LMFDB - Embedded newform 405.4.b.f.244.8

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 91 x^{14} + 3268 x^{12} + 59128 x^{10} + 571975 x^{8} + 2881141 x^{6} + 6555196 x^{4} + \cdots + 614656 $ x^16 + 91x^14 + 3268x^12 + 59128x^10 + 571975x^8 + 2881141x^6 + 6555196x^4 + 4069504*x^2 + 614656
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{12}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			244.8
Root			$-0.473990i$ of defining polynomial
Character	$\chi$	$=$	405.244
Dual form			405.4.b.f.244.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.473990i q^{2} +7.77533 q^{4} +(-10.6248 + 3.48041i) q^{5} -8.20657i q^{7} -7.47735i q^{8} +O(q^{10})$	\(q-0.473990i q^{2} +7.77533 q^{4} +(-10.6248 + 3.48041i) q^{5} -8.20657i q^{7} -7.47735i q^{8} +(1.64968 + 5.03606i) q^{10} -5.26583 q^{11} +77.0342i q^{13} -3.88983 q^{14} +58.6585 q^{16} -88.9397i q^{17} +91.7358 q^{19} +(-82.6115 + 27.0613i) q^{20} +2.49595i q^{22} +154.441i q^{23} +(100.774 - 73.9574i) q^{25} +36.5135 q^{26} -63.8088i q^{28} +168.303 q^{29} +72.7885 q^{31} -87.6223i q^{32} -42.1565 q^{34} +(28.5622 + 87.1933i) q^{35} +154.836i q^{37} -43.4818i q^{38} +(26.0242 + 79.4455i) q^{40} -7.95157 q^{41} -23.2014i q^{43} -40.9436 q^{44} +73.2034 q^{46} +301.398i q^{47} +275.652 q^{49} +(-35.0551 - 47.7657i) q^{50} +598.967i q^{52} +344.542i q^{53} +(55.9485 - 18.3272i) q^{55} -61.3634 q^{56} -79.7741i q^{58} +251.558 q^{59} +272.902 q^{61} -34.5010i q^{62} +427.736 q^{64} +(-268.110 - 818.475i) q^{65} -835.467i q^{67} -691.535i q^{68} +(41.3288 - 13.5382i) q^{70} -351.152 q^{71} +522.749i q^{73} +73.3909 q^{74} +713.276 q^{76} +43.2144i q^{77} +700.107 q^{79} +(-623.236 + 204.155i) q^{80} +3.76897i q^{82} -241.303i q^{83} +(309.546 + 944.968i) q^{85} -10.9972 q^{86} +39.3744i q^{88} +1021.39 q^{89} +632.187 q^{91} +1200.83i q^{92} +142.860 q^{94} +(-974.676 + 319.278i) q^{95} +389.284i q^{97} -130.656i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 54 q^{4} + 3 q^{5}+O(q^{10}) $ 16 * q - 54 * q^4 + 3 * q^5	$ 16 q - 54 q^{4} + 3 q^{5} - 10 q^{10} + 90 q^{11} - 102 q^{14} + 146 q^{16} - 4 q^{19} - 6 q^{20} - 71 q^{25} + 468 q^{26} - 516 q^{29} + 38 q^{31} - 212 q^{34} + 267 q^{35} - 44 q^{40} + 576 q^{41} - 1644 q^{44} - 290 q^{46} + 4 q^{49} + 558 q^{50} + 15 q^{55} + 2430 q^{56} - 2202 q^{59} + 20 q^{61} + 322 q^{64} + 339 q^{65} - 636 q^{70} + 2952 q^{71} - 4080 q^{74} - 396 q^{76} + 218 q^{79} - 1266 q^{80} + 704 q^{85} + 6108 q^{86} - 4074 q^{89} - 942 q^{91} + 1078 q^{94} - 1692 q^{95}+O(q^{100}) $ 16 * q - 54 * q^4 + 3 * q^5 - 10 * q^10 + 90 * q^11 - 102 * q^14 + 146 * q^16 - 4 * q^19 - 6 * q^20 - 71 * q^25 + 468 * q^26 - 516 * q^29 + 38 * q^31 - 212 * q^34 + 267 * q^35 - 44 * q^40 + 576 * q^41 - 1644 * q^44 - 290 * q^46 + 4 * q^49 + 558 * q^50 + 15 * q^55 + 2430 * q^56 - 2202 * q^59 + 20 * q^61 + 322 * q^64 + 339 * q^65 - 636 * q^70 + 2952 * q^71 - 4080 * q^74 - 396 * q^76 + 218 * q^79 - 1266 * q^80 + 704 * q^85 + 6108 * q^86 - 4074 * q^89 - 942 * q^91 + 1078 * q^94 - 1692 * 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 $

$2$		−	0.473990i		−	0.167581i	−0.996483	−	0.0837904i	$-0.973297\pi$
$2$		−	0.473990i		−	0.167581i	0.996483	−	0.0837904i	$-0.0267026\pi$
$3$		0			0
$3$		0			0
$4$	7.77533			0.971917
$5$	−10.6248	+	3.48041i	−0.950313	+	0.311297i
$5$	−10.6248	+	3.48041i	−0.950313	+	0.311297i
$6$		0			0
$7$		−	8.20657i		−	0.443113i	−0.975148	−	0.221557i	$-0.928886\pi$
$7$		−	8.20657i		−	0.443113i	0.975148	−	0.221557i	$-0.0711138\pi$
$8$		−	7.47735i		−	0.330455i
$9$		0			0
$10$	1.64968	+	5.03606i	0.0521674	+	0.159254i
$11$	−5.26583			−0.144337			−0.0721685	−	0.997392i	$-0.522992\pi$
$11$	−5.26583			−0.144337			−0.0721685	+	0.997392i	$0.522992\pi$
$12$		0			0
$13$			77.0342i			1.64350i	0.569851	+	0.821748i	$0.307001\pi$
$13$			77.0342i			1.64350i	−0.569851	+	0.821748i	$0.692999\pi$
$14$	−3.88983			−0.0742573
$15$		0			0
$16$	58.6585			0.916539
$17$		−	88.9397i		−	1.26888i	−0.772970	−	0.634442i	$-0.781230\pi$
$17$		−	88.9397i		−	1.26888i	0.772970	−	0.634442i	$-0.218770\pi$
$18$		0			0
$19$	91.7358			1.10766			0.553832	−	0.832628i	$-0.313165\pi$
$19$	91.7358			1.10766			0.553832	+	0.832628i	$0.313165\pi$
$20$	−82.6115	+	27.0613i	−0.923625	+	0.302555i
$21$		0			0
$22$			2.49595i			0.0241881i
$23$			154.441i			1.40014i	0.714076	+	0.700068i	$0.246847\pi$
$23$			154.441i			1.40014i	−0.714076	+	0.700068i	$0.753153\pi$
$24$		0			0
$25$	100.774	−	73.9574i	0.806188	−	0.591659i
$26$	36.5135			0.275418
$27$		0			0
$28$		−	63.8088i		−	0.430669i
$29$	168.303			1.07769			0.538847	−	0.842403i	$-0.318860\pi$
$29$	168.303			1.07769			0.538847	+	0.842403i	$0.318860\pi$
$30$		0			0
$31$	72.7885			0.421716			0.210858	−	0.977517i	$-0.432374\pi$
$31$	72.7885			0.421716			0.210858	+	0.977517i	$0.432374\pi$
$32$		−	87.6223i		−	0.484050i
$33$		0			0
$34$	−42.1565			−0.212641
$35$	28.5622	+	87.1933i	0.137940	+	0.421096i
$36$		0			0
$37$			154.836i			0.687971i	0.938975	+	0.343986i	$0.111777\pi$
$37$			154.836i			0.687971i	−0.938975	+	0.343986i	$0.888223\pi$
$38$		−	43.4818i		−	0.185623i
$39$		0			0
$40$	26.0242	+	79.4455i	0.102870	+	0.314036i
$41$	−7.95157			−0.0302885			−0.0151442	−	0.999885i	$-0.504821\pi$
$41$	−7.95157			−0.0302885			−0.0151442	+	0.999885i	$0.504821\pi$
$42$		0			0
$43$		−	23.2014i		−	0.0822833i	−0.999153	−	0.0411416i	$-0.986901\pi$
$43$		−	23.2014i		−	0.0822833i	0.999153	−	0.0411416i	$-0.0130995\pi$
$44$	−40.9436			−0.140284
$45$		0			0
$46$	73.2034			0.234636
$47$			301.398i			0.935392i	0.883889	+	0.467696i	$0.154916\pi$
$47$			301.398i			0.935392i	−0.883889	+	0.467696i	$0.845084\pi$
$48$		0			0
$49$	275.652			0.803651
$50$	−35.0551	−	47.7657i	−0.0991507	−	0.135102i
$51$		0			0
$52$			598.967i			1.59734i
$53$			344.542i			0.892952i	0.894796	+	0.446476i	$0.147321\pi$
$53$			344.542i			0.892952i	−0.894796	+	0.446476i	$0.852679\pi$
$54$		0			0
$55$	55.9485	−	18.3272i	0.137165	−	0.0449317i
$56$	−61.3634			−0.146429
$57$		0			0
$58$		−	79.7741i		−	0.180601i
$59$	251.558			0.555085			0.277542	−	0.960713i	$-0.410480\pi$
$59$	251.558			0.555085			0.277542	+	0.960713i	$0.410480\pi$
$60$		0			0
$61$	272.902			0.572811			0.286406	−	0.958108i	$-0.407539\pi$
$61$	272.902			0.572811			0.286406	+	0.958108i	$0.407539\pi$
$62$		−	34.5010i		−	0.0706715i
$63$		0			0
$64$	427.736			0.835421
$65$	−268.110	−	818.475i	−0.511616	−	1.56184i
$66$		0			0
$67$		−	835.467i		−	1.52341i	−0.647923	−	0.761706i	$-0.724362\pi$
$67$		−	835.467i		−	1.52341i	0.647923	−	0.761706i	$-0.275638\pi$
$68$		−	691.535i		−	1.23325i
$69$		0			0
$70$	41.3288	−	13.5382i	0.0705676	−	0.0231161i
$71$	−351.152			−0.586959			−0.293479	−	0.955965i	$-0.594813\pi$
$71$	−351.152			−0.586959			−0.293479	+	0.955965i	$0.594813\pi$
$72$		0			0
$73$			522.749i			0.838126i	0.907957	+	0.419063i	$0.137641\pi$
$73$			522.749i			0.838126i	−0.907957	+	0.419063i	$0.862359\pi$
$74$	73.3909			0.115291
$75$		0			0
$76$	713.276			1.07656
$77$			43.2144i			0.0639576i
$78$		0			0
$79$	700.107			0.997065			0.498533	−	0.866871i	$-0.333872\pi$
$79$	700.107			0.997065			0.498533	+	0.866871i	$0.333872\pi$
$80$	−623.236	+	204.155i	−0.870998	+	0.285316i
$81$		0			0
$82$			3.76897i			0.00507577i
$83$		−	241.303i		−	0.319113i	−0.987189	−	0.159557i	$-0.948994\pi$
$83$		−	241.303i		−	0.319113i	0.987189	−	0.159557i	$-0.0510065\pi$
$84$		0			0
$85$	309.546	+	944.968i	0.395000	+	1.20584i
$86$	−10.9972			−0.0137891
$87$		0			0
$88$			39.3744i			0.0476969i
$89$	1021.39			1.21648			0.608242	−	0.793752i	$-0.291875\pi$
$89$	1021.39			1.21648			0.608242	+	0.793752i	$0.291875\pi$
$90$		0			0
$91$	632.187			0.728255
$92$			1200.83i			1.36082i
$93$		0			0
$94$	142.860			0.156754
$95$	−974.676	+	319.278i	−1.05263	+	0.344813i
$96$		0			0
$97$			389.284i			0.407483i	0.979025	+	0.203741i	$0.0653102\pi$
$97$			389.284i			0.407483i	−0.979025	+	0.203741i	$0.934690\pi$
$98$		−	130.656i		−	0.134676i
$99$		0			0
$100$	783.548	−	575.043i	0.783548	−	0.575043i
$101$	864.519			0.851712			0.425856	−	0.904791i	$-0.359973\pi$
$101$	864.519			0.851712			0.425856	+	0.904791i	$0.359973\pi$
$102$		0			0
$103$			1228.21i			1.17494i	0.809246	+	0.587470i	$0.199876\pi$
$103$			1228.21i			1.17494i	−0.809246	+	0.587470i	$0.800124\pi$
$104$	576.012			0.543102
$105$		0			0
$106$	163.309			0.149642
$107$		−	1860.34i		−	1.68080i	−0.541963	−	0.840402i	$-0.682319\pi$
$107$		−	1860.34i		−	1.68080i	0.541963	−	0.840402i	$-0.317681\pi$
$108$		0			0
$109$	−1037.31			−0.911524			−0.455762	−	0.890102i	$-0.650633\pi$
$109$	−1037.31			−0.911524			−0.455762	+	0.890102i	$0.650633\pi$
$110$	−8.68692	−	26.5190i	−0.00752969	−	0.0229863i
$111$		0			0
$112$		−	481.385i		−	0.406130i
$113$			1407.73i			1.17193i	0.810335	+	0.585967i	$0.199285\pi$
$113$			1407.73i			1.17193i	−0.810335	+	0.585967i	$0.800715\pi$
$114$		0			0
$115$	−537.517	−	1640.90i	−0.435858	−	1.33057i
$116$	1308.61			1.04743
$117$		0			0
$118$		−	119.236i		−	0.0930215i
$119$	−729.890			−0.562259
$120$		0			0
$121$	−1303.27			−0.979167
$122$		−	129.353i		−	0.0959922i
$123$		0			0
$124$	565.955			0.409873
$125$	−813.299	+	1136.52i	−0.581949	+	0.813225i
$126$		0			0
$127$			198.654i			0.138801i	0.997589	+	0.0694005i	$0.0221086\pi$
$127$			198.654i			0.138801i	−0.997589	+	0.0694005i	$0.977891\pi$
$128$		−	903.721i		−	0.624050i
$129$		0			0
$130$	−387.949	+	127.082i	−0.261734	+	0.0857369i
$131$	1452.18			0.968532			0.484266	−	0.874921i	$-0.339087\pi$
$131$	1452.18			0.968532			0.484266	+	0.874921i	$0.339087\pi$
$132$		0			0
$133$		−	752.836i		−	0.490821i
$134$	−396.003			−0.255295
$135$		0			0
$136$	−665.033			−0.419310
$137$			636.656i			0.397030i	0.980098	+	0.198515i	$0.0636119\pi$
$137$			636.656i			0.397030i	−0.980098	+	0.198515i	$0.936388\pi$
$138$		0			0
$139$	−1523.24			−0.929494			−0.464747	−	0.885444i	$-0.653855\pi$
$139$	−1523.24			−0.929494			−0.464747	+	0.885444i	$0.653855\pi$
$140$	222.081	+	677.957i	0.134066	+	0.409270i
$141$		0			0
$142$			166.443i			0.0983630i
$143$		−	405.649i		−	0.237217i
$144$		0			0
$145$	−1788.19	+	585.764i	−1.02415	+	0.335483i
$146$	247.778			0.140454
$147$		0			0
$148$			1203.90i			0.668651i
$149$	−2102.60			−1.15605			−0.578027	−	0.816018i	$-0.696177\pi$
$149$	−2102.60			−1.15605			−0.578027	+	0.816018i	$0.696177\pi$
$150$		0			0
$151$	−2360.50			−1.27215			−0.636074	−	0.771628i	$-0.719443\pi$
$151$	−2360.50			−1.27215			−0.636074	+	0.771628i	$0.719443\pi$
$152$		−	685.940i		−	0.366034i
$153$		0			0
$154$	20.4832			0.0107181
$155$	−773.365	+	253.334i	−0.400762	+	0.131279i
$156$		0			0
$157$		−	2004.19i		−	1.01880i	−0.860530	−	0.509400i	$-0.829868\pi$
$157$		−	2004.19i		−	1.01880i	0.860530	−	0.509400i	$-0.170132\pi$
$158$		−	331.844i		−	0.167089i
$159$		0			0
$160$	304.961	+	930.971i	0.150683	+	0.459998i
$161$	1267.43			0.620419
$162$		0			0
$163$		−	2772.38i		−	1.33220i	−0.745861	−	0.666102i	$-0.767962\pi$
$163$		−	2772.38i		−	1.33220i	0.745861	−	0.666102i	$-0.232038\pi$
$164$	−61.8261			−0.0294379
$165$		0			0
$166$	−114.375			−0.0534773
$167$		−	1846.33i		−	0.855529i	−0.903890	−	0.427764i	$-0.859301\pi$
$167$		−	1846.33i		−	0.855529i	0.903890	−	0.427764i	$-0.140699\pi$
$168$		0			0
$169$	−3737.27			−1.70108
$170$	447.905	−	146.722i	0.202075	−	0.0661944i
$171$		0			0
$172$		−	180.399i		−	0.0799725i
$173$		−	193.172i		−	0.0848937i	−0.999099	−	0.0424469i	$-0.986485\pi$
$173$		−	193.172i		−	0.0848937i	0.999099	−	0.0424469i	$-0.0135153\pi$
$174$		0			0
$175$	−606.936	−	827.005i	−0.262172	−	0.357233i
$176$	−308.886			−0.132290
$177$		0			0
$178$		−	484.128i		−	0.203859i
$179$	−3931.26			−1.64154			−0.820770	−	0.571258i	$-0.806456\pi$
$179$	−3931.26			−1.64154			−0.820770	+	0.571258i	$0.806456\pi$
$180$		0			0
$181$	−1921.32			−0.789007			−0.394503	−	0.918894i	$-0.629083\pi$
$181$	−1921.32			−0.789007			−0.394503	+	0.918894i	$0.629083\pi$
$182$		−	299.650i		−	0.122042i
$183$		0			0
$184$	1154.81			0.462682
$185$	−538.894	−	1645.11i	−0.214163	−	0.653788i
$186$		0			0
$187$			468.341i			0.183147i
$188$			2343.47i			0.909123i
$189$		0			0
$190$	151.334	+	461.987i	0.0577840	+	0.176400i
$191$	1507.83			0.571219			0.285609	−	0.958346i	$-0.407804\pi$
$191$	1507.83			0.571219			0.285609	+	0.958346i	$0.407804\pi$
$192$		0			0
$193$		−	4065.65i		−	1.51633i	−0.652061	−	0.758166i	$-0.726096\pi$
$193$		−	4065.65i		−	1.51633i	0.652061	−	0.758166i	$-0.273904\pi$
$194$	184.517			0.0682863
$195$		0			0
$196$	2143.29			0.781082
$197$			3916.67i			1.41650i	0.705960	+	0.708252i	$0.250516\pi$
$197$			3916.67i			1.41650i	−0.705960	+	0.708252i	$0.749484\pi$
$198$		0			0
$199$	−615.676			−0.219317			−0.109659	−	0.993969i	$-0.534976\pi$
$199$	−615.676			−0.219317			−0.109659	+	0.993969i	$0.534976\pi$
$200$	−553.005	−	753.519i	−0.195517	−	0.266409i
$201$		0			0
$202$		−	409.773i		−	0.142730i
$203$		−	1381.19i		−	0.477541i
$204$		0			0
$205$	84.4840	−	27.6747i	0.0287835	−	0.00942871i
$206$	582.157			0.196897
$207$		0			0
$208$			4518.71i			1.50633i
$209$	−483.065			−0.159877
$210$		0			0
$211$	−2233.49			−0.728719			−0.364360	−	0.931258i	$-0.618712\pi$
$211$	−2233.49			−0.728719			−0.364360	+	0.931258i	$0.618712\pi$
$212$			2678.93i			0.867875i
$213$		0			0
$214$	−881.784			−0.281670
$215$	80.7503	+	246.511i	0.0256145	+	0.0781948i
$216$		0			0
$217$		−	597.344i		−	0.186868i
$218$			491.674i			0.152754i
$219$		0			0
$220$	435.018	−	142.500i	0.133313	−	0.0436698i
$221$	6851.40			2.08541
$222$		0			0
$223$			4630.44i			1.39048i	0.718778	+	0.695240i	$0.244702\pi$
$223$			4630.44i			1.39048i	−0.718778	+	0.695240i	$0.755298\pi$
$224$	−719.079			−0.214489
$225$		0			0
$226$	667.252			0.196394
$227$		−	467.693i		−	0.136748i	−0.997660	−	0.0683741i	$-0.978219\pi$
$227$		−	467.693i		−	0.136748i	0.997660	−	0.0683741i	$-0.0217812\pi$
$228$		0			0
$229$	−1457.15			−0.420485			−0.210242	−	0.977649i	$-0.567425\pi$
$229$	−1457.15			−0.420485			−0.210242	+	0.977649i	$0.567425\pi$
$230$	−777.773	+	254.777i	−0.222977	+	0.0730414i
$231$		0			0
$232$		−	1258.46i		−	0.356130i
$233$		−	2617.43i		−	0.735938i	−0.929838	−	0.367969i	$-0.880053\pi$
$233$		−	2617.43i		−	0.735938i	0.929838	−	0.367969i	$-0.119947\pi$
$234$		0			0
$235$	−1048.99	−	3202.30i	−0.291185	−	0.888915i
$236$	1955.94			0.539496
$237$		0			0
$238$			345.960i			0.0942239i
$239$	−3711.16			−1.00441			−0.502207	−	0.864748i	$-0.667478\pi$
$239$	−3711.16			−1.00441			−0.502207	+	0.864748i	$0.667478\pi$
$240$		0			0
$241$	−1844.11			−0.492902			−0.246451	−	0.969155i	$-0.579264\pi$
$241$	−1844.11			−0.492902			−0.246451	+	0.969155i	$0.579264\pi$
$242$			617.737i			0.164090i
$243$		0			0
$244$	2121.90			0.556725
$245$	−2928.75	+	959.382i	−0.763719	+	0.250174i
$246$		0			0
$247$			7066.80i			1.82044i
$248$		−	544.265i		−	0.139358i
$249$		0			0
$250$	538.698	+	385.496i	0.136281	+	0.0975235i
$251$	7204.10			1.81163			0.905815	−	0.423674i	$-0.139260\pi$
$251$	7204.10			1.81163			0.905815	+	0.423674i	$0.139260\pi$
$252$		0			0
$253$		−	813.259i		−	0.202091i
$254$	94.1602			0.0232604
$255$		0			0
$256$	2993.53			0.730842
$257$			1757.15i			0.426489i	0.976999	+	0.213245i	$0.0684031\pi$
$257$			1757.15i			0.426489i	−0.976999	+	0.213245i	$0.931597\pi$
$258$		0			0
$259$	1270.68			0.304849
$260$	−2084.65	−	6363.92i	−0.497248	−	1.51797i
$261$		0			0
$262$		−	688.319i		−	0.162307i
$263$			5356.91i			1.25597i	0.778224	+	0.627987i	$0.216121\pi$
$263$			5356.91i			1.25597i	−0.778224	+	0.627987i	$0.783879\pi$
$264$		0			0
$265$	−1199.15	−	3660.69i	−0.277973	−	0.848584i
$266$	−356.837			−0.0822521
$267$		0			0
$268$		−	6496.04i		−	1.48063i
$269$	−1040.60			−0.235860			−0.117930	−	0.993022i	$-0.537626\pi$
$269$	−1040.60			−0.235860			−0.117930	+	0.993022i	$0.537626\pi$
$270$		0			0
$271$	−1531.38			−0.343265			−0.171632	−	0.985161i	$-0.554904\pi$
$271$	−1531.38			−0.343265			−0.171632	+	0.985161i	$0.554904\pi$
$272$		−	5217.06i		−	1.16298i
$273$		0			0
$274$	301.768			0.0665347
$275$	−530.656	+	389.447i	−0.116363	+	0.0853983i
$276$		0			0
$277$		−	6231.09i		−	1.35159i	−0.737090	−	0.675794i	$-0.763801\pi$
$277$		−	6231.09i		−	1.35159i	0.737090	−	0.675794i	$-0.236199\pi$
$278$			722.001i			0.155765i
$279$		0			0
$280$	651.975	−	213.570i	0.139153	−	0.0455829i
$281$	9113.95			1.93485			0.967425	−	0.253157i	$-0.0814690\pi$
$281$	9113.95			1.93485			0.967425	+	0.253157i	$0.0814690\pi$
$282$		0			0
$283$			1515.09i			0.318243i	0.987259	+	0.159122i	$0.0508662\pi$
$283$			1515.09i			0.318243i	−0.987259	+	0.159122i	$0.949134\pi$
$284$	−2730.32			−0.570475
$285$		0			0
$286$	−192.274			−0.0397531
$287$			65.2552i			0.0134212i
$288$		0			0
$289$	−2997.26			−0.610067
$290$	277.646	+	847.586i	0.0562205	+	0.171627i
$291$		0			0
$292$			4064.55i			0.814588i
$293$			1384.30i			0.276013i	0.990431	+	0.138006i	$0.0440694\pi$
$293$			1384.30i			0.276013i	−0.990431	+	0.138006i	$0.955931\pi$
$294$		0			0
$295$	−2672.75	+	875.522i	−0.527504	+	0.172796i
$296$	1157.77			0.227344
$297$		0			0
$298$			996.613i			0.193732i
$299$	−11897.2			−2.30112
$300$		0			0
$301$	−190.404			−0.0364608
$302$			1118.85i			0.213188i
$303$		0			0
$304$	5381.08			1.01522
$305$	−2899.53	+	949.809i	−0.544350	+	0.178314i
$306$		0			0
$307$		−	1628.16i		−	0.302685i	−0.988481	−	0.151342i	$-0.951640\pi$
$307$		−	1628.16i		−	0.302685i	0.988481	−	0.151342i	$-0.0483596\pi$
$308$			336.006i			0.0621615i
$309$		0			0
$310$	120.078	+	366.567i	0.0219998	+	0.0671600i
$311$	1406.63			0.256471			0.128236	−	0.991744i	$-0.459069\pi$
$311$	1406.63			0.256471			0.128236	+	0.991744i	$0.459069\pi$
$312$		0			0
$313$			5431.54i			0.980858i	0.871481	+	0.490429i	$0.163160\pi$
$313$			5431.54i			0.980858i	−0.871481	+	0.490429i	$0.836840\pi$
$314$	−949.964			−0.170731
$315$		0			0
$316$	5443.56			0.969065
$317$		−	7681.60i		−	1.36101i	−0.732741	−	0.680507i	$-0.761760\pi$
$317$		−	7681.60i		−	1.36101i	0.732741	−	0.680507i	$-0.238240\pi$
$318$		0			0
$319$	−886.257			−0.155551
$320$	−4544.61	+	1488.69i	−0.793911	+	0.260064i
$321$		0			0
$322$		−	600.749i		−	0.103970i
$323$		−	8158.95i		−	1.40550i
$324$		0			0
$325$	5697.25	+	7763.01i	0.972389	+	1.32497i
$326$	−1314.08			−0.223252
$327$		0			0
$328$			59.4567i			0.0100090i
$329$	2473.44			0.414485
$330$		0			0
$331$	4128.88			0.685630			0.342815	−	0.939403i	$-0.388620\pi$
$331$	4128.88			0.685630			0.342815	+	0.939403i	$0.388620\pi$
$332$		−	1876.21i		−	0.310152i
$333$		0			0
$334$	−875.142			−0.143370
$335$	2907.77	+	8876.69i	0.474234	+	1.44772i
$336$		0			0
$337$			2624.15i			0.424174i	0.977251	+	0.212087i	$0.0680260\pi$
$337$			2624.15i			0.424174i	−0.977251	+	0.212087i	$0.931974\pi$
$338$			1771.43i			0.285068i
$339$		0			0
$340$	2406.82	+	7347.44i	0.383907	+	1.17197i
$341$	−383.292			−0.0608693
$342$		0			0
$343$		−	5077.01i		−	0.799221i
$344$	−173.485			−0.0271909
$345$		0			0
$346$	−91.5617			−0.0142266
$347$		−	6914.81i		−	1.06976i	−0.844928	−	0.534880i	$-0.820357\pi$
$347$		−	6914.81i		−	1.06976i	0.844928	−	0.534880i	$-0.179643\pi$
$348$		0			0
$349$	2660.41			0.408047			0.204024	−	0.978966i	$-0.434598\pi$
$349$	2660.41			0.408047			0.204024	+	0.978966i	$0.434598\pi$
$350$	−391.992	+	287.682i	−0.0598653	+	0.0439350i
$351$		0			0
$352$			461.404i			0.0698663i
$353$		−	4517.49i		−	0.681139i	−0.940219	−	0.340569i	$-0.889380\pi$
$353$		−	4517.49i		−	0.681139i	0.940219	−	0.340569i	$-0.110620\pi$
$354$		0			0
$355$	3730.93	−	1222.15i	0.557794	−	0.182719i
$356$	7941.64			1.18232
$357$		0			0
$358$			1863.38i			0.275091i
$359$	2702.34			0.397281			0.198640	−	0.980072i	$-0.436347\pi$
$359$	2702.34			0.397281			0.198640	+	0.980072i	$0.436347\pi$
$360$		0			0
$361$	1556.45			0.226921
$362$			910.684i			0.132222i
$363$		0			0
$364$	4915.46			0.707803
$365$	−1819.38	−	5554.12i	−0.260906	−	0.796481i
$366$		0			0
$367$		−	3538.31i		−	0.503265i	−0.967823	−	0.251632i	$-0.919033\pi$
$367$		−	3538.31i		−	0.503265i	0.967823	−	0.251632i	$-0.0809674\pi$
$368$			9059.26i			1.28328i
$369$		0			0
$370$	−779.765	+	255.430i	−0.109562	+	0.0358897i
$371$	2827.51			0.395679
$372$		0			0
$373$		−	5729.89i		−	0.795395i	−0.917517	−	0.397697i	$-0.869809\pi$
$373$		−	5729.89i		−	0.795395i	0.917517	−	0.397697i	$-0.130191\pi$
$374$	221.989			0.0306919
$375$		0			0
$376$	2253.66			0.309105
$377$			12965.1i			1.77119i
$378$		0			0
$379$	8727.59			1.18287			0.591433	−	0.806354i	$-0.298563\pi$
$379$	8727.59			1.18287			0.591433	+	0.806354i	$0.298563\pi$
$380$	−7578.43	+	2482.49i	−1.02307	+	0.335129i
$381$		0			0
$382$		−	714.696i		−	0.0957252i
$383$			4562.80i			0.608742i	0.952554	+	0.304371i	$0.0984462\pi$
$383$			4562.80i			0.608742i	−0.952554	+	0.304371i	$0.901554\pi$
$384$		0			0
$385$	−150.404	−	459.145i	−0.0199098	−	0.0607797i
$386$	−1927.08			−0.254108
$387$		0			0
$388$			3026.82i			0.396039i
$389$	12084.4			1.57507			0.787536	−	0.616269i	$-0.211357\pi$
$389$	12084.4			1.57507			0.787536	+	0.616269i	$0.211357\pi$
$390$		0			0
$391$	13735.9			1.77661
$392$		−	2061.15i		−	0.265571i
$393$		0			0
$394$	1856.46			0.237379
$395$	−7438.51	+	2436.66i	−0.947524	+	0.310383i
$396$		0			0
$397$			5880.63i			0.743426i	0.928348	+	0.371713i	$0.121230\pi$
$397$			5880.63i			0.743426i	−0.928348	+	0.371713i	$0.878770\pi$
$398$			291.824i			0.0367533i
$399$		0			0
$400$	5911.22	−	4338.23i	0.738903	−	0.542278i
$401$	−3295.11			−0.410350			−0.205175	−	0.978725i	$-0.565776\pi$
$401$	−3295.11			−0.410350			−0.205175	+	0.978725i	$0.565776\pi$
$402$		0			0
$403$			5607.21i			0.693089i
$404$	6721.92			0.827793
$405$		0			0
$406$	−654.672			−0.0800267
$407$		−	815.342i		−	0.0992997i
$408$		0			0
$409$	6217.42			0.751667			0.375834	−	0.926687i	$-0.377356\pi$
$409$	6217.42			0.751667			0.375834	+	0.926687i	$0.377356\pi$
$410$	−13.1175	−	40.0446i	−0.00158007	−	0.00482356i
$411$		0			0
$412$			9549.71i			1.14194i
$413$		−	2064.42i		−	0.245965i
$414$		0			0
$415$	839.831	+	2563.80i	0.0993390	+	0.303257i
$416$	6749.92			0.795534
$417$		0			0
$418$			228.968i			0.0267923i
$419$	1740.28			0.202908			0.101454	−	0.994840i	$-0.467651\pi$
$419$	1740.28			0.202908			0.101454	+	0.994840i	$0.467651\pi$
$420$		0			0
$421$	−1284.58			−0.148709			−0.0743544	−	0.997232i	$-0.523690\pi$
$421$	−1284.58			−0.148709			−0.0743544	+	0.997232i	$0.523690\pi$
$422$			1058.65i			0.122119i
$423$		0			0
$424$	2576.26			0.295081
$425$	−6577.74	−	8962.76i	−0.750747	−	1.02296i
$426$		0			0
$427$		−	2239.59i		−	0.253820i
$428$		−	14464.8i		−	1.63360i
$429$		0			0
$430$	116.844	−	38.2748i	0.0131040	−	0.00429250i
$431$	11917.4			1.33189			0.665943	−	0.746003i	$-0.268030\pi$
$431$	11917.4			1.33189			0.665943	+	0.746003i	$0.268030\pi$
$432$		0			0
$433$			1603.61i			0.177978i	0.996033	+	0.0889889i	$0.0283636\pi$
$433$			1603.61i			0.177978i	−0.996033	+	0.0889889i	$0.971636\pi$
$434$	−283.135			−0.0313155
$435$		0			0
$436$	−8065.42			−0.885925
$437$			14167.7i			1.55088i
$438$		0			0
$439$	−1816.45			−0.197482			−0.0987409	−	0.995113i	$-0.531482\pi$
$439$	−1816.45			−0.197482			−0.0987409	+	0.995113i	$0.531482\pi$
$440$	−137.039	−	418.346i	−0.0148479	−	0.0453270i
$441$		0			0
$442$		−	3247.49i		−	0.349474i
$443$			2265.28i			0.242950i	0.992594	+	0.121475i	$0.0387624\pi$
$443$			2265.28i			0.242950i	−0.992594	+	0.121475i	$0.961238\pi$
$444$		0			0
$445$	−10852.1	+	3554.85i	−1.15604	+	0.378688i
$446$	2194.78			0.233018
$447$		0			0
$448$		−	3510.24i		−	0.370186i
$449$	−13259.9			−1.39371			−0.696854	−	0.717213i	$-0.745417\pi$
$449$	−13259.9			−1.39371			−0.696854	+	0.717213i	$0.745417\pi$
$450$		0			0
$451$	41.8716			0.00437175
$452$			10945.6i			1.13902i
$453$		0			0
$454$	−221.682			−0.0229164
$455$	−6716.87	+	2200.27i	−0.692070	+	0.226704i
$456$		0			0
$457$		−	131.858i		−	0.0134968i	−0.999977	−	0.00674840i	$-0.997852\pi$
$457$		−	131.858i		−	0.0134968i	0.999977	−	0.00674840i	$-0.00214810\pi$
$458$			690.673i			0.0704652i
$459$		0			0
$460$	−4179.37	−	12758.6i	−0.423618	−	1.29320i
$461$	−12782.5			−1.29141			−0.645705	−	0.763587i	$-0.723436\pi$
$461$	−12782.5			−1.29141			−0.645705	+	0.763587i	$0.723436\pi$
$462$		0			0
$463$			4621.41i			0.463877i	0.972730	+	0.231939i	$0.0745069\pi$
$463$			4621.41i			0.463877i	−0.972730	+	0.231939i	$0.925493\pi$
$464$	9872.42			0.987749
$465$		0			0
$466$	−1240.64			−0.123329
$467$		−	5550.68i		−	0.550010i	−0.961443	−	0.275005i	$-0.911320\pi$
$467$		−	5550.68i		−	0.550010i	0.961443	−	0.275005i	$-0.0886795\pi$
$468$		0			0
$469$	−6856.32			−0.675044
$470$	−1517.86	+	497.210i	−0.148965	+	0.0487970i
$471$		0			0
$472$		−	1880.98i		−	0.183431i
$473$			122.175i			0.0118765i
$474$		0			0
$475$	9244.54	−	6784.54i	0.892986	−	0.655360i
$476$	−5675.13			−0.546469
$477$		0			0
$478$			1759.05i			0.168320i
$479$	1941.39			0.185186			0.0925932	−	0.995704i	$-0.470484\pi$
$479$	1941.39			0.185186			0.0925932	+	0.995704i	$0.470484\pi$
$480$		0			0
$481$	−11927.7			−1.13068
$482$			874.088i			0.0826008i
$483$		0			0
$484$	−10133.4			−0.951669
$485$	−1354.87	−	4136.08i	−0.126848	−	0.387236i
$486$		0			0
$487$		−	178.136i		−	0.0165752i	−0.999966	−	0.00828761i	$-0.997362\pi$
$487$		−	178.136i		−	0.0165752i	0.999966	−	0.00828761i	$-0.00263806\pi$
$488$		−	2040.58i		−	0.189289i
$489$		0			0
$490$	454.737	+	1388.20i	0.0419244	+	0.127985i
$491$	5591.59			0.513941			0.256971	−	0.966419i	$-0.417276\pi$
$491$	5591.59			0.513941			0.256971	+	0.966419i	$0.417276\pi$
$492$		0			0
$493$		−	14968.8i		−	1.36747i
$494$	3349.59			0.305071
$495$		0			0
$496$	4269.66			0.386519
$497$			2881.75i			0.260089i
$498$		0			0
$499$	−11367.0			−1.01976			−0.509879	−	0.860246i	$-0.670310\pi$
$499$	−11367.0			−1.01976			−0.509879	+	0.860246i	$0.670310\pi$
$500$	−6323.67	+	8836.80i	−0.565606	+	0.790387i
$501$		0			0
$502$		−	3414.67i		−	0.303594i
$503$		−	9225.98i		−	0.817825i	−0.912574	−	0.408913i	$-0.865908\pi$
$503$		−	9225.98i		−	0.817825i	0.912574	−	0.408913i	$-0.134092\pi$
$504$		0			0
$505$	−9185.36	+	3008.88i	−0.809392	+	0.265135i
$506$	−385.476			−0.0338666
$507$		0			0
$508$			1544.60i			0.134903i
$509$	−18313.4			−1.59475			−0.797376	−	0.603483i	$-0.793779\pi$
$509$	−18313.4			−1.59475			−0.797376	+	0.603483i	$0.793779\pi$
$510$		0			0
$511$	4289.98			0.371384
$512$		−	8648.67i		−	0.746525i
$513$		0			0
$514$	832.870			0.0714714
$515$	−4274.66	−	13049.5i	−0.365755	−	1.11656i
$516$		0			0
$517$		−	1587.11i		−	0.135012i
$518$		−	602.288i		−	0.0510869i
$519$		0			0
$520$	−6120.02	+	2004.76i	−0.516117	+	0.169066i
$521$	11305.4			0.950670			0.475335	−	0.879805i	$-0.342327\pi$
$521$	11305.4			0.950670			0.475335	+	0.879805i	$0.342327\pi$
$522$		0			0
$523$		−	2825.92i		−	0.236269i	−0.992998	−	0.118135i	$-0.962309\pi$
$523$		−	2825.92i		−	0.236269i	0.992998	−	0.118135i	$-0.0376914\pi$
$524$	11291.2			0.941332
$525$		0			0
$526$	2539.12			0.210477
$527$		−	6473.78i		−	0.535109i
$528$		0			0
$529$	−11684.9			−0.960380
$530$	−1735.13	+	568.383i	−0.142206	+	0.0465830i
$531$		0			0
$532$		−	5853.55i		−	0.477037i
$533$		−	612.543i		−	0.0497790i
$534$		0			0
$535$	6474.75	+	19765.8i	0.523229	+	1.59729i
$536$	−6247.08			−0.503420
$537$		0			0
$538$			493.233i			0.0395256i
$539$	−1451.54			−0.115997
$540$		0			0
$541$	−20588.2			−1.63615			−0.818075	−	0.575112i	$-0.804959\pi$
$541$	−20588.2			−1.63615			−0.818075	+	0.575112i	$0.804959\pi$
$542$			725.859i			0.0575246i
$543$		0			0
$544$	−7793.10			−0.614203
$545$	11021.2	−	3610.25i	0.866233	−	0.283755i
$546$		0			0
$547$			4353.95i			0.340332i	0.985415	+	0.170166i	$0.0544303\pi$
$547$			4353.95i			0.340332i	−0.985415	+	0.170166i	$0.945570\pi$
$548$			4950.21i			0.385881i
$549$		0			0
$550$	184.594	+	251.526i	0.0143111	+	0.0195002i
$551$	15439.4			1.19372
$552$		0			0
$553$		−	5745.48i		−	0.441813i
$554$	−2953.48			−0.226500
$555$		0			0
$556$	−11843.7			−0.903391
$557$		−	22267.5i		−	1.69390i	−0.531672	−	0.846951i	$-0.678436\pi$
$557$		−	22267.5i		−	1.69390i	0.531672	−	0.846951i	$-0.321564\pi$
$558$		0			0
$559$	1787.30			0.135232
$560$	1675.42	+	5114.63i	0.126427	+	0.385951i
$561$		0			0
$562$		−	4319.92i		−	0.324244i
$563$			23010.8i			1.72254i	0.508148	+	0.861270i	$0.330330\pi$
$563$			23010.8i			1.72254i	−0.508148	+	0.861270i	$0.669670\pi$
$564$		0			0
$565$	−4899.49	−	14956.9i	−0.364820	−	1.11370i
$566$	718.139			0.0533315
$567$		0			0
$568$			2625.69i			0.193964i
$569$	−14244.0			−1.04945			−0.524726	−	0.851271i	$-0.675832\pi$
$569$	−14244.0			−1.04945			−0.524726	+	0.851271i	$0.675832\pi$
$570$		0			0
$571$	−4793.42			−0.351310			−0.175655	−	0.984452i	$-0.556204\pi$
$571$	−4793.42			−0.351310			−0.175655	+	0.984452i	$0.556204\pi$
$572$		−	3154.06i		−	0.230556i
$573$		0			0
$574$	30.9303			0.00224914
$575$	11422.0	+	15563.5i	0.828403	+	1.12877i
$576$		0			0
$577$			6617.46i			0.477450i	0.971087	+	0.238725i	$0.0767294\pi$
$577$			6617.46i			0.477450i	−0.971087	+	0.238725i	$0.923271\pi$
$578$			1420.67i			0.102236i
$579$		0			0
$580$	−13903.8	+	4554.51i	−0.995386	+	0.326062i
$581$	−1980.27			−0.141403
$582$		0			0
$583$		−	1814.30i		−	0.128886i
$584$	3908.78			0.276963
$585$		0			0
$586$	656.145			0.0462544
$587$			16455.4i			1.15705i	0.815665	+	0.578524i	$0.196371\pi$
$587$			16455.4i			1.15705i	−0.815665	+	0.578524i	$0.803629\pi$
$588$		0			0
$589$	6677.31			0.467120
$590$	414.989	+	1266.86i	0.0289573	+	0.0883996i
$591$		0			0
$592$			9082.47i			0.630552i
$593$			15990.1i			1.10731i	0.832746	+	0.553655i	$0.186767\pi$
$593$			15990.1i			1.10731i	−0.832746	+	0.553655i	$0.813233\pi$
$594$		0			0
$595$	7754.94	−	2540.31i	0.534322	−	0.175030i
$596$	−16348.4			−1.12359
$597$		0			0
$598$			5639.17i			0.385623i
$599$	20460.9			1.39567			0.697836	−	0.716257i	$-0.254146\pi$
$599$	20460.9			1.39567			0.697836	+	0.716257i	$0.254146\pi$
$600$		0			0
$601$	−641.739			−0.0435559			−0.0217779	−	0.999763i	$-0.506933\pi$
$601$	−641.739			−0.0435559			−0.0217779	+	0.999763i	$0.506933\pi$
$602$			90.2496i			0.00611013i
$603$		0			0
$604$	−18353.6			−1.23642
$605$	13847.0	−	4535.91i	0.930515	−	0.304812i
$606$		0			0
$607$		−	18751.3i		−	1.25386i	−0.779077	−	0.626928i	$-0.784312\pi$
$607$		−	18751.3i		−	1.25386i	0.779077	−	0.626928i	$-0.215688\pi$
$608$		−	8038.10i		−	0.536165i
$609$		0			0
$610$	450.200	+	1374.35i	0.0298821	+	0.0912226i
$611$	−23218.0			−1.53731
$612$		0			0
$613$		−	24213.4i		−	1.59538i	−0.603067	−	0.797690i	$-0.706055\pi$
$613$		−	24213.4i		−	1.59538i	0.603067	−	0.797690i	$-0.293945\pi$
$614$	−771.733			−0.0507241
$615$		0			0
$616$	323.129			0.0211351
$617$		−	15870.9i		−	1.03556i	−0.855515	−	0.517779i	$-0.826759\pi$
$617$		−	15870.9i		−	1.03556i	0.855515	−	0.517779i	$-0.173241\pi$
$618$		0			0
$619$	24434.2			1.58658			0.793289	−	0.608846i	$-0.208367\pi$
$619$	24434.2			1.58658			0.793289	+	0.608846i	$0.208367\pi$
$620$	−6013.17	+	1969.75i	−0.389508	+	0.127592i
$621$		0			0
$622$		−	666.727i		−	0.0429796i
$623$		−	8382.11i		−	0.539040i
$624$		0			0
$625$	4685.61	−	14905.9i	0.299879	−	0.953977i
$626$	2574.49			0.164373
$627$		0			0
$628$		−	15583.2i		−	0.990188i
$629$	13771.1			0.872956
$630$		0			0
$631$	22573.7			1.42416			0.712079	−	0.702099i	$-0.247754\pi$
$631$	22573.7			1.42416			0.712079	+	0.702099i	$0.247754\pi$
$632$		−	5234.94i		−	0.329486i
$633$		0			0
$634$	−3641.00			−0.228080
$635$	−691.398	−	2110.67i	−0.0432083	−	0.131904i
$636$		0			0
$637$			21234.7i			1.32080i
$638$			420.077i			0.0260674i
$639$		0			0
$640$	3145.32	+	9601.87i	0.194265	+	0.593043i
$641$	3722.30			0.229364			0.114682	−	0.993402i	$-0.463415\pi$
$641$	3722.30			0.229364			0.114682	+	0.993402i	$0.463415\pi$
$642$		0			0
$643$			2998.76i			0.183919i	0.995763	+	0.0919593i	$0.0293130\pi$
$643$			2998.76i			0.183919i	−0.995763	+	0.0919593i	$0.970687\pi$
$644$	9854.68			0.602995
$645$		0			0
$646$	−3867.26			−0.235535
$647$			4302.38i			0.261428i	0.991420	+	0.130714i	$0.0417270\pi$
$647$			4302.38i			0.261428i	−0.991420	+	0.130714i	$0.958273\pi$
$648$		0			0
$649$	−1324.66			−0.0801193
$650$	3679.59	−	2700.44i	0.222039	−	0.162954i
$651$		0			0
$652$		−	21556.1i		−	1.29479i
$653$			801.281i			0.0480192i	0.999712	+	0.0240096i	$0.00764323\pi$
$653$			801.281i			0.0480192i	−0.999712	+	0.0240096i	$0.992357\pi$
$654$		0			0
$655$	−15429.2	+	5054.18i	−0.920408	+	0.301501i
$656$	−466.427			−0.0277606
$657$		0			0
$658$		−	1172.39i		−	0.0694596i
$659$	−7456.39			−0.440759			−0.220379	−	0.975414i	$-0.570730\pi$
$659$	−7456.39			−0.440759			−0.220379	+	0.975414i	$0.570730\pi$
$660$		0			0
$661$	2665.06			0.156821			0.0784106	−	0.996921i	$-0.475015\pi$
$661$	2665.06			0.156821			0.0784106	+	0.996921i	$0.475015\pi$
$662$		−	1957.05i		−	0.114898i
$663$		0			0
$664$	−1804.30			−0.105453
$665$	2620.18	+	7998.75i	0.152791	+	0.466433i
$666$		0			0
$667$			25992.9i			1.50892i
$668$		−	14355.8i		−	0.831502i
$669$		0			0
$670$	4207.46	−	1378.25i	0.242610	−	0.0794724i
$671$	−1437.05			−0.0826779
$672$		0			0
$673$			23549.8i			1.34885i	0.738342	+	0.674427i	$0.235609\pi$
$673$			23549.8i			1.34885i	−0.738342	+	0.674427i	$0.764391\pi$
$674$	1243.82			0.0710834
$675$		0			0
$676$	−29058.6			−1.65331
$677$			9939.71i			0.564275i	0.959374	+	0.282138i	$0.0910434\pi$
$677$			9939.71i			0.564275i	−0.959374	+	0.282138i	$0.908957\pi$
$678$		0			0
$679$	3194.69			0.180561
$680$	7065.85	−	2314.58i	0.398475	−	0.130530i
$681$		0			0
$682$			181.676i			0.0102005i
$683$			11680.3i			0.654371i	0.944960	+	0.327185i	$0.106100\pi$
$683$			11680.3i			0.654371i	−0.944960	+	0.327185i	$0.893900\pi$
$684$		0			0
$685$	−2215.82	−	6764.35i	−0.123594	−	0.377303i
$686$	−2406.45			−0.133934
$687$		0			0
$688$		−	1360.96i		−	0.0754158i
$689$	−26541.5			−1.46756
$690$		0			0
$691$	−12356.9			−0.680290			−0.340145	−	0.940373i	$-0.610476\pi$
$691$	−12356.9			−0.680290			−0.340145	+	0.940373i	$0.610476\pi$
$692$		−	1501.98i		−	0.0825096i
$693$		0			0
$694$	−3277.55			−0.179271
$695$	16184.2	−	5301.50i	0.883310	−	0.289349i
$696$		0			0
$697$			707.210i			0.0384326i
$698$		−	1261.01i		−	0.0683808i
$699$		0			0
$700$	−4719.13	−	6430.24i	−0.254809	−	0.347200i
$701$	−10659.6			−0.574333			−0.287167	−	0.957881i	$-0.592713\pi$
$701$	−10659.6			−0.574333			−0.287167	+	0.957881i	$0.592713\pi$
$702$		0			0
$703$			14204.0i			0.762042i
$704$	−2252.38			−0.120582
$705$		0			0
$706$	−2141.25			−0.114146
$707$		−	7094.74i		−	0.377405i
$708$		0			0
$709$	−19338.6			−1.02437			−0.512183	−	0.858877i	$-0.671163\pi$
$709$	−19338.6			−1.02437			−0.512183	+	0.858877i	$0.671163\pi$
$710$	−579.288	−	1768.42i	−0.0306201	−	0.0934756i
$711$		0			0
$712$		−	7637.29i		−	0.401994i
$713$			11241.5i			0.590460i
$714$		0			0
$715$	1411.82	+	4309.95i	0.0738451	+	0.225431i
$716$	−30566.8			−1.59544
$717$		0			0
$718$		−	1280.88i		−	0.0665766i
$719$	−27300.1			−1.41603			−0.708013	−	0.706199i	$-0.750408\pi$
$719$	−27300.1			−1.41603			−0.708013	+	0.706199i	$0.750408\pi$
$720$		0			0
$721$	10079.4			0.520631
$722$		−	737.742i		−	0.0380276i
$723$		0			0
$724$	−14938.9			−0.766849
$725$	16960.5	−	12447.3i	0.868825	−	0.637628i
$726$		0			0
$727$		−	1764.93i		−	0.0900381i	−0.998986	−	0.0450191i	$-0.985665\pi$
$727$		−	1764.93i		−	0.0900381i	0.998986	−	0.0450191i	$-0.0143349\pi$
$728$		−	4727.08i		−	0.240656i
$729$		0			0
$730$	−2632.60	+	862.368i	−0.133475	+	0.0437228i
$731$	−2063.52			−0.104408
$732$		0			0
$733$			17305.5i			0.872024i	0.899941	+	0.436012i	$0.143609\pi$
$733$			17305.5i			0.872024i	−0.899941	+	0.436012i	$0.856391\pi$
$734$	−1677.12			−0.0843375
$735$		0			0
$736$	13532.5			0.677735
$737$			4399.43i			0.219885i
$738$		0			0
$739$	1456.60			0.0725058			0.0362529	−	0.999343i	$-0.488458\pi$
$739$	1456.60			0.0725058			0.0362529	+	0.999343i	$0.488458\pi$
$740$	−4190.08	−	12791.3i	−0.208149	−	0.635427i
$741$		0			0
$742$		−	1340.21i		−	0.0663082i
$743$		−	16937.1i		−	0.836288i	−0.908381	−	0.418144i	$-0.862681\pi$
$743$		−	16937.1i		−	0.836288i	0.908381	−	0.418144i	$-0.137319\pi$
$744$		0			0
$745$	22339.8	−	7317.91i	1.09861	−	0.359876i
$746$	−2715.91			−0.133293
$747$		0			0
$748$			3641.51i			0.178004i
$749$	−15267.0			−0.744787
$750$		0			0
$751$	12935.6			0.628531			0.314265	−	0.949335i	$-0.398242\pi$
$751$	12935.6			0.628531			0.314265	+	0.949335i	$0.398242\pi$
$752$			17679.6i			0.857323i
$753$		0			0
$754$	6145.34			0.296817
$755$	25079.8	−	8215.48i	1.20894	−	0.396016i
$756$		0			0
$757$		−	37371.9i		−	1.79433i	−0.441697	−	0.897164i	$-0.645623\pi$
$757$		−	37371.9i		−	1.79433i	0.441697	−	0.897164i	$-0.354377\pi$
$758$		−	4136.79i		−	0.198226i
$759$		0			0
$760$	2387.35	+	7287.99i	0.113945	+	0.347846i
$761$	13999.3			0.666854			0.333427	−	0.942776i	$-0.391795\pi$
$761$	13999.3			0.666854			0.333427	+	0.942776i	$0.391795\pi$
$762$		0			0
$763$			8512.74i			0.403908i
$764$	11723.9			0.555177
$765$		0			0
$766$	2162.72			0.102013
$767$			19378.5i			0.912280i
$768$		0			0
$769$	10442.5			0.489684			0.244842	−	0.969563i	$-0.421264\pi$
$769$	10442.5			0.489684			0.244842	+	0.969563i	$0.421264\pi$
$770$	−217.630	+	71.2898i	−0.0101855	+	0.00333650i
$771$		0			0
$772$		−	31611.8i		−	1.47375i
$773$		−	23092.7i		−	1.07450i	−0.843424	−	0.537249i	$-0.819463\pi$
$773$		−	23092.7i		−	1.07450i	0.843424	−	0.537249i	$-0.180537\pi$
$774$		0			0
$775$	7335.16	−	5383.25i	0.339983	−	0.249512i
$776$	2910.82			0.134655
$777$		0			0
$778$		−	5727.88i		−	0.263952i
$779$	−729.444			−0.0335495
$780$		0			0
$781$	1849.11			0.0847199
$782$		−	6510.68i		−	0.297726i
$783$		0			0
$784$	16169.3			0.736577
$785$	6975.38	+	21294.1i	0.317149	+	0.968178i
$786$		0			0
$787$			8516.24i			0.385732i	0.981225	+	0.192866i	$0.0617783\pi$
$787$			8516.24i			0.385732i	−0.981225	+	0.192866i	$0.938222\pi$
$788$			30453.4i			1.37672i
$789$		0			0
$790$	1154.95	+	3525.78i	0.0520143	+	0.158787i
$791$	11552.7			0.519299
$792$		0			0
$793$			21022.8i			0.941413i
$794$	2787.36			0.124584
$795$		0			0
$796$	−4787.08			−0.213158
$797$			29402.3i			1.30675i	0.757033	+	0.653376i	$0.226648\pi$
$797$			29402.3i			1.30675i	−0.757033	+	0.653376i	$0.773352\pi$
$798$		0			0
$799$	26806.2			1.18690
$800$	−6480.32	−	8830.01i	−0.286392	−	0.390235i
$801$		0			0
$802$			1561.85i			0.0687667i
$803$		−	2752.71i		−	0.120973i
$804$		0			0
$805$	−13466.2	+	4411.17i	−0.589592	+	0.193134i
$806$	2657.76			0.116148
$807$		0			0
$808$		−	6464.31i		−	0.281453i
$809$	−7358.17			−0.319777			−0.159888	−	0.987135i	$-0.551113\pi$
$809$	−7358.17			−0.319777			−0.159888	+	0.987135i	$0.551113\pi$
$810$		0			0
$811$	14069.2			0.609168			0.304584	−	0.952485i	$-0.401483\pi$
$811$	14069.2			0.609168			0.304584	+	0.952485i	$0.401483\pi$
$812$		−	10739.2i		−	0.464130i
$813$		0			0
$814$	−386.464			−0.0166407
$815$	9648.99	+	29456.0i	0.414711	+	1.26601i
$816$		0			0
$817$		−	2128.40i		−	0.0911423i
$818$		−	2947.00i		−	0.125965i
$819$		0			0
$820$	656.892	−	215.180i	0.0279752	−	0.00916392i
$821$	−26680.0			−1.13415			−0.567076	−	0.823665i	$-0.691925\pi$
$821$	−26680.0			−1.13415			−0.567076	+	0.823665i	$0.691925\pi$
$822$		0			0
$823$		−	7614.37i		−	0.322503i	−0.986913	−	0.161252i	$-0.948447\pi$
$823$		−	7614.37i		−	0.322503i	0.986913	−	0.161252i	$-0.0515531\pi$
$824$	9183.73			0.388265
$825$		0			0
$826$	−978.517			−0.0412191
$827$			24854.1i			1.04505i	0.852623	+	0.522527i	$0.175011\pi$
$827$			24854.1i			1.04505i	−0.852623	+	0.522527i	$0.824989\pi$
$828$		0			0
$829$	17292.6			0.724482			0.362241	−	0.932084i	$-0.382012\pi$
$829$	17292.6			0.724482			0.362241	+	0.932084i	$0.382012\pi$
$830$	1215.21	−	398.072i	0.0508201	−	0.0166473i
$831$		0			0
$832$			32950.3i			1.37301i
$833$		−	24516.4i		−	1.01974i
$834$		0			0
$835$	6425.98	+	19616.9i	0.266323	+	0.813020i
$836$	−3755.99			−0.155387
$837$		0			0
$838$		−	824.876i		−	0.0340034i
$839$	7049.18			0.290065			0.145033	−	0.989427i	$-0.453671\pi$
$839$	7049.18			0.290065			0.145033	+	0.989427i	$0.453671\pi$
$840$		0			0
$841$	3937.02			0.161426
$842$			608.876i			0.0249207i
$843$		0			0
$844$	−17366.1			−0.708254
$845$	39707.9	−	13007.2i	1.61656	−	0.529541i
$846$		0			0
$847$			10695.4i			0.433882i
$848$			20210.3i			0.818425i
$849$		0			0
$850$	−4248.26	+	3117.78i	−0.171428	+	0.125811i
$851$	−23913.0			−0.963253
$852$		0			0
$853$		−	11269.1i		−	0.452342i	−0.974088	−	0.226171i	$-0.927379\pi$
$853$		−	11269.1i		−	0.452342i	0.974088	−	0.226171i	$-0.0726209\pi$
$854$	−1061.54			−0.0425354
$855$		0			0
$856$	−13910.4			−0.555431
$857$		−	34154.0i		−	1.36135i	−0.732584	−	0.680676i	$-0.761686\pi$
$857$		−	34154.0i		−	1.36135i	0.732584	−	0.680676i	$-0.238314\pi$
$858$		0			0
$859$	42734.3			1.69741			0.848704	−	0.528868i	$-0.177383\pi$
$859$	42734.3			1.69741			0.848704	+	0.528868i	$0.177383\pi$
$860$	627.861	+	1916.70i	0.0248952	+	0.0759989i
$861$		0			0
$862$		−	5648.75i		−	0.223198i
$863$		−	6893.93i		−	0.271926i	−0.990714	−	0.135963i	$-0.956587\pi$
$863$		−	6893.93i		−	0.271926i	0.990714	−	0.135963i	$-0.0434128\pi$
$864$		0			0
$865$	672.318	+	2052.42i	0.0264272	+	0.0806756i
$866$	760.093			0.0298256
$867$		0			0
$868$		−	4644.55i		−	0.181620i
$869$	−3686.64			−0.143913
$870$		0			0
$871$	64359.6			2.50372
$872$			7756.32i			0.301218i
$873$		0			0
$874$	6715.37			0.259898
$875$	9326.90	+	6674.40i	0.360351	+	0.257869i
$876$		0			0
$877$			47938.4i			1.84580i	0.385040	+	0.922900i	$0.374188\pi$
$877$			47938.4i			1.84580i	−0.385040	+	0.922900i	$0.625812\pi$
$878$			860.980i			0.0330942i
$879$		0			0
$880$	3281.85	−	1075.05i	0.125717	−	0.0411816i
$881$	−35033.8			−1.33975			−0.669875	−	0.742474i	$-0.733652\pi$
$881$	−35033.8			−1.33975			−0.669875	+	0.742474i	$0.733652\pi$
$882$		0			0
$883$		−	155.057i		−	0.00590951i	−0.999996	−	0.00295475i	$-0.999059\pi$
$883$		−	155.057i		−	0.00590951i	0.999996	−	0.00295475i	$-0.000940528\pi$
$884$	53271.9			2.02684
$885$		0			0
$886$	1073.72			0.0407137
$887$			4476.49i			0.169454i	0.996404	+	0.0847272i	$0.0270019\pi$
$887$			4476.49i			0.169454i	−0.996404	+	0.0847272i	$0.972998\pi$
$888$		0			0
$889$	1630.27			0.0615045
$890$	1684.96	+	5143.78i	0.0634608	+	0.193730i
$891$		0			0
$892$			36003.2i			1.35143i
$893$			27649.0i			1.03610i
$894$		0			0
$895$	41768.9	−	13682.4i	1.55998	−	0.511007i
$896$	−7416.45			−0.276525
$897$		0			0
$898$			6285.07i			0.233559i
$899$	12250.6			0.454481
$900$		0			0
$901$	30643.4			1.13305
$902$		−	19.8467i		−	0.000732621i
$903$		0			0
$904$	10526.1			0.387272
$905$	20413.6	−	6686.96i	0.749803	−	0.245615i
$906$		0			0
$907$			12588.2i			0.460843i	0.973091	+	0.230422i	$0.0740106\pi$
$907$			12588.2i			0.460843i	−0.973091	+	0.230422i	$0.925989\pi$
$908$		−	3636.47i		−	0.132908i
$909$		0			0
$910$	1042.90	+	3183.73i	0.0379912	+	0.115978i
$911$	−39913.0			−1.45157			−0.725784	−	0.687923i	$-0.758523\pi$
$911$	−39913.0			−1.45157			−0.725784	+	0.687923i	$0.758523\pi$
$912$		0			0
$913$			1270.66i			0.0460599i
$914$	−62.4992			−0.00226180
$915$		0			0
$916$	−11329.8			−0.408676
$917$		−	11917.4i		−	0.429169i
$918$		0			0
$919$	36540.8			1.31161			0.655804	−	0.754931i	$-0.272330\pi$
$919$	36540.8			1.31161			0.655804	+	0.754931i	$0.272330\pi$
$920$	−12269.6	+	4019.20i	−0.439693	+	0.144032i
$921$		0			0
$922$			6058.77i			0.216416i
$923$		−	27050.7i		−	0.964665i
$924$		0			0
$925$	11451.3	+	15603.4i	0.407044	+	0.554635i
$926$	2190.50			0.0777369
$927$		0			0
$928$		−	14747.1i		−	0.521658i
$929$	18732.5			0.661566			0.330783	−	0.943707i	$-0.392687\pi$
$929$	18732.5			0.661566			0.330783	+	0.943707i	$0.392687\pi$
$930$		0			0
$931$	25287.2			0.890175
$932$		−	20351.4i		−	0.715270i
$933$		0			0
$934$	−2630.97			−0.0921711
$935$	−1630.02	−	4976.04i	−0.0570131	−	0.174047i
$936$		0			0
$937$		−	41501.3i		−	1.44695i	−0.690353	−	0.723473i	$-0.742545\pi$
$937$		−	41501.3i		−	1.44695i	0.690353	−	0.723473i	$-0.257455\pi$
$938$			3249.83i			0.113124i
$939$		0			0
$940$	−8156.23	−	24898.9i	−0.283007	−	0.863951i
$941$	−27366.5			−0.948058			−0.474029	−	0.880509i	$-0.657201\pi$
$941$	−27366.5			−0.948058			−0.474029	+	0.880509i	$0.657201\pi$
$942$		0			0
$943$		−	1228.05i		−	0.0424080i
$944$	14756.0			0.508757
$945$		0			0
$946$	57.9095			0.00199028
$947$		−	51887.7i		−	1.78049i	−0.455482	−	0.890245i	$-0.650533\pi$
$947$		−	51887.7i		−	1.78049i	0.455482	−	0.890245i	$-0.349467\pi$
$948$		0			0
$949$	−40269.6			−1.37746
$950$	−3215.80	−	4381.82i	−0.109826	−	0.149647i
$951$		0			0
$952$			5457.64i			0.185802i
$953$			10624.9i			0.361150i	0.983561	+	0.180575i	$0.0577958\pi$
$953$			10624.9i			0.361150i	−0.983561	+	0.180575i	$0.942204\pi$
$954$		0			0
$955$	−16020.4	+	5247.86i	−0.542836	+	0.177819i
$956$	−28855.5			−0.976206
$957$		0			0
$958$		−	920.198i		−	0.0310337i
$959$	5224.76			0.175929
$960$		0			0
$961$	−24492.8			−0.822155
$962$			5653.61i			0.189480i
$963$		0			0
$964$	−14338.5			−0.479059
$965$	14150.1	+	43196.8i	0.472030	+	1.44099i
$966$		0			0
$967$		−	22458.3i		−	0.746857i	−0.927659	−	0.373429i	$-0.878182\pi$
$967$		−	22458.3i		−	0.746857i	0.927659	−	0.373429i	$-0.121818\pi$
$968$			9745.01i			0.323571i
$969$		0			0
$970$	−1960.46	+	642.194i	−0.0648933	+	0.0212573i
$971$	−42186.8			−1.39427			−0.697137	−	0.716938i	$-0.745543\pi$
$971$	−42186.8			−1.39427			−0.697137	+	0.716938i	$0.745543\pi$
$972$		0			0
$973$			12500.6i			0.411871i
$974$	−84.4349			−0.00277769
$975$		0			0
$976$	16008.0			0.525004
$977$		−	28679.4i		−	0.939135i	−0.882897	−	0.469568i	$-0.844410\pi$
$977$		−	28679.4i		−	0.939135i	0.882897	−	0.469568i	$-0.155590\pi$
$978$		0			0
$979$	−5378.46			−0.175584
$980$	−22772.0	+	7459.51i	−0.742272	+	0.243148i
$981$		0			0
$982$		−	2650.36i		−	0.0861267i
$983$		−	53382.5i		−	1.73208i	−0.499973	−	0.866041i	$-0.666657\pi$
$983$		−	53382.5i		−	1.73208i	0.499973	−	0.866041i	$-0.333343\pi$
$984$		0			0
$985$	−13631.6	−	41613.9i	−0.440954	−	1.34612i
$986$	−7095.08			−0.229162
$987$		0			0
$988$			54946.7i			1.76932i
$989$	3583.24			0.115208
$990$		0			0
$991$	−1505.90			−0.0482708			−0.0241354	−	0.999709i	$-0.507683\pi$
$991$	−1505.90			−0.0482708			−0.0241354	+	0.999709i	$0.507683\pi$
$992$		−	6377.90i		−	0.204132i
$993$		0			0
$994$	1365.92			0.0435859
$995$	6541.44	−	2142.80i	0.208420	−	0.0682727i
$996$		0			0
$997$		−	35521.0i		−	1.12835i	−0.825656	−	0.564174i	$-0.809195\pi$
$997$		−	35521.0i		−	1.12835i	0.825656	−	0.564174i	$-0.190805\pi$
$998$			5387.87i			0.170892i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.b.f.244.8		16
3.2	odd	2		405.4.b.e.244.9		16
5.2	odd	4		2025.4.a.bl.1.9		16
5.3	odd	4		2025.4.a.bl.1.8		16
5.4	even	2	inner	405.4.b.f.244.9		16
9.2	odd	6		45.4.j.a.4.9	yes	32
9.4	even	3		135.4.j.a.19.9		32
9.5	odd	6		45.4.j.a.34.8	yes	32
9.7	even	3		135.4.j.a.64.8		32
15.2	even	4		2025.4.a.bk.1.8		16
15.8	even	4		2025.4.a.bk.1.9		16
15.14	odd	2		405.4.b.e.244.8		16
45.2	even	12		225.4.e.g.76.9		32
45.4	even	6		135.4.j.a.19.8		32
45.14	odd	6		45.4.j.a.34.9	yes	32
45.23	even	12		225.4.e.g.151.8		32
45.29	odd	6		45.4.j.a.4.8	✓	32
45.32	even	12		225.4.e.g.151.9		32
45.34	even	6		135.4.j.a.64.9		32
45.38	even	12		225.4.e.g.76.8		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.j.a.4.8	✓	32	45.29	odd	6
45.4.j.a.4.9	yes	32	9.2	odd	6
45.4.j.a.34.8	yes	32	9.5	odd	6
45.4.j.a.34.9	yes	32	45.14	odd	6
135.4.j.a.19.8		32	45.4	even	6
135.4.j.a.19.9		32	9.4	even	3
135.4.j.a.64.8		32	9.7	even	3
135.4.j.a.64.9		32	45.34	even	6
225.4.e.g.76.8		32	45.38	even	12
225.4.e.g.76.9		32	45.2	even	12
225.4.e.g.151.8		32	45.23	even	12
225.4.e.g.151.9		32	45.32	even	12
405.4.b.e.244.8		16	15.14	odd	2
405.4.b.e.244.9		16	3.2	odd	2
405.4.b.f.244.8		16	1.1	even	1	trivial
405.4.b.f.244.9		16	5.4	even	2	inner
2025.4.a.bk.1.8		16	15.2	even	4
2025.4.a.bk.1.9		16	15.8	even	4
2025.4.a.bl.1.8		16	5.3	odd	4
2025.4.a.bl.1.9		16	5.2	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)

\(f(q)\)	\(=\)	\(q-0.473990i q^{2} +7.77533 q^{4} +(-10.6248 + 3.48041i) q^{5} -8.20657i q^{7} -7.47735i q^{8} +O(q^{10})\)	\(q-0.473990i q^{2} +7.77533 q^{4} +(-10.6248 + 3.48041i) q^{5} -8.20657i q^{7} -7.47735i q^{8} +(1.64968 + 5.03606i) q^{10} -5.26583 q^{11} +77.0342i q^{13} -3.88983 q^{14} +58.6585 q^{16} -88.9397i q^{17} +91.7358 q^{19} +(-82.6115 + 27.0613i) q^{20} +2.49595i q^{22} +154.441i q^{23} +(100.774 - 73.9574i) q^{25} +36.5135 q^{26} -63.8088i q^{28} +168.303 q^{29} +72.7885 q^{31} -87.6223i q^{32} -42.1565 q^{34} +(28.5622 + 87.1933i) q^{35} +154.836i q^{37} -43.4818i q^{38} +(26.0242 + 79.4455i) q^{40} -7.95157 q^{41} -23.2014i q^{43} -40.9436 q^{44} +73.2034 q^{46} +301.398i q^{47} +275.652 q^{49} +(-35.0551 - 47.7657i) q^{50} +598.967i q^{52} +344.542i q^{53} +(55.9485 - 18.3272i) q^{55} -61.3634 q^{56} -79.7741i q^{58} +251.558 q^{59} +272.902 q^{61} -34.5010i q^{62} +427.736 q^{64} +(-268.110 - 818.475i) q^{65} -835.467i q^{67} -691.535i q^{68} +(41.3288 - 13.5382i) q^{70} -351.152 q^{71} +522.749i q^{73} +73.3909 q^{74} +713.276 q^{76} +43.2144i q^{77} +700.107 q^{79} +(-623.236 + 204.155i) q^{80} +3.76897i q^{82} -241.303i q^{83} +(309.546 + 944.968i) q^{85} -10.9972 q^{86} +39.3744i q^{88} +1021.39 q^{89} +632.187 q^{91} +1200.83i q^{92} +142.860 q^{94} +(-974.676 + 319.278i) q^{95} +389.284i q^{97} -130.656i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 54 q^{4} + 3 q^{5}+O(q^{10}) \) 16 * q - 54 * q^4 + 3 * q^5	\( 16 q - 54 q^{4} + 3 q^{5} - 10 q^{10} + 90 q^{11} - 102 q^{14} + 146 q^{16} - 4 q^{19} - 6 q^{20} - 71 q^{25} + 468 q^{26} - 516 q^{29} + 38 q^{31} - 212 q^{34} + 267 q^{35} - 44 q^{40} + 576 q^{41} - 1644 q^{44} - 290 q^{46} + 4 q^{49} + 558 q^{50} + 15 q^{55} + 2430 q^{56} - 2202 q^{59} + 20 q^{61} + 322 q^{64} + 339 q^{65} - 636 q^{70} + 2952 q^{71} - 4080 q^{74} - 396 q^{76} + 218 q^{79} - 1266 q^{80} + 704 q^{85} + 6108 q^{86} - 4074 q^{89} - 942 q^{91} + 1078 q^{94} - 1692 q^{95}+O(q^{100}) \) 16 * q - 54 * q^4 + 3 * q^5 - 10 * q^10 + 90 * q^11 - 102 * q^14 + 146 * q^16 - 4 * q^19 - 6 * q^20 - 71 * q^25 + 468 * q^26 - 516 * q^29 + 38 * q^31 - 212 * q^34 + 267 * q^35 - 44 * q^40 + 576 * q^41 - 1644 * q^44 - 290 * q^46 + 4 * q^49 + 558 * q^50 + 15 * q^55 + 2430 * q^56 - 2202 * q^59 + 20 * q^61 + 322 * q^64 + 339 * q^65 - 636 * q^70 + 2952 * q^71 - 4080 * q^74 - 396 * q^76 + 218 * q^79 - 1266 * q^80 + 704 * q^85 + 6108 * q^86 - 4074 * q^89 - 942 * q^91 + 1078 * q^94 - 1692 * q^95

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