LMFDB - Embedded newform 150.8.a.i.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:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
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)

$f(q)$

$=$

$q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -216.000 q^{6} +988.000 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})$

\(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -216.000 q^{6} +988.000 q^{7} -512.000 q^{8} +729.000 q^{9} -8040.00 q^{11} +1728.00 q^{12} +3334.00 q^{13} -7904.00 q^{14} +4096.00 q^{16} -6582.00 q^{17} -5832.00 q^{18} -27436.0 q^{19} +26676.0 q^{21} +64320.0 q^{22} -48600.0 q^{23} -13824.0 q^{24} -26672.0 q^{26} +19683.0 q^{27} +63232.0 q^{28} -132414. q^{29} +254408. q^{31} -32768.0 q^{32} -217080. q^{33} +52656.0 q^{34} +46656.0 q^{36} -519434. q^{37} +219488. q^{38} +90018.0 q^{39} +92394.0 q^{41} -213408. q^{42} +234532. q^{43} -514560. q^{44} +388800. q^{46} +1.27764e6 q^{47} +110592. q^{48} +152601. q^{49} -177714. q^{51} +213376. q^{52} +835278. q^{53} -157464. q^{54} -505856. q^{56} -740772. q^{57} +1.05931e6 q^{58} -3.06876e6 q^{59} -1.00933e6 q^{61} -2.03526e6 q^{62} +720252. q^{63} +262144. q^{64} +1.73664e6 q^{66} -3.08217e6 q^{67} -421248. q^{68} -1.31220e6 q^{69} -3.66672e6 q^{71} -373248. q^{72} -1.12287e6 q^{73} +4.15547e6 q^{74} -1.75590e6 q^{76} -7.94352e6 q^{77} -720144. q^{78} -4.12881e6 q^{79} +531441. q^{81} -739152. q^{82} -4.58656e6 q^{83} +1.70726e6 q^{84} -1.87626e6 q^{86} -3.57518e6 q^{87} +4.11648e6 q^{88} -5.76368e6 q^{89} +3.29399e6 q^{91} -3.11040e6 q^{92} +6.86902e6 q^{93} -1.02211e7 q^{94} -884736. q^{96} -6.74755e6 q^{97} -1.22081e6 q^{98} -5.86116e6 q^{99} +O(q^{100})\)

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
$7$	988.000	1.08871	0.544357	−	0.838854i	$-0.316774\pi$
$7$	988.000	1.08871	0.544357	+	0.838854i	$0.316774\pi$
$8$	−512.000	−0.353553
$9$	729.000	0.333333
$10$	0	0
$11$	−8040.00	−1.82130	−0.910650	−	0.413178i	$-0.864419\pi$
$11$	−8040.00	−1.82130	−0.910650	+	0.413178i	$0.864419\pi$
$12$	1728.00	0.288675
$13$	3334.00	0.420885	0.210443	−	0.977606i	$-0.432509\pi$
$13$	3334.00	0.420885	0.210443	+	0.977606i	$0.432509\pi$
$14$	−7904.00	−0.769837
$15$	0	0
$16$	4096.00	0.250000
$17$	−6582.00	−0.324928	−0.162464	−	0.986715i	$-0.551944\pi$
$17$	−6582.00	−0.324928	−0.162464	+	0.986715i	$0.551944\pi$
$18$	−5832.00	−0.235702
$19$	−27436.0	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−27436.0	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	26676.0	0.628569
$22$	64320.0	1.28785
$23$	−48600.0	−0.832892	−0.416446	−	0.909160i	$-0.636725\pi$
$23$	−48600.0	−0.832892	−0.416446	+	0.909160i	$0.636725\pi$
$24$	−13824.0	−0.204124
$25$	0	0
$26$	−26672.0	−0.297611
$27$	19683.0	0.192450
$28$	63232.0	0.544357
$29$	−132414.	−1.00819	−0.504093	−	0.863649i	$-0.668173\pi$
$29$	−132414.	−1.00819	−0.504093	+	0.863649i	$0.668173\pi$
$30$	0	0
$31$	254408.	1.53379	0.766893	−	0.641775i	$-0.221802\pi$
$31$	254408.	1.53379	0.766893	+	0.641775i	$0.221802\pi$
$32$	−32768.0	−0.176777
$33$	−217080.	−1.05153
$34$	52656.0	0.229759
$35$	0	0
$36$	46656.0	0.166667
$37$	−519434.	−1.68587	−0.842935	−	0.538015i	$-0.819175\pi$
$37$	−519434.	−1.68587	−0.842935	+	0.538015i	$0.819175\pi$
$38$	219488.	0.648886
$39$	90018.0	0.242998
$40$	0	0
$41$	92394.0	0.209363	0.104682	−	0.994506i	$-0.466618\pi$
$41$	92394.0	0.209363	0.104682	+	0.994506i	$0.466618\pi$
$42$	−213408.	−0.444466
$43$	234532.	0.449845	0.224922	−	0.974377i	$-0.427787\pi$
$43$	234532.	0.449845	0.224922	+	0.974377i	$0.427787\pi$
$44$	−514560.	−0.910650
$45$	0	0
$46$	388800.	0.588944
$47$	1.27764e6	1.79501	0.897503	−	0.441008i	$-0.145379\pi$
$47$	1.27764e6	1.79501	0.897503	+	0.441008i	$0.145379\pi$
$48$	110592.	0.144338
$49$	152601.	0.185298
$50$	0	0
$51$	−177714.	−0.187597
$52$	213376.	0.210443
$53$	835278.	0.770665	0.385332	−	0.922778i	$-0.374087\pi$
$53$	835278.	0.770665	0.385332	+	0.922778i	$0.374087\pi$
$54$	−157464.	−0.136083
$55$	0	0
$56$	−505856.	−0.384919
$57$	−740772.	−0.529813
$58$	1.05931e6	0.712896
$59$	−3.06876e6	−1.94527	−0.972637	−	0.232329i	$-0.925365\pi$
$59$	−3.06876e6	−1.94527	−0.972637	+	0.232329i	$0.925365\pi$
$60$	0	0
$61$	−1.00933e6	−0.569349	−0.284675	−	0.958624i	$-0.591886\pi$
$61$	−1.00933e6	−0.569349	−0.284675	+	0.958624i	$0.591886\pi$
$62$	−2.03526e6	−1.08455
$63$	720252.	0.362905
$64$	262144.	0.125000
$65$	0	0
$66$	1.73664e6	0.743543
$67$	−3.08217e6	−1.25197	−0.625987	−	0.779834i	$-0.715304\pi$
$67$	−3.08217e6	−1.25197	−0.625987	+	0.779834i	$0.715304\pi$
$68$	−421248.	−0.162464
$69$	−1.31220e6	−0.480871
$70$	0	0
$71$	−3.66672e6	−1.21583	−0.607916	−	0.794001i	$-0.707994\pi$
$71$	−3.66672e6	−1.21583	−0.607916	+	0.794001i	$0.707994\pi$
$72$	−373248.	−0.117851
$73$	−1.12287e6	−0.337830	−0.168915	−	0.985631i	$-0.554026\pi$
$73$	−1.12287e6	−0.337830	−0.168915	+	0.985631i	$0.554026\pi$
$74$	4.15547e6	1.19209
$75$	0	0
$76$	−1.75590e6	−0.458831
$77$	−7.94352e6	−1.98288
$78$	−720144.	−0.171826
$79$	−4.12881e6	−0.942171	−0.471086	−	0.882087i	$-0.656138\pi$
$79$	−4.12881e6	−0.942171	−0.471086	+	0.882087i	$0.656138\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	−739152.	−0.148042
$83$	−4.58656e6	−0.880468	−0.440234	−	0.897883i	$-0.645104\pi$
$83$	−4.58656e6	−0.880468	−0.440234	+	0.897883i	$0.645104\pi$
$84$	1.70726e6	0.314285
$85$	0	0
$86$	−1.87626e6	−0.318088
$87$	−3.57518e6	−0.582077
$88$	4.11648e6	0.643927
$89$	−5.76368e6	−0.866632	−0.433316	−	0.901242i	$-0.642657\pi$
$89$	−5.76368e6	−0.866632	−0.433316	+	0.901242i	$0.642657\pi$
$90$	0	0
$91$	3.29399e6	0.458224
$92$	−3.11040e6	−0.416446
$93$	6.86902e6	0.885532
$94$	−1.02211e7	−1.26926
$95$	0	0
$96$	−884736.	−0.102062
$97$	−6.74755e6	−0.750663	−0.375332	−	0.926891i	$-0.622471\pi$
$97$	−6.74755e6	−0.750663	−0.375332	+	0.926891i	$0.622471\pi$
$98$	−1.22081e6	−0.131026
$99$	−5.86116e6	−0.607100
$100$	0	0
$101$	5.70974e6	0.551431	0.275716	−	0.961239i	$-0.411085\pi$
$101$	5.70974e6	0.551431	0.275716	+	0.961239i	$0.411085\pi$
$102$	1.42171e6	0.132651
$103$	6.76769e6	0.610254	0.305127	−	0.952312i	$-0.401301\pi$
$103$	6.76769e6	0.610254	0.305127	+	0.952312i	$0.401301\pi$
$104$	−1.70701e6	−0.148805
$105$	0	0
$106$	−6.68222e6	−0.544942
$107$	−1.76452e6	−0.139246	−0.0696229	−	0.997573i	$-0.522180\pi$
$107$	−1.76452e6	−0.139246	−0.0696229	+	0.997573i	$0.522180\pi$
$108$	1.25971e6	0.0962250
$109$	643790.	0.0476158	0.0238079	−	0.999717i	$-0.492421\pi$
$109$	643790.	0.0476158	0.0238079	+	0.999717i	$0.492421\pi$
$110$	0	0
$111$	−1.40247e7	−0.973338
$112$	4.04685e6	0.272178
$113$	−1.42571e7	−0.929515	−0.464757	−	0.885438i	$-0.653858\pi$
$113$	−1.42571e7	−0.929515	−0.464757	+	0.885438i	$0.653858\pi$
$114$	5.92618e6	0.374634
$115$	0	0
$116$	−8.47450e6	−0.504093
$117$	2.43049e6	0.140295
$118$	2.45501e7	1.37552
$119$	−6.50302e6	−0.353753
$120$	0	0
$121$	4.51544e7	2.31714
$122$	8.07464e6	0.402591
$123$	2.49464e6	0.120876
$124$	1.62821e7	0.766893
$125$	0	0
$126$	−5.76202e6	−0.256612
$127$	4.41682e6	0.191336	0.0956680	−	0.995413i	$-0.469501\pi$
$127$	4.41682e6	0.191336	0.0956680	+	0.995413i	$0.469501\pi$
$128$	−2.09715e6	−0.0883883
$129$	6.33236e6	0.259718
$130$	0	0
$131$	−3.58480e7	−1.39320	−0.696602	−	0.717457i	$-0.745306\pi$
$131$	−3.58480e7	−1.39320	−0.696602	+	0.717457i	$0.745306\pi$
$132$	−1.38931e7	−0.525764
$133$	−2.71068e7	−0.999072
$134$	2.46574e7	0.885279
$135$	0	0
$136$	3.36998e6	0.114879
$137$	1.54837e7	0.514463	0.257231	−	0.966350i	$-0.417190\pi$
$137$	1.54837e7	0.514463	0.257231	+	0.966350i	$0.417190\pi$
$138$	1.04976e7	0.340027
$139$	−4.61360e7	−1.45710	−0.728548	−	0.684995i	$-0.759804\pi$
$139$	−4.61360e7	−1.45710	−0.728548	+	0.684995i	$0.759804\pi$
$140$	0	0
$141$	3.44963e7	1.03635
$142$	2.93338e7	0.859723
$143$	−2.68054e7	−0.766559
$144$	2.98598e6	0.0833333
$145$	0	0
$146$	8.98293e6	0.238882
$147$	4.12023e6	0.106982
$148$	−3.32438e7	−0.842935
$149$	4.89613e7	1.21255	0.606276	−	0.795254i	$-0.292662\pi$
$149$	4.89613e7	1.21255	0.606276	+	0.795254i	$0.292662\pi$
$150$	0	0
$151$	6.36953e7	1.50553	0.752763	−	0.658292i	$-0.228721\pi$
$151$	6.36953e7	1.50553	0.752763	+	0.658292i	$0.228721\pi$
$152$	1.40472e7	0.324443
$153$	−4.79828e6	−0.108309
$154$	6.35482e7	1.40210
$155$	0	0
$156$	5.76115e6	0.121499
$157$	9.10586e7	1.87790	0.938949	−	0.344056i	$-0.111801\pi$
$157$	9.10586e7	1.87790	0.938949	+	0.344056i	$0.111801\pi$
$158$	3.30305e7	0.666216
$159$	2.25525e7	0.444944
$160$	0	0
$161$	−4.80168e7	−0.906781
$162$	−4.25153e6	−0.0785674
$163$	3.38098e7	0.611485	0.305743	−	0.952114i	$-0.401095\pi$
$163$	3.38098e7	0.611485	0.305743	+	0.952114i	$0.401095\pi$
$164$	5.91322e6	0.104682
$165$	0	0
$166$	3.66924e7	0.622585
$167$	6.76806e7	1.12449	0.562246	−	0.826970i	$-0.309937\pi$
$167$	6.76806e7	1.12449	0.562246	+	0.826970i	$0.309937\pi$
$168$	−1.36581e7	−0.222233
$169$	−5.16330e7	−0.822855
$170$	0	0
$171$	−2.00008e7	−0.305888
$172$	1.50100e7	0.224922
$173$	−4.02346e7	−0.590796	−0.295398	−	0.955374i	$-0.595452\pi$
$173$	−4.02346e7	−0.590796	−0.295398	+	0.955374i	$0.595452\pi$
$174$	2.86014e7	0.411590
$175$	0	0
$176$	−3.29318e7	−0.455325
$177$	−8.28565e7	−1.12310
$178$	4.61094e7	0.612801
$179$	−1.27582e8	−1.66266	−0.831329	−	0.555781i	$-0.812419\pi$
$179$	−1.27582e8	−1.66266	−0.831329	+	0.555781i	$0.812419\pi$
$180$	0	0
$181$	−7.47231e7	−0.936656	−0.468328	−	0.883555i	$-0.655143\pi$
$181$	−7.47231e7	−0.936656	−0.468328	+	0.883555i	$0.655143\pi$
$182$	−2.63519e7	−0.324013
$183$	−2.72519e7	−0.328714
$184$	2.48832e7	0.294472
$185$	0	0
$186$	−5.49521e7	−0.626166
$187$	5.29193e7	0.591791
$188$	8.17690e7	0.897503
$189$	1.94468e7	0.209523
$190$	0	0
$191$	5.82415e7	0.604806	0.302403	−	0.953180i	$-0.402211\pi$
$191$	5.82415e7	0.604806	0.302403	+	0.953180i	$0.402211\pi$
$192$	7.07789e6	0.0721688
$193$	−1.89230e8	−1.89470	−0.947348	−	0.320207i	$-0.896248\pi$
$193$	−1.89230e8	−1.89470	−0.947348	+	0.320207i	$0.896248\pi$
$194$	5.39804e7	0.530799
$195$	0	0
$196$	9.76646e6	0.0926491
$197$	−6.15975e7	−0.574025	−0.287013	−	0.957927i	$-0.592662\pi$
$197$	−6.15975e7	−0.574025	−0.287013	+	0.957927i	$0.592662\pi$
$198$	4.68893e7	0.429285
$199$	1.35983e8	1.22321	0.611604	−	0.791164i	$-0.290525\pi$
$199$	1.35983e8	1.22321	0.611604	+	0.791164i	$0.290525\pi$
$200$	0	0
$201$	−8.32186e7	−0.722827
$202$	−4.56779e7	−0.389921
$203$	−1.30825e8	−1.09763
$204$	−1.13737e7	−0.0937985
$205$	0	0
$206$	−5.41415e7	−0.431514
$207$	−3.54294e7	−0.277631
$208$	1.36561e7	0.105221
$209$	2.20585e8	1.67134
$210$	0	0
$211$	2.50058e8	1.83254	0.916268	−	0.400565i	$-0.131186\pi$
$211$	2.50058e8	1.83254	0.916268	+	0.400565i	$0.131186\pi$
$212$	5.34578e7	0.385332
$213$	−9.90014e7	−0.701961
$214$	1.41161e7	0.0984617
$215$	0	0
$216$	−1.00777e7	−0.0680414
$217$	2.51355e8	1.66985
$218$	−5.15032e6	−0.0336695
$219$	−3.03174e7	−0.195046
$220$	0	0
$221$	−2.19444e7	−0.136757
$222$	1.12198e8	0.688254
$223$	2.34027e8	1.41319	0.706594	−	0.707619i	$-0.250231\pi$
$223$	2.34027e8	1.41319	0.706594	+	0.707619i	$0.250231\pi$
$224$	−3.23748e7	−0.192459
$225$	0	0
$226$	1.14057e8	0.657266
$227$	−3.27245e8	−1.85687	−0.928437	−	0.371489i	$-0.878847\pi$
$227$	−3.27245e8	−1.85687	−0.928437	+	0.371489i	$0.878847\pi$
$228$	−4.74094e7	−0.264906
$229$	−2.42853e8	−1.33635	−0.668174	−	0.744005i	$-0.732924\pi$
$229$	−2.42853e8	−1.33635	−0.668174	+	0.744005i	$0.732924\pi$
$230$	0	0
$231$	−2.14475e8	−1.14481
$232$	6.77960e7	0.356448
$233$	1.35169e8	0.700052	0.350026	−	0.936740i	$-0.386173\pi$
$233$	1.35169e8	0.700052	0.350026	+	0.936740i	$0.386173\pi$
$234$	−1.94439e7	−0.0992036
$235$	0	0
$236$	−1.96401e8	−0.972637
$237$	−1.11478e8	−0.543963
$238$	5.20241e7	0.250141
$239$	6.79202e7	0.321815	0.160908	−	0.986969i	$-0.448558\pi$
$239$	6.79202e7	0.321815	0.160908	+	0.986969i	$0.448558\pi$
$240$	0	0
$241$	2.14382e7	0.0986574	0.0493287	−	0.998783i	$-0.484292\pi$
$241$	2.14382e7	0.0986574	0.0493287	+	0.998783i	$0.484292\pi$
$242$	−3.61235e8	−1.63846
$243$	1.43489e7	0.0641500
$244$	−6.45971e7	−0.284675
$245$	0	0
$246$	−1.99571e7	−0.0854722
$247$	−9.14716e7	−0.386231
$248$	−1.30257e8	−0.542275
$249$	−1.23837e8	−0.508338
$250$	0	0
$251$	8.01294e7	0.319841	0.159920	−	0.987130i	$-0.448876\pi$
$251$	8.01294e7	0.319841	0.159920	+	0.987130i	$0.448876\pi$
$252$	4.60961e7	0.181452
$253$	3.90744e8	1.51695
$254$	−3.53346e7	−0.135295
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	1.83529e7	0.0674432	0.0337216	−	0.999431i	$-0.489264\pi$
$257$	1.83529e7	0.0674432	0.0337216	+	0.999431i	$0.489264\pi$
$258$	−5.06589e7	−0.183648
$259$	−5.13201e8	−1.83543
$260$	0	0
$261$	−9.65298e7	−0.336062
$262$	2.86784e8	0.985144
$263$	2.18329e8	0.740060	0.370030	−	0.929020i	$-0.379347\pi$
$263$	2.18329e8	0.740060	0.370030	+	0.929020i	$0.379347\pi$
$264$	1.11145e8	0.371771
$265$	0	0
$266$	2.16854e8	0.706451
$267$	−1.55619e8	−0.500350
$268$	−1.97259e8	−0.625987
$269$	−9.06729e7	−0.284017	−0.142009	−	0.989865i	$-0.545356\pi$
$269$	−9.06729e7	−0.284017	−0.142009	+	0.989865i	$0.545356\pi$
$270$	0	0
$271$	2.14335e8	0.654185	0.327092	−	0.944992i	$-0.393931\pi$
$271$	2.14335e8	0.654185	0.327092	+	0.944992i	$0.393931\pi$
$272$	−2.69599e7	−0.0812319
$273$	8.89378e7	0.264556
$274$	−1.23870e8	−0.363780
$275$	0	0
$276$	−8.39808e7	−0.240435
$277$	5.21702e7	0.147483	0.0737417	−	0.997277i	$-0.476506\pi$
$277$	5.21702e7	0.147483	0.0737417	+	0.997277i	$0.476506\pi$
$278$	3.69088e8	1.03032
$279$	1.85463e8	0.511262
$280$	0	0
$281$	2.69514e8	0.724619	0.362310	−	0.932058i	$-0.381988\pi$
$281$	2.69514e8	0.724619	0.362310	+	0.932058i	$0.381988\pi$
$282$	−2.75970e8	−0.732808
$283$	−5.34114e8	−1.40082	−0.700408	−	0.713743i	$-0.746999\pi$
$283$	−5.34114e8	−1.40082	−0.700408	+	0.713743i	$0.746999\pi$
$284$	−2.34670e8	−0.607916
$285$	0	0
$286$	2.14443e8	0.542039
$287$	9.12853e7	0.227937
$288$	−2.38879e7	−0.0589256
$289$	−3.67016e8	−0.894422
$290$	0	0
$291$	−1.82184e8	−0.433396
$292$	−7.18634e7	−0.168915
$293$	4.12548e8	0.958159	0.479079	−	0.877772i	$-0.340971\pi$
$293$	4.12548e8	0.958159	0.479079	+	0.877772i	$0.340971\pi$
$294$	−3.29618e7	−0.0756477
$295$	0	0
$296$	2.65950e8	0.596045
$297$	−1.58251e8	−0.350509
$298$	−3.91690e8	−0.857404
$299$	−1.62032e8	−0.350552
$300$	0	0
$301$	2.31718e8	0.489752
$302$	−5.09562e8	−1.06457
$303$	1.54163e8	0.318369
$304$	−1.12378e8	−0.229416
$305$	0	0
$306$	3.83862e7	0.0765862
$307$	−3.01332e8	−0.594375	−0.297187	−	0.954819i	$-0.596049\pi$
$307$	−3.01332e8	−0.594375	−0.297187	+	0.954819i	$0.596049\pi$
$308$	−5.08385e8	−0.991438
$309$	1.82728e8	0.352330
$310$	0	0
$311$	−5.89748e8	−1.11175	−0.555873	−	0.831268i	$-0.687616\pi$
$311$	−5.89748e8	−1.11175	−0.555873	+	0.831268i	$0.687616\pi$
$312$	−4.60892e7	−0.0859129
$313$	2.16634e8	0.399320	0.199660	−	0.979865i	$-0.436016\pi$
$313$	2.16634e8	0.399320	0.199660	+	0.979865i	$0.436016\pi$
$314$	−7.28469e8	−1.32787
$315$	0	0
$316$	−2.64244e8	−0.471086
$317$	1.47956e8	0.260871	0.130436	−	0.991457i	$-0.458362\pi$
$317$	1.47956e8	0.260871	0.130436	+	0.991457i	$0.458362\pi$
$318$	−1.80420e8	−0.314623
$319$	1.06461e9	1.83621
$320$	0	0
$321$	−4.76419e7	−0.0803937
$322$	3.84134e8	0.641191
$323$	1.80584e8	0.298174
$324$	3.40122e7	0.0555556
$325$	0	0
$326$	−2.70478e8	−0.432386
$327$	1.73823e7	0.0274910
$328$	−4.73057e7	−0.0740211
$329$	1.26231e9	1.95425
$330$	0	0
$331$	3.67114e8	0.556421	0.278210	−	0.960520i	$-0.410259\pi$
$331$	3.67114e8	0.556421	0.278210	+	0.960520i	$0.410259\pi$
$332$	−2.93540e8	−0.440234
$333$	−3.78667e8	−0.561957
$334$	−5.41445e8	−0.795136
$335$	0	0
$336$	1.09265e8	0.157142
$337$	−6.12445e8	−0.871691	−0.435845	−	0.900022i	$-0.643551\pi$
$337$	−6.12445e8	−0.871691	−0.435845	+	0.900022i	$0.643551\pi$
$338$	4.13064e8	0.581847
$339$	−3.84941e8	−0.536656
$340$	0	0
$341$	−2.04544e9	−2.79349
$342$	1.60007e8	0.216295
$343$	−6.62891e8	−0.886977
$344$	−1.20080e8	−0.159044
$345$	0	0
$346$	3.21876e8	0.417756
$347$	5.51097e8	0.708067	0.354034	−	0.935233i	$-0.384810\pi$
$347$	5.51097e8	0.708067	0.354034	+	0.935233i	$0.384810\pi$
$348$	−2.28811e8	−0.291038
$349$	−7.09429e6	−0.00893347	−0.00446673	−	0.999990i	$-0.501422\pi$
$349$	−7.09429e6	−0.00893347	−0.00446673	+	0.999990i	$0.501422\pi$
$350$	0	0
$351$	6.56231e7	0.0809994
$352$	2.63455e8	0.321964
$353$	−1.31396e8	−0.158990	−0.0794952	−	0.996835i	$-0.525331\pi$
$353$	−1.31396e8	−0.158990	−0.0794952	+	0.996835i	$0.525331\pi$
$354$	6.62852e8	0.794155
$355$	0	0
$356$	−3.68875e8	−0.433316
$357$	−1.75581e8	−0.204240
$358$	1.02065e9	1.17568
$359$	3.06664e8	0.349810	0.174905	−	0.984585i	$-0.444038\pi$
$359$	3.06664e8	0.349810	0.174905	+	0.984585i	$0.444038\pi$
$360$	0	0
$361$	−1.41138e8	−0.157895
$362$	5.97785e8	0.662316
$363$	1.21917e9	1.33780
$364$	2.10815e8	0.229112
$365$	0	0
$366$	2.18015e8	0.232436
$367$	1.32444e9	1.39863	0.699313	−	0.714815i	$-0.253489\pi$
$367$	1.32444e9	1.39863	0.699313	+	0.714815i	$0.253489\pi$
$368$	−1.99066e8	−0.208223
$369$	6.73552e7	0.0697877
$370$	0	0
$371$	8.25255e8	0.839034
$372$	4.39617e8	0.442766
$373$	1.76524e9	1.76126	0.880630	−	0.473805i	$-0.157120\pi$
$373$	1.76524e9	1.76126	0.880630	+	0.473805i	$0.157120\pi$
$374$	−4.23354e8	−0.418459
$375$	0	0
$376$	−6.54152e8	−0.634631
$377$	−4.41468e8	−0.424331
$378$	−1.55574e8	−0.148155
$379$	9.79148e8	0.923871	0.461935	−	0.886914i	$-0.347155\pi$
$379$	9.79148e8	0.923871	0.461935	+	0.886914i	$0.347155\pi$
$380$	0	0
$381$	1.19254e8	0.110468
$382$	−4.65932e8	−0.427662
$383$	−1.53116e9	−1.39259	−0.696297	−	0.717754i	$-0.745170\pi$
$383$	−1.53116e9	−1.39259	−0.696297	+	0.717754i	$0.745170\pi$
$384$	−5.66231e7	−0.0510310
$385$	0	0
$386$	1.51384e9	1.33975
$387$	1.70974e8	0.149948
$388$	−4.31843e8	−0.375332
$389$	1.13175e9	0.974825	0.487413	−	0.873172i	$-0.337941\pi$
$389$	1.13175e9	0.974825	0.487413	+	0.873172i	$0.337941\pi$
$390$	0	0
$391$	3.19885e8	0.270630
$392$	−7.81317e7	−0.0655128
$393$	−9.67895e8	−0.804367
$394$	4.92780e8	0.405897
$395$	0	0
$396$	−3.75114e8	−0.303550
$397$	2.19817e8	0.176317	0.0881584	−	0.996106i	$-0.471902\pi$
$397$	2.19817e8	0.176317	0.0881584	+	0.996106i	$0.471902\pi$
$398$	−1.08787e9	−0.864939
$399$	−7.31883e8	−0.576815
$400$	0	0
$401$	9.88276e8	0.765373	0.382686	−	0.923878i	$-0.374999\pi$
$401$	9.88276e8	0.765373	0.382686	+	0.923878i	$0.374999\pi$
$402$	6.65749e8	0.511116
$403$	8.48196e8	0.645548
$404$	3.65423e8	0.275716
$405$	0	0
$406$	1.04660e9	0.776139
$407$	4.17625e9	3.07048
$408$	9.09896e7	0.0663256
$409$	1.94597e9	1.40638	0.703192	−	0.711000i	$-0.251757\pi$
$409$	1.94597e9	1.40638	0.703192	+	0.711000i	$0.251757\pi$
$410$	0	0
$411$	4.18061e8	0.297025
$412$	4.33132e8	0.305127
$413$	−3.03193e9	−2.11785
$414$	2.83435e8	0.196315
$415$	0	0
$416$	−1.09249e8	−0.0744027
$417$	−1.24567e9	−0.841255
$418$	−1.76468e9	−1.18182
$419$	1.07893e9	0.716549	0.358275	−	0.933616i	$-0.383365\pi$
$419$	1.07893e9	0.716549	0.358275	+	0.933616i	$0.383365\pi$
$420$	0	0
$421$	−2.53361e9	−1.65483	−0.827414	−	0.561592i	$-0.810189\pi$
$421$	−2.53361e9	−1.65483	−0.827414	+	0.561592i	$0.810189\pi$
$422$	−2.00047e9	−1.29580
$423$	9.31400e8	0.598335
$424$	−4.27662e8	−0.272471
$425$	0	0
$426$	7.92012e8	0.496361
$427$	−9.97218e8	−0.619859
$428$	−1.12929e8	−0.0696229
$429$	−7.23745e8	−0.442573
$430$	0	0
$431$	1.84499e9	1.11000	0.555000	−	0.831850i	$-0.312718\pi$
$431$	1.84499e9	1.11000	0.555000	+	0.831850i	$0.312718\pi$
$432$	8.06216e7	0.0481125
$433$	−6.06893e8	−0.359256	−0.179628	−	0.983735i	$-0.557489\pi$
$433$	−6.06893e8	−0.359256	−0.179628	+	0.983735i	$0.557489\pi$
$434$	−2.01084e9	−1.18077
$435$	0	0
$436$	4.12026e7	0.0238079
$437$	1.33339e9	0.764314
$438$	2.42539e8	0.137918
$439$	−2.71714e9	−1.53280	−0.766400	−	0.642364i	$-0.777954\pi$
$439$	−2.71714e9	−1.53280	−0.766400	+	0.642364i	$0.777954\pi$
$440$	0	0
$441$	1.11246e8	0.0617661
$442$	1.75555e8	0.0967020
$443$	2.85842e9	1.56212	0.781058	−	0.624458i	$-0.214680\pi$
$443$	2.85842e9	1.56212	0.781058	+	0.624458i	$0.214680\pi$
$444$	−8.97582e8	−0.486669
$445$	0	0
$446$	−1.87222e9	−0.999274
$447$	1.32195e9	0.700068
$448$	2.58998e8	0.136089
$449$	−7.25038e7	−0.0378006	−0.0189003	−	0.999821i	$-0.506017\pi$
$449$	−7.25038e7	−0.0378006	−0.0189003	+	0.999821i	$0.506017\pi$
$450$	0	0
$451$	−7.42848e8	−0.381313
$452$	−9.12454e8	−0.464757
$453$	1.71977e9	0.869215
$454$	2.61796e9	1.31301
$455$	0	0
$456$	3.79275e8	0.187317
$457$	−1.29110e9	−0.632778	−0.316389	−	0.948629i	$-0.602471\pi$
$457$	−1.29110e9	−0.632778	−0.316389	+	0.948629i	$0.602471\pi$
$458$	1.94283e9	0.944941
$459$	−1.29554e8	−0.0625324
$460$	0	0
$461$	1.25755e9	0.597822	0.298911	−	0.954281i	$-0.403377\pi$
$461$	1.25755e9	0.597822	0.298911	+	0.954281i	$0.403377\pi$
$462$	1.71580e9	0.809506
$463$	−1.06437e9	−0.498380	−0.249190	−	0.968455i	$-0.580164\pi$
$463$	−1.06437e9	−0.498380	−0.249190	+	0.968455i	$0.580164\pi$
$464$	−5.42368e8	−0.252047
$465$	0	0
$466$	−1.08135e9	−0.495011
$467$	2.43398e9	1.10588	0.552940	−	0.833221i	$-0.313506\pi$
$467$	2.43398e9	1.10588	0.552940	+	0.833221i	$0.313506\pi$
$468$	1.55551e8	0.0701476
$469$	−3.04519e9	−1.36304
$470$	0	0
$471$	2.45858e9	1.08420
$472$	1.57121e9	0.687758
$473$	−1.88564e9	−0.819302
$474$	8.91823e8	0.384640
$475$	0	0
$476$	−4.16193e8	−0.176877
$477$	6.08918e8	0.256888
$478$	−5.43362e8	−0.227558
$479$	−1.38642e9	−0.576396	−0.288198	−	0.957571i	$-0.593056\pi$
$479$	−1.38642e9	−0.576396	−0.288198	+	0.957571i	$0.593056\pi$
$480$	0	0
$481$	−1.73179e9	−0.709558
$482$	−1.71506e8	−0.0697613
$483$	−1.29645e9	−0.523531
$484$	2.88988e9	1.15857
$485$	0	0
$486$	−1.14791e8	−0.0453609
$487$	−2.08323e9	−0.817310	−0.408655	−	0.912689i	$-0.634002\pi$
$487$	−2.08323e9	−0.817310	−0.408655	+	0.912689i	$0.634002\pi$
$488$	5.16777e8	0.201295
$489$	9.12865e8	0.353041
$490$	0	0
$491$	3.32501e8	0.126767	0.0633837	−	0.997989i	$-0.479811\pi$
$491$	3.32501e8	0.126767	0.0633837	+	0.997989i	$0.479811\pi$
$492$	1.59657e8	0.0604379
$493$	8.71549e8	0.327588
$494$	7.31773e8	0.273107
$495$	0	0
$496$	1.04206e9	0.383447
$497$	−3.62272e9	−1.32369
$498$	9.90696e8	0.359449
$499$	4.21560e8	0.151882	0.0759412	−	0.997112i	$-0.475804\pi$
$499$	4.21560e8	0.151882	0.0759412	+	0.997112i	$0.475804\pi$
$500$	0	0
$501$	1.82738e9	0.649226
$502$	−6.41035e8	−0.226162
$503$	8.08436e8	0.283242	0.141621	−	0.989921i	$-0.454769\pi$
$503$	8.08436e8	0.283242	0.141621	+	0.989921i	$0.454769\pi$
$504$	−3.68769e8	−0.128306
$505$	0	0
$506$	−3.12595e9	−1.07264
$507$	−1.39409e9	−0.475076
$508$	2.82676e8	0.0956680
$509$	−1.45321e8	−0.0488445	−0.0244223	−	0.999702i	$-0.507775\pi$
$509$	−1.45321e8	−0.0488445	−0.0244223	+	0.999702i	$0.507775\pi$
$510$	0	0
$511$	−1.10939e9	−0.367800
$512$	−1.34218e8	−0.0441942
$513$	−5.40023e8	−0.176604
$514$	−1.46823e8	−0.0476895
$515$	0	0
$516$	4.05271e8	0.129859
$517$	−1.02722e10	−3.26925
$518$	4.10561e9	1.29785
$519$	−1.08633e9	−0.341096
$520$	0	0
$521$	−1.93158e8	−0.0598383	−0.0299192	−	0.999552i	$-0.509525\pi$
$521$	−1.93158e8	−0.0598383	−0.0299192	+	0.999552i	$0.509525\pi$
$522$	7.72238e8	0.237632
$523$	2.19458e9	0.670802	0.335401	−	0.942075i	$-0.391128\pi$
$523$	2.19458e9	0.670802	0.335401	+	0.942075i	$0.391128\pi$
$524$	−2.29427e9	−0.696602
$525$	0	0
$526$	−1.74663e9	−0.523301
$527$	−1.67451e9	−0.498370
$528$	−8.89160e8	−0.262882
$529$	−1.04287e9	−0.306290
$530$	0	0
$531$	−2.23713e9	−0.648425
$532$	−1.73483e9	−0.499536
$533$	3.08042e8	0.0881179
$534$	1.24495e9	0.353801
$535$	0	0
$536$	1.57807e9	0.442639
$537$	−3.44471e9	−0.959936
$538$	7.25383e8	0.200830
$539$	−1.22691e9	−0.337484
$540$	0	0
$541$	−6.29137e9	−1.70826	−0.854131	−	0.520058i	$-0.825910\pi$
$541$	−6.29137e9	−1.70826	−0.854131	+	0.520058i	$0.825910\pi$
$542$	−1.71468e9	−0.462578
$543$	−2.01752e9	−0.540779
$544$	2.15679e8	0.0574396
$545$	0	0
$546$	−7.11502e8	−0.187069
$547$	−2.61205e9	−0.682379	−0.341190	−	0.939994i	$-0.610830\pi$
$547$	−2.61205e9	−0.682379	−0.341190	+	0.939994i	$0.610830\pi$
$548$	9.90959e8	0.257231
$549$	−7.35802e8	−0.189783
$550$	0	0
$551$	3.63291e9	0.925175
$552$	6.71846e8	0.170013
$553$	−4.07926e9	−1.02576
$554$	−4.17361e8	−0.104286
$555$	0	0
$556$	−2.95270e9	−0.728548
$557$	6.03816e9	1.48051	0.740255	−	0.672327i	$-0.234705\pi$
$557$	6.03816e9	1.48051	0.740255	+	0.672327i	$0.234705\pi$
$558$	−1.48371e9	−0.361517
$559$	7.81930e8	0.189333
$560$	0	0
$561$	1.42882e9	0.341671
$562$	−2.15612e9	−0.512383
$563$	−8.16188e8	−0.192757	−0.0963787	−	0.995345i	$-0.530726\pi$
$563$	−8.16188e8	−0.192757	−0.0963787	+	0.995345i	$0.530726\pi$
$564$	2.20776e9	0.518174
$565$	0	0
$566$	4.27291e9	0.990527
$567$	5.25064e8	0.120968
$568$	1.87736e9	0.429861
$569$	−3.35516e9	−0.763520	−0.381760	−	0.924261i	$-0.624682\pi$
$569$	−3.35516e9	−0.763520	−0.381760	+	0.924261i	$0.624682\pi$
$570$	0	0
$571$	6.86780e9	1.54380	0.771900	−	0.635744i	$-0.219306\pi$
$571$	6.86780e9	1.54380	0.771900	+	0.635744i	$0.219306\pi$
$572$	−1.71554e9	−0.383279
$573$	1.57252e9	0.349185
$574$	−7.30282e8	−0.161176
$575$	0	0
$576$	1.91103e8	0.0416667
$577$	2.92115e9	0.633051	0.316525	−	0.948584i	$-0.397484\pi$
$577$	2.92115e9	0.633051	0.316525	+	0.948584i	$0.397484\pi$
$578$	2.93613e9	0.632452
$579$	−5.10921e9	−1.09390
$580$	0	0
$581$	−4.53152e9	−0.958577
$582$	1.45747e9	0.306457
$583$	−6.71564e9	−1.40361
$584$	5.74907e8	0.119441
$585$	0	0
$586$	−3.30038e9	−0.677520
$587$	5.24861e9	1.07105	0.535526	−	0.844519i	$-0.320113\pi$
$587$	5.24861e9	1.07105	0.535526	+	0.844519i	$0.320113\pi$
$588$	2.63695e8	0.0534910
$589$	−6.97994e9	−1.40750
$590$	0	0
$591$	−1.66313e9	−0.331414
$592$	−2.12760e9	−0.421468
$593$	3.02365e9	0.595443	0.297722	−	0.954653i	$-0.403773\pi$
$593$	3.02365e9	0.595443	0.297722	+	0.954653i	$0.403773\pi$
$594$	1.26601e9	0.247848
$595$	0	0
$596$	3.13352e9	0.606276
$597$	3.67155e9	0.706219
$598$	1.29626e9	0.247878
$599$	−8.11227e8	−0.154223	−0.0771114	−	0.997022i	$-0.524570\pi$
$599$	−8.11227e8	−0.154223	−0.0771114	+	0.997022i	$0.524570\pi$
$600$	0	0
$601$	−7.55714e9	−1.42003	−0.710014	−	0.704188i	$-0.751311\pi$
$601$	−7.55714e9	−1.42003	−0.710014	+	0.704188i	$0.751311\pi$
$602$	−1.85374e9	−0.346307
$603$	−2.24690e9	−0.417324
$604$	4.07650e9	0.752763
$605$	0	0
$606$	−1.23330e9	−0.225121
$607$	3.10246e9	0.563048	0.281524	−	0.959554i	$-0.409160\pi$
$607$	3.10246e9	0.563048	0.281524	+	0.959554i	$0.409160\pi$
$608$	8.99023e8	0.162221
$609$	−3.53228e9	−0.633715
$610$	0	0
$611$	4.25965e9	0.755492
$612$	−3.07090e8	−0.0541546
$613$	1.82934e9	0.320762	0.160381	−	0.987055i	$-0.448728\pi$
$613$	1.82934e9	0.320762	0.160381	+	0.987055i	$0.448728\pi$
$614$	2.41065e9	0.420286
$615$	0	0
$616$	4.06708e9	0.701052
$617$	−1.14156e10	−1.95659	−0.978294	−	0.207220i	$-0.933558\pi$
$617$	−1.14156e10	−1.95659	−0.978294	+	0.207220i	$0.933558\pi$
$618$	−1.46182e9	−0.249135
$619$	−5.65578e9	−0.958462	−0.479231	−	0.877689i	$-0.659084\pi$
$619$	−5.65578e9	−0.958462	−0.479231	+	0.877689i	$0.659084\pi$
$620$	0	0
$621$	−9.56594e8	−0.160290
$622$	4.71799e9	0.786123
$623$	−5.69451e9	−0.943514
$624$	3.68714e8	0.0607496
$625$	0	0
$626$	−1.73307e9	−0.282362
$627$	5.95581e9	0.964949
$628$	5.82775e9	0.938949
$629$	3.41891e9	0.547786
$630$	0	0
$631$	3.93985e9	0.624277	0.312138	−	0.950037i	$-0.398955\pi$
$631$	3.93985e9	0.624277	0.312138	+	0.950037i	$0.398955\pi$
$632$	2.11395e9	0.333108
$633$	6.75157e9	1.05802
$634$	−1.18365e9	−0.184464
$635$	0	0
$636$	1.44336e9	0.222472
$637$	5.08772e8	0.0779893
$638$	−8.51687e9	−1.29840
$639$	−2.67304e9	−0.405277
$640$	0	0
$641$	5.45282e9	0.817745	0.408873	−	0.912591i	$-0.365922\pi$
$641$	5.45282e9	0.817745	0.408873	+	0.912591i	$0.365922\pi$
$642$	3.81135e8	0.0568469
$643$	−6.55463e7	−0.00972322	−0.00486161	−	0.999988i	$-0.501548\pi$
$643$	−6.55463e7	−0.00972322	−0.00486161	+	0.999988i	$0.501548\pi$
$644$	−3.07308e9	−0.453391
$645$	0	0
$646$	−1.44467e9	−0.210841
$647$	5.34631e9	0.776048	0.388024	−	0.921649i	$-0.373158\pi$
$647$	5.34631e9	0.776048	0.388024	+	0.921649i	$0.373158\pi$
$648$	−2.72098e8	−0.0392837
$649$	2.46728e10	3.54293
$650$	0	0
$651$	6.78659e9	0.964091
$652$	2.16383e9	0.305743
$653$	3.07490e9	0.432151	0.216075	−	0.976377i	$-0.430674\pi$
$653$	3.07490e9	0.432151	0.216075	+	0.976377i	$0.430674\pi$
$654$	−1.39059e8	−0.0194391
$655$	0	0
$656$	3.78446e8	0.0523408
$657$	−8.18569e8	−0.112610
$658$	−1.00985e10	−1.38186
$659$	9.34559e9	1.27206	0.636031	−	0.771664i	$-0.280575\pi$
$659$	9.34559e9	1.27206	0.636031	+	0.771664i	$0.280575\pi$
$660$	0	0
$661$	−3.45515e9	−0.465331	−0.232666	−	0.972557i	$-0.574745\pi$
$661$	−3.45515e9	−0.465331	−0.232666	+	0.972557i	$0.574745\pi$
$662$	−2.93691e9	−0.393449
$663$	−5.92498e8	−0.0789569
$664$	2.34832e9	0.311292
$665$	0	0
$666$	3.02934e9	0.397364
$667$	6.43532e9	0.839711
$668$	4.33156e9	0.562246
$669$	6.31874e9	0.815904
$670$	0	0
$671$	8.11501e9	1.03696
$672$	−8.74119e8	−0.111116
$673$	−4.36251e9	−0.551676	−0.275838	−	0.961204i	$-0.588955\pi$
$673$	−4.36251e9	−0.551676	−0.275838	+	0.961204i	$0.588955\pi$
$674$	4.89956e9	0.616379
$675$	0	0
$676$	−3.30451e9	−0.411428
$677$	1.58108e9	0.195836	0.0979182	−	0.995194i	$-0.468782\pi$
$677$	1.58108e9	0.195836	0.0979182	+	0.995194i	$0.468782\pi$
$678$	3.07953e9	0.379473
$679$	−6.66658e9	−0.817258
$680$	0	0
$681$	−8.83562e9	−1.07207
$682$	1.63635e10	1.97529
$683$	6.54644e9	0.786200	0.393100	−	0.919496i	$-0.371403\pi$
$683$	6.54644e9	0.786200	0.393100	+	0.919496i	$0.371403\pi$
$684$	−1.28005e9	−0.152944
$685$	0	0
$686$	5.30313e9	0.627188
$687$	−6.55704e9	−0.771541
$688$	9.60643e8	0.112461
$689$	2.78482e9	0.324362
$690$	0	0
$691$	1.13916e10	1.31345	0.656724	−	0.754131i	$-0.271942\pi$
$691$	1.13916e10	1.31345	0.656724	+	0.754131i	$0.271942\pi$
$692$	−2.57501e9	−0.295398
$693$	−5.79083e9	−0.660959
$694$	−4.40877e9	−0.500679
$695$	0	0
$696$	1.83049e9	0.205795
$697$	−6.08137e8	−0.0680279
$698$	5.67543e7	0.00631691
$699$	3.64955e9	0.404175
$700$	0	0
$701$	−9.60390e9	−1.05301	−0.526507	−	0.850171i	$-0.676499\pi$
$701$	−9.60390e9	−1.05301	−0.526507	+	0.850171i	$0.676499\pi$
$702$	−5.24985e8	−0.0572752
$703$	1.42512e10	1.54706
$704$	−2.10764e9	−0.227663
$705$	0	0
$706$	1.05117e9	0.112423
$707$	5.64122e9	0.600351
$708$	−5.30282e9	−0.561552
$709$	−1.93713e9	−0.204126	−0.102063	−	0.994778i	$-0.532544\pi$
$709$	−1.93713e9	−0.204126	−0.102063	+	0.994778i	$0.532544\pi$
$710$	0	0
$711$	−3.00990e9	−0.314057
$712$	2.95100e9	0.306401
$713$	−1.23642e10	−1.27748
$714$	1.40465e9	0.144419
$715$	0	0
$716$	−8.16523e9	−0.831329
$717$	1.83385e9	0.185800
$718$	−2.45331e9	−0.247353
$719$	7.40065e9	0.742538	0.371269	−	0.928525i	$-0.378923\pi$
$719$	7.40065e9	0.742538	0.371269	+	0.928525i	$0.378923\pi$
$720$	0	0
$721$	6.68648e9	0.664392
$722$	1.12910e9	0.111648
$723$	5.78833e8	0.0569599
$724$	−4.78228e9	−0.468328
$725$	0	0
$726$	−9.75336e9	−0.945967
$727$	−4.30049e8	−0.0415095	−0.0207547	−	0.999785i	$-0.506607\pi$
$727$	−4.30049e8	−0.0415095	−0.0207547	+	0.999785i	$0.506607\pi$
$728$	−1.68652e9	−0.162007
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−1.54369e9	−0.146167
$732$	−1.74412e9	−0.164357
$733$	1.13354e8	0.0106309	0.00531547	−	0.999986i	$-0.498308\pi$
$733$	1.13354e8	0.0106309	0.00531547	+	0.999986i	$0.498308\pi$
$734$	−1.05955e10	−0.988978
$735$	0	0
$736$	1.59252e9	0.147236
$737$	2.47807e10	2.28022
$738$	−5.38842e8	−0.0493474
$739$	5.90138e9	0.537896	0.268948	−	0.963155i	$-0.413324\pi$
$739$	5.90138e9	0.537896	0.268948	+	0.963155i	$0.413324\pi$
$740$	0	0
$741$	−2.46973e9	−0.222991
$742$	−6.60204e9	−0.593286
$743$	−2.64871e9	−0.236905	−0.118453	−	0.992960i	$-0.537793\pi$
$743$	−2.64871e9	−0.236905	−0.118453	+	0.992960i	$0.537793\pi$
$744$	−3.51694e9	−0.313083
$745$	0	0
$746$	−1.41219e10	−1.24540
$747$	−3.34360e9	−0.293489
$748$	3.38683e9	0.295895
$749$	−1.74334e9	−0.151599
$750$	0	0
$751$	−1.44933e10	−1.24861	−0.624305	−	0.781181i	$-0.714618\pi$
$751$	−1.44933e10	−1.24861	−0.624305	+	0.781181i	$0.714618\pi$
$752$	5.23321e9	0.448752
$753$	2.16349e9	0.184660
$754$	3.53175e9	0.300047
$755$	0	0
$756$	1.24460e9	0.104762
$757$	2.02665e9	0.169802	0.0849012	−	0.996389i	$-0.472943\pi$
$757$	2.02665e9	0.169802	0.0849012	+	0.996389i	$0.472943\pi$
$758$	−7.83319e9	−0.653275
$759$	1.05501e10	0.875810
$760$	0	0
$761$	−1.92349e10	−1.58214	−0.791069	−	0.611727i	$-0.790475\pi$
$761$	−1.92349e10	−1.58214	−0.791069	+	0.611727i	$0.790475\pi$
$762$	−9.54033e8	−0.0781126
$763$	6.36065e8	0.0518400
$764$	3.72746e9	0.302403
$765$	0	0
$766$	1.22493e10	0.984712
$767$	−1.02312e10	−0.818738
$768$	4.52985e8	0.0360844
$769$	6.35399e9	0.503854	0.251927	−	0.967746i	$-0.418936\pi$
$769$	6.35399e9	0.503854	0.251927	+	0.967746i	$0.418936\pi$
$770$	0	0
$771$	4.95527e8	0.0389383
$772$	−1.21107e10	−0.947348
$773$	3.27598e9	0.255102	0.127551	−	0.991832i	$-0.459288\pi$
$773$	3.27598e9	0.255102	0.127551	+	0.991832i	$0.459288\pi$
$774$	−1.36779e9	−0.106029
$775$	0	0
$776$	3.45475e9	0.265400
$777$	−1.38564e10	−1.05969
$778$	−9.05400e9	−0.689305
$779$	−2.53492e9	−0.192125
$780$	0	0
$781$	2.94804e10	2.21440
$782$	−2.55908e9	−0.191364
$783$	−2.60630e9	−0.194026
$784$	6.25054e8	0.0463245
$785$	0	0
$786$	7.74316e9	0.568773
$787$	1.21823e10	0.890875	0.445437	−	0.895313i	$-0.353048\pi$
$787$	1.21823e10	0.890875	0.445437	+	0.895313i	$0.353048\pi$
$788$	−3.94224e9	−0.287013
$789$	5.89489e9	0.427274
$790$	0	0
$791$	−1.40860e10	−1.01198
$792$	3.00091e9	0.214642
$793$	−3.36511e9	−0.239631
$794$	−1.75853e9	−0.124675
$795$	0	0
$796$	8.70294e9	0.611604
$797$	−2.40548e10	−1.68306	−0.841528	−	0.540214i	$-0.818343\pi$
$797$	−2.40548e10	−1.68306	−0.841528	+	0.540214i	$0.818343\pi$
$798$	5.85506e9	0.407870
$799$	−8.40943e9	−0.583247
$800$	0	0
$801$	−4.20172e9	−0.288877
$802$	−7.90621e9	−0.541200
$803$	9.02784e9	0.615290
$804$	−5.32599e9	−0.361414
$805$	0	0
$806$	−6.78557e9	−0.456472
$807$	−2.44817e9	−0.163977
$808$	−2.92339e9	−0.194960
$809$	7.44389e8	0.0494288	0.0247144	−	0.999695i	$-0.492132\pi$
$809$	7.44389e8	0.0494288	0.0247144	+	0.999695i	$0.492132\pi$
$810$	0	0
$811$	1.18071e10	0.777264	0.388632	−	0.921393i	$-0.372948\pi$
$811$	1.18071e10	0.777264	0.388632	+	0.921393i	$0.372948\pi$
$812$	−8.37280e9	−0.548813
$813$	5.78704e9	0.377694
$814$	−3.34100e10	−2.17116
$815$	0	0
$816$	−7.27917e8	−0.0468993
$817$	−6.43462e9	−0.412806
$818$	−1.55677e10	−0.994464
$819$	2.40132e9	0.152741
$820$	0	0
$821$	−2.93131e10	−1.84867	−0.924336	−	0.381579i	$-0.875381\pi$
$821$	−2.93131e10	−1.84867	−0.924336	+	0.381579i	$0.875381\pi$
$822$	−3.34449e9	−0.210028
$823$	2.03889e10	1.27496	0.637478	−	0.770469i	$-0.279978\pi$
$823$	2.03889e10	1.27496	0.637478	+	0.770469i	$0.279978\pi$
$824$	−3.46506e9	−0.215757
$825$	0	0
$826$	2.42555e10	1.49754
$827$	−2.41235e10	−1.48310	−0.741550	−	0.670898i	$-0.765909\pi$
$827$	−2.41235e10	−1.48310	−0.741550	+	0.670898i	$0.765909\pi$
$828$	−2.26748e9	−0.138815
$829$	−6.32929e9	−0.385846	−0.192923	−	0.981214i	$-0.561797\pi$
$829$	−6.32929e9	−0.385846	−0.192923	+	0.981214i	$0.561797\pi$
$830$	0	0
$831$	1.40859e9	0.0851496
$832$	8.73988e8	0.0526107
$833$	−1.00442e9	−0.0602085
$834$	9.96538e9	0.594857
$835$	0	0
$836$	1.41175e10	0.835670
$837$	5.00751e9	0.295177
$838$	−8.63148e9	−0.506677
$839$	−2.01943e10	−1.18049	−0.590243	−	0.807225i	$-0.700968\pi$
$839$	−2.01943e10	−1.18049	−0.590243	+	0.807225i	$0.700968\pi$
$840$	0	0
$841$	2.83591e8	0.0164402
$842$	2.02689e10	1.17014
$843$	7.27689e9	0.418359
$844$	1.60037e10	0.916268
$845$	0	0
$846$	−7.45120e9	−0.423087
$847$	4.46126e10	2.52270
$848$	3.42130e9	0.192666
$849$	−1.44211e10	−0.808762
$850$	0	0
$851$	2.52445e10	1.40415
$852$	−6.33609e9	−0.350980
$853$	1.07171e10	0.591229	0.295614	−	0.955307i	$-0.404476\pi$
$853$	1.07171e10	0.591229	0.295614	+	0.955307i	$0.404476\pi$
$854$	7.97774e9	0.438306
$855$	0	0
$856$	9.03432e8	0.0492309
$857$	−8.85297e9	−0.480458	−0.240229	−	0.970716i	$-0.577223\pi$
$857$	−8.85297e9	−0.480458	−0.240229	+	0.970716i	$0.577223\pi$
$858$	5.78996e9	0.312946
$859$	−2.37823e10	−1.28020	−0.640100	−	0.768292i	$-0.721107\pi$
$859$	−2.37823e10	−1.28020	−0.640100	+	0.768292i	$0.721107\pi$
$860$	0	0
$861$	2.46470e9	0.131599
$862$	−1.47599e10	−0.784889
$863$	2.99137e10	1.58428	0.792140	−	0.610340i	$-0.208967\pi$
$863$	2.99137e10	1.58428	0.792140	+	0.610340i	$0.208967\pi$
$864$	−6.44973e8	−0.0340207
$865$	0	0
$866$	4.85514e9	0.254032
$867$	−9.90943e9	−0.516395
$868$	1.60867e10	0.834927
$869$	3.31956e10	1.71598
$870$	0	0
$871$	−1.02760e10	−0.526937
$872$	−3.29620e8	−0.0168347
$873$	−4.91897e9	−0.250221
$874$	−1.06671e10	−0.540452
$875$	0	0
$876$	−1.94031e9	−0.0975231
$877$	−9.04648e9	−0.452878	−0.226439	−	0.974025i	$-0.572708\pi$
$877$	−9.04648e9	−0.452878	−0.226439	+	0.974025i	$0.572708\pi$
$878$	2.17371e10	1.08385
$879$	1.11388e10	0.553193
$880$	0	0
$881$	−4.04413e9	−0.199255	−0.0996275	−	0.995025i	$-0.531765\pi$
$881$	−4.04413e9	−0.199255	−0.0996275	+	0.995025i	$0.531765\pi$
$882$	−8.89969e8	−0.0436752
$883$	−2.37982e10	−1.16328	−0.581638	−	0.813448i	$-0.697588\pi$
$883$	−2.37982e10	−1.16328	−0.581638	+	0.813448i	$0.697588\pi$
$884$	−1.40444e9	−0.0683786
$885$	0	0
$886$	−2.28674e10	−1.10458
$887$	−7.17840e9	−0.345378	−0.172689	−	0.984976i	$-0.555246\pi$
$887$	−7.17840e9	−0.345378	−0.172689	+	0.984976i	$0.555246\pi$
$888$	7.18066e9	0.344127
$889$	4.36382e9	0.208310
$890$	0	0
$891$	−4.27279e9	−0.202367
$892$	1.49778e10	0.706594
$893$	−3.50533e10	−1.64721
$894$	−1.05756e10	−0.495023
$895$	0	0
$896$	−2.07199e9	−0.0962296
$897$	−4.37487e9	−0.202391
$898$	5.80031e8	0.0267291
$899$	−3.36872e10	−1.54634
$900$	0	0
$901$	−5.49780e9	−0.250410
$902$	5.94278e9	0.269629
$903$	6.25638e9	0.282759
$904$	7.29963e9	0.328633
$905$	0	0
$906$	−1.37582e10	−0.614628
$907$	1.61336e10	0.717969	0.358985	−	0.933343i	$-0.383123\pi$
$907$	1.61336e10	0.717969	0.358985	+	0.933343i	$0.383123\pi$
$908$	−2.09437e10	−0.928437
$909$	4.16240e9	0.183810
$910$	0	0
$911$	9.90491e9	0.434047	0.217023	−	0.976166i	$-0.430365\pi$
$911$	9.90491e9	0.434047	0.217023	+	0.976166i	$0.430365\pi$
$912$	−3.03420e9	−0.132453
$913$	3.68759e10	1.60360
$914$	1.03288e10	0.447442
$915$	0	0
$916$	−1.55426e10	−0.668174
$917$	−3.54178e10	−1.51680
$918$	1.03643e9	0.0442171
$919$	−1.07988e10	−0.458958	−0.229479	−	0.973314i	$-0.573702\pi$
$919$	−1.07988e10	−0.458958	−0.229479	+	0.973314i	$0.573702\pi$
$920$	0	0
$921$	−8.13595e9	−0.343162
$922$	−1.00604e10	−0.422724
$923$	−1.22248e10	−0.511726
$924$	−1.37264e10	−0.572407
$925$	0	0
$926$	8.51498e9	0.352408
$927$	4.93365e9	0.203418
$928$	4.33894e9	0.178224
$929$	−1.24984e10	−0.511444	−0.255722	−	0.966750i	$-0.582313\pi$
$929$	−1.24984e10	−0.511444	−0.255722	+	0.966750i	$0.582313\pi$
$930$	0	0
$931$	−4.18676e9	−0.170041
$932$	8.65079e9	0.350026
$933$	−1.59232e10	−0.641866
$934$	−1.94718e10	−0.781975
$935$	0	0
$936$	−1.24441e9	−0.0496018
$937$	−4.89405e9	−0.194348	−0.0971739	−	0.995267i	$-0.530980\pi$
$937$	−4.89405e9	−0.194348	−0.0971739	+	0.995267i	$0.530980\pi$
$938$	2.43615e10	0.963815
$939$	5.84912e9	0.230548
$940$	0	0
$941$	1.32199e10	0.517209	0.258604	−	0.965983i	$-0.416737\pi$
$941$	1.32199e10	0.517209	0.258604	+	0.965983i	$0.416737\pi$
$942$	−1.96687e10	−0.766649
$943$	−4.49035e9	−0.174377
$944$	−1.25696e10	−0.486319
$945$	0	0
$946$	1.50851e10	0.579334
$947$	−6.88319e9	−0.263369	−0.131684	−	0.991292i	$-0.542039\pi$
$947$	−6.88319e9	−0.263369	−0.131684	+	0.991292i	$0.542039\pi$
$948$	−7.13458e9	−0.271981
$949$	−3.74364e9	−0.142188
$950$	0	0
$951$	3.99482e9	0.150614
$952$	3.32954e9	0.125071
$953$	2.56888e10	0.961433	0.480717	−	0.876876i	$-0.340377\pi$
$953$	2.56888e10	0.961433	0.480717	+	0.876876i	$0.340377\pi$
$954$	−4.87134e9	−0.181647
$955$	0	0
$956$	4.34689e9	0.160908
$957$	2.87444e10	1.06014
$958$	1.10914e10	0.407573
$959$	1.52979e10	0.560103
$960$	0	0
$961$	3.72108e10	1.35250
$962$	1.38543e10	0.501734
$963$	−1.28633e9	−0.0464153
$964$	1.37205e9	0.0493287
$965$	0	0
$966$	1.03716e10	0.370192
$967$	1.17717e9	0.0418647	0.0209323	−	0.999781i	$-0.493337\pi$
$967$	1.17717e9	0.0418647	0.0209323	+	0.999781i	$0.493337\pi$
$968$	−2.31191e10	−0.819231
$969$	4.87576e9	0.172151
$970$	0	0
$971$	−3.53867e9	−0.124043	−0.0620215	−	0.998075i	$-0.519755\pi$
$971$	−3.53867e9	−0.124043	−0.0620215	+	0.998075i	$0.519755\pi$
$972$	9.18330e8	0.0320750
$973$	−4.55824e10	−1.58636
$974$	1.66659e10	0.577925
$975$	0	0
$976$	−4.13422e9	−0.142337
$977$	9.98892e9	0.342679	0.171340	−	0.985212i	$-0.445190\pi$
$977$	9.98892e9	0.342679	0.171340	+	0.985212i	$0.445190\pi$
$978$	−7.30292e9	−0.249638
$979$	4.63400e10	1.57840
$980$	0	0
$981$	4.69323e8	0.0158719
$982$	−2.66001e9	−0.0896381
$983$	−4.58129e10	−1.53833	−0.769167	−	0.639048i	$-0.779329\pi$
$983$	−4.58129e10	−1.53833	−0.769167	+	0.639048i	$0.779329\pi$
$984$	−1.27725e9	−0.0427361
$985$	0	0
$986$	−6.97239e9	−0.231639
$987$	3.40823e10	1.12829
$988$	−5.85418e9	−0.193115
$989$	−1.13983e10	−0.374672
$990$	0	0
$991$	2.31266e10	0.754840	0.377420	−	0.926042i	$-0.376811\pi$
$991$	2.31266e10	0.754840	0.377420	+	0.926042i	$0.376811\pi$
$992$	−8.33644e9	−0.271138
$993$	9.91208e9	0.321250
$994$	2.89818e10	0.935992
$995$	0	0
$996$	−7.92557e9	−0.254169
$997$	−4.79092e10	−1.53104	−0.765519	−	0.643413i	$-0.777518\pi$
$997$	−4.79092e10	−1.53104	−0.765519	+	0.643413i	$0.777518\pi$
$998$	−3.37248e9	−0.107397
$999$	−1.02240e10	−0.324446

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.8.a.i.1.1		1
3.2	odd	2		450.8.a.x.1.1		1
5.2	odd	4		150.8.c.a.49.1		2
5.3	odd	4		150.8.c.a.49.2		2
5.4	even	2		30.8.a.d.1.1	✓	1
15.2	even	4		450.8.c.r.199.2		2
15.8	even	4		450.8.c.r.199.1		2
15.14	odd	2		90.8.a.c.1.1		1
20.19	odd	2		240.8.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.8.a.d.1.1	✓	1	5.4	even	2
90.8.a.c.1.1		1	15.14	odd	2
150.8.a.i.1.1		1	1.1	even	1	trivial
150.8.c.a.49.1		2	5.2	odd	4
150.8.c.a.49.2		2	5.3	odd	4
240.8.a.j.1.1		1	20.19	odd	2
450.8.a.x.1.1		1	3.2	odd	2
450.8.c.r.199.1		2	15.8	even	4
450.8.c.r.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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