LMFDB - Embedded newform 25.12.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [25,12,Mod(1,25)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(25, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("25.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 25 = 5^{2} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	25.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$19.2085795140$
Analytic rank:	$1$
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} - 415x^{2} + 3184 $ x^4 - 415*x^2 + 3184
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 5)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$20.1787$ of defining polynomial
Character	$\chi$	$=$	25.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-27.1071 q^{2} -524.040 q^{3} -1313.20 q^{4} +14205.2 q^{6} +66589.5 q^{7} +91112.6 q^{8} +97470.8 q^{9} +O(q^{10})$	\(q-27.1071 q^{2} -524.040 q^{3} -1313.20 q^{4} +14205.2 q^{6} +66589.5 q^{7} +91112.6 q^{8} +97470.8 q^{9} -374453. q^{11} +688170. q^{12} +1.12751e6 q^{13} -1.80505e6 q^{14} +219635. q^{16} -5.82894e6 q^{17} -2.64216e6 q^{18} +6.03360e6 q^{19} -3.48955e7 q^{21} +1.01503e7 q^{22} +7.51593e6 q^{23} -4.77466e7 q^{24} -3.05636e7 q^{26} +4.17535e7 q^{27} -8.74454e7 q^{28} -4.87039e7 q^{29} -2.71596e8 q^{31} -1.92552e8 q^{32} +1.96228e8 q^{33} +1.58006e8 q^{34} -1.27999e8 q^{36} +542545. q^{37} -1.63554e8 q^{38} -5.90860e8 q^{39} +8.20963e8 q^{41} +9.45918e8 q^{42} +6.03499e8 q^{43} +4.91732e8 q^{44} -2.03735e8 q^{46} -4.69204e8 q^{47} -1.15098e8 q^{48} +2.45683e9 q^{49} +3.05460e9 q^{51} -1.48065e9 q^{52} -3.82491e9 q^{53} -1.13182e9 q^{54} +6.06714e9 q^{56} -3.16185e9 q^{57} +1.32022e9 q^{58} -2.19696e9 q^{59} -7.28175e9 q^{61} +7.36220e9 q^{62} +6.49053e9 q^{63} +4.76973e9 q^{64} -5.31918e9 q^{66} +2.11343e9 q^{67} +7.65458e9 q^{68} -3.93865e9 q^{69} -4.86982e9 q^{71} +8.88082e9 q^{72} -1.92542e10 q^{73} -1.47068e7 q^{74} -7.92334e9 q^{76} -2.49346e10 q^{77} +1.60165e10 q^{78} -4.54858e10 q^{79} -3.91472e10 q^{81} -2.22540e10 q^{82} -3.61471e10 q^{83} +4.58249e10 q^{84} -1.63591e10 q^{86} +2.55228e10 q^{87} -3.41174e10 q^{88} +3.43834e9 q^{89} +7.50802e10 q^{91} -9.86994e9 q^{92} +1.42327e11 q^{93} +1.27188e10 q^{94} +1.00905e11 q^{96} +1.57518e11 q^{97} -6.65976e10 q^{98} -3.64982e10 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 72 q^{4} - 1752 q^{6} - 25452 q^{9}+O(q^{10}) $ 4 * q + 72 * q^4 - 1752 * q^6 - 25452 * q^9	$ 4 q + 72 q^{4} - 1752 q^{6} - 25452 q^{9} - 326352 q^{11} - 1080696 q^{14} - 9834976 q^{16} - 15460880 q^{19} - 81019872 q^{21} - 74415840 q^{24} - 325970832 q^{26} - 216242520 q^{29} - 684043072 q^{31} - 265782016 q^{34} - 553353336 q^{36} - 5997024 q^{39} + 1012873368 q^{41} + 1553573664 q^{44} + 5241789688 q^{46} + 1900646372 q^{49} + 8691953088 q^{51} + 6403356720 q^{54} + 10366738080 q^{56} - 3200971440 q^{59} - 2310471352 q^{61} + 5401150592 q^{64} - 17010985824 q^{66} - 32956101984 q^{69} - 60335466912 q^{71} + 5525992944 q^{74} - 52987638240 q^{76} - 74637768320 q^{79} - 77727716316 q^{81} + 76499865504 q^{84} + 39045421128 q^{86} - 118272499560 q^{89} + 51565095648 q^{91} + 266098749224 q^{94} + 313591828608 q^{96} - 119560366224 q^{99}+O(q^{100}) $ 4 * q + 72 * q^4 - 1752 * q^6 - 25452 * q^9 - 326352 * q^11 - 1080696 * q^14 - 9834976 * q^16 - 15460880 * q^19 - 81019872 * q^21 - 74415840 * q^24 - 325970832 * q^26 - 216242520 * q^29 - 684043072 * q^31 - 265782016 * q^34 - 553353336 * q^36 - 5997024 * q^39 + 1012873368 * q^41 + 1553573664 * q^44 + 5241789688 * q^46 + 1900646372 * q^49 + 8691953088 * q^51 + 6403356720 * q^54 + 10366738080 * q^56 - 3200971440 * q^59 - 2310471352 * q^61 + 5401150592 * q^64 - 17010985824 * q^66 - 32956101984 * q^69 - 60335466912 * q^71 + 5525992944 * q^74 - 52987638240 * q^76 - 74637768320 * q^79 - 77727716316 * q^81 + 76499865504 * q^84 + 39045421128 * q^86 - 118272499560 * q^89 + 51565095648 * q^91 + 266098749224 * q^94 + 313591828608 * q^96 - 119560366224 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−27.1071	−0.598989	−0.299495	−	0.954098i	$-0.596818\pi$
$2$	−27.1071	−0.598989	−0.299495	+	0.954098i	$0.596818\pi$
$3$	−524.040	−1.24508	−0.622540	−	0.782588i	$-0.713899\pi$
$3$	−524.040	−1.24508	−0.622540	+	0.782588i	$0.713899\pi$
$4$	−1313.20	−0.641212
$5$	0	0
$5$	0	0
$6$	14205.2	0.745790
$7$	66589.5	1.49750	0.748749	−	0.662854i	$-0.230655\pi$
$7$	66589.5	1.49750	0.748749	+	0.662854i	$0.230655\pi$
$8$	91112.6	0.983068
$9$	97470.8	0.550225
$10$	0	0
$11$	−374453.	−0.701031	−0.350515	−	0.936557i	$-0.613994\pi$
$11$	−374453.	−0.701031	−0.350515	+	0.936557i	$0.613994\pi$
$12$	688170.	0.798361
$13$	1.12751e6	0.842232	0.421116	−	0.907007i	$-0.361639\pi$
$13$	1.12751e6	0.842232	0.421116	+	0.907007i	$0.361639\pi$
$14$	−1.80505e6	−0.896985
$15$	0	0
$16$	219635.	0.0523651
$17$	−5.82894e6	−0.995681	−0.497841	−	0.867269i	$-0.665874\pi$
$17$	−5.82894e6	−0.995681	−0.497841	+	0.867269i	$0.665874\pi$
$18$	−2.64216e6	−0.329579
$19$	6.03360e6	0.559026	0.279513	−	0.960142i	$-0.409827\pi$
$19$	6.03360e6	0.559026	0.279513	+	0.960142i	$0.409827\pi$
$20$	0	0
$21$	−3.48955e7	−1.86451
$22$	1.01503e7	0.419910
$23$	7.51593e6	0.243489	0.121745	−	0.992561i	$-0.461151\pi$
$23$	7.51593e6	0.243489	0.121745	+	0.992561i	$0.461151\pi$
$24$	−4.77466e7	−1.22400
$25$	0	0
$26$	−3.05636e7	−0.504488
$27$	4.17535e7	0.560005
$28$	−8.74454e7	−0.960214
$29$	−4.87039e7	−0.440935	−0.220467	−	0.975394i	$-0.570758\pi$
$29$	−4.87039e7	−0.440935	−0.220467	+	0.975394i	$0.570758\pi$
$30$	0	0
$31$	−2.71596e8	−1.70386	−0.851932	−	0.523653i	$-0.824569\pi$
$31$	−2.71596e8	−1.70386	−0.851932	+	0.523653i	$0.824569\pi$
$32$	−1.92552e8	−1.01443
$33$	1.96228e8	0.872840
$34$	1.58006e8	0.596402
$35$	0	0
$36$	−1.27999e8	−0.352811
$37$	542545.	0.00128625	0.000643126	−	1.00000i	$-0.499795\pi$
$37$	542545.	0.00128625	0.000643126		1.00000i	$0.499795\pi$
$38$	−1.63554e8	−0.334850
$39$	−5.90860e8	−1.04865
$40$	0	0
$41$	8.20963e8	1.10666	0.553328	−	0.832964i	$-0.313358\pi$
$41$	8.20963e8	1.10666	0.553328	+	0.832964i	$0.313358\pi$
$42$	9.45918e8	1.11682
$43$	6.03499e8	0.626037	0.313018	−	0.949747i	$-0.398660\pi$
$43$	6.03499e8	0.626037	0.313018	+	0.949747i	$0.398660\pi$
$44$	4.91732e8	0.449509
$45$	0	0
$46$	−2.03735e8	−0.145847
$47$	−4.69204e8	−0.298417	−0.149208	−	0.988806i	$-0.547673\pi$
$47$	−4.69204e8	−0.298417	−0.149208	+	0.988806i	$0.547673\pi$
$48$	−1.15098e8	−0.0651988
$49$	2.45683e9	1.24250
$50$	0	0
$51$	3.05460e9	1.23970
$52$	−1.48065e9	−0.540049
$53$	−3.82491e9	−1.25633	−0.628166	−	0.778079i	$-0.716194\pi$
$53$	−3.82491e9	−1.25633	−0.628166	+	0.778079i	$0.716194\pi$
$54$	−1.13182e9	−0.335437
$55$	0	0
$56$	6.06714e9	1.47214
$57$	−3.16185e9	−0.696032
$58$	1.32022e9	0.264115
$59$	−2.19696e9	−0.400069	−0.200035	−	0.979789i	$-0.564105\pi$
$59$	−2.19696e9	−0.400069	−0.200035	+	0.979789i	$0.564105\pi$
$60$	0	0
$61$	−7.28175e9	−1.10388	−0.551940	−	0.833884i	$-0.686112\pi$
$61$	−7.28175e9	−1.10388	−0.551940	+	0.833884i	$0.686112\pi$
$62$	7.36220e9	1.02060
$63$	6.49053e9	0.823962
$64$	4.76973e9	0.555270
$65$	0	0
$66$	−5.31918e9	−0.522821
$67$	2.11343e9	0.191239	0.0956197	−	0.995418i	$-0.469517\pi$
$67$	2.11343e9	0.191239	0.0956197	+	0.995418i	$0.469517\pi$
$68$	7.65458e9	0.638443
$69$	−3.93865e9	−0.303164
$70$	0	0
$71$	−4.86982e9	−0.320326	−0.160163	−	0.987091i	$-0.551202\pi$
$71$	−4.86982e9	−0.320326	−0.160163	+	0.987091i	$0.551202\pi$
$72$	8.88082e9	0.540909
$73$	−1.92542e10	−1.08705	−0.543524	−	0.839393i	$-0.682910\pi$
$73$	−1.92542e10	−1.08705	−0.543524	+	0.839393i	$0.682910\pi$
$74$	−1.47068e7	−0.000770451 0
$75$	0	0
$76$	−7.92334e9	−0.358454
$77$	−2.49346e10	−1.04979
$78$	1.60165e10	0.628128
$79$	−4.54858e10	−1.66313	−0.831566	−	0.555426i	$-0.812555\pi$
$79$	−4.54858e10	−1.66313	−0.831566	+	0.555426i	$0.812555\pi$
$80$	0	0
$81$	−3.91472e10	−1.24748
$82$	−2.22540e10	−0.662874
$83$	−3.61471e10	−1.00726	−0.503632	−	0.863918i	$-0.668003\pi$
$83$	−3.61471e10	−1.00726	−0.503632	+	0.863918i	$0.668003\pi$
$84$	4.58249e10	1.19554
$85$	0	0
$86$	−1.63591e10	−0.374989
$87$	2.55228e10	0.548999
$88$	−3.41174e10	−0.689161
$89$	3.43834e9	0.0652685	0.0326342	−	0.999467i	$-0.489610\pi$
$89$	3.43834e9	0.0652685	0.0326342	+	0.999467i	$0.489610\pi$
$90$	0	0
$91$	7.50802e10	1.26124
$92$	−9.86994e9	−0.156128
$93$	1.42327e11	2.12145
$94$	1.27188e10	0.178748
$95$	0	0
$96$	1.00905e11	1.26305
$97$	1.57518e11	1.86246	0.931228	−	0.364436i	$-0.118738\pi$
$97$	1.57518e11	1.86246	0.931228	+	0.364436i	$0.118738\pi$
$98$	−6.65976e10	−0.744244
$99$	−3.64982e10	−0.385725
$100$	0	0
$101$	−3.14292e10	−0.297554	−0.148777	−	0.988871i	$-0.547534\pi$
$101$	−3.14292e10	−0.297554	−0.148777	+	0.988871i	$0.547534\pi$
$102$	−8.28014e10	−0.742569
$103$	3.42578e10	0.291176	0.145588	−	0.989345i	$-0.453493\pi$
$103$	3.42578e10	0.291176	0.145588	+	0.989345i	$0.453493\pi$
$104$	1.02730e11	0.827971
$105$	0	0
$106$	1.03683e11	0.752529
$107$	−2.10883e11	−1.45355	−0.726776	−	0.686874i	$-0.758982\pi$
$107$	−2.10883e11	−1.45355	−0.726776	+	0.686874i	$0.758982\pi$
$108$	−5.48308e10	−0.359082
$109$	1.44958e10	0.0902394	0.0451197	−	0.998982i	$-0.485633\pi$
$109$	1.44958e10	0.0902394	0.0451197	+	0.998982i	$0.485633\pi$
$110$	0	0
$111$	−2.84315e8	−0.00160149
$112$	1.46254e10	0.0784167
$113$	−2.14619e9	−0.0109581	−0.00547906	−	0.999985i	$-0.501744\pi$
$113$	−2.14619e9	−0.0109581	−0.00547906	+	0.999985i	$0.501744\pi$
$114$	8.57087e10	0.416916
$115$	0	0
$116$	6.39580e10	0.282733
$117$	1.09899e11	0.463417
$118$	5.95532e10	0.239637
$119$	−3.88146e11	−1.49103
$120$	0	0
$121$	−1.45097e11	−0.508556
$122$	1.97388e11	0.661212
$123$	−4.30217e11	−1.37787
$124$	3.56661e11	1.09254
$125$	0	0
$126$	−1.75940e11	−0.493544
$127$	−2.28294e8	−0.000613160 0	−0.000306580	−	1.00000i	$-0.500098\pi$
$127$	−2.28294e8	−0.000613160 0	−0.000306580		1.00000i	$0.500098\pi$
$128$	2.65053e11	0.681834
$129$	−3.16257e11	−0.779466
$130$	0	0
$131$	−5.23011e11	−1.18446	−0.592228	−	0.805771i	$-0.701751\pi$
$131$	−5.23011e11	−1.18446	−0.592228	+	0.805771i	$0.701751\pi$
$132$	−2.57687e11	−0.559675
$133$	4.01774e11	0.837140
$134$	−5.72892e10	−0.114550
$135$	0	0
$136$	−5.31090e11	−0.978822
$137$	2.04327e11	0.361712	0.180856	−	0.983510i	$-0.442113\pi$
$137$	2.04327e11	0.361712	0.180856	+	0.983510i	$0.442113\pi$
$138$	1.06766e11	0.181592
$139$	−1.10282e11	−0.180270	−0.0901348	−	0.995930i	$-0.528730\pi$
$139$	−1.10282e11	−0.180270	−0.0901348	+	0.995930i	$0.528730\pi$
$140$	0	0
$141$	2.45881e11	0.371553
$142$	1.32007e11	0.191872
$143$	−4.22199e11	−0.590430
$144$	2.14080e10	0.0288126
$145$	0	0
$146$	5.21926e11	0.651130
$147$	−1.28748e12	−1.54701
$148$	−7.12471e8	−0.000824761 0
$149$	−5.11067e11	−0.570102	−0.285051	−	0.958512i	$-0.592011\pi$
$149$	−5.11067e11	−0.570102	−0.285051	+	0.958512i	$0.592011\pi$
$150$	0	0
$151$	−3.26638e10	−0.0338605	−0.0169303	−	0.999857i	$-0.505389\pi$
$151$	−3.26638e10	−0.0338605	−0.0169303	+	0.999857i	$0.505389\pi$
$152$	5.49737e11	0.549560
$153$	−5.68151e11	−0.547849
$154$	6.75906e11	0.628814
$155$	0	0
$156$	7.75919e11	0.672405
$157$	1.29615e12	1.08444	0.542221	−	0.840236i	$-0.317584\pi$
$157$	1.29615e12	1.08444	0.542221	+	0.840236i	$0.317584\pi$
$158$	1.23299e12	0.996198
$159$	2.00441e12	1.56423
$160$	0	0
$161$	5.00482e11	0.364625
$162$	1.06117e12	0.747225
$163$	9.53484e11	0.649055	0.324527	−	0.945876i	$-0.394795\pi$
$163$	9.53484e11	0.649055	0.324527	+	0.945876i	$0.394795\pi$
$164$	−1.07809e12	−0.709601
$165$	0	0
$166$	9.79844e11	0.603340
$167$	−2.16450e12	−1.28949	−0.644745	−	0.764398i	$-0.723036\pi$
$167$	−2.16450e12	−1.28949	−0.644745	+	0.764398i	$0.723036\pi$
$168$	−3.17942e12	−1.83294
$169$	−5.20883e11	−0.290645
$170$	0	0
$171$	5.88100e11	0.307590
$172$	−7.92516e11	−0.401422
$173$	−3.76047e11	−0.184497	−0.0922483	−	0.995736i	$-0.529405\pi$
$173$	−3.76047e11	−0.184497	−0.0922483	+	0.995736i	$0.529405\pi$
$174$	−6.91850e11	−0.328845
$175$	0	0
$176$	−8.22430e10	−0.0367096
$177$	1.15129e12	0.498119
$178$	−9.32035e10	−0.0390951
$179$	9.97885e11	0.405872	0.202936	−	0.979192i	$-0.434952\pi$
$179$	9.97885e11	0.405872	0.202936	+	0.979192i	$0.434952\pi$
$180$	0	0
$181$	3.32751e12	1.27317	0.636587	−	0.771205i	$-0.280346\pi$
$181$	3.32751e12	1.27317	0.636587	+	0.771205i	$0.280346\pi$
$182$	−2.03521e12	−0.755469
$183$	3.81593e12	1.37442
$184$	6.84796e11	0.239366
$185$	0	0
$186$	−3.85809e12	−1.27072
$187$	2.18266e12	0.698003
$188$	6.16160e11	0.191348
$189$	2.78034e12	0.838607
$190$	0	0
$191$	−4.02856e12	−1.14674	−0.573372	−	0.819295i	$-0.694365\pi$
$191$	−4.02856e12	−1.14674	−0.573372	+	0.819295i	$0.694365\pi$
$192$	−2.49953e12	−0.691356
$193$	−1.01858e12	−0.273798	−0.136899	−	0.990585i	$-0.543714\pi$
$193$	−1.01858e12	−0.273798	−0.136899	+	0.990585i	$0.543714\pi$
$194$	−4.26987e12	−1.11559
$195$	0	0
$196$	−3.22631e12	−0.796706
$197$	4.56750e12	1.09677	0.548384	−	0.836227i	$-0.315243\pi$
$197$	4.56750e12	1.09677	0.548384	+	0.836227i	$0.315243\pi$
$198$	9.89362e11	0.231045
$199$	−2.19872e12	−0.499435	−0.249717	−	0.968319i	$-0.580338\pi$
$199$	−2.19872e12	−0.499435	−0.249717	+	0.968319i	$0.580338\pi$
$200$	0	0
$201$	−1.10752e12	−0.238108
$202$	8.51955e11	0.178231
$203$	−3.24316e12	−0.660299
$204$	−4.01130e12	−0.794913
$205$	0	0
$206$	−9.28632e11	−0.174411
$207$	7.32584e11	0.133974
$208$	2.47641e11	0.0441036
$209$	−2.25930e12	−0.391894
$210$	0	0
$211$	8.39469e11	0.138182	0.0690910	−	0.997610i	$-0.477990\pi$
$211$	8.39469e11	0.138182	0.0690910	+	0.997610i	$0.477990\pi$
$212$	5.02289e12	0.805575
$213$	2.55198e12	0.398831
$214$	5.71644e12	0.870662
$215$	0	0
$216$	3.80427e12	0.550524
$217$	−1.80855e13	−2.55153
$218$	−3.92940e11	−0.0540524
$219$	1.00900e13	1.35346
$220$	0	0
$221$	−6.57218e12	−0.838595
$222$	7.70697e9	0.000959274 0
$223$	−1.61958e13	−1.96665	−0.983323	−	0.181867i	$-0.941786\pi$
$223$	−1.61958e13	−1.96665	−0.983323	+	0.181867i	$0.941786\pi$
$224$	−1.28220e13	−1.51911
$225$	0	0
$226$	5.81770e10	0.00656379
$227$	6.08481e11	0.0670046	0.0335023	−	0.999439i	$-0.489334\pi$
$227$	6.08481e11	0.0670046	0.0335023	+	0.999439i	$0.489334\pi$
$228$	4.15215e12	0.446304
$229$	1.59740e13	1.67617	0.838083	−	0.545542i	$-0.183676\pi$
$229$	1.59740e13	1.67617	0.838083	+	0.545542i	$0.183676\pi$
$230$	0	0
$231$	1.30667e13	1.30708
$232$	−4.43754e12	−0.433469
$233$	1.09497e13	1.04459	0.522296	−	0.852764i	$-0.325076\pi$
$233$	1.09497e13	1.04459	0.522296	+	0.852764i	$0.325076\pi$
$234$	−2.97905e12	−0.277582
$235$	0	0
$236$	2.88505e12	0.256529
$237$	2.38364e13	2.07073
$238$	1.05215e13	0.893111
$239$	−2.11532e13	−1.75464	−0.877321	−	0.479905i	$-0.840671\pi$
$239$	−2.11532e13	−1.75464	−0.877321	+	0.479905i	$0.840671\pi$
$240$	0	0
$241$	1.41468e13	1.12089	0.560447	−	0.828190i	$-0.310629\pi$
$241$	1.41468e13	1.12089	0.560447	+	0.828190i	$0.310629\pi$
$242$	3.93317e12	0.304619
$243$	1.31182e13	0.993204
$244$	9.56242e12	0.707821
$245$	0	0
$246$	1.16620e13	0.825332
$247$	6.80294e12	0.470829
$248$	−2.47459e13	−1.67501
$249$	1.89425e13	1.25413
$250$	0	0
$251$	−1.64508e12	−0.104227	−0.0521136	−	0.998641i	$-0.516596\pi$
$251$	−1.64508e12	−0.104227	−0.0521136	+	0.998641i	$0.516596\pi$
$252$	−8.52338e12	−0.528334
$253$	−2.81436e12	−0.170693
$254$	6.18840e9	0.000367276 0
$255$	0	0
$256$	−1.69533e13	−0.963681
$257$	−3.16010e13	−1.75820	−0.879101	−	0.476635i	$-0.841856\pi$
$257$	−3.16010e13	−1.75820	−0.879101	+	0.476635i	$0.841856\pi$
$258$	8.57284e12	0.466892
$259$	3.61278e10	0.00192616
$260$	0	0
$261$	−4.74720e12	−0.242614
$262$	1.41773e13	0.709476
$263$	1.50443e12	0.0737250	0.0368625	−	0.999320i	$-0.488264\pi$
$263$	1.50443e12	0.0737250	0.0368625	+	0.999320i	$0.488264\pi$
$264$	1.78789e13	0.858061
$265$	0	0
$266$	−1.08910e13	−0.501438
$267$	−1.80183e12	−0.0812645
$268$	−2.77537e12	−0.122625
$269$	−3.90666e13	−1.69110	−0.845548	−	0.533900i	$-0.820726\pi$
$269$	−3.90666e13	−1.69110	−0.845548	+	0.533900i	$0.820726\pi$
$270$	0	0
$271$	−4.54729e12	−0.188982	−0.0944912	−	0.995526i	$-0.530122\pi$
$271$	−4.54729e12	−0.188982	−0.0944912	+	0.995526i	$0.530122\pi$
$272$	−1.28024e12	−0.0521390
$273$	−3.93450e13	−1.57035
$274$	−5.53873e12	−0.216662
$275$	0	0
$276$	5.17224e12	0.194392
$277$	−2.22049e13	−0.818105	−0.409053	−	0.912511i	$-0.634141\pi$
$277$	−2.22049e13	−0.818105	−0.409053	+	0.912511i	$0.634141\pi$
$278$	2.98943e12	0.107980
$279$	−2.64727e13	−0.937509
$280$	0	0
$281$	−2.26709e13	−0.771941	−0.385970	−	0.922511i	$-0.626133\pi$
$281$	−2.26709e13	−0.771941	−0.385970	+	0.922511i	$0.626133\pi$
$282$	−6.66515e12	−0.222556
$283$	2.94259e13	0.963617	0.481809	−	0.876276i	$-0.339980\pi$
$283$	2.94259e13	0.963617	0.481809	+	0.876276i	$0.339980\pi$
$284$	6.39505e12	0.205397
$285$	0	0
$286$	1.14446e13	0.353661
$287$	5.46675e13	1.65721
$288$	−1.87682e13	−0.558168
$289$	−2.95383e11	−0.00861880
$290$	0	0
$291$	−8.25458e13	−2.31891
$292$	2.52846e13	0.697029
$293$	4.02146e13	1.08796	0.543979	−	0.839099i	$-0.316917\pi$
$293$	4.02146e13	1.08796	0.543979	+	0.839099i	$0.316917\pi$
$294$	3.48998e13	0.926644
$295$	0	0
$296$	4.94327e10	0.00126447
$297$	−1.56347e13	−0.392581
$298$	1.38536e13	0.341485
$299$	8.47428e12	0.205074
$300$	0	0
$301$	4.01866e13	0.937489
$302$	8.85423e11	0.0202821
$303$	1.64701e13	0.370478
$304$	1.32519e12	0.0292735
$305$	0	0
$306$	1.54010e13	0.328156
$307$	1.17567e13	0.246050	0.123025	−	0.992404i	$-0.460740\pi$
$307$	1.17567e13	0.246050	0.123025	+	0.992404i	$0.460740\pi$
$308$	3.27442e13	0.673139
$309$	−1.79525e13	−0.362537
$310$	0	0
$311$	9.66888e12	0.188449	0.0942245	−	0.995551i	$-0.469963\pi$
$311$	9.66888e12	0.188449	0.0942245	+	0.995551i	$0.469963\pi$
$312$	−5.38348e13	−1.03089
$313$	−7.95938e13	−1.49756	−0.748782	−	0.662817i	$-0.769361\pi$
$313$	−7.95938e13	−1.49756	−0.748782	+	0.662817i	$0.769361\pi$
$314$	−3.51349e13	−0.649569
$315$	0	0
$316$	5.97321e13	1.06642
$317$	−3.01135e13	−0.528366	−0.264183	−	0.964473i	$-0.585102\pi$
$317$	−3.01135e13	−0.528366	−0.264183	+	0.964473i	$0.585102\pi$
$318$	−5.43338e13	−0.936959
$319$	1.82373e13	0.309109
$320$	0	0
$321$	1.10511e14	1.80979
$322$	−1.35666e13	−0.218406
$323$	−3.51695e13	−0.556611
$324$	5.14082e13	0.799898
$325$	0	0
$326$	−2.58462e13	−0.388777
$327$	−7.59637e12	−0.112355
$328$	7.48001e13	1.08792
$329$	−3.12440e13	−0.446879
$330$	0	0
$331$	−9.09460e13	−1.25814	−0.629071	−	0.777348i	$-0.716564\pi$
$331$	−9.09460e13	−1.25814	−0.629071	+	0.777348i	$0.716564\pi$
$332$	4.74684e13	0.645870
$333$	5.28823e10	0.000707729 0
$334$	5.86735e13	0.772390
$335$	0	0
$336$	−7.66429e12	−0.0976351
$337$	6.71550e13	0.841616	0.420808	−	0.907150i	$-0.361747\pi$
$337$	6.71550e13	0.841616	0.420808	+	0.907150i	$0.361747\pi$
$338$	1.41197e13	0.174093
$339$	1.12469e12	0.0136437
$340$	0	0
$341$	1.01700e14	1.19446
$342$	−1.59417e13	−0.184243
$343$	3.19298e13	0.363144
$344$	5.49863e13	0.615437
$345$	0	0
$346$	1.01936e13	0.110511
$347$	−1.00178e14	−1.06895	−0.534476	−	0.845183i	$-0.679491\pi$
$347$	−1.00178e14	−1.06895	−0.534476	+	0.845183i	$0.679491\pi$
$348$	−3.35166e13	−0.352025
$349$	3.57336e12	0.0369434	0.0184717	−	0.999829i	$-0.494120\pi$
$349$	3.57336e12	0.0369434	0.0184717	+	0.999829i	$0.494120\pi$
$350$	0	0
$351$	4.70775e13	0.471655
$352$	7.21017e13	0.711149
$353$	−4.27043e13	−0.414677	−0.207339	−	0.978269i	$-0.566480\pi$
$353$	−4.27043e13	−0.414677	−0.207339	+	0.978269i	$0.566480\pi$
$354$	−3.12083e13	−0.298368
$355$	0	0
$356$	−4.51523e12	−0.0418509
$357$	2.03404e14	1.85645
$358$	−2.70498e13	−0.243113
$359$	−1.34889e14	−1.19387	−0.596934	−	0.802290i	$-0.703615\pi$
$359$	−1.34889e14	−1.19387	−0.596934	+	0.802290i	$0.703615\pi$
$360$	0	0
$361$	−8.00859e13	−0.687490
$362$	−9.01994e13	−0.762617
$363$	7.60366e13	0.633193
$364$	−9.85956e13	−0.808723
$365$	0	0
$366$	−1.03439e14	−0.823262
$367$	−1.04526e14	−0.819519	−0.409760	−	0.912193i	$-0.634387\pi$
$367$	−1.04526e14	−0.819519	−0.409760	+	0.912193i	$0.634387\pi$
$368$	1.65076e12	0.0127503
$369$	8.00199e13	0.608910
$370$	0	0
$371$	−2.54699e14	−1.88135
$372$	−1.86905e14	−1.36030
$373$	2.42814e14	1.74131	0.870654	−	0.491896i	$-0.163696\pi$
$373$	2.42814e14	1.74131	0.870654	+	0.491896i	$0.163696\pi$
$374$	−5.91657e13	−0.418096
$375$	0	0
$376$	−4.27504e13	−0.293364
$377$	−5.49141e13	−0.371369
$378$	−7.53672e13	−0.502316
$379$	1.40487e14	0.922827	0.461413	−	0.887185i	$-0.347343\pi$
$379$	1.40487e14	0.922827	0.461413	+	0.887185i	$0.347343\pi$
$380$	0	0
$381$	1.19635e11	0.000763434 0
$382$	1.09203e14	0.686886
$383$	−2.17176e14	−1.34654	−0.673269	−	0.739398i	$-0.735110\pi$
$383$	−2.17176e14	−1.34654	−0.673269	+	0.739398i	$0.735110\pi$
$384$	−1.38899e14	−0.848938
$385$	0	0
$386$	2.76108e13	0.164002
$387$	5.88235e13	0.344461
$388$	−2.06853e14	−1.19423
$389$	2.28055e14	1.29812	0.649062	−	0.760735i	$-0.275161\pi$
$389$	2.28055e14	1.29812	0.649062	+	0.760735i	$0.275161\pi$
$390$	0	0
$391$	−4.38099e13	−0.242438
$392$	2.23848e14	1.22146
$393$	2.74079e14	1.47474
$394$	−1.23812e14	−0.656951
$395$	0	0
$396$	4.79295e13	0.247332
$397$	−7.68462e13	−0.391088	−0.195544	−	0.980695i	$-0.562647\pi$
$397$	−7.68462e13	−0.391088	−0.195544	+	0.980695i	$0.562647\pi$
$398$	5.96011e13	0.299156
$399$	−2.10546e14	−1.04231
$400$	0	0
$401$	2.93708e14	1.41456	0.707282	−	0.706932i	$-0.249921\pi$
$401$	2.93708e14	1.41456	0.707282	+	0.706932i	$0.249921\pi$
$402$	3.00218e13	0.142624
$403$	−3.06228e14	−1.43505
$404$	4.12728e13	0.190795
$405$	0	0
$406$	8.79129e13	0.395512
$407$	−2.03157e11	−0.000901702 0
$408$	2.78312e14	1.21871
$409$	1.31901e14	0.569861	0.284930	−	0.958548i	$-0.408030\pi$
$409$	1.31901e14	0.569861	0.284930	+	0.958548i	$0.408030\pi$
$410$	0	0
$411$	−1.07076e14	−0.450361
$412$	−4.49875e13	−0.186705
$413$	−1.46294e14	−0.599103
$414$	−1.98583e13	−0.0802489
$415$	0	0
$416$	−2.17105e14	−0.854389
$417$	5.77921e13	0.224450
$418$	6.12431e13	0.234740
$419$	3.47913e14	1.31612	0.658058	−	0.752967i	$-0.271378\pi$
$419$	3.47913e14	1.31612	0.658058	+	0.752967i	$0.271378\pi$
$420$	0	0
$421$	3.60019e14	1.32670	0.663351	−	0.748308i	$-0.269134\pi$
$421$	3.60019e14	1.32670	0.663351	+	0.748308i	$0.269134\pi$
$422$	−2.27556e13	−0.0827695
$423$	−4.57337e13	−0.164197
$424$	−3.48498e14	−1.23506
$425$	0	0
$426$	−6.91768e13	−0.238896
$427$	−4.84888e14	−1.65306
$428$	2.76932e14	0.932035
$429$	2.21249e14	0.735133
$430$	0	0
$431$	4.03045e14	1.30535	0.652677	−	0.757636i	$-0.273646\pi$
$431$	4.03045e14	1.30535	0.652677	+	0.757636i	$0.273646\pi$
$432$	9.17055e12	0.0293248
$433$	−3.79440e14	−1.19801	−0.599004	−	0.800746i	$-0.704437\pi$
$433$	−3.79440e14	−1.19801	−0.599004	+	0.800746i	$0.704437\pi$
$434$	4.90245e14	1.52834
$435$	0	0
$436$	−1.90359e13	−0.0578626
$437$	4.53481e13	0.136117
$438$	−2.73510e14	−0.810710
$439$	−5.70014e14	−1.66852	−0.834259	−	0.551373i	$-0.814104\pi$
$439$	−5.70014e14	−1.66852	−0.834259	+	0.551373i	$0.814104\pi$
$440$	0	0
$441$	2.39469e14	0.683655
$442$	1.78153e14	0.502309
$443$	6.54692e14	1.82312	0.911562	−	0.411162i	$-0.134877\pi$
$443$	6.54692e14	1.82312	0.911562	+	0.411162i	$0.134877\pi$
$444$	3.73363e11	0.00102689
$445$	0	0
$446$	4.39023e14	1.17800
$447$	2.67819e14	0.709824
$448$	3.17614e14	0.831515
$449$	2.16742e14	0.560515	0.280258	−	0.959925i	$-0.409580\pi$
$449$	2.16742e14	0.560515	0.280258	+	0.959925i	$0.409580\pi$
$450$	0	0
$451$	−3.07412e14	−0.775799
$452$	2.81838e12	0.00702648
$453$	1.71171e13	0.0421591
$454$	−1.64942e13	−0.0401350
$455$	0	0
$456$	−2.88084e14	−0.684247
$457$	4.04825e14	0.950011	0.475006	−	0.879983i	$-0.342446\pi$
$457$	4.04825e14	0.950011	0.475006	+	0.879983i	$0.342446\pi$
$458$	−4.33008e14	−1.00401
$459$	−2.43379e14	−0.557587
$460$	0	0
$461$	−4.78062e14	−1.06937	−0.534686	−	0.845051i	$-0.679570\pi$
$461$	−4.78062e14	−1.06937	−0.534686	+	0.845051i	$0.679570\pi$
$462$	−3.54202e14	−0.782924
$463$	−7.90859e13	−0.172744	−0.0863721	−	0.996263i	$-0.527527\pi$
$463$	−7.90859e13	−0.172744	−0.0863721	+	0.996263i	$0.527527\pi$
$464$	−1.06971e13	−0.0230896
$465$	0	0
$466$	−2.96816e14	−0.625699
$467$	3.83987e14	0.799970	0.399985	−	0.916522i	$-0.369015\pi$
$467$	3.83987e14	0.799970	0.399985	+	0.916522i	$0.369015\pi$
$468$	−1.44320e14	−0.297149
$469$	1.40732e14	0.286381
$470$	0	0
$471$	−6.79233e14	−1.35022
$472$	−2.00170e14	−0.393295
$473$	−2.25982e14	−0.438871
$474$	−6.46136e14	−1.24035
$475$	0	0
$476$	5.09714e14	0.956067
$477$	−3.72817e14	−0.691266
$478$	5.73404e14	1.05101
$479$	4.33999e14	0.786400	0.393200	−	0.919453i	$-0.371368\pi$
$479$	4.33999e14	0.786400	0.393200	+	0.919453i	$0.371368\pi$
$480$	0	0
$481$	6.11724e11	0.00108332
$482$	−3.83480e14	−0.671404
$483$	−2.62272e14	−0.453987
$484$	1.90542e14	0.326092
$485$	0	0
$486$	−3.55596e14	−0.594918
$487$	−3.35969e14	−0.555763	−0.277881	−	0.960615i	$-0.589632\pi$
$487$	−3.35969e14	−0.555763	−0.277881	+	0.960615i	$0.589632\pi$
$488$	−6.63460e14	−1.08519
$489$	−4.99663e14	−0.808126
$490$	0	0
$491$	−6.44062e14	−1.01854	−0.509272	−	0.860606i	$-0.670085\pi$
$491$	−6.44062e14	−1.01854	−0.509272	+	0.860606i	$0.670085\pi$
$492$	5.64962e14	0.883510
$493$	2.83892e14	0.439031
$494$	−1.84408e14	−0.282022
$495$	0	0
$496$	−5.96522e13	−0.0892231
$497$	−3.24278e14	−0.479687
$498$	−5.13477e14	−0.751207
$499$	−6.77869e14	−0.980828	−0.490414	−	0.871490i	$-0.663154\pi$
$499$	−6.77869e14	−0.980828	−0.490414	+	0.871490i	$0.663154\pi$
$500$	0	0
$501$	1.13429e15	1.60552
$502$	4.45934e13	0.0624310
$503$	9.66327e14	1.33814	0.669068	−	0.743202i	$-0.266694\pi$
$503$	9.66327e14	1.33814	0.669068	+	0.743202i	$0.266694\pi$
$504$	5.91369e14	0.810010
$505$	0	0
$506$	7.62893e13	0.102243
$507$	2.72963e14	0.361877
$508$	2.99796e11	0.000393166 0
$509$	1.19572e13	0.0155125	0.00775626	−	0.999970i	$-0.497531\pi$
$509$	1.19572e13	0.0155125	0.00775626	+	0.999970i	$0.497531\pi$
$510$	0	0
$511$	−1.28212e15	−1.62785
$512$	−8.32750e13	−0.104600
$513$	2.51924e14	0.313057
$514$	8.56614e14	1.05314
$515$	0	0
$516$	4.15310e14	0.499803
$517$	1.75695e14	0.209199
$518$	−9.79321e11	−0.00115375
$519$	1.97064e14	0.229713
$520$	0	0
$521$	1.26458e15	1.44324	0.721618	−	0.692292i	$-0.243399\pi$
$521$	1.26458e15	1.44324	0.721618	+	0.692292i	$0.243399\pi$
$522$	1.28683e14	0.145323
$523$	7.08792e13	0.0792062	0.0396031	−	0.999215i	$-0.487391\pi$
$523$	7.08792e13	0.0792062	0.0396031	+	0.999215i	$0.487391\pi$
$524$	6.86820e14	0.759487
$525$	0	0
$526$	−4.07807e13	−0.0441605
$527$	1.58312e15	1.69650
$528$	4.30986e13	0.0457064
$529$	−8.96321e14	−0.940713
$530$	0	0
$531$	−2.14139e14	−0.220128
$532$	−5.27611e14	−0.536784
$533$	9.25643e14	0.932060
$534$	4.88424e13	0.0486765
$535$	0	0
$536$	1.92560e14	0.188001
$537$	−5.22932e14	−0.505343
$538$	1.05898e15	1.01295
$539$	−9.19966e14	−0.871031
$540$	0	0
$541$	1.90242e15	1.76491	0.882453	−	0.470400i	$-0.155890\pi$
$541$	1.90242e15	1.76491	0.882453	+	0.470400i	$0.155890\pi$
$542$	1.23264e14	0.113198
$543$	−1.74375e15	−1.58520
$544$	1.12238e15	1.01005
$545$	0	0
$546$	1.06653e15	0.940620
$547$	1.94867e13	0.0170141	0.00850704	−	0.999964i	$-0.497292\pi$
$547$	1.94867e13	0.0170141	0.00850704	+	0.999964i	$0.497292\pi$
$548$	−2.68323e14	−0.231934
$549$	−7.09758e14	−0.607383
$550$	0	0
$551$	−2.93860e14	−0.246494
$552$	−3.58861e14	−0.298031
$553$	−3.02888e15	−2.49054
$554$	6.01910e14	0.490036
$555$	0	0
$556$	1.44822e14	0.115591
$557$	−1.67247e15	−1.32177	−0.660883	−	0.750489i	$-0.729818\pi$
$557$	−1.67247e15	−1.32177	−0.660883	+	0.750489i	$0.729818\pi$
$558$	7.17600e14	0.561558
$559$	6.80450e14	0.527268
$560$	0	0
$561$	−1.14380e15	−0.869070
$562$	6.14544e14	0.462384
$563$	7.96866e13	0.0593730	0.0296865	−	0.999559i	$-0.490549\pi$
$563$	7.96866e13	0.0593730	0.0296865	+	0.999559i	$0.490549\pi$
$564$	−3.22892e14	−0.238244
$565$	0	0
$566$	−7.97653e14	−0.577196
$567$	−2.60679e15	−1.86810
$568$	−4.43702e14	−0.314902
$569$	2.11263e15	1.48493	0.742467	−	0.669883i	$-0.233656\pi$
$569$	2.11263e15	1.48493	0.742467	+	0.669883i	$0.233656\pi$
$570$	0	0
$571$	1.05231e15	0.725513	0.362756	−	0.931884i	$-0.381836\pi$
$571$	1.05231e15	0.725513	0.362756	+	0.931884i	$0.381836\pi$
$572$	5.54432e14	0.378591
$573$	2.11113e15	1.42779
$574$	−1.48188e15	−0.992653
$575$	0	0
$576$	4.64910e14	0.305524
$577$	1.13168e15	0.736642	0.368321	−	0.929699i	$-0.379933\pi$
$577$	1.13168e15	0.736642	0.368321	+	0.929699i	$0.379933\pi$
$578$	8.00698e12	0.00516257
$579$	5.33776e14	0.340900
$580$	0	0
$581$	−2.40701e15	−1.50838
$582$	2.23758e15	1.38900
$583$	1.43225e15	0.880727
$584$	−1.75430e15	−1.06864
$585$	0	0
$586$	−1.09010e15	−0.651675
$587$	8.84276e14	0.523695	0.261847	−	0.965109i	$-0.415668\pi$
$587$	8.84276e14	0.523695	0.261847	+	0.965109i	$0.415668\pi$
$588$	1.69072e15	0.991964
$589$	−1.63870e15	−0.952503
$590$	0	0
$591$	−2.39355e15	−1.36556
$592$	1.19162e11	6.73548e−5 0
$593$	−2.89873e15	−1.62333	−0.811665	−	0.584123i	$-0.801438\pi$
$593$	−2.89873e15	−1.62333	−0.811665	+	0.584123i	$0.801438\pi$
$594$	4.23812e14	0.235152
$595$	0	0
$596$	6.71134e14	0.365557
$597$	1.15222e15	0.621836
$598$	−2.29714e14	−0.122837
$599$	−2.28284e15	−1.20956	−0.604781	−	0.796392i	$-0.706739\pi$
$599$	−2.28284e15	−1.20956	−0.604781	+	0.796392i	$0.706739\pi$
$600$	0	0
$601$	1.28437e15	0.668159	0.334080	−	0.942545i	$-0.391575\pi$
$601$	1.28437e15	0.668159	0.334080	+	0.942545i	$0.391575\pi$
$602$	−1.08935e15	−0.561545
$603$	2.05998e14	0.105225
$604$	4.28942e13	0.0217118
$605$	0	0
$606$	−4.46458e14	−0.221912
$607$	5.70541e14	0.281027	0.140514	−	0.990079i	$-0.455125\pi$
$607$	5.70541e14	0.281027	0.140514	+	0.990079i	$0.455125\pi$
$608$	−1.16178e15	−0.567095
$609$	1.69955e15	0.822126
$610$	0	0
$611$	−5.29032e14	−0.251336
$612$	7.46098e14	0.351288
$613$	3.14628e15	1.46813	0.734065	−	0.679080i	$-0.237621\pi$
$613$	3.14628e15	1.46813	0.734065	+	0.679080i	$0.237621\pi$
$614$	−3.18690e14	−0.147381
$615$	0	0
$616$	−2.27186e15	−1.03202
$617$	−3.60874e15	−1.62475	−0.812375	−	0.583135i	$-0.801826\pi$
$617$	−3.60874e15	−1.62475	−0.812375	+	0.583135i	$0.801826\pi$
$618$	4.86640e14	0.217156
$619$	2.19239e15	0.969660	0.484830	−	0.874608i	$-0.338882\pi$
$619$	2.19239e15	0.969660	0.484830	+	0.874608i	$0.338882\pi$
$620$	0	0
$621$	3.13817e14	0.136355
$622$	−2.62096e14	−0.112879
$623$	2.28957e14	0.0977394
$624$	−1.29774e14	−0.0549125
$625$	0	0
$626$	2.15756e15	0.897024
$627$	1.18396e15	0.487940
$628$	−1.70210e15	−0.695357
$629$	−3.16246e12	−0.00128070
$630$	0	0
$631$	3.99257e14	0.158888	0.0794441	−	0.996839i	$-0.474685\pi$
$631$	3.99257e14	0.158888	0.0794441	+	0.996839i	$0.474685\pi$
$632$	−4.14433e15	−1.63497
$633$	−4.39915e14	−0.172048
$634$	8.16291e14	0.316486
$635$	0	0
$636$	−2.63219e15	−1.00301
$637$	2.77010e15	1.04647
$638$	−4.94361e14	−0.185153
$639$	−4.74665e14	−0.176251
$640$	0	0
$641$	−1.23875e15	−0.452132	−0.226066	−	0.974112i	$-0.572586\pi$
$641$	−1.23875e15	−0.452132	−0.226066	+	0.974112i	$0.572586\pi$
$642$	−2.99564e15	−1.08404
$643$	−2.78159e15	−0.998005	−0.499003	−	0.866601i	$-0.666300\pi$
$643$	−2.78159e15	−0.998005	−0.499003	+	0.866601i	$0.666300\pi$
$644$	−6.57234e14	−0.233802
$645$	0	0
$646$	9.53345e14	0.333404
$647$	3.99567e15	1.38553	0.692766	−	0.721163i	$-0.256392\pi$
$647$	3.99567e15	1.38553	0.692766	+	0.721163i	$0.256392\pi$
$648$	−3.56680e15	−1.22636
$649$	8.22656e14	0.280461
$650$	0	0
$651$	9.47750e15	3.17686
$652$	−1.25212e15	−0.416182
$653$	1.37144e15	0.452016	0.226008	−	0.974125i	$-0.427432\pi$
$653$	1.37144e15	0.452016	0.226008	+	0.974125i	$0.427432\pi$
$654$	2.05916e14	0.0672996
$655$	0	0
$656$	1.80312e14	0.0579502
$657$	−1.87672e15	−0.598122
$658$	8.46936e14	0.267675
$659$	−4.53043e15	−1.41994	−0.709969	−	0.704233i	$-0.751291\pi$
$659$	−4.53043e15	−1.41994	−0.709969	+	0.704233i	$0.751291\pi$
$660$	0	0
$661$	−8.53788e13	−0.0263174	−0.0131587	−	0.999913i	$-0.504189\pi$
$661$	−8.53788e13	−0.0263174	−0.0131587	+	0.999913i	$0.504189\pi$
$662$	2.46529e15	0.753613
$663$	3.44409e15	1.04412
$664$	−3.29345e15	−0.990209
$665$	0	0
$666$	−1.43349e12	−0.000423922 0
$667$	−3.66055e14	−0.107363
$668$	2.84243e15	0.826836
$669$	8.48726e15	2.44863
$670$	0	0
$671$	2.72667e15	0.773854
$672$	6.71922e15	1.89142
$673$	−2.76196e13	−0.00771143	−0.00385571	−	0.999993i	$-0.501227\pi$
$673$	−2.76196e13	−0.00771143	−0.00385571	+	0.999993i	$0.501227\pi$
$674$	−1.82038e15	−0.504119
$675$	0	0
$676$	6.84025e14	0.186365
$677$	5.40053e15	1.45948	0.729741	−	0.683724i	$-0.239641\pi$
$677$	5.40053e15	1.45948	0.729741	+	0.683724i	$0.239641\pi$
$678$	−3.04871e13	−0.00817245
$679$	1.04891e16	2.78903
$680$	0	0
$681$	−3.18868e14	−0.0834261
$682$	−2.75680e15	−0.715469
$683$	1.45094e15	0.373539	0.186769	−	0.982404i	$-0.440198\pi$
$683$	1.45094e15	0.373539	0.186769	+	0.982404i	$0.440198\pi$
$684$	−7.72294e14	−0.197231
$685$	0	0
$686$	−8.65527e14	−0.217519
$687$	−8.37099e15	−2.08696
$688$	1.32550e14	0.0327825
$689$	−4.31263e15	−1.05812
$690$	0	0
$691$	−3.79473e15	−0.916330	−0.458165	−	0.888867i	$-0.651493\pi$
$691$	−3.79473e15	−0.916330	−0.458165	+	0.888867i	$0.651493\pi$
$692$	4.93826e14	0.118302
$693$	−2.43039e15	−0.577622
$694$	2.71553e15	0.640291
$695$	0	0
$696$	2.32545e15	0.539704
$697$	−4.78534e15	−1.10188
$698$	−9.68636e13	−0.0221287
$699$	−5.73810e15	−1.30060
$700$	0	0
$701$	−3.00014e15	−0.669410	−0.334705	−	0.942323i	$-0.608637\pi$
$701$	−3.00014e15	−0.669410	−0.334705	+	0.942323i	$0.608637\pi$
$702$	−1.27614e15	−0.282516
$703$	3.27350e12	0.000719048 0
$704$	−1.78604e15	−0.389261
$705$	0	0
$706$	1.15759e15	0.248387
$707$	−2.09285e15	−0.445586
$708$	−1.51188e15	−0.319400
$709$	−3.33135e14	−0.0698338	−0.0349169	−	0.999390i	$-0.511117\pi$
$709$	−3.33135e14	−0.0698338	−0.0349169	+	0.999390i	$0.511117\pi$
$710$	0	0
$711$	−4.43354e15	−0.915098
$712$	3.13276e14	0.0641634
$713$	−2.04130e15	−0.414872
$714$	−5.51370e15	−1.11200
$715$	0	0
$716$	−1.31043e15	−0.260250
$717$	1.10851e16	2.18467
$718$	3.65645e15	0.715114
$719$	1.88706e15	0.366249	0.183124	−	0.983090i	$-0.441379\pi$
$719$	1.88706e15	0.366249	0.183124	+	0.983090i	$0.441379\pi$
$720$	0	0
$721$	2.28121e15	0.436035
$722$	2.17090e15	0.411799
$723$	−7.41349e15	−1.39560
$724$	−4.36970e15	−0.816374
$725$	0	0
$726$	−2.06114e15	−0.379276
$727$	−3.20690e15	−0.585660	−0.292830	−	0.956164i	$-0.594597\pi$
$727$	−3.20690e15	−0.585660	−0.292830	+	0.956164i	$0.594597\pi$
$728$	6.84076e15	1.23989
$729$	6.03607e13	0.0108581
$730$	0	0
$731$	−3.51776e15	−0.623333
$732$	−5.01109e15	−0.881294
$733$	4.94602e15	0.863344	0.431672	−	0.902031i	$-0.357924\pi$
$733$	4.94602e15	0.863344	0.431672	+	0.902031i	$0.357924\pi$
$734$	2.83339e15	0.490883
$735$	0	0
$736$	−1.44721e15	−0.247004
$737$	−7.91381e14	−0.134065
$738$	−2.16911e15	−0.364730
$739$	2.24992e15	0.375511	0.187756	−	0.982216i	$-0.439879\pi$
$739$	2.24992e15	0.375511	0.187756	+	0.982216i	$0.439879\pi$
$740$	0	0
$741$	−3.56501e15	−0.586220
$742$	6.90416e15	1.12691
$743$	−6.59378e15	−1.06831	−0.534154	−	0.845387i	$-0.679370\pi$
$743$	−6.59378e15	−1.06831	−0.534154	+	0.845387i	$0.679370\pi$
$744$	1.29678e16	2.08553
$745$	0	0
$746$	−6.58200e15	−1.04302
$747$	−3.52328e15	−0.554222
$748$	−2.86628e15	−0.447568
$749$	−1.40426e16	−2.17669
$750$	0	0
$751$	4.76790e15	0.728295	0.364147	−	0.931341i	$-0.381360\pi$
$751$	4.76790e15	0.728295	0.364147	+	0.931341i	$0.381360\pi$
$752$	−1.03054e14	−0.0156266
$753$	8.62087e14	0.129771
$754$	1.48856e15	0.222446
$755$	0	0
$756$	−3.65115e15	−0.537725
$757$	3.60332e15	0.526836	0.263418	−	0.964682i	$-0.415150\pi$
$757$	3.60332e15	0.526836	0.263418	+	0.964682i	$0.415150\pi$
$758$	−3.80820e15	−0.552763
$759$	1.47484e15	0.212527
$760$	0	0
$761$	−1.00138e16	−1.42227	−0.711136	−	0.703054i	$-0.751819\pi$
$761$	−1.00138e16	−1.42227	−0.711136	+	0.703054i	$0.751819\pi$
$762$	−3.24297e12	−0.000457289 0
$763$	9.65267e14	0.135133
$764$	5.29031e15	0.735306
$765$	0	0
$766$	5.88702e15	0.806561
$767$	−2.47709e15	−0.336951
$768$	8.88418e15	1.19986
$769$	−1.14898e16	−1.54070	−0.770351	−	0.637620i	$-0.779919\pi$
$769$	−1.14898e16	−1.54070	−0.770351	+	0.637620i	$0.779919\pi$
$770$	0	0
$771$	1.65602e16	2.18910
$772$	1.33760e15	0.175562
$773$	1.24078e16	1.61699	0.808497	−	0.588501i	$-0.200282\pi$
$773$	1.24078e16	1.61699	0.808497	+	0.588501i	$0.200282\pi$
$774$	−1.59454e15	−0.206329
$775$	0	0
$776$	1.43519e16	1.83092
$777$	−1.89324e13	−0.00239822
$778$	−6.18192e15	−0.777562
$779$	4.95336e15	0.618649
$780$	0	0
$781$	1.82352e15	0.224558
$782$	1.18756e15	0.145217
$783$	−2.03356e15	−0.246926
$784$	5.39607e14	0.0650637
$785$	0	0
$786$	−7.42949e15	−0.883355
$787$	−3.57890e15	−0.422560	−0.211280	−	0.977426i	$-0.567763\pi$
$787$	−3.57890e15	−0.422560	−0.211280	+	0.977426i	$0.567763\pi$
$788$	−5.99805e15	−0.703260
$789$	−7.88380e14	−0.0917935
$790$	0	0
$791$	−1.42913e14	−0.0164098
$792$	−3.32545e15	−0.379194
$793$	−8.21024e15	−0.929723
$794$	2.08308e15	0.234258
$795$	0	0
$796$	2.88737e15	0.320244
$797$	−1.25596e16	−1.38343	−0.691714	−	0.722172i	$-0.743144\pi$
$797$	−1.25596e16	−1.38343	−0.691714	+	0.722172i	$0.743144\pi$
$798$	5.70729e15	0.624330
$799$	2.73496e15	0.297128
$800$	0	0
$801$	3.35137e14	0.0359124
$802$	−7.96160e15	−0.847308
$803$	7.20977e15	0.762055
$804$	1.45440e15	0.152678
$805$	0	0
$806$	8.30095e15	0.859578
$807$	2.04725e16	2.10555
$808$	−2.86359e15	−0.292515
$809$	8.95686e15	0.908738	0.454369	−	0.890813i	$-0.349865\pi$
$809$	8.95686e15	0.908738	0.454369	+	0.890813i	$0.349865\pi$
$810$	0	0
$811$	−8.33940e15	−0.834680	−0.417340	−	0.908750i	$-0.637038\pi$
$811$	−8.33940e15	−0.834680	−0.417340	+	0.908750i	$0.637038\pi$
$812$	4.25893e15	0.423392
$813$	2.38296e15	0.235298
$814$	5.50702e12	0.000540110 0
$815$	0	0
$816$	6.70897e14	0.0649173
$817$	3.64127e15	0.349971
$818$	−3.57545e15	−0.341340
$819$	7.31813e15	0.693967
$820$	0	0
$821$	3.07549e15	0.287757	0.143879	−	0.989595i	$-0.454042\pi$
$821$	3.07549e15	0.287757	0.143879	+	0.989595i	$0.454042\pi$
$822$	2.90251e15	0.269761
$823$	1.67606e16	1.54736	0.773678	−	0.633579i	$-0.218415\pi$
$823$	1.67606e16	1.54736	0.773678	+	0.633579i	$0.218415\pi$
$824$	3.12132e15	0.286246
$825$	0	0
$826$	3.96562e15	0.358856
$827$	1.18413e16	1.06443	0.532217	−	0.846608i	$-0.321359\pi$
$827$	1.18413e16	1.06443	0.532217	+	0.846608i	$0.321359\pi$
$828$	−9.62031e14	−0.0859057
$829$	−3.10234e15	−0.275195	−0.137597	−	0.990488i	$-0.543938\pi$
$829$	−3.10234e15	−0.275195	−0.137597	+	0.990488i	$0.543938\pi$
$830$	0	0
$831$	1.16362e16	1.01861
$832$	5.37792e15	0.467666
$833$	−1.43207e16	−1.23713
$834$	−1.56658e15	−0.134443
$835$	0	0
$836$	2.96691e15	0.251287
$837$	−1.13401e16	−0.954173
$838$	−9.43094e15	−0.788339
$839$	−2.08909e16	−1.73487	−0.867436	−	0.497549i	$-0.834233\pi$
$839$	−2.08909e16	−1.73487	−0.867436	+	0.497549i	$0.834233\pi$
$840$	0	0
$841$	−9.82844e15	−0.805576
$842$	−9.75908e15	−0.794680
$843$	1.18805e16	0.961129
$844$	−1.10239e15	−0.0886039
$845$	0	0
$846$	1.23971e15	0.0983519
$847$	−9.66193e15	−0.761562
$848$	−8.40086e14	−0.0657880
$849$	−1.54204e16	−1.19978
$850$	0	0
$851$	4.07773e12	0.000313189 0
$852$	−3.35126e15	−0.255735
$853$	−2.42451e16	−1.83825	−0.919124	−	0.393968i	$-0.871102\pi$
$853$	−2.42451e16	−1.83825	−0.919124	+	0.393968i	$0.871102\pi$
$854$	1.31439e16	0.990164
$855$	0	0
$856$	−1.92141e16	−1.42894
$857$	9.53835e15	0.704821	0.352410	−	0.935846i	$-0.385362\pi$
$857$	9.53835e15	0.704821	0.352410	+	0.935846i	$0.385362\pi$
$858$	−5.99743e15	−0.440337
$859$	4.70173e15	0.343001	0.171500	−	0.985184i	$-0.445139\pi$
$859$	4.70173e15	0.343001	0.171500	+	0.985184i	$0.445139\pi$
$860$	0	0
$861$	−2.86479e16	−2.06336
$862$	−1.09254e16	−0.781893
$863$	−1.12556e16	−0.800401	−0.400201	−	0.916428i	$-0.631060\pi$
$863$	−1.12556e16	−0.800401	−0.400201	+	0.916428i	$0.631060\pi$
$864$	−8.03974e15	−0.568089
$865$	0	0
$866$	1.02855e16	0.717593
$867$	1.54792e14	0.0107311
$868$	2.37499e16	1.63607
$869$	1.70323e16	1.16591
$870$	0	0
$871$	2.38292e15	0.161068
$872$	1.32075e15	0.0887115
$873$	1.53534e16	1.02477
$874$	−1.22926e15	−0.0815324
$875$	0	0
$876$	−1.32501e16	−0.867857
$877$	−9.19338e15	−0.598381	−0.299190	−	0.954193i	$-0.596717\pi$
$877$	−9.19338e15	−0.598381	−0.299190	+	0.954193i	$0.596717\pi$
$878$	1.54515e16	0.999424
$879$	−2.10741e16	−1.35460
$880$	0	0
$881$	−1.98624e16	−1.26085	−0.630425	−	0.776250i	$-0.717120\pi$
$881$	−1.98624e16	−1.26085	−0.630425	+	0.776250i	$0.717120\pi$
$882$	−6.49133e15	−0.409502
$883$	5.79702e15	0.363430	0.181715	−	0.983351i	$-0.441835\pi$
$883$	5.79702e15	0.363430	0.181715	+	0.983351i	$0.441835\pi$
$884$	8.63061e15	0.537717
$885$	0	0
$886$	−1.77468e16	−1.09203
$887$	8.29351e15	0.507176	0.253588	−	0.967312i	$-0.418389\pi$
$887$	8.29351e15	0.507176	0.253588	+	0.967312i	$0.418389\pi$
$888$	−2.59047e13	−0.00157437
$889$	−1.52020e13	−0.000918206 0
$890$	0	0
$891$	1.46588e16	0.874520
$892$	2.12684e16	1.26104
$893$	−2.83099e15	−0.166823
$894$	−7.25982e15	−0.425176
$895$	0	0
$896$	1.76498e16	1.02104
$897$	−4.44086e15	−0.255334
$898$	−5.87525e15	−0.335742
$899$	1.32278e16	0.751293
$900$	0	0
$901$	2.22952e16	1.25091
$902$	8.33305e15	0.464695
$903$	−2.10594e16	−1.16725
$904$	−1.95545e14	−0.0107726
$905$	0	0
$906$	−4.63997e14	−0.0252528
$907$	2.89763e16	1.56748	0.783741	−	0.621088i	$-0.213309\pi$
$907$	2.89763e16	1.56748	0.783741	+	0.621088i	$0.213309\pi$
$908$	−7.99059e14	−0.0429642
$909$	−3.06342e15	−0.163722
$910$	0	0
$911$	1.91910e16	1.01332	0.506659	−	0.862147i	$-0.330880\pi$
$911$	1.91910e16	1.01332	0.506659	+	0.862147i	$0.330880\pi$
$912$	−6.94454e14	−0.0364478
$913$	1.35354e16	0.706123
$914$	−1.09737e16	−0.569046
$915$	0	0
$916$	−2.09770e16	−1.07478
$917$	−3.48270e16	−1.77372
$918$	6.59730e15	0.333988
$919$	−5.30946e15	−0.267187	−0.133593	−	0.991036i	$-0.542652\pi$
$919$	−5.30946e15	−0.267187	−0.133593	+	0.991036i	$0.542652\pi$
$920$	0	0
$921$	−6.16096e15	−0.306352
$922$	1.29589e16	0.640542
$923$	−5.49076e15	−0.269788
$924$	−1.71593e16	−0.838113
$925$	0	0
$926$	2.14379e15	0.103472
$927$	3.33914e15	0.160212
$928$	9.37804e15	0.447299
$929$	3.41655e16	1.61995	0.809974	−	0.586466i	$-0.199481\pi$
$929$	3.41655e16	1.61995	0.809974	+	0.586466i	$0.199481\pi$
$930$	0	0
$931$	1.48235e16	0.694590
$932$	−1.43792e16	−0.669805
$933$	−5.06688e15	−0.234634
$934$	−1.04088e16	−0.479174
$935$	0	0
$936$	1.00132e16	0.455571
$937$	1.15303e16	0.521524	0.260762	−	0.965403i	$-0.416026\pi$
$937$	1.15303e16	0.521524	0.260762	+	0.965403i	$0.416026\pi$
$938$	−3.81485e15	−0.171539
$939$	4.17103e16	1.86459
$940$	0	0
$941$	−3.10666e16	−1.37262	−0.686310	−	0.727309i	$-0.740771\pi$
$941$	−3.10666e16	−1.37262	−0.686310	+	0.727309i	$0.740771\pi$
$942$	1.84121e16	0.808766
$943$	6.17030e15	0.269459
$944$	−4.82529e14	−0.0209497
$945$	0	0
$946$	6.12572e15	0.262879
$947$	−3.79319e16	−1.61838	−0.809188	−	0.587550i	$-0.800092\pi$
$947$	−3.79319e16	−1.61838	−0.809188	+	0.587550i	$0.800092\pi$
$948$	−3.13020e16	−1.32778
$949$	−2.17093e16	−0.915547
$950$	0	0
$951$	1.57807e16	0.657858
$952$	−3.53650e16	−1.46578
$953$	1.52578e16	0.628756	0.314378	−	0.949298i	$-0.398204\pi$
$953$	1.52578e16	0.628756	0.314378	+	0.949298i	$0.398204\pi$
$954$	1.01060e16	0.414061
$955$	0	0
$956$	2.77785e16	1.12510
$957$	−9.55707e15	−0.384865
$958$	−1.17645e16	−0.471045
$959$	1.36060e16	0.541663
$960$	0	0
$961$	4.83561e16	1.90315
$962$	−1.65821e13	−0.000648898 0
$963$	−2.05549e16	−0.799782
$964$	−1.85776e16	−0.718731
$965$	0	0
$966$	7.10946e15	0.271933
$967$	8.70387e15	0.331029	0.165515	−	0.986207i	$-0.447071\pi$
$967$	8.70387e15	0.331029	0.165515	+	0.986207i	$0.447071\pi$
$968$	−1.32202e16	−0.499945
$969$	1.84302e16	0.693026
$970$	0	0
$971$	−6.67861e15	−0.248302	−0.124151	−	0.992263i	$-0.539621\pi$
$971$	−6.67861e15	−0.248302	−0.124151	+	0.992263i	$0.539621\pi$
$972$	−1.72268e16	−0.636855
$973$	−7.34361e15	−0.269953
$974$	9.10715e15	0.332896
$975$	0	0
$976$	−1.59933e15	−0.0578048
$977$	3.64352e16	1.30949	0.654744	−	0.755851i	$-0.272777\pi$
$977$	3.64352e16	1.30949	0.654744	+	0.755851i	$0.272777\pi$
$978$	1.35445e16	0.484058
$979$	−1.28749e15	−0.0457552
$980$	0	0
$981$	1.41292e15	0.0496520
$982$	1.74587e16	0.610097
$983$	3.91117e16	1.35913	0.679567	−	0.733613i	$-0.262168\pi$
$983$	3.91117e16	1.35913	0.679567	+	0.733613i	$0.262168\pi$
$984$	−3.91982e16	−1.35454
$985$	0	0
$986$	−7.69550e15	−0.262975
$987$	1.63731e16	0.556400
$988$	−8.93364e15	−0.301901
$989$	4.53585e15	0.152433
$990$	0	0
$991$	−3.51778e15	−0.116913	−0.0584566	−	0.998290i	$-0.518618\pi$
$991$	−3.51778e15	−0.116913	−0.0584566	+	0.998290i	$0.518618\pi$
$992$	5.22965e16	1.72846
$993$	4.76593e16	1.56649
$994$	8.79026e15	0.287327
$995$	0	0
$996$	−2.48753e16	−0.804160
$997$	2.71445e15	0.0872687	0.0436343	−	0.999048i	$-0.486106\pi$
$997$	2.71445e15	0.0872687	0.0436343	+	0.999048i	$0.486106\pi$
$998$	1.83751e16	0.587505
$999$	2.26532e13	0.000720308 0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.12.a.e.1.2		4
3.2	odd	2		225.12.a.r.1.3		4
5.2	odd	4		5.12.b.a.4.2	✓	4
5.3	odd	4		5.12.b.a.4.3	yes	4
5.4	even	2	inner	25.12.a.e.1.3		4
15.2	even	4		45.12.b.b.19.3		4
15.8	even	4		45.12.b.b.19.2		4
15.14	odd	2		225.12.a.r.1.2		4
20.3	even	4		80.12.c.a.49.4		4
20.7	even	4		80.12.c.a.49.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.12.b.a.4.2	✓	4	5.2	odd	4
5.12.b.a.4.3	yes	4	5.3	odd	4
25.12.a.e.1.2		4	1.1	even	1	trivial
25.12.a.e.1.3		4	5.4	even	2	inner
45.12.b.b.19.2		4	15.8	even	4
45.12.b.b.19.3		4	15.2	even	4
80.12.c.a.49.1		4	20.7	even	4
80.12.c.a.49.4		4	20.3	even	4
225.12.a.r.1.2		4	15.14	odd	2
225.12.a.r.1.3		4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	25.a (trivial)

\(f(q)\)	\(=\)	\(q-27.1071 q^{2} -524.040 q^{3} -1313.20 q^{4} +14205.2 q^{6} +66589.5 q^{7} +91112.6 q^{8} +97470.8 q^{9} +O(q^{10})\)	\(q-27.1071 q^{2} -524.040 q^{3} -1313.20 q^{4} +14205.2 q^{6} +66589.5 q^{7} +91112.6 q^{8} +97470.8 q^{9} -374453. q^{11} +688170. q^{12} +1.12751e6 q^{13} -1.80505e6 q^{14} +219635. q^{16} -5.82894e6 q^{17} -2.64216e6 q^{18} +6.03360e6 q^{19} -3.48955e7 q^{21} +1.01503e7 q^{22} +7.51593e6 q^{23} -4.77466e7 q^{24} -3.05636e7 q^{26} +4.17535e7 q^{27} -8.74454e7 q^{28} -4.87039e7 q^{29} -2.71596e8 q^{31} -1.92552e8 q^{32} +1.96228e8 q^{33} +1.58006e8 q^{34} -1.27999e8 q^{36} +542545. q^{37} -1.63554e8 q^{38} -5.90860e8 q^{39} +8.20963e8 q^{41} +9.45918e8 q^{42} +6.03499e8 q^{43} +4.91732e8 q^{44} -2.03735e8 q^{46} -4.69204e8 q^{47} -1.15098e8 q^{48} +2.45683e9 q^{49} +3.05460e9 q^{51} -1.48065e9 q^{52} -3.82491e9 q^{53} -1.13182e9 q^{54} +6.06714e9 q^{56} -3.16185e9 q^{57} +1.32022e9 q^{58} -2.19696e9 q^{59} -7.28175e9 q^{61} +7.36220e9 q^{62} +6.49053e9 q^{63} +4.76973e9 q^{64} -5.31918e9 q^{66} +2.11343e9 q^{67} +7.65458e9 q^{68} -3.93865e9 q^{69} -4.86982e9 q^{71} +8.88082e9 q^{72} -1.92542e10 q^{73} -1.47068e7 q^{74} -7.92334e9 q^{76} -2.49346e10 q^{77} +1.60165e10 q^{78} -4.54858e10 q^{79} -3.91472e10 q^{81} -2.22540e10 q^{82} -3.61471e10 q^{83} +4.58249e10 q^{84} -1.63591e10 q^{86} +2.55228e10 q^{87} -3.41174e10 q^{88} +3.43834e9 q^{89} +7.50802e10 q^{91} -9.86994e9 q^{92} +1.42327e11 q^{93} +1.27188e10 q^{94} +1.00905e11 q^{96} +1.57518e11 q^{97} -6.65976e10 q^{98} -3.64982e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 72 q^{4} - 1752 q^{6} - 25452 q^{9}+O(q^{10}) \) 4 * q + 72 * q^4 - 1752 * q^6 - 25452 * q^9	\( 4 q + 72 q^{4} - 1752 q^{6} - 25452 q^{9} - 326352 q^{11} - 1080696 q^{14} - 9834976 q^{16} - 15460880 q^{19} - 81019872 q^{21} - 74415840 q^{24} - 325970832 q^{26} - 216242520 q^{29} - 684043072 q^{31} - 265782016 q^{34} - 553353336 q^{36} - 5997024 q^{39} + 1012873368 q^{41} + 1553573664 q^{44} + 5241789688 q^{46} + 1900646372 q^{49} + 8691953088 q^{51} + 6403356720 q^{54} + 10366738080 q^{56} - 3200971440 q^{59} - 2310471352 q^{61} + 5401150592 q^{64} - 17010985824 q^{66} - 32956101984 q^{69} - 60335466912 q^{71} + 5525992944 q^{74} - 52987638240 q^{76} - 74637768320 q^{79} - 77727716316 q^{81} + 76499865504 q^{84} + 39045421128 q^{86} - 118272499560 q^{89} + 51565095648 q^{91} + 266098749224 q^{94} + 313591828608 q^{96} - 119560366224 q^{99}+O(q^{100}) \) 4 * q + 72 * q^4 - 1752 * q^6 - 25452 * q^9 - 326352 * q^11 - 1080696 * q^14 - 9834976 * q^16 - 15460880 * q^19 - 81019872 * q^21 - 74415840 * q^24 - 325970832 * q^26 - 216242520 * q^29 - 684043072 * q^31 - 265782016 * q^34 - 553353336 * q^36 - 5997024 * q^39 + 1012873368 * q^41 + 1553573664 * q^44 + 5241789688 * q^46 + 1900646372 * q^49 + 8691953088 * q^51 + 6403356720 * q^54 + 10366738080 * q^56 - 3200971440 * q^59 - 2310471352 * q^61 + 5401150592 * q^64 - 17010985824 * q^66 - 32956101984 * q^69 - 60335466912 * q^71 + 5525992944 * q^74 - 52987638240 * q^76 - 74637768320 * q^79 - 77727716316 * q^81 + 76499865504 * q^84 + 39045421128 * q^86 - 118272499560 * q^89 + 51565095648 * q^91 + 266098749224 * q^94 + 313591828608 * q^96 - 119560366224 * q^99

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