0.456691
$210$	0	0
$211$	1.09214e8	0.800367	0.400183	−	0.916435i	$-0.368946\pi$
$211$	1.09214e8	0.800367	0.400183	+	0.916435i	$0.368946\pi$
$212$	5.70324e7	0.411099
$213$	−7.37605e7	−0.522993
$214$	4.63130e7	0.323039
$215$	0	0
$216$	1.00777e7	0.0680414
$217$	−3.88431e7	−0.258051
$218$	−4.22141e7	−0.275969
$219$	−1.35773e8	−0.873491
$220$	0	0
$221$	1.03354e8	0.644099
$222$	3.49760e7	0.214553
$223$	1.08024e8	0.652307	0.326154	−	0.945317i	$-0.394247\pi$
$223$	1.08024e8	0.652307	0.326154	+	0.945317i	$0.394247\pi$
$224$	−1.28123e7	−0.0761656
$225$	0	0
$226$	1.80477e8	1.04002
$227$	2.99005e8	1.69663	0.848315	−	0.529491i	$-0.177617\pi$
$227$	2.99005e8	1.69663	0.848315	+	0.529491i	$0.177617\pi$
$228$	2.36822e7	0.132328
$229$	1.61024e8	0.886065	0.443032	−	0.896506i	$-0.353903\pi$
$229$	1.61024e8	0.886065	0.443032	+	0.896506i	$0.353903\pi$
$230$	0	0
$231$	4.64297e7	0.247830
$232$	8.06246e7	0.423897
$233$	9.08599e7	0.470572	0.235286	−	0.971926i	$-0.424397\pi$
$233$	9.08599e7	0.470572	0.235286	+	0.971926i	$0.424397\pi$
$234$	−7.84229e7	−0.400117
$235$	0	0
$236$	−8.60294e7	−0.426045
$237$	−4.24300e7	−0.207040
$238$	−2.40418e7	−0.115597
$239$	−3.44068e8	−1.63024	−0.815121	−	0.579290i	$-0.803330\pi$
$239$	−3.44068e8	−1.63024	−0.815121	+	0.579290i	$0.803330\pi$
$240$	0	0
$241$	8.00430e7	0.368353	0.184176	−	0.982893i	$-0.441038\pi$
$241$	8.00430e7	0.368353	0.184176	+	0.982893i	$0.441038\pi$
$242$	1.15814e6	0.00525298
$243$	−1.43489e7	−0.0641500
$244$	2.17224e8	0.957290
$245$	0	0
$246$	1.12742e8	0.482849
$247$	−1.84291e8	−0.778153
$248$	5.08636e7	0.211751
$249$	−2.10410e8	−0.863711
$250$	0	0
$251$	−337188.	−0.00134590	−0.000672952	−	1.00000i	$-0.500214\pi$
$251$	−337188.	−0.00134590	−0.000672952		1.00000i	$0.500214\pi$
$252$	1.82425e7	0.0718096
$253$	1.56032e8	0.605749
$254$	−2.11346e8	−0.809238
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	4.49342e8	1.65124	0.825622	−	0.564224i	$-0.190824\pi$
$257$	4.49342e8	1.65124	0.825622	+	0.564224i	$0.190824\pi$
$258$	−7.36502e7	−0.266996
$259$	6.33131e7	0.226435
$260$	0	0
$261$	−1.14796e8	−0.399653
$262$	−2.64696e8	−0.909270
$263$	2.61487e8	0.886350	0.443175	−	0.896435i	$-0.353852\pi$
$263$	2.61487e8	0.886350	0.443175	+	0.896435i	$0.353852\pi$
$264$	−6.07980e7	−0.203365
$265$	0	0
$266$	4.28692e7	0.139656
$267$	1.56660e8	0.503698
$268$	1.43933e8	0.456760
$269$	−1.90967e8	−0.598170	−0.299085	−	0.954227i	$-0.596681\pi$
$269$	−1.90967e8	−0.598170	−0.299085	+	0.954227i	$0.596681\pi$
$270$	0	0
$271$	−1.25436e7	−0.0382850	−0.0191425	−	0.999817i	$-0.506094\pi$
$271$	−1.25436e7	−0.0382850	−0.0191425	+	0.999817i	$0.506094\pi$
$272$	3.14819e7	0.0948570
$273$	−1.41960e8	−0.422276
$274$	−4.25042e8	−1.24826
$275$	0	0
$276$	6.13060e7	0.175518
$277$	−2.37038e8	−0.670099	−0.335049	−	0.942201i	$-0.608753\pi$
$277$	−2.37038e8	−0.670099	−0.335049	+	0.942201i	$0.608753\pi$
$278$	−3.81685e8	−1.06549
$279$	−7.24210e7	−0.199641
$280$	0	0
$281$	8.65943e7	0.232818	0.116409	−	0.993201i	$-0.462862\pi$
$281$	8.65943e7	0.232818	0.116409	+	0.993201i	$0.462862\pi$
$282$	1.09914e7	0.0291864
$283$	4.33034e8	1.13572	0.567858	−	0.823126i	$-0.307772\pi$
$283$	4.33034e8	1.13572	0.567858	+	0.823126i	$0.307772\pi$
$284$	1.74840e8	0.452925
$285$	0	0
$286$	4.73119e8	1.19589
$287$	2.04083e8	0.509590
$288$	−2.38879e7	−0.0589256
$289$	−3.51264e8	−0.856035
$290$	0	0
$291$	−6.74544e7	−0.160467
$292$	3.21832e8	0.756465
$293$	7.48968e8	1.73951	0.869754	−	0.493485i	$-0.164277\pi$
$293$	7.48968e8	1.73951	0.869754	+	0.493485i	$0.164277\pi$
$294$	−1.44863e8	−0.332462
$295$	0	0
$296$	−8.29061e7	−0.185808
$297$	8.65658e7	0.191734
$298$	5.03222e8	1.10155
$299$	−4.77073e8	−1.03213
$300$	0	0
$301$	−1.33320e8	−0.281782
$302$	1.91294e8	0.399646
$303$	−3.07933e8	−0.635927
$304$	−5.61357e7	−0.114599
$305$	0	0
$306$	−4.48248e7	−0.0894320
$307$	−4.07994e8	−0.804766	−0.402383	−	0.915471i	$-0.631818\pi$
$307$	−4.07994e8	−0.804766	−0.402383	+	0.915471i	$0.631818\pi$
$308$	−1.10056e8	−0.214627
$309$	−3.14071e8	−0.605582
$310$	0	0
$311$	−7.67102e8	−1.44608	−0.723039	−	0.690808i	$-0.757255\pi$
$311$	−7.67102e8	−1.44608	−0.723039	+	0.690808i	$0.757255\pi$
$312$	1.85891e8	0.346512
$313$	5.21907e8	0.962028	0.481014	−	0.876713i	$-0.340269\pi$
$313$	5.21907e8	0.962028	0.481014	+	0.876713i	$0.340269\pi$
$314$	6.68198e8	1.21801
$315$	0	0
$316$	1.00575e8	0.179302
$317$	−4.62217e8	−0.814965	−0.407482	−	0.913213i	$-0.633593\pi$
$317$	−4.62217e8	−0.814965	−0.407482	+	0.913213i	$0.633593\pi$
$318$	1.92485e8	0.335661
$319$	6.92553e8	1.19450
$320$	0	0
$321$	1.56306e8	0.263760
$322$	1.10975e8	0.185238
$323$	−1.05337e8	−0.173928
$324$	3.40122e7	0.0555556
$325$	0	0
$326$	7.85801e8	1.25618
$327$	−1.42473e8	−0.225328
$328$	−2.67239e8	−0.418160
$329$	1.98964e7	0.0308027
$330$	0	0
$331$	−1.29664e8	−0.196527	−0.0982637	−	0.995160i	$-0.531329\pi$
$331$	−1.29664e8	−0.196527	−0.0982637	+	0.995160i	$0.531329\pi$
$332$	4.98750e8	0.747996
$333$	1.18044e8	0.175182
$334$	6.51184e8	0.956293
$335$	0	0
$336$	−4.32415e7	−0.0621889
$337$	−6.32060e8	−0.899609	−0.449804	−	0.893127i	$-0.648506\pi$
$337$	−6.32060e8	−0.899609	−0.449804	+	0.893127i	$0.648506\pi$
$338$	−9.44586e8	−1.33056
$339$	6.09110e8	0.849174
$340$	0	0
$341$	4.36911e8	0.596695
$342$	7.99276e7	0.108045
$343$	−5.84234e8	−0.781731
$344$	1.74578e8	0.231225
$345$	0	0
$346$	2.67689e8	0.347427
$347$	4.73147e8	0.607915	0.303957	−	0.952686i	$-0.401692\pi$
$347$	4.73147e8	0.607915	0.303957	+	0.952686i	$0.401692\pi$
$348$	2.72108e8	0.346110
$349$	1.34069e9	1.68827	0.844133	−	0.536134i	$-0.180116\pi$
$349$	1.34069e9	1.68827	0.844133	+	0.536134i	$0.180116\pi$
$350$	0	0
$351$	−2.64677e8	−0.326694
$352$	1.44114e8	0.176119
$353$	1.60419e8	0.194108	0.0970541	−	0.995279i	$-0.469058\pi$
$353$	1.60419e8	0.194108	0.0970541	+	0.995279i	$0.469058\pi$
$354$	−2.90349e8	−0.347864
$355$	0	0
$356$	−3.71343e8	−0.436215
$357$	−8.11411e7	−0.0943848
$358$	−4.70826e8	−0.542338
$359$	−1.04189e9	−1.18848	−0.594239	−	0.804288i	$-0.702547\pi$
$359$	−1.04189e9	−1.18848	−0.594239	+	0.804288i	$0.702547\pi$
$360$	0	0
$361$	−7.06045e8	−0.789873
$362$	−1.18707e9	−1.31521
$363$	3.90871e6	0.00428904
$364$	3.36498e8	0.365702
$365$	0	0
$366$	7.33131e8	0.781624
$367$	−1.29681e8	−0.136944	−0.0684721	−	0.997653i	$-0.521812\pi$
$367$	−1.29681e8	−0.136944	−0.0684721	+	0.997653i	$0.521812\pi$
$368$	−1.45318e8	−0.152003
$369$	3.80503e8	0.394245
$370$	0	0
$371$	3.48433e8	0.354250
$372$	1.71665e8	0.172894
$373$	−5.67328e8	−0.566048	−0.283024	−	0.959113i	$-0.591338\pi$
$373$	−5.67328e8	−0.566048	−0.283024	+	0.959113i	$0.591338\pi$
$374$	2.70424e8	0.267298
$375$	0	0
$376$	−2.60536e7	−0.0252761
$377$	−2.11750e9	−2.03530
$378$	6.15684e7	0.0586323
$379$	−1.43047e9	−1.34971	−0.674857	−	0.737948i	$-0.735795\pi$
$379$	−1.43047e9	−1.34971	−0.674857	+	0.737948i	$0.735795\pi$
$380$	0	0
$381$	−7.13293e8	−0.660740
$382$	6.57872e8	0.603837
$383$	−1.27053e8	−0.115555	−0.0577776	−	0.998329i	$-0.518401\pi$
$383$	−1.27053e8	−0.115555	−0.0577776	+	0.998329i	$0.518401\pi$
$384$	5.66231e7	0.0510310
$385$	0	0
$386$	−7.45788e8	−0.660024
$387$	−2.48569e8	−0.218001
$388$	1.59892e8	0.138968
$389$	1.05093e9	0.905212	0.452606	−	0.891711i	$-0.350494\pi$
$389$	1.05093e9	0.905212	0.452606	+	0.891711i	$0.350494\pi$
$390$	0	0
$391$	−2.72684e8	−0.230696
$392$	3.43379e8	0.287920
$393$	−8.93349e8	−0.742416
$394$	−2.79677e8	−0.230367
$395$	0	0
$396$	−2.05193e8	−0.166046
$397$	8.46196e8	0.678741	0.339371	−	0.940653i	$-0.389786\pi$
$397$	8.46196e8	0.678741	0.339371	+	0.940653i	$0.389786\pi$
$398$	−1.22814e9	−0.976468
$399$	1.44684e8	0.114029
$400$	0	0
$401$	2.76860e8	0.214415	0.107208	−	0.994237i	$-0.465809\pi$
$401$	2.76860e8	0.214415	0.107208	+	0.994237i	$0.465809\pi$
$402$	4.85773e8	0.372943
$403$	−1.33587e9	−1.01671
$404$	7.29916e8	0.550729
$405$	0	0
$406$	4.92566e8	0.365278
$407$	−7.12151e8	−0.523590
$408$	1.06251e8	0.0774504
$409$	1.10876e9	0.801322	0.400661	−	0.916226i	$-0.368781\pi$
$409$	1.10876e9	0.801322	0.400661	+	0.916226i	$0.368781\pi$
$410$	0	0
$411$	−1.43452e9	−1.01920
$412$	7.44464e8	0.524449
$413$	−5.25586e8	−0.367129
$414$	2.06908e8	0.143310
$415$	0	0
$416$	−4.40631e8	−0.300088
$417$	−1.28819e9	−0.869966
$418$	−4.82197e8	−0.322929
$419$	−9.77258e8	−0.649023	−0.324512	−	0.945882i	$-0.605200\pi$
$419$	−9.77258e8	−0.649023	−0.324512	+	0.945882i	$0.605200\pi$
$420$	0	0
$421$	−2.26975e9	−1.48248	−0.741242	−	0.671238i	$-0.765763\pi$
$421$	−2.26975e9	−1.48248	−0.741242	+	0.671238i	$0.765763\pi$
$422$	−8.73711e8	−0.565945
$423$	3.70959e7	0.0238306
$424$	−4.56260e8	−0.290691
$425$	0	0
$426$	5.90084e8	0.369812
$427$	1.32710e9	0.824911
$428$	−3.70504e8	−0.228423
$429$	1.59678e9	0.976436
$430$	0	0
$431$	6.00005e7	0.0360981	0.0180490	−	0.999837i	$-0.494254\pi$
$431$	6.00005e7	0.0360981	0.0180490	+	0.999837i	$0.494254\pi$
$432$	−8.06216e7	−0.0481125
$433$	−8.11589e8	−0.480428	−0.240214	−	0.970720i	$-0.577218\pi$
$433$	−8.11589e8	−0.480428	−0.240214	+	0.970720i	$0.577218\pi$
$434$	3.10745e8	0.182469
$435$	0	0
$436$	3.37713e8	0.195139
$437$	4.86226e8	0.278710
$438$	1.08618e9	0.617651
$439$	−2.52075e9	−1.42201	−0.711006	−	0.703185i	$-0.751760\pi$
$439$	−2.52075e9	−1.42201	−0.711006	+	0.703185i	$0.751760\pi$
$440$	0	0
$441$	−4.88913e8	−0.271454
$442$	−8.26829e8	−0.455447
$443$	2.11506e9	1.15587	0.577936	−	0.816082i	$-0.303858\pi$
$443$	2.11506e9	1.15587	0.577936	+	0.816082i	$0.303858\pi$
$444$	−2.79808e8	−0.151712
$445$	0	0
$446$	−8.64190e8	−0.461251
$447$	1.69837e9	0.899408
$448$	1.02498e8	0.0538572
$449$	7.45547e7	0.0388699	0.0194349	−	0.999811i	$-0.493813\pi$
$449$	7.45547e7	0.0388699	0.0194349	+	0.999811i	$0.493813\pi$
$450$	0	0
$451$	−2.29554e9	−1.17833
$452$	−1.44382e9	−0.735406
$453$	6.45616e8	0.326310
$454$	−2.39204e9	−1.19970
$455$	0	0
$456$	−1.89458e8	−0.0935698
$457$	−6.94773e7	−0.0340515	−0.0170257	−	0.999855i	$-0.505420\pi$
$457$	−6.94773e7	−0.0340515	−0.0170257	+	0.999855i	$0.505420\pi$
$458$	−1.28819e9	−0.626542
$459$	−1.51284e8	−0.0730209
$460$	0	0
$461$	6.30756e8	0.299853	0.149927	−	0.988697i	$-0.452096\pi$
$461$	6.30756e8	0.299853	0.149927	+	0.988697i	$0.452096\pi$
$462$	−3.71437e8	−0.175242
$463$	−4.00480e9	−1.87520	−0.937600	−	0.347715i	$-0.886958\pi$
$463$	−4.00480e9	−1.87520	−0.937600	+	0.347715i	$0.886958\pi$
$464$	−6.44997e8	−0.299740
$465$	0	0
$466$	−7.26879e8	−0.332745
$467$	1.55339e9	0.705782	0.352891	−	0.935664i	$-0.385199\pi$
$467$	1.55339e9	0.705782	0.352891	+	0.935664i	$0.385199\pi$
$468$	6.27383e8	0.282926
$469$	8.79340e8	0.393597
$470$	0	0
$471$	2.25517e9	0.994502
$472$	6.88236e8	0.301259
$473$	1.49960e9	0.651570
$474$	3.39440e8	0.146399
$475$	0	0
$476$	1.92334e8	0.0817397
$477$	6.49635e8	0.274066
$478$	2.75255e9	1.15276
$479$	1.86723e8	0.0776288	0.0388144	−	0.999246i	$-0.487642\pi$
$479$	1.86723e8	0.0776288	0.0388144	+	0.999246i	$0.487642\pi$
$480$	0	0
$481$	2.17742e9	0.892142
$482$	−6.40344e8	−0.260465
$483$	3.74541e8	0.151246
$484$	−9.26509e6	−0.00371442
$485$	0	0
$486$	1.14791e8	0.0453609
$487$	3.00114e9	1.17743	0.588714	−	0.808341i	$-0.299634\pi$
$487$	3.00114e9	1.17743	0.588714	+	0.808341i	$0.299634\pi$
$488$	−1.73779e9	−0.676906
$489$	2.65208e9	1.02566
$490$	0	0
$491$	4.51257e9	1.72044	0.860218	−	0.509927i	$-0.170328\pi$
$491$	4.51257e9	1.72044	0.860218	+	0.509927i	$0.170328\pi$
$492$	−9.01933e8	−0.341426
$493$	−1.21031e9	−0.454919
$494$	1.47433e9	0.550237
$495$	0	0
$496$	−4.06909e8	−0.149731
$497$	1.06816e9	0.390292
$498$	1.68328e9	0.610736
$499$	−3.03177e9	−1.09231	−0.546153	−	0.837685i	$-0.683908\pi$
$499$	−3.03177e9	−1.09231	−0.546153	+	0.837685i	$0.683908\pi$
$500$	0	0
$501$	2.19775e9	0.780810
$502$	2.69750e6	0.000951698 0
$503$	3.58716e9	1.25679	0.628395	−	0.777894i	$-0.283712\pi$
$503$	3.58716e9	1.25679	0.628395	+	0.777894i	$0.283712\pi$
$504$	−1.45940e8	−0.0507770
$505$	0	0
$506$	−1.24826e9	−0.428329
$507$	−3.18798e9	−1.08639
$508$	1.69077e9	0.572217
$509$	−1.19095e9	−0.400296	−0.200148	−	0.979766i	$-0.564142\pi$
$509$	−1.19095e9	−0.400296	−0.200148	+	0.979766i	$0.564142\pi$
$510$	0	0
$511$	1.96619e9	0.651857
$512$	−1.34218e8	−0.0441942
$513$	2.69756e8	0.0882185
$514$	−3.59474e9	−1.16761
$515$	0	0
$516$	5.89201e8	0.188795
$517$	−2.23797e8	−0.0712257
$518$	−5.06505e8	−0.160114
$519$	9.03450e8	0.283673
$520$	0	0
$521$	3.98654e9	1.23499	0.617495	−	0.786574i	$-0.288147\pi$
$521$	3.98654e9	1.23499	0.617495	+	0.786574i	$0.288147\pi$
$522$	9.18365e8	0.282598
$523$	2.99646e9	0.915911	0.457955	−	0.888975i	$-0.348582\pi$
$523$	2.99646e9	0.915911	0.457955	+	0.888975i	$0.348582\pi$
$524$	2.11757e9	0.642951
$525$	0	0
$526$	−2.09190e9	−0.626744
$527$	−7.63550e8	−0.227248
$528$	4.86384e8	0.143800
$529$	−2.14614e9	−0.630322
$530$	0	0
$531$	−9.79929e8	−0.284030
$532$	−3.42954e8	−0.0987518
$533$	7.01869e9	2.00776
$534$	−1.25328e9	−0.356168
$535$	0	0
$536$	−1.15146e9	−0.322978
$537$	−1.58904e9	−0.442817
$538$	1.52773e9	0.422970
$539$	2.94957e9	0.811331
$540$	0	0
$541$	4.03189e9	1.09476	0.547379	−	0.836885i	$-0.315626\pi$
$541$	4.03189e9	1.09476	0.547379	+	0.836885i	$0.315626\pi$
$542$	1.00349e8	0.0270716
$543$	−4.00635e9	−1.07386
$544$	−2.51855e8	−0.0670740
$545$	0	0
$546$	1.13568e9	0.298594
$547$	4.17631e9	1.09103	0.545516	−	0.838101i	$-0.316334\pi$
$547$	4.17631e9	1.09103	0.545516	+	0.838101i	$0.316334\pi$
$548$	3.40034e9	0.882654
$549$	2.47432e9	0.638194
$550$	0	0
$551$	2.15813e9	0.549599
$552$	−4.90448e8	−0.124110
$553$	6.14449e8	0.154507
$554$	1.89630e9	0.473831
$555$	0	0
$556$	3.05348e9	0.753413
$557$	9.00894e8	0.220892	0.110446	−	0.993882i	$-0.464772\pi$
$557$	9.00894e8	0.220892	0.110446	+	0.993882i	$0.464772\pi$
$558$	5.79368e8	0.141168
$559$	−4.58506e9	−1.11021
$560$	0	0
$561$	9.12682e8	0.218248
$562$	−6.92754e8	−0.164627
$563$	−3.52184e9	−0.831746	−0.415873	−	0.909423i	$-0.636524\pi$
$563$	−3.52184e9	−0.831746	−0.415873	+	0.909423i	$0.636524\pi$
$564$	−8.79310e7	−0.0206379
$565$	0	0
$566$	−3.46427e9	−0.803073
$567$	2.07793e8	0.0478731
$568$	−1.39872e9	−0.320266
$569$	5.76276e9	1.31141	0.655704	−	0.755018i	$-0.272372\pi$
$569$	5.76276e9	1.31141	0.655704	+	0.755018i	$0.272372\pi$
$570$	0	0
$571$	−7.28513e9	−1.63761	−0.818806	−	0.574071i	$-0.805363\pi$
$571$	−7.28513e9	−1.63761	−0.818806	+	0.574071i	$0.805363\pi$
$572$	−3.78495e9	−0.845619
$573$	2.22032e9	0.493031
$574$	−1.63267e9	−0.360334
$575$	0	0
$576$	1.91103e8	0.0416667
$577$	−9.21977e8	−0.199804	−0.0999022	−	0.994997i	$-0.531853\pi$
$577$	−9.21977e8	−0.199804	−0.0999022	+	0.994997i	$0.531853\pi$
$578$	2.81011e9	0.605308
$579$	−2.51703e9	−0.538907
$580$	0	0
$581$	3.04705e9	0.644559
$582$	5.39635e8	0.113467
$583$	−3.91920e9	−0.819138
$584$	−2.57465e9	−0.534902
$585$	0	0
$586$	−5.99174e9	−1.23002
$587$	9.20186e9	1.87777	0.938885	−	0.344231i	$-0.111860\pi$
$587$	9.20186e9	1.87777	0.938885	+	0.344231i	$0.111860\pi$
$588$	1.15890e9	0.235086
$589$	1.36150e9	0.274545
$590$	0	0
$591$	−9.43911e8	−0.188094
$592$	6.63249e8	0.131386
$593$	5.52826e9	1.08867	0.544336	−	0.838867i	$-0.316782\pi$
$593$	5.52826e9	1.08867	0.544336	+	0.838867i	$0.316782\pi$
$594$	−6.92527e8	−0.135576
$595$	0	0
$596$	−4.02577e9	−0.778910
$597$	−4.14498e9	−0.797283
$598$	3.81658e9	0.729828
$599$	−8.92445e9	−1.69663	−0.848316	−	0.529490i	$-0.822383\pi$
$599$	−8.92445e9	−1.69663	−0.848316	+	0.529490i	$0.822383\pi$
$600$	0	0
$601$	1.13084e9	0.212491	0.106245	−	0.994340i	$-0.466117\pi$
$601$	1.13084e9	0.212491	0.106245	+	0.994340i	$0.466117\pi$
$602$	1.06656e9	0.199250
$603$	1.63949e9	0.304507
$604$	−1.53035e9	−0.282593
$605$	0	0
$606$	2.46347e9	0.449668
$607$	−3.49658e8	−0.0634575	−0.0317287	−	0.999497i	$-0.510101\pi$
$607$	−3.49658e8	−0.0634575	−0.0317287	+	0.999497i	$0.510101\pi$
$608$	4.49085e8	0.0810338
$609$	1.66241e9	0.298248
$610$	0	0
$611$	6.84264e8	0.121361
$612$	3.58598e8	0.0632380
$613$	−4.23756e9	−0.743026	−0.371513	−	0.928428i	$-0.621161\pi$
$613$	−4.23756e9	−0.743026	−0.371513	+	0.928428i	$0.621161\pi$
$614$	3.26395e9	0.569056
$615$	0	0
$616$	8.80444e8	0.151764
$617$	−5.97299e8	−0.102375	−0.0511875	−	0.998689i	$-0.516301\pi$
$617$	−5.97299e8	−0.102375	−0.0511875	+	0.998689i	$0.516301\pi$
$618$	2.51257e9	0.428211
$619$	−3.63076e9	−0.615291	−0.307645	−	0.951501i	$-0.599541\pi$
$619$	−3.63076e9	−0.615291	−0.307645	+	0.951501i	$0.599541\pi$
$620$	0	0
$621$	6.98313e8	0.117012
$622$	6.13681e9	1.02253
$623$	−2.26868e9	−0.375893
$624$	−1.48713e9	−0.245021
$625$	0	0
$626$	−4.17525e9	−0.680256
$627$	−1.62741e9	−0.263671
$628$	−5.34558e9	−0.861264
$629$	1.24456e9	0.199407
$630$	0	0
$631$	7.86274e9	1.24587	0.622933	−	0.782275i	$-0.285941\pi$
$631$	7.86274e9	1.24587	0.622933	+	0.782275i	$0.285941\pi$
$632$	−8.04598e8	−0.126785
$633$	−2.94877e9	−0.462092
$634$	3.69774e9	0.576267
$635$	0	0
$636$	−1.53988e9	−0.237348
$637$	−9.01839e9	−1.38242
$638$	−5.54042e9	−0.844638
$639$	1.99153e9	0.301950
$640$	0	0
$641$	−5.70348e9	−0.855336	−0.427668	−	0.903936i	$-0.640665\pi$
$641$	−5.70348e9	−0.855336	−0.427668	+	0.903936i	$0.640665\pi$
$642$	−1.25045e9	−0.186507
$643$	−1.08513e10	−1.60969	−0.804846	−	0.593483i	$-0.797752\pi$
$643$	−1.08513e10	−1.60969	−0.804846	+	0.593483i	$0.797752\pi$
$644$	−8.87801e8	−0.130983
$645$	0	0
$646$	8.42693e8	0.122986
$647$	6.28626e9	0.912488	0.456244	−	0.889855i	$-0.349194\pi$
$647$	6.28626e9	0.912488	0.456244	+	0.889855i	$0.349194\pi$
$648$	−2.72098e8	−0.0392837
$649$	5.91184e9	0.848918
$650$	0	0
$651$	1.04876e9	0.148986
$652$	−6.28640e9	−0.888251
$653$	−4.34354e9	−0.610447	−0.305224	−	0.952281i	$-0.598731\pi$
$653$	−4.34354e9	−0.610447	−0.305224	+	0.952281i	$0.598731\pi$
$654$	1.13978e9	0.159331
$655$	0	0
$656$	2.13792e9	0.295683
$657$	3.66587e9	0.504310
$658$	−1.59171e8	−0.0217808
$659$	−2.57954e9	−0.351110	−0.175555	−	0.984470i	$-0.556172\pi$
$659$	−2.57954e9	−0.351110	−0.175555	+	0.984470i	$0.556172\pi$
$660$	0	0
$661$	−6.55087e8	−0.0882255	−0.0441127	−	0.999027i	$-0.514046\pi$
$661$	−6.55087e8	−0.0882255	−0.0441127	+	0.999027i	$0.514046\pi$
$662$	1.03732e9	0.138966
$663$	−2.79055e9	−0.371871
$664$	−3.99000e9	−0.528913
$665$	0	0
$666$	−9.44352e8	−0.123872
$667$	5.58672e9	0.728982
$668$	−5.20947e9	−0.676201
$669$	−2.91664e9	−0.376610
$670$	0	0
$671$	−1.49274e10	−1.90746
$672$	3.45932e8	0.0439742
$673$	4.69700e9	0.593975	0.296988	−	0.954881i	$-0.404018\pi$
$673$	4.69700e9	0.593975	0.296988	+	0.954881i	$0.404018\pi$
$674$	5.05648e9	0.636120
$675$	0	0
$676$	7.55669e9	0.940845
$677$	−1.27894e10	−1.58413	−0.792064	−	0.610438i	$-0.790994\pi$
$677$	−1.27894e10	−1.58413	−0.792064	+	0.610438i	$0.790994\pi$
$678$	−4.87288e9	−0.600457
$679$	9.76840e8	0.119751
$680$	0	0
$681$	−8.07312e9	−0.979550
$682$	−3.49528e9	−0.421927
$683$	−6.86870e9	−0.824902	−0.412451	−	0.910980i	$-0.635327\pi$
$683$	−6.86870e9	−0.824902	−0.412451	+	0.910980i	$0.635327\pi$
$684$	−6.39420e8	−0.0763994
$685$	0	0
$686$	4.67387e9	0.552768
$687$	−4.34764e9	−0.511570
$688$	−1.39663e9	−0.163501
$689$	1.19831e10	1.39573
$690$	0	0
$691$	−1.03089e10	−1.18861	−0.594306	−	0.804239i	$-0.702573\pi$
$691$	−1.03089e10	−1.18861	−0.594306	+	0.804239i	$0.702573\pi$
$692$	−2.14151e9	−0.245668
$693$	−1.25360e9	−0.143085
$694$	−3.78517e9	−0.429861
$695$	0	0
$696$	−2.17687e9	−0.244737
$697$	4.01172e9	0.448762
$698$	−1.07256e10	−1.19378
$699$	−2.45322e9	−0.271685
$700$	0	0
$701$	4.93389e9	0.540974	0.270487	−	0.962724i	$-0.412815\pi$
$701$	4.93389e9	0.540974	0.270487	+	0.962724i	$0.412815\pi$
$702$	2.11742e9	0.231008
$703$	−2.21920e9	−0.240908
$704$	−1.15291e9	−0.124535
$705$	0	0
$706$	−1.28335e9	−0.137255
$707$	4.45933e9	0.474571
$708$	2.32279e9	0.245977
$709$	1.27656e10	1.34517	0.672587	−	0.740018i	$-0.265183\pi$
$709$	1.27656e10	1.34517	0.672587	+	0.740018i	$0.265183\pi$
$710$	0	0
$711$	1.14561e9	0.119534
$712$	2.97075e9	0.308451
$713$	3.52449e9	0.364152
$714$	6.49129e8	0.0667401
$715$	0	0
$716$	3.76661e9	0.383491
$717$	9.28985e9	0.941221
$718$	8.33512e9	0.840381
$719$	−6.60270e9	−0.662477	−0.331238	−	0.943547i	$-0.607466\pi$
$719$	−6.60270e9	−0.662477	−0.331238	+	0.943547i	$0.607466\pi$
$720$	0	0
$721$	4.54821e9	0.451926
$722$	5.64836e9	0.558524
$723$	−2.16116e9	−0.212669
$724$	9.49654e9	0.929994
$725$	0	0
$726$	−3.12697e7	−0.00303281
$727$	−8.09480e9	−0.781332	−0.390666	−	0.920532i	$-0.627755\pi$
$727$	−8.09480e9	−0.781332	−0.390666	+	0.920532i	$0.627755\pi$
$728$	−2.69198e9	−0.258590
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−2.62072e9	−0.248147
$732$	−5.86505e9	−0.552692
$733$	1.34431e9	0.126076	0.0630382	−	0.998011i	$-0.479921\pi$
$733$	1.34431e9	0.126076	0.0630382	+	0.998011i	$0.479921\pi$
$734$	1.03745e9	0.0968342
$735$	0	0
$736$	1.16254e9	0.107482
$737$	−9.89089e9	−0.910121
$738$	−3.04402e9	−0.278773
$739$	1.61473e10	1.47179	0.735895	−	0.677096i	$-0.236762\pi$
$739$	1.61473e10	1.47179	0.735895	+	0.677096i	$0.236762\pi$
$740$	0	0
$741$	4.97586e9	0.449267
$742$	−2.78746e9	−0.250493
$743$	1.40013e9	0.125230	0.0626149	−	0.998038i	$-0.480056\pi$
$743$	1.40013e9	0.125230	0.0626149	+	0.998038i	$0.480056\pi$
$744$	−1.37332e9	−0.122255
$745$	0	0
$746$	4.53863e9	0.400257
$747$	5.68107e9	0.498664
$748$	−2.16339e9	−0.189008
$749$	−2.26355e9	−0.196835
$750$	0	0
$751$	−1.57372e10	−1.35577	−0.677887	−	0.735166i	$-0.737104\pi$
$751$	−1.57372e10	−1.35577	−0.677887	+	0.735166i	$0.737104\pi$
$752$	2.08429e8	0.0178729
$753$	9.10408e6	0.000777058 0
$754$	1.69400e10	1.43917
$755$	0	0
$756$	−4.92547e8	−0.0414593
$757$	−1.02021e10	−0.854777	−0.427389	−	0.904068i	$-0.640566\pi$
$757$	−1.02021e10	−0.854777	−0.427389	+	0.904068i	$0.640566\pi$
$758$	1.14438e10	0.954392
$759$	−4.21287e9	−0.349729
$760$	0	0
$761$	1.16793e9	0.0960661	0.0480330	−	0.998846i	$-0.484705\pi$
$761$	1.16793e9	0.0960661	0.0480330	+	0.998846i	$0.484705\pi$
$762$	5.70634e9	0.467214
$763$	2.06322e9	0.168155
$764$	−5.26297e9	−0.426977
$765$	0	0
$766$	1.01642e9	0.0817099
$767$	−1.80756e10	−1.44647
$768$	−4.52985e8	−0.0360844
$769$	4.80308e9	0.380871	0.190435	−	0.981700i	$-0.439010\pi$
$769$	4.80308e9	0.380871	0.190435	+	0.981700i	$0.439010\pi$
$770$	0	0
$771$	−1.21322e10	−0.953346
$772$	5.96630e9	0.466707
$773$	9.04745e9	0.704527	0.352264	−	0.935901i	$-0.385412\pi$
$773$	9.04745e9	0.704527	0.352264	+	0.935901i	$0.385412\pi$
$774$	1.98855e9	0.154150
$775$	0	0
$776$	−1.27914e9	−0.0982654
$777$	−1.70945e9	−0.130732
$778$	−8.40744e9	−0.640082
$779$	−7.15335e9	−0.542161
$780$	0	0
$781$	−1.20148e10	−0.902479
$782$	2.18147e9	0.163127
$783$	3.09948e9	0.230740
$784$	−2.74703e9	−0.203590
$785$	0	0
$786$	7.14679e9	0.524967
$787$	−2.03500e10	−1.48817	−0.744086	−	0.668084i	$-0.767115\pi$
$787$	−2.03500e10	−1.48817	−0.744086	+	0.668084i	$0.767115\pi$
$788$	2.23742e9	0.162894
$789$	−7.06015e9	−0.511734
$790$	0	0
$791$	−8.82081e9	−0.633711
$792$	1.64154e9	0.117413
$793$	4.56408e10	3.25010
$794$	−6.76957e9	−0.479943
$795$	0	0
$796$	9.82514e9	0.690467
$797$	1.08233e10	0.757276	0.378638	−	0.925545i	$-0.376393\pi$
$797$	1.08233e10	0.757276	0.378638	+	0.925545i	$0.376393\pi$
$798$	−1.15747e9	−0.0806305
$799$	3.91110e8	0.0271260
$800$	0	0
$801$	−4.22983e9	−0.290810
$802$	−2.21488e9	−0.151614
$803$	−2.21159e10	−1.50730
$804$	−3.88619e9	−0.263711
$805$	0	0
$806$	1.06869e10	0.718919
$807$	5.15610e9	0.345353
$808$	−5.83933e9	−0.389424
$809$	−1.04227e10	−0.692090	−0.346045	−	0.938218i	$-0.612476\pi$
$809$	−1.04227e10	−0.692090	−0.346045	+	0.938218i	$0.612476\pi$
$810$	0	0
$811$	1.53451e10	1.01018	0.505089	−	0.863067i	$-0.331460\pi$
$811$	1.53451e10	1.01018	0.505089	+	0.863067i	$0.331460\pi$
$812$	−3.94053e9	−0.258291
$813$	3.38676e8	0.0221039
$814$	5.69720e9	0.370234
$815$	0	0
$816$	−8.50010e8	−0.0547657
$817$	4.67303e9	0.299793
$818$	−8.87010e9	−0.566620
$819$	3.83292e9	0.243801
$820$	0	0
$821$	1.20228e10	0.758236	0.379118	−	0.925348i	$-0.376227\pi$
$821$	1.20228e10	0.758236	0.379118	+	0.925348i	$0.376227\pi$
$822$	1.14761e10	0.720684
$823$	1.70599e10	1.06678	0.533392	−	0.845868i	$-0.320917\pi$
$823$	1.70599e10	1.06678	0.533392	+	0.845868i	$0.320917\pi$
$824$	−5.95571e9	−0.370842
$825$	0	0
$826$	4.20469e9	0.259599
$827$	1.60661e10	0.987737	0.493869	−	0.869537i	$-0.335582\pi$
$827$	1.60661e10	0.987737	0.493869	+	0.869537i	$0.335582\pi$
$828$	−1.65526e9	−0.101335
$829$	−2.81007e8	−0.0171307	−0.00856536	−	0.999963i	$-0.502726\pi$
$829$	−2.81007e8	−0.0171307	−0.00856536	+	0.999963i	$0.502726\pi$
$830$	0	0
$831$	6.40003e9	0.386882
$832$	3.52505e9	0.212194
$833$	−5.15471e9	−0.308992
$834$	1.03055e10	0.615159
$835$	0	0
$836$	3.85757e9	0.228345
$837$	1.95537e9	0.115263
$838$	7.81807e9	0.458929
$839$	−2.52384e10	−1.47535	−0.737674	−	0.675157i	$-0.764076\pi$
$839$	−2.52384e10	−1.47535	−0.737674	+	0.675157i	$0.764076\pi$
$840$	0	0
$841$	7.54692e9	0.437506
$842$	1.81580e10	1.04827
$843$	−2.33805e9	−0.134418
$844$	6.98969e9	0.400183
$845$	0	0
$846$	−2.96767e8	−0.0168508
$847$	−5.66039e7	−0.00320077
$848$	3.65008e9	0.205550
$849$	−1.16919e10	−0.655706
$850$	0	0
$851$	−5.74481e9	−0.319538
$852$	−4.72067e9	−0.261496
$853$	1.64023e10	0.904865	0.452433	−	0.891799i	$-0.350556\pi$
$853$	1.64023e10	0.904865	0.452433	+	0.891799i	$0.350556\pi$
$854$	−1.06168e10	−0.583300
$855$	0	0
$856$	2.96403e9	0.161519
$857$	−2.71192e10	−1.47178	−0.735892	−	0.677099i	$-0.763237\pi$
$857$	−2.71192e10	−1.47178	−0.735892	+	0.677099i	$0.763237\pi$
$858$	−1.27742e10	−0.690445
$859$	−1.92449e10	−1.03595	−0.517977	−	0.855394i	$-0.673315\pi$
$859$	−1.92449e10	−1.03595	−0.517977	+	0.855394i	$0.673315\pi$
$860$	0	0
$861$	−5.51025e9	−0.294212
$862$	−4.80004e8	−0.0255252
$863$	−1.33344e10	−0.706215	−0.353107	−	0.935583i	$-0.614875\pi$
$863$	−1.33344e10	−0.706215	−0.353107	+	0.935583i	$0.614875\pi$
$864$	6.44973e8	0.0340207
$865$	0	0
$866$	6.49271e9	0.339714
$867$	9.48413e9	0.494232
$868$	−2.48596e9	−0.129025
$869$	−6.91137e9	−0.357269
$870$	0	0
$871$	3.02416e10	1.55075
$872$	−2.70170e9	−0.137984
$873$	1.82127e9	0.0926455
$874$	−3.88981e9	−0.197078
$875$	0	0
$876$	−8.68946e9	−0.436745
$877$	5.75811e9	0.288258	0.144129	−	0.989559i	$-0.453962\pi$
$877$	5.75811e9	0.288258	0.144129	+	0.989559i	$0.453962\pi$
$878$	2.01660e10	1.00551
$879$	−2.02221e10	−1.00431
$880$	0	0
$881$	8.99848e9	0.443357	0.221678	−	0.975120i	$-0.428846\pi$
$881$	8.99848e9	0.443357	0.221678	+	0.975120i	$0.428846\pi$
$882$	3.91130e9	0.191947
$883$	2.60967e10	1.27562	0.637811	−	0.770193i	$-0.279840\pi$
$883$	2.60967e10	1.27562	0.637811	+	0.770193i	$0.279840\pi$
$884$	6.61463e9	0.322050
$885$	0	0
$886$	−1.69205e10	−0.817325
$887$	8.96544e9	0.431359	0.215680	−	0.976464i	$-0.430803\pi$
$887$	8.96544e9	0.431359	0.215680	+	0.976464i	$0.430803\pi$
$888$	2.23847e9	0.107277
$889$	1.03295e10	0.493088
$890$	0	0
$891$	−2.33728e9	−0.110698
$892$	6.91352e9	0.326154
$893$	−6.97393e8	−0.0327716
$894$	−1.35870e10	−0.635978
$895$	0	0
$896$	−8.19986e8	−0.0380828
$897$	1.28810e10	0.595902
$898$	−5.96438e8	−0.0274851
$899$	1.56435e10	0.718086
$900$	0	0
$901$	6.84924e9	0.311965
$902$	1.83644e10	0.833207
$903$	3.59965e9	0.162687
$904$	1.15505e10	0.520011
$905$	0	0
$906$	−5.16493e9	−0.230736
$907$	1.27630e10	0.567973	0.283987	−	0.958828i	$-0.408343\pi$
$907$	1.27630e10	0.567973	0.283987	+	0.958828i	$0.408343\pi$
$908$	1.91363e10	0.848315
$909$	8.31420e9	0.367153
$910$	0	0
$911$	−1.24867e10	−0.547185	−0.273593	−	0.961846i	$-0.588212\pi$
$911$	−1.24867e10	−0.547185	−0.273593	+	0.961846i	$0.588212\pi$
$912$	1.51566e9	0.0661639
$913$	−3.42734e10	−1.49042
$914$	5.55818e8	0.0240780
$915$	0	0
$916$	1.03055e10	0.443032
$917$	1.29370e10	0.554041
$918$	1.21027e9	0.0516336
$919$	−3.66199e10	−1.55637	−0.778186	−	0.628034i	$-0.783860\pi$
$919$	−3.66199e10	−1.55637	−0.778186	+	0.628034i	$0.783860\pi$
$920$	0	0
$921$	1.10158e10	0.464632
$922$	−5.04605e9	−0.212028
$923$	3.67355e10	1.53773
$924$	2.97150e9	0.123915
$925$	0	0
$926$	3.20384e10	1.32597
$927$	8.47991e9	0.349633
$928$	5.15998e9	0.211948
$929$	−2.08997e10	−0.855235	−0.427618	−	0.903960i	$-0.640647\pi$
$929$	−2.08997e10	−0.855235	−0.427618	+	0.903960i	$0.640647\pi$
$930$	0	0
$931$	9.19142e9	0.373301
$932$	5.81503e9	0.235286
$933$	2.07117e10	0.834893
$934$	−1.24271e10	−0.499063
$935$	0	0
$936$	−5.01907e9	−0.200059
$937$	−8.14500e9	−0.323447	−0.161723	−	0.986836i	$-0.551705\pi$
$937$	−8.14500e9	−0.323447	−0.161723	+	0.986836i	$0.551705\pi$
$938$	−7.03472e9	−0.278315
$939$	−1.40915e10	−0.555427
$940$	0	0
$941$	−1.84860e10	−0.723234	−0.361617	−	0.932327i	$-0.617775\pi$
$941$	−1.84860e10	−0.723234	−0.361617	+	0.932327i	$0.617775\pi$
$942$	−1.80413e10	−0.703219
$943$	−1.85178e10	−0.719116
$944$	−5.50588e9	−0.213022
$945$	0	0
$946$	−1.19968e10	−0.460730
$947$	2.31553e10	0.885985	0.442992	−	0.896525i	$-0.353917\pi$
$947$	2.31553e10	0.885985	0.442992	+	0.896525i	$0.353917\pi$
$948$	−2.71552e9	−0.103520
$949$	6.76199e10	2.56828
$950$	0	0
$951$	1.24799e10	0.470520
$952$	−1.53868e9	−0.0577987
$953$	−3.64771e10	−1.36520	−0.682599	−	0.730793i	$-0.739150\pi$
$953$	−3.64771e10	−1.36520	−0.682599	+	0.730793i	$0.739150\pi$
$954$	−5.19708e9	−0.193794
$955$	0	0
$956$	−2.20204e10	−0.815121
$957$	−1.86989e10	−0.689644
$958$	−1.49378e9	−0.0548919
$959$	2.07739e10	0.760596
$960$	0	0
$961$	−1.76436e10	−0.641291
$962$	−1.74194e10	−0.630840
$963$	−4.22027e9	−0.152282
$964$	5.12275e9	0.184176
$965$	0	0
$966$	−2.99633e9	−0.106947
$967$	−5.58270e9	−0.198542	−0.0992708	−	0.995060i	$-0.531651\pi$
$967$	−5.58270e9	−0.198542	−0.0992708	+	0.995060i	$0.531651\pi$
$968$	7.41207e7	0.00262649
$969$	2.84409e9	0.100418
$970$	0	0
$971$	−3.32789e10	−1.16655	−0.583274	−	0.812276i	$-0.698228\pi$
$971$	−3.32789e10	−1.16655	−0.583274	+	0.812276i	$0.698228\pi$
$972$	−9.18330e8	−0.0320750
$973$	1.86548e10	0.649227
$974$	−2.40091e10	−0.832567
$975$	0	0
$976$	1.39023e10	0.478645
$977$	−2.95598e10	−1.01407	−0.507037	−	0.861924i	$-0.669259\pi$
$977$	−2.95598e10	−1.01407	−0.507037	+	0.861924i	$0.669259\pi$
$978$	−2.12166e10	−0.725254
$979$	2.55183e10	0.869184
$980$	0	0
$981$	3.84676e9	0.130093
$982$	−3.61005e10	−1.21653
$983$	1.66893e10	0.560404	0.280202	−	0.959941i	$-0.409599\pi$
$983$	1.66893e10	0.560404	0.280202	+	0.959941i	$0.409599\pi$
$984$	7.21546e9	0.241425
$985$	0	0
$986$	9.68252e9	0.321676
$987$	−5.37204e8	−0.0177840
$988$	−1.17946e10	−0.389077
$989$	1.20970e10	0.397642
$990$	0	0
$991$	−5.29073e9	−0.172686	−0.0863431	−	0.996265i	$-0.527518\pi$
$991$	−5.29073e9	−0.172686	−0.0863431	+	0.996265i	$0.527518\pi$
$992$	3.25527e9	0.105876
$993$	3.50094e9	0.113465
$994$	−8.54530e9	−0.275978
$995$	0	0
$996$	−1.34662e10	−0.431856
$997$	2.57597e9	0.0823204	0.0411602	−	0.999153i	$-0.486895\pi$
$997$	2.57597e9	0.0823204	0.0411602	+	0.999153i	$0.486895\pi$
$998$	2.42542e10	0.772377
$999$	−3.18719e9	−0.101141

Twists

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

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