LMFDB - Embedded newform 72.22.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,22,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, 22, names="a")

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

chi := DirichletCharacter("72.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 22 $
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:	$201.223687887$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4963x + 96223 $ x^3 - x^2 - 4963*x + 96223
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{21}\cdot 3^{5}\cdot 5\cdot 7 $
Twist minimal:	no (minimal twist has level 8)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$57.9766$ 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)

$f(q)$	$=$	$q-3.09730e7 q^{5} +9.17385e7 q^{7} +O(q^{10})$	$q-3.09730e7 q^{5} +9.17385e7 q^{7} -8.72158e10 q^{11} -2.36850e11 q^{13} -7.42283e12 q^{17} +4.68680e9 q^{19} -3.33703e14 q^{23} +4.82489e14 q^{25} -3.23982e15 q^{29} +6.40473e15 q^{31} -2.84142e15 q^{35} -1.61009e16 q^{37} +5.77168e16 q^{41} -2.01468e17 q^{43} -6.62056e17 q^{47} -5.50130e17 q^{49} -4.62651e17 q^{53} +2.70133e18 q^{55} -7.39502e18 q^{59} -5.50188e18 q^{61} +7.33594e18 q^{65} +6.03520e18 q^{67} +4.43147e19 q^{71} -2.48622e19 q^{73} -8.00105e18 q^{77} +5.70948e19 q^{79} +1.31820e20 q^{83} +2.29907e20 q^{85} -3.97023e20 q^{89} -2.17282e19 q^{91} -1.45164e17 q^{95} -9.80402e20 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 24111774 q^{5} + 295988280 q^{7}+O(q^{10}) $ 3 * q + 24111774 * q^5 + 295988280 * q^7	$ 3 q + 24111774 q^{5} + 295988280 q^{7} + 40335108684 q^{11} + 133734425946 q^{13} - 7797732274422 q^{17} + 35788199781996 q^{19} - 193770761479080 q^{23} + 11\!\cdots\!01 q^{25}+ \cdots - 32\!\cdots\!42 q^{97}+O(q^{100}) $ 3 * q + 24111774 * q^5 + 295988280 * q^7 + 40335108684 * q^11 + 133734425946 * q^13 - 7797732274422 * q^17 + 35788199781996 * q^19 - 193770761479080 * q^23 + 1127564438439501 * q^25 - 5607343422466122 * q^29 + 11246757871503072 * q^31 - 5274251425350096 * q^35 - 24272499791100606 * q^37 + 298159108991869602 * q^41 - 33333932139754860 * q^43 + 120874283547603888 * q^47 - 850403331975639477 * q^49 + 1138443393004854222 * q^53 + 6957552484263571704 * q^55 - 9225624498709937412 * q^59 - 6554902294063924182 * q^61 + 20714581819561144452 * q^65 - 15793054074531629124 * q^67 + 41139582493467997704 * q^71 - 19422167949903851970 * q^73 - 68380237365358617888 * q^77 - 131735321299806049488 * q^79 - 64013993832679681068 * q^83 + 390258202763001297252 * q^85 - 429891446897537246766 * q^89 - 297181701588021496176 * q^91 + 1036372406649019824696 * q^95 - 324059336514148638042 * 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
$5$	−3.09730e7	−1.41840	−0.709199	−	0.705008i	$-0.750943\pi$
$5$	−3.09730e7	−1.41840	−0.709199	+	0.705008i	$0.750943\pi$
$6$	0	0
$7$	9.17385e7	0.122750	0.0613751	−	0.998115i	$-0.480451\pi$
$7$	9.17385e7	0.122750	0.0613751	+	0.998115i	$0.480451\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−8.72158e10	−1.01385	−0.506923	−	0.861991i	$-0.669217\pi$
$11$	−8.72158e10	−1.01385	−0.506923	+	0.861991i	$0.669217\pi$
$12$	0	0
$13$	−2.36850e11	−0.476505	−0.238253	−	0.971203i	$-0.576575\pi$
$13$	−2.36850e11	−0.476505	−0.238253	+	0.971203i	$0.576575\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.42283e12	−0.893009	−0.446505	−	0.894781i	$-0.647331\pi$
$17$	−7.42283e12	−0.893009	−0.446505	+	0.894781i	$0.647331\pi$
$18$	0	0
$19$	4.68680e9	0.000175373 0	8.76867e−5	−	1.00000i	$-0.499972\pi$
$19$	4.68680e9	0.000175373 0	8.76867e−5		1.00000i	$0.499972\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.33703e14	−1.67965	−0.839823	−	0.542860i	$-0.817341\pi$
$23$	−3.33703e14	−1.67965	−0.839823	+	0.542860i	$0.817341\pi$
$24$	0	0
$25$	4.82489e14	1.01185
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.23982e15	−1.43002	−0.715010	−	0.699114i	$-0.753578\pi$
$29$	−3.23982e15	−1.43002	−0.715010	+	0.699114i	$0.753578\pi$
$30$	0	0
$31$	6.40473e15	1.40347	0.701735	−	0.712438i	$-0.252409\pi$
$31$	6.40473e15	1.40347	0.701735	+	0.712438i	$0.252409\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.84142e15	−0.174109
$36$	0	0
$37$	−1.61009e16	−0.550467	−0.275234	−	0.961377i	$-0.588755\pi$
$37$	−1.61009e16	−0.550467	−0.275234	+	0.961377i	$0.588755\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.77168e16	0.671540	0.335770	−	0.941944i	$-0.391003\pi$
$41$	5.77168e16	0.671540	0.335770	+	0.941944i	$0.391003\pi$
$42$	0	0
$43$	−2.01468e17	−1.42163	−0.710817	−	0.703377i	$-0.751675\pi$
$43$	−2.01468e17	−1.42163	−0.710817	+	0.703377i	$0.751675\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.62056e17	−1.83598	−0.917989	−	0.396607i	$-0.870188\pi$
$47$	−6.62056e17	−1.83598	−0.917989	+	0.396607i	$0.870188\pi$
$48$	0	0
$49$	−5.50130e17	−0.984932
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.62651e17	−0.363376	−0.181688	−	0.983356i	$-0.558156\pi$
$53$	−4.62651e17	−0.363376	−0.181688	+	0.983356i	$0.558156\pi$
$54$	0	0
$55$	2.70133e18	1.43804
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−7.39502e18	−1.88362	−0.941809	−	0.336148i	$-0.890876\pi$
$59$	−7.39502e18	−1.88362	−0.941809	+	0.336148i	$0.890876\pi$
$60$	0	0
$61$	−5.50188e18	−0.987524	−0.493762	−	0.869597i	$-0.664379\pi$
$61$	−5.50188e18	−0.987524	−0.493762	+	0.869597i	$0.664379\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	7.33594e18	0.675874
$66$	0	0
$67$	6.03520e18	0.404489	0.202244	−	0.979335i	$-0.435176\pi$
$67$	6.03520e18	0.404489	0.202244	+	0.979335i	$0.435176\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.43147e19	1.61561	0.807803	−	0.589453i	$-0.200657\pi$
$71$	4.43147e19	1.61561	0.807803	+	0.589453i	$0.200657\pi$
$72$	0	0
$73$	−2.48622e19	−0.677095	−0.338548	−	0.940949i	$-0.609936\pi$
$73$	−2.48622e19	−0.677095	−0.338548	+	0.940949i	$0.609936\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−8.00105e18	−0.124450
$78$	0	0
$79$	5.70948e19	0.678441	0.339220	−	0.940707i	$-0.389837\pi$
$79$	5.70948e19	0.678441	0.339220	+	0.940707i	$0.389837\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.31820e20	0.932528	0.466264	−	0.884646i	$-0.345600\pi$
$83$	1.31820e20	0.932528	0.466264	+	0.884646i	$0.345600\pi$
$84$	0	0
$85$	2.29907e20	1.26664
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3.97023e20	−1.34965	−0.674823	−	0.737979i	$-0.735780\pi$
$89$	−3.97023e20	−1.34965	−0.674823	+	0.737979i	$0.735780\pi$
$90$	0	0
$91$	−2.17282e19	−0.0584911
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.45164e17	−0.000248749 0
$96$	0	0
$97$	−9.80402e20	−1.34990	−0.674949	−	0.737864i	$-0.735835\pi$
$97$	−9.80402e20	−1.34990	−0.674949	+	0.737864i	$0.735835\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.46191e19	−0.0582085	−0.0291042	−	0.999576i	$-0.509265\pi$
$101$	−6.46191e19	−0.0582085	−0.0291042	+	0.999576i	$0.509265\pi$
$102$	0	0
$103$	−7.39481e19	−0.0542171	−0.0271085	−	0.999632i	$-0.508630\pi$
$103$	−7.39481e19	−0.0542171	−0.0271085	+	0.999632i	$0.508630\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.69439e20	−0.378133	−0.189066	−	0.981964i	$-0.560546\pi$
$107$	−7.69439e20	−0.378133	−0.189066	+	0.981964i	$0.560546\pi$
$108$	0	0
$109$	−3.27500e21	−1.32505	−0.662526	−	0.749039i	$-0.730516\pi$
$109$	−3.27500e21	−1.32505	−0.662526	+	0.749039i	$0.730516\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.53293e20	0.0701938	0.0350969	−	0.999384i	$-0.488826\pi$
$113$	2.53293e20	0.0701938	0.0350969	+	0.999384i	$0.488826\pi$
$114$	0	0
$115$	1.03358e22	2.38241
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.80959e20	−0.109617
$120$	0	0
$121$	2.06345e20	0.0278836
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.75063e20	−0.0168128
$126$	0	0
$127$	−3.02098e21	−0.245589	−0.122795	−	0.992432i	$-0.539186\pi$
$127$	−3.02098e21	−0.245589	−0.122795	+	0.992432i	$0.539186\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.70668e22	1.00185	0.500925	−	0.865491i	$-0.332993\pi$
$131$	1.70668e22	1.00185	0.500925	+	0.865491i	$0.332993\pi$
$132$	0	0
$133$	4.29960e17	2.15271e−5 0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.34811e22	0.861296	0.430648	−	0.902520i	$-0.358285\pi$
$137$	2.34811e22	0.861296	0.430648	+	0.902520i	$0.358285\pi$
$138$	0	0
$139$	−4.53132e22	−1.42748	−0.713738	−	0.700413i	$-0.752999\pi$
$139$	−4.53132e22	−1.42748	−0.713738	+	0.700413i	$0.752999\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.06570e22	0.483103
$144$	0	0
$145$	1.00347e23	2.02834
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−6.98352e22	−1.06076	−0.530381	−	0.847759i	$-0.677951\pi$
$149$	−6.98352e22	−1.06076	−0.530381	+	0.847759i	$0.677951\pi$
$150$	0	0
$151$	−4.73560e22	−0.625341	−0.312671	−	0.949862i	$-0.601224\pi$
$151$	−4.73560e22	−0.625341	−0.312671	+	0.949862i	$0.601224\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.98374e23	−1.99068
$156$	0	0
$157$	−2.22945e21	−0.0195547	−0.00977733	−	0.999952i	$-0.503112\pi$
$157$	−2.22945e21	−0.0195547	−0.00977733	+	0.999952i	$0.503112\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.06134e22	−0.206177
$162$	0	0
$163$	1.48202e23	0.876768	0.438384	−	0.898788i	$-0.355551\pi$
$163$	1.48202e23	0.876768	0.438384	+	0.898788i	$0.355551\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.25766e23	1.03547	0.517733	−	0.855542i	$-0.326776\pi$
$167$	2.25766e23	1.03547	0.517733	+	0.855542i	$0.326776\pi$
$168$	0	0
$169$	−1.90967e23	−0.772943
$170$	0	0
$171$	0	0
$172$	0	0
$173$	5.37523e23	1.70182	0.850909	−	0.525313i	$-0.176052\pi$
$173$	5.37523e23	1.70182	0.850909	+	0.525313i	$0.176052\pi$
$174$	0	0
$175$	4.42628e22	0.124205
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1.13222e23	−0.250597	−0.125299	−	0.992119i	$-0.539989\pi$
$179$	−1.13222e23	−0.250597	−0.125299	+	0.992119i	$0.539989\pi$
$180$	0	0
$181$	2.55968e23	0.504151	0.252075	−	0.967708i	$-0.418887\pi$
$181$	2.55968e23	0.504151	0.252075	+	0.967708i	$0.418887\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4.98692e23	0.780781
$186$	0	0
$187$	6.47388e23	0.905374
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7.48387e23	0.838062	0.419031	−	0.907972i	$-0.362370\pi$
$191$	7.48387e23	0.838062	0.419031	+	0.907972i	$0.362370\pi$
$192$	0	0
$193$	1.77062e24	1.77735	0.888674	−	0.458539i	$-0.151627\pi$
$193$	1.77062e24	1.77735	0.888674	+	0.458539i	$0.151627\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.04387e24	−1.65409	−0.827044	−	0.562137i	$-0.809979\pi$
$197$	−2.04387e24	−1.65409	−0.827044	+	0.562137i	$0.809979\pi$
$198$	0	0
$199$	−8.13486e22	−0.0592097	−0.0296048	−	0.999562i	$-0.509425\pi$
$199$	−8.13486e22	−0.0592097	−0.0296048	+	0.999562i	$0.509425\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.97216e23	−0.175535
$204$	0	0
$205$	−1.78766e24	−0.952511
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.08763e20	−0.000177802 0
$210$	0	0
$211$	3.52722e24	1.38825	0.694123	−	0.719856i	$-0.255792\pi$
$211$	3.52722e24	1.38825	0.694123	+	0.719856i	$0.255792\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	6.24008e24	2.01644
$216$	0	0
$217$	5.87560e23	0.172276
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.75809e24	0.425523
$222$	0	0
$223$	−2.37573e24	−0.523114	−0.261557	−	0.965188i	$-0.584236\pi$
$223$	−2.37573e24	−0.523114	−0.261557	+	0.965188i	$0.584236\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−6.71337e24	−1.22650	−0.613252	−	0.789887i	$-0.710139\pi$
$227$	−6.71337e24	−1.22650	−0.613252	+	0.789887i	$0.710139\pi$
$228$	0	0
$229$	3.10829e24	0.517903	0.258952	−	0.965890i	$-0.416623\pi$
$229$	3.10829e24	0.517903	0.258952	+	0.965890i	$0.416623\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3.90742e24	0.542817	0.271408	−	0.962464i	$-0.412511\pi$
$233$	3.90742e24	0.542817	0.271408	+	0.962464i	$0.412511\pi$
$234$	0	0
$235$	2.05059e25	2.60415
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2.38784e23	−0.0253997	−0.0126998	−	0.999919i	$-0.504043\pi$
$239$	−2.38784e23	−0.0253997	−0.0126998	+	0.999919i	$0.504043\pi$
$240$	0	0
$241$	1.14005e25	1.11107	0.555537	−	0.831492i	$-0.312513\pi$
$241$	1.14005e25	1.11107	0.555537	+	0.831492i	$0.312513\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.70392e25	1.39703
$246$	0	0
$247$	−1.11007e21	−8.35663e−5 0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	5.80122e24	0.368931	0.184466	−	0.982839i	$-0.440945\pi$
$251$	5.80122e24	0.368931	0.184466	+	0.982839i	$0.440945\pi$
$252$	0	0
$253$	2.91042e25	1.70290
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.21415e25	1.09878	0.549388	−	0.835567i	$-0.314861\pi$
$257$	2.21415e25	1.09878	0.549388	+	0.835567i	$0.314861\pi$
$258$	0	0
$259$	−1.47707e24	−0.0675699
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3.74763e25	1.45956	0.729779	−	0.683683i	$-0.239623\pi$
$263$	3.74763e25	1.45956	0.729779	+	0.683683i	$0.239623\pi$
$264$	0	0
$265$	1.43297e25	0.515412
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.64824e25	−0.506549	−0.253274	−	0.967394i	$-0.581508\pi$
$269$	−1.64824e25	−0.506549	−0.253274	+	0.967394i	$0.581508\pi$
$270$	0	0
$271$	−2.86168e25	−0.813660	−0.406830	−	0.913504i	$-0.633366\pi$
$271$	−2.86168e25	−0.813660	−0.406830	+	0.913504i	$0.633366\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.20807e25	−1.02586
$276$	0	0
$277$	5.26085e25	1.18855	0.594277	−	0.804261i	$-0.297438\pi$
$277$	5.26085e25	1.18855	0.594277	+	0.804261i	$0.297438\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−6.45378e25	−1.25429	−0.627145	−	0.778903i	$-0.715777\pi$
$281$	−6.45378e25	−1.25429	−0.627145	+	0.778903i	$0.715777\pi$
$282$	0	0
$283$	4.56042e25	0.822711	0.411355	−	0.911475i	$-0.365056\pi$
$283$	4.56042e25	0.822711	0.411355	+	0.911475i	$0.365056\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.29485e24	0.0824317
$288$	0	0
$289$	−1.39935e25	−0.202535
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5.87863e25	0.736489	0.368244	−	0.929729i	$-0.379959\pi$
$293$	5.87863e25	0.736489	0.368244	+	0.929729i	$0.379959\pi$
$294$	0	0
$295$	2.29046e26	2.67172
$296$	0	0
$297$	0	0
$298$	0	0
$299$	7.90374e25	0.800360
$300$	0	0
$301$	−1.84824e25	−0.174506
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.70410e26	1.40070
$306$	0	0
$307$	−1.65443e26	−1.26969	−0.634843	−	0.772641i	$-0.718935\pi$
$307$	−1.65443e26	−1.26969	−0.634843	+	0.772641i	$0.718935\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	7.32700e24	0.0490842	0.0245421	−	0.999699i	$-0.492187\pi$
$311$	7.32700e24	0.0490842	0.0245421	+	0.999699i	$0.492187\pi$
$312$	0	0
$313$	−7.64901e25	−0.479059	−0.239530	−	0.970889i	$-0.576993\pi$
$313$	−7.64901e25	−0.479059	−0.239530	+	0.970889i	$0.576993\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9.82997e25	0.538803	0.269402	−	0.963028i	$-0.413174\pi$
$317$	9.82997e25	0.538803	0.269402	+	0.963028i	$0.413174\pi$
$318$	0	0
$319$	2.82564e26	1.44982
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.47893e22	−0.000156610 0
$324$	0	0
$325$	−1.14277e26	−0.482153
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−6.07360e25	−0.225367
$330$	0	0
$331$	−2.46189e26	−0.857187	−0.428593	−	0.903498i	$-0.640991\pi$
$331$	−2.46189e26	−0.857187	−0.428593	+	0.903498i	$0.640991\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.86928e26	−0.573726
$336$	0	0
$337$	4.89205e26	1.41051	0.705257	−	0.708952i	$-0.250832\pi$
$337$	4.89205e26	1.41051	0.705257	+	0.708952i	$0.250832\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−5.58594e26	−1.42290
$342$	0	0
$343$	−1.01708e26	−0.243651
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.00972e26	−0.214161	−0.107080	−	0.994250i	$-0.534150\pi$
$347$	−1.00972e26	−0.214161	−0.107080	+	0.994250i	$0.534150\pi$
$348$	0	0
$349$	−4.95706e25	−0.0989822	−0.0494911	−	0.998775i	$-0.515760\pi$
$349$	−4.95706e25	−0.0989822	−0.0494911	+	0.998775i	$0.515760\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1.88459e26	−0.333873	−0.166937	−	0.985968i	$-0.553387\pi$
$353$	−1.88459e26	−0.333873	−0.166937	+	0.985968i	$0.553387\pi$
$354$	0	0
$355$	−1.37256e27	−2.29157
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5.21557e26	−0.774123	−0.387062	−	0.922054i	$-0.626510\pi$
$359$	−5.21557e26	−0.774123	−0.387062	+	0.922054i	$0.626510\pi$
$360$	0	0
$361$	−7.14209e26	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	7.70057e26	0.960390
$366$	0	0
$367$	−4.38478e25	−0.0516362	−0.0258181	−	0.999667i	$-0.508219\pi$
$367$	−4.38478e25	−0.0516362	−0.0258181	+	0.999667i	$0.508219\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−4.24429e25	−0.0446045
$372$	0	0
$373$	1.92501e25	0.0191201	0.00956004	−	0.999954i	$-0.496957\pi$
$373$	1.92501e25	0.0191201	0.00956004	+	0.999954i	$0.496957\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.67351e26	0.681412
$378$	0	0
$379$	−2.04774e27	−1.72014	−0.860070	−	0.510175i	$-0.829581\pi$
$379$	−2.04774e27	−1.72014	−0.860070	+	0.510175i	$0.829581\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.75914e27	−1.32347	−0.661733	−	0.749740i	$-0.730179\pi$
$383$	−1.75914e27	−1.32347	−0.661733	+	0.749740i	$0.730179\pi$
$384$	0	0
$385$	2.47816e26	0.176519
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.73236e26	0.174609	0.0873046	−	0.996182i	$-0.472175\pi$
$389$	2.73236e26	0.174609	0.0873046	+	0.996182i	$0.472175\pi$
$390$	0	0
$391$	2.47702e27	1.49994
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.76840e27	−0.962299
$396$	0	0
$397$	−1.21691e27	−0.627998	−0.313999	−	0.949423i	$-0.601669\pi$
$397$	−1.21691e27	−0.627998	−0.313999	+	0.949423i	$0.601669\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.15061e27	0.534455	0.267228	−	0.963633i	$-0.413892\pi$
$401$	1.15061e27	0.534455	0.267228	+	0.963633i	$0.413892\pi$
$402$	0	0
$403$	−1.51696e27	−0.668761
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.40425e27	0.558089
$408$	0	0
$409$	−1.54628e26	−0.0583705	−0.0291852	−	0.999574i	$-0.509291\pi$
$409$	−1.54628e26	−0.0583705	−0.0291852	+	0.999574i	$0.509291\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−6.78408e26	−0.231215
$414$	0	0
$415$	−4.08287e27	−1.32270
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4.10695e27	−1.20302	−0.601510	−	0.798866i	$-0.705434\pi$
$419$	−4.10695e27	−1.20302	−0.601510	+	0.798866i	$0.705434\pi$
$420$	0	0
$421$	−4.80891e27	−1.33994	−0.669969	−	0.742389i	$-0.733693\pi$
$421$	−4.80891e27	−1.33994	−0.669969	+	0.742389i	$0.733693\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−3.58144e27	−0.903594
$426$	0	0
$427$	−5.04734e26	−0.121219
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4.51446e27	0.983092	0.491546	−	0.870852i	$-0.336432\pi$
$431$	4.51446e27	0.983092	0.491546	+	0.870852i	$0.336432\pi$
$432$	0	0
$433$	−4.04253e27	−0.838552	−0.419276	−	0.907859i	$-0.637716\pi$
$433$	−4.04253e27	−0.838552	−0.419276	+	0.907859i	$0.637716\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.56400e24	−0.000294565 0
$438$	0	0
$439$	3.73290e26	0.0670145	0.0335072	−	0.999438i	$-0.489332\pi$
$439$	3.73290e26	0.0670145	0.0335072	+	0.999438i	$0.489332\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−8.67248e27	−1.41548	−0.707741	−	0.706472i	$-0.750286\pi$
$443$	−8.67248e27	−1.41548	−0.707741	+	0.706472i	$0.750286\pi$
$444$	0	0
$445$	1.22970e28	1.91434
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4.22534e27	0.598790	0.299395	−	0.954129i	$-0.403215\pi$
$449$	4.22534e27	0.598790	0.299395	+	0.954129i	$0.403215\pi$
$450$	0	0
$451$	−5.03382e27	−0.680838
$452$	0	0
$453$	0	0
$454$	0	0
$455$	6.72988e26	0.0829637
$456$	0	0
$457$	8.06069e27	0.948969	0.474484	−	0.880264i	$-0.342635\pi$
$457$	8.06069e27	0.948969	0.474484	+	0.880264i	$0.342635\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	9.88705e27	1.06220	0.531101	−	0.847309i	$-0.321779\pi$
$461$	9.88705e27	1.06220	0.531101	+	0.847309i	$0.321779\pi$
$462$	0	0
$463$	1.00097e28	1.02759	0.513794	−	0.857913i	$-0.328239\pi$
$463$	1.00097e28	1.02759	0.513794	+	0.857913i	$0.328239\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.60212e28	−1.50268	−0.751342	−	0.659913i	$-0.770593\pi$
$467$	−1.60212e28	−1.50268	−0.751342	+	0.659913i	$0.770593\pi$
$468$	0	0
$469$	5.53660e26	0.0496511
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.75712e28	1.44132
$474$	0	0
$475$	2.26133e24	0.000177452 0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	9.14679e27	0.657273	0.328636	−	0.944457i	$-0.393411\pi$
$479$	9.14679e27	0.657273	0.328636	+	0.944457i	$0.393411\pi$
$480$	0	0
$481$	3.81349e27	0.262300
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3.03660e28	1.91469
$486$	0	0
$487$	−1.63338e28	−0.986357	−0.493179	−	0.869928i	$-0.664165\pi$
$487$	−1.63338e28	−0.986357	−0.493179	+	0.869928i	$0.664165\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−1.10020e28	−0.609702	−0.304851	−	0.952400i	$-0.598607\pi$
$491$	−1.10020e28	−0.609702	−0.304851	+	0.952400i	$0.598607\pi$
$492$	0	0
$493$	2.40487e28	1.27702
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.06537e27	0.198316
$498$	0	0
$499$	2.04342e28	0.955657	0.477829	−	0.878453i	$-0.341424\pi$
$499$	2.04342e28	0.955657	0.477829	+	0.878453i	$0.341424\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1.67038e28	−0.718375	−0.359188	−	0.933265i	$-0.616946\pi$
$503$	−1.67038e28	−0.718375	−0.359188	+	0.933265i	$0.616946\pi$
$504$	0	0
$505$	2.00145e27	0.0825628
$506$	0	0
$507$	0	0
$508$	0	0
$509$	5.42146e27	0.205864	0.102932	−	0.994688i	$-0.467178\pi$
$509$	5.42146e27	0.205864	0.102932	+	0.994688i	$0.467178\pi$
$510$	0	0
$511$	−2.28082e27	−0.0831136
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2.29039e27	0.0769014
$516$	0	0
$517$	5.77418e28	1.86140
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.37414e28	−0.408541	−0.204271	−	0.978914i	$-0.565482\pi$
$521$	−1.37414e28	−0.408541	−0.204271	+	0.978914i	$0.565482\pi$
$522$	0	0
$523$	−3.33678e28	−0.952927	−0.476463	−	0.879194i	$-0.658081\pi$
$523$	−3.33678e28	−0.952927	−0.476463	+	0.879194i	$0.658081\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.75412e28	−1.25331
$528$	0	0
$529$	7.18860e28	1.82121
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.36702e28	−0.319992
$534$	0	0
$535$	2.38318e28	0.536343
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.79800e28	0.998570
$540$	0	0
$541$	−9.34113e28	−1.86994	−0.934971	−	0.354723i	$-0.884575\pi$
$541$	−9.34113e28	−1.86994	−0.934971	+	0.354723i	$0.884575\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1.01437e29	1.87945
$546$	0	0
$547$	1.90655e28	0.339923	0.169961	−	0.985451i	$-0.445636\pi$
$547$	1.90655e28	0.339923	0.169961	+	0.985451i	$0.445636\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.51844e25	−0.000250788 0
$552$	0	0
$553$	5.23779e27	0.0832787
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.85895e28	0.716247	0.358124	−	0.933674i	$-0.383417\pi$
$557$	4.85895e28	0.716247	0.358124	+	0.933674i	$0.383417\pi$
$558$	0	0
$559$	4.77177e28	0.677416
$560$	0	0
$561$	0	0
$562$	0	0
$563$	4.82251e28	0.635236	0.317618	−	0.948219i	$-0.397117\pi$
$563$	4.82251e28	0.635236	0.317618	+	0.948219i	$0.397117\pi$
$564$	0	0
$565$	−7.84523e27	−0.0995628
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8.25601e28	0.972952	0.486476	−	0.873694i	$-0.338282\pi$
$569$	8.25601e28	0.972952	0.486476	+	0.873694i	$0.338282\pi$
$570$	0	0
$571$	−3.00348e28	−0.341150	−0.170575	−	0.985345i	$-0.554563\pi$
$571$	−3.00348e28	−0.341150	−0.170575	+	0.985345i	$0.554563\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.61008e29	−1.69956
$576$	0	0
$577$	7.52665e28	0.766047	0.383024	−	0.923739i	$-0.374883\pi$
$577$	7.52665e28	0.766047	0.383024	+	0.923739i	$0.374883\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.20930e28	0.114468
$582$	0	0
$583$	4.03504e28	0.368408
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.52918e29	−1.29945	−0.649723	−	0.760171i	$-0.725115\pi$
$587$	−1.52918e29	−1.29945	−0.649723	+	0.760171i	$0.725115\pi$
$588$	0	0
$589$	3.00177e25	0.000246131 0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−3.89818e28	−0.297706	−0.148853	−	0.988859i	$-0.547558\pi$
$593$	−3.89818e28	−0.297706	−0.148853	+	0.988859i	$0.547558\pi$
$594$	0	0
$595$	2.10913e28	0.155481
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−1.28684e29	−0.884186	−0.442093	−	0.896969i	$-0.645764\pi$
$599$	−1.28684e29	−0.884186	−0.442093	+	0.896969i	$0.645764\pi$
$600$	0	0
$601$	−1.81454e29	−1.20388	−0.601942	−	0.798540i	$-0.705606\pi$
$601$	−1.81454e29	−1.20388	−0.601942	+	0.798540i	$0.705606\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−6.39114e27	−0.0395500
$606$	0	0
$607$	−2.81972e29	−1.68549	−0.842743	−	0.538316i	$-0.819061\pi$
$607$	−2.81972e29	−1.68549	−0.842743	+	0.538316i	$0.819061\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.56808e29	0.874853
$612$	0	0
$613$	8.09843e28	0.436582	0.218291	−	0.975884i	$-0.429952\pi$
$613$	8.09843e28	0.436582	0.218291	+	0.975884i	$0.429952\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−3.47454e29	−1.74946	−0.874728	−	0.484614i	$-0.838960\pi$
$617$	−3.47454e29	−1.74946	−0.874728	+	0.484614i	$0.838960\pi$
$618$	0	0
$619$	−2.90076e29	−1.41176	−0.705878	−	0.708333i	$-0.749447\pi$
$619$	−2.90076e29	−1.41176	−0.705878	+	0.708333i	$0.749447\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3.64223e28	−0.165669
$624$	0	0
$625$	−2.24647e29	−0.988006
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.19514e29	0.491572
$630$	0	0
$631$	−4.23594e29	−1.68516	−0.842581	−	0.538570i	$-0.818965\pi$
$631$	−4.23594e29	−1.68516	−0.842581	+	0.538570i	$0.818965\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	9.35689e28	0.348343
$636$	0	0
$637$	1.30298e29	0.469325
$638$	0	0
$639$	0	0
$640$	0	0
$641$	3.43690e29	1.15920	0.579598	−	0.814902i	$-0.303209\pi$
$641$	3.43690e29	1.15920	0.579598	+	0.814902i	$0.303209\pi$
$642$	0	0
$643$	1.89646e29	0.619055	0.309527	−	0.950891i	$-0.399829\pi$
$643$	1.89646e29	0.619055	0.309527	+	0.950891i	$0.399829\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−3.06810e29	−0.938370	−0.469185	−	0.883100i	$-0.655452\pi$
$647$	−3.06810e29	−0.938370	−0.469185	+	0.883100i	$0.655452\pi$
$648$	0	0
$649$	6.44962e29	1.90970
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.17693e29	−0.326710	−0.163355	−	0.986567i	$-0.552232\pi$
$653$	−1.17693e29	−0.326710	−0.163355	+	0.986567i	$0.552232\pi$
$654$	0	0
$655$	−5.28609e29	−1.42102
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−3.42901e29	−0.864712	−0.432356	−	0.901703i	$-0.642318\pi$
$659$	−3.42901e29	−0.864712	−0.432356	+	0.901703i	$0.642318\pi$
$660$	0	0
$661$	−3.95818e29	−0.966897	−0.483448	−	0.875373i	$-0.660616\pi$
$661$	−3.95818e29	−0.966897	−0.483448	+	0.875373i	$0.660616\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.33172e25	−3.05340e−5 0
$666$	0	0
$667$	1.08114e30	2.40193
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4.79851e29	1.00120
$672$	0	0
$673$	−9.58339e29	−1.93803	−0.969017	−	0.246996i	$-0.920557\pi$
$673$	−9.58339e29	−1.93803	−0.969017	+	0.246996i	$0.920557\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−5.12797e29	−0.974461	−0.487230	−	0.873273i	$-0.661993\pi$
$677$	−5.12797e29	−0.974461	−0.487230	+	0.873273i	$0.661993\pi$
$678$	0	0
$679$	−8.99406e28	−0.165700
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−5.75298e29	−0.996495	−0.498248	−	0.867035i	$-0.666023\pi$
$683$	−5.75298e29	−0.996495	−0.498248	+	0.867035i	$0.666023\pi$
$684$	0	0
$685$	−7.27280e29	−1.22166
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.09579e29	0.173151
$690$	0	0
$691$	2.46471e29	0.377787	0.188893	−	0.981998i	$-0.439510\pi$
$691$	2.46471e29	0.377787	0.188893	+	0.981998i	$0.439510\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.40348e30	2.02473
$696$	0	0
$697$	−4.28422e29	−0.599691
$698$	0	0
$699$	0	0
$700$	0	0
$701$	8.94607e29	1.17922	0.589608	−	0.807690i	$-0.299282\pi$
$701$	8.94607e29	1.17922	0.589608	+	0.807690i	$0.299282\pi$
$702$	0	0
$703$	−7.54616e25	−9.65373e−5 0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5.92806e27	−0.00714510
$708$	0	0
$709$	7.52975e29	0.881039	0.440520	−	0.897743i	$-0.354794\pi$
$709$	7.52975e29	0.881039	0.440520	+	0.897743i	$0.354794\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2.13728e30	−2.35733
$714$	0	0
$715$	−6.39810e29	−0.685232
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−4.85535e29	−0.490419	−0.245210	−	0.969470i	$-0.578857\pi$
$719$	−4.85535e29	−0.490419	−0.245210	+	0.969470i	$0.578857\pi$
$720$	0	0
$721$	−6.78389e27	−0.00665516
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.56318e30	−1.44697
$726$	0	0
$727$	1.86351e30	1.67579	0.837895	−	0.545831i	$-0.183786\pi$
$727$	1.86351e30	1.67579	0.837895	+	0.545831i	$0.183786\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.49546e30	1.26953
$732$	0	0
$733$	−3.73906e29	−0.308440	−0.154220	−	0.988037i	$-0.549286\pi$
$733$	−3.73906e29	−0.308440	−0.154220	+	0.988037i	$0.549286\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.26365e29	−0.410089
$738$	0	0
$739$	8.38257e29	0.634762	0.317381	−	0.948298i	$-0.397197\pi$
$739$	8.38257e29	0.634762	0.317381	+	0.948298i	$0.397197\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	7.60139e29	0.543889	0.271945	−	0.962313i	$-0.412333\pi$
$743$	7.60139e29	0.543889	0.271945	+	0.962313i	$0.412333\pi$
$744$	0	0
$745$	2.16300e30	1.50458
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−7.05871e28	−0.0464159
$750$	0	0
$751$	−1.08974e30	−0.696794	−0.348397	−	0.937347i	$-0.613274\pi$
$751$	−1.08974e30	−0.696794	−0.348397	+	0.937347i	$0.613274\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.46676e30	0.886983
$756$	0	0
$757$	3.06224e30	1.80108	0.900540	−	0.434774i	$-0.143172\pi$
$757$	3.06224e30	1.80108	0.900540	+	0.434774i	$0.143172\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	3.46912e30	1.93055	0.965274	−	0.261241i	$-0.0841318\pi$
$761$	3.46912e30	1.93055	0.965274	+	0.261241i	$0.0841318\pi$
$762$	0	0
$763$	−3.00444e29	−0.162650
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.75151e30	0.897554
$768$	0	0
$769$	−7.20768e29	−0.359392	−0.179696	−	0.983722i	$-0.557511\pi$
$769$	−7.20768e29	−0.359392	−0.179696	+	0.983722i	$0.557511\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.63916e30	0.773990	0.386995	−	0.922082i	$-0.373513\pi$
$773$	1.63916e30	0.773990	0.386995	+	0.922082i	$0.373513\pi$
$774$	0	0
$775$	3.09021e30	1.42011
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.70507e26	0.000117770 0
$780$	0	0
$781$	−3.86494e30	−1.63798
$782$	0	0
$783$	0	0
$784$	0	0
$785$	6.90526e28	0.0277363
$786$	0	0
$787$	1.22284e30	0.478227	0.239114	−	0.970992i	$-0.423143\pi$
$787$	1.22284e30	0.478227	0.239114	+	0.970992i	$0.423143\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.32367e28	0.00861630
$792$	0	0
$793$	1.30312e30	0.470560
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.32063e30	−0.452345	−0.226173	−	0.974087i	$-0.572621\pi$
$797$	−1.32063e30	−0.452345	−0.226173	+	0.974087i	$0.572621\pi$
$798$	0	0
$799$	4.91433e30	1.63954
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.16838e30	0.686470
$804$	0	0
$805$	9.48189e29	0.292441
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−5.54086e30	−1.62225	−0.811124	−	0.584874i	$-0.801144\pi$
$809$	−5.54086e30	−1.62225	−0.811124	+	0.584874i	$0.801144\pi$
$810$	0	0
$811$	−4.51703e30	−1.28865	−0.644323	−	0.764753i	$-0.722861\pi$
$811$	−4.51703e30	−1.28865	−0.644323	+	0.764753i	$0.722861\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−4.59026e30	−1.24361
$816$	0	0
$817$	−9.44242e26	−0.000249317 0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−6.49500e30	−1.62921	−0.814603	−	0.580019i	$-0.803045\pi$
$821$	−6.49500e30	−1.62921	−0.814603	+	0.580019i	$0.803045\pi$
$822$	0	0
$823$	6.58977e30	1.61128	0.805642	−	0.592402i	$-0.201820\pi$
$823$	6.58977e30	1.61128	0.805642	+	0.592402i	$0.201820\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.99217e29	0.0695309	0.0347655	−	0.999395i	$-0.488932\pi$
$827$	2.99217e29	0.0695309	0.0347655	+	0.999395i	$0.488932\pi$
$828$	0	0
$829$	−3.02375e30	−0.685053	−0.342526	−	0.939508i	$-0.611283\pi$
$829$	−3.02375e30	−0.685053	−0.342526	+	0.939508i	$0.611283\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	4.08352e30	0.879554
$834$	0	0
$835$	−6.99266e30	−1.46870
$836$	0	0
$837$	0	0
$838$	0	0
$839$	6.48734e30	1.29588	0.647942	−	0.761690i	$-0.275630\pi$
$839$	6.48734e30	1.29588	0.647942	+	0.761690i	$0.275630\pi$
$840$	0	0
$841$	5.36360e30	1.04496
$842$	0	0
$843$	0	0
$844$	0	0
$845$	5.91481e30	1.09634
$846$	0	0
$847$	1.89298e28	0.00342272
$848$	0	0
$849$	0	0
$850$	0	0
$851$	5.37291e30	0.924590
$852$	0	0
$853$	1.83172e30	0.307535	0.153767	−	0.988107i	$-0.450859\pi$
$853$	1.83172e30	0.307535	0.153767	+	0.988107i	$0.450859\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.04074e31	−1.66358	−0.831792	−	0.555088i	$-0.812685\pi$
$857$	−1.04074e31	−1.66358	−0.831792	+	0.555088i	$0.812685\pi$
$858$	0	0
$859$	−8.40237e29	−0.131061	−0.0655306	−	0.997851i	$-0.520874\pi$
$859$	−8.40237e29	−0.131061	−0.0655306	+	0.997851i	$0.520874\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	6.66021e30	0.989406	0.494703	−	0.869062i	$-0.335277\pi$
$863$	6.66021e30	0.989406	0.494703	+	0.869062i	$0.335277\pi$
$864$	0	0
$865$	−1.66487e31	−2.41386
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−4.97957e30	−0.687835
$870$	0	0
$871$	−1.42944e30	−0.192741
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.60600e28	−0.00206377
$876$	0	0
$877$	−1.99416e30	−0.250187	−0.125093	−	0.992145i	$-0.539923\pi$
$877$	−1.99416e30	−0.250187	−0.125093	+	0.992145i	$0.539923\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	7.25457e30	0.867690	0.433845	−	0.900987i	$-0.357157\pi$
$881$	7.25457e30	0.867690	0.433845	+	0.900987i	$0.357157\pi$
$882$	0	0
$883$	7.43622e30	0.868491	0.434245	−	0.900795i	$-0.357015\pi$
$883$	7.43622e30	0.868491	0.434245	+	0.900795i	$0.357015\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−5.32005e30	−0.592540	−0.296270	−	0.955104i	$-0.595743\pi$
$887$	−5.32005e30	−0.592540	−0.296270	+	0.955104i	$0.595743\pi$
$888$	0	0
$889$	−2.77140e29	−0.0301461
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−3.10293e27	−0.000321982 0
$894$	0	0
$895$	3.50684e30	0.355446
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.07502e31	−2.00699
$900$	0	0
$901$	3.43418e30	0.324498
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−7.92809e30	−0.715086
$906$	0	0
$907$	−3.82478e30	−0.337077	−0.168539	−	0.985695i	$-0.553905\pi$
$907$	−3.82478e30	−0.337077	−0.168539	+	0.985695i	$0.553905\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	3.98935e30	0.335706	0.167853	−	0.985812i	$-0.446317\pi$
$911$	3.98935e30	0.335706	0.167853	+	0.985812i	$0.446317\pi$
$912$	0	0
$913$	−1.14968e31	−0.945440
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.56568e30	0.122977
$918$	0	0
$919$	−2.24335e31	−1.72220	−0.861102	−	0.508432i	$-0.830225\pi$
$919$	−2.24335e31	−1.72220	−0.861102	+	0.508432i	$0.830225\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1.04959e31	−0.769844
$924$	0	0
$925$	−7.76850e30	−0.556992
$926$	0	0
$927$	0	0
$928$	0	0
$929$	5.26649e30	0.360875	0.180437	−	0.983586i	$-0.442249\pi$
$929$	5.26649e30	0.360875	0.180437	+	0.983586i	$0.442249\pi$
$930$	0	0
$931$	−2.57835e27	−0.000172731 0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2.00515e31	−1.28418
$936$	0	0
$937$	−3.29722e30	−0.206482	−0.103241	−	0.994656i	$-0.532921\pi$
$937$	−3.29722e30	−0.206482	−0.103241	+	0.994656i	$0.532921\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2.14709e31	1.28576	0.642879	−	0.765968i	$-0.277740\pi$
$941$	2.14709e31	1.28576	0.642879	+	0.765968i	$0.277740\pi$
$942$	0	0
$943$	−1.92603e31	−1.12795
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.56465e31	0.876483	0.438242	−	0.898857i	$-0.355601\pi$
$947$	1.56465e31	0.876483	0.438242	+	0.898857i	$0.355601\pi$
$948$	0	0
$949$	5.88861e30	0.322639
$950$	0	0
$951$	0	0
$952$	0	0
$953$	4.71750e30	0.247307	0.123654	−	0.992325i	$-0.460539\pi$
$953$	4.71750e30	0.247307	0.123654	+	0.992325i	$0.460539\pi$
$954$	0	0
$955$	−2.31798e31	−1.18871
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.15412e30	0.105724
$960$	0	0
$961$	2.01951e31	0.969728
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−5.48413e31	−2.52099
$966$	0	0
$967$	−3.43670e31	−1.54584	−0.772919	−	0.634505i	$-0.781204\pi$
$967$	−3.43670e31	−1.54584	−0.772919	+	0.634505i	$0.781204\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−4.39842e31	−1.89450	−0.947249	−	0.320497i	$-0.896150\pi$
$971$	−4.39842e31	−1.89450	−0.947249	+	0.320497i	$0.896150\pi$
$972$	0	0
$973$	−4.15696e30	−0.175223
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.85934e31	0.750700	0.375350	−	0.926883i	$-0.377523\pi$
$977$	1.85934e31	0.750700	0.375350	+	0.926883i	$0.377523\pi$
$978$	0	0
$979$	3.46266e31	1.36833
$980$	0	0
$981$	0	0
$982$	0	0
$983$	2.18619e31	0.827704	0.413852	−	0.910344i	$-0.364183\pi$
$983$	2.18619e31	0.827704	0.413852	+	0.910344i	$0.364183\pi$
$984$	0	0
$985$	6.33049e31	2.34616
$986$	0	0
$987$	0	0
$988$	0	0
$989$	6.72306e31	2.38784
$990$	0	0
$991$	−8.68993e30	−0.302164	−0.151082	−	0.988521i	$-0.548276\pi$
$991$	−8.68993e30	−0.302164	−0.151082	+	0.988521i	$0.548276\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	2.51961e30	0.0839829
$996$	0	0
$997$	1.31572e31	0.429403	0.214701	−	0.976680i	$-0.431122\pi$
$997$	1.31572e31	0.429403	0.214701	+	0.976680i	$0.431122\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.22.a.f.1.1		3
3.2	odd	2		8.22.a.b.1.2	✓	3
12.11	even	2		16.22.a.f.1.2		3
24.5	odd	2		64.22.a.l.1.2		3
24.11	even	2		64.22.a.m.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.22.a.b.1.2	✓	3	3.2	odd	2
16.22.a.f.1.2		3	12.11	even	2
64.22.a.l.1.2		3	24.5	odd	2
64.22.a.m.1.2		3	24.11	even	2
72.22.a.f.1.1		3	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:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	72.a (trivial)

\(f(q)\)	\(=\)	\(q-3.09730e7 q^{5} +9.17385e7 q^{7} +O(q^{10})\)	\(q-3.09730e7 q^{5} +9.17385e7 q^{7} -8.72158e10 q^{11} -2.36850e11 q^{13} -7.42283e12 q^{17} +4.68680e9 q^{19} -3.33703e14 q^{23} +4.82489e14 q^{25} -3.23982e15 q^{29} +6.40473e15 q^{31} -2.84142e15 q^{35} -1.61009e16 q^{37} +5.77168e16 q^{41} -2.01468e17 q^{43} -6.62056e17 q^{47} -5.50130e17 q^{49} -4.62651e17 q^{53} +2.70133e18 q^{55} -7.39502e18 q^{59} -5.50188e18 q^{61} +7.33594e18 q^{65} +6.03520e18 q^{67} +4.43147e19 q^{71} -2.48622e19 q^{73} -8.00105e18 q^{77} +5.70948e19 q^{79} +1.31820e20 q^{83} +2.29907e20 q^{85} -3.97023e20 q^{89} -2.17282e19 q^{91} -1.45164e17 q^{95} -9.80402e20 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 24111774 q^{5} + 295988280 q^{7}+O(q^{10}) \) 3 * q + 24111774 * q^5 + 295988280 * q^7	\( 3 q + 24111774 q^{5} + 295988280 q^{7} + 40335108684 q^{11} + 133734425946 q^{13} - 7797732274422 q^{17} + 35788199781996 q^{19} - 193770761479080 q^{23} + 11\!\cdots\!01 q^{25}+ \cdots - 32\!\cdots\!42 q^{97}+O(q^{100}) \) 3 * q + 24111774 * q^5 + 295988280 * q^7 + 40335108684 * q^11 + 133734425946 * q^13 - 7797732274422 * q^17 + 35788199781996 * q^19 - 193770761479080 * q^23 + 1127564438439501 * q^25 - 5607343422466122 * q^29 + 11246757871503072 * q^31 - 5274251425350096 * q^35 - 24272499791100606 * q^37 + 298159108991869602 * q^41 - 33333932139754860 * q^43 + 120874283547603888 * q^47 - 850403331975639477 * q^49 + 1138443393004854222 * q^53 + 6957552484263571704 * q^55 - 9225624498709937412 * q^59 - 6554902294063924182 * q^61 + 20714581819561144452 * q^65 - 15793054074531629124 * q^67 + 41139582493467997704 * q^71 - 19422167949903851970 * q^73 - 68380237365358617888 * q^77 - 131735321299806049488 * q^79 - 64013993832679681068 * q^83 + 390258202763001297252 * q^85 - 429891446897537246766 * q^89 - 297181701588021496176 * q^91 + 1036372406649019824696 * q^95 - 324059336514148638042 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more