LMFDB - Embedded newform 405.3.l.g.298.1

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			298.1
Root			$2.15988 - 0.578737i$ of defining polynomial
Character	$\chi$	$=$	405.298
Dual form			405.3.l.g.352.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.578737 - 2.15988i) q^{2} +(-0.866025 + 0.500000i) q^{4} +(4.89849 - 1.00240i) q^{5} +(-1.83013 - 6.83013i) q^{7} +(-4.74342 - 4.74342i) q^{8} +(-5.00000 - 10.0000i) q^{10} +(-7.90569 + 13.6931i) q^{11} +(3.66025 - 13.6603i) q^{13} +(-13.6931 + 7.90569i) q^{14} +(-9.50000 + 16.4545i) q^{16} +(3.16228 - 3.16228i) q^{17} -18.0000i q^{19} +(-3.74101 + 3.31735i) q^{20} +(34.1506 + 9.15064i) q^{22} +(1.15747 - 4.31975i) q^{23} +(22.9904 - 9.82051i) q^{25} -31.6228 q^{26} +(5.00000 + 5.00000i) q^{28} +(-41.0792 - 23.7171i) q^{29} +(-4.00000 - 6.92820i) q^{31} +(15.1191 + 4.05116i) q^{32} +(-8.66025 - 5.00000i) q^{34} +(-15.8114 - 31.6228i) q^{35} +(10.0000 - 10.0000i) q^{37} +(-38.8778 + 10.4173i) q^{38} +(-27.9904 - 18.4808i) q^{40} +(15.8114 + 27.3861i) q^{41} +(-13.6603 + 3.66025i) q^{43} -15.8114i q^{44} -10.0000 q^{46} +(15.0472 + 56.1568i) q^{47} +(-0.866025 + 0.500000i) q^{49} +(-34.5165 - 43.9729i) q^{50} +(3.66025 + 13.6603i) q^{52} +(-25.2982 - 25.2982i) q^{53} +(-25.0000 + 75.0000i) q^{55} +(-23.7171 + 41.0792i) q^{56} +(-27.4519 + 102.452i) q^{58} +(-41.0792 + 23.7171i) q^{59} +(29.0000 - 50.2295i) q^{61} +(-12.6491 + 12.6491i) q^{62} +41.0000i q^{64} +(4.23665 - 70.5836i) q^{65} +(-95.6218 - 25.6218i) q^{67} +(-1.15747 + 4.31975i) q^{68} +(-59.1506 + 52.4519i) q^{70} +63.2456 q^{71} +(55.0000 + 55.0000i) q^{73} +(-27.3861 - 15.8114i) q^{74} +(9.00000 + 15.5885i) q^{76} +(107.994 + 28.9368i) q^{77} +(10.3923 + 6.00000i) q^{79} +(-30.0416 + 90.1249i) q^{80} +(50.0000 - 50.0000i) q^{82} +(73.4358 - 19.6771i) q^{83} +(12.3205 - 18.6603i) q^{85} +(15.8114 + 27.3861i) q^{86} +(102.452 - 27.4519i) q^{88} -100.000 q^{91} +(1.15747 + 4.31975i) q^{92} +(112.583 - 65.0000i) q^{94} +(-18.0432 - 88.1728i) q^{95} +(-1.83013 - 6.83013i) q^{97} +(1.58114 + 1.58114i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

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

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 $

$2$	−0.578737	−	2.15988i	−0.289368	−	1.07994i	−0.945588	−	0.325368i	$-0.894512\pi$
$2$	−0.578737	−	2.15988i	−0.289368	−	1.07994i	0.656219	−	0.754570i	$-0.272155\pi$
$3$		0			0
$3$		0			0
$4$	−0.866025	+	0.500000i	−0.216506	+	0.125000i
$5$	4.89849	−	1.00240i	0.979698	−	0.200480i
$5$	4.89849	−	1.00240i	0.979698	−	0.200480i
$6$		0			0
$7$	−1.83013	−	6.83013i	−0.261447	−	0.975732i	−0.964390	−	0.264486i	$-0.914798\pi$
$7$	−1.83013	−	6.83013i	−0.261447	−	0.975732i	0.702943	−	0.711246i	$-0.251869\pi$
$8$	−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
$13$	3.66025	−	13.6603i	0.281558	−	1.05079i	−0.669760	−	0.742578i	$-0.733603\pi$
$13$	3.66025	−	13.6603i	0.281558	−	1.05079i	0.951318	−	0.308211i	$-0.0997303\pi$
$14$	−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	−0.607955	−	0.793971i	$-0.708010\pi$
$17$	3.16228	−	3.16228i	0.186016	−	0.186016i	0.793971	+	0.607955i	$0.208010\pi$
$18$		0			0
$19$		−	18.0000i		−	0.947368i	−0.880695	−	0.473684i	$-0.842924\pi$
$19$		−	18.0000i		−	0.947368i	0.880695	−	0.473684i	$-0.157076\pi$
$20$	−3.74101	+	3.31735i	−0.187051	+	0.165867i
$21$		0			0
$22$	34.1506	+	9.15064i	1.55230	+	0.415938i
$23$	1.15747	−	4.31975i	0.0503250	−	0.187815i	−0.936188	−	0.351501i	$-0.885672\pi$
$23$	1.15747	−	4.31975i	0.0503250	−	0.187815i	0.986513	+	0.163685i	$0.0523382\pi$
$24$		0			0
$25$	22.9904	−	9.82051i	0.919615	−	0.392820i
$26$	−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	−0.420532	−	0.907278i	$-0.638157\pi$
$29$	−41.0792	−	23.7171i	−1.41652	−	0.817830i	−0.995992	+	0.0894471i	$0.971490\pi$
$30$		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	$0.874520\pi$
$32$	15.1191	+	4.05116i	0.472473	+	0.126599i
$33$		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	−0.558939	−	0.829209i	$-0.688791\pi$
$37$	10.0000	−	10.0000i	0.270270	−	0.270270i	0.829209	+	0.558939i	$0.188791\pi$
$38$	−38.8778	+	10.4173i	−1.02310	+	0.274139i
$39$		0			0
$40$	−27.9904	−	18.4808i	−0.699760	−	0.462019i
$41$	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
$43$	−13.6603	+	3.66025i	−0.317680	+	0.0851222i	−0.414136	−	0.910215i	$-0.635916\pi$
$43$	−13.6603	+	3.66025i	−0.317680	+	0.0851222i	0.0964555	+	0.995337i	$0.469249\pi$
$44$		−	15.8114i		−	0.359350i
$45$		0			0
$46$	−10.0000			−0.217391
$47$	15.0472	+	56.1568i	0.320152	+	1.19482i	0.919096	+	0.394034i	$0.128921\pi$
$47$	15.0472	+	56.1568i	0.320152	+	1.19482i	−0.598944	+	0.800791i	$0.704413\pi$
$48$		0			0
$49$	−0.866025	+	0.500000i	−0.0176740	+	0.0102041i
$50$	−34.5165	−	43.9729i	−0.690329	−	0.879458i
$51$		0			0
$52$	3.66025	+	13.6603i	0.0703895	+	0.262697i
$53$	−25.2982	−	25.2982i	−0.477325	−	0.477325i	0.426950	−	0.904275i	$-0.359588\pi$
$53$	−25.2982	−	25.2982i	−0.477325	−	0.477325i	−0.904275	+	0.426950i	$0.859588\pi$
$54$		0			0
$55$	−25.0000	+	75.0000i	−0.454545	+	1.36364i
$56$	−23.7171	+	41.0792i	−0.423519	+	0.733557i
$57$		0			0
$58$	−27.4519	+	102.452i	−0.473309	+	1.76641i
$59$	−41.0792	+	23.7171i	−0.696257	+	0.401984i	−0.805952	−	0.591981i	$-0.798346\pi$
$59$	−41.0792	+	23.7171i	−0.696257	+	0.401984i	0.109695	+	0.993965i	$0.465013\pi$
$60$		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
$65$	4.23665	−	70.5836i	0.0651792	−	1.08590i
$66$		0			0
$67$	−95.6218	−	25.6218i	−1.42719	−	0.382415i	−0.539162	−	0.842202i	$-0.681259\pi$
$67$	−95.6218	−	25.6218i	−1.42719	−	0.382415i	−0.888029	+	0.459788i	$0.847926\pi$
$68$	−1.15747	+	4.31975i	−0.0170217	+	0.0635258i
$69$		0			0
$70$	−59.1506	+	52.4519i	−0.845009	+	0.749313i
$71$	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	0.975117	−	0.221692i	$-0.0711580\pi$
$73$	55.0000	+	55.0000i	0.753425	+	0.753425i	−0.221692	+	0.975117i	$0.571158\pi$
$74$	−27.3861	−	15.8114i	−0.370083	−	0.213667i
$75$		0			0
$76$	9.00000	+	15.5885i	0.118421	+	0.205111i
$77$	107.994	+	28.9368i	1.40252	+	0.375803i
$78$		0			0
$79$	10.3923	+	6.00000i	0.131548	+	0.0759494i	0.564330	−	0.825549i	$-0.309135\pi$
$79$	10.3923	+	6.00000i	0.131548	+	0.0759494i	−0.432782	+	0.901499i	$0.642468\pi$
$80$	−30.0416	+	90.1249i	−0.375520	+	1.12656i
$81$		0			0
$82$	50.0000	−	50.0000i	0.609756	−	0.609756i
$83$	73.4358	−	19.6771i	0.884768	−	0.237073i	0.212305	−	0.977203i	$-0.431903\pi$
$83$	73.4358	−	19.6771i	0.884768	−	0.237073i	0.672463	+	0.740130i	$0.265236\pi$
$84$		0			0
$85$	12.3205	−	18.6603i	0.144947	−	0.219532i
$86$	15.8114	+	27.3861i	0.183853	+	0.318443i
$87$		0			0
$88$	102.452	−	27.4519i	1.16423	−	0.311953i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$	−100.000			−1.09890
$92$	1.15747	+	4.31975i	0.0125812	+	0.0469538i
$93$		0			0
$94$	112.583	−	65.0000i	1.19769	−	0.691489i
$95$	−18.0432	−	88.1728i	−0.189929	−	0.928135i
$96$		0			0
$97$	−1.83013	−	6.83013i	−0.0188673	−	0.0704137i	0.955850	−	0.293854i	$-0.0949379\pi$
$97$	−1.83013	−	6.83013i	−0.0188673	−	0.0704137i	−0.974718	+	0.223440i	$0.928271\pi$
$98$	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
$103$	−12.8109	+	47.8109i	−0.124378	+	0.464183i	−0.999817	−	0.0191450i	$-0.993906\pi$
$103$	−12.8109	+	47.8109i	−0.124378	+	0.464183i	0.875439	+	0.483328i	$0.160572\pi$
$104$	−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	−0.368215	−	0.929741i	$-0.620031\pi$
$107$	60.0833	−	60.0833i	0.561526	−	0.561526i	0.929741	+	0.368215i	$0.120031\pi$
$108$		0			0
$109$		−	162.000i		−	1.48624i	−0.669159	−	0.743119i	$-0.733345\pi$
$109$		−	162.000i		−	1.48624i	0.669159	−	0.743119i	$-0.266655\pi$
$110$	176.459	+	10.5916i	1.60417	+	0.0962875i
$111$		0			0
$112$	129.772	+	34.7724i	1.15868	+	0.310468i
$113$	42.8265	−	159.831i	0.378996	−	1.41443i	−0.468422	−	0.883505i	$-0.655177\pi$
$113$	42.8265	−	159.831i	0.378996	−	1.41443i	0.847418	−	0.530927i	$-0.178156\pi$
$114$		0			0
$115$	1.33975	−	22.3205i	0.0116500	−	0.194091i
$116$	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
$122$	−125.273	−	33.5667i	−1.02683	−	0.275137i
$123$		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	−0.456601	−	0.889672i	$-0.650933\pi$
$127$	55.0000	−	55.0000i	0.433071	−	0.433071i	0.889672	+	0.456601i	$0.150933\pi$
$128$	149.031	−	39.9329i	1.16431	−	0.311975i
$129$		0			0
$130$	−154.904	+	31.6987i	−1.19157	+	0.243836i
$131$	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
$133$	−122.942	+	32.9423i	−0.924378	+	0.247686i
$134$			221.359i			1.65194i
$135$		0			0
$136$	−30.0000			−0.220588
$137$	−5.78737	−	21.5988i	−0.0422436	−	0.157655i	0.941582	−	0.336783i	$-0.109339\pi$
$137$	−5.78737	−	21.5988i	−0.0422436	−	0.157655i	−0.983826	+	0.179128i	$0.942672\pi$
$138$		0			0
$139$	88.3346	−	51.0000i	0.635501	−	0.366906i	−0.147379	−	0.989080i	$-0.547084\pi$
$139$	88.3346	−	51.0000i	0.635501	−	0.366906i	0.782879	+	0.622174i	$0.213750\pi$
$140$	29.5045	+	19.4804i	0.210746	+	0.139146i
$141$		0			0
$142$	−36.6025	−	136.603i	−0.257764	−	0.961990i
$143$	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
$148$	−3.66025	+	13.6603i	−0.0247314	+	0.0922990i
$149$	−41.0792	+	23.7171i	−0.275699	+	0.159175i	−0.631475	−	0.775396i	$-0.717550\pi$
$149$	−41.0792	+	23.7171i	−0.275699	+	0.159175i	0.355776	+	0.934571i	$0.384217\pi$
$150$		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
$155$	−26.5388	−	29.9281i	−0.171218	−	0.193085i
$156$		0			0
$157$	273.205	+	73.2051i	1.74016	+	0.466274i	0.982482	−	0.186355i	$-0.0596675\pi$
$157$	273.205	+	73.2051i	1.74016	+	0.466274i	0.757677	+	0.652629i	$0.226334\pi$
$158$	6.94484	−	25.9185i	0.0439547	−	0.164041i
$159$		0			0
$160$	78.1218	+	4.68911i	0.488261	+	0.0293069i
$161$	−31.6228			−0.196415
$162$		0			0
$163$	100.000	+	100.000i	0.613497	+	0.613497i	0.943856	−	0.330359i	$-0.107170\pi$
$163$	100.000	+	100.000i	0.613497	+	0.613497i	−0.330359	+	0.943856i	$0.607170\pi$
$164$	−27.3861	−	15.8114i	−0.166989	−	0.0964109i
$165$		0			0
$166$	−85.0000	−	147.224i	−0.512048	−	0.886893i
$167$	203.028	+	54.4013i	1.21574	+	0.325756i	0.809011	−	0.587793i	$-0.200003\pi$
$167$	203.028	+	54.4013i	1.21574	+	0.325756i	0.406727	+	0.913550i	$0.366670\pi$
$168$		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
$173$	151.191	−	40.5116i	0.873938	−	0.234171i	0.206148	−	0.978521i	$-0.433907\pi$
$173$	151.191	−	40.5116i	0.873938	−	0.234171i	0.667790	+	0.744350i	$0.267240\pi$
$174$		0			0
$175$	−109.151	−	139.054i	−0.623718	−	0.794597i
$176$	−150.208	−	260.168i	−0.853456	−	1.47823i
$177$		0			0
$178$		0			0
$179$		−	142.302i		−	0.794986i	−0.917605	−	0.397493i	$-0.869880\pi$
$179$		−	142.302i		−	0.794986i	0.917605	−	0.397493i	$-0.130120\pi$
$180$		0			0
$181$	218.000			1.20442			0.602210	−	0.798338i	$-0.294287\pi$
$181$	218.000			1.20442			0.602210	+	0.798338i	$0.294287\pi$
$182$	57.8737	+	215.988i	0.317987	+	1.18674i
$183$		0			0
$184$	−25.9808	+	15.0000i	−0.141200	+	0.0815217i
$185$	38.9609	−	59.0089i	0.210599	−	0.318967i
$186$		0			0
$187$	18.3013	+	68.3013i	0.0978678	+	0.365247i
$188$	−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
$193$	−45.7532	+	170.753i	−0.237063	+	0.884731i	0.740145	+	0.672447i	$0.234757\pi$
$193$	−45.7532	+	170.753i	−0.237063	+	0.884731i	−0.977208	+	0.212284i	$0.931910\pi$
$194$	−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	−0.233868	−	0.972268i	$-0.575138\pi$
$197$	145.465	−	145.465i	0.738400	−	0.738400i	0.972268	+	0.233868i	$0.0751385\pi$
$198$		0			0
$199$			18.0000i			0.0904523i	0.998977	+	0.0452261i	$0.0144008\pi$
$199$			18.0000i			0.0904523i	−0.998977	+	0.0452261i	$0.985599\pi$
$200$	−155.636	−	62.4702i	−0.778179	−	0.312351i
$201$		0			0
$202$	34.1506	+	9.15064i	0.169063	+	0.0453002i
$203$	−86.8105	+	323.981i	−0.427638	+	1.59597i
$204$		0			0
$205$	104.904	+	118.301i	0.511726	+	0.577079i
$206$	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	$0.583759\pi$
$212$	34.5580	+	9.25979i	0.163009	+	0.0436783i
$213$		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
$218$	−349.900	+	93.7554i	−1.60505	+	0.430071i
$219$		0			0
$220$	−15.8494	−	77.4519i	−0.0720426	−	0.352054i
$221$	−31.6228	−	54.7723i	−0.143089	−	0.247838i
$222$		0			0
$223$	293.695	−	78.6955i	1.31702	−	0.352894i	0.469159	−	0.883114i	$-0.344557\pi$
$223$	293.695	−	78.6955i	1.31702	−	0.352894i	0.847861	+	0.530219i	$0.177890\pi$
$224$		−	110.680i		−	0.494106i
$225$		0			0
$226$	−370.000			−1.63717
$227$	−99.5428	−	371.499i	−0.438514	−	1.63656i	−0.732514	−	0.680752i	$-0.761653\pi$
$227$	−99.5428	−	371.499i	−0.438514	−	1.63656i	0.293999	−	0.955806i	$-0.405014\pi$
$228$		0			0
$229$	−67.5500	+	39.0000i	−0.294978	+	0.170306i	−0.640185	−	0.768221i	$-0.721142\pi$
$229$	−67.5500	+	39.0000i	−0.294978	+	0.170306i	0.345207	+	0.938527i	$0.387809\pi$
$230$	−48.9849	+	10.0240i	−0.212978	+	0.0435827i
$231$		0			0
$232$	82.3557	+	307.356i	0.354982	+	1.32481i
$233$	−110.680	−	110.680i	−0.475020	−	0.475020i	0.428515	−	0.903535i	$-0.359037\pi$
$233$	−110.680	−	110.680i	−0.475020	−	0.475020i	−0.903535	+	0.428515i	$0.859037\pi$
$234$		0			0
$235$	130.000	+	260.000i	0.553191	+	1.10638i
$236$	23.7171	−	41.0792i	0.100496	−	0.174064i
$237$		0			0
$238$	−18.3013	+	68.3013i	−0.0768961	+	0.286980i
$239$	328.634	−	189.737i	1.37504	−	0.793877i	0.383479	−	0.923550i	$-0.374726\pi$
$239$	328.634	−	189.737i	1.37504	−	0.793877i	0.991557	+	0.129672i	$0.0413926\pi$
$240$		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
$245$	−3.74101	+	3.31735i	−0.0152694	+	0.0135402i
$246$		0			0
$247$	−245.885	−	65.8846i	−0.995484	−	0.266739i
$248$	−13.8897	+	51.8370i	−0.0560068	+	0.209020i
$249$		0			0
$250$	−213.157	−	180.801i	−0.852628	−	0.723205i
$251$	−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
$257$	−419.016	−	112.275i	−1.63041	−	0.436868i	−0.676375	−	0.736557i	$-0.736450\pi$
$257$	−419.016	−	112.275i	−1.63041	−	0.436868i	−0.954037	+	0.299690i	$0.903117\pi$
$258$		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
$263$	−393.097	+	105.330i	−1.49467	+	0.400495i	−0.911310	−	0.411721i	$-0.864928\pi$
$263$	−393.097	+	105.330i	−1.49467	+	0.400495i	−0.583357	+	0.812216i	$0.698261\pi$
$264$		0			0
$265$	−149.282	−	98.5641i	−0.563328	−	0.371940i
$266$	142.302	+	246.475i	0.534972	+	0.926598i
$267$		0			0
$268$	95.6218	−	25.6218i	0.356798	−	0.0956037i
$269$			142.302i			0.529006i	0.964385	+	0.264503i	$0.0852078\pi$
$269$			142.302i			0.529006i	−0.964385	+	0.264503i	$0.914792\pi$
$270$		0			0
$271$	−178.000			−0.656827			−0.328413	−	0.944534i	$-0.606514\pi$
$271$	−178.000			−0.656827			−0.328413	+	0.944534i	$0.606514\pi$
$272$	21.9920	+	82.0753i	0.0808530	+	0.301747i
$273$		0			0
$274$	−43.3013	+	25.0000i	−0.158034	+	0.0912409i
$275$	−47.2821	+	392.447i	−0.171935	+	1.42708i
$276$		0			0
$277$	−84.1858	−	314.186i	−0.303920	−	1.13424i	−0.933871	−	0.357610i	$-0.883592\pi$
$277$	−84.1858	−	314.186i	−0.303920	−	1.13424i	0.629951	−	0.776635i	$-0.283075\pi$
$278$	−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
$283$	−128.109	+	478.109i	−0.452682	+	1.68943i	0.242134	+	0.970243i	$0.422153\pi$
$283$	−128.109	+	478.109i	−0.452682	+	1.68943i	−0.694816	+	0.719188i	$0.744514\pi$
$284$	−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
$290$	−31.7749	+	529.377i	−0.109569	+	1.82544i
$291$		0			0
$292$	−75.1314	−	20.1314i	−0.257299	−	0.0689431i
$293$	74.0783	−	276.464i	0.252827	−	0.943563i	−0.716459	−	0.697629i	$-0.754238\pi$
$293$	74.0783	−	276.464i	0.252827	−	0.943563i	0.969286	−	0.245935i	$-0.0790949\pi$
$294$		0			0
$295$	−177.452	+	157.356i	−0.601532	+	0.533409i
$296$	−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
$302$	−47.5173	−	12.7322i	−0.157342	−	0.0421596i
$303$		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	−0.326355	−	0.945247i	$-0.605820\pi$
$307$	190.000	−	190.000i	0.618893	−	0.618893i	0.945247	+	0.326355i	$0.105820\pi$
$308$	−107.994	+	28.9368i	−0.350629	+	0.0939508i
$309$		0			0
$310$	−49.2820	+	74.6410i	−0.158974	+	0.240777i
$311$	−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
$313$	−198.074	+	53.0737i	−0.632823	+	0.169564i	−0.560951	−	0.827849i	$-0.689564\pi$
$313$	−198.074	+	53.0737i	−0.632823	+	0.169564i	−0.0718727	+	0.997414i	$0.522898\pi$
$314$		−	632.456i		−	2.01419i
$315$		0			0
$316$	−12.0000			−0.0379747
$317$	−5.78737	−	21.5988i	−0.0182567	−	0.0681349i	0.956197	−	0.292725i	$-0.0945622\pi$
$317$	−5.78737	−	21.5988i	−0.0182567	−	0.0681349i	−0.974453	+	0.224591i	$0.927896\pi$
$318$		0			0
$319$	649.519	−	375.000i	2.03611	−	1.17555i
$320$	41.0985	+	200.838i	0.128433	+	0.627619i
$321$		0			0
$322$	18.3013	+	68.3013i	0.0568362	+	0.212116i
$323$	−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
$328$	54.9038	−	204.904i	0.167390	−	0.624707i
$329$	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
$335$	−494.086	−	29.6565i	−1.47488	−	0.0885270i
$336$		0			0
$337$	211.734	+	56.7339i	0.628291	+	0.168350i	0.558894	−	0.829239i	$-0.311226\pi$
$337$	211.734	+	56.7339i	0.628291	+	0.168350i	0.0693967	+	0.997589i	$0.477893\pi$
$338$	−17.9408	+	66.9562i	−0.0530794	+	0.198095i
$339$		0			0
$340$	−1.33975	+	22.3205i	−0.00394043	+	0.0656486i
$341$	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
$347$	−224.627	−	60.1886i	−0.647340	−	0.173454i	−0.0798143	−	0.996810i	$-0.525433\pi$
$347$	−224.627	−	60.1886i	−0.647340	−	0.173454i	−0.567526	+	0.823355i	$0.692099\pi$
$348$		0			0
$349$	275.396	+	159.000i	0.789101	+	0.455587i	0.839646	−	0.543134i	$-0.182763\pi$
$349$	275.396	+	159.000i	0.789101	+	0.455587i	−0.0505453	+	0.998722i	$0.516096\pi$
$350$	−237.171	+	316.228i	−0.677631	+	0.903508i
$351$		0			0
$352$	−175.000	+	175.000i	−0.497159	+	0.497159i
$353$	306.702	−	82.1807i	0.868845	−	0.232806i	0.203257	−	0.979125i	$-0.434847\pi$
$353$	306.702	−	82.1807i	0.868845	−	0.232806i	0.665588	+	0.746319i	$0.268181\pi$
$354$		0			0
$355$	309.808	−	63.3975i	0.872698	−	0.178584i
$356$		0			0
$357$		0			0
$358$	−307.356	+	82.3557i	−0.858536	+	0.230044i
$359$			284.605i			0.792772i	0.918084	+	0.396386i	$0.129736\pi$
$359$			284.605i			0.792772i	−0.918084	+	0.396386i	$0.870264\pi$
$360$		0			0
$361$	37.0000			0.102493
$362$	−126.165	−	470.853i	−0.348521	−	1.30070i
$363$		0			0
$364$	86.6025	−	50.0000i	0.237919	−	0.137363i
$365$	324.549	+	214.285i	0.889175	+	0.587082i
$366$		0			0
$367$	−67.7147	−	252.715i	−0.184509	−	0.688596i	−0.994735	−	0.102479i	$-0.967323\pi$
$367$	−67.7147	−	252.715i	−0.184509	−	0.688596i	0.810226	−	0.586117i	$-0.199344\pi$
$368$	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
$373$	36.6025	−	136.603i	0.0981301	−	0.366227i	−0.899345	−	0.437239i	$-0.855956\pi$
$373$	36.6025	−	136.603i	0.0981301	−	0.366227i	0.997475	+	0.0710125i	$0.0226230\pi$
$374$	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	$-0.736643\pi$
$379$		−	558.000i		−	1.47230i	0.676821	−	0.736148i	$-0.263357\pi$
$380$	59.7123	+	67.3383i	0.157138	+	0.177206i
$381$		0			0
$382$	341.506	+	91.5064i	0.893996	+	0.239545i
$383$	−103.015	+	384.458i	−0.268969	+	1.00381i	0.690807	+	0.723039i	$0.257256\pi$
$383$	−103.015	+	384.458i	−0.268969	+	1.00381i	−0.959776	+	0.280767i	$0.909411\pi$
$384$		0			0
$385$	558.013	+	33.4936i	1.44938	+	0.0869965i
$386$	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	0.951692	−	0.307053i	$-0.0993430\pi$
$389$	451.871	+	260.888i	1.16162	+	0.670663i	0.209930	+	0.977716i	$0.432676\pi$
$390$		0			0
$391$	−10.0000	−	17.3205i	−0.0255754	−	0.0442980i
$392$	6.47963	+	1.73621i	0.0165297	+	0.00442911i
$393$		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	−0.954172	−	0.299260i	$-0.903260\pi$
$397$	−260.000	+	260.000i	−0.654912	+	0.654912i	0.299260	+	0.954172i	$0.403260\pi$
$398$	38.8778	−	10.4173i	0.0976828	−	0.0261740i
$399$		0			0
$400$	−56.8172	+	471.590i	−0.142043	+	1.17897i
$401$	−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
$403$	−109.282	+	29.2820i	−0.271171	+	0.0726601i
$404$		−	15.8114i		−	0.0391371i
$405$		0			0
$406$	750.000			1.84729
$407$	57.8737	+	215.988i	0.142196	+	0.530682i
$408$		0			0
$409$	−301.377	+	174.000i	−0.736863	+	0.425428i	−0.820928	−	0.571032i	$-0.806543\pi$
$409$	−301.377	+	174.000i	−0.736863	+	0.425428i	0.0840648	+	0.996460i	$0.473210\pi$
$410$	194.804	−	295.045i	0.475133	−	0.719621i
$411$		0			0
$412$	−12.8109	−	47.8109i	−0.0310944	−	0.116046i
$413$	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
$418$	164.711	−	614.711i	0.394046	−	1.47060i
$419$	−534.029	+	308.322i	−1.27453	+	0.735852i	−0.975838	−	0.218496i	$-0.929885\pi$
$419$	−534.029	+	308.322i	−1.27453	+	0.735852i	−0.298696	+	0.954348i	$0.596552\pi$
$420$		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
$425$	41.6468	−	103.757i	0.0979925	−	0.244134i
$426$		0			0
$427$	−396.147	−	106.147i	−0.927746	−	0.248589i
$428$	−21.9920	+	82.0753i	−0.0513832	+	0.191765i
$429$		0			0
$430$	104.904	+	118.301i	0.243962	+	0.275119i
$431$	−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	0.854434	−	0.519561i	$-0.173904\pi$
$433$	145.000	+	145.000i	0.334873	+	0.334873i	−0.519561	+	0.854434i	$0.673904\pi$
$434$	109.545	+	63.2456i	0.252407	+	0.145727i
$435$		0			0
$436$	81.0000	+	140.296i	0.185780	+	0.321780i
$437$	−77.7555	−	20.8345i	−0.177930	−	0.0476763i
$438$		0			0
$439$	−67.5500	−	39.0000i	−0.153872	−	0.0888383i	0.421087	−	0.907020i	$-0.361649\pi$
$439$	−67.5500	−	39.0000i	−0.153872	−	0.0888383i	−0.574959	+	0.818182i	$0.694982\pi$
$440$	474.342	−	237.171i	1.07805	−	0.539025i
$441$		0			0
$442$	−100.000	+	100.000i	−0.226244	+	0.226244i
$443$	267.825	−	71.7634i	0.604570	−	0.161994i	0.0564665	−	0.998404i	$-0.482017\pi$
$443$	267.825	−	71.7634i	0.604570	−	0.161994i	0.548104	+	0.836410i	$0.315350\pi$
$444$		0			0
$445$		0			0
$446$	−339.945	−	588.802i	−0.762208	−	1.32018i
$447$		0			0
$448$	280.035	−	75.0352i	0.625079	−	0.167489i
$449$			284.605i			0.633864i	0.948448	+	0.316932i	$0.102653\pi$
$449$			284.605i			0.633864i	−0.948448	+	0.316932i	$0.897347\pi$
$450$		0			0
$451$	−500.000			−1.10865
$452$	42.8265	+	159.831i	0.0947490	+	0.353608i
$453$		0			0
$454$	−744.782	+	430.000i	−1.64049	+	0.947137i
$455$	−489.849	+	100.240i	−1.07659	+	0.220308i
$456$		0			0
$457$	−133.599	−	498.599i	−0.292340	−	1.09103i	−0.943307	−	0.331921i	$-0.892303\pi$
$457$	−133.599	−	498.599i	−0.292340	−	1.09103i	0.650967	−	0.759106i	$-0.274363\pi$
$458$	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
$463$	−12.8109	+	47.8109i	−0.0276693	+	0.103263i	−0.978380	−	0.206817i	$-0.933689\pi$
$463$	−12.8109	+	47.8109i	−0.0276693	+	0.103263i	0.950710	+	0.310081i	$0.100356\pi$
$464$	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	−0.570649	−	0.821194i	$-0.693308\pi$
$467$	117.004	−	117.004i	0.250544	−	0.250544i	0.821194	+	0.570649i	$0.193308\pi$
$468$		0			0
$469$			700.000i			1.49254i
$470$	486.332	−	431.255i	1.03475	−	0.917565i
$471$		0			0
$472$	307.356	+	82.3557i	0.651177	+	0.174482i
$473$	57.8737	−	215.988i	0.122355	−	0.456633i
$474$		0			0
$475$	−176.769	−	413.827i	−0.372146	−	0.871214i
$476$	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	0.802143	−	0.597132i	$-0.203693\pi$
$479$	328.634	+	189.737i	0.686083	+	0.396110i	−0.116060	+	0.993242i	$0.537027\pi$
$480$		0			0
$481$	−100.000	−	173.205i	−0.207900	−	0.360094i
$482$	457.894	+	122.692i	0.949987	+	0.254548i
$483$		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	−0.823700	−	0.567026i	$-0.808094\pi$
$487$	−125.000	+	125.000i	−0.256674	+	0.256674i	0.567026	+	0.823700i	$0.308094\pi$
$488$	−375.818	+	100.700i	−0.770120	+	0.206353i
$489$		0			0
$490$	9.33013	+	6.16025i	0.0190411	+	0.0125719i
$491$	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
$493$	−204.904	+	54.9038i	−0.415626	+	0.111367i
$494$			569.210i			1.15225i
$495$		0			0
$496$	152.000			0.306452
$497$	−115.747	−	431.975i	−0.232892	−	0.869165i
$498$		0			0
$499$	−192.258	+	111.000i	−0.385286	+	0.222445i	−0.680116	−	0.733105i	$-0.738070\pi$
$499$	−192.258	+	111.000i	−0.385286	+	0.222445i	0.294830	+	0.955550i	$0.404737\pi$
$500$	−53.4293	+	113.006i	−0.106859	+	0.226012i
$501$		0			0
$502$	210.465	+	785.465i	0.419252	+	1.56467i
$503$	458.530	+	458.530i	0.911591	+	0.911591i	0.996397	−	0.0848065i	$-0.0270272\pi$
$503$	458.530	+	458.530i	0.911591	+	0.911591i	−0.0848065	+	0.996397i	$0.527027\pi$
$504$		0			0
$505$	−25.0000	+	75.0000i	−0.0495050	+	0.148515i
$506$	79.0569	−	136.931i	0.156239	−	0.270614i
$507$		0			0
$508$	−20.1314	+	75.1314i	−0.0396287	+	0.147896i
$509$	205.396	−	118.585i	0.403528	−	0.232977i	−0.284477	−	0.958683i	$-0.591820\pi$
$509$	205.396	−	118.585i	0.403528	−	0.232977i	0.688005	+	0.725706i	$0.258487\pi$
$510$		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
$515$	−14.8283	+	247.043i	−0.0287928	+	0.479695i
$516$		0			0
$517$	−887.917	−	237.917i	−1.71744	−	0.460187i
$518$	−57.8737	+	215.988i	−0.111725	+	0.416964i
$519$		0			0
$520$	−354.904	+	314.711i	−0.682507	+	0.605214i
$521$	−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	0.966000	−	0.258543i	$-0.0832423\pi$
$523$	370.000	+	370.000i	0.707457	+	0.707457i	−0.258543	+	0.966000i	$0.583242\pi$
$524$	−150.624	−	86.9626i	−0.287450	−	0.165959i
$525$		0			0
$526$	455.000	+	788.083i	0.865019	+	1.49826i
$527$	−34.5580	−	9.25979i	−0.0655750	−	0.0175708i
$528$		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
$533$	431.975	−	115.747i	0.810460	−	0.217162i
$534$		0			0
$535$	234.090	−	354.545i	0.437551	−	0.662701i
$536$	332.039	+	575.109i	0.619476	+	1.07296i
$537$		0			0
$538$	307.356	−	82.3557i	0.571293	−	0.153078i
$539$		−	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	$0.391410\pi$
$542$	103.015	+	384.458i	0.190065	+	0.709332i
$543$		0			0
$544$	60.6218	−	35.0000i	0.111437	−	0.0643382i
$545$	−162.389	−	793.555i	−0.297962	−	1.45606i
$546$		0			0
$547$	179.352	+	669.352i	0.327884	+	1.22368i	0.911381	+	0.411564i	$0.135017\pi$
$547$	179.352	+	669.352i	0.327884	+	1.22368i	−0.583497	+	0.812115i	$0.698316\pi$
$548$	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
$553$	21.9615	−	81.9615i	0.0397134	−	0.148213i
$554$	−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	−0.896742	−	0.442555i	$-0.854072\pi$
$557$	−252.982	+	252.982i	−0.454187	+	0.454187i	0.442555	+	0.896742i	$0.354072\pi$
$558$		0			0
$559$			200.000i			0.357782i
$560$	670.545	+	40.2482i	1.19740	+	0.0718717i
$561$		0			0
$562$	341.506	+	91.5064i	0.607663	+	0.162823i
$563$	74.0783	−	276.464i	0.131578	−	0.491055i	−0.868411	−	0.495846i	$-0.834858\pi$
$563$	74.0783	−	276.464i	0.131578	−	0.491055i	0.999989	+	0.00479040i	$0.00152484\pi$
$564$		0			0
$565$	49.5706	−	825.859i	0.0877356	−	1.46170i
$566$	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	0.348611	−	0.937268i	$-0.386654\pi$
$569$	−164.317	−	94.8683i	−0.288782	−	0.166728i	−0.637392	+	0.770539i	$0.719987\pi$
$570$		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	$0.637875\pi$
$572$	−215.988	−	57.8737i	−0.377601	−	0.101178i
$573$		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	−0.736785	−	0.676127i	$-0.763657\pi$
$577$	−35.0000	+	35.0000i	−0.0606586	+	0.0606586i	0.676127	+	0.736785i	$0.263657\pi$
$578$	581.007	−	155.680i	1.00520	−	0.269343i
$579$		0			0
$580$	232.356	−	47.5481i	0.400613	−	0.0819795i
$581$	−268.794	−	465.564i	−0.462640	−	0.801315i
$582$		0			0
$583$	546.410	−	146.410i	0.937239	−	0.251132i
$584$		−	521.776i		−	0.893452i
$585$		0			0
$586$	−640.000			−1.09215
$587$	56.7162	+	211.668i	0.0966205	+	0.360593i	0.997260	−	0.0739741i	$-0.0235682\pi$
$587$	56.7162	+	211.668i	0.0966205	+	0.360593i	−0.900640	+	0.434567i	$0.856902\pi$
$588$		0			0
$589$	−124.708	+	72.0000i	−0.211728	+	0.122241i
$590$	442.567	+	292.207i	0.750113	+	0.495265i
$591$		0			0
$592$	69.5448	+	259.545i	0.117474	+	0.438420i
$593$	−167.601	−	167.601i	−0.282632	−	0.282632i	0.551526	−	0.834158i	$-0.314046\pi$
$593$	−167.601	−	167.601i	−0.282632	−	0.282632i	−0.834158	+	0.551526i	$0.814046\pi$
$594$		0			0
$595$	−150.000	−	50.0000i	−0.252101	−	0.0840336i
$596$	23.7171	−	41.0792i	0.0397938	−	0.0689248i
$597$		0			0
$598$	−36.6025	+	136.603i	−0.0612083	+	0.228432i
$599$	−903.742	+	521.776i	−1.50875	+	0.871078i	−0.508804	+	0.860883i	$0.669912\pi$
$599$	−903.742	+	521.776i	−1.50875	+	0.871078i	−0.999948	+	0.0101955i	$0.996755\pi$
$600$		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
$605$	−427.938	−	482.591i	−0.707336	−	0.797671i
$606$		0			0
$607$	−894.747	−	239.747i	−1.47405	−	0.394970i	−0.569730	−	0.821832i	$-0.692952\pi$
$607$	−894.747	−	239.747i	−1.47405	−	0.394970i	−0.904317	+	0.426862i	$0.859619\pi$
$608$	72.9209	−	272.144i	0.119936	−	0.447606i
$609$		0			0
$610$	−647.295	−	38.8526i	−1.06114	−	0.0636928i
$611$	822.192			1.34565
$612$		0			0
$613$	−620.000	−	620.000i	−1.01142	−	1.01142i	−0.999934	−	0.0114852i	$-0.996344\pi$
$613$	−620.000	−	620.000i	−1.01142	−	1.01142i	−0.0114852	−	0.999934i	$-0.503656\pi$
$614$	−520.336	−	300.416i	−0.847453	−	0.489277i
$615$		0			0
$616$	−375.000	−	649.519i	−0.608766	−	1.05441i
$617$	47.5173	+	12.7322i	0.0770134	+	0.0206357i	0.297120	−	0.954840i	$-0.403974\pi$
$617$	47.5173	+	12.7322i	0.0770134	+	0.0206357i	−0.220107	+	0.975476i	$0.570641\pi$
$618$		0			0
$619$	−223.435	−	129.000i	−0.360961	−	0.208401i	0.308542	−	0.951211i	$-0.400159\pi$
$619$	−223.435	−	129.000i	−0.360961	−	0.208401i	−0.669502	+	0.742810i	$0.733492\pi$
$620$	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
$625$	432.115	−	451.554i	0.691384	−	0.722487i
$626$	229.265	+	397.099i	0.366238	+	0.634343i
$627$		0			0
$628$	−273.205	+	73.2051i	−0.435040	+	0.116569i
$629$		−	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	$0.277514\pi$
$632$	−20.8345	−	77.7555i	−0.0329660	−	0.123031i
$633$		0			0
$634$	−43.3013	+	25.0000i	−0.0682985	+	0.0394322i
$635$	214.285	−	324.549i	0.337456	−	0.511101i
$636$		0			0
$637$	3.66025	+	13.6603i	0.00574608	+	0.0214447i
$638$	−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
$643$	300.141	−	1120.14i	0.466782	−	1.74205i	−0.184131	−	0.982902i	$-0.558947\pi$
$643$	300.141	−	1120.14i	0.466782	−	1.74205i	0.650913	−	0.759152i	$-0.274386\pi$
$644$	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	−0.0521974	+	0.998637i	$0.516622\pi$
$647$	−679.890	+	679.890i	−1.05083	+	1.05083i	−0.998637	+	0.0521974i	$0.983378\pi$
$648$		0			0
$649$		−	750.000i		−	1.15562i
$650$	−727.020	+	310.552i	−1.11849	+	0.477772i
$651$		0			0
$652$	−136.603	−	36.6025i	−0.209513	−	0.0561389i
$653$	188.668	−	704.120i	0.288925	−	1.07828i	−0.656998	−	0.753893i	$-0.728174\pi$
$653$	188.668	−	704.120i	0.288925	−	1.07828i	0.945923	−	0.324391i	$-0.105159\pi$
$654$		0			0
$655$	576.971	+	650.657i	0.880872	+	0.993369i
$656$	−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	−0.227354	−	0.973812i	$-0.573008\pi$
$659$	−780.505	−	450.625i	−1.18438	−	0.683801i	−0.957023	+	0.290012i	$0.906341\pi$
$660$		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	$0.625844\pi$
$662$	1041.06	+	278.951i	1.57260	+	0.421376i
$663$		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
$668$	−203.028	+	54.4013i	−0.303935	+	0.0814390i
$669$		0			0
$670$	221.891	+	1084.33i	0.331181	+	1.61840i
$671$	458.530	+	794.198i	0.683354	+	1.18360i
$672$		0			0
$673$	−1058.67	+	283.670i	−1.57306	+	0.421500i	−0.936769	−	0.349948i	$-0.886199\pi$
$673$	−1058.67	+	283.670i	−1.57306	+	0.421500i	−0.636291	+	0.771449i	$0.719532\pi$
$674$		−	490.153i		−	0.727230i
$675$		0			0
$676$	31.0000			0.0458580
$677$	181.723	+	678.201i	0.268425	+	1.00177i	0.960121	+	0.279585i	$0.0901970\pi$
$677$	181.723	+	678.201i	0.268425	+	1.00177i	−0.691696	+	0.722189i	$0.743136\pi$
$678$		0			0
$679$	−43.3013	+	25.0000i	−0.0637721	+	0.0368189i
$680$	−146.955	+	30.0721i	−0.216110	+	0.0442236i
$681$		0			0
$682$	−73.2051	−	273.205i	−0.107339	−	0.400594i
$683$	60.0833	+	60.0833i	0.0879697	+	0.0879697i	0.749722	−	0.661753i	$-0.230187\pi$
$683$	60.0833	+	60.0833i	0.0879697	+	0.0879697i	−0.661753	+	0.749722i	$0.730187\pi$
$684$		0			0
$685$	−50.0000	−	100.000i	−0.0729927	−	0.145985i
$686$	−379.473	+	657.267i	−0.553168	+	0.958115i
$687$		0			0
$688$	69.5448	−	259.545i	0.101083	−	0.377245i
$689$	−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
$695$	381.583	−	338.370i	0.549041	−	0.486863i
$696$		0			0
$697$	136.603	+	36.6025i	0.195986	+	0.0525144i
$698$	184.038	−	686.841i	0.263665	−	0.984012i
$699$		0			0
$700$	164.054	+	65.8494i	0.234363	+	0.0940705i
$701$	−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
$707$	107.994	+	28.9368i	0.152749	+	0.0409291i
$708$		0			0
$709$	431.281	+	249.000i	0.608294	+	0.351199i	0.772298	−	0.635261i	$-0.219107\pi$
$709$	431.281	+	249.000i	0.608294	+	0.351199i	−0.164003	+	0.986460i	$0.552441\pi$
$710$	−316.228	−	632.456i	−0.445391	−	0.890782i
$711$		0			0
$712$		0			0
$713$	−34.5580	+	9.25979i	−0.0484685	+	0.0129871i
$714$		0			0
$715$	933.013	+	616.025i	1.30491	+	0.861574i
$716$	71.1512	+	123.238i	0.0993733	+	0.172120i
$717$		0			0
$718$	614.711	−	164.711i	0.856144	−	0.229403i
$719$			569.210i			0.791669i	0.918322	+	0.395834i	$0.129545\pi$
$719$			569.210i			0.791669i	−0.918322	+	0.395834i	$0.870455\pi$
$720$		0			0
$721$	350.000			0.485437
$722$	−21.4133	−	79.9154i	−0.0296583	−	0.110686i
$723$		0			0
$724$	−188.794	+	109.000i	−0.260765	+	0.150552i
$725$	−1177.34	−	141.846i	−1.62392	−	0.195650i
$726$		0			0
$727$	−331.253	−	1236.25i	−0.455644	−	1.70049i	−0.686188	−	0.727424i	$-0.740717\pi$
$727$	−331.253	−	1236.25i	−0.455644	−	1.70049i	0.230545	−	0.973062i	$-0.425949\pi$
$728$	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
$733$	201.314	−	751.314i	0.274644	−	1.02498i	−0.681436	−	0.731878i	$-0.738644\pi$
$733$	201.314	−	751.314i	0.274644	−	1.02498i	0.956080	−	0.293107i	$-0.0946892\pi$
$734$	−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	$-0.957229\pi$
$739$		−	198.000i		−	0.267930i	0.990986	−	0.133965i	$-0.0427709\pi$
$740$	−4.23665	+	70.5836i	−0.00572520	+	0.0953833i
$741$		0			0
$742$	546.410	+	146.410i	0.736402	+	0.197318i
$743$	−353.030	+	1317.52i	−0.475141	+	1.77325i	0.145717	+	0.989326i	$0.453451\pi$
$743$	−353.030	+	1317.52i	−0.475141	+	1.77325i	−0.620858	+	0.783923i	$0.713216\pi$
$744$		0			0
$745$	−177.452	+	157.356i	−0.238190	+	0.211216i
$746$	−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	$0.994400\pi$
$752$	−1066.98	−	285.896i	−1.41885	−	0.380181i
$753$		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	−0.810420	−	0.585849i	$-0.800761\pi$
$757$	−170.000	+	170.000i	−0.224571	+	0.224571i	0.585849	+	0.810420i	$0.300761\pi$
$758$	−1205.21	+	322.935i	−1.58999	+	0.426036i
$759$		0			0
$760$	−332.654	+	503.827i	−0.437702	+	0.662930i
$761$	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
$763$	−1106.48	+	296.481i	−1.45017	+	0.388572i
$764$		−	158.114i		−	0.206955i
$765$		0			0
$766$	890.000			1.16188
$767$	173.621	+	647.963i	0.226364	+	0.844801i
$768$		0			0
$769$	275.396	−	159.000i	0.358122	−	0.206762i	−0.310134	−	0.950693i	$-0.600374\pi$
$769$	275.396	−	159.000i	0.358122	−	0.206762i	0.668257	+	0.743931i	$0.267041\pi$
$770$	−250.600	−	1224.62i	−0.325455	−	1.59042i
$771$		0			0
$772$	−45.7532	−	170.753i	−0.0592658	−	0.221183i
$773$	−452.206	−	452.206i	−0.585001	−	0.585001i	0.351272	−	0.936273i	$-0.385749\pi$
$773$	−452.206	−	452.206i	−0.585001	−	0.585001i	−0.936273	+	0.351272i	$0.885749\pi$
$774$		0			0
$775$	−160.000	−	120.000i	−0.206452	−	0.154839i
$776$	−23.7171	+	41.0792i	−0.0305633	+	0.0529371i
$777$		0			0
$778$	301.971	−	1126.97i	0.388137	−	1.44855i
$779$	492.950	−	284.605i	0.632799	−	0.365347i
$780$		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)

\(f(q)\)	\(=\)	\(q+(-0.578737 - 2.15988i) q^{2} +(-0.866025 + 0.500000i) q^{4} +(4.89849 - 1.00240i) q^{5} +(-1.83013 - 6.83013i) q^{7} +(-4.74342 - 4.74342i) q^{8} +(-5.00000 - 10.0000i) q^{10} +(-7.90569 + 13.6931i) q^{11} +(3.66025 - 13.6603i) q^{13} +(-13.6931 + 7.90569i) q^{14} +(-9.50000 + 16.4545i) q^{16} +(3.16228 - 3.16228i) q^{17} -18.0000i q^{19} +(-3.74101 + 3.31735i) q^{20} +(34.1506 + 9.15064i) q^{22} +(1.15747 - 4.31975i) q^{23} +(22.9904 - 9.82051i) q^{25} -31.6228 q^{26} +(5.00000 + 5.00000i) q^{28} +(-41.0792 - 23.7171i) q^{29} +(-4.00000 - 6.92820i) q^{31} +(15.1191 + 4.05116i) q^{32} +(-8.66025 - 5.00000i) q^{34} +(-15.8114 - 31.6228i) q^{35} +(10.0000 - 10.0000i) q^{37} +(-38.8778 + 10.4173i) q^{38} +(-27.9904 - 18.4808i) q^{40} +(15.8114 + 27.3861i) q^{41} +(-13.6603 + 3.66025i) q^{43} -15.8114i q^{44} -10.0000 q^{46} +(15.0472 + 56.1568i) q^{47} +(-0.866025 + 0.500000i) q^{49} +(-34.5165 - 43.9729i) q^{50} +(3.66025 + 13.6603i) q^{52} +(-25.2982 - 25.2982i) q^{53} +(-25.0000 + 75.0000i) q^{55} +(-23.7171 + 41.0792i) q^{56} +(-27.4519 + 102.452i) q^{58} +(-41.0792 + 23.7171i) q^{59} +(29.0000 - 50.2295i) q^{61} +(-12.6491 + 12.6491i) q^{62} +41.0000i q^{64} +(4.23665 - 70.5836i) q^{65} +(-95.6218 - 25.6218i) q^{67} +(-1.15747 + 4.31975i) q^{68} +(-59.1506 + 52.4519i) q^{70} +63.2456 q^{71} +(55.0000 + 55.0000i) q^{73} +(-27.3861 - 15.8114i) q^{74} +(9.00000 + 15.5885i) q^{76} +(107.994 + 28.9368i) q^{77} +(10.3923 + 6.00000i) q^{79} +(-30.0416 + 90.1249i) q^{80} +(50.0000 - 50.0000i) q^{82} +(73.4358 - 19.6771i) q^{83} +(12.3205 - 18.6603i) q^{85} +(15.8114 + 27.3861i) q^{86} +(102.452 - 27.4519i) q^{88} -100.000 q^{91} +(1.15747 + 4.31975i) q^{92} +(112.583 - 65.0000i) q^{94} +(-18.0432 - 88.1728i) q^{95} +(-1.83013 - 6.83013i) q^{97} +(1.58114 + 1.58114i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 20 q^{7} - 40 q^{10} - 40 q^{13} - 76 q^{16} + 100 q^{22} + 80 q^{25} + 40 q^{28} - 32 q^{31} + 80 q^{37} - 120 q^{40} - 40 q^{43} - 80 q^{46} - 40 q^{52} - 200 q^{55} + 300 q^{58} + 232 q^{61} - 280 q^{67}+ \cdots + 20 q^{97}+O(q^{100}) \) 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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