LMFDB - Embedded newform 117.4.g.e.100.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,4,Mod(55,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("117.55");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	117.g (of order $3$, degree $2$, 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:	$6.90322347067$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 29x^{6} + 2x^{5} + 595x^{4} - 288x^{3} + 2526x^{2} + 1872x + 6084 $ x^8 - 2x^7 + 29x^6 + 2x^5 + 595x^4 - 288x^3 + 2526x^2 + 1872*x + 6084
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			100.4
Root			$2.66520 - 4.61626i$ of defining polynomial
Character	$\chi$	$=$	117.100
Dual form			117.4.g.e.55.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.66520 - 4.61626i) q^{2} +(-10.2065 - 17.6783i) q^{4} +16.4131 q^{5} +(-4.83984 - 8.38285i) q^{7} -66.1667 q^{8} +O(q^{10})$	\(q+(2.66520 - 4.61626i) q^{2} +(-10.2065 - 17.6783i) q^{4} +16.4131 q^{5} +(-4.83984 - 8.38285i) q^{7} -66.1667 q^{8} +(43.7441 - 75.7670i) q^{10} +(13.7941 - 23.8921i) q^{11} +(-37.3033 + 28.3807i) q^{13} -51.5965 q^{14} +(-94.6948 + 164.016i) q^{16} +(53.9641 + 93.4685i) q^{17} +(1.12362 + 1.94616i) q^{19} +(-167.521 - 290.155i) q^{20} +(-73.5279 - 127.354i) q^{22} +(20.9045 - 36.2077i) q^{23} +144.390 q^{25} +(31.5919 + 247.842i) q^{26} +(-98.7961 + 171.120i) q^{28} +(30.8106 - 53.3656i) q^{29} +191.932 q^{31} +(240.094 + 415.855i) q^{32} +575.300 q^{34} +(-79.4368 - 137.589i) q^{35} +(-49.2118 + 85.2373i) q^{37} +11.9786 q^{38} -1086.00 q^{40} +(-15.3726 + 26.6261i) q^{41} +(-119.163 - 206.396i) q^{43} -563.160 q^{44} +(-111.429 - 193.001i) q^{46} +511.482 q^{47} +(124.652 - 215.903i) q^{49} +(384.826 - 666.539i) q^{50} +(882.459 + 369.788i) q^{52} -492.825 q^{53} +(226.404 - 392.142i) q^{55} +(320.236 + 554.665i) q^{56} +(-164.233 - 284.460i) q^{58} +(242.089 + 419.311i) q^{59} +(222.011 + 384.534i) q^{61} +(511.536 - 886.007i) q^{62} +1044.47 q^{64} +(-612.262 + 465.815i) q^{65} +(-95.0568 + 164.643i) q^{67} +(1101.57 - 1907.98i) q^{68} -846.858 q^{70} +(242.392 + 419.836i) q^{71} -957.780 q^{73} +(262.318 + 454.348i) q^{74} +(22.9365 - 39.7271i) q^{76} -267.045 q^{77} -375.216 q^{79} +(-1554.23 + 2692.01i) q^{80} +(81.9421 + 141.928i) q^{82} +715.765 q^{83} +(885.717 + 1534.11i) q^{85} -1270.37 q^{86} +(-912.708 + 1580.86i) q^{88} +(-519.076 + 899.066i) q^{89} +(418.453 + 175.350i) q^{91} -853.451 q^{92} +(1363.20 - 2361.13i) q^{94} +(18.4420 + 31.9425i) q^{95} +(-32.7818 - 56.7797i) q^{97} +(-664.443 - 1150.85i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{2} - 22 q^{4} + 12 q^{5} + 14 q^{7} - 108 q^{8}+O(q^{10}) $ 8 * q + 2 * q^2 - 22 * q^4 + 12 * q^5 + 14 * q^7 - 108 * q^8	$ 8 q + 2 q^{2} - 22 q^{4} + 12 q^{5} + 14 q^{7} - 108 q^{8} + 62 q^{10} + 40 q^{11} - 60 q^{13} - 80 q^{14} - 122 q^{16} + 98 q^{17} - 124 q^{19} - 466 q^{20} - 220 q^{22} + 104 q^{23} - 116 q^{25} - 14 q^{26} + 144 q^{28} + 194 q^{29} + 52 q^{31} + 654 q^{32} + 2124 q^{34} + 88 q^{35} - 102 q^{37} - 664 q^{38} - 1996 q^{40} - 1054 q^{41} - 450 q^{43} + 88 q^{44} + 172 q^{46} + 192 q^{47} - 1070 q^{49} + 996 q^{50} + 2280 q^{52} - 524 q^{53} - 204 q^{55} + 2164 q^{56} - 722 q^{58} + 308 q^{59} + 928 q^{61} + 2780 q^{62} + 2052 q^{64} - 2346 q^{65} + 1134 q^{67} + 1786 q^{68} - 4648 q^{70} + 1064 q^{71} + 1904 q^{73} + 1158 q^{74} + 1708 q^{76} - 5016 q^{77} - 1492 q^{79} - 2922 q^{80} - 1734 q^{82} + 808 q^{83} + 1394 q^{85} - 6336 q^{86} - 3060 q^{88} + 1620 q^{89} + 3278 q^{91} - 664 q^{92} + 772 q^{94} + 2204 q^{95} - 2166 q^{97} - 1906 q^{98}+O(q^{100}) $ 8 * q + 2 * q^2 - 22 * q^4 + 12 * q^5 + 14 * q^7 - 108 * q^8 + 62 * q^10 + 40 * q^11 - 60 * q^13 - 80 * q^14 - 122 * q^16 + 98 * q^17 - 124 * q^19 - 466 * q^20 - 220 * q^22 + 104 * q^23 - 116 * q^25 - 14 * q^26 + 144 * q^28 + 194 * q^29 + 52 * q^31 + 654 * q^32 + 2124 * q^34 + 88 * q^35 - 102 * q^37 - 664 * q^38 - 1996 * q^40 - 1054 * q^41 - 450 * q^43 + 88 * q^44 + 172 * q^46 + 192 * q^47 - 1070 * q^49 + 996 * q^50 + 2280 * q^52 - 524 * q^53 - 204 * q^55 + 2164 * q^56 - 722 * q^58 + 308 * q^59 + 928 * q^61 + 2780 * q^62 + 2052 * q^64 - 2346 * q^65 + 1134 * q^67 + 1786 * q^68 - 4648 * q^70 + 1064 * q^71 + 1904 * q^73 + 1158 * q^74 + 1708 * q^76 - 5016 * q^77 - 1492 * q^79 - 2922 * q^80 - 1734 * q^82 + 808 * q^83 + 1394 * q^85 - 6336 * q^86 - 3060 * q^88 + 1620 * q^89 + 3278 * q^91 - 664 * q^92 + 772 * q^94 + 2204 * q^95 - 2166 * q^97 - 1906 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

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 $

$2$	2.66520	−	4.61626i	0.942289	−	1.63209i	0.181199	−	0.983446i	$-0.442002\pi$
$2$	2.66520	−	4.61626i	0.942289	−	1.63209i	0.761090	−	0.648647i	$-0.224665\pi$
$3$		0			0
$3$		0			0
$4$	−10.2065	−	17.6783i	−1.27582	−	2.20978i
$5$	16.4131			1.46803			0.734016	−	0.679132i	$-0.237644\pi$
$5$	16.4131			1.46803			0.734016	+	0.679132i	$0.237644\pi$
$6$		0			0
$7$	−4.83984	−	8.38285i	−0.261327	−	0.452631i	0.705268	−	0.708941i	$-0.250827\pi$
$7$	−4.83984	−	8.38285i	−0.261327	−	0.452631i	−0.966595	+	0.256309i	$0.917493\pi$
$8$	−66.1667			−2.92418
$9$		0			0
$10$	43.7441	−	75.7670i	1.38331	−	2.39596i
$11$	13.7941	−	23.8921i	0.378098	−	0.654884i	−0.612688	−	0.790325i	$-0.709912\pi$
$11$	13.7941	−	23.8921i	0.378098	−	0.654884i	0.990785	+	0.135441i	$0.0432451\pi$
$12$		0			0
$13$	−37.3033	+	28.3807i	−0.795852	+	0.605491i
$13$	−37.3033	+	28.3807i	−0.795852	+	0.605491i
$14$	−51.5965			−0.984982
$15$		0			0
$16$	−94.6948	+	164.016i	−1.47961	+	2.56275i
$17$	53.9641	+	93.4685i	0.769895	+	1.33350i	0.937619	+	0.347663i	$0.113025\pi$
$17$	53.9641	+	93.4685i	0.769895	+	1.33350i	−0.167725	+	0.985834i	$0.553642\pi$
$18$		0			0
$19$	1.12362	+	1.94616i	0.0135671	+	0.0234989i	0.872729	−	0.488205i	$-0.162348\pi$
$19$	1.12362	+	1.94616i	0.0135671	+	0.0234989i	−0.859162	+	0.511703i	$0.829015\pi$
$20$	−167.521	−	290.155i	−1.87294	−	3.24403i
$21$		0			0
$22$	−73.5279	−	127.354i	−0.712554	−	1.23418i
$23$	20.9045	−	36.2077i	0.189517	−	0.328253i	−0.755572	−	0.655065i	$-0.772641\pi$
$23$	20.9045	−	36.2077i	0.189517	−	0.328253i	0.945089	+	0.326812i	$0.105974\pi$
$24$		0			0
$25$	144.390			1.15512
$26$	31.5919	+	247.842i	0.238296	+	1.86945i
$27$		0			0
$28$	−98.7961	+	171.120i	−0.666811	+	1.15495i
$29$	30.8106	−	53.3656i	0.197289	−	0.341715i	−0.750359	−	0.661030i	$-0.770119\pi$
$29$	30.8106	−	53.3656i	0.197289	−	0.341715i	0.947649	+	0.319315i	$0.103453\pi$
$30$		0			0
$31$	191.932			1.11200			0.556000	−	0.831182i	$-0.312335\pi$
$31$	191.932			1.11200			0.556000	+	0.831182i	$0.312335\pi$
$32$	240.094	+	415.855i	1.32634	+	2.29729i
$33$		0			0
$34$	575.300			2.90185
$35$	−79.4368	−	137.589i	−0.383636	−	0.664477i
$36$		0			0
$37$	−49.2118	+	85.2373i	−0.218659	+	0.378728i	−0.954398	−	0.298537i	$-0.903501\pi$
$37$	−49.2118	+	85.2373i	−0.218659	+	0.378728i	0.735740	+	0.677265i	$0.236835\pi$
$38$	11.9786			0.0511366
$39$		0			0
$40$	−1086.00			−4.29279
$41$	−15.3726	+	26.6261i	−0.0585561	+	0.101422i	−0.893817	−	0.448431i	$-0.851983\pi$
$41$	−15.3726	+	26.6261i	−0.0585561	+	0.101422i	0.835261	+	0.549853i	$0.185316\pi$
$42$		0			0
$43$	−119.163	−	206.396i	−0.422608	−	0.731978i	0.573586	−	0.819145i	$-0.305552\pi$
$43$	−119.163	−	206.396i	−0.422608	−	0.731978i	−0.996194	+	0.0871672i	$0.972219\pi$
$44$	−563.160			−1.92953
$45$		0			0
$46$	−111.429	−	193.001i	−0.357160	−	0.618618i
$47$	511.482			1.58739			0.793695	−	0.608316i	$-0.208155\pi$
$47$	511.482			1.58739			0.793695	+	0.608316i	$0.208155\pi$
$48$		0			0
$49$	124.652	−	215.903i	0.363416	−	0.629456i
$50$	384.826	−	666.539i	1.08845	−	1.88526i
$51$		0			0
$52$	882.459	+	369.788i	2.35337	+	0.986162i
$53$	−492.825			−1.27726			−0.638630	−	0.769514i	$-0.720498\pi$
$53$	−492.825			−1.27726			−0.638630	+	0.769514i	$0.720498\pi$
$54$		0			0
$55$	226.404	−	392.142i	0.555059	−	0.961390i
$56$	320.236	+	554.665i	0.764167	+	1.32358i
$57$		0			0
$58$	−164.233	−	284.460i	−0.371807	−	0.643989i
$59$	242.089	+	419.311i	0.534192	+	0.925248i	0.999202	+	0.0399427i	$0.0127175\pi$
$59$	242.089	+	419.311i	0.534192	+	0.925248i	−0.465010	+	0.885306i	$0.653949\pi$
$60$		0			0
$61$	222.011	+	384.534i	0.465993	+	0.807123i	0.999246	−	0.0388329i	$-0.0123640\pi$
$61$	222.011	+	384.534i	0.465993	+	0.807123i	−0.533253	+	0.845956i	$0.679031\pi$
$62$	511.536	−	886.007i	1.04783	−	1.81489i
$63$		0			0
$64$	1044.47			2.03998
$65$	−612.262	+	465.815i	−1.16834	+	0.888880i
$66$		0			0
$67$	−95.0568	+	164.643i	−0.173329	+	0.300215i	−0.939582	−	0.342325i	$-0.888786\pi$
$67$	−95.0568	+	164.643i	−0.173329	+	0.300215i	0.766253	+	0.642539i	$0.222119\pi$
$68$	1101.57	−	1907.98i	1.96449	−	3.40260i
$69$		0			0
$70$	−846.858			−1.44598
$71$	242.392	+	419.836i	0.405164	+	0.701765i	0.994341	−	0.106239i	$-0.0338810\pi$
$71$	242.392	+	419.836i	0.405164	+	0.701765i	−0.589176	+	0.808005i	$0.700548\pi$
$72$		0			0
$73$	−957.780			−1.53561			−0.767806	−	0.640683i	$-0.778651\pi$
$73$	−957.780			−1.53561			−0.767806	+	0.640683i	$0.778651\pi$
$74$	262.318	+	454.348i	0.412079	+	0.713742i
$75$		0			0
$76$	22.9365	−	39.7271i	0.0346183	−	0.0599607i
$77$	−267.045			−0.395228
$78$		0			0
$79$	−375.216			−0.534368			−0.267184	−	0.963646i	$-0.586093\pi$
$79$	−375.216			−0.534368			−0.267184	+	0.963646i	$0.586093\pi$
$80$	−1554.23	+	2692.01i	−2.17211	+	3.76220i
$81$		0			0
$82$	81.9421	+	141.928i	0.110354	+	0.191138i
$83$	715.765			0.946571			0.473286	−	0.880909i	$-0.343068\pi$
$83$	715.765			0.946571			0.473286	+	0.880909i	$0.343068\pi$
$84$		0			0
$85$	885.717	+	1534.11i	1.13023	+	1.95762i
$86$	−1270.37			−1.59288
$87$		0			0
$88$	−912.708	+	1580.86i	−1.10563	+	1.91500i
$89$	−519.076	+	899.066i	−0.618224	+	1.07080i	0.371585	+	0.928399i	$0.378814\pi$
$89$	−519.076	+	899.066i	−0.618224	+	1.07080i	−0.989810	+	0.142397i	$0.954519\pi$
$90$		0			0
$91$	418.453	+	175.350i	0.482042	+	0.201996i
$92$	−853.451			−0.967157
$93$		0			0
$94$	1363.20	−	2361.13i	1.49578	−	2.59077i
$95$	18.4420	+	31.9425i	0.0199169	+	0.0344972i
$96$		0			0
$97$	−32.7818	−	56.7797i	−0.0343143	−	0.0594341i	0.848358	−	0.529423i	$-0.177591\pi$
$97$	−32.7818	−	56.7797i	−0.0343143	−	0.0594341i	−0.882673	+	0.469989i	$0.844258\pi$
$98$	−664.443	−	1150.85i	−0.684887	−	1.18626i
$99$		0			0
$100$	−1473.72	−	2552.56i	−1.47372	−	2.55256i
$101$	265.899	−	460.551i	0.261960	−	0.453728i	−0.704803	−	0.709403i	$-0.748965\pi$
$101$	265.899	−	460.551i	0.261960	−	0.453728i	0.966763	+	0.255676i	$0.0822979\pi$
$102$		0			0
$103$	−735.984			−0.704064			−0.352032	−	0.935988i	$-0.614509\pi$
$103$	−735.984			−0.704064			−0.352032	+	0.935988i	$0.614509\pi$
$104$	2468.23	−	1877.86i	2.32721	−	1.77057i
$105$		0			0
$106$	−1313.48	+	2275.01i	−1.20355	+	2.08461i
$107$	−391.632	+	678.327i	−0.353837	+	0.612863i	−0.986918	−	0.161222i	$-0.948456\pi$
$107$	−391.632	+	678.327i	−0.353837	+	0.612863i	0.633081	+	0.774085i	$0.281790\pi$
$108$		0			0
$109$	−532.339			−0.467788			−0.233894	−	0.972262i	$-0.575147\pi$
$109$	−532.339			−0.467788			−0.233894	+	0.972262i	$0.575147\pi$
$110$	−1206.82	−	2090.27i	−1.04605	−	1.81182i
$111$		0			0
$112$	1833.23			1.54664
$113$	−90.2946	−	156.395i	−0.0751699	−	0.130198i	0.825990	−	0.563684i	$-0.190617\pi$
$113$	−90.2946	−	156.395i	−0.0751699	−	0.130198i	−0.901160	+	0.433486i	$0.857283\pi$
$114$		0			0
$115$	343.107	−	594.280i	0.278217	−	0.481886i
$116$	−1257.88			−1.00682
$117$		0			0
$118$	2580.86			2.01346
$119$	522.355	−	904.746i	0.402389	−	0.696957i
$120$		0			0
$121$	284.947	+	493.542i	0.214085	+	0.370805i
$122$	2366.81			1.75640
$123$		0			0
$124$	−1958.96	−	3393.02i	−1.41871	−	2.45728i
$125$	318.242			0.227716
$126$		0			0
$127$	−715.817	+	1239.83i	−0.500146	+	0.866278i	0.499854	+	0.866110i	$0.333387\pi$
$127$	−715.817	+	1239.83i	−0.500146	+	0.866278i	−1.00000		0.000168331i	$0.999946\pi$
$128$	862.972	−	1494.71i	0.595912	−	1.03215i
$129$		0			0
$130$	518.521	+	4067.85i	0.349825	+	2.74441i
$131$	−2067.32			−1.37880			−0.689400	−	0.724381i	$-0.742126\pi$
$131$	−2067.32			−1.37880			−0.689400	+	0.724381i	$0.742126\pi$
$132$		0			0
$133$	10.8762	−	18.8382i	0.00709090	−	0.0122818i
$134$	506.690	+	877.613i	0.326652	+	0.565778i
$135$		0			0
$136$	−3570.62	−	6184.50i	−2.25131	−	3.89939i
$137$	−193.756	−	335.595i	−0.120830	−	0.209283i	0.799265	−	0.600978i	$-0.205222\pi$
$137$	−193.756	−	335.595i	−0.120830	−	0.209283i	−0.920095	+	0.391695i	$0.871889\pi$
$138$		0			0
$139$	−376.284	−	651.743i	−0.229611	−	0.397699i	0.728082	−	0.685491i	$-0.240412\pi$
$139$	−376.284	−	651.743i	−0.229611	−	0.397699i	−0.957693	+	0.287792i	$0.907079\pi$
$140$	−1621.55	+	2808.61i	−0.978900	+	1.69550i
$141$		0			0
$142$	2584.09			1.52713
$143$	163.508	+	1282.74i	0.0956171	+	0.750126i
$144$		0			0
$145$	505.698	−	875.894i	0.289627	−	0.501649i
$146$	−2552.67	+	4421.36i	−1.44699	+	2.50626i
$147$		0			0
$148$	2009.13			1.11587
$149$	−1318.36	−	2283.47i	−0.724862	−	1.25550i	−0.959031	−	0.283301i	$-0.908570\pi$
$149$	−1318.36	−	2283.47i	−0.724862	−	1.25550i	0.234169	−	0.972196i	$-0.424763\pi$
$150$		0			0
$151$	−3332.42			−1.79595			−0.897975	−	0.440046i	$-0.854962\pi$
$151$	−3332.42			−1.79595			−0.897975	+	0.440046i	$0.854962\pi$
$152$	−74.3459	−	128.771i	−0.0396727	−	0.0687151i
$153$		0			0
$154$	−711.727	+	1232.75i	−0.372419	+	0.645049i
$155$	3150.20			1.63245
$156$		0			0
$157$	−1625.26			−0.826179			−0.413089	−	0.910690i	$-0.635550\pi$
$157$	−1625.26			−0.826179			−0.413089	+	0.910690i	$0.635550\pi$
$158$	−1000.02	+	1732.09i	−0.503529	+	0.872138i
$159$		0			0
$160$	3940.68	+	6825.46i	1.94711	+	3.37250i
$161$	−404.698			−0.198104
$162$		0			0
$163$	917.683	+	1589.47i	0.440972	+	0.763786i	0.997762	−	0.0668673i	$-0.0213004\pi$
$163$	917.683	+	1589.47i	0.440972	+	0.763786i	−0.556790	+	0.830653i	$0.687967\pi$
$164$	627.605			0.298828
$165$		0			0
$166$	1907.65	−	3304.15i	0.891944	−	1.54489i
$167$	972.498	−	1684.42i	0.450624	−	0.780503i	−0.547801	−	0.836609i	$-0.684535\pi$
$167$	972.498	−	1684.42i	0.450624	−	0.780503i	0.998425	+	0.0561052i	$0.0178682\pi$
$168$		0			0
$169$	586.072	−	2117.39i	0.266760	−	0.963763i
$170$	9442.44			4.26001
$171$		0			0
$172$	−2432.48	+	4213.18i	−1.07834	+	1.86774i
$173$	1265.81	+	2192.45i	0.556289	+	0.963522i	0.997802	+	0.0662666i	$0.0211088\pi$
$173$	1265.81	+	2192.45i	0.556289	+	0.963522i	−0.441512	+	0.897255i	$0.645558\pi$
$174$		0			0
$175$	−698.823	−	1210.40i	−0.301863	−	0.522842i
$176$	2612.46	+	4524.90i	1.11887	+	1.93794i
$177$		0			0
$178$	2766.88	+	4792.38i	1.16509	+	2.01800i
$179$	2131.51	−	3691.88i	0.890035	−	1.54159i	0.0502037	−	0.998739i	$-0.484013\pi$
$179$	2131.51	−	3691.88i	0.890035	−	1.54159i	0.839831	−	0.542847i	$-0.182654\pi$
$180$		0			0
$181$	3944.61			1.61989			0.809946	−	0.586504i	$-0.199496\pi$
$181$	3944.61			1.61989			0.809946	+	0.586504i	$0.199496\pi$
$182$	1924.72	−	1464.35i	0.783900	−	0.596398i
$183$		0			0
$184$	−1383.18	+	2395.74i	−0.554182	+	0.959871i
$185$	−807.717	+	1399.01i	−0.320998	+	0.555984i
$186$		0			0
$187$	2977.54			1.16438
$188$	−5220.46	−	9042.11i	−2.02522	−	3.50779i
$189$		0			0
$190$	196.606			0.0750701
$191$	−107.054	−	185.424i	−0.0405559	−	0.0702449i	0.845035	−	0.534711i	$-0.179580\pi$
$191$	−107.054	−	185.424i	−0.0405559	−	0.0702449i	−0.885591	+	0.464466i	$0.846246\pi$
$192$		0			0
$193$	603.593	−	1045.45i	0.225117	−	0.389914i	−0.731238	−	0.682123i	$-0.761057\pi$
$193$	603.593	−	1045.45i	0.225117	−	0.389914i	0.956355	+	0.292209i	$0.0943902\pi$
$194$	−349.480			−0.129336
$195$		0			0
$196$	−5089.06			−1.85461
$197$	−463.816	+	803.352i	−0.167744	+	0.290541i	−0.937626	−	0.347645i	$-0.886981\pi$
$197$	−463.816	+	803.352i	−0.167744	+	0.290541i	0.769883	+	0.638186i	$0.220315\pi$
$198$		0			0
$199$	−239.476	−	414.784i	−0.0853064	−	0.147755i	0.820215	−	0.572055i	$-0.193854\pi$
$199$	−239.476	−	414.784i	−0.0853064	−	0.147755i	−0.905522	+	0.424300i	$0.860520\pi$
$200$	−9553.77			−3.37777
$201$		0			0
$202$	−1417.35	−	2454.92i	−0.493684	−	0.855086i
$203$	−596.474			−0.206228
$204$		0			0
$205$	−252.312	+	437.017i	−0.0859621	+	0.148891i
$206$	−1961.54	+	3397.49i	−0.663432	+	1.14910i
$207$		0			0
$208$	−1122.46	−	8805.85i	−0.374178	−	2.93546i
$209$	61.9970			0.0205188
$210$		0			0
$211$	725.477	−	1256.56i	0.236701	−	0.409978i	−0.723065	−	0.690780i	$-0.757267\pi$
$211$	725.477	−	1256.56i	0.236701	−	0.409978i	0.959766	+	0.280802i	$0.0906005\pi$
$212$	5030.04	+	8712.29i	1.62955	+	2.82246i
$213$		0			0
$214$	2087.55	+	3615.75i	0.666833	+	1.15499i
$215$	−1955.83	−	3387.59i	−0.620402	−	1.07457i
$216$		0			0
$217$	−928.920	−	1608.94i	−0.290595	−	0.503326i
$218$	−1418.79	+	2457.41i	−0.440791	+	0.763473i
$219$		0			0
$220$	−9243.19			−2.83262
$221$	−4665.74	−	1955.15i	−1.42014	−	0.595101i
$222$		0			0
$223$	1029.89	−	1783.83i	0.309268	−	0.535668i	−0.668935	−	0.743321i	$-0.733249\pi$
$223$	1029.89	−	1783.83i	0.309268	−	0.535668i	0.978202	+	0.207654i	$0.0665827\pi$
$224$	2324.03	−	4025.34i	0.693218	−	1.20069i
$225$		0			0
$226$	−962.612			−0.283327
$227$	2241.23	+	3881.93i	0.655311	+	1.13503i	0.981816	+	0.189837i	$0.0607959\pi$
$227$	2241.23	+	3881.93i	0.655311	+	1.13503i	−0.326504	+	0.945196i	$0.605871\pi$
$228$		0			0
$229$	−1630.39			−0.470477			−0.235239	−	0.971938i	$-0.575587\pi$
$229$	−1630.39			−0.470477			−0.235239	+	0.971938i	$0.575587\pi$
$230$	−1828.90	−	3167.74i	−0.524321	−	0.908151i
$231$		0			0
$232$	−2038.64	+	3531.02i	−0.576910	+	0.999237i
$233$	1903.69			0.535258			0.267629	−	0.963522i	$-0.413760\pi$
$233$	1903.69			0.535258			0.267629	+	0.963522i	$0.413760\pi$
$234$		0			0
$235$	8395.00			2.33034
$236$	4941.79	−	8559.44i	1.36306	−	2.36090i
$237$		0			0
$238$	−2784.36	−	4822.65i	−0.758333	−	1.31347i
$239$	−3763.79			−1.01866			−0.509328	−	0.860572i	$-0.670106\pi$
$239$	−3763.79			−1.01866			−0.509328	+	0.860572i	$0.670106\pi$
$240$		0			0
$241$	1807.37	+	3130.46i	0.483083	+	0.836724i	0.999811	−	0.0194250i	$-0.00618357\pi$
$241$	1807.37	+	3130.46i	0.483083	+	0.836724i	−0.516728	+	0.856149i	$0.672850\pi$
$242$	3037.75			0.806918
$243$		0			0
$244$	4531.92	−	7849.52i	1.18904	−	2.05948i
$245$	2045.92	−	3543.64i	0.533507	−	0.924061i
$246$		0			0
$247$	−97.1479	−	40.7092i	−0.0250258	−	0.0104869i
$248$	−12699.5			−3.25169
$249$		0			0
$250$	848.178	−	1469.09i	0.214574	−	0.371653i
$251$	−2864.88	−	4962.12i	−0.720438	−	1.24783i	−0.960824	−	0.277158i	$-0.910608\pi$
$251$	−2864.88	−	4962.12i	−0.720438	−	1.24783i	0.240387	−	0.970677i	$-0.422726\pi$
$252$		0			0
$253$	−576.717	−	998.903i	−0.143312	−	0.248223i
$254$	3815.59	+	6608.79i	0.942564	+	1.63257i
$255$		0			0
$256$	−422.095	−	731.091i	−0.103051	−	0.178489i
$257$	−2762.89	+	4785.47i	−0.670602	+	1.16152i	0.307132	+	0.951667i	$0.400631\pi$
$257$	−2762.89	+	4785.47i	−0.670602	+	1.16152i	−0.977734	+	0.209849i	$0.932703\pi$
$258$		0			0
$259$	952.709			0.228565
$260$	14483.9	+	6069.37i	3.45482	+	1.44772i
$261$		0			0
$262$	−5509.82	+	9543.28i	−1.29923	+	2.25033i
$263$	2611.60	−	4523.43i	0.612313	−	1.06056i	−0.378536	−	0.925586i	$-0.623573\pi$
$263$	2611.60	−	4523.43i	0.612313	−	1.06056i	0.990850	−	0.134971i	$-0.0430941\pi$
$264$		0			0
$265$	−8088.78			−1.87506
$266$	−57.9747	−	100.415i	−0.0133634	−	0.0231460i
$267$		0			0
$268$	3880.81			0.884545
$269$	3601.94	+	6238.75i	0.816410	+	1.41406i	0.908311	+	0.418295i	$0.137372\pi$
$269$	3601.94	+	6238.75i	0.816410	+	1.41406i	−0.0919010	+	0.995768i	$0.529294\pi$
$270$		0			0
$271$	4288.84	−	7428.49i	0.961360	−	1.66512i	0.242269	−	0.970209i	$-0.422108\pi$
$271$	4288.84	−	7428.49i	0.961360	−	1.66512i	0.719091	−	0.694916i	$-0.244558\pi$
$272$	−20440.5			−4.55656
$273$		0			0
$274$	−2065.59			−0.455427
$275$	1991.72	−	3449.76i	0.436747	−	0.756467i
$276$		0			0
$277$	−3584.60	−	6208.70i	−0.777536	−	1.34673i	−0.933358	−	0.358947i	$-0.883136\pi$
$277$	−3584.60	−	6208.70i	−0.777536	−	1.34673i	0.155822	−	0.987785i	$-0.450197\pi$
$278$	−4011.48			−0.865442
$279$		0			0
$280$	5256.06	+	9103.77i	1.12182	+	1.94305i
$281$	−849.157			−0.180272			−0.0901360	−	0.995929i	$-0.528730\pi$
$281$	−849.157			−0.180272			−0.0901360	+	0.995929i	$0.528730\pi$
$282$		0			0
$283$	−557.686	+	965.941i	−0.117141	+	0.202895i	−0.918634	−	0.395110i	$-0.870706\pi$
$283$	−557.686	+	965.941i	−0.117141	+	0.202895i	0.801492	+	0.598005i	$0.204040\pi$
$284$	4947.97	−	8570.14i	1.03383	−	1.79065i
$285$		0			0
$286$	6357.23	+	2663.95i	1.31437	+	0.550779i
$287$	297.604			0.0612091
$288$		0			0
$289$	−3367.75	+	5833.11i	−0.685476	+	1.18728i
$290$	−2695.57	−	4668.86i	−0.545825	−	0.945396i
$291$		0			0
$292$	9775.62	+	16931.9i	1.95916	+	3.39337i
$293$	−931.764	−	1613.86i	−0.185782	−	0.321784i	0.758058	−	0.652188i	$-0.226149\pi$
$293$	−931.764	−	1613.86i	−0.185782	−	0.321784i	−0.943840	+	0.330403i	$0.892815\pi$
$294$		0			0
$295$	3973.43	+	6882.19i	0.784211	+	1.35829i
$296$	3256.18	−	5639.87i	0.639397	−	1.10747i
$297$		0			0
$298$	−14054.8			−2.73212
$299$	247.792	+	1943.95i	0.0479269	+	0.375992i
$300$		0			0
$301$	−1153.46	+	1997.85i	−0.220878	+	0.382571i
$302$	−8881.55	+	15383.3i	−1.69230	+	2.93116i
$303$		0			0
$304$	−425.602			−0.0802959
$305$	3643.88	+	6311.39i	0.684092	+	1.18488i
$306$		0			0
$307$	−6387.50			−1.18747			−0.593736	−	0.804660i	$-0.702348\pi$
$307$	−6387.50			−1.18747			−0.593736	+	0.804660i	$0.702348\pi$
$308$	2725.60	+	4720.89i	0.504239	+	0.873368i
$309$		0			0
$310$	8395.89	−	14542.1i	1.53824	−	2.66431i
$311$	3492.59			0.636806			0.318403	−	0.947955i	$-0.396853\pi$
$311$	3492.59			0.636806			0.318403	+	0.947955i	$0.396853\pi$
$312$		0			0
$313$	−5912.01			−1.06762			−0.533812	−	0.845603i	$-0.679241\pi$
$313$	−5912.01			−1.06762			−0.533812	+	0.845603i	$0.679241\pi$
$314$	−4331.64	+	7502.63i	−0.778499	+	1.34840i
$315$		0			0
$316$	3829.66	+	6633.16i	0.681756	+	1.18084i
$317$	1677.54			0.297224			0.148612	−	0.988896i	$-0.452519\pi$
$317$	1677.54			0.297224			0.148612	+	0.988896i	$0.452519\pi$
$318$		0			0
$319$	−850.009	−	1472.26i	−0.149189	−	0.258403i
$320$	17143.0			2.99476
$321$		0			0
$322$	−1078.60	+	1868.19i	−0.186671	+	0.323323i
$323$	−121.270	+	210.045i	−0.0208905	+	0.0361834i
$324$		0			0
$325$	−5386.21	+	4097.88i	−0.919301	+	0.699413i
$326$	9783.22			1.66209
$327$		0			0
$328$	1017.15	−	1761.76i	0.171228	−	0.296576i
$329$	−2475.49	−	4287.68i	−0.414828	−	0.718503i
$330$		0			0
$331$	−1005.15	−	1740.98i	−0.166913	−	0.289102i	0.770420	−	0.637537i	$-0.220047\pi$
$331$	−1005.15	−	1740.98i	−0.166913	−	0.289102i	−0.937333	+	0.348435i	$0.886713\pi$
$332$	−7305.49	−	12653.5i	−1.20765	−	2.09172i
$333$		0			0
$334$	−5183.80	−	8978.60i	−0.849236	−	1.47092i
$335$	−1560.18	+	2702.30i	−0.254452	+	0.440724i
$336$		0			0
$337$	7139.24			1.15400			0.577002	−	0.816743i	$-0.304222\pi$
$337$	7139.24			1.15400			0.577002	+	0.816743i	$0.304222\pi$
$338$	−8212.40	−	8348.71i	−1.32159	−	1.34352i
$339$		0			0
$340$	18080.2	−	31315.9i	2.88394	−	4.99512i
$341$	2647.53	−	4585.65i	0.420444	−	0.728231i
$342$		0			0
$343$	−5733.31			−0.902536
$344$	7884.60	+	13656.5i	1.23578	+	2.14044i
$345$		0			0
$346$	13494.6			2.09674
$347$	0.569949	+	0.987181i	8.81743e−5	+	0.000152722i	0.866069	−	0.499924i	$-0.166639\pi$
$347$	0.569949	+	0.987181i	8.81743e−5	+	0.000152722i	−0.865981	+	0.500076i	$0.833305\pi$
$348$		0			0
$349$	−6099.55	+	10564.7i	−0.935535	+	1.62039i	−0.161857	+	0.986814i	$0.551748\pi$
$349$	−6099.55	+	10564.7i	−0.935535	+	1.62039i	−0.773678	+	0.633579i	$0.781585\pi$
$350$	−7450.00			−1.13777
$351$		0			0
$352$	13247.5			2.00595
$353$	−5446.15	+	9433.01i	−0.821160	+	1.42229i	0.0836595	+	0.996494i	$0.473339\pi$
$353$	−5446.15	+	9433.01i	−0.821160	+	1.42229i	−0.904819	+	0.425796i	$0.859994\pi$
$354$		0			0
$355$	3978.41	+	6890.80i	0.594794	+	1.03021i
$356$	21191.9			3.15497
$357$		0			0
$358$	−11361.8	−	19679.2i	−1.67734	−	2.90524i
$359$	3525.78			0.518339			0.259169	−	0.965832i	$-0.416551\pi$
$359$	3525.78			0.518339			0.259169	+	0.965832i	$0.416551\pi$
$360$		0			0
$361$	3426.97	−	5935.69i	0.499632	−	0.865388i
$362$	10513.2	−	18209.3i	1.52641	−	2.64382i
$363$		0			0
$364$	−1171.08	−	9187.24i	−0.168630	−	1.32292i
$365$	−15720.1			−2.25433
$366$		0			0
$367$	−1191.88	+	2064.39i	−0.169525	+	0.293625i	−0.938253	−	0.345950i	$-0.887557\pi$
$367$	−1191.88	+	2064.39i	−0.169525	+	0.293625i	0.768728	+	0.639576i	$0.220890\pi$
$368$	3959.09	+	6857.35i	0.560821	+	0.971370i
$369$		0			0
$370$	4305.45	+	7457.26i	0.604945	+	1.04780i
$371$	2385.20	+	4131.28i	0.333782	+	0.578128i
$372$		0			0
$373$	6641.10	+	11502.7i	0.921885	+	1.59675i	0.796495	+	0.604645i	$0.206685\pi$
$373$	6641.10	+	11502.7i	0.921885	+	1.59675i	0.125390	+	0.992108i	$0.459982\pi$
$374$	7935.73	−	13745.1i	1.09718	−	1.90038i
$375$		0			0
$376$	−33843.1			−4.64181
$377$	365.214	+	2865.14i	0.0498925	+	0.391412i
$378$		0			0
$379$	2218.36	−	3842.32i	0.300659	−	0.520756i	−0.675627	−	0.737244i	$-0.736127\pi$
$379$	2218.36	−	3842.32i	0.300659	−	0.520756i	0.976285	+	0.216488i	$0.0694601\pi$
$380$	376.458	−	652.045i	0.0508208	−	0.0880242i
$381$		0			0
$382$	−1141.28			−0.152862
$383$	−405.206	−	701.838i	−0.0540602	−	0.0936351i	0.837729	−	0.546086i	$-0.183883\pi$
$383$	−405.206	−	701.838i	−0.0540602	−	0.0936351i	−0.891789	+	0.452451i	$0.850550\pi$
$384$		0			0
$385$	−4383.03			−0.580207
$386$	−3217.39	−	5572.68i	−0.424251	−	0.734824i
$387$		0			0
$388$	−669.177	+	1159.05i	−0.0875576	+	0.151654i
$389$	−3463.79			−0.451469			−0.225734	−	0.974189i	$-0.572478\pi$
$389$	−3463.79			−0.451469			−0.225734	+	0.974189i	$0.572478\pi$
$390$		0			0
$391$	4512.37			0.583633
$392$	−8247.80	+	14285.6i	−1.06270	+	1.84064i
$393$		0			0
$394$	2472.32	+	4282.18i	0.316126	+	0.547546i
$395$	−6158.45			−0.784469
$396$		0			0
$397$	−212.703	−	368.412i	−0.0268898	−	0.0465745i	0.852267	−	0.523106i	$-0.175227\pi$
$397$	−212.703	−	368.412i	−0.0268898	−	0.0465745i	−0.879157	+	0.476532i	$0.841894\pi$
$398$	−2553.00			−0.321533
$399$		0			0
$400$	−13672.9	+	23682.2i	−1.70912	+	2.96028i
$401$	−593.424	+	1027.84i	−0.0739007	+	0.128000i	−0.900608	−	0.434633i	$-0.856878\pi$
$401$	−593.424	+	1027.84i	−0.0739007	+	0.128000i	0.826707	+	0.562633i	$0.190211\pi$
$402$		0			0
$403$	−7159.70	+	5447.16i	−0.884987	+	0.673306i
$404$	−10855.6			−1.33685
$405$		0			0
$406$	−1589.72	+	2753.48i	−0.194327	+	0.336583i
$407$	1357.66	+	2351.54i	0.165349	+	0.286392i
$408$		0			0
$409$	−4003.71	−	6934.63i	−0.484036	−	0.838375i	0.515796	−	0.856711i	$-0.327496\pi$
$409$	−4003.71	−	6934.63i	−0.484036	−	0.838375i	−0.999832	+	0.0183369i	$0.994163\pi$
$410$	1344.92	+	2329.47i	0.162002	+	0.280596i
$411$		0			0
$412$	7511.85	+	13010.9i	0.898258	+	1.55583i
$413$	2343.35	−	4058.80i	0.279198	−	0.483585i
$414$		0			0
$415$	11747.9			1.38960
$416$	−20758.5	−	8698.72i	−2.44656	−	1.02522i
$417$		0			0
$418$	165.234	−	286.194i	0.0193346	−	0.0334885i
$419$	3416.23	−	5917.08i	0.398314	−	0.689901i	−0.595204	−	0.803575i	$-0.702929\pi$
$419$	3416.23	−	5917.08i	0.398314	−	0.689901i	0.993518	+	0.113674i	$0.0362620\pi$
$420$		0			0
$421$	10739.6			1.24326			0.621632	−	0.783309i	$-0.286470\pi$
$421$	10739.6			1.24326			0.621632	+	0.783309i	$0.286470\pi$
$422$	−3867.08	−	6697.98i	−0.446082	−	0.772636i
$423$		0			0
$424$	32608.6			3.73494
$425$	7791.85	+	13495.9i	0.889318	+	1.54034i
$426$		0			0
$427$	2148.99	−	3722.16i	0.243553	−	0.421846i
$428$	15988.9			1.80573
$429$		0			0
$430$	−20850.7			−2.33839
$431$	2607.22	−	4515.84i	0.291382	−	0.504688i	−0.682755	−	0.730647i	$-0.739218\pi$
$431$	2607.22	−	4515.84i	0.291382	−	0.504688i	0.974137	+	0.225959i	$0.0725517\pi$
$432$		0			0
$433$	−4321.12	−	7484.40i	−0.479584	−	0.830664i	0.520142	−	0.854080i	$-0.325879\pi$
$433$	−4321.12	−	7484.40i	−0.479584	−	0.830664i	−0.999726	+	0.0234161i	$0.992546\pi$
$434$	−9903.02			−1.09530
$435$		0			0
$436$	5433.34	+	9410.83i	0.596812	+	1.03371i
$437$	93.9545			0.0102848
$438$		0			0
$439$	6513.12	−	11281.1i	0.708097	−	1.22646i	−0.257466	−	0.966287i	$-0.582887\pi$
$439$	6513.12	−	11281.1i	0.708097	−	1.22646i	0.965562	−	0.260172i	$-0.0837793\pi$
$440$	−14980.4	+	25946.8i	−1.62309	+	2.81128i
$441$		0			0
$442$	−21460.6	+	16327.4i	−2.30945	+	1.75705i
$443$	11533.0			1.23690			0.618450	−	0.785824i	$-0.287761\pi$
$443$	11533.0			1.23690			0.618450	+	0.785824i	$0.287761\pi$
$444$		0			0
$445$	−8519.64	+	14756.5i	−0.907573	+	1.57196i
$446$	−5489.74	−	9508.50i	−0.582840	−	1.00951i
$447$		0			0
$448$	−5055.08	−	8755.65i	−0.533103	−	0.923361i
$449$	4941.37	+	8558.71i	0.519372	+	0.899578i	0.999747	+	0.0225149i	$0.00716731\pi$
$449$	4941.37	+	8558.71i	0.519372	+	0.899578i	−0.480375	+	0.877063i	$0.659499\pi$
$450$		0			0
$451$	424.102	+	734.566i	0.0442798	+	0.0766949i
$452$	−1843.19	+	3192.50i	−0.191806	+	0.332218i
$453$		0			0
$454$	23893.3			2.46997
$455$	6868.11	+	2878.04i	0.707653	+	0.296537i
$456$		0			0
$457$	7814.04	−	13534.3i	0.799836	−	1.38536i	−0.119886	−	0.992788i	$-0.538253\pi$
$457$	7814.04	−	13534.3i	0.799836	−	1.38536i	0.919722	−	0.392569i	$-0.128414\pi$
$458$	−4345.31	+	7526.31i	−0.443326	+	0.767863i
$459$		0			0
$460$	−14007.8			−1.41982
$461$	−3873.73	−	6709.50i	−0.391361	−	0.677858i	0.601268	−	0.799047i	$-0.294662\pi$
$461$	−3873.73	−	6709.50i	−0.391361	−	0.677858i	−0.992629	+	0.121190i	$0.961329\pi$
$462$		0			0
$463$	−333.422			−0.0334675			−0.0167337	−	0.999860i	$-0.505327\pi$
$463$	−333.422			−0.0334675			−0.0167337	+	0.999860i	$0.505327\pi$
$464$	5835.21	+	10106.9i	0.583821	+	1.01121i
$465$		0			0
$466$	5073.71	−	8787.93i	0.504368	−	0.873590i
$467$	−8198.33			−0.812363			−0.406182	−	0.913792i	$-0.633140\pi$
$467$	−8198.33			−0.812363			−0.406182	+	0.913792i	$0.633140\pi$
$468$		0			0
$469$	1840.24			0.181182
$470$	22374.3	−	38753.5i	2.19585	−	3.80333i
$471$		0			0
$472$	−16018.2	−	27744.4i	−1.56208	−	2.70559i
$473$	−6574.96			−0.639148
$474$		0			0
$475$	162.238	+	281.005i	0.0156716	+	0.0271440i
$476$	−21325.8			−2.05350
$477$		0			0
$478$	−10031.2	+	17374.6i	−0.959870	+	1.66254i
$479$	−3217.94	+	5573.64i	−0.306955	+	0.531662i	−0.977695	−	0.210031i	$-0.932643\pi$
$479$	−3217.94	+	5573.64i	−0.306955	+	0.531662i	0.670740	+	0.741693i	$0.265977\pi$
$480$		0			0
$481$	−583.332	−	4576.30i	−0.0552966	−	0.433807i
$482$	19268.0			1.82082
$483$		0			0
$484$	5816.64	−	10074.7i	0.546266	−	0.946160i
$485$	−538.050	−	931.931i	−0.0503745	−	0.0872511i
$486$		0			0
$487$	−4047.69	−	7010.80i	−0.376629	−	0.652340i	0.613941	−	0.789352i	$-0.289583\pi$
$487$	−4047.69	−	7010.80i	−0.376629	−	0.652340i	−0.990569	+	0.137012i	$0.956250\pi$
$488$	−14689.7	−	25443.3i	−1.36265	−	2.36017i
$489$		0			0
$490$	−10905.6	−	18889.0i	−1.00544	−	1.74147i
$491$	2558.23	−	4430.99i	0.235135	−	0.407266i	−0.724177	−	0.689614i	$-0.757780\pi$
$491$	2558.23	−	4430.99i	0.235135	−	0.407266i	0.959312	+	0.282348i	$0.0911134\pi$
$492$		0			0
$493$	6650.67			0.607568
$494$	−446.842	+	339.962i	−0.0406971	+	0.0309628i
$495$		0			0
$496$	−18175.0	+	31479.9i	−1.64532	+	2.84978i
$497$	2346.28	−	4063.88i	0.211761	−	0.366780i
$498$		0			0
$499$	−18050.7			−1.61936			−0.809682	−	0.586870i	$-0.800360\pi$
$499$	−18050.7			−1.61936			−0.809682	+	0.586870i	$0.800360\pi$
$500$	−3248.15	−	5625.97i	−0.290524	−	0.503202i
$501$		0			0
$502$	−30541.9			−2.71544
$503$	5265.53	+	9120.16i	0.466756	+	0.808445i	0.999279	−	0.0379705i	$-0.0120893\pi$
$503$	5265.53	+	9120.16i	0.466756	+	0.808445i	−0.532523	+	0.846416i	$0.678756\pi$
$504$		0			0
$505$	4364.22	−	7559.06i	0.384565	−	0.666087i
$506$	−6148.25			−0.540165
$507$		0			0
$508$	29224.1			2.55238
$509$	−981.654	+	1700.27i	−0.0854834	+	0.148062i	−0.905597	−	0.424139i	$-0.860577\pi$
$509$	−981.654	+	1700.27i	−0.0854834	+	0.148062i	0.820114	+	0.572201i	$0.193910\pi$
$510$		0			0
$511$	4635.50	+	8028.93i	0.401297	+	0.695066i
$512$	9307.69			0.803409
$513$		0			0
$514$	14727.3	+	25508.5i	1.26380	+	2.18897i
$515$	−12079.8			−1.03359
$516$		0			0
$517$	7055.43	−	12220.4i	0.600188	−	1.03956i
$518$	2539.16	−	4397.95i	0.215375	−	0.373040i
$519$		0			0
$520$	40511.4	−	30821.4i	3.41642	−	2.59925i
$521$	7044.93			0.592407			0.296203	−	0.955125i	$-0.404279\pi$
$521$	7044.93			0.592407			0.296203	+	0.955125i	$0.404279\pi$
$522$		0			0
$523$	1606.65	−	2782.79i	0.134328	−	0.232664i	−0.791012	−	0.611800i	$-0.790446\pi$
$523$	1606.65	−	2782.79i	0.134328	−	0.232664i	0.925341	+	0.379137i	$0.123779\pi$
$524$	21100.2	+	36546.6i	1.75910	+	3.04685i
$525$		0			0
$526$	−13920.9	−	24111.7i	−1.15395	−	1.99870i
$527$	10357.4	+	17939.6i	0.856123	+	1.48285i
$528$		0			0
$529$	5209.50	+	9023.13i	0.428167	+	0.741606i
$530$	−21558.2	+	37339.9i	−1.76685	+	3.06027i
$531$		0			0
$532$	−444.036			−0.0361868
$533$	−182.219	−	1429.53i	−0.0148082	−	0.116172i
$534$		0			0
$535$	−6427.90	+	11133.4i	−0.519443	+	0.899702i
$536$	6289.59	−	10893.9i	0.506845	−	0.877882i
$537$		0			0
$538$	38399.5			3.07718
$539$	−3438.92	−	5956.38i	−0.274814	−	0.475991i
$540$		0			0
$541$	11251.4			0.894150			0.447075	−	0.894497i	$-0.352466\pi$
$541$	11251.4			0.894150			0.447075	+	0.894497i	$0.352466\pi$
$542$	−22861.2	−	39596.8i	−1.81176	−	3.13806i
$543$		0			0
$544$	−25912.9	+	44882.4i	−2.04229	+	3.53735i
$545$	−8737.33			−0.686727
$546$		0			0
$547$	1533.54			0.119871			0.0599353	−	0.998202i	$-0.480911\pi$
$547$	1533.54			0.119871			0.0599353	+	0.998202i	$0.480911\pi$
$548$	−3955.16	+	6850.53i	−0.308314	+	0.534015i
$549$		0			0
$550$	−10616.7	−	18388.6i	−0.823083	−	1.42562i
$551$	138.477			0.0107066
$552$		0			0
$553$	1815.98	+	3145.38i	0.139645	+	0.241872i
$554$	−38214.6			−2.93066
$555$		0			0
$556$	−7681.12	+	13304.1i	−0.585885	+	1.01478i
$557$	8422.84	−	14588.8i	0.640731	−	1.10978i	−0.344539	−	0.938772i	$-0.611965\pi$
$557$	8422.84	−	14588.8i	0.640731	−	1.10978i	0.985270	−	0.171006i	$-0.0547019\pi$
$558$		0			0
$559$	10302.8	+	4317.33i	0.779540	+	0.326661i
$560$	30089.0			2.27052
$561$		0			0
$562$	−2263.17	+	3919.92i	−0.169868	+	0.294221i
$563$	−10410.0	−	18030.7i	−0.779273	−	1.34974i	−0.932361	−	0.361528i	$-0.882255\pi$
$563$	−10410.0	−	18030.7i	−0.779273	−	1.34974i	0.153089	−	0.988212i	$-0.451078\pi$
$564$		0			0
$565$	−1482.01	−	2566.92i	−0.110352	−	0.191135i
$566$	2972.69	+	5148.84i	0.220762	+	0.382371i
$567$		0			0
$568$	−16038.3	−	27779.1i	−1.18477	−	2.05209i
$569$	11818.3	−	20469.9i	0.870735	−	1.50816i	0.00949803	−	0.999955i	$-0.496977\pi$
$569$	11818.3	−	20469.9i	0.870735	−	1.50816i	0.861237	−	0.508203i	$-0.169690\pi$
$570$		0			0
$571$	−26955.1			−1.97554			−0.987771	−	0.155913i	$-0.950168\pi$
$571$	−26955.1			−1.97554			−0.987771	+	0.155913i	$0.950168\pi$
$572$	21007.7	−	15982.9i	1.53562	−	1.16832i
$573$		0			0
$574$	793.173	−	1373.82i	0.0576767	−	0.0998989i
$575$	3018.39	−	5228.01i	0.218914	−	0.379170i
$576$		0			0
$577$	23499.8			1.69551			0.847755	−	0.530388i	$-0.177954\pi$
$577$	23499.8			1.69551			0.847755	+	0.530388i	$0.177954\pi$
$578$	17951.4	+	31092.8i	1.29183	+	2.23752i
$579$		0			0
$580$	−20645.7			−1.47805
$581$	−3464.19	−	6000.15i	−0.247365	−	0.428448i
$582$		0			0
$583$	−6798.07	+	11774.6i	−0.482929	+	0.836457i
$584$	63373.1			4.49040
$585$		0			0
$586$	−9933.33			−0.700243
$587$	2318.75	−	4016.19i	0.163041	−	0.282395i	−0.772917	−	0.634507i	$-0.781203\pi$
$587$	2318.75	−	4016.19i	0.163041	−	0.282395i	0.935958	+	0.352112i	$0.114536\pi$
$588$		0			0
$589$	215.658	+	373.530i	0.0150866	+	0.0261308i
$590$	42359.9			2.95582
$591$		0			0
$592$	−9320.20	−	16143.1i	−0.647057	−	1.12074i
$593$	−12633.5			−0.874869			−0.437434	−	0.899250i	$-0.644113\pi$
$593$	−12633.5			−0.874869			−0.437434	+	0.899250i	$0.644113\pi$
$594$		0			0
$595$	8573.47	−	14849.7i	0.590719	−	1.02316i
$596$	−26911.8	+	46612.7i	−1.84958	+	3.20357i
$597$		0			0
$598$	9634.18	+	4037.14i	0.658814	+	0.276072i
$599$	18757.1			1.27946			0.639730	−	0.768600i	$-0.279046\pi$
$599$	18757.1			1.27946			0.639730	+	0.768600i	$0.279046\pi$
$600$		0			0
$601$	1816.49	−	3146.25i	0.123288	−	0.213541i	−0.797774	−	0.602956i	$-0.793989\pi$
$601$	1816.49	−	3146.25i	0.123288	−	0.213541i	0.921062	+	0.389415i	$0.127323\pi$
$602$	6148.38	+	10649.3i	0.416261	+	0.720986i
$603$		0			0
$604$	34012.5	+	58911.4i	2.29131	+	3.96866i
$605$	4676.85	+	8100.55i	0.314283	+	0.544354i
$606$		0			0
$607$	6349.99	+	10998.5i	0.424610	+	0.735445i	0.996384	−	0.0849656i	$-0.0270781\pi$
$607$	6349.99	+	10998.5i	0.424610	+	0.735445i	−0.571774	+	0.820411i	$0.693745\pi$
$608$	−539.546	+	934.522i	−0.0359893	+	0.0623353i
$609$		0			0
$610$	38846.6			2.57845
$611$	−19080.0	+	14516.2i	−1.26333	+	0.961151i
$612$		0			0
$613$	−10820.1	+	18740.9i	−0.712918	+	1.23481i	0.250839	+	0.968029i	$0.419293\pi$
$613$	−10820.1	+	18740.9i	−0.712918	+	1.23481i	−0.963757	+	0.266781i	$0.914040\pi$
$614$	−17023.9	+	29486.3i	−1.11894	+	1.93806i
$615$		0			0
$616$	17669.5			1.15572
$617$	−8270.85	−	14325.5i	−0.539663	−	0.934723i	−0.998922	−	0.0464208i	$-0.985219\pi$
$617$	−8270.85	−	14325.5i	−0.539663	−	0.934723i	0.459259	−	0.888302i	$-0.348115\pi$
$618$		0			0
$619$	−21138.9			−1.37261			−0.686303	−	0.727316i	$-0.740767\pi$
$619$	−21138.9			−1.37261			−0.686303	+	0.727316i	$0.740767\pi$
$620$	−32152.6	−	55690.0i	−2.08271	−	3.60736i
$621$		0			0
$622$	9308.44	−	16122.7i	0.600055	−	1.03933i
$623$	10049.0			0.646235
$624$		0			0
$625$	−12825.4			−0.820823
$626$	−15756.7	+	27291.3i	−1.00601	+	1.74246i
$627$		0			0
$628$	16588.3	+	28731.8i	1.05405	+	1.82568i
$629$	−10622.7			−0.673377
$630$		0			0
$631$	−2744.90	−	4754.31i	−0.173174	−	0.299946i	0.766354	−	0.642419i	$-0.222069\pi$
$631$	−2744.90	−	4754.31i	−0.173174	−	0.299946i	−0.939528	+	0.342473i	$0.888736\pi$
$632$	24826.8			1.56259
$633$		0			0
$634$	4470.97	−	7743.94i	0.280071	−	0.485097i
$635$	−11748.8	+	20349.5i	−0.734230	+	1.27172i
$636$		0			0
$637$	1477.56	+	11591.6i	0.0919044	+	0.720999i
$638$	−9061.76			−0.562318
$639$		0			0
$640$	14164.0	−	24532.8i	0.874817	−	1.51523i
$641$	2148.52	+	3721.35i	0.132389	+	0.229305i	0.924597	−	0.380946i	$-0.124402\pi$
$641$	2148.52	+	3721.35i	0.132389	+	0.229305i	−0.792208	+	0.610251i	$0.791068\pi$
$642$		0			0
$643$	−12848.5	−	22254.2i	−0.788016	−	1.36488i	−0.927181	−	0.374615i	$-0.877775\pi$
$643$	−12848.5	−	22254.2i	−0.788016	−	1.36488i	0.139164	−	0.990269i	$-0.455558\pi$
$644$	4130.57	+	7154.35i	0.252744	+	0.437766i
$645$		0			0
$646$	646.416	+	1119.62i	0.0393698	+	0.0681905i
$647$	1087.49	−	1883.59i	0.0660798	−	0.114454i	−0.831093	−	0.556134i	$-0.812284\pi$
$647$	1087.49	−	1883.59i	0.0660798	−	0.114454i	0.897172	+	0.441680i	$0.145617\pi$
$648$		0			0
$649$	13357.6			0.807907
$650$	4561.54	+	35785.7i	0.275259	+	2.15943i
$651$		0			0
$652$	18732.7	−	32446.1i	1.12520	−	1.94890i
$653$	−7727.27	+	13384.0i	−0.463080	+	0.802078i	−0.999113	−	0.0421191i	$-0.986589\pi$
$653$	−7727.27	+	13384.0i	−0.463080	+	0.802078i	0.536032	+	0.844197i	$0.319922\pi$
$654$		0			0
$655$	−33931.1			−2.02412
$656$	−2911.41	−	5042.71i	−0.173280	−	0.300129i
$657$		0			0
$658$	−26390.7			−1.56355
$659$	−1574.39	−	2726.92i	−0.0930643	−	0.161192i	0.815735	−	0.578426i	$-0.196333\pi$
$659$	−1574.39	−	2726.92i	−0.0930643	−	0.161192i	−0.908799	+	0.417234i	$0.863000\pi$
$660$		0			0
$661$	1049.85	−	1818.39i	0.0617767	−	0.107000i	−0.833483	−	0.552545i	$-0.813657\pi$
$661$	1049.85	−	1818.39i	0.0617767	−	0.107000i	0.895260	+	0.445545i	$0.146990\pi$
$662$	−10715.7			−0.629122
$663$		0			0
$664$	−47359.8			−2.76795
$665$	178.513	−	309.193i	0.0104097	−	0.0180301i
$666$		0			0
$667$	−1288.16	−	2231.16i	−0.0747794	−	0.129522i
$668$	−39703.4			−2.29966
$669$		0			0
$670$	8316.35	+	14404.3i	0.479535	+	0.830580i
$671$	12249.7			0.704763
$672$		0			0
$673$	−15485.4	+	26821.5i	−0.886950	+	1.53624i	−0.0434884	+	0.999054i	$0.513847\pi$
$673$	−15485.4	+	26821.5i	−0.886950	+	1.53624i	−0.843462	+	0.537189i	$0.819486\pi$
$674$	19027.5	−	32956.6i	1.08741	−	1.88344i
$675$		0			0
$676$	−43413.5	+	11250.5i	−2.47004	+	0.640104i
$677$	−14640.6			−0.831141			−0.415570	−	0.909561i	$-0.636418\pi$
$677$	−14640.6			−0.831141			−0.415570	+	0.909561i	$0.636418\pi$
$678$		0			0
$679$	−317.317	+	549.610i	−0.0179345	+	0.0310635i
$680$	−58605.0	−	101507.i	−3.30500	−	5.72442i
$681$		0			0
$682$	−14112.4	−	24443.3i	−0.792360	−	1.37241i
$683$	−3342.92	−	5790.10i	−0.187281	−	0.324381i	0.757062	−	0.653343i	$-0.226634\pi$
$683$	−3342.92	−	5790.10i	−0.187281	−	0.324381i	−0.944343	+	0.328963i	$0.893301\pi$
$684$		0			0
$685$	−3180.13	−	5508.15i	−0.177382	−	0.307235i
$686$	−15280.4	+	26466.4i	−0.850450	+	1.47302i
$687$		0			0
$688$	45136.3			2.50117
$689$	18384.0	−	13986.7i	1.01651	−	0.773370i
$690$		0			0
$691$	15097.0	−	26148.8i	0.831141	−	1.43958i	−0.0659934	−	0.997820i	$-0.521022\pi$
$691$	15097.0	−	26148.8i	0.831141	−	1.43958i	0.897134	−	0.441758i	$-0.145645\pi$
$692$	25839.2	−	44754.8i	1.41945	−	2.45856i
$693$		0			0
$694$	6.07611			0.000332343
$695$	−6175.98	−	10697.1i	−0.337077	−	0.583834i
$696$		0			0
$697$	−3318.28			−0.180328
$698$	32513.0	+	56314.2i	1.76309	+	3.05376i
$699$		0			0
$700$	−14265.1	+	24707.9i	−0.770245	+	1.33410i
$701$	30300.9			1.63260			0.816298	−	0.577631i	$-0.196023\pi$
$701$	30300.9			1.63260			0.816298	+	0.577631i	$0.196023\pi$
$702$		0			0
$703$	−221.181			−0.0118663
$704$	14407.5	−	24954.6i	0.771313	−	1.33595i
$705$		0			0
$706$	29030.1	+	50281.7i	1.54754	+	2.68042i
$707$	−5147.64			−0.273829
$708$		0			0
$709$	−13061.6	−	22623.4i	−0.691875	−	1.19836i	−0.971223	−	0.238173i	$-0.923452\pi$
$709$	−13061.6	−	22623.4i	−0.691875	−	1.19836i	0.279348	−	0.960190i	$-0.409882\pi$
$710$	42412.9			2.24187
$711$		0			0
$712$	34345.5	−	59488.2i	1.80780	−	3.13120i
$713$	4012.24	−	6949.41i	0.210743	−	0.365017i
$714$		0			0
$715$	2683.68	+	21053.7i	0.140369	+	1.10121i
$716$	−87021.3			−4.54209
$717$		0			0
$718$	9396.90	−	16275.9i	0.488425	−	0.845977i
$719$	9662.83	+	16736.5i	0.501200	+	0.868104i	0.999999	+	0.00138631i	$0.000441276\pi$
$719$	9662.83	+	16736.5i	0.501200	+	0.868104i	−0.498799	+	0.866718i	$0.666225\pi$
$720$		0			0
$721$	3562.05	+	6169.64i	0.183991	+	0.318682i
$722$	−18267.1	−	31639.6i	−0.941595	−	1.63089i
$723$		0			0
$724$	−40260.8	−	69733.8i	−2.06669	−	3.57961i
$725$	4448.73	−	7705.43i	0.227892	−	0.394721i
$726$		0			0
$727$	26065.8			1.32975			0.664875	−	0.746954i	$-0.268485\pi$
$727$	26065.8			1.32975			0.664875	+	0.746954i	$0.268485\pi$
$728$	−27687.7	−	11602.3i	−1.40958	−	0.590674i
$729$		0			0
$730$	−41897.2	+	72568.1i	−2.12423	+	3.67927i
$731$	12861.0	−	22275.9i	0.650727	−	1.12709i
$732$		0			0
$733$	1055.45			0.0531843			0.0265921	−	0.999646i	$-0.491534\pi$
$733$	1055.45			0.0531843			0.0265921	+	0.999646i	$0.491534\pi$
$734$	6353.18	+	11004.0i	0.319482	+	0.553359i
$735$		0			0
$736$	20076.2			1.00546
$737$	2622.44	+	4542.21i	0.131070	+	0.227021i
$738$		0			0
$739$	−4705.20	+	8149.64i	−0.234213	+	0.405669i	−0.959044	−	0.283258i	$-0.908585\pi$
$739$	−4705.20	+	8149.64i	−0.234213	+	0.405669i	0.724831	+	0.688927i	$0.241918\pi$
$740$	32976.0			1.63814
$741$		0			0
$742$	25428.1			1.25808
$743$	−3761.85	+	6515.72i	−0.185746	+	0.321721i	−0.943827	−	0.330439i	$-0.892803\pi$
$743$	−3761.85	+	6515.72i	−0.185746	+	0.321721i	0.758082	+	0.652159i	$0.226137\pi$
$744$		0			0
$745$	−21638.4	−	37478.8i	−1.06412	−	1.84311i
$746$	70799.4			3.47473
$747$		0			0
$748$	−30390.4	−	52637.7i	−1.48554	−	2.57303i
$749$	7581.76			0.369868
$750$		0			0
$751$	6492.03	−	11244.5i	0.315443	−	0.546363i	−0.664089	−	0.747654i	$-0.731180\pi$
$751$	6492.03	−	11244.5i	0.315443	−	0.546363i	0.979532	+	0.201291i	$0.0645136\pi$
$752$	−48434.7	+	83891.3i	−2.34871	+	4.06809i
$753$		0			0
$754$	14199.6	+	5950.24i	0.685833	+	0.287394i
$755$	−54695.3			−2.63651
$756$		0			0
$757$	13967.3	−	24192.1i	0.670609	−	1.16153i	−0.307123	−	0.951670i	$-0.599366\pi$
$757$	13967.3	−	24192.1i	0.670609	−	1.16153i	0.977732	−	0.209859i	$-0.0673004\pi$
$758$	−11824.7	−	20481.1i	−0.566615	−	0.981406i
$759$		0			0
$760$	−1220.25	−	2113.53i	−0.0582408	−	0.100876i
$761$	−7759.63	−	13440.1i	−0.369627	−	0.640214i	0.619880	−	0.784697i	$-0.287181\pi$
$761$	−7759.63	−	13440.1i	−0.369627	−	0.640214i	−0.989507	+	0.144483i	$0.953848\pi$
$762$		0			0
$763$	2576.44	+	4462.52i	0.122246	+	0.211735i
$764$	−2185.31	+	3785.07i	−0.103484	+	0.179240i
$765$		0			0
$766$	−4319.82			−0.203761
$767$	−20931.1	−	8771.02i	−0.985368	−	0.412912i
$768$		0			0
$769$	−6442.59	+	11158.9i	−0.302114	+	0.523277i	−0.976615	−	0.214997i	$-0.931026\pi$
$769$	−6442.59	+	11158.9i	−0.302114	+	0.523277i	0.674501	+	0.738274i	$0.264359\pi$
$770$	−11681.6	+	20233.2i	−0.546723	+	0.946952i
$771$		0			0
$772$	−24642.4			−1.14883
$773$	2946.02	+	5102.66i	0.137078	+	0.237425i	0.926389	−	0.376567i	$-0.122896\pi$
$773$	2946.02	+	5102.66i	0.137078	+	0.237425i	−0.789312	+	0.613993i	$0.789562\pi$
$774$		0			0
$775$	27713.0			1.28449
$776$	2169.06	+	3756.92i	0.100341	+	0.173796i
$777$		0			0
$778$	−9231.69	+	15989.8i	−0.425414	+	0.736839i
$779$	−69.0916			−0.00317775
$780$		0			0

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:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	117.g (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.66520 - 4.61626i) q^{2} +(-10.2065 - 17.6783i) q^{4} +16.4131 q^{5} +(-4.83984 - 8.38285i) q^{7} -66.1667 q^{8} +O(q^{10})\)	\(q+(2.66520 - 4.61626i) q^{2} +(-10.2065 - 17.6783i) q^{4} +16.4131 q^{5} +(-4.83984 - 8.38285i) q^{7} -66.1667 q^{8} +(43.7441 - 75.7670i) q^{10} +(13.7941 - 23.8921i) q^{11} +(-37.3033 + 28.3807i) q^{13} -51.5965 q^{14} +(-94.6948 + 164.016i) q^{16} +(53.9641 + 93.4685i) q^{17} +(1.12362 + 1.94616i) q^{19} +(-167.521 - 290.155i) q^{20} +(-73.5279 - 127.354i) q^{22} +(20.9045 - 36.2077i) q^{23} +144.390 q^{25} +(31.5919 + 247.842i) q^{26} +(-98.7961 + 171.120i) q^{28} +(30.8106 - 53.3656i) q^{29} +191.932 q^{31} +(240.094 + 415.855i) q^{32} +575.300 q^{34} +(-79.4368 - 137.589i) q^{35} +(-49.2118 + 85.2373i) q^{37} +11.9786 q^{38} -1086.00 q^{40} +(-15.3726 + 26.6261i) q^{41} +(-119.163 - 206.396i) q^{43} -563.160 q^{44} +(-111.429 - 193.001i) q^{46} +511.482 q^{47} +(124.652 - 215.903i) q^{49} +(384.826 - 666.539i) q^{50} +(882.459 + 369.788i) q^{52} -492.825 q^{53} +(226.404 - 392.142i) q^{55} +(320.236 + 554.665i) q^{56} +(-164.233 - 284.460i) q^{58} +(242.089 + 419.311i) q^{59} +(222.011 + 384.534i) q^{61} +(511.536 - 886.007i) q^{62} +1044.47 q^{64} +(-612.262 + 465.815i) q^{65} +(-95.0568 + 164.643i) q^{67} +(1101.57 - 1907.98i) q^{68} -846.858 q^{70} +(242.392 + 419.836i) q^{71} -957.780 q^{73} +(262.318 + 454.348i) q^{74} +(22.9365 - 39.7271i) q^{76} -267.045 q^{77} -375.216 q^{79} +(-1554.23 + 2692.01i) q^{80} +(81.9421 + 141.928i) q^{82} +715.765 q^{83} +(885.717 + 1534.11i) q^{85} -1270.37 q^{86} +(-912.708 + 1580.86i) q^{88} +(-519.076 + 899.066i) q^{89} +(418.453 + 175.350i) q^{91} -853.451 q^{92} +(1363.20 - 2361.13i) q^{94} +(18.4420 + 31.9425i) q^{95} +(-32.7818 - 56.7797i) q^{97} +(-664.443 - 1150.85i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} - 22 q^{4} + 12 q^{5} + 14 q^{7} - 108 q^{8}+O(q^{10}) \) 8 * q + 2 * q^2 - 22 * q^4 + 12 * q^5 + 14 * q^7 - 108 * q^8	\( 8 q + 2 q^{2} - 22 q^{4} + 12 q^{5} + 14 q^{7} - 108 q^{8} + 62 q^{10} + 40 q^{11} - 60 q^{13} - 80 q^{14} - 122 q^{16} + 98 q^{17} - 124 q^{19} - 466 q^{20} - 220 q^{22} + 104 q^{23} - 116 q^{25} - 14 q^{26} + 144 q^{28} + 194 q^{29} + 52 q^{31} + 654 q^{32} + 2124 q^{34} + 88 q^{35} - 102 q^{37} - 664 q^{38} - 1996 q^{40} - 1054 q^{41} - 450 q^{43} + 88 q^{44} + 172 q^{46} + 192 q^{47} - 1070 q^{49} + 996 q^{50} + 2280 q^{52} - 524 q^{53} - 204 q^{55} + 2164 q^{56} - 722 q^{58} + 308 q^{59} + 928 q^{61} + 2780 q^{62} + 2052 q^{64} - 2346 q^{65} + 1134 q^{67} + 1786 q^{68} - 4648 q^{70} + 1064 q^{71} + 1904 q^{73} + 1158 q^{74} + 1708 q^{76} - 5016 q^{77} - 1492 q^{79} - 2922 q^{80} - 1734 q^{82} + 808 q^{83} + 1394 q^{85} - 6336 q^{86} - 3060 q^{88} + 1620 q^{89} + 3278 q^{91} - 664 q^{92} + 772 q^{94} + 2204 q^{95} - 2166 q^{97} - 1906 q^{98}+O(q^{100}) \) 8 * q + 2 * q^2 - 22 * q^4 + 12 * q^5 + 14 * q^7 - 108 * q^8 + 62 * q^10 + 40 * q^11 - 60 * q^13 - 80 * q^14 - 122 * q^16 + 98 * q^17 - 124 * q^19 - 466 * q^20 - 220 * q^22 + 104 * q^23 - 116 * q^25 - 14 * q^26 + 144 * q^28 + 194 * q^29 + 52 * q^31 + 654 * q^32 + 2124 * q^34 + 88 * q^35 - 102 * q^37 - 664 * q^38 - 1996 * q^40 - 1054 * q^41 - 450 * q^43 + 88 * q^44 + 172 * q^46 + 192 * q^47 - 1070 * q^49 + 996 * q^50 + 2280 * q^52 - 524 * q^53 - 204 * q^55 + 2164 * q^56 - 722 * q^58 + 308 * q^59 + 928 * q^61 + 2780 * q^62 + 2052 * q^64 - 2346 * q^65 + 1134 * q^67 + 1786 * q^68 - 4648 * q^70 + 1064 * q^71 + 1904 * q^73 + 1158 * q^74 + 1708 * q^76 - 5016 * q^77 - 1492 * q^79 - 2922 * q^80 - 1734 * q^82 + 808 * q^83 + 1394 * q^85 - 6336 * q^86 - 3060 * q^88 + 1620 * q^89 + 3278 * q^91 - 664 * q^92 + 772 * q^94 + 2204 * q^95 - 2166 * q^97 - 1906 * q^98

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