LMFDB - Embedded newform 72.14.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,14,Mod(1,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 14, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("72.1");

S:= CuspForms(chi, 14);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	72.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:	$77.2062688454$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 8)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	72.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4330.00 q^{5} -139992. q^{7} +O(q^{10})$

$q+4330.00 q^{5} -139992. q^{7} +6.48432e6 q^{11} -2.25880e7 q^{13} +2.37323e7 q^{17} +3.25345e8 q^{19} -9.21601e8 q^{23} -1.20195e9 q^{25} +3.86588e9 q^{29} -2.25340e9 q^{31} -6.06165e8 q^{35} +1.82504e10 q^{37} -3.44228e10 q^{41} -1.71925e10 q^{43} +6.73717e10 q^{47} -7.72913e10 q^{49} +8.72812e10 q^{53} +2.80771e10 q^{55} -5.40215e11 q^{59} -5.12766e10 q^{61} -9.78062e10 q^{65} +2.55199e10 q^{67} +1.38750e12 q^{71} -8.19049e11 q^{73} -9.07753e11 q^{77} -4.03094e12 q^{79} -4.18082e12 q^{83} +1.02761e11 q^{85} -2.67703e12 q^{89} +3.16214e12 q^{91} +1.40874e12 q^{95} -1.40395e13 q^{97} +O(q^{100})$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	4330.00	0.123932	0.0619659	−	0.998078i	$-0.480263\pi$
$5$	4330.00	0.123932	0.0619659	+	0.998078i	$0.480263\pi$
$6$	0	0
$7$	−139992.	−0.449745	−0.224872	−	0.974388i	$-0.572196\pi$
$7$	−139992.	−0.449745	−0.224872	+	0.974388i	$0.572196\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	6.48432e6	1.10360	0.551801	−	0.833976i	$-0.313941\pi$
$11$	6.48432e6	1.10360	0.551801	+	0.833976i	$0.313941\pi$
$12$	0	0
$13$	−2.25880e7	−1.29792	−0.648958	−	0.760824i	$-0.724795\pi$
$13$	−2.25880e7	−1.29792	−0.648958	+	0.760824i	$0.724795\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.37323e7	0.238463	0.119232	−	0.992866i	$-0.461957\pi$
$17$	2.37323e7	0.238463	0.119232	+	0.992866i	$0.461957\pi$
$18$	0	0
$19$	3.25345e8	1.58652	0.793260	−	0.608883i	$-0.208382\pi$
$19$	3.25345e8	1.58652	0.793260	+	0.608883i	$0.208382\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−9.21601e8	−1.29811	−0.649055	−	0.760741i	$-0.724836\pi$
$23$	−9.21601e8	−1.29811	−0.649055	+	0.760741i	$0.724836\pi$
$24$	0	0
$25$	−1.20195e9	−0.984641
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.86588e9	1.20687	0.603436	−	0.797411i	$-0.293798\pi$
$29$	3.86588e9	1.20687	0.603436	+	0.797411i	$0.293798\pi$
$30$	0	0
$31$	−2.25340e9	−0.456024	−0.228012	−	0.973658i	$-0.573223\pi$
$31$	−2.25340e9	−0.456024	−0.228012	+	0.973658i	$0.573223\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−6.06165e8	−0.0557377
$36$	0	0
$37$	1.82504e10	1.16939	0.584697	−	0.811252i	$-0.301213\pi$
$37$	1.82504e10	1.16939	0.584697	+	0.811252i	$0.301213\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.44228e10	−1.13175	−0.565877	−	0.824490i	$-0.691462\pi$
$41$	−3.44228e10	−1.13175	−0.565877	+	0.824490i	$0.691462\pi$
$42$	0	0
$43$	−1.71925e10	−0.414757	−0.207379	−	0.978261i	$-0.566493\pi$
$43$	−1.71925e10	−0.414757	−0.207379	+	0.978261i	$0.566493\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.73717e10	0.911679	0.455839	−	0.890062i	$-0.349339\pi$
$47$	6.73717e10	0.911679	0.455839	+	0.890062i	$0.349339\pi$
$48$	0	0
$49$	−7.72913e10	−0.797730
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.72812e10	0.540913	0.270457	−	0.962732i	$-0.412825\pi$
$53$	8.72812e10	0.540913	0.270457	+	0.962732i	$0.412825\pi$
$54$	0	0
$55$	2.80771e10	0.136771
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5.40215e11	−1.66736	−0.833678	−	0.552251i	$-0.813769\pi$
$59$	−5.40215e11	−1.66736	−0.833678	+	0.552251i	$0.813769\pi$
$60$	0	0
$61$	−5.12766e10	−0.127431	−0.0637155	−	0.997968i	$-0.520295\pi$
$61$	−5.12766e10	−0.127431	−0.0637155	+	0.997968i	$0.520295\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−9.78062e10	−0.160853
$66$	0	0
$67$	2.55199e10	0.0344662	0.0172331	−	0.999851i	$-0.494514\pi$
$67$	2.55199e10	0.0344662	0.0172331	+	0.999851i	$0.494514\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.38750e12	1.28545	0.642723	−	0.766099i	$-0.277805\pi$
$71$	1.38750e12	1.28545	0.642723	+	0.766099i	$0.277805\pi$
$72$	0	0
$73$	−8.19049e11	−0.633449	−0.316724	−	0.948518i	$-0.602583\pi$
$73$	−8.19049e11	−0.633449	−0.316724	+	0.948518i	$0.602583\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−9.07753e11	−0.496339
$78$	0	0
$79$	−4.03094e12	−1.86565	−0.932824	−	0.360332i	$-0.882663\pi$
$79$	−4.03094e12	−1.86565	−0.932824	+	0.360332i	$0.882663\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−4.18082e12	−1.40364	−0.701818	−	0.712357i	$-0.747628\pi$
$83$	−4.18082e12	−1.40364	−0.701818	+	0.712357i	$0.747628\pi$
$84$	0	0
$85$	1.02761e11	0.0295532
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.67703e12	−0.570976	−0.285488	−	0.958382i	$-0.592156\pi$
$89$	−2.67703e12	−0.570976	−0.285488	+	0.958382i	$0.592156\pi$
$90$	0	0
$91$	3.16214e12	0.583731
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.40874e12	0.196620
$96$	0	0
$97$	−1.40395e13	−1.71133	−0.855666	−	0.517528i	$-0.826852\pi$
$97$	−1.40395e13	−1.71133	−0.855666	+	0.517528i	$0.826852\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.22392e12	0.583411	0.291706	−	0.956508i	$-0.405777\pi$
$101$	6.22392e12	0.583411	0.291706	+	0.956508i	$0.405777\pi$
$102$	0	0
$103$	−2.11756e13	−1.74740	−0.873702	−	0.486461i	$-0.838288\pi$
$103$	−2.11756e13	−1.74740	−0.873702	+	0.486461i	$0.838288\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.87895e13	−1.21038	−0.605188	−	0.796083i	$-0.706902\pi$
$107$	−1.87895e13	−1.21038	−0.605188	+	0.796083i	$0.706902\pi$
$108$	0	0
$109$	−9.95159e12	−0.568356	−0.284178	−	0.958772i	$-0.591721\pi$
$109$	−9.95159e12	−0.568356	−0.284178	+	0.958772i	$0.591721\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.08879e13	−0.943812	−0.471906	−	0.881649i	$-0.656434\pi$
$113$	−2.08879e13	−0.943812	−0.471906	+	0.881649i	$0.656434\pi$
$114$	0	0
$115$	−3.99053e12	−0.160877
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.32233e12	−0.107248
$120$	0	0
$121$	7.52375e12	0.217936
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.04901e13	−0.245960
$126$	0	0
$127$	−6.26814e13	−1.32561	−0.662803	−	0.748794i	$-0.730633\pi$
$127$	−6.26814e13	−1.32561	−0.662803	+	0.748794i	$0.730633\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	6.97773e12	0.120629	0.0603144	−	0.998179i	$-0.480790\pi$
$131$	6.97773e12	0.120629	0.0603144	+	0.998179i	$0.480790\pi$
$132$	0	0
$133$	−4.55457e13	−0.713529
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.78591e13	0.747632	0.373816	−	0.927503i	$-0.378049\pi$
$137$	5.78591e13	0.747632	0.373816	+	0.927503i	$0.378049\pi$
$138$	0	0
$139$	−3.93498e13	−0.462750	−0.231375	−	0.972865i	$-0.574322\pi$
$139$	−3.93498e13	−0.462750	−0.231375	+	0.972865i	$0.574322\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.46468e14	−1.43238
$144$	0	0
$145$	1.67393e13	0.149570
$146$	0	0
$147$	0	0
$148$	0	0
$149$	4.31370e13	0.322953	0.161476	−	0.986877i	$-0.448374\pi$
$149$	4.31370e13	0.322953	0.161476	+	0.986877i	$0.448374\pi$
$150$	0	0
$151$	2.19599e14	1.50758	0.753788	−	0.657117i	$-0.228224\pi$
$151$	2.19599e14	1.50758	0.753788	+	0.657117i	$0.228224\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−9.75723e12	−0.0565159
$156$	0	0
$157$	6.61866e13	0.352714	0.176357	−	0.984326i	$-0.443569\pi$
$157$	6.61866e13	0.352714	0.176357	+	0.984326i	$0.443569\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.29017e14	0.583818
$162$	0	0
$163$	−2.47622e14	−1.03412	−0.517058	−	0.855950i	$-0.672973\pi$
$163$	−2.47622e14	−1.03412	−0.517058	+	0.855950i	$0.672973\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.05226e13	0.215904	0.107952	−	0.994156i	$-0.465571\pi$
$167$	6.05226e13	0.215904	0.107952	+	0.994156i	$0.465571\pi$
$168$	0	0
$169$	2.07344e14	0.684586
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.82357e14	0.800752	0.400376	−	0.916351i	$-0.368879\pi$
$173$	2.82357e14	0.800752	0.400376	+	0.916351i	$0.368879\pi$
$174$	0	0
$175$	1.68264e14	0.442837
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.65618e14	0.376324	0.188162	−	0.982138i	$-0.439747\pi$
$179$	1.65618e14	0.376324	0.188162	+	0.982138i	$0.439747\pi$
$180$	0	0
$181$	8.52134e14	1.80135	0.900673	−	0.434497i	$-0.143074\pi$
$181$	8.52134e14	1.80135	0.900673	+	0.434497i	$0.143074\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.90242e13	0.144925
$186$	0	0
$187$	1.53888e14	0.263168
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−9.30990e14	−1.38748	−0.693742	−	0.720223i	$-0.744039\pi$
$191$	−9.30990e14	−1.38748	−0.693742	+	0.720223i	$0.744039\pi$
$192$	0	0
$193$	4.17870e14	0.581994	0.290997	−	0.956724i	$-0.406013\pi$
$193$	4.17870e14	0.581994	0.290997	+	0.956724i	$0.406013\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.17420e15	1.43124	0.715621	−	0.698489i	$-0.246144\pi$
$197$	1.17420e15	1.43124	0.715621	+	0.698489i	$0.246144\pi$
$198$	0	0
$199$	9.52478e13	0.108720	0.0543601	−	0.998521i	$-0.482688\pi$
$199$	9.52478e13	0.108720	0.0543601	+	0.998521i	$0.482688\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5.41192e14	−0.542784
$204$	0	0
$205$	−1.49051e14	−0.140260
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.10964e15	1.75089
$210$	0	0
$211$	7.61637e14	0.594172	0.297086	−	0.954851i	$-0.403985\pi$
$211$	7.61637e14	0.594172	0.297086	+	0.954851i	$0.403985\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−7.44435e13	−0.0514016
$216$	0	0
$217$	3.15458e14	0.205094
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−5.36065e14	−0.309505
$222$	0	0
$223$	−1.62673e15	−0.885796	−0.442898	−	0.896572i	$-0.646050\pi$
$223$	−1.62673e15	−0.885796	−0.442898	+	0.896572i	$0.646050\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.50895e15	−0.731991	−0.365996	−	0.930617i	$-0.619271\pi$
$227$	−1.50895e15	−0.731991	−0.365996	+	0.930617i	$0.619271\pi$
$228$	0	0
$229$	1.69644e15	0.777334	0.388667	−	0.921378i	$-0.372936\pi$
$229$	1.69644e15	0.777334	0.388667	+	0.921378i	$0.372936\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.27819e15	−0.523339	−0.261670	−	0.965158i	$-0.584273\pi$
$233$	−1.27819e15	−0.523339	−0.261670	+	0.965158i	$0.584273\pi$
$234$	0	0
$235$	2.91720e14	0.112986
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.10138e15	0.382252	0.191126	−	0.981566i	$-0.438786\pi$
$239$	1.10138e15	0.382252	0.191126	+	0.981566i	$0.438786\pi$
$240$	0	0
$241$	9.15325e14	0.300929	0.150465	−	0.988615i	$-0.451923\pi$
$241$	9.15325e14	0.300929	0.150465	+	0.988615i	$0.451923\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.34671e14	−0.0988641
$246$	0	0
$247$	−7.34890e15	−2.05917
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−5.68301e15	−1.43450	−0.717248	−	0.696818i	$-0.754598\pi$
$251$	−5.68301e15	−1.43450	−0.717248	+	0.696818i	$0.754598\pi$
$252$	0	0
$253$	−5.97596e15	−1.43260
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7.40938e15	1.60404	0.802022	−	0.597294i	$-0.203758\pi$
$257$	7.40938e15	1.60404	0.802022	+	0.597294i	$0.203758\pi$
$258$	0	0
$259$	−2.55491e15	−0.525929
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4.44830e15	−0.828861	−0.414431	−	0.910081i	$-0.636019\pi$
$263$	−4.44830e15	−0.828861	−0.414431	+	0.910081i	$0.636019\pi$
$264$	0	0
$265$	3.77928e14	0.0670364
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3.99525e15	0.642916	0.321458	−	0.946924i	$-0.395827\pi$
$269$	3.99525e15	0.642916	0.321458	+	0.946924i	$0.395827\pi$
$270$	0	0
$271$	5.58778e15	0.856917	0.428458	−	0.903562i	$-0.359057\pi$
$271$	5.58778e15	0.856917	0.428458	+	0.903562i	$0.359057\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−7.79386e15	−1.08665
$276$	0	0
$277$	−9.01343e14	−0.119887	−0.0599435	−	0.998202i	$-0.519092\pi$
$277$	−9.01343e14	−0.119887	−0.0599435	+	0.998202i	$0.519092\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1.24836e16	−1.51268	−0.756342	−	0.654176i	$-0.773015\pi$
$281$	−1.24836e16	−1.51268	−0.756342	+	0.654176i	$0.773015\pi$
$282$	0	0
$283$	−5.47980e15	−0.634093	−0.317046	−	0.948410i	$-0.602691\pi$
$283$	−5.47980e15	−0.634093	−0.317046	+	0.948410i	$0.602691\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.81892e15	0.509000
$288$	0	0
$289$	−9.34136e15	−0.943135
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−4.61630e15	−0.426240	−0.213120	−	0.977026i	$-0.568363\pi$
$293$	−4.61630e15	−0.426240	−0.213120	+	0.977026i	$0.568363\pi$
$294$	0	0
$295$	−2.33913e15	−0.206638
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.08171e16	1.68484
$300$	0	0
$301$	2.40681e15	0.186535
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.22028e14	−0.0157928
$306$	0	0
$307$	−2.22392e16	−1.51607	−0.758037	−	0.652211i	$-0.773842\pi$
$307$	−2.22392e16	−1.51607	−0.758037	+	0.652211i	$0.773842\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.51173e16	−0.947393	−0.473697	−	0.880688i	$-0.657081\pi$
$311$	−1.51173e16	−0.947393	−0.473697	+	0.880688i	$0.657081\pi$
$312$	0	0
$313$	8.36531e15	0.502856	0.251428	−	0.967876i	$-0.419100\pi$
$313$	8.36531e15	0.502856	0.251428	+	0.967876i	$0.419100\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9.34425e15	0.517201	0.258600	−	0.965984i	$-0.416739\pi$
$317$	9.34425e15	0.517201	0.258600	+	0.965984i	$0.416739\pi$
$318$	0	0
$319$	2.50676e16	1.33191
$320$	0	0
$321$	0	0
$322$	0	0
$323$	7.72117e15	0.378327
$324$	0	0
$325$	2.71498e16	1.27798
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−9.43151e15	−0.410023
$330$	0	0
$331$	3.92466e16	1.64029	0.820144	−	0.572157i	$-0.193893\pi$
$331$	3.92466e16	1.64029	0.820144	+	0.572157i	$0.193893\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.10501e14	0.00427146
$336$	0	0
$337$	−3.01727e16	−1.12207	−0.561036	−	0.827792i	$-0.689597\pi$
$337$	−3.01727e16	−1.12207	−0.561036	+	0.827792i	$0.689597\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.46118e16	−0.503269
$342$	0	0
$343$	2.43838e16	0.808519
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.02701e16	1.23834	0.619172	−	0.785255i	$-0.287468\pi$
$347$	4.02701e16	1.23834	0.619172	+	0.785255i	$0.287468\pi$
$348$	0	0
$349$	−4.36418e16	−1.29282	−0.646409	−	0.762991i	$-0.723730\pi$
$349$	−4.36418e16	−1.29282	−0.646409	+	0.762991i	$0.723730\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.08122e16	0.847593	0.423796	−	0.905758i	$-0.360697\pi$
$353$	3.08122e16	0.847593	0.423796	+	0.905758i	$0.360697\pi$
$354$	0	0
$355$	6.00788e15	0.159308
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.06926e16	1.00323	0.501616	−	0.865090i	$-0.332739\pi$
$359$	4.06926e16	1.00323	0.501616	+	0.865090i	$0.332739\pi$
$360$	0	0
$361$	6.37963e16	1.51705
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3.54648e15	−0.0785045
$366$	0	0
$367$	−1.06190e16	−0.226857	−0.113429	−	0.993546i	$-0.536183\pi$
$367$	−1.06190e16	−0.226857	−0.113429	+	0.993546i	$0.536183\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.22187e16	−0.243273
$372$	0	0
$373$	7.41221e16	1.42508	0.712542	−	0.701630i	$-0.247544\pi$
$373$	7.41221e16	1.42508	0.712542	+	0.701630i	$0.247544\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−8.73226e16	−1.56642
$378$	0	0
$379$	4.60131e16	0.797493	0.398746	−	0.917061i	$-0.369445\pi$
$379$	4.60131e16	0.797493	0.398746	+	0.917061i	$0.369445\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7.36970e16	1.19305	0.596524	−	0.802595i	$-0.296548\pi$
$383$	7.36970e16	1.19305	0.596524	+	0.802595i	$0.296548\pi$
$384$	0	0
$385$	−3.93057e15	−0.0615122
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−7.03726e16	−1.02975	−0.514874	−	0.857266i	$-0.672161\pi$
$389$	−7.03726e16	−1.02975	−0.514874	+	0.857266i	$0.672161\pi$
$390$	0	0
$391$	−2.18717e16	−0.309552
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.74540e16	−0.231213
$396$	0	0
$397$	−1.07601e17	−1.37936	−0.689680	−	0.724115i	$-0.742249\pi$
$397$	−1.07601e17	−1.37936	−0.689680	+	0.724115i	$0.742249\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.85722e16	0.463272	0.231636	−	0.972802i	$-0.425592\pi$
$401$	3.85722e16	0.463272	0.231636	+	0.972802i	$0.425592\pi$
$402$	0	0
$403$	5.08999e16	0.591881
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.18341e17	1.29054
$408$	0	0
$409$	−5.34639e16	−0.564753	−0.282377	−	0.959304i	$-0.591123\pi$
$409$	−5.34639e16	−0.564753	−0.282377	+	0.959304i	$0.591123\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7.56257e16	0.749884
$414$	0	0
$415$	−1.81030e16	−0.173955
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.04340e17	−0.942016	−0.471008	−	0.882129i	$-0.656110\pi$
$419$	−1.04340e17	−0.942016	−0.471008	+	0.882129i	$0.656110\pi$
$420$	0	0
$421$	−7.03173e16	−0.615500	−0.307750	−	0.951467i	$-0.599576\pi$
$421$	−7.03173e16	−0.615500	−0.307750	+	0.951467i	$0.599576\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.85251e16	−0.234801
$426$	0	0
$427$	7.17831e15	0.0573114
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−6.40437e16	−0.481254	−0.240627	−	0.970618i	$-0.577353\pi$
$431$	−6.40437e16	−0.481254	−0.240627	+	0.970618i	$0.577353\pi$
$432$	0	0
$433$	−1.12484e17	−0.820201	−0.410101	−	0.912040i	$-0.634506\pi$
$433$	−1.12484e17	−0.820201	−0.410101	+	0.912040i	$0.634506\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.99838e17	−2.05948
$438$	0	0
$439$	−3.14978e16	−0.210020	−0.105010	−	0.994471i	$-0.533487\pi$
$439$	−3.14978e16	−0.210020	−0.105010	+	0.994471i	$0.533487\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1.14104e17	−0.717261	−0.358631	−	0.933480i	$-0.616756\pi$
$443$	−1.14104e17	−0.717261	−0.358631	+	0.933480i	$0.616756\pi$
$444$	0	0
$445$	−1.15915e16	−0.0707621
$446$	0	0
$447$	0	0
$448$	0	0
$449$	7.86758e16	0.453148	0.226574	−	0.973994i	$-0.427247\pi$
$449$	7.86758e16	0.453148	0.226574	+	0.973994i	$0.427247\pi$
$450$	0	0
$451$	−2.23209e17	−1.24900
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.36921e16	0.0723428
$456$	0	0
$457$	2.35679e17	1.21022	0.605112	−	0.796140i	$-0.293128\pi$
$457$	2.35679e17	1.21022	0.605112	+	0.796140i	$0.293128\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2.33465e17	1.13283	0.566415	−	0.824120i	$-0.308330\pi$
$461$	2.33465e17	1.13283	0.566415	+	0.824120i	$0.308330\pi$
$462$	0	0
$463$	−1.43911e17	−0.678920	−0.339460	−	0.940620i	$-0.610244\pi$
$463$	−1.43911e17	−0.678920	−0.339460	+	0.940620i	$0.610244\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.72838e17	1.21715	0.608576	−	0.793496i	$-0.291741\pi$
$467$	2.72838e17	1.21715	0.608576	+	0.793496i	$0.291741\pi$
$468$	0	0
$469$	−3.57259e15	−0.0155010
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.11482e17	−0.457727
$474$	0	0
$475$	−3.91050e17	−1.56215
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−3.61844e17	−1.36880	−0.684401	−	0.729106i	$-0.739936\pi$
$479$	−3.61844e17	−1.36880	−0.684401	+	0.729106i	$0.739936\pi$
$480$	0	0
$481$	−4.12240e17	−1.51778
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.07909e16	−0.212089
$486$	0	0
$487$	3.33934e17	1.13429	0.567143	−	0.823619i	$-0.308049\pi$
$487$	3.33934e17	1.13429	0.567143	+	0.823619i	$0.308049\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.79534e16	0.0578251	0.0289125	−	0.999582i	$-0.490796\pi$
$491$	1.79534e16	0.0578251	0.0289125	+	0.999582i	$0.490796\pi$
$492$	0	0
$493$	9.17461e16	0.287795
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.94239e17	−0.578123
$498$	0	0
$499$	4.21623e17	1.22256	0.611281	−	0.791414i	$-0.290655\pi$
$499$	4.21623e17	1.22256	0.611281	+	0.791414i	$0.290655\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−3.53513e17	−0.973225	−0.486612	−	0.873618i	$-0.661768\pi$
$503$	−3.53513e17	−0.973225	−0.486612	+	0.873618i	$0.661768\pi$
$504$	0	0
$505$	2.69496e16	0.0723032
$506$	0	0
$507$	0	0
$508$	0	0
$509$	5.95547e17	1.51792	0.758962	−	0.651135i	$-0.225707\pi$
$509$	5.95547e17	1.51792	0.758962	+	0.651135i	$0.225707\pi$
$510$	0	0
$511$	1.14660e17	0.284890
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−9.16903e16	−0.216559
$516$	0	0
$517$	4.36860e17	1.00613
$518$	0	0
$519$	0	0
$520$	0	0
$521$	6.18807e17	1.35553	0.677767	−	0.735277i	$-0.262948\pi$
$521$	6.18807e17	1.35553	0.677767	+	0.735277i	$0.262948\pi$
$522$	0	0
$523$	−3.97661e17	−0.849674	−0.424837	−	0.905270i	$-0.639669\pi$
$523$	−3.97661e17	−0.849674	−0.424837	+	0.905270i	$0.639669\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.34783e16	−0.108745
$528$	0	0
$529$	3.45311e17	0.685092
$530$	0	0
$531$	0	0
$532$	0	0
$533$	7.77544e17	1.46892
$534$	0	0
$535$	−8.13584e16	−0.150004
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−5.01182e17	−0.880376
$540$	0	0
$541$	4.06842e17	0.697660	0.348830	−	0.937186i	$-0.386579\pi$
$541$	4.06842e17	0.697660	0.348830	+	0.937186i	$0.386579\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−4.30904e16	−0.0704374
$546$	0	0
$547$	−2.81721e17	−0.449678	−0.224839	−	0.974396i	$-0.572186\pi$
$547$	−2.81721e17	−0.449678	−0.224839	+	0.974396i	$0.572186\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.25774e18	1.91473
$552$	0	0
$553$	5.64299e17	0.839065
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.23400e18	−1.75088	−0.875442	−	0.483324i	$-0.839429\pi$
$557$	−1.23400e18	−1.75088	−0.875442	+	0.483324i	$0.839429\pi$
$558$	0	0
$559$	3.88345e17	0.538320
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−1.42261e18	−1.88270	−0.941348	−	0.337437i	$-0.890440\pi$
$563$	−1.42261e18	−1.88270	−0.941348	+	0.337437i	$0.890440\pi$
$564$	0	0
$565$	−9.04448e16	−0.116968
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4.50797e17	0.556866	0.278433	−	0.960456i	$-0.410185\pi$
$569$	4.50797e17	0.556866	0.278433	+	0.960456i	$0.410185\pi$
$570$	0	0
$571$	7.13748e17	0.861807	0.430904	−	0.902398i	$-0.358195\pi$
$571$	7.13748e17	0.861807	0.430904	+	0.902398i	$0.358195\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.10772e18	1.27817
$576$	0	0
$577$	1.28447e18	1.44905	0.724524	−	0.689250i	$-0.242060\pi$
$577$	1.28447e18	1.44905	0.724524	+	0.689250i	$0.242060\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	5.85282e17	0.631277
$582$	0	0
$583$	5.65960e17	0.596953
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.80690e16	0.0585864	0.0292932	−	0.999571i	$-0.490674\pi$
$587$	5.80690e16	0.0585864	0.0292932	+	0.999571i	$0.490674\pi$
$588$	0	0
$589$	−7.33133e17	−0.723491
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.39716e18	1.31944	0.659722	−	0.751509i	$-0.270674\pi$
$593$	1.39716e18	1.31944	0.659722	+	0.751509i	$0.270674\pi$
$594$	0	0
$595$	−1.43857e16	−0.0132914
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−5.23231e17	−0.462827	−0.231414	−	0.972855i	$-0.574335\pi$
$599$	−5.23231e17	−0.462827	−0.231414	+	0.972855i	$0.574335\pi$
$600$	0	0
$601$	−1.51221e18	−1.30897	−0.654484	−	0.756076i	$-0.727114\pi$
$601$	−1.51221e18	−1.30897	−0.654484	+	0.756076i	$0.727114\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3.25778e16	0.0270092
$606$	0	0
$607$	−1.70314e18	−1.38205	−0.691024	−	0.722832i	$-0.742840\pi$
$607$	−1.70314e18	−1.38205	−0.691024	+	0.722832i	$0.742840\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.52180e18	−1.18328
$612$	0	0
$613$	−6.08885e17	−0.463492	−0.231746	−	0.972776i	$-0.574444\pi$
$613$	−6.08885e17	−0.463492	−0.231746	+	0.972776i	$0.574444\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−5.93917e17	−0.433383	−0.216692	−	0.976240i	$-0.569527\pi$
$617$	−5.93917e17	−0.433383	−0.216692	+	0.976240i	$0.569527\pi$
$618$	0	0
$619$	1.00496e18	0.718055	0.359028	−	0.933327i	$-0.383108\pi$
$619$	1.00496e18	0.718055	0.359028	+	0.933327i	$0.383108\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3.74762e17	0.256793
$624$	0	0
$625$	1.42181e18	0.954159
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4.33123e17	0.278857
$630$	0	0
$631$	−7.35349e16	−0.0463770	−0.0231885	−	0.999731i	$-0.507382\pi$
$631$	−7.35349e16	−0.0463770	−0.0231885	+	0.999731i	$0.507382\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−2.71410e17	−0.164285
$636$	0	0
$637$	1.74586e18	1.03539
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2.82655e18	−1.60946	−0.804728	−	0.593644i	$-0.797689\pi$
$641$	−2.82655e18	−1.60946	−0.804728	+	0.593644i	$0.797689\pi$
$642$	0	0
$643$	−1.57731e18	−0.880126	−0.440063	−	0.897967i	$-0.645044\pi$
$643$	−1.57731e18	−0.880126	−0.440063	+	0.897967i	$0.645044\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.78710e18	0.957789	0.478895	−	0.877872i	$-0.341038\pi$
$647$	1.78710e18	0.957789	0.478895	+	0.877872i	$0.341038\pi$
$648$	0	0
$649$	−3.50293e18	−1.84010
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−3.34506e18	−1.68837	−0.844185	−	0.536051i	$-0.819915\pi$
$653$	−3.34506e18	−1.68837	−0.844185	+	0.536051i	$0.819915\pi$
$654$	0	0
$655$	3.02136e16	0.0149497
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.69802e18	0.807584	0.403792	−	0.914851i	$-0.367692\pi$
$659$	1.69802e18	0.807584	0.403792	+	0.914851i	$0.367692\pi$
$660$	0	0
$661$	−1.72815e18	−0.805883	−0.402942	−	0.915226i	$-0.632012\pi$
$661$	−1.72815e18	−0.805883	−0.402942	+	0.915226i	$0.632012\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.97213e17	−0.0884289
$666$	0	0
$667$	−3.56280e18	−1.56665
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−3.32494e17	−0.140633
$672$	0	0
$673$	−1.82377e18	−0.756608	−0.378304	−	0.925681i	$-0.623493\pi$
$673$	−1.82377e18	−0.756608	−0.378304	+	0.925681i	$0.623493\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.09138e18	1.23403	0.617014	−	0.786952i	$-0.288342\pi$
$677$	3.09138e18	1.23403	0.617014	+	0.786952i	$0.288342\pi$
$678$	0	0
$679$	1.96541e18	0.769662
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−3.14496e17	−0.118544	−0.0592722	−	0.998242i	$-0.518878\pi$
$683$	−3.14496e17	−0.118544	−0.0592722	+	0.998242i	$0.518878\pi$
$684$	0	0
$685$	2.50530e17	0.0926554
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1.97151e18	−0.702060
$690$	0	0
$691$	−4.48311e17	−0.156665	−0.0783325	−	0.996927i	$-0.524960\pi$
$691$	−4.48311e17	−0.156665	−0.0783325	+	0.996927i	$0.524960\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.70385e17	−0.0573494
$696$	0	0
$697$	−8.16932e17	−0.269881
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−3.21704e18	−1.02398	−0.511988	−	0.858992i	$-0.671091\pi$
$701$	−3.21704e18	−1.02398	−0.511988	+	0.858992i	$0.671091\pi$
$702$	0	0
$703$	5.93767e18	1.85527
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−8.71299e17	−0.262386
$708$	0	0
$709$	−3.21328e18	−0.950053	−0.475026	−	0.879971i	$-0.657562\pi$
$709$	−3.21328e18	−0.950053	−0.475026	+	0.879971i	$0.657562\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.07674e18	0.591970
$714$	0	0
$715$	−6.34207e17	−0.177518
$716$	0	0
$717$	0	0
$718$	0	0
$719$	2.51628e18	0.679237	0.339619	−	0.940563i	$-0.389702\pi$
$719$	2.51628e18	0.679237	0.339619	+	0.940563i	$0.389702\pi$
$720$	0	0
$721$	2.96441e18	0.785886
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4.64661e18	−1.18834
$726$	0	0
$727$	−1.20284e18	−0.302157	−0.151078	−	0.988522i	$-0.548275\pi$
$727$	−1.20284e18	−0.302157	−0.151078	+	0.988522i	$0.548275\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−4.08017e17	−0.0989044
$732$	0	0
$733$	−1.79874e18	−0.428345	−0.214173	−	0.976796i	$-0.568705\pi$
$733$	−1.79874e18	−0.428345	−0.214173	+	0.976796i	$0.568705\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.65479e17	0.0380369
$738$	0	0
$739$	−2.14261e18	−0.483898	−0.241949	−	0.970289i	$-0.577787\pi$
$739$	−2.14261e18	−0.483898	−0.241949	+	0.970289i	$0.577787\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	5.34550e18	1.16563	0.582816	−	0.812604i	$-0.301951\pi$
$743$	5.34550e18	1.16563	0.582816	+	0.812604i	$0.301951\pi$
$744$	0	0
$745$	1.86783e17	0.0400241
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2.63038e18	0.544360
$750$	0	0
$751$	3.42693e18	0.697021	0.348511	−	0.937305i	$-0.386688\pi$
$751$	3.42693e18	0.697021	0.348511	+	0.937305i	$0.386688\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	9.50862e17	0.186837
$756$	0	0
$757$	1.77769e18	0.343347	0.171674	−	0.985154i	$-0.445083\pi$
$757$	1.77769e18	0.343347	0.171674	+	0.985154i	$0.445083\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−4.61017e18	−0.860432	−0.430216	−	0.902726i	$-0.641563\pi$
$761$	−4.61017e18	−0.860432	−0.430216	+	0.902726i	$0.641563\pi$
$762$	0	0
$763$	1.39314e18	0.255615
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.22024e19	2.16409
$768$	0	0
$769$	4.06261e18	0.708409	0.354204	−	0.935168i	$-0.384752\pi$
$769$	4.06261e18	0.708409	0.354204	+	0.935168i	$0.384752\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	2.77091e18	0.467149	0.233574	−	0.972339i	$-0.424958\pi$
$773$	2.77091e18	0.467149	0.233574	+	0.972339i	$0.424958\pi$
$774$	0	0
$775$	2.70849e18	0.449020
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.11993e19	−1.79555
$780$	0	0
$781$	8.99700e18	1.41862
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.86588e17	0.0437125
$786$	0	0
$787$	4.81906e18	0.722980	0.361490	−	0.932376i	$-0.382268\pi$
$787$	4.81906e18	0.722980	0.361490	+	0.932376i	$0.382268\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.92414e18	0.424475
$792$	0	0
$793$	1.15824e18	0.165395
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8.13421e17	0.112418	0.0562090	−	0.998419i	$-0.482099\pi$
$797$	8.13421e17	0.112418	0.0562090	+	0.998419i	$0.482099\pi$
$798$	0	0
$799$	1.59888e18	0.217402
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−5.31098e18	−0.699075
$804$	0	0
$805$	5.58642e17	0.0723537
$806$	0	0
$807$	0	0
$808$	0	0
$809$	6.03069e18	0.756313	0.378156	−	0.925742i	$-0.376558\pi$
$809$	6.03069e18	0.756313	0.378156	+	0.925742i	$0.376558\pi$
$810$	0	0
$811$	3.20788e18	0.395897	0.197948	−	0.980212i	$-0.436572\pi$
$811$	3.20788e18	0.395897	0.197948	+	0.980212i	$0.436572\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1.07220e18	−0.128160
$816$	0	0
$817$	−5.59349e18	−0.658021
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.08418e19	1.23558	0.617789	−	0.786344i	$-0.288029\pi$
$821$	1.08418e19	1.23558	0.617789	+	0.786344i	$0.288029\pi$
$822$	0	0
$823$	−3.62368e18	−0.406491	−0.203246	−	0.979128i	$-0.565149\pi$
$823$	−3.62368e18	−0.406491	−0.203246	+	0.979128i	$0.565149\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−5.78670e18	−0.628992	−0.314496	−	0.949259i	$-0.601836\pi$
$827$	−5.78670e18	−0.628992	−0.314496	+	0.949259i	$0.601836\pi$
$828$	0	0
$829$	−8.10871e18	−0.867655	−0.433828	−	0.900996i	$-0.642837\pi$
$829$	−8.10871e18	−0.867655	−0.433828	+	0.900996i	$0.642837\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.83430e18	−0.190229
$834$	0	0
$835$	2.62063e17	0.0267573
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.53236e19	1.51673	0.758365	−	0.651830i	$-0.225998\pi$
$839$	1.53236e19	1.51673	0.758365	+	0.651830i	$0.225998\pi$
$840$	0	0
$841$	4.68439e18	0.456541
$842$	0	0
$843$	0	0
$844$	0	0
$845$	8.97800e17	0.0848420
$846$	0	0
$847$	−1.05326e18	−0.0980156
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1.68196e19	−1.51800
$852$	0	0
$853$	9.91139e18	0.880979	0.440490	−	0.897758i	$-0.354805\pi$
$853$	9.91139e18	0.880979	0.440490	+	0.897758i	$0.354805\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.72370e18	0.321069	0.160535	−	0.987030i	$-0.448678\pi$
$857$	3.72370e18	0.321069	0.160535	+	0.987030i	$0.448678\pi$
$858$	0	0
$859$	−3.07811e18	−0.261414	−0.130707	−	0.991421i	$-0.541725\pi$
$859$	−3.07811e18	−0.261414	−0.130707	+	0.991421i	$0.541725\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	4.68138e18	0.385748	0.192874	−	0.981224i	$-0.438219\pi$
$863$	4.68138e18	0.385748	0.192874	+	0.981224i	$0.438219\pi$
$864$	0	0
$865$	1.22260e18	0.0992387
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2.61379e19	−2.05893
$870$	0	0
$871$	−5.76445e17	−0.0447342
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.46853e18	0.110619
$876$	0	0
$877$	−3.27869e18	−0.243334	−0.121667	−	0.992571i	$-0.538824\pi$
$877$	−3.27869e18	−0.243334	−0.121667	+	0.992571i	$0.538824\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	8.02437e18	0.578186	0.289093	−	0.957301i	$-0.406646\pi$
$881$	8.02437e18	0.578186	0.289093	+	0.957301i	$0.406646\pi$
$882$	0	0
$883$	−2.07925e19	−1.47626	−0.738128	−	0.674661i	$-0.764290\pi$
$883$	−2.07925e19	−1.47626	−0.738128	+	0.674661i	$0.764290\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.61191e19	−1.80075	−0.900376	−	0.435112i	$-0.856709\pi$
$887$	−2.61191e19	−1.80075	−0.900376	+	0.435112i	$0.856709\pi$
$888$	0	0
$889$	8.77489e18	0.596184
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2.19191e19	1.44640
$894$	0	0
$895$	7.17124e17	0.0466385
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−8.71138e18	−0.550363
$900$	0	0
$901$	2.07138e18	0.128988
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3.68974e18	0.223244
$906$	0	0
$907$	−2.55190e19	−1.52201	−0.761004	−	0.648747i	$-0.775293\pi$
$907$	−2.55190e19	−1.52201	−0.761004	+	0.648747i	$0.775293\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.62951e19	0.944466	0.472233	−	0.881474i	$-0.343448\pi$
$911$	1.62951e19	0.944466	0.472233	+	0.881474i	$0.343448\pi$
$912$	0	0
$913$	−2.71098e19	−1.54905
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−9.76826e17	−0.0542521
$918$	0	0
$919$	−2.25963e18	−0.123733	−0.0618667	−	0.998084i	$-0.519705\pi$
$919$	−2.25963e18	−0.123733	−0.0618667	+	0.998084i	$0.519705\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−3.13409e19	−1.66840
$924$	0	0
$925$	−2.19361e19	−1.15143
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.38768e19	0.708250	0.354125	−	0.935198i	$-0.384779\pi$
$929$	1.38768e19	0.708250	0.354125	+	0.935198i	$0.384779\pi$
$930$	0	0
$931$	−2.51463e19	−1.26561
$932$	0	0
$933$	0	0
$934$	0	0
$935$	6.66334e17	0.0326149
$936$	0	0
$937$	1.27938e19	0.617578	0.308789	−	0.951131i	$-0.400076\pi$
$937$	1.27938e19	0.617578	0.308789	+	0.951131i	$0.400076\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.00758e19	0.473093	0.236547	−	0.971620i	$-0.423984\pi$
$941$	1.00758e19	0.473093	0.236547	+	0.971620i	$0.423984\pi$
$942$	0	0
$943$	3.17241e19	1.46914
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.66672e19	−0.750912	−0.375456	−	0.926840i	$-0.622514\pi$
$947$	−1.66672e19	−0.750912	−0.375456	+	0.926840i	$0.622514\pi$
$948$	0	0
$949$	1.85007e19	0.822163
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−2.79060e19	−1.20668	−0.603341	−	0.797483i	$-0.706164\pi$
$953$	−2.79060e19	−1.20668	−0.603341	+	0.797483i	$0.706164\pi$
$954$	0	0
$955$	−4.03119e18	−0.171954
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−8.09981e18	−0.336243
$960$	0	0
$961$	−1.93397e19	−0.792042
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.80938e18	0.0721276
$966$	0	0
$967$	4.06377e19	1.59829	0.799147	−	0.601136i	$-0.205285\pi$
$967$	4.06377e19	1.59829	0.799147	+	0.601136i	$0.205285\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−3.91648e18	−0.149958	−0.0749792	−	0.997185i	$-0.523889\pi$
$971$	−3.91648e18	−0.149958	−0.0749792	+	0.997185i	$0.523889\pi$
$972$	0	0
$973$	5.50865e18	0.208119
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−8.91758e18	−0.328044	−0.164022	−	0.986457i	$-0.552447\pi$
$977$	−8.91758e18	−0.328044	−0.164022	+	0.986457i	$0.552447\pi$
$978$	0	0
$979$	−1.73587e19	−0.630130
$980$	0	0
$981$	0	0
$982$	0	0
$983$	3.03354e19	1.07239	0.536194	−	0.844095i	$-0.319862\pi$
$983$	3.03354e19	1.07239	0.536194	+	0.844095i	$0.319862\pi$
$984$	0	0
$985$	5.08430e18	0.177376
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.58446e19	0.538401
$990$	0	0
$991$	−4.90085e19	−1.64359	−0.821794	−	0.569785i	$-0.807027\pi$
$991$	−4.90085e19	−1.64359	−0.821794	+	0.569785i	$0.807027\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	4.12423e17	0.0134739
$996$	0	0
$997$	−1.57448e19	−0.507715	−0.253857	−	0.967242i	$-0.581699\pi$
$997$	−1.57448e19	−0.507715	−0.253857	+	0.967242i	$0.581699\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.14.a.a.1.1		1
3.2	odd	2		8.14.a.a.1.1	✓	1
4.3	odd	2		144.14.a.f.1.1		1
12.11	even	2		16.14.a.c.1.1		1
15.2	even	4		200.14.c.a.49.2		2
15.8	even	4		200.14.c.a.49.1		2
15.14	odd	2		200.14.a.a.1.1		1
24.5	odd	2		64.14.a.f.1.1		1
24.11	even	2		64.14.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.14.a.a.1.1	✓	1	3.2	odd	2
16.14.a.c.1.1		1	12.11	even	2
64.14.a.d.1.1		1	24.11	even	2
64.14.a.f.1.1		1	24.5	odd	2
72.14.a.a.1.1		1	1.1	even	1	trivial
144.14.a.f.1.1		1	4.3	odd	2
200.14.a.a.1.1		1	15.14	odd	2
200.14.c.a.49.1		2	15.8	even	4
200.14.c.a.49.2		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:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	72.a (trivial)

Introduction

L-functions

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Fields

Representations

Groups

Database

Properties

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