LMFDB - Embedded newform 405.4.b.f.244.16

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			244.16
Root			$5.05435i$ of defining polynomial
Character	$\chi$	$=$	405.244
Dual form			405.4.b.f.244.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.05435i q^{2} -17.5465 q^{4} +(7.77185 - 8.03731i) q^{5} +21.0117i q^{7} -48.2513i q^{8} +O(q^{10})$	\(q+5.05435i q^{2} -17.5465 q^{4} +(7.77185 - 8.03731i) q^{5} +21.0117i q^{7} -48.2513i q^{8} +(40.6234 + 39.2817i) q^{10} +28.5629 q^{11} +10.0704i q^{13} -106.201 q^{14} +103.507 q^{16} +82.7541i q^{17} -1.91981 q^{19} +(-136.369 + 141.026i) q^{20} +144.367i q^{22} +170.564i q^{23} +(-4.19662 - 124.930i) q^{25} -50.8993 q^{26} -368.682i q^{28} -256.428 q^{29} +48.0525 q^{31} +137.151i q^{32} -418.268 q^{34} +(168.878 + 163.300i) q^{35} +161.834i q^{37} -9.70338i q^{38} +(-387.810 - 375.002i) q^{40} -279.348 q^{41} -269.073i q^{43} -501.179 q^{44} -862.088 q^{46} +9.58841i q^{47} -98.4924 q^{49} +(631.438 - 21.2112i) q^{50} -176.700i q^{52} +35.7716i q^{53} +(221.987 - 229.569i) q^{55} +1013.84 q^{56} -1296.08i q^{58} -562.771 q^{59} -79.3242 q^{61} +242.874i q^{62} +134.846 q^{64} +(80.9389 + 78.2656i) q^{65} +466.752i q^{67} -1452.04i q^{68} +(-825.376 + 853.567i) q^{70} -316.854 q^{71} +633.515i q^{73} -817.968 q^{74} +33.6858 q^{76} +600.156i q^{77} +791.315 q^{79} +(804.442 - 831.918i) q^{80} -1411.92i q^{82} +228.078i q^{83} +(665.120 + 643.152i) q^{85} +1359.99 q^{86} -1378.20i q^{88} +53.9091 q^{89} -211.596 q^{91} -2992.79i q^{92} -48.4632 q^{94} +(-14.9205 + 15.4301i) q^{95} -96.6940i q^{97} -497.815i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$			5.05435i			1.78698i	0.449080	+	0.893492i	$0.351752\pi$
$2$			5.05435i			1.78698i	−0.449080	+	0.893492i	$0.648248\pi$
$3$		0			0
$3$		0			0
$4$	−17.5465			−2.19331
$5$	7.77185	−	8.03731i	0.695136	−	0.718879i
$5$	7.77185	−	8.03731i	0.695136	−	0.718879i
$6$		0			0
$7$			21.0117i			1.13453i	0.823537	+	0.567263i	$0.191998\pi$
$7$			21.0117i			1.13453i	−0.823537	+	0.567263i	$0.808002\pi$
$8$		−	48.2513i		−	2.13243i
$9$		0			0
$10$	40.6234	+	39.2817i	1.28462	+	1.24220i
$11$	28.5629			0.782914			0.391457	−	0.920196i	$-0.371971\pi$
$11$	28.5629			0.782914			0.391457	+	0.920196i	$0.371971\pi$
$12$		0			0
$13$			10.0704i			0.214848i	0.994213	+	0.107424i	$0.0342603\pi$
$13$			10.0704i			0.214848i	−0.994213	+	0.107424i	$0.965740\pi$
$14$	−106.201			−2.02738
$15$		0			0
$16$	103.507			1.61730
$17$			82.7541i			1.18064i	0.807171	+	0.590318i	$0.200998\pi$
$17$			82.7541i			1.18064i	−0.807171	+	0.590318i	$0.799002\pi$
$18$		0			0
$19$	−1.91981			−0.0231807			−0.0115904	−	0.999933i	$-0.503689\pi$
$19$	−1.91981			−0.0231807			−0.0115904	+	0.999933i	$0.503689\pi$
$20$	−136.369	+	141.026i	−1.52465	+	1.57672i
$21$		0			0
$22$			144.367i			1.39905i
$23$			170.564i			1.54630i	0.634222	+	0.773151i	$0.281321\pi$
$23$			170.564i			1.54630i	−0.634222	+	0.773151i	$0.718679\pi$
$24$		0			0
$25$	−4.19662	−	124.930i	−0.0335730	−	0.999436i
$26$	−50.8993			−0.383930
$27$		0			0
$28$		−	368.682i		−	2.48837i
$29$	−256.428			−1.64199			−0.820993	−	0.570939i	$-0.806579\pi$
$29$	−256.428			−1.64199			−0.820993	+	0.570939i	$0.806579\pi$
$30$		0			0
$31$	48.0525			0.278403			0.139201	−	0.990264i	$-0.455546\pi$
$31$	48.0525			0.278403			0.139201	+	0.990264i	$0.455546\pi$
$32$			137.151i			0.757660i
$33$		0			0
$34$	−418.268			−2.10978
$35$	168.878	+	163.300i	0.815587	+	0.788650i
$36$		0			0
$37$			161.834i			0.719065i	0.933132	+	0.359533i	$0.117064\pi$
$37$			161.834i			0.719065i	−0.933132	+	0.359533i	$0.882936\pi$
$38$		−	9.70338i		−	0.0414236i
$39$		0			0
$40$	−387.810	−	375.002i	−1.53295	−	1.48232i
$41$	−279.348			−1.06407			−0.532034	−	0.846723i	$-0.678572\pi$
$41$	−279.348			−1.06407			−0.532034	+	0.846723i	$0.678572\pi$
$42$		0			0
$43$		−	269.073i		−	0.954261i	−0.878833	−	0.477130i	$-0.841677\pi$
$43$		−	269.073i		−	0.954261i	0.878833	−	0.477130i	$-0.158323\pi$
$44$	−501.179			−1.71717
$45$		0			0
$46$	−862.088			−2.76322
$47$			9.58841i			0.0297577i	0.999889	+	0.0148789i	$0.00473626\pi$
$47$			9.58841i			0.0297577i	−0.999889	+	0.0148789i	$0.995264\pi$
$48$		0			0
$49$	−98.4924			−0.287150
$50$	631.438	−	21.2112i	1.78598	−	0.0599944i
$51$		0			0
$52$		−	176.700i		−	0.471229i
$53$			35.7716i			0.0927095i	0.998925	+	0.0463547i	$0.0147605\pi$
$53$			35.7716i			0.0927095i	−0.998925	+	0.0463547i	$0.985240\pi$
$54$		0			0
$55$	221.987	−	229.569i	0.544231	−	0.562820i
$56$	1013.84			2.41929
$57$		0			0
$58$		−	1296.08i		−	2.93420i
$59$	−562.771			−1.24181			−0.620903	−	0.783888i	$-0.713234\pi$
$59$	−562.771			−1.24181			−0.620903	+	0.783888i	$0.713234\pi$
$60$		0			0
$61$	−79.3242			−0.166499			−0.0832494	−	0.996529i	$-0.526530\pi$
$61$	−79.3242			−0.166499			−0.0832494	+	0.996529i	$0.526530\pi$
$62$			242.874i			0.497501i
$63$		0			0
$64$	134.846			0.263372
$65$	80.9389	+	78.2656i	0.154450	+	0.149349i
$66$		0			0
$67$			466.752i			0.851087i	0.904938	+	0.425544i	$0.139917\pi$
$67$			466.752i			0.851087i	−0.904938	+	0.425544i	$0.860083\pi$
$68$		−	1452.04i		−	2.58950i
$69$		0			0
$70$	−825.376	+	853.567i	−1.40930	+	1.45744i
$71$	−316.854			−0.529628			−0.264814	−	0.964299i	$-0.585311\pi$
$71$	−316.854			−0.529628			−0.264814	+	0.964299i	$0.585311\pi$
$72$		0			0
$73$			633.515i			1.01572i	0.861440	+	0.507859i	$0.169563\pi$
$73$			633.515i			1.01572i	−0.861440	+	0.507859i	$0.830437\pi$
$74$	−817.968			−1.28496
$75$		0			0
$76$	33.6858			0.0508425
$77$			600.156i			0.888236i
$78$		0			0
$79$	791.315			1.12696			0.563481	−	0.826129i	$-0.309462\pi$
$79$	791.315			1.12696			0.563481	+	0.826129i	$0.309462\pi$
$80$	804.442	−	831.918i	1.12424	−	1.16264i
$81$		0			0
$82$		−	1411.92i		−	1.90147i
$83$			228.078i			0.301624i	0.988562	+	0.150812i	$0.0481888\pi$
$83$			228.078i			0.301624i	−0.988562	+	0.150812i	$0.951811\pi$
$84$		0			0
$85$	665.120	+	643.152i	0.848734	+	0.820702i
$86$	1359.99			1.70525
$87$		0			0
$88$		−	1378.20i		−	1.66950i
$89$	53.9091			0.0642062			0.0321031	−	0.999485i	$-0.489780\pi$
$89$	53.9091			0.0642062			0.0321031	+	0.999485i	$0.489780\pi$
$90$		0			0
$91$	−211.596			−0.243751
$92$		−	2992.79i		−	3.39152i
$93$		0			0
$94$	−48.4632			−0.0531766
$95$	−14.9205	+	15.4301i	−0.0161137	+	0.0166641i
$96$		0			0
$97$		−	96.6940i		−	0.101214i	−0.998719	−	0.0506071i	$-0.983884\pi$
$97$		−	96.6940i		−	0.101214i	0.998719	−	0.0506071i	$-0.0161156\pi$
$98$		−	497.815i		−	0.513132i
$99$		0			0
$100$	73.6360	+	2192.07i	0.0736360	+	2.19207i
$101$	1083.42			1.06737			0.533686	−	0.845683i	$-0.320807\pi$
$101$	1083.42			1.06737			0.533686	+	0.845683i	$0.320807\pi$
$102$		0			0
$103$		−	430.322i		−	0.411659i	−0.978588	−	0.205830i	$-0.934011\pi$
$103$		−	430.322i		−	0.411659i	0.978588	−	0.205830i	$-0.0659892\pi$
$104$	485.909			0.458148
$105$		0			0
$106$	−180.802			−0.165670
$107$		−	12.9348i		−	0.0116865i	−0.999983	−	0.00584323i	$-0.998140\pi$
$107$		−	12.9348i		−	0.0116865i	0.999983	−	0.00584323i	$-0.00185997\pi$
$108$		0			0
$109$	20.1087			0.0176703			0.00883517	−	0.999961i	$-0.497188\pi$
$109$	20.1087			0.0176703			0.00883517	+	0.999961i	$0.497188\pi$
$110$	1160.32	+	1122.00i	1.00575	+	0.972532i
$111$		0			0
$112$			2174.86i			1.83487i
$113$			1130.64i			0.941254i	0.882332	+	0.470627i	$0.155972\pi$
$113$			1130.64i			0.941254i	−0.882332	+	0.470627i	$0.844028\pi$
$114$		0			0
$115$	1370.87	+	1325.59i	1.11160	+	1.07489i
$116$	4499.42			3.60138
$117$		0			0
$118$		−	2844.44i		−	2.21909i
$119$	−1738.81			−1.33946
$120$		0			0
$121$	−515.159			−0.387046
$122$		−	400.933i		−	0.297531i
$123$		0			0
$124$	−843.152			−0.610623
$125$	−1036.71	−	937.204i	−0.741811	−	0.670609i
$126$		0			0
$127$		−	277.149i		−	0.193646i	−0.995302	−	0.0968228i	$-0.969132\pi$
$127$		−	277.149i		−	0.193646i	0.995302	−	0.0968228i	$-0.0308680\pi$
$128$			1778.77i			1.22830i
$129$		0			0
$130$	−395.582	+	409.094i	−0.266883	+	0.275999i
$131$	502.599			0.335208			0.167604	−	0.985854i	$-0.446397\pi$
$131$	502.599			0.335208			0.167604	+	0.985854i	$0.446397\pi$
$132$		0			0
$133$		−	40.3384i		−	0.0262991i
$134$	−2359.13			−1.52088
$135$		0			0
$136$	3992.99			2.51762
$137$			2221.12i			1.38513i	0.721354	+	0.692566i	$0.243520\pi$
$137$			2221.12i			1.38513i	−0.721354	+	0.692566i	$0.756480\pi$
$138$		0			0
$139$	−2632.79			−1.60655			−0.803273	−	0.595611i	$-0.796910\pi$
$139$	−2632.79			−1.60655			−0.803273	+	0.595611i	$0.796910\pi$
$140$	−2963.21	−	2865.34i	−1.78883	−	1.72975i
$141$		0			0
$142$		−	1601.49i		−	0.946437i
$143$			287.640i			0.168208i
$144$		0			0
$145$	−1992.92	+	2060.99i	−1.14140	+	1.18039i
$146$	−3202.01			−1.81507
$147$		0			0
$148$		−	2839.62i		−	1.57713i
$149$	−2565.71			−1.41068			−0.705340	−	0.708869i	$-0.749206\pi$
$149$	−2565.71			−1.41068			−0.705340	+	0.708869i	$0.749206\pi$
$150$		0			0
$151$	−1504.49			−0.810818			−0.405409	−	0.914135i	$-0.632871\pi$
$151$	−1504.49			−0.810818			−0.405409	+	0.914135i	$0.632871\pi$
$152$			92.6331i			0.0494312i
$153$		0			0
$154$	−3033.40			−1.58726
$155$	373.457	−	386.213i	0.193528	−	0.200138i
$156$		0			0
$157$		−	1198.19i		−	0.609082i	−0.952499	−	0.304541i	$-0.901497\pi$
$157$		−	1198.19i		−	0.609082i	0.952499	−	0.304541i	$-0.0985031\pi$
$158$			3999.59i			2.01386i
$159$		0			0
$160$	1102.33	+	1065.92i	0.544666	+	0.526677i
$161$	−3583.83			−1.75432
$162$		0			0
$163$		−	2204.91i		−	1.05952i	−0.848148	−	0.529760i	$-0.822282\pi$
$163$		−	2204.91i		−	1.05952i	0.848148	−	0.529760i	$-0.177718\pi$
$164$	4901.57			2.33383
$165$		0			0
$166$	−1152.78			−0.538997
$167$			1132.11i			0.524584i	0.964989	+	0.262292i	$0.0844784\pi$
$167$			1132.11i			0.524584i	−0.964989	+	0.262292i	$0.915522\pi$
$168$		0			0
$169$	2095.59			0.953840
$170$	−3250.72	+	3361.75i	−1.46658	+	1.51667i
$171$		0			0
$172$			4721.28i			2.09299i
$173$		−	1854.00i		−	0.814778i	−0.913255	−	0.407389i	$-0.866439\pi$
$173$		−	1854.00i		−	0.814778i	0.913255	−	0.407389i	$-0.133561\pi$
$174$		0			0
$175$	2624.98	−	88.1783i	1.13389	−	0.0380894i
$176$	2956.47			1.26620
$177$		0			0
$178$			272.476i			0.114735i
$179$	3205.81			1.33862			0.669312	−	0.742981i	$-0.266589\pi$
$179$	3205.81			1.33862			0.669312	+	0.742981i	$0.266589\pi$
$180$		0			0
$181$	2278.79			0.935806			0.467903	−	0.883780i	$-0.345010\pi$
$181$	2278.79			0.935806			0.467903	+	0.883780i	$0.345010\pi$
$182$		−	1069.48i		−	0.435579i
$183$		0			0
$184$	8229.91			3.29737
$185$	1300.71	+	1257.75i	0.516921	+	0.499848i
$186$		0			0
$187$			2363.70i			0.924336i
$188$		−	168.243i		−	0.0652679i
$189$		0			0
$190$	−77.9890	−	75.4132i	−0.0297785	−	0.0287950i
$191$	−3388.65			−1.28374			−0.641870	−	0.766814i	$-0.721841\pi$
$191$	−3388.65			−1.28374			−0.641870	+	0.766814i	$0.721841\pi$
$192$		0			0
$193$			2778.22i			1.03617i	0.855329	+	0.518085i	$0.173355\pi$
$193$			2778.22i			1.03617i	−0.855329	+	0.518085i	$0.826645\pi$
$194$	488.725			0.180868
$195$		0			0
$196$	1728.19			0.629808
$197$			2941.71i			1.06390i	0.846776	+	0.531950i	$0.178541\pi$
$197$			2941.71i			1.06390i	−0.846776	+	0.531950i	$0.821459\pi$
$198$		0			0
$199$	4929.28			1.75592			0.877959	−	0.478737i	$-0.158905\pi$
$199$	4929.28			1.75592			0.877959	+	0.478737i	$0.158905\pi$
$200$	−6028.01	+	202.492i	−2.13122	+	0.0715919i
$201$		0			0
$202$			5475.99i			1.90737i
$203$		−	5388.00i		−	1.86288i
$204$		0			0
$205$	−2171.05	+	2245.20i	−0.739672	+	0.764936i
$206$	2175.00			0.735628
$207$		0			0
$208$			1042.36i			0.347474i
$209$	−54.8353			−0.0181485
$210$		0			0
$211$	5418.51			1.76789			0.883947	−	0.467587i	$-0.154877\pi$
$211$	5418.51			1.76789			0.883947	+	0.467587i	$0.154877\pi$
$212$		−	627.665i		−	0.203341i
$213$		0			0
$214$	65.3768			0.0208835
$215$	−2162.62	−	2091.19i	−0.685998	−	0.663341i
$216$		0			0
$217$			1009.67i			0.315855i
$218$			101.637i			0.0315766i
$219$		0			0
$220$	−3895.09	+	4028.13i	−1.19367	+	1.23444i
$221$	−833.366			−0.253657
$222$		0			0
$223$		−	4084.35i		−	1.22650i	−0.789891	−	0.613248i	$-0.789863\pi$
$223$		−	4084.35i		−	1.22650i	0.789891	−	0.613248i	$-0.210137\pi$
$224$	−2881.78			−0.859586
$225$		0			0
$226$	−5714.65			−1.68201
$227$			221.829i			0.0648604i	0.999474	+	0.0324302i	$0.0103247\pi$
$227$			221.829i			0.0648604i	−0.999474	+	0.0324302i	$0.989675\pi$
$228$		0			0
$229$	1598.07			0.461152			0.230576	−	0.973054i	$-0.425939\pi$
$229$	1598.07			0.461152			0.230576	+	0.973054i	$0.425939\pi$
$230$	−6700.02	+	6928.87i	−1.92081	+	1.98642i
$231$		0			0
$232$			12373.0i			3.50141i
$233$			6620.40i			1.86144i	0.365727	+	0.930722i	$0.380820\pi$
$233$			6620.40i			1.86144i	−0.365727	+	0.930722i	$0.619180\pi$
$234$		0			0
$235$	77.0650	+	74.5197i	0.0213922	+	0.0206857i
$236$	9874.65			2.72366
$237$		0			0
$238$		−	8788.53i		−	2.39360i
$239$	3557.20			0.962744			0.481372	−	0.876516i	$-0.340139\pi$
$239$	3557.20			0.962744			0.481372	+	0.876516i	$0.340139\pi$
$240$		0			0
$241$	2967.34			0.793126			0.396563	−	0.918007i	$-0.370203\pi$
$241$	2967.34			0.793126			0.396563	+	0.918007i	$0.370203\pi$
$242$		−	2603.79i		−	0.691645i
$243$		0			0
$244$	1391.86			0.365183
$245$	−765.468	+	791.613i	−0.199608	+	0.206426i
$246$		0			0
$247$		−	19.3332i		−	0.00498034i
$248$		−	2318.59i		−	0.593673i
$249$		0			0
$250$	4736.96	−	5239.91i	1.19837	−	1.32560i
$251$	903.562			0.227220			0.113610	−	0.993525i	$-0.463759\pi$
$251$	903.562			0.227220			0.113610	+	0.993525i	$0.463759\pi$
$252$		0			0
$253$			4871.79i			1.21062i
$254$	1400.81			0.346041
$255$		0			0
$256$	−7911.76			−1.93158
$257$			1484.49i			0.360312i	0.983638	+	0.180156i	$0.0576602\pi$
$257$			1484.49i			0.360312i	−0.983638	+	0.180156i	$0.942340\pi$
$258$		0			0
$259$	−3400.42			−0.815799
$260$	−1420.19	−	1373.29i	−0.338756	−	0.327568i
$261$		0			0
$262$			2540.31i			0.599011i
$263$			2716.65i			0.636943i	0.947932	+	0.318472i	$0.103170\pi$
$263$			2716.65i			0.636943i	−0.947932	+	0.318472i	$0.896830\pi$
$264$		0			0
$265$	287.507	+	278.011i	0.0666469	+	0.0644456i
$266$	203.885			0.0469961
$267$		0			0
$268$		−	8189.85i		−	1.86670i
$269$	7492.51			1.69824			0.849120	−	0.528201i	$-0.177133\pi$
$269$	7492.51			1.69824			0.849120	+	0.528201i	$0.177133\pi$
$270$		0			0
$271$	2013.02			0.451226			0.225613	−	0.974217i	$-0.427561\pi$
$271$	2013.02			0.451226			0.225613	+	0.974217i	$0.427561\pi$
$272$			8565.63i			1.90944i
$273$		0			0
$274$	−11226.3			−2.47521
$275$	−119.868	−	3568.35i	−0.0262848	−	0.782472i
$276$		0			0
$277$		−	3570.97i		−	0.774580i	−0.921958	−	0.387290i	$-0.873411\pi$
$277$		−	3570.97i		−	0.774580i	0.921958	−	0.387290i	$-0.126589\pi$
$278$		−	13307.0i		−	2.87087i
$279$		0			0
$280$	7879.43	−	8148.56i	1.68174	−	1.73918i
$281$	6761.40			1.43541			0.717707	−	0.696346i	$-0.245192\pi$
$281$	6761.40			1.43541			0.717707	+	0.696346i	$0.245192\pi$
$282$		0			0
$283$			4012.29i			0.842776i	0.906880	+	0.421388i	$0.138457\pi$
$283$			4012.29i			0.842776i	−0.906880	+	0.421388i	$0.861543\pi$
$284$	5559.66			1.16164
$285$		0			0
$286$	−1453.83			−0.300584
$287$		−	5869.58i		−	1.20721i
$288$		0			0
$289$	−1935.23			−0.393901
$290$	−10417.0	−	10072.9i	−2.10933	−	2.03967i
$291$		0			0
$292$		−	11116.0i		−	2.22778i
$293$		−	2446.05i		−	0.487712i	−0.969811	−	0.243856i	$-0.921588\pi$
$293$		−	2446.05i		−	0.487712i	0.969811	−	0.243856i	$-0.0784125\pi$
$294$		0			0
$295$	−4373.77	+	4523.16i	−0.863223	+	0.892707i
$296$	7808.72			1.53335
$297$		0			0
$298$		−	12968.0i		−	2.52086i
$299$	−1717.64			−0.332220
$300$		0			0
$301$	5653.68			1.08263
$302$		−	7604.22i		−	1.44892i
$303$		0			0
$304$	−198.714			−0.0374902
$305$	−616.496	+	637.553i	−0.115739	+	0.119692i
$306$		0			0
$307$		−	5539.06i		−	1.02974i	−0.857268	−	0.514871i	$-0.827840\pi$
$307$		−	5539.06i		−	1.02974i	0.857268	−	0.514871i	$-0.172160\pi$
$308$		−	10530.6i		−	1.94818i
$309$		0			0
$310$	1952.05	+	1887.58i	0.357643	+	0.345831i
$311$	3525.20			0.642752			0.321376	−	0.946952i	$-0.395855\pi$
$311$	3525.20			0.642752			0.321376	+	0.946952i	$0.395855\pi$
$312$		0			0
$313$			6910.59i			1.24795i	0.781443	+	0.623977i	$0.214484\pi$
$313$			6910.59i			1.24795i	−0.781443	+	0.623977i	$0.785516\pi$
$314$	6056.07			1.08842
$315$		0			0
$316$	−13884.8			−2.47177
$317$		−	4209.66i		−	0.745861i	−0.927859	−	0.372930i	$-0.878353\pi$
$317$		−	4209.66i		−	0.745861i	0.927859	−	0.372930i	$-0.121647\pi$
$318$		0			0
$319$	−7324.35			−1.28553
$320$	1048.01	−	1083.80i	0.183079	−	0.189332i
$321$		0			0
$322$		−	18114.0i		−	3.13494i
$323$		−	158.872i		−	0.0273680i
$324$		0			0
$325$	1258.09	−	42.2617i	0.214727	−	0.00721310i
$326$	11144.4			1.89334
$327$		0			0
$328$			13478.9i			2.26905i
$329$	−201.469			−0.0337609
$330$		0			0
$331$	−1696.24			−0.281673			−0.140836	−	0.990033i	$-0.544979\pi$
$331$	−1696.24			−0.281673			−0.140836	+	0.990033i	$0.544979\pi$
$332$		−	4001.96i		−	0.661554i
$333$		0			0
$334$	−5722.10			−0.937423
$335$	3751.43	+	3627.53i	0.611828	+	0.591621i
$336$		0			0
$337$		−	7205.22i		−	1.16467i	−0.812950	−	0.582334i	$-0.802140\pi$
$337$		−	7205.22i		−	1.16467i	0.812950	−	0.582334i	$-0.197860\pi$
$338$			10591.8i			1.70450i
$339$		0			0
$340$	−11670.5	−	11285.1i	−1.86154	−	1.80005i
$341$	1372.52			0.217965
$342$		0			0
$343$			5137.53i			0.808747i
$344$	−12983.1			−2.03489
$345$		0			0
$346$	9370.75			1.45600
$347$		−	6496.64i		−	1.00507i	−0.864558	−	0.502533i	$-0.832401\pi$
$347$		−	6496.64i		−	1.00507i	0.864558	−	0.502533i	$-0.167599\pi$
$348$		0			0
$349$	6464.51			0.991510			0.495755	−	0.868462i	$-0.334891\pi$
$349$	6464.51			0.991510			0.495755	+	0.868462i	$0.334891\pi$
$350$	445.684	+	13267.6i	0.0680652	+	2.02624i
$351$		0			0
$352$			3917.44i			0.593183i
$353$		−	11849.0i		−	1.78657i	−0.449493	−	0.893284i	$-0.648395\pi$
$353$		−	11849.0i		−	1.78657i	0.449493	−	0.893284i	$-0.351605\pi$
$354$		0			0
$355$	−2462.54	+	2546.65i	−0.368163	+	0.380738i
$356$	−945.915			−0.140824
$357$		0			0
$358$			16203.3i			2.39210i
$359$	−11702.9			−1.72048			−0.860242	−	0.509886i	$-0.829688\pi$
$359$	−11702.9			−1.72048			−0.860242	+	0.509886i	$0.829688\pi$
$360$		0			0
$361$	−6855.31			−0.999463
$362$			11517.8i			1.67227i
$363$		0			0
$364$	3712.77			0.534621
$365$	5091.76	+	4923.59i	0.730177	+	0.706061i
$366$		0			0
$367$			2580.51i			0.367034i	0.983017	+	0.183517i	$0.0587483\pi$
$367$			2580.51i			0.367034i	−0.983017	+	0.183517i	$0.941252\pi$
$368$			17654.5i			2.50083i
$369$		0			0
$370$	−6357.13	+	6574.26i	−0.893220	+	0.923729i
$371$	−751.622			−0.105181
$372$		0			0
$373$			4499.56i			0.624607i	0.949982	+	0.312304i	$0.101101\pi$
$373$			4499.56i			0.624607i	−0.949982	+	0.312304i	$0.898899\pi$
$374$	−11947.0			−1.65177
$375$		0			0
$376$	462.653			0.0634561
$377$		−	2582.34i		−	0.352777i
$378$		0			0
$379$	5888.24			0.798044			0.399022	−	0.916941i	$-0.369350\pi$
$379$	5888.24			0.798044			0.399022	+	0.916941i	$0.369350\pi$
$380$	261.801	−	270.743i	0.0353424	−	0.0365496i
$381$		0			0
$382$		−	17127.4i		−	2.29402i
$383$			8020.16i			1.07000i	0.844851	+	0.535001i	$0.179689\pi$
$383$			8020.16i			1.07000i	−0.844851	+	0.535001i	$0.820311\pi$
$384$		0			0
$385$	4823.64	+	4664.33i	0.638534	+	0.617444i
$386$	−14042.1			−1.85162
$387$		0			0
$388$			1696.64i			0.221994i
$389$	−167.380			−0.0218162			−0.0109081	−	0.999941i	$-0.503472\pi$
$389$	−167.380			−0.0218162			−0.0109081	+	0.999941i	$0.503472\pi$
$390$		0			0
$391$	−14114.8			−1.82562
$392$			4752.38i			0.612325i
$393$		0			0
$394$	−14868.4			−1.90117
$395$	6149.99	−	6360.04i	0.783391	−	0.810148i
$396$		0			0
$397$		−	5893.09i		−	0.745002i	−0.928032	−	0.372501i	$-0.878500\pi$
$397$		−	5893.09i		−	0.745002i	0.928032	−	0.372501i	$-0.121500\pi$
$398$			24914.3i			3.13779i
$399$		0			0
$400$	−434.380	−	12931.1i	−0.0542975	−	1.61639i
$401$	2632.86			0.327877			0.163938	−	0.986471i	$-0.447580\pi$
$401$	2632.86			0.327877			0.163938	+	0.986471i	$0.447580\pi$
$402$		0			0
$403$			483.908i			0.0598143i
$404$	−19010.2			−2.34108
$405$		0			0
$406$	27232.9			3.32893
$407$			4622.47i			0.562966i
$408$		0			0
$409$	13690.3			1.65512			0.827558	−	0.561381i	$-0.189730\pi$
$409$	13690.3			1.65512			0.827558	+	0.561381i	$0.189730\pi$
$410$	−11348.1	−	10973.3i	−1.36693	−	1.32178i
$411$		0			0
$412$			7550.64i			0.902896i
$413$		−	11824.8i		−	1.40886i
$414$		0			0
$415$	1833.13	+	1772.59i	0.216831	+	0.209669i
$416$	−1381.17			−0.162782
$417$		0			0
$418$		−	277.157i		−	0.0324311i
$419$	258.973			0.0301949			0.0150974	−	0.999886i	$-0.495194\pi$
$419$	258.973			0.0301949			0.0150974	+	0.999886i	$0.495194\pi$
$420$		0			0
$421$	13643.1			1.57939			0.789696	−	0.613499i	$-0.210238\pi$
$421$	13643.1			1.57939			0.789696	+	0.613499i	$0.210238\pi$
$422$			27387.1i			3.15920i
$423$		0			0
$424$	1726.02			0.197696
$425$	10338.4	−	347.288i	1.17997	−	0.0396375i
$426$		0			0
$427$		−	1666.74i		−	0.188897i
$428$			226.959i			0.0256320i
$429$		0			0
$430$	10569.6	−	10930.6i	1.18538	−	1.22587i
$431$	13258.5			1.48176			0.740881	−	0.671636i	$-0.234408\pi$
$431$	13258.5			1.48176			0.740881	+	0.671636i	$0.234408\pi$
$432$		0			0
$433$			14980.2i			1.66259i	0.555832	+	0.831295i	$0.312400\pi$
$433$			14980.2i			1.66259i	−0.555832	+	0.831295i	$0.687600\pi$
$434$	−5103.20			−0.564428
$435$		0			0
$436$	−352.837			−0.0387565
$437$		−	327.449i		−	0.0358444i
$438$		0			0
$439$	7362.59			0.800450			0.400225	−	0.916417i	$-0.368932\pi$
$439$	7362.59			0.800450			0.400225	+	0.916417i	$0.368932\pi$
$440$	−11077.0	−	10711.2i	−1.20017	−	1.16053i
$441$		0			0
$442$		−	4212.13i		−	0.453282i
$443$			1587.81i			0.170291i	0.996369	+	0.0851456i	$0.0271355\pi$
$443$			1587.81i			0.170291i	−0.996369	+	0.0851456i	$0.972864\pi$
$444$		0			0
$445$	418.974	−	433.284i	0.0446320	−	0.0461565i
$446$	20643.8			2.19173
$447$		0			0
$448$			2833.35i			0.298802i
$449$	−7331.63			−0.770604			−0.385302	−	0.922791i	$-0.625903\pi$
$449$	−7331.63			−0.770604			−0.385302	+	0.922791i	$0.625903\pi$
$450$		0			0
$451$	−7978.99			−0.833074
$452$		−	19838.8i		−	2.06446i
$453$		0			0
$454$	−1121.20			−0.115905
$455$	−1644.50	+	1700.67i	−0.169440	+	0.175227i
$456$		0			0
$457$		−	6296.83i		−	0.644537i	−0.946648	−	0.322268i	$-0.895555\pi$
$457$		−	6296.83i		−	0.644537i	0.946648	−	0.322268i	$-0.104445\pi$
$458$			8077.23i			0.824070i
$459$		0			0
$460$	−24054.0	−	23259.5i	−2.43809	−	2.35757i
$461$	−6335.46			−0.640068			−0.320034	−	0.947406i	$-0.603694\pi$
$461$	−6335.46			−0.640068			−0.320034	+	0.947406i	$0.603694\pi$
$462$		0			0
$463$		−	2295.72i		−	0.230435i	−0.993340	−	0.115217i	$-0.963244\pi$
$463$		−	2295.72i		−	0.230435i	0.993340	−	0.115217i	$-0.0367565\pi$
$464$	−26542.2			−2.65558
$465$		0			0
$466$	−33461.8			−3.32637
$467$		−	6182.82i		−	0.612648i	−0.951927	−	0.306324i	$-0.900901\pi$
$467$		−	6182.82i		−	0.612648i	0.951927	−	0.306324i	$-0.0990991\pi$
$468$		0			0
$469$	−9807.26			−0.965581
$470$	−376.649	+	389.514i	−0.0369649	+	0.0382275i
$471$		0			0
$472$			27154.4i			2.64806i
$473$		−	7685.51i		−	0.747104i
$474$		0			0
$475$	8.05671	+	239.841i	0.000778246	+	0.0231677i
$476$	30509.9			2.93786
$477$		0			0
$478$			17979.3i			1.72041i
$479$	3898.36			0.371860			0.185930	−	0.982563i	$-0.440470\pi$
$479$	3898.36			0.371860			0.185930	+	0.982563i	$0.440470\pi$
$480$		0			0
$481$	−1629.74			−0.154490
$482$			14998.0i			1.41730i
$483$		0			0
$484$	9039.22			0.848913
$485$	−777.159	−	751.491i	−0.0727608	−	0.0703576i
$486$		0			0
$487$		−	14776.3i		−	1.37490i	−0.726232	−	0.687450i	$-0.758730\pi$
$487$		−	14776.3i		−	1.37490i	0.726232	−	0.687450i	$-0.241270\pi$
$488$			3827.49i			0.355046i
$489$		0			0
$490$	−4001.09	−	3868.95i	−0.368879	−	0.356696i
$491$	−3637.24			−0.334310			−0.167155	−	0.985931i	$-0.553458\pi$
$491$	−3637.24			−0.334310			−0.167155	+	0.985931i	$0.553458\pi$
$492$		0			0
$493$		−	21220.5i		−	1.93859i
$494$	97.7169			0.00889978
$495$		0			0
$496$	4973.77			0.450260
$497$		−	6657.64i		−	0.600877i
$498$		0			0
$499$	3257.75			0.292258			0.146129	−	0.989266i	$-0.453319\pi$
$499$	3257.75			0.292258			0.146129	+	0.989266i	$0.453319\pi$
$500$	18190.7	+	16444.6i	1.62702	+	1.47085i
$501$		0			0
$502$			4566.92i			0.406039i
$503$			20857.4i			1.84888i	0.381327	+	0.924440i	$0.375467\pi$
$503$			20857.4i			1.84888i	−0.381327	+	0.924440i	$0.624533\pi$
$504$		0			0
$505$	8420.19	−	8707.79i	0.741968	−	0.767310i
$506$	−24623.8			−2.16336
$507$		0			0
$508$			4862.99i			0.424725i
$509$	−19907.5			−1.73356			−0.866781	−	0.498688i	$-0.833815\pi$
$509$	−19907.5			−1.73356			−0.866781	+	0.498688i	$0.833815\pi$
$510$		0			0
$511$	−13311.2			−1.15236
$512$		−	25758.7i		−	2.22340i
$513$		0			0
$514$	−7503.15			−0.643871
$515$	−3458.63	−	3344.40i	−0.295933	−	0.286159i
$516$		0			0
$517$			273.873i			0.0232977i
$518$		−	17186.9i		−	1.45782i
$519$		0			0
$520$	3776.42	−	3905.40i	0.318475	−	0.329352i
$521$	4264.69			0.358617			0.179308	−	0.983793i	$-0.442614\pi$
$521$	4264.69			0.358617			0.179308	+	0.983793i	$0.442614\pi$
$522$		0			0
$523$		−	3687.86i		−	0.308334i	−0.988045	−	0.154167i	$-0.950731\pi$
$523$		−	3687.86i		−	0.308334i	0.988045	−	0.154167i	$-0.0492694\pi$
$524$	−8818.84			−0.735215
$525$		0			0
$526$	−13730.9			−1.13821
$527$			3976.54i			0.328692i
$528$		0			0
$529$	−16924.9			−1.39105
$530$	−1405.17	+	1453.16i	−0.115163	+	0.119097i
$531$		0			0
$532$			707.798i			0.0576822i
$533$		−	2813.14i		−	0.228613i
$534$		0			0
$535$	−103.961	−	100.527i	−0.00840114	−	0.00812367i
$536$	22521.4			1.81488
$537$		0			0
$538$			37869.8i			3.03473i
$539$	−2813.23			−0.224813
$540$		0			0
$541$	21551.0			1.71266			0.856331	−	0.516428i	$-0.172739\pi$
$541$	21551.0			1.71266			0.856331	+	0.516428i	$0.172739\pi$
$542$			10174.5i			0.806334i
$543$		0			0
$544$	−11349.8			−0.894521
$545$	156.282	−	161.620i	0.0122833	−	0.0127028i
$546$		0			0
$547$		−	19907.0i		−	1.55606i	−0.628229	−	0.778029i	$-0.716220\pi$
$547$		−	19907.0i		−	1.55606i	0.628229	−	0.778029i	$-0.283780\pi$
$548$		−	38972.8i		−	3.03802i
$549$		0			0
$550$	18035.7	−	605.855i	1.39826	−	0.0469704i
$551$	492.293			0.0380624
$552$		0			0
$553$			16626.9i			1.27857i
$554$	18048.9			1.38416
$555$		0			0
$556$	46196.1			3.52365
$557$		−	1747.00i		−	0.132896i	−0.997790	−	0.0664479i	$-0.978833\pi$
$557$		−	1747.00i		−	0.132896i	0.997790	−	0.0664479i	$-0.0211666\pi$
$558$		0			0
$559$	2709.67			0.205021
$560$	17480.0	+	16902.7i	1.31905	+	1.27548i
$561$		0			0
$562$			34174.5i			2.56506i
$563$		−	2122.04i		−	0.158851i	−0.996841	−	0.0794257i	$-0.974691\pi$
$563$		−	2122.04i		−	0.158851i	0.996841	−	0.0794257i	$-0.0253086\pi$
$564$		0			0
$565$	9087.30	+	8787.17i	0.676647	+	0.654299i
$566$	−20279.5			−1.50603
$567$		0			0
$568$			15288.6i			1.12939i
$569$	10753.1			0.792254			0.396127	−	0.918196i	$-0.370354\pi$
$569$	10753.1			0.792254			0.396127	+	0.918196i	$0.370354\pi$
$570$		0			0
$571$	725.745			0.0531900			0.0265950	−	0.999646i	$-0.491534\pi$
$571$	725.745			0.0531900			0.0265950	+	0.999646i	$0.491534\pi$
$572$		−	5047.07i		−	0.368931i
$573$		0			0
$574$	29666.9			2.15727
$575$	21308.4	−	715.791i	1.54543	−	0.0519140i
$576$		0			0
$577$			23839.3i			1.72001i	0.510290	+	0.860003i	$0.329538\pi$
$577$			23839.3i			1.72001i	−0.510290	+	0.860003i	$0.670462\pi$
$578$		−	9781.36i		−	0.703894i
$579$		0			0
$580$	34968.8	−	36163.2i	2.50345	−	2.58896i
$581$	−4792.30			−0.342200
$582$		0			0
$583$			1021.74i			0.0725835i
$584$	30567.9			2.16594
$585$		0			0
$586$	12363.2			0.871534
$587$		−	7777.68i		−	0.546881i	−0.961889	−	0.273440i	$-0.911838\pi$
$587$		−	7777.68i		−	0.546881i	0.961889	−	0.273440i	$-0.0881616\pi$
$588$		0			0
$589$	−92.2515			−0.00645357
$590$	−22861.7	−	22106.6i	−1.59525	−	1.54257i
$591$		0			0
$592$			16751.0i			1.16294i
$593$			6677.79i			0.462435i	0.972902	+	0.231217i	$0.0742709\pi$
$593$			6677.79i			0.462435i	−0.972902	+	0.231217i	$0.925729\pi$
$594$		0			0
$595$	−13513.7	+	13975.3i	−0.931108	+	0.962911i
$596$	45019.2			3.09406
$597$		0			0
$598$		−	8681.57i		−	0.593672i
$599$	12946.9			0.883132			0.441566	−	0.897229i	$-0.354423\pi$
$599$	12946.9			0.883132			0.441566	+	0.897229i	$0.354423\pi$
$600$		0			0
$601$	−4366.09			−0.296334			−0.148167	−	0.988962i	$-0.547337\pi$
$601$	−4366.09			−0.296334			−0.148167	+	0.988962i	$0.547337\pi$
$602$			28575.7i			1.93465i
$603$		0			0
$604$	26398.5			1.77838
$605$	−4003.74	+	4140.49i	−0.269050	+	0.278239i
$606$		0			0
$607$		−	24177.5i		−	1.61669i	−0.588706	−	0.808347i	$-0.700362\pi$
$607$		−	24177.5i		−	1.61669i	0.588706	−	0.808347i	$-0.299638\pi$
$608$		−	263.304i		−	0.0175631i
$609$		0			0
$610$	−3222.42	−	3115.99i	−0.213888	−	0.206824i
$611$	−96.5591			−0.00639339
$612$		0			0
$613$		−	7772.63i		−	0.512127i	−0.966660	−	0.256063i	$-0.917574\pi$
$613$		−	7772.63i		−	0.512127i	0.966660	−	0.256063i	$-0.0824256\pi$
$614$	27996.3			1.84013
$615$		0			0
$616$	28958.3			1.89410
$617$		−	626.260i		−	0.0408627i	−0.999791	−	0.0204313i	$-0.993496\pi$
$617$		−	626.260i		−	0.0408627i	0.999791	−	0.0204313i	$-0.00650395\pi$
$618$		0			0
$619$	−22835.1			−1.48275			−0.741373	−	0.671093i	$-0.765825\pi$
$619$	−22835.1			−1.48275			−0.741373	+	0.671093i	$0.765825\pi$
$620$	−6552.85	+	6776.67i	−0.424466	+	0.438964i
$621$		0			0
$622$			17817.6i			1.14859i
$623$			1132.72i			0.0728436i
$624$		0			0
$625$	−15589.8	+	1048.56i	−0.997746	+	0.0671081i
$626$	−34928.6			−2.23007
$627$		0			0
$628$			21024.0i			1.33591i
$629$	−13392.5			−0.848954
$630$		0			0
$631$	−26862.1			−1.69471			−0.847355	−	0.531027i	$-0.821806\pi$
$631$	−26862.1			−1.69471			−0.847355	+	0.531027i	$0.821806\pi$
$632$		−	38182.0i		−	2.40316i
$633$		0			0
$634$	21277.1			1.33284
$635$	−2227.53	−	2153.96i	−0.139208	−	0.134610i
$636$		0			0
$637$		−	991.857i		−	0.0616936i
$638$		−	37019.8i		−	2.29723i
$639$		0			0
$640$	14296.5	+	13824.3i	0.883000	+	0.853836i
$641$	−8609.98			−0.530536			−0.265268	−	0.964175i	$-0.585460\pi$
$641$	−8609.98			−0.530536			−0.265268	+	0.964175i	$0.585460\pi$
$642$		0			0
$643$			17173.4i			1.05327i	0.850091	+	0.526636i	$0.176547\pi$
$643$			17173.4i			1.05327i	−0.850091	+	0.526636i	$0.823453\pi$
$644$	62883.6			3.84777
$645$		0			0
$646$	802.994			0.0489062
$647$		−	1258.94i		−	0.0764978i	−0.999268	−	0.0382489i	$-0.987822\pi$
$647$		−	1258.94i		−	0.0764978i	0.999268	−	0.0382489i	$-0.0121780\pi$
$648$		0			0
$649$	−16074.4			−0.972226
$650$	213.605	+	6358.83i	0.0128897	+	0.383714i
$651$		0			0
$652$			38688.4i			2.32386i
$653$		−	2540.41i		−	0.152242i	−0.997099	−	0.0761209i	$-0.975746\pi$
$653$		−	2540.41i		−	0.152242i	0.997099	−	0.0761209i	$-0.0242535\pi$
$654$		0			0
$655$	3906.12	−	4039.54i	0.233015	−	0.240974i
$656$	−28914.5			−1.72092
$657$		0			0
$658$		−	1018.29i		−	0.0603302i
$659$	−26199.8			−1.54871			−0.774356	−	0.632750i	$-0.781926\pi$
$659$	−26199.8			−1.54871			−0.774356	+	0.632750i	$0.781926\pi$
$660$		0			0
$661$	−5443.77			−0.320330			−0.160165	−	0.987090i	$-0.551203\pi$
$661$	−5443.77			−0.320330			−0.160165	+	0.987090i	$0.551203\pi$
$662$		−	8573.39i		−	0.503345i
$663$		0			0
$664$	11005.0			0.643190
$665$	−324.212	−	313.504i	−0.0189059	−	0.0182815i
$666$		0			0
$667$		−	43737.3i		−	2.53901i
$668$		−	19864.6i		−	1.15058i
$669$		0			0
$670$	−18334.8	+	18961.0i	−1.05722	+	1.09333i
$671$	−2265.73			−0.130354
$672$		0			0
$673$			16820.8i			0.963440i	0.876325	+	0.481720i	$0.159988\pi$
$673$			16820.8i			0.963440i	−0.876325	+	0.481720i	$0.840012\pi$
$674$	36417.7			2.08124
$675$		0			0
$676$	−36770.2			−2.09207
$677$			23199.5i			1.31703i	0.752568	+	0.658515i	$0.228815\pi$
$677$			23199.5i			1.31703i	−0.752568	+	0.658515i	$0.771185\pi$
$678$		0			0
$679$	2031.71			0.114830
$680$	31032.9	−	32092.9i	1.75009	−	1.80986i
$681$		0			0
$682$			6937.20i			0.389500i
$683$			24902.4i			1.39512i	0.716528	+	0.697558i	$0.245730\pi$
$683$			24902.4i			1.39512i	−0.716528	+	0.697558i	$0.754270\pi$
$684$		0			0
$685$	17851.8	+	17262.2i	0.995742	+	0.962855i
$686$	−25966.9			−1.44522
$687$		0			0
$688$		−	27850.9i		−	1.54332i
$689$	−360.234			−0.0199185
$690$		0			0
$691$	8467.45			0.466161			0.233080	−	0.972457i	$-0.425119\pi$
$691$	8467.45			0.466161			0.233080	+	0.972457i	$0.425119\pi$
$692$			32531.1i			1.78706i
$693$		0			0
$694$	32836.3			1.79604
$695$	−20461.6	+	21160.5i	−1.11677	+	1.15491i
$696$		0			0
$697$		−	23117.2i		−	1.25628i
$698$			32673.9i			1.77181i
$699$		0			0
$700$	−46059.2	+	1547.22i	−2.48696	+	0.0835419i
$701$	−5972.30			−0.321784			−0.160892	−	0.986972i	$-0.551437\pi$
$701$	−5972.30			−0.321784			−0.160892	+	0.986972i	$0.551437\pi$
$702$		0			0
$703$		−	310.691i		−	0.0166685i
$704$	3851.61			0.206197
$705$		0			0
$706$	59889.0			3.19257
$707$			22764.6i			1.21096i
$708$		0			0
$709$	7549.78			0.399913			0.199956	−	0.979805i	$-0.435920\pi$
$709$	7549.78			0.399913			0.199956	+	0.979805i	$0.435920\pi$
$710$	−12871.7	−	12446.5i	−0.680373	−	0.657902i
$711$		0			0
$712$		−	2601.18i		−	0.136915i
$713$			8196.00i			0.430495i
$714$		0			0
$715$	2311.85	+	2235.50i	0.120921	+	0.116927i
$716$	−56250.7			−2.93602
$717$		0			0
$718$		−	59150.4i		−	3.07448i
$719$	−31449.7			−1.63126			−0.815631	−	0.578573i	$-0.803610\pi$
$719$	−31449.7			−1.63126			−0.815631	+	0.578573i	$0.803610\pi$
$720$		0			0
$721$	9041.81			0.467038
$722$		−	34649.2i		−	1.78602i
$723$		0			0
$724$	−39984.7			−2.05251
$725$	1076.13	+	32035.5i	0.0551264	+	1.64106i
$726$		0			0
$727$		−	7962.03i		−	0.406183i	−0.979160	−	0.203092i	$-0.934901\pi$
$727$		−	7962.03i		−	0.406183i	0.979160	−	0.203092i	$-0.0650990\pi$
$728$			10209.8i			0.519780i
$729$		0			0
$730$	−24885.5	+	25735.5i	−1.26172	+	1.30481i
$731$	22266.9			1.12663
$732$		0			0
$733$		−	11003.2i		−	0.554451i	−0.960805	−	0.277226i	$-0.910585\pi$
$733$		−	11003.2i		−	0.554451i	0.960805	−	0.277226i	$-0.0894150\pi$
$734$	−13042.8			−0.655884
$735$		0			0
$736$	−23393.0			−1.17157
$737$			13331.8i			0.666328i
$738$		0			0
$739$	−24719.1			−1.23046			−0.615228	−	0.788349i	$-0.710936\pi$
$739$	−24719.1			−1.23046			−0.615228	+	0.788349i	$0.710936\pi$
$740$	−22822.9	−	22069.1i	−1.13377	−	1.09632i
$741$		0			0
$742$		−	3798.96i		−	0.187957i
$743$			13326.8i			0.658024i	0.944326	+	0.329012i	$0.106716\pi$
$743$			13326.8i			0.658024i	−0.944326	+	0.329012i	$0.893284\pi$
$744$		0			0
$745$	−19940.3	+	20621.4i	−0.980614	+	1.01411i
$746$	−22742.4			−1.11616
$747$		0			0
$748$		−	41474.6i		−	2.02735i
$749$	271.781			0.0132586
$750$		0			0
$751$	−23800.9			−1.15647			−0.578234	−	0.815871i	$-0.696258\pi$
$751$	−23800.9			−1.15647			−0.578234	+	0.815871i	$0.696258\pi$
$752$			992.468i			0.0481271i
$753$		0			0
$754$	13052.0			0.630407
$755$	−11692.7	+	12092.0i	−0.563629	+	0.582880i
$756$		0			0
$757$		−	20867.5i		−	1.00191i	−0.865474	−	0.500953i	$-0.832983\pi$
$757$		−	20867.5i		−	1.00191i	0.865474	−	0.500953i	$-0.167017\pi$
$758$			29761.2i			1.42609i
$759$		0			0
$760$	744.521	+	719.931i	0.0355350	+	0.0343614i
$761$	15487.5			0.737742			0.368871	−	0.929481i	$-0.379744\pi$
$761$	15487.5			0.737742			0.368871	+	0.929481i	$0.379744\pi$
$762$		0			0
$763$			422.519i			0.0200475i
$764$	59458.9			2.81564
$765$		0			0
$766$	−40536.7			−1.91208
$767$		−	5667.33i		−	0.266800i
$768$		0			0
$769$	18468.0			0.866024			0.433012	−	0.901388i	$-0.357451\pi$
$769$	18468.0			0.866024			0.433012	+	0.901388i	$0.357451\pi$
$770$	−23575.2	+	24380.4i	−1.10336	+	1.14105i
$771$		0			0
$772$		−	48748.0i		−	2.27264i
$773$		−	15379.9i		−	0.715621i	−0.933794	−	0.357810i	$-0.883523\pi$
$773$		−	15379.9i		−	0.715621i	0.933794	−	0.357810i	$-0.116477\pi$
$774$		0			0
$775$	−201.658	−	6003.17i	−0.00934681	−	0.278246i
$776$	−4665.61			−0.215832
$777$		0			0
$778$		−	845.997i		−	0.0389852i
$779$	536.294			0.0246659
$780$		0			0
$781$	−9050.27			−0.414653
$782$		−	71341.3i		−	3.26235i
$783$		0			0
$784$	−10194.7			−0.464407
$785$	−9630.21	−	9312.15i	−0.437856	−	0.423395i
$786$		0			0
$787$		−	38251.8i		−	1.73257i	−0.499552	−	0.866284i	$-0.666502\pi$
$787$		−	38251.8i		−	1.73257i	0.499552	−	0.866284i	$-0.333498\pi$
$788$		−	51616.7i		−	2.33346i
$789$		0			0
$790$	32145.9	+	31084.2i	1.44772	+	1.39991i
$791$	−23756.7			−1.06788
$792$		0			0
$793$		−	798.826i		−	0.0357720i
$794$	29785.8			1.33131
$795$		0			0
$796$	−86491.5			−3.85127
$797$		−	38981.7i		−	1.73250i	−0.499612	−	0.866249i	$-0.666524\pi$
$797$		−	38981.7i		−	1.73250i	0.499612	−	0.866249i	$-0.333476\pi$
$798$		0			0
$799$	−793.480			−0.0351330
$800$	17134.2	−	575.572i	0.757233	−	0.0254369i
$801$		0			0
$802$			13307.4i			0.585911i
$803$			18095.1i			0.795219i
$804$		0			0
$805$	−27853.0	+	28804.4i	−1.21949	+	1.26114i
$806$	−2445.84			−0.106887
$807$		0			0
$808$		−	52276.5i		−	2.27609i
$809$	−4490.50			−0.195152			−0.0975758	−	0.995228i	$-0.531109\pi$
$809$	−4490.50			−0.195152			−0.0975758	+	0.995228i	$0.531109\pi$
$810$		0			0
$811$	33791.9			1.46312			0.731562	−	0.681775i	$-0.238792\pi$
$811$	33791.9			1.46312			0.731562	+	0.681775i	$0.238792\pi$
$812$			94540.5i			4.08586i
$813$		0			0
$814$	−23363.6			−1.00601
$815$	−17721.5	−	17136.2i	−0.761666	−	0.736510i
$816$		0			0
$817$			516.568i			0.0221205i
$818$			69195.7i			2.95766i
$819$		0			0
$820$	38094.3	−	39395.4i	1.62233	−	1.67774i
$821$	−6765.56			−0.287600			−0.143800	−	0.989607i	$-0.545932\pi$
$821$	−6765.56			−0.287600			−0.143800	+	0.989607i	$0.545932\pi$
$822$		0			0
$823$		−	39347.9i		−	1.66656i	−0.552848	−	0.833282i	$-0.686459\pi$
$823$		−	39347.9i		−	1.66656i	0.552848	−	0.833282i	$-0.313541\pi$
$824$	−20763.6			−0.877832
$825$		0			0
$826$	59766.6			2.51761
$827$		−	632.071i		−	0.0265771i	−0.999912	−	0.0132885i	$-0.995770\pi$
$827$		−	632.071i		−	0.0265771i	0.999912	−	0.0132885i	$-0.00423000\pi$
$828$		0			0
$829$	19585.8			0.820560			0.410280	−	0.911959i	$-0.365431\pi$
$829$	19585.8			0.820560			0.410280	+	0.911959i	$0.365431\pi$
$830$	−8959.27	+	9265.28i	−0.374676	+	0.387473i
$831$		0			0
$832$			1357.96i			0.0565849i
$833$		−	8150.64i		−	0.339019i
$834$		0			0
$835$	9099.14	+	8798.62i	0.377112	+	0.364657i
$836$	962.167			0.0398053
$837$		0			0
$838$			1308.94i			0.0539577i
$839$	11672.6			0.480315			0.240157	−	0.970734i	$-0.422801\pi$
$839$	11672.6			0.480315			0.240157	+	0.970734i	$0.422801\pi$
$840$		0			0
$841$	41366.5			1.69611
$842$			68957.0i			2.82235i
$843$		0			0
$844$	−95075.8			−3.87754
$845$	16286.6	−	16842.9i	0.663048	−	0.685695i
$846$		0			0
$847$		−	10824.4i		−	0.439114i
$848$			3702.61i			0.149939i
$849$		0			0
$850$	1755.31	+	52254.1i	0.0708315	+	2.10859i
$851$	−27603.1			−1.11189
$852$		0			0
$853$			22100.9i			0.887129i	0.896242	+	0.443565i	$0.146286\pi$
$853$			22100.9i			0.887129i	−0.896242	+	0.443565i	$0.853714\pi$
$854$	8424.28			0.337556
$855$		0			0
$856$	−624.118			−0.0249205
$857$			29897.9i			1.19171i	0.803093	+	0.595853i	$0.203186\pi$
$857$			29897.9i			1.19171i	−0.803093	+	0.595853i	$0.796814\pi$
$858$		0			0
$859$	−14083.5			−0.559397			−0.279698	−	0.960088i	$-0.590234\pi$
$859$	−14083.5			−0.559397			−0.279698	+	0.960088i	$0.590234\pi$
$860$	37946.4	+	36693.1i	1.50461	+	1.45491i
$861$		0			0
$862$			67013.1i			2.64788i
$863$		−	11367.8i		−	0.448396i	−0.974544	−	0.224198i	$-0.928024\pi$
$863$		−	11367.8i		−	0.448396i	0.974544	−	0.224198i	$-0.0719762\pi$
$864$		0			0
$865$	−14901.1	−	14409.0i	−0.585727	−	0.566381i
$866$	−75715.1			−2.97102
$867$		0			0
$868$		−	17716.1i		−	0.692768i
$869$	22602.3			0.882313
$870$		0			0
$871$	−4700.38			−0.182854
$872$		−	970.272i		−	0.0376807i
$873$		0			0
$874$	1655.04			0.0640534
$875$	19692.3	−	21783.1i	0.760823	−	0.841604i
$876$		0			0
$877$			44820.8i			1.72576i	0.505411	+	0.862879i	$0.331341\pi$
$877$			44820.8i			1.72576i	−0.505411	+	0.862879i	$0.668659\pi$
$878$			37213.1i			1.43039i
$879$		0			0
$880$	22977.2	−	23762.0i	0.880184	−	0.910248i
$881$	−11986.7			−0.458391			−0.229196	−	0.973380i	$-0.573610\pi$
$881$	−11986.7			−0.458391			−0.229196	+	0.973380i	$0.573610\pi$
$882$		0			0
$883$			14223.4i			0.542078i	0.962568	+	0.271039i	$0.0873672\pi$
$883$			14223.4i			0.542078i	−0.962568	+	0.271039i	$0.912633\pi$
$884$	14622.6			0.556349
$885$		0			0
$886$	−8025.33			−0.304307
$887$			23140.0i			0.875947i	0.898988	+	0.437973i	$0.144304\pi$
$887$			23140.0i			0.875947i	−0.898988	+	0.437973i	$0.855696\pi$
$888$		0			0
$889$	5823.38			0.219696
$890$	2189.97	+	2117.64i	0.0824809	+	0.0797567i
$891$		0			0
$892$			71666.0i			2.69008i
$893$		−	18.4079i		−	0.000689806i
$894$		0			0
$895$	24915.1	−	25766.1i	0.930526	−	0.962309i
$896$	−37375.0			−1.39354
$897$		0			0
$898$		−	37056.7i		−	1.37706i
$899$	−12322.0			−0.457133
$900$		0			0
$901$	−2960.24			−0.109456
$902$		−	40328.6i		−	1.48869i
$903$		0			0
$904$	54554.8			2.00715
$905$	17710.4	−	18315.3i	0.650512	−	0.672731i
$906$		0			0
$907$		−	24525.8i		−	0.897867i	−0.893565	−	0.448934i	$-0.851804\pi$
$907$		−	24525.8i		−	0.897867i	0.893565	−	0.448934i	$-0.148196\pi$
$908$		−	3892.32i		−	0.142259i
$909$		0			0
$910$	−8595.76	−	8311.86i	−0.313128	−	0.302786i
$911$	8453.86			0.307452			0.153726	−	0.988114i	$-0.450873\pi$
$911$	8453.86			0.307452			0.153726	+	0.988114i	$0.450873\pi$
$912$		0			0
$913$			6514.57i			0.236145i
$914$	31826.4			1.15178
$915$		0			0
$916$	−28040.6			−1.01145
$917$			10560.5i			0.380302i
$918$		0			0
$919$	−29199.7			−1.04811			−0.524053	−	0.851686i	$-0.675581\pi$
$919$	−29199.7			−1.04811			−0.524053	+	0.851686i	$0.675581\pi$
$920$	63961.6	−	66146.3i	2.29212	−	2.37041i
$921$		0			0
$922$		−	32021.6i		−	1.14379i
$923$		−	3190.84i		−	0.113790i
$924$		0			0
$925$	20217.9	−	679.158i	0.718660	−	0.0241412i
$926$	11603.4			0.411783
$927$		0			0
$928$		−	35169.5i		−	1.24407i
$929$	−50606.8			−1.78725			−0.893624	−	0.448816i	$-0.851846\pi$
$929$	−50606.8			−1.78725			−0.893624	+	0.448816i	$0.851846\pi$
$930$		0			0
$931$	189.086			0.00665634
$932$		−	116165.i		−	4.08272i
$933$		0			0
$934$	31250.1			1.09479
$935$	18997.8	+	18370.3i	0.664485	+	0.642539i
$936$		0			0
$937$			30351.9i			1.05822i	0.848553	+	0.529110i	$0.177474\pi$
$937$			30351.9i			1.05822i	−0.848553	+	0.529110i	$0.822526\pi$
$938$		−	49569.4i		−	1.72548i
$939$		0			0
$940$	−1352.22	−	1307.56i	−0.0469197	−	0.0453700i
$941$	22500.5			0.779486			0.389743	−	0.920924i	$-0.372564\pi$
$941$	22500.5			0.779486			0.389743	+	0.920924i	$0.372564\pi$
$942$		0			0
$943$		−	47646.5i		−	1.64537i
$944$	−58250.8			−2.00837
$945$		0			0
$946$	38845.3			1.33506
$947$		−	36629.5i		−	1.25691i	−0.777844	−	0.628457i	$-0.783687\pi$
$947$		−	36629.5i		−	1.25691i	0.777844	−	0.628457i	$-0.216313\pi$
$948$		0			0
$949$	−6379.75			−0.218225
$950$	−1212.24	+	40.7214i	−0.0414002	+	0.00139071i
$951$		0			0
$952$			83899.6i			2.85630i
$953$		−	29632.2i		−	1.00722i	−0.863931	−	0.503611i	$-0.832005\pi$
$953$		−	29632.2i		−	1.00722i	0.863931	−	0.503611i	$-0.167995\pi$
$954$		0			0
$955$	−26336.1	+	27235.6i	−0.892373	+	0.922853i
$956$	−62416.3			−2.11160
$957$		0			0
$958$			19703.7i			0.664507i
$959$	−46669.6			−1.57147
$960$		0			0
$961$	−27482.0			−0.922492
$962$		−	8237.27i		−	0.276071i
$963$		0			0
$964$	−52066.4			−1.73957
$965$	22329.4	+	21591.9i	0.744880	+	0.720278i
$966$		0			0
$967$			49714.8i			1.65328i	0.562733	+	0.826639i	$0.309750\pi$
$967$			49714.8i			1.65328i	−0.562733	+	0.826639i	$0.690250\pi$
$968$			24857.1i			0.825347i
$969$		0			0
$970$	3798.30	−	3928.04i	0.125728	−	0.130022i
$971$	22201.1			0.733745			0.366872	−	0.930271i	$-0.380429\pi$
$971$	22201.1			0.733745			0.366872	+	0.930271i	$0.380429\pi$
$972$		0			0
$973$		−	55319.4i		−	1.82267i
$974$	74684.4			2.45692
$975$		0			0
$976$	−8210.62			−0.269278
$977$		−	7907.35i		−	0.258934i	−0.991584	−	0.129467i	$-0.958673\pi$
$977$		−	7907.35i		−	0.258934i	0.991584	−	0.129467i	$-0.0413267\pi$
$978$		0			0
$979$	1539.80			0.0502679
$980$	13431.3	−	13890.0i	0.437802	−	0.452756i
$981$		0			0
$982$		−	18383.9i		−	0.597406i
$983$		−	8184.84i		−	0.265571i	−0.991145	−	0.132785i	$-0.957608\pi$
$983$		−	8184.84i		−	0.265571i	0.991145	−	0.132785i	$-0.0423921\pi$
$984$		0			0
$985$	23643.4	+	22862.5i	0.764815	+	0.739555i
$986$	107256.			3.46422
$987$		0			0
$988$			339.230i			0.0109234i
$989$	45894.0			1.47558
$990$		0			0
$991$	−4964.85			−0.159146			−0.0795729	−	0.996829i	$-0.525356\pi$
$991$	−4964.85			−0.159146			−0.0795729	+	0.996829i	$0.525356\pi$
$992$			6590.46i			0.210935i
$993$		0			0
$994$	33650.0			1.07376
$995$	38309.6	−	39618.2i	1.22060	−	1.26229i
$996$		0			0
$997$			7098.29i			0.225481i	0.993624	+	0.112741i	$0.0359630\pi$
$997$			7098.29i			0.225481i	−0.993624	+	0.112741i	$0.964037\pi$
$998$			16465.8i			0.522261i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.b.f.244.16		16
3.2	odd	2		405.4.b.e.244.1		16
5.2	odd	4		2025.4.a.bl.1.1		16
5.3	odd	4		2025.4.a.bl.1.16		16
5.4	even	2	inner	405.4.b.f.244.1		16
9.2	odd	6		45.4.j.a.4.1	✓	32
9.4	even	3		135.4.j.a.19.1		32
9.5	odd	6		45.4.j.a.34.16	yes	32
9.7	even	3		135.4.j.a.64.16		32
15.2	even	4		2025.4.a.bk.1.16		16
15.8	even	4		2025.4.a.bk.1.1		16
15.14	odd	2		405.4.b.e.244.16		16
45.2	even	12		225.4.e.g.76.1		32
45.4	even	6		135.4.j.a.19.16		32
45.14	odd	6		45.4.j.a.34.1	yes	32
45.23	even	12		225.4.e.g.151.16		32
45.29	odd	6		45.4.j.a.4.16	yes	32
45.32	even	12		225.4.e.g.151.1		32
45.34	even	6		135.4.j.a.64.1		32
45.38	even	12		225.4.e.g.76.16		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.j.a.4.1	✓	32	9.2	odd	6
45.4.j.a.4.16	yes	32	45.29	odd	6
45.4.j.a.34.1	yes	32	45.14	odd	6
45.4.j.a.34.16	yes	32	9.5	odd	6
135.4.j.a.19.1		32	9.4	even	3
135.4.j.a.19.16		32	45.4	even	6
135.4.j.a.64.1		32	45.34	even	6
135.4.j.a.64.16		32	9.7	even	3
225.4.e.g.76.1		32	45.2	even	12
225.4.e.g.76.16		32	45.38	even	12
225.4.e.g.151.1		32	45.32	even	12
225.4.e.g.151.16		32	45.23	even	12
405.4.b.e.244.1		16	3.2	odd	2
405.4.b.e.244.16		16	15.14	odd	2
405.4.b.f.244.1		16	5.4	even	2	inner
405.4.b.f.244.16		16	1.1	even	1	trivial
2025.4.a.bk.1.1		16	15.8	even	4
2025.4.a.bk.1.16		16	15.2	even	4
2025.4.a.bl.1.1		16	5.2	odd	4
2025.4.a.bl.1.16		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)

\(f(q)\)	\(=\)	\(q+5.05435i q^{2} -17.5465 q^{4} +(7.77185 - 8.03731i) q^{5} +21.0117i q^{7} -48.2513i q^{8} +O(q^{10})\)	\(q+5.05435i q^{2} -17.5465 q^{4} +(7.77185 - 8.03731i) q^{5} +21.0117i q^{7} -48.2513i q^{8} +(40.6234 + 39.2817i) q^{10} +28.5629 q^{11} +10.0704i q^{13} -106.201 q^{14} +103.507 q^{16} +82.7541i q^{17} -1.91981 q^{19} +(-136.369 + 141.026i) q^{20} +144.367i q^{22} +170.564i q^{23} +(-4.19662 - 124.930i) q^{25} -50.8993 q^{26} -368.682i q^{28} -256.428 q^{29} +48.0525 q^{31} +137.151i q^{32} -418.268 q^{34} +(168.878 + 163.300i) q^{35} +161.834i q^{37} -9.70338i q^{38} +(-387.810 - 375.002i) q^{40} -279.348 q^{41} -269.073i q^{43} -501.179 q^{44} -862.088 q^{46} +9.58841i q^{47} -98.4924 q^{49} +(631.438 - 21.2112i) q^{50} -176.700i q^{52} +35.7716i q^{53} +(221.987 - 229.569i) q^{55} +1013.84 q^{56} -1296.08i q^{58} -562.771 q^{59} -79.3242 q^{61} +242.874i q^{62} +134.846 q^{64} +(80.9389 + 78.2656i) q^{65} +466.752i q^{67} -1452.04i q^{68} +(-825.376 + 853.567i) q^{70} -316.854 q^{71} +633.515i q^{73} -817.968 q^{74} +33.6858 q^{76} +600.156i q^{77} +791.315 q^{79} +(804.442 - 831.918i) q^{80} -1411.92i q^{82} +228.078i q^{83} +(665.120 + 643.152i) q^{85} +1359.99 q^{86} -1378.20i q^{88} +53.9091 q^{89} -211.596 q^{91} -2992.79i q^{92} -48.4632 q^{94} +(-14.9205 + 15.4301i) q^{95} -96.6940i q^{97} -497.815i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 54 q^{4} + 3 q^{5}+O(q^{10}) \) 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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