LMFDB - Embedded newform 475.2.e.a.26.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(26,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("475.26");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	475.e (of order $3$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$3.79289409601$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			26.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	475.26
Dual form			475.2.e.a.201.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 1.73205i) q^{2} +(-1.00000 + 1.73205i) q^{4} -4.00000 q^{7} +(1.50000 - 2.59808i) q^{9} +O(q^{10})$	\(q+(-1.00000 - 1.73205i) q^{2} +(-1.00000 + 1.73205i) q^{4} -4.00000 q^{7} +(1.50000 - 2.59808i) q^{9} -1.00000 q^{11} +(1.00000 - 1.73205i) q^{13} +(4.00000 + 6.92820i) q^{14} +(2.00000 + 3.46410i) q^{16} +(1.00000 + 1.73205i) q^{17} -6.00000 q^{18} +(-3.50000 + 2.59808i) q^{19} +(1.00000 + 1.73205i) q^{22} +(-3.00000 + 5.19615i) q^{23} -4.00000 q^{26} +(4.00000 - 6.92820i) q^{28} +(-4.50000 + 7.79423i) q^{29} -7.00000 q^{31} +(4.00000 - 6.92820i) q^{32} +(2.00000 - 3.46410i) q^{34} +(3.00000 + 5.19615i) q^{36} +2.00000 q^{37} +(8.00000 + 3.46410i) q^{38} +(-1.00000 - 1.73205i) q^{41} +(-1.00000 - 1.73205i) q^{43} +(1.00000 - 1.73205i) q^{44} +12.0000 q^{46} +(-3.00000 + 5.19615i) q^{47} +9.00000 q^{49} +(2.00000 + 3.46410i) q^{52} +(-2.00000 + 3.46410i) q^{53} +18.0000 q^{58} +(-4.50000 - 7.79423i) q^{59} +(3.50000 - 6.06218i) q^{61} +(7.00000 + 12.1244i) q^{62} +(-6.00000 + 10.3923i) q^{63} -8.00000 q^{64} +(5.00000 - 8.66025i) q^{67} -4.00000 q^{68} +(-0.500000 - 0.866025i) q^{71} +(-5.00000 - 8.66025i) q^{73} +(-2.00000 - 3.46410i) q^{74} +(-1.00000 - 8.66025i) q^{76} +4.00000 q^{77} +(-0.500000 - 0.866025i) q^{79} +(-4.50000 - 7.79423i) q^{81} +(-2.00000 + 3.46410i) q^{82} -6.00000 q^{83} +(-2.00000 + 3.46410i) q^{86} +(5.50000 - 9.52628i) q^{89} +(-4.00000 + 6.92820i) q^{91} +(-6.00000 - 10.3923i) q^{92} +12.0000 q^{94} +(3.00000 + 5.19615i) q^{97} +(-9.00000 - 15.5885i) q^{98} +(-1.50000 + 2.59808i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{4} - 8 q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^4 - 8 * q^7 + 3 * q^9	$ 2 q - 2 q^{2} - 2 q^{4} - 8 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} + 8 q^{14} + 4 q^{16} + 2 q^{17} - 12 q^{18} - 7 q^{19} + 2 q^{22} - 6 q^{23} - 8 q^{26} + 8 q^{28} - 9 q^{29} - 14 q^{31} + 8 q^{32} + 4 q^{34} + 6 q^{36} + 4 q^{37} + 16 q^{38} - 2 q^{41} - 2 q^{43} + 2 q^{44} + 24 q^{46} - 6 q^{47} + 18 q^{49} + 4 q^{52} - 4 q^{53} + 36 q^{58} - 9 q^{59} + 7 q^{61} + 14 q^{62} - 12 q^{63} - 16 q^{64} + 10 q^{67} - 8 q^{68} - q^{71} - 10 q^{73} - 4 q^{74} - 2 q^{76} + 8 q^{77} - q^{79} - 9 q^{81} - 4 q^{82} - 12 q^{83} - 4 q^{86} + 11 q^{89} - 8 q^{91} - 12 q^{92} + 24 q^{94} + 6 q^{97} - 18 q^{98} - 3 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^4 - 8 * q^7 + 3 * q^9 - 2 * q^11 + 2 * q^13 + 8 * q^14 + 4 * q^16 + 2 * q^17 - 12 * q^18 - 7 * q^19 + 2 * q^22 - 6 * q^23 - 8 * q^26 + 8 * q^28 - 9 * q^29 - 14 * q^31 + 8 * q^32 + 4 * q^34 + 6 * q^36 + 4 * q^37 + 16 * q^38 - 2 * q^41 - 2 * q^43 + 2 * q^44 + 24 * q^46 - 6 * q^47 + 18 * q^49 + 4 * q^52 - 4 * q^53 + 36 * q^58 - 9 * q^59 + 7 * q^61 + 14 * q^62 - 12 * q^63 - 16 * q^64 + 10 * q^67 - 8 * q^68 - q^71 - 10 * q^73 - 4 * q^74 - 2 * q^76 + 8 * q^77 - q^79 - 9 * q^81 - 4 * q^82 - 12 * q^83 - 4 * q^86 + 11 * q^89 - 8 * q^91 - 12 * q^92 + 24 * q^94 + 6 * q^97 - 18 * q^98 - 3 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$

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 $

$2$	−1.00000	−	1.73205i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$2$	−1.00000	−	1.73205i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$3$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$3$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$4$	−1.00000	+	1.73205i	−0.500000	+	0.866025i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−4.00000			−1.51186			−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000			−1.51186			−0.755929	+	0.654654i	$0.772814\pi$
$8$		0			0
$9$	1.50000	−	2.59808i	0.500000	−	0.866025i
$10$		0			0
$11$	−1.00000			−0.301511			−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000			−0.301511			−0.150756	+	0.988571i	$0.548171\pi$
$12$		0			0
$13$	1.00000	−	1.73205i	0.277350	−	0.480384i	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	1.00000	−	1.73205i	0.277350	−	0.480384i	0.970725	+	0.240192i	$0.0772105\pi$
$14$	4.00000	+	6.92820i	1.06904	+	1.85164i
$15$		0			0
$16$	2.00000	+	3.46410i	0.500000	+	0.866025i
$17$	1.00000	+	1.73205i	0.242536	+	0.420084i	0.961436	−	0.275029i	$-0.0886875\pi$
$17$	1.00000	+	1.73205i	0.242536	+	0.420084i	−0.718900	+	0.695113i	$0.755354\pi$
$18$	−6.00000			−1.41421
$19$	−3.50000	+	2.59808i	−0.802955	+	0.596040i
$19$	−3.50000	+	2.59808i	−0.802955	+	0.596040i
$20$		0			0
$21$		0			0
$22$	1.00000	+	1.73205i	0.213201	+	0.369274i
$23$	−3.00000	+	5.19615i	−0.625543	+	1.08347i	0.362892	+	0.931831i	$0.381789\pi$
$23$	−3.00000	+	5.19615i	−0.625543	+	1.08347i	−0.988436	+	0.151642i	$0.951544\pi$
$24$		0			0
$25$		0			0
$26$	−4.00000			−0.784465
$27$		0			0
$28$	4.00000	−	6.92820i	0.755929	−	1.30931i
$29$	−4.50000	+	7.79423i	−0.835629	+	1.44735i	0.0578882	+	0.998323i	$0.481563\pi$
$29$	−4.50000	+	7.79423i	−0.835629	+	1.44735i	−0.893517	+	0.449029i	$0.851770\pi$
$30$		0			0
$31$	−7.00000			−1.25724			−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000			−1.25724			−0.628619	+	0.777714i	$0.716379\pi$
$32$	4.00000	−	6.92820i	0.707107	−	1.22474i
$33$		0			0
$34$	2.00000	−	3.46410i	0.342997	−	0.594089i
$35$		0			0
$36$	3.00000	+	5.19615i	0.500000	+	0.866025i
$37$	2.00000			0.328798			0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000			0.328798			0.164399	+	0.986394i	$0.447432\pi$
$38$	8.00000	+	3.46410i	1.29777	+	0.561951i
$39$		0			0
$40$		0			0
$41$	−1.00000	−	1.73205i	−0.156174	−	0.270501i	0.777312	−	0.629115i	$-0.216583\pi$
$41$	−1.00000	−	1.73205i	−0.156174	−	0.270501i	−0.933486	+	0.358614i	$0.883249\pi$
$42$		0			0
$43$	−1.00000	−	1.73205i	−0.152499	−	0.264135i	0.779647	−	0.626219i	$-0.215399\pi$
$43$	−1.00000	−	1.73205i	−0.152499	−	0.264135i	−0.932145	+	0.362084i	$0.882065\pi$
$44$	1.00000	−	1.73205i	0.150756	−	0.261116i
$45$		0			0
$46$	12.0000			1.76930
$47$	−3.00000	+	5.19615i	−0.437595	+	0.757937i	−0.997503	−	0.0706177i	$-0.977503\pi$
$47$	−3.00000	+	5.19615i	−0.437595	+	0.757937i	0.559908	+	0.828554i	$0.310836\pi$
$48$		0			0
$49$	9.00000			1.28571
$50$		0			0
$51$		0			0
$52$	2.00000	+	3.46410i	0.277350	+	0.480384i
$53$	−2.00000	+	3.46410i	−0.274721	+	0.475831i	−0.970065	−	0.242846i	$-0.921919\pi$
$53$	−2.00000	+	3.46410i	−0.274721	+	0.475831i	0.695344	+	0.718677i	$0.255252\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	18.0000			2.36352
$59$	−4.50000	−	7.79423i	−0.585850	−	1.01472i	−0.994769	−	0.102151i	$-0.967427\pi$
$59$	−4.50000	−	7.79423i	−0.585850	−	1.01472i	0.408919	−	0.912571i	$-0.365906\pi$
$60$		0			0
$61$	3.50000	−	6.06218i	0.448129	−	0.776182i	−0.550135	−	0.835076i	$-0.685424\pi$
$61$	3.50000	−	6.06218i	0.448129	−	0.776182i	0.998264	+	0.0588933i	$0.0187572\pi$
$62$	7.00000	+	12.1244i	0.889001	+	1.53979i
$63$	−6.00000	+	10.3923i	−0.755929	+	1.30931i
$64$	−8.00000			−1.00000
$65$		0			0
$66$		0			0
$67$	5.00000	−	8.66025i	0.610847	−	1.05802i	−0.380251	−	0.924883i	$-0.624162\pi$
$67$	5.00000	−	8.66025i	0.610847	−	1.05802i	0.991098	−	0.133135i	$-0.0425044\pi$
$68$	−4.00000			−0.485071
$69$		0			0
$70$		0			0
$71$	−0.500000	−	0.866025i	−0.0593391	−	0.102778i	0.834830	−	0.550508i	$-0.185566\pi$
$71$	−0.500000	−	0.866025i	−0.0593391	−	0.102778i	−0.894169	+	0.447730i	$0.852233\pi$
$72$		0			0
$73$	−5.00000	−	8.66025i	−0.585206	−	1.01361i	−0.994850	−	0.101361i	$-0.967680\pi$
$73$	−5.00000	−	8.66025i	−0.585206	−	1.01361i	0.409644	−	0.912245i	$-0.365653\pi$
$74$	−2.00000	−	3.46410i	−0.232495	−	0.402694i
$75$		0			0
$76$	−1.00000	−	8.66025i	−0.114708	−	0.993399i
$77$	4.00000			0.455842
$78$		0			0
$79$	−0.500000	−	0.866025i	−0.0562544	−	0.0974355i	0.836527	−	0.547926i	$-0.184582\pi$
$79$	−0.500000	−	0.866025i	−0.0562544	−	0.0974355i	−0.892781	+	0.450490i	$0.851249\pi$
$80$		0			0
$81$	−4.50000	−	7.79423i	−0.500000	−	0.866025i
$82$	−2.00000	+	3.46410i	−0.220863	+	0.382546i
$83$	−6.00000			−0.658586			−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000			−0.658586			−0.329293	+	0.944228i	$0.606810\pi$
$84$		0			0
$85$		0			0
$86$	−2.00000	+	3.46410i	−0.215666	+	0.373544i
$87$		0			0
$88$		0			0
$89$	5.50000	−	9.52628i	0.582999	−	1.00978i	−0.412123	−	0.911128i	$-0.635213\pi$
$89$	5.50000	−	9.52628i	0.582999	−	1.00978i	0.995122	−	0.0986553i	$-0.0314541\pi$
$90$		0			0
$91$	−4.00000	+	6.92820i	−0.419314	+	0.726273i
$92$	−6.00000	−	10.3923i	−0.625543	−	1.08347i
$93$		0			0
$94$	12.0000			1.23771
$95$		0			0
$96$		0			0
$97$	3.00000	+	5.19615i	0.304604	+	0.527589i	0.977173	−	0.212445i	$-0.0681426\pi$
$97$	3.00000	+	5.19615i	0.304604	+	0.527589i	−0.672569	+	0.740034i	$0.734809\pi$
$98$	−9.00000	−	15.5885i	−0.909137	−	1.57467i
$99$	−1.50000	+	2.59808i	−0.150756	+	0.261116i
$100$		0			0
$101$	−7.50000	+	12.9904i	−0.746278	+	1.29259i	0.203317	+	0.979113i	$0.434828\pi$
$101$	−7.50000	+	12.9904i	−0.746278	+	1.29259i	−0.949595	+	0.313478i	$0.898506\pi$
$102$		0			0
$103$	16.0000			1.57653			0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000			1.57653			0.788263	+	0.615338i	$0.210980\pi$
$104$		0			0
$105$		0			0
$106$	8.00000			0.777029
$107$	−10.0000			−0.966736			−0.483368	−	0.875417i	$-0.660587\pi$
$107$	−10.0000			−0.966736			−0.483368	+	0.875417i	$0.660587\pi$
$108$		0			0
$109$	−7.50000	−	12.9904i	−0.718370	−	1.24425i	−0.961645	−	0.274296i	$-0.911555\pi$
$109$	−7.50000	−	12.9904i	−0.718370	−	1.24425i	0.243276	−	0.969957i	$-0.421778\pi$
$110$		0			0
$111$		0			0
$112$	−8.00000	−	13.8564i	−0.755929	−	1.30931i
$113$	−12.0000			−1.12887			−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000			−1.12887			−0.564433	+	0.825479i	$0.690905\pi$
$114$		0			0
$115$		0			0
$116$	−9.00000	−	15.5885i	−0.835629	−	1.44735i
$117$	−3.00000	−	5.19615i	−0.277350	−	0.480384i
$118$	−9.00000	+	15.5885i	−0.828517	+	1.43503i
$119$	−4.00000	−	6.92820i	−0.366679	−	0.635107i
$120$		0			0
$121$	−10.0000			−0.909091
$122$	−14.0000			−1.26750
$123$		0			0
$124$	7.00000	−	12.1244i	0.628619	−	1.08880i
$125$		0			0
$126$	24.0000			2.13809
$127$	3.00000	−	5.19615i	0.266207	−	0.461084i	−0.701672	−	0.712500i	$-0.747563\pi$
$127$	3.00000	−	5.19615i	0.266207	−	0.461084i	0.967879	+	0.251416i	$0.0808962\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6.00000	−	10.3923i	−0.524222	−	0.907980i	−0.999602	−	0.0281993i	$-0.991023\pi$
$131$	−6.00000	−	10.3923i	−0.524222	−	0.907980i	0.475380	−	0.879781i	$-0.342311\pi$
$132$		0			0
$133$	14.0000	−	10.3923i	1.21395	−	0.901127i
$134$	−20.0000			−1.72774
$135$		0			0
$136$		0			0
$137$	−6.00000	+	10.3923i	−0.512615	+	0.887875i	0.487278	+	0.873247i	$0.337990\pi$
$137$	−6.00000	+	10.3923i	−0.512615	+	0.887875i	−0.999893	+	0.0146279i	$0.995344\pi$
$138$		0			0
$139$	−10.0000	+	17.3205i	−0.848189	+	1.46911i	0.0346338	+	0.999400i	$0.488974\pi$
$139$	−10.0000	+	17.3205i	−0.848189	+	1.46911i	−0.882823	+	0.469706i	$0.844360\pi$
$140$		0			0
$141$		0			0
$142$	−1.00000	+	1.73205i	−0.0839181	+	0.145350i
$143$	−1.00000	+	1.73205i	−0.0836242	+	0.144841i
$144$	12.0000			1.00000
$145$		0			0
$146$	−10.0000	+	17.3205i	−0.827606	+	1.43346i
$147$		0			0
$148$	−2.00000	+	3.46410i	−0.164399	+	0.284747i
$149$	0.500000	+	0.866025i	0.0409616	+	0.0709476i	0.885779	−	0.464107i	$-0.153625\pi$
$149$	0.500000	+	0.866025i	0.0409616	+	0.0709476i	−0.844818	+	0.535054i	$0.820291\pi$
$150$		0			0
$151$	9.00000			0.732410			0.366205	−	0.930534i	$-0.380657\pi$
$151$	9.00000			0.732410			0.366205	+	0.930534i	$0.380657\pi$
$152$		0			0
$153$	6.00000			0.485071
$154$	−4.00000	−	6.92820i	−0.322329	−	0.558291i
$155$		0			0
$156$		0			0
$157$	−2.00000	−	3.46410i	−0.159617	−	0.276465i	0.775113	−	0.631822i	$-0.217693\pi$
$157$	−2.00000	−	3.46410i	−0.159617	−	0.276465i	−0.934731	+	0.355357i	$0.884359\pi$
$158$	−1.00000	+	1.73205i	−0.0795557	+	0.137795i
$159$		0			0
$160$		0			0
$161$	12.0000	−	20.7846i	0.945732	−	1.63806i
$162$	−9.00000	+	15.5885i	−0.707107	+	1.22474i
$163$	4.00000			0.313304			0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000			0.313304			0.156652	+	0.987654i	$0.449930\pi$
$164$	4.00000			0.312348
$165$		0			0
$166$	6.00000	+	10.3923i	0.465690	+	0.806599i
$167$	6.00000	−	10.3923i	0.464294	−	0.804181i	−0.534875	−	0.844931i	$-0.679641\pi$
$167$	6.00000	−	10.3923i	0.464294	−	0.804181i	0.999169	+	0.0407502i	$0.0129748\pi$
$168$		0			0
$169$	4.50000	+	7.79423i	0.346154	+	0.599556i
$170$		0			0
$171$	1.50000	+	12.9904i	0.114708	+	0.993399i
$172$	4.00000			0.304997
$173$	12.0000	+	20.7846i	0.912343	+	1.58022i	0.810745	+	0.585399i	$0.199062\pi$
$173$	12.0000	+	20.7846i	0.912343	+	1.58022i	0.101598	+	0.994826i	$0.467605\pi$
$174$		0			0
$175$		0			0
$176$	−2.00000	−	3.46410i	−0.150756	−	0.261116i
$177$		0			0
$178$	−22.0000			−1.64897
$179$	15.0000			1.12115			0.560576	−	0.828103i	$-0.310580\pi$
$179$	15.0000			1.12115			0.560576	+	0.828103i	$0.310580\pi$
$180$		0			0
$181$	3.00000	−	5.19615i	0.222988	−	0.386227i	−0.732726	−	0.680524i	$-0.761752\pi$
$181$	3.00000	−	5.19615i	0.222988	−	0.386227i	0.955714	+	0.294297i	$0.0950855\pi$
$182$	16.0000			1.18600
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−1.00000	−	1.73205i	−0.0731272	−	0.126660i
$188$	−6.00000	−	10.3923i	−0.437595	−	0.757937i
$189$		0			0
$190$		0			0
$191$	−3.00000			−0.217072			−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000			−0.217072			−0.108536	+	0.994092i	$0.534616\pi$
$192$		0			0
$193$	−8.00000	−	13.8564i	−0.575853	−	0.997406i	−0.995948	−	0.0899262i	$-0.971337\pi$
$193$	−8.00000	−	13.8564i	−0.575853	−	0.997406i	0.420096	−	0.907480i	$-0.361996\pi$
$194$	6.00000	−	10.3923i	0.430775	−	0.746124i
$195$		0			0
$196$	−9.00000	+	15.5885i	−0.642857	+	1.11346i
$197$	−18.0000			−1.28245			−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000			−1.28245			−0.641223	+	0.767354i	$0.721573\pi$
$198$	6.00000			0.426401
$199$	−6.50000	+	11.2583i	−0.460773	+	0.798082i	−0.999000	−	0.0447181i	$-0.985761\pi$
$199$	−6.50000	+	11.2583i	−0.460773	+	0.798082i	0.538227	+	0.842800i	$0.319094\pi$
$200$		0			0
$201$		0			0
$202$	30.0000			2.11079
$203$	18.0000	−	31.1769i	1.26335	−	2.18819i
$204$		0			0
$205$		0			0
$206$	−16.0000	−	27.7128i	−1.11477	−	1.93084i
$207$	9.00000	+	15.5885i	0.625543	+	1.08347i
$208$	8.00000			0.554700
$209$	3.50000	−	2.59808i	0.242100	−	0.179713i
$210$		0			0
$211$	−2.50000	−	4.33013i	−0.172107	−	0.298098i	0.767049	−	0.641588i	$-0.221724\pi$
$211$	−2.50000	−	4.33013i	−0.172107	−	0.298098i	−0.939156	+	0.343490i	$0.888391\pi$
$212$	−4.00000	−	6.92820i	−0.274721	−	0.475831i
$213$		0			0
$214$	10.0000	+	17.3205i	0.683586	+	1.18401i
$215$		0			0
$216$		0			0
$217$	28.0000			1.90076
$218$	−15.0000	+	25.9808i	−1.01593	+	1.75964i
$219$		0			0
$220$		0			0
$221$	4.00000			0.269069
$222$		0			0
$223$	1.00000	+	1.73205i	0.0669650	+	0.115987i	0.897564	−	0.440884i	$-0.145335\pi$
$223$	1.00000	+	1.73205i	0.0669650	+	0.115987i	−0.830599	+	0.556871i	$0.812002\pi$
$224$	−16.0000	+	27.7128i	−1.06904	+	1.85164i
$225$		0			0
$226$	12.0000	+	20.7846i	0.798228	+	1.38257i
$227$	14.0000			0.929213			0.464606	−	0.885517i	$-0.346196\pi$
$227$	14.0000			0.929213			0.464606	+	0.885517i	$0.346196\pi$
$228$		0			0
$229$	17.0000			1.12339			0.561696	−	0.827344i	$-0.310149\pi$
$229$	17.0000			1.12339			0.561696	+	0.827344i	$0.310149\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	4.00000	+	6.92820i	0.262049	+	0.453882i	0.966786	−	0.255586i	$-0.0822686\pi$
$233$	4.00000	+	6.92820i	0.262049	+	0.453882i	−0.704737	+	0.709468i	$0.748935\pi$
$234$	−6.00000	+	10.3923i	−0.392232	+	0.679366i
$235$		0			0
$236$	18.0000			1.17170
$237$		0			0
$238$	−8.00000	+	13.8564i	−0.518563	+	0.898177i
$239$	−19.0000			−1.22901			−0.614504	−	0.788914i	$-0.710644\pi$
$239$	−19.0000			−1.22901			−0.614504	+	0.788914i	$0.710644\pi$
$240$		0			0
$241$	−0.500000	+	0.866025i	−0.0322078	+	0.0557856i	−0.881680	−	0.471848i	$-0.843587\pi$
$241$	−0.500000	+	0.866025i	−0.0322078	+	0.0557856i	0.849472	+	0.527633i	$0.176921\pi$
$242$	10.0000	+	17.3205i	0.642824	+	1.11340i
$243$		0			0
$244$	7.00000	+	12.1244i	0.448129	+	0.776182i
$245$		0			0
$246$		0			0
$247$	1.00000	+	8.66025i	0.0636285	+	0.551039i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−6.50000	+	11.2583i	−0.410276	+	0.710620i	−0.994920	−	0.100671i	$-0.967901\pi$
$251$	−6.50000	+	11.2583i	−0.410276	+	0.710620i	0.584643	+	0.811290i	$0.301234\pi$
$252$	−12.0000	−	20.7846i	−0.755929	−	1.30931i
$253$	3.00000	−	5.19615i	0.188608	−	0.326679i
$254$	−12.0000			−0.752947
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$	3.00000	−	5.19615i	0.187135	−	0.324127i	−0.757159	−	0.653231i	$-0.773413\pi$
$257$	3.00000	−	5.19615i	0.187135	−	0.324127i	0.944294	+	0.329104i	$0.106747\pi$
$258$		0			0
$259$	−8.00000			−0.497096
$260$		0			0
$261$	13.5000	+	23.3827i	0.835629	+	1.44735i
$262$	−12.0000	+	20.7846i	−0.741362	+	1.28408i
$263$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$263$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$264$		0			0
$265$		0			0
$266$	−32.0000	−	13.8564i	−1.96205	−	0.849591i
$267$		0			0
$268$	10.0000	+	17.3205i	0.610847	+	1.05802i
$269$	−1.50000	−	2.59808i	−0.0914566	−	0.158408i	0.816668	−	0.577108i	$-0.195819\pi$
$269$	−1.50000	−	2.59808i	−0.0914566	−	0.158408i	−0.908124	+	0.418701i	$0.862486\pi$
$270$		0			0
$271$	−1.50000	−	2.59808i	−0.0911185	−	0.157822i	0.816864	−	0.576831i	$-0.195711\pi$
$271$	−1.50000	−	2.59808i	−0.0911185	−	0.157822i	−0.907982	+	0.419009i	$0.862378\pi$
$272$	−4.00000	+	6.92820i	−0.242536	+	0.420084i
$273$		0			0
$274$	24.0000			1.44989
$275$		0			0
$276$		0			0
$277$	−28.0000			−1.68236			−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000			−1.68236			−0.841178	+	0.540758i	$0.818138\pi$
$278$	40.0000			2.39904
$279$	−10.5000	+	18.1865i	−0.628619	+	1.08880i
$280$		0			0
$281$	−5.00000	+	8.66025i	−0.298275	+	0.516627i	−0.975741	−	0.218926i	$-0.929745\pi$
$281$	−5.00000	+	8.66025i	−0.298275	+	0.516627i	0.677466	+	0.735554i	$0.263078\pi$
$282$		0			0
$283$	−7.00000	−	12.1244i	−0.416107	−	0.720718i	0.579437	−	0.815017i	$-0.303272\pi$
$283$	−7.00000	−	12.1244i	−0.416107	−	0.720718i	−0.995544	+	0.0942988i	$0.969939\pi$
$284$	2.00000			0.118678
$285$		0			0
$286$	4.00000			0.236525
$287$	4.00000	+	6.92820i	0.236113	+	0.408959i
$288$	−12.0000	−	20.7846i	−0.707107	−	1.22474i
$289$	6.50000	−	11.2583i	0.382353	−	0.662255i
$290$		0			0
$291$		0			0
$292$	20.0000			1.17041
$293$	4.00000			0.233682			0.116841	−	0.993151i	$-0.462723\pi$
$293$	4.00000			0.233682			0.116841	+	0.993151i	$0.462723\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$	1.00000	−	1.73205i	0.0579284	−	0.100335i
$299$	6.00000	+	10.3923i	0.346989	+	0.601003i
$300$		0			0
$301$	4.00000	+	6.92820i	0.230556	+	0.399335i
$302$	−9.00000	−	15.5885i	−0.517892	−	0.897015i
$303$		0			0
$304$	−16.0000	−	6.92820i	−0.917663	−	0.397360i
$305$		0			0
$306$	−6.00000	−	10.3923i	−0.342997	−	0.594089i
$307$	−8.00000	−	13.8564i	−0.456584	−	0.790827i	0.542194	−	0.840254i	$-0.317594\pi$
$307$	−8.00000	−	13.8564i	−0.456584	−	0.790827i	−0.998778	+	0.0494267i	$0.984261\pi$
$308$	−4.00000	+	6.92820i	−0.227921	+	0.394771i
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	−15.0000	+	25.9808i	−0.847850	+	1.46852i	0.0352727	+	0.999378i	$0.488770\pi$
$313$	−15.0000	+	25.9808i	−0.847850	+	1.46852i	−0.883123	+	0.469142i	$0.844563\pi$
$314$	−4.00000	+	6.92820i	−0.225733	+	0.390981i
$315$		0			0
$316$	2.00000			0.112509
$317$	1.00000	−	1.73205i	0.0561656	−	0.0972817i	−0.836576	−	0.547852i	$-0.815446\pi$
$317$	1.00000	−	1.73205i	0.0561656	−	0.0972817i	0.892741	+	0.450570i	$0.148779\pi$
$318$		0			0
$319$	4.50000	−	7.79423i	0.251952	−	0.436393i
$320$		0			0
$321$		0			0
$322$	−48.0000			−2.67494
$323$	−8.00000	−	3.46410i	−0.445132	−	0.192748i
$324$	18.0000			1.00000
$325$		0			0
$326$	−4.00000	−	6.92820i	−0.221540	−	0.383718i
$327$		0			0
$328$		0			0
$329$	12.0000	−	20.7846i	0.661581	−	1.14589i
$330$		0			0
$331$	20.0000			1.09930			0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000			1.09930			0.549650	+	0.835395i	$0.314761\pi$
$332$	6.00000	−	10.3923i	0.329293	−	0.570352i
$333$	3.00000	−	5.19615i	0.164399	−	0.284747i
$334$	−24.0000			−1.31322
$335$		0			0
$336$		0			0
$337$	−17.0000	−	29.4449i	−0.926049	−	1.60396i	−0.789865	−	0.613280i	$-0.789850\pi$
$337$	−17.0000	−	29.4449i	−0.926049	−	1.60396i	−0.136184	−	0.990684i	$-0.543484\pi$
$338$	9.00000	−	15.5885i	0.489535	−	0.847900i
$339$		0			0
$340$		0			0
$341$	7.00000			0.379071
$342$	21.0000	−	15.5885i	1.13555	−	0.842927i
$343$	−8.00000			−0.431959
$344$		0			0
$345$		0			0
$346$	24.0000	−	41.5692i	1.29025	−	2.23478i
$347$	6.00000	+	10.3923i	0.322097	+	0.557888i	0.980921	−	0.194409i	$-0.0622790\pi$
$347$	6.00000	+	10.3923i	0.322097	+	0.557888i	−0.658824	+	0.752297i	$0.728946\pi$
$348$		0			0
$349$	14.0000			0.749403			0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000			0.749403			0.374701	+	0.927146i	$0.377745\pi$
$350$		0			0
$351$		0			0
$352$	−4.00000	+	6.92820i	−0.213201	+	0.369274i
$353$	−8.00000			−0.425797			−0.212899	−	0.977074i	$-0.568290\pi$
$353$	−8.00000			−0.425797			−0.212899	+	0.977074i	$0.568290\pi$
$354$		0			0
$355$		0			0
$356$	11.0000	+	19.0526i	0.582999	+	1.00978i
$357$		0			0
$358$	−15.0000	−	25.9808i	−0.792775	−	1.37313i
$359$	10.0000	+	17.3205i	0.527780	+	0.914141i	0.999476	+	0.0323801i	$0.0103087\pi$
$359$	10.0000	+	17.3205i	0.527780	+	0.914141i	−0.471696	+	0.881761i	$0.656358\pi$
$360$		0			0
$361$	5.50000	−	18.1865i	0.289474	−	0.957186i
$362$	−12.0000			−0.630706
$363$		0			0
$364$	−8.00000	−	13.8564i	−0.419314	−	0.726273i
$365$		0			0
$366$		0			0
$367$	8.00000	−	13.8564i	0.417597	−	0.723299i	−0.578101	−	0.815966i	$-0.696206\pi$
$367$	8.00000	−	13.8564i	0.417597	−	0.723299i	0.995697	+	0.0926670i	$0.0295392\pi$
$368$	−24.0000			−1.25109
$369$	−6.00000			−0.312348
$370$		0			0
$371$	8.00000	−	13.8564i	0.415339	−	0.719389i
$372$		0			0
$373$	−12.0000			−0.621336			−0.310668	−	0.950518i	$-0.600553\pi$
$373$	−12.0000			−0.621336			−0.310668	+	0.950518i	$0.600553\pi$
$374$	−2.00000	+	3.46410i	−0.103418	+	0.179124i
$375$		0			0
$376$		0			0
$377$	9.00000	+	15.5885i	0.463524	+	0.802846i
$378$		0			0
$379$	−29.0000			−1.48963			−0.744815	−	0.667271i	$-0.767462\pi$
$379$	−29.0000			−1.48963			−0.744815	+	0.667271i	$0.767462\pi$
$380$		0			0
$381$		0			0
$382$	3.00000	+	5.19615i	0.153493	+	0.265858i
$383$	−3.00000	−	5.19615i	−0.153293	−	0.265511i	0.779143	−	0.626846i	$-0.215654\pi$
$383$	−3.00000	−	5.19615i	−0.153293	−	0.265511i	−0.932436	+	0.361335i	$0.882321\pi$
$384$		0			0
$385$		0			0
$386$	−16.0000	+	27.7128i	−0.814379	+	1.41055i
$387$	−6.00000			−0.304997
$388$	−12.0000			−0.609208
$389$	16.5000	−	28.5788i	0.836583	−	1.44900i	−0.0561516	−	0.998422i	$-0.517883\pi$
$389$	16.5000	−	28.5788i	0.836583	−	1.44900i	0.892735	−	0.450582i	$-0.148784\pi$
$390$		0			0
$391$	−12.0000			−0.606866
$392$		0			0
$393$		0			0
$394$	18.0000	+	31.1769i	0.906827	+	1.57067i
$395$		0			0
$396$	−3.00000	−	5.19615i	−0.150756	−	0.261116i
$397$	4.00000	+	6.92820i	0.200754	+	0.347717i	0.948772	−	0.315963i	$-0.102327\pi$
$397$	4.00000	+	6.92820i	0.200754	+	0.347717i	−0.748017	+	0.663679i	$0.768994\pi$
$398$	26.0000			1.30326
$399$		0			0
$400$		0			0
$401$	−1.50000	−	2.59808i	−0.0749064	−	0.129742i	0.826139	−	0.563466i	$-0.190532\pi$
$401$	−1.50000	−	2.59808i	−0.0749064	−	0.129742i	−0.901046	+	0.433724i	$0.857199\pi$
$402$		0			0
$403$	−7.00000	+	12.1244i	−0.348695	+	0.603957i
$404$	−15.0000	−	25.9808i	−0.746278	−	1.29259i
$405$		0			0
$406$	−72.0000			−3.57330
$407$	−2.00000			−0.0991363
$408$		0			0
$409$	2.50000	−	4.33013i	0.123617	−	0.214111i	−0.797574	−	0.603220i	$-0.793884\pi$
$409$	2.50000	−	4.33013i	0.123617	−	0.214111i	0.921192	+	0.389109i	$0.127217\pi$
$410$		0			0
$411$		0			0
$412$	−16.0000	+	27.7128i	−0.788263	+	1.36531i
$413$	18.0000	+	31.1769i	0.885722	+	1.53412i
$414$	18.0000	−	31.1769i	0.884652	−	1.53226i
$415$		0			0
$416$	−8.00000	−	13.8564i	−0.392232	−	0.679366i
$417$		0			0
$418$	−8.00000	−	3.46410i	−0.391293	−	0.169435i
$419$	9.00000			0.439679			0.219839	−	0.975536i	$-0.429447\pi$
$419$	9.00000			0.439679			0.219839	+	0.975536i	$0.429447\pi$
$420$		0			0
$421$	7.50000	+	12.9904i	0.365528	+	0.633112i	0.988861	−	0.148844i	$-0.0475552\pi$
$421$	7.50000	+	12.9904i	0.365528	+	0.633112i	−0.623333	+	0.781956i	$0.714222\pi$
$422$	−5.00000	+	8.66025i	−0.243396	+	0.421575i
$423$	9.00000	+	15.5885i	0.437595	+	0.757937i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−14.0000	+	24.2487i	−0.677507	+	1.17348i
$428$	10.0000	−	17.3205i	0.483368	−	0.837218i
$429$		0			0
$430$		0			0
$431$	−10.5000	+	18.1865i	−0.505767	+	0.876014i	0.494211	+	0.869342i	$0.335457\pi$
$431$	−10.5000	+	18.1865i	−0.505767	+	0.876014i	−0.999978	+	0.00667224i	$0.997876\pi$
$432$		0			0
$433$	2.00000	−	3.46410i	0.0961139	−	0.166474i	−0.813959	−	0.580922i	$-0.802692\pi$
$433$	2.00000	−	3.46410i	0.0961139	−	0.166474i	0.910073	+	0.414448i	$0.136025\pi$
$434$	−28.0000	−	48.4974i	−1.34404	−	2.32795i
$435$		0			0
$436$	30.0000			1.43674
$437$	−3.00000	−	25.9808i	−0.143509	−	1.24283i
$438$		0			0
$439$	6.50000	+	11.2583i	0.310228	+	0.537331i	0.978412	−	0.206666i	$-0.0662612\pi$
$439$	6.50000	+	11.2583i	0.310228	+	0.537331i	−0.668184	+	0.743996i	$0.732928\pi$
$440$		0			0
$441$	13.5000	−	23.3827i	0.642857	−	1.11346i
$442$	−4.00000	−	6.92820i	−0.190261	−	0.329541i
$443$	−2.00000	+	3.46410i	−0.0950229	+	0.164584i	−0.909618	−	0.415445i	$-0.863626\pi$
$443$	−2.00000	+	3.46410i	−0.0950229	+	0.164584i	0.814595	+	0.580030i	$0.196959\pi$
$444$		0			0
$445$		0			0
$446$	2.00000	−	3.46410i	0.0947027	−	0.164030i
$447$		0			0
$448$	32.0000			1.51186
$449$	31.0000			1.46298			0.731490	−	0.681852i	$-0.238825\pi$
$449$	31.0000			1.46298			0.731490	+	0.681852i	$0.238825\pi$
$450$		0			0
$451$	1.00000	+	1.73205i	0.0470882	+	0.0815591i
$452$	12.0000	−	20.7846i	0.564433	−	0.977626i
$453$		0			0
$454$	−14.0000	−	24.2487i	−0.657053	−	1.13805i
$455$		0			0
$456$		0			0
$457$	34.0000			1.59045			0.795226	−	0.606313i	$-0.207352\pi$
$457$	34.0000			1.59045			0.795226	+	0.606313i	$0.207352\pi$
$458$	−17.0000	−	29.4449i	−0.794358	−	1.37587i
$459$		0			0
$460$		0			0
$461$	12.5000	+	21.6506i	0.582183	+	1.00837i	0.995220	+	0.0976564i	$0.0311346\pi$
$461$	12.5000	+	21.6506i	0.582183	+	1.00837i	−0.413037	+	0.910714i	$0.635532\pi$
$462$		0			0
$463$	−4.00000			−0.185896			−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000			−0.185896			−0.0929479	+	0.995671i	$0.529629\pi$
$464$	−36.0000			−1.67126
$465$		0			0
$466$	8.00000	−	13.8564i	0.370593	−	0.641886i
$467$	10.0000			0.462745			0.231372	−	0.972865i	$-0.425678\pi$
$467$	10.0000			0.462745			0.231372	+	0.972865i	$0.425678\pi$
$468$	12.0000			0.554700
$469$	−20.0000	+	34.6410i	−0.923514	+	1.59957i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.00000	+	1.73205i	0.0459800	+	0.0796398i
$474$		0			0
$475$		0			0
$476$	16.0000			0.733359
$477$	6.00000	+	10.3923i	0.274721	+	0.475831i
$478$	19.0000	+	32.9090i	0.869040	+	1.50522i
$479$	−7.50000	+	12.9904i	−0.342684	+	0.593546i	−0.984930	−	0.172953i	$-0.944669\pi$
$479$	−7.50000	+	12.9904i	−0.342684	+	0.593546i	0.642246	+	0.766498i	$0.278003\pi$
$480$		0			0
$481$	2.00000	−	3.46410i	0.0911922	−	0.157949i
$482$	2.00000			0.0910975
$483$		0			0
$484$	10.0000	−	17.3205i	0.454545	−	0.787296i
$485$		0			0
$486$		0			0
$487$	−38.0000			−1.72194			−0.860972	−	0.508652i	$-0.830144\pi$
$487$	−38.0000			−1.72194			−0.860972	+	0.508652i	$0.830144\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−6.50000	−	11.2583i	−0.293341	−	0.508081i	0.681257	−	0.732045i	$-0.261434\pi$
$491$	−6.50000	−	11.2583i	−0.293341	−	0.508081i	−0.974598	+	0.223963i	$0.928100\pi$
$492$		0			0
$493$	−18.0000			−0.810679
$494$	14.0000	−	10.3923i	0.629890	−	0.467572i
$495$		0			0
$496$	−14.0000	−	24.2487i	−0.628619	−	1.08880i
$497$	2.00000	+	3.46410i	0.0897123	+	0.155386i
$498$		0			0
$499$	−14.0000	−	24.2487i	−0.626726	−	1.08552i	−0.988204	−	0.153141i	$-0.951061\pi$
$499$	−14.0000	−	24.2487i	−0.626726	−	1.08552i	0.361478	−	0.932381i	$-0.382272\pi$
$500$		0			0
$501$		0			0
$502$	26.0000			1.16044
$503$	13.0000	−	22.5167i	0.579641	−	1.00397i	−0.415879	−	0.909420i	$-0.636526\pi$
$503$	13.0000	−	22.5167i	0.579641	−	1.00397i	0.995520	−	0.0945483i	$-0.0301407\pi$
$504$		0			0
$505$		0			0
$506$	−12.0000			−0.533465
$507$		0			0
$508$	6.00000	+	10.3923i	0.266207	+	0.461084i
$509$	−9.00000	+	15.5885i	−0.398918	+	0.690946i	−0.993593	−	0.113020i	$-0.963948\pi$
$509$	−9.00000	+	15.5885i	−0.398918	+	0.690946i	0.594675	+	0.803966i	$0.297281\pi$
$510$		0			0
$511$	20.0000	+	34.6410i	0.884748	+	1.53243i
$512$	32.0000			1.41421
$513$		0			0
$514$	−12.0000			−0.529297
$515$		0			0
$516$		0			0
$517$	3.00000	−	5.19615i	0.131940	−	0.228527i
$518$	8.00000	+	13.8564i	0.351500	+	0.608816i
$519$		0			0
$520$		0			0
$521$	15.0000			0.657162			0.328581	−	0.944476i	$-0.393430\pi$
$521$	15.0000			0.657162			0.328581	+	0.944476i	$0.393430\pi$
$522$	27.0000	−	46.7654i	1.18176	−	2.04686i
$523$	−8.00000	+	13.8564i	−0.349816	+	0.605898i	−0.986216	−	0.165460i	$-0.947089\pi$
$523$	−8.00000	+	13.8564i	−0.349816	+	0.605898i	0.636401	+	0.771358i	$0.280422\pi$
$524$	24.0000			1.04844
$525$		0			0
$526$		0			0
$527$	−7.00000	−	12.1244i	−0.304925	−	0.528145i
$528$		0			0
$529$	−6.50000	−	11.2583i	−0.282609	−	0.489493i
$530$		0			0
$531$	−27.0000			−1.17170
$532$	4.00000	+	34.6410i	0.173422	+	1.50188i
$533$	−4.00000			−0.173259
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−3.00000	+	5.19615i	−0.129339	+	0.224022i
$539$	−9.00000			−0.387657
$540$		0			0
$541$	−11.5000	+	19.9186i	−0.494424	+	0.856367i	−0.999979	−	0.00642713i	$-0.997954\pi$
$541$	−11.5000	+	19.9186i	−0.494424	+	0.856367i	0.505556	+	0.862794i	$0.331288\pi$
$542$	−3.00000	+	5.19615i	−0.128861	+	0.223194i
$543$		0			0
$544$	16.0000			0.685994
$545$		0			0
$546$		0			0
$547$	−11.0000	+	19.0526i	−0.470326	+	0.814629i	−0.999424	−	0.0339321i	$-0.989197\pi$
$547$	−11.0000	+	19.0526i	−0.470326	+	0.814629i	0.529098	+	0.848561i	$0.322530\pi$
$548$	−12.0000	−	20.7846i	−0.512615	−	0.887875i
$549$	−10.5000	−	18.1865i	−0.448129	−	0.776182i
$550$		0			0
$551$	−4.50000	−	38.9711i	−0.191706	−	1.66023i
$552$		0			0
$553$	2.00000	+	3.46410i	0.0850487	+	0.147309i
$554$	28.0000	+	48.4974i	1.18961	+	2.06046i
$555$		0			0
$556$	−20.0000	−	34.6410i	−0.848189	−	1.46911i
$557$	−7.00000	+	12.1244i	−0.296600	+	0.513725i	−0.975356	−	0.220638i	$-0.929186\pi$
$557$	−7.00000	+	12.1244i	−0.296600	+	0.513725i	0.678756	+	0.734364i	$0.262519\pi$
$558$	42.0000			1.77800
$559$	−4.00000			−0.169182
$560$		0			0
$561$		0			0
$562$	20.0000			0.843649
$563$	−24.0000			−1.01148			−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000			−1.01148			−0.505740	+	0.862686i	$0.668780\pi$
$564$		0			0
$565$		0			0
$566$	−14.0000	+	24.2487i	−0.588464	+	1.01925i
$567$	18.0000	+	31.1769i	0.755929	+	1.30931i
$568$		0			0
$569$	15.0000			0.628833			0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000			0.628833			0.314416	+	0.949285i	$0.398191\pi$
$570$		0			0
$571$	13.0000			0.544033			0.272017	−	0.962293i	$-0.412309\pi$
$571$	13.0000			0.544033			0.272017	+	0.962293i	$0.412309\pi$
$572$	−2.00000	−	3.46410i	−0.0836242	−	0.144841i
$573$		0			0
$574$	8.00000	−	13.8564i	0.333914	−	0.578355i
$575$		0			0
$576$	−12.0000	+	20.7846i	−0.500000	+	0.866025i
$577$	6.00000			0.249783			0.124892	−	0.992170i	$-0.460142\pi$
$577$	6.00000			0.249783			0.124892	+	0.992170i	$0.460142\pi$
$578$	−26.0000			−1.08146
$579$		0			0
$580$		0			0
$581$	24.0000			0.995688
$582$		0			0
$583$	2.00000	−	3.46410i	0.0828315	−	0.143468i
$584$		0			0
$585$		0			0
$586$	−4.00000	−	6.92820i	−0.165238	−	0.286201i
$587$	12.0000	+	20.7846i	0.495293	+	0.857873i	0.999985	−	0.00542667i	$-0.00172737\pi$
$587$	12.0000	+	20.7846i	0.495293	+	0.857873i	−0.504692	+	0.863299i	$0.668394\pi$
$588$		0			0
$589$	24.5000	−	18.1865i	1.00950	−	0.749363i
$590$		0			0
$591$		0			0
$592$	4.00000	+	6.92820i	0.164399	+	0.284747i
$593$	16.0000	−	27.7128i	0.657041	−	1.13803i	−0.324337	−	0.945942i	$-0.605141\pi$
$593$	16.0000	−	27.7128i	0.657041	−	1.13803i	0.981378	−	0.192087i	$-0.0615256\pi$
$594$		0			0
$595$		0			0
$596$	−2.00000			−0.0819232
$597$		0			0
$598$	12.0000	−	20.7846i	0.490716	−	0.849946i
$599$	6.00000	−	10.3923i	0.245153	−	0.424618i	−0.717021	−	0.697051i	$-0.754495\pi$
$599$	6.00000	−	10.3923i	0.245153	−	0.424618i	0.962175	+	0.272433i	$0.0878284\pi$
$600$		0			0
$601$	17.0000			0.693444			0.346722	−	0.937968i	$-0.387295\pi$
$601$	17.0000			0.693444			0.346722	+	0.937968i	$0.387295\pi$
$602$	8.00000	−	13.8564i	0.326056	−	0.564745i
$603$	−15.0000	−	25.9808i	−0.610847	−	1.05802i
$604$	−9.00000	+	15.5885i	−0.366205	+	0.634285i
$605$		0			0
$606$		0			0
$607$	−14.0000			−0.568242			−0.284121	−	0.958788i	$-0.591702\pi$
$607$	−14.0000			−0.568242			−0.284121	+	0.958788i	$0.591702\pi$
$608$	4.00000	+	34.6410i	0.162221	+	1.40488i
$609$		0			0
$610$		0			0
$611$	6.00000	+	10.3923i	0.242734	+	0.420428i
$612$	−6.00000	+	10.3923i	−0.242536	+	0.420084i
$613$	−12.0000	−	20.7846i	−0.484675	−	0.839482i	0.515170	−	0.857088i	$-0.327729\pi$
$613$	−12.0000	−	20.7846i	−0.484675	−	0.839482i	−0.999845	+	0.0176058i	$0.994396\pi$
$614$	−16.0000	+	27.7128i	−0.645707	+	1.11840i
$615$		0			0
$616$		0			0
$617$	−9.00000	+	15.5885i	−0.362326	+	0.627568i	−0.988343	−	0.152242i	$-0.951351\pi$
$617$	−9.00000	+	15.5885i	−0.362326	+	0.627568i	0.626017	+	0.779809i	$0.284684\pi$
$618$		0			0
$619$	4.00000			0.160774			0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000			0.160774			0.0803868	+	0.996764i	$0.474384\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−22.0000	+	38.1051i	−0.881411	+	1.52665i
$624$		0			0
$625$		0			0
$626$	60.0000			2.39808
$627$		0			0
$628$	8.00000			0.319235
$629$	2.00000	+	3.46410i	0.0797452	+	0.138123i
$630$		0			0
$631$	−0.500000	+	0.866025i	−0.0199047	+	0.0344759i	−0.875806	−	0.482663i	$-0.839670\pi$
$631$	−0.500000	+	0.866025i	−0.0199047	+	0.0344759i	0.855901	+	0.517139i	$0.173003\pi$
$632$		0			0
$633$		0			0
$634$	−4.00000			−0.158860
$635$		0			0
$636$		0			0
$637$	9.00000	−	15.5885i	0.356593	−	0.617637i
$638$	−18.0000			−0.712627
$639$	−3.00000			−0.118678
$640$		0			0
$641$	−10.5000	−	18.1865i	−0.414725	−	0.718325i	0.580674	−	0.814136i	$-0.302789\pi$
$641$	−10.5000	−	18.1865i	−0.414725	−	0.718325i	−0.995400	+	0.0958109i	$0.969456\pi$
$642$		0			0
$643$	23.0000	+	39.8372i	0.907031	+	1.57102i	0.818167	+	0.574981i	$0.194991\pi$
$643$	23.0000	+	39.8372i	0.907031	+	1.57102i	0.0888646	+	0.996044i	$0.471676\pi$
$644$	24.0000	+	41.5692i	0.945732	+	1.63806i
$645$		0			0
$646$	2.00000	+	17.3205i	0.0786889	+	0.681466i
$647$	−6.00000			−0.235884			−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000			−0.235884			−0.117942	+	0.993020i	$0.537630\pi$
$648$		0			0
$649$	4.50000	+	7.79423i	0.176640	+	0.305950i
$650$		0			0
$651$		0			0
$652$	−4.00000	+	6.92820i	−0.156652	+	0.271329i
$653$	10.0000			0.391330			0.195665	−	0.980671i	$-0.437313\pi$
$653$	10.0000			0.391330			0.195665	+	0.980671i	$0.437313\pi$
$654$		0			0
$655$		0			0
$656$	4.00000	−	6.92820i	0.156174	−	0.270501i
$657$	−30.0000			−1.17041
$658$	−48.0000			−1.87123
$659$	−10.0000	+	17.3205i	−0.389545	+	0.674711i	−0.992388	−	0.123148i	$-0.960701\pi$
$659$	−10.0000	+	17.3205i	−0.389545	+	0.674711i	0.602844	+	0.797859i	$0.294034\pi$
$660$		0			0
$661$	7.50000	−	12.9904i	0.291716	−	0.505267i	−0.682499	−	0.730886i	$-0.739107\pi$
$661$	7.50000	−	12.9904i	0.291716	−	0.505267i	0.974216	+	0.225619i	$0.0724404\pi$
$662$	−20.0000	−	34.6410i	−0.777322	−	1.34636i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−12.0000			−0.464991
$667$	−27.0000	−	46.7654i	−1.04544	−	1.81076i
$668$	12.0000	+	20.7846i	0.464294	+	0.804181i
$669$		0			0
$670$		0			0
$671$	−3.50000	+	6.06218i	−0.135116	+	0.234028i
$672$		0			0
$673$	−20.0000			−0.770943			−0.385472	−	0.922720i	$-0.625961\pi$
$673$	−20.0000			−0.770943			−0.385472	+	0.922720i	$0.625961\pi$
$674$	−34.0000	+	58.8897i	−1.30963	+	2.26835i
$675$		0			0
$676$	−18.0000			−0.692308
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$	−12.0000	−	20.7846i	−0.460518	−	0.797640i
$680$		0			0
$681$		0			0
$682$	−7.00000	−	12.1244i	−0.268044	−	0.464266i
$683$	30.0000			1.14792			0.573959	−	0.818884i	$-0.305407\pi$
$683$	30.0000			1.14792			0.573959	+	0.818884i	$0.305407\pi$
$684$	−24.0000	−	10.3923i	−0.917663	−	0.397360i
$685$		0			0
$686$	8.00000	+	13.8564i	0.305441	+	0.529040i
$687$		0			0
$688$	4.00000	−	6.92820i	0.152499	−	0.264135i
$689$	4.00000	+	6.92820i	0.152388	+	0.263944i
$690$		0			0
$691$	3.00000			0.114125			0.0570627	−	0.998371i	$-0.481827\pi$
$691$	3.00000			0.114125			0.0570627	+	0.998371i	$0.481827\pi$
$692$	−48.0000			−1.82469
$693$	6.00000	−	10.3923i	0.227921	−	0.394771i
$694$	12.0000	−	20.7846i	0.455514	−	0.788973i
$695$		0			0
$696$		0			0
$697$	2.00000	−	3.46410i	0.0757554	−	0.131212i
$698$	−14.0000	−	24.2487i	−0.529908	−	0.917827i
$699$		0			0
$700$		0			0
$701$	−3.00000	−	5.19615i	−0.113308	−	0.196256i	0.803794	−	0.594908i	$-0.202811\pi$
$701$	−3.00000	−	5.19615i	−0.113308	−	0.196256i	−0.917102	+	0.398652i	$0.869478\pi$
$702$		0			0
$703$	−7.00000	+	5.19615i	−0.264010	+	0.195977i
$704$	8.00000			0.301511
$705$		0			0
$706$	8.00000	+	13.8564i	0.301084	+	0.521493i
$707$	30.0000	−	51.9615i	1.12827	−	1.95421i
$708$		0			0
$709$	−12.5000	+	21.6506i	−0.469447	+	0.813107i	−0.999390	−	0.0349269i	$-0.988880\pi$
$709$	−12.5000	+	21.6506i	−0.469447	+	0.813107i	0.529943	+	0.848034i	$0.322213\pi$
$710$		0			0
$711$	−3.00000			−0.112509
$712$		0			0
$713$	21.0000	−	36.3731i	0.786456	−	1.36218i
$714$		0			0
$715$		0			0
$716$	−15.0000	+	25.9808i	−0.560576	+	0.970947i
$717$		0			0
$718$	20.0000	−	34.6410i	0.746393	−	1.29279i
$719$	10.5000	+	18.1865i	0.391584	+	0.678243i	0.992659	−	0.120950i	$-0.0385939\pi$
$719$	10.5000	+	18.1865i	0.391584	+	0.678243i	−0.601075	+	0.799193i	$0.705261\pi$
$720$		0			0
$721$	−64.0000			−2.38348
$722$	−37.0000	+	8.66025i	−1.37700	+	0.322301i
$723$		0			0
$724$	6.00000	+	10.3923i	0.222988	+	0.386227i
$725$		0			0
$726$		0			0
$727$	−4.00000	−	6.92820i	−0.148352	−	0.256953i	0.782267	−	0.622944i	$-0.214063\pi$
$727$	−4.00000	−	6.92820i	−0.148352	−	0.256953i	−0.930618	+	0.365991i	$0.880730\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$	2.00000	−	3.46410i	0.0739727	−	0.128124i
$732$		0			0
$733$	−6.00000			−0.221615			−0.110808	−	0.993842i	$-0.535344\pi$
$733$	−6.00000			−0.221615			−0.110808	+	0.993842i	$0.535344\pi$
$734$	−32.0000			−1.18114
$735$		0			0
$736$	24.0000	+	41.5692i	0.884652	+	1.53226i
$737$	−5.00000	+	8.66025i	−0.184177	+	0.319005i
$738$	6.00000	+	10.3923i	0.220863	+	0.382546i
$739$	14.5000	+	25.1147i	0.533391	+	0.923861i	0.999239	+	0.0389959i	$0.0124159\pi$
$739$	14.5000	+	25.1147i	0.533391	+	0.923861i	−0.465848	+	0.884865i	$0.654251\pi$
$740$		0			0
$741$		0			0
$742$	−32.0000			−1.17476
$743$	−25.0000	−	43.3013i	−0.917161	−	1.58857i	−0.803706	−	0.595026i	$-0.797142\pi$
$743$	−25.0000	−	43.3013i	−0.917161	−	1.58857i	−0.113455	−	0.993543i	$-0.536192\pi$
$744$		0			0
$745$		0			0
$746$	12.0000	+	20.7846i	0.439351	+	0.760979i
$747$	−9.00000	+	15.5885i	−0.329293	+	0.570352i
$748$	4.00000			0.146254
$749$	40.0000			1.46157
$750$		0			0
$751$	−20.5000	+	35.5070i	−0.748056	+	1.29567i	0.200698	+	0.979653i	$0.435679\pi$
$751$	−20.5000	+	35.5070i	−0.748056	+	1.29567i	−0.948753	+	0.316017i	$0.897654\pi$
$752$	−24.0000			−0.875190
$753$		0			0
$754$	18.0000	−	31.1769i	0.655521	−	1.13540i
$755$		0			0
$756$		0			0
$757$	−13.0000	−	22.5167i	−0.472493	−	0.818382i	0.527011	−	0.849858i	$-0.323312\pi$
$757$	−13.0000	−	22.5167i	−0.472493	−	0.818382i	−0.999505	+	0.0314762i	$0.989979\pi$
$758$	29.0000	+	50.2295i	1.05333	+	1.82442i
$759$		0			0
$760$		0			0
$761$	−18.0000			−0.652499			−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000			−0.652499			−0.326250	+	0.945284i	$0.605785\pi$
$762$		0			0
$763$	30.0000	+	51.9615i	1.08607	+	1.88113i
$764$	3.00000	−	5.19615i	0.108536	−	0.187990i
$765$		0			0
$766$	−6.00000	+	10.3923i	−0.216789	+	0.375489i
$767$	−18.0000			−0.649942
$768$		0			0
$769$	−2.50000	+	4.33013i	−0.0901523	+	0.156148i	−0.907575	−	0.419890i	$-0.862069\pi$
$769$	−2.50000	+	4.33013i	−0.0901523	+	0.156148i	0.817423	+	0.576038i	$0.195402\pi$
$770$		0			0
$771$		0			0
$772$	32.0000			1.15171
$773$	−17.0000	+	29.4449i	−0.611448	+	1.05906i	0.379549	+	0.925172i	$0.376079\pi$
$773$	−17.0000	+	29.4449i	−0.611448	+	1.05906i	−0.990997	+	0.133887i	$0.957254\pi$
$774$	6.00000	+	10.3923i	0.215666	+	0.373544i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−66.0000			−2.36621
$779$	8.00000	+	3.46410i	0.286630	+	0.124114i
$780$		0			0
$781$	0.500000	+	0.866025i	0.0178914	+	0.0309888i
$782$	12.0000	+	20.7846i	0.429119	+	0.743256i
$783$		0			0
$784$	18.0000	+	31.1769i	0.642857	+	1.11346i
$785$		0			0
$786$		0			0
$787$	22.0000

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:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.e (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.73205i) q^{2} +(-1.00000 + 1.73205i) q^{4} -4.00000 q^{7} +(1.50000 - 2.59808i) q^{9} +O(q^{10})\)	\(q+(-1.00000 - 1.73205i) q^{2} +(-1.00000 + 1.73205i) q^{4} -4.00000 q^{7} +(1.50000 - 2.59808i) q^{9} -1.00000 q^{11} +(1.00000 - 1.73205i) q^{13} +(4.00000 + 6.92820i) q^{14} +(2.00000 + 3.46410i) q^{16} +(1.00000 + 1.73205i) q^{17} -6.00000 q^{18} +(-3.50000 + 2.59808i) q^{19} +(1.00000 + 1.73205i) q^{22} +(-3.00000 + 5.19615i) q^{23} -4.00000 q^{26} +(4.00000 - 6.92820i) q^{28} +(-4.50000 + 7.79423i) q^{29} -7.00000 q^{31} +(4.00000 - 6.92820i) q^{32} +(2.00000 - 3.46410i) q^{34} +(3.00000 + 5.19615i) q^{36} +2.00000 q^{37} +(8.00000 + 3.46410i) q^{38} +(-1.00000 - 1.73205i) q^{41} +(-1.00000 - 1.73205i) q^{43} +(1.00000 - 1.73205i) q^{44} +12.0000 q^{46} +(-3.00000 + 5.19615i) q^{47} +9.00000 q^{49} +(2.00000 + 3.46410i) q^{52} +(-2.00000 + 3.46410i) q^{53} +18.0000 q^{58} +(-4.50000 - 7.79423i) q^{59} +(3.50000 - 6.06218i) q^{61} +(7.00000 + 12.1244i) q^{62} +(-6.00000 + 10.3923i) q^{63} -8.00000 q^{64} +(5.00000 - 8.66025i) q^{67} -4.00000 q^{68} +(-0.500000 - 0.866025i) q^{71} +(-5.00000 - 8.66025i) q^{73} +(-2.00000 - 3.46410i) q^{74} +(-1.00000 - 8.66025i) q^{76} +4.00000 q^{77} +(-0.500000 - 0.866025i) q^{79} +(-4.50000 - 7.79423i) q^{81} +(-2.00000 + 3.46410i) q^{82} -6.00000 q^{83} +(-2.00000 + 3.46410i) q^{86} +(5.50000 - 9.52628i) q^{89} +(-4.00000 + 6.92820i) q^{91} +(-6.00000 - 10.3923i) q^{92} +12.0000 q^{94} +(3.00000 + 5.19615i) q^{97} +(-9.00000 - 15.5885i) q^{98} +(-1.50000 + 2.59808i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{4} - 8 q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^4 - 8 * q^7 + 3 * q^9	\( 2 q - 2 q^{2} - 2 q^{4} - 8 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} + 8 q^{14} + 4 q^{16} + 2 q^{17} - 12 q^{18} - 7 q^{19} + 2 q^{22} - 6 q^{23} - 8 q^{26} + 8 q^{28} - 9 q^{29} - 14 q^{31} + 8 q^{32} + 4 q^{34} + 6 q^{36} + 4 q^{37} + 16 q^{38} - 2 q^{41} - 2 q^{43} + 2 q^{44} + 24 q^{46} - 6 q^{47} + 18 q^{49} + 4 q^{52} - 4 q^{53} + 36 q^{58} - 9 q^{59} + 7 q^{61} + 14 q^{62} - 12 q^{63} - 16 q^{64} + 10 q^{67} - 8 q^{68} - q^{71} - 10 q^{73} - 4 q^{74} - 2 q^{76} + 8 q^{77} - q^{79} - 9 q^{81} - 4 q^{82} - 12 q^{83} - 4 q^{86} + 11 q^{89} - 8 q^{91} - 12 q^{92} + 24 q^{94} + 6 q^{97} - 18 q^{98} - 3 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^4 - 8 * q^7 + 3 * q^9 - 2 * q^11 + 2 * q^13 + 8 * q^14 + 4 * q^16 + 2 * q^17 - 12 * q^18 - 7 * q^19 + 2 * q^22 - 6 * q^23 - 8 * q^26 + 8 * q^28 - 9 * q^29 - 14 * q^31 + 8 * q^32 + 4 * q^34 + 6 * q^36 + 4 * q^37 + 16 * q^38 - 2 * q^41 - 2 * q^43 + 2 * q^44 + 24 * q^46 - 6 * q^47 + 18 * q^49 + 4 * q^52 - 4 * q^53 + 36 * q^58 - 9 * q^59 + 7 * q^61 + 14 * q^62 - 12 * q^63 - 16 * q^64 + 10 * q^67 - 8 * q^68 - q^71 - 10 * q^73 - 4 * q^74 - 2 * q^76 + 8 * q^77 - q^79 - 9 * q^81 - 4 * q^82 - 12 * q^83 - 4 * q^86 + 11 * q^89 - 8 * q^91 - 12 * q^92 + 24 * q^94 + 6 * q^97 - 18 * q^98 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

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