LMFDB - Embedded newform 300.8.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,8,Mod(1,300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(300, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("300.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 300 = 2^{2} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	300.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:	$93.7155076452$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 60)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	300.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-27.0000 q^{3} +1408.00 q^{7} +729.000 q^{9} +O(q^{10})$

$q-27.0000 q^{3} +1408.00 q^{7} +729.000 q^{9} -4044.00 q^{11} +5890.00 q^{13} -31002.0 q^{17} -40300.0 q^{19} -38016.0 q^{21} +78912.0 q^{23} -19683.0 q^{27} -157194. q^{29} +114824. q^{31} +109188. q^{33} +471994. q^{37} -159030. q^{39} -404310. q^{41} +253852. q^{43} -437688. q^{47} +1.15892e6 q^{49} +837054. q^{51} -334926. q^{53} +1.08810e6 q^{57} +562596. q^{59} +3.24666e6 q^{61} +1.02643e6 q^{63} -3.89515e6 q^{67} -2.13062e6 q^{69} -2.34516e6 q^{71} -5.72695e6 q^{73} -5.69395e6 q^{77} -5.22201e6 q^{79} +531441. q^{81} +2.92813e6 q^{83} +4.24424e6 q^{87} -3.16023e6 q^{89} +8.29312e6 q^{91} -3.10025e6 q^{93} +1.89869e6 q^{97} -2.94808e6 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1408.00	1.55153	0.775764	−	0.631023i	$-0.217365\pi$
$7$	1408.00	1.55153	0.775764	+	0.631023i	$0.217365\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	−4044.00	−0.916087	−0.458044	−	0.888930i	$-0.651450\pi$
$11$	−4044.00	−0.916087	−0.458044	+	0.888930i	$0.651450\pi$
$12$	0	0
$13$	5890.00	0.743556	0.371778	−	0.928322i	$-0.378748\pi$
$13$	5890.00	0.743556	0.371778	+	0.928322i	$0.378748\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−31002.0	−1.53045	−0.765224	−	0.643764i	$-0.777372\pi$
$17$	−31002.0	−1.53045	−0.765224	+	0.643764i	$0.777372\pi$
$18$	0	0
$19$	−40300.0	−1.34793	−0.673965	−	0.738763i	$-0.735410\pi$
$19$	−40300.0	−1.34793	−0.673965	+	0.738763i	$0.735410\pi$
$20$	0	0
$21$	−38016.0	−0.895775
$22$	0	0
$23$	78912.0	1.35237	0.676185	−	0.736732i	$-0.263632\pi$
$23$	78912.0	1.35237	0.676185	+	0.736732i	$0.263632\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−19683.0	−0.192450
$28$	0	0
$29$	−157194.	−1.19686	−0.598429	−	0.801175i	$-0.704208\pi$
$29$	−157194.	−1.19686	−0.598429	+	0.801175i	$0.704208\pi$
$30$	0	0
$31$	114824.	0.692256	0.346128	−	0.938187i	$-0.387496\pi$
$31$	114824.	0.692256	0.346128	+	0.938187i	$0.387496\pi$
$32$	0	0
$33$	109188.	0.528903
$34$	0	0
$35$	0	0
$36$	0	0
$37$	471994.	1.53190	0.765950	−	0.642900i	$-0.222269\pi$
$37$	471994.	1.53190	0.765950	+	0.642900i	$0.222269\pi$
$38$	0	0
$39$	−159030.	−0.429292
$40$	0	0
$41$	−404310.	−0.916159	−0.458080	−	0.888911i	$-0.651463\pi$
$41$	−404310.	−0.916159	−0.458080	+	0.888911i	$0.651463\pi$
$42$	0	0
$43$	253852.	0.486901	0.243451	−	0.969913i	$-0.421721\pi$
$43$	253852.	0.486901	0.243451	+	0.969913i	$0.421721\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−437688.	−0.614925	−0.307462	−	0.951560i	$-0.599480\pi$
$47$	−437688.	−0.614925	−0.307462	+	0.951560i	$0.599480\pi$
$48$	0	0
$49$	1.15892e6	1.40724
$50$	0	0
$51$	837054.	0.883604
$52$	0	0
$53$	−334926.	−0.309018	−0.154509	−	0.987991i	$-0.549380\pi$
$53$	−334926.	−0.309018	−0.154509	+	0.987991i	$0.549380\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.08810e6	0.778228
$58$	0	0
$59$	562596.	0.356627	0.178314	−	0.983974i	$-0.442936\pi$
$59$	562596.	0.356627	0.178314	+	0.983974i	$0.442936\pi$
$60$	0	0
$61$	3.24666e6	1.83140	0.915699	−	0.401865i	$-0.131638\pi$
$61$	3.24666e6	1.83140	0.915699	+	0.401865i	$0.131638\pi$
$62$	0	0
$63$	1.02643e6	0.517176
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−3.89515e6	−1.58220	−0.791101	−	0.611685i	$-0.790492\pi$
$67$	−3.89515e6	−1.58220	−0.791101	+	0.611685i	$0.790492\pi$
$68$	0	0
$69$	−2.13062e6	−0.780791
$70$	0	0
$71$	−2.34516e6	−0.777621	−0.388811	−	0.921318i	$-0.627114\pi$
$71$	−2.34516e6	−0.777621	−0.388811	+	0.921318i	$0.627114\pi$
$72$	0	0
$73$	−5.72695e6	−1.72303	−0.861517	−	0.507729i	$-0.830485\pi$
$73$	−5.72695e6	−1.72303	−0.861517	+	0.507729i	$0.830485\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−5.69395e6	−1.42133
$78$	0	0
$79$	−5.22201e6	−1.19163	−0.595817	−	0.803120i	$-0.703172\pi$
$79$	−5.22201e6	−1.19163	−0.595817	+	0.803120i	$0.703172\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	2.92813e6	0.562105	0.281052	−	0.959692i	$-0.409317\pi$
$83$	2.92813e6	0.562105	0.281052	+	0.959692i	$0.409317\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.24424e6	0.691007
$88$	0	0
$89$	−3.16023e6	−0.475175	−0.237588	−	0.971366i	$-0.576357\pi$
$89$	−3.16023e6	−0.475175	−0.237588	+	0.971366i	$0.576357\pi$
$90$	0	0
$91$	8.29312e6	1.15365
$92$	0	0
$93$	−3.10025e6	−0.399674
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1.89869e6	0.211228	0.105614	−	0.994407i	$-0.466319\pi$
$97$	1.89869e6	0.211228	0.105614	+	0.994407i	$0.466319\pi$
$98$	0	0
$99$	−2.94808e6	−0.305362
$100$	0	0
$101$	−9.82984e6	−0.949340	−0.474670	−	0.880164i	$-0.657433\pi$
$101$	−9.82984e6	−0.949340	−0.474670	+	0.880164i	$0.657433\pi$
$102$	0	0
$103$	−5.89107e6	−0.531207	−0.265604	−	0.964082i	$-0.585571\pi$
$103$	−5.89107e6	−0.531207	−0.265604	+	0.964082i	$0.585571\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.25372e7	1.77851	0.889254	−	0.457413i	$-0.151224\pi$
$107$	2.25372e7	1.77851	0.889254	+	0.457413i	$0.151224\pi$
$108$	0	0
$109$	3.29178e6	0.243466	0.121733	−	0.992563i	$-0.461155\pi$
$109$	3.29178e6	0.243466	0.121733	+	0.992563i	$0.461155\pi$
$110$	0	0
$111$	−1.27438e7	−0.884443
$112$	0	0
$113$	−2.33746e7	−1.52394	−0.761972	−	0.647610i	$-0.775768\pi$
$113$	−2.33746e7	−1.52394	−0.761972	+	0.647610i	$0.775768\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.29381e6	0.247852
$118$	0	0
$119$	−4.36508e7	−2.37453
$120$	0	0
$121$	−3.13323e6	−0.160784
$122$	0	0
$123$	1.09164e7	0.528945
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1.65642e7	−0.717560	−0.358780	−	0.933422i	$-0.616807\pi$
$127$	−1.65642e7	−0.717560	−0.358780	+	0.933422i	$0.616807\pi$
$128$	0	0
$129$	−6.85400e6	−0.281113
$130$	0	0
$131$	3.40056e7	1.32160	0.660802	−	0.750561i	$-0.270216\pi$
$131$	3.40056e7	1.32160	0.660802	+	0.750561i	$0.270216\pi$
$132$	0	0
$133$	−5.67424e7	−2.09135
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.40863e6	−0.0468030	−0.0234015	−	0.999726i	$-0.507450\pi$
$137$	−1.40863e6	−0.0468030	−0.0234015	+	0.999726i	$0.507450\pi$
$138$	0	0
$139$	−2.99496e7	−0.945886	−0.472943	−	0.881093i	$-0.656808\pi$
$139$	−2.99496e7	−0.945886	−0.472943	+	0.881093i	$0.656808\pi$
$140$	0	0
$141$	1.18176e7	0.355027
$142$	0	0
$143$	−2.38192e7	−0.681162
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−3.12909e7	−0.812469
$148$	0	0
$149$	−3.93727e7	−0.975088	−0.487544	−	0.873099i	$-0.662107\pi$
$149$	−3.93727e7	−0.975088	−0.487544	+	0.873099i	$0.662107\pi$
$150$	0	0
$151$	−1.26282e7	−0.298486	−0.149243	−	0.988801i	$-0.547684\pi$
$151$	−1.26282e7	−0.298486	−0.149243	+	0.988801i	$0.547684\pi$
$152$	0	0
$153$	−2.26005e7	−0.510149
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−3.16107e7	−0.651906	−0.325953	−	0.945386i	$-0.605685\pi$
$157$	−3.16107e7	−0.651906	−0.325953	+	0.945386i	$0.605685\pi$
$158$	0	0
$159$	9.04300e6	0.178411
$160$	0	0
$161$	1.11108e8	2.09824
$162$	0	0
$163$	1.74658e6	0.0315887	0.0157944	−	0.999875i	$-0.494972\pi$
$163$	1.74658e6	0.0315887	0.0157944	+	0.999875i	$0.494972\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−7.41416e7	−1.23184	−0.615920	−	0.787809i	$-0.711216\pi$
$167$	−7.41416e7	−1.23184	−0.615920	+	0.787809i	$0.711216\pi$
$168$	0	0
$169$	−2.80564e7	−0.447125
$170$	0	0
$171$	−2.93787e7	−0.449310
$172$	0	0
$173$	3.28050e7	0.481703	0.240851	−	0.970562i	$-0.422573\pi$
$173$	3.28050e7	0.481703	0.240851	+	0.970562i	$0.422573\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1.51901e7	−0.205899
$178$	0	0
$179$	−8.37478e7	−1.09141	−0.545705	−	0.837977i	$-0.683738\pi$
$179$	−8.37478e7	−1.09141	−0.545705	+	0.837977i	$0.683738\pi$
$180$	0	0
$181$	9.32111e7	1.16840	0.584201	−	0.811609i	$-0.301408\pi$
$181$	9.32111e7	1.16840	0.584201	+	0.811609i	$0.301408\pi$
$182$	0	0
$183$	−8.76599e7	−1.05736
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.25372e8	1.40202
$188$	0	0
$189$	−2.77137e7	−0.298592
$190$	0	0
$191$	−4.80707e7	−0.499187	−0.249593	−	0.968351i	$-0.580297\pi$
$191$	−4.80707e7	−0.499187	−0.249593	+	0.968351i	$0.580297\pi$
$192$	0	0
$193$	−1.72226e7	−0.172444	−0.0862222	−	0.996276i	$-0.527480\pi$
$193$	−1.72226e7	−0.172444	−0.0862222	+	0.996276i	$0.527480\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.07288e7	0.286361	0.143180	−	0.989697i	$-0.454267\pi$
$197$	3.07288e7	0.286361	0.143180	+	0.989697i	$0.454267\pi$
$198$	0	0
$199$	−1.81386e8	−1.63161	−0.815807	−	0.578324i	$-0.803707\pi$
$199$	−1.81386e8	−1.63161	−0.815807	+	0.578324i	$0.803707\pi$
$200$	0	0
$201$	1.05169e8	0.913485
$202$	0	0
$203$	−2.21329e8	−1.85696
$204$	0	0
$205$	0	0
$206$	0	0
$207$	5.75268e7	0.450790
$208$	0	0
$209$	1.62973e8	1.23482
$210$	0	0
$211$	−1.08304e8	−0.793700	−0.396850	−	0.917884i	$-0.629897\pi$
$211$	−1.08304e8	−0.793700	−0.396850	+	0.917884i	$0.629897\pi$
$212$	0	0
$213$	6.33193e7	0.448960
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.61672e8	1.07405
$218$	0	0
$219$	1.54628e8	0.994794
$220$	0	0
$221$	−1.82602e8	−1.13797
$222$	0	0
$223$	2.25721e8	1.36303	0.681516	−	0.731803i	$-0.261321\pi$
$223$	2.25721e8	1.36303	0.681516	+	0.731803i	$0.261321\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2.71001e8	−1.53773	−0.768866	−	0.639410i	$-0.779179\pi$
$227$	−2.71001e8	−1.53773	−0.768866	+	0.639410i	$0.779179\pi$
$228$	0	0
$229$	−2.76697e8	−1.52258	−0.761291	−	0.648411i	$-0.775434\pi$
$229$	−2.76697e8	−1.52258	−0.761291	+	0.648411i	$0.775434\pi$
$230$	0	0
$231$	1.53737e8	0.820608
$232$	0	0
$233$	2.15356e8	1.11535	0.557675	−	0.830059i	$-0.311693\pi$
$233$	2.15356e8	1.11535	0.557675	+	0.830059i	$0.311693\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.40994e8	0.687990
$238$	0	0
$239$	−4.12012e8	−1.95217	−0.976083	−	0.217397i	$-0.930243\pi$
$239$	−4.12012e8	−1.95217	−0.976083	+	0.217397i	$0.930243\pi$
$240$	0	0
$241$	−4.24902e7	−0.195537	−0.0977686	−	0.995209i	$-0.531171\pi$
$241$	−4.24902e7	−0.195537	−0.0977686	+	0.995209i	$0.531171\pi$
$242$	0	0
$243$	−1.43489e7	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.37367e8	−1.00226
$248$	0	0
$249$	−7.90596e7	−0.324531
$250$	0	0
$251$	5.73048e7	0.228735	0.114368	−	0.993439i	$-0.463516\pi$
$251$	5.73048e7	0.228735	0.114368	+	0.993439i	$0.463516\pi$
$252$	0	0
$253$	−3.19120e8	−1.23889
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.76317e8	−0.647932	−0.323966	−	0.946069i	$-0.605016\pi$
$257$	−1.76317e8	−0.647932	−0.323966	+	0.946069i	$0.605016\pi$
$258$	0	0
$259$	6.64568e8	2.37679
$260$	0	0
$261$	−1.14594e8	−0.398953
$262$	0	0
$263$	−2.37053e8	−0.803525	−0.401763	−	0.915744i	$-0.631602\pi$
$263$	−2.37053e8	−0.803525	−0.401763	+	0.915744i	$0.631602\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	8.53262e7	0.274342
$268$	0	0
$269$	1.00666e8	0.315320	0.157660	−	0.987493i	$-0.449605\pi$
$269$	1.00666e8	0.315320	0.157660	+	0.987493i	$0.449605\pi$
$270$	0	0
$271$	2.89760e7	0.0884395	0.0442197	−	0.999022i	$-0.485920\pi$
$271$	2.89760e7	0.0884395	0.0442197	+	0.999022i	$0.485920\pi$
$272$	0	0
$273$	−2.23914e8	−0.666059
$274$	0	0
$275$	0	0
$276$	0	0
$277$	3.38055e8	0.955670	0.477835	−	0.878450i	$-0.341422\pi$
$277$	3.38055e8	0.955670	0.477835	+	0.878450i	$0.341422\pi$
$278$	0	0
$279$	8.37067e7	0.230752
$280$	0	0
$281$	−3.36322e8	−0.904240	−0.452120	−	0.891957i	$-0.649332\pi$
$281$	−3.36322e8	−0.904240	−0.452120	+	0.891957i	$0.649332\pi$
$282$	0	0
$283$	−3.31324e7	−0.0868961	−0.0434481	−	0.999056i	$-0.513834\pi$
$283$	−3.31324e7	−0.0868961	−0.0434481	+	0.999056i	$0.513834\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.69268e8	−1.42145
$288$	0	0
$289$	5.50785e8	1.34227
$290$	0	0
$291$	−5.12645e7	−0.121953
$292$	0	0
$293$	5.02960e8	1.16815	0.584073	−	0.811701i	$-0.301458\pi$
$293$	5.02960e8	1.16815	0.584073	+	0.811701i	$0.301458\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	7.95981e7	0.176301
$298$	0	0
$299$	4.64792e8	1.00556
$300$	0	0
$301$	3.57424e8	0.755441
$302$	0	0
$303$	2.65406e8	0.548102
$304$	0	0
$305$	0	0
$306$	0	0
$307$	6.19829e8	1.22261	0.611305	−	0.791395i	$-0.290645\pi$
$307$	6.19829e8	1.22261	0.611305	+	0.791395i	$0.290645\pi$
$308$	0	0
$309$	1.59059e8	0.306693
$310$	0	0
$311$	3.27308e8	0.617014	0.308507	−	0.951222i	$-0.400171\pi$
$311$	3.27308e8	0.617014	0.308507	+	0.951222i	$0.400171\pi$
$312$	0	0
$313$	−7.57552e7	−0.139639	−0.0698196	−	0.997560i	$-0.522242\pi$
$313$	−7.57552e7	−0.139639	−0.0698196	+	0.997560i	$0.522242\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.89915e8	1.56906	0.784532	−	0.620088i	$-0.212903\pi$
$317$	8.89915e8	1.56906	0.784532	+	0.620088i	$0.212903\pi$
$318$	0	0
$319$	6.35693e8	1.09643
$320$	0	0
$321$	−6.08503e8	−1.02682
$322$	0	0
$323$	1.24938e9	2.06294
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−8.88781e7	−0.140565
$328$	0	0
$329$	−6.16265e8	−0.954073
$330$	0	0
$331$	−4.75587e8	−0.720829	−0.360414	−	0.932792i	$-0.617365\pi$
$331$	−4.75587e8	−0.720829	−0.360414	+	0.932792i	$0.617365\pi$
$332$	0	0
$333$	3.44084e8	0.510633
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−7.78711e8	−1.10834	−0.554168	−	0.832405i	$-0.686963\pi$
$337$	−7.78711e8	−1.10834	−0.554168	+	0.832405i	$0.686963\pi$
$338$	0	0
$339$	6.31113e8	0.879849
$340$	0	0
$341$	−4.64348e8	−0.634167
$342$	0	0
$343$	4.72212e8	0.631841
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2.99149e8	−0.384357	−0.192179	−	0.981360i	$-0.561555\pi$
$347$	−2.99149e8	−0.384357	−0.192179	+	0.981360i	$0.561555\pi$
$348$	0	0
$349$	1.28192e9	1.61426	0.807129	−	0.590375i	$-0.201020\pi$
$349$	1.28192e9	1.61426	0.807129	+	0.590375i	$0.201020\pi$
$350$	0	0
$351$	−1.15933e8	−0.143097
$352$	0	0
$353$	1.06589e9	1.28974	0.644871	−	0.764292i	$-0.276911\pi$
$353$	1.06589e9	1.28974	0.644871	+	0.764292i	$0.276911\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.17857e9	1.37094
$358$	0	0
$359$	−7.78278e8	−0.887777	−0.443889	−	0.896082i	$-0.646401\pi$
$359$	−7.78278e8	−0.887777	−0.443889	+	0.896082i	$0.646401\pi$
$360$	0	0
$361$	7.30218e8	0.816916
$362$	0	0
$363$	8.45973e7	0.0928290
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−7.66622e8	−0.809562	−0.404781	−	0.914414i	$-0.632652\pi$
$367$	−7.66622e8	−0.809562	−0.404781	+	0.914414i	$0.632652\pi$
$368$	0	0
$369$	−2.94742e8	−0.305386
$370$	0	0
$371$	−4.71576e8	−0.479449
$372$	0	0
$373$	7.65361e8	0.763634	0.381817	−	0.924238i	$-0.375298\pi$
$373$	7.65361e8	0.763634	0.381817	+	0.924238i	$0.375298\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−9.25873e8	−0.889931
$378$	0	0
$379$	−8.08441e8	−0.762800	−0.381400	−	0.924410i	$-0.624558\pi$
$379$	−8.08441e8	−0.762800	−0.381400	+	0.924410i	$0.624558\pi$
$380$	0	0
$381$	4.47234e8	0.414284
$382$	0	0
$383$	2.42686e8	0.220724	0.110362	−	0.993891i	$-0.464799\pi$
$383$	2.42686e8	0.220724	0.110362	+	0.993891i	$0.464799\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.85058e8	0.162300
$388$	0	0
$389$	−1.22399e9	−1.05428	−0.527138	−	0.849780i	$-0.676735\pi$
$389$	−1.22399e9	−1.05428	−0.527138	+	0.849780i	$0.676735\pi$
$390$	0	0
$391$	−2.44643e9	−2.06973
$392$	0	0
$393$	−9.18152e8	−0.763028
$394$	0	0
$395$	0	0
$396$	0	0
$397$	1.14698e8	0.0920002	0.0460001	−	0.998941i	$-0.485353\pi$
$397$	1.14698e8	0.0920002	0.0460001	+	0.998941i	$0.485353\pi$
$398$	0	0
$399$	1.53204e9	1.20744
$400$	0	0
$401$	2.38822e9	1.84956	0.924780	−	0.380502i	$-0.124249\pi$
$401$	2.38822e9	1.84956	0.924780	+	0.380502i	$0.124249\pi$
$402$	0	0
$403$	6.76313e8	0.514731
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.90874e9	−1.40335
$408$	0	0
$409$	2.73427e9	1.97610	0.988051	−	0.154130i	$-0.0492575\pi$
$409$	2.73427e9	1.97610	0.988051	+	0.154130i	$0.0492575\pi$
$410$	0	0
$411$	3.80329e7	0.0270217
$412$	0	0
$413$	7.92135e8	0.553317
$414$	0	0
$415$	0	0
$416$	0	0
$417$	8.08638e8	0.546107
$418$	0	0
$419$	2.13873e9	1.42039	0.710193	−	0.704007i	$-0.248607\pi$
$419$	2.13873e9	1.42039	0.710193	+	0.704007i	$0.248607\pi$
$420$	0	0
$421$	1.46225e9	0.955071	0.477535	−	0.878613i	$-0.341530\pi$
$421$	1.46225e9	0.955071	0.477535	+	0.878613i	$0.341530\pi$
$422$	0	0
$423$	−3.19075e8	−0.204975
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.57130e9	2.84146
$428$	0	0
$429$	6.43117e8	0.393269
$430$	0	0
$431$	−2.29783e9	−1.38245	−0.691223	−	0.722641i	$-0.742928\pi$
$431$	−2.29783e9	−1.38245	−0.691223	+	0.722641i	$0.742928\pi$
$432$	0	0
$433$	−2.51171e9	−1.48683	−0.743415	−	0.668830i	$-0.766795\pi$
$433$	−2.51171e9	−1.48683	−0.743415	+	0.668830i	$0.766795\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.18015e9	−1.82290
$438$	0	0
$439$	8.85269e8	0.499401	0.249700	−	0.968323i	$-0.419668\pi$
$439$	8.85269e8	0.499401	0.249700	+	0.968323i	$0.419668\pi$
$440$	0	0
$441$	8.44853e8	0.469079
$442$	0	0
$443$	−2.54765e9	−1.39228	−0.696141	−	0.717905i	$-0.745101\pi$
$443$	−2.54765e9	−1.39228	−0.696141	+	0.717905i	$0.745101\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1.06306e9	0.562967
$448$	0	0
$449$	−1.47885e9	−0.771011	−0.385506	−	0.922705i	$-0.625973\pi$
$449$	−1.47885e9	−0.771011	−0.385506	+	0.922705i	$0.625973\pi$
$450$	0	0
$451$	1.63503e9	0.839282
$452$	0	0
$453$	3.40962e8	0.172331
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2.49375e9	1.22221	0.611105	−	0.791550i	$-0.290725\pi$
$457$	2.49375e9	1.22221	0.611105	+	0.791550i	$0.290725\pi$
$458$	0	0
$459$	6.10212e8	0.294535
$460$	0	0
$461$	−2.78024e9	−1.32169	−0.660845	−	0.750523i	$-0.729802\pi$
$461$	−2.78024e9	−1.32169	−0.660845	+	0.750523i	$0.729802\pi$
$462$	0	0
$463$	−2.43950e9	−1.14227	−0.571134	−	0.820857i	$-0.693496\pi$
$463$	−2.43950e9	−1.14227	−0.571134	+	0.820857i	$0.693496\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3.09202e9	−1.40486	−0.702431	−	0.711752i	$-0.747902\pi$
$467$	−3.09202e9	−1.40486	−0.702431	+	0.711752i	$0.747902\pi$
$468$	0	0
$469$	−5.48437e9	−2.45483
$470$	0	0
$471$	8.53488e8	0.376378
$472$	0	0
$473$	−1.02658e9	−0.446044
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2.44161e8	−0.103006
$478$	0	0
$479$	−4.53752e9	−1.88644	−0.943222	−	0.332162i	$-0.892222\pi$
$479$	−4.53752e9	−1.88644	−0.943222	+	0.332162i	$0.892222\pi$
$480$	0	0
$481$	2.78004e9	1.13905
$482$	0	0
$483$	−2.99992e9	−1.21142
$484$	0	0
$485$	0	0
$486$	0	0
$487$	1.00925e9	0.395956	0.197978	−	0.980206i	$-0.436562\pi$
$487$	1.00925e9	0.395956	0.197978	+	0.980206i	$0.436562\pi$
$488$	0	0
$489$	−4.71577e7	−0.0182377
$490$	0	0
$491$	−2.81299e9	−1.07246	−0.536232	−	0.844071i	$-0.680153\pi$
$491$	−2.81299e9	−1.07246	−0.536232	+	0.844071i	$0.680153\pi$
$492$	0	0
$493$	4.87333e9	1.83173
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.30199e9	−1.20650
$498$	0	0
$499$	−2.57229e9	−0.926761	−0.463381	−	0.886159i	$-0.653364\pi$
$499$	−2.57229e9	−0.926761	−0.463381	+	0.886159i	$0.653364\pi$
$500$	0	0
$501$	2.00182e9	0.711203
$502$	0	0
$503$	3.11257e9	1.09052	0.545258	−	0.838269i	$-0.316432\pi$
$503$	3.11257e9	1.09052	0.545258	+	0.838269i	$0.316432\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	7.57523e8	0.258148
$508$	0	0
$509$	−3.29985e9	−1.10913	−0.554564	−	0.832141i	$-0.687115\pi$
$509$	−3.29985e9	−1.10913	−0.554564	+	0.832141i	$0.687115\pi$
$510$	0	0
$511$	−8.06355e9	−2.67333
$512$	0	0
$513$	7.93225e8	0.259409
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.77001e9	0.563325
$518$	0	0
$519$	−8.85735e8	−0.278111
$520$	0	0
$521$	4.62318e9	1.43222	0.716109	−	0.697988i	$-0.245921\pi$
$521$	4.62318e9	1.43222	0.716109	+	0.697988i	$0.245921\pi$
$522$	0	0
$523$	−2.14178e9	−0.654665	−0.327333	−	0.944909i	$-0.606150\pi$
$523$	−2.14178e9	−0.654665	−0.327333	+	0.944909i	$0.606150\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.55977e9	−1.05946
$528$	0	0
$529$	2.82228e9	0.828905
$530$	0	0
$531$	4.10132e8	0.118876
$532$	0	0
$533$	−2.38139e9	−0.681216
$534$	0	0
$535$	0	0
$536$	0	0
$537$	2.26119e9	0.630126
$538$	0	0
$539$	−4.68668e9	−1.28915
$540$	0	0
$541$	2.22351e8	0.0603737	0.0301869	−	0.999544i	$-0.490390\pi$
$541$	2.22351e8	0.0603737	0.0301869	+	0.999544i	$0.490390\pi$
$542$	0	0
$543$	−2.51670e9	−0.674578
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−3.09922e9	−0.809648	−0.404824	−	0.914395i	$-0.632667\pi$
$547$	−3.09922e9	−0.809648	−0.404824	+	0.914395i	$0.632667\pi$
$548$	0	0
$549$	2.36682e9	0.610466
$550$	0	0
$551$	6.33492e9	1.61328
$552$	0	0
$553$	−7.35259e9	−1.84885
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5.67375e9	1.39116	0.695580	−	0.718449i	$-0.255148\pi$
$557$	5.67375e9	1.39116	0.695580	+	0.718449i	$0.255148\pi$
$558$	0	0
$559$	1.49519e9	0.362038
$560$	0	0
$561$	−3.38505e9	−0.809459
$562$	0	0
$563$	2.92089e9	0.689821	0.344910	−	0.938636i	$-0.387909\pi$
$563$	2.92089e9	0.689821	0.344910	+	0.938636i	$0.387909\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	7.48269e8	0.172392
$568$	0	0
$569$	−8.37219e9	−1.90522	−0.952612	−	0.304188i	$-0.901615\pi$
$569$	−8.37219e9	−1.90522	−0.952612	+	0.304188i	$0.901615\pi$
$570$	0	0
$571$	5.15075e9	1.15783	0.578914	−	0.815389i	$-0.303477\pi$
$571$	5.15075e9	1.15783	0.578914	+	0.815389i	$0.303477\pi$
$572$	0	0
$573$	1.29791e9	0.288206
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1.04947e9	0.227435	0.113717	−	0.993513i	$-0.463724\pi$
$577$	1.04947e9	0.227435	0.113717	+	0.993513i	$0.463724\pi$
$578$	0	0
$579$	4.65011e8	0.0995608
$580$	0	0
$581$	4.12281e9	0.872121
$582$	0	0
$583$	1.35444e9	0.283087
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−5.34026e9	−1.08976	−0.544878	−	0.838515i	$-0.683424\pi$
$587$	−5.34026e9	−1.08976	−0.544878	+	0.838515i	$0.683424\pi$
$588$	0	0
$589$	−4.62741e9	−0.933113
$590$	0	0
$591$	−8.29678e8	−0.165331
$592$	0	0
$593$	−1.57140e9	−0.309454	−0.154727	−	0.987957i	$-0.549450\pi$
$593$	−1.57140e9	−0.309454	−0.154727	+	0.987957i	$0.549450\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.89741e9	0.942013
$598$	0	0
$599$	6.49327e9	1.23444	0.617220	−	0.786791i	$-0.288259\pi$
$599$	6.49327e9	1.23444	0.617220	+	0.786791i	$0.288259\pi$
$600$	0	0
$601$	6.40533e9	1.20360	0.601798	−	0.798649i	$-0.294451\pi$
$601$	6.40533e9	1.20360	0.601798	+	0.798649i	$0.294451\pi$
$602$	0	0
$603$	−2.83956e9	−0.527401
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−7.67730e9	−1.39331	−0.696656	−	0.717406i	$-0.745329\pi$
$607$	−7.67730e9	−1.39331	−0.696656	+	0.717406i	$0.745329\pi$
$608$	0	0
$609$	5.97589e9	1.07212
$610$	0	0
$611$	−2.57798e9	−0.457231
$612$	0	0
$613$	−7.85022e9	−1.37648	−0.688241	−	0.725482i	$-0.741617\pi$
$613$	−7.85022e9	−1.37648	−0.688241	+	0.725482i	$0.741617\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−2.18100e8	−0.0373816	−0.0186908	−	0.999825i	$-0.505950\pi$
$617$	−2.18100e8	−0.0373816	−0.0186908	+	0.999825i	$0.505950\pi$
$618$	0	0
$619$	9.02304e9	1.52910	0.764549	−	0.644565i	$-0.222962\pi$
$619$	9.02304e9	1.52910	0.764549	+	0.644565i	$0.222962\pi$
$620$	0	0
$621$	−1.55322e9	−0.260264
$622$	0	0
$623$	−4.44960e9	−0.737247
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−4.40028e9	−0.712925
$628$	0	0
$629$	−1.46328e10	−2.34449
$630$	0	0
$631$	5.58704e9	0.885276	0.442638	−	0.896700i	$-0.354043\pi$
$631$	5.58704e9	0.885276	0.442638	+	0.896700i	$0.354043\pi$
$632$	0	0
$633$	2.92421e9	0.458243
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.82604e9	1.04636
$638$	0	0
$639$	−1.70962e9	−0.259207
$640$	0	0
$641$	3.29846e9	0.494662	0.247331	−	0.968931i	$-0.420447\pi$
$641$	3.29846e9	0.494662	0.247331	+	0.968931i	$0.420447\pi$
$642$	0	0
$643$	1.92439e9	0.285466	0.142733	−	0.989761i	$-0.454411\pi$
$643$	1.92439e9	0.285466	0.142733	+	0.989761i	$0.454411\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.72440e8	0.0830932	0.0415466	−	0.999137i	$-0.486772\pi$
$647$	5.72440e8	0.0830932	0.0415466	+	0.999137i	$0.486772\pi$
$648$	0	0
$649$	−2.27514e9	−0.326702
$650$	0	0
$651$	−4.36515e9	−0.620106
$652$	0	0
$653$	1.54763e9	0.217506	0.108753	−	0.994069i	$-0.465314\pi$
$653$	1.54763e9	0.217506	0.108753	+	0.994069i	$0.465314\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−4.17495e9	−0.574344
$658$	0	0
$659$	3.70783e9	0.504685	0.252343	−	0.967638i	$-0.418799\pi$
$659$	3.70783e9	0.504685	0.252343	+	0.967638i	$0.418799\pi$
$660$	0	0
$661$	−2.93524e9	−0.395311	−0.197656	−	0.980272i	$-0.563333\pi$
$661$	−2.93524e9	−0.395311	−0.197656	+	0.980272i	$0.563333\pi$
$662$	0	0
$663$	4.93025e9	0.657009
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.24045e10	−1.61860
$668$	0	0
$669$	−6.09448e9	−0.786947
$670$	0	0
$671$	−1.31295e10	−1.67772
$672$	0	0
$673$	7.09060e9	0.896665	0.448332	−	0.893867i	$-0.352018\pi$
$673$	7.09060e9	0.896665	0.448332	+	0.893867i	$0.352018\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−3.69689e8	−0.0457906	−0.0228953	−	0.999738i	$-0.507288\pi$
$677$	−3.69689e8	−0.0457906	−0.0228953	+	0.999738i	$0.507288\pi$
$678$	0	0
$679$	2.67335e9	0.327727
$680$	0	0
$681$	7.31703e9	0.887810
$682$	0	0
$683$	1.41015e10	1.69353	0.846763	−	0.531970i	$-0.178548\pi$
$683$	1.41015e10	1.69353	0.846763	+	0.531970i	$0.178548\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	7.47082e9	0.879063
$688$	0	0
$689$	−1.97271e9	−0.229772
$690$	0	0
$691$	−8.61216e9	−0.992976	−0.496488	−	0.868044i	$-0.665377\pi$
$691$	−8.61216e9	−0.992976	−0.496488	+	0.868044i	$0.665377\pi$
$692$	0	0
$693$	−4.15089e9	−0.473778
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.25344e10	1.40213
$698$	0	0
$699$	−5.81461e9	−0.643948
$700$	0	0
$701$	−1.00744e10	−1.10460	−0.552299	−	0.833646i	$-0.686249\pi$
$701$	−1.00744e10	−1.10460	−0.552299	+	0.833646i	$0.686249\pi$
$702$	0	0
$703$	−1.90214e10	−2.06489
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.38404e10	−1.47293
$708$	0	0
$709$	1.81914e10	1.91692	0.958460	−	0.285226i	$-0.0920687\pi$
$709$	1.81914e10	1.91692	0.958460	+	0.285226i	$0.0920687\pi$
$710$	0	0
$711$	−3.80684e9	−0.397211
$712$	0	0
$713$	9.06099e9	0.936186
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.11243e10	1.12708
$718$	0	0
$719$	−6.30763e8	−0.0632871	−0.0316435	−	0.999499i	$-0.510074\pi$
$719$	−6.30763e8	−0.0632871	−0.0316435	+	0.999499i	$0.510074\pi$
$720$	0	0
$721$	−8.29463e9	−0.824183
$722$	0	0
$723$	1.14724e9	0.112893
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.20944e10	1.16738	0.583691	−	0.811976i	$-0.301608\pi$
$727$	1.20944e10	1.16738	0.583691	+	0.811976i	$0.301608\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−7.86992e9	−0.745177
$732$	0	0
$733$	1.78931e10	1.67811	0.839056	−	0.544045i	$-0.183108\pi$
$733$	1.78931e10	1.67811	0.839056	+	0.544045i	$0.183108\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.57520e10	1.44944
$738$	0	0
$739$	−1.23755e10	−1.12800	−0.563999	−	0.825775i	$-0.690738\pi$
$739$	−1.23755e10	−1.12800	−0.563999	+	0.825775i	$0.690738\pi$
$740$	0	0
$741$	6.40891e9	0.578656
$742$	0	0
$743$	−1.30742e10	−1.16938	−0.584690	−	0.811257i	$-0.698784\pi$
$743$	−1.30742e10	−1.16938	−0.584690	+	0.811257i	$0.698784\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.13461e9	0.187368
$748$	0	0
$749$	3.17323e10	2.75941
$750$	0	0
$751$	5.29032e9	0.455766	0.227883	−	0.973688i	$-0.426820\pi$
$751$	5.29032e9	0.455766	0.227883	+	0.973688i	$0.426820\pi$
$752$	0	0
$753$	−1.54723e9	−0.132060
$754$	0	0
$755$	0	0
$756$	0	0
$757$	8.94480e9	0.749437	0.374718	−	0.927139i	$-0.377739\pi$
$757$	8.94480e9	0.749437	0.374718	+	0.927139i	$0.377739\pi$
$758$	0	0
$759$	8.61624e9	0.715273
$760$	0	0
$761$	2.25775e10	1.85708	0.928540	−	0.371234i	$-0.121065\pi$
$761$	2.25775e10	1.85708	0.928540	+	0.371234i	$0.121065\pi$
$762$	0	0
$763$	4.63483e9	0.377744
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.31369e9	0.265172
$768$	0	0
$769$	−7.86783e8	−0.0623897	−0.0311949	−	0.999513i	$-0.509931\pi$
$769$	−7.86783e8	−0.0623897	−0.0311949	+	0.999513i	$0.509931\pi$
$770$	0	0
$771$	4.76057e9	0.374084
$772$	0	0
$773$	−2.85766e9	−0.222527	−0.111264	−	0.993791i	$-0.535490\pi$
$773$	−2.85766e9	−0.222527	−0.111264	+	0.993791i	$0.535490\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1.79433e10	−1.37224
$778$	0	0
$779$	1.62937e10	1.23492
$780$	0	0
$781$	9.48383e9	0.712369
$782$	0	0
$783$	3.09405e9	0.230336
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−1.09177e10	−0.798400	−0.399200	−	0.916864i	$-0.630712\pi$
$787$	−1.09177e10	−0.798400	−0.399200	+	0.916864i	$0.630712\pi$
$788$	0	0
$789$	6.40042e9	0.463916
$790$	0	0
$791$	−3.29114e10	−2.36444
$792$	0	0
$793$	1.91228e10	1.36175
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.44170e10	−1.00872	−0.504360	−	0.863494i	$-0.668271\pi$
$797$	−1.44170e10	−1.00872	−0.504360	+	0.863494i	$0.668271\pi$
$798$	0	0
$799$	1.35692e10	0.941111
$800$	0	0
$801$	−2.30381e9	−0.158392
$802$	0	0
$803$	2.31598e10	1.57845
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−2.71799e9	−0.182050
$808$	0	0
$809$	1.09536e10	0.727340	0.363670	−	0.931528i	$-0.381524\pi$
$809$	1.09536e10	0.727340	0.363670	+	0.931528i	$0.381524\pi$
$810$	0	0
$811$	−1.61876e10	−1.06564	−0.532820	−	0.846229i	$-0.678868\pi$
$811$	−1.61876e10	−1.06564	−0.532820	+	0.846229i	$0.678868\pi$
$812$	0	0
$813$	−7.82353e8	−0.0510606
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−1.02302e10	−0.656309
$818$	0	0
$819$	6.04568e9	0.384549
$820$	0	0
$821$	3.29633e9	0.207888	0.103944	−	0.994583i	$-0.466854\pi$
$821$	3.29633e9	0.207888	0.103944	+	0.994583i	$0.466854\pi$
$822$	0	0
$823$	5.47351e9	0.342268	0.171134	−	0.985248i	$-0.445257\pi$
$823$	5.47351e9	0.342268	0.171134	+	0.985248i	$0.445257\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.64804e9	0.162800	0.0814002	−	0.996681i	$-0.474061\pi$
$827$	2.64804e9	0.162800	0.0814002	+	0.996681i	$0.474061\pi$
$828$	0	0
$829$	−9.41293e9	−0.573831	−0.286916	−	0.957956i	$-0.592630\pi$
$829$	−9.41293e9	−0.573831	−0.286916	+	0.957956i	$0.592630\pi$
$830$	0	0
$831$	−9.12748e9	−0.551756
$832$	0	0
$833$	−3.59289e10	−2.15370
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−2.26008e9	−0.133225
$838$	0	0
$839$	6.45231e8	0.0377180	0.0188590	−	0.999822i	$-0.493997\pi$
$839$	6.45231e8	0.0377180	0.0188590	+	0.999822i	$0.493997\pi$
$840$	0	0
$841$	7.46008e9	0.432471
$842$	0	0
$843$	9.08070e9	0.522063
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−4.41159e9	−0.249462
$848$	0	0
$849$	8.94575e8	0.0501695
$850$	0	0
$851$	3.72460e10	2.07170
$852$	0	0
$853$	2.24452e10	1.23823	0.619116	−	0.785300i	$-0.287491\pi$
$853$	2.24452e10	1.23823	0.619116	+	0.785300i	$0.287491\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.37440e10	0.745897	0.372949	−	0.927852i	$-0.378347\pi$
$857$	1.37440e10	0.745897	0.372949	+	0.927852i	$0.378347\pi$
$858$	0	0
$859$	−3.19910e9	−0.172207	−0.0861036	−	0.996286i	$-0.527442\pi$
$859$	−3.19910e9	−0.172207	−0.0861036	+	0.996286i	$0.527442\pi$
$860$	0	0
$861$	1.53702e10	0.820672
$862$	0	0
$863$	8.08687e9	0.428295	0.214147	−	0.976801i	$-0.431303\pi$
$863$	8.08687e9	0.428295	0.214147	+	0.976801i	$0.431303\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−1.48712e10	−0.774960
$868$	0	0
$869$	2.11178e10	1.09164
$870$	0	0
$871$	−2.29424e10	−1.17646
$872$	0	0
$873$	1.38414e9	0.0704094
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.25367e10	−0.627603	−0.313801	−	0.949489i	$-0.601603\pi$
$877$	−1.25367e10	−0.627603	−0.313801	+	0.949489i	$0.601603\pi$
$878$	0	0
$879$	−1.35799e10	−0.674429
$880$	0	0
$881$	−5.17644e9	−0.255044	−0.127522	−	0.991836i	$-0.540702\pi$
$881$	−5.17644e9	−0.255044	−0.127522	+	0.991836i	$0.540702\pi$
$882$	0	0
$883$	2.85719e10	1.39661	0.698306	−	0.715799i	$-0.253937\pi$
$883$	2.85719e10	1.39661	0.698306	+	0.715799i	$0.253937\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.46756e10	0.706096	0.353048	−	0.935605i	$-0.385145\pi$
$887$	1.46756e10	0.706096	0.353048	+	0.935605i	$0.385145\pi$
$888$	0	0
$889$	−2.33224e10	−1.11331
$890$	0	0
$891$	−2.14915e9	−0.101787
$892$	0	0
$893$	1.76388e10	0.828876
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.25494e10	−0.580562
$898$	0	0
$899$	−1.80496e10	−0.828533
$900$	0	0
$901$	1.03834e10	0.472935
$902$	0	0
$903$	−9.65044e9	−0.436154
$904$	0	0
$905$	0	0
$906$	0	0
$907$	2.65921e9	0.118339	0.0591694	−	0.998248i	$-0.481155\pi$
$907$	2.65921e9	0.118339	0.0591694	+	0.998248i	$0.481155\pi$
$908$	0	0
$909$	−7.16595e9	−0.316447
$910$	0	0
$911$	3.50467e10	1.53579	0.767897	−	0.640574i	$-0.221303\pi$
$911$	3.50467e10	1.53579	0.767897	+	0.640574i	$0.221303\pi$
$912$	0	0
$913$	−1.18414e10	−0.514937
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.78799e10	2.05050
$918$	0	0
$919$	2.39541e10	1.01807	0.509033	−	0.860747i	$-0.330003\pi$
$919$	2.39541e10	1.01807	0.509033	+	0.860747i	$0.330003\pi$
$920$	0	0
$921$	−1.67354e10	−0.705874
$922$	0	0
$923$	−1.38130e10	−0.578205
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−4.29459e9	−0.177069
$928$	0	0
$929$	2.55929e10	1.04728	0.523642	−	0.851938i	$-0.324573\pi$
$929$	2.55929e10	1.04728	0.523642	+	0.851938i	$0.324573\pi$
$930$	0	0
$931$	−4.67045e10	−1.89686
$932$	0	0
$933$	−8.83731e9	−0.356233
$934$	0	0
$935$	0	0
$936$	0	0
$937$	4.51255e9	0.179198	0.0895991	−	0.995978i	$-0.471441\pi$
$937$	4.51255e9	0.179198	0.0895991	+	0.995978i	$0.471441\pi$
$938$	0	0
$939$	2.04539e9	0.0806207
$940$	0	0
$941$	5.61581e9	0.219709	0.109855	−	0.993948i	$-0.464961\pi$
$941$	5.61581e9	0.219709	0.109855	+	0.993948i	$0.464961\pi$
$942$	0	0
$943$	−3.19049e10	−1.23899
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.52152e10	1.34743	0.673713	−	0.738993i	$-0.264698\pi$
$947$	3.52152e10	1.34743	0.673713	+	0.738993i	$0.264698\pi$
$948$	0	0
$949$	−3.37318e10	−1.28117
$950$	0	0
$951$	−2.40277e10	−0.905900
$952$	0	0
$953$	2.36572e10	0.885396	0.442698	−	0.896671i	$-0.354021\pi$
$953$	2.36572e10	0.885396	0.442698	+	0.896671i	$0.354021\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−1.71637e10	−0.633022
$958$	0	0
$959$	−1.98335e9	−0.0726162
$960$	0	0
$961$	−1.43281e10	−0.520782
$962$	0	0
$963$	1.64296e10	0.592836
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1.27863e10	−0.454730	−0.227365	−	0.973810i	$-0.573011\pi$
$967$	−1.27863e10	−0.454730	−0.227365	+	0.973810i	$0.573011\pi$
$968$	0	0
$969$	−3.37333e10	−1.19104
$970$	0	0
$971$	−8.30143e9	−0.290995	−0.145498	−	0.989359i	$-0.546478\pi$
$971$	−8.30143e9	−0.290995	−0.145498	+	0.989359i	$0.546478\pi$
$972$	0	0
$973$	−4.21690e10	−1.46757
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.82338e10	1.65470	0.827352	−	0.561683i	$-0.189846\pi$
$977$	4.82338e10	1.65470	0.827352	+	0.561683i	$0.189846\pi$
$978$	0	0
$979$	1.27800e10	0.435302
$980$	0	0
$981$	2.39971e9	0.0811553
$982$	0	0
$983$	−1.91644e10	−0.643512	−0.321756	−	0.946823i	$-0.604273\pi$
$983$	−1.91644e10	−0.643512	−0.321756	+	0.946823i	$0.604273\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.66391e10	0.550834
$988$	0	0
$989$	2.00320e10	0.658471
$990$	0	0
$991$	5.44569e10	1.77744	0.888720	−	0.458451i	$-0.151596\pi$
$991$	5.44569e10	1.77744	0.888720	+	0.458451i	$0.151596\pi$
$992$	0	0
$993$	1.28408e10	0.416171
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−2.23881e10	−0.715457	−0.357728	−	0.933826i	$-0.616449\pi$
$997$	−2.23881e10	−0.715457	−0.357728	+	0.933826i	$0.616449\pi$
$998$	0	0
$999$	−9.29026e9	−0.294814

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.8.a.d.1.1		1
5.2	odd	4		300.8.d.b.49.2		2
5.3	odd	4		300.8.d.b.49.1		2
5.4	even	2		60.8.a.d.1.1	✓	1
15.14	odd	2		180.8.a.a.1.1		1
20.19	odd	2		240.8.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.8.a.d.1.1	✓	1	5.4	even	2
180.8.a.a.1.1		1	15.14	odd	2
240.8.a.g.1.1		1	20.19	odd	2
300.8.a.d.1.1		1	1.1	even	1	trivial
300.8.d.b.49.1		2	5.3	odd	4
300.8.d.b.49.2		2	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	300.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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