LMFDB - Embedded newform 72.18.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,18,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, 18, names="a")

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

chi := DirichletCharacter("72.1");

S:= CuspForms(chi, 18);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 18 $
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:	$131.919902888$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{2146}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2146 $ x^2 - 2146
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8}\cdot 3 $
Twist minimal:	no (minimal twist has level 8)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-46.3249$ of defining polynomial
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)

$f(q)$	$=$	$q-684741. q^{5} +1.14317e7 q^{7} +O(q^{10})$	$q-684741. q^{5} +1.14317e7 q^{7} +6.71354e8 q^{11} +5.24288e9 q^{13} -1.10085e10 q^{17} +1.02767e11 q^{19} -4.80722e11 q^{23} -2.94069e11 q^{25} +2.77099e12 q^{29} +1.01908e12 q^{31} -7.82775e12 q^{35} -4.01373e13 q^{37} -5.54603e13 q^{41} +7.56717e13 q^{43} +9.85390e13 q^{47} -1.01947e14 q^{49} +6.37330e14 q^{53} -4.59704e14 q^{55} +1.65484e15 q^{59} +2.22251e14 q^{61} -3.59002e15 q^{65} +1.72692e15 q^{67} -1.21864e15 q^{71} -9.87996e15 q^{73} +7.67472e15 q^{77} +2.90668e15 q^{79} -1.24773e16 q^{83} +7.53795e15 q^{85} +2.06746e16 q^{89} +5.99350e16 q^{91} -7.03688e16 q^{95} +5.68657e16 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 53620 q^{5} - 333168 q^{7}+O(q^{10}) $ 2 * q + 53620 * q^5 - 333168 * q^7	$ 2 q + 53620 q^{5} - 333168 q^{7} - 430974680 q^{11} + 2667521948 q^{13} - 60673503268 q^{17} + 178629960040 q^{19} - 528756594608 q^{23} - 511831510850 q^{25} + 7240660091460 q^{29} - 1878351140288 q^{31} - 16514472678240 q^{35} - 20332464566580 q^{37} + 1763041905324 q^{41} + 193394525968664 q^{43} - 100763837765472 q^{47} - 196165218276494 q^{49} + 317818146060052 q^{53} - 12\!\cdots\!20 q^{55}+ \cdots + 11\!\cdots\!08 q^{97}+O(q^{100}) $ 2 * q + 53620 * q^5 - 333168 * q^7 - 430974680 * q^11 + 2667521948 * q^13 - 60673503268 * q^17 + 178629960040 * q^19 - 528756594608 * q^23 - 511831510850 * q^25 + 7240660091460 * q^29 - 1878351140288 * q^31 - 16514472678240 * q^35 - 20332464566580 * q^37 + 1763041905324 * q^41 + 193394525968664 * q^43 - 100763837765472 * q^47 - 196165218276494 * q^49 + 317818146060052 * q^53 - 1273620709243120 * q^55 + 1262050788321736 * q^59 + 2765859692723708 * q^61 - 5491559693146280 * q^65 + 1963006544550088 * q^67 - 483639107104528 * q^71 - 2176892348591212 * q^73 + 20643473818671936 * q^77 - 5295839905627744 * q^79 - 9972518018887144 * q^83 - 29132785359362600 * q^85 + 711099036813900 * q^89 + 90233771615940576 * q^91 - 14354471748500080 * q^95 + 114870546609971908 * q^97

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
$5$	−684741.	−0.783937	−0.391969	−	0.919979i	$-0.628206\pi$
$5$	−684741.	−0.783937	−0.391969	+	0.919979i	$0.628206\pi$
$6$	0	0
$7$	1.14317e7	0.749510	0.374755	−	0.927124i	$-0.377727\pi$
$7$	1.14317e7	0.749510	0.374755	+	0.927124i	$0.377727\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	6.71354e8	0.944309	0.472154	−	0.881516i	$-0.343477\pi$
$11$	6.71354e8	0.944309	0.472154	+	0.881516i	$0.343477\pi$
$12$	0	0
$13$	5.24288e9	1.78259	0.891295	−	0.453424i	$-0.149798\pi$
$13$	5.24288e9	1.78259	0.891295	+	0.453424i	$0.149798\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.10085e10	−0.382746	−0.191373	−	0.981517i	$-0.561294\pi$
$17$	−1.10085e10	−0.382746	−0.191373	+	0.981517i	$0.561294\pi$
$18$	0	0
$19$	1.02767e11	1.38819	0.694093	−	0.719885i	$-0.255805\pi$
$19$	1.02767e11	1.38819	0.694093	+	0.719885i	$0.255805\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.80722e11	−1.27999	−0.639997	−	0.768378i	$-0.721064\pi$
$23$	−4.80722e11	−1.27999	−0.639997	+	0.768378i	$0.721064\pi$
$24$	0	0
$25$	−2.94069e11	−0.385442
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.77099e12	1.02861	0.514306	−	0.857607i	$-0.328050\pi$
$29$	2.77099e12	1.02861	0.514306	+	0.857607i	$0.328050\pi$
$30$	0	0
$31$	1.01908e12	0.214601	0.107301	−	0.994227i	$-0.465779\pi$
$31$	1.01908e12	0.214601	0.107301	+	0.994227i	$0.465779\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−7.82775e12	−0.587569
$36$	0	0
$37$	−4.01373e13	−1.87860	−0.939298	−	0.343102i	$-0.888522\pi$
$37$	−4.01373e13	−1.87860	−0.939298	+	0.343102i	$0.888522\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.54603e13	−1.08472	−0.542362	−	0.840145i	$-0.682470\pi$
$41$	−5.54603e13	−1.08472	−0.542362	+	0.840145i	$0.682470\pi$
$42$	0	0
$43$	7.56717e13	0.987305	0.493653	−	0.869659i	$-0.335661\pi$
$43$	7.56717e13	0.987305	0.493653	+	0.869659i	$0.335661\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.85390e13	0.603638	0.301819	−	0.953365i	$-0.402406\pi$
$47$	9.85390e13	0.603638	0.301819	+	0.953365i	$0.402406\pi$
$48$	0	0
$49$	−1.01947e14	−0.438235
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.37330e14	1.40611	0.703056	−	0.711135i	$-0.251819\pi$
$53$	6.37330e14	1.40611	0.703056	+	0.711135i	$0.251819\pi$
$54$	0	0
$55$	−4.59704e14	−0.740279
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.65484e15	1.46728	0.733640	−	0.679538i	$-0.237820\pi$
$59$	1.65484e15	1.46728	0.733640	+	0.679538i	$0.237820\pi$
$60$	0	0
$61$	2.22251e14	0.148436	0.0742182	−	0.997242i	$-0.476354\pi$
$61$	2.22251e14	0.148436	0.0742182	+	0.997242i	$0.476354\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.59002e15	−1.39744
$66$	0	0
$67$	1.72692e15	0.519560	0.259780	−	0.965668i	$-0.416350\pi$
$67$	1.72692e15	0.519560	0.259780	+	0.965668i	$0.416350\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.21864e15	−0.223965	−0.111983	−	0.993710i	$-0.535720\pi$
$71$	−1.21864e15	−0.223965	−0.111983	+	0.993710i	$0.535720\pi$
$72$	0	0
$73$	−9.87996e15	−1.43387	−0.716937	−	0.697138i	$-0.754456\pi$
$73$	−9.87996e15	−1.43387	−0.716937	+	0.697138i	$0.754456\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.67472e15	0.707769
$78$	0	0
$79$	2.90668e15	0.215559	0.107780	−	0.994175i	$-0.465626\pi$
$79$	2.90668e15	0.215559	0.107780	+	0.994175i	$0.465626\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1.24773e16	−0.608075	−0.304037	−	0.952660i	$-0.598335\pi$
$83$	−1.24773e16	−0.608075	−0.304037	+	0.952660i	$0.598335\pi$
$84$	0	0
$85$	7.53795e15	0.300049
$86$	0	0
$87$	0	0
$88$	0	0
$89$	2.06746e16	0.556701	0.278351	−	0.960480i	$-0.410212\pi$
$89$	2.06746e16	0.556701	0.278351	+	0.960480i	$0.410212\pi$
$90$	0	0
$91$	5.99350e16	1.33607
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−7.03688e16	−1.08825
$96$	0	0
$97$	5.68657e16	0.736700	0.368350	−	0.929687i	$-0.379923\pi$
$97$	5.68657e16	0.736700	0.368350	+	0.929687i	$0.379923\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.05844e15	0.0189150	0.00945751	−	0.999955i	$-0.496990\pi$
$101$	2.05844e15	0.0189150	0.00945751	+	0.999955i	$0.496990\pi$
$102$	0	0
$103$	−9.19881e16	−0.715509	−0.357754	−	0.933816i	$-0.616458\pi$
$103$	−9.19881e16	−0.715509	−0.357754	+	0.933816i	$0.616458\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.56515e17	1.44328	0.721640	−	0.692268i	$-0.243388\pi$
$107$	2.56515e17	1.44328	0.721640	+	0.692268i	$0.243388\pi$
$108$	0	0
$109$	−3.89834e16	−0.187393	−0.0936967	−	0.995601i	$-0.529868\pi$
$109$	−3.89834e16	−0.187393	−0.0936967	+	0.995601i	$0.529868\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.03583e17	−0.366542	−0.183271	−	0.983062i	$-0.558669\pi$
$113$	−1.03583e17	−0.366542	−0.183271	+	0.983062i	$0.558669\pi$
$114$	0	0
$115$	3.29170e17	1.00343
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.25845e17	−0.286872
$120$	0	0
$121$	−5.47304e16	−0.108281
$122$	0	0
$123$	0	0
$124$	0	0
$125$	7.23777e17	1.08610
$126$	0	0
$127$	−1.46041e17	−0.191488	−0.0957441	−	0.995406i	$-0.530523\pi$
$127$	−1.46041e17	−0.191488	−0.0957441	+	0.995406i	$0.530523\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.43335e18	−1.44393	−0.721967	−	0.691928i	$-0.756762\pi$
$131$	−1.43335e18	−1.44393	−0.721967	+	0.691928i	$0.756762\pi$
$132$	0	0
$133$	1.17480e18	1.04046
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.63665e18	1.81521	0.907606	−	0.419822i	$-0.137908\pi$
$137$	2.63665e18	1.81521	0.907606	+	0.419822i	$0.137908\pi$
$138$	0	0
$139$	2.11287e18	1.28602	0.643009	−	0.765858i	$-0.277686\pi$
$139$	2.11287e18	1.28602	0.643009	+	0.765858i	$0.277686\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.51983e18	1.68332
$144$	0	0
$145$	−1.89741e18	−0.806367
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.47689e18	−0.498041	−0.249021	−	0.968498i	$-0.580109\pi$
$149$	−1.47689e18	−0.498041	−0.249021	+	0.968498i	$0.580109\pi$
$150$	0	0
$151$	2.16874e18	0.652985	0.326492	−	0.945200i	$-0.394133\pi$
$151$	2.16874e18	0.652985	0.326492	+	0.945200i	$0.394133\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−6.97803e17	−0.168234
$156$	0	0
$157$	6.31830e18	1.36601	0.683004	−	0.730415i	$-0.260673\pi$
$157$	6.31830e18	1.36601	0.683004	+	0.730415i	$0.260673\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−5.49547e18	−0.959368
$162$	0	0
$163$	1.04630e19	1.64461	0.822306	−	0.569046i	$-0.192687\pi$
$163$	1.04630e19	1.64461	0.822306	+	0.569046i	$0.192687\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.20560e19	1.54210	0.771052	−	0.636773i	$-0.219731\pi$
$167$	1.20560e19	1.54210	0.771052	+	0.636773i	$0.219731\pi$
$168$	0	0
$169$	1.88374e19	2.17763
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3.42778e18	0.324804	0.162402	−	0.986725i	$-0.448076\pi$
$173$	3.42778e18	0.324804	0.162402	+	0.986725i	$0.448076\pi$
$174$	0	0
$175$	−3.36171e18	−0.288893
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1.72370e19	−1.22239	−0.611197	−	0.791479i	$-0.709312\pi$
$179$	−1.72370e19	−1.22239	−0.611197	+	0.791479i	$0.709312\pi$
$180$	0	0
$181$	−1.00941e19	−0.651331	−0.325665	−	0.945485i	$-0.605588\pi$
$181$	−1.00941e19	−0.651331	−0.325665	+	0.945485i	$0.605588\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.74837e19	1.47270
$186$	0	0
$187$	−7.39058e18	−0.361431
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.43562e19	0.995005	0.497503	−	0.867463i	$-0.334251\pi$
$191$	2.43562e19	0.995005	0.497503	+	0.867463i	$0.334251\pi$
$192$	0	0
$193$	3.53853e19	1.32308	0.661539	−	0.749911i	$-0.269904\pi$
$193$	3.53853e19	1.32308	0.661539	+	0.749911i	$0.269904\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−8.04294e18	−0.252611	−0.126305	−	0.991991i	$-0.540312\pi$
$197$	−8.04294e18	−0.252611	−0.126305	+	0.991991i	$0.540312\pi$
$198$	0	0
$199$	4.98342e19	1.43640	0.718201	−	0.695835i	$-0.244966\pi$
$199$	4.98342e19	1.43640	0.718201	+	0.695835i	$0.244966\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.16771e19	0.770955
$204$	0	0
$205$	3.79759e19	0.850356
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6.89930e19	1.31088
$210$	0	0
$211$	−1.14353e19	−0.200376	−0.100188	−	0.994969i	$-0.531944\pi$
$211$	−1.14353e19	−0.200376	−0.100188	+	0.994969i	$0.531944\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5.18155e19	−0.773985
$216$	0	0
$217$	1.16498e19	0.160846
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−5.77161e19	−0.682279
$222$	0	0
$223$	−9.70378e19	−1.06255	−0.531275	−	0.847199i	$-0.678287\pi$
$223$	−9.70378e19	−1.06255	−0.531275	+	0.847199i	$0.678287\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.56045e19	0.335186	0.167593	−	0.985856i	$-0.446401\pi$
$227$	3.56045e19	0.335186	0.167593	+	0.985856i	$0.446401\pi$
$228$	0	0
$229$	−2.56164e19	−0.223829	−0.111914	−	0.993718i	$-0.535698\pi$
$229$	−2.56164e19	−0.223829	−0.111914	+	0.993718i	$0.535698\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.28548e19	−0.0969483	−0.0484741	−	0.998824i	$-0.515436\pi$
$233$	−1.28548e19	−0.0969483	−0.0484741	+	0.998824i	$0.515436\pi$
$234$	0	0
$235$	−6.74737e19	−0.473214
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2.31860e20	−1.40878	−0.704390	−	0.709813i	$-0.748779\pi$
$239$	−2.31860e20	−1.40878	−0.704390	+	0.709813i	$0.748779\pi$
$240$	0	0
$241$	−2.61676e17	−0.00148122	−0.000740609	−	1.00000i	$-0.500236\pi$
$241$	−2.61676e17	−0.00148122	−0.000740609		1.00000i	$0.500236\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.98072e19	0.343549
$246$	0	0
$247$	5.38795e20	2.47457
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−7.78336e19	−0.311846	−0.155923	−	0.987769i	$-0.549835\pi$
$251$	−7.78336e19	−0.311846	−0.155923	+	0.987769i	$0.549835\pi$
$252$	0	0
$253$	−3.22735e20	−1.20871
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7.89910e19	−0.258909	−0.129454	−	0.991585i	$-0.541323\pi$
$257$	−7.89910e19	−0.258909	−0.129454	+	0.991585i	$0.541323\pi$
$258$	0	0
$259$	−4.58837e20	−1.40803
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2.40675e20	−0.648348	−0.324174	−	0.945998i	$-0.605086\pi$
$263$	−2.40675e20	−0.648348	−0.324174	+	0.945998i	$0.605086\pi$
$264$	0	0
$265$	−4.36406e20	−1.10230
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5.42480e20	1.20639	0.603197	−	0.797592i	$-0.293893\pi$
$269$	5.42480e20	1.20639	0.603197	+	0.797592i	$0.293893\pi$
$270$	0	0
$271$	−4.29207e20	−0.896248	−0.448124	−	0.893971i	$-0.647908\pi$
$271$	−4.29207e20	−0.896248	−0.448124	+	0.893971i	$0.647908\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.97425e20	−0.363976
$276$	0	0
$277$	6.37181e20	1.10455	0.552275	−	0.833662i	$-0.313760\pi$
$277$	6.37181e20	1.10455	0.552275	+	0.833662i	$0.313760\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5.23193e20	0.802894	0.401447	−	0.915882i	$-0.368507\pi$
$281$	5.23193e20	0.802894	0.401447	+	0.915882i	$0.368507\pi$
$282$	0	0
$283$	−1.19945e21	−1.73300	−0.866499	−	0.499180i	$-0.833635\pi$
$283$	−1.19945e21	−1.73300	−0.866499	+	0.499180i	$0.833635\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.34005e20	−0.813012
$288$	0	0
$289$	−7.06054e20	−0.853505
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−6.90842e20	−0.743026	−0.371513	−	0.928428i	$-0.621161\pi$
$293$	−6.90842e20	−0.743026	−0.371513	+	0.928428i	$0.621161\pi$
$294$	0	0
$295$	−1.13313e21	−1.15026
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.52037e21	−2.28170
$300$	0	0
$301$	8.65056e20	0.739995
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.52184e20	−0.116365
$306$	0	0
$307$	−1.67762e21	−1.21343	−0.606717	−	0.794918i	$-0.707514\pi$
$307$	−1.67762e21	−1.21343	−0.606717	+	0.794918i	$0.707514\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2.35150e21	1.52364	0.761818	−	0.647791i	$-0.224307\pi$
$311$	2.35150e21	1.52364	0.761818	+	0.647791i	$0.224307\pi$
$312$	0	0
$313$	5.33805e20	0.327533	0.163767	−	0.986499i	$-0.447636\pi$
$313$	5.33805e20	0.327533	0.163767	+	0.986499i	$0.447636\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.01032e21	1.10729	0.553645	−	0.832753i	$-0.313237\pi$
$317$	2.01032e21	1.10729	0.553645	+	0.832753i	$0.313237\pi$
$318$	0	0
$319$	1.86031e21	0.971327
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.13131e21	−0.531323
$324$	0	0
$325$	−1.54177e21	−0.687086
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.12647e21	0.452432
$330$	0	0
$331$	3.24927e21	1.23950	0.619752	−	0.784798i	$-0.287233\pi$
$331$	3.24927e21	1.23950	0.619752	+	0.784798i	$0.287233\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.18249e21	−0.407303
$336$	0	0
$337$	2.35844e21	0.772272	0.386136	−	0.922442i	$-0.373810\pi$
$337$	2.35844e21	0.772272	0.386136	+	0.922442i	$0.373810\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	6.84161e20	0.202650
$342$	0	0
$343$	−3.82479e21	−1.07797
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4.44155e21	−1.13431	−0.567157	−	0.823609i	$-0.691957\pi$
$347$	−4.44155e21	−1.13431	−0.567157	+	0.823609i	$0.691957\pi$
$348$	0	0
$349$	3.88021e21	0.943711	0.471855	−	0.881676i	$-0.343584\pi$
$349$	3.88021e21	0.943711	0.471855	+	0.881676i	$0.343584\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1.29170e20	−0.0285152	−0.0142576	−	0.999898i	$-0.504538\pi$
$353$	−1.29170e20	−0.0285152	−0.0142576	+	0.999898i	$0.504538\pi$
$354$	0	0
$355$	8.34455e20	0.175575
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.74029e21	−0.524197	−0.262098	−	0.965041i	$-0.584415\pi$
$359$	−2.74029e21	−0.524197	−0.262098	+	0.965041i	$0.584415\pi$
$360$	0	0
$361$	5.08066e21	0.927063
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6.76521e21	1.12407
$366$	0	0
$367$	−2.61315e21	−0.414479	−0.207239	−	0.978290i	$-0.566448\pi$
$367$	−2.61315e21	−0.414479	−0.207239	+	0.978290i	$0.566448\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	7.28577e21	1.05389
$372$	0	0
$373$	6.19272e21	0.855768	0.427884	−	0.903834i	$-0.359259\pi$
$373$	6.19272e21	0.855768	0.427884	+	0.903834i	$0.359259\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.45280e22	1.83359
$378$	0	0
$379$	1.47374e21	0.177823	0.0889113	−	0.996040i	$-0.471661\pi$
$379$	1.47374e21	0.177823	0.0889113	+	0.996040i	$0.471661\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3.39392e21	−0.374552	−0.187276	−	0.982307i	$-0.559966\pi$
$383$	−3.39392e21	−0.374552	−0.187276	+	0.982307i	$0.559966\pi$
$384$	0	0
$385$	−5.25520e21	−0.554846
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−5.88689e21	−0.569265	−0.284632	−	0.958637i	$-0.591872\pi$
$389$	−5.88689e21	−0.569265	−0.284632	+	0.958637i	$0.591872\pi$
$390$	0	0
$391$	5.29201e21	0.489913
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.99032e21	−0.168985
$396$	0	0
$397$	1.80920e22	1.47153	0.735763	−	0.677239i	$-0.236824\pi$
$397$	1.80920e22	1.47153	0.735763	+	0.677239i	$0.236824\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.37678e22	1.77526	0.887629	−	0.460558i	$-0.152351\pi$
$401$	2.37678e22	1.77526	0.887629	+	0.460558i	$0.152351\pi$
$402$	0	0
$403$	5.34289e21	0.382546
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.69463e22	−1.77397
$408$	0	0
$409$	1.01542e21	0.0641205	0.0320603	−	0.999486i	$-0.489793\pi$
$409$	1.01542e21	0.0641205	0.0320603	+	0.999486i	$0.489793\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.89176e22	1.09974
$414$	0	0
$415$	8.54373e21	0.476693
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.19062e22	−0.612287	−0.306144	−	0.951985i	$-0.599039\pi$
$419$	−1.19062e22	−0.612287	−0.306144	+	0.951985i	$0.599039\pi$
$420$	0	0
$421$	−5.70081e21	−0.281539	−0.140770	−	0.990042i	$-0.544958\pi$
$421$	−5.70081e21	−0.281539	−0.140770	+	0.990042i	$0.544958\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.23725e21	0.147527
$426$	0	0
$427$	2.54071e21	0.111255
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.33091e22	0.942906	0.471453	−	0.881891i	$-0.343730\pi$
$431$	2.33091e22	0.942906	0.471453	+	0.881891i	$0.343730\pi$
$432$	0	0
$433$	−5.82694e21	−0.226618	−0.113309	−	0.993560i	$-0.536145\pi$
$433$	−5.82694e21	−0.226618	−0.113309	+	0.993560i	$0.536145\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.94024e22	−1.77687
$438$	0	0
$439$	−1.39338e22	−0.482081	−0.241041	−	0.970515i	$-0.577489\pi$
$439$	−1.39338e22	−0.482081	−0.241041	+	0.970515i	$0.577489\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3.73936e22	1.19775	0.598874	−	0.800843i	$-0.295615\pi$
$443$	3.73936e22	1.19775	0.598874	+	0.800843i	$0.295615\pi$
$444$	0	0
$445$	−1.41568e22	−0.436419
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−3.61756e22	−1.03353	−0.516764	−	0.856128i	$-0.672863\pi$
$449$	−3.61756e22	−1.03353	−0.516764	+	0.856128i	$0.672863\pi$
$450$	0	0
$451$	−3.72335e22	−1.02431
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4.10400e22	−1.04739
$456$	0	0
$457$	4.49241e22	1.10457	0.552283	−	0.833657i	$-0.313757\pi$
$457$	4.49241e22	1.10457	0.552283	+	0.833657i	$0.313757\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−6.69974e20	−0.0152968	−0.00764839	−	0.999971i	$-0.502435\pi$
$461$	−6.69974e20	−0.0152968	−0.00764839	+	0.999971i	$0.502435\pi$
$462$	0	0
$463$	1.32337e22	0.291234	0.145617	−	0.989341i	$-0.453483\pi$
$463$	1.32337e22	0.291234	0.145617	+	0.989341i	$0.453483\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.65052e22	−1.36039	−0.680193	−	0.733033i	$-0.738104\pi$
$467$	−6.65052e22	−1.36039	−0.680193	+	0.733033i	$0.738104\pi$
$468$	0	0
$469$	1.97416e22	0.389416
$470$	0	0
$471$	0	0
$472$	0	0
$473$	5.08025e22	0.932321
$474$	0	0
$475$	−3.02206e22	−0.535066
$476$	0	0
$477$	0	0
$478$	0	0
$479$	5.70928e22	0.941303	0.470651	−	0.882319i	$-0.344019\pi$
$479$	5.70928e22	0.941303	0.470651	+	0.882319i	$0.344019\pi$
$480$	0	0
$481$	−2.10435e23	−3.34877
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.89383e22	−0.577526
$486$	0	0
$487$	−5.38038e22	−0.770579	−0.385289	−	0.922796i	$-0.625898\pi$
$487$	−5.38038e22	−0.770579	−0.385289	+	0.922796i	$0.625898\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	8.23196e22	1.09979	0.549896	−	0.835233i	$-0.314667\pi$
$491$	8.23196e22	1.09979	0.549896	+	0.835233i	$0.314667\pi$
$492$	0	0
$493$	−3.05043e22	−0.393697
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.39312e22	−0.167864
$498$	0	0
$499$	1.30825e22	0.152348	0.0761739	−	0.997095i	$-0.475730\pi$
$499$	1.30825e22	0.152348	0.0761739	+	0.997095i	$0.475730\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.91572e22	−0.317262	−0.158631	−	0.987338i	$-0.550708\pi$
$503$	−2.91572e22	−0.317262	−0.158631	+	0.987338i	$0.550708\pi$
$504$	0	0
$505$	−1.40950e21	−0.0148282
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−2.99375e22	−0.294520	−0.147260	−	0.989098i	$-0.547045\pi$
$509$	−2.99375e22	−0.294520	−0.147260	+	0.989098i	$0.547045\pi$
$510$	0	0
$511$	−1.12945e23	−1.07470
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6.29880e22	0.560914
$516$	0	0
$517$	6.61546e22	0.570020
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.35616e23	−1.09444	−0.547219	−	0.836989i	$-0.684314\pi$
$521$	−1.35616e23	−1.09444	−0.547219	+	0.836989i	$0.684314\pi$
$522$	0	0
$523$	2.21394e23	1.72942	0.864711	−	0.502271i	$-0.167502\pi$
$523$	2.21394e23	1.72942	0.864711	+	0.502271i	$0.167502\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.12185e22	−0.0821377
$528$	0	0
$529$	9.00439e22	0.638383
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2.90772e23	−1.93362
$534$	0	0
$535$	−1.75646e23	−1.13144
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−6.84424e22	−0.413829
$540$	0	0
$541$	−1.87467e23	−1.09837	−0.549186	−	0.835700i	$-0.685062\pi$
$541$	−1.87467e23	−1.09837	−0.549186	+	0.835700i	$0.685062\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.66935e22	0.146905
$546$	0	0
$547$	−8.76202e22	−0.467425	−0.233713	−	0.972306i	$-0.575087\pi$
$547$	−8.76202e22	−0.467425	−0.233713	+	0.972306i	$0.575087\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2.84766e23	1.42791
$552$	0	0
$553$	3.32283e22	0.161564
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.52062e23	1.15276	0.576378	−	0.817183i	$-0.304465\pi$
$557$	2.52062e23	1.15276	0.576378	+	0.817183i	$0.304465\pi$
$558$	0	0
$559$	3.96738e23	1.75996
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.31530e22	0.138421	0.0692105	−	0.997602i	$-0.477952\pi$
$563$	3.31530e22	0.138421	0.0692105	+	0.997602i	$0.477952\pi$
$564$	0	0
$565$	7.09278e22	0.287346
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.02213e23	−0.389990	−0.194995	−	0.980804i	$-0.562469\pi$
$569$	−1.02213e23	−0.389990	−0.194995	+	0.980804i	$0.562469\pi$
$570$	0	0
$571$	2.69197e23	0.996928	0.498464	−	0.866910i	$-0.333898\pi$
$571$	2.69197e23	0.996928	0.498464	+	0.866910i	$0.333898\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.41366e23	0.493364
$576$	0	0
$577$	−2.40015e23	−0.813286	−0.406643	−	0.913587i	$-0.633301\pi$
$577$	−2.40015e23	−0.813286	−0.406643	+	0.913587i	$0.633301\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1.42637e23	−0.455758
$582$	0	0
$583$	4.27874e23	1.32780
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.35931e23	−0.398011	−0.199005	−	0.979998i	$-0.563771\pi$
$587$	−1.35931e23	−0.398011	−0.199005	+	0.979998i	$0.563771\pi$
$588$	0	0
$589$	1.04727e23	0.297906
$590$	0	0
$591$	0	0
$592$	0	0
$593$	4.20710e23	1.12984	0.564921	−	0.825145i	$-0.308907\pi$
$593$	4.20710e23	1.12984	0.564921	+	0.825145i	$0.308907\pi$
$594$	0	0
$595$	8.61715e22	0.224890
$596$	0	0
$597$	0	0
$598$	0	0
$599$	3.37249e23	0.831425	0.415712	−	0.909496i	$-0.363532\pi$
$599$	3.37249e23	0.831425	0.415712	+	0.909496i	$0.363532\pi$
$600$	0	0
$601$	−7.45318e22	−0.178611	−0.0893056	−	0.996004i	$-0.528465\pi$
$601$	−7.45318e22	−0.178611	−0.0893056	+	0.996004i	$0.528465\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3.74761e22	0.0848856
$606$	0	0
$607$	5.68051e23	1.25108	0.625538	−	0.780194i	$-0.284880\pi$
$607$	5.68051e23	1.25108	0.625538	+	0.780194i	$0.284880\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.16628e23	1.07604
$612$	0	0
$613$	5.66099e23	1.14677	0.573387	−	0.819284i	$-0.305629\pi$
$613$	5.66099e23	1.14677	0.573387	+	0.819284i	$0.305629\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−7.70454e23	−1.47680	−0.738402	−	0.674361i	$-0.764419\pi$
$617$	−7.70454e23	−1.47680	−0.738402	+	0.674361i	$0.764419\pi$
$618$	0	0
$619$	−3.95903e23	−0.738275	−0.369138	−	0.929375i	$-0.620347\pi$
$619$	−3.95903e23	−0.738275	−0.369138	+	0.929375i	$0.620347\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.36346e23	0.417253
$624$	0	0
$625$	−2.71243e23	−0.465992
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4.41850e23	0.719026
$630$	0	0
$631$	−2.38852e23	−0.378337	−0.189168	−	0.981945i	$-0.560579\pi$
$631$	−2.38852e23	−0.378337	−0.189168	+	0.981945i	$0.560579\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.00000e23	0.150115
$636$	0	0
$637$	−5.34495e23	−0.781193
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2.88419e23	−0.399697	−0.199848	−	0.979827i	$-0.564045\pi$
$641$	−2.88419e23	−0.399697	−0.199848	+	0.979827i	$0.564045\pi$
$642$	0	0
$643$	−7.78497e23	−1.05066	−0.525332	−	0.850897i	$-0.676059\pi$
$643$	−7.78497e23	−1.05066	−0.525332	+	0.850897i	$0.676059\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−3.76233e23	−0.481693	−0.240847	−	0.970563i	$-0.577425\pi$
$647$	−3.76233e23	−0.481693	−0.240847	+	0.970563i	$0.577425\pi$
$648$	0	0
$649$	1.11098e24	1.38557
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.76191e22	0.0563663	0.0281831	−	0.999603i	$-0.491028\pi$
$653$	4.76191e22	0.0563663	0.0281831	+	0.999603i	$0.491028\pi$
$654$	0	0
$655$	9.81476e23	1.13195
$656$	0	0
$657$	0	0
$658$	0	0
$659$	2.28586e23	0.250336	0.125168	−	0.992136i	$-0.460053\pi$
$659$	2.28586e23	0.250336	0.125168	+	0.992136i	$0.460053\pi$
$660$	0	0
$661$	1.23156e24	1.31445	0.657224	−	0.753695i	$-0.271731\pi$
$661$	1.23156e24	1.31445	0.657224	+	0.753695i	$0.271731\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−8.04434e23	−0.815655
$666$	0	0
$667$	−1.33208e24	−1.31662
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.49209e23	0.140170
$672$	0	0
$673$	3.22012e23	0.294947	0.147474	−	0.989066i	$-0.452886\pi$
$673$	3.22012e23	0.294947	0.147474	+	0.989066i	$0.452886\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−6.06657e23	−0.528371	−0.264186	−	0.964472i	$-0.585103\pi$
$677$	−6.06657e23	−0.528371	−0.264186	+	0.964472i	$0.585103\pi$
$678$	0	0
$679$	6.50071e23	0.552164
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−7.97688e23	−0.644551	−0.322276	−	0.946646i	$-0.604448\pi$
$683$	−7.97688e23	−0.644551	−0.322276	+	0.946646i	$0.604448\pi$
$684$	0	0
$685$	−1.80542e24	−1.42301
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3.34145e24	2.50652
$690$	0	0
$691$	2.47998e24	1.81503	0.907516	−	0.420018i	$-0.137976\pi$
$691$	2.47998e24	1.81503	0.907516	+	0.420018i	$0.137976\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.44677e24	−1.00816
$696$	0	0
$697$	6.10533e23	0.415174
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−6.61477e23	−0.428462	−0.214231	−	0.976783i	$-0.568724\pi$
$701$	−6.61477e23	−0.428462	−0.214231	+	0.976783i	$0.568724\pi$
$702$	0	0
$703$	−4.12479e24	−2.60784
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.35315e22	0.0141770
$708$	0	0
$709$	−2.60176e24	−1.53029	−0.765147	−	0.643856i	$-0.777334\pi$
$709$	−2.60176e24	−1.53029	−0.765147	+	0.643856i	$0.777334\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−4.89892e23	−0.274688
$714$	0	0
$715$	−2.41017e24	−1.31961
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−2.59675e24	−1.35592	−0.677962	−	0.735097i	$-0.737137\pi$
$719$	−2.59675e24	−1.35592	−0.677962	+	0.735097i	$0.737137\pi$
$720$	0	0
$721$	−1.05158e24	−0.536281
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−8.14862e23	−0.396471
$726$	0	0
$727$	−2.73046e24	−1.29776	−0.648879	−	0.760892i	$-0.724762\pi$
$727$	−2.73046e24	−1.29776	−0.648879	+	0.760892i	$0.724762\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−8.33029e23	−0.377887
$732$	0	0
$733$	3.56499e23	0.158006	0.0790031	−	0.996874i	$-0.474826\pi$
$733$	3.56499e23	0.158006	0.0790031	+	0.996874i	$0.474826\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.15937e24	0.490625
$738$	0	0
$739$	2.56674e24	1.06146	0.530730	−	0.847541i	$-0.321918\pi$
$739$	2.56674e24	1.06146	0.530730	+	0.847541i	$0.321918\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.57755e24	−0.623128	−0.311564	−	0.950225i	$-0.600853\pi$
$743$	−1.57755e24	−0.623128	−0.311564	+	0.950225i	$0.600853\pi$
$744$	0	0
$745$	1.01129e24	0.390433
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2.93240e24	1.08175
$750$	0	0
$751$	5.71108e23	0.205958	0.102979	−	0.994684i	$-0.467163\pi$
$751$	5.71108e23	0.205958	0.102979	+	0.994684i	$0.467163\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1.48502e24	−0.511899
$756$	0	0
$757$	−6.66627e23	−0.224682	−0.112341	−	0.993670i	$-0.535835\pi$
$757$	−6.66627e23	−0.224682	−0.112341	+	0.993670i	$0.535835\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−3.50345e24	−1.12908	−0.564542	−	0.825404i	$-0.690947\pi$
$761$	−3.50345e24	−1.12908	−0.564542	+	0.825404i	$0.690947\pi$
$762$	0	0
$763$	−4.45646e23	−0.140453
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.67611e24	2.61556
$768$	0	0
$769$	−1.55948e24	−0.459840	−0.229920	−	0.973210i	$-0.573846\pi$
$769$	−1.55948e24	−0.459840	−0.229920	+	0.973210i	$0.573846\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	4.68122e24	1.32079	0.660394	−	0.750919i	$-0.270389\pi$
$773$	4.68122e24	1.32079	0.660394	+	0.750919i	$0.270389\pi$
$774$	0	0
$775$	−2.99679e23	−0.0827163
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5.69949e24	−1.50580
$780$	0	0
$781$	−8.18141e23	−0.211492
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−4.32640e24	−1.07086
$786$	0	0
$787$	−3.84015e24	−0.930172	−0.465086	−	0.885265i	$-0.653977\pi$
$787$	−3.84015e24	−0.930172	−0.465086	+	0.885265i	$0.653977\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.18413e24	−0.274727
$792$	0	0
$793$	1.16524e24	0.264601
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7.56582e24	1.64611	0.823057	−	0.567959i	$-0.192267\pi$
$797$	7.56582e24	1.64611	0.823057	+	0.567959i	$0.192267\pi$
$798$	0	0
$799$	−1.08476e24	−0.231040
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−6.63295e24	−1.35402
$804$	0	0
$805$	3.76298e24	0.752084
$806$	0	0
$807$	0	0
$808$	0	0
$809$	7.52328e24	1.44160	0.720800	−	0.693143i	$-0.243774\pi$
$809$	7.52328e24	1.44160	0.720800	+	0.693143i	$0.243774\pi$
$810$	0	0
$811$	1.00203e25	1.88020	0.940102	−	0.340893i	$-0.110729\pi$
$811$	1.00203e25	1.88020	0.940102	+	0.340893i	$0.110729\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−7.16447e24	−1.28927
$816$	0	0
$817$	7.77655e24	1.37056
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.21167e24	0.373941	0.186970	−	0.982366i	$-0.440133\pi$
$821$	2.21167e24	0.373941	0.186970	+	0.982366i	$0.440133\pi$
$822$	0	0
$823$	2.54721e24	0.421858	0.210929	−	0.977501i	$-0.432351\pi$
$823$	2.54721e24	0.421858	0.210929	+	0.977501i	$0.432351\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.70056e24	−0.429197	−0.214599	−	0.976702i	$-0.568844\pi$
$827$	−2.70056e24	−0.429197	−0.214599	+	0.976702i	$0.568844\pi$
$828$	0	0
$829$	6.08133e24	0.946859	0.473430	−	0.880832i	$-0.343016\pi$
$829$	6.08133e24	0.946859	0.473430	+	0.880832i	$0.343016\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.12228e24	0.167733
$834$	0	0
$835$	−8.25524e24	−1.20891
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−3.18225e24	−0.447464	−0.223732	−	0.974651i	$-0.571824\pi$
$839$	−3.18225e24	−0.447464	−0.223732	+	0.974651i	$0.571824\pi$
$840$	0	0
$841$	4.21223e23	0.0580425
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.28987e25	−1.70712
$846$	0	0
$847$	−6.25661e23	−0.0811578
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.92949e25	2.40459
$852$	0	0
$853$	2.43556e24	0.297531	0.148766	−	0.988872i	$-0.452470\pi$
$853$	2.43556e24	0.297531	0.148766	+	0.988872i	$0.452470\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.08693e23	−0.0127604	−0.00638022	−	0.999980i	$-0.502031\pi$
$857$	−1.08693e23	−0.0127604	−0.00638022	+	0.999980i	$0.502031\pi$
$858$	0	0
$859$	−2.41916e24	−0.278435	−0.139217	−	0.990262i	$-0.544459\pi$
$859$	−2.41916e24	−0.278435	−0.139217	+	0.990262i	$0.544459\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.43640e25	1.58922	0.794608	−	0.607123i	$-0.207677\pi$
$863$	1.43640e25	1.58922	0.794608	+	0.607123i	$0.207677\pi$
$864$	0	0
$865$	−2.34714e24	−0.254626
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.95141e24	0.203555
$870$	0	0
$871$	9.05402e24	0.926163
$872$	0	0
$873$	0	0
$874$	0	0
$875$	8.27400e24	0.814043
$876$	0	0
$877$	4.42394e24	0.426887	0.213443	−	0.976955i	$-0.431532\pi$
$877$	4.42394e24	0.426887	0.213443	+	0.976955i	$0.431532\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−7.22889e24	−0.671084	−0.335542	−	0.942025i	$-0.608919\pi$
$881$	−7.22889e24	−0.671084	−0.335542	+	0.942025i	$0.608919\pi$
$882$	0	0
$883$	1.56510e25	1.42520	0.712599	−	0.701572i	$-0.247518\pi$
$883$	1.56510e25	1.42520	0.712599	+	0.701572i	$0.247518\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	9.58184e24	0.839650	0.419825	−	0.907605i	$-0.362092\pi$
$887$	9.58184e24	0.839650	0.419825	+	0.907605i	$0.362092\pi$
$888$	0	0
$889$	−1.66949e24	−0.143522
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.01266e25	0.837962
$894$	0	0
$895$	1.18029e25	0.958280
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.82384e24	0.220741
$900$	0	0
$901$	−7.01603e24	−0.538184
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6.91188e24	0.510603
$906$	0	0
$907$	−1.51534e25	−1.09863	−0.549313	−	0.835617i	$-0.685110\pi$
$907$	−1.51534e25	−1.09863	−0.549313	+	0.835617i	$0.685110\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−2.15196e25	−1.50289	−0.751445	−	0.659795i	$-0.770643\pi$
$911$	−2.15196e25	−1.50289	−0.751445	+	0.659795i	$0.770643\pi$
$912$	0	0
$913$	−8.37669e24	−0.574210
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.63857e25	−1.08224
$918$	0	0
$919$	−5.75945e24	−0.373422	−0.186711	−	0.982415i	$-0.559783\pi$
$919$	−5.75945e24	−0.373422	−0.186711	+	0.982415i	$0.559783\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−6.38920e24	−0.399238
$924$	0	0
$925$	1.18031e25	0.724090
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.80184e25	−1.06557	−0.532784	−	0.846251i	$-0.678854\pi$
$929$	−1.80184e25	−1.06557	−0.532784	+	0.846251i	$0.678854\pi$
$930$	0	0
$931$	−1.04768e25	−0.608352
$932$	0	0
$933$	0	0
$934$	0	0
$935$	5.06063e24	0.283339
$936$	0	0
$937$	1.21756e25	0.669428	0.334714	−	0.942320i	$-0.391360\pi$
$937$	1.21756e25	0.669428	0.334714	+	0.942320i	$0.391360\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	4.16970e24	0.221102	0.110551	−	0.993870i	$-0.464738\pi$
$941$	4.16970e24	0.221102	0.110551	+	0.993870i	$0.464738\pi$
$942$	0	0
$943$	2.66610e25	1.38844
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.68683e24	−0.235453	−0.117726	−	0.993046i	$-0.537561\pi$
$947$	−4.68683e24	−0.235453	−0.117726	+	0.993046i	$0.537561\pi$
$948$	0	0
$949$	−5.17994e25	−2.55601
$950$	0	0
$951$	0	0
$952$	0	0
$953$	2.40507e25	1.14509	0.572543	−	0.819875i	$-0.305957\pi$
$953$	2.40507e25	1.14509	0.572543	+	0.819875i	$0.305957\pi$
$954$	0	0
$955$	−1.66777e25	−0.780022
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3.01414e25	1.36052
$960$	0	0
$961$	−2.15116e25	−0.953946
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−2.42298e25	−1.03721
$966$	0	0
$967$	2.05735e25	0.865331	0.432665	−	0.901555i	$-0.357573\pi$
$967$	2.05735e25	0.865331	0.432665	+	0.901555i	$0.357573\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−1.34864e25	−0.547687	−0.273844	−	0.961774i	$-0.588295\pi$
$971$	−1.34864e25	−0.547687	−0.273844	+	0.961774i	$0.588295\pi$
$972$	0	0
$973$	2.41537e25	0.963884
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4.43897e25	−1.71072	−0.855359	−	0.518036i	$-0.826663\pi$
$977$	−4.43897e25	−1.71072	−0.855359	+	0.518036i	$0.826663\pi$
$978$	0	0
$979$	1.38800e25	0.525698
$980$	0	0
$981$	0	0
$982$	0	0
$983$	2.03497e23	0.00744481	0.00372240	−	0.999993i	$-0.498815\pi$
$983$	2.03497e23	0.00744481	0.00372240	+	0.999993i	$0.498815\pi$
$984$	0	0
$985$	5.50733e24	0.198031
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3.63771e25	−1.26374
$990$	0	0
$991$	−5.54664e25	−1.89411	−0.947053	−	0.321077i	$-0.895955\pi$
$991$	−5.54664e25	−1.89411	−0.947053	+	0.321077i	$0.895955\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−3.41235e25	−1.12605
$996$	0	0
$997$	−1.47645e25	−0.478971	−0.239485	−	0.970900i	$-0.576979\pi$
$997$	−1.47645e25	−0.478971	−0.239485	+	0.970900i	$0.576979\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.18.a.c.1.1		2
3.2	odd	2		8.18.a.a.1.2	✓	2
12.11	even	2		16.18.a.d.1.1		2
24.5	odd	2		64.18.a.k.1.1		2
24.11	even	2		64.18.a.h.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.18.a.a.1.2	✓	2	3.2	odd	2
16.18.a.d.1.1		2	12.11	even	2
64.18.a.h.1.2		2	24.11	even	2
64.18.a.k.1.1		2	24.5	odd	2
72.18.a.c.1.1		2	1.1	even	1	trivial

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:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	72.a (trivial)

\(f(q)\)	\(=\)	\(q-684741. q^{5} +1.14317e7 q^{7} +O(q^{10})\)	\(q-684741. q^{5} +1.14317e7 q^{7} +6.71354e8 q^{11} +5.24288e9 q^{13} -1.10085e10 q^{17} +1.02767e11 q^{19} -4.80722e11 q^{23} -2.94069e11 q^{25} +2.77099e12 q^{29} +1.01908e12 q^{31} -7.82775e12 q^{35} -4.01373e13 q^{37} -5.54603e13 q^{41} +7.56717e13 q^{43} +9.85390e13 q^{47} -1.01947e14 q^{49} +6.37330e14 q^{53} -4.59704e14 q^{55} +1.65484e15 q^{59} +2.22251e14 q^{61} -3.59002e15 q^{65} +1.72692e15 q^{67} -1.21864e15 q^{71} -9.87996e15 q^{73} +7.67472e15 q^{77} +2.90668e15 q^{79} -1.24773e16 q^{83} +7.53795e15 q^{85} +2.06746e16 q^{89} +5.99350e16 q^{91} -7.03688e16 q^{95} +5.68657e16 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 53620 q^{5} - 333168 q^{7}+O(q^{10}) \) 2 * q + 53620 * q^5 - 333168 * q^7	\( 2 q + 53620 q^{5} - 333168 q^{7} - 430974680 q^{11} + 2667521948 q^{13} - 60673503268 q^{17} + 178629960040 q^{19} - 528756594608 q^{23} - 511831510850 q^{25} + 7240660091460 q^{29} - 1878351140288 q^{31} - 16514472678240 q^{35} - 20332464566580 q^{37} + 1763041905324 q^{41} + 193394525968664 q^{43} - 100763837765472 q^{47} - 196165218276494 q^{49} + 317818146060052 q^{53} - 12\!\cdots\!20 q^{55}+ \cdots + 11\!\cdots\!08 q^{97}+O(q^{100}) \) 2 * q + 53620 * q^5 - 333168 * q^7 - 430974680 * q^11 + 2667521948 * q^13 - 60673503268 * q^17 + 178629960040 * q^19 - 528756594608 * q^23 - 511831510850 * q^25 + 7240660091460 * q^29 - 1878351140288 * q^31 - 16514472678240 * q^35 - 20332464566580 * q^37 + 1763041905324 * q^41 + 193394525968664 * q^43 - 100763837765472 * q^47 - 196165218276494 * q^49 + 317818146060052 * q^53 - 1273620709243120 * q^55 + 1262050788321736 * q^59 + 2765859692723708 * q^61 - 5491559693146280 * q^65 + 1963006544550088 * q^67 - 483639107104528 * q^71 - 2176892348591212 * q^73 + 20643473818671936 * q^77 - 5295839905627744 * q^79 - 9972518018887144 * q^83 - 29132785359362600 * q^85 + 711099036813900 * q^89 + 90233771615940576 * q^91 - 14354471748500080 * q^95 + 114870546609971908 * q^97

Introduction

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