LMFDB - Embedded newform 405.3.l.g.217.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,3,Mod(28,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([4, 9]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("405.28");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	405.l (of order $12$, degree $4$, not 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:	$11.0354507066$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{12})$
Coefficient field:	8.0.3317760000.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 25x^{4} + 625 $ x^8 - 25*x^4 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			217.2
Root			$-0.578737 - 2.15988i$ of defining polynomial
Character	$\chi$	$=$	405.217
Dual form			405.3.l.g.28.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(2.15988 - 0.578737i) q^{2} +(0.866025 - 0.500000i) q^{4} +(-3.31735 + 3.74101i) q^{5} +(6.83013 - 1.83013i) q^{7} +(-4.74342 + 4.74342i) q^{8} +(-5.00000 + 10.0000i) q^{10} +(-7.90569 + 13.6931i) q^{11} +(-13.6603 - 3.66025i) q^{13} +(13.6931 - 7.90569i) q^{14} +(-9.50000 + 16.4545i) q^{16} +(3.16228 + 3.16228i) q^{17} +18.0000i q^{19} +(-1.00240 + 4.89849i) q^{20} +(-9.15064 + 34.1506i) q^{22} +(-4.31975 - 1.15747i) q^{23} +(-2.99038 - 24.8205i) q^{25} -31.6228 q^{26} +(5.00000 - 5.00000i) q^{28} +(41.0792 + 23.7171i) q^{29} +(-4.00000 - 6.92820i) q^{31} +(-4.05116 + 15.1191i) q^{32} +(8.66025 + 5.00000i) q^{34} +(-15.8114 + 31.6228i) q^{35} +(10.0000 + 10.0000i) q^{37} +(10.4173 + 38.8778i) q^{38} +(-2.00962 - 33.4808i) q^{40} +(15.8114 + 27.3861i) q^{41} +(3.66025 + 13.6603i) q^{43} +15.8114i q^{44} -10.0000 q^{46} +(-56.1568 + 15.0472i) q^{47} +(0.866025 - 0.500000i) q^{49} +(-20.8234 - 51.8786i) q^{50} +(-13.6603 + 3.66025i) q^{52} +(-25.2982 + 25.2982i) q^{53} +(-25.0000 - 75.0000i) q^{55} +(-23.7171 + 41.0792i) q^{56} +(102.452 + 27.4519i) q^{58} +(41.0792 - 23.7171i) q^{59} +(29.0000 - 50.2295i) q^{61} +(-12.6491 - 12.6491i) q^{62} -41.0000i q^{64} +(59.0089 - 38.9609i) q^{65} +(25.6218 - 95.6218i) q^{67} +(4.31975 + 1.15747i) q^{68} +(-15.8494 + 77.4519i) q^{70} +63.2456 q^{71} +(55.0000 - 55.0000i) q^{73} +(27.3861 + 15.8114i) q^{74} +(9.00000 + 15.5885i) q^{76} +(-28.9368 + 107.994i) q^{77} +(-10.3923 - 6.00000i) q^{79} +(-30.0416 - 90.1249i) q^{80} +(50.0000 + 50.0000i) q^{82} +(-19.6771 - 73.4358i) q^{83} +(-22.3205 + 1.33975i) q^{85} +(15.8114 + 27.3861i) q^{86} +(-27.4519 - 102.452i) q^{88} -100.000 q^{91} +(-4.31975 + 1.15747i) q^{92} +(-112.583 + 65.0000i) q^{94} +(-67.3383 - 59.7123i) q^{95} +(6.83013 - 1.83013i) q^{97} +(1.58114 - 1.58114i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 20 q^{7} - 40 q^{10} - 40 q^{13} - 76 q^{16} + 100 q^{22} + 80 q^{25} + 40 q^{28} - 32 q^{31} + 80 q^{37} - 120 q^{40} - 40 q^{43} - 80 q^{46} - 40 q^{52} - 200 q^{55} + 300 q^{58} + 232 q^{61} - 280 q^{67}+ \cdots + 20 q^{97}+O(q^{100}) $ 8 * q + 20 * q^7 - 40 * q^10 - 40 * q^13 - 76 * q^16 + 100 * q^22 + 80 * q^25 + 40 * q^28 - 32 * q^31 + 80 * q^37 - 120 * q^40 - 40 * q^43 - 80 * q^46 - 40 * q^52 - 200 * q^55 + 300 * q^58 + 232 * q^61 - 280 * q^67 - 300 * q^70 + 440 * q^73 + 72 * q^76 + 400 * q^82 - 40 * q^85 + 300 * q^88 - 800 * q^91 + 20 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$e\left(\frac{2}{3}\right)$

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 $

$2$	2.15988	−	0.578737i	1.07994	−	0.289368i	0.325368	−	0.945588i	$-0.394512\pi$
$2$	2.15988	−	0.578737i	1.07994	−	0.289368i	0.754570	+	0.656219i	$0.227845\pi$
$3$		0			0
$3$		0			0
$4$	0.866025	−	0.500000i	0.216506	−	0.125000i
$5$	−3.31735	+	3.74101i	−0.663470	+	0.748203i
$5$	−3.31735	+	3.74101i	−0.663470	+	0.748203i
$6$		0			0
$7$	6.83013	−	1.83013i	0.975732	−	0.261447i	0.264486	−	0.964390i	$-0.414798\pi$
$7$	6.83013	−	1.83013i	0.975732	−	0.261447i	0.711246	+	0.702943i	$0.248131\pi$
$8$	−4.74342	+	4.74342i	−0.592927	+	0.592927i
$9$		0			0
$10$	−5.00000	+	10.0000i	−0.500000	+	1.00000i
$11$	−7.90569	+	13.6931i	−0.718699	+	1.24482i	0.242816	+	0.970072i	$0.421929\pi$
$11$	−7.90569	+	13.6931i	−0.718699	+	1.24482i	−0.961515	+	0.274752i	$0.911404\pi$
$12$		0			0
$13$	−13.6603	−	3.66025i	−1.05079	−	0.281558i	−0.308211	−	0.951318i	$-0.599730\pi$
$13$	−13.6603	−	3.66025i	−1.05079	−	0.281558i	−0.742578	+	0.669760i	$0.766397\pi$
$14$	13.6931	−	7.90569i	0.978076	−	0.564692i
$15$		0			0
$16$	−9.50000	+	16.4545i	−0.593750	+	1.02841i
$17$	3.16228	+	3.16228i	0.186016	+	0.186016i	0.793971	−	0.607955i	$-0.208010\pi$
$17$	3.16228	+	3.16228i	0.186016	+	0.186016i	−0.607955	+	0.793971i	$0.708010\pi$
$18$		0			0
$19$			18.0000i			0.947368i	0.880695	+	0.473684i	$0.157076\pi$
$19$			18.0000i			0.947368i	−0.880695	+	0.473684i	$0.842924\pi$
$20$	−1.00240	+	4.89849i	−0.0501201	+	0.244924i
$21$		0			0
$22$	−9.15064	+	34.1506i	−0.415938	+	1.55230i
$23$	−4.31975	−	1.15747i	−0.187815	−	0.0503250i	0.163685	−	0.986513i	$-0.447662\pi$
$23$	−4.31975	−	1.15747i	−0.187815	−	0.0503250i	−0.351501	+	0.936188i	$0.614328\pi$
$24$		0			0
$25$	−2.99038	−	24.8205i	−0.119615	−	0.992820i
$26$	−31.6228			−1.21626
$27$		0			0
$28$	5.00000	−	5.00000i	0.178571	−	0.178571i
$29$	41.0792	+	23.7171i	1.41652	+	0.817830i	0.995992	−	0.0894471i	$-0.0285100\pi$
$29$	41.0792	+	23.7171i	1.41652	+	0.817830i	0.420532	+	0.907278i	$0.361843\pi$
$30$		0			0
$31$	−4.00000	−	6.92820i	−0.129032	−	0.223490i	0.794270	−	0.607565i	$-0.207854\pi$
$31$	−4.00000	−	6.92820i	−0.129032	−	0.223490i	−0.923302	+	0.384075i	$0.874520\pi$
$32$	−4.05116	+	15.1191i	−0.126599	+	0.472473i
$33$		0			0
$34$	8.66025	+	5.00000i	0.254713	+	0.147059i
$35$	−15.8114	+	31.6228i	−0.451754	+	0.903508i
$36$		0			0
$37$	10.0000	+	10.0000i	0.270270	+	0.270270i	0.829209	−	0.558939i	$-0.188791\pi$
$37$	10.0000	+	10.0000i	0.270270	+	0.270270i	−0.558939	+	0.829209i	$0.688791\pi$
$38$	10.4173	+	38.8778i	0.274139	+	1.02310i
$39$		0			0
$40$	−2.00962	−	33.4808i	−0.0502405	−	0.837019i
$41$	15.8114	+	27.3861i	0.385644	+	0.667954i	0.991858	−	0.127347i	$-0.0406461\pi$
$41$	15.8114	+	27.3861i	0.385644	+	0.667954i	−0.606215	+	0.795301i	$0.707313\pi$
$42$		0			0
$43$	3.66025	+	13.6603i	0.0851222	+	0.317680i	0.995337	−	0.0964555i	$-0.0307505\pi$
$43$	3.66025	+	13.6603i	0.0851222	+	0.317680i	−0.910215	+	0.414136i	$0.864084\pi$
$44$			15.8114i			0.359350i
$45$		0			0
$46$	−10.0000			−0.217391
$47$	−56.1568	+	15.0472i	−1.19482	+	0.320152i	−0.800791	−	0.598944i	$-0.795587\pi$
$47$	−56.1568	+	15.0472i	−1.19482	+	0.320152i	−0.394034	+	0.919096i	$0.628921\pi$
$48$		0			0
$49$	0.866025	−	0.500000i	0.0176740	−	0.0102041i
$50$	−20.8234	−	51.8786i	−0.416468	−	1.03757i
$51$		0			0
$52$	−13.6603	+	3.66025i	−0.262697	+	0.0703895i
$53$	−25.2982	+	25.2982i	−0.477325	+	0.477325i	−0.904275	−	0.426950i	$-0.859588\pi$
$53$	−25.2982	+	25.2982i	−0.477325	+	0.477325i	0.426950	+	0.904275i	$0.359588\pi$
$54$		0			0
$55$	−25.0000	−	75.0000i	−0.454545	−	1.36364i
$56$	−23.7171	+	41.0792i	−0.423519	+	0.733557i
$57$		0			0
$58$	102.452	+	27.4519i	1.76641	+	0.473309i
$59$	41.0792	−	23.7171i	0.696257	−	0.401984i	−0.109695	−	0.993965i	$-0.534987\pi$
$59$	41.0792	−	23.7171i	0.696257	−	0.401984i	0.805952	+	0.591981i	$0.201654\pi$
$60$		0			0
$61$	29.0000	−	50.2295i	0.475410	−	0.823434i	−0.524193	−	0.851599i	$-0.675633\pi$
$61$	29.0000	−	50.2295i	0.475410	−	0.823434i	0.999603	+	0.0281652i	$0.00896646\pi$
$62$	−12.6491	−	12.6491i	−0.204018	−	0.204018i
$63$		0			0
$64$		−	41.0000i		−	0.640625i
$65$	59.0089	−	38.9609i	0.907829	−	0.599398i
$66$		0			0
$67$	25.6218	−	95.6218i	0.382415	−	1.42719i	−0.459788	−	0.888029i	$-0.652074\pi$
$67$	25.6218	−	95.6218i	0.382415	−	1.42719i	0.842202	−	0.539162i	$-0.181259\pi$
$68$	4.31975	+	1.15747i	0.0635258	+	0.0170217i
$69$		0			0
$70$	−15.8494	+	77.4519i	−0.226419	+	1.10646i
$71$	63.2456			0.890782			0.445391	−	0.895336i	$-0.353065\pi$
$71$	63.2456			0.890782			0.445391	+	0.895336i	$0.353065\pi$
$72$		0			0
$73$	55.0000	−	55.0000i	0.753425	−	0.753425i	−0.221692	−	0.975117i	$-0.571158\pi$
$73$	55.0000	−	55.0000i	0.753425	−	0.753425i	0.975117	+	0.221692i	$0.0711580\pi$
$74$	27.3861	+	15.8114i	0.370083	+	0.213667i
$75$		0			0
$76$	9.00000	+	15.5885i	0.118421	+	0.205111i
$77$	−28.9368	+	107.994i	−0.375803	+	1.40252i
$78$		0			0
$79$	−10.3923	−	6.00000i	−0.131548	−	0.0759494i	0.432782	−	0.901499i	$-0.357532\pi$
$79$	−10.3923	−	6.00000i	−0.131548	−	0.0759494i	−0.564330	+	0.825549i	$0.690865\pi$
$80$	−30.0416	−	90.1249i	−0.375520	−	1.12656i
$81$		0			0
$82$	50.0000	+	50.0000i	0.609756	+	0.609756i
$83$	−19.6771	−	73.4358i	−0.237073	−	0.884768i	−0.977203	−	0.212305i	$-0.931903\pi$
$83$	−19.6771	−	73.4358i	−0.237073	−	0.884768i	0.740130	−	0.672463i	$-0.234764\pi$
$84$		0			0
$85$	−22.3205	+	1.33975i	−0.262594	+	0.0157617i
$86$	15.8114	+	27.3861i	0.183853	+	0.318443i
$87$		0			0
$88$	−27.4519	−	102.452i	−0.311953	−	1.16423i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$	−100.000			−1.09890
$92$	−4.31975	+	1.15747i	−0.0469538	+	0.0125812i
$93$		0			0
$94$	−112.583	+	65.0000i	−1.19769	+	0.691489i
$95$	−67.3383	−	59.7123i	−0.708824	−	0.628550i
$96$		0			0
$97$	6.83013	−	1.83013i	0.0704137	−	0.0188673i	−0.223440	−	0.974718i	$-0.571729\pi$
$97$	6.83013	−	1.83013i	0.0704137	−	0.0188673i	0.293854	+	0.955850i	$0.405062\pi$
$98$	1.58114	−	1.58114i	0.0161341	−	0.0161341i
$99$		0			0
$100$	−15.0000	−	20.0000i	−0.150000	−	0.200000i
$101$	−7.90569	+	13.6931i	−0.0782742	+	0.135575i	−0.902505	−	0.430678i	$-0.858274\pi$
$101$	−7.90569	+	13.6931i	−0.0782742	+	0.135575i	0.824231	+	0.566253i	$0.191608\pi$
$102$		0			0
$103$	47.8109	+	12.8109i	0.464183	+	0.124378i	0.483328	−	0.875439i	$-0.339428\pi$
$103$	47.8109	+	12.8109i	0.464183	+	0.124378i	−0.0191450	+	0.999817i	$0.506094\pi$
$104$	82.1584	−	47.4342i	0.789984	−	0.456098i
$105$		0			0
$106$	−40.0000	+	69.2820i	−0.377358	+	0.653604i
$107$	60.0833	+	60.0833i	0.561526	+	0.561526i	0.929741	−	0.368215i	$-0.120031\pi$
$107$	60.0833	+	60.0833i	0.561526	+	0.561526i	−0.368215	+	0.929741i	$0.620031\pi$
$108$		0			0
$109$			162.000i			1.48624i	0.669159	+	0.743119i	$0.266655\pi$
$109$			162.000i			1.48624i	−0.669159	+	0.743119i	$0.733345\pi$
$110$	−97.4022	−	147.522i	−0.885474	−	1.34111i
$111$		0			0
$112$	−34.7724	+	129.772i	−0.310468	+	1.15868i
$113$	−159.831	−	42.8265i	−1.41443	−	0.378996i	−0.530927	−	0.847418i	$-0.678156\pi$
$113$	−159.831	−	42.8265i	−1.41443	−	0.378996i	−0.883505	+	0.468422i	$0.844823\pi$
$114$		0			0
$115$	18.6603	−	12.3205i	0.162263	−	0.107135i
$116$	47.4342			0.408915
$117$		0			0
$118$	75.0000	−	75.0000i	0.635593	−	0.635593i
$119$	27.3861	+	15.8114i	0.230136	+	0.132869i
$120$		0			0
$121$	−64.5000	−	111.717i	−0.533058	−	0.923283i
$122$	33.5667	−	125.273i	0.275137	−	1.02683i
$123$		0			0
$124$	−6.92820	−	4.00000i	−0.0558726	−	0.0322581i
$125$	102.774	+	71.1512i	0.822192	+	0.569210i
$126$		0			0
$127$	55.0000	+	55.0000i	0.433071	+	0.433071i	0.889672	−	0.456601i	$-0.150933\pi$
$127$	55.0000	+	55.0000i	0.433071	+	0.433071i	−0.456601	+	0.889672i	$0.650933\pi$
$128$	−39.9329	−	149.031i	−0.311975	−	1.16431i
$129$		0			0
$130$	104.904	−	118.301i	0.806952	−	0.910010i
$131$	86.9626	+	150.624i	0.663837	+	1.14980i	0.979599	+	0.200961i	$0.0644064\pi$
$131$	86.9626	+	150.624i	0.663837	+	1.14980i	−0.315762	+	0.948838i	$0.602260\pi$
$132$		0			0
$133$	32.9423	+	122.942i	0.247686	+	0.924378i
$134$		−	221.359i		−	1.65194i
$135$		0			0
$136$	−30.0000			−0.220588
$137$	21.5988	−	5.78737i	0.157655	−	0.0422436i	−0.179128	−	0.983826i	$-0.557328\pi$
$137$	21.5988	−	5.78737i	0.157655	−	0.0422436i	0.336783	+	0.941582i	$0.390661\pi$
$138$		0			0
$139$	−88.3346	+	51.0000i	−0.635501	+	0.366906i	−0.782879	−	0.622174i	$-0.786250\pi$
$139$	−88.3346	+	51.0000i	−0.635501	+	0.366906i	0.147379	+	0.989080i	$0.452916\pi$
$140$	2.11832	+	35.2918i	0.0151309	+	0.252084i
$141$		0			0
$142$	136.603	−	36.6025i	0.961990	−	0.257764i
$143$	158.114	−	158.114i	1.10569	−	1.10569i
$144$		0			0
$145$	−225.000	+	75.0000i	−1.55172	+	0.517241i
$146$	86.9626	−	150.624i	0.595634	−	1.03167i
$147$		0			0
$148$	13.6603	+	3.66025i	0.0922990	+	0.0247314i
$149$	41.0792	−	23.7171i	0.275699	−	0.159175i	−0.355776	−	0.934571i	$-0.615783\pi$
$149$	41.0792	−	23.7171i	0.275699	−	0.159175i	0.631475	+	0.775396i	$0.282450\pi$
$150$		0			0
$151$	11.0000	−	19.0526i	0.0728477	−	0.126176i	−0.827301	−	0.561759i	$-0.810125\pi$
$151$	11.0000	−	19.0526i	0.0728477	−	0.126176i	0.900148	+	0.435584i	$0.143458\pi$
$152$	−85.3815	−	85.3815i	−0.561720	−	0.561720i
$153$		0			0
$154$			250.000i			1.62338i
$155$	39.1879	+	8.01921i	0.252825	+	0.0517369i
$156$		0			0
$157$	−73.2051	+	273.205i	−0.466274	+	1.74016i	0.186355	+	0.982482i	$0.440333\pi$
$157$	−73.2051	+	273.205i	−0.466274	+	1.74016i	−0.652629	+	0.757677i	$0.726334\pi$
$158$	−25.9185	−	6.94484i	−0.164041	−	0.0439547i
$159$		0			0
$160$	−43.1218	−	65.3109i	−0.269511	−	0.408193i
$161$	−31.6228			−0.196415
$162$		0			0
$163$	100.000	−	100.000i	0.613497	−	0.613497i	−0.330359	−	0.943856i	$-0.607170\pi$
$163$	100.000	−	100.000i	0.613497	−	0.613497i	0.943856	+	0.330359i	$0.107170\pi$
$164$	27.3861	+	15.8114i	0.166989	+	0.0964109i
$165$		0			0
$166$	−85.0000	−	147.224i	−0.512048	−	0.886893i
$167$	−54.4013	+	203.028i	−0.325756	+	1.21574i	0.587793	+	0.809011i	$0.299997\pi$
$167$	−54.4013	+	203.028i	−0.325756	+	1.21574i	−0.913550	+	0.406727i	$0.866670\pi$
$168$		0			0
$169$	26.8468	+	15.5000i	0.158857	+	0.0917160i
$170$	−47.4342	+	15.8114i	−0.279024	+	0.0930082i
$171$		0			0
$172$	10.0000	+	10.0000i	0.0581395	+	0.0581395i
$173$	−40.5116	−	151.191i	−0.234171	−	0.873938i	−0.978521	−	0.206148i	$-0.933907\pi$
$173$	−40.5116	−	151.191i	−0.234171	−	0.873938i	0.744350	−	0.667790i	$-0.232760\pi$
$174$		0			0
$175$	−65.8494	−	164.054i	−0.376282	−	0.937454i
$176$	−150.208	−	260.168i	−0.853456	−	1.47823i
$177$		0			0
$178$		0			0
$179$			142.302i			0.794986i	0.917605	+	0.397493i	$0.130120\pi$
$179$			142.302i			0.794986i	−0.917605	+	0.397493i	$0.869880\pi$
$180$		0			0
$181$	218.000			1.20442			0.602210	−	0.798338i	$-0.294287\pi$
$181$	218.000			1.20442			0.602210	+	0.798338i	$0.294287\pi$
$182$	−215.988	+	57.8737i	−1.18674	+	0.317987i
$183$		0			0
$184$	25.9808	−	15.0000i	0.141200	−	0.0815217i
$185$	−70.5836	+	4.23665i	−0.381533	+	0.0229008i
$186$		0			0
$187$	−68.3013	+	18.3013i	−0.365247	+	0.0978678i
$188$	−41.1096	+	41.1096i	−0.218668	+	0.218668i
$189$		0			0
$190$	−180.000	−	90.0000i	−0.947368	−	0.473684i
$191$	−79.0569	+	136.931i	−0.413911	+	0.716914i	−0.995313	−	0.0967016i	$-0.969171\pi$
$191$	−79.0569	+	136.931i	−0.413911	+	0.716914i	0.581403	+	0.813616i	$0.302504\pi$
$192$		0			0
$193$	170.753	+	45.7532i	0.884731	+	0.237063i	0.672447	−	0.740145i	$-0.265243\pi$
$193$	170.753	+	45.7532i	0.884731	+	0.237063i	0.212284	+	0.977208i	$0.431910\pi$
$194$	13.6931	−	7.90569i	0.0705828	−	0.0407510i
$195$		0			0
$196$	0.500000	−	0.866025i	0.00255102	−	0.00441850i
$197$	145.465	+	145.465i	0.738400	+	0.738400i	0.972268	−	0.233868i	$-0.0751385\pi$
$197$	145.465	+	145.465i	0.738400	+	0.738400i	−0.233868	+	0.972268i	$0.575138\pi$
$198$		0			0
$199$		−	18.0000i		−	0.0904523i	−0.998977	−	0.0452261i	$-0.985599\pi$
$199$		−	18.0000i		−	0.0904523i	0.998977	−	0.0452261i	$-0.0144008\pi$
$200$	131.919	+	103.549i	0.659593	+	0.517747i
$201$		0			0
$202$	−9.15064	+	34.1506i	−0.0453002	+	0.169063i
$203$	323.981	+	86.8105i	1.59597	+	0.427638i
$204$		0			0
$205$	−154.904	−	31.6987i	−0.755628	−	0.154628i
$206$	110.680			0.537280
$207$		0			0
$208$	190.000	−	190.000i	0.913462	−	0.913462i
$209$	−246.475	−	142.302i	−1.17931	−	0.680873i
$210$		0			0
$211$	149.000	+	258.076i	0.706161	+	1.22311i	0.966271	+	0.257528i	$0.0829079\pi$
$211$	149.000	+	258.076i	0.706161	+	1.22311i	−0.260110	+	0.965579i	$0.583759\pi$
$212$	−9.25979	+	34.5580i	−0.0436783	+	0.163009i
$213$		0			0
$214$	164.545	+	95.0000i	0.768901	+	0.443925i
$215$	−63.2456	−	31.6228i	−0.294165	−	0.147083i
$216$		0			0
$217$	−40.0000	−	40.0000i	−0.184332	−	0.184332i
$218$	93.7554	+	349.900i	0.430071	+	1.60505i
$219$		0			0
$220$	−59.1506	−	52.4519i	−0.268867	−	0.238418i
$221$	−31.6228	−	54.7723i	−0.143089	−	0.247838i
$222$		0			0
$223$	−78.6955	−	293.695i	−0.352894	−	1.31702i	−0.883114	−	0.469159i	$-0.844557\pi$
$223$	−78.6955	−	293.695i	−0.352894	−	1.31702i	0.530219	−	0.847861i	$-0.322110\pi$
$224$			110.680i			0.494106i
$225$		0			0
$226$	−370.000			−1.63717
$227$	371.499	−	99.5428i	1.63656	−	0.438514i	0.680752	−	0.732514i	$-0.261653\pi$
$227$	371.499	−	99.5428i	1.63656	−	0.438514i	0.955806	+	0.293999i	$0.0949864\pi$
$228$		0			0
$229$	67.5500	−	39.0000i	0.294978	−	0.170306i	−0.345207	−	0.938527i	$-0.612191\pi$
$229$	67.5500	−	39.0000i	0.294978	−	0.170306i	0.640185	+	0.768221i	$0.278858\pi$
$230$	33.1735	−	37.4101i	0.144233	−	0.162653i
$231$		0			0
$232$	−307.356	+	82.3557i	−1.32481	+	0.354982i
$233$	−110.680	+	110.680i	−0.475020	+	0.475020i	−0.903535	−	0.428515i	$-0.859037\pi$
$233$	−110.680	+	110.680i	−0.475020	+	0.475020i	0.428515	+	0.903535i	$0.359037\pi$
$234$		0			0
$235$	130.000	−	260.000i	0.553191	−	1.10638i
$236$	23.7171	−	41.0792i	0.100496	−	0.174064i
$237$		0			0
$238$	68.3013	+	18.3013i	0.286980	+	0.0768961i
$239$	−328.634	+	189.737i	−1.37504	+	0.793877i	−0.991557	−	0.129672i	$-0.958607\pi$
$239$	−328.634	+	189.737i	−1.37504	+	0.793877i	−0.383479	+	0.923550i	$0.625274\pi$
$240$		0			0
$241$	−106.000	+	183.597i	−0.439834	+	0.761815i	−0.997676	−	0.0681321i	$-0.978296\pi$
$241$	−106.000	+	183.597i	−0.439834	+	0.761815i	0.557842	+	0.829947i	$0.311629\pi$
$242$	−203.967	−	203.967i	−0.842838	−	0.842838i
$243$		0			0
$244$		−	58.0000i		−	0.237705i
$245$	−1.00240	+	4.89849i	−0.00409144	+	0.0199938i
$246$		0			0
$247$	65.8846	−	245.885i	0.266739	−	0.995484i
$248$	51.8370	+	13.8897i	0.209020	+	0.0560068i
$249$		0			0
$250$	263.157	+	94.1987i	1.05263	+	0.376795i
$251$	−363.662			−1.44885			−0.724426	−	0.689352i	$-0.757895\pi$
$251$	−363.662			−1.44885			−0.724426	+	0.689352i	$0.757895\pi$
$252$		0			0
$253$	50.0000	−	50.0000i	0.197628	−	0.197628i
$254$	150.624	+	86.9626i	0.593007	+	0.342373i
$255$		0			0
$256$	−90.5000	−	156.751i	−0.353516	−	0.612307i
$257$	112.275	−	419.016i	0.436868	−	1.63041i	−0.299690	−	0.954037i	$-0.596883\pi$
$257$	112.275	−	419.016i	0.436868	−	1.63041i	0.736557	−	0.676375i	$-0.236450\pi$
$258$		0			0
$259$	86.6025	+	50.0000i	0.334373	+	0.193050i
$260$	31.6228	−	63.2456i	0.121626	−	0.243252i
$261$		0			0
$262$	275.000	+	275.000i	1.04962	+	1.04962i
$263$	105.330	+	393.097i	0.400495	+	1.49467i	0.812216	+	0.583357i	$0.198261\pi$
$263$	105.330	+	393.097i	0.400495	+	1.49467i	−0.411721	+	0.911310i	$0.635072\pi$
$264$		0			0
$265$	−10.7180	−	178.564i	−0.0404452	−	0.673827i
$266$	142.302	+	246.475i	0.534972	+	0.926598i
$267$		0			0
$268$	−25.6218	−	95.6218i	−0.0956037	−	0.356798i
$269$		−	142.302i		−	0.529006i	−0.964385	−	0.264503i	$-0.914792\pi$
$269$		−	142.302i		−	0.529006i	0.964385	−	0.264503i	$-0.0852078\pi$
$270$		0			0
$271$	−178.000			−0.656827			−0.328413	−	0.944534i	$-0.606514\pi$
$271$	−178.000			−0.656827			−0.328413	+	0.944534i	$0.606514\pi$
$272$	−82.0753	+	21.9920i	−0.301747	+	0.0808530i
$273$		0			0
$274$	43.3013	−	25.0000i	0.158034	−	0.0912409i
$275$	363.510	+	155.276i	1.32185	+	0.564640i
$276$		0			0
$277$	314.186	−	84.1858i	1.13424	−	0.303920i	0.357610	−	0.933871i	$-0.383592\pi$
$277$	314.186	−	84.1858i	1.13424	−	0.303920i	0.776635	+	0.629951i	$0.216925\pi$
$278$	−161.276	+	161.276i	−0.580130	+	0.580130i
$279$		0			0
$280$	−75.0000	−	225.000i	−0.267857	−	0.803571i
$281$	−79.0569	+	136.931i	−0.281341	+	0.487298i	−0.971715	−	0.236155i	$-0.924113\pi$
$281$	−79.0569	+	136.931i	−0.281341	+	0.487298i	0.690374	+	0.723453i	$0.257446\pi$
$282$		0			0
$283$	478.109	+	128.109i	1.68943	+	0.452682i	0.970243	−	0.242134i	$-0.0778474\pi$
$283$	478.109	+	128.109i	1.68943	+	0.452682i	0.719188	+	0.694816i	$0.244514\pi$
$284$	54.7723	−	31.6228i	0.192860	−	0.111348i
$285$		0			0
$286$	250.000	−	433.013i	0.874126	−	1.51403i
$287$	158.114	+	158.114i	0.550919	+	0.550919i
$288$		0			0
$289$		−	269.000i		−	0.930796i
$290$	−442.567	+	292.207i	−1.52609	+	1.00761i
$291$		0			0
$292$	20.1314	−	75.1314i	0.0689431	−	0.257299i
$293$	−276.464	−	74.0783i	−0.943563	−	0.252827i	−0.245935	−	0.969286i	$-0.579095\pi$
$293$	−276.464	−	74.0783i	−0.943563	−	0.252827i	−0.697629	+	0.716459i	$0.745762\pi$
$294$		0			0
$295$	−47.5481	+	232.356i	−0.161180	+	0.787646i
$296$	−94.8683			−0.320501
$297$		0			0
$298$	75.0000	−	75.0000i	0.251678	−	0.251678i
$299$	54.7723	+	31.6228i	0.183185	+	0.105762i
$300$		0			0
$301$	50.0000	+	86.6025i	0.166113	+	0.287716i
$302$	12.7322	−	47.5173i	0.0421596	−	0.157342i
$303$		0			0
$304$	−296.181	−	171.000i	−0.974279	−	0.562500i
$305$	91.7061	+	275.118i	0.300676	+	0.902027i
$306$		0			0
$307$	190.000	+	190.000i	0.618893	+	0.618893i	0.945247	−	0.326355i	$-0.105820\pi$
$307$	190.000	+	190.000i	0.618893	+	0.618893i	−0.326355	+	0.945247i	$0.605820\pi$
$308$	28.9368	+	107.994i	0.0939508	+	0.350629i
$309$		0			0
$310$	89.2820	−	5.35898i	0.288007	−	0.0172870i
$311$	−126.491	−	219.089i	−0.406724	−	0.704466i	0.587797	−	0.809009i	$-0.299996\pi$
$311$	−126.491	−	219.089i	−0.406724	−	0.704466i	−0.994520	+	0.104542i	$0.966662\pi$
$312$		0			0
$313$	53.0737	+	198.074i	0.169564	+	0.632823i	0.997414	+	0.0718727i	$0.0228975\pi$
$313$	53.0737	+	198.074i	0.169564	+	0.632823i	−0.827849	+	0.560951i	$0.810436\pi$
$314$			632.456i			2.01419i
$315$		0			0
$316$	−12.0000			−0.0379747
$317$	21.5988	−	5.78737i	0.0681349	−	0.0182567i	−0.224591	−	0.974453i	$-0.572104\pi$
$317$	21.5988	−	5.78737i	0.0681349	−	0.0182567i	0.292725	+	0.956197i	$0.405438\pi$
$318$		0			0
$319$	−649.519	+	375.000i	−2.03611	+	1.17555i
$320$	153.382	+	136.011i	0.479318	+	0.425035i
$321$		0			0
$322$	−68.3013	+	18.3013i	−0.212116	+	0.0568362i
$323$	−56.9210	+	56.9210i	−0.176226	+	0.176226i
$324$		0			0
$325$	−50.0000	+	350.000i	−0.153846	+	1.07692i
$326$	158.114	−	273.861i	0.485012	−	0.840065i
$327$		0			0
$328$	−204.904	−	54.9038i	−0.624707	−	0.167390i
$329$	−356.020	+	205.548i	−1.08213	+	0.624766i
$330$		0			0
$331$	−241.000	+	417.424i	−0.728097	+	1.26110i	0.229590	+	0.973287i	$0.426261\pi$
$331$	−241.000	+	417.424i	−0.728097	+	1.26110i	−0.957687	+	0.287813i	$0.907072\pi$
$332$	−53.7587	−	53.7587i	−0.161924	−	0.161924i
$333$		0			0
$334$			470.000i			1.40719i
$335$	272.726	+	413.062i	0.814108	+	1.23302i
$336$		0			0
$337$	−56.7339	+	211.734i	−0.168350	+	0.628291i	0.829239	+	0.558894i	$0.188774\pi$
$337$	−56.7339	+	211.734i	−0.168350	+	0.628291i	−0.997589	+	0.0693967i	$0.977893\pi$
$338$	66.9562	+	17.9408i	0.198095	+	0.0530794i
$339$		0			0
$340$	−18.6603	+	12.3205i	−0.0548831	+	0.0362368i
$341$	126.491			0.370942
$342$		0			0
$343$	−240.000	+	240.000i	−0.699708	+	0.699708i
$344$	−82.1584	−	47.4342i	−0.238833	−	0.137890i
$345$		0			0
$346$	−175.000	−	303.109i	−0.505780	−	0.876037i
$347$	60.1886	−	224.627i	0.173454	−	0.647340i	−0.823355	−	0.567526i	$-0.807901\pi$
$347$	60.1886	−	224.627i	0.173454	−	0.647340i	0.996810	−	0.0798143i	$-0.0254327\pi$
$348$		0			0
$349$	−275.396	−	159.000i	−0.789101	−	0.455587i	0.0505453	−	0.998722i	$-0.483904\pi$
$349$	−275.396	−	159.000i	−0.789101	−	0.455587i	−0.839646	+	0.543134i	$0.817237\pi$
$350$	−237.171	−	316.228i	−0.677631	−	0.903508i
$351$		0			0
$352$	−175.000	−	175.000i	−0.497159	−	0.497159i
$353$	−82.1807	−	306.702i	−0.232806	−	0.868845i	−0.979125	−	0.203257i	$-0.934847\pi$
$353$	−82.1807	−	306.702i	−0.232806	−	0.868845i	0.746319	−	0.665588i	$-0.231819\pi$
$354$		0			0
$355$	−209.808	+	236.603i	−0.591007	+	0.666486i
$356$		0			0
$357$		0			0
$358$	82.3557	+	307.356i	0.230044	+	0.858536i
$359$		−	284.605i		−	0.792772i	−0.918084	−	0.396386i	$-0.870264\pi$
$359$		−	284.605i		−	0.792772i	0.918084	−	0.396386i	$-0.129736\pi$
$360$		0			0
$361$	37.0000			0.102493
$362$	470.853	−	126.165i	1.30070	−	0.348521i
$363$		0			0
$364$	−86.6025	+	50.0000i	−0.237919	+	0.137363i
$365$	23.3016	+	388.210i	0.0638399	+	1.06359i
$366$		0			0
$367$	252.715	−	67.7147i	0.688596	−	0.184509i	0.102479	−	0.994735i	$-0.467323\pi$
$367$	252.715	−	67.7147i	0.688596	−	0.184509i	0.586117	+	0.810226i	$0.300656\pi$
$368$	60.0833	−	60.0833i	0.163270	−	0.163270i
$369$		0			0
$370$	−150.000	+	50.0000i	−0.405405	+	0.135135i
$371$	−126.491	+	219.089i	−0.340946	+	0.590536i
$372$		0			0
$373$	−136.603	−	36.6025i	−0.366227	−	0.0981301i	0.0710125	−	0.997475i	$-0.477377\pi$
$373$	−136.603	−	36.6025i	−0.366227	−	0.0981301i	−0.437239	+	0.899345i	$0.644044\pi$
$374$	−136.931	+	79.0569i	−0.366125	+	0.211382i
$375$		0			0
$376$	195.000	−	337.750i	0.518617	−	0.898271i
$377$	−474.342	−	474.342i	−1.25820	−	1.25820i
$378$		0			0
$379$			558.000i			1.47230i	0.676821	+	0.736148i	$0.263357\pi$
$379$			558.000i			1.47230i	−0.676821	+	0.736148i	$0.736643\pi$
$380$	−88.1728	−	18.0432i	−0.232034	−	0.0474822i
$381$		0			0
$382$	−91.5064	+	341.506i	−0.239545	+	0.893996i
$383$	384.458	+	103.015i	1.00381	+	0.268969i	0.723039	−	0.690807i	$-0.242744\pi$
$383$	384.458	+	103.015i	1.00381	+	0.268969i	0.280767	+	0.959776i	$0.409411\pi$
$384$		0			0
$385$	−308.013	−	466.506i	−0.800033	−	1.21170i
$386$	395.285			1.02405
$387$		0			0
$388$	5.00000	−	5.00000i	0.0128866	−	0.0128866i
$389$	−451.871	−	260.888i	−1.16162	−	0.670663i	−0.209930	−	0.977716i	$-0.567324\pi$
$389$	−451.871	−	260.888i	−1.16162	−	0.670663i	−0.951692	+	0.307053i	$0.900657\pi$
$390$		0			0
$391$	−10.0000	−	17.3205i	−0.0255754	−	0.0442980i
$392$	−1.73621	+	6.47963i	−0.00442911	+	0.0165297i
$393$		0			0
$394$	398.372	+	230.000i	1.01110	+	0.583756i
$395$	56.9210	−	18.9737i	0.144104	−	0.0480346i
$396$		0			0
$397$	−260.000	−	260.000i	−0.654912	−	0.654912i	0.299260	−	0.954172i	$-0.403260\pi$
$397$	−260.000	−	260.000i	−0.654912	−	0.654912i	−0.954172	+	0.299260i	$0.903260\pi$
$398$	−10.4173	−	38.8778i	−0.0261740	−	0.0976828i
$399$		0			0
$400$	436.817	+	186.590i	1.09204	+	0.466474i
$401$	−126.491	−	219.089i	−0.315439	−	0.546357i	0.664092	−	0.747651i	$-0.268818\pi$
$401$	−126.491	−	219.089i	−0.315439	−	0.546357i	−0.979531	+	0.201295i	$0.935485\pi$
$402$		0			0
$403$	29.2820	+	109.282i	0.0726601	+	0.271171i
$404$			15.8114i			0.0391371i
$405$		0			0
$406$	750.000			1.84729
$407$	−215.988	+	57.8737i	−0.530682	+	0.142196i
$408$		0			0
$409$	301.377	−	174.000i	0.736863	−	0.425428i	−0.0840648	−	0.996460i	$-0.526790\pi$
$409$	301.377	−	174.000i	0.736863	−	0.425428i	0.820928	+	0.571032i	$0.193457\pi$
$410$	−352.918	+	21.1832i	−0.860776	+	0.0516664i
$411$		0			0
$412$	47.8109	−	12.8109i	0.116046	−	0.0310944i
$413$	237.171	−	237.171i	0.574263	−	0.574263i
$414$		0			0
$415$	340.000	+	170.000i	0.819277	+	0.409639i
$416$	110.680	−	191.703i	0.266057	−	0.460824i
$417$		0			0
$418$	−614.711	−	164.711i	−1.47060	−	0.394046i
$419$	534.029	−	308.322i	1.27453	−	0.735852i	0.298696	−	0.954348i	$-0.403448\pi$
$419$	534.029	−	308.322i	1.27453	−	0.735852i	0.975838	+	0.218496i	$0.0701151\pi$
$420$		0			0
$421$	−169.000	+	292.717i	−0.401425	+	0.695289i	−0.993898	−	0.110302i	$-0.964818\pi$
$421$	−169.000	+	292.717i	−0.401425	+	0.695289i	0.592473	+	0.805590i	$0.298152\pi$
$422$	471.179	+	471.179i	1.11654	+	1.11654i
$423$		0			0
$424$		−	240.000i		−	0.566038i
$425$	69.0329	−	87.9458i	0.162430	−	0.206931i
$426$		0			0
$427$	106.147	−	396.147i	0.248589	−	0.927746i
$428$	82.0753	+	21.9920i	0.191765	+	0.0513832i
$429$		0			0
$430$	−154.904	−	31.6987i	−0.360241	−	0.0737180i
$431$	−221.359			−0.513595			−0.256797	−	0.966465i	$-0.582667\pi$
$431$	−221.359			−0.513595			−0.256797	+	0.966465i	$0.582667\pi$
$432$		0			0
$433$	145.000	−	145.000i	0.334873	−	0.334873i	−0.519561	−	0.854434i	$-0.673904\pi$
$433$	145.000	−	145.000i	0.334873	−	0.334873i	0.854434	+	0.519561i	$0.173904\pi$
$434$	−109.545	−	63.2456i	−0.252407	−	0.145727i
$435$		0			0
$436$	81.0000	+	140.296i	0.185780	+	0.321780i
$437$	20.8345	−	77.7555i	0.0476763	−	0.177930i
$438$		0			0
$439$	67.5500	+	39.0000i	0.153872	+	0.0888383i	0.574959	−	0.818182i	$-0.305018\pi$
$439$	67.5500	+	39.0000i	0.153872	+	0.0888383i	−0.421087	+	0.907020i	$0.638351\pi$
$440$	474.342	+	237.171i	1.07805	+	0.539025i
$441$		0			0
$442$	−100.000	−	100.000i	−0.226244	−	0.226244i
$443$	−71.7634	−	267.825i	−0.161994	−	0.604570i	−0.998404	−	0.0564665i	$-0.982017\pi$
$443$	−71.7634	−	267.825i	−0.161994	−	0.604570i	0.836410	−	0.548104i	$-0.184650\pi$
$444$		0			0
$445$		0			0
$446$	−339.945	−	588.802i	−0.762208	−	1.32018i
$447$		0			0
$448$	−75.0352	−	280.035i	−0.167489	−	0.625079i
$449$		−	284.605i		−	0.633864i	−0.948448	−	0.316932i	$-0.897347\pi$
$449$		−	284.605i		−	0.633864i	0.948448	−	0.316932i	$-0.102653\pi$
$450$		0			0
$451$	−500.000			−1.10865
$452$	−159.831	+	42.8265i	−0.353608	+	0.0947490i
$453$		0			0
$454$	744.782	−	430.000i	1.64049	−	0.947137i
$455$	331.735	−	374.101i	0.729088	−	0.822201i
$456$		0			0
$457$	498.599	−	133.599i	1.09103	−	0.292340i	0.331921	−	0.943307i	$-0.392303\pi$
$457$	498.599	−	133.599i	1.09103	−	0.292340i	0.759106	+	0.650967i	$0.225637\pi$
$458$	123.329	−	123.329i	0.269277	−	0.269277i
$459$		0			0
$460$	10.0000	−	20.0000i	0.0217391	−	0.0434783i
$461$	419.002	−	725.732i	0.908898	−	1.57426i	0.0932994	−	0.995638i	$-0.470259\pi$
$461$	419.002	−	725.732i	0.908898	−	1.57426i	0.815598	−	0.578619i	$-0.196408\pi$
$462$		0			0
$463$	47.8109	+	12.8109i	0.103263	+	0.0276693i	0.310081	−	0.950710i	$-0.399644\pi$
$463$	47.8109	+	12.8109i	0.103263	+	0.0276693i	−0.206817	+	0.978380i	$0.566311\pi$
$464$	−780.505	+	450.625i	−1.68212	+	0.971174i
$465$		0			0
$466$	−175.000	+	303.109i	−0.375536	+	0.650448i
$467$	117.004	+	117.004i	0.250544	+	0.250544i	0.821194	−	0.570649i	$-0.193308\pi$
$467$	117.004	+	117.004i	0.250544	+	0.250544i	−0.570649	+	0.821194i	$0.693308\pi$
$468$		0			0
$469$		−	700.000i		−	1.49254i
$470$	130.312	−	636.804i	0.277260	−	1.35490i
$471$		0			0
$472$	−82.3557	+	307.356i	−0.174482	+	0.651177i
$473$	−215.988	−	57.8737i	−0.456633	−	0.122355i
$474$		0			0
$475$	446.769	−	53.8269i	0.940567	−	0.113320i
$476$	31.6228			0.0664344
$477$		0			0
$478$	−600.000	+	600.000i	−1.25523	+	1.25523i
$479$	−328.634	−	189.737i	−0.686083	−	0.396110i	0.116060	−	0.993242i	$-0.462973\pi$
$479$	−328.634	−	189.737i	−0.686083	−	0.396110i	−0.802143	+	0.597132i	$0.796307\pi$
$480$		0			0
$481$	−100.000	−	173.205i	−0.207900	−	0.360094i
$482$	−122.692	+	457.894i	−0.254548	+	0.949987i
$483$		0			0
$484$	−111.717	−	64.5000i	−0.230821	−	0.133264i
$485$	−15.8114	+	31.6228i	−0.0326008	+	0.0652016i
$486$		0			0
$487$	−125.000	−	125.000i	−0.256674	−	0.256674i	0.567026	−	0.823700i	$-0.308094\pi$
$487$	−125.000	−	125.000i	−0.256674	−	0.256674i	−0.823700	+	0.567026i	$0.808094\pi$
$488$	100.700	+	375.818i	0.206353	+	0.770120i
$489$		0			0
$490$	0.669873	+	11.1603i	0.00136709	+	0.0227760i
$491$	229.265	+	397.099i	0.466935	+	0.808755i	0.999287	−	0.0377681i	$-0.0120248\pi$
$491$	229.265	+	397.099i	0.466935	+	0.808755i	−0.532351	+	0.846523i	$0.678692\pi$
$492$		0			0
$493$	54.9038	+	204.904i	0.111367	+	0.415626i
$494$		−	569.210i		−	1.15225i
$495$		0			0
$496$	152.000			0.306452
$497$	431.975	−	115.747i	0.869165	−	0.232892i
$498$		0			0
$499$	192.258	−	111.000i	0.385286	−	0.222445i	−0.294830	−	0.955550i	$-0.595263\pi$
$499$	192.258	−	111.000i	0.385286	−	0.222445i	0.680116	+	0.733105i	$0.261930\pi$
$500$	124.581	+	10.2318i	0.249161	+	0.0204636i
$501$		0			0
$502$	−785.465	+	210.465i	−1.56467	+	0.419252i
$503$	458.530	−	458.530i	0.911591	−	0.911591i	−0.0848065	−	0.996397i	$-0.527027\pi$
$503$	458.530	−	458.530i	0.911591	−	0.911591i	0.996397	+	0.0848065i	$0.0270272\pi$
$504$		0			0
$505$	−25.0000	−	75.0000i	−0.0495050	−	0.148515i
$506$	79.0569	−	136.931i	0.156239	−	0.270614i
$507$		0			0
$508$	75.1314	+	20.1314i	0.147896	+	0.0396287i
$509$	−205.396	+	118.585i	−0.403528	+	0.232977i	−0.688005	−	0.725706i	$-0.741513\pi$
$509$	−205.396	+	118.585i	−0.403528	+	0.232977i	0.284477	+	0.958683i	$0.408180\pi$
$510$		0			0
$511$	275.000	−	476.314i	0.538160	−	0.932121i
$512$	150.208	+	150.208i	0.293375	+	0.293375i
$513$		0			0
$514$		−	970.000i		−	1.88716i
$515$	−206.531	+	136.363i	−0.401031	+	0.264783i
$516$		0			0
$517$	237.917	−	887.917i	0.460187	−	1.71744i
$518$	215.988	+	57.8737i	0.416964	+	0.111725i
$519$		0			0
$520$	−95.0962	+	464.711i	−0.182877	+	0.893676i
$521$	−790.569			−1.51741			−0.758704	−	0.651436i	$-0.774167\pi$
$521$	−790.569			−1.51741			−0.758704	+	0.651436i	$0.774167\pi$
$522$		0			0
$523$	370.000	−	370.000i	0.707457	−	0.707457i	−0.258543	−	0.966000i	$-0.583242\pi$
$523$	370.000	−	370.000i	0.707457	−	0.707457i	0.966000	+	0.258543i	$0.0832423\pi$
$524$	150.624	+	86.9626i	0.287450	+	0.165959i
$525$		0			0
$526$	455.000	+	788.083i	0.865019	+	1.49826i
$527$	9.25979	−	34.5580i	0.0175708	−	0.0655750i
$528$		0			0
$529$	−440.807	−	254.500i	−0.833283	−	0.481096i
$530$	−126.491	−	379.473i	−0.238662	−	0.715987i
$531$		0			0
$532$	90.0000	+	90.0000i	0.169173	+	0.169173i
$533$	−115.747	−	431.975i	−0.217162	−	0.810460i
$534$		0			0
$535$	−424.090	+	25.4552i	−0.792691	+	0.0475798i
$536$	332.039	+	575.109i	0.619476	+	1.07296i
$537$		0			0
$538$	−82.3557	−	307.356i	−0.153078	−	0.571293i
$539$			15.8114i			0.0293347i
$540$		0			0
$541$	362.000			0.669131			0.334566	−	0.942372i	$-0.391410\pi$
$541$	362.000			0.669131			0.334566	+	0.942372i	$0.391410\pi$
$542$	−384.458	+	103.015i	−0.709332	+	0.190065i
$543$		0			0
$544$	−60.6218	+	35.0000i	−0.111437	+	0.0643382i
$545$	−606.044	−	537.411i	−1.11201	−	0.986075i
$546$		0			0
$547$	−669.352	+	179.352i	−1.22368	+	0.327884i	−0.812115	−	0.583497i	$-0.801684\pi$
$547$	−669.352	+	179.352i	−1.22368	+	0.327884i	−0.411564	+	0.911381i	$0.635017\pi$
$548$	15.8114	−	15.8114i	0.0288529	−	0.0288529i
$549$		0			0
$550$	875.000	+	125.000i	1.59091	+	0.227273i
$551$	−426.907	+	739.425i	−0.774787	+	1.34197i
$552$		0			0
$553$	−81.9615	−	21.9615i	−0.148213	−	0.0397134i
$554$	629.881	−	363.662i	1.13697	−	0.656429i
$555$		0			0
$556$	−51.0000	+	88.3346i	−0.0917266	+	0.158875i
$557$	−252.982	−	252.982i	−0.454187	−	0.454187i	0.442555	−	0.896742i	$-0.354072\pi$
$557$	−252.982	−	252.982i	−0.454187	−	0.454187i	−0.896742	+	0.442555i	$0.854072\pi$
$558$		0			0
$559$		−	200.000i		−	0.357782i
$560$	−370.128	−	560.585i	−0.660943	−	1.00104i
$561$		0			0
$562$	−91.5064	+	341.506i	−0.162823	+	0.607663i
$563$	−276.464	−	74.0783i	−0.491055	−	0.131578i	0.00479040	−	0.999989i	$-0.498475\pi$
$563$	−276.464	−	74.0783i	−0.491055	−	0.131578i	−0.495846	+	0.868411i	$0.665142\pi$
$564$		0			0
$565$	690.429	−	455.859i	1.22200	−	0.806830i
$566$	1106.80			1.95547
$567$		0			0
$568$	−300.000	+	300.000i	−0.528169	+	0.528169i
$569$	164.317	+	94.8683i	0.288782	+	0.166728i	0.637392	−	0.770539i	$-0.280013\pi$
$569$	164.317	+	94.8683i	0.288782	+	0.166728i	−0.348611	+	0.937268i	$0.613346\pi$
$570$		0			0
$571$	329.000	+	569.845i	0.576182	+	0.997977i	0.995912	+	0.0903277i	$0.0287914\pi$
$571$	329.000	+	569.845i	0.576182	+	0.997977i	−0.419730	+	0.907649i	$0.637875\pi$
$572$	57.8737	−	215.988i	0.101178	−	0.377601i
$573$		0			0
$574$	433.013	+	250.000i	0.754378	+	0.435540i
$575$	−15.8114	+	110.680i	−0.0274981	+	0.192486i
$576$		0			0
$577$	−35.0000	−	35.0000i	−0.0606586	−	0.0606586i	0.676127	−	0.736785i	$-0.263657\pi$
$577$	−35.0000	−	35.0000i	−0.0606586	−	0.0606586i	−0.736785	+	0.676127i	$0.763657\pi$
$578$	−155.680	−	581.007i	−0.269343	−	1.00520i
$579$		0			0
$580$	−157.356	+	177.452i	−0.271303	+	0.305952i
$581$	−268.794	−	465.564i	−0.462640	−	0.801315i
$582$		0			0
$583$	−146.410	−	546.410i	−0.251132	−	0.937239i
$584$			521.776i			0.893452i
$585$		0			0
$586$	−640.000			−1.09215
$587$	−211.668	+	56.7162i	−0.360593	+	0.0966205i	−0.434567	−	0.900640i	$-0.643098\pi$
$587$	−211.668	+	56.7162i	−0.360593	+	0.0966205i	0.0739741	+	0.997260i	$0.476432\pi$
$588$		0			0
$589$	124.708	−	72.0000i	0.211728	−	0.122241i
$590$	31.7749	+	529.377i	0.0538557	+	0.897250i
$591$		0			0
$592$	−259.545	+	69.5448i	−0.438420	+	0.117474i
$593$	−167.601	+	167.601i	−0.282632	+	0.282632i	−0.834158	−	0.551526i	$-0.814046\pi$
$593$	−167.601	+	167.601i	−0.282632	+	0.282632i	0.551526	+	0.834158i	$0.314046\pi$
$594$		0			0
$595$	−150.000	+	50.0000i	−0.252101	+	0.0840336i
$596$	23.7171	−	41.0792i	0.0397938	−	0.0689248i
$597$		0			0
$598$	136.603	+	36.6025i	0.228432	+	0.0612083i
$599$	903.742	−	521.776i	1.50875	−	0.871078i	0.508804	−	0.860883i	$-0.330088\pi$
$599$	903.742	−	521.776i	1.50875	−	0.871078i	0.999948	−	0.0101955i	$-0.00324539\pi$
$600$		0			0
$601$	191.000	−	330.822i	0.317804	−	0.550452i	−0.662226	−	0.749304i	$-0.730388\pi$
$601$	191.000	−	330.822i	0.317804	−	0.550452i	0.980029	+	0.198852i	$0.0637214\pi$
$602$	158.114	+	158.114i	0.262648	+	0.262648i
$603$		0			0
$604$		−	22.0000i		−	0.0364238i
$605$	631.905	+	129.310i	1.04447	+	0.213735i
$606$		0			0
$607$	239.747	−	894.747i	0.394970	−	1.47405i	−0.426862	−	0.904317i	$-0.640381\pi$
$607$	239.747	−	894.747i	0.394970	−	1.47405i	0.821832	−	0.569730i	$-0.192952\pi$
$608$	−272.144	−	72.9209i	−0.447606	−	0.119936i
$609$		0			0
$610$	357.295	+	541.147i	0.585729	+	0.887127i
$611$	822.192			1.34565
$612$		0			0
$613$	−620.000	+	620.000i	−1.01142	+	1.01142i	−0.0114852	+	0.999934i	$0.503656\pi$
$613$	−620.000	+	620.000i	−1.01142	+	1.01142i	−0.999934	+	0.0114852i	$0.996344\pi$
$614$	520.336	+	300.416i	0.847453	+	0.489277i
$615$		0			0
$616$	−375.000	−	649.519i	−0.608766	−	1.05441i
$617$	−12.7322	+	47.5173i	−0.0206357	+	0.0770134i	−0.975476	−	0.220107i	$-0.929359\pi$
$617$	−12.7322	+	47.5173i	−0.0206357	+	0.0770134i	0.954840	+	0.297120i	$0.0960261\pi$
$618$		0			0
$619$	223.435	+	129.000i	0.360961	+	0.208401i	0.669502	−	0.742810i	$-0.266508\pi$
$619$	223.435	+	129.000i	0.360961	+	0.208401i	−0.308542	+	0.951211i	$0.599841\pi$
$620$	37.9473	−	12.6491i	0.0612054	−	0.0204018i
$621$		0			0
$622$	−400.000	−	400.000i	−0.643087	−	0.643087i
$623$		0			0
$624$		0			0
$625$	−607.115	+	148.446i	−0.971384	+	0.237513i
$626$	229.265	+	397.099i	0.366238	+	0.634343i
$627$		0			0
$628$	73.2051	+	273.205i	0.116569	+	0.435040i
$629$			63.2456i			0.100549i
$630$		0			0
$631$	812.000			1.28685			0.643423	−	0.765511i	$-0.277514\pi$
$631$	812.000			1.28685			0.643423	+	0.765511i	$0.277514\pi$
$632$	77.7555	−	20.8345i	0.123031	−	0.0329660i
$633$		0			0
$634$	43.3013	−	25.0000i	0.0682985	−	0.0394322i
$635$	−388.210	+	23.3016i	−0.611354	+	0.0366954i
$636$		0			0
$637$	−13.6603	+	3.66025i	−0.0214447	+	0.00574608i
$638$	−1185.85	+	1185.85i	−1.85871	+	1.85871i
$639$		0			0
$640$	690.000	+	345.000i	1.07812	+	0.539062i
$641$	−221.359	+	383.406i	−0.345335	+	0.598137i	−0.985414	−	0.170171i	$-0.945568\pi$
$641$	−221.359	+	383.406i	−0.345335	+	0.598137i	0.640080	+	0.768308i	$0.278901\pi$
$642$		0			0
$643$	−1120.14	−	300.141i	−1.74205	−	0.466782i	−0.759152	−	0.650913i	$-0.774386\pi$
$643$	−1120.14	−	300.141i	−1.74205	−	0.466782i	−0.982902	+	0.184131i	$0.941053\pi$
$644$	−27.3861	+	15.8114i	−0.0425250	+	0.0245518i
$645$		0			0
$646$	−90.0000	+	155.885i	−0.139319	+	0.241307i
$647$	−679.890	−	679.890i	−1.05083	−	1.05083i	−0.998637	−	0.0521974i	$-0.983378\pi$
$647$	−679.890	−	679.890i	−1.05083	−	1.05083i	−0.0521974	−	0.998637i	$-0.516622\pi$
$648$		0			0
$649$			750.000i			1.15562i
$650$	94.5642	+	784.893i	0.145483	+	1.20753i
$651$		0			0
$652$	36.6025	−	136.603i	0.0561389	−	0.209513i
$653$	−704.120	−	188.668i	−1.07828	−	0.288925i	−0.324391	−	0.945923i	$-0.605159\pi$
$653$	−704.120	−	188.668i	−1.07828	−	0.288925i	−0.753893	+	0.656998i	$0.771826\pi$
$654$		0			0
$655$	−851.971	−	174.343i	−1.30072	−	0.266173i
$656$	−600.833			−0.915904
$657$		0			0
$658$	−650.000	+	650.000i	−0.987842	+	0.987842i
$659$	780.505	+	450.625i	1.18438	+	0.683801i	0.957023	−	0.290012i	$-0.0936592\pi$
$659$	780.505	+	450.625i	1.18438	+	0.683801i	0.227354	+	0.973812i	$0.426992\pi$
$660$		0			0
$661$	401.000	+	694.552i	0.606657	+	1.05076i	0.991787	+	0.127898i	$0.0408230\pi$
$661$	401.000	+	694.552i	0.606657	+	1.05076i	−0.385131	+	0.922862i	$0.625844\pi$
$662$	−278.951	+	1041.06i	−0.421376	+	1.57260i
$663$		0			0
$664$	441.673	+	255.000i	0.665170	+	0.384036i
$665$	−569.210	−	284.605i	−0.855955	−	0.427977i
$666$		0			0
$667$	−150.000	−	150.000i	−0.224888	−	0.224888i
$668$	54.4013	+	203.028i	0.0814390	+	0.303935i
$669$		0			0
$670$	828.109	+	734.327i	1.23598	+	1.09601i
$671$	458.530	+	794.198i	0.683354	+	1.18360i
$672$		0			0
$673$	283.670	+	1058.67i	0.421500	+	1.57306i	0.771449	+	0.636291i	$0.219532\pi$
$673$	283.670	+	1058.67i	0.421500	+	1.57306i	−0.349948	+	0.936769i	$0.613801\pi$
$674$			490.153i			0.727230i
$675$		0			0
$676$	31.0000			0.0458580
$677$	−678.201	+	181.723i	−1.00177	+	0.268425i	−0.722189	−	0.691696i	$-0.756864\pi$
$677$	−678.201	+	181.723i	−1.00177	+	0.268425i	−0.279585	+	0.960121i	$0.590197\pi$
$678$		0			0
$679$	43.3013	−	25.0000i	0.0637721	−	0.0368189i
$680$	99.5205	−	112.230i	0.146354	−	0.165045i
$681$		0			0
$682$	273.205	−	73.2051i	0.400594	−	0.107339i
$683$	60.0833	−	60.0833i	0.0879697	−	0.0879697i	−0.661753	−	0.749722i	$-0.730187\pi$
$683$	60.0833	−	60.0833i	0.0879697	−	0.0879697i	0.749722	+	0.661753i	$0.230187\pi$
$684$		0			0
$685$	−50.0000	+	100.000i	−0.0729927	+	0.145985i
$686$	−379.473	+	657.267i	−0.553168	+	0.958115i
$687$		0			0
$688$	−259.545	−	69.5448i	−0.377245	−	0.101083i
$689$	438.178	−	252.982i	0.635962	−	0.367173i
$690$		0			0
$691$	281.000	−	486.706i	0.406657	−	0.704351i	−0.587856	−	0.808966i	$-0.700028\pi$
$691$	281.000	−	486.706i	0.406657	−	0.704351i	0.994513	+	0.104615i	$0.0333611\pi$
$692$	−110.680	−	110.680i	−0.159942	−	0.159942i
$693$		0			0
$694$		−	520.000i		−	0.749280i
$695$	102.245	−	499.646i	0.147115	−	0.718915i
$696$		0			0
$697$	−36.6025	+	136.603i	−0.0525144	+	0.195986i
$698$	−686.841	−	184.038i	−0.984012	−	0.263665i
$699$		0			0
$700$	−139.054	−	109.151i	−0.198649	−	0.155929i
$701$	−363.662			−0.518776			−0.259388	−	0.965773i	$-0.583521\pi$
$701$	−363.662			−0.518776			−0.259388	+	0.965773i	$0.583521\pi$
$702$		0			0
$703$	−180.000	+	180.000i	−0.256046	+	0.256046i
$704$	561.416	+	324.133i	0.797465	+	0.460417i
$705$		0			0
$706$	−355.000	−	614.878i	−0.502833	−	0.870932i
$707$	−28.9368	+	107.994i	−0.0409291	+	0.152749i
$708$		0			0
$709$	−431.281	−	249.000i	−0.608294	−	0.351199i	0.164003	−	0.986460i	$-0.447559\pi$
$709$	−431.281	−	249.000i	−0.608294	−	0.351199i	−0.772298	+	0.635261i	$0.780893\pi$
$710$	−316.228	+	632.456i	−0.445391	+	0.890782i
$711$		0			0
$712$		0			0
$713$	9.25979	+	34.5580i	0.0129871	+	0.0484685i
$714$		0			0
$715$	66.9873	+	1116.03i	0.0936885	+	1.56087i
$716$	71.1512	+	123.238i	0.0993733	+	0.172120i
$717$		0			0
$718$	−164.711	−	614.711i	−0.229403	−	0.856144i
$719$		−	569.210i		−	0.791669i	−0.918322	−	0.395834i	$-0.870455\pi$
$719$		−	569.210i		−	0.791669i	0.918322	−	0.395834i	$-0.129545\pi$
$720$		0			0
$721$	350.000			0.485437
$722$	79.9154	−	21.4133i	0.110686	−	0.0296583i
$723$		0			0
$724$	188.794	−	109.000i	0.260765	−	0.150552i
$725$	465.828	−	1090.53i	0.642521	−	1.50418i
$726$		0			0
$727$	1236.25	−	331.253i	1.70049	−	0.455644i	0.727424	−	0.686188i	$-0.240717\pi$
$727$	1236.25	−	331.253i	1.70049	−	0.455644i	0.973062	+	0.230545i	$0.0740508\pi$
$728$	474.342	−	474.342i	0.651568	−	0.651568i
$729$		0			0
$730$	275.000	+	825.000i	0.376712	+	1.13014i
$731$	−31.6228	+	54.7723i	−0.0432596	+	0.0749278i
$732$		0			0
$733$	−751.314	−	201.314i	−1.02498	−	0.274644i	−0.293107	−	0.956080i	$-0.594689\pi$
$733$	−751.314	−	201.314i	−1.02498	−	0.274644i	−0.731878	+	0.681436i	$0.761356\pi$
$734$	506.643	−	292.511i	0.690250	−	0.398516i
$735$		0			0
$736$	35.0000	−	60.6218i	0.0475543	−	0.0823665i
$737$	1106.80	+	1106.80i	1.50176	+	1.50176i
$738$		0			0
$739$			198.000i			0.267930i	0.990986	+	0.133965i	$0.0427709\pi$
$739$			198.000i			0.267930i	−0.990986	+	0.133965i	$0.957229\pi$
$740$	−59.0089	+	38.9609i	−0.0797418	+	0.0526498i
$741$		0			0
$742$	−146.410	+	546.410i	−0.197318	+	0.736402i
$743$	1317.52	+	353.030i	1.77325	+	0.475141i	0.989326	−	0.145717i	$-0.0465490\pi$
$743$	1317.52	+	353.030i	1.77325	+	0.475141i	0.783923	+	0.620858i	$0.213216\pi$
$744$		0			0
$745$	−47.5481	+	232.356i	−0.0638229	+	0.311887i
$746$	−316.228			−0.423898
$747$		0			0
$748$	−50.0000	+	50.0000i	−0.0668449	+	0.0668449i
$749$	520.336	+	300.416i	0.694708	+	0.401090i
$750$		0			0
$751$	−364.000	−	630.466i	−0.484687	−	0.839503i	0.515158	−	0.857095i	$-0.327733\pi$
$751$	−364.000	−	630.466i	−0.484687	−	0.839503i	−0.999845	+	0.0175925i	$0.994400\pi$
$752$	285.896	−	1066.98i	0.380181	−	1.41885i
$753$		0			0
$754$	−1299.04	−	750.000i	−1.72286	−	0.994695i
$755$	34.7851	+	104.355i	0.0460729	+	0.138219i
$756$		0			0
$757$	−170.000	−	170.000i	−0.224571	−	0.224571i	0.585849	−	0.810420i	$-0.300761\pi$
$757$	−170.000	−	170.000i	−0.224571	−	0.224571i	−0.810420	+	0.585849i	$0.800761\pi$
$758$	322.935	+	1205.21i	0.426036	+	1.58999i
$759$		0			0
$760$	602.654	−	36.1731i	0.792965	−	0.0475962i
$761$	15.8114	+	27.3861i	0.0207771	+	0.0359870i	0.876227	−	0.481899i	$-0.160053\pi$
$761$	15.8114	+	27.3861i	0.0207771	+	0.0359870i	−0.855450	+	0.517886i	$0.826719\pi$
$762$		0			0
$763$	296.481	+	1106.48i	0.388572	+	1.45017i
$764$			158.114i			0.206955i
$765$		0			0
$766$	890.000			1.16188
$767$	−647.963	+	173.621i	−0.844801	+	0.226364i
$768$		0			0
$769$	−275.396	+	159.000i	−0.358122	+	0.206762i	−0.668257	−	0.743931i	$-0.732959\pi$
$769$	−275.396	+	159.000i	−0.358122	+	0.206762i	0.310134	+	0.950693i	$0.399626\pi$
$770$	−935.254	−	829.337i	−1.21462	−	1.07706i
$771$		0			0
$772$	170.753	−	45.7532i	0.221183	−	0.0592658i
$773$	−452.206	+	452.206i	−0.585001	+	0.585001i	−0.936273	−	0.351272i	$-0.885749\pi$
$773$	−452.206	+	452.206i	−0.585001	+	0.585001i	0.351272	+	0.936273i	$0.385749\pi$
$774$		0			0
$775$	−160.000	+	120.000i	−0.206452	+	0.154839i
$776$	−23.7171	+	41.0792i	−0.0305633	+	0.0529371i
$777$		0			0
$778$	−1126.97	−	301.971i	−1.44855	−	0.388137i
$779$	−492.950	+	284.605i	−0.632799	+	0.365347i
$780$		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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	405.l (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.15988 - 0.578737i) q^{2} +(0.866025 - 0.500000i) q^{4} +(-3.31735 + 3.74101i) q^{5} +(6.83013 - 1.83013i) q^{7} +(-4.74342 + 4.74342i) q^{8} +(-5.00000 + 10.0000i) q^{10} +(-7.90569 + 13.6931i) q^{11} +(-13.6603 - 3.66025i) q^{13} +(13.6931 - 7.90569i) q^{14} +(-9.50000 + 16.4545i) q^{16} +(3.16228 + 3.16228i) q^{17} +18.0000i q^{19} +(-1.00240 + 4.89849i) q^{20} +(-9.15064 + 34.1506i) q^{22} +(-4.31975 - 1.15747i) q^{23} +(-2.99038 - 24.8205i) q^{25} -31.6228 q^{26} +(5.00000 - 5.00000i) q^{28} +(41.0792 + 23.7171i) q^{29} +(-4.00000 - 6.92820i) q^{31} +(-4.05116 + 15.1191i) q^{32} +(8.66025 + 5.00000i) q^{34} +(-15.8114 + 31.6228i) q^{35} +(10.0000 + 10.0000i) q^{37} +(10.4173 + 38.8778i) q^{38} +(-2.00962 - 33.4808i) q^{40} +(15.8114 + 27.3861i) q^{41} +(3.66025 + 13.6603i) q^{43} +15.8114i q^{44} -10.0000 q^{46} +(-56.1568 + 15.0472i) q^{47} +(0.866025 - 0.500000i) q^{49} +(-20.8234 - 51.8786i) q^{50} +(-13.6603 + 3.66025i) q^{52} +(-25.2982 + 25.2982i) q^{53} +(-25.0000 - 75.0000i) q^{55} +(-23.7171 + 41.0792i) q^{56} +(102.452 + 27.4519i) q^{58} +(41.0792 - 23.7171i) q^{59} +(29.0000 - 50.2295i) q^{61} +(-12.6491 - 12.6491i) q^{62} -41.0000i q^{64} +(59.0089 - 38.9609i) q^{65} +(25.6218 - 95.6218i) q^{67} +(4.31975 + 1.15747i) q^{68} +(-15.8494 + 77.4519i) q^{70} +63.2456 q^{71} +(55.0000 - 55.0000i) q^{73} +(27.3861 + 15.8114i) q^{74} +(9.00000 + 15.5885i) q^{76} +(-28.9368 + 107.994i) q^{77} +(-10.3923 - 6.00000i) q^{79} +(-30.0416 - 90.1249i) q^{80} +(50.0000 + 50.0000i) q^{82} +(-19.6771 - 73.4358i) q^{83} +(-22.3205 + 1.33975i) q^{85} +(15.8114 + 27.3861i) q^{86} +(-27.4519 - 102.452i) q^{88} -100.000 q^{91} +(-4.31975 + 1.15747i) q^{92} +(-112.583 + 65.0000i) q^{94} +(-67.3383 - 59.7123i) q^{95} +(6.83013 - 1.83013i) q^{97} +(1.58114 - 1.58114i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 20 q^{7} - 40 q^{10} - 40 q^{13} - 76 q^{16} + 100 q^{22} + 80 q^{25} + 40 q^{28} - 32 q^{31} + 80 q^{37} - 120 q^{40} - 40 q^{43} - 80 q^{46} - 40 q^{52} - 200 q^{55} + 300 q^{58} + 232 q^{61} - 280 q^{67}+ \cdots + 20 q^{97}+O(q^{100}) \) 8 * q + 20 * q^7 - 40 * q^10 - 40 * q^13 - 76 * q^16 + 100 * q^22 + 80 * q^25 + 40 * q^28 - 32 * q^31 + 80 * q^37 - 120 * q^40 - 40 * q^43 - 80 * q^46 - 40 * q^52 - 200 * q^55 + 300 * q^58 + 232 * q^61 - 280 * q^67 - 300 * q^70 + 440 * q^73 + 72 * q^76 + 400 * q^82 - 40 * q^85 + 300 * q^88 - 800 * q^91 + 20 * q^97

Introduction

L-functions

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