LMFDB - Embedded newform 405.4.b.f.244.14

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			244.14
Root			$4.07626i$ of defining polynomial
Character	$\chi$	$=$	405.244
Dual form			405.4.b.f.244.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.07626i q^{2} -8.61587 q^{4} +(-10.3994 - 4.10522i) q^{5} +13.3430i q^{7} -2.51043i q^{8} +O(q^{10})$	\(q+4.07626i q^{2} -8.61587 q^{4} +(-10.3994 - 4.10522i) q^{5} +13.3430i q^{7} -2.51043i q^{8} +(16.7339 - 42.3906i) q^{10} -11.5617 q^{11} +40.0796i q^{13} -54.3893 q^{14} -58.6938 q^{16} -93.3392i q^{17} -75.1002 q^{19} +(89.5997 + 35.3700i) q^{20} -47.1284i q^{22} -142.001i q^{23} +(91.2943 + 85.3835i) q^{25} -163.375 q^{26} -114.961i q^{28} -174.047 q^{29} +248.868 q^{31} -259.334i q^{32} +380.475 q^{34} +(54.7758 - 138.758i) q^{35} +82.9086i q^{37} -306.128i q^{38} +(-10.3059 + 26.1069i) q^{40} +449.454 q^{41} -279.850i q^{43} +99.6140 q^{44} +578.831 q^{46} -33.8764i q^{47} +164.966 q^{49} +(-348.045 + 372.139i) q^{50} -345.320i q^{52} -423.321i q^{53} +(120.234 + 47.4633i) q^{55} +33.4966 q^{56} -709.461i q^{58} -615.628 q^{59} -502.608 q^{61} +1014.45i q^{62} +587.563 q^{64} +(164.535 - 416.803i) q^{65} -57.7537i q^{67} +804.198i q^{68} +(565.615 + 223.280i) q^{70} +252.598 q^{71} -823.934i q^{73} -337.957 q^{74} +647.054 q^{76} -154.267i q^{77} +205.565 q^{79} +(610.379 + 240.951i) q^{80} +1832.09i q^{82} +646.559i q^{83} +(-383.178 + 970.670i) q^{85} +1140.74 q^{86} +29.0248i q^{88} -1324.94 q^{89} -534.780 q^{91} +1223.46i q^{92} +138.089 q^{94} +(780.996 + 308.303i) q^{95} -563.768i q^{97} +672.442i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$			4.07626i			1.44117i	0.693364	+	0.720587i	$0.256128\pi$
$2$			4.07626i			1.44117i	−0.693364	+	0.720587i	$0.743872\pi$
$3$		0			0
$3$		0			0
$4$	−8.61587			−1.07698
$5$	−10.3994	−	4.10522i	−0.930149	−	0.367182i
$5$	−10.3994	−	4.10522i	−0.930149	−	0.367182i
$6$		0			0
$7$			13.3430i			0.720452i	0.932865	+	0.360226i	$0.117300\pi$
$7$			13.3430i			0.720452i	−0.932865	+	0.360226i	$0.882700\pi$
$8$		−	2.51043i		−	0.110946i
$9$		0			0
$10$	16.7339	−	42.3906i	0.529173	−	1.34051i
$11$	−11.5617			−0.316907			−0.158454	−	0.987366i	$-0.550651\pi$
$11$	−11.5617			−0.316907			−0.158454	+	0.987366i	$0.550651\pi$
$12$		0			0
$13$			40.0796i			0.855083i	0.903996	+	0.427541i	$0.140620\pi$
$13$			40.0796i			0.855083i	−0.903996	+	0.427541i	$0.859380\pi$
$14$	−54.3893			−1.03830
$15$		0			0
$16$	−58.6938			−0.917090
$17$		−	93.3392i		−	1.33165i	−0.746107	−	0.665826i	$-0.768079\pi$
$17$		−	93.3392i		−	1.33165i	0.746107	−	0.665826i	$-0.231921\pi$
$18$		0			0
$19$	−75.1002			−0.906799			−0.453399	−	0.891307i	$-0.649789\pi$
$19$	−75.1002			−0.906799			−0.453399	+	0.891307i	$0.649789\pi$
$20$	89.5997	+	35.3700i	1.00176	+	0.395449i
$21$		0			0
$22$		−	47.1284i		−	0.456719i
$23$		−	142.001i		−	1.28735i	−0.765297	−	0.643677i	$-0.777408\pi$
$23$		−	142.001i		−	1.28735i	0.765297	−	0.643677i	$-0.222592\pi$
$24$		0			0
$25$	91.2943	+	85.3835i	0.730355	+	0.683068i
$26$	−163.375			−1.23232
$27$		0			0
$28$		−	114.961i		−	0.775915i
$29$	−174.047			−1.11447			−0.557237	−	0.830354i	$-0.688138\pi$
$29$	−174.047			−1.11447			−0.557237	+	0.830354i	$0.688138\pi$
$30$		0			0
$31$	248.868			1.44187			0.720937	−	0.693001i	$-0.243712\pi$
$31$	248.868			1.44187			0.720937	+	0.693001i	$0.243712\pi$
$32$		−	259.334i		−	1.43263i
$33$		0			0
$34$	380.475			1.91914
$35$	54.7758	−	138.758i	0.264537	−	0.670128i
$36$		0			0
$37$			82.9086i			0.368381i	0.982891	+	0.184190i	$0.0589663\pi$
$37$			82.9086i			0.368381i	−0.982891	+	0.184190i	$0.941034\pi$
$38$		−	306.128i		−	1.30686i
$39$		0			0
$40$	−10.3059	+	26.1069i	−0.0407376	+	0.103197i
$41$	449.454			1.71202			0.856011	−	0.516957i	$-0.172935\pi$
$41$	449.454			1.71202			0.856011	+	0.516957i	$0.172935\pi$
$42$		0			0
$43$		−	279.850i		−	0.992481i	−0.868185	−	0.496241i	$-0.834713\pi$
$43$		−	279.850i		−	0.992481i	0.868185	−	0.496241i	$-0.165287\pi$
$44$	99.6140			0.341304
$45$		0			0
$46$	578.831			1.85530
$47$		−	33.8764i		−	0.105136i	−0.998617	−	0.0525678i	$-0.983259\pi$
$47$		−	33.8764i		−	0.105136i	0.998617	−	0.0525678i	$-0.0167406\pi$
$48$		0			0
$49$	164.966			0.480949
$50$	−348.045	+	372.139i	−0.984420	+	1.05257i
$51$		0			0
$52$		−	345.320i		−	0.920910i
$53$		−	423.321i		−	1.09712i	−0.836110	−	0.548562i	$-0.815175\pi$
$53$		−	423.321i		−	1.09712i	0.836110	−	0.548562i	$-0.184825\pi$
$54$		0			0
$55$	120.234	+	47.4633i	0.294771	+	0.116363i
$56$	33.4966			0.0799316
$57$		0			0
$58$		−	709.461i		−	1.60615i
$59$	−615.628			−1.35844			−0.679220	−	0.733935i	$-0.737682\pi$
$59$	−615.628			−1.35844			−0.679220	+	0.733935i	$0.737682\pi$
$60$		0			0
$61$	−502.608			−1.05496			−0.527479	−	0.849568i	$-0.676863\pi$
$61$	−502.608			−1.05496			−0.527479	+	0.849568i	$0.676863\pi$
$62$			1014.45i			2.07799i
$63$		0			0
$64$	587.563			1.14758
$65$	164.535	−	416.803i	0.313971	−	0.795354i
$66$		0			0
$67$		−	57.7537i		−	0.105310i	−0.998613	−	0.0526548i	$-0.983232\pi$
$67$		−	57.7537i		−	0.105310i	0.998613	−	0.0526548i	$-0.0167683\pi$
$68$			804.198i			1.43417i
$69$		0			0
$70$	565.615	+	223.280i	0.965771	+	0.381244i
$71$	252.598			0.422224			0.211112	−	0.977462i	$-0.432292\pi$
$71$	252.598			0.422224			0.211112	+	0.977462i	$0.432292\pi$
$72$		0			0
$73$		−	823.934i		−	1.32102i	−0.750819	−	0.660508i	$-0.770341\pi$
$73$		−	823.934i		−	1.32102i	0.750819	−	0.660508i	$-0.229659\pi$
$74$	−337.957			−0.530901
$75$		0			0
$76$	647.054			0.976607
$77$		−	154.267i		−	0.228317i
$78$		0			0
$79$	205.565			0.292758			0.146379	−	0.989229i	$-0.453238\pi$
$79$	205.565			0.292758			0.146379	+	0.989229i	$0.453238\pi$
$80$	610.379	+	240.951i	0.853031	+	0.336739i
$81$		0			0
$82$			1832.09i			2.46732i
$83$			646.559i			0.855049i	0.904004	+	0.427525i	$0.140614\pi$
$83$			646.559i			0.855049i	−0.904004	+	0.427525i	$0.859386\pi$
$84$		0			0
$85$	−383.178	+	970.670i	−0.488959	+	1.23864i
$86$	1140.74			1.43034
$87$		0			0
$88$			29.0248i			0.0351598i
$89$	−1324.94			−1.57802			−0.789010	−	0.614380i	$-0.789406\pi$
$89$	−1324.94			−1.57802			−0.789010	+	0.614380i	$0.789406\pi$
$90$		0			0
$91$	−534.780			−0.616046
$92$			1223.46i			1.38646i
$93$		0			0
$94$	138.089			0.151519
$95$	780.996	+	308.303i	0.843458	+	0.332960i
$96$		0			0
$97$		−	563.768i		−	0.590124i	−0.955478	−	0.295062i	$-0.904660\pi$
$97$		−	563.768i		−	0.590124i	0.955478	−	0.295062i	$-0.0953403\pi$
$98$			672.442i			0.693132i
$99$		0			0
$100$	−786.580	−	735.653i	−0.786580	−	0.735653i
$101$	311.591			0.306975			0.153487	−	0.988151i	$-0.450950\pi$
$101$	311.591			0.306975			0.153487	+	0.988151i	$0.450950\pi$
$102$		0			0
$103$		−	81.5226i		−	0.0779870i	−0.999239	−	0.0389935i	$-0.987585\pi$
$103$		−	81.5226i		−	0.0779870i	0.999239	−	0.0389935i	$-0.0124152\pi$
$104$	100.617			0.0948684
$105$		0			0
$106$	1725.56			1.58115
$107$		−	145.138i		−	0.131131i	−0.997848	−	0.0655655i	$-0.979115\pi$
$107$		−	145.138i		−	0.131131i	0.997848	−	0.0655655i	$-0.0208851\pi$
$108$		0			0
$109$	−326.484			−0.286894			−0.143447	−	0.989658i	$-0.545819\pi$
$109$	−326.484			−0.286894			−0.143447	+	0.989658i	$0.545819\pi$
$110$	−193.473	+	490.107i	−0.167699	+	0.424817i
$111$		0			0
$112$		−	783.148i		−	0.660719i
$113$			52.0212i			0.0433074i	0.999766	+	0.0216537i	$0.00689313\pi$
$113$			52.0212i			0.0433074i	−0.999766	+	0.0216537i	$0.993107\pi$
$114$		0			0
$115$	−582.943	+	1476.72i	−0.472694	+	1.19743i
$116$	1499.57			1.20027
$117$		0			0
$118$		−	2509.46i		−	1.95775i
$119$	1245.42			0.959391
$120$		0			0
$121$	−1197.33			−0.899570
$122$		−	2048.76i		−	1.52038i
$123$		0			0
$124$	−2144.22			−1.55287
$125$	−598.887	−	1262.72i	−0.428528	−	0.903528i
$126$		0			0
$127$		−	266.008i		−	0.185862i	−0.995673	−	0.0929309i	$-0.970376\pi$
$127$		−	266.008i		−	0.185862i	0.995673	−	0.0929309i	$-0.0296236\pi$
$128$			320.383i			0.221235i
$129$		0			0
$130$	1699.00	+	670.689i	1.14624	+	0.452487i
$131$	−2542.39			−1.69565			−0.847825	−	0.530277i	$-0.822088\pi$
$131$	−2542.39			−1.69565			−0.847825	+	0.530277i	$0.822088\pi$
$132$		0			0
$133$		−	1002.06i		−	0.653305i
$134$	235.419			0.151769
$135$		0			0
$136$	−234.322			−0.147742
$137$		−	123.495i		−	0.0770140i	−0.999258	−	0.0385070i	$-0.987740\pi$
$137$		−	123.495i		−	0.0770140i	0.999258	−	0.0385070i	$-0.0122602\pi$
$138$		0			0
$139$	−215.123			−0.131270			−0.0656348	−	0.997844i	$-0.520907\pi$
$139$	−215.123			−0.131270			−0.0656348	+	0.997844i	$0.520907\pi$
$140$	−471.941	+	1195.52i	−0.284902	+	0.721716i
$141$		0			0
$142$			1029.66i			0.608499i
$143$		−	463.388i		−	0.270982i
$144$		0			0
$145$	1809.98	+	714.502i	1.03663	+	0.409215i
$146$	3358.57			1.90382
$147$		0			0
$148$		−	714.329i		−	0.396740i
$149$	−221.877			−0.121993			−0.0609963	−	0.998138i	$-0.519428\pi$
$149$	−221.877			−0.121993			−0.0609963	+	0.998138i	$0.519428\pi$
$150$		0			0
$151$	495.348			0.266959			0.133480	−	0.991052i	$-0.457385\pi$
$151$	495.348			0.266959			0.133480	+	0.991052i	$0.457385\pi$
$152$			188.534i			0.100606i
$153$		0			0
$154$	628.832			0.329044
$155$	−2588.08	−	1021.66i	−1.34116	−	0.529430i
$156$		0			0
$157$			2206.41i			1.12160i	0.827952	+	0.560798i	$0.189506\pi$
$157$			2206.41i			1.12160i	−0.827952	+	0.560798i	$0.810494\pi$
$158$			837.935i			0.421915i
$159$		0			0
$160$	−1064.62	+	2696.92i	−0.526037	+	1.33256i
$161$	1894.71			0.927477
$162$		0			0
$163$			931.108i			0.447423i	0.974655	+	0.223712i	$0.0718174\pi$
$163$			931.108i			0.447423i	−0.974655	+	0.223712i	$0.928183\pi$
$164$	−3872.44			−1.84382
$165$		0			0
$166$	−2635.54			−1.23227
$167$		−	1724.43i		−	0.799045i	−0.916723	−	0.399522i	$-0.869176\pi$
$167$		−	1724.43i		−	0.799045i	0.916723	−	0.399522i	$-0.130824\pi$
$168$		0			0
$169$	590.628			0.268834
$170$	−3956.70	−	1561.93i	−1.78509	−	0.704675i
$171$		0			0
$172$			2411.15i			1.06889i
$173$		−	2451.99i		−	1.07758i	−0.842440	−	0.538790i	$-0.818881\pi$
$173$		−	2451.99i		−	1.07758i	0.842440	−	0.538790i	$-0.181119\pi$
$174$		0			0
$175$	−1139.27	+	1218.14i	−0.492118	+	0.526185i
$176$	678.599			0.290633
$177$		0			0
$178$		−	5400.81i		−	2.27420i
$179$	−1051.07			−0.438885			−0.219443	−	0.975625i	$-0.570424\pi$
$179$	−1051.07			−0.438885			−0.219443	+	0.975625i	$0.570424\pi$
$180$		0			0
$181$	−2663.06			−1.09361			−0.546805	−	0.837260i	$-0.684156\pi$
$181$	−2663.06			−1.09361			−0.546805	+	0.837260i	$0.684156\pi$
$182$		−	2179.90i		−	0.887829i
$183$		0			0
$184$	−356.483			−0.142827
$185$	340.358	−	862.198i	0.135263	−	0.342649i
$186$		0			0
$187$			1079.16i			0.422010i
$188$			291.874i			0.113229i
$189$		0			0
$190$	−1256.72	+	3183.54i	−0.479854	+	1.21557i
$191$	−2223.59			−0.842374			−0.421187	−	0.906974i	$-0.638386\pi$
$191$	−2223.59			−0.842374			−0.421187	+	0.906974i	$0.638386\pi$
$192$		0			0
$193$		−	4850.19i		−	1.80894i	−0.426542	−	0.904468i	$-0.640268\pi$
$193$		−	4850.19i		−	1.80894i	0.426542	−	0.904468i	$-0.359732\pi$
$194$	2298.06			0.850471
$195$		0			0
$196$	−1421.32			−0.517974
$197$			372.662i			0.134777i	0.997727	+	0.0673884i	$0.0214667\pi$
$197$			372.662i			0.134777i	−0.997727	+	0.0673884i	$0.978533\pi$
$198$		0			0
$199$	−4996.59			−1.77990			−0.889948	−	0.456063i	$-0.849259\pi$
$199$	−4996.59			−1.77990			−0.889948	+	0.456063i	$0.849259\pi$
$200$	214.350	−	229.188i	0.0757840	−	0.0810303i
$201$		0			0
$202$			1270.12i			0.442404i
$203$		−	2322.30i		−	0.802925i
$204$		0			0
$205$	−4674.04	−	1845.11i	−1.59244	−	0.628624i
$206$	332.307			0.112393
$207$		0			0
$208$		−	2352.42i		−	0.784188i
$209$	868.286			0.287371
$210$		0			0
$211$	−563.402			−0.183821			−0.0919104	−	0.995767i	$-0.529297\pi$
$211$	−563.402			−0.183821			−0.0919104	+	0.995767i	$0.529297\pi$
$212$			3647.28i			1.18158i
$213$		0			0
$214$	591.619			0.188982
$215$	−1148.84	+	2910.26i	−0.364421	+	0.923155i
$216$		0			0
$217$			3320.64i			1.03880i
$218$		−	1330.83i		−	0.413465i
$219$		0			0
$220$	−1035.92	−	408.937i	−0.317464	−	0.125321i
$221$	3741.00			1.13867
$222$		0			0
$223$			5374.35i			1.61387i	0.590639	+	0.806936i	$0.298876\pi$
$223$			5374.35i			1.61387i	−0.590639	+	0.806936i	$0.701124\pi$
$224$	3460.29			1.03214
$225$		0			0
$226$	−212.052			−0.0624136
$227$		−	1111.67i		−	0.325040i	−0.986705	−	0.162520i	$-0.948038\pi$
$227$		−	1111.67i		−	0.325040i	0.986705	−	0.162520i	$-0.0519621\pi$
$228$		0			0
$229$	−2037.75			−0.588027			−0.294013	−	0.955801i	$-0.594991\pi$
$229$	−2037.75			−0.588027			−0.294013	+	0.955801i	$0.594991\pi$
$230$	−6019.48	−	2376.23i	−1.72571	−	0.681234i
$231$		0			0
$232$			436.933i			0.123647i
$233$		−	2127.08i		−	0.598066i	−0.954243	−	0.299033i	$-0.903336\pi$
$233$		−	2127.08i		−	0.598066i	0.954243	−	0.299033i	$-0.0966641\pi$
$234$		0			0
$235$	−139.070	+	352.293i	−0.0386039	+	0.0977918i
$236$	5304.17			1.46302
$237$		0			0
$238$			5076.66i			1.38265i
$239$	−1982.88			−0.536659			−0.268330	−	0.963327i	$-0.586472\pi$
$239$	−1982.88			−0.536659			−0.268330	+	0.963327i	$0.586472\pi$
$240$		0			0
$241$	−4219.68			−1.12786			−0.563929	−	0.825823i	$-0.690711\pi$
$241$	−4219.68			−1.12786			−0.563929	+	0.825823i	$0.690711\pi$
$242$		−	4880.61i		−	1.29644i
$243$		0			0
$244$	4330.41			1.13617
$245$	−1715.54	−	677.220i	−0.447354	−	0.176596i
$246$		0			0
$247$		−	3009.99i		−	0.775388i
$248$		−	624.767i		−	0.159971i
$249$		0			0
$250$	5147.17	−	2441.22i	1.30214	−	0.617584i
$251$	1638.58			0.412057			0.206028	−	0.978546i	$-0.433946\pi$
$251$	1638.58			0.412057			0.206028	+	0.978546i	$0.433946\pi$
$252$		0			0
$253$			1641.77i			0.407972i
$254$	1084.32			0.267859
$255$		0			0
$256$	3394.54			0.828745
$257$			6167.31i			1.49691i	0.663184	+	0.748456i	$0.269205\pi$
$257$			6167.31i			1.49691i	−0.663184	+	0.748456i	$0.730795\pi$
$258$		0			0
$259$	−1106.25			−0.265401
$260$	−1417.62	+	3591.12i	−0.338142	+	0.856583i
$261$		0			0
$262$		−	10363.5i		−	2.44373i
$263$			6361.88i			1.49160i	0.666171	+	0.745799i	$0.267932\pi$
$263$			6361.88i			1.49160i	−0.666171	+	0.745799i	$0.732068\pi$
$264$		0			0
$265$	−1737.82	+	4402.27i	−0.402844	+	1.02049i
$266$	4084.65			0.941526
$267$		0			0
$268$			497.598i			0.113417i
$269$	2404.84			0.545077			0.272539	−	0.962145i	$-0.412137\pi$
$269$	2404.84			0.545077			0.272539	+	0.962145i	$0.412137\pi$
$270$		0			0
$271$	6986.46			1.56604			0.783021	−	0.621995i	$-0.213678\pi$
$271$	6986.46			1.56604			0.783021	+	0.621995i	$0.213678\pi$
$272$			5478.43i			1.22125i
$273$		0			0
$274$	503.398			0.110991
$275$	−1055.52	−	987.178i	−0.231455	−	0.216469i
$276$		0			0
$277$			1502.01i			0.325803i	0.986642	+	0.162901i	$0.0520852\pi$
$277$			1502.01i			0.325803i	−0.986642	+	0.162901i	$0.947915\pi$
$278$		−	876.896i		−	0.189182i
$279$		0			0
$280$	−348.344	−	137.511i	−0.0743483	−	0.0293494i
$281$	1443.06			0.306355			0.153178	−	0.988199i	$-0.451049\pi$
$281$	1443.06			0.306355			0.153178	+	0.988199i	$0.451049\pi$
$282$		0			0
$283$		−	4662.08i		−	0.979264i	−0.871929	−	0.489632i	$-0.837131\pi$
$283$		−	4662.08i		−	0.979264i	0.871929	−	0.489632i	$-0.162869\pi$
$284$	−2176.35			−0.454728
$285$		0			0
$286$	1888.89			0.390532
$287$			5997.04i			1.23343i
$288$		0			0
$289$	−3799.21			−0.773297
$290$	−2912.49	+	7377.95i	−0.589750	+	1.49396i
$291$		0			0
$292$			7098.91i			1.42271i
$293$		−	1105.95i		−	0.220513i	−0.993903	−	0.110256i	$-0.964833\pi$
$293$		−	1105.95i		−	0.220513i	0.993903	−	0.110256i	$-0.0351672\pi$
$294$		0			0
$295$	6402.16	+	2527.29i	1.26355	+	0.498795i
$296$	208.136			0.0408705
$297$		0			0
$298$		−	904.429i		−	0.175813i
$299$	5691.32			1.10079
$300$		0			0
$301$	3734.02			0.715035
$302$			2019.16i			0.384734i
$303$		0			0
$304$	4407.92			0.831616
$305$	5226.82	+	2063.32i	0.981268	+	0.387361i
$306$		0			0
$307$		−	8585.39i		−	1.59607i	−0.602609	−	0.798036i	$-0.705872\pi$
$307$		−	8585.39i		−	1.59607i	0.602609	−	0.798036i	$-0.294128\pi$
$308$			1329.15i			0.245893i
$309$		0			0
$310$	4164.55	−	10549.7i	0.763001	−	1.93284i
$311$	7039.00			1.28342			0.641712	−	0.766945i	$-0.278224\pi$
$311$	7039.00			1.28342			0.641712	+	0.766945i	$0.278224\pi$
$312$		0			0
$313$		−	5196.42i		−	0.938399i	−0.883092	−	0.469199i	$-0.844543\pi$
$313$		−	5196.42i		−	0.938399i	0.883092	−	0.469199i	$-0.155457\pi$
$314$	−8993.89			−1.61642
$315$		0			0
$316$	−1771.12			−0.315295
$317$		−	556.011i		−	0.0985133i	−0.998786	−	0.0492566i	$-0.984315\pi$
$317$		−	556.011i		−	0.0985133i	0.998786	−	0.0492566i	$-0.0156852\pi$
$318$		0			0
$319$	2012.28			0.353185
$320$	−6110.29	−	2412.08i	−1.06742	−	0.421372i
$321$		0			0
$322$			7723.31i			1.33666i
$323$			7009.80i			1.20754i
$324$		0			0
$325$	−3422.13	+	3659.04i	−0.584080	+	0.624514i
$326$	−3795.43			−0.644815
$327$		0			0
$328$		−	1128.32i		−	0.189943i
$329$	452.011			0.0757452
$330$		0			0
$331$	−509.573			−0.0846184			−0.0423092	−	0.999105i	$-0.513471\pi$
$331$	−509.573			−0.0846184			−0.0423092	+	0.999105i	$0.513471\pi$
$332$		−	5570.67i		−	0.920874i
$333$		0			0
$334$	7029.22			1.15156
$335$	−237.092	+	600.603i	−0.0386678	+	0.0979536i
$336$		0			0
$337$			783.515i			0.126649i	0.997993	+	0.0633246i	$0.0201703\pi$
$337$			783.515i			0.126649i	−0.997993	+	0.0633246i	$0.979830\pi$
$338$			2407.55i			0.387436i
$339$		0			0
$340$	3301.41	−	8363.17i	0.526600	−	1.33399i
$341$	−2877.34			−0.456940
$342$		0			0
$343$			6777.76i			1.06695i
$344$	−702.544			−0.110112
$345$		0			0
$346$	9994.95			1.55298
$347$			630.674i			0.0975688i	0.998809	+	0.0487844i	$0.0155347\pi$
$347$			630.674i			0.0975688i	−0.998809	+	0.0487844i	$0.984465\pi$
$348$		0			0
$349$	−387.854			−0.0594882			−0.0297441	−	0.999558i	$-0.509469\pi$
$349$	−387.854			−0.0594882			−0.0297441	+	0.999558i	$0.509469\pi$
$350$	−4965.44	−	4643.95i	−0.758325	−	0.709227i
$351$		0			0
$352$			2998.34i			0.454012i
$353$			2140.74i			0.322776i	0.986891	+	0.161388i	$0.0515971\pi$
$353$			2140.74i			0.322776i	−0.986891	+	0.161388i	$0.948403\pi$
$354$		0			0
$355$	−2626.87	−	1036.97i	−0.392731	−	0.155033i
$356$	11415.5			1.69950
$357$		0			0
$358$		−	4284.42i		−	0.632510i
$359$	1180.69			0.173578			0.0867889	−	0.996227i	$-0.472339\pi$
$359$	1180.69			0.173578			0.0867889	+	0.996227i	$0.472339\pi$
$360$		0			0
$361$	−1218.95			−0.177716
$362$		−	10855.3i		−	1.57608i
$363$		0			0
$364$	4607.59			0.663471
$365$	−3382.43	+	8568.41i	−0.485054	+	1.22874i
$366$		0			0
$367$		−	8058.99i		−	1.14626i	−0.819466	−	0.573128i	$-0.805730\pi$
$367$		−	8058.99i		−	1.14626i	0.819466	−	0.573128i	$-0.194270\pi$
$368$			8334.55i			1.18062i
$369$		0			0
$370$	3514.54	+	1387.39i	0.493817	+	0.194937i
$371$	5648.35			0.790425
$372$		0			0
$373$			9866.56i			1.36963i	0.728718	+	0.684813i	$0.240116\pi$
$373$			9866.56i			1.36963i	−0.728718	+	0.684813i	$0.759884\pi$
$374$	−4398.93			−0.608191
$375$		0			0
$376$	−85.0443			−0.0116644
$377$		−	6975.73i		−	0.952967i
$378$		0			0
$379$	4799.17			0.650440			0.325220	−	0.945638i	$-0.394562\pi$
$379$	4799.17			0.650440			0.325220	+	0.945638i	$0.394562\pi$
$380$	−6728.96	−	2656.30i	−0.908390	−	0.358593i
$381$		0			0
$382$		−	9063.93i		−	1.21401i
$383$		−	5701.76i		−	0.760695i	−0.924844	−	0.380347i	$-0.875804\pi$
$383$		−	5701.76i		−	0.760695i	0.924844	−	0.380347i	$-0.124196\pi$
$384$		0			0
$385$	−633.301	+	1604.28i	−0.0838337	+	0.212368i
$386$	19770.6			2.60699
$387$		0			0
$388$			4857.35i			0.635553i
$389$	−8912.17			−1.16161			−0.580804	−	0.814044i	$-0.697262\pi$
$389$	−8912.17			−1.16161			−0.580804	+	0.814044i	$0.697262\pi$
$390$		0			0
$391$	−13254.2			−1.71431
$392$		−	414.135i		−	0.0533596i
$393$		0			0
$394$	−1519.06			−0.194237
$395$	−2137.75	−	843.889i	−0.272308	−	0.107495i
$396$		0			0
$397$		−	9971.00i		−	1.26053i	−0.776381	−	0.630264i	$-0.782946\pi$
$397$		−	9971.00i		−	1.26053i	0.776381	−	0.630264i	$-0.217054\pi$
$398$		−	20367.4i		−	2.56514i
$399$		0			0
$400$	−5358.41	−	5011.48i	−0.669801	−	0.626435i
$401$	710.271			0.0884520			0.0442260	−	0.999022i	$-0.485918\pi$
$401$	710.271			0.0884520			0.0442260	+	0.999022i	$0.485918\pi$
$402$		0			0
$403$			9974.54i			1.23292i
$404$	−2684.62			−0.330606
$405$		0			0
$406$	9466.30			1.15715
$407$		−	958.563i		−	0.116743i
$408$		0			0
$409$	−6162.68			−0.745048			−0.372524	−	0.928023i	$-0.621508\pi$
$409$	−6162.68			−0.745048			−0.372524	+	0.928023i	$0.621508\pi$
$410$	7521.13	−	19052.6i	0.905957	−	2.29498i
$411$		0			0
$412$			702.388i			0.0839907i
$413$		−	8214.30i		−	0.978691i
$414$		0			0
$415$	2654.27	−	6723.81i	0.313959	−	0.795323i
$416$	10394.0			1.22502
$417$		0			0
$418$			3539.36i			0.414152i
$419$	−13217.7			−1.54111			−0.770556	−	0.637372i	$-0.780022\pi$
$419$	−13217.7			−1.54111			−0.770556	+	0.637372i	$0.780022\pi$
$420$		0			0
$421$	2760.80			0.319603			0.159802	−	0.987149i	$-0.448915\pi$
$421$	2760.80			0.319603			0.159802	+	0.987149i	$0.448915\pi$
$422$		−	2296.57i		−	0.264918i
$423$		0			0
$424$	−1062.72			−0.121722
$425$	7969.63	−	8521.34i	0.909609	−	0.972578i
$426$		0			0
$427$		−	6706.28i		−	0.760046i
$428$			1250.49i			0.141226i
$429$		0			0
$430$	−11863.0	−	4682.99i	−1.33043	−	0.525195i
$431$	3057.57			0.341712			0.170856	−	0.985296i	$-0.445347\pi$
$431$	3057.57			0.341712			0.170856	+	0.985296i	$0.445347\pi$
$432$		0			0
$433$			14533.5i			1.61302i	0.591221	+	0.806510i	$0.298646\pi$
$433$			14533.5i			1.61302i	−0.591221	+	0.806510i	$0.701354\pi$
$434$	−13535.8			−1.49709
$435$		0			0
$436$	2812.94			0.308980
$437$			10664.3i			1.16737i
$438$		0			0
$439$	1497.89			0.162848			0.0814240	−	0.996680i	$-0.474053\pi$
$439$	1497.89			0.162848			0.0814240	+	0.996680i	$0.474053\pi$
$440$	119.153	−	301.841i	0.0129100	−	0.0327038i
$441$		0			0
$442$			15249.3i			1.64103i
$443$			13363.3i			1.43320i	0.697484	+	0.716600i	$0.254303\pi$
$443$			13363.3i			1.43320i	−0.697484	+	0.716600i	$0.745697\pi$
$444$		0			0
$445$	13778.6	+	5439.19i	1.46779	+	0.579421i
$446$	−21907.2			−2.32587
$447$		0			0
$448$			7839.83i			0.826779i
$449$	−15023.0			−1.57902			−0.789509	−	0.613739i	$-0.789665\pi$
$449$	−15023.0			−1.57902			−0.789509	+	0.613739i	$0.789665\pi$
$450$		0			0
$451$	−5196.45			−0.542553
$452$		−	448.207i		−	0.0466414i
$453$		0			0
$454$	4531.44			0.468439
$455$	5561.38	+	2195.39i	0.573015	+	0.226201i
$456$		0			0
$457$		−	14460.7i		−	1.48018i	−0.672507	−	0.740091i	$-0.734782\pi$
$457$		−	14460.7i		−	1.48018i	0.672507	−	0.740091i	$-0.265218\pi$
$458$		−	8306.38i		−	0.847449i
$459$		0			0
$460$	5022.56	−	12723.2i	0.509083	−	1.28961i
$461$	6397.52			0.646339			0.323170	−	0.946341i	$-0.395252\pi$
$461$	6397.52			0.646339			0.323170	+	0.946341i	$0.395252\pi$
$462$		0			0
$463$			6228.94i			0.625234i	0.949879	+	0.312617i	$0.101206\pi$
$463$			6228.94i			0.625234i	−0.949879	+	0.312617i	$0.898794\pi$
$464$	10215.5			1.02207
$465$		0			0
$466$	8670.51			0.861917
$467$			1434.12i			0.142105i	0.997473	+	0.0710526i	$0.0226358\pi$
$467$			1434.12i			0.142105i	−0.997473	+	0.0710526i	$0.977364\pi$
$468$		0			0
$469$	770.605			0.0758705
$470$	−1436.04	−	566.885i	−0.140935	−	0.0556350i
$471$		0			0
$472$			1545.49i			0.150714i
$473$			3235.54i			0.314525i
$474$		0			0
$475$	−6856.23	−	6412.32i	−0.662285	−	0.619405i
$476$	−10730.4			−1.03325
$477$		0			0
$478$		−	8082.71i		−	0.773419i
$479$	−12095.5			−1.15378			−0.576888	−	0.816823i	$-0.695733\pi$
$479$	−12095.5			−1.15378			−0.576888	+	0.816823i	$0.695733\pi$
$480$		0			0
$481$	−3322.94			−0.314996
$482$		−	17200.5i		−	1.62544i
$483$		0			0
$484$	10316.0			0.968822
$485$	−2314.39	+	5862.84i	−0.216683	+	0.548903i
$486$		0			0
$487$		−	9600.57i		−	0.893313i	−0.894706	−	0.446656i	$-0.852615\pi$
$487$		−	9600.57i		−	0.893313i	0.894706	−	0.446656i	$-0.147385\pi$
$488$			1261.76i			0.117044i
$489$		0			0
$490$	2760.52	−	6992.98i	0.254505	−	0.644716i
$491$	2091.41			0.192228			0.0961141	−	0.995370i	$-0.469359\pi$
$491$	2091.41			0.192228			0.0961141	+	0.995370i	$0.469359\pi$
$492$		0			0
$493$			16245.4i			1.48409i
$494$	12269.5			1.11747
$495$		0			0
$496$	−14607.0			−1.32233
$497$			3370.41i			0.304192i
$498$		0			0
$499$	18595.9			1.66827			0.834133	−	0.551563i	$-0.185968\pi$
$499$	18595.9			1.66827			0.834133	+	0.551563i	$0.185968\pi$
$500$	5159.93	+	10879.4i	0.461518	+	0.973085i
$501$		0			0
$502$			6679.27i			0.593845i
$503$		−	5038.79i		−	0.446657i	−0.974743	−	0.223329i	$-0.928308\pi$
$503$		−	5038.79i		−	0.446657i	0.974743	−	0.223329i	$-0.0716923\pi$
$504$		0			0
$505$	−3240.35	−	1279.15i	−0.285532	−	0.112716i
$506$	−6692.26			−0.587959
$507$		0			0
$508$			2291.89i			0.200170i
$509$	6501.99			0.566200			0.283100	−	0.959090i	$-0.408637\pi$
$509$	6501.99			0.566200			0.283100	+	0.959090i	$0.408637\pi$
$510$		0			0
$511$	10993.7			0.951729
$512$			16400.1i			1.41560i
$513$		0			0
$514$	−25139.6			−2.15731
$515$	−334.668	+	847.785i	−0.0286354	+	0.0725395i
$516$		0			0
$517$			391.668i			0.0333183i
$518$		−	4509.34i		−	0.382488i
$519$		0			0
$520$	−1046.36	−	413.055i	−0.0882418	−	0.0348340i
$521$	3689.43			0.310244			0.155122	−	0.987895i	$-0.450423\pi$
$521$	3689.43			0.310244			0.155122	+	0.987895i	$0.450423\pi$
$522$		0			0
$523$		−	5040.68i		−	0.421441i	−0.977546	−	0.210720i	$-0.932419\pi$
$523$		−	5040.68i		−	0.421441i	0.977546	−	0.210720i	$-0.0675810\pi$
$524$	21904.9			1.82619
$525$		0			0
$526$	−25932.7			−2.14965
$527$		−	23229.2i		−	1.92007i
$528$		0			0
$529$	−7997.15			−0.657282
$530$	−17944.8	−	7083.82i	−1.47070	−	0.580569i
$531$		0			0
$532$			8633.61i			0.703599i
$533$			18013.9i			1.46392i
$534$		0			0
$535$	−595.823	+	1509.34i	−0.0481489	+	0.121971i
$536$	−144.987			−0.0116837
$537$		0			0
$538$			9802.75i			0.785552i
$539$	−1907.28			−0.152416
$540$		0			0
$541$	744.536			0.0591684			0.0295842	−	0.999562i	$-0.490582\pi$
$541$	744.536			0.0591684			0.0295842	+	0.999562i	$0.490582\pi$
$542$			28478.6i			2.25694i
$543$		0			0
$544$	−24206.1			−1.90777
$545$	3395.23	+	1340.29i	0.266855	+	0.105342i
$546$		0			0
$547$		−	6401.69i		−	0.500396i	−0.968195	−	0.250198i	$-0.919504\pi$
$547$		−	6401.69i		−	0.500396i	0.968195	−	0.250198i	$-0.0804957\pi$
$548$			1064.02i			0.0829428i
$549$		0			0
$550$	4023.99	−	4302.56i	0.311970	−	0.333567i
$551$	13071.0			1.01060
$552$		0			0
$553$			2742.84i			0.210918i
$554$	−6122.60			−0.469538
$555$		0			0
$556$	1853.47			0.141375
$557$			6623.77i			0.503874i	0.967744	+	0.251937i	$0.0810676\pi$
$557$			6623.77i			0.503874i	−0.967744	+	0.251937i	$0.918932\pi$
$558$		0			0
$559$	11216.3			0.848653
$560$	−3215.00	+	8144.26i	−0.242604	+	0.614567i
$561$		0			0
$562$			5882.29i			0.441512i
$563$			13355.7i			0.999778i	0.866089	+	0.499889i	$0.166626\pi$
$563$			13355.7i			0.999778i	−0.866089	+	0.499889i	$0.833374\pi$
$564$		0			0
$565$	213.558	−	540.988i	0.0159017	−	0.0402824i
$566$	19003.8			1.41129
$567$		0			0
$568$		−	634.131i		−	0.0468443i
$569$	−20926.1			−1.54177			−0.770887	−	0.636972i	$-0.780187\pi$
$569$	−20926.1			−1.54177			−0.770887	+	0.636972i	$0.780187\pi$
$570$		0			0
$571$	25369.8			1.85936			0.929679	−	0.368370i	$-0.120084\pi$
$571$	25369.8			1.85936			0.929679	+	0.368370i	$0.120084\pi$
$572$			3992.49i			0.291843i
$573$		0			0
$574$	−24445.5			−1.77759
$575$	12124.5	−	12963.8i	0.879351	−	0.940226i
$576$		0			0
$577$		−	2474.30i		−	0.178521i	−0.996008	−	0.0892605i	$-0.971550\pi$
$577$		−	2474.30i		−	0.178521i	0.996008	−	0.0892605i	$-0.0284504\pi$
$578$		−	15486.6i		−	1.11446i
$579$		0			0
$580$	−15594.6	−	6156.05i	−1.11643	−	0.440717i
$581$	−8627.01			−0.616022
$582$		0			0
$583$			4894.30i			0.347687i
$584$	−2068.43			−0.146562
$585$		0			0
$586$	4508.13			0.317797
$587$		−	15752.6i		−	1.10763i	−0.832639	−	0.553817i	$-0.813171\pi$
$587$		−	15752.6i		−	1.10763i	0.832639	−	0.553817i	$-0.186829\pi$
$588$		0			0
$589$	−18690.1			−1.30749
$590$	−10301.9	+	26096.8i	−0.718850	+	1.82100i
$591$		0			0
$592$		−	4866.22i		−	0.337838i
$593$			6162.03i			0.426719i	0.976974	+	0.213360i	$0.0684406\pi$
$593$			6162.03i			0.426719i	−0.976974	+	0.213360i	$0.931559\pi$
$594$		0			0
$595$	−12951.6	−	5112.73i	−0.892377	−	0.352271i
$596$	1911.67			0.131384
$597$		0			0
$598$			23199.3i			1.58644i
$599$	7814.07			0.533012			0.266506	−	0.963833i	$-0.414131\pi$
$599$	7814.07			0.533012			0.266506	+	0.963833i	$0.414131\pi$
$600$		0			0
$601$	−24315.4			−1.65033			−0.825164	−	0.564893i	$-0.808917\pi$
$601$	−24315.4			−1.65033			−0.825164	+	0.564893i	$0.808917\pi$
$602$			15220.8i			1.03049i
$603$		0			0
$604$	−4267.85			−0.287510
$605$	12451.5	+	4915.29i	0.836734	+	0.330306i
$606$		0			0
$607$		−	1586.65i		−	0.106096i	−0.998592	−	0.0530478i	$-0.983106\pi$
$607$		−	1586.65i		−	0.106096i	0.998592	−	0.0530478i	$-0.0168936\pi$
$608$			19476.1i			1.29911i
$609$		0			0
$610$	−8410.61	+	21305.8i	−0.558255	+	1.41418i
$611$	1357.75			0.0898997
$612$		0			0
$613$			12856.4i			0.847089i	0.905875	+	0.423545i	$0.139214\pi$
$613$			12856.4i			0.847089i	−0.905875	+	0.423545i	$0.860786\pi$
$614$	34996.3			2.30022
$615$		0			0
$616$	−387.277			−0.0253309
$617$			3854.46i			0.251499i	0.992062	+	0.125749i	$0.0401335\pi$
$617$			3854.46i			0.251499i	−0.992062	+	0.125749i	$0.959867\pi$
$618$		0			0
$619$	−5644.03			−0.366482			−0.183241	−	0.983068i	$-0.558659\pi$
$619$	−5644.03			−0.366482			−0.183241	+	0.983068i	$0.558659\pi$
$620$	22298.5	+	8802.48i	1.44440	+	0.570187i
$621$		0			0
$622$			28692.8i			1.84964i
$623$		−	17678.7i		−	1.13689i
$624$		0			0
$625$	1044.31	+	15590.1i	0.0668360	+	0.997764i
$626$	21181.9			1.35240
$627$		0			0
$628$		−	19010.1i		−	1.20794i
$629$	7738.62			0.490555
$630$		0			0
$631$	−6379.75			−0.402494			−0.201247	−	0.979541i	$-0.564499\pi$
$631$	−6379.75			−0.402494			−0.201247	+	0.979541i	$0.564499\pi$
$632$		−	516.057i		−	0.0324804i
$633$		0			0
$634$	2266.44			0.141975
$635$	−1092.02	+	2766.32i	−0.0682451	+	0.172879i
$636$		0			0
$637$			6611.75i			0.411251i
$638$			8202.57i			0.509001i
$639$		0			0
$640$	1315.24	−	3331.79i	0.0812337	−	0.205782i
$641$	−19543.9			−1.20427			−0.602135	−	0.798395i	$-0.705683\pi$
$641$	−19543.9			−1.20427			−0.602135	+	0.798395i	$0.705683\pi$
$642$		0			0
$643$		−	26175.3i		−	1.60537i	−0.596401	−	0.802686i	$-0.703403\pi$
$643$		−	26175.3i		−	1.60537i	0.596401	−	0.802686i	$-0.296597\pi$
$644$	−16324.5			−0.998877
$645$		0			0
$646$	−28573.7			−1.74028
$647$		−	20349.8i		−	1.23653i	−0.785971	−	0.618264i	$-0.787836\pi$
$647$		−	20349.8i		−	1.23653i	0.785971	−	0.618264i	$-0.212164\pi$
$648$		0			0
$649$	7117.71			0.430500
$650$	−14915.2	−	13949.5i	−0.900033	−	0.841761i
$651$		0			0
$652$		−	8022.30i		−	0.481867i
$653$		−	21661.4i		−	1.29813i	−0.760734	−	0.649064i	$-0.775161\pi$
$653$		−	21661.4i		−	1.29813i	0.760734	−	0.649064i	$-0.224839\pi$
$654$		0			0
$655$	26439.3	+	10437.1i	1.57721	+	0.622612i
$656$	−26380.2			−1.57008
$657$		0			0
$658$			1842.51i			0.109162i
$659$	9447.97			0.558484			0.279242	−	0.960221i	$-0.409917\pi$
$659$	9447.97			0.558484			0.279242	+	0.960221i	$0.409917\pi$
$660$		0			0
$661$	−14301.8			−0.841569			−0.420784	−	0.907161i	$-0.638245\pi$
$661$	−14301.8			−0.841569			−0.420784	+	0.907161i	$0.638245\pi$
$662$		−	2077.15i		−	0.121950i
$663$		0			0
$664$	1623.14			0.0948647
$665$	−4113.67	+	10420.8i	−0.239882	+	0.607671i
$666$		0			0
$667$			24714.8i			1.43472i
$668$			14857.5i			0.860558i
$669$		0			0
$670$	−2448.21	−	966.447i	−0.141168	−	0.0557270i
$671$	5811.00			0.334324
$672$		0			0
$673$		−	13078.6i		−	0.749096i	−0.927208	−	0.374548i	$-0.877798\pi$
$673$		−	13078.6i		−	0.749096i	0.927208	−	0.374548i	$-0.122202\pi$
$674$	−3193.81			−0.182524
$675$		0			0
$676$	−5088.77			−0.289529
$677$			15622.6i			0.886891i	0.896301	+	0.443446i	$0.146244\pi$
$677$			15622.6i			0.886891i	−0.896301	+	0.443446i	$0.853756\pi$
$678$		0			0
$679$	7522.33			0.425156
$680$	2436.80	+	961.942i	0.137422	+	0.0542483i
$681$		0			0
$682$		−	11728.8i		−	0.658531i
$683$		−	3176.18i		−	0.177940i	−0.996034	−	0.0889701i	$-0.971642\pi$
$683$		−	3176.18i		−	0.177940i	0.996034	−	0.0889701i	$-0.0283576\pi$
$684$		0			0
$685$	−506.975	+	1284.27i	−0.0282781	+	0.0716345i
$686$	−27627.9			−1.53766
$687$		0			0
$688$			16425.4i			0.910195i
$689$	16966.5			0.938132
$690$		0			0
$691$	30731.4			1.69186			0.845931	−	0.533293i	$-0.179046\pi$
$691$	30731.4			1.69186			0.845931	+	0.533293i	$0.179046\pi$
$692$			21126.0i			1.16054i
$693$		0			0
$694$	−2570.79			−0.140614
$695$	2237.14	+	883.126i	0.122100	+	0.0481998i
$696$		0			0
$697$		−	41951.7i		−	2.27982i
$698$		−	1580.99i		−	0.0857328i
$699$		0			0
$700$	9815.78	−	10495.3i	0.530003	−	0.566693i
$701$	−18536.2			−0.998718			−0.499359	−	0.866395i	$-0.666431\pi$
$701$	−18536.2			−0.998718			−0.499359	+	0.866395i	$0.666431\pi$
$702$		0			0
$703$		−	6226.45i		−	0.334047i
$704$	−6793.22			−0.363678
$705$		0			0
$706$	−8726.19			−0.465176
$707$			4157.54i			0.221160i
$708$		0			0
$709$	16619.3			0.880325			0.440162	−	0.897918i	$-0.354921\pi$
$709$	16619.3			0.880325			0.440162	+	0.897918i	$0.354921\pi$
$710$	4226.96	−	10707.8i	0.223430	−	0.565994i
$711$		0			0
$712$			3326.18i			0.175076i
$713$		−	35339.4i		−	1.85620i
$714$		0			0
$715$	−1902.31	+	4818.95i	−0.0994997	+	0.252054i
$716$	9055.86			0.472672
$717$		0			0
$718$			4812.80i			0.250156i
$719$	10245.8			0.531441			0.265720	−	0.964050i	$-0.414390\pi$
$719$	10245.8			0.531441			0.265720	+	0.964050i	$0.414390\pi$
$720$		0			0
$721$	1087.75			0.0561859
$722$		−	4968.77i		−	0.256120i
$723$		0			0
$724$	22944.6			1.17780
$725$	−15889.5	−	14860.8i	−0.813961	−	0.761261i
$726$		0			0
$727$		−	30936.5i		−	1.57823i	−0.614247	−	0.789114i	$-0.710540\pi$
$727$		−	30936.5i		−	1.57823i	0.614247	−	0.789114i	$-0.289460\pi$
$728$			1342.53i			0.0683481i
$729$		0			0
$730$	−34927.0	−	13787.7i	−1.77083	−	0.699047i
$731$	−26121.0			−1.32164
$732$		0			0
$733$			3336.77i			0.168140i	0.996460	+	0.0840698i	$0.0267919\pi$
$733$			3336.77i			0.168140i	−0.996460	+	0.0840698i	$0.973208\pi$
$734$	32850.5			1.65195
$735$		0			0
$736$	−36825.6			−1.84431
$737$			667.731i			0.0333734i
$738$		0			0
$739$	23494.6			1.16950			0.584752	−	0.811212i	$-0.301192\pi$
$739$	23494.6			1.16950			0.584752	+	0.811212i	$0.301192\pi$
$740$	−2932.48	+	7428.58i	−0.145676	+	0.369027i
$741$		0			0
$742$			23024.1i			1.13914i
$743$			2905.75i			0.143475i	0.997424	+	0.0717373i	$0.0228543\pi$
$743$			2905.75i			0.143475i	−0.997424	+	0.0717373i	$0.977146\pi$
$744$		0			0
$745$	2307.39	+	910.856i	0.113471	+	0.0447935i
$746$	−40218.6			−1.97387
$747$		0			0
$748$		−	9297.89i		−	0.454498i
$749$	1936.57			0.0944735
$750$		0			0
$751$	30691.2			1.49126			0.745632	−	0.666358i	$-0.232148\pi$
$751$	30691.2			1.49126			0.745632	+	0.666358i	$0.232148\pi$
$752$			1988.33i			0.0964189i
$753$		0			0
$754$	28434.9			1.37339
$755$	−5151.31	−	2033.51i	−0.248312	−	0.0980226i
$756$		0			0
$757$		−	35875.7i		−	1.72249i	−0.508192	−	0.861244i	$-0.669686\pi$
$757$		−	35875.7i		−	1.72249i	0.508192	−	0.861244i	$-0.330314\pi$
$758$			19562.6i			0.937397i
$759$		0			0
$760$	773.974	−	1960.64i	0.0369408	−	0.0935787i
$761$	−26842.1			−1.27861			−0.639307	−	0.768951i	$-0.720779\pi$
$761$	−26842.1			−1.27861			−0.639307	+	0.768951i	$0.720779\pi$
$762$		0			0
$763$		−	4356.26i		−	0.206694i
$764$	19158.2			0.907223
$765$		0			0
$766$	23241.8			1.09629
$767$		−	24674.1i		−	1.16158i
$768$		0			0
$769$	−12755.2			−0.598133			−0.299067	−	0.954232i	$-0.596675\pi$
$769$	−12755.2			−0.598133			−0.299067	+	0.954232i	$0.596675\pi$
$770$	−6539.47	−	2581.50i	−0.306060	−	0.120819i
$771$		0			0
$772$			41788.6i			1.94819i
$773$		−	9984.62i		−	0.464582i	−0.972646	−	0.232291i	$-0.925378\pi$
$773$		−	9984.62i		−	0.464582i	0.972646	−	0.232291i	$-0.0746222\pi$
$774$		0			0
$775$	22720.3	+	21249.3i	1.05308	+	0.984898i
$776$	−1415.30			−0.0654721
$777$		0			0
$778$		−	36328.3i		−	1.67408i
$779$	−33754.1			−1.55246
$780$		0			0
$781$	−2920.46			−0.133806
$782$		−	54027.6i		−	2.47062i
$783$		0			0
$784$	−9682.45			−0.441074
$785$	9057.80	−	22945.3i	0.411830	−	1.04325i
$786$		0			0
$787$			35676.2i			1.61591i	0.589245	+	0.807954i	$0.299425\pi$
$787$			35676.2i			1.61591i	−0.589245	+	0.807954i	$0.700575\pi$
$788$		−	3210.80i		−	0.145152i
$789$		0			0
$790$	3439.91	−	8714.01i	0.154920	−	0.392444i
$791$	−694.116			−0.0312009
$792$		0			0
$793$		−	20144.3i		−	0.902076i
$794$	40644.3			1.81664
$795$		0			0
$796$	43050.0			1.91692
$797$		−	41560.5i		−	1.84711i	−0.383465	−	0.923555i	$-0.625269\pi$
$797$		−	41560.5i		−	1.84711i	0.383465	−	0.923555i	$-0.374731\pi$
$798$		0			0
$799$	−3161.99			−0.140004
$800$	22142.9	−	23675.8i	0.978586	−	1.04633i
$801$		0			0
$802$			2895.25i			0.127475i
$803$			9526.08i			0.418640i
$804$		0			0
$805$	−19703.8	−	7778.19i	−0.862692	−	0.340553i
$806$	−40658.8			−1.77685
$807$		0			0
$808$		−	782.227i		−	0.0340577i
$809$	12100.2			0.525861			0.262931	−	0.964815i	$-0.415311\pi$
$809$	12100.2			0.525861			0.262931	+	0.964815i	$0.415311\pi$
$810$		0			0
$811$	30794.9			1.33336			0.666680	−	0.745344i	$-0.267715\pi$
$811$	30794.9			1.33336			0.666680	+	0.745344i	$0.267715\pi$
$812$			20008.7i			0.864736i
$813$		0			0
$814$	3907.35			0.168246
$815$	3822.40	−	9682.95i	0.164286	−	0.416170i
$816$		0			0
$817$			21016.8i			0.899981i
$818$		−	25120.7i		−	1.07374i
$819$		0			0
$820$	40270.9	+	15897.2i	1.71503	+	0.677018i
$821$	32746.1			1.39202			0.696009	−	0.718033i	$-0.254958\pi$
$821$	32746.1			1.39202			0.696009	+	0.718033i	$0.254958\pi$
$822$		0			0
$823$		−	19816.1i		−	0.839303i	−0.907685	−	0.419652i	$-0.862152\pi$
$823$		−	19816.1i		−	0.839303i	0.907685	−	0.419652i	$-0.137848\pi$
$824$	−204.657			−0.00865238
$825$		0			0
$826$	33483.6			1.41046
$827$			37093.9i			1.55971i	0.625960	+	0.779855i	$0.284707\pi$
$827$			37093.9i			1.55971i	−0.625960	+	0.779855i	$0.715293\pi$
$828$		0			0
$829$	2960.66			0.124039			0.0620193	−	0.998075i	$-0.480246\pi$
$829$	2960.66			0.124039			0.0620193	+	0.998075i	$0.480246\pi$
$830$	27408.0	+	10819.5i	1.14620	+	0.452469i
$831$		0			0
$832$			23549.3i			0.981279i
$833$		−	15397.8i		−	0.640457i
$834$		0			0
$835$	−7079.17	+	17933.0i	−0.293395	+	0.743231i
$836$	−7481.04			−0.309494
$837$		0			0
$838$		−	53878.7i		−	2.22101i
$839$	−8121.13			−0.334175			−0.167087	−	0.985942i	$-0.553436\pi$
$839$	−8121.13			−0.334175			−0.167087	+	0.985942i	$0.553436\pi$
$840$		0			0
$841$	5903.40			0.242052
$842$			11253.7i			0.460604i
$843$		0			0
$844$	4854.20			0.197972
$845$	−6142.16	−	2424.66i	−0.250055	−	0.0987109i
$846$		0			0
$847$		−	15975.9i		−	0.648097i
$848$			24846.3i			1.00616i
$849$		0			0
$850$	34735.2	+	32486.3i	1.40165	+	1.31091i
$851$	11773.1			0.474237
$852$		0			0
$853$			13277.5i			0.532957i	0.963841	+	0.266478i	$0.0858601\pi$
$853$			13277.5i			0.532957i	−0.963841	+	0.266478i	$0.914140\pi$
$854$	27336.5			1.09536
$855$		0			0
$856$	−364.359			−0.0145485
$857$		−	17722.7i		−	0.706411i	−0.935546	−	0.353206i	$-0.885092\pi$
$857$		−	17722.7i		−	0.706411i	0.935546	−	0.353206i	$-0.114908\pi$
$858$		0			0
$859$	−9026.72			−0.358542			−0.179271	−	0.983800i	$-0.557374\pi$
$859$	−9026.72			−0.358542			−0.179271	+	0.983800i	$0.557374\pi$
$860$	9898.29	−	25074.5i	0.392476	−	0.994223i
$861$		0			0
$862$			12463.4i			0.492467i
$863$		−	10197.6i		−	0.402235i	−0.979567	−	0.201117i	$-0.935543\pi$
$863$		−	10197.6i		−	0.402235i	0.979567	−	0.201117i	$-0.0644573\pi$
$864$		0			0
$865$	−10066.0	+	25499.2i	−0.395668	+	1.00231i
$866$	−59242.4			−2.32464
$867$		0			0
$868$		−	28610.2i		−	1.11877i
$869$	−2376.68			−0.0927771
$870$		0			0
$871$	2314.74			0.0900484
$872$			819.616i			0.0318299i
$873$		0			0
$874$	−43470.3			−1.68239
$875$	16848.4	−	7990.92i	0.650949	−	0.308734i
$876$		0			0
$877$		−	13326.7i		−	0.513127i	−0.966527	−	0.256563i	$-0.917410\pi$
$877$		−	13326.7i		−	0.513127i	0.966527	−	0.256563i	$-0.0825902\pi$
$878$			6105.78i			0.234692i
$879$		0			0
$880$	−7057.01	−	2785.80i	−0.270332	−	0.106715i
$881$	15016.4			0.574252			0.287126	−	0.957893i	$-0.407300\pi$
$881$	15016.4			0.574252			0.287126	+	0.957893i	$0.407300\pi$
$882$		0			0
$883$		−	6981.59i		−	0.266081i	−0.991111	−	0.133040i	$-0.957526\pi$
$883$		−	6981.59i		−	0.266081i	0.991111	−	0.133040i	$-0.0424740\pi$
$884$	−32231.9			−1.22633
$885$		0			0
$886$	−54472.1			−2.06549
$887$			25728.2i			0.973920i	0.873424	+	0.486960i	$0.161894\pi$
$887$			25728.2i			0.973920i	−0.873424	+	0.486960i	$0.838106\pi$
$888$		0			0
$889$	3549.34			0.133904
$890$	−22171.5	+	56165.1i	−0.835046	+	2.11535i
$891$		0			0
$892$		−	46304.7i		−	1.73811i
$893$			2544.12i			0.0953369i
$894$		0			0
$895$	10930.5	+	4314.86i	0.408229	+	0.161151i
$896$	−4274.86			−0.159389
$897$		0			0
$898$		−	61237.6i		−	2.27564i
$899$	−43314.8			−1.60693
$900$		0			0
$901$	−39512.4			−1.46099
$902$		−	21182.1i		−	0.781913i
$903$		0			0
$904$	130.596			0.00480481
$905$	27694.2	+	10932.4i	1.01722	+	0.401554i
$906$		0			0
$907$		−	17952.3i		−	0.657218i	−0.944466	−	0.328609i	$-0.893420\pi$
$907$		−	17952.3i		−	0.657218i	0.944466	−	0.328609i	$-0.106580\pi$
$908$			9577.98i			0.350062i
$909$		0			0
$910$	−8948.97	+	22669.6i	−0.325995	+	0.825814i
$911$	12037.2			0.437773			0.218886	−	0.975750i	$-0.429758\pi$
$911$	12037.2			0.437773			0.218886	+	0.975750i	$0.429758\pi$
$912$		0			0
$913$		−	7475.32i		−	0.270971i
$914$	58945.5			2.13320
$915$		0			0
$916$	17557.0			0.633295
$917$		−	33923.1i		−	1.22163i
$918$		0			0
$919$	14182.6			0.509075			0.254537	−	0.967063i	$-0.418077\pi$
$919$	14182.6			0.509075			0.254537	+	0.967063i	$0.418077\pi$
$920$	3707.20	+	1463.44i	0.132851	+	0.0524437i
$921$		0			0
$922$			26077.9i			0.931487i
$923$			10124.0i			0.361037i
$924$		0			0
$925$	−7079.02	+	7569.08i	−0.251629	+	0.269049i
$926$	−25390.8			−0.901071
$927$		0			0
$928$			45136.4i			1.59663i
$929$	−30537.9			−1.07849			−0.539244	−	0.842150i	$-0.681290\pi$
$929$	−30537.9			−1.07849			−0.539244	+	0.842150i	$0.681290\pi$
$930$		0			0
$931$	−12389.0			−0.436124
$932$			18326.6i			0.644107i
$933$		0			0
$934$	−5845.84			−0.204798
$935$	4430.19	−	11222.6i	0.154955	−	0.392533i
$936$		0			0
$937$			26958.0i			0.939892i	0.882695	+	0.469946i	$0.155727\pi$
$937$			26958.0i			0.939892i	−0.882695	+	0.469946i	$0.844273\pi$
$938$			3141.18i			0.109343i
$939$		0			0
$940$	1198.21	−	3035.31i	0.0415758	−	0.105320i
$941$	−16063.4			−0.556486			−0.278243	−	0.960511i	$-0.589752\pi$
$941$	−16063.4			−0.556486			−0.278243	+	0.960511i	$0.589752\pi$
$942$		0			0
$943$		−	63822.7i		−	2.20398i
$944$	36133.6			1.24581
$945$		0			0
$946$	−13188.9			−0.453285
$947$			38504.8i			1.32127i	0.750709	+	0.660633i	$0.229712\pi$
$947$			38504.8i			1.32127i	−0.750709	+	0.660633i	$0.770288\pi$
$948$		0			0
$949$	33022.9			1.12958
$950$	26138.3	−	27947.7i	0.892671	−	0.954468i
$951$		0			0
$952$		−	3126.55i		−	0.106441i
$953$			48509.6i			1.64888i	0.565952	+	0.824438i	$0.308509\pi$
$953$			48509.6i			1.64888i	−0.565952	+	0.824438i	$0.691491\pi$
$954$		0			0
$955$	23124.0	+	9128.33i	0.783533	+	0.309305i
$956$	17084.2			0.577973
$957$		0			0
$958$		−	49304.5i		−	1.66279i
$959$	1647.79			0.0554849
$960$		0			0
$961$	32144.5			1.07900
$962$		−	13545.2i		−	0.453964i
$963$		0			0
$964$	36356.2			1.21468
$965$	−19911.1	+	50439.0i	−0.664209	+	1.68258i
$966$		0			0
$967$		−	30364.0i		−	1.00976i	−0.863188	−	0.504882i	$-0.831536\pi$
$967$		−	30364.0i		−	1.00976i	0.863188	−	0.504882i	$-0.168464\pi$
$968$			3005.81i			0.0998041i
$969$		0			0
$970$	−23898.4	−	9434.06i	−0.791065	−	0.312278i
$971$	15199.6			0.502346			0.251173	−	0.967942i	$-0.419184\pi$
$971$	15199.6			0.502346			0.251173	+	0.967942i	$0.419184\pi$
$972$		0			0
$973$		−	2870.37i		−	0.0945734i
$974$	39134.4			1.28742
$975$		0			0
$976$	29500.0			0.967491
$977$			59956.5i			1.96333i	0.190605	+	0.981667i	$0.438955\pi$
$977$			59956.5i			1.96333i	−0.190605	+	0.981667i	$0.561045\pi$
$978$		0			0
$979$	15318.6			0.500086
$980$	14780.9	+	5834.84i	0.481793	+	0.190191i
$981$		0			0
$982$			8525.13i			0.277034i
$983$			10733.0i			0.348250i	0.984724	+	0.174125i	$0.0557097\pi$
$983$			10733.0i			0.348250i	−0.984724	+	0.174125i	$0.944290\pi$
$984$		0			0
$985$	1529.86	−	3875.45i	0.0494877	−	0.125363i
$986$	−66220.5			−2.13883
$987$		0			0
$988$			25933.6i			0.835080i
$989$	−39738.8			−1.27768
$990$		0			0
$991$	30040.3			0.962928			0.481464	−	0.876466i	$-0.340105\pi$
$991$	30040.3			0.962928			0.481464	+	0.876466i	$0.340105\pi$
$992$		−	64540.1i		−	2.06568i
$993$		0			0
$994$	−13738.7			−0.438394
$995$	51961.5	+	20512.1i	1.65557	+	0.653546i
$996$		0			0
$997$			59325.1i			1.88450i	0.334912	+	0.942249i	$0.391293\pi$
$997$			59325.1i			1.88450i	−0.334912	+	0.942249i	$0.608707\pi$
$998$			75801.5i			2.40426i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.b.f.244.14		16
3.2	odd	2		405.4.b.e.244.3		16
5.2	odd	4		2025.4.a.bl.1.3		16
5.3	odd	4		2025.4.a.bl.1.14		16
5.4	even	2	inner	405.4.b.f.244.3		16
9.2	odd	6		45.4.j.a.4.3	✓	32
9.4	even	3		135.4.j.a.19.3		32
9.5	odd	6		45.4.j.a.34.14	yes	32
9.7	even	3		135.4.j.a.64.14		32
15.2	even	4		2025.4.a.bk.1.14		16
15.8	even	4		2025.4.a.bk.1.3		16
15.14	odd	2		405.4.b.e.244.14		16
45.2	even	12		225.4.e.g.76.3		32
45.4	even	6		135.4.j.a.19.14		32
45.14	odd	6		45.4.j.a.34.3	yes	32
45.23	even	12		225.4.e.g.151.14		32
45.29	odd	6		45.4.j.a.4.14	yes	32
45.32	even	12		225.4.e.g.151.3		32
45.34	even	6		135.4.j.a.64.3		32
45.38	even	12		225.4.e.g.76.14		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.j.a.4.3	✓	32	9.2	odd	6
45.4.j.a.4.14	yes	32	45.29	odd	6
45.4.j.a.34.3	yes	32	45.14	odd	6
45.4.j.a.34.14	yes	32	9.5	odd	6
135.4.j.a.19.3		32	9.4	even	3
135.4.j.a.19.14		32	45.4	even	6
135.4.j.a.64.3		32	45.34	even	6
135.4.j.a.64.14		32	9.7	even	3
225.4.e.g.76.3		32	45.2	even	12
225.4.e.g.76.14		32	45.38	even	12
225.4.e.g.151.3		32	45.32	even	12
225.4.e.g.151.14		32	45.23	even	12
405.4.b.e.244.3		16	3.2	odd	2
405.4.b.e.244.14		16	15.14	odd	2
405.4.b.f.244.3		16	5.4	even	2	inner
405.4.b.f.244.14		16	1.1	even	1	trivial
2025.4.a.bk.1.3		16	15.8	even	4
2025.4.a.bk.1.14		16	15.2	even	4
2025.4.a.bl.1.3		16	5.2	odd	4
2025.4.a.bl.1.14		16	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+4.07626i q^{2} -8.61587 q^{4} +(-10.3994 - 4.10522i) q^{5} +13.3430i q^{7} -2.51043i q^{8} +O(q^{10})\)	\(q+4.07626i q^{2} -8.61587 q^{4} +(-10.3994 - 4.10522i) q^{5} +13.3430i q^{7} -2.51043i q^{8} +(16.7339 - 42.3906i) q^{10} -11.5617 q^{11} +40.0796i q^{13} -54.3893 q^{14} -58.6938 q^{16} -93.3392i q^{17} -75.1002 q^{19} +(89.5997 + 35.3700i) q^{20} -47.1284i q^{22} -142.001i q^{23} +(91.2943 + 85.3835i) q^{25} -163.375 q^{26} -114.961i q^{28} -174.047 q^{29} +248.868 q^{31} -259.334i q^{32} +380.475 q^{34} +(54.7758 - 138.758i) q^{35} +82.9086i q^{37} -306.128i q^{38} +(-10.3059 + 26.1069i) q^{40} +449.454 q^{41} -279.850i q^{43} +99.6140 q^{44} +578.831 q^{46} -33.8764i q^{47} +164.966 q^{49} +(-348.045 + 372.139i) q^{50} -345.320i q^{52} -423.321i q^{53} +(120.234 + 47.4633i) q^{55} +33.4966 q^{56} -709.461i q^{58} -615.628 q^{59} -502.608 q^{61} +1014.45i q^{62} +587.563 q^{64} +(164.535 - 416.803i) q^{65} -57.7537i q^{67} +804.198i q^{68} +(565.615 + 223.280i) q^{70} +252.598 q^{71} -823.934i q^{73} -337.957 q^{74} +647.054 q^{76} -154.267i q^{77} +205.565 q^{79} +(610.379 + 240.951i) q^{80} +1832.09i q^{82} +646.559i q^{83} +(-383.178 + 970.670i) q^{85} +1140.74 q^{86} +29.0248i q^{88} -1324.94 q^{89} -534.780 q^{91} +1223.46i q^{92} +138.089 q^{94} +(780.996 + 308.303i) q^{95} -563.768i q^{97} +672.442i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 54 q^{4} + 3 q^{5}+O(q^{10}) \) 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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