LMFDB - Embedded newform 405.2.b.b.244.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,2,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, 2, names="a")

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

chi := DirichletCharacter("405.244");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
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:	$3.23394128186$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x^{2} - 3x + 9 $ x^4 - x^3 - 2x^2 - 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			244.3
Root			$1.68614 - 0.396143i$ of defining polynomial
Character	$\chi$	$=$	405.244
Dual form			405.2.b.b.244.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.792287i q^{2} +1.37228 q^{4} +(-0.686141 + 2.12819i) q^{5} -3.46410i q^{7} +2.67181i q^{8} +O(q^{10})$	\(q+0.792287i q^{2} +1.37228 q^{4} +(-0.686141 + 2.12819i) q^{5} -3.46410i q^{7} +2.67181i q^{8} +(-1.68614 - 0.543620i) q^{10} +4.37228 q^{11} +5.84096i q^{13} +2.74456 q^{14} +0.627719 q^{16} +0.792287i q^{17} +0.372281 q^{19} +(-0.941578 + 2.92048i) q^{20} +3.46410i q^{22} -1.58457i q^{23} +(-4.05842 - 2.92048i) q^{25} -4.62772 q^{26} -4.75372i q^{28} -5.74456 q^{29} +6.37228 q^{31} +5.84096i q^{32} -0.627719 q^{34} +(7.37228 + 2.37686i) q^{35} +2.37686i q^{37} +0.294954i q^{38} +(-5.68614 - 1.83324i) q^{40} -4.37228 q^{41} -3.46410i q^{43} +6.00000 q^{44} +1.25544 q^{46} +1.87953i q^{47} -5.00000 q^{49} +(2.31386 - 3.21543i) q^{50} +8.01544i q^{52} -11.9769i q^{53} +(-3.00000 + 9.30506i) q^{55} +9.25544 q^{56} -4.55134i q^{58} +1.62772 q^{59} +9.37228 q^{61} +5.04868i q^{62} -3.37228 q^{64} +(-12.4307 - 4.00772i) q^{65} -11.6819i q^{67} +1.08724i q^{68} +(-1.88316 + 5.84096i) q^{70} -13.1168 q^{71} -2.37686i q^{73} -1.88316 q^{74} +0.510875 q^{76} -15.1460i q^{77} -6.74456 q^{79} +(-0.430703 + 1.33591i) q^{80} -3.46410i q^{82} -11.9769i q^{83} +(-1.68614 - 0.543620i) q^{85} +2.74456 q^{86} +11.6819i q^{88} -3.00000 q^{89} +20.2337 q^{91} -2.17448i q^{92} -1.48913 q^{94} +(-0.255437 + 0.792287i) q^{95} +1.28962i q^{97} -3.96143i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{4} + 3 q^{5}+O(q^{10}) $ 4 * q - 6 * q^4 + 3 * q^5	$ 4 q - 6 q^{4} + 3 q^{5} - q^{10} + 6 q^{11} - 12 q^{14} + 14 q^{16} - 10 q^{19} - 21 q^{20} + q^{25} - 30 q^{26} + 14 q^{31} - 14 q^{34} + 18 q^{35} - 17 q^{40} - 6 q^{41} + 24 q^{44} + 28 q^{46} - 20 q^{49} + 15 q^{50} - 12 q^{55} + 60 q^{56} + 18 q^{59} + 26 q^{61} - 2 q^{64} - 21 q^{65} - 42 q^{70} - 18 q^{71} - 42 q^{74} + 48 q^{76} - 4 q^{79} + 27 q^{80} - q^{85} - 12 q^{86} - 12 q^{89} + 12 q^{91} + 40 q^{94} - 24 q^{95}+O(q^{100}) $ 4 * q - 6 * q^4 + 3 * q^5 - q^10 + 6 * q^11 - 12 * q^14 + 14 * q^16 - 10 * q^19 - 21 * q^20 + q^25 - 30 * q^26 + 14 * q^31 - 14 * q^34 + 18 * q^35 - 17 * q^40 - 6 * q^41 + 24 * q^44 + 28 * q^46 - 20 * q^49 + 15 * q^50 - 12 * q^55 + 60 * q^56 + 18 * q^59 + 26 * q^61 - 2 * q^64 - 21 * q^65 - 42 * q^70 - 18 * q^71 - 42 * q^74 + 48 * q^76 - 4 * q^79 + 27 * q^80 - q^85 - 12 * q^86 - 12 * q^89 + 12 * q^91 + 40 * q^94 - 24 * 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 $

$2$			0.792287i			0.560232i	0.959966	+	0.280116i	$0.0903729\pi$
$2$			0.792287i			0.560232i	−0.959966	+	0.280116i	$0.909627\pi$
$3$		0			0
$3$		0			0
$4$	1.37228			0.686141
$5$	−0.686141	+	2.12819i	−0.306851	+	0.951757i
$5$	−0.686141	+	2.12819i	−0.306851	+	0.951757i
$6$		0			0
$7$		−	3.46410i		−	1.30931i	−0.755929	−	0.654654i	$-0.772814\pi$
$7$		−	3.46410i		−	1.30931i	0.755929	−	0.654654i	$-0.227186\pi$
$8$			2.67181i			0.944629i
$9$		0			0
$10$	−1.68614	−	0.543620i	−0.533204	−	0.171908i
$11$	4.37228			1.31829			0.659146	−	0.752015i	$-0.270918\pi$
$11$	4.37228			1.31829			0.659146	+	0.752015i	$0.270918\pi$
$12$		0			0
$13$			5.84096i			1.61999i	0.586436	+	0.809996i	$0.300531\pi$
$13$			5.84096i			1.61999i	−0.586436	+	0.809996i	$0.699469\pi$
$14$	2.74456			0.733515
$15$		0			0
$16$	0.627719			0.156930
$17$			0.792287i			0.192158i	0.995374	+	0.0960789i	$0.0306301\pi$
$17$			0.792287i			0.192158i	−0.995374	+	0.0960789i	$0.969370\pi$
$18$		0			0
$19$	0.372281			0.0854072			0.0427036	−	0.999088i	$-0.486403\pi$
$19$	0.372281			0.0854072			0.0427036	+	0.999088i	$0.486403\pi$
$20$	−0.941578	+	2.92048i	−0.210543	+	0.653039i
$21$		0			0
$22$			3.46410i			0.738549i
$23$		−	1.58457i		−	0.330407i	−0.986260	−	0.165203i	$-0.947172\pi$
$23$		−	1.58457i		−	0.330407i	0.986260	−	0.165203i	$-0.0528280\pi$
$24$		0			0
$25$	−4.05842	−	2.92048i	−0.811684	−	0.584096i
$26$	−4.62772			−0.907570
$27$		0			0
$28$		−	4.75372i		−	0.898369i
$29$	−5.74456			−1.06674			−0.533369	−	0.845883i	$-0.679074\pi$
$29$	−5.74456			−1.06674			−0.533369	+	0.845883i	$0.679074\pi$
$30$		0			0
$31$	6.37228			1.14450			0.572248	−	0.820081i	$-0.306072\pi$
$31$	6.37228			1.14450			0.572248	+	0.820081i	$0.306072\pi$
$32$			5.84096i			1.03255i
$33$		0			0
$34$	−0.627719			−0.107653
$35$	7.37228	+	2.37686i	1.24614	+	0.401763i
$36$		0			0
$37$			2.37686i			0.390754i	0.980728	+	0.195377i	$0.0625930\pi$
$37$			2.37686i			0.390754i	−0.980728	+	0.195377i	$0.937407\pi$
$38$			0.294954i			0.0478478i
$39$		0			0
$40$	−5.68614	−	1.83324i	−0.899058	−	0.289861i
$41$	−4.37228			−0.682836			−0.341418	−	0.939912i	$-0.610907\pi$
$41$	−4.37228			−0.682836			−0.341418	+	0.939912i	$0.610907\pi$
$42$		0			0
$43$		−	3.46410i		−	0.528271i	−0.964486	−	0.264135i	$-0.914913\pi$
$43$		−	3.46410i		−	0.528271i	0.964486	−	0.264135i	$-0.0850865\pi$
$44$	6.00000			0.904534
$45$		0			0
$46$	1.25544			0.185104
$47$			1.87953i			0.274157i	0.990560	+	0.137079i	$0.0437713\pi$
$47$			1.87953i			0.274157i	−0.990560	+	0.137079i	$0.956229\pi$
$48$		0			0
$49$	−5.00000			−0.714286
$50$	2.31386	−	3.21543i	0.327229	−	0.454731i
$51$		0			0
$52$			8.01544i			1.11154i
$53$		−	11.9769i		−	1.64515i	−0.568656	−	0.822575i	$-0.692536\pi$
$53$		−	11.9769i		−	1.64515i	0.568656	−	0.822575i	$-0.307464\pi$
$54$		0			0
$55$	−3.00000	+	9.30506i	−0.404520	+	1.25469i
$56$	9.25544			1.23681
$57$		0			0
$58$		−	4.55134i		−	0.597621i
$59$	1.62772			0.211911			0.105955	−	0.994371i	$-0.466210\pi$
$59$	1.62772			0.211911			0.105955	+	0.994371i	$0.466210\pi$
$60$		0			0
$61$	9.37228			1.20000			0.599999	−	0.800001i	$-0.295168\pi$
$61$	9.37228			1.20000			0.599999	+	0.800001i	$0.295168\pi$
$62$			5.04868i			0.641182i
$63$		0			0
$64$	−3.37228			−0.421535
$65$	−12.4307	−	4.00772i	−1.54184	−	0.497097i
$66$		0			0
$67$		−	11.6819i		−	1.42717i	−0.700566	−	0.713587i	$-0.747069\pi$
$67$		−	11.6819i		−	1.42717i	0.700566	−	0.713587i	$-0.252931\pi$
$68$			1.08724i			0.131847i
$69$		0			0
$70$	−1.88316	+	5.84096i	−0.225080	+	0.698129i
$71$	−13.1168			−1.55668			−0.778341	−	0.627841i	$-0.783939\pi$
$71$	−13.1168			−1.55668			−0.778341	+	0.627841i	$0.783939\pi$
$72$		0			0
$73$		−	2.37686i		−	0.278191i	−0.990279	−	0.139095i	$-0.955581\pi$
$73$		−	2.37686i		−	0.278191i	0.990279	−	0.139095i	$-0.0444194\pi$
$74$	−1.88316			−0.218912
$75$		0			0
$76$	0.510875			0.0586013
$77$		−	15.1460i		−	1.72605i
$78$		0			0
$79$	−6.74456			−0.758823			−0.379411	−	0.925228i	$-0.623873\pi$
$79$	−6.74456			−0.758823			−0.379411	+	0.925228i	$0.623873\pi$
$80$	−0.430703	+	1.33591i	−0.0481541	+	0.149359i
$81$		0			0
$82$		−	3.46410i		−	0.382546i
$83$		−	11.9769i		−	1.31463i	−0.753615	−	0.657317i	$-0.771691\pi$
$83$		−	11.9769i		−	1.31463i	0.753615	−	0.657317i	$-0.228309\pi$
$84$		0			0
$85$	−1.68614	−	0.543620i	−0.182888	−	0.0589639i
$86$	2.74456			0.295954
$87$		0			0
$88$			11.6819i			1.24530i
$89$	−3.00000			−0.317999			−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000			−0.317999			−0.159000	+	0.987279i	$0.550827\pi$
$90$		0			0
$91$	20.2337			2.12107
$92$		−	2.17448i		−	0.226705i
$93$		0			0
$94$	−1.48913			−0.153592
$95$	−0.255437	+	0.792287i	−0.0262073	+	0.0812869i
$96$		0			0
$97$			1.28962i			0.130941i	0.997855	+	0.0654706i	$0.0208548\pi$
$97$			1.28962i			0.130941i	−0.997855	+	0.0654706i	$0.979145\pi$
$98$		−	3.96143i		−	0.400165i
$99$		0			0
$100$	−5.56930	−	4.00772i	−0.556930	−	0.400772i
$101$	16.3723			1.62910			0.814551	−	0.580091i	$-0.196983\pi$
$101$	16.3723			1.62910			0.814551	+	0.580091i	$0.196983\pi$
$102$		0			0
$103$		−	6.92820i		−	0.682656i	−0.939944	−	0.341328i	$-0.889123\pi$
$103$		−	6.92820i		−	0.682656i	0.939944	−	0.341328i	$-0.110877\pi$
$104$	−15.6060			−1.53029
$105$		0			0
$106$	9.48913			0.921665
$107$		−	13.2665i		−	1.28252i	−0.767323	−	0.641260i	$-0.778412\pi$
$107$		−	13.2665i		−	1.28252i	0.767323	−	0.641260i	$-0.221588\pi$
$108$		0			0
$109$	−9.74456			−0.933360			−0.466680	−	0.884426i	$-0.654550\pi$
$109$	−9.74456			−0.933360			−0.466680	+	0.884426i	$0.654550\pi$
$110$	−7.37228	−	2.37686i	−0.702919	−	0.226625i
$111$		0			0
$112$		−	2.17448i		−	0.205469i
$113$		−	6.13592i		−	0.577218i	−0.957447	−	0.288609i	$-0.906807\pi$
$113$		−	6.13592i		−	0.577218i	0.957447	−	0.288609i	$-0.0931928\pi$
$114$		0			0
$115$	3.37228	+	1.08724i	0.314467	+	0.101386i
$116$	−7.88316			−0.731933
$117$		0			0
$118$			1.28962i			0.118719i
$119$	2.74456			0.251594
$120$		0			0
$121$	8.11684			0.737895
$122$			7.42554i			0.672276i
$123$		0			0
$124$	8.74456			0.785285
$125$	9.00000	−	6.63325i	0.804984	−	0.593296i
$126$		0			0
$127$			10.3923i			0.922168i	0.887357	+	0.461084i	$0.152539\pi$
$127$			10.3923i			0.922168i	−0.887357	+	0.461084i	$0.847461\pi$
$128$			9.01011i			0.796389i
$129$		0			0
$130$	3.17527	−	9.84868i	0.278489	−	0.863787i
$131$	−4.37228			−0.382008			−0.191004	−	0.981589i	$-0.561174\pi$
$131$	−4.37228			−0.382008			−0.191004	+	0.981589i	$0.561174\pi$
$132$		0			0
$133$		−	1.28962i		−	0.111824i
$134$	9.25544			0.799548
$135$		0			0
$136$	−2.11684			−0.181518
$137$		−	14.3537i		−	1.22632i	−0.789958	−	0.613161i	$-0.789898\pi$
$137$		−	14.3537i		−	1.22632i	0.789958	−	0.613161i	$-0.210102\pi$
$138$		0			0
$139$	−11.1168			−0.942918			−0.471459	−	0.881888i	$-0.656273\pi$
$139$	−11.1168			−0.942918			−0.471459	+	0.881888i	$0.656273\pi$
$140$	10.1168	+	3.26172i	0.855029	+	0.275666i
$141$		0			0
$142$		−	10.3923i		−	0.872103i
$143$			25.5383i			2.13562i
$144$		0			0
$145$	3.94158	−	12.2255i	0.327330	−	1.01528i
$146$	1.88316			0.155851
$147$		0			0
$148$			3.26172i			0.268112i
$149$	10.6277			0.870657			0.435328	−	0.900272i	$-0.356632\pi$
$149$	10.6277			0.870657			0.435328	+	0.900272i	$0.356632\pi$
$150$		0			0
$151$	0.883156			0.0718702			0.0359351	−	0.999354i	$-0.488559\pi$
$151$	0.883156			0.0718702			0.0359351	+	0.999354i	$0.488559\pi$
$152$			0.994667i			0.0806781i
$153$		0			0
$154$	12.0000			0.966988
$155$	−4.37228	+	13.5615i	−0.351190	+	1.08928i
$156$		0			0
$157$			11.4795i			0.916167i	0.888909	+	0.458084i	$0.151464\pi$
$157$			11.4795i			0.916167i	−0.888909	+	0.458084i	$0.848536\pi$
$158$		−	5.34363i		−	0.425116i
$159$		0			0
$160$	−12.4307	−	4.00772i	−0.982733	−	0.316838i
$161$	−5.48913			−0.432604
$162$		0			0
$163$			15.1460i			1.18633i	0.805082	+	0.593164i	$0.202122\pi$
$163$			15.1460i			1.18633i	−0.805082	+	0.593164i	$0.797878\pi$
$164$	−6.00000			−0.468521
$165$		0			0
$166$	9.48913			0.736499
$167$			13.5615i			1.04942i	0.851282	+	0.524708i	$0.175826\pi$
$167$			13.5615i			1.04942i	−0.851282	+	0.524708i	$0.824174\pi$
$168$		0			0
$169$	−21.1168			−1.62437
$170$	0.430703	−	1.33591i	0.0330334	−	0.102459i
$171$		0			0
$172$		−	4.75372i		−	0.362468i
$173$			11.1846i			0.850349i	0.905111	+	0.425174i	$0.139787\pi$
$173$			11.1846i			0.850349i	−0.905111	+	0.425174i	$0.860213\pi$
$174$		0			0
$175$	−10.1168	+	14.0588i	−0.764762	+	1.06274i
$176$	2.74456			0.206879
$177$		0			0
$178$		−	2.37686i		−	0.178153i
$179$	−4.88316			−0.364984			−0.182492	−	0.983207i	$-0.558416\pi$
$179$	−4.88316			−0.364984			−0.182492	+	0.983207i	$0.558416\pi$
$180$		0			0
$181$	−13.8614			−1.03031			−0.515155	−	0.857097i	$-0.672266\pi$
$181$	−13.8614			−1.03031			−0.515155	+	0.857097i	$0.672266\pi$
$182$			16.0309i			1.18829i
$183$		0			0
$184$	4.23369			0.312112
$185$	−5.05842	−	1.63086i	−0.371903	−	0.119903i
$186$		0			0
$187$			3.46410i			0.253320i
$188$			2.57924i			0.188110i
$189$		0			0
$190$	−0.627719	−	0.202380i	−0.0455395	−	0.0146822i
$191$	−7.62772			−0.551922			−0.275961	−	0.961169i	$-0.588996\pi$
$191$	−7.62772			−0.551922			−0.275961	+	0.961169i	$0.588996\pi$
$192$		0			0
$193$			11.4795i			0.826316i	0.910659	+	0.413158i	$0.135574\pi$
$193$			11.4795i			0.826316i	−0.910659	+	0.413158i	$0.864426\pi$
$194$	−1.02175			−0.0733573
$195$		0			0
$196$	−6.86141			−0.490100
$197$			6.43087i			0.458181i	0.973405	+	0.229090i	$0.0735751\pi$
$197$			6.43087i			0.458181i	−0.973405	+	0.229090i	$0.926425\pi$
$198$		0			0
$199$	−16.0000			−1.13421			−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000			−1.13421			−0.567105	+	0.823646i	$0.691937\pi$
$200$	7.80298	−	10.8434i	0.551754	−	0.766741i
$201$		0			0
$202$			12.9715i			0.912675i
$203$			19.8997i			1.39669i
$204$		0			0
$205$	3.00000	−	9.30506i	0.209529	−	0.649894i
$206$	5.48913			0.382445
$207$		0			0
$208$			3.66648i			0.254225i
$209$	1.62772			0.112592
$210$		0			0
$211$	−1.86141			−0.128145			−0.0640723	−	0.997945i	$-0.520409\pi$
$211$	−1.86141			−0.128145			−0.0640723	+	0.997945i	$0.520409\pi$
$212$		−	16.4356i		−	1.12880i
$213$		0			0
$214$	10.5109			0.718509
$215$	7.37228	+	2.37686i	0.502785	+	0.162101i
$216$		0			0
$217$		−	22.0742i		−	1.49850i
$218$		−	7.72049i		−	0.522898i
$219$		0			0
$220$	−4.11684	+	12.7692i	−0.277558	+	0.860897i
$221$	−4.62772			−0.311294
$222$		0			0
$223$		−	18.6101i		−	1.24623i	−0.782132	−	0.623113i	$-0.785868\pi$
$223$		−	18.6101i		−	1.24623i	0.782132	−	0.623113i	$-0.214132\pi$
$224$	20.2337			1.35192
$225$		0			0
$226$	4.86141			0.323376
$227$			13.5615i			0.900105i	0.893002	+	0.450053i	$0.148595\pi$
$227$			13.5615i			0.900105i	−0.893002	+	0.450053i	$0.851405\pi$
$228$		0			0
$229$	17.6060			1.16344			0.581718	−	0.813391i	$-0.302381\pi$
$229$	17.6060			1.16344			0.581718	+	0.813391i	$0.302381\pi$
$230$	−0.861407	+	2.67181i	−0.0567995	+	0.176174i
$231$		0			0
$232$		−	15.3484i		−	1.00767i
$233$		−	6.13592i		−	0.401977i	−0.979594	−	0.200989i	$-0.935585\pi$
$233$		−	6.13592i		−	0.401977i	0.979594	−	0.200989i	$-0.0644154\pi$
$234$		0			0
$235$	−4.00000	−	1.28962i	−0.260931	−	0.0841256i
$236$	2.23369			0.145401
$237$		0			0
$238$			2.17448i			0.140951i
$239$	22.9783			1.48634			0.743170	−	0.669103i	$-0.233321\pi$
$239$	22.9783			1.48634			0.743170	+	0.669103i	$0.233321\pi$
$240$		0			0
$241$	−1.51087			−0.0973240			−0.0486620	−	0.998815i	$-0.515496\pi$
$241$	−1.51087			−0.0973240			−0.0486620	+	0.998815i	$0.515496\pi$
$242$			6.43087i			0.413392i
$243$		0			0
$244$	12.8614			0.823367
$245$	3.43070	−	10.6410i	0.219180	−	0.679827i
$246$		0			0
$247$			2.17448i			0.138359i
$248$			17.0256i			1.08112i
$249$		0			0
$250$	5.25544	+	7.13058i	0.332383	+	0.450978i
$251$	20.2337			1.27714			0.638570	−	0.769564i	$-0.279526\pi$
$251$	20.2337			1.27714			0.638570	+	0.769564i	$0.279526\pi$
$252$		0			0
$253$		−	6.92820i		−	0.435572i
$254$	−8.23369			−0.516628
$255$		0			0
$256$	−13.8832			−0.867697
$257$			6.43087i			0.401147i	0.979679	+	0.200573i	$0.0642805\pi$
$257$			6.43087i			0.401147i	−0.979679	+	0.200573i	$0.935720\pi$
$258$		0			0
$259$	8.23369			0.511616
$260$	−17.0584	−	5.49972i	−1.05792	−	0.341078i
$261$		0			0
$262$		−	3.46410i		−	0.214013i
$263$			26.5330i			1.63609i	0.575151	+	0.818047i	$0.304943\pi$
$263$			26.5330i			1.63609i	−0.575151	+	0.818047i	$0.695057\pi$
$264$		0			0
$265$	25.4891	+	8.21782i	1.56578	+	0.504817i
$266$	1.02175			0.0626475
$267$		0			0
$268$		−	16.0309i		−	0.979242i
$269$	−29.2337			−1.78241			−0.891205	−	0.453601i	$-0.850139\pi$
$269$	−29.2337			−1.78241			−0.891205	+	0.453601i	$0.850139\pi$
$270$		0			0
$271$	5.25544			0.319245			0.159623	−	0.987178i	$-0.448972\pi$
$271$	5.25544			0.319245			0.159623	+	0.987178i	$0.448972\pi$
$272$			0.497333i			0.0301553i
$273$		0			0
$274$	11.3723			0.687025
$275$	−17.7446	−	12.7692i	−1.07004	−	0.770010i
$276$		0			0
$277$			22.0742i			1.32631i	0.748481	+	0.663156i	$0.230783\pi$
$277$			22.0742i			1.32631i	−0.748481	+	0.663156i	$0.769217\pi$
$278$		−	8.80773i		−	0.528253i
$279$		0			0
$280$	−6.35053	+	19.6974i	−0.379517	+	1.17714i
$281$	1.37228			0.0818634			0.0409317	−	0.999162i	$-0.486967\pi$
$281$	1.37228			0.0818634			0.0409317	+	0.999162i	$0.486967\pi$
$282$		0			0
$283$		−	27.7128i		−	1.64736i	−0.567058	−	0.823678i	$-0.691918\pi$
$283$		−	27.7128i		−	1.64736i	0.567058	−	0.823678i	$-0.308082\pi$
$284$	−18.0000			−1.06810
$285$		0			0
$286$	−20.2337			−1.19644
$287$			15.1460i			0.894042i
$288$		0			0
$289$	16.3723			0.963075
$290$	9.68614	+	3.12286i	0.568790	+	0.183381i
$291$		0			0
$292$		−	3.26172i		−	0.190878i
$293$			0.792287i			0.0462859i	0.999732	+	0.0231430i	$0.00736729\pi$
$293$			0.792287i			0.0462859i	−0.999732	+	0.0231430i	$0.992633\pi$
$294$		0			0
$295$	−1.11684	+	3.46410i	−0.0650252	+	0.201688i
$296$	−6.35053			−0.369117
$297$		0			0
$298$			8.42020i			0.487769i
$299$	9.25544			0.535256
$300$		0			0
$301$	−12.0000			−0.691669
$302$			0.699713i			0.0402640i
$303$		0			0
$304$	0.233688			0.0134029
$305$	−6.43070	+	19.9460i	−0.368221	+	1.14211i
$306$		0			0
$307$			15.1460i			0.864429i	0.901771	+	0.432215i	$0.142268\pi$
$307$			15.1460i			0.864429i	−0.901771	+	0.432215i	$0.857732\pi$
$308$		−	20.7846i		−	1.18431i
$309$		0			0
$310$	−10.7446	−	3.46410i	−0.610250	−	0.196748i
$311$	15.8614			0.899418			0.449709	−	0.893175i	$-0.351528\pi$
$311$	15.8614			0.899418			0.449709	+	0.893175i	$0.351528\pi$
$312$		0			0
$313$			24.4511i			1.38206i	0.722827	+	0.691029i	$0.242842\pi$
$313$			24.4511i			1.38206i	−0.722827	+	0.691029i	$0.757158\pi$
$314$	−9.09509			−0.513266
$315$		0			0
$316$	−9.25544			−0.520659
$317$		−	29.4998i		−	1.65687i	−0.560084	−	0.828436i	$-0.689231\pi$
$317$		−	29.4998i		−	1.65687i	0.560084	−	0.828436i	$-0.310769\pi$
$318$		0			0
$319$	−25.1168			−1.40627
$320$	2.31386	−	7.17687i	0.129349	−	0.401199i
$321$		0			0
$322$		−	4.34896i		−	0.242358i
$323$			0.294954i			0.0164117i
$324$		0			0
$325$	17.0584	−	23.7051i	0.946231	−	1.31492i
$326$	−12.0000			−0.664619
$327$		0			0
$328$		−	11.6819i		−	0.645026i
$329$	6.51087			0.358956
$330$		0			0
$331$	9.11684			0.501107			0.250554	−	0.968103i	$-0.419387\pi$
$331$	9.11684			0.501107			0.250554	+	0.968103i	$0.419387\pi$
$332$		−	16.4356i		−	0.902023i
$333$		0			0
$334$	−10.7446			−0.587916
$335$	24.8614	+	8.01544i	1.35832	+	0.437930i
$336$		0			0
$337$		−	6.92820i		−	0.377403i	−0.982034	−	0.188702i	$-0.939572\pi$
$337$		−	6.92820i		−	0.377403i	0.982034	−	0.188702i	$-0.0604279\pi$
$338$		−	16.7306i		−	0.910025i
$339$		0			0
$340$	−2.31386	−	0.746000i	−0.125487	−	0.0404575i
$341$	27.8614			1.50878
$342$		0			0
$343$		−	6.92820i		−	0.374088i
$344$	9.25544			0.499020
$345$		0			0
$346$	−8.86141			−0.476392
$347$		−	2.87419i		−	0.154295i	−0.997020	−	0.0771474i	$-0.975419\pi$
$347$		−	2.87419i		−	0.154295i	0.997020	−	0.0771474i	$-0.0245812\pi$
$348$		0			0
$349$	26.6060			1.42418			0.712092	−	0.702086i	$-0.247748\pi$
$349$	26.6060			1.42418			0.712092	+	0.702086i	$0.247748\pi$
$350$	−11.1386	−	8.01544i	−0.595383	−	0.428443i
$351$		0			0
$352$			25.5383i			1.36120i
$353$			1.87953i			0.100037i	0.998748	+	0.0500186i	$0.0159281\pi$
$353$			1.87953i			0.100037i	−0.998748	+	0.0500186i	$0.984072\pi$
$354$		0			0
$355$	9.00000	−	27.9152i	0.477670	−	1.48158i
$356$	−4.11684			−0.218192
$357$		0			0
$358$		−	3.86886i		−	0.204476i
$359$	−10.8832			−0.574391			−0.287196	−	0.957872i	$-0.592723\pi$
$359$	−10.8832			−0.574391			−0.287196	+	0.957872i	$0.592723\pi$
$360$		0			0
$361$	−18.8614			−0.992706
$362$		−	10.9822i		−	0.577212i
$363$		0			0
$364$	27.7663			1.45535
$365$	5.05842	+	1.63086i	0.264770	+	0.0853632i
$366$		0			0
$367$			16.4356i			0.857934i	0.903320	+	0.428967i	$0.141122\pi$
$367$			16.4356i			0.857934i	−0.903320	+	0.428967i	$0.858878\pi$
$368$		−	0.994667i		−	0.0518506i
$369$		0			0
$370$	1.29211	−	4.00772i	0.0671736	−	0.208352i
$371$	−41.4891			−2.15401
$372$		0			0
$373$			8.21782i			0.425503i	0.977106	+	0.212751i	$0.0682424\pi$
$373$			8.21782i			0.425503i	−0.977106	+	0.212751i	$0.931758\pi$
$374$	−2.74456			−0.141918
$375$		0			0
$376$	−5.02175			−0.258977
$377$		−	33.5538i		−	1.72811i
$378$		0			0
$379$	1.48913			0.0764912			0.0382456	−	0.999268i	$-0.487823\pi$
$379$	1.48913			0.0764912			0.0382456	+	0.999268i	$0.487823\pi$
$380$	−0.350532	+	1.08724i	−0.0179819	+	0.0557743i
$381$		0			0
$382$		−	6.04334i		−	0.309204i
$383$		−	11.0920i		−	0.566776i	−0.959005	−	0.283388i	$-0.908542\pi$
$383$		−	11.0920i		−	0.566776i	0.959005	−	0.283388i	$-0.0914584\pi$
$384$		0			0
$385$	32.2337	+	10.3923i	1.64278	+	0.529641i
$386$	−9.09509			−0.462928
$387$		0			0
$388$			1.76972i			0.0898440i
$389$	−0.510875			−0.0259024			−0.0129512	−	0.999916i	$-0.504123\pi$
$389$	−0.510875			−0.0259024			−0.0129512	+	0.999916i	$0.504123\pi$
$390$		0			0
$391$	1.25544			0.0634902
$392$		−	13.3591i		−	0.674735i
$393$		0			0
$394$	−5.09509			−0.256687
$395$	4.62772	−	14.3537i	0.232846	−	0.722215i
$396$		0			0
$397$		−	17.5229i		−	0.879449i	−0.898133	−	0.439724i	$-0.855076\pi$
$397$		−	17.5229i		−	0.879449i	0.898133	−	0.439724i	$-0.144924\pi$
$398$		−	12.6766i		−	0.635420i
$399$		0			0
$400$	−2.54755	−	1.83324i	−0.127377	−	0.0916620i
$401$	−25.3723			−1.26703			−0.633516	−	0.773730i	$-0.718389\pi$
$401$	−25.3723			−1.26703			−0.633516	+	0.773730i	$0.718389\pi$
$402$		0			0
$403$			37.2203i			1.85407i
$404$	22.4674			1.11779
$405$		0			0
$406$	−15.7663			−0.782469
$407$			10.3923i			0.515127i
$408$		0			0
$409$	−22.8614			−1.13042			−0.565212	−	0.824946i	$-0.691206\pi$
$409$	−22.8614			−1.13042			−0.565212	+	0.824946i	$0.691206\pi$
$410$	7.37228	+	2.37686i	0.364091	+	0.117385i
$411$		0			0
$412$		−	9.50744i		−	0.468398i
$413$		−	5.63858i		−	0.277457i
$414$		0			0
$415$	25.4891	+	8.21782i	1.25121	+	0.403397i
$416$	−34.1168			−1.67272
$417$		0			0
$418$			1.28962i			0.0630774i
$419$	−17.4891			−0.854400			−0.427200	−	0.904157i	$-0.640500\pi$
$419$	−17.4891			−0.854400			−0.427200	+	0.904157i	$0.640500\pi$
$420$		0			0
$421$	−21.7446			−1.05977			−0.529883	−	0.848071i	$-0.677764\pi$
$421$	−21.7446			−1.05977			−0.529883	+	0.848071i	$0.677764\pi$
$422$		−	1.47477i		−	0.0717906i
$423$		0			0
$424$	32.0000			1.55406
$425$	2.31386	−	3.21543i	0.112239	−	0.155972i
$426$		0			0
$427$		−	32.4665i		−	1.57117i
$428$		−	18.2054i		−	0.879990i
$429$		0			0
$430$	−1.88316	+	5.84096i	−0.0908138	+	0.281676i
$431$	33.3505			1.60644			0.803219	−	0.595683i	$-0.203119\pi$
$431$	33.3505			1.60644			0.803219	+	0.595683i	$0.203119\pi$
$432$		0			0
$433$		−	12.7692i		−	0.613647i	−0.951766	−	0.306823i	$-0.900734\pi$
$433$		−	12.7692i		−	0.613647i	0.951766	−	0.306823i	$-0.0992661\pi$
$434$	17.4891			0.839505
$435$		0			0
$436$	−13.3723			−0.640416
$437$		−	0.589907i		−	0.0282191i
$438$		0			0
$439$	−31.8614			−1.52066			−0.760331	−	0.649536i	$-0.774963\pi$
$439$	−31.8614			−1.52066			−0.760331	+	0.649536i	$0.774963\pi$
$440$	−24.8614	−	8.01544i	−1.18522	−	0.382121i
$441$		0			0
$442$		−	3.66648i		−	0.174397i
$443$		−	21.4843i		−	1.02075i	−0.859952	−	0.510375i	$-0.829506\pi$
$443$		−	21.4843i		−	1.02075i	0.859952	−	0.510375i	$-0.170494\pi$
$444$		0			0
$445$	2.05842	−	6.38458i	0.0975786	−	0.302658i
$446$	14.7446			0.698175
$447$		0			0
$448$			11.6819i			0.551919i
$449$	−10.8832			−0.513608			−0.256804	−	0.966464i	$-0.582669\pi$
$449$	−10.8832			−0.513608			−0.256804	+	0.966464i	$0.582669\pi$
$450$		0			0
$451$	−19.1168			−0.900177
$452$		−	8.42020i		−	0.396053i
$453$		0			0
$454$	−10.7446			−0.504267
$455$	−13.8832	+	43.0612i	−0.650852	+	2.01874i
$456$		0			0
$457$		−	20.9870i		−	0.981730i	−0.871236	−	0.490865i	$-0.836681\pi$
$457$		−	20.9870i		−	0.981730i	0.871236	−	0.490865i	$-0.163319\pi$
$458$			13.9490i			0.651793i
$459$		0			0
$460$	4.62772	+	1.49200i	0.215768	+	0.0695649i
$461$	−24.0951			−1.12222			−0.561110	−	0.827741i	$-0.689626\pi$
$461$	−24.0951			−1.12222			−0.561110	+	0.827741i	$0.689626\pi$
$462$		0			0
$463$			13.8564i			0.643962i	0.946746	+	0.321981i	$0.104349\pi$
$463$			13.8564i			0.643962i	−0.946746	+	0.321981i	$0.895651\pi$
$464$	−3.60597			−0.167403
$465$		0			0
$466$	4.86141			0.225200
$467$			18.3152i			0.847525i	0.905773	+	0.423763i	$0.139291\pi$
$467$			18.3152i			0.847525i	−0.905773	+	0.423763i	$0.860709\pi$
$468$		0			0
$469$	−40.4674			−1.86861
$470$	1.02175	−	3.16915i	0.0471298	−	0.146182i
$471$		0			0
$472$			4.34896i			0.200177i
$473$		−	15.1460i		−	0.696415i
$474$		0			0
$475$	−1.51087	−	1.08724i	−0.0693237	−	0.0498860i
$476$	3.76631			0.172629
$477$		0			0
$478$			18.2054i			0.832694i
$479$	18.6060			0.850128			0.425064	−	0.905163i	$-0.360252\pi$
$479$	18.6060			0.850128			0.425064	+	0.905163i	$0.360252\pi$
$480$		0			0
$481$	−13.8832			−0.633017
$482$		−	1.19705i		−	0.0545240i
$483$		0			0
$484$	11.1386			0.506300
$485$	−2.74456	−	0.884861i	−0.124624	−	0.0401795i
$486$		0			0
$487$			9.50744i			0.430823i	0.976523	+	0.215412i	$0.0691093\pi$
$487$			9.50744i			0.430823i	−0.976523	+	0.215412i	$0.930891\pi$
$488$			25.0410i			1.13355i
$489$		0			0
$490$	8.43070	+	2.71810i	0.380860	+	0.122791i
$491$	31.6277			1.42734			0.713669	−	0.700483i	$-0.247032\pi$
$491$	31.6277			1.42734			0.713669	+	0.700483i	$0.247032\pi$
$492$		0			0
$493$		−	4.55134i		−	0.204982i
$494$	−1.72281			−0.0775130
$495$		0			0
$496$	4.00000			0.179605
$497$			45.4381i			2.03818i
$498$		0			0
$499$	24.3723			1.09105			0.545527	−	0.838094i	$-0.316330\pi$
$499$	24.3723			1.09105			0.545527	+	0.838094i	$0.316330\pi$
$500$	12.3505	−	9.10268i	0.552333	−	0.407084i
$501$		0			0
$502$			16.0309i			0.715494i
$503$			3.16915i			0.141305i	0.997501	+	0.0706527i	$0.0225082\pi$
$503$			3.16915i			0.141305i	−0.997501	+	0.0706527i	$0.977492\pi$
$504$		0			0
$505$	−11.2337	+	34.8434i	−0.499893	+	1.55051i
$506$	5.48913			0.244021
$507$		0			0
$508$			14.2612i			0.632737i
$509$	0.510875			0.0226441			0.0113221	−	0.999936i	$-0.496396\pi$
$509$	0.510875			0.0226441			0.0113221	+	0.999936i	$0.496396\pi$
$510$		0			0
$511$	−8.23369			−0.364237
$512$			7.02078i			0.310277i
$513$		0			0
$514$	−5.09509			−0.224735
$515$	14.7446	+	4.75372i	0.649723	+	0.209474i
$516$		0			0
$517$			8.21782i			0.361419i
$518$			6.52344i			0.286624i
$519$		0			0
$520$	10.7079	−	33.2125i	0.469572	−	1.45647i
$521$	18.0000			0.788594			0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000			0.788594			0.394297	+	0.918983i	$0.370988\pi$
$522$		0			0
$523$		−	4.75372i		−	0.207866i	−0.994584	−	0.103933i	$-0.966857\pi$
$523$		−	4.75372i		−	0.207866i	0.994584	−	0.103933i	$-0.0331427\pi$
$524$	−6.00000			−0.262111
$525$		0			0
$526$	−21.0217			−0.916592
$527$			5.04868i			0.219924i
$528$		0			0
$529$	20.4891			0.890832
$530$	−6.51087	+	20.1947i	−0.282814	+	0.877202i
$531$		0			0
$532$		−	1.76972i		−	0.0767272i
$533$		−	25.5383i		−	1.10619i
$534$		0			0
$535$	28.2337	+	9.10268i	1.22065	+	0.393543i
$536$	31.2119			1.34815
$537$		0			0
$538$		−	23.1615i		−	0.998562i
$539$	−21.8614			−0.941637
$540$		0			0
$541$	19.2337			0.826921			0.413460	−	0.910522i	$-0.364320\pi$
$541$	19.2337			0.826921			0.413460	+	0.910522i	$0.364320\pi$
$542$			4.16381i			0.178851i
$543$		0			0
$544$	−4.62772			−0.198412
$545$	6.68614	−	20.7383i	0.286403	−	0.888332i
$546$		0			0
$547$			6.92820i			0.296229i	0.988970	+	0.148114i	$0.0473203\pi$
$547$			6.92820i			0.296229i	−0.988970	+	0.148114i	$0.952680\pi$
$548$		−	19.6974i		−	0.841430i
$549$		0			0
$550$	10.1168	−	14.0588i	0.431384	−	0.599469i
$551$	−2.13859			−0.0911071
$552$		0			0
$553$			23.3639i			0.993532i
$554$	−17.4891			−0.743042
$555$		0			0
$556$	−15.2554			−0.646975
$557$			25.0410i			1.06102i	0.847678	+	0.530511i	$0.178000\pi$
$557$			25.0410i			1.06102i	−0.847678	+	0.530511i	$0.822000\pi$
$558$		0			0
$559$	20.2337			0.855794
$560$	4.62772	+	1.49200i	0.195557	+	0.0630485i
$561$		0			0
$562$			1.08724i			0.0458625i
$563$			32.1716i			1.35587i	0.735122	+	0.677935i	$0.237125\pi$
$563$			32.1716i			1.35587i	−0.735122	+	0.677935i	$0.762875\pi$
$564$		0			0
$565$	13.0584	+	4.21010i	0.549372	+	0.177120i
$566$	21.9565			0.922901
$567$		0			0
$568$		−	35.0458i		−	1.47049i
$569$	20.4891			0.858949			0.429474	−	0.903079i	$-0.358699\pi$
$569$	20.4891			0.858949			0.429474	+	0.903079i	$0.358699\pi$
$570$		0			0
$571$	4.13859			0.173195			0.0865974	−	0.996243i	$-0.472401\pi$
$571$	4.13859			0.173195			0.0865974	+	0.996243i	$0.472401\pi$
$572$			35.0458i			1.46534i
$573$		0			0
$574$	−12.0000			−0.500870
$575$	−4.62772	+	6.43087i	−0.192989	+	0.268186i
$576$		0			0
$577$		−	18.4077i		−	0.766325i	−0.923681	−	0.383162i	$-0.874835\pi$
$577$		−	18.4077i		−	0.766325i	0.923681	−	0.383162i	$-0.125165\pi$
$578$			12.9715i			0.539545i
$579$		0			0
$580$	5.40895	−	16.7769i	0.224595	−	0.696622i
$581$	−41.4891			−1.72126
$582$		0			0
$583$		−	52.3663i		−	2.16879i
$584$	6.35053			0.262787
$585$		0			0
$586$	−0.627719			−0.0259308
$587$		−	16.7306i		−	0.690546i	−0.938502	−	0.345273i	$-0.887786\pi$
$587$		−	16.7306i		−	0.690546i	0.938502	−	0.345273i	$-0.112214\pi$
$588$		0			0
$589$	2.37228			0.0977481
$590$	−2.74456	−	0.884861i	−0.112992	−	0.0364291i
$591$		0			0
$592$			1.49200i			0.0613208i
$593$			1.67715i			0.0688722i	0.999407	+	0.0344361i	$0.0109635\pi$
$593$			1.67715i			0.0688722i	−0.999407	+	0.0344361i	$0.989036\pi$
$594$		0			0
$595$	−1.88316	+	5.84096i	−0.0772019	+	0.239456i
$596$	14.5842			0.597393
$597$		0			0
$598$			7.33296i			0.299867i
$599$	1.62772			0.0665068			0.0332534	−	0.999447i	$-0.489413\pi$
$599$	1.62772			0.0665068			0.0332534	+	0.999447i	$0.489413\pi$
$600$		0			0
$601$	−17.9783			−0.733348			−0.366674	−	0.930349i	$-0.619504\pi$
$601$	−17.9783			−0.733348			−0.366674	+	0.930349i	$0.619504\pi$
$602$		−	9.50744i		−	0.387494i
$603$		0			0
$604$	1.21194			0.0493131
$605$	−5.56930	+	17.2742i	−0.226424	+	0.702297i
$606$		0			0
$607$			43.2636i			1.75602i	0.478647	+	0.878008i	$0.341128\pi$
$607$			43.2636i			1.75602i	−0.478647	+	0.878008i	$0.658872\pi$
$608$			2.17448i			0.0881869i
$609$		0			0
$610$	−15.8030	−	5.09496i	−0.639844	−	0.206289i
$611$	−10.9783			−0.444132
$612$		0			0
$613$		−	30.2921i		−	1.22348i	−0.791057	−	0.611742i	$-0.790469\pi$
$613$		−	30.2921i		−	1.22348i	0.791057	−	0.611742i	$-0.209531\pi$
$614$	−12.0000			−0.484281
$615$		0			0
$616$	40.4674			1.63048
$617$			20.6920i			0.833030i	0.909129	+	0.416515i	$0.136749\pi$
$617$			20.6920i			0.833030i	−0.909129	+	0.416515i	$0.863251\pi$
$618$		0			0
$619$	−4.00000			−0.160774			−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000			−0.160774			−0.0803868	+	0.996764i	$0.525616\pi$
$620$	−6.00000	+	18.6101i	−0.240966	+	0.747401i
$621$		0			0
$622$			12.5668i			0.503882i
$623$			10.3923i			0.416359i
$624$		0			0
$625$	7.94158	+	23.7051i	0.317663	+	0.948204i
$626$	−19.3723			−0.774272
$627$		0			0
$628$			15.7532i			0.628620i
$629$	−1.88316			−0.0750863
$630$		0			0
$631$	6.37228			0.253677			0.126838	−	0.991923i	$-0.459517\pi$
$631$	6.37228			0.253677			0.126838	+	0.991923i	$0.459517\pi$
$632$		−	18.0202i		−	0.716806i
$633$		0			0
$634$	23.3723			0.928232
$635$	−22.1168	−	7.13058i	−0.877680	−	0.282969i
$636$		0			0
$637$		−	29.2048i		−	1.15714i
$638$		−	19.8997i		−	0.787839i
$639$		0			0
$640$	−19.1753	−	6.18220i	−0.757969	−	0.244373i
$641$	−7.97825			−0.315122			−0.157561	−	0.987509i	$-0.550363\pi$
$641$	−7.97825			−0.315122			−0.157561	+	0.987509i	$0.550363\pi$
$642$		0			0
$643$		−	1.28962i		−	0.0508577i	−0.999677	−	0.0254288i	$-0.991905\pi$
$643$		−	1.28962i		−	0.0508577i	0.999677	−	0.0254288i	$-0.00809512\pi$
$644$	−7.53262			−0.296827
$645$		0			0
$646$	−0.233688			−0.00919433
$647$			28.7075i			1.12861i	0.825567	+	0.564304i	$0.190855\pi$
$647$			28.7075i			1.12861i	−0.825567	+	0.564304i	$0.809145\pi$
$648$		0			0
$649$	7.11684			0.279361
$650$	18.7812	+	13.5152i	0.736661	+	0.530108i
$651$		0			0
$652$			20.7846i			0.813988i
$653$		−	6.33830i		−	0.248037i	−0.992280	−	0.124018i	$-0.960422\pi$
$653$		−	6.33830i		−	0.248037i	0.992280	−	0.124018i	$-0.0395782\pi$
$654$		0			0
$655$	3.00000	−	9.30506i	0.117220	−	0.363579i
$656$	−2.74456			−0.107157
$657$		0			0
$658$			5.15848i			0.201099i
$659$	21.2554			0.827994			0.413997	−	0.910278i	$-0.364132\pi$
$659$	21.2554			0.827994			0.413997	+	0.910278i	$0.364132\pi$
$660$		0			0
$661$	−15.2337			−0.592522			−0.296261	−	0.955107i	$-0.595740\pi$
$661$	−15.2337			−0.592522			−0.296261	+	0.955107i	$0.595740\pi$
$662$			7.22316i			0.280736i
$663$		0			0
$664$	32.0000			1.24184
$665$	2.74456	+	0.884861i	0.106430	+	0.0343134i
$666$		0			0
$667$			9.10268i			0.352457i
$668$			18.6101i			0.720047i
$669$		0			0
$670$	−6.35053	+	19.6974i	−0.245342	+	0.760976i
$671$	40.9783			1.58195
$672$		0			0
$673$		−	10.5947i		−	0.408395i	−0.978930	−	0.204198i	$-0.934542\pi$
$673$		−	10.5947i		−	0.408395i	0.978930	−	0.204198i	$-0.0654585\pi$
$674$	5.48913			0.211433
$675$		0			0
$676$	−28.9783			−1.11455
$677$			26.5330i			1.01975i	0.860250	+	0.509873i	$0.170308\pi$
$677$			26.5330i			1.01975i	−0.860250	+	0.509873i	$0.829692\pi$
$678$		0			0
$679$	4.46738			0.171442
$680$	1.45245	−	4.50506i	0.0556990	−	0.172761i
$681$		0			0
$682$			22.0742i			0.845266i
$683$		−	27.1229i		−	1.03783i	−0.854826	−	0.518915i	$-0.826336\pi$
$683$		−	27.1229i		−	1.03783i	0.854826	−	0.518915i	$-0.173664\pi$
$684$		0			0
$685$	30.5475	+	9.84868i	1.16716	+	0.376299i
$686$	5.48913			0.209576
$687$		0			0
$688$		−	2.17448i		−	0.0829013i
$689$	69.9565			2.66513
$690$		0			0
$691$	45.7228			1.73938			0.869689	−	0.493600i	$-0.164319\pi$
$691$	45.7228			1.73938			0.869689	+	0.493600i	$0.164319\pi$
$692$			15.3484i			0.583459i
$693$		0			0
$694$	2.27719			0.0864408
$695$	7.62772	−	23.6588i	0.289336	−	0.897430i
$696$		0			0
$697$		−	3.46410i		−	0.131212i
$698$			21.0796i			0.797873i
$699$		0			0
$700$	−13.8832	+	19.2926i	−0.524734	+	0.729192i
$701$	−17.2337			−0.650907			−0.325454	−	0.945558i	$-0.605517\pi$
$701$	−17.2337			−0.650907			−0.325454	+	0.945558i	$0.605517\pi$
$702$		0			0
$703$			0.884861i			0.0333732i
$704$	−14.7446			−0.555707
$705$		0			0
$706$	−1.48913			−0.0560440
$707$		−	56.7152i		−	2.13300i
$708$		0			0
$709$	−52.3505			−1.96607			−0.983033	−	0.183430i	$-0.941280\pi$
$709$	−52.3505			−1.96607			−0.983033	+	0.183430i	$0.941280\pi$
$710$	22.1168	+	7.13058i	0.830030	+	0.267606i
$711$		0			0
$712$		−	8.01544i		−	0.300391i
$713$		−	10.0974i		−	0.378149i
$714$		0			0
$715$	−54.3505	−	17.5229i	−2.03259	−	0.655319i
$716$	−6.70106			−0.250431
$717$		0			0
$718$		−	8.62258i		−	0.321792i
$719$	10.8832			0.405873			0.202937	−	0.979192i	$-0.434951\pi$
$719$	10.8832			0.405873			0.202937	+	0.979192i	$0.434951\pi$
$720$		0			0
$721$	−24.0000			−0.893807
$722$		−	14.9436i		−	0.556145i
$723$		0			0
$724$	−19.0217			−0.706938
$725$	23.3139	+	16.7769i	0.865855	+	0.623078i
$726$		0			0
$727$		−	48.9022i		−	1.81368i	−0.421473	−	0.906841i	$-0.638487\pi$
$727$		−	48.9022i		−	1.81368i	0.421473	−	0.906841i	$-0.361513\pi$
$728$			54.0607i			2.00362i
$729$		0			0
$730$	−1.29211	+	4.00772i	−0.0478231	+	0.148332i
$731$	2.74456			0.101511
$732$		0			0
$733$		−	16.4356i		−	0.607064i	−0.952821	−	0.303532i	$-0.901834\pi$
$733$		−	16.4356i		−	0.607064i	0.952821	−	0.303532i	$-0.0981660\pi$
$734$	−13.0217			−0.480642
$735$		0			0
$736$	9.25544			0.341160
$737$		−	51.0767i		−	1.88143i
$738$		0			0
$739$	−5.62772			−0.207019			−0.103509	−	0.994628i	$-0.533007\pi$
$739$	−5.62772			−0.207019			−0.103509	+	0.994628i	$0.533007\pi$
$740$	−6.94158	−	2.23800i	−0.255177	−	0.0822705i
$741$		0			0
$742$		−	32.8713i		−	1.20674i
$743$		−	9.39764i		−	0.344766i	−0.985030	−	0.172383i	$-0.944853\pi$
$743$		−	9.39764i		−	0.344766i	0.985030	−	0.172383i	$-0.0551467\pi$
$744$		0			0
$745$	−7.29211	+	22.6179i	−0.267162	+	0.828654i
$746$	−6.51087			−0.238380
$747$		0			0
$748$			4.75372i			0.173813i
$749$	−45.9565			−1.67921
$750$		0			0
$751$	21.7228			0.792677			0.396338	−	0.918105i	$-0.370281\pi$
$751$	21.7228			0.792677			0.396338	+	0.918105i	$0.370281\pi$
$752$			1.17981i			0.0430234i
$753$		0			0
$754$	26.5842			0.968140
$755$	−0.605969	+	1.87953i	−0.0220535	+	0.0684030i
$756$		0			0
$757$			41.5692i			1.51086i	0.655230	+	0.755429i	$0.272572\pi$
$757$			41.5692i			1.51086i	−0.655230	+	0.755429i	$0.727428\pi$
$758$			1.17981i			0.0428528i
$759$		0			0
$760$	−2.11684	−	0.682481i	−0.0767860	−	0.0247562i
$761$	−32.4891			−1.17773			−0.588865	−	0.808231i	$-0.700425\pi$
$761$	−32.4891			−1.17773			−0.588865	+	0.808231i	$0.700425\pi$
$762$		0			0
$763$			33.7562i			1.22205i
$764$	−10.4674			−0.378696
$765$		0			0
$766$	8.78806			0.317526
$767$			9.50744i			0.343294i
$768$		0			0
$769$	16.4891			0.594613			0.297307	−	0.954782i	$-0.403912\pi$
$769$	16.4891			0.594613			0.297307	+	0.954782i	$0.403912\pi$
$770$	−8.23369	+	25.5383i	−0.296722	+	0.920338i
$771$		0			0
$772$			15.7532i			0.566969i
$773$			21.5769i			0.776067i	0.921645	+	0.388034i	$0.126846\pi$
$773$			21.5769i			0.776067i	−0.921645	+	0.388034i	$0.873154\pi$
$774$		0			0
$775$	−25.8614	−	18.6101i	−0.928969	−	0.668496i
$776$	−3.44563			−0.123691
$777$		0			0
$778$		−	0.404759i		−	0.0145113i
$779$	−1.62772			−0.0583191
$780$		0			0
$781$	−57.3505			−2.05216
$782$			0.994667i			0.0355692i
$783$		0			0
$784$	−3.13859			−0.112093
$785$	−24.4307	−	7.87658i	−0.871969	−	0.281127i
$786$		0			0
$787$		−	17.3205i		−	0.617409i	−0.951158	−	0.308705i	$-0.900105\pi$
$787$		−	17.3205i		−	0.617409i	0.951158	−	0.308705i	$-0.0998955\pi$
$788$			8.82496i			0.314376i
$789$		0			0
$790$	11.3723	+	3.66648i	0.404608	+	0.130448i
$791$	−21.2554			−0.755756
$792$		0			0
$793$			54.7431i			1.94399i
$794$	13.8832			0.492695
$795$		0			0
$796$	−21.9565			−0.778228
$797$		−	25.6309i		−	0.907893i	−0.891029	−	0.453947i	$-0.850016\pi$
$797$		−	25.6309i		−	0.907893i	0.891029	−	0.453947i	$-0.149984\pi$
$798$		0			0
$799$	−1.48913			−0.0526815
$800$	17.0584	−	23.7051i	0.603106	−	0.838102i
$801$		0			0
$802$		−	20.1021i		−	0.709831i
$803$		−	10.3923i		−	0.366736i
$804$		0			0
$805$	3.76631	−	11.6819i	0.132745	−	0.411734i
$806$	−29.4891			−1.03871
$807$		0			0
$808$			43.7437i			1.53890i
$809$	21.0000			0.738321			0.369160	−	0.929366i	$-0.379645\pi$
$809$	21.0000			0.738321			0.369160	+	0.929366i	$0.379645\pi$
$810$		0			0
$811$	24.8832			0.873766			0.436883	−	0.899518i	$-0.356082\pi$
$811$	24.8832			0.873766			0.436883	+	0.899518i	$0.356082\pi$
$812$			27.3081i			0.958325i
$813$		0			0
$814$	−8.23369			−0.288591
$815$	−32.2337	−	10.3923i	−1.12910	−	0.364027i
$816$		0			0
$817$		−	1.28962i		−	0.0451181i
$818$		−	18.1128i		−	0.633299i
$819$		0			0
$820$	4.11684	−	12.7692i	0.143766	−	0.445919i
$821$	40.7228			1.42124			0.710618	−	0.703578i	$-0.248415\pi$
$821$	40.7228			1.42124			0.710618	+	0.703578i	$0.248415\pi$
$822$		0			0
$823$			4.34896i			0.151595i	0.997123	+	0.0757977i	$0.0241503\pi$
$823$			4.34896i			0.151595i	−0.997123	+	0.0757977i	$0.975850\pi$
$824$	18.5109			0.644857
$825$		0			0
$826$	4.46738			0.155440
$827$		−	22.3692i		−	0.777853i	−0.921269	−	0.388926i	$-0.872846\pi$
$827$		−	22.3692i		−	0.777853i	0.921269	−	0.388926i	$-0.127154\pi$
$828$		0			0
$829$	23.3505			0.810997			0.405499	−	0.914096i	$-0.367098\pi$
$829$	23.3505			0.810997			0.405499	+	0.914096i	$0.367098\pi$
$830$	−6.51087	+	20.1947i	−0.225996	+	0.700968i
$831$		0			0
$832$		−	19.6974i		−	0.682883i
$833$		−	3.96143i		−	0.137256i
$834$		0			0
$835$	−28.8614	−	9.30506i	−0.998790	−	0.322015i
$836$	2.23369			0.0772537
$837$		0			0
$838$		−	13.8564i		−	0.478662i
$839$	−45.8614			−1.58331			−0.791656	−	0.610967i	$-0.790781\pi$
$839$	−45.8614			−1.58331			−0.791656	+	0.610967i	$0.790781\pi$
$840$		0			0
$841$	4.00000			0.137931
$842$		−	17.2279i		−	0.593714i
$843$		0			0
$844$	−2.55437			−0.0879252
$845$	14.4891	−	44.9407i	0.498441	−	1.54601i
$846$		0			0
$847$		−	28.1176i		−	0.966131i
$848$		−	7.51811i		−	0.258173i
$849$		0			0
$850$	2.54755	+	1.83324i	0.0873802	+	0.0628796i
$851$	3.76631			0.129108
$852$		0			0
$853$			10.7971i			0.369684i	0.982768	+	0.184842i	$0.0591774\pi$
$853$			10.7971i			0.369684i	−0.982768	+	0.184842i	$0.940823\pi$
$854$	25.7228			0.880217
$855$		0			0
$856$	35.4456			1.21151
$857$		−	49.3995i		−	1.68746i	−0.536772	−	0.843728i	$-0.680356\pi$
$857$		−	49.3995i		−	1.68746i	0.536772	−	0.843728i	$-0.319644\pi$
$858$		0			0
$859$	−48.3288			−1.64896			−0.824478	−	0.565893i	$-0.808531\pi$
$859$	−48.3288			−1.64896			−0.824478	+	0.565893i	$0.808531\pi$
$860$	10.1168	+	3.26172i	0.344982	+	0.111224i
$861$		0			0
$862$			26.4232i			0.899978i
$863$		−	16.7306i		−	0.569516i	−0.958599	−	0.284758i	$-0.908087\pi$
$863$		−	16.7306i		−	0.569516i	0.958599	−	0.284758i	$-0.0919133\pi$
$864$		0			0
$865$	−23.8030	−	7.67420i	−0.809326	−	0.260931i
$866$	10.1168			0.343784
$867$		0			0
$868$		−	30.2921i		−	1.02818i
$869$	−29.4891			−1.00035
$870$		0			0
$871$	68.2337			2.31201
$872$		−	26.0357i		−	0.881679i
$873$		0			0
$874$	0.467376			0.0158092
$875$	−22.9783	−	31.1769i	−0.776807	−	1.05397i
$876$		0			0
$877$			1.08724i			0.0367135i	0.999832	+	0.0183568i	$0.00584346\pi$
$877$			1.08724i			0.0367135i	−0.999832	+	0.0183568i	$0.994157\pi$
$878$		−	25.2434i		−	0.851923i
$879$		0			0
$880$	−1.88316	+	5.84096i	−0.0634812	+	0.196899i
$881$	−10.8832			−0.366663			−0.183331	−	0.983051i	$-0.558688\pi$
$881$	−10.8832			−0.366663			−0.183331	+	0.983051i	$0.558688\pi$
$882$		0			0
$883$			54.9455i			1.84906i	0.381104	+	0.924532i	$0.375544\pi$
$883$			54.9455i			1.84906i	−0.381104	+	0.924532i	$0.624456\pi$
$884$	−6.35053			−0.213592
$885$		0			0
$886$	17.0217			0.571857
$887$			43.8535i			1.47246i	0.676733	+	0.736228i	$0.263395\pi$
$887$			43.8535i			1.47246i	−0.676733	+	0.736228i	$0.736605\pi$
$888$		0			0
$889$	36.0000			1.20740
$890$	5.05842	+	1.63086i	0.169559	+	0.0546666i
$891$		0			0
$892$		−	25.5383i		−	0.855087i
$893$			0.699713i			0.0234150i
$894$		0			0
$895$	3.35053	−	10.3923i	0.111996	−	0.347376i
$896$	31.2119			1.04272
$897$		0			0
$898$		−	8.62258i		−	0.287739i
$899$	−36.6060			−1.22088
$900$		0			0
$901$	9.48913			0.316129
$902$		−	15.1460i		−	0.504308i
$903$		0			0
$904$	16.3940			0.545257
$905$	9.51087	−	29.4998i	0.316152	−	0.980605i
$906$		0			0
$907$			18.2054i			0.604499i	0.953229	+	0.302250i	$0.0977376\pi$
$907$			18.2054i			0.604499i	−0.953229	+	0.302250i	$0.902262\pi$
$908$			18.6101i			0.617599i
$909$		0			0
$910$	−34.1168	−	10.9994i	−1.13096	−	0.364628i
$911$	18.6060			0.616443			0.308222	−	0.951315i	$-0.400266\pi$
$911$	18.6060			0.616443			0.308222	+	0.951315i	$0.400266\pi$
$912$		0			0
$913$		−	52.3663i		−	1.73307i
$914$	16.6277			0.549996
$915$		0			0
$916$	24.1603			0.798280
$917$			15.1460i			0.500166i
$918$		0			0
$919$	−55.3505			−1.82585			−0.912923	−	0.408132i	$-0.866180\pi$
$919$	−55.3505			−1.82585			−0.912923	+	0.408132i	$0.866180\pi$
$920$	−2.90491	+	9.01011i	−0.0957719	+	0.297055i
$921$		0			0
$922$		−	19.0902i		−	0.628703i
$923$		−	76.6150i		−	2.52181i
$924$		0			0
$925$	6.94158	−	9.64630i	0.228238	−	0.317169i
$926$	−10.9783			−0.360768
$927$		0			0
$928$		−	33.5538i		−	1.10146i
$929$	−3.51087			−0.115188			−0.0575940	−	0.998340i	$-0.518343\pi$
$929$	−3.51087			−0.115188			−0.0575940	+	0.998340i	$0.518343\pi$
$930$		0			0
$931$	−1.86141			−0.0610051
$932$		−	8.42020i		−	0.275813i
$933$		0			0
$934$	−14.5109			−0.474810
$935$	−7.37228	−	2.37686i	−0.241099	−	0.0777317i
$936$		0			0
$937$			22.2766i			0.727745i	0.931449	+	0.363873i	$0.118546\pi$
$937$			22.2766i			0.727745i	−0.931449	+	0.363873i	$0.881454\pi$
$938$		−	32.0618i		−	1.04685i
$939$		0			0
$940$	−5.48913	−	1.76972i	−0.179036	−	0.0577220i
$941$	6.86141			0.223675			0.111838	−	0.993726i	$-0.464326\pi$
$941$	6.86141			0.223675			0.111838	+	0.993726i	$0.464326\pi$
$942$		0			0
$943$			6.92820i			0.225613i
$944$	1.02175			0.0332551
$945$		0			0
$946$	12.0000			0.390154
$947$		−	43.1538i		−	1.40231i	−0.713009	−	0.701155i	$-0.752668\pi$
$947$		−	43.1538i		−	1.40231i	0.713009	−	0.701155i	$-0.247332\pi$
$948$		0			0
$949$	13.8832			0.450666
$950$	0.861407	−	1.19705i	0.0279477	−	0.0388373i
$951$		0			0
$952$			7.33296i			0.237663i
$953$			25.0410i			0.811158i	0.914060	+	0.405579i	$0.132930\pi$
$953$			25.0410i			0.811158i	−0.914060	+	0.405579i	$0.867070\pi$
$954$		0			0
$955$	5.23369	−	16.2333i	0.169358	−	0.525296i
$956$	31.5326			1.01984
$957$		0			0
$958$			14.7413i			0.476269i
$959$	−49.7228			−1.60563
$960$		0			0
$961$	9.60597			0.309870
$962$		−	10.9994i		−	0.354636i
$963$		0			0
$964$	−2.07335			−0.0667780
$965$	−24.4307	−	7.87658i	−0.786452	−	0.253556i
$966$		0			0
$967$			43.2636i			1.39126i	0.718399	+	0.695632i	$0.244875\pi$
$967$			43.2636i			1.39126i	−0.718399	+	0.695632i	$0.755125\pi$
$968$			21.6867i			0.697037i
$969$		0			0
$970$	0.701064	−	2.17448i	0.0225098	−	0.0698184i
$971$	−41.5842			−1.33450			−0.667251	−	0.744833i	$-0.732529\pi$
$971$	−41.5842			−1.33450			−0.667251	+	0.744833i	$0.732529\pi$
$972$		0			0
$973$			38.5099i			1.23457i
$974$	−7.53262			−0.241361
$975$		0			0
$976$	5.88316			0.188315
$977$			28.3027i			0.905484i	0.891642	+	0.452742i	$0.149554\pi$
$977$			28.3027i			0.905484i	−0.891642	+	0.452742i	$0.850446\pi$
$978$		0			0
$979$	−13.1168			−0.419216
$980$	4.70789	−	14.6024i	0.150388	−	0.466457i
$981$		0			0
$982$			25.0582i			0.799640i
$983$			50.3770i			1.60678i	0.595456	+	0.803388i	$0.296971\pi$
$983$			50.3770i			1.60678i	−0.595456	+	0.803388i	$0.703029\pi$
$984$		0			0
$985$	−13.6861	−	4.41248i	−0.436077	−	0.140593i
$986$	3.60597			0.114837
$987$		0			0
$988$			2.98400i			0.0949337i
$989$	−5.48913			−0.174544
$990$		0			0
$991$	38.6060			1.22636			0.613180	−	0.789944i	$-0.289890\pi$
$991$	38.6060			1.22636			0.613180	+	0.789944i	$0.289890\pi$
$992$			37.2203i			1.18174i
$993$		0			0
$994$	−36.0000			−1.14185
$995$	10.9783	−	34.0511i	0.348034	−	1.07949i
$996$		0			0
$997$			60.3817i			1.91231i	0.292864	+	0.956154i	$0.405392\pi$
$997$			60.3817i			1.91231i	−0.292864	+	0.956154i	$0.594608\pi$
$998$			19.3098i			0.611242i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.2.b.b.244.3	yes	4
3.2	odd	2		405.2.b.a.244.2	✓	4
5.2	odd	4		2025.2.a.x.1.2		4
5.3	odd	4		2025.2.a.x.1.3		4
5.4	even	2	inner	405.2.b.b.244.2	yes	4
9.2	odd	6		405.2.j.e.109.1		4
9.4	even	3		405.2.j.d.379.1		4
9.5	odd	6		405.2.j.b.379.2		4
9.7	even	3		405.2.j.a.109.2		4
15.2	even	4		2025.2.a.w.1.3		4
15.8	even	4		2025.2.a.w.1.2		4
15.14	odd	2		405.2.b.a.244.3	yes	4
45.4	even	6		405.2.j.a.379.2		4
45.14	odd	6		405.2.j.e.379.1		4
45.29	odd	6		405.2.j.b.109.2		4
45.34	even	6		405.2.j.d.109.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.2.b.a.244.2	✓	4	3.2	odd	2
405.2.b.a.244.3	yes	4	15.14	odd	2
405.2.b.b.244.2	yes	4	5.4	even	2	inner
405.2.b.b.244.3	yes	4	1.1	even	1	trivial
405.2.j.a.109.2		4	9.7	even	3
405.2.j.a.379.2		4	45.4	even	6
405.2.j.b.109.2		4	45.29	odd	6
405.2.j.b.379.2		4	9.5	odd	6
405.2.j.d.109.1		4	45.34	even	6
405.2.j.d.379.1		4	9.4	even	3
405.2.j.e.109.1		4	9.2	odd	6
405.2.j.e.379.1		4	45.14	odd	6
2025.2.a.w.1.2		4	15.8	even	4
2025.2.a.w.1.3		4	15.2	even	4
2025.2.a.x.1.2		4	5.2	odd	4
2025.2.a.x.1.3		4	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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	405.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.792287i q^{2} +1.37228 q^{4} +(-0.686141 + 2.12819i) q^{5} -3.46410i q^{7} +2.67181i q^{8} +O(q^{10})\)	\(q+0.792287i q^{2} +1.37228 q^{4} +(-0.686141 + 2.12819i) q^{5} -3.46410i q^{7} +2.67181i q^{8} +(-1.68614 - 0.543620i) q^{10} +4.37228 q^{11} +5.84096i q^{13} +2.74456 q^{14} +0.627719 q^{16} +0.792287i q^{17} +0.372281 q^{19} +(-0.941578 + 2.92048i) q^{20} +3.46410i q^{22} -1.58457i q^{23} +(-4.05842 - 2.92048i) q^{25} -4.62772 q^{26} -4.75372i q^{28} -5.74456 q^{29} +6.37228 q^{31} +5.84096i q^{32} -0.627719 q^{34} +(7.37228 + 2.37686i) q^{35} +2.37686i q^{37} +0.294954i q^{38} +(-5.68614 - 1.83324i) q^{40} -4.37228 q^{41} -3.46410i q^{43} +6.00000 q^{44} +1.25544 q^{46} +1.87953i q^{47} -5.00000 q^{49} +(2.31386 - 3.21543i) q^{50} +8.01544i q^{52} -11.9769i q^{53} +(-3.00000 + 9.30506i) q^{55} +9.25544 q^{56} -4.55134i q^{58} +1.62772 q^{59} +9.37228 q^{61} +5.04868i q^{62} -3.37228 q^{64} +(-12.4307 - 4.00772i) q^{65} -11.6819i q^{67} +1.08724i q^{68} +(-1.88316 + 5.84096i) q^{70} -13.1168 q^{71} -2.37686i q^{73} -1.88316 q^{74} +0.510875 q^{76} -15.1460i q^{77} -6.74456 q^{79} +(-0.430703 + 1.33591i) q^{80} -3.46410i q^{82} -11.9769i q^{83} +(-1.68614 - 0.543620i) q^{85} +2.74456 q^{86} +11.6819i q^{88} -3.00000 q^{89} +20.2337 q^{91} -2.17448i q^{92} -1.48913 q^{94} +(-0.255437 + 0.792287i) q^{95} +1.28962i q^{97} -3.96143i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{4} + 3 q^{5}+O(q^{10}) \) 4 * q - 6 * q^4 + 3 * q^5	\( 4 q - 6 q^{4} + 3 q^{5} - q^{10} + 6 q^{11} - 12 q^{14} + 14 q^{16} - 10 q^{19} - 21 q^{20} + q^{25} - 30 q^{26} + 14 q^{31} - 14 q^{34} + 18 q^{35} - 17 q^{40} - 6 q^{41} + 24 q^{44} + 28 q^{46} - 20 q^{49} + 15 q^{50} - 12 q^{55} + 60 q^{56} + 18 q^{59} + 26 q^{61} - 2 q^{64} - 21 q^{65} - 42 q^{70} - 18 q^{71} - 42 q^{74} + 48 q^{76} - 4 q^{79} + 27 q^{80} - q^{85} - 12 q^{86} - 12 q^{89} + 12 q^{91} + 40 q^{94} - 24 q^{95}+O(q^{100}) \) 4 * q - 6 * q^4 + 3 * q^5 - q^10 + 6 * q^11 - 12 * q^14 + 14 * q^16 - 10 * q^19 - 21 * q^20 + q^25 - 30 * q^26 + 14 * q^31 - 14 * q^34 + 18 * q^35 - 17 * q^40 - 6 * q^41 + 24 * q^44 + 28 * q^46 - 20 * q^49 + 15 * q^50 - 12 * q^55 + 60 * q^56 + 18 * q^59 + 26 * q^61 - 2 * q^64 - 21 * q^65 - 42 * q^70 - 18 * q^71 - 42 * q^74 + 48 * q^76 - 4 * q^79 + 27 * q^80 - q^85 - 12 * q^86 - 12 * q^89 + 12 * q^91 + 40 * q^94 - 24 * q^95

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