LMFDB - Embedded newform 405.4.b.c.244.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,4,Mod(244,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("405.244");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	405.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$23.8957735523$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 44x^{6} + 567x^{4} + 2024x^{2} + 1900 $ x^8 + 44x^6 + 567x^4 + 2024*x^2 + 1900
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			244.5
Root			$1.22227i$ of defining polynomial
Character	$\chi$	$=$	405.244
Dual form			405.4.b.c.244.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.22227i q^{2} +6.50606 q^{4} +(10.5103 + 3.81233i) q^{5} -33.1668i q^{7} +17.7303i q^{8} +O(q^{10})$	\(q+1.22227i q^{2} +6.50606 q^{4} +(10.5103 + 3.81233i) q^{5} -33.1668i q^{7} +17.7303i q^{8} +(-4.65969 + 12.8464i) q^{10} +23.6760 q^{11} -30.9094i q^{13} +40.5387 q^{14} +30.3772 q^{16} -48.7539i q^{17} -34.3436 q^{19} +(68.3805 + 24.8032i) q^{20} +28.9385i q^{22} -181.561i q^{23} +(95.9323 + 80.1373i) q^{25} +37.7796 q^{26} -215.785i q^{28} +4.29174 q^{29} -270.521 q^{31} +178.972i q^{32} +59.5905 q^{34} +(126.443 - 348.592i) q^{35} +163.355i q^{37} -41.9772i q^{38} +(-67.5938 + 186.351i) q^{40} +7.40836 q^{41} -378.631i q^{43} +154.038 q^{44} +221.917 q^{46} +177.089i q^{47} -757.035 q^{49} +(-97.9494 + 117.255i) q^{50} -201.098i q^{52} +587.283i q^{53} +(248.842 + 90.2608i) q^{55} +588.057 q^{56} +5.24566i q^{58} +844.470 q^{59} +590.383 q^{61} -330.649i q^{62} +24.2662 q^{64} +(117.837 - 324.867i) q^{65} +494.174i q^{67} -317.196i q^{68} +(426.074 + 154.547i) q^{70} +53.0514 q^{71} -427.365i q^{73} -199.664 q^{74} -223.441 q^{76} -785.258i q^{77} +708.463 q^{79} +(319.273 + 115.808i) q^{80} +9.05501i q^{82} -243.242i q^{83} +(185.866 - 512.418i) q^{85} +462.790 q^{86} +419.784i q^{88} +558.336 q^{89} -1025.17 q^{91} -1181.25i q^{92} -216.450 q^{94} +(-360.961 - 130.929i) q^{95} +471.121i q^{97} -925.301i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{4} + 15 q^{5}+O(q^{10}) $ 8 * q - 24 * q^4 + 15 * q^5	$ 8 q - 24 q^{4} + 15 q^{5} + 7 q^{10} - 42 q^{14} + 4 q^{16} + 118 q^{19} - 129 q^{20} + 17 q^{25} - 36 q^{26} + 318 q^{29} - 416 q^{31} + 638 q^{34} + 192 q^{35} - 265 q^{40} + 486 q^{41} + 852 q^{44} - 598 q^{46} + 350 q^{49} + 1143 q^{50} - 162 q^{55} + 1530 q^{56} - 1146 q^{59} - 398 q^{61} + 1640 q^{64} - 1833 q^{65} + 630 q^{70} - 1728 q^{71} + 1218 q^{74} - 3498 q^{76} + 2596 q^{79} + 1923 q^{80} - 233 q^{85} + 480 q^{86} + 1086 q^{89} - 2574 q^{91} + 1238 q^{94} + 1674 q^{95}+O(q^{100}) $ 8 * q - 24 * q^4 + 15 * q^5 + 7 * q^10 - 42 * q^14 + 4 * q^16 + 118 * q^19 - 129 * q^20 + 17 * q^25 - 36 * q^26 + 318 * q^29 - 416 * q^31 + 638 * q^34 + 192 * q^35 - 265 * q^40 + 486 * q^41 + 852 * q^44 - 598 * q^46 + 350 * q^49 + 1143 * q^50 - 162 * q^55 + 1530 * q^56 - 1146 * q^59 - 398 * q^61 + 1640 * q^64 - 1833 * q^65 + 630 * q^70 - 1728 * q^71 + 1218 * q^74 - 3498 * q^76 + 2596 * q^79 + 1923 * q^80 - 233 * q^85 + 480 * q^86 + 1086 * q^89 - 2574 * q^91 + 1238 * q^94 + 1674 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$-1$	$1$

Coefficient data

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

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

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

$2$			1.22227i			0.432138i	0.976378	+	0.216069i	$0.0693235\pi$
$2$			1.22227i			0.432138i	−0.976378	+	0.216069i	$0.930676\pi$
$3$		0			0
$3$		0			0
$4$	6.50606			0.813257
$5$	10.5103	+	3.81233i	0.940069	+	0.340985i
$5$	10.5103	+	3.81233i	0.940069	+	0.340985i
$6$		0			0
$7$		−	33.1668i		−	1.79084i	−0.445225	−	0.895419i	$-0.646876\pi$
$7$		−	33.1668i		−	1.79084i	0.445225	−	0.895419i	$-0.353124\pi$
$8$			17.7303i			0.783577i
$9$		0			0
$10$	−4.65969	+	12.8464i	−0.147352	+	0.406239i
$11$	23.6760			0.648963			0.324482	−	0.945892i	$-0.394810\pi$
$11$	23.6760			0.648963			0.324482	+	0.945892i	$0.394810\pi$
$12$		0			0
$13$		−	30.9094i		−	0.659440i	−0.944079	−	0.329720i	$-0.893046\pi$
$13$		−	30.9094i		−	0.659440i	0.944079	−	0.329720i	$-0.106954\pi$
$14$	40.5387			0.773888
$15$		0			0
$16$	30.3772			0.474644
$17$		−	48.7539i		−	0.695563i	−0.937576	−	0.347781i	$-0.886935\pi$
$17$		−	48.7539i		−	0.695563i	0.937576	−	0.347781i	$-0.113065\pi$
$18$		0			0
$19$	−34.3436			−0.414682			−0.207341	−	0.978269i	$-0.566481\pi$
$19$	−34.3436			−0.414682			−0.207341	+	0.978269i	$0.566481\pi$
$20$	68.3805	+	24.8032i	0.764518	+	0.277308i
$21$		0			0
$22$			28.9385i			0.280441i
$23$		−	181.561i		−	1.64601i	−0.568036	−	0.823004i	$-0.692297\pi$
$23$		−	181.561i		−	1.64601i	0.568036	−	0.823004i	$-0.307703\pi$
$24$		0			0
$25$	95.9323	+	80.1373i	0.767459	+	0.641099i
$26$	37.7796			0.284969
$27$		0			0
$28$		−	215.785i		−	1.45641i
$29$	4.29174			0.0274812			0.0137406	−	0.999906i	$-0.495626\pi$
$29$	4.29174			0.0274812			0.0137406	+	0.999906i	$0.495626\pi$
$30$		0			0
$31$	−270.521			−1.56732			−0.783660	−	0.621190i	$-0.786650\pi$
$31$	−270.521			−1.56732			−0.783660	+	0.621190i	$0.786650\pi$
$32$			178.972i			0.988688i
$33$		0			0
$34$	59.5905			0.300579
$35$	126.443	−	348.592i	0.610649	−	1.68351i
$36$		0			0
$37$			163.355i			0.725821i	0.931824	+	0.362910i	$0.118217\pi$
$37$			163.355i			0.725821i	−0.931824	+	0.362910i	$0.881783\pi$
$38$		−	41.9772i		−	0.179200i
$39$		0			0
$40$	−67.5938	+	186.351i	−0.267188	+	0.736616i
$41$	7.40836			0.0282193			0.0141097	−	0.999900i	$-0.495509\pi$
$41$	7.40836			0.0282193			0.0141097	+	0.999900i	$0.495509\pi$
$42$		0			0
$43$		−	378.631i		−	1.34281i	−0.741091	−	0.671404i	$-0.765691\pi$
$43$		−	378.631i		−	1.34281i	0.741091	−	0.671404i	$-0.234309\pi$
$44$	154.038			0.527774
$45$		0			0
$46$	221.917			0.711302
$47$			177.089i			0.549596i	0.961502	+	0.274798i	$0.0886110\pi$
$47$			177.089i			0.549596i	−0.961502	+	0.274798i	$0.911389\pi$
$48$		0			0
$49$	−757.035			−2.20710
$50$	−97.9494	+	117.255i	−0.277043	+	0.331648i
$51$		0			0
$52$		−	201.098i		−	0.536294i
$53$			587.283i			1.52207i	0.648713	+	0.761033i	$0.275308\pi$
$53$			587.283i			1.52207i	−0.648713	+	0.761033i	$0.724692\pi$
$54$		0			0
$55$	248.842	+	90.2608i	0.610070	+	0.221287i
$56$	588.057			1.40326
$57$		0			0
$58$			5.24566i			0.0118757i
$59$	844.470			1.86340			0.931700	−	0.363228i	$-0.118325\pi$
$59$	844.470			1.86340			0.931700	+	0.363228i	$0.118325\pi$
$60$		0			0
$61$	590.383			1.23919			0.619596	−	0.784921i	$-0.287296\pi$
$61$	590.383			1.23919			0.619596	+	0.784921i	$0.287296\pi$
$62$		−	330.649i		−	0.677298i
$63$		0			0
$64$	24.2662			0.0473949
$65$	117.837	−	324.867i	0.224859	−	0.619919i
$66$		0			0
$67$			494.174i			0.901090i	0.892754	+	0.450545i	$0.148770\pi$
$67$			494.174i			0.901090i	−0.892754	+	0.450545i	$0.851230\pi$
$68$		−	317.196i		−	0.565671i
$69$		0			0
$70$	426.074	+	154.547i	0.727508	+	0.263884i
$71$	53.0514			0.0886767			0.0443383	−	0.999017i	$-0.485882\pi$
$71$	53.0514			0.0886767			0.0443383	+	0.999017i	$0.485882\pi$
$72$		0			0
$73$		−	427.365i		−	0.685196i	−0.939482	−	0.342598i	$-0.888693\pi$
$73$		−	427.365i		−	0.685196i	0.939482	−	0.342598i	$-0.111307\pi$
$74$	−199.664			−0.313654
$75$		0			0
$76$	−223.441			−0.337243
$77$		−	785.258i		−	1.16219i
$78$		0			0
$79$	708.463			1.00897			0.504483	−	0.863422i	$-0.331683\pi$
$79$	708.463			1.00897			0.504483	+	0.863422i	$0.331683\pi$
$80$	319.273	+	115.808i	0.446198	+	0.161847i
$81$		0			0
$82$			9.05501i			0.0121946i
$83$		−	243.242i		−	0.321678i	−0.986981	−	0.160839i	$-0.948580\pi$
$83$		−	243.242i		−	0.321678i	0.986981	−	0.160839i	$-0.0514199\pi$
$84$		0			0
$85$	185.866	−	512.418i	0.237176	−	0.653877i
$86$	462.790			0.580278
$87$		0			0
$88$			419.784i			0.508512i
$89$	558.336			0.664983			0.332491	−	0.943106i	$-0.392111\pi$
$89$	558.336			0.664983			0.332491	+	0.943106i	$0.392111\pi$
$90$		0			0
$91$	−1025.17			−1.18095
$92$		−	1181.25i		−	1.33863i
$93$		0			0
$94$	−216.450			−0.237501
$95$	−360.961	−	130.929i	−0.389830	−	0.141400i
$96$		0			0
$97$			471.121i			0.493145i	0.969124	+	0.246572i	$0.0793043\pi$
$97$			471.121i			0.493145i	−0.969124	+	0.246572i	$0.920696\pi$
$98$		−	925.301i		−	0.953771i
$99$		0			0
$100$	624.141	+	521.378i	0.624141	+	0.521378i
$101$	1173.91			1.15652			0.578258	−	0.815854i	$-0.303733\pi$
$101$	1173.91			1.15652			0.578258	+	0.815854i	$0.303733\pi$
$102$		0			0
$103$			1495.18i			1.43034i	0.698952	+	0.715169i	$0.253650\pi$
$103$			1495.18i			1.43034i	−0.698952	+	0.715169i	$0.746350\pi$
$104$	548.033			0.516722
$105$		0			0
$106$	−717.818			−0.657742
$107$			717.918i			0.648633i	0.945949	+	0.324317i	$0.105134\pi$
$107$			717.918i			0.648633i	−0.945949	+	0.324317i	$0.894866\pi$
$108$		0			0
$109$	−282.308			−0.248075			−0.124038	−	0.992278i	$-0.539584\pi$
$109$	−282.308			−0.248075			−0.124038	+	0.992278i	$0.539584\pi$
$110$	−110.323	+	304.152i	−0.0956263	+	0.263634i
$111$		0			0
$112$		−	1007.51i		−	0.850011i
$113$			2328.58i			1.93854i	0.246002	+	0.969269i	$0.420883\pi$
$113$			2328.58i			1.93854i	−0.246002	+	0.969269i	$0.579117\pi$
$114$		0			0
$115$	692.171	−	1908.26i	0.561264	−	1.54736i
$116$	27.9223			0.0223493
$117$		0			0
$118$			1032.17i			0.805246i
$119$	−1617.01			−1.24564
$120$		0			0
$121$	−770.445			−0.578847
$122$			721.607i			0.535502i
$123$		0			0
$124$	−1760.02			−1.27463
$125$	702.767	+	1207.99i	0.502859	+	0.864369i
$126$		0			0
$127$		−	162.215i		−	0.113341i	−0.998393	−	0.0566704i	$-0.981952\pi$
$127$		−	162.215i		−	0.113341i	0.998393	−	0.0566704i	$-0.0180484\pi$
$128$			1461.43i			1.00917i
$129$		0			0
$130$	397.075	+	144.028i	0.267890	+	0.0971701i
$131$	−2239.21			−1.49344			−0.746720	−	0.665138i	$-0.768373\pi$
$131$	−2239.21			−1.49344			−0.746720	+	0.665138i	$0.768373\pi$
$132$		0			0
$133$			1139.07i			0.742629i
$134$	−604.014			−0.389395
$135$		0			0
$136$	864.423			0.545027
$137$			1001.81i			0.624748i	0.949959	+	0.312374i	$0.101124\pi$
$137$			1001.81i			0.624748i	−0.949959	+	0.312374i	$0.898876\pi$
$138$		0			0
$139$	1793.56			1.09445			0.547224	−	0.836986i	$-0.315685\pi$
$139$	1793.56			1.09445			0.547224	+	0.836986i	$0.315685\pi$
$140$	822.643	−	2267.96i	0.496614	−	1.36913i
$141$		0			0
$142$			64.8431i			0.0383205i
$143$		−	731.812i		−	0.427952i
$144$		0			0
$145$	45.1074	+	16.3615i	0.0258343	+	0.00937069i
$146$	522.355			0.296099
$147$		0			0
$148$			1062.80i			0.590279i
$149$	−1986.20			−1.09205			−0.546027	−	0.837768i	$-0.683860\pi$
$149$	−1986.20			−1.09205			−0.546027	+	0.837768i	$0.683860\pi$
$150$		0			0
$151$	−1868.36			−1.00692			−0.503460	−	0.864018i	$-0.667940\pi$
$151$	−1868.36			−1.00692			−0.503460	+	0.864018i	$0.667940\pi$
$152$		−	608.923i		−	0.324935i
$153$		0			0
$154$	959.797			0.502225
$155$	−2843.25	−	1031.31i	−1.47339	−	0.534433i
$156$		0			0
$157$		−	1961.80i		−	0.997255i	−0.866816	−	0.498627i	$-0.833838\pi$
$157$		−	1961.80i		−	0.997255i	0.866816	−	0.498627i	$-0.166162\pi$
$158$			865.933i			0.436012i
$159$		0			0
$160$	−682.299	+	1881.04i	−0.337128	+	0.929435i
$161$	−6021.81			−2.94773
$162$		0			0
$163$		−	1586.28i		−	0.762254i	−0.924523	−	0.381127i	$-0.875536\pi$
$163$		−	1586.28i		−	0.762254i	0.924523	−	0.381127i	$-0.124464\pi$
$164$	48.1992			0.0229495
$165$		0			0
$166$	297.307			0.139009
$167$		−	2442.69i		−	1.13186i	−0.824452	−	0.565932i	$-0.808516\pi$
$167$		−	2442.69i		−	1.13186i	0.824452	−	0.565932i	$-0.191484\pi$
$168$		0			0
$169$	1241.61			0.565139
$170$	626.313	+	227.178i	0.282565	+	0.102493i
$171$		0			0
$172$		−	2463.40i		−	1.09205i
$173$			1520.90i			0.668391i	0.942504	+	0.334196i	$0.108465\pi$
$173$			1520.90i			0.668391i	−0.942504	+	0.334196i	$0.891535\pi$
$174$		0			0
$175$	2657.90	−	3181.77i	1.14810	−	1.37439i
$176$	719.212			0.308027
$177$		0			0
$178$			682.437i			0.287364i
$179$	−902.128			−0.376694			−0.188347	−	0.982103i	$-0.560313\pi$
$179$	−902.128			−0.376694			−0.188347	+	0.982103i	$0.560313\pi$
$180$		0			0
$181$	−3472.27			−1.42592			−0.712962	−	0.701203i	$-0.752647\pi$
$181$	−3472.27			−1.42592			−0.712962	+	0.701203i	$0.752647\pi$
$182$		−	1253.03i		−	0.510333i
$183$		0			0
$184$	3219.14			1.28977
$185$	−622.762	+	1716.91i	−0.247494	+	0.682321i
$186$		0			0
$187$		−	1154.30i		−	0.451394i
$188$			1152.15i			0.446963i
$189$		0			0
$190$	160.031	−	441.192i	0.0611044	−	0.168460i
$191$	1160.63			0.439688			0.219844	−	0.975535i	$-0.429445\pi$
$191$	1160.63			0.439688			0.219844	+	0.975535i	$0.429445\pi$
$192$		0			0
$193$		−	2061.38i		−	0.768815i	−0.923164	−	0.384408i	$-0.874406\pi$
$193$		−	2061.38i		−	0.768815i	0.923164	−	0.384408i	$-0.125594\pi$
$194$	−575.837			−0.213106
$195$		0			0
$196$	−4925.31			−1.79494
$197$			743.457i			0.268879i	0.990922	+	0.134439i	$0.0429233\pi$
$197$			743.457i			0.268879i	−0.990922	+	0.134439i	$0.957077\pi$
$198$		0			0
$199$	−350.592			−0.124888			−0.0624442	−	0.998048i	$-0.519890\pi$
$199$	−350.592			−0.124888			−0.0624442	+	0.998048i	$0.519890\pi$
$200$	−1420.86	+	1700.91i	−0.502350	+	0.601363i
$201$		0			0
$202$			1434.83i			0.499774i
$203$		−	142.343i		−	0.0492144i
$204$		0			0
$205$	77.8640	+	28.2431i	0.0265281	+	0.00962236i
$206$	−1827.52			−0.618103
$207$		0			0
$208$		−	938.942i		−	0.313000i
$209$	−813.121			−0.269114
$210$		0			0
$211$	151.733			0.0495059			0.0247529	−	0.999694i	$-0.492120\pi$
$211$	151.733			0.0495059			0.0247529	+	0.999694i	$0.492120\pi$
$212$			3820.90i			1.23783i
$213$		0			0
$214$	−877.490			−0.280299
$215$	1443.47	−	3979.53i	0.457877	−	1.26233i
$216$		0			0
$217$			8972.30i			2.80682i
$218$		−	345.057i		−	0.107203i
$219$		0			0
$220$	1618.98	+	587.242i	0.496144	+	0.179963i
$221$	−1506.95			−0.458682
$222$		0			0
$223$		−	2528.95i		−	0.759423i	−0.925105	−	0.379711i	$-0.876023\pi$
$223$		−	2528.95i		−	0.759423i	0.925105	−	0.379711i	$-0.123977\pi$
$224$	5935.91			1.77058
$225$		0			0
$226$	−2846.16			−0.837715
$227$			4898.39i			1.43224i	0.697979	+	0.716118i	$0.254083\pi$
$227$			4898.39i			1.43224i	−0.697979	+	0.716118i	$0.745917\pi$
$228$		0			0
$229$	2865.32			0.826838			0.413419	−	0.910541i	$-0.364334\pi$
$229$	2865.32			0.826838			0.413419	+	0.910541i	$0.364334\pi$
$230$	2332.41	+	846.020i	0.668672	+	0.242543i
$231$		0			0
$232$			76.0939i			0.0215337i
$233$			4330.31i			1.21754i	0.793345	+	0.608772i	$0.208338\pi$
$233$			4330.31i			1.21754i	−0.793345	+	0.608772i	$0.791662\pi$
$234$		0			0
$235$	−675.120	+	1861.25i	−0.187404	+	0.516658i
$236$	5494.17			1.51542
$237$		0			0
$238$		−	1976.42i		−	0.538288i
$239$	−3760.32			−1.01772			−0.508860	−	0.860850i	$-0.669933\pi$
$239$	−3760.32			−1.01772			−0.508860	+	0.860850i	$0.669933\pi$
$240$		0			0
$241$	−3105.62			−0.830087			−0.415043	−	0.909802i	$-0.636234\pi$
$241$	−3105.62			−0.830087			−0.415043	+	0.909802i	$0.636234\pi$
$242$		−	941.692i		−	0.250142i
$243$		0			0
$244$	3841.06			1.00778
$245$	−7956.66	−	2886.07i	−2.07483	−	0.752588i
$246$		0			0
$247$			1061.54i			0.273458i
$248$		−	4796.41i		−	1.22812i
$249$		0			0
$250$	−1476.49	+	858.970i	−0.373526	+	0.217304i
$251$	−2618.73			−0.658536			−0.329268	−	0.944236i	$-0.606802\pi$
$251$	−2618.73			−0.658536			−0.329268	+	0.944236i	$0.606802\pi$
$252$		0			0
$253$		−	4298.65i		−	1.06820i
$254$	198.271			0.0489788
$255$		0			0
$256$	−1592.14			−0.388705
$257$			1144.76i			0.277853i	0.990303	+	0.138926i	$0.0443652\pi$
$257$			1144.76i			0.277853i	−0.990303	+	0.138926i	$0.955635\pi$
$258$		0			0
$259$	5417.95			1.29983
$260$	766.652	−	2113.60i	0.182868	−	0.504154i
$261$		0			0
$262$		−	2736.92i		−	0.645372i
$263$		−	2950.25i		−	0.691713i	−0.938287	−	0.345856i	$-0.887588\pi$
$263$		−	2950.25i		−	0.691713i	0.938287	−	0.345856i	$-0.112412\pi$
$264$		0			0
$265$	−2238.92	+	6172.51i	−0.519002	+	1.43085i
$266$	−1392.25			−0.320918
$267$		0			0
$268$			3215.13i			0.732818i
$269$	−3069.15			−0.695647			−0.347824	−	0.937560i	$-0.613079\pi$
$269$	−3069.15			−0.695647			−0.347824	+	0.937560i	$0.613079\pi$
$270$		0			0
$271$	−6181.73			−1.38566			−0.692829	−	0.721102i	$-0.743636\pi$
$271$	−6181.73			−1.38566			−0.692829	+	0.721102i	$0.743636\pi$
$272$		−	1481.01i		−	0.330145i
$273$		0			0
$274$	−1224.48			−0.269977
$275$	2271.30	+	1897.33i	0.498052	+	0.416049i
$276$		0			0
$277$			5044.68i			1.09424i	0.837053	+	0.547122i	$0.184276\pi$
$277$			5044.68i			1.09424i	−0.837053	+	0.547122i	$0.815724\pi$
$278$			2192.22i			0.472952i
$279$		0			0
$280$	6180.65	+	2241.87i	1.31916	+	0.478490i
$281$	6352.35			1.34857			0.674287	−	0.738469i	$-0.264451\pi$
$281$	6352.35			1.34857			0.674287	+	0.738469i	$0.264451\pi$
$282$		0			0
$283$			1303.15i			0.273726i	0.990590	+	0.136863i	$0.0437020\pi$
$283$			1303.15i			0.273726i	−0.990590	+	0.136863i	$0.956298\pi$
$284$	345.156			0.0721169
$285$		0			0
$286$	894.472			0.184934
$287$		−	245.711i		−	0.0505362i
$288$		0			0
$289$	2536.05			0.516193
$290$	−19.9982	+	55.1334i	−0.00404943	+	0.0111640i
$291$		0			0
$292$		−	2780.46i		−	0.557240i
$293$		−	2433.97i		−	0.485303i	−0.970113	−	0.242652i	$-0.921983\pi$
$293$		−	2433.97i		−	0.485303i	0.970113	−	0.242652i	$-0.0780172\pi$
$294$		0			0
$295$	8875.62	+	3219.40i	1.75172	+	0.635392i
$296$	−2896.33			−0.568736
$297$		0			0
$298$		−	2427.67i		−	0.471917i
$299$	−5611.95			−1.08544
$300$		0			0
$301$	−12558.0			−2.40475
$302$		−	2283.64i		−	0.435128i
$303$		0			0
$304$	−1043.26			−0.196827
$305$	6205.09	+	2250.73i	1.16493	+	0.422546i
$306$		0			0
$307$		−	5629.59i		−	1.04657i	−0.852157	−	0.523286i	$-0.824706\pi$
$307$		−	5629.59i		−	1.04657i	0.852157	−	0.523286i	$-0.175294\pi$
$308$		−	5108.93i		−	0.945157i
$309$		0			0
$310$	1260.54	−	3475.22i	0.230948	−	0.636707i
$311$	4607.52			0.840092			0.420046	−	0.907503i	$-0.362014\pi$
$311$	4607.52			0.840092			0.420046	+	0.907503i	$0.362014\pi$
$312$		0			0
$313$		−	7214.85i		−	1.30290i	−0.758692	−	0.651450i	$-0.774161\pi$
$313$		−	7214.85i		−	1.30290i	0.758692	−	0.651450i	$-0.225839\pi$
$314$	2397.85			0.430951
$315$		0			0
$316$	4609.30			0.820549
$317$		−	6183.53i		−	1.09559i	−0.836613	−	0.547794i	$-0.815468\pi$
$317$		−	6183.53i		−	1.09559i	0.836613	−	0.547794i	$-0.184532\pi$
$318$		0			0
$319$	101.611			0.0178343
$320$	255.045	+	92.5106i	0.0445545	+	0.0161609i
$321$		0			0
$322$		−	7360.27i		−	1.27383i
$323$			1674.39i			0.288438i
$324$		0			0
$325$	2477.00	−	2965.21i	0.422766	−	0.506093i
$326$	1938.87			0.329398
$327$		0			0
$328$			131.353i			0.0221120i
$329$	5873.46			0.984238
$330$		0			0
$331$	1657.31			0.275208			0.137604	−	0.990487i	$-0.456060\pi$
$331$	1657.31			0.275208			0.137604	+	0.990487i	$0.456060\pi$
$332$		−	1582.54i		−	0.261607i
$333$		0			0
$334$	2985.63			0.489121
$335$	−1883.95	+	5193.92i	−0.307258	+	0.847086i
$336$		0			0
$337$			8802.58i			1.42287i	0.702752	+	0.711435i	$0.251955\pi$
$337$			8802.58i			1.42287i	−0.702752	+	0.711435i	$0.748045\pi$
$338$			1517.58i			0.244218i
$339$		0			0
$340$	1209.25	−	3333.82i	0.192885	−	0.531770i
$341$	−6404.86			−1.01713
$342$		0			0
$343$			13732.2i			2.16172i
$344$	6713.26			1.05219
$345$		0			0
$346$	−1858.95			−0.288837
$347$			3024.89i			0.467967i	0.972241	+	0.233983i	$0.0751762\pi$
$347$			3024.89i			0.467967i	−0.972241	+	0.233983i	$0.924824\pi$
$348$		0			0
$349$	8247.55			1.26499			0.632494	−	0.774565i	$-0.282031\pi$
$349$	8247.55			1.26499			0.632494	+	0.774565i	$0.282031\pi$
$350$	3888.98	+	3248.67i	0.593927	+	0.496139i
$351$		0			0
$352$			4237.34i			0.641622i
$353$			10224.9i			1.54169i	0.637021	+	0.770847i	$0.280167\pi$
$353$			10224.9i			1.54169i	−0.637021	+	0.770847i	$0.719833\pi$
$354$		0			0
$355$	557.586	+	202.249i	0.0833622	+	0.0302374i
$356$	3632.56			0.540802
$357$		0			0
$358$		−	1102.64i		−	0.162784i
$359$	9178.35			1.34934			0.674672	−	0.738118i	$-0.264285\pi$
$359$	9178.35			1.34934			0.674672	+	0.738118i	$0.264285\pi$
$360$		0			0
$361$	−5679.52			−0.828039
$362$		−	4244.05i		−	0.616195i
$363$		0			0
$364$	−6669.78			−0.960416
$365$	1629.26	−	4491.73i	0.233641	−	0.644131i
$366$		0			0
$367$			640.156i			0.0910514i	0.998963	+	0.0455257i	$0.0144963\pi$
$367$			640.156i			0.0910514i	−0.998963	+	0.0455257i	$0.985504\pi$
$368$		−	5515.33i		−	0.781268i
$369$		0			0
$370$	−2098.52	−	761.183i	−0.294857	−	0.106951i
$371$	19478.3			2.72577
$372$		0			0
$373$			4807.89i			0.667408i	0.942678	+	0.333704i	$0.108299\pi$
$373$			4807.89i			0.667408i	−0.942678	+	0.333704i	$0.891701\pi$
$374$	1410.87			0.195065
$375$		0			0
$376$	−3139.84			−0.430651
$377$		−	132.655i		−	0.0181222i
$378$		0			0
$379$	−4449.52			−0.603051			−0.301526	−	0.953458i	$-0.597496\pi$
$379$	−4449.52			−0.603051			−0.301526	+	0.953458i	$0.597496\pi$
$380$	−2348.43	−	851.832i	−0.317032	−	0.114995i
$381$		0			0
$382$			1418.61i			0.190006i
$383$			6289.84i			0.839153i	0.907720	+	0.419577i	$0.137821\pi$
$383$			6289.84i			0.839153i	−0.907720	+	0.419577i	$0.862179\pi$
$384$		0			0
$385$	2993.66	−	8253.29i	0.396288	−	1.09254i
$386$	2519.56			0.332234
$387$		0			0
$388$			3065.14i			0.401054i
$389$	−5650.19			−0.736442			−0.368221	−	0.929738i	$-0.620033\pi$
$389$	−5650.19			−0.736442			−0.368221	+	0.929738i	$0.620033\pi$
$390$		0			0
$391$	−8851.83			−1.14490
$392$		−	13422.5i		−	1.72943i
$393$		0			0
$394$	−908.705			−0.116193
$395$	7446.15	+	2700.89i	0.948498	+	0.344042i
$396$		0			0
$397$		−	11922.6i		−	1.50725i	−0.657307	−	0.753623i	$-0.728304\pi$
$397$		−	11922.6i		−	1.50725i	0.657307	−	0.753623i	$-0.271696\pi$
$398$		−	428.517i		−	0.0539689i
$399$		0			0
$400$	2914.16	+	2434.35i	0.364270	+	0.304294i
$401$	−14042.9			−1.74881			−0.874403	−	0.485201i	$-0.838746\pi$
$401$	−14042.9			−1.74881			−0.874403	+	0.485201i	$0.838746\pi$
$402$		0			0
$403$			8361.63i			1.03355i
$404$	7637.50			0.940544
$405$		0			0
$406$	173.982			0.0212674
$407$			3867.59i			0.471031i
$408$		0			0
$409$	−4015.80			−0.485498			−0.242749	−	0.970089i	$-0.578049\pi$
$409$	−4015.80			−0.485498			−0.242749	+	0.970089i	$0.578049\pi$
$410$	−34.5207	+	95.1708i	−0.00415818	+	0.0114638i
$411$		0			0
$412$			9727.75i			1.16323i
$413$		−	28008.4i		−	3.33705i
$414$		0			0
$415$	927.317	−	2556.54i	0.109687	−	0.302399i
$416$	5531.91			0.651981
$417$		0			0
$418$		−	993.853i		−	0.116294i
$419$	15881.9			1.85175			0.925873	−	0.377835i	$-0.123331\pi$
$419$	15881.9			1.85175			0.925873	+	0.377835i	$0.123331\pi$
$420$		0			0
$421$	3401.87			0.393817			0.196908	−	0.980422i	$-0.436910\pi$
$421$	3401.87			0.393817			0.196908	+	0.980422i	$0.436910\pi$
$422$			185.459i			0.0213934i
$423$		0			0
$424$	−10412.7			−1.19266
$425$	3907.01	−	4677.08i	0.445924	−	0.533816i
$426$		0			0
$427$		−	19581.1i		−	2.21919i
$428$			4670.82i			0.527506i
$429$		0			0
$430$	4864.05	+	1764.31i	0.545501	+	0.197866i
$431$	−6067.90			−0.678145			−0.339072	−	0.940760i	$-0.610113\pi$
$431$	−6067.90			−0.678145			−0.339072	+	0.940760i	$0.610113\pi$
$432$		0			0
$433$			2368.69i			0.262891i	0.991323	+	0.131446i	$0.0419619\pi$
$433$			2368.69i			0.262891i	−0.991323	+	0.131446i	$0.958038\pi$
$434$	−10966.6			−1.21293
$435$		0			0
$436$	−1836.71			−0.201749
$437$			6235.47i			0.682570i
$438$		0			0
$439$	3731.54			0.405687			0.202844	−	0.979211i	$-0.434982\pi$
$439$	3731.54			0.405687			0.202844	+	0.979211i	$0.434982\pi$
$440$	−1600.35	+	4412.05i	−0.173395	+	0.478036i
$441$		0			0
$442$		−	1841.90i		−	0.198214i
$443$		−	15917.5i		−	1.70714i	−0.520975	−	0.853572i	$-0.674431\pi$
$443$		−	15917.5i		−	1.70714i	0.520975	−	0.853572i	$-0.325569\pi$
$444$		0			0
$445$	5868.27	+	2128.56i	0.625129	+	0.226749i
$446$	3091.06			0.328175
$447$		0			0
$448$		−	804.831i		−	0.0848766i
$449$	−9088.41			−0.955252			−0.477626	−	0.878563i	$-0.658503\pi$
$449$	−9088.41			−0.955252			−0.477626	+	0.878563i	$0.658503\pi$
$450$		0			0
$451$	175.401			0.0183133
$452$			15149.9i			1.57653i
$453$		0			0
$454$	−5987.16			−0.618923
$455$	−10774.8	−	3908.26i	−1.11017	−	0.402686i
$456$		0			0
$457$		−	11877.8i		−	1.21580i	−0.794013	−	0.607901i	$-0.792012\pi$
$457$		−	11877.8i		−	1.21580i	0.794013	−	0.607901i	$-0.207988\pi$
$458$			3502.20i			0.357308i
$459$		0			0
$460$	4503.31	−	12415.3i	0.456452	−	1.25840i
$461$	−7251.15			−0.732581			−0.366291	−	0.930501i	$-0.619372\pi$
$461$	−7251.15			−0.732581			−0.366291	+	0.930501i	$0.619372\pi$
$462$		0			0
$463$		−	1778.78i		−	0.178547i	−0.996007	−	0.0892733i	$-0.971546\pi$
$463$		−	1778.78i		−	0.178547i	0.996007	−	0.0892733i	$-0.0284545\pi$
$464$	130.371			0.0130438
$465$		0			0
$466$	−5292.80			−0.526147
$467$			5153.79i			0.510683i	0.966851	+	0.255342i	$0.0821879\pi$
$467$			5153.79i			0.510683i	−0.966851	+	0.255342i	$0.917812\pi$
$468$		0			0
$469$	16390.2			1.61371
$470$	−2274.95	−	825.178i	−0.223267	−	0.0809843i
$471$		0			0
$472$			14972.7i			1.46012i
$473$		−	8964.49i		−	0.871433i
$474$		0			0
$475$	−3294.66	−	2752.20i	−0.318252	−	0.265852i
$476$	−10520.4			−1.01303
$477$		0			0
$478$		−	4596.13i		−	0.439795i
$479$	7850.07			0.748808			0.374404	−	0.927266i	$-0.377847\pi$
$479$	7850.07			0.748808			0.374404	+	0.927266i	$0.377847\pi$
$480$		0			0
$481$	5049.20			0.478635
$482$		−	3795.91i		−	0.358712i
$483$		0			0
$484$	−5012.56			−0.470751
$485$	−1796.07	+	4951.61i	−0.168155	+	0.463590i
$486$		0			0
$487$			8659.46i			0.805744i	0.915256	+	0.402872i	$0.131988\pi$
$487$			8659.46i			0.805744i	−0.915256	+	0.402872i	$0.868012\pi$
$488$			10467.7i			0.971002i
$489$		0			0
$490$	3527.55	−	9725.18i	0.325221	−	0.896610i
$491$	1990.75			0.182977			0.0914883	−	0.995806i	$-0.470838\pi$
$491$	1990.75			0.182977			0.0914883	+	0.995806i	$0.470838\pi$
$492$		0			0
$493$		−	209.239i		−	0.0191149i
$494$	−1297.49			−0.118172
$495$		0			0
$496$	−8217.67			−0.743920
$497$		−	1759.54i		−	0.158806i
$498$		0			0
$499$	11262.0			1.01033			0.505166	−	0.863022i	$-0.331431\pi$
$499$	11262.0			1.01033			0.505166	+	0.863022i	$0.331431\pi$
$500$	4572.24	+	7859.26i	0.408954	+	0.702954i
$501$		0			0
$502$		−	3200.79i		−	0.284578i
$503$			9837.79i			0.872058i	0.899933	+	0.436029i	$0.143616\pi$
$503$			9837.79i			0.872058i	−0.899933	+	0.436029i	$0.856384\pi$
$504$		0			0
$505$	12338.1	+	4475.31i	1.08720	+	0.394354i
$506$	5254.11			0.461608
$507$		0			0
$508$		−	1055.38i		−	0.0921753i
$509$	−9403.17			−0.818837			−0.409419	−	0.912347i	$-0.634268\pi$
$509$	−9403.17			−0.818837			−0.409419	+	0.912347i	$0.634268\pi$
$510$		0			0
$511$	−14174.3			−1.22707
$512$			9745.45i			0.841195i
$513$		0			0
$514$	−1399.21			−0.120071
$515$	−5700.13	+	15714.8i	−0.487723	+	1.34462i
$516$		0			0
$517$			4192.76i			0.356668i
$518$			6622.20i			0.561704i
$519$		0			0
$520$	5759.99	+	2089.28i	0.485754	+	0.176194i
$521$	6915.28			0.581505			0.290752	−	0.956798i	$-0.406094\pi$
$521$	6915.28			0.581505			0.290752	+	0.956798i	$0.406094\pi$
$522$		0			0
$523$			9137.40i			0.763959i	0.924171	+	0.381980i	$0.124758\pi$
$523$			9137.40i			0.763959i	−0.924171	+	0.381980i	$0.875242\pi$
$524$	−14568.4			−1.21455
$525$		0			0
$526$	3606.01			0.298915
$527$			13188.9i			1.09017i
$528$		0			0
$529$	−20797.5			−1.70934
$530$	−7544.48	−	2736.56i	−0.618323	−	0.224280i
$531$		0			0
$532$			7410.83i			0.603948i
$533$		−	228.988i		−	0.0186089i
$534$		0			0
$535$	−2736.94	+	7545.53i	−0.221174	+	0.609760i
$536$	−8761.87			−0.706073
$537$		0			0
$538$		−	3751.32i		−	0.300615i
$539$	−17923.6			−1.43233
$540$		0			0
$541$	4862.56			0.386429			0.193214	−	0.981157i	$-0.438109\pi$
$541$	4862.56			0.386429			0.193214	+	0.981157i	$0.438109\pi$
$542$		−	7555.74i		−	0.598795i
$543$		0			0
$544$	8725.57			0.687695
$545$	−2967.14	−	1076.25i	−0.233208	−	0.0845900i
$546$		0			0
$547$			1037.26i			0.0810791i	0.999178	+	0.0405395i	$0.0129077\pi$
$547$			1037.26i			0.0810791i	−0.999178	+	0.0405395i	$0.987092\pi$
$548$			6517.84i			0.508081i
$549$		0			0
$550$	−2319.05	+	2776.14i	−0.179791	+	0.215227i
$551$	−147.394			−0.0113960
$552$		0			0
$553$		−	23497.4i		−	1.80689i
$554$	−6165.96			−0.472864
$555$		0			0
$556$	11669.0			0.890067
$557$		−	86.7829i		−	0.00660164i	−0.999995	−	0.00330082i	$-0.998949\pi$
$557$		−	86.7829i		−	0.00660164i	0.999995	−	0.00330082i	$-0.00105068\pi$
$558$		0			0
$559$	−11703.3			−0.885502
$560$	3840.98	−	10589.3i	0.289841	−	0.799069i
$561$		0			0
$562$			7764.28i			0.582770i
$563$		−	15914.5i		−	1.19133i	−0.803234	−	0.595664i	$-0.796889\pi$
$563$		−	15914.5i		−	1.19133i	0.803234	−	0.595664i	$-0.203111\pi$
$564$		0			0
$565$	−8877.33	+	24474.1i	−0.661012	+	1.82236i
$566$	−1592.81			−0.118287
$567$		0			0
$568$			940.618i			0.0694850i
$569$	−14177.4			−1.04454			−0.522272	−	0.852779i	$-0.674916\pi$
$569$	−14177.4			−1.04454			−0.522272	+	0.852779i	$0.674916\pi$
$570$		0			0
$571$	21676.4			1.58867			0.794333	−	0.607483i	$-0.207821\pi$
$571$	21676.4			1.58867			0.794333	+	0.607483i	$0.207821\pi$
$572$		−	4761.21i		−	0.348035i
$573$		0			0
$574$	300.326			0.0218386
$575$	14549.8	−	17417.6i	1.05525	−	1.26324i
$576$		0			0
$577$		−	6083.49i		−	0.438924i	−0.975621	−	0.219462i	$-0.929570\pi$
$577$		−	6083.49i		−	0.438924i	0.975621	−	0.219462i	$-0.0704302\pi$
$578$			3099.74i			0.223066i
$579$		0			0
$580$	293.471	+	106.449i	0.0210099	+	0.00762078i
$581$	−8067.54			−0.576072
$582$		0			0
$583$			13904.5i			0.987765i
$584$	7577.32			0.536903
$585$		0			0
$586$	2974.96			0.209718
$587$		−	8266.51i		−	0.581253i	−0.956837	−	0.290626i	$-0.906136\pi$
$587$		−	8266.51i		−	0.581253i	0.956837	−	0.290626i	$-0.0938636\pi$
$588$		0			0
$589$	9290.65			0.649940
$590$	−3934.97	+	10848.4i	−0.274577	+	0.756986i
$591$		0			0
$592$			4962.27i			0.344507i
$593$			18155.4i			1.25725i	0.777707	+	0.628627i	$0.216383\pi$
$593$			18155.4i			1.25725i	−0.777707	+	0.628627i	$0.783617\pi$
$594$		0			0
$595$	−16995.3	−	6164.57i	−1.17099	−	0.424744i
$596$	−12922.3			−0.888120
$597$		0			0
$598$		−	6859.32i		−	0.469061i
$599$	6899.65			0.470637			0.235319	−	0.971918i	$-0.424387\pi$
$599$	6899.65			0.470637			0.235319	+	0.971918i	$0.424387\pi$
$600$		0			0
$601$	−8402.89			−0.570318			−0.285159	−	0.958480i	$-0.592046\pi$
$601$	−8402.89			−0.570318			−0.285159	+	0.958480i	$0.592046\pi$
$602$		−	15349.2i		−	1.03918i
$603$		0			0
$604$	−12155.7			−0.818886
$605$	−8097.60	−	2937.19i	−0.544156	−	0.197378i
$606$		0			0
$607$		−	16342.9i		−	1.09281i	−0.837520	−	0.546407i	$-0.815995\pi$
$607$		−	16342.9i		−	1.09281i	0.837520	−	0.546407i	$-0.184005\pi$
$608$		−	6146.53i		−	0.409992i
$609$		0			0
$610$	−2751.00	+	7584.30i	−0.182598	+	0.503409i
$611$	5473.70			0.362426
$612$		0			0
$613$		−	15671.0i		−	1.03254i	−0.856427	−	0.516268i	$-0.827321\pi$
$613$		−	15671.0i		−	1.03254i	0.856427	−	0.516268i	$-0.172679\pi$
$614$	6880.87			0.452263
$615$		0			0
$616$	13922.9			0.910663
$617$			30226.8i			1.97226i	0.165962	+	0.986132i	$0.446927\pi$
$617$			30226.8i			1.97226i	−0.165962	+	0.986132i	$0.553073\pi$
$618$		0			0
$619$	−4890.59			−0.317560			−0.158780	−	0.987314i	$-0.550756\pi$
$619$	−4890.59			−0.317560			−0.158780	+	0.987314i	$0.550756\pi$
$620$	−18498.3	−	6709.78i	−1.19824	−	0.434631i
$621$		0			0
$622$			5631.64i			0.363035i
$623$		−	18518.2i		−	1.19088i
$624$		0			0
$625$	2781.02	+	15375.5i	0.177985	+	0.984033i
$626$	8818.49			0.563032
$627$		0			0
$628$		−	12763.6i		−	0.811025i
$629$	7964.19			0.504854
$630$		0			0
$631$	−18164.8			−1.14600			−0.573001	−	0.819554i	$-0.694221\pi$
$631$	−18164.8			−1.14600			−0.573001	+	0.819554i	$0.694221\pi$
$632$			12561.3i			0.790602i
$633$		0			0
$634$	7557.94			0.473445
$635$	618.418	−	1704.93i	0.0386475	−	0.106548i
$636$		0			0
$637$			23399.5i			1.45545i
$638$			124.197i			0.00770688i
$639$		0			0
$640$	−5571.46	+	15360.1i	−0.344112	+	0.948689i
$641$	−17122.9			−1.05509			−0.527546	−	0.849526i	$-0.676888\pi$
$641$	−17122.9			−1.05509			−0.527546	+	0.849526i	$0.676888\pi$
$642$		0			0
$643$			17652.2i			1.08263i	0.840819	+	0.541317i	$0.182074\pi$
$643$			17652.2i			1.08263i	−0.840819	+	0.541317i	$0.817926\pi$
$644$	−39178.2			−2.39726
$645$		0			0
$646$	−2046.55			−0.124645
$647$		−	6134.20i		−	0.372736i	−0.982480	−	0.186368i	$-0.940328\pi$
$647$		−	6134.20i		−	0.372736i	0.982480	−	0.186368i	$-0.0596717\pi$
$648$		0			0
$649$	19993.7			1.20928
$650$	3624.29	+	3027.56i	0.218702	+	0.182693i
$651$		0			0
$652$		−	10320.5i		−	0.619908i
$653$		−	12879.6i		−	0.771850i	−0.922530	−	0.385925i	$-0.873882\pi$
$653$		−	12879.6i		−	0.771850i	0.922530	−	0.385925i	$-0.126118\pi$
$654$		0			0
$655$	−23534.8	−	8536.60i	−1.40394	−	0.509241i
$656$	225.045			0.0133941
$657$		0			0
$658$			7178.95i			0.425326i
$659$	28522.0			1.68598			0.842990	−	0.537929i	$-0.180793\pi$
$659$	28522.0			1.68598			0.842990	+	0.537929i	$0.180793\pi$
$660$		0			0
$661$	−3738.93			−0.220011			−0.110006	−	0.993931i	$-0.535087\pi$
$661$	−3738.93			−0.220011			−0.110006	+	0.993931i	$0.535087\pi$
$662$			2025.67i			0.118928i
$663$		0			0
$664$	4312.75			0.252059
$665$	−4342.50	+	11971.9i	−0.253225	+	0.698122i
$666$		0			0
$667$		−	779.214i		−	0.0452343i
$668$		−	15892.3i		−	0.920497i
$669$		0			0
$670$	−6348.37	−	2302.70i	−0.366058	−	0.132778i
$671$	13977.9			0.804190
$672$		0			0
$673$		−	22142.8i		−	1.26827i	−0.773224	−	0.634133i	$-0.781357\pi$
$673$		−	22142.8i		−	1.26827i	0.773224	−	0.634133i	$-0.218643\pi$
$674$	−10759.1			−0.614876
$675$		0			0
$676$	8077.98			0.459603
$677$			11631.7i			0.660329i	0.943923	+	0.330164i	$0.107104\pi$
$677$			11631.7i			0.660329i	−0.943923	+	0.330164i	$0.892896\pi$
$678$		0			0
$679$	15625.6			0.883143
$680$	9085.33	+	3295.46i	0.512362	+	0.185846i
$681$		0			0
$682$		−	7828.46i		−	0.439541i
$683$			28361.6i			1.58891i	0.607321	+	0.794456i	$0.292244\pi$
$683$			28361.6i			1.58891i	−0.607321	+	0.794456i	$0.707756\pi$
$684$		0			0
$685$	−3819.23	+	10529.3i	−0.213030	+	0.587306i
$686$	−16784.5			−0.934160
$687$		0			0
$688$		−	11501.8i		−	0.637356i
$689$	18152.6			1.00371
$690$		0			0
$691$	−8292.49			−0.456529			−0.228264	−	0.973599i	$-0.573305\pi$
$691$	−8292.49			−0.456529			−0.228264	+	0.973599i	$0.573305\pi$
$692$			9895.05i			0.543574i
$693$		0			0
$694$	−3697.23			−0.202226
$695$	18850.9	+	6837.66i	1.02886	+	0.373190i
$696$		0			0
$697$		−	361.187i		−	0.0196283i
$698$			10080.7i			0.546649i
$699$		0			0
$700$	17292.4	−	20700.8i	0.933703	−	1.11774i
$701$	−586.036			−0.0315753			−0.0157876	−	0.999875i	$-0.505026\pi$
$701$	−586.036			−0.0315753			−0.0157876	+	0.999875i	$0.505026\pi$
$702$		0			0
$703$		−	5610.19i		−	0.300985i
$704$	574.527			0.0307575
$705$		0			0
$706$	−12497.6			−0.666224
$707$		−	38934.7i		−	2.07113i
$708$		0			0
$709$	17673.2			0.936149			0.468075	−	0.883689i	$-0.344948\pi$
$709$	17673.2			0.936149			0.468075	+	0.883689i	$0.344948\pi$
$710$	−247.203	+	681.520i	−0.0130667	+	0.0360239i
$711$		0			0
$712$			9899.46i			0.521065i
$713$			49116.1i			2.57982i
$714$		0			0
$715$	2789.91	−	7691.55i	0.145925	−	0.402305i
$716$	−5869.30			−0.306349
$717$		0			0
$718$			11218.4i			0.583102i
$719$	−2714.68			−0.140807			−0.0704037	−	0.997519i	$-0.522429\pi$
$719$	−2714.68			−0.140807			−0.0704037	+	0.997519i	$0.522429\pi$
$720$		0			0
$721$	49590.4			2.56150
$722$		−	6941.90i		−	0.357827i
$723$		0			0
$724$	−22590.8			−1.15964
$725$	411.717	+	343.929i	0.0210907	+	0.0176182i
$726$		0			0
$727$		−	2595.19i		−	0.132394i	−0.997807	−	0.0661969i	$-0.978913\pi$
$727$		−	2595.19i		−	0.132394i	0.997807	−	0.0661969i	$-0.0210865\pi$
$728$		−	18176.5i		−	0.925365i
$729$		0			0
$730$	5490.11	+	1991.39i	0.278353	+	0.100965i
$731$	−18459.8			−0.934007
$732$		0			0
$733$			8334.71i			0.419986i	0.977703	+	0.209993i	$0.0673441\pi$
$733$			8334.71i			0.419986i	−0.977703	+	0.209993i	$0.932656\pi$
$734$	−782.443			−0.0393467
$735$		0			0
$736$	32494.3			1.62739
$737$			11700.1i			0.584774i
$738$		0			0
$739$	18970.8			0.944317			0.472159	−	0.881514i	$-0.343475\pi$
$739$	18970.8			0.944317			0.472159	+	0.881514i	$0.343475\pi$
$740$	−4051.73	+	11170.3i	−0.201276	+	0.554903i
$741$		0			0
$742$			23807.7i			1.17791i
$743$			15043.1i			0.742770i	0.928479	+	0.371385i	$0.121117\pi$
$743$			15043.1i			0.742770i	−0.928479	+	0.371385i	$0.878883\pi$
$744$		0			0
$745$	−20875.5	−	7572.05i	−1.02661	−	0.372374i
$746$	−5876.54			−0.288412
$747$		0			0
$748$		−	7509.94i		−	0.367100i
$749$	23811.0			1.16160
$750$		0			0
$751$	7098.11			0.344892			0.172446	−	0.985019i	$-0.444833\pi$
$751$	7098.11			0.344892			0.172446	+	0.985019i	$0.444833\pi$
$752$			5379.46i			0.260863i
$753$		0			0
$754$	162.140			0.00783130
$755$	−19637.0	−	7122.80i	−0.946575	−	0.343345i
$756$		0			0
$757$		−	25376.1i		−	1.21837i	−0.793027	−	0.609187i	$-0.791496\pi$
$757$		−	25376.1i		−	1.21837i	0.793027	−	0.609187i	$-0.208504\pi$
$758$		−	5438.51i		−	0.260601i
$759$		0			0
$760$	2321.41	−	6399.96i	0.110798	−	0.305462i
$761$	7489.36			0.356753			0.178377	−	0.983962i	$-0.442915\pi$
$761$	7489.36			0.356753			0.178377	+	0.983962i	$0.442915\pi$
$762$		0			0
$763$			9363.25i			0.444263i
$764$	7551.14			0.357580
$765$		0			0
$766$	−7687.88			−0.362630
$767$		−	26102.1i		−	1.22880i
$768$		0			0
$769$	−35506.8			−1.66503			−0.832515	−	0.554002i	$-0.813100\pi$
$769$	−35506.8			−1.66503			−0.832515	+	0.554002i	$0.813100\pi$
$770$	10087.7	+	3659.06i	0.472126	+	0.171251i
$771$		0			0
$772$		−	13411.4i		−	0.625244i
$773$		−	37535.4i		−	1.74651i	−0.487259	−	0.873257i	$-0.662003\pi$
$773$		−	37535.4i		−	1.74651i	0.487259	−	0.873257i	$-0.337997\pi$
$774$		0			0
$775$	−25951.7	−	21678.8i	−1.20285	−	1.00481i
$776$	−8353.12			−0.386417
$777$		0			0
$778$		−	6906.05i		−	0.318244i
$779$	−254.430			−0.0117020
$780$		0			0
$781$	1256.05			0.0575479
$782$		−	10819.3i		−	0.494755i
$783$		0			0
$784$	−22996.6			−1.04759
$785$	7479.04	−	20619.1i	0.340049	−	0.937488i
$786$		0			0
$787$			2496.63i			0.113082i	0.998400	+	0.0565408i	$0.0180071\pi$
$787$			2496.63i			0.113082i	−0.998400	+	0.0565408i	$0.981993\pi$
$788$			4836.97i			0.218668i
$789$		0			0
$790$	−3301.22	+	9101.21i	−0.148674	+	0.409882i
$791$	77231.7			3.47161
$792$		0			0
$793$		−	18248.4i		−	0.817174i
$794$	14572.6			0.651338
$795$		0			0
$796$	−2280.97			−0.101566
$797$		−	31119.0i		−	1.38305i	−0.722352	−	0.691526i	$-0.756939\pi$
$797$		−	31119.0i		−	1.38305i	0.722352	−	0.691526i	$-0.243061\pi$
$798$		0			0
$799$	8633.77			0.382279
$800$	−14342.3	+	17169.2i	−0.633847	+	0.758777i
$801$		0			0
$802$		−	17164.3i		−	0.755724i
$803$		−	10118.3i		−	0.444667i
$804$		0			0
$805$	−63290.9	−	22957.1i	−2.77107	−	1.00513i
$806$	−10220.2			−0.446638
$807$		0			0
$808$			20813.7i			0.906218i
$809$	−26410.8			−1.14778			−0.573890	−	0.818932i	$-0.694566\pi$
$809$	−26410.8			−1.14778			−0.573890	+	0.818932i	$0.694566\pi$
$810$		0			0
$811$	2617.02			0.113312			0.0566559	−	0.998394i	$-0.481956\pi$
$811$	2617.02			0.113312			0.0566559	+	0.998394i	$0.481956\pi$
$812$		−	926.093i		−	0.0400240i
$813$		0			0
$814$	−4727.24			−0.203550
$815$	6047.43	−	16672.3i	0.259917	−	0.716571i
$816$		0			0
$817$			13003.6i			0.556839i
$818$		−	4908.40i		−	0.209802i
$819$		0			0
$820$	506.588	+	183.751i	0.0215742	+	0.00782545i
$821$	−11728.9			−0.498589			−0.249295	−	0.968428i	$-0.580199\pi$
$821$	−11728.9			−0.498589			−0.249295	+	0.968428i	$0.580199\pi$
$822$		0			0
$823$			33171.7i			1.40497i	0.711698	+	0.702486i	$0.247926\pi$
$823$			33171.7i			1.40497i	−0.711698	+	0.702486i	$0.752074\pi$
$824$	−26510.1			−1.12078
$825$		0			0
$826$	34233.8			1.44206
$827$		−	37989.2i		−	1.59736i	−0.601757	−	0.798679i	$-0.705532\pi$
$827$		−	37989.2i		−	1.59736i	0.601757	−	0.798679i	$-0.294468\pi$
$828$		0			0
$829$	9051.33			0.379211			0.189605	−	0.981860i	$-0.439279\pi$
$829$	9051.33			0.379211			0.189605	+	0.981860i	$0.439279\pi$
$830$	3124.78	+	1133.43i	0.130678	+	0.0474000i
$831$		0			0
$832$		−	750.053i		−	0.0312541i
$833$			36908.4i			1.53518i
$834$		0			0
$835$	9312.35	−	25673.4i	0.385949	−	1.06403i
$836$	−5290.21			−0.218858
$837$		0			0
$838$			19412.0i			0.800209i
$839$	29458.7			1.21219			0.606094	−	0.795393i	$-0.292735\pi$
$839$	29458.7			1.21219			0.606094	+	0.795393i	$0.292735\pi$
$840$		0			0
$841$	−24370.6			−0.999245
$842$			4158.00i			0.170183i
$843$		0			0
$844$	987.185			0.0402610
$845$	13049.7	+	4733.42i	0.531269	+	0.192704i
$846$		0			0
$847$			25553.2i			1.03662i
$848$			17840.0i			0.722440i
$849$		0			0
$850$	5716.65	+	4775.42i	0.230682	+	0.192701i
$851$	29658.9			1.19471
$852$		0			0
$853$		−	10163.5i		−	0.407960i	−0.978975	−	0.203980i	$-0.934612\pi$
$853$		−	10163.5i		−	0.407960i	0.978975	−	0.203980i	$-0.0653878\pi$
$854$	23933.4			0.958997
$855$		0			0
$856$	−12728.9			−0.508254
$857$			19520.2i			0.778059i	0.921225	+	0.389030i	$0.127190\pi$
$857$			19520.2i			0.778059i	−0.921225	+	0.389030i	$0.872810\pi$
$858$		0			0
$859$	23043.9			0.915308			0.457654	−	0.889130i	$-0.348690\pi$
$859$	23043.9			0.915308			0.457654	+	0.889130i	$0.348690\pi$
$860$	9391.28	−	25891.0i	0.372372	−	1.02660i
$861$		0			0
$862$		−	7416.61i		−	0.293052i
$863$		−	32489.6i		−	1.28153i	−0.767738	−	0.640764i	$-0.778618\pi$
$863$		−	32489.6i		−	1.28153i	0.767738	−	0.640764i	$-0.221382\pi$
$864$		0			0
$865$	−5798.16	+	15985.1i	−0.227911	+	0.628334i
$866$	−2895.18			−0.113605
$867$		0			0
$868$			58374.3i			2.28266i
$869$	16773.6			0.654782
$870$		0			0
$871$	15274.6			0.594215
$872$		−	5005.41i		−	0.194386i
$873$		0			0
$874$	−7621.43			−0.294964
$875$	40065.2	−	23308.5i	1.54794	−	0.900539i
$876$		0			0
$877$		−	40252.9i		−	1.54988i	−0.632036	−	0.774939i	$-0.717780\pi$
$877$		−	40252.9i		−	1.54988i	0.632036	−	0.774939i	$-0.282220\pi$
$878$			4560.95i			0.175313i
$879$		0			0
$880$	7559.13	+	2741.87i	0.289566	+	0.105032i
$881$	49327.8			1.88637			0.943187	−	0.332263i	$-0.107812\pi$
$881$	49327.8			1.88637			0.943187	+	0.332263i	$0.107812\pi$
$882$		0			0
$883$			31936.4i			1.21715i	0.793495	+	0.608577i	$0.208259\pi$
$883$			31936.4i			1.21715i	−0.793495	+	0.608577i	$0.791741\pi$
$884$	−9804.33			−0.373026
$885$		0			0
$886$	19455.5			0.737721
$887$		−	3174.52i		−	0.120169i	−0.998193	−	0.0600845i	$-0.980863\pi$
$887$		−	3174.52i		−	0.120169i	0.998193	−	0.0600845i	$-0.0191370\pi$
$888$		0			0
$889$	−5380.16			−0.202975
$890$	−2601.67	+	7172.61i	−0.0979868	+	0.270142i
$891$		0			0
$892$		−	16453.5i		−	0.617606i
$893$		−	6081.86i		−	0.227908i
$894$		0			0
$895$	−9481.63	−	3439.21i	−0.354118	−	0.128447i
$896$	48471.0			1.80726
$897$		0			0
$898$		−	11108.5i		−	0.412801i
$899$	−1161.00			−0.0430719
$900$		0			0
$901$	28632.4			1.05869
$902$			214.387i			0.00791386i
$903$		0			0
$904$	−41286.5			−1.51899
$905$	−36494.6	−	13237.4i	−1.34047	−	0.486218i
$906$		0			0
$907$		−	27550.1i		−	1.00858i	−0.863533	−	0.504292i	$-0.831754\pi$
$907$		−	27550.1i		−	1.00858i	0.863533	−	0.504292i	$-0.168246\pi$
$908$			31869.2i			1.16478i
$909$		0			0
$910$	4776.95	−	13169.7i	0.174016	−	0.479748i
$911$	−5634.64			−0.204922			−0.102461	−	0.994737i	$-0.532672\pi$
$911$	−5634.64			−0.204922			−0.102461	+	0.994737i	$0.532672\pi$
$912$		0			0
$913$		−	5759.00i		−	0.208757i
$914$	14517.9			0.525394
$915$		0			0
$916$	18642.0			0.672432
$917$			74267.4i			2.67451i
$918$		0			0
$919$	−8518.91			−0.305781			−0.152891	−	0.988243i	$-0.548858\pi$
$919$	−8518.91			−0.305781			−0.152891	+	0.988243i	$0.548858\pi$
$920$	33834.1	+	12272.4i	1.21248	+	0.439793i
$921$		0			0
$922$		−	8862.87i		−	0.316576i
$923$		−	1639.79i		−	0.0584770i
$924$		0			0
$925$	−13090.8	+	15671.0i	−0.465323	+	0.557037i
$926$	2174.15			0.0771567
$927$		0			0
$928$			768.100i			0.0271704i
$929$	6640.06			0.234503			0.117252	−	0.993102i	$-0.462592\pi$
$929$	6640.06			0.234503			0.117252	+	0.993102i	$0.462592\pi$
$930$		0			0
$931$	25999.3			0.915245
$932$			28173.2i			0.990176i
$933$		0			0
$934$	−6299.32			−0.220685
$935$	4400.57	−	12132.0i	0.153919	−	0.424342i
$936$		0			0
$937$			44972.1i			1.56795i	0.620789	+	0.783977i	$0.286812\pi$
$937$			44972.1i			1.56795i	−0.620789	+	0.783977i	$0.713188\pi$
$938$			20033.2i			0.697343i
$939$		0			0
$940$	−4392.37	+	12109.4i	−0.152408	+	0.420176i
$941$	−6264.93			−0.217036			−0.108518	−	0.994094i	$-0.534611\pi$
$941$	−6264.93			−0.217036			−0.108518	+	0.994094i	$0.534611\pi$
$942$		0			0
$943$		−	1345.07i		−	0.0464492i
$944$	25652.7			0.884452
$945$		0			0
$946$	10957.0			0.376579
$947$			9672.34i			0.331899i	0.986134	+	0.165950i	$0.0530689\pi$
$947$			9672.34i			0.331899i	−0.986134	+	0.165950i	$0.946931\pi$
$948$		0			0
$949$	−13209.6			−0.451846
$950$	3363.94	−	4026.97i	0.114885	−	0.137528i
$951$		0			0
$952$		−	28670.1i		−	0.976054i
$953$		−	14173.3i		−	0.481761i	−0.970555	−	0.240880i	$-0.922564\pi$
$953$		−	14173.3i		−	0.481761i	0.970555	−	0.240880i	$-0.0774362\pi$
$954$		0			0
$955$	12198.6	+	4424.71i	0.413337	+	0.149927i
$956$	−24464.9			−0.827667
$957$		0			0
$958$			9594.91i			0.323588i
$959$	33226.8			1.11882
$960$		0			0
$961$	43390.4			1.45649
$962$			6171.48i			0.206836i
$963$		0			0
$964$	−20205.4			−0.675074
$965$	7858.65	−	21665.7i	0.262154	−	0.722739i
$966$		0			0
$967$			43588.7i			1.44955i	0.688984	+	0.724776i	$0.258057\pi$
$967$			43588.7i			1.44955i	−0.688984	+	0.724776i	$0.741943\pi$
$968$		−	13660.2i		−	0.453571i
$969$		0			0
$970$	−6052.21	−	2195.28i	−0.200335	−	0.0726661i
$971$	20974.8			0.693216			0.346608	−	0.938010i	$-0.387333\pi$
$971$	20974.8			0.693216			0.346608	+	0.938010i	$0.387333\pi$
$972$		0			0
$973$		−	59486.8i		−	1.95998i
$974$	−10584.2			−0.348192
$975$		0			0
$976$	17934.2			0.588176
$977$			8765.10i			0.287022i	0.989649	+	0.143511i	$0.0458392\pi$
$977$			8765.10i			0.287022i	−0.989649	+	0.143511i	$0.954161\pi$
$978$		0			0
$979$	13219.2			0.431549
$980$	−51766.5	−	18776.9i	−1.68737	−	0.612047i
$981$		0			0
$982$			2433.24i			0.0790710i
$983$		−	39341.3i		−	1.27649i	−0.769832	−	0.638247i	$-0.779660\pi$
$983$		−	39341.3i		−	1.27649i	0.769832	−	0.638247i	$-0.220340\pi$
$984$		0			0
$985$	−2834.30	+	7813.95i	−0.0916836	+	0.252764i
$986$	255.747			0.00826028
$987$		0			0
$988$			6906.44i			0.222392i
$989$	−68744.9			−2.21027
$990$		0			0
$991$	5063.64			0.162313			0.0811563	−	0.996701i	$-0.474139\pi$
$991$	5063.64			0.162313			0.0811563	+	0.996701i	$0.474139\pi$
$992$		−	48415.5i		−	1.54959i
$993$		0			0
$994$	2150.64			0.0686258
$995$	−3684.82	−	1336.57i	−0.117404	−	0.0425850i
$996$		0			0
$997$			29030.8i			0.922181i	0.887353	+	0.461090i	$0.152542\pi$
$997$			29030.8i			0.922181i	−0.887353	+	0.461090i	$0.847458\pi$
$998$			13765.2i			0.436603i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.b.c.244.5	yes	8
3.2	odd	2		405.4.b.b.244.4	✓	8
5.2	odd	4		2025.4.a.bd.1.4		8
5.3	odd	4		2025.4.a.bd.1.5		8
5.4	even	2	inner	405.4.b.c.244.4	yes	8
15.2	even	4		2025.4.a.bc.1.5		8
15.8	even	4		2025.4.a.bc.1.4		8
15.14	odd	2		405.4.b.b.244.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.4.b.b.244.4	✓	8	3.2	odd	2
405.4.b.b.244.5	yes	8	15.14	odd	2
405.4.b.c.244.4	yes	8	5.4	even	2	inner
405.4.b.c.244.5	yes	8	1.1	even	1	trivial
2025.4.a.bc.1.4		8	15.8	even	4
2025.4.a.bc.1.5		8	15.2	even	4
2025.4.a.bd.1.4		8	5.2	odd	4
2025.4.a.bd.1.5		8	5.3	odd	4

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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	405.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.22227i q^{2} +6.50606 q^{4} +(10.5103 + 3.81233i) q^{5} -33.1668i q^{7} +17.7303i q^{8} +O(q^{10})\)	\(q+1.22227i q^{2} +6.50606 q^{4} +(10.5103 + 3.81233i) q^{5} -33.1668i q^{7} +17.7303i q^{8} +(-4.65969 + 12.8464i) q^{10} +23.6760 q^{11} -30.9094i q^{13} +40.5387 q^{14} +30.3772 q^{16} -48.7539i q^{17} -34.3436 q^{19} +(68.3805 + 24.8032i) q^{20} +28.9385i q^{22} -181.561i q^{23} +(95.9323 + 80.1373i) q^{25} +37.7796 q^{26} -215.785i q^{28} +4.29174 q^{29} -270.521 q^{31} +178.972i q^{32} +59.5905 q^{34} +(126.443 - 348.592i) q^{35} +163.355i q^{37} -41.9772i q^{38} +(-67.5938 + 186.351i) q^{40} +7.40836 q^{41} -378.631i q^{43} +154.038 q^{44} +221.917 q^{46} +177.089i q^{47} -757.035 q^{49} +(-97.9494 + 117.255i) q^{50} -201.098i q^{52} +587.283i q^{53} +(248.842 + 90.2608i) q^{55} +588.057 q^{56} +5.24566i q^{58} +844.470 q^{59} +590.383 q^{61} -330.649i q^{62} +24.2662 q^{64} +(117.837 - 324.867i) q^{65} +494.174i q^{67} -317.196i q^{68} +(426.074 + 154.547i) q^{70} +53.0514 q^{71} -427.365i q^{73} -199.664 q^{74} -223.441 q^{76} -785.258i q^{77} +708.463 q^{79} +(319.273 + 115.808i) q^{80} +9.05501i q^{82} -243.242i q^{83} +(185.866 - 512.418i) q^{85} +462.790 q^{86} +419.784i q^{88} +558.336 q^{89} -1025.17 q^{91} -1181.25i q^{92} -216.450 q^{94} +(-360.961 - 130.929i) q^{95} +471.121i q^{97} -925.301i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{4} + 15 q^{5}+O(q^{10}) \) 8 * q - 24 * q^4 + 15 * q^5	\( 8 q - 24 q^{4} + 15 q^{5} + 7 q^{10} - 42 q^{14} + 4 q^{16} + 118 q^{19} - 129 q^{20} + 17 q^{25} - 36 q^{26} + 318 q^{29} - 416 q^{31} + 638 q^{34} + 192 q^{35} - 265 q^{40} + 486 q^{41} + 852 q^{44} - 598 q^{46} + 350 q^{49} + 1143 q^{50} - 162 q^{55} + 1530 q^{56} - 1146 q^{59} - 398 q^{61} + 1640 q^{64} - 1833 q^{65} + 630 q^{70} - 1728 q^{71} + 1218 q^{74} - 3498 q^{76} + 2596 q^{79} + 1923 q^{80} - 233 q^{85} + 480 q^{86} + 1086 q^{89} - 2574 q^{91} + 1238 q^{94} + 1674 q^{95}+O(q^{100}) \) 8 * q - 24 * q^4 + 15 * q^5 + 7 * q^10 - 42 * q^14 + 4 * q^16 + 118 * q^19 - 129 * q^20 + 17 * q^25 - 36 * q^26 + 318 * q^29 - 416 * q^31 + 638 * q^34 + 192 * q^35 - 265 * q^40 + 486 * q^41 + 852 * q^44 - 598 * q^46 + 350 * q^49 + 1143 * q^50 - 162 * q^55 + 1530 * q^56 - 1146 * q^59 - 398 * q^61 + 1640 * q^64 - 1833 * q^65 + 630 * q^70 - 1728 * q^71 + 1218 * q^74 - 3498 * q^76 + 2596 * q^79 + 1923 * q^80 - 233 * q^85 + 480 * q^86 + 1086 * q^89 - 2574 * q^91 + 1238 * q^94 + 1674 * q^95

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