LMFDB - Embedded newform 405.4.b.e.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.e.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	$0.0711138\pi$
$7$			8.20657i			0.443113i	−0.975148	+	0.221557i	$0.928886\pi$
$8$		−	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	$-0.477008\pi$
$11$	5.26583			0.144337			0.0721685	+	0.997392i	$0.477008\pi$
$12$		0			0
$13$		−	77.0342i		−	1.64350i	−0.569851	−	0.821748i	$-0.692999\pi$
$13$		−	77.0342i		−	1.64350i	0.569851	−	0.821748i	$-0.307001\pi$
$14$	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	$-0.681140\pi$
$29$	−168.303			−1.07769			−0.538847	+	0.842403i	$0.681140\pi$
$30$		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	$-0.888223\pi$
$37$		−	154.836i		−	0.687971i	0.938975	−	0.343986i	$-0.111777\pi$
$38$		−	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	$-0.495179\pi$
$41$	7.95157			0.0302885			0.0151442	+	0.999885i	$0.495179\pi$
$42$		0			0
$43$			23.2014i			0.0822833i	0.999153	+	0.0411416i	$0.0130995\pi$
$43$			23.2014i			0.0822833i	−0.999153	+	0.0411416i	$0.986901\pi$
$44$	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	$-0.589520\pi$
$59$	−251.558			−0.555085			−0.277542	+	0.960713i	$0.589520\pi$
$60$		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	$0.275638\pi$
$67$			835.467i			1.52341i	−0.647923	+	0.761706i	$0.724362\pi$
$68$		−	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	$-0.405187\pi$
$71$	351.152			0.586959			0.293479	+	0.955965i	$0.405187\pi$
$72$		0			0
$73$		−	522.749i		−	0.838126i	−0.907957	−	0.419063i	$-0.862359\pi$
$73$		−	522.749i		−	0.838126i	0.907957	−	0.419063i	$-0.137641\pi$
$74$	−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	$-0.708125\pi$
$89$	−1021.39			−1.21648			−0.608242	+	0.793752i	$0.708125\pi$
$90$		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	$-0.934690\pi$
$97$		−	389.284i		−	0.407483i	0.979025	−	0.203741i	$-0.0653102\pi$
$98$		−	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	$-0.640027\pi$
$101$	−864.519			−0.851712			−0.425856	+	0.904791i	$0.640027\pi$
$102$		0			0
$103$		−	1228.21i		−	1.17494i	−0.809246	−	0.587470i	$-0.800124\pi$
$103$		−	1228.21i		−	1.17494i	0.809246	−	0.587470i	$-0.199876\pi$
$104$	−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	$-0.977891\pi$
$127$		−	198.654i		−	0.138801i	0.997589	−	0.0694005i	$-0.0221086\pi$
$128$		−	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	$-0.660913\pi$
$131$	−1452.18			−0.968532			−0.484266	+	0.874921i	$0.660913\pi$
$132$		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	$-0.303823\pi$
$149$	2102.60			1.15605			0.578027	+	0.816018i	$0.303823\pi$
$150$		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	$0.170132\pi$
$157$			2004.19i			1.01880i	−0.860530	+	0.509400i	$0.829868\pi$
$158$		−	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	$0.232038\pi$
$163$			2772.38i			1.33220i	−0.745861	+	0.666102i	$0.767962\pi$
$164$	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	$-0.193544\pi$
$179$	3931.26			1.64154			0.820770	+	0.571258i	$0.193544\pi$
$180$		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	$-0.592196\pi$
$191$	−1507.83			−0.571219			−0.285609	+	0.958346i	$0.592196\pi$
$192$		0			0
$193$			4065.65i			1.51633i	0.652061	+	0.758166i	$0.273904\pi$
$193$			4065.65i			1.51633i	−0.652061	+	0.758166i	$0.726096\pi$
$194$	−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	$-0.755298\pi$
$223$		−	4630.44i		−	1.39048i	0.718778	−	0.695240i	$-0.244702\pi$
$224$	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	$-0.332522\pi$
$239$	3711.16			1.00441			0.502207	+	0.864748i	$0.332522\pi$
$240$		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	$-0.860740\pi$
$251$	−7204.10			−1.81163			−0.905815	+	0.423674i	$0.860740\pi$
$252$		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	$-0.462374\pi$
$269$	1040.60			0.235860			0.117930	+	0.993022i	$0.462374\pi$
$270$		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	$0.236199\pi$
$277$			6231.09i			1.35159i	−0.737090	+	0.675794i	$0.763801\pi$
$278$			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	$-0.918531\pi$
$281$	−9113.95			−1.93485			−0.967425	+	0.253157i	$0.918531\pi$
$282$		0			0
$283$		−	1515.09i		−	0.318243i	−0.987259	−	0.159122i	$-0.949134\pi$
$283$		−	1515.09i		−	0.318243i	0.987259	−	0.159122i	$-0.0508662\pi$
$284$	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	$0.0483596\pi$
$307$			1628.16i			0.302685i	−0.988481	+	0.151342i	$0.951640\pi$
$308$			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	$-0.540931\pi$
$311$	−1406.63			−0.256471			−0.128236	+	0.991744i	$0.540931\pi$
$312$		0			0
$313$		−	5431.54i		−	0.980858i	−0.871481	−	0.490429i	$-0.836840\pi$
$313$		−	5431.54i		−	0.980858i	0.871481	−	0.490429i	$-0.163160\pi$
$314$	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	$-0.931974\pi$
$337$		−	2624.15i		−	0.424174i	0.977251	−	0.212087i	$-0.0680260\pi$
$338$			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	$-0.563653\pi$
$359$	−2702.34			−0.397281			−0.198640	+	0.980072i	$0.563653\pi$
$360$		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	$0.0809674\pi$
$367$			3538.31i			0.503265i	−0.967823	+	0.251632i	$0.919033\pi$
$368$			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	$0.130191\pi$
$373$			5729.89i			0.795395i	−0.917517	+	0.397697i	$0.869809\pi$
$374$	−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	$-0.788643\pi$
$389$	−12084.4			−1.57507			−0.787536	+	0.616269i	$0.788643\pi$
$390$		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	$-0.878770\pi$
$397$		−	5880.63i		−	0.743426i	0.928348	−	0.371713i	$-0.121230\pi$
$398$			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	$-0.434224\pi$
$401$	3295.11			0.410350			0.205175	+	0.978725i	$0.434224\pi$
$402$		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	$-0.532349\pi$
$419$	−1740.28			−0.202908			−0.101454	+	0.994840i	$0.532349\pi$
$420$		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	$-0.731970\pi$
$431$	−11917.4			−1.33189			−0.665943	+	0.746003i	$0.731970\pi$
$432$		0			0
$433$		−	1603.61i		−	0.177978i	−0.996033	−	0.0889889i	$-0.971636\pi$
$433$		−	1603.61i		−	0.177978i	0.996033	−	0.0889889i	$-0.0283636\pi$
$434$	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	$-0.254583\pi$
$449$	13259.9			1.39371			0.696854	+	0.717213i	$0.254583\pi$
$450$		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	$0.00214810\pi$
$457$			131.858i			0.0134968i	−0.999977	+	0.00674840i	$0.997852\pi$
$458$			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	$-0.276564\pi$
$461$	12782.5			1.29141			0.645705	+	0.763587i	$0.276564\pi$
$462$		0			0
$463$		−	4621.41i		−	0.463877i	−0.972730	−	0.231939i	$-0.925493\pi$
$463$		−	4621.41i		−	0.463877i	0.972730	−	0.231939i	$-0.0745069\pi$
$464$	−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	$-0.529516\pi$
$479$	−1941.39			−0.185186			−0.0925932	+	0.995704i	$0.529516\pi$
$480$		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	$0.00263806\pi$
$487$			178.136i			0.0165752i	−0.999966	+	0.00828761i	$0.997362\pi$
$488$		−	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	$-0.582724\pi$
$491$	−5591.59			−0.513941			−0.256971	+	0.966419i	$0.582724\pi$
$492$		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	$-0.206221\pi$
$509$	18313.4			1.59475			0.797376	+	0.603483i	$0.206221\pi$
$510$		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	$-0.657673\pi$
$521$	−11305.4			−0.950670			−0.475335	+	0.879805i	$0.657673\pi$
$522$		0			0
$523$			2825.92i			0.236269i	0.992998	+	0.118135i	$0.0376914\pi$
$523$			2825.92i			0.236269i	−0.992998	+	0.118135i	$0.962309\pi$
$524$	−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	$-0.945570\pi$
$547$		−	4353.95i		−	0.340332i	0.985415	−	0.170166i	$-0.0544303\pi$
$548$			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	$-0.324168\pi$
$569$	14244.0			1.04945			0.524726	+	0.851271i	$0.324168\pi$
$570$		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	$-0.923271\pi$
$577$		−	6617.46i		−	0.477450i	0.971087	−	0.238725i	$-0.0767294\pi$
$578$			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	$-0.745854\pi$
$599$	−20460.9			−1.39567			−0.697836	+	0.716257i	$0.745854\pi$
$600$		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	$0.215688\pi$
$607$			18751.3i			1.25386i	−0.779077	+	0.626928i	$0.784312\pi$
$608$		−	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	$0.293945\pi$
$613$			24213.4i			1.59538i	−0.603067	+	0.797690i	$0.706055\pi$
$614$	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	$-0.536585\pi$
$641$	−3722.30			−0.229364			−0.114682	+	0.993402i	$0.536585\pi$
$642$		0			0
$643$		−	2998.76i		−	0.183919i	−0.995763	−	0.0919593i	$-0.970687\pi$
$643$		−	2998.76i		−	0.183919i	0.995763	−	0.0919593i	$-0.0293130\pi$
$644$	−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	$-0.429270\pi$
$659$	7456.39			0.440759			0.220379	+	0.975414i	$0.429270\pi$
$660$		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	$-0.764391\pi$
$673$		−	23549.8i		−	1.34885i	0.738342	−	0.674427i	$-0.235609\pi$
$674$	−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	$-0.407287\pi$
$701$	10659.6			0.574333			0.287167	+	0.957881i	$0.407287\pi$
$702$		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	$-0.249592\pi$
$719$	27300.1			1.41603			0.708013	+	0.706199i	$0.249592\pi$
$720$		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	$0.0143349\pi$
$727$			1764.93i			0.0900381i	−0.998986	+	0.0450191i	$0.985665\pi$
$728$		−	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	$-0.856391\pi$
$733$		−	17305.5i		−	0.872024i	0.899941	−	0.436012i	$-0.143609\pi$
$734$	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	$0.354377\pi$
$757$			37371.9i			1.79433i	−0.441697	+	0.897164i	$0.645623\pi$
$758$		−	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	$-0.608205\pi$
$761$	−13999.3			−0.666854			−0.333427	+	0.942776i	$0.608205\pi$
$762$		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	$-0.938222\pi$
$787$		−	8516.24i		−	0.385732i	0.981225	−	0.192866i	$-0.0617783\pi$
$788$			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	$-0.448887\pi$
$809$	7358.17			0.319777			0.159888	+	0.987135i	$0.448887\pi$
$810$		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	$-0.308075\pi$
$821$	26680.0			1.13415			0.567076	+	0.823665i	$0.308075\pi$
$822$		0			0
$823$			7614.37i			0.322503i	0.986913	+	0.161252i	$0.0515531\pi$
$823$			7614.37i			0.322503i	−0.986913	+	0.161252i	$0.948447\pi$
$824$	−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	$-0.546329\pi$
$839$	−7049.18			−0.290065			−0.145033	+	0.989427i	$0.546329\pi$
$840$		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	$0.0726209\pi$
$853$			11269.1i			0.452342i	−0.974088	+	0.226171i	$0.927379\pi$
$854$	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	$-0.625812\pi$
$877$		−	47938.4i		−	1.84580i	0.385040	−	0.922900i	$-0.374188\pi$
$878$			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	$-0.266348\pi$
$881$	35033.8			1.33975			0.669875	+	0.742474i	$0.266348\pi$
$882$		0			0
$883$			155.057i			0.00590951i	0.999996	+	0.00295475i	$0.000940528\pi$
$883$			155.057i			0.00590951i	−0.999996	+	0.00295475i	$0.999059\pi$
$884$	−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	$-0.925989\pi$
$907$		−	12588.2i		−	0.460843i	0.973091	−	0.230422i	$-0.0740106\pi$
$908$		−	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	$-0.241477\pi$
$911$	39913.0			1.45157			0.725784	+	0.687923i	$0.241477\pi$
$912$		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	$-0.607313\pi$
$929$	−18732.5			−0.661566			−0.330783	+	0.943707i	$0.607313\pi$
$930$		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	$0.257455\pi$
$937$			41501.3i			1.44695i	−0.690353	+	0.723473i	$0.742545\pi$
$938$			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	$-0.342799\pi$
$941$	27366.5			0.948058			0.474029	+	0.880509i	$0.342799\pi$
$942$		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	$0.121818\pi$
$967$			22458.3i			0.746857i	−0.927659	+	0.373429i	$0.878182\pi$
$968$			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	$-0.254457\pi$
$971$	42186.8			1.39427			0.697137	+	0.716938i	$0.254457\pi$
$972$		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	$0.190805\pi$
$997$			35521.0i			1.12835i	−0.825656	+	0.564174i	$0.809195\pi$
$998$			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.e.244.8		16
3.2	odd	2		405.4.b.f.244.9		16
5.2	odd	4		2025.4.a.bk.1.9		16
5.3	odd	4		2025.4.a.bk.1.8		16
5.4	even	2	inner	405.4.b.e.244.9		16
9.2	odd	6		135.4.j.a.64.9		32
9.4	even	3		45.4.j.a.34.9	yes	32
9.5	odd	6		135.4.j.a.19.8		32
9.7	even	3		45.4.j.a.4.8	✓	32
15.2	even	4		2025.4.a.bl.1.8		16
15.8	even	4		2025.4.a.bl.1.9		16
15.14	odd	2		405.4.b.f.244.8		16
45.4	even	6		45.4.j.a.34.8	yes	32
45.7	odd	12		225.4.e.g.76.8		32
45.13	odd	12		225.4.e.g.151.9		32
45.14	odd	6		135.4.j.a.19.9		32
45.22	odd	12		225.4.e.g.151.8		32
45.29	odd	6		135.4.j.a.64.8		32
45.34	even	6		45.4.j.a.4.9	yes	32
45.43	odd	12		225.4.e.g.76.9		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.j.a.4.8	✓	32	9.7	even	3
45.4.j.a.4.9	yes	32	45.34	even	6
45.4.j.a.34.8	yes	32	45.4	even	6
45.4.j.a.34.9	yes	32	9.4	even	3
135.4.j.a.19.8		32	9.5	odd	6
135.4.j.a.19.9		32	45.14	odd	6
135.4.j.a.64.8		32	45.29	odd	6
135.4.j.a.64.9		32	9.2	odd	6
225.4.e.g.76.8		32	45.7	odd	12
225.4.e.g.76.9		32	45.43	odd	12
225.4.e.g.151.8		32	45.22	odd	12
225.4.e.g.151.9		32	45.13	odd	12
405.4.b.e.244.8		16	1.1	even	1	trivial
405.4.b.e.244.9		16	5.4	even	2	inner
405.4.b.f.244.8		16	15.14	odd	2
405.4.b.f.244.9		16	3.2	odd	2
2025.4.a.bk.1.8		16	5.3	odd	4
2025.4.a.bk.1.9		16	5.2	odd	4
2025.4.a.bl.1.8		16	15.2	even	4
2025.4.a.bl.1.9		16	15.8	even	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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