LMFDB - Embedded newform 81.12.a.c.1.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [81,12,Mod(1,81)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(81, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 12, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("81.1"); S:= CuspForms(chi, 12); N := Newforms(S);

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	81.a (trivial)

Newform invariants

comment:select newform

sage:traces = [10,-33] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$62.2357976253$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 3 x^{9} - 14790 x^{8} + 93060 x^{7} + 72223254 x^{6} - 592709562 x^{5} - 132941711592 x^{4} + \cdots - 18\!\cdots\!74 $ x^10 - 3x^9 - 14790x^8 + 93060x^7 + 72223254x^6 - 592709562x^5 - 132941711592x^4 + 1226670836724x^3 + 71629903044513x^2 - 424311619946427*x - 1872029164523274
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{30} $
Twist minimal:	no (minimal twist has level 9)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-64.1286$ of defining polynomial
Character	$\chi$	$=$	81.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+61.1286 q^{2} +1688.71 q^{4} -3368.11 q^{5} +10796.9 q^{7} -21963.1 q^{8} -205888. q^{10} -23907.9 q^{11} +2.37145e6 q^{13} +659999. q^{14} -4.80105e6 q^{16} -8.22600e6 q^{17} +6.96671e6 q^{19} -5.68775e6 q^{20} -1.46146e6 q^{22} -3.92161e7 q^{23} -3.74840e7 q^{25} +1.44964e8 q^{26} +1.82328e7 q^{28} -1.43223e8 q^{29} -1.40238e8 q^{31} -2.48501e8 q^{32} -5.02844e8 q^{34} -3.63651e7 q^{35} -9.06766e7 q^{37} +4.25865e8 q^{38} +7.39741e7 q^{40} +1.06810e9 q^{41} -1.16886e9 q^{43} -4.03735e7 q^{44} -2.39723e9 q^{46} -6.65214e8 q^{47} -1.86075e9 q^{49} -2.29134e9 q^{50} +4.00469e9 q^{52} -2.63429e9 q^{53} +8.05245e7 q^{55} -2.37133e8 q^{56} -8.75505e9 q^{58} +2.57374e9 q^{59} -2.31915e9 q^{61} -8.57257e9 q^{62} -5.35796e9 q^{64} -7.98731e9 q^{65} +1.80212e10 q^{67} -1.38913e10 q^{68} -2.22295e9 q^{70} -8.01761e9 q^{71} +6.36413e9 q^{73} -5.54293e9 q^{74} +1.17647e10 q^{76} -2.58131e8 q^{77} +3.50524e9 q^{79} +1.61704e10 q^{80} +6.52913e10 q^{82} -2.23844e10 q^{83} +2.77061e10 q^{85} -7.14509e10 q^{86} +5.25093e8 q^{88} +2.24189e10 q^{89} +2.56043e10 q^{91} -6.62245e10 q^{92} -4.06636e10 q^{94} -2.34646e10 q^{95} +1.08894e11 q^{97} -1.13745e11 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 33 q^{2} + 9217 q^{4} - 7230 q^{5} - 8512 q^{7} + 14559 q^{8} + 2046 q^{10} - 112776 q^{11} - 279706 q^{13} - 3901584 q^{14} + 7342081 q^{16} - 13882896 q^{17} + 3514700 q^{19} - 34163508 q^{20}+ \cdots - 655061802039 q^{98}+O(q^{100}) $ 10 * q - 33 * q^2 + 9217 * q^4 - 7230 * q^5 - 8512 * q^7 + 14559 * q^8 + 2046 * q^10 - 112776 * q^11 - 279706 * q^13 - 3901584 * q^14 + 7342081 * q^16 - 13882896 * q^17 + 3514700 * q^19 - 34163508 * q^20 - 2274591 * q^22 - 69371616 * q^23 + 45286204 * q^25 - 240964572 * q^26 - 30672898 * q^28 - 25437246 * q^29 - 114575368 * q^31 + 80396559 * q^32 + 243855063 * q^34 + 89073732 * q^35 - 67109164 * q^37 + 489799995 * q^38 - 107425416 * q^40 + 331873026 * q^41 + 1118847584 * q^43 - 139238637 * q^44 + 1441268796 * q^46 - 1469650704 * q^47 + 3553434720 * q^49 - 6643771701 * q^50 - 3632448874 * q^52 - 7457130972 * q^53 + 2490724992 * q^55 - 27669139026 * q^56 + 1387480560 * q^58 - 26505032592 * q^59 - 990409066 * q^61 - 45522498090 * q^62 - 3758354783 * q^64 - 39045315390 * q^65 - 6557215720 * q^67 - 77299152993 * q^68 + 785437278 * q^70 - 61026859872 * q^71 - 6306446968 * q^73 - 109519086216 * q^74 - 14574055597 * q^76 - 88616208018 * q^77 + 7621233248 * q^79 - 199583341536 * q^80 - 29584238667 * q^82 - 99007044180 * q^83 - 12911595156 * q^85 - 214357830519 * q^86 + 54423523605 * q^88 - 238298852412 * q^89 + 69105826108 * q^91 - 461776423998 * q^92 - 13393667064 * q^94 - 418952909328 * q^95 - 123483551938 * q^97 - 655061802039 * q^98

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 $

$2$	61.1286	1.35076	0.675382	−	0.737468i	$-0.263979\pi$
$2$	61.1286	1.35076	0.675382	+	0.737468i	$0.263979\pi$
$3$	0	0
$3$	0	0
$4$	1688.71	0.824564
$5$	−3368.11	−0.482004	−0.241002	−	0.970525i	$-0.577476\pi$
$5$	−3368.11	−0.482004	−0.241002	+	0.970525i	$0.577476\pi$
$6$	0	0
$7$	10796.9	0.242806	0.121403	−	0.992603i	$-0.461261\pi$
$7$	10796.9	0.242806	0.121403	+	0.992603i	$0.461261\pi$
$8$	−21963.1	−0.236973
$9$	0	0
$10$	−205888.	−0.651074
$11$	−23907.9	−0.0447592	−0.0223796	−	0.999750i	$-0.507124\pi$
$11$	−23907.9	−0.0447592	−0.0223796	+	0.999750i	$0.507124\pi$
$12$	0	0
$13$	2.37145e6	1.77144	0.885719	−	0.464221i	$-0.153666\pi$
$13$	2.37145e6	1.77144	0.885719	+	0.464221i	$0.153666\pi$
$14$	659999.	0.327974
$15$	0	0
$16$	−4.80105e6	−1.14466
$17$	−8.22600e6	−1.40514	−0.702570	−	0.711615i	$-0.747964\pi$
$17$	−8.22600e6	−1.40514	−0.702570	+	0.711615i	$0.747964\pi$
$18$	0	0
$19$	6.96671e6	0.645480	0.322740	−	0.946488i	$-0.395396\pi$
$19$	6.96671e6	0.645480	0.322740	+	0.946488i	$0.395396\pi$
$20$	−5.68775e6	−0.397443
$21$	0	0
$22$	−1.46146e6	−0.0604591
$23$	−3.92161e7	−1.27046	−0.635230	−	0.772323i	$-0.719095\pi$
$23$	−3.92161e7	−1.27046	−0.635230	+	0.772323i	$0.719095\pi$
$24$	0	0
$25$	−3.74840e7	−0.767672
$26$	1.44964e8	2.39280
$27$	0	0
$28$	1.82328e7	0.200209
$29$	−1.43223e8	−1.29666	−0.648329	−	0.761361i	$-0.724532\pi$
$29$	−1.43223e8	−1.29666	−0.648329	+	0.761361i	$0.724532\pi$
$30$	0	0
$31$	−1.40238e8	−0.879786	−0.439893	−	0.898050i	$-0.644984\pi$
$31$	−1.40238e8	−0.879786	−0.439893	+	0.898050i	$0.644984\pi$
$32$	−2.48501e8	−1.30919
$33$	0	0
$34$	−5.02844e8	−1.89801
$35$	−3.63651e7	−0.117034
$36$	0	0
$37$	−9.06766e7	−0.214974	−0.107487	−	0.994206i	$-0.534280\pi$
$37$	−9.06766e7	−0.214974	−0.107487	+	0.994206i	$0.534280\pi$
$38$	4.25865e8	0.871892
$39$	0	0
$40$	7.39741e7	0.114222
$41$	1.06810e9	1.43979	0.719896	−	0.694082i	$-0.244190\pi$
$41$	1.06810e9	1.43979	0.719896	+	0.694082i	$0.244190\pi$
$42$	0	0
$43$	−1.16886e9	−1.21251	−0.606257	−	0.795269i	$-0.707330\pi$
$43$	−1.16886e9	−1.21251	−0.606257	+	0.795269i	$0.707330\pi$
$44$	−4.03735e7	−0.0369068
$45$	0	0
$46$	−2.39723e9	−1.71609
$47$	−6.65214e8	−0.423081	−0.211540	−	0.977369i	$-0.567848\pi$
$47$	−6.65214e8	−0.423081	−0.211540	+	0.977369i	$0.567848\pi$
$48$	0	0
$49$	−1.86075e9	−0.941045
$50$	−2.29134e9	−1.03694
$51$	0	0
$52$	4.00469e9	1.46066
$53$	−2.63429e9	−0.865259	−0.432629	−	0.901572i	$-0.642414\pi$
$53$	−2.63429e9	−0.865259	−0.432629	+	0.901572i	$0.642414\pi$
$54$	0	0
$55$	8.05245e7	0.0215741
$56$	−2.37133e8	−0.0575385
$57$	0	0
$58$	−8.75505e9	−1.75148
$59$	2.57374e9	0.468682	0.234341	−	0.972154i	$-0.424707\pi$
$59$	2.57374e9	0.468682	0.234341	+	0.972154i	$0.424707\pi$
$60$	0	0
$61$	−2.31915e9	−0.351573	−0.175786	−	0.984428i	$-0.556247\pi$
$61$	−2.31915e9	−0.351573	−0.175786	+	0.984428i	$0.556247\pi$
$62$	−8.57257e9	−1.18838
$63$	0	0
$64$	−5.35796e9	−0.623749
$65$	−7.98731e9	−0.853841
$66$	0	0
$67$	1.80212e10	1.63069	0.815345	−	0.578976i	$-0.196547\pi$
$67$	1.80212e10	1.63069	0.815345	+	0.578976i	$0.196547\pi$
$68$	−1.38913e10	−1.15863
$69$	0	0
$70$	−2.22295e9	−0.158085
$71$	−8.01761e9	−0.527380	−0.263690	−	0.964607i	$-0.584940\pi$
$71$	−8.01761e9	−0.527380	−0.263690	+	0.964607i	$0.584940\pi$
$72$	0	0
$73$	6.36413e9	0.359305	0.179653	−	0.983730i	$-0.442503\pi$
$73$	6.36413e9	0.359305	0.179653	+	0.983730i	$0.442503\pi$
$74$	−5.54293e9	−0.290379
$75$	0	0
$76$	1.17647e10	0.532240
$77$	−2.58131e8	−0.0108678
$78$	0	0
$79$	3.50524e9	0.128165	0.0640824	−	0.997945i	$-0.479588\pi$
$79$	3.50524e9	0.128165	0.0640824	+	0.997945i	$0.479588\pi$
$80$	1.61704e10	0.551730
$81$	0	0
$82$	6.52913e10	1.94482
$83$	−2.23844e10	−0.623757	−0.311879	−	0.950122i	$-0.600958\pi$
$83$	−2.23844e10	−0.623757	−0.311879	+	0.950122i	$0.600958\pi$
$84$	0	0
$85$	2.77061e10	0.677284
$86$	−7.14509e10	−1.63782
$87$	0	0
$88$	5.25093e8	0.0106067
$89$	2.24189e10	0.425568	0.212784	−	0.977099i	$-0.431747\pi$
$89$	2.24189e10	0.425568	0.212784	+	0.977099i	$0.431747\pi$
$90$	0	0
$91$	2.56043e10	0.430116
$92$	−6.62245e10	−1.04758
$93$	0	0
$94$	−4.06636e10	−0.571482
$95$	−2.34646e10	−0.311124
$96$	0	0
$97$	1.08894e11	1.28754	0.643769	−	0.765220i	$-0.277370\pi$
$97$	1.08894e11	1.28754	0.643769	+	0.765220i	$0.277370\pi$
$98$	−1.13745e11	−1.27113
$99$	0	0
$100$	−6.32994e10	−0.632994
$101$	−2.79733e10	−0.264836	−0.132418	−	0.991194i	$-0.542274\pi$
$101$	−2.79733e10	−0.264836	−0.132418	+	0.991194i	$0.542274\pi$
$102$	0	0
$103$	4.60630e10	0.391514	0.195757	−	0.980652i	$-0.437284\pi$
$103$	4.60630e10	0.391514	0.195757	+	0.980652i	$0.437284\pi$
$104$	−5.20845e10	−0.419783
$105$	0	0
$106$	−1.61030e11	−1.16876
$107$	−2.52397e11	−1.73970	−0.869849	−	0.493319i	$-0.835784\pi$
$107$	−2.52397e11	−1.73970	−0.869849	+	0.493319i	$0.835784\pi$
$108$	0	0
$109$	−2.77857e11	−1.72972	−0.864859	−	0.502015i	$-0.832592\pi$
$109$	−2.77857e11	−1.72972	−0.864859	+	0.502015i	$0.832592\pi$
$110$	4.92235e9	0.0291416
$111$	0	0
$112$	−5.18364e10	−0.277930
$113$	1.72897e11	0.882785	0.441392	−	0.897314i	$-0.354485\pi$
$113$	1.72897e11	0.882785	0.441392	+	0.897314i	$0.354485\pi$
$114$	0	0
$115$	1.32084e11	0.612368
$116$	−2.41862e11	−1.06918
$117$	0	0
$118$	1.57329e11	0.633078
$119$	−8.88152e10	−0.341176
$120$	0	0
$121$	−2.84740e11	−0.997997
$122$	−1.41767e11	−0.474892
$123$	0	0
$124$	−2.36821e11	−0.725440
$125$	2.90708e11	0.852026
$126$	0	0
$127$	−2.78560e11	−0.748167	−0.374084	−	0.927395i	$-0.622043\pi$
$127$	−2.78560e11	−0.748167	−0.374084	+	0.927395i	$0.622043\pi$
$128$	1.81405e11	0.466653
$129$	0	0
$130$	−4.88253e11	−1.15334
$131$	−1.80696e11	−0.409219	−0.204610	−	0.978844i	$-0.565592\pi$
$131$	−1.80696e11	−0.409219	−0.204610	+	0.978844i	$0.565592\pi$
$132$	0	0
$133$	7.52188e10	0.156727
$134$	1.10161e12	2.20268
$135$	0	0
$136$	1.80669e11	0.332980
$137$	−2.28140e11	−0.403866	−0.201933	−	0.979399i	$-0.564722\pi$
$137$	−2.28140e11	−0.403866	−0.201933	+	0.979399i	$0.564722\pi$
$138$	0	0
$139$	9.06508e11	1.48180	0.740901	−	0.671614i	$-0.234399\pi$
$139$	9.06508e11	1.48180	0.740901	+	0.671614i	$0.234399\pi$
$140$	−6.14100e10	−0.0965016
$141$	0	0
$142$	−4.90105e11	−0.712366
$143$	−5.66965e10	−0.0792882
$144$	0	0
$145$	4.82392e11	0.624994
$146$	3.89030e11	0.485336
$147$	0	0
$148$	−1.53126e11	−0.177260
$149$	9.38804e11	1.04725	0.523625	−	0.851949i	$-0.324579\pi$
$149$	9.38804e11	1.04725	0.523625	+	0.851949i	$0.324579\pi$
$150$	0	0
$151$	−9.60559e11	−0.995751	−0.497876	−	0.867248i	$-0.665886\pi$
$151$	−9.60559e11	−0.995751	−0.497876	+	0.867248i	$0.665886\pi$
$152$	−1.53011e11	−0.152961
$153$	0	0
$154$	−1.57792e10	−0.0146798
$155$	4.72338e11	0.424061
$156$	0	0
$157$	1.86770e12	1.56264	0.781318	−	0.624133i	$-0.214548\pi$
$157$	1.86770e12	1.56264	0.781318	+	0.624133i	$0.214548\pi$
$158$	2.14270e11	0.173120
$159$	0	0
$160$	8.36977e11	0.631036
$161$	−4.23412e11	−0.308475
$162$	0	0
$163$	1.51923e12	1.03417	0.517085	−	0.855934i	$-0.327017\pi$
$163$	1.51923e12	1.03417	0.517085	+	0.855934i	$0.327017\pi$
$164$	1.80370e12	1.18720
$165$	0	0
$166$	−1.36833e12	−0.842549
$167$	8.50407e11	0.506624	0.253312	−	0.967385i	$-0.418480\pi$
$167$	8.50407e11	0.506624	0.253312	+	0.967385i	$0.418480\pi$
$168$	0	0
$169$	3.83163e12	2.13799
$170$	1.69363e12	0.914850
$171$	0	0
$172$	−1.97386e12	−0.999795
$173$	−2.11572e12	−1.03802	−0.519010	−	0.854768i	$-0.673699\pi$
$173$	−2.11572e12	−1.03802	−0.519010	+	0.854768i	$0.673699\pi$
$174$	0	0
$175$	−4.04710e11	−0.186395
$176$	1.14783e11	0.0512340
$177$	0	0
$178$	1.37043e12	0.574842
$179$	8.66770e11	0.352543	0.176271	−	0.984342i	$-0.443596\pi$
$179$	8.66770e11	0.352543	0.176271	+	0.984342i	$0.443596\pi$
$180$	0	0
$181$	1.19641e12	0.457772	0.228886	−	0.973453i	$-0.426492\pi$
$181$	1.19641e12	0.457772	0.228886	+	0.973453i	$0.426492\pi$
$182$	1.56516e12	0.580985
$183$	0	0
$184$	8.61308e11	0.301065
$185$	3.05409e11	0.103618
$186$	0	0
$187$	1.96667e11	0.0628929
$188$	−1.12335e12	−0.348857
$189$	0	0
$190$	−1.43436e12	−0.420256
$191$	4.23566e12	1.20570	0.602848	−	0.797856i	$-0.294033\pi$
$191$	4.23566e12	1.20570	0.602848	+	0.797856i	$0.294033\pi$
$192$	0	0
$193$	−7.72433e11	−0.207633	−0.103816	−	0.994596i	$-0.533105\pi$
$193$	−7.72433e11	−0.207633	−0.103816	+	0.994596i	$0.533105\pi$
$194$	6.65654e12	1.73916
$195$	0	0
$196$	−3.14227e12	−0.775952
$197$	−2.24808e12	−0.539819	−0.269909	−	0.962886i	$-0.586994\pi$
$197$	−2.24808e12	−0.539819	−0.269909	+	0.962886i	$0.586994\pi$
$198$	0	0
$199$	4.46460e12	1.01412	0.507062	−	0.861910i	$-0.330732\pi$
$199$	4.46460e12	1.01412	0.507062	+	0.861910i	$0.330732\pi$
$200$	8.23265e11	0.181918
$201$	0	0
$202$	−1.70997e12	−0.357730
$203$	−1.54637e12	−0.314836
$204$	0	0
$205$	−3.59747e12	−0.693986
$206$	2.81577e12	0.528843
$207$	0	0
$208$	−1.13855e13	−2.02769
$209$	−1.66560e11	−0.0288912
$210$	0	0
$211$	2.70478e12	0.445224	0.222612	−	0.974907i	$-0.428542\pi$
$211$	2.70478e12	0.445224	0.222612	+	0.974907i	$0.428542\pi$
$212$	−4.44854e12	−0.713461
$213$	0	0
$214$	−1.54287e13	−2.34992
$215$	3.93685e12	0.584437
$216$	0	0
$217$	−1.51414e12	−0.213617
$218$	−1.69850e13	−2.33644
$219$	0	0
$220$	1.35982e11	0.0177892
$221$	−1.95076e13	−2.48912
$222$	0	0
$223$	7.20279e12	0.874629	0.437315	−	0.899309i	$-0.355930\pi$
$223$	7.20279e12	0.874629	0.437315	+	0.899309i	$0.355930\pi$
$224$	−2.68304e12	−0.317879
$225$	0	0
$226$	1.05689e13	1.19243
$227$	−3.22339e12	−0.354953	−0.177476	−	0.984125i	$-0.556793\pi$
$227$	−3.22339e12	−0.354953	−0.177476	+	0.984125i	$0.556793\pi$
$228$	0	0
$229$	−1.46406e13	−1.53625	−0.768126	−	0.640299i	$-0.778810\pi$
$229$	−1.46406e13	−1.53625	−0.768126	+	0.640299i	$0.778810\pi$
$230$	8.07411e12	0.827164
$231$	0	0
$232$	3.14563e12	0.307273
$233$	2.97867e12	0.284162	0.142081	−	0.989855i	$-0.454621\pi$
$233$	2.97867e12	0.284162	0.142081	+	0.989855i	$0.454621\pi$
$234$	0	0
$235$	2.24051e12	0.203927
$236$	4.34629e12	0.386458
$237$	0	0
$238$	−5.42915e12	−0.460849
$239$	−9.31747e12	−0.772876	−0.386438	−	0.922315i	$-0.626295\pi$
$239$	−9.31747e12	−0.772876	−0.386438	+	0.922315i	$0.626295\pi$
$240$	0	0
$241$	−4.54727e11	−0.0360294	−0.0180147	−	0.999838i	$-0.505735\pi$
$241$	−4.54727e11	−0.0360294	−0.0180147	+	0.999838i	$0.505735\pi$
$242$	−1.74058e13	−1.34806
$243$	0	0
$244$	−3.91637e12	−0.289894
$245$	6.26722e12	0.453588
$246$	0	0
$247$	1.65212e13	1.14343
$248$	3.08007e12	0.208486
$249$	0	0
$250$	1.77706e13	1.15089
$251$	−1.22730e13	−0.777582	−0.388791	−	0.921326i	$-0.627107\pi$
$251$	−1.22730e13	−0.777582	−0.388791	+	0.921326i	$0.627107\pi$
$252$	0	0
$253$	9.37576e11	0.0568648
$254$	−1.70280e13	−1.01060
$255$	0	0
$256$	2.20621e13	1.25409
$257$	1.04580e13	0.581859	0.290930	−	0.956744i	$-0.406035\pi$
$257$	1.04580e13	0.581859	0.290930	+	0.956744i	$0.406035\pi$
$258$	0	0
$259$	−9.79025e11	−0.0521970
$260$	−1.34882e13	−0.704046
$261$	0	0
$262$	−1.10457e13	−0.552759
$263$	−1.07970e13	−0.529111	−0.264555	−	0.964370i	$-0.585225\pi$
$263$	−1.07970e13	−0.529111	−0.264555	+	0.964370i	$0.585225\pi$
$264$	0	0
$265$	8.87257e12	0.417058
$266$	4.59802e12	0.211701
$267$	0	0
$268$	3.04324e13	1.34461
$269$	−3.26649e13	−1.41398	−0.706992	−	0.707222i	$-0.749948\pi$
$269$	−3.26649e13	−1.41398	−0.706992	+	0.707222i	$0.749948\pi$
$270$	0	0
$271$	7.15587e12	0.297393	0.148697	−	0.988883i	$-0.452492\pi$
$271$	7.15587e12	0.297393	0.148697	+	0.988883i	$0.452492\pi$
$272$	3.94934e13	1.60841
$273$	0	0
$274$	−1.39459e13	−0.545528
$275$	8.96164e11	0.0343604
$276$	0	0
$277$	−7.86922e12	−0.289930	−0.144965	−	0.989437i	$-0.546307\pi$
$277$	−7.86922e12	−0.289930	−0.144965	+	0.989437i	$0.546307\pi$
$278$	5.54136e13	2.00156
$279$	0	0
$280$	7.98691e11	0.0277338
$281$	−1.85782e13	−0.632585	−0.316293	−	0.948662i	$-0.602438\pi$
$281$	−1.85782e13	−0.632585	−0.316293	+	0.948662i	$0.602438\pi$
$282$	0	0
$283$	−2.05083e13	−0.671589	−0.335795	−	0.941935i	$-0.609005\pi$
$283$	−2.05083e13	−0.671589	−0.335795	+	0.941935i	$0.609005\pi$
$284$	−1.35394e13	−0.434859
$285$	0	0
$286$	−3.46578e12	−0.107100
$287$	1.15321e13	0.349590
$288$	0	0
$289$	3.33952e13	0.974419
$290$	2.94880e13	0.844220
$291$	0	0
$292$	1.07471e13	0.296270
$293$	−6.56039e12	−0.177483	−0.0887417	−	0.996055i	$-0.528285\pi$
$293$	−6.56039e12	−0.177483	−0.0887417	+	0.996055i	$0.528285\pi$
$294$	0	0
$295$	−8.66862e12	−0.225907
$296$	1.99154e12	0.0509430
$297$	0	0
$298$	5.73878e13	1.41459
$299$	−9.29991e13	−2.25054
$300$	0	0
$301$	−1.26201e13	−0.294406
$302$	−5.87176e13	−1.34502
$303$	0	0
$304$	−3.34475e13	−0.738855
$305$	7.81116e12	0.169460
$306$	0	0
$307$	−8.74222e13	−1.82962	−0.914810	−	0.403885i	$-0.867660\pi$
$307$	−8.74222e13	−1.82962	−0.914810	+	0.403885i	$0.867660\pi$
$308$	−4.35908e11	−0.00896120
$309$	0	0
$310$	2.88733e13	0.572806
$311$	6.50721e13	1.26827	0.634136	−	0.773221i	$-0.281356\pi$
$311$	6.50721e13	1.26827	0.634136	+	0.773221i	$0.281356\pi$
$312$	0	0
$313$	−3.90600e13	−0.734916	−0.367458	−	0.930040i	$-0.619772\pi$
$313$	−3.90600e13	−0.734916	−0.367458	+	0.930040i	$0.619772\pi$
$314$	1.14170e14	2.11075
$315$	0	0
$316$	5.91932e12	0.105680
$317$	1.02156e14	1.79241	0.896205	−	0.443641i	$-0.146313\pi$
$317$	1.02156e14	1.79241	0.896205	+	0.443641i	$0.146313\pi$
$318$	0	0
$319$	3.42418e12	0.0580373
$320$	1.80462e13	0.300650
$321$	0	0
$322$	−2.58826e13	−0.416678
$323$	−5.73082e13	−0.906990
$324$	0	0
$325$	−8.88915e13	−1.35988
$326$	9.28685e13	1.39692
$327$	0	0
$328$	−2.34587e13	−0.341192
$329$	−7.18224e12	−0.102727
$330$	0	0
$331$	−5.18135e13	−0.716786	−0.358393	−	0.933571i	$-0.616675\pi$
$331$	−5.18135e13	−0.716786	−0.358393	+	0.933571i	$0.616675\pi$
$332$	−3.78007e13	−0.514328
$333$	0	0
$334$	5.19842e13	0.684330
$335$	−6.06972e13	−0.785999
$336$	0	0
$337$	−1.73666e13	−0.217646	−0.108823	−	0.994061i	$-0.534708\pi$
$337$	−1.73666e13	−0.217646	−0.108823	+	0.994061i	$0.534708\pi$
$338$	2.34222e14	2.88793
$339$	0	0
$340$	4.67874e13	0.558464
$341$	3.35281e12	0.0393785
$342$	0	0
$343$	−4.14393e13	−0.471297
$344$	2.56719e13	0.287333
$345$	0	0
$346$	−1.29331e14	−1.40212
$347$	−1.08458e14	−1.15731	−0.578654	−	0.815573i	$-0.696422\pi$
$347$	−1.08458e14	−1.15731	−0.578654	+	0.815573i	$0.696422\pi$
$348$	0	0
$349$	6.00573e13	0.620906	0.310453	−	0.950589i	$-0.399519\pi$
$349$	6.00573e13	0.620906	0.310453	+	0.950589i	$0.399519\pi$
$350$	−2.47394e13	−0.251776
$351$	0	0
$352$	5.94114e12	0.0585983
$353$	5.15538e13	0.500610	0.250305	−	0.968167i	$-0.419469\pi$
$353$	5.15538e13	0.500610	0.250305	+	0.968167i	$0.419469\pi$
$354$	0	0
$355$	2.70042e13	0.254200
$356$	3.78589e13	0.350908
$357$	0	0
$358$	5.29844e13	0.476202
$359$	1.25903e14	1.11434	0.557169	−	0.830399i	$-0.311887\pi$
$359$	1.25903e14	1.11434	0.557169	+	0.830399i	$0.311887\pi$
$360$	0	0
$361$	−6.79552e13	−0.583355
$362$	7.31351e13	0.618342
$363$	0	0
$364$	4.32382e13	0.354658
$365$	−2.14351e13	−0.173187
$366$	0	0
$367$	1.65007e14	1.29372	0.646859	−	0.762609i	$-0.276082\pi$
$367$	1.65007e14	1.29372	0.646859	+	0.762609i	$0.276082\pi$
$368$	1.88278e14	1.45424
$369$	0	0
$370$	1.86692e13	0.139964
$371$	−2.84421e13	−0.210090
$372$	0	0
$373$	−1.50728e14	−1.08093	−0.540463	−	0.841368i	$-0.681751\pi$
$373$	−1.50728e14	−1.08093	−0.540463	+	0.841368i	$0.681751\pi$
$374$	1.20220e13	0.0849535
$375$	0	0
$376$	1.46102e13	0.100259
$377$	−3.39648e14	−2.29695
$378$	0	0
$379$	1.00251e14	0.658524	0.329262	−	0.944239i	$-0.393200\pi$
$379$	1.00251e14	0.658524	0.329262	+	0.944239i	$0.393200\pi$
$380$	−3.96249e13	−0.256542
$381$	0	0
$382$	2.58920e14	1.62861
$383$	−7.83312e13	−0.485670	−0.242835	−	0.970068i	$-0.578077\pi$
$383$	−7.83312e13	−0.485670	−0.242835	+	0.970068i	$0.578077\pi$
$384$	0	0
$385$	8.69414e11	0.00523833
$386$	−4.72177e13	−0.280463
$387$	0	0
$388$	1.83890e14	1.06166
$389$	−3.44673e14	−1.96193	−0.980966	−	0.194179i	$-0.937796\pi$
$389$	−3.44673e14	−1.96193	−0.980966	+	0.194179i	$0.937796\pi$
$390$	0	0
$391$	3.22592e14	1.78518
$392$	4.08680e13	0.223002
$393$	0	0
$394$	−1.37422e14	−0.729167
$395$	−1.18060e13	−0.0617760
$396$	0	0
$397$	1.28797e14	0.655477	0.327739	−	0.944768i	$-0.393713\pi$
$397$	1.28797e14	0.655477	0.327739	+	0.944768i	$0.393713\pi$
$398$	2.72915e14	1.36984
$399$	0	0
$400$	1.79962e14	0.878722
$401$	−2.14946e14	−1.03523	−0.517613	−	0.855615i	$-0.673179\pi$
$401$	−2.14946e14	−1.03523	−0.517613	+	0.855615i	$0.673179\pi$
$402$	0	0
$403$	−3.32568e14	−1.55849
$404$	−4.72387e13	−0.218374
$405$	0	0
$406$	−9.45273e13	−0.425269
$407$	2.16789e12	0.00962206
$408$	0	0
$409$	−3.11149e14	−1.34428	−0.672140	−	0.740424i	$-0.734625\pi$
$409$	−3.11149e14	−1.34428	−0.672140	+	0.740424i	$0.734625\pi$
$410$	−2.19908e14	−0.937411
$411$	0	0
$412$	7.77868e13	0.322828
$413$	2.77883e13	0.113799
$414$	0	0
$415$	7.53930e13	0.300654
$416$	−5.89308e14	−2.31915
$417$	0	0
$418$	−1.01816e13	−0.0390252
$419$	2.34977e14	0.888889	0.444445	−	0.895806i	$-0.353401\pi$
$419$	2.34977e14	0.888889	0.444445	+	0.895806i	$0.353401\pi$
$420$	0	0
$421$	2.30362e14	0.848907	0.424454	−	0.905450i	$-0.360466\pi$
$421$	2.30362e14	0.848907	0.424454	+	0.905450i	$0.360466\pi$
$422$	1.65340e14	0.601393
$423$	0	0
$424$	5.78572e13	0.205043
$425$	3.08343e14	1.07869
$426$	0	0
$427$	−2.50396e13	−0.0853640
$428$	−4.26225e14	−1.43449
$429$	0	0
$430$	2.40654e14	0.789437
$431$	4.11119e13	0.133151	0.0665753	−	0.997781i	$-0.478793\pi$
$431$	4.11119e13	0.133151	0.0665753	+	0.997781i	$0.478793\pi$
$432$	0	0
$433$	2.71750e14	0.857999	0.428999	−	0.903305i	$-0.358866\pi$
$433$	2.71750e14	0.857999	0.428999	+	0.903305i	$0.358866\pi$
$434$	−9.25571e13	−0.288547
$435$	0	0
$436$	−4.69219e14	−1.42626
$437$	−2.73207e14	−0.820058
$438$	0	0
$439$	−4.37073e14	−1.27938	−0.639689	−	0.768634i	$-0.720937\pi$
$439$	−4.37073e14	−1.27938	−0.639689	+	0.768634i	$0.720937\pi$
$440$	−1.76857e12	−0.00511249
$441$	0	0
$442$	−1.19247e15	−3.36221
$443$	−3.50717e14	−0.976644	−0.488322	−	0.872664i	$-0.662391\pi$
$443$	−3.50717e14	−0.976644	−0.488322	+	0.872664i	$0.662391\pi$
$444$	0	0
$445$	−7.55092e13	−0.205125
$446$	4.40297e14	1.18142
$447$	0	0
$448$	−5.78493e13	−0.151450
$449$	6.70356e13	0.173361	0.0866804	−	0.996236i	$-0.472374\pi$
$449$	6.70356e13	0.173361	0.0866804	+	0.996236i	$0.472374\pi$
$450$	0	0
$451$	−2.55360e13	−0.0644439
$452$	2.91971e14	0.727912
$453$	0	0
$454$	−1.97041e14	−0.479457
$455$	−8.62381e13	−0.207318
$456$	0	0
$457$	1.28524e14	0.301609	0.150804	−	0.988564i	$-0.451814\pi$
$457$	1.28524e14	0.301609	0.150804	+	0.988564i	$0.451814\pi$
$458$	−8.94956e14	−2.07511
$459$	0	0
$460$	2.23051e14	0.504936
$461$	2.49143e14	0.557306	0.278653	−	0.960392i	$-0.410112\pi$
$461$	2.49143e14	0.557306	0.278653	+	0.960392i	$0.410112\pi$
$462$	0	0
$463$	2.82966e14	0.618071	0.309036	−	0.951050i	$-0.399994\pi$
$463$	2.82966e14	0.618071	0.309036	+	0.951050i	$0.399994\pi$
$464$	6.87622e14	1.48423
$465$	0	0
$466$	1.82082e14	0.383835
$467$	−6.12139e14	−1.27529	−0.637643	−	0.770332i	$-0.720090\pi$
$467$	−6.12139e14	−1.27529	−0.637643	+	0.770332i	$0.720090\pi$
$468$	0	0
$469$	1.94572e14	0.395941
$470$	1.36959e14	0.275457
$471$	0	0
$472$	−5.65273e13	−0.111065
$473$	2.79451e13	0.0542712
$474$	0	0
$475$	−2.61140e14	−0.495517
$476$	−1.49983e14	−0.281322
$477$	0	0
$478$	−5.69564e14	−1.04397
$479$	−3.14429e14	−0.569741	−0.284870	−	0.958566i	$-0.591951\pi$
$479$	−3.14429e14	−0.569741	−0.284870	+	0.958566i	$0.591951\pi$
$480$	0	0
$481$	−2.15035e14	−0.380813
$482$	−2.77968e13	−0.0486672
$483$	0	0
$484$	−4.80842e14	−0.822912
$485$	−3.66767e14	−0.620599
$486$	0	0
$487$	−3.34356e14	−0.553095	−0.276548	−	0.961000i	$-0.589190\pi$
$487$	−3.34356e14	−0.553095	−0.276548	+	0.961000i	$0.589190\pi$
$488$	5.09358e13	0.0833133
$489$	0	0
$490$	3.83106e14	0.612690
$491$	−1.06996e15	−1.69207	−0.846033	−	0.533130i	$-0.821015\pi$
$491$	−1.06996e15	−1.69207	−0.846033	+	0.533130i	$0.821015\pi$
$492$	0	0
$493$	1.17816e15	1.82198
$494$	1.00992e15	1.54450
$495$	0	0
$496$	6.73290e14	1.00705
$497$	−8.65652e13	−0.128051
$498$	0	0
$499$	5.42607e14	0.785114	0.392557	−	0.919728i	$-0.371591\pi$
$499$	5.42607e14	0.785114	0.392557	+	0.919728i	$0.371591\pi$
$500$	4.90921e14	0.702549
$501$	0	0
$502$	−7.50233e14	−1.05033
$503$	1.33845e15	1.85344	0.926720	−	0.375753i	$-0.122616\pi$
$503$	1.33845e15	1.85344	0.926720	+	0.375753i	$0.122616\pi$
$504$	0	0
$505$	9.42171e13	0.127652
$506$	5.73127e13	0.0768109
$507$	0	0
$508$	−4.70407e14	−0.616912
$509$	−9.30841e13	−0.120761	−0.0603806	−	0.998175i	$-0.519231\pi$
$509$	−9.30841e13	−0.120761	−0.0603806	+	0.998175i	$0.519231\pi$
$510$	0	0
$511$	6.87128e13	0.0872414
$512$	9.77110e14	1.22732
$513$	0	0
$514$	6.39285e14	0.785955
$515$	−1.55145e14	−0.188711
$516$	0	0
$517$	1.59039e13	0.0189368
$518$	−5.98465e13	−0.0705058
$519$	0	0
$520$	1.75426e14	0.202337
$521$	2.45282e14	0.279935	0.139968	−	0.990156i	$-0.455300\pi$
$521$	2.45282e14	0.279935	0.139968	+	0.990156i	$0.455300\pi$
$522$	0	0
$523$	9.73813e14	1.08822	0.544109	−	0.839014i	$-0.316868\pi$
$523$	9.73813e14	1.08822	0.544109	+	0.839014i	$0.316868\pi$
$524$	−3.05142e14	−0.337427
$525$	0	0
$526$	−6.60006e14	−0.714704
$527$	1.15360e15	1.23622
$528$	0	0
$529$	5.85092e14	0.614070
$530$	5.42368e14	0.563348
$531$	0	0
$532$	1.27023e14	0.129231
$533$	2.53294e15	2.55050
$534$	0	0
$535$	8.50101e14	0.838542
$536$	−3.95801e14	−0.386429
$537$	0	0
$538$	−1.99676e15	−1.90996
$539$	4.44868e13	0.0421204
$540$	0	0
$541$	9.56240e14	0.887119	0.443560	−	0.896245i	$-0.353715\pi$
$541$	9.56240e14	0.887119	0.443560	+	0.896245i	$0.353715\pi$
$542$	4.37428e14	0.401708
$543$	0	0
$544$	2.04417e15	1.83960
$545$	9.35852e14	0.833732
$546$	0	0
$547$	1.03639e15	0.904881	0.452440	−	0.891795i	$-0.350554\pi$
$547$	1.03639e15	0.904881	0.452440	+	0.891795i	$0.350554\pi$
$548$	−3.85261e14	−0.333013
$549$	0	0
$550$	5.47813e13	0.0464128
$551$	−9.97797e14	−0.836967
$552$	0	0
$553$	3.78457e13	0.0311192
$554$	−4.81034e14	−0.391627
$555$	0	0
$556$	1.53083e15	1.22184
$557$	−8.16738e14	−0.645475	−0.322737	−	0.946489i	$-0.604603\pi$
$557$	−8.16738e14	−0.645475	−0.322737	+	0.946489i	$0.604603\pi$
$558$	0	0
$559$	−2.77190e15	−2.14789
$560$	1.74590e14	0.133963
$561$	0	0
$562$	−1.13566e15	−0.854474
$563$	2.06254e15	1.53676	0.768382	−	0.639992i	$-0.221062\pi$
$563$	2.06254e15	1.53676	0.768382	+	0.639992i	$0.221062\pi$
$564$	0	0
$565$	−5.82334e14	−0.425506
$566$	−1.25364e15	−0.907159
$567$	0	0
$568$	1.76092e14	0.124975
$569$	−4.97519e14	−0.349697	−0.174849	−	0.984595i	$-0.555944\pi$
$569$	−4.97519e14	−0.349697	−0.174849	+	0.984595i	$0.555944\pi$
$570$	0	0
$571$	5.65558e14	0.389923	0.194961	−	0.980811i	$-0.437542\pi$
$571$	5.65558e14	0.389923	0.194961	+	0.980811i	$0.437542\pi$
$572$	−9.57438e13	−0.0653781
$573$	0	0
$574$	7.04943e14	0.472214
$575$	1.46997e15	0.975297
$576$	0	0
$577$	−2.22207e14	−0.144641	−0.0723205	−	0.997381i	$-0.523040\pi$
$577$	−2.22207e14	−0.144641	−0.0723205	+	0.997381i	$0.523040\pi$
$578$	2.04140e15	1.31621
$579$	0	0
$580$	8.14619e14	0.515348
$581$	−2.41682e14	−0.151452
$582$	0	0
$583$	6.29804e13	0.0387283
$584$	−1.39776e14	−0.0851456
$585$	0	0
$586$	−4.01027e14	−0.239738
$587$	−2.31383e15	−1.37032	−0.685159	−	0.728393i	$-0.740267\pi$
$587$	−2.31383e15	−1.37032	−0.685159	+	0.728393i	$0.740267\pi$
$588$	0	0
$589$	−9.77000e14	−0.567885
$590$	−5.29901e14	−0.305147
$591$	0	0
$592$	4.35342e14	0.246072
$593$	−1.19808e15	−0.670940	−0.335470	−	0.942051i	$-0.608895\pi$
$593$	−1.19808e15	−0.670940	−0.335470	+	0.942051i	$0.608895\pi$
$594$	0	0
$595$	2.99139e14	0.164449
$596$	1.58536e15	0.863524
$597$	0	0
$598$	−5.68491e15	−3.03995
$599$	2.04717e13	0.0108469	0.00542345	−	0.999985i	$-0.498274\pi$
$599$	2.04717e13	0.0108469	0.00542345	+	0.999985i	$0.498274\pi$
$600$	0	0
$601$	−9.23345e14	−0.480346	−0.240173	−	0.970730i	$-0.577204\pi$
$601$	−9.23345e14	−0.480346	−0.240173	+	0.970730i	$0.577204\pi$
$602$	−7.71448e14	−0.397673
$603$	0	0
$604$	−1.62210e15	−0.821060
$605$	9.59035e14	0.481039
$606$	0	0
$607$	−1.53032e15	−0.753780	−0.376890	−	0.926258i	$-0.623007\pi$
$607$	−1.53032e15	−0.753780	−0.376890	+	0.926258i	$0.623007\pi$
$608$	−1.73123e15	−0.845057
$609$	0	0
$610$	4.77485e14	0.228900
$611$	−1.57752e15	−0.749461
$612$	0	0
$613$	−1.36125e15	−0.635194	−0.317597	−	0.948226i	$-0.602876\pi$
$613$	−1.36125e15	−0.635194	−0.317597	+	0.948226i	$0.602876\pi$
$614$	−5.34400e15	−2.47138
$615$	0	0
$616$	5.66937e12	0.00257538
$617$	−6.20749e14	−0.279478	−0.139739	−	0.990188i	$-0.544626\pi$
$617$	−6.20749e14	−0.279478	−0.139739	+	0.990188i	$0.544626\pi$
$618$	0	0
$619$	−3.74605e15	−1.65682	−0.828408	−	0.560125i	$-0.810753\pi$
$619$	−3.74605e15	−1.65682	−0.828408	+	0.560125i	$0.810753\pi$
$620$	7.97640e14	0.349665
$621$	0	0
$622$	3.97777e15	1.71314
$623$	2.42054e14	0.103330
$624$	0	0
$625$	8.51135e14	0.356992
$626$	−2.38768e15	−0.992699
$627$	0	0
$628$	3.15399e15	1.28849
$629$	7.45906e14	0.302068
$630$	0	0
$631$	−2.91198e15	−1.15885	−0.579424	−	0.815026i	$-0.696723\pi$
$631$	−2.91198e15	−1.15885	−0.579424	+	0.815026i	$0.696723\pi$
$632$	−7.69860e13	−0.0303716
$633$	0	0
$634$	6.24464e15	2.42112
$635$	9.38221e14	0.360620
$636$	0	0
$637$	−4.41269e15	−1.66700
$638$	2.09315e14	0.0783948
$639$	0	0
$640$	−6.10991e14	−0.224929
$641$	1.82042e15	0.664435	0.332218	−	0.943203i	$-0.392203\pi$
$641$	1.82042e15	0.664435	0.332218	+	0.943203i	$0.392203\pi$
$642$	0	0
$643$	−8.04673e14	−0.288708	−0.144354	−	0.989526i	$-0.546110\pi$
$643$	−8.04673e14	−0.288708	−0.144354	+	0.989526i	$0.546110\pi$
$644$	−7.15018e14	−0.254358
$645$	0	0
$646$	−3.50317e15	−1.22513
$647$	4.39292e15	1.52328	0.761641	−	0.648000i	$-0.224394\pi$
$647$	4.39292e15	1.52328	0.761641	+	0.648000i	$0.224394\pi$
$648$	0	0
$649$	−6.15327e13	−0.0209778
$650$	−5.43381e15	−1.83688
$651$	0	0
$652$	2.56554e15	0.852739
$653$	2.14802e15	0.707972	0.353986	−	0.935251i	$-0.384826\pi$
$653$	2.14802e15	0.707972	0.353986	+	0.935251i	$0.384826\pi$
$654$	0	0
$655$	6.08603e14	0.197245
$656$	−5.12798e15	−1.64807
$657$	0	0
$658$	−4.39041e14	−0.138759
$659$	3.77504e15	1.18318	0.591591	−	0.806238i	$-0.298500\pi$
$659$	3.77504e15	1.18318	0.591591	+	0.806238i	$0.298500\pi$
$660$	0	0
$661$	3.46636e14	0.106848	0.0534240	−	0.998572i	$-0.482987\pi$
$661$	3.46636e14	0.106848	0.0534240	+	0.998572i	$0.482987\pi$
$662$	−3.16729e15	−0.968208
$663$	0	0
$664$	4.91631e14	0.147814
$665$	−2.53345e14	−0.0755429
$666$	0	0
$667$	5.61666e15	1.64735
$668$	1.43609e15	0.417744
$669$	0	0
$670$	−3.71034e15	−1.06170
$671$	5.54462e13	0.0157361
$672$	0	0
$673$	6.21195e15	1.73438	0.867191	−	0.497975i	$-0.165923\pi$
$673$	6.21195e15	1.73438	0.867191	+	0.497975i	$0.165923\pi$
$674$	−1.06160e15	−0.293988
$675$	0	0
$676$	6.47050e15	1.76291
$677$	−2.69108e15	−0.727257	−0.363629	−	0.931544i	$-0.618462\pi$
$677$	−2.69108e15	−0.727257	−0.363629	+	0.931544i	$0.618462\pi$
$678$	0	0
$679$	1.17572e15	0.312622
$680$	−6.08511e14	−0.160498
$681$	0	0
$682$	2.04952e14	0.0531911
$683$	−1.01372e15	−0.260978	−0.130489	−	0.991450i	$-0.541655\pi$
$683$	−1.01372e15	−0.260978	−0.130489	+	0.991450i	$0.541655\pi$
$684$	0	0
$685$	7.68399e14	0.194665
$686$	−2.53313e15	−0.636612
$687$	0	0
$688$	5.61176e15	1.38791
$689$	−6.24709e15	−1.53275
$690$	0	0
$691$	4.21887e15	1.01875	0.509374	−	0.860545i	$-0.329877\pi$
$691$	4.21887e15	1.01875	0.509374	+	0.860545i	$0.329877\pi$
$692$	−3.57284e15	−0.855913
$693$	0	0
$694$	−6.62988e15	−1.56325
$695$	−3.05322e15	−0.714235
$696$	0	0
$697$	−8.78617e15	−2.02311
$698$	3.67122e15	0.838697
$699$	0	0
$700$	−6.83437e14	−0.153695
$701$	5.50609e15	1.22855	0.614277	−	0.789090i	$-0.289448\pi$
$701$	5.50609e15	1.22855	0.614277	+	0.789090i	$0.289448\pi$
$702$	0	0
$703$	−6.31718e14	−0.138761
$704$	1.28098e14	0.0279185
$705$	0	0
$706$	3.15141e15	0.676207
$707$	−3.02025e14	−0.0643037
$708$	0	0
$709$	9.90050e14	0.207540	0.103770	−	0.994601i	$-0.466909\pi$
$709$	9.90050e14	0.207540	0.103770	+	0.994601i	$0.466909\pi$
$710$	1.65073e15	0.343364
$711$	0	0
$712$	−4.92388e14	−0.100848
$713$	5.49960e15	1.11773
$714$	0	0
$715$	1.90960e14	0.0382172
$716$	1.46372e15	0.290694
$717$	0	0
$718$	7.69629e15	1.50521
$719$	−5.12704e15	−0.995079	−0.497539	−	0.867441i	$-0.665763\pi$
$719$	−5.12704e15	−0.995079	−0.497539	+	0.867441i	$0.665763\pi$
$720$	0	0
$721$	4.97337e14	0.0950619
$722$	−4.15400e15	−0.787975
$723$	0	0
$724$	2.02039e15	0.377462
$725$	5.36858e15	0.995407
$726$	0	0
$727$	−5.38362e15	−0.983184	−0.491592	−	0.870826i	$-0.663585\pi$
$727$	−5.38362e15	−0.983184	−0.491592	+	0.870826i	$0.663585\pi$
$728$	−5.62351e14	−0.101926
$729$	0	0
$730$	−1.31030e15	−0.233934
$731$	9.61506e15	1.70375
$732$	0	0
$733$	−5.27269e14	−0.0920366	−0.0460183	−	0.998941i	$-0.514653\pi$
$733$	−5.27269e14	−0.0920366	−0.0460183	+	0.998941i	$0.514653\pi$
$734$	1.00867e16	1.74751
$735$	0	0
$736$	9.74523e15	1.66327
$737$	−4.30849e14	−0.0729884
$738$	0	0
$739$	−4.81946e15	−0.804366	−0.402183	−	0.915559i	$-0.631749\pi$
$739$	−4.81946e15	−0.804366	−0.402183	+	0.915559i	$0.631749\pi$
$740$	5.15745e14	0.0854399
$741$	0	0
$742$	−1.73863e15	−0.283782
$743$	−1.16330e16	−1.88474	−0.942370	−	0.334572i	$-0.891408\pi$
$743$	−1.16330e16	−1.88474	−0.942370	+	0.334572i	$0.891408\pi$
$744$	0	0
$745$	−3.16199e15	−0.504779
$746$	−9.21381e15	−1.46008
$747$	0	0
$748$	3.32112e14	0.0518592
$749$	−2.72510e15	−0.422409
$750$	0	0
$751$	−3.03780e14	−0.0464023	−0.0232012	−	0.999731i	$-0.507386\pi$
$751$	−3.03780e14	−0.0464023	−0.0232012	+	0.999731i	$0.507386\pi$
$752$	3.19372e15	0.484283
$753$	0	0
$754$	−2.07622e16	−3.10264
$755$	3.23527e15	0.479956
$756$	0	0
$757$	9.51119e14	0.139062	0.0695308	−	0.997580i	$-0.477850\pi$
$757$	9.51119e14	0.139062	0.0695308	+	0.997580i	$0.477850\pi$
$758$	6.12819e15	0.889511
$759$	0	0
$760$	5.15357e14	0.0737281
$761$	7.46397e15	1.06012	0.530059	−	0.847961i	$-0.322170\pi$
$761$	7.46397e15	1.06012	0.530059	+	0.847961i	$0.322170\pi$
$762$	0	0
$763$	−2.99999e15	−0.419986
$764$	7.15278e15	0.994172
$765$	0	0
$766$	−4.78828e15	−0.656026
$767$	6.10349e15	0.830241
$768$	0	0
$769$	1.26726e16	1.69930	0.849650	−	0.527348i	$-0.176813\pi$
$769$	1.26726e16	1.69930	0.849650	+	0.527348i	$0.176813\pi$
$770$	5.31461e13	0.00707575
$771$	0	0
$772$	−1.30441e15	−0.171206
$773$	8.00391e15	1.04307	0.521537	−	0.853229i	$-0.325359\pi$
$773$	8.00391e15	1.04307	0.521537	+	0.853229i	$0.325359\pi$
$774$	0	0
$775$	5.25669e15	0.675387
$776$	−2.39165e15	−0.305112
$777$	0	0
$778$	−2.10694e16	−2.65011
$779$	7.44113e15	0.929357
$780$	0	0
$781$	1.91684e14	0.0236051
$782$	1.97196e16	2.41135
$783$	0	0
$784$	8.93356e15	1.07718
$785$	−6.29060e15	−0.753198
$786$	0	0
$787$	−8.83933e15	−1.04366	−0.521829	−	0.853050i	$-0.674750\pi$
$787$	−8.83933e15	−1.04366	−0.521829	+	0.853050i	$0.674750\pi$
$788$	−3.79635e15	−0.445115
$789$	0	0
$790$	−7.21686e14	−0.0834448
$791$	1.86674e15	0.214345
$792$	0	0
$793$	−5.49976e15	−0.622790
$794$	7.87318e15	0.885395
$795$	0	0
$796$	7.53940e15	0.836209
$797$	2.43818e15	0.268563	0.134281	−	0.990943i	$-0.457127\pi$
$797$	2.43818e15	0.268563	0.134281	+	0.990943i	$0.457127\pi$
$798$	0	0
$799$	5.47205e15	0.594488
$800$	9.31480e15	1.00503
$801$	0	0
$802$	−1.31393e16	−1.39835
$803$	−1.52153e14	−0.0160822
$804$	0	0
$805$	1.42610e15	0.148687
$806$	−2.03294e16	−2.10515
$807$	0	0
$808$	6.14381e14	0.0627589
$809$	1.43706e16	1.45800	0.729001	−	0.684513i	$-0.239985\pi$
$809$	1.43706e16	1.45800	0.729001	+	0.684513i	$0.239985\pi$
$810$	0	0
$811$	1.77414e16	1.77572	0.887858	−	0.460117i	$-0.152193\pi$
$811$	1.77414e16	1.77572	0.887858	+	0.460117i	$0.152193\pi$
$812$	−2.61136e15	−0.259602
$813$	0	0
$814$	1.32520e14	0.0129971
$815$	−5.11693e15	−0.498475
$816$	0	0
$817$	−8.14313e15	−0.782654
$818$	−1.90201e16	−1.81581
$819$	0	0
$820$	−6.07507e15	−0.572235
$821$	−3.97601e15	−0.372015	−0.186007	−	0.982548i	$-0.559555\pi$
$821$	−3.97601e15	−0.372015	−0.186007	+	0.982548i	$0.559555\pi$
$822$	0	0
$823$	2.27431e15	0.209967	0.104983	−	0.994474i	$-0.466521\pi$
$823$	2.27431e15	0.209967	0.104983	+	0.994474i	$0.466521\pi$
$824$	−1.01169e15	−0.0927783
$825$	0	0
$826$	1.69866e15	0.153715
$827$	−2.06363e16	−1.85503	−0.927516	−	0.373783i	$-0.878061\pi$
$827$	−2.06363e16	−1.85503	−0.927516	+	0.373783i	$0.878061\pi$
$828$	0	0
$829$	−1.41555e16	−1.25567	−0.627834	−	0.778348i	$-0.716058\pi$
$829$	−1.41555e16	−1.25567	−0.627834	+	0.778348i	$0.716058\pi$
$830$	4.60867e15	0.406112
$831$	0	0
$832$	−1.27062e16	−1.10493
$833$	1.53066e16	1.32230
$834$	0	0
$835$	−2.86426e15	−0.244195
$836$	−2.81270e14	−0.0238226
$837$	0	0
$838$	1.43638e16	1.20068
$839$	−1.59310e16	−1.32298	−0.661490	−	0.749954i	$-0.730076\pi$
$839$	−1.59310e16	−1.32298	−0.661490	+	0.749954i	$0.730076\pi$
$840$	0	0
$841$	8.31245e15	0.681320
$842$	1.40817e16	1.14667
$843$	0	0
$844$	4.56758e15	0.367116
$845$	−1.29053e16	−1.03052
$846$	0	0
$847$	−3.07431e15	−0.242320
$848$	1.26473e16	0.990426
$849$	0	0
$850$	1.88486e16	1.45705
$851$	3.55598e15	0.273116
$852$	0	0
$853$	3.14749e14	0.0238641	0.0119320	−	0.999929i	$-0.496202\pi$
$853$	3.14749e14	0.0238641	0.0119320	+	0.999929i	$0.496202\pi$
$854$	−1.53064e15	−0.115307
$855$	0	0
$856$	5.54343e15	0.412261
$857$	1.30216e15	0.0962210	0.0481105	−	0.998842i	$-0.484680\pi$
$857$	1.30216e15	0.0962210	0.0481105	+	0.998842i	$0.484680\pi$
$858$	0	0
$859$	5.97763e15	0.436081	0.218040	−	0.975940i	$-0.430034\pi$
$859$	5.97763e15	0.436081	0.218040	+	0.975940i	$0.430034\pi$
$860$	6.64819e15	0.481906
$861$	0	0
$862$	2.51312e15	0.179855
$863$	−2.42290e16	−1.72296	−0.861480	−	0.507791i	$-0.830462\pi$
$863$	−2.42290e16	−1.72296	−0.861480	+	0.507791i	$0.830462\pi$
$864$	0	0
$865$	7.12598e15	0.500330
$866$	1.66117e16	1.15895
$867$	0	0
$868$	−2.55693e15	−0.176141
$869$	−8.38030e13	−0.00573655
$870$	0	0
$871$	4.27363e16	2.88867
$872$	6.10260e15	0.409897
$873$	0	0
$874$	−1.67008e16	−1.10770
$875$	3.13875e15	0.206877
$876$	0	0
$877$	1.42759e16	0.929193	0.464596	−	0.885523i	$-0.346199\pi$
$877$	1.42759e16	0.929193	0.464596	+	0.885523i	$0.346199\pi$
$878$	−2.67177e16	−1.72814
$879$	0	0
$880$	−3.86602e14	−0.0246950
$881$	6.74661e15	0.428271	0.214135	−	0.976804i	$-0.431307\pi$
$881$	6.74661e15	0.428271	0.214135	+	0.976804i	$0.431307\pi$
$882$	0	0
$883$	−5.83016e15	−0.365508	−0.182754	−	0.983159i	$-0.558501\pi$
$883$	−5.83016e15	−0.365508	−0.182754	+	0.983159i	$0.558501\pi$
$884$	−3.29426e16	−2.05244
$885$	0	0
$886$	−2.14388e16	−1.31922
$887$	−1.08682e16	−0.664625	−0.332312	−	0.943169i	$-0.607829\pi$
$887$	−1.08682e16	−0.664625	−0.332312	+	0.943169i	$0.607829\pi$
$888$	0	0
$889$	−3.00758e15	−0.181660
$890$	−4.61577e15	−0.277076
$891$	0	0
$892$	1.21634e16	0.721188
$893$	−4.63436e15	−0.273090
$894$	0	0
$895$	−2.91937e15	−0.169927
$896$	1.95861e15	0.113306
$897$	0	0
$898$	4.09779e15	0.234170
$899$	2.00854e16	1.14078
$900$	0	0
$901$	2.16696e16	1.21581
$902$	−1.56098e15	−0.0870485
$903$	0	0
$904$	−3.79735e15	−0.209196
$905$	−4.02965e15	−0.220648
$906$	0	0
$907$	2.17653e16	1.17740	0.588701	−	0.808351i	$-0.299640\pi$
$907$	2.17653e16	1.17740	0.588701	+	0.808351i	$0.299640\pi$
$908$	−5.44336e15	−0.292681
$909$	0	0
$910$	−5.27161e15	−0.280037
$911$	2.21539e14	0.0116977	0.00584883	−	0.999983i	$-0.498138\pi$
$911$	2.21539e14	0.0116977	0.00584883	+	0.999983i	$0.498138\pi$
$912$	0	0
$913$	5.35164e14	0.0279189
$914$	7.85647e15	0.407402
$915$	0	0
$916$	−2.47236e16	−1.26674
$917$	−1.95095e15	−0.0993609
$918$	0	0
$919$	−2.25201e15	−0.113327	−0.0566637	−	0.998393i	$-0.518046\pi$
$919$	−2.25201e15	−0.113327	−0.0566637	+	0.998393i	$0.518046\pi$
$920$	−2.90098e15	−0.145115
$921$	0	0
$922$	1.52298e16	0.752788
$923$	−1.90134e16	−0.934222
$924$	0	0
$925$	3.39892e15	0.165029
$926$	1.72973e16	0.834869
$927$	0	0
$928$	3.55911e16	1.69757
$929$	−2.48808e16	−1.17972	−0.589860	−	0.807506i	$-0.700817\pi$
$929$	−2.48808e16	−1.17972	−0.589860	+	0.807506i	$0.700817\pi$
$930$	0	0
$931$	−1.29633e16	−0.607426
$932$	5.03011e15	0.234309
$933$	0	0
$934$	−3.74192e16	−1.72261
$935$	−6.62395e14	−0.0303147
$936$	0	0
$937$	−3.66946e16	−1.65972	−0.829859	−	0.557974i	$-0.811579\pi$
$937$	−3.66946e16	−1.65972	−0.829859	+	0.557974i	$0.811579\pi$
$938$	1.18939e16	0.534823
$939$	0	0
$940$	3.78357e15	0.168151
$941$	−1.86688e16	−0.824846	−0.412423	−	0.910992i	$-0.635317\pi$
$941$	−1.86688e16	−0.824846	−0.412423	+	0.910992i	$0.635317\pi$
$942$	0	0
$943$	−4.18866e16	−1.82920
$944$	−1.23566e16	−0.536480
$945$	0	0
$946$	1.70824e15	0.0733075
$947$	7.29010e15	0.311034	0.155517	−	0.987833i	$-0.450296\pi$
$947$	7.29010e15	0.311034	0.155517	+	0.987833i	$0.450296\pi$
$948$	0	0
$949$	1.50922e16	0.636487
$950$	−1.59631e16	−0.669327
$951$	0	0
$952$	1.95066e15	0.0808496
$953$	−2.29869e16	−0.947261	−0.473630	−	0.880724i	$-0.657057\pi$
$953$	−2.29869e16	−0.947261	−0.473630	+	0.880724i	$0.657057\pi$
$954$	0	0
$955$	−1.42662e16	−0.581150
$956$	−1.57345e16	−0.637285
$957$	0	0
$958$	−1.92206e16	−0.769585
$959$	−2.46320e15	−0.0980612
$960$	0	0
$961$	−5.74171e15	−0.225976
$962$	−1.31448e16	−0.514389
$963$	0	0
$964$	−7.67900e14	−0.0297085
$965$	2.60164e15	0.100080
$966$	0	0
$967$	−1.95063e16	−0.741872	−0.370936	−	0.928658i	$-0.620963\pi$
$967$	−1.95063e16	−0.741872	−0.370936	+	0.928658i	$0.620963\pi$
$968$	6.25378e15	0.236498
$969$	0	0
$970$	−2.24200e16	−0.838282
$971$	−2.10598e16	−0.782976	−0.391488	−	0.920183i	$-0.628040\pi$
$971$	−2.10598e16	−0.782976	−0.391488	+	0.920183i	$0.628040\pi$
$972$	0	0
$973$	9.78746e15	0.359790
$974$	−2.04387e16	−0.747101
$975$	0	0
$976$	1.11344e16	0.402431
$977$	−1.89495e16	−0.681049	−0.340524	−	0.940236i	$-0.610605\pi$
$977$	−1.89495e16	−0.681049	−0.340524	+	0.940236i	$0.610605\pi$
$978$	0	0
$979$	−5.35989e14	−0.0190481
$980$	1.05835e16	0.374012
$981$	0	0
$982$	−6.54049e16	−2.28558
$983$	−1.35763e15	−0.0471779	−0.0235889	−	0.999722i	$-0.507509\pi$
$983$	−1.35763e15	−0.0471779	−0.0235889	+	0.999722i	$0.507509\pi$
$984$	0	0
$985$	7.57178e15	0.260195
$986$	7.20190e16	2.46107
$987$	0	0
$988$	2.78995e16	0.942830
$989$	4.58382e16	1.54045
$990$	0	0
$991$	1.54216e16	0.512537	0.256268	−	0.966606i	$-0.417507\pi$
$991$	1.54216e16	0.512537	0.256268	+	0.966606i	$0.417507\pi$
$992$	3.48493e16	1.15181
$993$	0	0
$994$	−5.29161e15	−0.172967
$995$	−1.50373e16	−0.488812
$996$	0	0
$997$	−3.69625e16	−1.18833	−0.594166	−	0.804342i	$-0.702518\pi$
$997$	−3.69625e16	−1.18833	−0.594166	+	0.804342i	$0.702518\pi$
$998$	3.31688e16	1.06050
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.12.a.c.1.9		10
3.2	odd	2		81.12.a.e.1.2		10
9.2	odd	6		9.12.c.a.4.9	✓	20
9.4	even	3		27.12.c.a.19.2		20
9.5	odd	6		9.12.c.a.7.9	yes	20
9.7	even	3		27.12.c.a.10.2		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.12.c.a.4.9	✓	20	9.2	odd	6
9.12.c.a.7.9	yes	20	9.5	odd	6
27.12.c.a.10.2		20	9.7	even	3
27.12.c.a.19.2		20	9.4	even	3
81.12.a.c.1.9		10	1.1	even	1	trivial
81.12.a.e.1.2		10	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:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	81.a (trivial)

\(f(q)\)	\(=\)	\(q+61.1286 q^{2} +1688.71 q^{4} -3368.11 q^{5} +10796.9 q^{7} -21963.1 q^{8} -205888. q^{10} -23907.9 q^{11} +2.37145e6 q^{13} +659999. q^{14} -4.80105e6 q^{16} -8.22600e6 q^{17} +6.96671e6 q^{19} -5.68775e6 q^{20} -1.46146e6 q^{22} -3.92161e7 q^{23} -3.74840e7 q^{25} +1.44964e8 q^{26} +1.82328e7 q^{28} -1.43223e8 q^{29} -1.40238e8 q^{31} -2.48501e8 q^{32} -5.02844e8 q^{34} -3.63651e7 q^{35} -9.06766e7 q^{37} +4.25865e8 q^{38} +7.39741e7 q^{40} +1.06810e9 q^{41} -1.16886e9 q^{43} -4.03735e7 q^{44} -2.39723e9 q^{46} -6.65214e8 q^{47} -1.86075e9 q^{49} -2.29134e9 q^{50} +4.00469e9 q^{52} -2.63429e9 q^{53} +8.05245e7 q^{55} -2.37133e8 q^{56} -8.75505e9 q^{58} +2.57374e9 q^{59} -2.31915e9 q^{61} -8.57257e9 q^{62} -5.35796e9 q^{64} -7.98731e9 q^{65} +1.80212e10 q^{67} -1.38913e10 q^{68} -2.22295e9 q^{70} -8.01761e9 q^{71} +6.36413e9 q^{73} -5.54293e9 q^{74} +1.17647e10 q^{76} -2.58131e8 q^{77} +3.50524e9 q^{79} +1.61704e10 q^{80} +6.52913e10 q^{82} -2.23844e10 q^{83} +2.77061e10 q^{85} -7.14509e10 q^{86} +5.25093e8 q^{88} +2.24189e10 q^{89} +2.56043e10 q^{91} -6.62245e10 q^{92} -4.06636e10 q^{94} -2.34646e10 q^{95} +1.08894e11 q^{97} -1.13745e11 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 33 q^{2} + 9217 q^{4} - 7230 q^{5} - 8512 q^{7} + 14559 q^{8} + 2046 q^{10} - 112776 q^{11} - 279706 q^{13} - 3901584 q^{14} + 7342081 q^{16} - 13882896 q^{17} + 3514700 q^{19} - 34163508 q^{20}+ \cdots - 655061802039 q^{98}+O(q^{100}) \) 10 * q - 33 * q^2 + 9217 * q^4 - 7230 * q^5 - 8512 * q^7 + 14559 * q^8 + 2046 * q^10 - 112776 * q^11 - 279706 * q^13 - 3901584 * q^14 + 7342081 * q^16 - 13882896 * q^17 + 3514700 * q^19 - 34163508 * q^20 - 2274591 * q^22 - 69371616 * q^23 + 45286204 * q^25 - 240964572 * q^26 - 30672898 * q^28 - 25437246 * q^29 - 114575368 * q^31 + 80396559 * q^32 + 243855063 * q^34 + 89073732 * q^35 - 67109164 * q^37 + 489799995 * q^38 - 107425416 * q^40 + 331873026 * q^41 + 1118847584 * q^43 - 139238637 * q^44 + 1441268796 * q^46 - 1469650704 * q^47 + 3553434720 * q^49 - 6643771701 * q^50 - 3632448874 * q^52 - 7457130972 * q^53 + 2490724992 * q^55 - 27669139026 * q^56 + 1387480560 * q^58 - 26505032592 * q^59 - 990409066 * q^61 - 45522498090 * q^62 - 3758354783 * q^64 - 39045315390 * q^65 - 6557215720 * q^67 - 77299152993 * q^68 + 785437278 * q^70 - 61026859872 * q^71 - 6306446968 * q^73 - 109519086216 * q^74 - 14574055597 * q^76 - 88616208018 * q^77 + 7621233248 * q^79 - 199583341536 * q^80 - 29584238667 * q^82 - 99007044180 * q^83 - 12911595156 * q^85 - 214357830519 * q^86 + 54423523605 * q^88 - 238298852412 * q^89 + 69105826108 * q^91 - 461776423998 * q^92 - 13393667064 * q^94 - 418952909328 * q^95 - 123483551938 * q^97 - 655061802039 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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