LMFDB - Embedded newform 405.4.b.e.244.7

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			244.7
Root			$-0.785333i$ of defining polynomial
Character	$\chi$	$=$	405.244
Dual form			405.4.b.e.244.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.785333i q^{2} +7.38325 q^{4} +(-3.61462 - 10.5799i) q^{5} +20.9136i q^{7} -12.0810i q^{8} +O(q^{10})$	\(q-0.785333i q^{2} +7.38325 q^{4} +(-3.61462 - 10.5799i) q^{5} +20.9136i q^{7} -12.0810i q^{8} +(-8.30875 + 2.83868i) q^{10} -59.3838 q^{11} +45.9320i q^{13} +16.4241 q^{14} +49.5784 q^{16} +43.0872i q^{17} -140.178 q^{19} +(-26.6877 - 78.1141i) q^{20} +46.6360i q^{22} +101.756i q^{23} +(-98.8690 + 76.4848i) q^{25} +36.0719 q^{26} +154.410i q^{28} -12.3918 q^{29} +71.9790 q^{31} -135.583i q^{32} +33.8378 q^{34} +(221.264 - 75.5948i) q^{35} -150.734i q^{37} +110.087i q^{38} +(-127.816 + 43.6682i) q^{40} +51.0821 q^{41} +34.4716i q^{43} -438.445 q^{44} +79.9121 q^{46} +117.976i q^{47} -94.3781 q^{49} +(60.0660 + 77.6451i) q^{50} +339.128i q^{52} +137.196i q^{53} +(214.650 + 628.275i) q^{55} +252.657 q^{56} +9.73173i q^{58} -496.314 q^{59} -247.700 q^{61} -56.5275i q^{62} +290.149 q^{64} +(485.956 - 166.027i) q^{65} +826.052i q^{67} +318.124i q^{68} +(-59.3671 - 173.766i) q^{70} -260.049 q^{71} +372.227i q^{73} -118.376 q^{74} -1034.97 q^{76} -1241.93i q^{77} -476.404 q^{79} +(-179.207 - 524.535i) q^{80} -40.1164i q^{82} -313.173i q^{83} +(455.859 - 155.744i) q^{85} +27.0717 q^{86} +717.414i q^{88} +817.628 q^{89} -960.603 q^{91} +751.288i q^{92} +92.6502 q^{94} +(506.692 + 1483.07i) q^{95} +941.296i q^{97} +74.1183i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	0.785333i		−	0.277657i	−0.990316	−	0.138829i	$-0.955666\pi$
$2$		−	0.785333i		−	0.277657i	0.990316	−	0.138829i	$-0.0443337\pi$
$3$		0			0
$3$		0			0
$4$	7.38325			0.922907
$5$	−3.61462	−	10.5799i	−0.323302	−	0.946296i
$5$	−3.61462	−	10.5799i	−0.323302	−	0.946296i
$6$		0			0
$7$			20.9136i			1.12923i	0.825355	+	0.564614i	$0.190975\pi$
$7$			20.9136i			1.12923i	−0.825355	+	0.564614i	$0.809025\pi$
$8$		−	12.0810i		−	0.533909i
$9$		0			0
$10$	−8.30875	+	2.83868i	−0.262746	+	0.0897671i
$11$	−59.3838			−1.62772			−0.813858	−	0.581064i	$-0.802637\pi$
$11$	−59.3838			−1.62772			−0.813858	+	0.581064i	$0.802637\pi$
$12$		0			0
$13$			45.9320i			0.979942i	0.871739	+	0.489971i	$0.162993\pi$
$13$			45.9320i			0.979942i	−0.871739	+	0.489971i	$0.837007\pi$
$14$	16.4241			0.313538
$15$		0			0
$16$	49.5784			0.774663
$17$			43.0872i			0.614717i	0.951594	+	0.307359i	$0.0994451\pi$
$17$			43.0872i			0.614717i	−0.951594	+	0.307359i	$0.900555\pi$
$18$		0			0
$19$	−140.178			−1.69258			−0.846292	−	0.532719i	$-0.821170\pi$
$19$	−140.178			−1.69258			−0.846292	+	0.532719i	$0.821170\pi$
$20$	−26.6877	−	78.1141i	−0.298377	−	0.873343i
$21$		0			0
$22$			46.6360i			0.451947i
$23$			101.756i			0.922501i	0.887270	+	0.461251i	$0.152599\pi$
$23$			101.756i			0.922501i	−0.887270	+	0.461251i	$0.847401\pi$
$24$		0			0
$25$	−98.8690	+	76.4848i	−0.790952	+	0.611878i
$26$	36.0719			0.272088
$27$		0			0
$28$			154.410i			1.04217i
$29$	−12.3918			−0.0793486			−0.0396743	−	0.999213i	$-0.512632\pi$
$29$	−12.3918			−0.0793486			−0.0396743	+	0.999213i	$0.512632\pi$
$30$		0			0
$31$	71.9790			0.417026			0.208513	−	0.978020i	$-0.433138\pi$
$31$	71.9790			0.417026			0.208513	+	0.978020i	$0.433138\pi$
$32$		−	135.583i		−	0.748999i
$33$		0			0
$34$	33.8378			0.170681
$35$	221.264	−	75.5948i	1.06858	−	0.365081i
$36$		0			0
$37$		−	150.734i		−	0.669742i	−0.942264	−	0.334871i	$-0.891307\pi$
$37$		−	150.734i		−	0.669742i	0.942264	−	0.334871i	$-0.108693\pi$
$38$			110.087i			0.469958i
$39$		0			0
$40$	−127.816	+	43.6682i	−0.505236	+	0.172614i
$41$	51.0821			0.194577			0.0972887	−	0.995256i	$-0.468983\pi$
$41$	51.0821			0.194577			0.0972887	+	0.995256i	$0.468983\pi$
$42$		0			0
$43$			34.4716i			0.122253i	0.998130	+	0.0611264i	$0.0194693\pi$
$43$			34.4716i			0.122253i	−0.998130	+	0.0611264i	$0.980531\pi$
$44$	−438.445			−1.50223
$45$		0			0
$46$	79.9121			0.256139
$47$			117.976i			0.366139i	0.983100	+	0.183069i	$0.0586033\pi$
$47$			117.976i			0.366139i	−0.983100	+	0.183069i	$0.941397\pi$
$48$		0			0
$49$	−94.3781			−0.275155
$50$	60.0660	+	77.6451i	0.169892	+	0.219613i
$51$		0			0
$52$			339.128i			0.904395i
$53$			137.196i			0.355573i	0.984069	+	0.177787i	$0.0568937\pi$
$53$			137.196i			0.355573i	−0.984069	+	0.177787i	$0.943106\pi$
$54$		0			0
$55$	214.650	+	628.275i	0.526244	+	1.54030i
$56$	252.657			0.602904
$57$		0			0
$58$			9.73173i			0.0220317i
$59$	−496.314			−1.09516			−0.547581	−	0.836753i	$-0.684451\pi$
$59$	−496.314			−1.09516			−0.547581	+	0.836753i	$0.684451\pi$
$60$		0			0
$61$	−247.700			−0.519914			−0.259957	−	0.965620i	$-0.583708\pi$
$61$	−247.700			−0.519914			−0.259957	+	0.965620i	$0.583708\pi$
$62$		−	56.5275i		−	0.115790i
$63$		0			0
$64$	290.149			0.566698
$65$	485.956	−	166.027i	0.927315	−	0.316817i
$66$		0			0
$67$			826.052i			1.50624i	0.657881	+	0.753121i	$0.271453\pi$
$67$			826.052i			1.50624i	−0.657881	+	0.753121i	$0.728547\pi$
$68$			318.124i			0.567326i
$69$		0			0
$70$	−59.3671	−	173.766i	−0.101367	−	0.296700i
$71$	−260.049			−0.434677			−0.217339	−	0.976096i	$-0.569738\pi$
$71$	−260.049			−0.434677			−0.217339	+	0.976096i	$0.569738\pi$
$72$		0			0
$73$			372.227i			0.596793i	0.954442	+	0.298396i	$0.0964518\pi$
$73$			372.227i			0.596793i	−0.954442	+	0.298396i	$0.903548\pi$
$74$	−118.376			−0.185959
$75$		0			0
$76$	−1034.97			−1.56210
$77$		−	1241.93i		−	1.83806i
$78$		0			0
$79$	−476.404			−0.678476			−0.339238	−	0.940701i	$-0.610169\pi$
$79$	−476.404			−0.678476			−0.339238	+	0.940701i	$0.610169\pi$
$80$	−179.207	−	524.535i	−0.250450	−	0.733060i
$81$		0			0
$82$		−	40.1164i		−	0.0540258i
$83$		−	313.173i		−	0.414160i	−0.978324	−	0.207080i	$-0.933604\pi$
$83$		−	313.173i		−	0.414160i	0.978324	−	0.207080i	$-0.0663960\pi$
$84$		0			0
$85$	455.859	−	155.744i	0.581704	−	0.198739i
$86$	27.0717			0.0339444
$87$		0			0
$88$			717.414i			0.869052i
$89$	817.628			0.973802			0.486901	−	0.873457i	$-0.338127\pi$
$89$	817.628			0.973802			0.486901	+	0.873457i	$0.338127\pi$
$90$		0			0
$91$	−960.603			−1.10658
$92$			751.288i			0.851382i
$93$		0			0
$94$	92.6502			0.101661
$95$	506.692	+	1483.07i	0.547216	+	1.60169i
$96$		0			0
$97$			941.296i			0.985300i	0.870227	+	0.492650i	$0.163972\pi$
$97$			941.296i			0.985300i	−0.870227	+	0.492650i	$0.836028\pi$
$98$			74.1183i			0.0763987i
$99$		0			0
$100$	−729.975	+	564.707i	−0.729975	+	0.564707i
$101$	−1002.87			−0.988017			−0.494008	−	0.869457i	$-0.664469\pi$
$101$	−1002.87			−0.988017			−0.494008	+	0.869457i	$0.664469\pi$
$102$		0			0
$103$		−	298.834i		−	0.285873i	−0.989732	−	0.142937i	$-0.954345\pi$
$103$		−	298.834i		−	0.285873i	0.989732	−	0.142937i	$-0.0456545\pi$
$104$	554.903			0.523199
$105$		0			0
$106$	107.745			0.0987275
$107$			984.533i			0.889518i	0.895650	+	0.444759i	$0.146711\pi$
$107$			984.533i			0.889518i	−0.895650	+	0.444759i	$0.853289\pi$
$108$		0			0
$109$	224.770			0.197514			0.0987572	−	0.995112i	$-0.468513\pi$
$109$	224.770			0.197514			0.0987572	+	0.995112i	$0.468513\pi$
$110$	493.405	−	168.572i	0.427675	−	0.146115i
$111$		0			0
$112$			1036.86i			0.874771i
$113$		−	137.764i		−	0.114688i	−0.998354	−	0.0573440i	$-0.981737\pi$
$113$		−	137.764i		−	0.114688i	0.998354	−	0.0573440i	$-0.0182632\pi$
$114$		0			0
$115$	1076.57	−	367.809i	0.872959	−	0.298246i
$116$	−91.4921			−0.0732313
$117$		0			0
$118$			389.772i			0.304080i
$119$	−901.109			−0.694155
$120$		0			0
$121$	2195.43			1.64946
$122$			194.527i			0.144358i
$123$		0			0
$124$	531.439			0.384876
$125$	1166.58	+	769.561i	0.834734	+	0.550653i
$126$		0			0
$127$		−	2742.19i		−	1.91598i	−0.286800	−	0.957991i	$-0.592591\pi$
$127$		−	2742.19i		−	1.91598i	0.286800	−	0.957991i	$-0.407409\pi$
$128$		−	1312.53i		−	0.906347i
$129$		0			0
$130$	−130.386	−	381.638i	−0.0879665	−	0.257476i
$131$	−960.401			−0.640539			−0.320270	−	0.947326i	$-0.603774\pi$
$131$	−960.401			−0.640539			−0.320270	+	0.947326i	$0.603774\pi$
$132$		0			0
$133$		−	2931.63i		−	1.91131i
$134$	648.726			0.418219
$135$		0			0
$136$	520.536			0.328203
$137$			726.517i			0.453070i	0.974003	+	0.226535i	$0.0727397\pi$
$137$			726.517i			0.453070i	−0.974003	+	0.226535i	$0.927260\pi$
$138$		0			0
$139$	−499.893			−0.305039			−0.152519	−	0.988300i	$-0.548739\pi$
$139$	−499.893			−0.305039			−0.152519	+	0.988300i	$0.548739\pi$
$140$	1633.65	−	558.135i	0.986203	−	0.336936i
$141$		0			0
$142$			204.225i			0.120691i
$143$		−	2727.61i		−	1.59507i
$144$		0			0
$145$	44.7919	+	131.105i	0.0256535	+	0.0750872i
$146$	292.322			0.165704
$147$		0			0
$148$		−	1112.90i		−	0.618109i
$149$	19.6970			0.0108298			0.00541492	−	0.999985i	$-0.498276\pi$
$149$	19.6970			0.0108298			0.00541492	+	0.999985i	$0.498276\pi$
$150$		0			0
$151$	655.417			0.353226			0.176613	−	0.984280i	$-0.443486\pi$
$151$	655.417			0.353226			0.176613	+	0.984280i	$0.443486\pi$
$152$			1693.49i			0.903686i
$153$		0			0
$154$	−975.326			−0.510351
$155$	−260.177	−	761.532i	−0.134825	−	0.394630i
$156$		0			0
$157$		−	1703.54i		−	0.865969i	−0.901401	−	0.432984i	$-0.857460\pi$
$157$		−	1703.54i		−	0.865969i	0.901401	−	0.432984i	$-0.142540\pi$
$158$			374.136i			0.188384i
$159$		0			0
$160$	−1434.46	+	490.083i	−0.708775	+	0.242153i
$161$	−2128.08			−1.04171
$162$		0			0
$163$		−	2606.84i		−	1.25266i	−0.779559	−	0.626329i	$-0.784557\pi$
$163$		−	2606.84i		−	1.25266i	0.779559	−	0.626329i	$-0.215443\pi$
$164$	377.152			0.179577
$165$		0			0
$166$	−245.945			−0.114994
$167$		−	1639.42i		−	0.759651i	−0.925058	−	0.379826i	$-0.875984\pi$
$167$		−	1639.42i		−	0.759651i	0.925058	−	0.379826i	$-0.124016\pi$
$168$		0			0
$169$	87.2514			0.0397139
$170$	−122.311	−	358.001i	−0.0551813	−	0.161514i
$171$		0			0
$172$			254.513i			0.112828i
$173$			1174.44i			0.516132i	0.966127	+	0.258066i	$0.0830852\pi$
$173$			1174.44i			0.516132i	−0.966127	+	0.258066i	$0.916915\pi$
$174$		0			0
$175$	−1599.57	−	2067.71i	−0.690950	−	0.893165i
$176$	−2944.15			−1.26093
$177$		0			0
$178$		−	642.110i		−	0.270383i
$179$	−2512.58			−1.04916			−0.524578	−	0.851362i	$-0.675777\pi$
$179$	−2512.58			−1.04916			−0.524578	+	0.851362i	$0.675777\pi$
$180$		0			0
$181$	3685.91			1.51366			0.756828	−	0.653614i	$-0.226748\pi$
$181$	3685.91			1.51366			0.756828	+	0.653614i	$0.226748\pi$
$182$			754.393i			0.307249i
$183$		0			0
$184$	1229.31			0.492531
$185$	−1594.75	+	544.846i	−0.633774	+	0.216529i
$186$		0			0
$187$		−	2558.68i		−	1.00058i
$188$			871.044i			0.337912i
$189$		0			0
$190$	1164.71	−	397.922i	0.444720	−	0.151938i
$191$	−2716.35			−1.02905			−0.514524	−	0.857476i	$-0.672031\pi$
$191$	−2716.35			−1.02905			−0.514524	+	0.857476i	$0.672031\pi$
$192$		0			0
$193$		−	1972.75i		−	0.735762i	−0.929873	−	0.367881i	$-0.880083\pi$
$193$		−	1972.75i		−	0.735762i	0.929873	−	0.367881i	$-0.119917\pi$
$194$	739.231			0.273576
$195$		0			0
$196$	−696.818			−0.253942
$197$		−	3406.43i		−	1.23197i	−0.787758	−	0.615985i	$-0.788758\pi$
$197$		−	3406.43i		−	1.23197i	0.787758	−	0.615985i	$-0.211242\pi$
$198$		0			0
$199$	981.722			0.349711			0.174855	−	0.984594i	$-0.444054\pi$
$199$	981.722			0.349711			0.174855	+	0.984594i	$0.444054\pi$
$200$	924.011	+	1194.43i	0.326687	+	0.422296i
$201$		0			0
$202$			787.590i			0.274330i
$203$		−	259.158i		−	0.0896026i
$204$		0			0
$205$	−184.642	−	540.444i	−0.0629073	−	0.184128i
$206$	−234.684			−0.0793748
$207$		0			0
$208$			2277.24i			0.759125i
$209$	8324.31			2.75505
$210$		0			0
$211$	−2017.52			−0.658254			−0.329127	−	0.944286i	$-0.606754\pi$
$211$	−2017.52			−0.658254			−0.329127	+	0.944286i	$0.606754\pi$
$212$			1012.96i			0.328161i
$213$		0			0
$214$	773.186			0.246981
$215$	364.706	−	124.602i	0.115687	−	0.0395246i
$216$		0			0
$217$			1505.34i			0.470918i
$218$		−	176.519i		−	0.0548413i
$219$		0			0
$220$	1584.81	+	4638.71i	0.485674	+	1.42155i
$221$	−1979.08			−0.602387
$222$		0			0
$223$			265.712i			0.0797911i	0.999204	+	0.0398956i	$0.0127025\pi$
$223$			265.712i			0.0797911i	−0.999204	+	0.0398956i	$0.987297\pi$
$224$	2835.53			0.845791
$225$		0			0
$226$	−108.191			−0.0318440
$227$		−	5124.95i		−	1.49848i	−0.662300	−	0.749239i	$-0.730419\pi$
$227$		−	5124.95i		−	1.49848i	0.662300	−	0.749239i	$-0.269581\pi$
$228$		0			0
$229$	−90.7643			−0.0261916			−0.0130958	−	0.999914i	$-0.504169\pi$
$229$	−90.7643			−0.0261916			−0.0130958	+	0.999914i	$0.504169\pi$
$230$	−288.852	−	845.463i	−0.0828102	−	0.242383i
$231$		0			0
$232$			149.706i			0.0423649i
$233$			5856.17i			1.64657i	0.567629	+	0.823284i	$0.307861\pi$
$233$			5856.17i			1.64657i	−0.567629	+	0.823284i	$0.692139\pi$
$234$		0			0
$235$	1248.17	−	426.438i	0.346475	−	0.118373i
$236$	−3664.41			−1.01073
$237$		0			0
$238$			707.670i			0.192737i
$239$	6868.82			1.85903			0.929513	−	0.368790i	$-0.120228\pi$
$239$	6868.82			1.85903			0.929513	+	0.368790i	$0.120228\pi$
$240$		0			0
$241$	−25.1374			−0.00671886			−0.00335943	−	0.999994i	$-0.501069\pi$
$241$	−25.1374			−0.00671886			−0.00335943	+	0.999994i	$0.501069\pi$
$242$		−	1724.14i		−	0.457984i
$243$		0			0
$244$	−1828.83			−0.479832
$245$	341.142	+	998.512i	0.0889581	+	0.260378i
$246$		0			0
$247$		−	6438.67i		−	1.65863i
$248$		−	869.577i		−	0.222654i
$249$		0			0
$250$	604.362	−	916.151i	0.152893	−	0.231770i
$251$	−5765.12			−1.44976			−0.724882	−	0.688873i	$-0.758106\pi$
$251$	−5765.12			−1.44976			−0.724882	+	0.688873i	$0.758106\pi$
$252$		0			0
$253$		−	6042.63i		−	1.50157i
$254$	−2153.53			−0.531986
$255$		0			0
$256$	1290.42			0.315044
$257$			5609.90i			1.36162i	0.732460	+	0.680810i	$0.238372\pi$
$257$			5609.90i			1.36162i	−0.732460	+	0.680810i	$0.761628\pi$
$258$		0			0
$259$	3152.38			0.756291
$260$	3587.94	−	1225.82i	0.855825	−	0.292392i
$261$		0			0
$262$			754.235i			0.177850i
$263$			5803.41i			1.36066i	0.732906	+	0.680330i	$0.238163\pi$
$263$			5803.41i			1.36066i	−0.732906	+	0.680330i	$0.761837\pi$
$264$		0			0
$265$	1451.53	−	495.914i	0.336478	−	0.114958i
$266$	−2302.31			−0.530690
$267$		0			0
$268$			6098.95i			1.39012i
$269$	7317.07			1.65847			0.829237	−	0.558897i	$-0.188775\pi$
$269$	7317.07			1.65847			0.829237	+	0.558897i	$0.188775\pi$
$270$		0			0
$271$	−8201.78			−1.83846			−0.919230	−	0.393722i	$-0.871187\pi$
$271$	−8201.78			−1.83846			−0.919230	+	0.393722i	$0.871187\pi$
$272$			2136.20i			0.476199i
$273$		0			0
$274$	570.558			0.125798
$275$	5871.21	−	4541.95i	1.28744	−	0.995964i
$276$		0			0
$277$		−	4598.40i		−	0.997441i	−0.866763	−	0.498720i	$-0.833803\pi$
$277$		−	4598.40i		−	0.997441i	0.866763	−	0.498720i	$-0.166197\pi$
$278$			392.582i			0.0846961i
$279$		0			0
$280$	−913.258	−	2673.08i	−0.194920	−	0.570526i
$281$	−3811.87			−0.809243			−0.404621	−	0.914484i	$-0.632597\pi$
$281$	−3811.87			−0.809243			−0.404621	+	0.914484i	$0.632597\pi$
$282$		0			0
$283$			4099.09i			0.861009i	0.902588	+	0.430504i	$0.141664\pi$
$283$			4099.09i			0.861009i	−0.902588	+	0.430504i	$0.858336\pi$
$284$	−1920.00			−0.401167
$285$		0			0
$286$	−2142.09			−0.442882
$287$			1068.31i			0.219722i
$288$		0			0
$289$	3056.49			0.622123
$290$	102.961	−	35.1765i	0.0208485	−	0.00712289i
$291$		0			0
$292$			2748.25i			0.550784i
$293$			573.453i			0.114340i	0.998364	+	0.0571698i	$0.0182076\pi$
$293$			573.453i			0.114340i	−0.998364	+	0.0571698i	$0.981792\pi$
$294$		0			0
$295$	1793.99	+	5250.96i	0.354068	+	1.03635i
$296$	−1821.01			−0.357581
$297$		0			0
$298$		−	15.4687i		−	0.00300698i
$299$	−4673.84			−0.903997
$300$		0			0
$301$	−720.925			−0.138051
$302$		−	514.721i		−	0.0980757i
$303$		0			0
$304$	−6949.82			−1.31118
$305$	895.342	+	2620.64i	0.168089	+	0.491992i
$306$		0			0
$307$			8610.98i			1.60083i	0.599447	+	0.800414i	$0.295387\pi$
$307$			8610.98i			1.60083i	−0.599447	+	0.800414i	$0.704613\pi$
$308$		−	9169.46i		−	1.69636i
$309$		0			0
$310$	−598.056	+	204.326i	−0.109572	+	0.0374352i
$311$	−3049.69			−0.556052			−0.278026	−	0.960574i	$-0.589680\pi$
$311$	−3049.69			−0.556052			−0.278026	+	0.960574i	$0.589680\pi$
$312$		0			0
$313$		−	4105.22i		−	0.741344i	−0.928764	−	0.370672i	$-0.879127\pi$
$313$		−	4105.22i		−	0.741344i	0.928764	−	0.370672i	$-0.120873\pi$
$314$	−1337.84			−0.240442
$315$		0			0
$316$	−3517.41			−0.626170
$317$		−	5420.67i		−	0.960427i	−0.877152	−	0.480214i	$-0.840559\pi$
$317$		−	5420.67i		−	0.960427i	0.877152	−	0.480214i	$-0.159441\pi$
$318$		0			0
$319$	735.874			0.129157
$320$	−1048.78	−	3069.75i	−0.183215	−	0.536264i
$321$		0			0
$322$			1671.25i			0.289239i
$323$		−	6039.90i		−	1.04046i
$324$		0			0
$325$	−3513.10	−	4541.25i	−0.599605	−	0.775087i
$326$	−2047.23			−0.347809
$327$		0			0
$328$		−	617.121i		−	0.103887i
$329$	−2467.29			−0.413454
$330$		0			0
$331$	5909.23			0.981272			0.490636	−	0.871365i	$-0.336765\pi$
$331$	5909.23			0.981272			0.490636	+	0.871365i	$0.336765\pi$
$332$		−	2312.24i		−	0.382231i
$333$		0			0
$334$	−1287.49			−0.210923
$335$	8739.55	−	2985.87i	1.42535	−	0.486971i
$336$		0			0
$337$			1324.01i			0.214016i	0.994258	+	0.107008i	$0.0341271\pi$
$337$			1324.01i			0.214016i	−0.994258	+	0.107008i	$0.965873\pi$
$338$		−	68.5214i		−	0.0110268i
$339$		0			0
$340$	3365.72	−	1149.90i	0.536859	−	0.183418i
$341$	−4274.39			−0.678800
$342$		0			0
$343$			5199.58i			0.818515i
$344$	416.451			0.0652718
$345$		0			0
$346$	922.324			0.143308
$347$			11227.7i			1.73699i	0.495697	+	0.868495i	$0.334913\pi$
$347$			11227.7i			1.73699i	−0.495697	+	0.868495i	$0.665087\pi$
$348$		0			0
$349$	−1042.88			−0.159954			−0.0799772	−	0.996797i	$-0.525485\pi$
$349$	−1042.88			−0.159954			−0.0799772	+	0.996797i	$0.525485\pi$
$350$	−1623.84	+	1256.20i	−0.247994	+	0.191847i
$351$		0			0
$352$			8051.45i			1.21916i
$353$			11522.2i			1.73730i	0.495426	+	0.868650i	$0.335012\pi$
$353$			11522.2i			1.73730i	−0.495426	+	0.868650i	$0.664988\pi$
$354$		0			0
$355$	939.978	+	2751.29i	0.140532	+	0.411333i
$356$	6036.75			0.898728
$357$		0			0
$358$			1973.21i			0.291306i
$359$	4751.13			0.698482			0.349241	−	0.937033i	$-0.386440\pi$
$359$	4751.13			0.698482			0.349241	+	0.937033i	$0.386440\pi$
$360$		0			0
$361$	12791.0			1.86484
$362$		−	2894.67i		−	0.420277i
$363$		0			0
$364$	−7092.37			−1.02127
$365$	3938.13	−	1345.46i	0.564743	−	0.192944i
$366$		0			0
$367$			11120.8i			1.58174i	0.611981	+	0.790872i	$0.290373\pi$
$367$			11120.8i			1.58174i	−0.611981	+	0.790872i	$0.709627\pi$
$368$			5044.89i			0.714627i
$369$		0			0
$370$	427.885	+	1252.41i	0.0601208	+	0.175972i
$371$	−2869.27			−0.401523
$372$		0			0
$373$			9512.54i			1.32048i	0.751053	+	0.660242i	$0.229546\pi$
$373$			9512.54i			1.32048i	−0.751053	+	0.660242i	$0.770454\pi$
$374$	−2009.42			−0.277819
$375$		0			0
$376$	1425.26			0.195485
$377$		−	569.182i		−	0.0777570i
$378$		0			0
$379$	3029.06			0.410534			0.205267	−	0.978706i	$-0.434194\pi$
$379$	3029.06			0.410534			0.205267	+	0.978706i	$0.434194\pi$
$380$	3741.03	+	10949.9i	0.505029	+	1.47821i
$381$		0			0
$382$			2133.24i			0.285722i
$383$		−	10405.3i		−	1.38822i	−0.719871	−	0.694108i	$-0.755799\pi$
$383$		−	10405.3i		−	1.38822i	0.719871	−	0.694108i	$-0.244201\pi$
$384$		0			0
$385$	−13139.5	+	4489.10i	−1.73935	+	0.594249i
$386$	−1549.27			−0.204289
$387$		0			0
$388$			6949.83i			0.909340i
$389$	9049.43			1.17950			0.589748	−	0.807587i	$-0.299227\pi$
$389$	9049.43			1.17950			0.589748	+	0.807587i	$0.299227\pi$
$390$		0			0
$391$	−4384.37			−0.567077
$392$			1140.18i			0.146908i
$393$		0			0
$394$	−2675.18			−0.342065
$395$	1722.02	+	5040.31i	0.219353	+	0.642039i
$396$		0			0
$397$		−	276.228i		−	0.0349206i	−0.999848	−	0.0174603i	$-0.994442\pi$
$397$		−	276.228i		−	0.0349206i	0.999848	−	0.0174603i	$-0.00555807\pi$
$398$		−	770.979i		−	0.0970997i
$399$		0			0
$400$	−4901.77	+	3792.00i	−0.612721	+	0.474000i
$401$	−14902.5			−1.85585			−0.927925	−	0.372767i	$-0.878409\pi$
$401$	−14902.5			−1.85585			−0.927925	+	0.372767i	$0.878409\pi$
$402$		0			0
$403$			3306.14i			0.408662i
$404$	−7404.47			−0.911847
$405$		0			0
$406$	−203.525			−0.0248788
$407$			8951.13i			1.09015i
$408$		0			0
$409$	1116.83			0.135021			0.0675104	−	0.997719i	$-0.478494\pi$
$409$	1116.83			0.135021			0.0675104	+	0.997719i	$0.478494\pi$
$410$	−424.428	+	145.006i	−0.0511244	+	0.0174666i
$411$		0			0
$412$		−	2206.36i		−	0.263834i
$413$		−	10379.7i		−	1.23669i
$414$		0			0
$415$	−3313.35	+	1132.00i	−0.391918	+	0.133899i
$416$	6227.62			0.733976
$417$		0			0
$418$		−	6537.36i		−	0.764958i
$419$	5086.23			0.593027			0.296514	−	0.955029i	$-0.404176\pi$
$419$	5086.23			0.593027			0.296514	+	0.955029i	$0.404176\pi$
$420$		0			0
$421$	−9640.06			−1.11598			−0.557990	−	0.829848i	$-0.688427\pi$
$421$	−9640.06			−1.11598			−0.557990	+	0.829848i	$0.688427\pi$
$422$			1584.42i			0.182769i
$423$		0			0
$424$	1657.47			0.189844
$425$	−3295.52	−	4259.99i	−0.376132	−	0.486212i
$426$		0			0
$427$		−	5180.30i		−	0.587101i
$428$			7269.06i			0.820942i
$429$		0			0
$430$	−97.8540	−	286.416i	−0.0109743	−	0.0321214i
$431$	−857.490			−0.0958326			−0.0479163	−	0.998851i	$-0.515258\pi$
$431$	−857.490			−0.0958326			−0.0479163	+	0.998851i	$0.515258\pi$
$432$		0			0
$433$		−	2306.40i		−	0.255978i	−0.991776	−	0.127989i	$-0.959148\pi$
$433$		−	2306.40i		−	0.255978i	0.991776	−	0.127989i	$-0.0408523\pi$
$434$	1182.19			0.130754
$435$		0			0
$436$	1659.53			0.182287
$437$		−	14263.9i		−	1.56141i
$438$		0			0
$439$	711.237			0.0773246			0.0386623	−	0.999252i	$-0.487690\pi$
$439$	711.237			0.0773246			0.0386623	+	0.999252i	$0.487690\pi$
$440$	7590.17	−	2593.18i	0.822380	−	0.280966i
$441$		0			0
$442$			1554.24i			0.167257i
$443$		−	15639.9i		−	1.67737i	−0.544618	−	0.838684i	$-0.683325\pi$
$443$		−	15639.9i		−	1.67737i	0.544618	−	0.838684i	$-0.316675\pi$
$444$		0			0
$445$	−2955.42	−	8650.43i	−0.314832	−	0.921505i
$446$	208.673			0.0221546
$447$		0			0
$448$			6068.06i			0.639931i
$449$	−8156.47			−0.857299			−0.428650	−	0.903471i	$-0.641011\pi$
$449$	−8156.47			−0.857299			−0.428650	+	0.903471i	$0.641011\pi$
$450$		0			0
$451$	−3033.44			−0.316717
$452$		−	1017.15i		−	0.105846i
$453$		0			0
$454$	−4024.79			−0.416063
$455$	3472.22	+	10163.1i	0.357759	+	1.04715i
$456$		0			0
$457$		−	14785.2i		−	1.51339i	−0.653766	−	0.756697i	$-0.726812\pi$
$457$		−	14785.2i		−	1.51339i	0.653766	−	0.756697i	$-0.273188\pi$
$458$			71.2802i			0.00727228i
$459$		0			0
$460$	7948.56	−	2715.62i	0.805660	−	0.275253i
$461$	−8293.15			−0.837853			−0.418927	−	0.908020i	$-0.637594\pi$
$461$	−8293.15			−0.837853			−0.418927	+	0.908020i	$0.637594\pi$
$462$		0			0
$463$			14357.0i			1.44109i	0.693407	+	0.720546i	$0.256109\pi$
$463$			14357.0i			1.44109i	−0.693407	+	0.720546i	$0.743891\pi$
$464$	−614.368			−0.0614684
$465$		0			0
$466$	4599.04			0.457181
$467$			4815.25i			0.477138i	0.971126	+	0.238569i	$0.0766782\pi$
$467$			4815.25i			0.477138i	−0.971126	+	0.238569i	$0.923322\pi$
$468$		0			0
$469$	−17275.7			−1.70089
$470$	−334.896	−	980.230i	−0.0328672	−	0.0962014i
$471$		0			0
$472$			5995.96i			0.584717i
$473$		−	2047.05i		−	0.198993i
$474$		0			0
$475$	13859.3	−	10721.5i	1.33875	−	1.03566i
$476$	−6653.11			−0.640641
$477$		0			0
$478$		−	5394.31i		−	0.516172i
$479$	−11827.7			−1.12823			−0.564116	−	0.825695i	$-0.690783\pi$
$479$	−11827.7			−1.12823			−0.564116	+	0.825695i	$0.690783\pi$
$480$		0			0
$481$	6923.50			0.656308
$482$			19.7413i			0.00186554i
$483$		0			0
$484$	16209.4			1.52230
$485$	9958.83	−	3402.43i	0.932386	−	0.318549i
$486$		0			0
$487$		−	13087.8i		−	1.21779i	−0.793251	−	0.608894i	$-0.791613\pi$
$487$		−	13087.8i		−	1.21779i	0.793251	−	0.608894i	$-0.208387\pi$
$488$			2992.46i			0.277586i
$489$		0			0
$490$	784.164	−	267.910i	0.0722958	−	0.0246998i
$491$	−6593.39			−0.606019			−0.303010	−	0.952988i	$-0.597991\pi$
$491$	−6593.39			−0.606019			−0.303010	+	0.952988i	$0.597991\pi$
$492$		0			0
$493$		−	533.930i		−	0.0487769i
$494$	−5056.50			−0.460532
$495$		0			0
$496$	3568.61			0.323055
$497$		−	5438.55i		−	0.490850i
$498$		0			0
$499$	16184.8			1.45196			0.725982	−	0.687713i	$-0.241385\pi$
$499$	16184.8			1.45196			0.725982	+	0.687713i	$0.241385\pi$
$500$	8613.13	+	5681.86i	0.770382	+	0.508201i
$501$		0			0
$502$			4527.54i			0.402537i
$503$			10342.9i			0.916833i	0.888738	+	0.458416i	$0.151583\pi$
$503$			10342.9i			0.916833i	−0.888738	+	0.458416i	$0.848417\pi$
$504$		0			0
$505$	3625.01	+	10610.3i	0.319428	+	0.934956i
$506$	−4745.48			−0.416921
$507$		0			0
$508$		−	20246.2i		−	1.76827i
$509$	−5648.61			−0.491887			−0.245943	−	0.969284i	$-0.579098\pi$
$509$	−5648.61			−0.491887			−0.245943	+	0.969284i	$0.579098\pi$
$510$		0			0
$511$	−7784.60			−0.673915
$512$		−	11513.7i		−	0.993821i
$513$		0			0
$514$	4405.64			0.378063
$515$	−3161.63	+	1080.17i	−0.270521	+	0.0924234i
$516$		0			0
$517$		−	7005.84i		−	0.595970i
$518$		−	2475.67i		−	0.209990i
$519$		0			0
$520$	−2005.77	−	5870.83i	−0.169151	−	0.495102i
$521$	15511.7			1.30438			0.652188	−	0.758058i	$-0.273851\pi$
$521$	15511.7			1.30438			0.652188	+	0.758058i	$0.273851\pi$
$522$		0			0
$523$			613.845i			0.0513223i	0.999671	+	0.0256612i	$0.00816910\pi$
$523$			613.845i			0.0513223i	−0.999671	+	0.0256612i	$0.991831\pi$
$524$	−7090.89			−0.591158
$525$		0			0
$526$	4557.61			0.377797
$527$			3101.38i			0.256353i
$528$		0			0
$529$	1812.78			0.148992
$530$	−389.457	−	1139.93i	−0.0319188	−	0.0934254i
$531$		0			0
$532$		−	21645.0i		−	1.76396i
$533$			2346.30i			0.190675i
$534$		0			0
$535$	10416.3	−	3558.72i	0.841747	−	0.287583i
$536$	9979.51			0.804196
$537$		0			0
$538$		−	5746.34i		−	0.460487i
$539$	5604.53			0.447874
$540$		0			0
$541$	−12701.1			−1.00936			−0.504678	−	0.863308i	$-0.668389\pi$
$541$	−12701.1			−1.00936			−0.504678	+	0.863308i	$0.668389\pi$
$542$			6441.12i			0.510461i
$543$		0			0
$544$	5841.91			0.460423
$545$	−812.459	−	2378.05i	−0.0638568	−	0.186907i
$546$		0			0
$547$			9413.79i			0.735840i	0.929857	+	0.367920i	$0.119930\pi$
$547$			9413.79i			0.735840i	−0.929857	+	0.367920i	$0.880070\pi$
$548$			5364.06i			0.418141i
$549$		0			0
$550$	−3566.95	−	4610.85i	−0.276537	−	0.357468i
$551$	1737.07			0.134304
$552$		0			0
$553$		−	9963.31i		−	0.766154i
$554$	−3611.27			−0.276947
$555$		0			0
$556$	−3690.83			−0.281522
$557$			14085.5i			1.07149i	0.844379	+	0.535746i	$0.179969\pi$
$557$			14085.5i			1.07149i	−0.844379	+	0.535746i	$0.820031\pi$
$558$		0			0
$559$	−1583.35			−0.119801
$560$	10969.9	−	3747.87i	0.827792	−	0.282815i
$561$		0			0
$562$			2993.59i			0.224692i
$563$			8442.04i			0.631953i	0.948767	+	0.315977i	$0.102332\pi$
$563$			8442.04i			0.631953i	−0.948767	+	0.315977i	$0.897668\pi$
$564$		0			0
$565$	−1457.53	+	497.965i	−0.108529	+	0.0370789i
$566$	3219.15			0.239065
$567$		0			0
$568$			3141.64i			0.232078i
$569$	3528.48			0.259968			0.129984	−	0.991516i	$-0.458507\pi$
$569$	3528.48			0.259968			0.129984	+	0.991516i	$0.458507\pi$
$570$		0			0
$571$	−14738.1			−1.08016			−0.540078	−	0.841615i	$-0.681605\pi$
$571$	−14738.1			−1.08016			−0.540078	+	0.841615i	$0.681605\pi$
$572$		−	20138.7i		−	1.47210i
$573$		0			0
$574$	838.978			0.0610074
$575$	−7782.76	−	10060.5i	−0.564458	−	0.729654i
$576$		0			0
$577$		−	17485.6i		−	1.26159i	−0.775951	−	0.630793i	$-0.782730\pi$
$577$		−	17485.6i		−	1.26159i	0.775951	−	0.630793i	$-0.217270\pi$
$578$		−	2400.36i		−	0.172737i
$579$		0			0
$580$	330.710	+	967.979i	0.0236758	+	0.0692985i
$581$	6549.58			0.467681
$582$		0			0
$583$		−	8147.24i		−	0.578772i
$584$	4496.87			0.318633
$585$		0			0
$586$	450.352			0.0317472
$587$			12494.4i			0.878536i	0.898356	+	0.439268i	$0.144762\pi$
$587$			12494.4i			0.878536i	−0.898356	+	0.439268i	$0.855238\pi$
$588$		0			0
$589$	−10089.9			−0.705853
$590$	4123.75	−	1408.88i	0.287749	−	0.0983095i
$591$		0			0
$592$		−	7473.14i		−	0.518824i
$593$		−	4044.65i		−	0.280091i	−0.990145	−	0.140046i	$-0.955275\pi$
$593$		−	4044.65i		−	0.280091i	0.990145	−	0.140046i	$-0.0447249\pi$
$594$		0			0
$595$	3257.17	+	9533.65i	0.224422	+	0.656876i
$596$	145.428			0.00999492
$597$		0			0
$598$			3670.52i			0.251001i
$599$	21563.1			1.47086			0.735431	−	0.677600i	$-0.236980\pi$
$599$	21563.1			1.47086			0.735431	+	0.677600i	$0.236980\pi$
$600$		0			0
$601$	20115.9			1.36530			0.682648	−	0.730747i	$-0.260828\pi$
$601$	20115.9			1.36530			0.682648	+	0.730747i	$0.260828\pi$
$602$			566.166i			0.0383309i
$603$		0			0
$604$	4839.11			0.325994
$605$	−7935.65	−	23227.5i	−0.533273	−	1.56088i
$606$		0			0
$607$			17200.6i			1.15017i	0.818094	+	0.575084i	$0.195031\pi$
$607$			17200.6i			1.15017i	−0.818094	+	0.575084i	$0.804969\pi$
$608$			19005.8i			1.26774i
$609$		0			0
$610$	2058.08	−	703.142i	0.136605	−	0.0466711i
$611$	−5418.86			−0.358795
$612$		0			0
$613$			894.343i			0.0589269i	0.999566	+	0.0294634i	$0.00937986\pi$
$613$			894.343i			0.0589269i	−0.999566	+	0.0294634i	$0.990620\pi$
$614$	6762.48			0.444481
$615$		0			0
$616$	−15003.7			−0.981357
$617$		−	6856.71i		−	0.447392i	−0.974659	−	0.223696i	$-0.928188\pi$
$617$		−	6856.71i		−	0.447392i	0.974659	−	0.223696i	$-0.0718122\pi$
$618$		0			0
$619$	−21165.2			−1.37431			−0.687157	−	0.726509i	$-0.741142\pi$
$619$	−21165.2			−1.37431			−0.687157	+	0.726509i	$0.741142\pi$
$620$	−1920.95	−	5622.58i	−0.124431	−	0.364207i
$621$		0			0
$622$			2395.02i			0.154392i
$623$			17099.5i			1.09964i
$624$		0			0
$625$	3925.15	−	15123.9i	0.251210	−	0.967933i
$626$	−3223.96			−0.205839
$627$		0			0
$628$		−	12577.6i		−	0.799208i
$629$	6494.70			0.411702
$630$		0			0
$631$	14889.2			0.939352			0.469676	−	0.882839i	$-0.344371\pi$
$631$	14889.2			0.939352			0.469676	+	0.882839i	$0.344371\pi$
$632$			5755.42i			0.362244i
$633$		0			0
$634$	−4257.03			−0.266669
$635$	−29012.1	+	9911.97i	−1.81309	+	0.619440i
$636$		0			0
$637$		−	4334.98i		−	0.269636i
$638$		−	577.906i		−	0.0358613i
$639$		0			0
$640$	−13886.5	+	4744.31i	−0.857673	+	0.293024i
$641$	22428.3			1.38200			0.691001	−	0.722854i	$-0.257170\pi$
$641$	22428.3			1.38200			0.691001	+	0.722854i	$0.257170\pi$
$642$		0			0
$643$			4262.51i			0.261426i	0.991420	+	0.130713i	$0.0417267\pi$
$643$			4262.51i			0.261426i	−0.991420	+	0.130713i	$0.958273\pi$
$644$	−15712.1			−0.961404
$645$		0			0
$646$	−4743.33			−0.288891
$647$			19619.7i			1.19216i	0.802924	+	0.596082i	$0.203277\pi$
$647$			19619.7i			1.19216i	−0.802924	+	0.596082i	$0.796723\pi$
$648$		0			0
$649$	29473.0			1.78261
$650$	−3566.39	+	2758.95i	−0.215208	+	0.166485i
$651$		0			0
$652$		−	19246.9i		−	1.15609i
$653$			4708.94i			0.282198i	0.989996	+	0.141099i	$0.0450635\pi$
$653$			4708.94i			0.282198i	−0.989996	+	0.141099i	$0.954936\pi$
$654$		0			0
$655$	3471.49	+	10161.0i	0.207088	+	0.606140i
$656$	2532.57			0.150732
$657$		0			0
$658$			1937.65i			0.114798i
$659$	24138.8			1.42688			0.713441	−	0.700715i	$-0.247135\pi$
$659$	24138.8			1.42688			0.713441	+	0.700715i	$0.247135\pi$
$660$		0			0
$661$	31706.5			1.86572			0.932859	−	0.360241i	$-0.117305\pi$
$661$	31706.5			1.86572			0.932859	+	0.360241i	$0.117305\pi$
$662$		−	4640.72i		−	0.272457i
$663$		0			0
$664$	−3783.44			−0.221123
$665$	−31016.4	+	10596.7i	−1.80867	+	0.617931i
$666$		0			0
$667$		−	1260.94i		−	0.0731991i
$668$		−	12104.2i		−	0.701087i
$669$		0			0
$670$	−2344.90	−	6863.46i	−0.135211	−	0.395759i
$671$	14709.4			0.846272
$672$		0			0
$673$		−	277.641i		−	0.0159023i	−0.999968	−	0.00795116i	$-0.997469\pi$
$673$		−	277.641i		−	0.0159023i	0.999968	−	0.00795116i	$-0.00253096\pi$
$674$	1039.79			0.0594231
$675$		0			0
$676$	644.199			0.0366522
$677$			8744.57i			0.496427i	0.968705	+	0.248213i	$0.0798434\pi$
$677$			8744.57i			0.496427i	−0.968705	+	0.248213i	$0.920157\pi$
$678$		0			0
$679$	−19685.9			−1.11263
$680$	−1881.54	−	5507.22i	−0.106109	−	0.310577i
$681$		0			0
$682$			3356.82i			0.188474i
$683$		−	34097.3i		−	1.91025i	−0.296210	−	0.955123i	$-0.595723\pi$
$683$		−	34097.3i		−	1.91025i	0.296210	−	0.955123i	$-0.404277\pi$
$684$		0			0
$685$	7686.49	−	2626.09i	0.428738	−	0.146478i
$686$	4083.40			0.227267
$687$		0			0
$688$			1709.05i			0.0947047i
$689$	−6301.71			−0.348441
$690$		0			0
$691$	−4982.20			−0.274286			−0.137143	−	0.990551i	$-0.543792\pi$
$691$	−4982.20			−0.274286			−0.137143	+	0.990551i	$0.543792\pi$
$692$			8671.17i			0.476341i
$693$		0			0
$694$	8817.50			0.482288
$695$	1806.92	+	5288.82i	0.0986195	+	0.288657i
$696$		0			0
$697$			2200.98i			0.119610i
$698$			819.008i			0.0444125i
$699$		0			0
$700$	−11810.0	−	15266.4i	−0.637682	−	0.824307i
$701$	−16450.0			−0.886315			−0.443158	−	0.896444i	$-0.646142\pi$
$701$	−16450.0			−0.886315			−0.443158	+	0.896444i	$0.646142\pi$
$702$		0			0
$703$			21129.6i			1.13360i
$704$	−17230.2			−0.922423
$705$		0			0
$706$	9048.79			0.482374
$707$		−	20973.7i		−	1.11570i
$708$		0			0
$709$	9139.71			0.484131			0.242066	−	0.970260i	$-0.422175\pi$
$709$	9139.71			0.484131			0.242066	+	0.970260i	$0.422175\pi$
$710$	2160.68	−	738.196i	0.114210	−	0.0390197i
$711$		0			0
$712$		−	9877.74i		−	0.519921i
$713$			7324.28i			0.384707i
$714$		0			0
$715$	−28857.9	+	9859.30i	−1.50941	+	0.515688i
$716$	−18551.0			−0.968273
$717$		0			0
$718$		−	3731.22i		−	0.193938i
$719$	28268.2			1.46624			0.733121	−	0.680098i	$-0.238063\pi$
$719$	28268.2			1.46624			0.733121	+	0.680098i	$0.238063\pi$
$720$		0			0
$721$	6249.68			0.322816
$722$		−	10045.2i		−	0.517787i
$723$		0			0
$724$	27214.0			1.39696
$725$	1225.17	−	947.788i	0.0627609	−	0.0485517i
$726$		0			0
$727$			32221.9i			1.64380i	0.569632	+	0.821900i	$0.307086\pi$
$727$			32221.9i			1.64380i	−0.569632	+	0.821900i	$0.692914\pi$
$728$			11605.0i			0.590811i
$729$		0			0
$730$	−1056.63	−	3092.74i	−0.0535723	−	0.156805i
$731$	−1485.29			−0.0751509
$732$		0			0
$733$		−	10764.2i		−	0.542408i	−0.962522	−	0.271204i	$-0.912578\pi$
$733$		−	10764.2i		−	0.542408i	0.962522	−	0.271204i	$-0.0874218\pi$
$734$	8733.52			0.439183
$735$		0			0
$736$	13796.4			0.690953
$737$		−	49054.0i		−	2.45174i
$738$		0			0
$739$	−21955.9			−1.09291			−0.546456	−	0.837488i	$-0.684023\pi$
$739$	−21955.9			−1.09291			−0.546456	+	0.837488i	$0.684023\pi$
$740$	−11774.4	+	4022.73i	−0.584914	+	0.199836i
$741$		0			0
$742$			2253.33i			0.111486i
$743$		−	5929.17i		−	0.292759i	−0.989228	−	0.146380i	$-0.953238\pi$
$743$		−	5929.17i		−	0.292759i	0.989228	−	0.146380i	$-0.0467621\pi$
$744$		0			0
$745$	−71.1974	−	208.393i	−0.00350130	−	0.0102482i
$746$	7470.51			0.366642
$747$		0			0
$748$		−	18891.4i		−	0.923446i
$749$	−20590.1			−1.00447
$750$		0			0
$751$	−14894.6			−0.723716			−0.361858	−	0.932233i	$-0.617857\pi$
$751$	−14894.6			−0.723716			−0.361858	+	0.932233i	$0.617857\pi$
$752$			5849.05i			0.283634i
$753$		0			0
$754$	−446.998			−0.0215898
$755$	−2369.09	−	6934.26i	−0.114199	−	0.334256i
$756$		0			0
$757$			11935.5i			0.573058i	0.958072	+	0.286529i	$0.0925014\pi$
$757$			11935.5i			0.573058i	−0.958072	+	0.286529i	$0.907499\pi$
$758$		−	2378.82i		−	0.113988i
$759$		0			0
$760$	17917.0	−	6121.33i	0.855154	−	0.292163i
$761$	28482.5			1.35675			0.678377	−	0.734714i	$-0.262683\pi$
$761$	28482.5			1.35675			0.678377	+	0.734714i	$0.262683\pi$
$762$		0			0
$763$			4700.75i			0.223039i
$764$	−20055.5			−0.949715
$765$		0			0
$766$	−8171.63			−0.385448
$767$		−	22796.7i		−	1.07320i
$768$		0			0
$769$	−13392.3			−0.628009			−0.314005	−	0.949421i	$-0.601671\pi$
$769$	−13392.3			−0.628009			−0.314005	+	0.949421i	$0.601671\pi$
$770$	3525.44	+	10318.9i	0.164997	+	0.482943i
$771$		0			0
$772$		−	14565.3i		−	0.679039i
$773$			37031.6i			1.72307i	0.507696	+	0.861536i	$0.330497\pi$
$773$			37031.6i			1.72307i	−0.507696	+	0.861536i	$0.669503\pi$
$774$		0			0
$775$	−7116.49	+	5505.30i	−0.329848	+	0.255169i
$776$	11371.8			0.526060
$777$		0			0
$778$		−	7106.81i		−	0.327496i
$779$	−7160.60			−0.329339
$780$		0			0
$781$	15442.7			0.707531
$782$			3443.19i			0.157453i
$783$		0			0
$784$	−4679.12			−0.213152
$785$	−18023.3	+	6157.65i	−0.819463	+	0.279969i
$786$		0			0
$787$			29263.5i			1.32545i	0.748862	+	0.662726i	$0.230601\pi$
$787$			29263.5i			1.32545i	−0.748862	+	0.662726i	$0.769399\pi$
$788$		−	25150.5i		−	1.13699i
$789$		0			0
$790$	3958.32	−	1352.36i	0.178267	−	0.0609048i
$791$	2881.14			0.129509
$792$		0			0
$793$		−	11377.4i		−	0.509485i
$794$	−216.931			−0.00969596
$795$		0			0
$796$	7248.30			0.322750
$797$			35700.2i			1.58666i	0.608793	+	0.793329i	$0.291654\pi$
$797$			35700.2i			1.58666i	−0.608793	+	0.793329i	$0.708346\pi$
$798$		0			0
$799$	−5083.24			−0.225072
$800$	10370.1	+	13405.0i	0.458297	+	0.592422i
$801$		0			0
$802$			11703.4i			0.515290i
$803$		−	22104.2i		−	0.971409i
$804$		0			0
$805$	7692.20	+	22514.9i	0.336788	+	0.985769i
$806$	2596.42			0.113468
$807$		0			0
$808$			12115.7i			0.527511i
$809$	−24027.7			−1.04421			−0.522106	−	0.852881i	$-0.674853\pi$
$809$	−24027.7			−1.04421			−0.522106	+	0.852881i	$0.674853\pi$
$810$		0			0
$811$	−26426.3			−1.14421			−0.572104	−	0.820181i	$-0.693873\pi$
$811$	−26426.3			−1.14421			−0.572104	+	0.820181i	$0.693873\pi$
$812$		−	1913.43i		−	0.0826948i
$813$		0			0
$814$	7029.62			0.302688
$815$	−27580.1	+	9422.73i	−1.18538	+	0.404986i
$816$		0			0
$817$		−	4832.17i		−	0.206923i
$818$		−	877.080i		−	0.0374895i
$819$		0			0
$820$	−1363.26	−	3990.23i	−0.0580575	−	0.169933i
$821$	−24496.7			−1.04134			−0.520669	−	0.853758i	$-0.674318\pi$
$821$	−24496.7			−1.04134			−0.520669	+	0.853758i	$0.674318\pi$
$822$		0			0
$823$			14743.0i			0.624434i	0.950011	+	0.312217i	$0.101072\pi$
$823$			14743.0i			0.624434i	−0.950011	+	0.312217i	$0.898928\pi$
$824$	−3610.20			−0.152630
$825$		0			0
$826$	−8151.52			−0.343375
$827$			27985.3i			1.17672i	0.808601	+	0.588358i	$0.200225\pi$
$827$			27985.3i			1.17672i	−0.808601	+	0.588358i	$0.799775\pi$
$828$		0			0
$829$	−39350.4			−1.64861			−0.824304	−	0.566147i	$-0.808433\pi$
$829$	−39350.4			−1.64861			−0.824304	+	0.566147i	$0.808433\pi$
$830$	889.000	+	2602.08i	0.0371779	+	0.108819i
$831$		0			0
$832$			13327.1i			0.555331i
$833$		−	4066.49i		−	0.169142i
$834$		0			0
$835$	−17344.9	+	5925.87i	−0.718855	+	0.245597i
$836$	61460.5			2.54265
$837$		0			0
$838$		−	3994.38i		−	0.164658i
$839$	22700.9			0.934113			0.467057	−	0.884227i	$-0.345314\pi$
$839$	22700.9			0.934113			0.467057	+	0.884227i	$0.345314\pi$
$840$		0			0
$841$	−24235.4			−0.993704
$842$			7570.65i			0.309860i
$843$		0			0
$844$	−14895.8			−0.607507
$845$	−315.381	−	923.112i	−0.0128396	−	0.0375811i
$846$		0			0
$847$			45914.3i			1.86261i
$848$			6801.99i			0.275449i
$849$		0			0
$850$	−3345.51	+	2588.08i	−0.135000	+	0.104436i
$851$	15338.0			0.617838
$852$		0			0
$853$			34055.9i			1.36700i	0.729951	+	0.683500i	$0.239543\pi$
$853$			34055.9i			1.36700i	−0.729951	+	0.683500i	$0.760457\pi$
$854$	−4068.26			−0.163013
$855$		0			0
$856$	11894.1			0.474921
$857$			22647.0i			0.902693i	0.892349	+	0.451347i	$0.149056\pi$
$857$			22647.0i			0.902693i	−0.892349	+	0.451347i	$0.850944\pi$
$858$		0			0
$859$	12157.1			0.482882			0.241441	−	0.970416i	$-0.422380\pi$
$859$	12157.1			0.482882			0.241441	+	0.970416i	$0.422380\pi$
$860$	2692.72	−	919.967i	0.106769	−	0.0364775i
$861$		0			0
$862$			673.415i			0.0266086i
$863$		−	11517.7i		−	0.454305i	−0.973859	−	0.227153i	$-0.927058\pi$
$863$		−	11517.7i		−	0.454305i	0.973859	−	0.227153i	$-0.0729417\pi$
$864$		0			0
$865$	12425.4	−	4245.15i	0.488413	−	0.166866i
$866$	−1811.29			−0.0710742
$867$		0			0
$868$			11114.3i			0.434613i
$869$	28290.6			1.10437
$870$		0			0
$871$	−37942.2			−1.47603
$872$		−	2715.44i		−	0.105455i
$873$		0			0
$874$	−11201.9			−0.433537
$875$	−16094.3	+	24397.3i	−0.621813	+	0.942605i
$876$		0			0
$877$		−	27266.1i		−	1.04984i	−0.851151	−	0.524921i	$-0.824095\pi$
$877$		−	27266.1i		−	1.04984i	0.851151	−	0.524921i	$-0.175905\pi$
$878$		−	558.558i		−	0.0214697i
$879$		0			0
$880$	10642.0	+	31148.9i	0.407661	+	1.19321i
$881$	9173.42			0.350806			0.175403	−	0.984497i	$-0.443877\pi$
$881$	9173.42			0.350806			0.175403	+	0.984497i	$0.443877\pi$
$882$		0			0
$883$			8150.59i			0.310633i	0.987865	+	0.155317i	$0.0496398\pi$
$883$			8150.59i			0.310633i	−0.987865	+	0.155317i	$0.950360\pi$
$884$	−14612.1			−0.555947
$885$		0			0
$886$	−12282.5			−0.465733
$887$		−	1442.34i		−	0.0545986i	−0.999627	−	0.0272993i	$-0.991309\pi$
$887$		−	1442.34i		−	0.0545986i	0.999627	−	0.0272993i	$-0.00869071\pi$
$888$		0			0
$889$	57348.9			2.16358
$890$	−6793.47	+	2320.99i	−0.255862	+	0.0874154i
$891$		0			0
$892$			1961.82i			0.0736398i
$893$		−	16537.6i		−	0.619721i
$894$		0			0
$895$	9082.03	+	26582.9i	0.339194	+	0.992812i
$896$	27449.7			1.02347
$897$		0			0
$898$			6405.54i			0.238035i
$899$	−891.953			−0.0330904
$900$		0			0
$901$	−5911.42			−0.218577
$902$			2382.26i			0.0879387i
$903$		0			0
$904$	−1664.32			−0.0612329
$905$	−13323.2	−	38996.6i	−0.489368	−	1.43237i
$906$		0			0
$907$			5472.41i			0.200340i	0.994970	+	0.100170i	$0.0319387\pi$
$907$			5472.41i			0.200340i	−0.994970	+	0.100170i	$0.968061\pi$
$908$		−	37838.8i		−	1.38296i
$909$		0			0
$910$	7981.41	−	2726.85i	0.290749	−	0.0993342i
$911$	23045.6			0.838130			0.419065	−	0.907956i	$-0.362358\pi$
$911$	23045.6			0.838130			0.419065	+	0.907956i	$0.362358\pi$
$912$		0			0
$913$			18597.4i			0.674134i
$914$	−11611.3			−0.420204
$915$		0			0
$916$	−670.136			−0.0241724
$917$		−	20085.4i		−	0.723315i
$918$		0			0
$919$	23130.3			0.830247			0.415124	−	0.909765i	$-0.363738\pi$
$919$	23130.3			0.830247			0.415124	+	0.909765i	$0.363738\pi$
$920$	−4443.49	−	13006.0i	−0.159236	−	0.466080i
$921$		0			0
$922$			6512.88i			0.232636i
$923$		−	11944.6i		−	0.425959i
$924$		0			0
$925$	11528.8	+	14902.9i	0.409801	+	0.529734i
$926$	11275.0			0.400130
$927$		0			0
$928$			1680.13i			0.0594320i
$929$	12258.9			0.432942			0.216471	−	0.976289i	$-0.430545\pi$
$929$	12258.9			0.432942			0.216471	+	0.976289i	$0.430545\pi$
$930$		0			0
$931$	13229.8			0.465723
$932$			43237.6i			1.51963i
$933$		0			0
$934$	3781.57			0.132481
$935$	−27070.6	+	9248.67i	−0.946849	+	0.323491i
$936$		0			0
$937$			37083.2i			1.29291i	0.762953	+	0.646454i	$0.223749\pi$
$937$			37083.2i			1.29291i	−0.762953	+	0.646454i	$0.776251\pi$
$938$			13567.2i			0.472264i
$939$		0			0
$940$	9215.57	−	3148.50i	0.319765	−	0.109247i
$941$	−2612.96			−0.0905206			−0.0452603	−	0.998975i	$-0.514412\pi$
$941$	−2612.96			−0.0905206			−0.0452603	+	0.998975i	$0.514412\pi$
$942$		0			0
$943$			5197.89i			0.179498i
$944$	−24606.5			−0.848382
$945$		0			0
$946$	−1607.62			−0.0552518
$947$		−	34322.4i		−	1.17775i	−0.808224	−	0.588875i	$-0.799571\pi$
$947$		−	34322.4i		−	1.17775i	0.808224	−	0.588875i	$-0.200429\pi$
$948$		0			0
$949$	−17097.1			−0.584822
$950$	−8419.95	−	10884.2i	−0.287557	−	0.371714i
$951$		0			0
$952$			10886.3i			0.370616i
$953$			33734.8i			1.14667i	0.819321	+	0.573335i	$0.194351\pi$
$953$			33734.8i			1.14667i	−0.819321	+	0.573335i	$0.805649\pi$
$954$		0			0
$955$	9818.58	+	28738.7i	0.332693	+	0.973783i
$956$	50714.2			1.71571
$957$		0			0
$958$			9288.72i			0.313262i
$959$	−15194.1			−0.511619
$960$		0			0
$961$	−24610.0			−0.826089
$962$		−	5437.25i		−	0.182229i
$963$		0			0
$964$	−185.596			−0.00620088
$965$	−20871.6	+	7130.77i	−0.696248	+	0.237873i
$966$		0			0
$967$		−	27695.9i		−	0.921033i	−0.887651	−	0.460517i	$-0.847664\pi$
$967$		−	27695.9i		−	0.921033i	0.887651	−	0.460517i	$-0.152336\pi$
$968$		−	26522.9i		−	0.880660i
$969$		0			0
$970$	−2672.04	−	7820.99i	−0.0884475	−	0.258883i
$971$	−3160.23			−0.104446			−0.0522228	−	0.998635i	$-0.516631\pi$
$971$	−3160.23			−0.104446			−0.0522228	+	0.998635i	$0.516631\pi$
$972$		0			0
$973$		−	10454.6i		−	0.344458i
$974$	−10278.3			−0.338128
$975$		0			0
$976$	−12280.6			−0.402758
$977$			31041.9i			1.01650i	0.861211	+	0.508248i	$0.169707\pi$
$977$			31041.9i			1.01650i	−0.861211	+	0.508248i	$0.830293\pi$
$978$		0			0
$979$	−48553.8			−1.58507
$980$	2518.73	+	7372.27i	0.0821000	+	0.240305i
$981$		0			0
$982$			5178.00i			0.168265i
$983$			7417.84i			0.240684i	0.992732	+	0.120342i	$0.0383991\pi$
$983$			7417.84i			0.240684i	−0.992732	+	0.120342i	$0.961601\pi$
$984$		0			0
$985$	−36039.7	+	12313.0i	−1.16581	+	0.398298i
$986$	−419.313			−0.0135433
$987$		0			0
$988$		−	47538.3i		−	1.53076i
$989$	−3507.68			−0.112778
$990$		0			0
$991$	13109.6			0.420221			0.210110	−	0.977678i	$-0.432618\pi$
$991$	13109.6			0.420221			0.210110	+	0.977678i	$0.432618\pi$
$992$		−	9759.16i		−	0.312352i
$993$		0			0
$994$	−4271.07			−0.136288
$995$	−3548.56	−	10386.5i	−0.113062	−	0.330930i
$996$		0			0
$997$		−	35703.9i		−	1.13416i	−0.823664	−	0.567079i	$-0.808074\pi$
$997$		−	35703.9i		−	1.13416i	0.823664	−	0.567079i	$-0.191926\pi$
$998$		−	12710.4i		−	0.403148i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.b.e.244.7		16
3.2	odd	2		405.4.b.f.244.10		16
5.2	odd	4		2025.4.a.bk.1.10		16
5.3	odd	4		2025.4.a.bk.1.7		16
5.4	even	2	inner	405.4.b.e.244.10		16
9.2	odd	6		135.4.j.a.64.10		32
9.4	even	3		45.4.j.a.34.10	yes	32
9.5	odd	6		135.4.j.a.19.7		32
9.7	even	3		45.4.j.a.4.7	✓	32
15.2	even	4		2025.4.a.bl.1.7		16
15.8	even	4		2025.4.a.bl.1.10		16
15.14	odd	2		405.4.b.f.244.7		16
45.4	even	6		45.4.j.a.34.7	yes	32
45.7	odd	12		225.4.e.g.76.7		32
45.13	odd	12		225.4.e.g.151.10		32
45.14	odd	6		135.4.j.a.19.10		32
45.22	odd	12		225.4.e.g.151.7		32
45.29	odd	6		135.4.j.a.64.7		32
45.34	even	6		45.4.j.a.4.10	yes	32
45.43	odd	12		225.4.e.g.76.10		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.j.a.4.7	✓	32	9.7	even	3
45.4.j.a.4.10	yes	32	45.34	even	6
45.4.j.a.34.7	yes	32	45.4	even	6
45.4.j.a.34.10	yes	32	9.4	even	3
135.4.j.a.19.7		32	9.5	odd	6
135.4.j.a.19.10		32	45.14	odd	6
135.4.j.a.64.7		32	45.29	odd	6
135.4.j.a.64.10		32	9.2	odd	6
225.4.e.g.76.7		32	45.7	odd	12
225.4.e.g.76.10		32	45.43	odd	12
225.4.e.g.151.7		32	45.22	odd	12
225.4.e.g.151.10		32	45.13	odd	12
405.4.b.e.244.7		16	1.1	even	1	trivial
405.4.b.e.244.10		16	5.4	even	2	inner
405.4.b.f.244.7		16	15.14	odd	2
405.4.b.f.244.10		16	3.2	odd	2
2025.4.a.bk.1.7		16	5.3	odd	4
2025.4.a.bk.1.10		16	5.2	odd	4
2025.4.a.bl.1.7		16	15.2	even	4
2025.4.a.bl.1.10		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)

\(f(q)\)	\(=\)	\(q-0.785333i q^{2} +7.38325 q^{4} +(-3.61462 - 10.5799i) q^{5} +20.9136i q^{7} -12.0810i q^{8} +O(q^{10})\)	\(q-0.785333i q^{2} +7.38325 q^{4} +(-3.61462 - 10.5799i) q^{5} +20.9136i q^{7} -12.0810i q^{8} +(-8.30875 + 2.83868i) q^{10} -59.3838 q^{11} +45.9320i q^{13} +16.4241 q^{14} +49.5784 q^{16} +43.0872i q^{17} -140.178 q^{19} +(-26.6877 - 78.1141i) q^{20} +46.6360i q^{22} +101.756i q^{23} +(-98.8690 + 76.4848i) q^{25} +36.0719 q^{26} +154.410i q^{28} -12.3918 q^{29} +71.9790 q^{31} -135.583i q^{32} +33.8378 q^{34} +(221.264 - 75.5948i) q^{35} -150.734i q^{37} +110.087i q^{38} +(-127.816 + 43.6682i) q^{40} +51.0821 q^{41} +34.4716i q^{43} -438.445 q^{44} +79.9121 q^{46} +117.976i q^{47} -94.3781 q^{49} +(60.0660 + 77.6451i) q^{50} +339.128i q^{52} +137.196i q^{53} +(214.650 + 628.275i) q^{55} +252.657 q^{56} +9.73173i q^{58} -496.314 q^{59} -247.700 q^{61} -56.5275i q^{62} +290.149 q^{64} +(485.956 - 166.027i) q^{65} +826.052i q^{67} +318.124i q^{68} +(-59.3671 - 173.766i) q^{70} -260.049 q^{71} +372.227i q^{73} -118.376 q^{74} -1034.97 q^{76} -1241.93i q^{77} -476.404 q^{79} +(-179.207 - 524.535i) q^{80} -40.1164i q^{82} -313.173i q^{83} +(455.859 - 155.744i) q^{85} +27.0717 q^{86} +717.414i q^{88} +817.628 q^{89} -960.603 q^{91} +751.288i q^{92} +92.6502 q^{94} +(506.692 + 1483.07i) q^{95} +941.296i q^{97} +74.1183i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 54 q^{4} - 3 q^{5}+O(q^{10}) \) 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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