LMFDB - Embedded newform 150.8.a.p.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)

$f(q)$

$=$

$q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +216.000 q^{6} +349.000 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})$

\(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +216.000 q^{6} +349.000 q^{7} +512.000 q^{8} +729.000 q^{9} +1182.00 q^{11} +1728.00 q^{12} +1723.00 q^{13} +2792.00 q^{14} +4096.00 q^{16} +7494.00 q^{17} +5832.00 q^{18} +12785.0 q^{19} +9423.00 q^{21} +9456.00 q^{22} -6402.00 q^{23} +13824.0 q^{24} +13784.0 q^{26} +19683.0 q^{27} +22336.0 q^{28} +108090. q^{29} +142427. q^{31} +32768.0 q^{32} +31914.0 q^{33} +59952.0 q^{34} +46656.0 q^{36} -276266. q^{37} +102280. q^{38} +46521.0 q^{39} +525072. q^{41} +75384.0 q^{42} +747013. q^{43} +75648.0 q^{44} -51216.0 q^{46} -571326. q^{47} +110592. q^{48} -701742. q^{49} +202338. q^{51} +110272. q^{52} +1.47203e6 q^{53} +157464. q^{54} +178688. q^{56} +345195. q^{57} +864720. q^{58} -1.58211e6 q^{59} -932893. q^{61} +1.13942e6 q^{62} +254421. q^{63} +262144. q^{64} +255312. q^{66} +1.68809e6 q^{67} +479616. q^{68} -172854. q^{69} +2.96275e6 q^{71} +373248. q^{72} +4.07880e6 q^{73} -2.21013e6 q^{74} +818240. q^{76} +412518. q^{77} +372168. q^{78} -5.63536e6 q^{79} +531441. q^{81} +4.20058e6 q^{82} +3.12032e6 q^{83} +603072. q^{84} +5.97610e6 q^{86} +2.91843e6 q^{87} +605184. q^{88} -9.15504e6 q^{89} +601327. q^{91} -409728. q^{92} +3.84553e6 q^{93} -4.57061e6 q^{94} +884736. q^{96} +1.00412e7 q^{97} -5.61394e6 q^{98} +861678. q^{99} +O(q^{100})\)

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
$7$	349.000	0.384576	0.192288	−	0.981339i	$-0.438409\pi$
$7$	349.000	0.384576	0.192288	+	0.981339i	$0.438409\pi$
$8$	512.000	0.353553
$9$	729.000	0.333333
$10$	0	0
$11$	1182.00	0.267758	0.133879	−	0.990998i	$-0.457257\pi$
$11$	1182.00	0.267758	0.133879	+	0.990998i	$0.457257\pi$
$12$	1728.00	0.288675
$13$	1723.00	0.217512	0.108756	−	0.994068i	$-0.465313\pi$
$13$	1723.00	0.217512	0.108756	+	0.994068i	$0.465313\pi$
$14$	2792.00	0.271936
$15$	0	0
$16$	4096.00	0.250000
$17$	7494.00	0.369950	0.184975	−	0.982743i	$-0.440780\pi$
$17$	7494.00	0.369950	0.184975	+	0.982743i	$0.440780\pi$
$18$	5832.00	0.235702
$19$	12785.0	0.427625	0.213813	−	0.976875i	$-0.431412\pi$
$19$	12785.0	0.427625	0.213813	+	0.976875i	$0.431412\pi$
$20$	0	0
$21$	9423.00	0.222035
$22$	9456.00	0.189334
$23$	−6402.00	−0.109716	−0.0548578	−	0.998494i	$-0.517471\pi$
$23$	−6402.00	−0.109716	−0.0548578	+	0.998494i	$0.517471\pi$
$24$	13824.0	0.204124
$25$	0	0
$26$	13784.0	0.153804
$27$	19683.0	0.192450
$28$	22336.0	0.192288
$29$	108090.	0.822986	0.411493	−	0.911413i	$-0.365007\pi$
$29$	108090.	0.822986	0.411493	+	0.911413i	$0.365007\pi$
$30$	0	0
$31$	142427.	0.858670	0.429335	−	0.903145i	$-0.358748\pi$
$31$	142427.	0.858670	0.429335	+	0.903145i	$0.358748\pi$
$32$	32768.0	0.176777
$33$	31914.0	0.154590
$34$	59952.0	0.261594
$35$	0	0
$36$	46656.0	0.166667
$37$	−276266.	−0.896647	−0.448323	−	0.893871i	$-0.647979\pi$
$37$	−276266.	−0.896647	−0.448323	+	0.893871i	$0.647979\pi$
$38$	102280.	0.302377
$39$	46521.0	0.125581
$40$	0	0
$41$	525072.	1.18980	0.594902	−	0.803798i	$-0.297191\pi$
$41$	525072.	1.18980	0.594902	+	0.803798i	$0.297191\pi$
$42$	75384.0	0.157003
$43$	747013.	1.43281	0.716405	−	0.697685i	$-0.245786\pi$
$43$	747013.	1.43281	0.716405	+	0.697685i	$0.245786\pi$
$44$	75648.0	0.133879
$45$	0	0
$46$	−51216.0	−0.0775806
$47$	−571326.	−0.802678	−0.401339	−	0.915930i	$-0.631455\pi$
$47$	−571326.	−0.802678	−0.401339	+	0.915930i	$0.631455\pi$
$48$	110592.	0.144338
$49$	−701742.	−0.852101
$50$	0	0
$51$	202338.	0.213590
$52$	110272.	0.108756
$53$	1.47203e6	1.35816	0.679079	−	0.734065i	$-0.262379\pi$
$53$	1.47203e6	1.35816	0.679079	+	0.734065i	$0.262379\pi$
$54$	157464.	0.136083
$55$	0	0
$56$	178688.	0.135968
$57$	345195.	0.246889
$58$	864720.	0.581939
$59$	−1.58211e6	−1.00289	−0.501447	−	0.865189i	$-0.667199\pi$
$59$	−1.58211e6	−1.00289	−0.501447	+	0.865189i	$0.667199\pi$
$60$	0	0
$61$	−932893.	−0.526232	−0.263116	−	0.964764i	$-0.584750\pi$
$61$	−932893.	−0.526232	−0.263116	+	0.964764i	$0.584750\pi$
$62$	1.13942e6	0.607172
$63$	254421.	0.128192
$64$	262144.	0.125000
$65$	0	0
$66$	255312.	0.109312
$67$	1.68809e6	0.685699	0.342850	−	0.939390i	$-0.388608\pi$
$67$	1.68809e6	0.685699	0.342850	+	0.939390i	$0.388608\pi$
$68$	479616.	0.184975
$69$	−172854.	−0.0633443
$70$	0	0
$71$	2.96275e6	0.982406	0.491203	−	0.871045i	$-0.336557\pi$
$71$	2.96275e6	0.982406	0.491203	+	0.871045i	$0.336557\pi$
$72$	373248.	0.117851
$73$	4.07880e6	1.22716	0.613581	−	0.789631i	$-0.289728\pi$
$73$	4.07880e6	1.22716	0.613581	+	0.789631i	$0.289728\pi$
$74$	−2.21013e6	−0.634025
$75$	0	0
$76$	818240.	0.213813
$77$	412518.	0.102973
$78$	372168.	0.0887990
$79$	−5.63536e6	−1.28596	−0.642979	−	0.765884i	$-0.722302\pi$
$79$	−5.63536e6	−1.28596	−0.642979	+	0.765884i	$0.722302\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	4.20058e6	0.841318
$83$	3.12032e6	0.598998	0.299499	−	0.954097i	$-0.403180\pi$
$83$	3.12032e6	0.598998	0.299499	+	0.954097i	$0.403180\pi$
$84$	603072.	0.111018
$85$	0	0
$86$	5.97610e6	1.01315
$87$	2.91843e6	0.475151
$88$	605184.	0.0946669
$89$	−9.15504e6	−1.37656	−0.688280	−	0.725445i	$-0.741634\pi$
$89$	−9.15504e6	−1.37656	−0.688280	+	0.725445i	$0.741634\pi$
$90$	0	0
$91$	601327.	0.0836500
$92$	−409728.	−0.0548578
$93$	3.84553e6	0.495753
$94$	−4.57061e6	−0.567579
$95$	0	0
$96$	884736.	0.102062
$97$	1.00412e7	1.11708	0.558540	−	0.829477i	$-0.311362\pi$
$97$	1.00412e7	1.11708	0.558540	+	0.829477i	$0.311362\pi$
$98$	−5.61394e6	−0.602527
$99$	861678.	0.0892528
$100$	0	0
$101$	4.29541e6	0.414839	0.207420	−	0.978252i	$-0.433493\pi$
$101$	4.29541e6	0.414839	0.207420	+	0.978252i	$0.433493\pi$
$102$	1.61870e6	0.151031
$103$	3.21099e6	0.289540	0.144770	−	0.989465i	$-0.453756\pi$
$103$	3.21099e6	0.289540	0.144770	+	0.989465i	$0.453756\pi$
$104$	882176.	0.0769022
$105$	0	0
$106$	1.17762e7	0.960363
$107$	−1.38940e7	−1.09644	−0.548219	−	0.836335i	$-0.684694\pi$
$107$	−1.38940e7	−1.09644	−0.548219	+	0.836335i	$0.684694\pi$
$108$	1.25971e6	0.0962250
$109$	−5.85654e6	−0.433160	−0.216580	−	0.976265i	$-0.569490\pi$
$109$	−5.85654e6	−0.433160	−0.216580	+	0.976265i	$0.569490\pi$
$110$	0	0
$111$	−7.45918e6	−0.517679
$112$	1.42950e6	0.0961440
$113$	1.26861e7	0.827095	0.413547	−	0.910483i	$-0.364290\pi$
$113$	1.26861e7	0.827095	0.413547	+	0.910483i	$0.364290\pi$
$114$	2.76156e6	0.174577
$115$	0	0
$116$	6.91776e6	0.411493
$117$	1.25607e6	0.0725041
$118$	−1.26569e7	−0.709152
$119$	2.61541e6	0.142274
$120$	0	0
$121$	−1.80900e7	−0.928305
$122$	−7.46314e6	−0.372102
$123$	1.41769e7	0.686934
$124$	9.11533e6	0.429335
$125$	0	0
$126$	2.03537e6	0.0906455
$127$	−924536.	−0.0400508	−0.0200254	−	0.999799i	$-0.506375\pi$
$127$	−924536.	−0.0400508	−0.0200254	+	0.999799i	$0.506375\pi$
$128$	2.09715e6	0.0883883
$129$	2.01694e7	0.827233
$130$	0	0
$131$	−2.36508e7	−0.919171	−0.459585	−	0.888134i	$-0.652002\pi$
$131$	−2.36508e7	−0.919171	−0.459585	+	0.888134i	$0.652002\pi$
$132$	2.04250e6	0.0772952
$133$	4.46196e6	0.164454
$134$	1.35047e7	0.484862
$135$	0	0
$136$	3.83693e6	0.130797
$137$	−4.24453e7	−1.41029	−0.705144	−	0.709064i	$-0.749118\pi$
$137$	−4.24453e7	−1.41029	−0.705144	+	0.709064i	$0.749118\pi$
$138$	−1.38283e6	−0.0447912
$139$	1.09283e7	0.345144	0.172572	−	0.984997i	$-0.444792\pi$
$139$	1.09283e7	0.345144	0.172572	+	0.984997i	$0.444792\pi$
$140$	0	0
$141$	−1.54258e7	−0.463426
$142$	2.37020e7	0.694666
$143$	2.03659e6	0.0582407
$144$	2.98598e6	0.0833333
$145$	0	0
$146$	3.26304e7	0.867735
$147$	−1.89470e7	−0.491961
$148$	−1.76810e7	−0.448323
$149$	7.99173e6	0.197920	0.0989598	−	0.995091i	$-0.468448\pi$
$149$	7.99173e6	0.197920	0.0989598	+	0.995091i	$0.468448\pi$
$150$	0	0
$151$	−6.15671e7	−1.45522	−0.727611	−	0.685990i	$-0.759369\pi$
$151$	−6.15671e7	−1.45522	−0.727611	+	0.685990i	$0.759369\pi$
$152$	6.54592e6	0.151188
$153$	5.46313e6	0.123317
$154$	3.30014e6	0.0728132
$155$	0	0
$156$	2.97734e6	0.0627904
$157$	71809.0	0.00148091	0.000740457	−	1.00000i	$-0.499764\pi$
$157$	71809.0	0.00148091	0.000740457		1.00000i	$0.499764\pi$
$158$	−4.50829e7	−0.909310
$159$	3.97448e7	0.784133
$160$	0	0
$161$	−2.23430e6	−0.0421940
$162$	4.25153e6	0.0785674
$163$	−9.34587e7	−1.69030	−0.845148	−	0.534532i	$-0.820488\pi$
$163$	−9.34587e7	−1.69030	−0.845148	+	0.534532i	$0.820488\pi$
$164$	3.36046e7	0.594902
$165$	0	0
$166$	2.49625e7	0.423556
$167$	−8.11209e7	−1.34780	−0.673899	−	0.738823i	$-0.735382\pi$
$167$	−8.11209e7	−1.34780	−0.673899	+	0.738823i	$0.735382\pi$
$168$	4.82458e6	0.0785013
$169$	−5.97798e7	−0.952688
$170$	0	0
$171$	9.32026e6	0.142542
$172$	4.78088e7	0.716405
$173$	−4.26584e7	−0.626388	−0.313194	−	0.949689i	$-0.601399\pi$
$173$	−4.26584e7	−0.626388	−0.313194	+	0.949689i	$0.601399\pi$
$174$	2.33474e7	0.335983
$175$	0	0
$176$	4.84147e6	0.0669396
$177$	−4.27170e7	−0.579021
$178$	−7.32403e7	−0.973375
$179$	4.23316e7	0.551670	0.275835	−	0.961205i	$-0.411046\pi$
$179$	4.23316e7	0.551670	0.275835	+	0.961205i	$0.411046\pi$
$180$	0	0
$181$	−8.80233e7	−1.10337	−0.551687	−	0.834051i	$-0.686016\pi$
$181$	−8.80233e7	−1.10337	−0.551687	+	0.834051i	$0.686016\pi$
$182$	4.81062e6	0.0591495
$183$	−2.51881e7	−0.303820
$184$	−3.27782e6	−0.0387903
$185$	0	0
$186$	3.07642e7	0.350551
$187$	8.85791e6	0.0990571
$188$	−3.65649e7	−0.401339
$189$	6.86937e6	0.0740117
$190$	0	0
$191$	−6.24994e7	−0.649022	−0.324511	−	0.945882i	$-0.605200\pi$
$191$	−6.24994e7	−0.649022	−0.324511	+	0.945882i	$0.605200\pi$
$192$	7.07789e6	0.0721688
$193$	−8.08338e7	−0.809361	−0.404681	−	0.914458i	$-0.632617\pi$
$193$	−8.08338e7	−0.809361	−0.404681	+	0.914458i	$0.632617\pi$
$194$	8.03296e7	0.789895
$195$	0	0
$196$	−4.49115e7	−0.426051
$197$	2.35939e7	0.219871	0.109936	−	0.993939i	$-0.464936\pi$
$197$	2.35939e7	0.219871	0.109936	+	0.993939i	$0.464936\pi$
$198$	6.89342e6	0.0631113
$199$	−1.71688e8	−1.54438	−0.772190	−	0.635391i	$-0.780839\pi$
$199$	−1.71688e8	−1.54438	−0.772190	+	0.635391i	$0.780839\pi$
$200$	0	0
$201$	4.55784e7	0.395889
$202$	3.43633e7	0.293336
$203$	3.77234e7	0.316501
$204$	1.29496e7	0.106795
$205$	0	0
$206$	2.56879e7	0.204736
$207$	−4.66706e6	−0.0365719
$208$	7.05741e6	0.0543780
$209$	1.51119e7	0.114500
$210$	0	0
$211$	−1.64995e8	−1.20916	−0.604579	−	0.796545i	$-0.706659\pi$
$211$	−1.64995e8	−1.20916	−0.604579	+	0.796545i	$0.706659\pi$
$212$	9.42098e7	0.679079
$213$	7.99943e7	0.567192
$214$	−1.11152e8	−0.775299
$215$	0	0
$216$	1.00777e7	0.0680414
$217$	4.97070e7	0.330224
$218$	−4.68523e7	−0.306290
$219$	1.10128e8	0.708503
$220$	0	0
$221$	1.29122e7	0.0804685
$222$	−5.96735e7	−0.366055
$223$	−7.51583e7	−0.453847	−0.226924	−	0.973913i	$-0.572867\pi$
$223$	−7.51583e7	−0.453847	−0.226924	+	0.973913i	$0.572867\pi$
$224$	1.14360e7	0.0679841
$225$	0	0
$226$	1.01489e8	0.584844
$227$	5.23842e7	0.297242	0.148621	−	0.988894i	$-0.452517\pi$
$227$	5.23842e7	0.297242	0.148621	+	0.988894i	$0.452517\pi$
$228$	2.20925e7	0.123445
$229$	−3.27581e8	−1.80258	−0.901291	−	0.433214i	$-0.857379\pi$
$229$	−3.27581e8	−1.80258	−0.901291	+	0.433214i	$0.857379\pi$
$230$	0	0
$231$	1.11380e7	0.0594518
$232$	5.53421e7	0.290970
$233$	3.73768e8	1.93578	0.967890	−	0.251374i	$-0.0808825\pi$
$233$	3.73768e8	1.93578	0.967890	+	0.251374i	$0.0808825\pi$
$234$	1.00485e7	0.0512681
$235$	0	0
$236$	−1.01255e8	−0.501447
$237$	−1.52155e8	−0.742448
$238$	2.09232e7	0.100603
$239$	2.51621e8	1.19221	0.596106	−	0.802905i	$-0.296714\pi$
$239$	2.51621e8	1.19221	0.596106	+	0.802905i	$0.296714\pi$
$240$	0	0
$241$	−1.96628e7	−0.0904870	−0.0452435	−	0.998976i	$-0.514406\pi$
$241$	−1.96628e7	−0.0904870	−0.0452435	+	0.998976i	$0.514406\pi$
$242$	−1.44720e8	−0.656411
$243$	1.43489e7	0.0641500
$244$	−5.97052e7	−0.263116
$245$	0	0
$246$	1.13416e8	0.485735
$247$	2.20286e7	0.0930136
$248$	7.29226e7	0.303586
$249$	8.42486e7	0.345832
$250$	0	0
$251$	1.47941e8	0.590515	0.295258	−	0.955418i	$-0.404595\pi$
$251$	1.47941e8	0.590515	0.295258	+	0.955418i	$0.404595\pi$
$252$	1.62829e7	0.0640960
$253$	−7.56716e6	−0.0293773
$254$	−7.39629e6	−0.0283202
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	1.64947e8	0.606146	0.303073	−	0.952967i	$-0.401987\pi$
$257$	1.64947e8	0.606146	0.303073	+	0.952967i	$0.401987\pi$
$258$	1.61355e8	0.584942
$259$	−9.64168e7	−0.344829
$260$	0	0
$261$	7.87976e7	0.274329
$262$	−1.89206e8	−0.649952
$263$	4.79094e8	1.62396	0.811980	−	0.583685i	$-0.198390\pi$
$263$	4.79094e8	1.62396	0.811980	+	0.583685i	$0.198390\pi$
$264$	1.63400e7	0.0546560
$265$	0	0
$266$	3.56957e7	0.116287
$267$	−2.47186e8	−0.794757
$268$	1.08038e8	0.342850
$269$	6.86987e7	0.215187	0.107593	−	0.994195i	$-0.465686\pi$
$269$	6.86987e7	0.215187	0.107593	+	0.994195i	$0.465686\pi$
$270$	0	0
$271$	4.05103e8	1.23644	0.618220	−	0.786005i	$-0.287854\pi$
$271$	4.05103e8	1.23644	0.618220	+	0.786005i	$0.287854\pi$
$272$	3.06954e7	0.0924874
$273$	1.62358e7	0.0482953
$274$	−3.39563e8	−0.997224
$275$	0	0
$276$	−1.10627e7	−0.0316722
$277$	4.88012e8	1.37959	0.689797	−	0.724003i	$-0.257700\pi$
$277$	4.88012e8	1.37959	0.689797	+	0.724003i	$0.257700\pi$
$278$	8.74264e7	0.244054
$279$	1.03829e8	0.286223
$280$	0	0
$281$	4.60265e8	1.23747	0.618736	−	0.785599i	$-0.287645\pi$
$281$	4.60265e8	1.23747	0.618736	+	0.785599i	$0.287645\pi$
$282$	−1.23406e8	−0.327692
$283$	−1.60766e8	−0.421639	−0.210819	−	0.977525i	$-0.567613\pi$
$283$	−1.60766e8	−0.421639	−0.210819	+	0.977525i	$0.567613\pi$
$284$	1.89616e8	0.491203
$285$	0	0
$286$	1.62927e7	0.0411824
$287$	1.83250e8	0.457570
$288$	2.38879e7	0.0589256
$289$	−3.54179e8	−0.863137
$290$	0	0
$291$	2.71112e8	0.644947
$292$	2.61043e8	0.613581
$293$	3.13326e8	0.727713	0.363856	−	0.931455i	$-0.381460\pi$
$293$	3.13326e8	0.727713	0.363856	+	0.931455i	$0.381460\pi$
$294$	−1.51576e8	−0.347869
$295$	0	0
$296$	−1.41448e8	−0.317013
$297$	2.32653e7	0.0515301
$298$	6.39338e7	0.139950
$299$	−1.10306e7	−0.0238645
$300$	0	0
$301$	2.60708e8	0.551024
$302$	−4.92537e8	−1.02900
$303$	1.15976e8	0.239508
$304$	5.23674e7	0.106906
$305$	0	0
$306$	4.37050e7	0.0871979
$307$	4.17748e8	0.824005	0.412003	−	0.911183i	$-0.364829\pi$
$307$	4.17748e8	0.824005	0.412003	+	0.911183i	$0.364829\pi$
$308$	2.64012e7	0.0514867
$309$	8.66967e7	0.167166
$310$	0	0
$311$	2.39978e8	0.452388	0.226194	−	0.974082i	$-0.427372\pi$
$311$	2.39978e8	0.452388	0.226194	+	0.974082i	$0.427372\pi$
$312$	2.38188e7	0.0443995
$313$	−3.80063e8	−0.700567	−0.350284	−	0.936644i	$-0.613915\pi$
$313$	−3.80063e8	−0.700567	−0.350284	+	0.936644i	$0.613915\pi$
$314$	574472.	0.00104716
$315$	0	0
$316$	−3.60663e8	−0.642979
$317$	4.53952e8	0.800392	0.400196	−	0.916430i	$-0.368942\pi$
$317$	4.53952e8	0.800392	0.400196	+	0.916430i	$0.368942\pi$
$318$	3.17958e8	0.554466
$319$	1.27762e8	0.220361
$320$	0	0
$321$	−3.75138e8	−0.633029
$322$	−1.78744e7	−0.0298357
$323$	9.58108e7	0.158200
$324$	3.40122e7	0.0555556
$325$	0	0
$326$	−7.47669e8	−1.19522
$327$	−1.58126e8	−0.250085
$328$	2.68837e8	0.420659
$329$	−1.99393e8	−0.308691
$330$	0	0
$331$	−1.16310e8	−0.176286	−0.0881429	−	0.996108i	$-0.528093\pi$
$331$	−1.16310e8	−0.176286	−0.0881429	+	0.996108i	$0.528093\pi$
$332$	1.99700e8	0.299499
$333$	−2.01398e8	−0.298882
$334$	−6.48967e8	−0.953037
$335$	0	0
$336$	3.85966e7	0.0555088
$337$	1.21556e8	0.173010	0.0865052	−	0.996251i	$-0.472430\pi$
$337$	1.21556e8	0.173010	0.0865052	+	0.996251i	$0.472430\pi$
$338$	−4.78238e8	−0.673652
$339$	3.42526e8	0.477523
$340$	0	0
$341$	1.68349e8	0.229916
$342$	7.45621e7	0.100792
$343$	−5.32324e8	−0.712274
$344$	3.82471e8	0.506575
$345$	0	0
$346$	−3.41267e8	−0.442923
$347$	−5.84598e8	−0.751111	−0.375555	−	0.926800i	$-0.622548\pi$
$347$	−5.84598e8	−0.751111	−0.375555	+	0.926800i	$0.622548\pi$
$348$	1.86780e8	0.237576
$349$	−1.97133e8	−0.248239	−0.124119	−	0.992267i	$-0.539611\pi$
$349$	−1.97133e8	−0.248239	−0.124119	+	0.992267i	$0.539611\pi$
$350$	0	0
$351$	3.39138e7	0.0418602
$352$	3.87318e7	0.0473334
$353$	−1.07669e9	−1.30281	−0.651403	−	0.758732i	$-0.725819\pi$
$353$	−1.07669e9	−1.30281	−0.651403	+	0.758732i	$0.725819\pi$
$354$	−3.41736e8	−0.409429
$355$	0	0
$356$	−5.85923e8	−0.688280
$357$	7.06160e7	0.0821418
$358$	3.38653e8	0.390090
$359$	−5.53446e8	−0.631313	−0.315657	−	0.948874i	$-0.602225\pi$
$359$	−5.53446e8	−0.631313	−0.315657	+	0.948874i	$0.602225\pi$
$360$	0	0
$361$	−7.30416e8	−0.817137
$362$	−7.04186e8	−0.780203
$363$	−4.88431e8	−0.535957
$364$	3.84849e7	0.0418250
$365$	0	0
$366$	−2.01505e8	−0.214833
$367$	−7.00931e8	−0.740191	−0.370096	−	0.928994i	$-0.620675\pi$
$367$	−7.00931e8	−0.740191	−0.370096	+	0.928994i	$0.620675\pi$
$368$	−2.62226e7	−0.0274289
$369$	3.82777e8	0.396601
$370$	0	0
$371$	5.13738e8	0.522315
$372$	2.46114e8	0.247877
$373$	−1.57875e9	−1.57519	−0.787593	−	0.616196i	$-0.788673\pi$
$373$	−1.57875e9	−1.57519	−0.787593	+	0.616196i	$0.788673\pi$
$374$	7.08633e7	0.0700439
$375$	0	0
$376$	−2.92519e8	−0.283790
$377$	1.86239e8	0.179009
$378$	5.49549e7	0.0523342
$379$	1.03913e9	0.980467	0.490233	−	0.871591i	$-0.336912\pi$
$379$	1.03913e9	0.980467	0.490233	+	0.871591i	$0.336912\pi$
$380$	0	0
$381$	−2.49625e7	−0.0231233
$382$	−4.99996e8	−0.458928
$383$	5.40835e7	0.0491891	0.0245946	−	0.999698i	$-0.492171\pi$
$383$	5.40835e7	0.0491891	0.0245946	+	0.999698i	$0.492171\pi$
$384$	5.66231e7	0.0510310
$385$	0	0
$386$	−6.46670e8	−0.572305
$387$	5.44572e8	0.477603
$388$	6.42637e8	0.558540
$389$	−8.84370e8	−0.761747	−0.380873	−	0.924627i	$-0.624377\pi$
$389$	−8.84370e8	−0.761747	−0.380873	+	0.924627i	$0.624377\pi$
$390$	0	0
$391$	−4.79766e7	−0.0405892
$392$	−3.59292e8	−0.301263
$393$	−6.38571e8	−0.530683
$394$	1.88751e8	0.155472
$395$	0	0
$396$	5.51474e7	0.0446264
$397$	−7.12864e8	−0.571794	−0.285897	−	0.958260i	$-0.592292\pi$
$397$	−7.12864e8	−0.571794	−0.285897	+	0.958260i	$0.592292\pi$
$398$	−1.37350e9	−1.09204
$399$	1.20473e8	0.0949478
$400$	0	0
$401$	8.47158e8	0.656084	0.328042	−	0.944663i	$-0.393611\pi$
$401$	8.47158e8	0.656084	0.328042	+	0.944663i	$0.393611\pi$
$402$	3.64627e8	0.279935
$403$	2.45402e8	0.186771
$404$	2.74906e8	0.207420
$405$	0	0
$406$	3.01787e8	0.223800
$407$	−3.26546e8	−0.240085
$408$	1.03597e8	0.0755156
$409$	1.83029e9	1.32278	0.661391	−	0.750041i	$-0.269966\pi$
$409$	1.83029e9	1.32278	0.661391	+	0.750041i	$0.269966\pi$
$410$	0	0
$411$	−1.14602e9	−0.814230
$412$	2.05503e8	0.144770
$413$	−5.52156e8	−0.385689
$414$	−3.73365e7	−0.0258602
$415$	0	0
$416$	5.64593e7	0.0384511
$417$	2.95064e8	0.199269
$418$	1.20895e8	0.0809639
$419$	1.72365e9	1.14472	0.572362	−	0.820001i	$-0.306027\pi$
$419$	1.72365e9	1.14472	0.572362	+	0.820001i	$0.306027\pi$
$420$	0	0
$421$	−1.08475e9	−0.708501	−0.354251	−	0.935151i	$-0.615264\pi$
$421$	−1.08475e9	−0.708501	−0.354251	+	0.935151i	$0.615264\pi$
$422$	−1.31996e9	−0.855004
$423$	−4.16497e8	−0.267559
$424$	7.53678e8	0.480182
$425$	0	0
$426$	6.39954e8	0.401066
$427$	−3.25580e8	−0.202376
$428$	−8.89216e8	−0.548219
$429$	5.49878e7	0.0336253
$430$	0	0
$431$	1.59920e9	0.962128	0.481064	−	0.876686i	$-0.340251\pi$
$431$	1.59920e9	0.962128	0.481064	+	0.876686i	$0.340251\pi$
$432$	8.06216e7	0.0481125
$433$	−8.34311e6	−0.00493879	−0.00246939	−	0.999997i	$-0.500786\pi$
$433$	−8.34311e6	−0.00493879	−0.00246939	+	0.999997i	$0.500786\pi$
$434$	3.97656e8	0.233504
$435$	0	0
$436$	−3.74818e8	−0.216580
$437$	−8.18496e7	−0.0469171
$438$	8.81020e8	0.500987
$439$	3.42364e9	1.93136	0.965679	−	0.259738i	$-0.0836363\pi$
$439$	3.42364e9	1.93136	0.965679	+	0.259738i	$0.0836363\pi$
$440$	0	0
$441$	−5.11570e8	−0.284034
$442$	1.03297e8	0.0568998
$443$	1.28184e9	0.700520	0.350260	−	0.936653i	$-0.386093\pi$
$443$	1.28184e9	0.700520	0.350260	+	0.936653i	$0.386093\pi$
$444$	−4.77388e8	−0.258840
$445$	0	0
$446$	−6.01266e8	−0.320918
$447$	2.15777e8	0.114269
$448$	9.14883e7	0.0480720
$449$	−2.00530e9	−1.04548	−0.522742	−	0.852491i	$-0.675091\pi$
$449$	−2.00530e9	−1.04548	−0.522742	+	0.852491i	$0.675091\pi$
$450$	0	0
$451$	6.20635e8	0.318580
$452$	8.11913e8	0.413547
$453$	−1.66231e9	−0.840173
$454$	4.19074e8	0.210182
$455$	0	0
$456$	1.76740e8	0.0872886
$457$	−2.75156e8	−0.134857	−0.0674284	−	0.997724i	$-0.521479\pi$
$457$	−2.75156e8	−0.134857	−0.0674284	+	0.997724i	$0.521479\pi$
$458$	−2.62065e9	−1.27462
$459$	1.47504e8	0.0711968
$460$	0	0
$461$	−1.50216e9	−0.714105	−0.357052	−	0.934084i	$-0.616218\pi$
$461$	−1.50216e9	−0.714105	−0.357052	+	0.934084i	$0.616218\pi$
$462$	8.91039e7	0.0420387
$463$	3.16499e9	1.48197	0.740984	−	0.671522i	$-0.234359\pi$
$463$	3.16499e9	1.48197	0.740984	+	0.671522i	$0.234359\pi$
$464$	4.42737e8	0.205747
$465$	0	0
$466$	2.99014e9	1.36880
$467$	6.12827e8	0.278438	0.139219	−	0.990262i	$-0.455541\pi$
$467$	6.12827e8	0.278438	0.139219	+	0.990262i	$0.455541\pi$
$468$	8.03883e7	0.0362520
$469$	5.89143e8	0.263703
$470$	0	0
$471$	1.93884e6	0.000855007 0
$472$	−8.10040e8	−0.354576
$473$	8.82969e8	0.383647
$474$	−1.21724e9	−0.524990
$475$	0	0
$476$	1.67386e8	0.0711369
$477$	1.07311e9	0.452720
$478$	2.01297e9	0.843022
$479$	−3.06649e9	−1.27487	−0.637436	−	0.770503i	$-0.720005\pi$
$479$	−3.06649e9	−1.27487	−0.637436	+	0.770503i	$0.720005\pi$
$480$	0	0
$481$	−4.76006e8	−0.195032
$482$	−1.57303e8	−0.0639840
$483$	−6.03260e7	−0.0243607
$484$	−1.15776e9	−0.464153
$485$	0	0
$486$	1.14791e8	0.0453609
$487$	−1.45331e9	−0.570172	−0.285086	−	0.958502i	$-0.592022\pi$
$487$	−1.45331e9	−0.570172	−0.285086	+	0.958502i	$0.592022\pi$
$488$	−4.77641e8	−0.186051
$489$	−2.52338e9	−0.975893
$490$	0	0
$491$	4.00759e8	0.152791	0.0763955	−	0.997078i	$-0.475659\pi$
$491$	4.00759e8	0.152791	0.0763955	+	0.997078i	$0.475659\pi$
$492$	9.07324e8	0.343467
$493$	8.10026e8	0.304463
$494$	1.76228e8	0.0657706
$495$	0	0
$496$	5.83381e8	0.214668
$497$	1.03400e9	0.377810
$498$	6.73989e8	0.244540
$499$	1.58817e9	0.572198	0.286099	−	0.958200i	$-0.407641\pi$
$499$	1.58817e9	0.572198	0.286099	+	0.958200i	$0.407641\pi$
$500$	0	0
$501$	−2.19026e9	−0.778152
$502$	1.18353e9	0.417557
$503$	4.36973e9	1.53097	0.765485	−	0.643454i	$-0.222499\pi$
$503$	4.36973e9	1.53097	0.765485	+	0.643454i	$0.222499\pi$
$504$	1.30264e8	0.0453227
$505$	0	0
$506$	−6.05373e7	−0.0207729
$507$	−1.61405e9	−0.550035
$508$	−5.91703e7	−0.0200254
$509$	−5.27342e9	−1.77248	−0.886238	−	0.463230i	$-0.846690\pi$
$509$	−5.27342e9	−1.77248	−0.886238	+	0.463230i	$0.846690\pi$
$510$	0	0
$511$	1.42350e9	0.471937
$512$	1.34218e8	0.0441942
$513$	2.51647e8	0.0822965
$514$	1.31957e9	0.428610
$515$	0	0
$516$	1.29084e9	0.413617
$517$	−6.75307e8	−0.214924
$518$	−7.71335e8	−0.243831
$519$	−1.15178e9	−0.361645
$520$	0	0
$521$	−3.18338e9	−0.986180	−0.493090	−	0.869978i	$-0.664133\pi$
$521$	−3.18338e9	−0.986180	−0.493090	+	0.869978i	$0.664133\pi$
$522$	6.30381e8	0.193980
$523$	3.43716e9	1.05062	0.525308	−	0.850912i	$-0.323950\pi$
$523$	3.43716e9	1.05062	0.525308	+	0.850912i	$0.323950\pi$
$524$	−1.51365e9	−0.459585
$525$	0	0
$526$	3.83275e9	1.14831
$527$	1.06735e9	0.317665
$528$	1.30720e8	0.0386476
$529$	−3.36384e9	−0.987962
$530$	0	0
$531$	−1.15336e9	−0.334298
$532$	2.85566e8	0.0822272
$533$	9.04699e8	0.258797
$534$	−1.97749e9	−0.561978
$535$	0	0
$536$	8.64302e8	0.242431
$537$	1.14295e9	0.318507
$538$	5.49590e8	0.152160
$539$	−8.29459e8	−0.228157
$540$	0	0
$541$	1.93045e9	0.524164	0.262082	−	0.965046i	$-0.415591\pi$
$541$	1.93045e9	0.524164	0.262082	+	0.965046i	$0.415591\pi$
$542$	3.24082e9	0.874295
$543$	−2.37663e9	−0.637033
$544$	2.45563e8	0.0653985
$545$	0	0
$546$	1.29887e8	0.0341500
$547$	−6.08023e9	−1.58842	−0.794208	−	0.607647i	$-0.792114\pi$
$547$	−6.08023e9	−1.58842	−0.794208	+	0.607647i	$0.792114\pi$
$548$	−2.71650e9	−0.705144
$549$	−6.80079e8	−0.175411
$550$	0	0
$551$	1.38193e9	0.351929
$552$	−8.85012e7	−0.0223956
$553$	−1.96674e9	−0.494549
$554$	3.90409e9	0.975520
$555$	0	0
$556$	6.99411e8	0.172572
$557$	5.30581e9	1.30094	0.650472	−	0.759530i	$-0.274571\pi$
$557$	5.30581e9	1.30094	0.650472	+	0.759530i	$0.274571\pi$
$558$	8.30634e8	0.202391
$559$	1.28710e9	0.311654
$560$	0	0
$561$	2.39164e8	0.0571906
$562$	3.68212e9	0.875025
$563$	7.98581e9	1.88599	0.942995	−	0.332806i	$-0.107995\pi$
$563$	7.98581e9	1.88599	0.942995	+	0.332806i	$0.107995\pi$
$564$	−9.87251e8	−0.231713
$565$	0	0
$566$	−1.28612e9	−0.298144
$567$	1.85473e8	0.0427307
$568$	1.51693e9	0.347333
$569$	−6.54893e9	−1.49031	−0.745156	−	0.666890i	$-0.767625\pi$
$569$	−6.54893e9	−1.49031	−0.745156	+	0.666890i	$0.767625\pi$
$570$	0	0
$571$	9.77321e7	0.0219690	0.0109845	−	0.999940i	$-0.496503\pi$
$571$	9.77321e7	0.0219690	0.0109845	+	0.999940i	$0.496503\pi$
$572$	1.30342e8	0.0291204
$573$	−1.68748e9	−0.374713
$574$	1.46600e9	0.323551
$575$	0	0
$576$	1.91103e8	0.0416667
$577$	2.63597e9	0.571249	0.285625	−	0.958342i	$-0.407799\pi$
$577$	2.63597e9	0.571249	0.285625	+	0.958342i	$0.407799\pi$
$578$	−2.83343e9	−0.610330
$579$	−2.18251e9	−0.467285
$580$	0	0
$581$	1.08899e9	0.230360
$582$	2.16890e9	0.456046
$583$	1.73994e9	0.363658
$584$	2.08834e9	0.433868
$585$	0	0
$586$	2.50661e9	0.514571
$587$	5.02492e9	1.02541	0.512703	−	0.858566i	$-0.328644\pi$
$587$	5.02492e9	1.02541	0.512703	+	0.858566i	$0.328644\pi$
$588$	−1.21261e9	−0.245980
$589$	1.82093e9	0.367189
$590$	0	0
$591$	6.37036e8	0.126943
$592$	−1.13159e9	−0.224162
$593$	2.75457e8	0.0542453	0.0271226	−	0.999632i	$-0.491366\pi$
$593$	2.75457e8	0.0542453	0.0271226	+	0.999632i	$0.491366\pi$
$594$	1.86122e8	0.0364373
$595$	0	0
$596$	5.11471e8	0.0989598
$597$	−4.63558e9	−0.891649
$598$	−8.82452e7	−0.0168747
$599$	5.39450e9	1.02555	0.512776	−	0.858523i	$-0.328617\pi$
$599$	5.39450e9	1.02555	0.512776	+	0.858523i	$0.328617\pi$
$600$	0	0
$601$	−1.54325e9	−0.289985	−0.144993	−	0.989433i	$-0.546316\pi$
$601$	−1.54325e9	−0.289985	−0.144993	+	0.989433i	$0.546316\pi$
$602$	2.08566e9	0.389633
$603$	1.23062e9	0.228566
$604$	−3.94029e9	−0.727611
$605$	0	0
$606$	9.27809e8	0.169357
$607$	−8.91297e9	−1.61757	−0.808783	−	0.588108i	$-0.799873\pi$
$607$	−8.91297e9	−1.61757	−0.808783	+	0.588108i	$0.799873\pi$
$608$	4.18939e8	0.0755941
$609$	1.01853e9	0.182732
$610$	0	0
$611$	−9.84395e8	−0.174592
$612$	3.49640e8	0.0616583
$613$	4.51765e9	0.792138	0.396069	−	0.918221i	$-0.370374\pi$
$613$	4.51765e9	0.792138	0.396069	+	0.918221i	$0.370374\pi$
$614$	3.34198e9	0.582660
$615$	0	0
$616$	2.11209e8	0.0364066
$617$	7.08126e8	0.121370	0.0606852	−	0.998157i	$-0.480671\pi$
$617$	7.08126e8	0.121370	0.0606852	+	0.998157i	$0.480671\pi$
$618$	6.93573e8	0.118204
$619$	6.51486e9	1.10405	0.552024	−	0.833828i	$-0.313856\pi$
$619$	6.51486e9	1.10405	0.552024	+	0.833828i	$0.313856\pi$
$620$	0	0
$621$	−1.26011e8	−0.0211148
$622$	1.91983e9	0.319886
$623$	−3.19511e9	−0.529392
$624$	1.90550e8	0.0313952
$625$	0	0
$626$	−3.04050e9	−0.495376
$627$	4.08020e8	0.0661067
$628$	4.59578e6	0.000740457 0
$629$	−2.07034e9	−0.331714
$630$	0	0
$631$	2.26508e9	0.358907	0.179453	−	0.983766i	$-0.442567\pi$
$631$	2.26508e9	0.358907	0.179453	+	0.983766i	$0.442567\pi$
$632$	−2.88530e9	−0.454655
$633$	−4.45487e9	−0.698108
$634$	3.63162e9	0.565963
$635$	0	0
$636$	2.54366e9	0.392067
$637$	−1.20910e9	−0.185342
$638$	1.02210e9	0.155819
$639$	2.15985e9	0.327469
$640$	0	0
$641$	−1.94542e9	−0.291750	−0.145875	−	0.989303i	$-0.546600\pi$
$641$	−1.94542e9	−0.291750	−0.145875	+	0.989303i	$0.546600\pi$
$642$	−3.00110e9	−0.447619
$643$	−2.51293e9	−0.372771	−0.186386	−	0.982477i	$-0.559677\pi$
$643$	−2.51293e9	−0.372771	−0.186386	+	0.982477i	$0.559677\pi$
$644$	−1.42995e8	−0.0210970
$645$	0	0
$646$	7.66486e8	0.111864
$647$	−1.12598e10	−1.63443	−0.817213	−	0.576336i	$-0.804482\pi$
$647$	−1.12598e10	−1.63443	−0.817213	+	0.576336i	$0.804482\pi$
$648$	2.72098e8	0.0392837
$649$	−1.87005e9	−0.268533
$650$	0	0
$651$	1.34209e9	0.190655
$652$	−5.98135e9	−0.845148
$653$	−8.35560e8	−0.117431	−0.0587154	−	0.998275i	$-0.518700\pi$
$653$	−8.35560e8	−0.117431	−0.0587154	+	0.998275i	$0.518700\pi$
$654$	−1.26501e9	−0.176837
$655$	0	0
$656$	2.15069e9	0.297451
$657$	2.97344e9	0.409054
$658$	−1.59514e9	−0.218277
$659$	8.86838e9	1.20711	0.603553	−	0.797323i	$-0.293751\pi$
$659$	8.86838e9	1.20711	0.603553	+	0.797323i	$0.293751\pi$
$660$	0	0
$661$	−9.48686e9	−1.27767	−0.638833	−	0.769345i	$-0.720583\pi$
$661$	−9.48686e9	−1.27767	−0.638833	+	0.769345i	$0.720583\pi$
$662$	−9.30476e8	−0.124653
$663$	3.48628e8	0.0464585
$664$	1.59760e9	0.211778
$665$	0	0
$666$	−1.61118e9	−0.211342
$667$	−6.91992e8	−0.0902944
$668$	−5.19174e9	−0.673899
$669$	−2.02927e9	−0.262029
$670$	0	0
$671$	−1.10268e9	−0.140903
$672$	3.08773e8	0.0392506
$673$	−1.02506e10	−1.29628	−0.648138	−	0.761523i	$-0.724452\pi$
$673$	−1.02506e10	−1.29628	−0.648138	+	0.761523i	$0.724452\pi$
$674$	9.72449e8	0.122337
$675$	0	0
$676$	−3.82591e9	−0.476344
$677$	1.85912e9	0.230275	0.115137	−	0.993350i	$-0.463269\pi$
$677$	1.85912e9	0.230275	0.115137	+	0.993350i	$0.463269\pi$
$678$	2.74021e9	0.337660
$679$	3.50438e9	0.429602
$680$	0	0
$681$	1.41437e9	0.171613
$682$	1.34679e9	0.162575
$683$	−5.56457e9	−0.668281	−0.334141	−	0.942523i	$-0.608446\pi$
$683$	−5.56457e9	−0.668281	−0.334141	+	0.942523i	$0.608446\pi$
$684$	5.96497e8	0.0712708
$685$	0	0
$686$	−4.25860e9	−0.503654
$687$	−8.84470e9	−1.04072
$688$	3.05977e9	0.358202
$689$	2.53630e9	0.295416
$690$	0	0
$691$	−8.11409e9	−0.935550	−0.467775	−	0.883848i	$-0.654944\pi$
$691$	−8.11409e9	−0.935550	−0.467775	+	0.883848i	$0.654944\pi$
$692$	−2.73014e9	−0.313194
$693$	3.00726e8	0.0343245
$694$	−4.67678e9	−0.531116
$695$	0	0
$696$	1.49424e9	0.167991
$697$	3.93489e9	0.440167
$698$	−1.57706e9	−0.175531
$699$	1.00917e10	1.11762
$700$	0	0
$701$	−3.84942e9	−0.422067	−0.211034	−	0.977479i	$-0.567683\pi$
$701$	−3.84942e9	−0.422067	−0.211034	+	0.977479i	$0.567683\pi$
$702$	2.71310e8	0.0295997
$703$	−3.53206e9	−0.383429
$704$	3.09854e8	0.0334698
$705$	0	0
$706$	−8.61354e9	−0.921224
$707$	1.49910e9	0.159537
$708$	−2.73389e9	−0.289510
$709$	−1.00515e10	−1.05918	−0.529590	−	0.848254i	$-0.677654\pi$
$709$	−1.00515e10	−1.05918	−0.529590	+	0.848254i	$0.677654\pi$
$710$	0	0
$711$	−4.10818e9	−0.428653
$712$	−4.68738e9	−0.486688
$713$	−9.11818e8	−0.0942095
$714$	5.64928e8	0.0580830
$715$	0	0
$716$	2.70923e9	0.275835
$717$	6.79376e9	0.688324
$718$	−4.42757e9	−0.446406
$719$	1.00523e10	1.00859	0.504295	−	0.863531i	$-0.331752\pi$
$719$	1.00523e10	1.00859	0.504295	+	0.863531i	$0.331752\pi$
$720$	0	0
$721$	1.12063e9	0.111350
$722$	−5.84332e9	−0.577803
$723$	−5.30896e8	−0.0522427
$724$	−5.63349e9	−0.551687
$725$	0	0
$726$	−3.90745e9	−0.378979
$727$	5.93791e9	0.573143	0.286571	−	0.958059i	$-0.407484\pi$
$727$	5.93791e9	0.573143	0.286571	+	0.958059i	$0.407484\pi$
$728$	3.07879e8	0.0295747
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	5.59812e9	0.530067
$732$	−1.61204e9	−0.151910
$733$	−1.65942e10	−1.55630	−0.778148	−	0.628081i	$-0.783841\pi$
$733$	−1.65942e10	−1.55630	−0.778148	+	0.628081i	$0.783841\pi$
$734$	−5.60745e9	−0.523394
$735$	0	0
$736$	−2.09781e8	−0.0193952
$737$	1.99532e9	0.183602
$738$	3.06222e9	0.280439
$739$	1.80688e9	0.164693	0.0823463	−	0.996604i	$-0.473759\pi$
$739$	1.80688e9	0.164693	0.0823463	+	0.996604i	$0.473759\pi$
$740$	0	0
$741$	5.94771e8	0.0537015
$742$	4.10990e9	0.369333
$743$	−7.03476e8	−0.0629199	−0.0314600	−	0.999505i	$-0.510016\pi$
$743$	−7.03476e8	−0.0629199	−0.0314600	+	0.999505i	$0.510016\pi$
$744$	1.96891e9	0.175275
$745$	0	0
$746$	−1.26300e10	−1.11382
$747$	2.27471e9	0.199666
$748$	5.66906e8	0.0495285
$749$	−4.84900e9	−0.421664
$750$	0	0
$751$	4.57834e9	0.394428	0.197214	−	0.980360i	$-0.436811\pi$
$751$	4.57834e9	0.394428	0.197214	+	0.980360i	$0.436811\pi$
$752$	−2.34015e9	−0.200670
$753$	3.99441e9	0.340934
$754$	1.48991e9	0.126579
$755$	0	0
$756$	4.39639e8	0.0370059
$757$	2.20398e10	1.84659	0.923296	−	0.384088i	$-0.125484\pi$
$757$	2.20398e10	1.84659	0.923296	+	0.384088i	$0.125484\pi$
$758$	8.31304e9	0.693295
$759$	−2.04313e8	−0.0169610
$760$	0	0
$761$	1.23897e10	1.01910	0.509549	−	0.860442i	$-0.329812\pi$
$761$	1.23897e10	1.01910	0.509549	+	0.860442i	$0.329812\pi$
$762$	−1.99700e8	−0.0163507
$763$	−2.04393e9	−0.166583
$764$	−3.99996e9	−0.324511
$765$	0	0
$766$	4.32668e8	0.0347820
$767$	−2.72598e9	−0.218141
$768$	4.52985e8	0.0360844
$769$	−1.19931e10	−0.951020	−0.475510	−	0.879710i	$-0.657736\pi$
$769$	−1.19931e10	−0.951020	−0.475510	+	0.879710i	$0.657736\pi$
$770$	0	0
$771$	4.45356e9	0.349959
$772$	−5.17336e9	−0.404681
$773$	−1.06439e10	−0.828842	−0.414421	−	0.910085i	$-0.636016\pi$
$773$	−1.06439e10	−0.828842	−0.414421	+	0.910085i	$0.636016\pi$
$774$	4.35658e9	0.337717
$775$	0	0
$776$	5.14109e9	0.394948
$777$	−2.60325e9	−0.199087
$778$	−7.07496e9	−0.538636
$779$	6.71305e9	0.508790
$780$	0	0
$781$	3.50197e9	0.263047
$782$	−3.83813e8	−0.0287009
$783$	2.12754e9	0.158384
$784$	−2.87434e9	−0.213025
$785$	0	0
$786$	−5.10857e9	−0.375250
$787$	1.55406e10	1.13646	0.568232	−	0.822868i	$-0.307627\pi$
$787$	1.55406e10	1.13646	0.568232	+	0.822868i	$0.307627\pi$
$788$	1.51001e9	0.109936
$789$	1.29355e10	0.937594
$790$	0	0
$791$	4.42747e9	0.318081
$792$	4.41179e8	0.0315556
$793$	−1.60737e9	−0.114462
$794$	−5.70291e9	−0.404320
$795$	0	0
$796$	−1.09880e10	−0.772190
$797$	−1.50320e10	−1.05175	−0.525876	−	0.850561i	$-0.676262\pi$
$797$	−1.50320e10	−1.05175	−0.525876	+	0.850561i	$0.676262\pi$
$798$	9.63784e8	0.0671382
$799$	−4.28152e9	−0.296950
$800$	0	0
$801$	−6.67402e9	−0.458853
$802$	6.77727e9	0.463921
$803$	4.82114e9	0.328583
$804$	2.91702e9	0.197944
$805$	0	0
$806$	1.96321e9	0.132067
$807$	1.85487e9	0.124238
$808$	2.19925e9	0.146668
$809$	−1.06635e10	−0.708076	−0.354038	−	0.935231i	$-0.615192\pi$
$809$	−1.06635e10	−0.708076	−0.354038	+	0.935231i	$0.615192\pi$
$810$	0	0
$811$	1.59006e10	1.04675	0.523373	−	0.852104i	$-0.324673\pi$
$811$	1.59006e10	1.04675	0.523373	+	0.852104i	$0.324673\pi$
$812$	2.41430e9	0.158250
$813$	1.09378e10	0.713859
$814$	−2.61237e9	−0.169766
$815$	0	0
$816$	8.28776e8	0.0533976
$817$	9.55056e9	0.612705
$818$	1.46423e10	0.935348
$819$	4.38367e8	0.0278833
$820$	0	0
$821$	1.59766e10	1.00759	0.503793	−	0.863824i	$-0.331937\pi$
$821$	1.59766e10	1.00759	0.503793	+	0.863824i	$0.331937\pi$
$822$	−9.16819e9	−0.575748
$823$	−1.94914e10	−1.21883	−0.609414	−	0.792852i	$-0.708595\pi$
$823$	−1.94914e10	−1.21883	−0.609414	+	0.792852i	$0.708595\pi$
$824$	1.64403e9	0.102368
$825$	0	0
$826$	−4.41725e9	−0.272723
$827$	−7.77419e9	−0.477954	−0.238977	−	0.971025i	$-0.576812\pi$
$827$	−7.77419e9	−0.477954	−0.238977	+	0.971025i	$0.576812\pi$
$828$	−2.98692e8	−0.0182859
$829$	−2.64927e10	−1.61505	−0.807523	−	0.589837i	$-0.799192\pi$
$829$	−2.64927e10	−1.61505	−0.807523	+	0.589837i	$0.799192\pi$
$830$	0	0
$831$	1.31763e10	0.796508
$832$	4.51674e8	0.0271890
$833$	−5.25885e9	−0.315234
$834$	2.36051e9	0.140905
$835$	0	0
$836$	9.67160e8	0.0572501
$837$	2.80339e9	0.165251
$838$	1.37892e10	0.809442
$839$	9.97382e9	0.583035	0.291518	−	0.956565i	$-0.405840\pi$
$839$	9.97382e9	0.583035	0.291518	+	0.956565i	$0.405840\pi$
$840$	0	0
$841$	−5.56643e9	−0.322694
$842$	−8.67796e9	−0.500986
$843$	1.24271e10	0.714455
$844$	−1.05597e10	−0.604579
$845$	0	0
$846$	−3.33197e9	−0.189193
$847$	−6.31343e9	−0.357004
$848$	6.02943e9	0.339540
$849$	−4.34067e9	−0.243433
$850$	0	0
$851$	1.76865e9	0.0983761
$852$	5.11964e9	0.283596
$853$	2.01214e10	1.11003	0.555017	−	0.831839i	$-0.312712\pi$
$853$	2.01214e10	1.11003	0.555017	+	0.831839i	$0.312712\pi$
$854$	−2.60464e9	−0.143102
$855$	0	0
$856$	−7.11373e9	−0.387649
$857$	−2.35382e9	−0.127744	−0.0638720	−	0.997958i	$-0.520345\pi$
$857$	−2.35382e9	−0.127744	−0.0638720	+	0.997958i	$0.520345\pi$
$858$	4.39903e8	0.0237767
$859$	3.35685e9	0.180699	0.0903494	−	0.995910i	$-0.471202\pi$
$859$	3.35685e9	0.180699	0.0903494	+	0.995910i	$0.471202\pi$
$860$	0	0
$861$	4.94775e9	0.264178
$862$	1.27936e10	0.680327
$863$	2.25282e10	1.19313	0.596567	−	0.802563i	$-0.296531\pi$
$863$	2.25282e10	1.19313	0.596567	+	0.802563i	$0.296531\pi$
$864$	6.44973e8	0.0340207
$865$	0	0
$866$	−6.67449e7	−0.00349225
$867$	−9.56282e9	−0.498333
$868$	3.18125e9	0.165112
$869$	−6.66100e9	−0.344326
$870$	0	0
$871$	2.90858e9	0.149148
$872$	−2.99855e9	−0.153145
$873$	7.32003e9	0.372360
$874$	−6.54797e8	−0.0331754
$875$	0	0
$876$	7.04816e9	0.354251
$877$	7.00472e9	0.350665	0.175332	−	0.984509i	$-0.443900\pi$
$877$	7.00472e9	0.350665	0.175332	+	0.984509i	$0.443900\pi$
$878$	2.73892e10	1.36568
$879$	8.45981e9	0.420145
$880$	0	0
$881$	1.26989e10	0.625680	0.312840	−	0.949806i	$-0.398720\pi$
$881$	1.26989e10	0.625680	0.312840	+	0.949806i	$0.398720\pi$
$882$	−4.09256e9	−0.200842
$883$	3.47081e10	1.69656	0.848279	−	0.529550i	$-0.177639\pi$
$883$	3.47081e10	1.69656	0.848279	+	0.529550i	$0.177639\pi$
$884$	8.26378e8	0.0402343
$885$	0	0
$886$	1.02547e10	0.495342
$887$	−1.05377e10	−0.507005	−0.253503	−	0.967335i	$-0.581583\pi$
$887$	−1.05377e10	−0.507005	−0.253503	+	0.967335i	$0.581583\pi$
$888$	−3.81910e9	−0.183027
$889$	−3.22663e8	−0.0154026
$890$	0	0
$891$	6.28163e8	0.0297509
$892$	−4.81013e9	−0.226924
$893$	−7.30440e9	−0.343245
$894$	1.72621e9	0.0808003
$895$	0	0
$896$	7.31906e8	0.0339920
$897$	−2.97827e8	−0.0137782
$898$	−1.60424e10	−0.739269
$899$	1.53949e10	0.706674
$900$	0	0
$901$	1.10314e10	0.502450
$902$	4.96508e9	0.225270
$903$	7.03910e9	0.318134
$904$	6.49531e9	0.292422
$905$	0	0
$906$	−1.32985e10	−0.594092
$907$	2.00148e10	0.890689	0.445344	−	0.895359i	$-0.353081\pi$
$907$	2.00148e10	0.890689	0.445344	+	0.895359i	$0.353081\pi$
$908$	3.35259e9	0.148621
$909$	3.13136e9	0.138280
$910$	0	0
$911$	5.11814e9	0.224284	0.112142	−	0.993692i	$-0.464229\pi$
$911$	5.11814e9	0.224284	0.112142	+	0.993692i	$0.464229\pi$
$912$	1.41392e9	0.0617224
$913$	3.68822e9	0.160387
$914$	−2.20125e9	−0.0953582
$915$	0	0
$916$	−2.09652e10	−0.901291
$917$	−8.25413e9	−0.353491
$918$	1.18004e9	0.0503438
$919$	−3.66676e10	−1.55840	−0.779198	−	0.626778i	$-0.784373\pi$
$919$	−3.66676e10	−1.55840	−0.779198	+	0.626778i	$0.784373\pi$
$920$	0	0
$921$	1.12792e10	0.475740
$922$	−1.20172e10	−0.504948
$923$	5.10482e9	0.213685
$924$	7.12831e8	0.0297259
$925$	0	0
$926$	2.53199e10	1.04791
$927$	2.34081e9	0.0965133
$928$	3.54189e9	0.145485
$929$	3.31004e10	1.35450	0.677249	−	0.735754i	$-0.263172\pi$
$929$	3.31004e10	1.35450	0.677249	+	0.735754i	$0.263172\pi$
$930$	0	0
$931$	−8.97177e9	−0.364380
$932$	2.39211e10	0.967890
$933$	6.47941e9	0.261186
$934$	4.90262e9	0.196886
$935$	0	0
$936$	6.43106e8	0.0256341
$937$	1.45346e10	0.577184	0.288592	−	0.957452i	$-0.406813\pi$
$937$	1.45346e10	0.577184	0.288592	+	0.957452i	$0.406813\pi$
$938$	4.71314e9	0.186467
$939$	−1.02617e10	−0.404473
$940$	0	0
$941$	−3.49536e10	−1.36750	−0.683751	−	0.729715i	$-0.739653\pi$
$941$	−3.49536e10	−1.36750	−0.683751	+	0.729715i	$0.739653\pi$
$942$	1.55107e7	0.000604581 0
$943$	−3.36151e9	−0.130540
$944$	−6.48032e9	−0.250723
$945$	0	0
$946$	7.06375e9	0.271279
$947$	1.76231e10	0.674308	0.337154	−	0.941450i	$-0.390536\pi$
$947$	1.76231e10	0.674308	0.337154	+	0.941450i	$0.390536\pi$
$948$	−9.73790e9	−0.371224
$949$	7.02777e9	0.266923
$950$	0	0
$951$	1.22567e10	0.462107
$952$	1.33909e9	0.0503014
$953$	3.58785e8	0.0134279	0.00671396	−	0.999977i	$-0.497863\pi$
$953$	3.58785e8	0.0134279	0.00671396	+	0.999977i	$0.497863\pi$
$954$	8.58487e9	0.320121
$955$	0	0
$956$	1.61037e10	0.596106
$957$	3.44958e9	0.127226
$958$	−2.45319e10	−0.901471
$959$	−1.48134e10	−0.542363
$960$	0	0
$961$	−7.22716e9	−0.262685
$962$	−3.80805e9	−0.137908
$963$	−1.01287e10	−0.365479
$964$	−1.25842e9	−0.0452435
$965$	0	0
$966$	−4.82608e8	−0.0172256
$967$	−3.58799e10	−1.27602	−0.638011	−	0.770027i	$-0.720243\pi$
$967$	−3.58799e10	−1.27602	−0.638011	+	0.770027i	$0.720243\pi$
$968$	−9.26210e9	−0.328206
$969$	2.58689e9	0.0913366
$970$	0	0
$971$	−3.23985e10	−1.13568	−0.567842	−	0.823137i	$-0.692222\pi$
$971$	−3.23985e10	−1.13568	−0.567842	+	0.823137i	$0.692222\pi$
$972$	9.18330e8	0.0320750
$973$	3.81398e9	0.132734
$974$	−1.16265e10	−0.403173
$975$	0	0
$976$	−3.82113e9	−0.131558
$977$	1.36456e10	0.468124	0.234062	−	0.972222i	$-0.424798\pi$
$977$	1.36456e10	0.468124	0.234062	+	0.972222i	$0.424798\pi$
$978$	−2.01871e10	−0.690061
$979$	−1.08213e10	−0.368586
$980$	0	0
$981$	−4.26941e9	−0.144387
$982$	3.20607e9	0.108040
$983$	−3.83603e10	−1.28808	−0.644042	−	0.764990i	$-0.722744\pi$
$983$	−3.83603e10	−1.28808	−0.644042	+	0.764990i	$0.722744\pi$
$984$	7.25860e9	0.242868
$985$	0	0
$986$	6.48021e9	0.215288
$987$	−5.38360e9	−0.178223
$988$	1.40983e9	0.0465068
$989$	−4.78238e9	−0.157202
$990$	0	0
$991$	2.11281e9	0.0689608	0.0344804	−	0.999405i	$-0.489022\pi$
$991$	2.11281e9	0.0689608	0.0344804	+	0.999405i	$0.489022\pi$
$992$	4.66705e9	0.151793
$993$	−3.14036e9	−0.101779
$994$	8.27200e9	0.267152
$995$	0	0
$996$	5.39191e9	0.172916
$997$	−2.51510e10	−0.803751	−0.401875	−	0.915694i	$-0.631642\pi$
$997$	−2.51510e10	−0.803751	−0.401875	+	0.915694i	$0.631642\pi$
$998$	1.27054e10	0.404605
$999$	−5.43774e9	−0.172560

Twists

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

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