LMFDB - Embedded newform 27.11.b.c.26.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [27,11,Mod(26,27)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(27, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 11, names="a")

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

chi := DirichletCharacter("27.26");

S:= CuspForms(chi, 11);

N := Newforms(S);

Level:	$ N $	$=$	$ 27 = 3^{3} $
Weight:	$ k $	$=$	$ 11 $
Character orbit:	$[\chi]$	$=$	27.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$17.1546458222$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 188x^{2} + 756 $ x^4 + 188*x^2 + 756
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{9} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.1
Root			$2.02760i$ of defining polynomial
Character	$\chi$	$=$	27.26
Dual form			27.11.b.c.26.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-42.0446i q^{2} -743.750 q^{4} -4853.52i q^{5} +31826.0 q^{7} -11783.0i q^{8} +O(q^{10})$	\(q-42.0446i q^{2} -743.750 q^{4} -4853.52i q^{5} +31826.0 q^{7} -11783.0i q^{8} -204064. q^{10} -114785. i q^{11} +266894. q^{13} -1.33811e6i q^{14} -1.25701e6 q^{16} +195246. i q^{17} -1.34484e6 q^{19} +3.60980e6i q^{20} -4.82607e6 q^{22} +9.10844e6i q^{23} -1.37910e7 q^{25} -1.12214e7i q^{26} -2.36706e7 q^{28} +9.95954e6i q^{29} -5.35116e6 q^{31} +4.07848e7i q^{32} +8.20904e6 q^{34} -1.54468e8i q^{35} +4.26102e7 q^{37} +5.65433e7i q^{38} -5.71891e7 q^{40} -1.71257e8i q^{41} -5.96642e7 q^{43} +8.53710e7i q^{44} +3.82961e8 q^{46} +3.45140e8i q^{47} +7.30418e8 q^{49} +5.79839e8i q^{50} -1.98502e8 q^{52} -3.02713e8i q^{53} -5.57109e8 q^{55} -3.75006e8i q^{56} +4.18745e8 q^{58} -4.87658e8i q^{59} -1.61436e8 q^{61} +2.24988e8i q^{62} +4.27600e8 q^{64} -1.29537e9i q^{65} -4.59991e8 q^{67} -1.45214e8i q^{68} -6.49455e9 q^{70} +2.12634e9i q^{71} +3.50692e9 q^{73} -1.79153e9i q^{74} +1.00023e9 q^{76} -3.65313e9i q^{77} -2.47582e8 q^{79} +6.10093e9i q^{80} -7.20044e9 q^{82} +5.42519e9i q^{83} +9.47630e8 q^{85} +2.50856e9i q^{86} -1.35251e9 q^{88} +8.33517e8i q^{89} +8.49415e9 q^{91} -6.77440e9i q^{92} +1.45113e10 q^{94} +6.52722e9i q^{95} -5.68524e9 q^{97} -3.07101e10i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 548 q^{4} + 20516 q^{7}+O(q^{10}) $ 4 * q - 548 * q^4 + 20516 * q^7	$ 4 q - 548 q^{4} + 20516 q^{7} - 199800 q^{10} - 262420 q^{13} - 3207800 q^{16} - 8233516 q^{19} - 21085704 q^{22} - 47203580 q^{25} - 67604212 q^{28} - 46994920 q^{31} - 78985368 q^{34} + 108086444 q^{37} + 196812720 q^{40} + 78251000 q^{43} + 882645768 q^{46} + 1826240448 q^{49} - 771022300 q^{52} + 1034575200 q^{55} + 1267544592 q^{58} - 3222389524 q^{61} - 1166263952 q^{64} + 1972286516 q^{67} - 17482328280 q^{70} + 4077598940 q^{73} - 603762388 q^{76} - 10402780252 q^{79} - 7945552656 q^{82} + 19825132320 q^{85} - 19783893168 q^{88} + 34160896780 q^{91} + 22345655544 q^{94} + 12969797468 q^{97}+O(q^{100}) $ 4 * q - 548 * q^4 + 20516 * q^7 - 199800 * q^10 - 262420 * q^13 - 3207800 * q^16 - 8233516 * q^19 - 21085704 * q^22 - 47203580 * q^25 - 67604212 * q^28 - 46994920 * q^31 - 78985368 * q^34 + 108086444 * q^37 + 196812720 * q^40 + 78251000 * q^43 + 882645768 * q^46 + 1826240448 * q^49 - 771022300 * q^52 + 1034575200 * q^55 + 1267544592 * q^58 - 3222389524 * q^61 - 1166263952 * q^64 + 1972286516 * q^67 - 17482328280 * q^70 + 4077598940 * q^73 - 603762388 * q^76 - 10402780252 * q^79 - 7945552656 * q^82 + 19825132320 * q^85 - 19783893168 * q^88 + 34160896780 * q^91 + 22345655544 * q^94 + 12969797468 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/27\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$	− 42.0446i	− 1.31389i	−0.753937	−	0.656947i	$-0.771847\pi$
$2$	− 42.0446i	− 1.31389i	0.753937	−	0.656947i	$-0.228153\pi$
$3$	0	0
$3$	0	0
$4$	−743.750	−0.726318
$5$	− 4853.52i	− 1.55313i	−0.630039	−	0.776563i	$-0.716961\pi$
$5$	− 4853.52i	− 1.55313i	0.630039	−	0.776563i	$-0.283039\pi$
$6$	0	0
$7$	31826.0	1.89361	0.946807	−	0.321801i	$-0.104288\pi$
$7$	31826.0	1.89361	0.946807	+	0.321801i	$0.104288\pi$
$8$	− 11783.0i	− 0.359589i
$9$	0	0
$10$	−204064.	−2.04064
$11$	− 114785.i	− 0.712722i	−0.934348	−	0.356361i	$-0.884017\pi$
$11$	− 114785.i	− 0.712722i	0.934348	−	0.356361i	$-0.115983\pi$
$12$	0	0
$13$	266894.	0.718822	0.359411	−	0.933179i	$-0.382978\pi$
$13$	266894.	0.718822	0.359411	+	0.933179i	$0.382978\pi$
$14$	− 1.33811e6i	− 2.48801i
$15$	0	0
$16$	−1.25701e6	−1.19878
$17$	195246.i	0.137511i	0.997634	+	0.0687555i	$0.0219028\pi$
$17$	195246.i	0.137511i	−0.997634	+	0.0687555i	$0.978097\pi$
$18$	0	0
$19$	−1.34484e6	−0.543129	−0.271565	−	0.962420i	$-0.587541\pi$
$19$	−1.34484e6	−0.543129	−0.271565	+	0.962420i	$0.587541\pi$
$20$	3.60980e6i	1.12806i
$21$	0	0
$22$	−4.82607e6	−0.936441
$23$	9.10844e6i	1.41516i	0.706634	+	0.707579i	$0.250213\pi$
$23$	9.10844e6i	1.41516i	−0.706634	+	0.707579i	$0.749787\pi$
$24$	0	0
$25$	−1.37910e7	−1.41220
$26$	− 1.12214e7i	− 0.944457i
$27$	0	0
$28$	−2.36706e7	−1.37537
$29$	9.95954e6i	0.485567i	0.970081	+	0.242783i	$0.0780605\pi$
$29$	9.95954e6i	0.485567i	−0.970081	+	0.242783i	$0.921940\pi$
$30$	0	0
$31$	−5.35116e6	−0.186913	−0.0934565	−	0.995623i	$-0.529792\pi$
$31$	−5.35116e6	−0.186913	−0.0934565	+	0.995623i	$0.529792\pi$
$32$	4.07848e7i	1.21548i
$33$	0	0
$34$	8.20904e6	0.180675
$35$	− 1.54468e8i	− 2.94102i
$36$	0	0
$37$	4.26102e7	0.614476	0.307238	−	0.951633i	$-0.400595\pi$
$37$	4.26102e7	0.614476	0.307238	+	0.951633i	$0.400595\pi$
$38$	5.65433e7i	0.713614i
$39$	0	0
$40$	−5.71891e7	−0.558488
$41$	− 1.71257e8i	− 1.47819i	−0.673603	−	0.739093i	$-0.735254\pi$
$41$	− 1.71257e8i	− 1.47819i	0.673603	−	0.739093i	$-0.264746\pi$
$42$	0	0
$43$	−5.96642e7	−0.405855	−0.202928	−	0.979194i	$-0.565046\pi$
$43$	−5.96642e7	−0.405855	−0.202928	+	0.979194i	$0.565046\pi$
$44$	8.53710e7i	0.517663i
$45$	0	0
$46$	3.82961e8	1.85937
$47$	3.45140e8i	1.50489i	0.658654	+	0.752446i	$0.271126\pi$
$47$	3.45140e8i	1.50489i	−0.658654	+	0.752446i	$0.728874\pi$
$48$	0	0
$49$	7.30418e8	2.58578
$50$	5.79839e8i	1.85548i
$51$	0	0
$52$	−1.98502e8	−0.522094
$53$	− 3.02713e8i	− 0.723856i	−0.932206	−	0.361928i	$-0.882119\pi$
$53$	− 3.02713e8i	− 0.723856i	0.932206	−	0.361928i	$-0.117881\pi$
$54$	0	0
$55$	−5.57109e8	−1.10695
$56$	− 3.75006e8i	− 0.680924i
$57$	0	0
$58$	4.18745e8	0.637984
$59$	− 4.87658e8i	− 0.682111i	−0.940043	−	0.341055i	$-0.889216\pi$
$59$	− 4.87658e8i	− 0.682111i	0.940043	−	0.341055i	$-0.110784\pi$
$60$	0	0
$61$	−1.61436e8	−0.191139	−0.0955697	−	0.995423i	$-0.530467\pi$
$61$	−1.61436e8	−0.191139	−0.0955697	+	0.995423i	$0.530467\pi$
$62$	2.24988e8i	0.245584i
$63$	0	0
$64$	4.27600e8	0.398233
$65$	− 1.29537e9i	− 1.11642i
$66$	0	0
$67$	−4.59991e8	−0.340703	−0.170351	−	0.985383i	$-0.554490\pi$
$67$	−4.59991e8	−0.340703	−0.170351	+	0.985383i	$0.554490\pi$
$68$	− 1.45214e8i	− 0.0998766i
$69$	0	0
$70$	−6.49455e9	−3.86419
$71$	2.12634e9i	1.17853i	0.807940	+	0.589265i	$0.200583\pi$
$71$	2.12634e9i	1.17853i	−0.807940	+	0.589265i	$0.799417\pi$
$72$	0	0
$73$	3.50692e9	1.69166	0.845828	−	0.533456i	$-0.179107\pi$
$73$	3.50692e9	1.69166	0.845828	+	0.533456i	$0.179107\pi$
$74$	− 1.79153e9i	− 0.807357i
$75$	0	0
$76$	1.00023e9	0.394484
$77$	− 3.65313e9i	− 1.34962i
$78$	0	0
$79$	−2.47582e8	−0.0804607	−0.0402303	−	0.999190i	$-0.512809\pi$
$79$	−2.47582e8	−0.0804607	−0.0402303	+	0.999190i	$0.512809\pi$
$80$	6.10093e9i	1.86186i
$81$	0	0
$82$	−7.20044e9	−1.94218
$83$	5.42519e9i	1.37729i	0.725101	+	0.688643i	$0.241793\pi$
$83$	5.42519e9i	1.37729i	−0.725101	+	0.688643i	$0.758207\pi$
$84$	0	0
$85$	9.47630e8	0.213572
$86$	2.50856e9i	0.533251i
$87$	0	0
$88$	−1.35251e9	−0.256287
$89$	8.33517e8i	0.149267i	0.997211	+	0.0746336i	$0.0237787\pi$
$89$	8.33517e8i	0.149267i	−0.997211	+	0.0746336i	$0.976221\pi$
$90$	0	0
$91$	8.49415e9	1.36117
$92$	− 6.77440e9i	− 1.02785i
$93$	0	0
$94$	1.45113e10	1.97727
$95$	6.52722e9i	0.843548i
$96$	0	0
$97$	−5.68524e9	−0.662049	−0.331024	−	0.943622i	$-0.607394\pi$
$97$	−5.68524e9	−0.662049	−0.331024	+	0.943622i	$0.607394\pi$
$98$	− 3.07101e10i	− 3.39744i
$99$	0	0
$100$	1.02571e10	1.02571
$101$	− 1.53576e8i	− 0.0146122i	−0.999973	−	0.00730612i	$-0.997674\pi$
$101$	− 1.53576e8i	− 0.0146122i	0.999973	−	0.00730612i	$-0.00232563\pi$
$102$	0	0
$103$	1.57719e10	1.36050	0.680248	−	0.732982i	$-0.261872\pi$
$103$	1.57719e10	1.36050	0.680248	+	0.732982i	$0.261872\pi$
$104$	− 3.14482e9i	− 0.258481i
$105$	0	0
$106$	−1.27275e10	−0.951070
$107$	5.53794e9i	0.394848i	0.980318	+	0.197424i	$0.0632575\pi$
$107$	5.53794e9i	0.394848i	−0.980318	+	0.197424i	$0.936743\pi$
$108$	0	0
$109$	3.62784e9	0.235785	0.117892	−	0.993026i	$-0.462386\pi$
$109$	3.62784e9	0.235785	0.117892	+	0.993026i	$0.462386\pi$
$110$	2.34234e10i	1.45441i
$111$	0	0
$112$	−4.00056e10	−2.27003
$113$	9.07477e9i	0.492542i	0.969201	+	0.246271i	$0.0792054\pi$
$113$	9.07477e9i	0.492542i	−0.969201	+	0.246271i	$0.920795\pi$
$114$	0	0
$115$	4.42080e10	2.19792
$116$	− 7.40740e9i	− 0.352676i
$117$	0	0
$118$	−2.05034e10	−0.896221
$119$	6.21389e9i	0.260393i
$120$	0	0
$121$	1.27619e10	0.492028
$122$	6.78750e9i	0.251137i
$123$	0	0
$124$	3.97992e9	0.135758
$125$	1.95374e10i	0.640202i
$126$	0	0
$127$	−3.98885e10	−1.20734	−0.603670	−	0.797235i	$-0.706295\pi$
$127$	−3.98885e10	−1.20734	−0.603670	+	0.797235i	$0.706295\pi$
$128$	2.37853e10i	0.692245i
$129$	0	0
$130$	−5.44635e10	−1.46686
$131$	− 7.31232e10i	− 1.89539i	−0.319177	−	0.947695i	$-0.603407\pi$
$131$	− 7.31232e10i	− 1.89539i	0.319177	−	0.947695i	$-0.396593\pi$
$132$	0	0
$133$	−4.28009e10	−1.02848
$134$	1.93401e10i	0.447647i
$135$	0	0
$136$	2.30059e9	0.0494475
$137$	− 5.17212e10i	− 1.07168i	−0.844319	−	0.535841i	$-0.819995\pi$
$137$	− 5.17212e10i	− 1.07168i	0.844319	−	0.535841i	$-0.180005\pi$
$138$	0	0
$139$	−5.04078e10	−0.971458	−0.485729	−	0.874109i	$-0.661446\pi$
$139$	−5.04078e10	−0.971458	−0.485729	+	0.874109i	$0.661446\pi$
$140$	1.14886e11i	2.13612i
$141$	0	0
$142$	8.94011e10	1.54846
$143$	− 3.06353e10i	− 0.512320i
$144$	0	0
$145$	4.83388e10	0.754147
$146$	− 1.47447e11i	− 2.22266i
$147$	0	0
$148$	−3.16913e10	−0.446305
$149$	− 9.01167e10i	− 1.22708i	−0.789663	−	0.613541i	$-0.789744\pi$
$149$	− 9.01167e10i	− 1.22708i	0.789663	−	0.613541i	$-0.210256\pi$
$150$	0	0
$151$	6.67438e10	0.850210	0.425105	−	0.905144i	$-0.360237\pi$
$151$	6.67438e10	0.850210	0.425105	+	0.905144i	$0.360237\pi$
$152$	1.58463e10i	0.195303i
$153$	0	0
$154$	−1.53594e11	−1.77326
$155$	2.59720e10i	0.290300i
$156$	0	0
$157$	7.09262e10	0.743547	0.371773	−	0.928324i	$-0.378750\pi$
$157$	7.09262e10	0.743547	0.371773	+	0.928324i	$0.378750\pi$
$158$	1.04095e10i	0.105717i
$159$	0	0
$160$	1.97950e11	1.88780
$161$	2.89885e11i	2.67976i
$162$	0	0
$163$	9.32173e10	0.810137	0.405068	−	0.914286i	$-0.367248\pi$
$163$	9.32173e10	0.810137	0.405068	+	0.914286i	$0.367248\pi$
$164$	1.27372e11i	1.07363i
$165$	0	0
$166$	2.28100e11	1.80961
$167$	− 9.22546e10i	− 0.710241i	−0.934821	−	0.355120i	$-0.884440\pi$
$167$	− 9.22546e10i	− 0.710241i	0.934821	−	0.355120i	$-0.115560\pi$
$168$	0	0
$169$	−6.66262e10	−0.483294
$170$	− 3.98427e10i	− 0.280611i
$171$	0	0
$172$	4.43752e10	0.294780
$173$	1.75298e11i	1.13122i	0.824673	+	0.565610i	$0.191359\pi$
$173$	1.75298e11i	1.13122i	−0.824673	+	0.565610i	$0.808641\pi$
$174$	0	0
$175$	−4.38913e11	−2.67417
$176$	1.44286e11i	0.854397i
$177$	0	0
$178$	3.50449e10	0.196121
$179$	− 2.28815e11i	− 1.24514i	−0.782563	−	0.622571i	$-0.786088\pi$
$179$	− 2.28815e11i	− 1.24514i	0.782563	−	0.622571i	$-0.213912\pi$
$180$	0	0
$181$	2.39660e10	0.123368	0.0616841	−	0.998096i	$-0.480353\pi$
$181$	2.39660e10	0.123368	0.0616841	+	0.998096i	$0.480353\pi$
$182$	− 3.57133e11i	− 1.78844i
$183$	0	0
$184$	1.07325e11	0.508876
$185$	− 2.06810e11i	− 0.954359i
$186$	0	0
$187$	2.24112e10	0.0980070
$188$	− 2.56697e11i	− 1.09303i
$189$	0	0
$190$	2.74434e11	1.10833
$191$	2.40981e11i	0.948016i	0.880521	+	0.474008i	$0.157193\pi$
$191$	2.40981e11i	0.948016i	−0.880521	+	0.474008i	$0.842807\pi$
$192$	0	0
$193$	−6.86116e9	−0.0256219	−0.0128109	−	0.999918i	$-0.504078\pi$
$193$	−6.86116e9	−0.0256219	−0.0128109	+	0.999918i	$0.504078\pi$
$194$	2.39034e11i	0.869862i
$195$	0	0
$196$	−5.43248e11	−1.87810
$197$	− 1.87585e11i	− 0.632218i	−0.948723	−	0.316109i	$-0.897624\pi$
$197$	− 1.87585e11i	− 0.632218i	0.948723	−	0.316109i	$-0.102376\pi$
$198$	0	0
$199$	−3.13244e11	−1.00373	−0.501866	−	0.864945i	$-0.667353\pi$
$199$	−3.13244e11	−1.00373	−0.501866	+	0.864945i	$0.667353\pi$
$200$	1.62500e11i	0.507813i
$201$	0	0
$202$	−6.45704e9	−0.0191989
$203$	3.16972e11i	0.919477i
$204$	0	0
$205$	−8.31200e11	−2.29581
$206$	− 6.63123e11i	− 1.78755i
$207$	0	0
$208$	−3.35489e11	−0.861710
$209$	1.54367e11i	0.387100i
$210$	0	0
$211$	6.16867e11	1.47496	0.737479	−	0.675370i	$-0.236016\pi$
$211$	6.16867e11	1.47496	0.737479	+	0.675370i	$0.236016\pi$
$212$	2.25143e11i	0.525750i
$213$	0	0
$214$	2.32841e11	0.518788
$215$	2.89581e11i	0.630345i
$216$	0	0
$217$	−1.70306e11	−0.353941
$218$	− 1.52531e11i	− 0.309796i
$219$	0	0
$220$	4.14350e11	0.803995
$221$	5.21099e10i	0.0988459i
$222$	0	0
$223$	−6.29102e10	−0.114077	−0.0570383	−	0.998372i	$-0.518166\pi$
$223$	−6.29102e10	−0.114077	−0.0570383	+	0.998372i	$0.518166\pi$
$224$	1.29802e12i	2.30165i
$225$	0	0
$226$	3.81545e11	0.647149
$227$	3.11463e11i	0.516746i	0.966045	+	0.258373i	$0.0831863\pi$
$227$	3.11463e11i	0.516746i	−0.966045	+	0.258373i	$0.916814\pi$
$228$	0	0
$229$	3.75521e10	0.0596289	0.0298145	−	0.999555i	$-0.490508\pi$
$229$	3.75521e10	0.0596289	0.0298145	+	0.999555i	$0.490508\pi$
$230$	− 1.85871e12i	− 2.88783i
$231$	0	0
$232$	1.17353e11	0.174605
$233$	8.71092e11i	1.26848i	0.773135	+	0.634241i	$0.218687\pi$
$233$	8.71092e11i	1.26848i	−0.773135	+	0.634241i	$0.781313\pi$
$234$	0	0
$235$	1.67514e12	2.33729
$236$	3.62695e11i	0.495429i
$237$	0	0
$238$	2.61261e11	0.342128
$239$	6.79757e11i	0.871694i	0.900021	+	0.435847i	$0.143551\pi$
$239$	6.79757e11i	0.871694i	−0.900021	+	0.435847i	$0.856449\pi$
$240$	0	0
$241$	−1.41486e12	−1.74032	−0.870160	−	0.492769i	$-0.835985\pi$
$241$	−1.41486e12	−1.74032	−0.870160	+	0.492769i	$0.835985\pi$
$242$	− 5.36571e11i	− 0.646472i
$243$	0	0
$244$	1.20068e11	0.138828
$245$	− 3.54510e12i	− 4.01604i
$246$	0	0
$247$	−3.58930e11	−0.390413
$248$	6.30529e10i	0.0672119i
$249$	0	0
$250$	8.21442e11	0.841157
$251$	− 9.30317e11i	− 0.933818i	−0.884305	−	0.466909i	$-0.845367\pi$
$251$	− 9.30317e11i	− 0.933818i	0.884305	−	0.466909i	$-0.154633\pi$
$252$	0	0
$253$	1.04551e12	1.00861
$254$	1.67710e12i	1.58632i
$255$	0	0
$256$	1.43791e12	1.30777
$257$	− 9.61711e11i	− 0.857786i	−0.903355	−	0.428893i	$-0.858904\pi$
$257$	− 9.61711e11i	− 0.857786i	0.903355	−	0.428893i	$-0.141096\pi$
$258$	0	0
$259$	1.35611e12	1.16358
$260$	9.63434e11i	0.810877i
$261$	0	0
$262$	−3.07444e12	−2.49034
$263$	− 3.37370e11i	− 0.268119i	−0.990973	−	0.134059i	$-0.957199\pi$
$263$	− 3.37370e11i	− 0.268119i	0.990973	−	0.134059i	$-0.0428013\pi$
$264$	0	0
$265$	−1.46923e12	−1.12424
$266$	1.79955e12i	1.35131i
$267$	0	0
$268$	3.42118e11	0.247458
$269$	5.96796e11i	0.423706i	0.977302	+	0.211853i	$0.0679498\pi$
$269$	5.96796e11i	0.423706i	−0.977302	+	0.211853i	$0.932050\pi$
$270$	0	0
$271$	−2.73730e11	−0.187273	−0.0936367	−	0.995606i	$-0.529849\pi$
$271$	−2.73730e11	−0.187273	−0.0936367	+	0.995606i	$0.529849\pi$
$272$	− 2.45426e11i	− 0.164845i
$273$	0	0
$274$	−2.17460e12	−1.40808
$275$	1.58300e12i	1.00651i
$276$	0	0
$277$	1.51777e12	0.930694	0.465347	−	0.885128i	$-0.345930\pi$
$277$	1.51777e12	0.930694	0.465347	+	0.885128i	$0.345930\pi$
$278$	2.11938e12i	1.27639i
$279$	0	0
$280$	−1.82010e12	−1.05756
$281$	3.31975e12i	1.89485i	0.319984	+	0.947423i	$0.396322\pi$
$281$	3.31975e12i	1.89485i	−0.319984	+	0.947423i	$0.603678\pi$
$282$	0	0
$283$	1.09147e12	0.601282	0.300641	−	0.953737i	$-0.402799\pi$
$283$	1.09147e12	0.601282	0.300641	+	0.953737i	$0.402799\pi$
$284$	− 1.58146e12i	− 0.855988i
$285$	0	0
$286$	−1.28805e12	−0.673135
$287$	− 5.45042e12i	− 2.79912i
$288$	0	0
$289$	1.97787e12	0.981091
$290$	− 2.03239e12i	− 0.990869i
$291$	0	0
$292$	−2.60827e12	−1.22868
$293$	7.89580e11i	0.365644i	0.983146	+	0.182822i	$0.0585232\pi$
$293$	7.89580e11i	0.365644i	−0.983146	+	0.182822i	$0.941477\pi$
$294$	0	0
$295$	−2.36686e12	−1.05940
$296$	− 5.02077e11i	− 0.220959i
$297$	0	0
$298$	−3.78892e12	−1.61226
$299$	2.43099e12i	1.01725i
$300$	0	0
$301$	−1.89887e12	−0.768534
$302$	− 2.80622e12i	− 1.11709i
$303$	0	0
$304$	1.69048e12	0.651092
$305$	7.83531e11i	0.296864i
$306$	0	0
$307$	3.49663e12	1.28221	0.641103	−	0.767454i	$-0.278477\pi$
$307$	3.49663e12	1.28221	0.641103	+	0.767454i	$0.278477\pi$
$308$	2.71701e12i	0.980253i
$309$	0	0
$310$	1.09198e12	0.381423
$311$	− 6.51859e11i	− 0.224053i	−0.993705	−	0.112027i	$-0.964266\pi$
$311$	− 6.51859e11i	− 0.224053i	0.993705	−	0.112027i	$-0.0357342\pi$
$312$	0	0
$313$	4.35319e12	1.44906	0.724529	−	0.689245i	$-0.242057\pi$
$313$	4.35319e12	1.44906	0.724529	+	0.689245i	$0.242057\pi$
$314$	− 2.98206e12i	− 0.976942i
$315$	0	0
$316$	1.84139e11	0.0584400
$317$	− 2.34536e12i	− 0.732677i	−0.930482	−	0.366338i	$-0.880611\pi$
$317$	− 2.34536e12i	− 0.732677i	0.930482	−	0.366338i	$-0.119389\pi$
$318$	0	0
$319$	1.14320e12	0.346074
$320$	− 2.07536e12i	− 0.618507i
$321$	0	0
$322$	1.21881e13	3.52093
$323$	− 2.62575e11i	− 0.0746862i
$324$	0	0
$325$	−3.68074e12	−1.01512
$326$	− 3.91928e12i	− 1.06443i
$327$	0	0
$328$	−2.01793e12	−0.531540
$329$	1.09844e13i	2.84969i
$330$	0	0
$331$	−3.62122e12	−0.911412	−0.455706	−	0.890130i	$-0.650613\pi$
$331$	−3.62122e12	−0.911412	−0.455706	+	0.890130i	$0.650613\pi$
$332$	− 4.03498e12i	− 1.00035i
$333$	0	0
$334$	−3.87881e12	−0.933181
$335$	2.23258e12i	0.529154i
$336$	0	0
$337$	−4.11719e12	−0.947220	−0.473610	−	0.880735i	$-0.657049\pi$
$337$	−4.11719e12	−0.947220	−0.473610	+	0.880735i	$0.657049\pi$
$338$	2.80127e12i	0.634998i
$339$	0	0
$340$	−7.04799e11	−0.155121
$341$	6.14231e11i	0.133217i
$342$	0	0
$343$	1.42562e13	3.00285
$344$	7.03024e11i	0.145941i
$345$	0	0
$346$	7.37034e12	1.48630
$347$	− 4.77044e12i	− 0.948224i	−0.880465	−	0.474112i	$-0.842769\pi$
$347$	− 4.77044e12i	− 0.948224i	0.880465	−	0.474112i	$-0.157231\pi$
$348$	0	0
$349$	−7.30065e12	−1.41005	−0.705024	−	0.709183i	$-0.749064\pi$
$349$	−7.30065e12	−1.41005	−0.705024	+	0.709183i	$0.749064\pi$
$350$	1.84539e13i	3.51357i
$351$	0	0
$352$	4.68146e12	0.866300
$353$	− 7.81345e11i	− 0.142551i	−0.997457	−	0.0712753i	$-0.977293\pi$
$353$	− 7.81345e11i	− 0.142551i	0.997457	−	0.0712753i	$-0.0227069\pi$
$354$	0	0
$355$	1.03202e13	1.83041
$356$	− 6.19928e11i	− 0.108415i
$357$	0	0
$358$	−9.62043e12	−1.63599
$359$	7.60588e12i	1.27549i	0.770247	+	0.637746i	$0.220133\pi$
$359$	7.60588e12i	1.27549i	−0.770247	+	0.637746i	$0.779867\pi$
$360$	0	0
$361$	−4.32247e12	−0.705011
$362$	− 1.00764e12i	− 0.162093i
$363$	0	0
$364$	−6.31752e12	−0.988644
$365$	− 1.70209e13i	− 2.62735i
$366$	0	0
$367$	1.03121e12	0.154887	0.0774437	−	0.996997i	$-0.475324\pi$
$367$	1.03121e12	0.154887	0.0774437	+	0.996997i	$0.475324\pi$
$368$	− 1.14494e13i	− 1.69646i
$369$	0	0
$370$	−8.69523e12	−1.25393
$371$	− 9.63415e12i	− 1.37070i
$372$	0	0
$373$	−7.79559e12	−1.07970	−0.539852	−	0.841760i	$-0.681520\pi$
$373$	−7.79559e12	−1.07970	−0.539852	+	0.841760i	$0.681520\pi$
$374$	− 9.42270e11i	− 0.128771i
$375$	0	0
$376$	4.06679e12	0.541143
$377$	2.65814e12i	0.349036i
$378$	0	0
$379$	−6.40957e12	−0.819658	−0.409829	−	0.912162i	$-0.634412\pi$
$379$	−6.40957e12	−0.819658	−0.409829	+	0.912162i	$0.634412\pi$
$380$	− 4.85461e12i	− 0.612684i
$381$	0	0
$382$	1.01319e13	1.24559
$383$	6.88415e12i	0.835327i	0.908602	+	0.417664i	$0.137151\pi$
$383$	6.88415e12i	0.835327i	−0.908602	+	0.417664i	$0.862849\pi$
$384$	0	0
$385$	−1.77305e13	−2.09613
$386$	2.88475e11i	0.0336644i
$387$	0	0
$388$	4.22839e12	0.480858
$389$	3.56571e12i	0.400312i	0.979764	+	0.200156i	$0.0641449\pi$
$389$	3.56571e12i	0.400312i	−0.979764	+	0.200156i	$0.935855\pi$
$390$	0	0
$391$	−1.77839e12	−0.194600
$392$	− 8.60653e12i	− 0.929817i
$393$	0	0
$394$	−7.88693e12	−0.830667
$395$	1.20164e12i	0.124966i
$396$	0	0
$397$	−1.15838e13	−1.17462	−0.587312	−	0.809361i	$-0.699814\pi$
$397$	−1.15838e13	−1.17462	−0.587312	+	0.809361i	$0.699814\pi$
$398$	1.31702e13i	1.31880i
$399$	0	0
$400$	1.73355e13	1.69292
$401$	3.47890e12i	0.335521i	0.985828	+	0.167760i	$0.0536535\pi$
$401$	3.47890e12i	0.335521i	−0.985828	+	0.167760i	$0.946346\pi$
$402$	0	0
$403$	−1.42819e12	−0.134357
$404$	1.14222e11i	0.0106131i
$405$	0	0
$406$	1.33270e13	1.20809
$407$	− 4.89099e12i	− 0.437951i
$408$	0	0
$409$	−1.20640e13	−1.05408	−0.527041	−	0.849840i	$-0.676698\pi$
$409$	−1.20640e13	−1.05408	−0.527041	+	0.849840i	$0.676698\pi$
$410$	3.49475e13i	3.01645i
$411$	0	0
$412$	−1.17303e13	−0.988153
$413$	− 1.55202e13i	− 1.29165i
$414$	0	0
$415$	2.63312e13	2.13910
$416$	1.08852e13i	0.873715i
$417$	0	0
$418$	6.49030e12	0.508608
$419$	4.66903e12i	0.361540i	0.983525	+	0.180770i	$0.0578590\pi$
$419$	4.66903e12i	0.361540i	−0.983525	+	0.180770i	$0.942141\pi$
$420$	0	0
$421$	7.42819e12	0.561658	0.280829	−	0.959758i	$-0.409391\pi$
$421$	7.42819e12	0.561658	0.280829	+	0.959758i	$0.409391\pi$
$422$	− 2.59360e13i	− 1.93794i
$423$	0	0
$424$	−3.56688e12	−0.260291
$425$	− 2.69264e12i	− 0.194193i
$426$	0	0
$427$	−5.13785e12	−0.361944
$428$	− 4.11884e12i	− 0.286785i
$429$	0	0
$430$	1.21753e13	0.828206
$431$	1.17229e13i	0.788223i	0.919063	+	0.394112i	$0.128948\pi$
$431$	1.17229e13i	0.788223i	−0.919063	+	0.394112i	$0.871052\pi$
$432$	0	0
$433$	1.35795e13	0.892160	0.446080	−	0.894993i	$-0.352820\pi$
$433$	1.35795e13	0.892160	0.446080	+	0.894993i	$0.352820\pi$
$434$	7.16045e12i	0.465041i
$435$	0	0
$436$	−2.69820e12	−0.171255
$437$	− 1.22494e13i	− 0.768613i
$438$	0	0
$439$	1.47686e13	0.905769	0.452884	−	0.891569i	$-0.350395\pi$
$439$	1.47686e13	0.905769	0.452884	+	0.891569i	$0.350395\pi$
$440$	6.56443e12i	0.398046i
$441$	0	0
$442$	2.19094e12	0.129873
$443$	1.64018e13i	0.961329i	0.876905	+	0.480664i	$0.159604\pi$
$443$	1.64018e13i	0.961329i	−0.876905	+	0.480664i	$0.840396\pi$
$444$	0	0
$445$	4.04549e12	0.231831
$446$	2.64503e12i	0.149885i
$447$	0	0
$448$	1.36088e13	0.754100
$449$	− 2.07647e13i	− 1.13787i	−0.822381	−	0.568937i	$-0.807355\pi$
$449$	− 2.07647e13i	− 1.13787i	0.822381	−	0.568937i	$-0.192645\pi$
$450$	0	0
$451$	−1.96577e13	−1.05354
$452$	− 6.74936e12i	− 0.357742i
$453$	0	0
$454$	1.30953e13	0.678949
$455$	− 4.12266e13i	− 2.11407i
$456$	0	0
$457$	−2.72870e13	−1.36891	−0.684455	−	0.729055i	$-0.739960\pi$
$457$	−2.72870e13	−1.36891	−0.684455	+	0.729055i	$0.739960\pi$
$458$	− 1.57886e12i	− 0.0783461i
$459$	0	0
$460$	−3.28797e13	−1.59639
$461$	1.88536e13i	0.905504i	0.891637	+	0.452752i	$0.149558\pi$
$461$	1.88536e13i	0.905504i	−0.891637	+	0.452752i	$0.850442\pi$
$462$	0	0
$463$	1.54771e13	0.727419	0.363710	−	0.931512i	$-0.381510\pi$
$463$	1.54771e13	0.727419	0.363710	+	0.931512i	$0.381510\pi$
$464$	− 1.25193e13i	− 0.582088i
$465$	0	0
$466$	3.66247e13	1.66665
$467$	2.77647e13i	1.24999i	0.780627	+	0.624997i	$0.214900\pi$
$467$	2.77647e13i	1.24999i	−0.780627	+	0.624997i	$0.785100\pi$
$468$	0	0
$469$	−1.46397e13	−0.645159
$470$	− 7.04307e13i	− 3.07095i
$471$	0	0
$472$	−5.74608e12	−0.245280
$473$	6.84853e12i	0.289262i
$474$	0	0
$475$	1.85468e13	0.767008
$476$	− 4.62158e12i	− 0.189128i
$477$	0	0
$478$	2.85801e13	1.14531
$479$	1.24270e12i	0.0492818i	0.999696	+	0.0246409i	$0.00784424\pi$
$479$	1.24270e12i	0.0492818i	−0.999696	+	0.0246409i	$0.992156\pi$
$480$	0	0
$481$	1.13724e13	0.441699
$482$	5.94874e13i	2.28660i
$483$	0	0
$484$	−9.49168e12	−0.357369
$485$	2.75934e13i	1.02825i
$486$	0	0
$487$	−2.86678e13	−1.04653	−0.523263	−	0.852171i	$-0.675286\pi$
$487$	−2.86678e13	−1.04653	−0.523263	+	0.852171i	$0.675286\pi$
$488$	1.90220e12i	0.0687317i
$489$	0	0
$490$	−1.49052e14	−5.27665
$491$	2.17841e12i	0.0763366i	0.999271	+	0.0381683i	$0.0121523\pi$
$491$	2.17841e12i	0.0763366i	−0.999271	+	0.0381683i	$0.987848\pi$
$492$	0	0
$493$	−1.94456e12	−0.0667708
$494$	1.50911e13i	0.512962i
$495$	0	0
$496$	6.72648e12	0.224068
$497$	6.76729e13i	2.23168i
$498$	0	0
$499$	−2.38184e13	−0.769858	−0.384929	−	0.922946i	$-0.625774\pi$
$499$	−2.38184e13	−0.769858	−0.384929	+	0.922946i	$0.625774\pi$
$500$	− 1.45309e13i	− 0.464990i
$501$	0	0
$502$	−3.91148e13	−1.22694
$503$	5.97407e13i	1.85537i	0.373365	+	0.927685i	$0.378204\pi$
$503$	5.97407e13i	1.85537i	−0.373365	+	0.927685i	$0.621796\pi$
$504$	0	0
$505$	−7.45384e11	−0.0226946
$506$	− 4.39580e13i	− 1.32521i
$507$	0	0
$508$	2.96671e13	0.876912
$509$	2.12758e13i	0.622727i	0.950291	+	0.311363i	$0.100786\pi$
$509$	2.12758e13i	0.622727i	−0.950291	+	0.311363i	$0.899214\pi$
$510$	0	0
$511$	1.11611e14	3.20334
$512$	− 3.61001e13i	− 1.02603i
$513$	0	0
$514$	−4.04347e13	−1.12704
$515$	− 7.65492e13i	− 2.11302i
$516$	0	0
$517$	3.96167e13	1.07257
$518$	− 5.70172e13i	− 1.52882i
$519$	0	0
$520$	−1.52634e13	−0.401453
$521$	− 4.71658e13i	− 1.22868i	−0.789041	−	0.614340i	$-0.789422\pi$
$521$	− 4.71658e13i	− 1.22868i	0.789041	−	0.614340i	$-0.210578\pi$
$522$	0	0
$523$	4.74916e13	1.21369	0.606846	−	0.794819i	$-0.292434\pi$
$523$	4.74916e13	1.21369	0.606846	+	0.794819i	$0.292434\pi$
$524$	5.43853e13i	1.37666i
$525$	0	0
$526$	−1.41846e13	−0.352280
$527$	− 1.04479e12i	− 0.0257026i
$528$	0	0
$529$	−4.15372e13	−1.00267
$530$	6.17730e13i	1.47713i
$531$	0	0
$532$	3.18331e13	0.747001
$533$	− 4.57074e13i	− 1.06255i
$534$	0	0
$535$	2.68785e13	0.613248
$536$	5.42009e12i	0.122513i
$537$	0	0
$538$	2.50920e13	0.556705
$539$	− 8.38407e13i	− 1.84294i
$540$	0	0
$541$	−9.22407e13	−1.99038	−0.995191	−	0.0979580i	$-0.968769\pi$
$541$	−9.22407e13	−1.99038	−0.995191	+	0.0979580i	$0.968769\pi$
$542$	1.15089e13i	0.246058i
$543$	0	0
$544$	−7.96306e12	−0.167142
$545$	− 1.76078e13i	− 0.366203i
$546$	0	0
$547$	4.00684e13	0.818211	0.409105	−	0.912487i	$-0.365841\pi$
$547$	4.00684e13	0.818211	0.409105	+	0.912487i	$0.365841\pi$
$548$	3.84676e13i	0.778381i
$549$	0	0
$550$	6.65565e13	1.32244
$551$	− 1.33940e13i	− 0.263726i
$552$	0	0
$553$	−7.87954e12	−0.152361
$554$	− 6.38140e13i	− 1.22283i
$555$	0	0
$556$	3.74908e13	0.705587
$557$	4.39550e13i	0.819847i	0.912120	+	0.409923i	$0.134445\pi$
$557$	4.39550e13i	0.819847i	−0.912120	+	0.409923i	$0.865555\pi$
$558$	0	0
$559$	−1.59240e13	−0.291738
$560$	1.94168e14i	3.52564i
$561$	0	0
$562$	1.39578e14	2.48963
$563$	1.67138e13i	0.295484i	0.989026	+	0.147742i	$0.0472005\pi$
$563$	1.67138e13i	0.295484i	−0.989026	+	0.147742i	$0.952799\pi$
$564$	0	0
$565$	4.40446e13	0.764981
$566$	− 4.58903e13i	− 0.790021i
$567$	0	0
$568$	2.50547e13	0.423787
$569$	− 3.30742e13i	− 0.554534i	−0.960793	−	0.277267i	$-0.910571\pi$
$569$	− 3.30742e13i	− 0.554534i	0.960793	−	0.277267i	$-0.0894286\pi$
$570$	0	0
$571$	1.28829e13	0.212243	0.106122	−	0.994353i	$-0.466157\pi$
$571$	1.28829e13	0.212243	0.106122	+	0.994353i	$0.466157\pi$
$572$	2.27850e13i	0.372107i
$573$	0	0
$574$	−2.29161e14	−3.67774
$575$	− 1.25615e14i	− 1.99849i
$576$	0	0
$577$	8.06724e13	1.26138	0.630690	−	0.776035i	$-0.282772\pi$
$577$	8.06724e13	1.26138	0.630690	+	0.776035i	$0.282772\pi$
$578$	− 8.31589e13i	− 1.28905i
$579$	0	0
$580$	−3.59520e13	−0.547750
$581$	1.72662e14i	2.60805i
$582$	0	0
$583$	−3.47468e13	−0.515908
$584$	− 4.13221e13i	− 0.608301i
$585$	0	0
$586$	3.31976e13	0.480417
$587$	1.00326e14i	1.43954i	0.694212	+	0.719770i	$0.255753\pi$
$587$	1.00326e14i	1.43954i	−0.694212	+	0.719770i	$0.744247\pi$
$588$	0	0
$589$	7.19647e12	0.101518
$590$	9.95135e13i	1.39195i
$591$	0	0
$592$	−5.35616e13	−0.736622
$593$	− 7.03769e13i	− 0.959747i	−0.877338	−	0.479873i	$-0.840683\pi$
$593$	− 7.03769e13i	− 0.959747i	0.877338	−	0.479873i	$-0.159317\pi$
$594$	0	0
$595$	3.01592e13	0.404423
$596$	6.70242e13i	0.891252i
$597$	0	0
$598$	1.02210e14	1.33656
$599$	− 4.06523e12i	− 0.0527171i	−0.999653	−	0.0263585i	$-0.991609\pi$
$599$	− 4.06523e12i	− 0.0527171i	0.999653	−	0.0263585i	$-0.00839115\pi$
$600$	0	0
$601$	−7.67643e13	−0.979010	−0.489505	−	0.872001i	$-0.662822\pi$
$601$	−7.67643e13	−0.979010	−0.489505	+	0.872001i	$0.662822\pi$
$602$	7.98373e13i	1.00977i
$603$	0	0
$604$	−4.96407e13	−0.617523
$605$	− 6.19403e13i	− 0.764181i
$606$	0	0
$607$	−4.92577e13	−0.597765	−0.298883	−	0.954290i	$-0.596614\pi$
$607$	−4.92577e13	−0.597765	−0.298883	+	0.954290i	$0.596614\pi$
$608$	− 5.48491e13i	− 0.660163i
$609$	0	0
$610$	3.29433e13	0.390048
$611$	9.21156e13i	1.08175i
$612$	0	0
$613$	−1.15889e14	−1.33887	−0.669437	−	0.742869i	$-0.733465\pi$
$613$	−1.15889e14	−1.33887	−0.669437	+	0.742869i	$0.733465\pi$
$614$	− 1.47015e14i	− 1.68468i
$615$	0	0
$616$	−4.30449e13	−0.485309
$617$	− 1.43634e14i	− 1.60632i	−0.595763	−	0.803160i	$-0.703150\pi$
$617$	− 1.43634e14i	− 1.60632i	0.595763	−	0.803160i	$-0.296850\pi$
$618$	0	0
$619$	−6.91730e13	−0.761173	−0.380586	−	0.924745i	$-0.624278\pi$
$619$	−6.91730e13	−0.761173	−0.380586	+	0.924745i	$0.624278\pi$
$620$	− 1.93166e13i	− 0.210850i
$621$	0	0
$622$	−2.74072e13	−0.294383
$623$	2.65275e13i	0.282654i
$624$	0	0
$625$	−3.98529e13	−0.417888
$626$	− 1.83028e14i	− 1.90391i
$627$	0	0
$628$	−5.27513e13	−0.540051
$629$	8.31947e12i	0.0844972i
$630$	0	0
$631$	−1.13985e14	−1.13946	−0.569731	−	0.821831i	$-0.692953\pi$
$631$	−1.13985e14	−1.13946	−0.569731	+	0.821831i	$0.692953\pi$
$632$	2.91726e12i	0.0289328i
$633$	0	0
$634$	−9.86096e13	−0.962660
$635$	1.93600e14i	1.87515i
$636$	0	0
$637$	1.94944e14	1.85871
$638$	− 4.80654e13i	− 0.454705i
$639$	0	0
$640$	1.15443e14	1.07514
$641$	− 1.97683e14i	− 1.82675i	−0.407117	−	0.913376i	$-0.633466\pi$
$641$	− 1.97683e14i	− 1.82675i	0.407117	−	0.913376i	$-0.366534\pi$
$642$	0	0
$643$	9.23072e13	0.839809	0.419905	−	0.907568i	$-0.362064\pi$
$643$	9.23072e13	0.839809	0.419905	+	0.907568i	$0.362064\pi$
$644$	− 2.15602e14i	− 1.94636i
$645$	0	0
$646$	−1.10399e13	−0.0981298
$647$	8.66533e13i	0.764300i	0.924100	+	0.382150i	$0.124816\pi$
$647$	8.66533e13i	0.764300i	−0.924100	+	0.382150i	$0.875184\pi$
$648$	0	0
$649$	−5.59756e13	−0.486155
$650$	1.54755e14i	1.33376i
$651$	0	0
$652$	−6.93303e13	−0.588417
$653$	− 3.15545e13i	− 0.265763i	−0.991132	−	0.132882i	$-0.957577\pi$
$653$	− 3.15545e13i	− 0.265763i	0.991132	−	0.132882i	$-0.0424230\pi$
$654$	0	0
$655$	−3.54905e14	−2.94378
$656$	2.15272e14i	1.77202i
$657$	0	0
$658$	4.61835e14	3.74419
$659$	− 5.34433e13i	− 0.429998i	−0.976614	−	0.214999i	$-0.931025\pi$
$659$	− 5.34433e13i	− 0.429998i	0.976614	−	0.214999i	$-0.0689748\pi$
$660$	0	0
$661$	2.21568e14	1.75590	0.877948	−	0.478755i	$-0.158912\pi$
$661$	2.21568e14	1.75590	0.877948	+	0.478755i	$0.158912\pi$
$662$	1.52253e14i	1.19750i
$663$	0	0
$664$	6.39251e13	0.495257
$665$	2.07735e14i	1.59736i
$666$	0	0
$667$	−9.07158e13	−0.687154
$668$	6.86143e13i	0.515861i
$669$	0	0
$670$	9.38678e13	0.695253
$671$	1.85303e13i	0.136229i
$672$	0	0
$673$	2.23899e14	1.62172	0.810862	−	0.585237i	$-0.198999\pi$
$673$	2.23899e14	1.62172	0.810862	+	0.585237i	$0.198999\pi$
$674$	1.73106e14i	1.24455i
$675$	0	0
$676$	4.95532e13	0.351025
$677$	2.70697e14i	1.90344i	0.306963	+	0.951721i	$0.400687\pi$
$677$	2.70697e14i	1.90344i	−0.306963	+	0.951721i	$0.599313\pi$
$678$	0	0
$679$	−1.80938e14	−1.25367
$680$	− 1.11659e13i	− 0.0767982i
$681$	0	0
$682$	2.58251e13	0.175033
$683$	− 2.28990e14i	− 1.54068i	−0.637632	−	0.770341i	$-0.720086\pi$
$683$	− 2.28990e14i	− 1.54068i	0.637632	−	0.770341i	$-0.279914\pi$
$684$	0	0
$685$	−2.51030e14	−1.66446
$686$	− 5.99397e14i	− 3.94543i
$687$	0	0
$688$	7.49986e13	0.486531
$689$	− 8.07923e13i	− 0.520324i
$690$	0	0
$691$	1.15663e14	0.734184	0.367092	−	0.930185i	$-0.380353\pi$
$691$	1.15663e14	0.734184	0.367092	+	0.930185i	$0.380353\pi$
$692$	− 1.30378e14i	− 0.821625i
$693$	0	0
$694$	−2.00571e14	−1.24587
$695$	2.44655e14i	1.50880i
$696$	0	0
$697$	3.34372e13	0.203267
$698$	3.06953e14i	1.85265i
$699$	0	0
$700$	3.26441e14	1.94229
$701$	− 5.38211e13i	− 0.317952i	−0.987282	−	0.158976i	$-0.949181\pi$
$701$	− 5.38211e13i	− 0.317952i	0.987282	−	0.158976i	$-0.0508193\pi$
$702$	0	0
$703$	−5.73040e13	−0.333740
$704$	− 4.90818e13i	− 0.283829i
$705$	0	0
$706$	−3.28513e13	−0.187296
$707$	− 4.88771e12i	− 0.0276699i
$708$	0	0
$709$	1.71797e14	0.958925	0.479462	−	0.877562i	$-0.340832\pi$
$709$	1.71797e14	0.958925	0.479462	+	0.877562i	$0.340832\pi$
$710$	− 4.33910e14i	− 2.40496i
$711$	0	0
$712$	9.82135e12	0.0536749
$713$	− 4.87408e13i	− 0.264512i
$714$	0	0
$715$	−1.48689e14	−0.795698
$716$	1.70181e14i	0.904369i
$717$	0	0
$718$	3.19786e14	1.67586
$719$	− 1.80527e13i	− 0.0939504i	−0.998896	−	0.0469752i	$-0.985042\pi$
$719$	− 1.80527e13i	− 0.0939504i	0.998896	−	0.0469752i	$-0.0149582\pi$
$720$	0	0
$721$	5.01956e14	2.57626
$722$	1.81736e14i	0.926309i
$723$	0	0
$724$	−1.78247e13	−0.0896045
$725$	− 1.37352e14i	− 0.685719i
$726$	0	0
$727$	5.34262e13	0.263077	0.131538	−	0.991311i	$-0.458008\pi$
$727$	5.34262e13	0.263077	0.131538	+	0.991311i	$0.458008\pi$
$728$	− 1.00087e14i	− 0.489463i
$729$	0	0
$730$	−7.15638e14	−3.45207
$731$	− 1.16492e13i	− 0.0558096i
$732$	0	0
$733$	−2.40751e14	−1.13775	−0.568877	−	0.822423i	$-0.692622\pi$
$733$	−2.40751e14	−1.13775	−0.568877	+	0.822423i	$0.692622\pi$
$734$	− 4.33568e13i	− 0.203506i
$735$	0	0
$736$	−3.71486e14	−1.72010
$737$	5.27999e13i	0.242826i
$738$	0	0
$739$	2.51761e14	1.14226	0.571132	−	0.820858i	$-0.306504\pi$
$739$	2.51761e14	1.14226	0.571132	+	0.820858i	$0.306504\pi$
$740$	1.53815e14i	0.693168i
$741$	0	0
$742$	−4.05064e14	−1.80096
$743$	− 3.95902e14i	− 1.74841i	−0.485555	−	0.874206i	$-0.661382\pi$
$743$	− 3.95902e14i	− 1.74841i	0.485555	−	0.874206i	$-0.338618\pi$
$744$	0	0
$745$	−4.37383e14	−1.90581
$746$	3.27762e14i	1.41862i
$747$	0	0
$748$	−1.66683e13	−0.0711843
$749$	1.76250e14i	0.747689i
$750$	0	0
$751$	−3.84509e13	−0.160956	−0.0804779	−	0.996756i	$-0.525645\pi$
$751$	−3.84509e13	−0.160956	−0.0804779	+	0.996756i	$0.525645\pi$
$752$	− 4.33845e14i	− 1.80404i
$753$	0	0
$754$	1.11760e14	0.458597
$755$	− 3.23942e14i	− 1.32048i
$756$	0	0
$757$	−1.90529e13	−0.0766445	−0.0383223	−	0.999265i	$-0.512201\pi$
$757$	−1.90529e13	−0.0766445	−0.0383223	+	0.999265i	$0.512201\pi$
$758$	2.69488e14i	1.07694i
$759$	0	0
$760$	7.69103e13	0.303331
$761$	2.52372e14i	0.988821i	0.869229	+	0.494410i	$0.164616\pi$
$761$	2.52372e14i	0.988821i	−0.869229	+	0.494410i	$0.835384\pi$
$762$	0	0
$763$	1.15459e14	0.446485
$764$	− 1.79229e14i	− 0.688561i
$765$	0	0
$766$	2.89442e14	1.09753
$767$	− 1.30153e14i	− 0.490317i
$768$	0	0
$769$	−1.86775e14	−0.694522	−0.347261	−	0.937768i	$-0.612888\pi$
$769$	−1.86775e14	−0.694522	−0.347261	+	0.937768i	$0.612888\pi$
$770$	7.45474e14i	2.75409i
$771$	0	0
$772$	5.10298e12	0.0186096
$773$	2.85152e14i	1.03319i	0.856231	+	0.516593i	$0.172800\pi$
$773$	2.85152e14i	1.03319i	−0.856231	+	0.516593i	$0.827200\pi$
$774$	0	0
$775$	7.37981e13	0.263959
$776$	6.69893e13i	0.238066i
$777$	0	0
$778$	1.49919e14	0.525967
$779$	2.30314e14i	0.802846i
$780$	0	0
$781$	2.44071e14	0.839965
$782$	7.47715e13i	0.255683i
$783$	0	0
$784$	−9.18144e14	−3.09978
$785$	− 3.44242e14i	− 1.15482i
$786$	0	0
$787$	−4.25310e14	−1.40874	−0.704371	−	0.709832i	$-0.748771\pi$
$787$	−4.25310e14	−1.40874	−0.704371	+	0.709832i	$0.748771\pi$
$788$	1.39516e14i	0.459191i
$789$	0	0
$790$	5.05227e13	0.164192
$791$	2.88814e14i	0.932685i
$792$	0	0
$793$	−4.30862e13	−0.137395
$794$	4.87037e14i	1.54333i
$795$	0	0
$796$	2.32975e14	0.729029
$797$	− 4.06659e14i	− 1.26456i	−0.774741	−	0.632279i	$-0.782120\pi$
$797$	− 4.06659e14i	− 1.26456i	0.774741	−	0.632279i	$-0.217880\pi$
$798$	0	0
$799$	−6.73871e13	−0.206939
$800$	− 5.62464e14i	− 1.71650i
$801$	0	0
$802$	1.46269e14	0.440839
$803$	− 4.02540e14i	− 1.20568i
$804$	0	0
$805$	1.40696e15	4.16201
$806$	6.00478e13i	0.176531i
$807$	0	0
$808$	−1.80959e12	−0.00525440
$809$	1.85066e14i	0.534053i	0.963689	+	0.267026i	$0.0860411\pi$
$809$	1.85066e14i	0.534053i	−0.963689	+	0.267026i	$0.913959\pi$
$810$	0	0
$811$	2.37324e13	0.0676454	0.0338227	−	0.999428i	$-0.489232\pi$
$811$	2.37324e13	0.0676454	0.0338227	+	0.999428i	$0.489232\pi$
$812$	− 2.35748e14i	− 0.667832i
$813$	0	0
$814$	−2.05640e14	−0.575421
$815$	− 4.52432e14i	− 1.25825i
$816$	0	0
$817$	8.02389e13	0.220432
$818$	5.07226e14i	1.38495i
$819$	0	0
$820$	6.18204e14	1.66749
$821$	− 3.93488e14i	− 1.05491i	−0.849582	−	0.527456i	$-0.823146\pi$
$821$	− 3.93488e14i	− 1.05491i	0.849582	−	0.527456i	$-0.176854\pi$
$822$	0	0
$823$	2.43584e14	0.645135	0.322567	−	0.946547i	$-0.395454\pi$
$823$	2.43584e14	0.645135	0.322567	+	0.946547i	$0.395454\pi$
$824$	− 1.85841e14i	− 0.489220i
$825$	0	0
$826$	−6.52540e14	−1.69710
$827$	− 1.86431e14i	− 0.481937i	−0.970533	−	0.240968i	$-0.922535\pi$
$827$	− 1.86431e14i	− 0.481937i	0.970533	−	0.240968i	$-0.0774650\pi$
$828$	0	0
$829$	1.90172e14	0.485706	0.242853	−	0.970063i	$-0.421917\pi$
$829$	1.90172e14	0.485706	0.242853	+	0.970063i	$0.421917\pi$
$830$	− 1.10709e15i	− 2.81055i
$831$	0	0
$832$	1.14124e14	0.286259
$833$	1.42611e14i	0.355572i
$834$	0	0
$835$	−4.47760e14	−1.10309
$836$	− 1.14810e14i	− 0.281158i
$837$	0	0
$838$	1.96308e14	0.475026
$839$	3.99766e14i	0.961603i	0.876830	+	0.480801i	$0.159654\pi$
$839$	3.99766e14i	0.961603i	−0.876830	+	0.480801i	$0.840346\pi$
$840$	0	0
$841$	3.21515e14	0.764225
$842$	− 3.12315e14i	− 0.737960i
$843$	0	0
$844$	−4.58795e14	−1.07129
$845$	3.23372e14i	0.750617i
$846$	0	0
$847$	4.06161e14	0.931711
$848$	3.80514e14i	0.867744i
$849$	0	0
$850$	−1.13211e14	−0.255149
$851$	3.88113e14i	0.869581i
$852$	0	0
$853$	1.21542e14	0.269141	0.134571	−	0.990904i	$-0.457034\pi$
$853$	1.21542e14	0.269141	0.134571	+	0.990904i	$0.457034\pi$
$854$	2.16019e14i	0.475557i
$855$	0	0
$856$	6.52537e13	0.141983
$857$	8.75936e14i	1.89482i	0.320022	+	0.947410i	$0.396310\pi$
$857$	8.75936e14i	1.89482i	−0.320022	+	0.947410i	$0.603690\pi$
$858$	0	0
$859$	−6.41455e14	−1.37151	−0.685757	−	0.727831i	$-0.740529\pi$
$859$	−6.41455e14	−1.37151	−0.685757	+	0.727831i	$0.740529\pi$
$860$	− 2.15376e14i	− 0.457831i
$861$	0	0
$862$	4.92885e14	1.03564
$863$	8.91689e14i	1.86277i	0.364034	+	0.931386i	$0.381399\pi$
$863$	8.91689e14i	1.86277i	−0.364034	+	0.931386i	$0.618601\pi$
$864$	0	0
$865$	8.50813e14	1.75693
$866$	− 5.70943e14i	− 1.17220i
$867$	0	0
$868$	1.26665e14	0.257074
$869$	2.84186e13i	0.0573461i
$870$	0	0
$871$	−1.22769e14	−0.244905
$872$	− 4.27469e13i	− 0.0847856i
$873$	0	0
$874$	−5.15022e14	−1.00988
$875$	6.21797e14i	1.21230i
$876$	0	0
$877$	4.93168e13	0.0950597	0.0475299	−	0.998870i	$-0.484865\pi$
$877$	4.93168e13	0.0950597	0.0475299	+	0.998870i	$0.484865\pi$
$878$	− 6.20941e14i	− 1.19008i
$879$	0	0
$880$	7.00293e14	1.32699
$881$	− 6.24033e13i	− 0.117578i	−0.998270	−	0.0587892i	$-0.981276\pi$
$881$	− 6.24033e13i	− 0.117578i	0.998270	−	0.0587892i	$-0.0187240\pi$
$882$	0	0
$883$	−7.70465e14	−1.43532	−0.717661	−	0.696393i	$-0.754787\pi$
$883$	−7.70465e14	−1.43532	−0.717661	+	0.696393i	$0.754787\pi$
$884$	− 3.87567e13i	− 0.0717936i
$885$	0	0
$886$	6.89606e14	1.26308
$887$	− 7.96141e13i	− 0.145001i	−0.997368	−	0.0725007i	$-0.976902\pi$
$887$	− 7.96141e13i	− 0.145001i	0.997368	−	0.0725007i	$-0.0230980\pi$
$888$	0	0
$889$	−1.26949e15	−2.28624
$890$	− 1.70091e14i	− 0.304601i
$891$	0	0
$892$	4.67894e13	0.0828559
$893$	− 4.64158e14i	− 0.817351i
$894$	0	0
$895$	−1.11056e15	−1.93386
$896$	7.56992e14i	1.31084i
$897$	0	0
$898$	−8.73045e14	−1.49505
$899$	− 5.32951e13i	− 0.0907588i
$900$	0	0
$901$	5.91035e13	0.0995381
$902$	8.26499e14i	1.38423i
$903$	0	0
$904$	1.06928e14	0.177113
$905$	− 1.16320e14i	− 0.191606i
$906$	0	0
$907$	−2.38098e14	−0.387900	−0.193950	−	0.981011i	$-0.562130\pi$
$907$	−2.38098e14	−0.387900	−0.193950	+	0.981011i	$0.562130\pi$
$908$	− 2.31650e14i	− 0.375322i
$909$	0	0
$910$	−1.73335e15	−2.77767
$911$	− 3.20993e14i	− 0.511569i	−0.966734	−	0.255784i	$-0.917666\pi$
$911$	− 3.20993e14i	− 0.511569i	0.966734	−	0.255784i	$-0.0823337\pi$
$912$	0	0
$913$	6.22727e14	0.981622
$914$	1.14727e15i	1.79860i
$915$	0	0
$916$	−2.79294e13	−0.0433095
$917$	− 2.32722e15i	− 3.58914i
$918$	0	0
$919$	8.57341e14	1.30790	0.653952	−	0.756536i	$-0.273110\pi$
$919$	8.57341e14	1.30790	0.653952	+	0.756536i	$0.273110\pi$
$920$	− 5.20904e14i	− 0.790348i
$921$	0	0
$922$	7.92694e14	1.18974
$923$	5.67507e14i	0.847154i
$924$	0	0
$925$	−5.87639e14	−0.867765
$926$	− 6.50729e14i	− 0.955752i
$927$	0	0
$928$	−4.06197e14	−0.590197
$929$	9.03445e14i	1.30564i	0.757513	+	0.652820i	$0.226414\pi$
$929$	9.03445e14i	1.30564i	−0.757513	+	0.652820i	$0.773586\pi$
$930$	0	0
$931$	−9.82296e14	−1.40441
$932$	− 6.47874e14i	− 0.921322i
$933$	0	0
$934$	1.16735e15	1.64236
$935$	− 1.08773e14i	− 0.152217i
$936$	0	0
$937$	−2.17810e14	−0.301564	−0.150782	−	0.988567i	$-0.548179\pi$
$937$	−2.17810e14	−0.301564	−0.150782	+	0.988567i	$0.548179\pi$
$938$	6.15519e14i	0.847671i
$939$	0	0
$940$	−1.24589e15	−1.69761
$941$	9.82201e14i	1.33123i	0.746296	+	0.665614i	$0.231830\pi$
$941$	9.82201e14i	1.33123i	−0.746296	+	0.665614i	$0.768170\pi$
$942$	0	0
$943$	1.55988e15	2.09187
$944$	6.12992e14i	0.817701i
$945$	0	0
$946$	2.87944e14	0.380060
$947$	− 8.85699e14i	− 1.16288i	−0.813588	−	0.581442i	$-0.802489\pi$
$947$	− 8.85699e14i	− 1.16288i	0.813588	−	0.581442i	$-0.197511\pi$
$948$	0	0
$949$	9.35976e14	1.21600
$950$	− 7.79791e14i	− 1.00777i
$951$	0	0
$952$	7.32184e13	0.0936344
$953$	− 9.78487e14i	− 1.24477i	−0.782710	−	0.622387i	$-0.786163\pi$
$953$	− 9.78487e14i	− 1.24477i	0.782710	−	0.622387i	$-0.213837\pi$
$954$	0	0
$955$	1.16961e15	1.47239
$956$	− 5.05569e14i	− 0.633127i
$957$	0	0
$958$	5.22486e13	0.0647511
$959$	− 1.64608e15i	− 2.02935i
$960$	0	0
$961$	−7.90993e14	−0.965063
$962$	− 4.78148e14i	− 0.580346i
$963$	0	0
$964$	1.05230e15	1.26403
$965$	3.33008e13i	0.0397940i
$966$	0	0
$967$	−1.25055e15	−1.47900	−0.739502	−	0.673154i	$-0.764939\pi$
$967$	−1.25055e15	−1.47900	−0.739502	+	0.673154i	$0.764939\pi$
$968$	− 1.50374e14i	− 0.176928i
$969$	0	0
$970$	1.16015e15	1.35101
$971$	1.04451e15i	1.21009i	0.796193	+	0.605043i	$0.206844\pi$
$971$	1.04451e15i	1.21009i	−0.796193	+	0.605043i	$0.793156\pi$
$972$	0	0
$973$	−1.60428e15	−1.83957
$974$	1.20533e15i	1.37502i
$975$	0	0
$976$	2.02927e14	0.229134
$977$	6.75052e14i	0.758341i	0.925327	+	0.379170i	$0.123791\pi$
$977$	6.75052e14i	0.758341i	−0.925327	+	0.379170i	$0.876209\pi$
$978$	0	0
$979$	9.56748e13	0.106386
$980$	2.63666e15i	2.91692i
$981$	0	0
$982$	9.15905e13	0.100298
$983$	5.60523e14i	0.610697i	0.952241	+	0.305348i	$0.0987729\pi$
$983$	5.60523e14i	0.610697i	−0.952241	+	0.305348i	$0.901227\pi$
$984$	0	0
$985$	−9.10447e14	−0.981914
$986$	8.17582e13i	0.0877297i
$987$	0	0
$988$	2.66954e14	0.283564
$989$	− 5.43448e14i	− 0.574349i
$990$	0	0
$991$	−6.89169e13	−0.0721037	−0.0360518	−	0.999350i	$-0.511478\pi$
$991$	−6.89169e13	−0.0721037	−0.0360518	+	0.999350i	$0.511478\pi$
$992$	− 2.18246e14i	− 0.227189i
$993$	0	0
$994$	2.84528e15	2.93220
$995$	1.52034e15i	1.55892i
$996$	0	0
$997$	6.28444e14	0.637956	0.318978	−	0.947762i	$-0.396660\pi$
$997$	6.28444e14	0.637956	0.318978	+	0.947762i	$0.396660\pi$
$998$	1.00144e15i	1.01151i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.11.b.c.26.1	✓	4
3.2	odd	2	inner	27.11.b.c.26.4	yes	4
9.2	odd	6		81.11.d.e.53.4		8
9.4	even	3		81.11.d.e.26.4		8
9.5	odd	6		81.11.d.e.26.1		8
9.7	even	3		81.11.d.e.53.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.11.b.c.26.1	✓	4	1.1	even	1	trivial
27.11.b.c.26.4	yes	4	3.2	odd	2	inner
81.11.d.e.26.1		8	9.5	odd	6
81.11.d.e.26.4		8	9.4	even	3
81.11.d.e.53.1		8	9.7	even	3
81.11.d.e.53.4		8	9.2	odd	6

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:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	27.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-42.0446i q^{2} -743.750 q^{4} -4853.52i q^{5} +31826.0 q^{7} -11783.0i q^{8} +O(q^{10})\)	\(q-42.0446i q^{2} -743.750 q^{4} -4853.52i q^{5} +31826.0 q^{7} -11783.0i q^{8} -204064. q^{10} -114785. i q^{11} +266894. q^{13} -1.33811e6i q^{14} -1.25701e6 q^{16} +195246. i q^{17} -1.34484e6 q^{19} +3.60980e6i q^{20} -4.82607e6 q^{22} +9.10844e6i q^{23} -1.37910e7 q^{25} -1.12214e7i q^{26} -2.36706e7 q^{28} +9.95954e6i q^{29} -5.35116e6 q^{31} +4.07848e7i q^{32} +8.20904e6 q^{34} -1.54468e8i q^{35} +4.26102e7 q^{37} +5.65433e7i q^{38} -5.71891e7 q^{40} -1.71257e8i q^{41} -5.96642e7 q^{43} +8.53710e7i q^{44} +3.82961e8 q^{46} +3.45140e8i q^{47} +7.30418e8 q^{49} +5.79839e8i q^{50} -1.98502e8 q^{52} -3.02713e8i q^{53} -5.57109e8 q^{55} -3.75006e8i q^{56} +4.18745e8 q^{58} -4.87658e8i q^{59} -1.61436e8 q^{61} +2.24988e8i q^{62} +4.27600e8 q^{64} -1.29537e9i q^{65} -4.59991e8 q^{67} -1.45214e8i q^{68} -6.49455e9 q^{70} +2.12634e9i q^{71} +3.50692e9 q^{73} -1.79153e9i q^{74} +1.00023e9 q^{76} -3.65313e9i q^{77} -2.47582e8 q^{79} +6.10093e9i q^{80} -7.20044e9 q^{82} +5.42519e9i q^{83} +9.47630e8 q^{85} +2.50856e9i q^{86} -1.35251e9 q^{88} +8.33517e8i q^{89} +8.49415e9 q^{91} -6.77440e9i q^{92} +1.45113e10 q^{94} +6.52722e9i q^{95} -5.68524e9 q^{97} -3.07101e10i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 548 q^{4} + 20516 q^{7}+O(q^{10}) \) 4 * q - 548 * q^4 + 20516 * q^7	\( 4 q - 548 q^{4} + 20516 q^{7} - 199800 q^{10} - 262420 q^{13} - 3207800 q^{16} - 8233516 q^{19} - 21085704 q^{22} - 47203580 q^{25} - 67604212 q^{28} - 46994920 q^{31} - 78985368 q^{34} + 108086444 q^{37} + 196812720 q^{40} + 78251000 q^{43} + 882645768 q^{46} + 1826240448 q^{49} - 771022300 q^{52} + 1034575200 q^{55} + 1267544592 q^{58} - 3222389524 q^{61} - 1166263952 q^{64} + 1972286516 q^{67} - 17482328280 q^{70} + 4077598940 q^{73} - 603762388 q^{76} - 10402780252 q^{79} - 7945552656 q^{82} + 19825132320 q^{85} - 19783893168 q^{88} + 34160896780 q^{91} + 22345655544 q^{94} + 12969797468 q^{97}+O(q^{100}) \) 4 * q - 548 * q^4 + 20516 * q^7 - 199800 * q^10 - 262420 * q^13 - 3207800 * q^16 - 8233516 * q^19 - 21085704 * q^22 - 47203580 * q^25 - 67604212 * q^28 - 46994920 * q^31 - 78985368 * q^34 + 108086444 * q^37 + 196812720 * q^40 + 78251000 * q^43 + 882645768 * q^46 + 1826240448 * q^49 - 771022300 * q^52 + 1034575200 * q^55 + 1267544592 * q^58 - 3222389524 * q^61 - 1166263952 * q^64 + 1972286516 * q^67 - 17482328280 * q^70 + 4077598940 * q^73 - 603762388 * q^76 - 10402780252 * q^79 - 7945552656 * q^82 + 19825132320 * q^85 - 19783893168 * q^88 + 34160896780 * q^91 + 22345655544 * q^94 + 12969797468 * q^97

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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