LMFDB - Embedded newform 45.4.b.b.19.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,4,Mod(19,45)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, 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("45.19");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	45.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:	$2.65508595026$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{41})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 21x^{2} + 100 $ x^4 + 21*x^2 + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			19.1
Root			$-3.70156i$ of defining polynomial
Character	$\chi$	$=$	45.19
Dual form			45.4.b.b.19.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.70156i q^{2} -14.1047 q^{4} +(-11.1047 - 1.29844i) q^{5} -16.2094i q^{7} +28.7016i q^{8} +O(q^{10})$	\(q-4.70156i q^{2} -14.1047 q^{4} +(-11.1047 - 1.29844i) q^{5} -16.2094i q^{7} +28.7016i q^{8} +(-6.10469 + 52.2094i) q^{10} +40.2094 q^{11} -19.7906i q^{13} -76.2094 q^{14} +22.1047 q^{16} -83.0156i q^{17} +48.8375 q^{19} +(156.628 + 18.3141i) q^{20} -189.047i q^{22} +1.61250i q^{23} +(121.628 + 28.8375i) q^{25} -93.0469 q^{26} +228.628i q^{28} -24.5344 q^{29} -12.4187 q^{31} +125.686i q^{32} -390.303 q^{34} +(-21.0469 + 180.000i) q^{35} +325.884i q^{37} -229.612i q^{38} +(37.2672 - 318.722i) q^{40} +242.419 q^{41} -367.350i q^{43} -567.141 q^{44} +7.58125 q^{46} +204.544i q^{47} +80.2562 q^{49} +(135.581 - 571.842i) q^{50} +279.141i q^{52} -61.5281i q^{53} +(-446.512 - 52.2094i) q^{55} +465.234 q^{56} +115.350i q^{58} -112.209 q^{59} +477.350 q^{61} +58.3875i q^{62} +767.758 q^{64} +(-25.6969 + 219.769i) q^{65} +558.094i q^{67} +1170.91i q^{68} +(846.281 + 98.9531i) q^{70} -558.281 q^{71} -1011.77i q^{73} +1532.17 q^{74} -688.837 q^{76} -651.769i q^{77} -1150.47 q^{79} +(-245.466 - 28.7016i) q^{80} -1139.75i q^{82} -1157.92i q^{83} +(-107.791 + 921.862i) q^{85} -1727.12 q^{86} +1154.07i q^{88} +96.9751 q^{89} -320.794 q^{91} -22.7438i q^{92} +961.675 q^{94} +(-542.325 - 63.4124i) q^{95} -1152.37i q^{97} -377.330i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 18 q^{4} - 6 q^{5}+O(q^{10}) $ 4 * q - 18 * q^4 - 6 * q^5	$ 4 q - 18 q^{4} - 6 q^{5} + 14 q^{10} + 84 q^{11} - 228 q^{14} + 50 q^{16} - 112 q^{19} + 396 q^{20} + 256 q^{25} + 12 q^{26} - 636 q^{29} + 104 q^{31} - 716 q^{34} + 300 q^{35} + 418 q^{40} + 816 q^{41} - 1116 q^{44} + 184 q^{46} - 140 q^{49} + 696 q^{50} - 864 q^{55} - 60 q^{56} - 372 q^{59} + 680 q^{61} + 958 q^{64} - 948 q^{65} + 1080 q^{70} + 72 q^{71} + 3132 q^{74} - 2448 q^{76} - 760 q^{79} - 444 q^{80} - 508 q^{85} - 4296 q^{86} + 2232 q^{89} + 1944 q^{91} + 3232 q^{94} - 2784 q^{95}+O(q^{100}) $ 4 * q - 18 * q^4 - 6 * q^5 + 14 * q^10 + 84 * q^11 - 228 * q^14 + 50 * q^16 - 112 * q^19 + 396 * q^20 + 256 * q^25 + 12 * q^26 - 636 * q^29 + 104 * q^31 - 716 * q^34 + 300 * q^35 + 418 * q^40 + 816 * q^41 - 1116 * q^44 + 184 * q^46 - 140 * q^49 + 696 * q^50 - 864 * q^55 - 60 * q^56 - 372 * q^59 + 680 * q^61 + 958 * q^64 - 948 * q^65 + 1080 * q^70 + 72 * q^71 + 3132 * q^74 - 2448 * q^76 - 760 * q^79 - 444 * q^80 - 508 * q^85 - 4296 * q^86 + 2232 * q^89 + 1944 * q^91 + 3232 * q^94 - 2784 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$1$	$-1$

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 $

$2$		−	4.70156i		−	1.66225i	−0.556083	−	0.831127i	$-0.687696\pi$
$2$		−	4.70156i		−	1.66225i	0.556083	−	0.831127i	$-0.312304\pi$
$3$		0			0
$3$		0			0
$4$	−14.1047			−1.76309
$5$	−11.1047	−	1.29844i	−0.993233	−	0.116136i
$5$	−11.1047	−	1.29844i	−0.993233	−	0.116136i
$6$		0			0
$7$		−	16.2094i		−	0.875224i	−0.899164	−	0.437612i	$-0.855824\pi$
$7$		−	16.2094i		−	0.875224i	0.899164	−	0.437612i	$-0.144176\pi$
$8$			28.7016i			1.26844i
$9$		0			0
$10$	−6.10469	+	52.2094i	−0.193047	+	1.65101i
$11$	40.2094			1.10214			0.551072	−	0.834458i	$-0.314219\pi$
$11$	40.2094			1.10214			0.551072	+	0.834458i	$0.314219\pi$
$12$		0			0
$13$		−	19.7906i		−	0.422226i	−0.977462	−	0.211113i	$-0.932291\pi$
$13$		−	19.7906i		−	0.422226i	0.977462	−	0.211113i	$-0.0677087\pi$
$14$	−76.2094			−1.45484
$15$		0			0
$16$	22.1047			0.345386
$17$		−	83.0156i		−	1.18437i	−0.805803	−	0.592184i	$-0.798266\pi$
$17$		−	83.0156i		−	1.18437i	0.805803	−	0.592184i	$-0.201734\pi$
$18$		0			0
$19$	48.8375			0.589689			0.294844	−	0.955545i	$-0.404732\pi$
$19$	48.8375			0.589689			0.294844	+	0.955545i	$0.404732\pi$
$20$	156.628	+	18.3141i	1.75116	+	0.204757i
$21$		0			0
$22$		−	189.047i		−	1.83204i
$23$			1.61250i			0.0146186i	0.999973	+	0.00730932i	$0.00232665\pi$
$23$			1.61250i			0.0146186i	−0.999973	+	0.00730932i	$0.997673\pi$
$24$		0			0
$25$	121.628	+	28.8375i	0.973025	+	0.230700i
$26$	−93.0469			−0.701846
$27$		0			0
$28$			228.628i			1.54309i
$29$	−24.5344			−0.157101			−0.0785504	−	0.996910i	$-0.525029\pi$
$29$	−24.5344			−0.157101			−0.0785504	+	0.996910i	$0.525029\pi$
$30$		0			0
$31$	−12.4187			−0.0719507			−0.0359754	−	0.999353i	$-0.511454\pi$
$31$	−12.4187			−0.0719507			−0.0359754	+	0.999353i	$0.511454\pi$
$32$			125.686i			0.694323i
$33$		0			0
$34$	−390.303			−1.96872
$35$	−21.0469	+	180.000i	−0.101645	+	0.869302i
$36$		0			0
$37$			325.884i			1.44797i	0.689813	+	0.723987i	$0.257693\pi$
$37$			325.884i			1.44797i	−0.689813	+	0.723987i	$0.742307\pi$
$38$		−	229.612i		−	0.980212i
$39$		0			0
$40$	37.2672	−	318.722i	0.147312	−	1.25986i
$41$	242.419			0.923401			0.461701	−	0.887036i	$-0.347239\pi$
$41$	242.419			0.923401			0.461701	+	0.887036i	$0.347239\pi$
$42$		0			0
$43$		−	367.350i		−	1.30280i	−0.758735	−	0.651399i	$-0.774182\pi$
$43$		−	367.350i		−	1.30280i	0.758735	−	0.651399i	$-0.225818\pi$
$44$	−567.141			−1.94317
$45$		0			0
$46$	7.58125			0.0242999
$47$			204.544i			0.634804i	0.948291	+	0.317402i	$0.102810\pi$
$47$			204.544i			0.634804i	−0.948291	+	0.317402i	$0.897190\pi$
$48$		0			0
$49$	80.2562			0.233983
$50$	135.581	−	571.842i	0.383482	−	1.61741i
$51$		0			0
$52$			279.141i			0.744420i
$53$		−	61.5281i		−	0.159463i	−0.996816	−	0.0797314i	$-0.974594\pi$
$53$		−	61.5281i		−	0.159463i	0.996816	−	0.0797314i	$-0.0254063\pi$
$54$		0			0
$55$	−446.512	−	52.2094i	−1.09469	−	0.127998i
$56$	465.234			1.11017
$57$		0			0
$58$			115.350i			0.261141i
$59$	−112.209			−0.247600			−0.123800	−	0.992307i	$-0.539508\pi$
$59$	−112.209			−0.247600			−0.123800	+	0.992307i	$0.539508\pi$
$60$		0			0
$61$	477.350			1.00194			0.500970	−	0.865464i	$-0.332977\pi$
$61$	477.350			1.00194			0.500970	+	0.865464i	$0.332977\pi$
$62$			58.3875i			0.119600i
$63$		0			0
$64$	767.758			1.49953
$65$	−25.6969	+	219.769i	−0.0490355	+	0.419369i
$66$		0			0
$67$			558.094i			1.01764i	0.860872	+	0.508821i	$0.169918\pi$
$67$			558.094i			1.01764i	−0.860872	+	0.508821i	$0.830082\pi$
$68$			1170.91i			2.08814i
$69$		0			0
$70$	846.281	+	98.9531i	1.44500	+	0.168959i
$71$	−558.281			−0.933180			−0.466590	−	0.884474i	$-0.654518\pi$
$71$	−558.281			−0.933180			−0.466590	+	0.884474i	$0.654518\pi$
$72$		0			0
$73$		−	1011.77i		−	1.62217i	−0.584927	−	0.811086i	$-0.698877\pi$
$73$		−	1011.77i		−	1.62217i	0.584927	−	0.811086i	$-0.301123\pi$
$74$	1532.17			2.40690
$75$		0			0
$76$	−688.837			−1.03967
$77$		−	651.769i		−	0.964623i
$78$		0			0
$79$	−1150.47			−1.63845			−0.819227	−	0.573470i	$-0.805597\pi$
$79$	−1150.47			−1.63845			−0.819227	+	0.573470i	$0.805597\pi$
$80$	−245.466	−	28.7016i	−0.343049	−	0.0401117i
$81$		0			0
$82$		−	1139.75i		−	1.53493i
$83$		−	1157.92i		−	1.53131i	−0.643251	−	0.765655i	$-0.722415\pi$
$83$		−	1157.92i		−	1.53131i	0.643251	−	0.765655i	$-0.277585\pi$
$84$		0			0
$85$	−107.791	+	921.862i	−0.137547	+	1.17635i
$86$	−1727.12			−2.16558
$87$		0			0
$88$			1154.07i			1.39801i
$89$	96.9751			0.115498			0.0577491	−	0.998331i	$-0.481608\pi$
$89$	96.9751			0.115498			0.0577491	+	0.998331i	$0.481608\pi$
$90$		0			0
$91$	−320.794			−0.369542
$92$		−	22.7438i		−	0.0257739i
$93$		0			0
$94$	961.675			1.05520
$95$	−542.325	−	63.4124i	−0.585699	−	0.0684840i
$96$		0			0
$97$		−	1152.37i		−	1.20625i	−0.797648	−	0.603123i	$-0.793923\pi$
$97$		−	1152.37i		−	1.20625i	0.797648	−	0.603123i	$-0.206077\pi$
$98$		−	377.330i		−	0.388939i
$99$		0			0
$100$	−1715.53	−	406.744i	−1.71553	−	0.406744i
$101$	1156.49			1.13936			0.569679	−	0.821867i	$-0.307068\pi$
$101$	1156.49			1.13936			0.569679	+	0.821867i	$0.307068\pi$
$102$		0			0
$103$			1333.70i			1.27585i	0.770096	+	0.637927i	$0.220208\pi$
$103$			1333.70i			1.27585i	−0.770096	+	0.637927i	$0.779792\pi$
$104$	568.022			0.535569
$105$		0			0
$106$	−289.278			−0.265068
$107$			798.263i			0.721224i	0.932716	+	0.360612i	$0.117432\pi$
$107$			798.263i			0.721224i	−0.932716	+	0.360612i	$0.882568\pi$
$108$		0			0
$109$	985.119			0.865663			0.432831	−	0.901475i	$-0.357515\pi$
$109$	985.119			0.865663			0.432831	+	0.901475i	$0.357515\pi$
$110$	−245.466	+	2099.31i	−0.212766	+	1.81965i
$111$		0			0
$112$		−	358.303i		−	0.302290i
$113$			1888.25i			1.57196i	0.618253	+	0.785979i	$0.287841\pi$
$113$			1888.25i			1.57196i	−0.618253	+	0.785979i	$0.712159\pi$
$114$		0			0
$115$	2.09373	−	17.9063i	0.00169775	−	0.0145197i
$116$	346.050			0.276982
$117$		0			0
$118$			527.559i			0.411574i
$119$	−1345.63			−1.03659
$120$		0			0
$121$	285.794			0.214721
$122$		−	2244.29i		−	1.66548i
$123$		0			0
$124$	175.163			0.126855
$125$	−1313.20	−	478.158i	−0.939648	−	0.342142i
$126$		0			0
$127$		−	620.859i		−	0.433798i	−0.976194	−	0.216899i	$-0.930406\pi$
$127$		−	620.859i		−	0.433798i	0.976194	−	0.216899i	$-0.0695943\pi$
$128$		−	2604.17i		−	1.79827i
$129$		0			0
$130$	1033.26	+	120.816i	0.697097	+	0.0815094i
$131$	2588.35			1.72630			0.863151	−	0.504947i	$-0.168488\pi$
$131$	2588.35			1.72630			0.863151	+	0.504947i	$0.168488\pi$
$132$		0			0
$133$		−	791.625i		−	0.516110i
$134$	2623.91			1.69158
$135$		0			0
$136$	2382.68			1.50230
$137$			1656.29i			1.03289i	0.856319	+	0.516447i	$0.172746\pi$
$137$			1656.29i			1.03289i	−0.856319	+	0.516447i	$0.827254\pi$
$138$		0			0
$139$	153.256			0.0935182			0.0467591	−	0.998906i	$-0.485111\pi$
$139$	153.256			0.0935182			0.0467591	+	0.998906i	$0.485111\pi$
$140$	296.859	−	2538.84i	0.179209	−	1.53265i
$141$		0			0
$142$			2624.79i			1.55118i
$143$		−	795.769i		−	0.465353i
$144$		0			0
$145$	272.447	+	31.8564i	0.156038	+	0.0182450i
$146$	−4756.89			−2.69646
$147$		0			0
$148$		−	4596.50i		−	2.55290i
$149$	1483.38			0.815591			0.407795	−	0.913073i	$-0.366298\pi$
$149$	1483.38			0.815591			0.407795	+	0.913073i	$0.366298\pi$
$150$		0			0
$151$	−394.281			−0.212491			−0.106246	−	0.994340i	$-0.533883\pi$
$151$	−394.281			−0.212491			−0.106246	+	0.994340i	$0.533883\pi$
$152$			1401.71i			0.747986i
$153$		0			0
$154$	−3064.33			−1.60345
$155$	137.906	+	16.1250i	0.0714639	+	0.00835606i
$156$		0			0
$157$			1727.05i			0.877922i	0.898506	+	0.438961i	$0.144653\pi$
$157$			1727.05i			0.877922i	−0.898506	+	0.438961i	$0.855347\pi$
$158$			5409.00i			2.72352i
$159$		0			0
$160$	163.195	−	1395.70i	0.0806358	−	0.689625i
$161$	26.1376			0.0127946
$162$		0			0
$163$			2034.28i			0.977529i	0.872416	+	0.488764i	$0.162552\pi$
$163$			2034.28i			0.977529i	−0.872416	+	0.488764i	$0.837448\pi$
$164$	−3419.24			−1.62804
$165$		0			0
$166$	−5444.06			−2.54543
$167$		−	192.900i		−	0.0893835i	−0.999001	−	0.0446918i	$-0.985769\pi$
$167$		−	192.900i		−	0.0893835i	0.999001	−	0.0446918i	$-0.0142306\pi$
$168$		0			0
$169$	1805.33			0.821726
$170$	4334.19	+	506.784i	1.95540	+	0.228639i
$171$		0			0
$172$			5181.36i			2.29695i
$173$		−	1239.91i		−	0.544905i	−0.962169	−	0.272452i	$-0.912165\pi$
$173$		−	1239.91i		−	0.544905i	0.962169	−	0.272452i	$-0.0878347\pi$
$174$		0			0
$175$	467.438	−	1971.52i	0.201914	−	0.851615i
$176$	888.816			0.380665
$177$		0			0
$178$		−	455.934i		−	0.191987i
$179$	−2636.86			−1.10105			−0.550525	−	0.834818i	$-0.685573\pi$
$179$	−2636.86			−1.10105			−0.550525	+	0.834818i	$0.685573\pi$
$180$		0			0
$181$	3317.58			1.36240			0.681199	−	0.732099i	$-0.261459\pi$
$181$	3317.58			1.36240			0.681199	+	0.732099i	$0.261459\pi$
$182$			1508.23i			0.614272i
$183$		0			0
$184$	−46.2812			−0.0185429
$185$	423.141	−	3618.84i	0.168162	−	1.43818i
$186$		0			0
$187$		−	3338.01i		−	1.30534i
$188$		−	2885.02i		−	1.11921i
$189$		0			0
$190$	−298.138	+	2549.77i	−0.113838	+	0.973579i
$191$	624.506			0.236585			0.118292	−	0.992979i	$-0.462258\pi$
$191$	624.506			0.236585			0.118292	+	0.992979i	$0.462258\pi$
$192$		0			0
$193$			436.144i			0.162665i	0.996687	+	0.0813324i	$0.0259175\pi$
$193$			436.144i			0.162665i	−0.996687	+	0.0813324i	$0.974082\pi$
$194$	−5417.96			−2.00509
$195$		0			0
$196$	−1131.99			−0.412532
$197$		−	3355.81i		−	1.21366i	−0.794831	−	0.606831i	$-0.792440\pi$
$197$		−	3355.81i		−	1.21366i	0.794831	−	0.606831i	$-0.207560\pi$
$198$		0			0
$199$	−3799.77			−1.35356			−0.676780	−	0.736185i	$-0.736625\pi$
$199$	−3799.77			−1.35356			−0.676780	+	0.736185i	$0.736625\pi$
$200$	−827.681	+	3490.92i	−0.292629	+	1.23423i
$201$		0			0
$202$		−	5437.31i		−	1.89390i
$203$			397.687i			0.137498i
$204$		0			0
$205$	−2691.98	−	314.766i	−0.917153	−	0.107240i
$206$	6270.46			2.12079
$207$		0			0
$208$		−	437.466i		−	0.145831i
$209$	1963.72			0.649922
$210$		0			0
$211$	2365.27			0.771715			0.385857	−	0.922558i	$-0.373906\pi$
$211$	2365.27			0.771715			0.385857	+	0.922558i	$0.373906\pi$
$212$			867.834i			0.281147i
$213$		0			0
$214$	3753.08			1.19886
$215$	−476.981	+	4079.31i	−0.151302	+	1.29398i
$216$		0			0
$217$			201.300i			0.0629730i
$218$		−	4631.60i		−	1.43895i
$219$		0			0
$220$	6297.92	+	736.397i	1.93003	+	0.225672i
$221$	−1642.93			−0.500070
$222$		0			0
$223$			3328.58i			0.999545i	0.866157	+	0.499772i	$0.166583\pi$
$223$			3328.58i			0.999545i	−0.866157	+	0.499772i	$0.833417\pi$
$224$	2037.29			0.607688
$225$		0			0
$226$	8877.71			2.61299
$227$			527.100i			0.154118i	0.997027	+	0.0770592i	$0.0245530\pi$
$227$			527.100i			0.154118i	−0.997027	+	0.0770592i	$0.975447\pi$
$228$		0			0
$229$	−2566.06			−0.740479			−0.370240	−	0.928936i	$-0.620724\pi$
$229$	−2566.06			−0.740479			−0.370240	+	0.928936i	$0.620724\pi$
$230$	−84.1875	−	9.84379i	−0.0241355	−	0.00282209i
$231$		0			0
$232$		−	704.175i		−	0.199273i
$233$			5534.99i			1.55626i	0.628101	+	0.778132i	$0.283832\pi$
$233$			5534.99i			1.55626i	−0.628101	+	0.778132i	$0.716168\pi$
$234$		0			0
$235$	265.587	−	2271.39i	0.0737234	−	0.630508i
$236$	1582.68			0.436541
$237$		0			0
$238$			6326.57i			1.72307i
$239$	1010.01			0.273355			0.136678	−	0.990616i	$-0.456358\pi$
$239$	1010.01			0.273355			0.136678	+	0.990616i	$0.456358\pi$
$240$		0			0
$241$	−4074.29			−1.08900			−0.544498	−	0.838762i	$-0.683280\pi$
$241$	−4074.29			−1.08900			−0.544498	+	0.838762i	$0.683280\pi$
$242$		−	1343.68i		−	0.356921i
$243$		0			0
$244$	−6732.87			−1.76651
$245$	−891.220	−	104.208i	−0.232400	−	0.0271738i
$246$		0			0
$247$		−	966.525i		−	0.248982i
$248$		−	356.437i		−	0.0912653i
$249$		0			0
$250$	−2248.09	+	6174.08i	−0.568726	+	1.56193i
$251$	−1773.98			−0.446107			−0.223054	−	0.974806i	$-0.571602\pi$
$251$	−1773.98			−0.446107			−0.223054	+	0.974806i	$0.571602\pi$
$252$		0			0
$253$			64.8375i			0.0161119i
$254$	−2919.01			−0.721082
$255$		0			0
$256$	−6101.62			−1.48965
$257$		−	662.784i		−	0.160869i	−0.996760	−	0.0804345i	$-0.974369\pi$
$257$		−	662.784i		−	0.160869i	0.996760	−	0.0804345i	$-0.0256308\pi$
$258$		0			0
$259$	5282.38			1.26730
$260$	362.447	−	3099.77i	0.0864538	−	0.739383i
$261$		0			0
$262$		−	12169.3i		−	2.86955i
$263$			712.312i			0.167008i	0.996507	+	0.0835039i	$0.0266111\pi$
$263$			712.312i			0.167008i	−0.996507	+	0.0835039i	$0.973389\pi$
$264$		0			0
$265$	−79.8904	+	683.250i	−0.0185194	+	0.158384i
$266$	−3721.87			−0.857905
$267$		0			0
$268$		−	7871.74i		−	1.79419i
$269$	−3136.41			−0.710894			−0.355447	−	0.934696i	$-0.615671\pi$
$269$	−3136.41			−0.710894			−0.355447	+	0.934696i	$0.615671\pi$
$270$		0			0
$271$	−2275.69			−0.510105			−0.255053	−	0.966927i	$-0.582093\pi$
$271$	−2275.69			−0.510105			−0.255053	+	0.966927i	$0.582093\pi$
$272$		−	1835.03i		−	0.409064i
$273$		0			0
$274$	7787.15			1.71693
$275$	4890.59	+	1159.54i	1.07241	+	0.254264i
$276$		0			0
$277$			5171.00i			1.12164i	0.827937	+	0.560821i	$0.189515\pi$
$277$			5171.00i			1.12164i	−0.827937	+	0.560821i	$0.810485\pi$
$278$		−	720.544i		−	0.155451i
$279$		0			0
$280$	−5166.28	−	604.078i	−1.10266	−	0.128931i
$281$	−2240.14			−0.475571			−0.237785	−	0.971318i	$-0.576422\pi$
$281$	−2240.14			−0.475571			−0.237785	+	0.971318i	$0.576422\pi$
$282$		0			0
$283$		−	225.244i		−	0.0473123i	−0.999720	−	0.0236561i	$-0.992469\pi$
$283$		−	225.244i		−	0.0473123i	0.999720	−	0.0236561i	$-0.00753068\pi$
$284$	7874.38			1.64528
$285$		0			0
$286$	−3741.36			−0.773535
$287$		−	3929.46i		−	0.808183i
$288$		0			0
$289$	−1978.59			−0.402726
$290$	149.775	−	1280.93i	0.0303279	−	0.259374i
$291$		0			0
$292$			14270.7i			2.86003i
$293$			1139.86i			0.227274i	0.993522	+	0.113637i	$0.0362501\pi$
$293$			1139.86i			0.227274i	−0.993522	+	0.113637i	$0.963750\pi$
$294$		0			0
$295$	1246.05	+	145.697i	0.245925	+	0.0287553i
$296$	−9353.39			−1.83667
$297$		0			0
$298$		−	6974.19i		−	1.35572i
$299$	31.9123			0.00617237
$300$		0			0
$301$	−5954.51			−1.14024
$302$			1853.74i			0.353214i
$303$		0			0
$304$	1079.54			0.203670
$305$	−5300.82	−	619.809i	−0.995161	−	0.116361i
$306$		0			0
$307$			5244.86i			0.975049i	0.873109	+	0.487525i	$0.162100\pi$
$307$			5244.86i			0.975049i	−0.873109	+	0.487525i	$0.837900\pi$
$308$			9192.99i			1.70071i
$309$		0			0
$310$	75.8125	−	648.375i	0.0138899	−	0.118791i
$311$	5188.26			0.945977			0.472989	−	0.881068i	$-0.343175\pi$
$311$	5188.26			0.945977			0.472989	+	0.881068i	$0.343175\pi$
$312$		0			0
$313$		−	486.656i		−	0.0878832i	−0.999034	−	0.0439416i	$-0.986008\pi$
$313$		−	486.656i		−	0.0878832i	0.999034	−	0.0439416i	$-0.0139915\pi$
$314$	8119.85			1.45933
$315$		0			0
$316$	16227.0			2.88873
$317$			4218.87i			0.747493i	0.927531	+	0.373747i	$0.121927\pi$
$317$			4218.87i			0.747493i	−0.927531	+	0.373747i	$0.878073\pi$
$318$		0			0
$319$	−986.512			−0.173148
$320$	−8525.71	−	996.886i	−1.48938	−	0.174149i
$321$		0			0
$322$		−	122.887i		−	0.0212678i
$323$		−	4054.27i		−	0.698408i
$324$		0			0
$325$	570.712	−	2407.10i	0.0974074	−	0.410836i
$326$	9564.30			1.62490
$327$		0			0
$328$			6957.80i			1.17128i
$329$	3315.53			0.555595
$330$		0			0
$331$	7439.94			1.23546			0.617728	−	0.786392i	$-0.288053\pi$
$331$	7439.94			1.23546			0.617728	+	0.786392i	$0.288053\pi$
$332$			16332.2i			2.69983i
$333$		0			0
$334$	−906.931			−0.148578
$335$	724.650	−	6197.46i	0.118185	−	1.01076i
$336$		0			0
$337$		−	6555.39i		−	1.05963i	−0.848113	−	0.529815i	$-0.822261\pi$
$337$		−	6555.39i		−	1.05963i	0.848113	−	0.529815i	$-0.177739\pi$
$338$		−	8487.88i		−	1.36592i
$339$		0			0
$340$	1520.35	−	13002.6i	0.242508	−	2.07401i
$341$	−499.350			−0.0793000
$342$		0			0
$343$		−	6860.72i		−	1.08001i
$344$	10543.5			1.65252
$345$		0			0
$346$	−5829.51			−0.905770
$347$		−	1950.56i		−	0.301763i	−0.988552	−	0.150881i	$-0.951789\pi$
$347$		−	1950.56i		−	0.301763i	0.988552	−	0.150881i	$-0.0482112\pi$
$348$		0			0
$349$	1426.74			0.218830			0.109415	−	0.993996i	$-0.465102\pi$
$349$	1426.74			0.218830			0.109415	+	0.993996i	$0.465102\pi$
$350$	−9269.20	−	2197.69i	−1.41560	−	0.335632i
$351$		0			0
$352$			5053.75i			0.765244i
$353$		−	7078.96i		−	1.06735i	−0.845689	−	0.533676i	$-0.820810\pi$
$353$		−	7078.96i		−	1.06735i	0.845689	−	0.533676i	$-0.179190\pi$
$354$		0			0
$355$	6199.54	+	724.893i	0.926866	+	0.108376i
$356$	−1367.80			−0.203633
$357$		0			0
$358$			12397.4i			1.83023i
$359$	−5409.79			−0.795314			−0.397657	−	0.917534i	$-0.630177\pi$
$359$	−5409.79			−0.795314			−0.397657	+	0.917534i	$0.630177\pi$
$360$		0			0
$361$	−4473.90			−0.652267
$362$		−	15597.8i		−	2.26465i
$363$		0			0
$364$	4524.69			0.651534
$365$	−1313.72	+	11235.4i	−0.188392	+	1.61120i
$366$		0			0
$367$			4940.09i			0.702645i	0.936255	+	0.351322i	$0.114268\pi$
$367$			4940.09i			0.702645i	−0.936255	+	0.351322i	$0.885732\pi$
$368$			35.6437i			0.00504907i
$369$		0			0
$370$	−17014.2	−	1989.42i	−2.39061	−	0.279527i
$371$	−997.332			−0.139566
$372$		0			0
$373$		−	12891.9i		−	1.78959i	−0.446473	−	0.894797i	$-0.647320\pi$
$373$		−	12891.9i		−	1.78959i	0.446473	−	0.894797i	$-0.352680\pi$
$374$	−15693.8			−2.16981
$375$		0			0
$376$	−5870.72			−0.805211
$377$			485.551i			0.0663320i
$378$		0			0
$379$	9475.15			1.28418			0.642092	−	0.766627i	$-0.278067\pi$
$379$	9475.15			1.28418			0.642092	+	0.766627i	$0.278067\pi$
$380$	7649.32	+	894.413i	1.03264	+	0.120743i
$381$		0			0
$382$		−	2936.16i		−	0.393264i
$383$		−	5800.97i		−	0.773931i	−0.922094	−	0.386966i	$-0.873523\pi$
$383$		−	5800.97i		−	0.773931i	0.922094	−	0.386966i	$-0.126477\pi$
$384$		0			0
$385$	−846.281	+	7237.69i	−0.112027	+	0.958095i
$386$	2050.56			0.270390
$387$		0			0
$388$			16253.9i			2.12672i
$389$	13779.7			1.79603			0.898016	−	0.439962i	$-0.145008\pi$
$389$	13779.7			1.79603			0.898016	+	0.439962i	$0.145008\pi$
$390$		0			0
$391$	133.862			0.0173138
$392$			2303.48i			0.296794i
$393$		0			0
$394$	−15777.5			−2.01741
$395$	12775.6	+	1493.81i	1.62737	+	0.190283i
$396$		0			0
$397$		−	2816.46i		−	0.356056i	−0.984025	−	0.178028i	$-0.943028\pi$
$397$		−	2816.46i		−	0.356056i	0.984025	−	0.178028i	$-0.0569718\pi$
$398$			17864.8i			2.24996i
$399$		0			0
$400$	2688.55	+	637.444i	0.336069	+	0.0796805i
$401$	−11986.4			−1.49270			−0.746352	−	0.665551i	$-0.768196\pi$
$401$	−11986.4			−1.49270			−0.746352	+	0.665551i	$0.768196\pi$
$402$		0			0
$403$			245.775i			0.0303794i
$404$	−16311.9			−2.00879
$405$		0			0
$406$	1869.75			0.228557
$407$			13103.6i			1.59588i
$408$		0			0
$409$	3339.07			0.403683			0.201841	−	0.979418i	$-0.435307\pi$
$409$	3339.07			0.403683			0.201841	+	0.979418i	$0.435307\pi$
$410$	−1479.89	+	12656.5i	−0.178260	+	1.52454i
$411$		0			0
$412$		−	18811.4i		−	2.24944i
$413$			1818.84i			0.216706i
$414$		0			0
$415$	−1503.49	+	12858.4i	−0.177840	+	1.52095i
$416$	2487.40			0.293161
$417$		0			0
$418$		−	9232.57i		−	1.08033i
$419$	1688.52			0.196873			0.0984363	−	0.995143i	$-0.468616\pi$
$419$	1688.52			0.196873			0.0984363	+	0.995143i	$0.468616\pi$
$420$		0			0
$421$	−2664.27			−0.308429			−0.154214	−	0.988037i	$-0.549285\pi$
$421$	−2664.27			−0.308429			−0.154214	+	0.988037i	$0.549285\pi$
$422$		−	11120.5i		−	1.28279i
$423$		0			0
$424$	1765.95			0.202269
$425$	2393.96	−	10097.0i	0.273233	−	1.15242i
$426$		0			0
$427$		−	7737.54i		−	0.876923i
$428$		−	11259.2i		−	1.27158i
$429$		0			0
$430$	19179.1	+	2242.56i	2.15093	+	0.251502i
$431$	−12266.0			−1.37084			−0.685420	−	0.728148i	$-0.740381\pi$
$431$	−12266.0			−1.37084			−0.685420	+	0.728148i	$0.740381\pi$
$432$		0			0
$433$			15647.3i			1.73664i	0.496008	+	0.868318i	$0.334799\pi$
$433$			15647.3i			1.73664i	−0.496008	+	0.868318i	$0.665201\pi$
$434$	946.425			0.104677
$435$		0			0
$436$	−13894.8			−1.52624
$437$			78.7503i			0.00862045i
$438$		0			0
$439$	−16131.0			−1.75373			−0.876867	−	0.480733i	$-0.840371\pi$
$439$	−16131.0			−1.75373			−0.876867	+	0.480733i	$0.840371\pi$
$440$	1498.49	−	12815.6i	0.162358	−	1.38855i
$441$		0			0
$442$			7724.34i			0.831243i
$443$			10053.7i			1.07825i	0.842225	+	0.539127i	$0.181246\pi$
$443$			10053.7i			1.07825i	−0.842225	+	0.539127i	$0.818754\pi$
$444$		0			0
$445$	−1076.88	−	125.916i	−0.114717	−	0.0134135i
$446$	15649.5			1.66150
$447$		0			0
$448$		−	12444.9i		−	1.31242i
$449$	7477.71			0.785957			0.392979	−	0.919548i	$-0.371445\pi$
$449$	7477.71			0.785957			0.392979	+	0.919548i	$0.371445\pi$
$450$		0			0
$451$	9747.51			1.01772
$452$		−	26633.1i		−	2.77150i
$453$		0			0
$454$	2478.19			0.256184
$455$	3562.31	+	416.531i	0.367041	+	0.0429170i
$456$		0			0
$457$			1363.46i			0.139562i	0.997562	+	0.0697812i	$0.0222301\pi$
$457$			1363.46i			0.139562i	−0.997562	+	0.0697812i	$0.977770\pi$
$458$			12064.5i			1.23086i
$459$		0			0
$460$	−29.5314	+	252.562i	−0.00299328	+	0.0255995i
$461$	−5276.77			−0.533109			−0.266555	−	0.963820i	$-0.585885\pi$
$461$	−5276.77			−0.533109			−0.266555	+	0.963820i	$0.585885\pi$
$462$		0			0
$463$		−	5740.02i		−	0.576159i	−0.957607	−	0.288079i	$-0.906983\pi$
$463$		−	5740.02i		−	0.576159i	0.957607	−	0.288079i	$-0.0930167\pi$
$464$	−542.325			−0.0542604
$465$		0			0
$466$	26023.1			2.58690
$467$		−	6233.36i		−	0.617657i	−0.951118	−	0.308828i	$-0.900063\pi$
$467$		−	6233.36i		−	0.617657i	0.951118	−	0.308828i	$-0.0999368\pi$
$468$		0			0
$469$	9046.35			0.890664
$470$	−10679.1	−	1248.68i	−1.04806	−	0.122547i
$471$		0			0
$472$		−	3220.58i		−	0.314067i
$473$		−	14770.9i		−	1.43587i
$474$		0			0
$475$	5940.01	+	1408.35i	0.573782	+	0.136041i
$476$	18979.7			1.82759
$477$		0			0
$478$		−	4748.61i		−	0.454385i
$479$	−19688.2			−1.87803			−0.939013	−	0.343881i	$-0.888258\pi$
$479$	−19688.2			−1.87803			−0.939013	+	0.343881i	$0.888258\pi$
$480$		0			0
$481$	6449.46			0.611372
$482$			19155.5i			1.81019i
$483$		0			0
$484$	−4031.03			−0.378572
$485$	−1496.29	+	12796.8i	−0.140088	+	1.19808i
$486$		0			0
$487$		−	3955.08i		−	0.368012i	−0.982925	−	0.184006i	$-0.941093\pi$
$487$		−	3955.08i		−	0.368012i	0.982925	−	0.184006i	$-0.0589065\pi$
$488$			13700.7i			1.27090i
$489$		0			0
$490$	−489.939	+	4190.13i	−0.0451698	+	0.386308i
$491$	13893.5			1.27699			0.638497	−	0.769624i	$-0.279557\pi$
$491$	13893.5			1.27699			0.638497	+	0.769624i	$0.279557\pi$
$492$		0			0
$493$			2036.74i			0.186065i
$494$	−4544.18			−0.413871
$495$		0			0
$496$	−274.512			−0.0248508
$497$			9049.39i			0.816741i
$498$		0			0
$499$	−13523.7			−1.21324			−0.606618	−	0.794993i	$-0.707474\pi$
$499$	−13523.7			−1.21324			−0.606618	+	0.794993i	$0.707474\pi$
$500$	18522.3	+	6744.27i	1.65668	+	0.603226i
$501$		0			0
$502$			8340.50i			0.741543i
$503$			13135.4i			1.16437i	0.813057	+	0.582184i	$0.197802\pi$
$503$			13135.4i			1.16437i	−0.813057	+	0.582184i	$0.802198\pi$
$504$		0			0
$505$	−12842.5	−	1501.63i	−1.13165	−	0.132320i
$506$	304.837			0.0267820
$507$		0			0
$508$			8757.03i			0.764823i
$509$	−2222.71			−0.193556			−0.0967778	−	0.995306i	$-0.530854\pi$
$509$	−2222.71			−0.193556			−0.0967778	+	0.995306i	$0.530854\pi$
$510$		0			0
$511$	−16400.1			−1.41976
$512$			7853.76i			0.677911i
$513$		0			0
$514$	−3116.12			−0.267405
$515$	1731.72	−	14810.3i	0.148172	−	1.26722i
$516$		0			0
$517$			8224.57i			0.699645i
$518$		−	24835.4i		−	2.10658i
$519$		0			0
$520$	−6307.71	−	737.541i	−0.531945	−	0.0621987i
$521$	4916.42			0.413421			0.206710	−	0.978402i	$-0.433724\pi$
$521$	4916.42			0.413421			0.206710	+	0.978402i	$0.433724\pi$
$522$		0			0
$523$		−	17743.4i		−	1.48349i	−0.670681	−	0.741746i	$-0.733998\pi$
$523$		−	17743.4i		−	1.48349i	0.670681	−	0.741746i	$-0.266002\pi$
$524$	−36507.9			−3.04362
$525$		0			0
$526$	3348.98			0.277609
$527$			1030.95i			0.0852161i
$528$		0			0
$529$	12164.4			0.999786
$530$	3212.34	+	375.610i	0.263274	+	0.0307839i
$531$		0			0
$532$			11165.6i			0.909946i
$533$		−	4797.62i		−	0.389884i
$534$		0			0
$535$	1036.49	−	8864.46i	0.0837599	−	0.716344i
$536$	−16018.2			−1.29082
$537$		0			0
$538$			14746.0i			1.18169i
$539$	3227.05			0.257883
$540$		0			0
$541$	−12671.3			−1.00699			−0.503495	−	0.863998i	$-0.667953\pi$
$541$	−12671.3			−1.00699			−0.503495	+	0.863998i	$0.667953\pi$
$542$			10699.3i			0.847924i
$543$		0			0
$544$	10433.9			0.822334
$545$	−10939.4	−	1279.12i	−0.859805	−	0.100534i
$546$		0			0
$547$			5250.90i			0.410443i	0.978716	+	0.205221i	$0.0657915\pi$
$547$			5250.90i			0.410443i	−0.978716	+	0.205221i	$0.934209\pi$
$548$		−	23361.5i		−	1.82108i
$549$		0			0
$550$	5451.64	−	22993.4i	0.422652	−	1.78262i
$551$	−1198.20			−0.0926406
$552$		0			0
$553$			18648.4i			1.43401i
$554$	24311.8			1.86445
$555$		0			0
$556$	−2161.63			−0.164881
$557$		−	25830.2i		−	1.96492i	−0.186465	−	0.982462i	$-0.559703\pi$
$557$		−	25830.2i		−	1.96492i	0.186465	−	0.982462i	$-0.440297\pi$
$558$		0			0
$559$	−7270.09			−0.550075
$560$	−465.234	+	3978.84i	−0.0351067	+	0.300244i
$561$		0			0
$562$			10532.1i			0.790519i
$563$			2021.14i			0.151298i	0.997135	+	0.0756490i	$0.0241029\pi$
$563$			2021.14i			0.151298i	−0.997135	+	0.0756490i	$0.975897\pi$
$564$		0			0
$565$	2451.77	−	20968.4i	0.182561	−	1.56132i
$566$	−1059.00			−0.0786450
$567$		0			0
$568$		−	16023.5i		−	1.18368i
$569$	8706.51			0.641469			0.320734	−	0.947169i	$-0.396070\pi$
$569$	8706.51			0.641469			0.320734	+	0.947169i	$0.396070\pi$
$570$		0			0
$571$	−12194.5			−0.893740			−0.446870	−	0.894599i	$-0.647461\pi$
$571$	−12194.5			−0.893740			−0.446870	+	0.894599i	$0.647461\pi$
$572$			11224.1i			0.820458i
$573$		0			0
$574$	−18474.6			−1.34340
$575$	−46.5004	+	196.125i	−0.00337252	+	0.0142243i
$576$		0			0
$577$		−	15264.0i		−	1.10130i	−0.834737	−	0.550649i	$-0.814380\pi$
$577$		−	15264.0i		−	1.10130i	0.834737	−	0.550649i	$-0.185620\pi$
$578$			9302.48i			0.669433i
$579$		0			0
$580$	−3842.78	−	449.324i	−0.275108	−	0.0321675i
$581$	−18769.2			−1.34024
$582$		0			0
$583$		−	2474.01i		−	0.175751i
$584$	29039.3			2.05763
$585$		0			0
$586$	5359.12			0.377787
$587$			8456.89i			0.594639i	0.954778	+	0.297319i	$0.0960926\pi$
$587$			8456.89i			0.594639i	−0.954778	+	0.297319i	$0.903907\pi$
$588$		0			0
$589$	−606.500			−0.0424285
$590$	685.003	−	5858.38i	0.0477985	−	0.408789i
$591$		0			0
$592$			7203.57i			0.500110i
$593$		−	1225.23i		−	0.0848467i	−0.999100	−	0.0424234i	$-0.986492\pi$
$593$		−	1225.23i		−	0.0848467i	0.999100	−	0.0424234i	$-0.0135078\pi$
$594$		0			0
$595$	14942.8	+	1747.22i	1.02957	+	0.120385i
$596$	−20922.6			−1.43796
$597$		0			0
$598$		−	150.038i		−	0.0102600i
$599$	−16060.0			−1.09548			−0.547741	−	0.836648i	$-0.684512\pi$
$599$	−16060.0			−1.09548			−0.547741	+	0.836648i	$0.684512\pi$
$600$		0			0
$601$	9699.93			0.658350			0.329175	−	0.944269i	$-0.393229\pi$
$601$	9699.93			0.658350			0.329175	+	0.944269i	$0.393229\pi$
$602$			27995.5i			1.89537i
$603$		0			0
$604$	5561.21			0.374640
$605$	−3173.65	−	371.085i	−0.213268	−	0.0249368i
$606$		0			0
$607$			23661.2i			1.58217i	0.611703	+	0.791087i	$0.290485\pi$
$607$			23661.2i			1.58217i	−0.611703	+	0.791087i	$0.709515\pi$
$608$			6138.19i			0.409435i
$609$		0			0
$610$	−2914.07	+	24922.1i	−0.193422	+	1.65421i
$611$	4048.05			0.268030
$612$		0			0
$613$			8085.63i			0.532749i	0.963870	+	0.266375i	$0.0858258\pi$
$613$			8085.63i			0.532749i	−0.963870	+	0.266375i	$0.914174\pi$
$614$	24659.0			1.62078
$615$		0			0
$616$	18706.8			1.22357
$617$			11035.1i			0.720029i	0.932947	+	0.360014i	$0.117228\pi$
$617$			11035.1i			0.720029i	−0.932947	+	0.360014i	$0.882772\pi$
$618$		0			0
$619$	16826.3			1.09258			0.546290	−	0.837596i	$-0.316040\pi$
$619$	16826.3			1.09258			0.546290	+	0.837596i	$0.316040\pi$
$620$	−1945.12	−	227.438i	−0.125997	−	0.0147324i
$621$		0			0
$622$		−	24392.9i		−	1.57245i
$623$		−	1571.90i		−	0.101087i
$624$		0			0
$625$	13961.8	+	7014.90i	0.893555	+	0.448954i
$626$	−2288.04			−0.146084
$627$		0			0
$628$		−	24359.5i		−	1.54785i
$629$	27053.5			1.71493
$630$		0			0
$631$	−3705.91			−0.233803			−0.116902	−	0.993144i	$-0.537296\pi$
$631$	−3705.91			−0.233803			−0.116902	+	0.993144i	$0.537296\pi$
$632$		−	33020.2i		−	2.07828i
$633$		0			0
$634$	19835.3			1.24252
$635$	−806.147	+	6894.45i	−0.0503795	+	0.430863i
$636$		0			0
$637$		−	1588.32i		−	0.0987937i
$638$			4638.15i			0.287815i
$639$		0			0
$640$	−3381.36	+	28918.5i	−0.208844	+	1.78610i
$641$	24597.4			1.51566			0.757829	−	0.652453i	$-0.226260\pi$
$641$	24597.4			1.51566			0.757829	+	0.652453i	$0.226260\pi$
$642$		0			0
$643$			21479.5i			1.31737i	0.752419	+	0.658685i	$0.228887\pi$
$643$			21479.5i			1.31737i	−0.752419	+	0.658685i	$0.771113\pi$
$644$	−368.662			−0.0225580
$645$		0			0
$646$	−19061.4			−1.16093
$647$		−	27119.7i		−	1.64789i	−0.566668	−	0.823946i	$-0.691768\pi$
$647$		−	27119.7i		−	1.64789i	0.566668	−	0.823946i	$-0.308232\pi$
$648$		0			0
$649$	−4511.87			−0.272891
$650$	−11317.1	−	2683.24i	−0.682913	−	0.161916i
$651$		0			0
$652$		−	28692.9i		−	1.72347i
$653$			18476.4i			1.10725i	0.832765	+	0.553627i	$0.186757\pi$
$653$			18476.4i			1.10725i	−0.832765	+	0.553627i	$0.813243\pi$
$654$		0			0
$655$	−28742.8	−	3360.82i	−1.71462	−	0.200485i
$656$	5358.59			0.318930
$657$		0			0
$658$		−	15588.1i		−	0.923540i
$659$	−19273.5			−1.13928			−0.569641	−	0.821894i	$-0.692918\pi$
$659$	−19273.5			−1.13928			−0.569641	+	0.821894i	$0.692918\pi$
$660$		0			0
$661$	25605.3			1.50670			0.753352	−	0.657618i	$-0.228436\pi$
$661$	25605.3			1.50670			0.753352	+	0.657618i	$0.228436\pi$
$662$		−	34979.3i		−	2.05364i
$663$		0			0
$664$	33234.3			1.94238
$665$	−1027.88	+	8790.75i	−0.0599388	+	0.512617i
$666$		0			0
$667$		−	39.5616i		−	0.00229660i
$668$			2720.79i			0.157591i
$669$		0			0
$670$	−29137.7	−	3406.99i	−1.68013	−	0.196453i
$671$	19193.9			1.10428
$672$		0			0
$673$		−	7855.52i		−	0.449938i	−0.974366	−	0.224969i	$-0.927772\pi$
$673$		−	7855.52i		−	0.449938i	0.974366	−	0.224969i	$-0.0722280\pi$
$674$	−30820.6			−1.76137
$675$		0			0
$676$	−25463.6			−1.44877
$677$			6763.09i			0.383939i	0.981401	+	0.191970i	$0.0614875\pi$
$677$			6763.09i			0.383939i	−0.981401	+	0.191970i	$0.938512\pi$
$678$		0			0
$679$	−18679.3			−1.05574
$680$	−26458.9	−	3093.76i	−1.49214	−	0.174471i
$681$		0			0
$682$			2347.72i			0.131817i
$683$			15608.6i			0.874447i	0.899353	+	0.437224i	$0.144038\pi$
$683$			15608.6i			0.874447i	−0.899353	+	0.437224i	$0.855962\pi$
$684$		0			0
$685$	2150.59	−	18392.6i	0.119956	−	1.02590i
$686$	−32256.1			−1.79525
$687$		0			0
$688$		−	8120.16i		−	0.449968i
$689$	−1217.68			−0.0673293
$690$		0			0
$691$	6203.15			0.341504			0.170752	−	0.985314i	$-0.445380\pi$
$691$	6203.15			0.341504			0.170752	+	0.985314i	$0.445380\pi$
$692$			17488.5i			0.960714i
$693$		0			0
$694$	−9170.69			−0.501606
$695$	−1701.86	−	198.994i	−0.0928854	−	0.0108608i
$696$		0			0
$697$		−	20124.5i		−	1.09365i
$698$		−	6707.90i		−	0.363750i
$699$		0			0
$700$	−6593.06	+	27807.6i	−0.355992	+	1.50147i
$701$	16507.9			0.889435			0.444718	−	0.895671i	$-0.353304\pi$
$701$	16507.9			0.889435			0.444718	+	0.895671i	$0.353304\pi$
$702$		0			0
$703$			15915.4i			0.853854i
$704$	30871.1			1.65269
$705$		0			0
$706$	−33282.2			−1.77421
$707$		−	18746.0i		−	0.997193i
$708$		0			0
$709$	25539.6			1.35283			0.676416	−	0.736520i	$-0.263532\pi$
$709$	25539.6			1.35283			0.676416	+	0.736520i	$0.263532\pi$
$710$	3408.13	−	29147.5i	0.180148	−	1.54069i
$711$		0			0
$712$			2783.34i			0.146503i
$713$		−	20.0252i		−	0.00105182i
$714$		0			0
$715$	−1033.26	+	8836.76i	−0.0540442	+	0.462204i
$716$	37192.1			1.94125
$717$		0			0
$718$			25434.4i			1.32201i
$719$	−7353.45			−0.381415			−0.190708	−	0.981647i	$-0.561078\pi$
$719$	−7353.45			−0.381415			−0.190708	+	0.981647i	$0.561078\pi$
$720$		0			0
$721$	21618.4			1.11666
$722$			21034.3i			1.08423i
$723$		0			0
$724$	−46793.4			−2.40202
$725$	−2984.07	−	707.510i	−0.152863	−	0.0362431i
$726$		0			0
$727$		−	21696.5i		−	1.10685i	−0.832900	−	0.553424i	$-0.813321\pi$
$727$		−	21696.5i		−	1.10685i	0.832900	−	0.553424i	$-0.186679\pi$
$728$		−	9207.28i		−	0.468742i
$729$		0			0
$730$	52823.8	+	6176.53i	2.67821	+	0.313156i
$731$	−30495.8			−1.54299
$732$		0			0
$733$		−	90.2714i		−	0.00454877i	−0.999997	−	0.00227439i	$-0.999276\pi$
$733$		−	90.2714i		−	0.00454877i	0.999997	−	0.00227439i	$-0.000723960\pi$
$734$	23226.1			1.16797
$735$		0			0
$736$	−202.668			−0.0101501
$737$			22440.6i			1.12159i
$738$		0			0
$739$	−14273.1			−0.710479			−0.355239	−	0.934775i	$-0.615601\pi$
$739$	−14273.1			−0.710479			−0.355239	+	0.934775i	$0.615601\pi$
$740$	−5968.27	+	51042.7i	−0.296484	+	2.53563i
$741$		0			0
$742$			4689.02i			0.231994i
$743$			15866.6i			0.783429i	0.920087	+	0.391715i	$0.128118\pi$
$743$			15866.6i			0.783429i	−0.920087	+	0.391715i	$0.871882\pi$
$744$		0			0
$745$	−16472.4	−	1926.07i	−0.810072	−	0.0947193i
$746$	−60612.2			−2.97476
$747$		0			0
$748$			47081.5i			2.30143i
$749$	12939.3			0.631232
$750$		0			0
$751$	26776.9			1.30107			0.650534	−	0.759477i	$-0.274545\pi$
$751$	26776.9			1.30107			0.650534	+	0.759477i	$0.274545\pi$
$752$			4521.37i			0.219252i
$753$		0			0
$754$	2282.85			0.110261
$755$	4378.37	+	511.950i	0.211053	+	0.0246778i
$756$		0			0
$757$			30478.0i			1.46333i	0.681663	+	0.731666i	$0.261257\pi$
$757$			30478.0i			1.46333i	−0.681663	+	0.731666i	$0.738743\pi$
$758$		−	44548.0i		−	2.13464i
$759$		0			0
$760$	1820.04	−	15565.6i	0.0868680	−	0.742925i
$761$	−29104.7			−1.38639			−0.693195	−	0.720750i	$-0.743798\pi$
$761$	−29104.7			−1.38639			−0.693195	+	0.720750i	$0.743798\pi$
$762$		0			0
$763$		−	15968.2i		−	0.757649i
$764$	−8808.47			−0.417119
$765$		0			0
$766$	−27273.6			−1.28647
$767$			2220.69i			0.104543i
$768$		0			0
$769$	−4170.65			−0.195575			−0.0977876	−	0.995207i	$-0.531177\pi$
$769$	−4170.65			−0.195575			−0.0977876	+	0.995207i	$0.531177\pi$
$770$	34028.4	+	3978.84i	1.59260	+	0.186218i
$771$		0			0
$772$		−	6151.67i		−	0.286792i
$773$		−	17738.5i		−	0.825367i	−0.910874	−	0.412684i	$-0.864591\pi$
$773$		−	17738.5i		−	0.825367i	0.910874	−	0.412684i	$-0.135409\pi$
$774$		0			0
$775$	−1510.47	−	358.125i	−0.0700099	−	0.0165990i
$776$	33075.0			1.53005
$777$		0			0
$778$		−	64786.0i		−	2.98546i
$779$	11839.1			0.544519
$780$		0			0
$781$	−22448.1			−1.02850
$782$		−	629.363i		−	0.0287800i
$783$		0			0
$784$	1774.04			0.0808145
$785$	2242.47	−	19178.4i	0.101958	−	0.871982i
$786$		0			0
$787$		−	3807.92i		−	0.172475i	−0.996275	−	0.0862374i	$-0.972516\pi$
$787$		−	3807.92i		−	0.172475i	0.996275	−	0.0862374i	$-0.0274844\pi$
$788$			47332.6i			2.13979i
$789$		0			0
$790$	7023.25	−	60065.2i	0.316299	−	2.70510i
$791$	30607.3			1.37582
$792$		0			0
$793$		−	9447.06i		−	0.423045i
$794$	−13241.8			−0.591856
$795$		0			0
$796$	53594.5			2.38644
$797$		−	23840.3i		−	1.05956i	−0.848136	−	0.529779i	$-0.822275\pi$
$797$		−	23840.3i		−	1.05956i	0.848136	−	0.529779i	$-0.177725\pi$
$798$		0			0
$799$	16980.3			0.751841
$800$	−3624.47	+	15286.9i	−0.160180	+	0.675594i
$801$		0			0
$802$			56355.0i			2.48125i
$803$		−	40682.6i		−	1.78787i
$804$		0			0
$805$	−290.249	−	33.9380i	−0.0127080	−	0.00148591i
$806$	1155.53			0.0504983
$807$		0			0
$808$			33193.1i			1.44521i
$809$	−1984.22			−0.0862316			−0.0431158	−	0.999070i	$-0.513728\pi$
$809$	−1984.22			−0.0862316			−0.0431158	+	0.999070i	$0.513728\pi$
$810$		0			0
$811$	−9713.78			−0.420588			−0.210294	−	0.977638i	$-0.567442\pi$
$811$	−9713.78			−0.420588			−0.210294	+	0.977638i	$0.567442\pi$
$812$		−	5609.25i		−	0.242421i
$813$		0			0
$814$	61607.4			2.65275
$815$	2641.39	−	22590.1i	0.113526	−	0.970914i
$816$		0			0
$817$		−	17940.5i		−	0.768246i
$818$		−	15698.8i		−	0.671023i
$819$		0			0
$820$	37969.6	+	4439.67i	1.61702	+	0.189073i
$821$	19235.4			0.817686			0.408843	−	0.912605i	$-0.365932\pi$
$821$	19235.4			0.817686			0.408843	+	0.912605i	$0.365932\pi$
$822$		0			0
$823$			12717.6i			0.538650i	0.963049	+	0.269325i	$0.0868005\pi$
$823$			12717.6i			0.538650i	−0.963049	+	0.269325i	$0.913199\pi$
$824$	−38279.2			−1.61835
$825$		0			0
$826$	8551.41			0.360220
$827$			6744.75i			0.283601i	0.989895	+	0.141800i	$0.0452891\pi$
$827$			6744.75i			0.283601i	−0.989895	+	0.141800i	$0.954711\pi$
$828$		0			0
$829$	−3404.22			−0.142622			−0.0713108	−	0.997454i	$-0.522718\pi$
$829$	−3404.22			−0.142622			−0.0713108	+	0.997454i	$0.522718\pi$
$830$	60454.5	+	7068.77i	2.52820	+	0.295615i
$831$		0			0
$832$		−	15194.4i		−	0.633139i
$833$		−	6662.52i		−	0.277122i
$834$		0			0
$835$	−250.469	+	2142.09i	−0.0103806	+	0.0887787i
$836$	−27697.7			−1.14587
$837$		0			0
$838$		−	7938.69i		−	0.327252i
$839$	−21361.9			−0.879015			−0.439508	−	0.898239i	$-0.644847\pi$
$839$	−21361.9			−0.879015			−0.439508	+	0.898239i	$0.644847\pi$
$840$		0			0
$841$	−23787.1			−0.975319
$842$			12526.2i			0.512687i
$843$		0			0
$844$	−33361.4			−1.36060
$845$	−20047.6	−	2344.11i	−0.816165	−	0.0954318i
$846$		0			0
$847$		−	4632.54i		−	0.187929i
$848$		−	1360.06i		−	0.0550762i
$849$		0			0
$850$	−47471.8	−	11255.4i	−1.91561	−	0.454183i
$851$	−525.488			−0.0211674
$852$		0			0
$853$			10728.9i			0.430657i	0.976542	+	0.215328i	$0.0690822\pi$
$853$			10728.9i			0.430657i	−0.976542	+	0.215328i	$0.930918\pi$
$854$	−36378.5			−1.45767
$855$		0			0
$856$	−22911.4			−0.914830
$857$			42895.2i			1.70977i	0.518817	+	0.854885i	$0.326373\pi$
$857$			42895.2i			1.70977i	−0.518817	+	0.854885i	$0.673627\pi$
$858$		0			0
$859$	−35530.5			−1.41127			−0.705637	−	0.708574i	$-0.749339\pi$
$859$	−35530.5			−1.41127			−0.705637	+	0.708574i	$0.749339\pi$
$860$	6727.67	−	57537.3i	0.266758	−	2.28140i
$861$		0			0
$862$			57669.3i			2.27868i
$863$			5704.35i			0.225004i	0.993652	+	0.112502i	$0.0358865\pi$
$863$			5704.35i			0.225004i	−0.993652	+	0.112502i	$0.964114\pi$
$864$		0			0
$865$	−1609.94	+	13768.8i	−0.0632830	+	0.541218i
$866$	73567.0			2.88673
$867$		0			0
$868$		−	2839.27i		−	0.111027i
$869$	−46259.6			−1.80581
$870$		0			0
$871$	11045.0			0.429674
$872$			28274.4i			1.09804i
$873$		0			0
$874$	370.249			0.0143294
$875$	−7750.64	+	21286.1i	−0.299451	+	0.822403i
$876$		0			0
$877$			50249.0i			1.93476i	0.253324	+	0.967382i	$0.418476\pi$
$877$			50249.0i			1.93476i	−0.253324	+	0.967382i	$0.581524\pi$
$878$			75840.7i			2.91515i
$879$		0			0
$880$	−9870.02	−	1154.07i	−0.378089	−	0.0442088i
$881$	26864.5			1.02734			0.513672	−	0.857987i	$-0.328285\pi$
$881$	26864.5			1.02734			0.513672	+	0.857987i	$0.328285\pi$
$882$		0			0
$883$		−	18942.1i		−	0.721918i	−0.932582	−	0.360959i	$-0.882449\pi$
$883$		−	18942.1i		−	0.721918i	0.932582	−	0.360959i	$-0.117551\pi$
$884$	23173.0			0.881667
$885$		0			0
$886$	47268.2			1.79233
$887$			25344.8i			0.959409i	0.877430	+	0.479705i	$0.159256\pi$
$887$			25344.8i			0.959409i	−0.877430	+	0.479705i	$0.840744\pi$
$888$		0			0
$889$	−10063.7			−0.379670
$890$	−592.002	+	5063.01i	−0.0222966	+	0.190688i
$891$		0			0
$892$		−	46948.6i		−	1.76228i
$893$			9989.40i			0.374337i
$894$		0			0
$895$	29281.5	+	3423.80i	1.09360	+	0.127871i
$896$	−42212.0			−1.57389
$897$		0			0
$898$		−	35156.9i		−	1.30646i
$899$	304.686			0.0113035
$900$		0			0
$901$	−5107.79			−0.188863
$902$		−	45828.5i		−	1.69171i
$903$		0			0
$904$	−54195.6			−1.99394
$905$	−36840.7	−	4307.67i	−1.35318	−	0.158223i
$906$		0			0
$907$			4800.11i			0.175728i	0.996132	+	0.0878639i	$0.0280041\pi$
$907$			4800.11i			0.175728i	−0.996132	+	0.0878639i	$0.971996\pi$
$908$		−	7434.58i		−	0.271724i
$909$		0			0
$910$	1958.34	−	16748.4i	0.0713390	−	0.610116i
$911$	25731.7			0.935819			0.467909	−	0.883776i	$-0.345007\pi$
$911$	25731.7			0.935819			0.467909	+	0.883776i	$0.345007\pi$
$912$		0			0
$913$		−	46559.4i		−	1.68772i
$914$	6410.40			0.231988
$915$		0			0
$916$	36193.4			1.30553
$917$		−	41955.6i		−	1.51090i
$918$		0			0
$919$	12751.9			0.457722			0.228861	−	0.973459i	$-0.426500\pi$
$919$	12751.9			0.457722			0.228861	+	0.973459i	$0.426500\pi$
$920$	513.938	+	60.0932i	0.0184174	+	0.00215350i
$921$		0			0
$922$			24809.0i			0.886163i
$923$			11048.7i			0.394012i
$924$		0			0
$925$	−9397.69	+	39636.7i	−0.334048	+	1.40892i
$926$	−26987.1			−0.957722
$927$		0			0
$928$		−	3083.63i		−	0.109079i
$929$	15557.8			0.549444			0.274722	−	0.961524i	$-0.411414\pi$
$929$	15557.8			0.549444			0.274722	+	0.961524i	$0.411414\pi$
$930$		0			0
$931$	3919.51			0.137977
$932$		−	78069.3i		−	2.74383i
$933$		0			0
$934$	−29306.5			−1.02670
$935$	−4334.19	+	37067.5i	−0.151597	+	1.29651i
$936$		0			0
$937$		−	23858.0i		−	0.831811i	−0.909408	−	0.415905i	$-0.863465\pi$
$937$		−	23858.0i		−	0.831811i	0.909408	−	0.415905i	$-0.136535\pi$
$938$		−	42532.0i		−	1.48051i
$939$		0			0
$940$	−3746.03	+	32037.3i	−0.129981	+	1.11164i
$941$	−9748.00			−0.337700			−0.168850	−	0.985642i	$-0.554005\pi$
$941$	−9748.00			−0.337700			−0.168850	+	0.985642i	$0.554005\pi$
$942$		0			0
$943$			390.899i			0.0134989i
$944$	−2480.35			−0.0855176
$945$		0			0
$946$	−69446.4			−2.38678
$947$			51537.0i			1.76845i	0.467057	+	0.884227i	$0.345314\pi$
$947$			51537.0i			1.76845i	−0.467057	+	0.884227i	$0.654686\pi$
$948$		0			0
$949$	−20023.5			−0.684923
$950$	6621.45	−	27927.3i	0.226135	−	0.953771i
$951$		0			0
$952$		−	38621.7i		−	1.31485i
$953$		−	5631.36i		−	0.191414i	−0.995410	−	0.0957071i	$-0.969489\pi$
$953$		−	5631.36i		−	0.191414i	0.995410	−	0.0957071i	$-0.0305112\pi$
$954$		0			0
$955$	−6934.95	−	810.883i	−0.234984	−	0.0274760i
$956$	−14245.8			−0.481948
$957$		0			0
$958$			92565.1i			3.12176i
$959$	26847.4			0.904013
$960$		0			0
$961$	−29636.8			−0.994823
$962$		−	30322.5i		−	1.01625i
$963$		0			0
$964$	57466.5			1.91999
$965$	566.305	−	4843.24i	0.0188912	−	0.161564i
$966$		0			0
$967$		−	43360.9i		−	1.44198i	−0.692946	−	0.720989i	$-0.743688\pi$
$967$		−	43360.9i		−	1.44198i	0.692946	−	0.720989i	$-0.256312\pi$
$968$			8202.72i			0.272361i
$969$		0			0
$970$	60164.8	+	7034.89i	1.99152	+	0.232862i
$971$	−12920.0			−0.427007			−0.213503	−	0.976942i	$-0.568487\pi$
$971$	−12920.0			−0.427007			−0.213503	+	0.976942i	$0.568487\pi$
$972$		0			0
$973$		−	2484.19i		−	0.0818493i
$974$	−18595.0			−0.611728
$975$		0			0
$976$	10551.7			0.346056
$977$			10650.4i			0.348759i	0.984679	+	0.174379i	$0.0557919\pi$
$977$			10650.4i			0.348759i	−0.984679	+	0.174379i	$0.944208\pi$
$978$		0			0
$979$	3899.31			0.127296
$980$	12570.4	+	1469.82i	0.409741	+	0.0479098i
$981$		0			0
$982$		−	65321.0i		−	2.12269i
$983$		−	49450.3i		−	1.60450i	−0.596991	−	0.802248i	$-0.703637\pi$
$983$		−	49450.3i		−	1.60450i	0.596991	−	0.802248i	$-0.296363\pi$
$984$		0			0
$985$	−4357.31	+	37265.2i	−0.140950	+	1.20545i
$986$	9575.85			0.309287
$987$		0			0
$988$			13632.5i			0.438976i
$989$	592.351			0.0190452
$990$		0			0
$991$	9410.47			0.301648			0.150824	−	0.988561i	$-0.451807\pi$
$991$	9410.47			0.301648			0.150824	+	0.988561i	$0.451807\pi$
$992$		−	1560.86i		−	0.0499571i
$993$		0			0
$994$	42546.3			1.35763
$995$	42195.2	+	4933.76i	1.34440	+	0.157197i
$996$		0			0
$997$			532.117i			0.0169030i	0.999964	+	0.00845151i	$0.00269023\pi$
$997$			532.117i			0.0169030i	−0.999964	+	0.00845151i	$0.997310\pi$
$998$			63582.6i			2.01671i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.4.b.b.19.1		4
3.2	odd	2		15.4.b.a.4.4	yes	4
4.3	odd	2		720.4.f.j.289.1		4
5.2	odd	4		225.4.a.o.1.2		2
5.3	odd	4		225.4.a.i.1.1		2
5.4	even	2	inner	45.4.b.b.19.4		4
12.11	even	2		240.4.f.f.49.4		4
15.2	even	4		75.4.a.c.1.1		2
15.8	even	4		75.4.a.f.1.2		2
15.14	odd	2		15.4.b.a.4.1	✓	4
20.19	odd	2		720.4.f.j.289.2		4
24.5	odd	2		960.4.f.q.769.3		4
24.11	even	2		960.4.f.p.769.1		4
60.23	odd	4		1200.4.a.bn.1.2		2
60.47	odd	4		1200.4.a.bt.1.1		2
60.59	even	2		240.4.f.f.49.2		4
120.29	odd	2		960.4.f.q.769.1		4
120.59	even	2		960.4.f.p.769.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.4.b.a.4.1	✓	4	15.14	odd	2
15.4.b.a.4.4	yes	4	3.2	odd	2
45.4.b.b.19.1		4	1.1	even	1	trivial
45.4.b.b.19.4		4	5.4	even	2	inner
75.4.a.c.1.1		2	15.2	even	4
75.4.a.f.1.2		2	15.8	even	4
225.4.a.i.1.1		2	5.3	odd	4
225.4.a.o.1.2		2	5.2	odd	4
240.4.f.f.49.2		4	60.59	even	2
240.4.f.f.49.4		4	12.11	even	2
720.4.f.j.289.1		4	4.3	odd	2
720.4.f.j.289.2		4	20.19	odd	2
960.4.f.p.769.1		4	24.11	even	2
960.4.f.p.769.3		4	120.59	even	2
960.4.f.q.769.1		4	120.29	odd	2
960.4.f.q.769.3		4	24.5	odd	2
1200.4.a.bn.1.2		2	60.23	odd	4
1200.4.a.bt.1.1		2	60.47	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:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	45.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.70156i q^{2} -14.1047 q^{4} +(-11.1047 - 1.29844i) q^{5} -16.2094i q^{7} +28.7016i q^{8} +O(q^{10})\)	\(q-4.70156i q^{2} -14.1047 q^{4} +(-11.1047 - 1.29844i) q^{5} -16.2094i q^{7} +28.7016i q^{8} +(-6.10469 + 52.2094i) q^{10} +40.2094 q^{11} -19.7906i q^{13} -76.2094 q^{14} +22.1047 q^{16} -83.0156i q^{17} +48.8375 q^{19} +(156.628 + 18.3141i) q^{20} -189.047i q^{22} +1.61250i q^{23} +(121.628 + 28.8375i) q^{25} -93.0469 q^{26} +228.628i q^{28} -24.5344 q^{29} -12.4187 q^{31} +125.686i q^{32} -390.303 q^{34} +(-21.0469 + 180.000i) q^{35} +325.884i q^{37} -229.612i q^{38} +(37.2672 - 318.722i) q^{40} +242.419 q^{41} -367.350i q^{43} -567.141 q^{44} +7.58125 q^{46} +204.544i q^{47} +80.2562 q^{49} +(135.581 - 571.842i) q^{50} +279.141i q^{52} -61.5281i q^{53} +(-446.512 - 52.2094i) q^{55} +465.234 q^{56} +115.350i q^{58} -112.209 q^{59} +477.350 q^{61} +58.3875i q^{62} +767.758 q^{64} +(-25.6969 + 219.769i) q^{65} +558.094i q^{67} +1170.91i q^{68} +(846.281 + 98.9531i) q^{70} -558.281 q^{71} -1011.77i q^{73} +1532.17 q^{74} -688.837 q^{76} -651.769i q^{77} -1150.47 q^{79} +(-245.466 - 28.7016i) q^{80} -1139.75i q^{82} -1157.92i q^{83} +(-107.791 + 921.862i) q^{85} -1727.12 q^{86} +1154.07i q^{88} +96.9751 q^{89} -320.794 q^{91} -22.7438i q^{92} +961.675 q^{94} +(-542.325 - 63.4124i) q^{95} -1152.37i q^{97} -377.330i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 18 q^{4} - 6 q^{5}+O(q^{10}) \) 4 * q - 18 * q^4 - 6 * q^5	\( 4 q - 18 q^{4} - 6 q^{5} + 14 q^{10} + 84 q^{11} - 228 q^{14} + 50 q^{16} - 112 q^{19} + 396 q^{20} + 256 q^{25} + 12 q^{26} - 636 q^{29} + 104 q^{31} - 716 q^{34} + 300 q^{35} + 418 q^{40} + 816 q^{41} - 1116 q^{44} + 184 q^{46} - 140 q^{49} + 696 q^{50} - 864 q^{55} - 60 q^{56} - 372 q^{59} + 680 q^{61} + 958 q^{64} - 948 q^{65} + 1080 q^{70} + 72 q^{71} + 3132 q^{74} - 2448 q^{76} - 760 q^{79} - 444 q^{80} - 508 q^{85} - 4296 q^{86} + 2232 q^{89} + 1944 q^{91} + 3232 q^{94} - 2784 q^{95}+O(q^{100}) \) 4 * q - 18 * q^4 - 6 * q^5 + 14 * q^10 + 84 * q^11 - 228 * q^14 + 50 * q^16 - 112 * q^19 + 396 * q^20 + 256 * q^25 + 12 * q^26 - 636 * q^29 + 104 * q^31 - 716 * q^34 + 300 * q^35 + 418 * q^40 + 816 * q^41 - 1116 * q^44 + 184 * q^46 - 140 * q^49 + 696 * q^50 - 864 * q^55 - 60 * q^56 - 372 * q^59 + 680 * q^61 + 958 * q^64 - 948 * q^65 + 1080 * q^70 + 72 * q^71 + 3132 * q^74 - 2448 * q^76 - 760 * q^79 - 444 * q^80 - 508 * q^85 - 4296 * q^86 + 2232 * q^89 + 1944 * q^91 + 3232 * q^94 - 2784 * q^95

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