LMFDB - Embedded newform 8.22.a.b.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8,22,Mod(1,8)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8.1"); S:= CuspForms(chi, 22); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 22, names="a")

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	8.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$22.3581875430$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{21}\cdot 3\cdot 5\cdot 7 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$57.9766$ of defining polynomial
Character	$\chi$	$=$	8.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-7983.67 q^{3} +3.09730e7 q^{5} +9.17385e7 q^{7} -1.03966e10 q^{9} +8.72158e10 q^{11} -2.36850e11 q^{13} -2.47278e11 q^{15} +7.42283e12 q^{17} +4.68680e9 q^{19} -7.32410e11 q^{21} +3.33703e14 q^{23} +4.82489e14 q^{25} +1.66515e14 q^{27} +3.23982e15 q^{29} +6.40473e15 q^{31} -6.96303e14 q^{33} +2.84142e15 q^{35} -1.61009e16 q^{37} +1.89093e15 q^{39} -5.77168e16 q^{41} -2.01468e17 q^{43} -3.22014e17 q^{45} +6.62056e17 q^{47} -5.50130e17 q^{49} -5.92615e16 q^{51} +4.62651e17 q^{53} +2.70133e18 q^{55} -3.74179e13 q^{57} +7.39502e18 q^{59} -5.50188e18 q^{61} -9.53770e17 q^{63} -7.33594e18 q^{65} +6.03520e18 q^{67} -2.66418e18 q^{69} -4.43147e19 q^{71} -2.48622e19 q^{73} -3.85204e18 q^{75} +8.00105e18 q^{77} +5.70948e19 q^{79} +1.07423e20 q^{81} -1.31820e20 q^{83} +2.29907e20 q^{85} -2.58657e19 q^{87} +3.97023e20 q^{89} -2.17282e19 q^{91} -5.11333e19 q^{93} +1.45164e17 q^{95} -9.80402e20 q^{97} -9.06749e20 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 96764 q^{3} - 24111774 q^{5} + 295988280 q^{7} + 18844697239 q^{9} - 40335108684 q^{11} + 133734425946 q^{13} - 1223136458200 q^{15} + 7797732274422 q^{17} + 35788199781996 q^{19} + 198539224853088 q^{21}+ \cdots - 94\!\cdots\!60 q^{99}+O(q^{100}) $ 3 * q + 96764 * q^3 - 24111774 * q^5 + 295988280 * q^7 + 18844697239 * q^9 - 40335108684 * q^11 + 133734425946 * q^13 - 1223136458200 * q^15 + 7797732274422 * q^17 + 35788199781996 * q^19 + 198539224853088 * q^21 + 193770761479080 * q^23 + 1127564438439501 * q^25 + 5282002293508952 * q^27 + 5607343422466122 * q^29 + 11246757871503072 * q^31 + 10014149026970384 * q^33 + 5274251425350096 * q^35 - 24272499791100606 * q^37 - 52925308377862264 * q^39 - 298159108991869602 * q^41 - 33333932139754860 * q^43 - 927411477977893478 * q^45 - 120874283547603888 * q^47 - 850403331975639477 * q^49 + 3954388789815661240 * q^51 - 1138443393004854222 * q^53 + 6957552484263571704 * q^55 + 681697547133656944 * q^57 + 9225624498709937412 * q^59 - 6554902294063924182 * q^61 + 21785214559631141976 * q^63 - 20714581819561144452 * q^65 - 15793054074531629124 * q^67 - 72988310309689939168 * q^69 - 41139582493467997704 * q^71 - 19422167949903851970 * q^73 - 75231657393126995900 * q^75 + 68380237365358617888 * q^77 - 131735321299806049488 * q^79 + 500425796062282339147 * q^81 + 64013993832679681068 * q^83 + 390258202763001297252 * q^85 + 101898953973185066568 * q^87 + 429891446897537246766 * q^89 - 297181701588021496176 * q^91 + 640035369009914700160 * q^93 - 1036372406649019824696 * q^95 - 324059336514148638042 * q^97 - 949982905688477352860 * q^99

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
$3$	−7983.67	−0.0780602	−0.0390301	−	0.999238i	$-0.512427\pi$
$3$	−7983.67	−0.0780602	−0.0390301	+	0.999238i	$0.512427\pi$
$4$	0	0
$5$	3.09730e7	1.41840	0.709199	−	0.705008i	$-0.249057\pi$
$5$	3.09730e7	1.41840	0.709199	+	0.705008i	$0.249057\pi$
$6$	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
$9$	−1.03966e10	−0.993907
$10$	0	0
$11$	8.72158e10	1.01385	0.506923	−	0.861991i	$-0.330783\pi$
$11$	8.72158e10	1.01385	0.506923	+	0.861991i	$0.330783\pi$
$12$	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
$15$	−2.47278e11	−0.110720
$16$	0	0
$17$	7.42283e12	0.893009	0.446505	−	0.894781i	$-0.352669\pi$
$17$	7.42283e12	0.893009	0.446505	+	0.894781i	$0.352669\pi$
$18$	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
$21$	−7.32410e11	−0.00958190
$22$	0	0
$23$	3.33703e14	1.67965	0.839823	−	0.542860i	$-0.182659\pi$
$23$	3.33703e14	1.67965	0.839823	+	0.542860i	$0.182659\pi$
$24$	0	0
$25$	4.82489e14	1.01185
$26$	0	0
$27$	1.66515e14	0.155645
$28$	0	0
$29$	3.23982e15	1.43002	0.715010	−	0.699114i	$-0.246422\pi$
$29$	3.23982e15	1.43002	0.715010	+	0.699114i	$0.246422\pi$
$30$	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
$33$	−6.96303e14	−0.0791410
$34$	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
$39$	1.89093e15	0.0371961
$40$	0	0
$41$	−5.77168e16	−0.671540	−0.335770	−	0.941944i	$-0.608997\pi$
$41$	−5.77168e16	−0.671540	−0.335770	+	0.941944i	$0.608997\pi$
$42$	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
$45$	−3.22014e17	−1.40976
$46$	0	0
$47$	6.62056e17	1.83598	0.917989	−	0.396607i	$-0.129812\pi$
$47$	6.62056e17	1.83598	0.917989	+	0.396607i	$0.129812\pi$
$48$	0	0
$49$	−5.50130e17	−0.984932
$50$	0	0
$51$	−5.92615e16	−0.0697085
$52$	0	0
$53$	4.62651e17	0.363376	0.181688	−	0.983356i	$-0.441844\pi$
$53$	4.62651e17	0.363376	0.181688	+	0.983356i	$0.441844\pi$
$54$	0	0
$55$	2.70133e18	1.43804
$56$	0	0
$57$	−3.74179e13	−1.36897e−5 0
$58$	0	0
$59$	7.39502e18	1.88362	0.941809	−	0.336148i	$-0.109124\pi$
$59$	7.39502e18	1.88362	0.941809	+	0.336148i	$0.109124\pi$
$60$	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
$63$	−9.53770e17	−0.122002
$64$	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
$69$	−2.66418e18	−0.131113
$70$	0	0
$71$	−4.43147e19	−1.61561	−0.807803	−	0.589453i	$-0.799343\pi$
$71$	−4.43147e19	−1.61561	−0.807803	+	0.589453i	$0.799343\pi$
$72$	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
$75$	−3.85204e18	−0.0789855
$76$	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
$81$	1.07423e20	0.981757
$82$	0	0
$83$	−1.31820e20	−0.932528	−0.466264	−	0.884646i	$-0.654400\pi$
$83$	−1.31820e20	−0.932528	−0.466264	+	0.884646i	$0.654400\pi$
$84$	0	0
$85$	2.29907e20	1.26664
$86$	0	0
$87$	−2.58657e19	−0.111628
$88$	0	0
$89$	3.97023e20	1.34965	0.674823	−	0.737979i	$-0.264220\pi$
$89$	3.97023e20	1.34965	0.674823	+	0.737979i	$0.264220\pi$
$90$	0	0
$91$	−2.17282e19	−0.0584911
$92$	0	0
$93$	−5.11333e19	−0.109555
$94$	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
$99$	−9.06749e20	−1.00767
$100$	0	0
$101$	6.46191e19	0.0582085	0.0291042	−	0.999576i	$-0.490735\pi$
$101$	6.46191e19	0.0582085	0.0291042	+	0.999576i	$0.490735\pi$
$102$	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
$105$	−2.26849e19	−0.0135910
$106$	0	0
$107$	7.69439e20	0.378133	0.189066	−	0.981964i	$-0.439454\pi$
$107$	7.69439e20	0.378133	0.189066	+	0.981964i	$0.439454\pi$
$108$	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
$111$	1.28544e20	0.0429696
$112$	0	0
$113$	−2.53293e20	−0.0701938	−0.0350969	−	0.999384i	$-0.511174\pi$
$113$	−2.53293e20	−0.0701938	−0.0350969	+	0.999384i	$0.511174\pi$
$114$	0	0
$115$	1.03358e22	2.38241
$116$	0	0
$117$	2.46243e21	0.473602
$118$	0	0
$119$	6.80959e20	0.109617
$120$	0	0
$121$	2.06345e20	0.0278836
$122$	0	0
$123$	4.60792e20	0.0524206
$124$	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
$129$	1.60846e21	0.110973
$130$	0	0
$131$	−1.70668e22	−1.00185	−0.500925	−	0.865491i	$-0.667007\pi$
$131$	−1.70668e22	−1.00185	−0.500925	+	0.865491i	$0.667007\pi$
$132$	0	0
$133$	4.29960e17	2.15271e−5 0
$134$	0	0
$135$	5.15748e21	0.220766
$136$	0	0
$137$	−2.34811e22	−0.861296	−0.430648	−	0.902520i	$-0.641715\pi$
$137$	−2.34811e22	−0.861296	−0.430648	+	0.902520i	$0.641715\pi$
$138$	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
$141$	−5.28564e21	−0.143317
$142$	0	0
$143$	−2.06570e22	−0.483103
$144$	0	0
$145$	1.00347e23	2.02834
$146$	0	0
$147$	4.39206e21	0.0768840
$148$	0	0
$149$	6.98352e22	1.06076	0.530381	−	0.847759i	$-0.322049\pi$
$149$	6.98352e22	1.06076	0.530381	+	0.847759i	$0.322049\pi$
$150$	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
$153$	−7.71723e22	−0.887568
$154$	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
$159$	−3.69365e21	−0.0283652
$160$	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
$165$	−2.15666e22	−0.112253
$166$	0	0
$167$	−2.25766e23	−1.03547	−0.517733	−	0.855542i	$-0.673224\pi$
$167$	−2.25766e23	−1.03547	−0.517733	+	0.855542i	$0.673224\pi$
$168$	0	0
$169$	−1.90967e23	−0.772943
$170$	0	0
$171$	−4.87269e19	−0.000174305 0
$172$	0	0
$173$	−5.37523e23	−1.70182	−0.850909	−	0.525313i	$-0.823948\pi$
$173$	−5.37523e23	−1.70182	−0.850909	+	0.525313i	$0.823948\pi$
$174$	0	0
$175$	4.42628e22	0.124205
$176$	0	0
$177$	−5.90394e22	−0.147036
$178$	0	0
$179$	1.13222e23	0.250597	0.125299	−	0.992119i	$-0.460011\pi$
$179$	1.13222e23	0.250597	0.125299	+	0.992119i	$0.460011\pi$
$180$	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
$183$	4.39252e22	0.0770863
$184$	0	0
$185$	−4.98692e23	−0.780781
$186$	0	0
$187$	6.47388e23	0.905374
$188$	0	0
$189$	1.52759e22	0.0191054
$190$	0	0
$191$	−7.48387e23	−0.838062	−0.419031	−	0.907972i	$-0.637630\pi$
$191$	−7.48387e23	−0.838062	−0.419031	+	0.907972i	$0.637630\pi$
$192$	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
$195$	5.85678e22	0.0527589
$196$	0	0
$197$	2.04387e24	1.65409	0.827044	−	0.562137i	$-0.190021\pi$
$197$	2.04387e24	1.65409	0.827044	+	0.562137i	$0.190021\pi$
$198$	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
$201$	−4.81831e22	−0.0315745
$202$	0	0
$203$	2.97216e23	0.175535
$204$	0	0
$205$	−1.78766e24	−0.952511
$206$	0	0
$207$	−3.46938e24	−1.66941
$208$	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
$213$	3.53794e23	0.126114
$214$	0	0
$215$	−6.24008e24	−2.01644
$216$	0	0
$217$	5.87560e23	0.172276
$218$	0	0
$219$	1.98492e23	0.0528542
$220$	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
$225$	−5.01625e24	−1.00569
$226$	0	0
$227$	6.71337e24	1.22650	0.613252	−	0.789887i	$-0.289861\pi$
$227$	6.71337e24	1.22650	0.613252	+	0.789887i	$0.289861\pi$
$228$	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
$231$	−6.38777e22	−0.00971457
$232$	0	0
$233$	−3.90742e24	−0.542817	−0.271408	−	0.962464i	$-0.587489\pi$
$233$	−3.90742e24	−0.542817	−0.271408	+	0.962464i	$0.587489\pi$
$234$	0	0
$235$	2.05059e25	2.60415
$236$	0	0
$237$	−4.55826e23	−0.0529592
$238$	0	0
$239$	2.38784e23	0.0253997	0.0126998	−	0.999919i	$-0.495957\pi$
$239$	2.38784e23	0.0253997	0.0126998	+	0.999919i	$0.495957\pi$
$240$	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
$243$	−2.59944e24	−0.232281
$244$	0	0
$245$	−1.70392e25	−1.39703
$246$	0	0
$247$	−1.11007e21	−8.35663e−5 0
$248$	0	0
$249$	1.05241e24	0.0727934
$250$	0	0
$251$	−5.80122e24	−0.368931	−0.184466	−	0.982839i	$-0.559055\pi$
$251$	−5.80122e24	−0.368931	−0.184466	+	0.982839i	$0.559055\pi$
$252$	0	0
$253$	2.91042e25	1.70290
$254$	0	0
$255$	−1.83551e24	−0.0988744
$256$	0	0
$257$	−2.21415e25	−1.09878	−0.549388	−	0.835567i	$-0.685139\pi$
$257$	−2.21415e25	−1.09878	−0.549388	+	0.835567i	$0.685139\pi$
$258$	0	0
$259$	−1.47707e24	−0.0675699
$260$	0	0
$261$	−3.36832e25	−1.42131
$262$	0	0
$263$	−3.74763e25	−1.45956	−0.729779	−	0.683683i	$-0.760377\pi$
$263$	−3.74763e25	−1.45956	−0.729779	+	0.683683i	$0.760377\pi$
$264$	0	0
$265$	1.43297e25	0.515412
$266$	0	0
$267$	−3.16970e24	−0.105354
$268$	0	0
$269$	1.64824e25	0.506549	0.253274	−	0.967394i	$-0.418492\pi$
$269$	1.64824e25	0.506549	0.253274	+	0.967394i	$0.418492\pi$
$270$	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
$273$	1.73471e23	0.00456583
$274$	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
$279$	−6.65875e25	−1.39492
$280$	0	0
$281$	6.45378e25	1.25429	0.627145	−	0.778903i	$-0.284223\pi$
$281$	6.45378e25	1.25429	0.627145	+	0.778903i	$0.284223\pi$
$282$	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
$285$	−1.15894e21	−1.94174e−5 0
$286$	0	0
$287$	−5.29485e24	−0.0824317
$288$	0	0
$289$	−1.39935e25	−0.202535
$290$	0	0
$291$	7.82721e24	0.105373
$292$	0	0
$293$	−5.87863e25	−0.736489	−0.368244	−	0.929729i	$-0.620041\pi$
$293$	−5.87863e25	−0.736489	−0.368244	+	0.929729i	$0.620041\pi$
$294$	0	0
$295$	2.29046e26	2.67172
$296$	0	0
$297$	1.45228e25	0.157800
$298$	0	0
$299$	−7.90374e25	−0.800360
$300$	0	0
$301$	−1.84824e25	−0.174506
$302$	0	0
$303$	−5.15898e23	−0.00454377
$304$	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
$309$	5.90377e23	0.00423220
$310$	0	0
$311$	−7.32700e24	−0.0490842	−0.0245421	−	0.999699i	$-0.507813\pi$
$311$	−7.32700e24	−0.0490842	−0.0245421	+	0.999699i	$0.507813\pi$
$312$	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
$315$	−2.95411e25	−0.173048
$316$	0	0
$317$	−9.82997e25	−0.538803	−0.269402	−	0.963028i	$-0.586826\pi$
$317$	−9.82997e25	−0.538803	−0.269402	+	0.963028i	$0.586826\pi$
$318$	0	0
$319$	2.82564e26	1.44982
$320$	0	0
$321$	−6.14295e24	−0.0295171
$322$	0	0
$323$	3.47893e22	0.000156610 0
$324$	0	0
$325$	−1.14277e26	−0.482153
$326$	0	0
$327$	2.61465e25	0.103434
$328$	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
$333$	1.67395e26	0.547113
$334$	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
$339$	2.02221e24	0.00547934
$340$	0	0
$341$	5.58594e26	1.42290
$342$	0	0
$343$	−1.01708e26	−0.243651
$344$	0	0
$345$	−8.25175e25	−0.185971
$346$	0	0
$347$	1.00972e26	0.214161	0.107080	−	0.994250i	$-0.465850\pi$
$347$	1.00972e26	0.214161	0.107080	+	0.994250i	$0.465850\pi$
$348$	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
$351$	−3.94391e25	−0.0741655
$352$	0	0
$353$	1.88459e26	0.333873	0.166937	−	0.985968i	$-0.446613\pi$
$353$	1.88459e26	0.333873	0.166937	+	0.985968i	$0.446613\pi$
$354$	0	0
$355$	−1.37256e27	−2.29157
$356$	0	0
$357$	−5.43656e24	−0.00855673
$358$	0	0
$359$	5.21557e26	0.774123	0.387062	−	0.922054i	$-0.373490\pi$
$359$	5.21557e26	0.774123	0.387062	+	0.922054i	$0.373490\pi$
$360$	0	0
$361$	−7.14209e26	−1.00000
$362$	0	0
$363$	−1.64740e24	−0.00217660
$364$	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
$369$	6.00060e26	0.667448
$370$	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
$375$	−1.39765e24	−0.00131241
$376$	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
$381$	2.41186e25	0.0191707
$382$	0	0
$383$	1.75914e27	1.32347	0.661733	−	0.749740i	$-0.269821\pi$
$383$	1.75914e27	1.32347	0.661733	+	0.749740i	$0.269821\pi$
$384$	0	0
$385$	2.47816e26	0.176519
$386$	0	0
$387$	2.09459e27	1.41297
$388$	0	0
$389$	−2.73236e26	−0.174609	−0.0873046	−	0.996182i	$-0.527825\pi$
$389$	−2.73236e26	−0.174609	−0.0873046	+	0.996182i	$0.527825\pi$
$390$	0	0
$391$	2.47702e27	1.49994
$392$	0	0
$393$	1.36256e26	0.0782046
$394$	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
$399$	−3.43266e21	−1.68041e−6 0
$400$	0	0
$401$	−1.15061e27	−0.534455	−0.267228	−	0.963633i	$-0.586108\pi$
$401$	−1.15061e27	−0.534455	−0.267228	+	0.963633i	$0.586108\pi$
$402$	0	0
$403$	−1.51696e27	−0.668761
$404$	0	0
$405$	3.32721e27	1.39252
$406$	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
$411$	1.87465e26	0.0672330
$412$	0	0
$413$	6.78408e26	0.231215
$414$	0	0
$415$	−4.08287e27	−1.32270
$416$	0	0
$417$	3.61766e26	0.111429
$418$	0	0
$419$	4.10695e27	1.20302	0.601510	−	0.798866i	$-0.294566\pi$
$419$	4.10695e27	1.20302	0.601510	+	0.798866i	$0.294566\pi$
$420$	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
$423$	−6.88314e27	−1.82479
$424$	0	0
$425$	3.58144e27	0.903594
$426$	0	0
$427$	−5.04734e26	−0.121219
$428$	0	0
$429$	1.64919e26	0.0377111
$430$	0	0
$431$	−4.51446e27	−0.983092	−0.491546	−	0.870852i	$-0.663568\pi$
$431$	−4.51446e27	−0.983092	−0.491546	+	0.870852i	$0.663568\pi$
$432$	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
$435$	−8.01138e26	−0.158332
$436$	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
$441$	5.71949e27	0.978931
$442$	0	0
$443$	8.67248e27	1.41548	0.707741	−	0.706472i	$-0.249714\pi$
$443$	8.67248e27	1.41548	0.707741	+	0.706472i	$0.249714\pi$
$444$	0	0
$445$	1.22970e28	1.91434
$446$	0	0
$447$	−5.57541e26	−0.0828034
$448$	0	0
$449$	−4.22534e27	−0.598790	−0.299395	−	0.954129i	$-0.596785\pi$
$449$	−4.22534e27	−0.598790	−0.299395	+	0.954129i	$0.596785\pi$
$450$	0	0
$451$	−5.03382e27	−0.680838
$452$	0	0
$453$	3.78075e26	0.0488143
$454$	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
$459$	1.23601e27	0.138992
$460$	0	0
$461$	−9.88705e27	−1.06220	−0.531101	−	0.847309i	$-0.678221\pi$
$461$	−9.88705e27	−1.06220	−0.531101	+	0.847309i	$0.678221\pi$
$462$	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
$465$	−1.58375e27	−0.155393
$466$	0	0
$467$	1.60212e28	1.50268	0.751342	−	0.659913i	$-0.229407\pi$
$467$	1.60212e28	1.50268	0.751342	+	0.659913i	$0.229407\pi$
$468$	0	0
$469$	5.53660e26	0.0496511
$470$	0	0
$471$	1.77992e25	0.00152644
$472$	0	0
$473$	−1.75712e28	−1.44132
$474$	0	0
$475$	2.26133e24	0.000177452 0
$476$	0	0
$477$	−4.81000e27	−0.361162
$478$	0	0
$479$	−9.14679e27	−0.657273	−0.328636	−	0.944457i	$-0.606589\pi$
$479$	−9.14679e27	−0.657273	−0.328636	+	0.944457i	$0.606589\pi$
$480$	0	0
$481$	3.81349e27	0.262300
$482$	0	0
$483$	−2.44407e26	−0.0160942
$484$	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
$489$	−1.18320e27	−0.0684407
$490$	0	0
$491$	1.10020e28	0.609702	0.304851	−	0.952400i	$-0.401393\pi$
$491$	1.10020e28	0.609702	0.304851	+	0.952400i	$0.401393\pi$
$492$	0	0
$493$	2.40487e28	1.27702
$494$	0	0
$495$	−2.80847e28	−1.42927
$496$	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
$501$	1.80244e27	0.0808288
$502$	0	0
$503$	1.67038e28	0.718375	0.359188	−	0.933265i	$-0.383054\pi$
$503$	1.67038e28	0.718375	0.359188	+	0.933265i	$0.383054\pi$
$504$	0	0
$505$	2.00145e27	0.0825628
$506$	0	0
$507$	1.52462e27	0.0603361
$508$	0	0
$509$	−5.42146e27	−0.205864	−0.102932	−	0.994688i	$-0.532822\pi$
$509$	−5.42146e27	−0.205864	−0.102932	+	0.994688i	$0.532822\pi$
$510$	0	0
$511$	−2.28082e27	−0.0831136
$512$	0	0
$513$	7.80424e23	2.72960e−5 0
$514$	0	0
$515$	−2.29039e27	−0.0769014
$516$	0	0
$517$	5.77418e28	1.86140
$518$	0	0
$519$	4.29141e27	0.132844
$520$	0	0
$521$	1.37414e28	0.408541	0.204271	−	0.978914i	$-0.434518\pi$
$521$	1.37414e28	0.408541	0.204271	+	0.978914i	$0.434518\pi$
$522$	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
$525$	−3.53380e26	−0.00969548
$526$	0	0
$527$	4.75412e28	1.25331
$528$	0	0
$529$	7.18860e28	1.82121
$530$	0	0
$531$	−7.68831e28	−1.87214
$532$	0	0
$533$	1.36702e28	0.319992
$534$	0	0
$535$	2.38318e28	0.536343
$536$	0	0
$537$	−9.03932e26	−0.0195617
$538$	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
$543$	−2.04356e27	−0.0393541
$544$	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
$549$	5.72009e28	0.981507
$550$	0	0
$551$	1.51844e25	0.000250788 0
$552$	0	0
$553$	5.23779e27	0.0832787
$554$	0	0
$555$	3.98140e27	0.0609480
$556$	0	0
$557$	−4.85895e28	−0.716247	−0.358124	−	0.933674i	$-0.616583\pi$
$557$	−4.85895e28	−0.716247	−0.358124	+	0.933674i	$0.616583\pi$
$558$	0	0
$559$	4.77177e28	0.677416
$560$	0	0
$561$	−5.16854e27	−0.0706736
$562$	0	0
$563$	−4.82251e28	−0.635236	−0.317618	−	0.948219i	$-0.602883\pi$
$563$	−4.82251e28	−0.635236	−0.317618	+	0.948219i	$0.602883\pi$
$564$	0	0
$565$	−7.84523e27	−0.0995628
$566$	0	0
$567$	9.85481e27	0.120511
$568$	0	0
$569$	−8.25601e28	−0.972952	−0.486476	−	0.873694i	$-0.661718\pi$
$569$	−8.25601e28	−0.972952	−0.486476	+	0.873694i	$0.661718\pi$
$570$	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
$573$	5.97488e27	0.0654193
$574$	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
$579$	−1.41360e28	−0.138740
$580$	0	0
$581$	−1.20930e28	−0.114468
$582$	0	0
$583$	4.03504e28	0.368408
$584$	0	0
$585$	7.62690e28	0.671756
$586$	0	0
$587$	1.52918e29	1.29945	0.649723	−	0.760171i	$-0.274885\pi$
$587$	1.52918e29	1.29945	0.649723	+	0.760171i	$0.274885\pi$
$588$	0	0
$589$	3.00177e25	0.000246131 0
$590$	0	0
$591$	−1.63176e28	−0.129118
$592$	0	0
$593$	3.89818e28	0.297706	0.148853	−	0.988859i	$-0.452442\pi$
$593$	3.89818e28	0.297706	0.148853	+	0.988859i	$0.452442\pi$
$594$	0	0
$595$	2.10913e28	0.155481
$596$	0	0
$597$	6.49461e26	0.00462192
$598$	0	0
$599$	1.28684e29	0.884186	0.442093	−	0.896969i	$-0.354236\pi$
$599$	1.28684e29	0.884186	0.442093	+	0.896969i	$0.354236\pi$
$600$	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
$603$	−6.27457e28	−0.402024
$604$	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
$609$	−2.37288e27	−0.0137023
$610$	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
$615$	1.42721e28	0.0743532
$616$	0	0
$617$	3.47454e29	1.74946	0.874728	−	0.484614i	$-0.161040\pi$
$617$	3.47454e29	1.74946	0.874728	+	0.484614i	$0.161040\pi$
$618$	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
$621$	5.55666e28	0.261428
$622$	0	0
$623$	3.64223e28	0.165669
$624$	0	0
$625$	−2.24647e29	−0.988006
$626$	0	0
$627$	−3.26343e24	−1.38792e−5 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
$633$	−2.81602e28	−0.108367
$634$	0	0
$635$	−9.35689e28	−0.348343
$636$	0	0
$637$	1.30298e29	0.469325
$638$	0	0
$639$	4.60723e29	1.60576
$640$	0	0
$641$	−3.43690e29	−1.15920	−0.579598	−	0.814902i	$-0.696791\pi$
$641$	−3.43690e29	−1.15920	−0.579598	+	0.814902i	$0.696791\pi$
$642$	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
$645$	4.98187e28	0.157404
$646$	0	0
$647$	3.06810e29	0.938370	0.469185	−	0.883100i	$-0.344548\pi$
$647$	3.06810e29	0.938370	0.469185	+	0.883100i	$0.344548\pi$
$648$	0	0
$649$	6.44962e29	1.90970
$650$	0	0
$651$	−4.69089e27	−0.0134479
$652$	0	0
$653$	1.17693e29	0.326710	0.163355	−	0.986567i	$-0.447768\pi$
$653$	1.17693e29	0.326710	0.163355	+	0.986567i	$0.447768\pi$
$654$	0	0
$655$	−5.28609e29	−1.42102
$656$	0	0
$657$	2.58483e29	0.672969
$658$	0	0
$659$	3.42901e29	0.864712	0.432356	−	0.901703i	$-0.357682\pi$
$659$	3.42901e29	0.864712	0.432356	+	0.901703i	$0.357682\pi$
$660$	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
$663$	1.40361e28	0.0332164
$664$	0	0
$665$	1.33172e25	3.05340e−5 0
$666$	0	0
$667$	1.08114e30	2.40193
$668$	0	0
$669$	1.89671e28	0.0408343
$670$	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
$675$	8.03418e28	0.157490
$676$	0	0
$677$	5.12797e29	0.974461	0.487230	−	0.873273i	$-0.338007\pi$
$677$	5.12797e29	0.974461	0.487230	+	0.873273i	$0.338007\pi$
$678$	0	0
$679$	−8.99406e28	−0.165700
$680$	0	0
$681$	−5.35973e28	−0.0957411
$682$	0	0
$683$	5.75298e29	0.996495	0.498248	−	0.867035i	$-0.333977\pi$
$683$	5.75298e29	0.996495	0.498248	+	0.867035i	$0.333977\pi$
$684$	0	0
$685$	−7.27280e29	−1.22166
$686$	0	0
$687$	−2.48156e28	−0.0404276
$688$	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
$693$	−8.31838e28	−0.123691
$694$	0	0
$695$	−1.40348e30	−2.02473
$696$	0	0
$697$	−4.28422e29	−0.599691
$698$	0	0
$699$	3.11956e28	0.0423724
$700$	0	0
$701$	−8.94607e29	−1.17922	−0.589608	−	0.807690i	$-0.700718\pi$
$701$	−8.94607e29	−1.17922	−0.589608	+	0.807690i	$0.700718\pi$
$702$	0	0
$703$	−7.54616e25	−9.65373e−5 0
$704$	0	0
$705$	−1.63712e29	−0.203280
$706$	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
$711$	−5.93592e29	−0.674307
$712$	0	0
$713$	2.13728e30	2.35733
$714$	0	0
$715$	−6.39810e29	−0.685232
$716$	0	0
$717$	−1.90638e27	−0.00198270
$718$	0	0
$719$	4.85535e29	0.490419	0.245210	−	0.969470i	$-0.421143\pi$
$719$	4.85535e29	0.490419	0.245210	+	0.969470i	$0.421143\pi$
$720$	0	0
$721$	−6.78389e27	−0.00665516
$722$	0	0
$723$	−9.10175e28	−0.0867307
$724$	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
$729$	−1.10293e30	−0.963625
$730$	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
$735$	1.36035e29	0.109052
$736$	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
$741$	8.86242e24	6.52321e−6 0
$742$	0	0
$743$	−7.60139e29	−0.543889	−0.271945	−	0.962313i	$-0.587667\pi$
$743$	−7.60139e29	−0.543889	−0.271945	+	0.962313i	$0.587667\pi$
$744$	0	0
$745$	2.16300e30	1.50458
$746$	0	0
$747$	1.37048e30	0.926846
$748$	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
$753$	4.63151e28	0.0287988
$754$	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
$759$	−2.32358e29	−0.132929
$760$	0	0
$761$	−3.46912e30	−1.93055	−0.965274	−	0.261241i	$-0.915868\pi$
$761$	−3.46912e30	−1.93055	−0.965274	+	0.261241i	$0.915868\pi$
$762$	0	0
$763$	−3.00444e29	−0.162650
$764$	0	0
$765$	−2.39026e30	−1.25892
$766$	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
$771$	1.76770e29	0.0857707
$772$	0	0
$773$	−1.63916e30	−0.773990	−0.386995	−	0.922082i	$-0.626487\pi$
$773$	−1.63916e30	−0.773990	−0.386995	+	0.922082i	$0.626487\pi$
$774$	0	0
$775$	3.09021e30	1.42011
$776$	0	0
$777$	1.17924e28	0.00527452
$778$	0	0
$779$	−2.70507e26	−0.000117770 0
$780$	0	0
$781$	−3.86494e30	−1.63798
$782$	0	0
$783$	5.39480e29	0.222575
$784$	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
$789$	2.99199e29	0.113933
$790$	0	0
$791$	−2.32367e28	−0.00861630
$792$	0	0
$793$	1.30312e30	0.470560
$794$	0	0
$795$	−1.14403e29	−0.0402332
$796$	0	0
$797$	1.32063e30	0.452345	0.226173	−	0.974087i	$-0.427379\pi$
$797$	1.32063e30	0.452345	0.226173	+	0.974087i	$0.427379\pi$
$798$	0	0
$799$	4.91433e30	1.63954
$800$	0	0
$801$	−4.12769e30	−1.34142
$802$	0	0
$803$	−2.16838e30	−0.686470
$804$	0	0
$805$	9.48189e29	0.292441
$806$	0	0
$807$	−1.31590e29	−0.0395413
$808$	0	0
$809$	5.54086e30	1.62225	0.811124	−	0.584874i	$-0.198856\pi$
$809$	5.54086e30	1.62225	0.811124	+	0.584874i	$0.198856\pi$
$810$	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
$813$	2.28467e29	0.0635145
$814$	0	0
$815$	4.59026e30	1.24361
$816$	0	0
$817$	−9.44242e26	−0.000249317 0
$818$	0	0
$819$	2.25900e29	0.0581347
$820$	0	0
$821$	6.49500e30	1.62921	0.814603	−	0.580019i	$-0.196955\pi$
$821$	6.49500e30	1.62921	0.814603	+	0.580019i	$0.196955\pi$
$822$	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
$825$	−3.35959e29	−0.0800791
$826$	0	0
$827$	−2.99217e29	−0.0695309	−0.0347655	−	0.999395i	$-0.511068\pi$
$827$	−2.99217e29	−0.0695309	−0.0347655	+	0.999395i	$0.511068\pi$
$828$	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
$831$	−4.20009e29	−0.0927787
$832$	0	0
$833$	−4.08352e30	−0.879554
$834$	0	0
$835$	−6.99266e30	−1.46870
$836$	0	0
$837$	1.06649e30	0.218443
$838$	0	0
$839$	−6.48734e30	−1.29588	−0.647942	−	0.761690i	$-0.724370\pi$
$839$	−6.48734e30	−1.29588	−0.647942	+	0.761690i	$0.724370\pi$
$840$	0	0
$841$	5.36360e30	1.04496
$842$	0	0
$843$	−5.15249e29	−0.0979101
$844$	0	0
$845$	−5.91481e30	−1.09634
$846$	0	0
$847$	1.89298e28	0.00342272
$848$	0	0
$849$	−3.64089e29	−0.0642210
$850$	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
$855$	−1.50922e27	−0.000247234 0
$856$	0	0
$857$	1.04074e31	1.66358	0.831792	−	0.555088i	$-0.187315\pi$
$857$	1.04074e31	1.66358	0.831792	+	0.555088i	$0.187315\pi$
$858$	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
$861$	4.22724e28	0.00643463
$862$	0	0
$863$	−6.66021e30	−0.989406	−0.494703	−	0.869062i	$-0.664723\pi$
$863$	−6.66021e30	−0.989406	−0.494703	+	0.869062i	$0.664723\pi$
$864$	0	0
$865$	−1.66487e31	−2.41386
$866$	0	0
$867$	1.11720e29	0.0158099
$868$	0	0
$869$	4.97957e30	0.687835
$870$	0	0
$871$	−1.42944e30	−0.192741
$872$	0	0
$873$	1.01929e31	1.34167
$874$	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
$879$	4.69331e29	0.0574905
$880$	0	0
$881$	−7.25457e30	−0.867690	−0.433845	−	0.900987i	$-0.642843\pi$
$881$	−7.25457e30	−0.867690	−0.433845	+	0.900987i	$0.642843\pi$
$882$	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
$885$	−1.82863e30	−0.208555
$886$	0	0
$887$	5.32005e30	0.592540	0.296270	−	0.955104i	$-0.404257\pi$
$887$	5.32005e30	0.592540	0.296270	+	0.955104i	$0.404257\pi$
$888$	0	0
$889$	−2.77140e29	−0.0301461
$890$	0	0
$891$	9.36897e30	0.995350
$892$	0	0
$893$	3.10293e27	0.000321982 0
$894$	0	0
$895$	3.50684e30	0.355446
$896$	0	0
$897$	6.31009e29	0.0624762
$898$	0	0
$899$	2.07502e31	2.00699
$900$	0	0
$901$	3.43418e30	0.324498
$902$	0	0
$903$	1.47557e29	0.0136220
$904$	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
$909$	−6.71820e29	−0.0578538
$910$	0	0
$911$	−3.98935e30	−0.335706	−0.167853	−	0.985812i	$-0.553683\pi$
$911$	−3.98935e30	−0.335706	−0.167853	+	0.985812i	$0.553683\pi$
$912$	0	0
$913$	−1.14968e31	−0.945440
$914$	0	0
$915$	1.36050e30	0.109339
$916$	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
$921$	1.32085e30	0.0991119
$922$	0	0
$923$	1.04959e31	0.769844
$924$	0	0
$925$	−7.76850e30	−0.556992
$926$	0	0
$927$	7.68810e29	0.0538867
$928$	0	0
$929$	−5.26649e30	−0.360875	−0.180437	−	0.983586i	$-0.557751\pi$
$929$	−5.26649e30	−0.360875	−0.180437	+	0.983586i	$0.557751\pi$
$930$	0	0
$931$	−2.57835e27	−0.000172731 0
$932$	0	0
$933$	5.84963e28	0.00383152
$934$	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
$939$	6.10672e29	0.0373955
$940$	0	0
$941$	−2.14709e31	−1.28576	−0.642879	−	0.765968i	$-0.722260\pi$
$941$	−2.14709e31	−1.28576	−0.642879	+	0.765968i	$0.722260\pi$
$942$	0	0
$943$	−1.92603e31	−1.12795
$944$	0	0
$945$	4.73139e29	0.0270991
$946$	0	0
$947$	−1.56465e31	−0.876483	−0.438242	−	0.898857i	$-0.644399\pi$
$947$	−1.56465e31	−0.876483	−0.438242	+	0.898857i	$0.644399\pi$
$948$	0	0
$949$	5.88861e30	0.322639
$950$	0	0
$951$	7.84793e29	0.0420591
$952$	0	0
$953$	−4.71750e30	−0.247307	−0.123654	−	0.992325i	$-0.539461\pi$
$953$	−4.71750e30	−0.247307	−0.123654	+	0.992325i	$0.539461\pi$
$954$	0	0
$955$	−2.31798e31	−1.18871
$956$	0	0
$957$	−2.25590e30	−0.113173
$958$	0	0
$959$	−2.15412e30	−0.105724
$960$	0	0
$961$	2.01951e31	0.969728
$962$	0	0
$963$	−7.99956e30	−0.375829
$964$	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
$969$	−2.77747e26	−1.22250e−5 0
$970$	0	0
$971$	4.39842e31	1.89450	0.947249	−	0.320497i	$-0.103850\pi$
$971$	4.39842e31	1.89450	0.947249	+	0.320497i	$0.103850\pi$
$972$	0	0
$973$	−4.15696e30	−0.175223
$974$	0	0
$975$	9.12354e29	0.0376370
$976$	0	0
$977$	−1.85934e31	−0.750700	−0.375350	−	0.926883i	$-0.622477\pi$
$977$	−1.85934e31	−0.750700	−0.375350	+	0.926883i	$0.622477\pi$
$978$	0	0
$979$	3.46266e31	1.36833
$980$	0	0
$981$	3.40489e31	1.31698
$982$	0	0
$983$	−2.18619e31	−0.827704	−0.413852	−	0.910344i	$-0.635817\pi$
$983$	−2.18619e31	−0.827704	−0.413852	+	0.910344i	$0.635817\pi$
$984$	0	0
$985$	6.33049e31	2.34616
$986$	0	0
$987$	−4.84897e29	−0.0175922
$988$	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
$993$	1.96550e30	0.0669122
$994$	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
$999$	−2.68104e30	−0.0856773

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.22.a.b.1.2	✓	3	1.1	even	1	trivial
16.22.a.f.1.2		3	4.3	odd	2
64.22.a.l.1.2		3	8.5	even	2
64.22.a.m.1.2		3	8.3	odd	2
72.22.a.f.1.1		3	3.2	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8 = 2^{3} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	8.a (trivial)

\(f(q)\)	\(=\)	\(q-7983.67 q^{3} +3.09730e7 q^{5} +9.17385e7 q^{7} -1.03966e10 q^{9} +8.72158e10 q^{11} -2.36850e11 q^{13} -2.47278e11 q^{15} +7.42283e12 q^{17} +4.68680e9 q^{19} -7.32410e11 q^{21} +3.33703e14 q^{23} +4.82489e14 q^{25} +1.66515e14 q^{27} +3.23982e15 q^{29} +6.40473e15 q^{31} -6.96303e14 q^{33} +2.84142e15 q^{35} -1.61009e16 q^{37} +1.89093e15 q^{39} -5.77168e16 q^{41} -2.01468e17 q^{43} -3.22014e17 q^{45} +6.62056e17 q^{47} -5.50130e17 q^{49} -5.92615e16 q^{51} +4.62651e17 q^{53} +2.70133e18 q^{55} -3.74179e13 q^{57} +7.39502e18 q^{59} -5.50188e18 q^{61} -9.53770e17 q^{63} -7.33594e18 q^{65} +6.03520e18 q^{67} -2.66418e18 q^{69} -4.43147e19 q^{71} -2.48622e19 q^{73} -3.85204e18 q^{75} +8.00105e18 q^{77} +5.70948e19 q^{79} +1.07423e20 q^{81} -1.31820e20 q^{83} +2.29907e20 q^{85} -2.58657e19 q^{87} +3.97023e20 q^{89} -2.17282e19 q^{91} -5.11333e19 q^{93} +1.45164e17 q^{95} -9.80402e20 q^{97} -9.06749e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 96764 q^{3} - 24111774 q^{5} + 295988280 q^{7} + 18844697239 q^{9} - 40335108684 q^{11} + 133734425946 q^{13} - 1223136458200 q^{15} + 7797732274422 q^{17} + 35788199781996 q^{19} + 198539224853088 q^{21}+ \cdots - 94\!\cdots\!60 q^{99}+O(q^{100}) \) 3 * q + 96764 * q^3 - 24111774 * q^5 + 295988280 * q^7 + 18844697239 * q^9 - 40335108684 * q^11 + 133734425946 * q^13 - 1223136458200 * q^15 + 7797732274422 * q^17 + 35788199781996 * q^19 + 198539224853088 * q^21 + 193770761479080 * q^23 + 1127564438439501 * q^25 + 5282002293508952 * q^27 + 5607343422466122 * q^29 + 11246757871503072 * q^31 + 10014149026970384 * q^33 + 5274251425350096 * q^35 - 24272499791100606 * q^37 - 52925308377862264 * q^39 - 298159108991869602 * q^41 - 33333932139754860 * q^43 - 927411477977893478 * q^45 - 120874283547603888 * q^47 - 850403331975639477 * q^49 + 3954388789815661240 * q^51 - 1138443393004854222 * q^53 + 6957552484263571704 * q^55 + 681697547133656944 * q^57 + 9225624498709937412 * q^59 - 6554902294063924182 * q^61 + 21785214559631141976 * q^63 - 20714581819561144452 * q^65 - 15793054074531629124 * q^67 - 72988310309689939168 * q^69 - 41139582493467997704 * q^71 - 19422167949903851970 * q^73 - 75231657393126995900 * q^75 + 68380237365358617888 * q^77 - 131735321299806049488 * q^79 + 500425796062282339147 * q^81 + 64013993832679681068 * q^83 + 390258202763001297252 * q^85 + 101898953973185066568 * q^87 + 429891446897537246766 * q^89 - 297181701588021496176 * q^91 + 640035369009914700160 * q^93 - 1036372406649019824696 * q^95 - 324059336514148638042 * q^97 - 949982905688477352860 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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