LMFDB - Embedded newform 476.2.i.a.137.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [476,2,Mod(137,476)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(476, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 0]))

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

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

chi := DirichletCharacter("476.137");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 476 = 2^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	476.i (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:	$3.80087913621$
Analytic rank:	$0$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			137.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	476.137
Dual form			476.2.i.a.205.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{3} +(2.00000 + 3.46410i) q^{5} +(-2.00000 + 1.73205i) q^{7} +(1.00000 + 1.73205i) q^{9} +O(q^{10})$	\(q+(-0.500000 + 0.866025i) q^{3} +(2.00000 + 3.46410i) q^{5} +(-2.00000 + 1.73205i) q^{7} +(1.00000 + 1.73205i) q^{9} +(1.50000 - 2.59808i) q^{11} -5.00000 q^{13} -4.00000 q^{15} +(0.500000 - 0.866025i) q^{17} +(-3.00000 - 5.19615i) q^{19} +(-0.500000 - 2.59808i) q^{21} +(2.00000 + 3.46410i) q^{23} +(-5.50000 + 9.52628i) q^{25} -5.00000 q^{27} +8.00000 q^{29} +(1.50000 + 2.59808i) q^{33} +(-10.0000 - 3.46410i) q^{35} +(4.00000 + 6.92820i) q^{37} +(2.50000 - 4.33013i) q^{39} -8.00000 q^{41} +10.0000 q^{43} +(-4.00000 + 6.92820i) q^{45} +(1.00000 + 1.73205i) q^{47} +(1.00000 - 6.92820i) q^{49} +(0.500000 + 0.866025i) q^{51} +(-1.50000 + 2.59808i) q^{53} +12.0000 q^{55} +6.00000 q^{57} +(-1.00000 + 1.73205i) q^{59} +(4.00000 + 6.92820i) q^{61} +(-5.00000 - 1.73205i) q^{63} +(-10.0000 - 17.3205i) q^{65} +(-7.00000 + 12.1244i) q^{67} -4.00000 q^{69} +7.00000 q^{71} +(4.00000 - 6.92820i) q^{73} +(-5.50000 - 9.52628i) q^{75} +(1.50000 + 7.79423i) q^{77} +(-7.50000 - 12.9904i) q^{79} +(-0.500000 + 0.866025i) q^{81} +12.0000 q^{83} +4.00000 q^{85} +(-4.00000 + 6.92820i) q^{87} +(-0.500000 - 0.866025i) q^{89} +(10.0000 - 8.66025i) q^{91} +(12.0000 - 20.7846i) q^{95} +18.0000 q^{97} +6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + 4 q^{5} - 4 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - q^3 + 4 * q^5 - 4 * q^7 + 2 * q^9	$ 2 q - q^{3} + 4 q^{5} - 4 q^{7} + 2 q^{9} + 3 q^{11} - 10 q^{13} - 8 q^{15} + q^{17} - 6 q^{19} - q^{21} + 4 q^{23} - 11 q^{25} - 10 q^{27} + 16 q^{29} + 3 q^{33} - 20 q^{35} + 8 q^{37} + 5 q^{39} - 16 q^{41} + 20 q^{43} - 8 q^{45} + 2 q^{47} + 2 q^{49} + q^{51} - 3 q^{53} + 24 q^{55} + 12 q^{57} - 2 q^{59} + 8 q^{61} - 10 q^{63} - 20 q^{65} - 14 q^{67} - 8 q^{69} + 14 q^{71} + 8 q^{73} - 11 q^{75} + 3 q^{77} - 15 q^{79} - q^{81} + 24 q^{83} + 8 q^{85} - 8 q^{87} - q^{89} + 20 q^{91} + 24 q^{95} + 36 q^{97} + 12 q^{99}+O(q^{100}) $ 2 * q - q^3 + 4 * q^5 - 4 * q^7 + 2 * q^9 + 3 * q^11 - 10 * q^13 - 8 * q^15 + q^17 - 6 * q^19 - q^21 + 4 * q^23 - 11 * q^25 - 10 * q^27 + 16 * q^29 + 3 * q^33 - 20 * q^35 + 8 * q^37 + 5 * q^39 - 16 * q^41 + 20 * q^43 - 8 * q^45 + 2 * q^47 + 2 * q^49 + q^51 - 3 * q^53 + 24 * q^55 + 12 * q^57 - 2 * q^59 + 8 * q^61 - 10 * q^63 - 20 * q^65 - 14 * q^67 - 8 * q^69 + 14 * q^71 + 8 * q^73 - 11 * q^75 + 3 * q^77 - 15 * q^79 - q^81 + 24 * q^83 + 8 * q^85 - 8 * q^87 - q^89 + 20 * q^91 + 24 * q^95 + 36 * q^97 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/476\mathbb{Z}\right)^\times$.

$n$	$239$	$309$	$409$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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			0
$2$		0			0
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i	−0.973494	−	0.228714i	$-0.926548\pi$
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i	0.684819	+	0.728714i	$0.259881\pi$
$4$		0			0
$5$	2.00000	+	3.46410i	0.894427	+	1.54919i	0.834512	+	0.550990i	$0.185750\pi$
$5$	2.00000	+	3.46410i	0.894427	+	1.54919i	0.0599153	+	0.998203i	$0.480917\pi$
$6$		0			0
$7$	−2.00000	+	1.73205i	−0.755929	+	0.654654i
$7$	−2.00000	+	1.73205i	−0.755929	+	0.654654i
$8$		0			0
$9$	1.00000	+	1.73205i	0.333333	+	0.577350i
$10$		0			0
$11$	1.50000	−	2.59808i	0.452267	−	0.783349i	−0.546259	−	0.837616i	$-0.683949\pi$
$11$	1.50000	−	2.59808i	0.452267	−	0.783349i	0.998526	+	0.0542666i	$0.0172821\pi$
$12$		0			0
$13$	−5.00000			−1.38675			−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000			−1.38675			−0.693375	+	0.720577i	$0.743877\pi$
$14$		0			0
$15$	−4.00000			−1.03280
$16$		0			0
$17$	0.500000	−	0.866025i	0.121268	−	0.210042i
$17$	0.500000	−	0.866025i	0.121268	−	0.210042i
$18$		0			0
$19$	−3.00000	−	5.19615i	−0.688247	−	1.19208i	−0.972404	−	0.233301i	$-0.925047\pi$
$19$	−3.00000	−	5.19615i	−0.688247	−	1.19208i	0.284157	−	0.958778i	$-0.408286\pi$
$20$		0			0
$21$	−0.500000	−	2.59808i	−0.109109	−	0.566947i
$22$		0			0
$23$	2.00000	+	3.46410i	0.417029	+	0.722315i	0.995639	−	0.0932891i	$-0.0297381\pi$
$23$	2.00000	+	3.46410i	0.417029	+	0.722315i	−0.578610	+	0.815604i	$0.696405\pi$
$24$		0			0
$25$	−5.50000	+	9.52628i	−1.10000	+	1.90526i
$26$		0			0
$27$	−5.00000			−0.962250
$28$		0			0
$29$	8.00000			1.48556			0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000			1.48556			0.742781	+	0.669534i	$0.233506\pi$
$30$		0			0
$31$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$31$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$32$		0			0
$33$	1.50000	+	2.59808i	0.261116	+	0.452267i
$34$		0			0
$35$	−10.0000	−	3.46410i	−1.69031	−	0.585540i
$36$		0			0
$37$	4.00000	+	6.92820i	0.657596	+	1.13899i	0.981236	+	0.192809i	$0.0617599\pi$
$37$	4.00000	+	6.92820i	0.657596	+	1.13899i	−0.323640	+	0.946180i	$0.604907\pi$
$38$		0			0
$39$	2.50000	−	4.33013i	0.400320	−	0.693375i
$40$		0			0
$41$	−8.00000			−1.24939			−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000			−1.24939			−0.624695	+	0.780869i	$0.714777\pi$
$42$		0			0
$43$	10.0000			1.52499			0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000			1.52499			0.762493	+	0.646997i	$0.223975\pi$
$44$		0			0
$45$	−4.00000	+	6.92820i	−0.596285	+	1.03280i
$46$		0			0
$47$	1.00000	+	1.73205i	0.145865	+	0.252646i	0.929695	−	0.368329i	$-0.120070\pi$
$47$	1.00000	+	1.73205i	0.145865	+	0.252646i	−0.783830	+	0.620975i	$0.786737\pi$
$48$		0			0
$49$	1.00000	−	6.92820i	0.142857	−	0.989743i
$50$		0			0
$51$	0.500000	+	0.866025i	0.0700140	+	0.121268i
$52$		0			0
$53$	−1.50000	+	2.59808i	−0.206041	+	0.356873i	−0.950464	−	0.310835i	$-0.899391\pi$
$53$	−1.50000	+	2.59808i	−0.206041	+	0.356873i	0.744423	+	0.667708i	$0.232725\pi$
$54$		0			0
$55$	12.0000			1.61808
$56$		0			0
$57$	6.00000			0.794719
$58$		0			0
$59$	−1.00000	+	1.73205i	−0.130189	+	0.225494i	−0.923749	−	0.382998i	$-0.874892\pi$
$59$	−1.00000	+	1.73205i	−0.130189	+	0.225494i	0.793560	+	0.608492i	$0.208225\pi$
$60$		0			0
$61$	4.00000	+	6.92820i	0.512148	+	0.887066i	0.999901	+	0.0140840i	$0.00448323\pi$
$61$	4.00000	+	6.92820i	0.512148	+	0.887066i	−0.487753	+	0.872982i	$0.662183\pi$
$62$		0			0
$63$	−5.00000	−	1.73205i	−0.629941	−	0.218218i
$64$		0			0
$65$	−10.0000	−	17.3205i	−1.24035	−	2.14834i
$66$		0			0
$67$	−7.00000	+	12.1244i	−0.855186	+	1.48123i	0.0212861	+	0.999773i	$0.493224\pi$
$67$	−7.00000	+	12.1244i	−0.855186	+	1.48123i	−0.876472	+	0.481452i	$0.840109\pi$
$68$		0			0
$69$	−4.00000			−0.481543
$70$		0			0
$71$	7.00000			0.830747			0.415374	−	0.909651i	$-0.363651\pi$
$71$	7.00000			0.830747			0.415374	+	0.909651i	$0.363651\pi$
$72$		0			0
$73$	4.00000	−	6.92820i	0.468165	−	0.810885i	−0.531174	−	0.847263i	$-0.678249\pi$
$73$	4.00000	−	6.92820i	0.468165	−	0.810885i	0.999338	+	0.0363782i	$0.0115821\pi$
$74$		0			0
$75$	−5.50000	−	9.52628i	−0.635085	−	1.10000i
$76$		0			0
$77$	1.50000	+	7.79423i	0.170941	+	0.888235i
$78$		0			0
$79$	−7.50000	−	12.9904i	−0.843816	−	1.46153i	−0.886646	−	0.462450i	$-0.846971\pi$
$79$	−7.50000	−	12.9904i	−0.843816	−	1.46153i	0.0428296	−	0.999082i	$-0.486363\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	12.0000			1.31717			0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000			1.31717			0.658586	+	0.752506i	$0.271155\pi$
$84$		0			0
$85$	4.00000			0.433861
$86$		0			0
$87$	−4.00000	+	6.92820i	−0.428845	+	0.742781i
$88$		0			0
$89$	−0.500000	−	0.866025i	−0.0529999	−	0.0917985i	0.838308	−	0.545197i	$-0.183545\pi$
$89$	−0.500000	−	0.866025i	−0.0529999	−	0.0917985i	−0.891308	+	0.453398i	$0.850212\pi$
$90$		0			0
$91$	10.0000	−	8.66025i	1.04828	−	0.907841i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	12.0000	−	20.7846i	1.23117	−	2.13246i
$96$		0			0
$97$	18.0000			1.82762			0.913812	−	0.406138i	$-0.133125\pi$
$97$	18.0000			1.82762			0.913812	+	0.406138i	$0.133125\pi$
$98$		0			0
$99$	6.00000			0.603023
$100$		0			0
$101$	3.00000	−	5.19615i	0.298511	−	0.517036i	−0.677284	−	0.735721i	$-0.736843\pi$
$101$	3.00000	−	5.19615i	0.298511	−	0.517036i	0.975796	+	0.218685i	$0.0701767\pi$
$102$		0			0
$103$	−2.00000	−	3.46410i	−0.197066	−	0.341328i	0.750510	−	0.660859i	$-0.229808\pi$
$103$	−2.00000	−	3.46410i	−0.197066	−	0.341328i	−0.947576	+	0.319531i	$0.896475\pi$
$104$		0			0
$105$	8.00000	−	6.92820i	0.780720	−	0.676123i
$106$		0			0
$107$	−0.500000	−	0.866025i	−0.0483368	−	0.0837218i	0.840845	−	0.541276i	$-0.182059\pi$
$107$	−0.500000	−	0.866025i	−0.0483368	−	0.0837218i	−0.889182	+	0.457555i	$0.848725\pi$
$108$		0			0
$109$	4.00000	−	6.92820i	0.383131	−	0.663602i	−0.608377	−	0.793648i	$-0.708179\pi$
$109$	4.00000	−	6.92820i	0.383131	−	0.663602i	0.991508	+	0.130046i	$0.0415126\pi$
$110$		0			0
$111$	−8.00000			−0.759326
$112$		0			0
$113$	−8.00000			−0.752577			−0.376288	−	0.926503i	$-0.622800\pi$
$113$	−8.00000			−0.752577			−0.376288	+	0.926503i	$0.622800\pi$
$114$		0			0
$115$	−8.00000	+	13.8564i	−0.746004	+	1.29212i
$116$		0			0
$117$	−5.00000	−	8.66025i	−0.462250	−	0.800641i
$118$		0			0
$119$	0.500000	+	2.59808i	0.0458349	+	0.238165i
$120$		0			0
$121$	1.00000	+	1.73205i	0.0909091	+	0.157459i
$122$		0			0
$123$	4.00000	−	6.92820i	0.360668	−	0.624695i
$124$		0			0
$125$	−24.0000			−2.14663
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$		0			0
$129$	−5.00000	+	8.66025i	−0.440225	+	0.762493i
$130$		0			0
$131$	2.00000	+	3.46410i	0.174741	+	0.302660i	0.940072	−	0.340977i	$-0.110758\pi$
$131$	2.00000	+	3.46410i	0.174741	+	0.302660i	−0.765331	+	0.643637i	$0.777425\pi$
$132$		0			0
$133$	15.0000	+	5.19615i	1.30066	+	0.450564i
$134$		0			0
$135$	−10.0000	−	17.3205i	−0.860663	−	1.49071i
$136$		0			0
$137$	3.50000	−	6.06218i	0.299025	−	0.517927i	−0.676888	−	0.736086i	$-0.736672\pi$
$137$	3.50000	−	6.06218i	0.299025	−	0.517927i	0.975913	+	0.218159i	$0.0700052\pi$
$138$		0			0
$139$	−1.00000			−0.0848189			−0.0424094	−	0.999100i	$-0.513503\pi$
$139$	−1.00000			−0.0848189			−0.0424094	+	0.999100i	$0.513503\pi$
$140$		0			0
$141$	−2.00000			−0.168430
$142$		0			0
$143$	−7.50000	+	12.9904i	−0.627182	+	1.08631i
$144$		0			0
$145$	16.0000	+	27.7128i	1.32873	+	2.30142i
$146$		0			0
$147$	5.50000	+	4.33013i	0.453632	+	0.357143i
$148$		0			0
$149$	5.50000	+	9.52628i	0.450578	+	0.780423i	0.998422	−	0.0561570i	$-0.0178847\pi$
$149$	5.50000	+	9.52628i	0.450578	+	0.780423i	−0.547844	+	0.836580i	$0.684551\pi$
$150$		0			0
$151$	−5.00000	+	8.66025i	−0.406894	+	0.704761i	−0.994540	−	0.104357i	$-0.966722\pi$
$151$	−5.00000	+	8.66025i	−0.406894	+	0.704761i	0.587646	+	0.809118i	$0.300055\pi$
$152$		0			0
$153$	2.00000			0.161690
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.500000	+	0.866025i	−0.0399043	+	0.0691164i	−0.885288	−	0.465044i	$-0.846039\pi$
$157$	−0.500000	+	0.866025i	−0.0399043	+	0.0691164i	0.845383	+	0.534160i	$0.179372\pi$
$158$		0			0
$159$	−1.50000	−	2.59808i	−0.118958	−	0.206041i
$160$		0			0
$161$	−10.0000	−	3.46410i	−0.788110	−	0.273009i
$162$		0			0
$163$	−2.00000	−	3.46410i	−0.156652	−	0.271329i	0.777007	−	0.629492i	$-0.216737\pi$
$163$	−2.00000	−	3.46410i	−0.156652	−	0.271329i	−0.933659	+	0.358162i	$0.883403\pi$
$164$		0			0
$165$	−6.00000	+	10.3923i	−0.467099	+	0.809040i
$166$		0			0
$167$	−19.0000			−1.47026			−0.735132	−	0.677924i	$-0.762880\pi$
$167$	−19.0000			−1.47026			−0.735132	+	0.677924i	$0.762880\pi$
$168$		0			0
$169$	12.0000			0.923077
$170$		0			0
$171$	6.00000	−	10.3923i	0.458831	−	0.794719i
$172$		0			0
$173$	1.00000	+	1.73205i	0.0760286	+	0.131685i	0.901533	−	0.432710i	$-0.142443\pi$
$173$	1.00000	+	1.73205i	0.0760286	+	0.131685i	−0.825505	+	0.564396i	$0.809109\pi$
$174$		0			0
$175$	−5.50000	−	28.5788i	−0.415761	−	2.16036i
$176$		0			0
$177$	−1.00000	−	1.73205i	−0.0751646	−	0.130189i
$178$		0			0
$179$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$179$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$180$		0			0
$181$	18.0000			1.33793			0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000			1.33793			0.668965	+	0.743294i	$0.266738\pi$
$182$		0			0
$183$	−8.00000			−0.591377
$184$		0			0
$185$	−16.0000	+	27.7128i	−1.17634	+	2.03749i
$186$		0			0
$187$	−1.50000	−	2.59808i	−0.109691	−	0.189990i
$188$		0			0
$189$	10.0000	−	8.66025i	0.727393	−	0.629941i
$190$		0			0
$191$	−6.00000	−	10.3923i	−0.434145	−	0.751961i	0.563081	−	0.826402i	$-0.309616\pi$
$191$	−6.00000	−	10.3923i	−0.434145	−	0.751961i	−0.997225	+	0.0744412i	$0.976283\pi$
$192$		0			0
$193$	5.00000	−	8.66025i	0.359908	−	0.623379i	−0.628037	−	0.778183i	$-0.716141\pi$
$193$	5.00000	−	8.66025i	0.359908	−	0.623379i	0.987945	+	0.154805i	$0.0494748\pi$
$194$		0			0
$195$	20.0000			1.43223
$196$		0			0
$197$	18.0000			1.28245			0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000			1.28245			0.641223	+	0.767354i	$0.278427\pi$
$198$		0			0
$199$	3.50000	−	6.06218i	0.248108	−	0.429736i	−0.714893	−	0.699234i	$-0.753524\pi$
$199$	3.50000	−	6.06218i	0.248108	−	0.429736i	0.963001	+	0.269498i	$0.0868577\pi$
$200$		0			0
$201$	−7.00000	−	12.1244i	−0.493742	−	0.855186i
$202$		0			0
$203$	−16.0000	+	13.8564i	−1.12298	+	0.972529i
$204$		0			0
$205$	−16.0000	−	27.7128i	−1.11749	−	1.93555i
$206$		0			0
$207$	−4.00000	+	6.92820i	−0.278019	+	0.481543i
$208$		0			0
$209$	−18.0000			−1.24509
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$		0			0
$213$	−3.50000	+	6.06218i	−0.239816	+	0.415374i
$214$		0			0
$215$	20.0000	+	34.6410i	1.36399	+	2.36250i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	4.00000	+	6.92820i	0.270295	+	0.468165i
$220$		0			0
$221$	−2.50000	+	4.33013i	−0.168168	+	0.291276i
$222$		0			0
$223$	−6.00000			−0.401790			−0.200895	−	0.979613i	$-0.564385\pi$
$223$	−6.00000			−0.401790			−0.200895	+	0.979613i	$0.564385\pi$
$224$		0			0
$225$	−22.0000			−1.46667
$226$		0			0
$227$	−12.5000	+	21.6506i	−0.829654	+	1.43700i	0.0686556	+	0.997640i	$0.478129\pi$
$227$	−12.5000	+	21.6506i	−0.829654	+	1.43700i	−0.898310	+	0.439363i	$0.855204\pi$
$228$		0			0
$229$	5.00000	+	8.66025i	0.330409	+	0.572286i	0.982592	−	0.185776i	$-0.0594799\pi$
$229$	5.00000	+	8.66025i	0.330409	+	0.572286i	−0.652183	+	0.758062i	$0.726147\pi$
$230$		0			0
$231$	−7.50000	−	2.59808i	−0.493464	−	0.170941i
$232$		0			0
$233$	−14.0000	−	24.2487i	−0.917170	−	1.58859i	−0.803692	−	0.595045i	$-0.797134\pi$
$233$	−14.0000	−	24.2487i	−0.917170	−	1.58859i	−0.113478	−	0.993540i	$-0.536199\pi$
$234$		0			0
$235$	−4.00000	+	6.92820i	−0.260931	+	0.451946i
$236$		0			0
$237$	15.0000			0.974355
$238$		0			0
$239$	2.00000			0.129369			0.0646846	−	0.997906i	$-0.479396\pi$
$239$	2.00000			0.129369			0.0646846	+	0.997906i	$0.479396\pi$
$240$		0			0
$241$	4.00000	−	6.92820i	0.257663	−	0.446285i	−0.707953	−	0.706260i	$-0.750381\pi$
$241$	4.00000	−	6.92820i	0.257663	−	0.446285i	0.965615	+	0.259975i	$0.0837143\pi$
$242$		0			0
$243$	−8.00000	−	13.8564i	−0.513200	−	0.888889i
$244$		0			0
$245$	26.0000	−	10.3923i	1.66108	−	0.663940i
$246$		0			0
$247$	15.0000	+	25.9808i	0.954427	+	1.65312i
$248$		0			0
$249$	−6.00000	+	10.3923i	−0.380235	+	0.658586i
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$	12.0000			0.754434
$254$		0			0
$255$	−2.00000	+	3.46410i	−0.125245	+	0.216930i
$256$		0			0
$257$	−1.50000	−	2.59808i	−0.0935674	−	0.162064i	0.815442	−	0.578838i	$-0.196494\pi$
$257$	−1.50000	−	2.59808i	−0.0935674	−	0.162064i	−0.909010	+	0.416775i	$0.863160\pi$
$258$		0			0
$259$	−20.0000	−	6.92820i	−1.24274	−	0.430498i
$260$		0			0
$261$	8.00000	+	13.8564i	0.495188	+	0.857690i
$262$		0			0
$263$	−2.00000	+	3.46410i	−0.123325	+	0.213606i	−0.921077	−	0.389380i	$-0.872689\pi$
$263$	−2.00000	+	3.46410i	−0.123325	+	0.213606i	0.797752	+	0.602986i	$0.206023\pi$
$264$		0			0
$265$	−12.0000			−0.737154
$266$		0			0
$267$	1.00000			0.0611990
$268$		0			0
$269$	9.00000	−	15.5885i	0.548740	−	0.950445i	−0.449622	−	0.893219i	$-0.648441\pi$
$269$	9.00000	−	15.5885i	0.548740	−	0.950445i	0.998361	−	0.0572259i	$-0.0182255\pi$
$270$		0			0
$271$	12.0000	+	20.7846i	0.728948	+	1.26258i	0.957328	+	0.289003i	$0.0933238\pi$
$271$	12.0000	+	20.7846i	0.728948	+	1.26258i	−0.228380	+	0.973572i	$0.573343\pi$
$272$		0			0
$273$	2.50000	+	12.9904i	0.151307	+	0.786214i
$274$		0			0
$275$	16.5000	+	28.5788i	0.994987	+	1.72337i
$276$		0			0
$277$	−4.00000	+	6.92820i	−0.240337	+	0.416275i	−0.960810	−	0.277207i	$-0.910591\pi$
$277$	−4.00000	+	6.92820i	−0.240337	+	0.416275i	0.720473	+	0.693482i	$0.243925\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−19.0000			−1.13344			−0.566722	−	0.823909i	$-0.691789\pi$
$281$	−19.0000			−1.13344			−0.566722	+	0.823909i	$0.691789\pi$
$282$		0			0
$283$	−7.50000	+	12.9904i	−0.445829	+	0.772198i	−0.998110	−	0.0614601i	$-0.980424\pi$
$283$	−7.50000	+	12.9904i	−0.445829	+	0.772198i	0.552281	+	0.833658i	$0.313758\pi$
$284$		0			0
$285$	12.0000	+	20.7846i	0.710819	+	1.23117i
$286$		0			0
$287$	16.0000	−	13.8564i	0.944450	−	0.817918i
$288$		0			0
$289$	−0.500000	−	0.866025i	−0.0294118	−	0.0509427i
$290$		0			0
$291$	−9.00000	+	15.5885i	−0.527589	+	0.913812i
$292$		0			0
$293$	−15.0000			−0.876309			−0.438155	−	0.898900i	$-0.644368\pi$
$293$	−15.0000			−0.876309			−0.438155	+	0.898900i	$0.644368\pi$
$294$		0			0
$295$	−8.00000			−0.465778
$296$		0			0
$297$	−7.50000	+	12.9904i	−0.435194	+	0.753778i
$298$		0			0
$299$	−10.0000	−	17.3205i	−0.578315	−	1.00167i
$300$		0			0
$301$	−20.0000	+	17.3205i	−1.15278	+	0.998337i
$302$		0			0
$303$	3.00000	+	5.19615i	0.172345	+	0.298511i
$304$		0			0
$305$	−16.0000	+	27.7128i	−0.916157	+	1.58683i
$306$		0			0
$307$	28.0000			1.59804			0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000			1.59804			0.799022	+	0.601302i	$0.205351\pi$
$308$		0			0
$309$	4.00000			0.227552
$310$		0			0
$311$	6.50000	−	11.2583i	0.368581	−	0.638401i	−0.620763	−	0.783998i	$-0.713177\pi$
$311$	6.50000	−	11.2583i	0.368581	−	0.638401i	0.989344	+	0.145597i	$0.0465103\pi$
$312$		0			0
$313$	−15.0000	−	25.9808i	−0.847850	−	1.46852i	−0.883123	−	0.469142i	$-0.844563\pi$
$313$	−15.0000	−	25.9808i	−0.847850	−	1.46852i	0.0352727	−	0.999378i	$-0.488770\pi$
$314$		0			0
$315$	−4.00000	−	20.7846i	−0.225374	−	1.17108i
$316$		0			0
$317$	1.00000	+	1.73205i	0.0561656	+	0.0972817i	0.892741	−	0.450570i	$-0.148779\pi$
$317$	1.00000	+	1.73205i	0.0561656	+	0.0972817i	−0.836576	+	0.547852i	$0.815446\pi$
$318$		0			0
$319$	12.0000	−	20.7846i	0.671871	−	1.16371i
$320$		0			0
$321$	1.00000			0.0558146
$322$		0			0
$323$	−6.00000			−0.333849
$324$		0			0
$325$	27.5000	−	47.6314i	1.52543	−	2.64211i
$326$		0			0
$327$	4.00000	+	6.92820i	0.221201	+	0.383131i
$328$		0			0
$329$	−5.00000	−	1.73205i	−0.275659	−	0.0954911i
$330$		0			0
$331$	3.00000	+	5.19615i	0.164895	+	0.285606i	0.936618	−	0.350352i	$-0.113938\pi$
$331$	3.00000	+	5.19615i	0.164895	+	0.285606i	−0.771723	+	0.635959i	$0.780605\pi$
$332$		0			0
$333$	−8.00000	+	13.8564i	−0.438397	+	0.759326i
$334$		0			0
$335$	−56.0000			−3.05961
$336$		0			0
$337$	8.00000			0.435788			0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000			0.435788			0.217894	+	0.975972i	$0.430081\pi$
$338$		0			0
$339$	4.00000	−	6.92820i	0.217250	−	0.376288i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	10.0000	+	15.5885i	0.539949	+	0.841698i
$344$		0			0
$345$	−8.00000	−	13.8564i	−0.430706	−	0.746004i
$346$		0			0
$347$	6.00000	−	10.3923i	0.322097	−	0.557888i	−0.658824	−	0.752297i	$-0.728946\pi$
$347$	6.00000	−	10.3923i	0.322097	−	0.557888i	0.980921	+	0.194409i	$0.0622790\pi$
$348$		0			0
$349$	−2.00000			−0.107058			−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000			−0.107058			−0.0535288	+	0.998566i	$0.517047\pi$
$350$		0			0
$351$	25.0000			1.33440
$352$		0			0
$353$	10.5000	−	18.1865i	0.558859	−	0.967972i	−0.438733	−	0.898617i	$-0.644573\pi$
$353$	10.5000	−	18.1865i	0.558859	−	0.967972i	0.997592	−	0.0693543i	$-0.0220939\pi$
$354$		0			0
$355$	14.0000	+	24.2487i	0.743043	+	1.28699i
$356$		0			0
$357$	−2.50000	−	0.866025i	−0.132314	−	0.0458349i
$358$		0			0
$359$	−11.0000	−	19.0526i	−0.580558	−	1.00556i	−0.995413	−	0.0956683i	$-0.969501\pi$
$359$	−11.0000	−	19.0526i	−0.580558	−	1.00556i	0.414855	−	0.909887i	$-0.363832\pi$
$360$		0			0
$361$	−8.50000	+	14.7224i	−0.447368	+	0.774865i
$362$		0			0
$363$	−2.00000			−0.104973
$364$		0			0
$365$	32.0000			1.67496
$366$		0			0
$367$	4.50000	−	7.79423i	0.234898	−	0.406855i	−0.724345	−	0.689438i	$-0.757858\pi$
$367$	4.50000	−	7.79423i	0.234898	−	0.406855i	0.959243	+	0.282582i	$0.0911910\pi$
$368$		0			0
$369$	−8.00000	−	13.8564i	−0.416463	−	0.721336i
$370$		0			0
$371$	−1.50000	−	7.79423i	−0.0778761	−	0.404656i
$372$		0			0
$373$	−15.5000	−	26.8468i	−0.802560	−	1.39007i	−0.917926	−	0.396751i	$-0.870138\pi$
$373$	−15.5000	−	26.8468i	−0.802560	−	1.39007i	0.115367	−	0.993323i	$-0.463196\pi$
$374$		0			0
$375$	12.0000	−	20.7846i	0.619677	−	1.07331i
$376$		0			0
$377$	−40.0000			−2.06010
$378$		0			0
$379$	1.00000			0.0513665			0.0256833	−	0.999670i	$-0.491824\pi$
$379$	1.00000			0.0513665			0.0256833	+	0.999670i	$0.491824\pi$
$380$		0			0
$381$	−4.00000	+	6.92820i	−0.204926	+	0.354943i
$382$		0			0
$383$	−1.00000	−	1.73205i	−0.0510976	−	0.0885037i	0.839345	−	0.543599i	$-0.182939\pi$
$383$	−1.00000	−	1.73205i	−0.0510976	−	0.0885037i	−0.890443	+	0.455095i	$0.849605\pi$
$384$		0			0
$385$	−24.0000	+	20.7846i	−1.22315	+	1.05928i
$386$		0			0
$387$	10.0000	+	17.3205i	0.508329	+	0.880451i
$388$		0			0
$389$	13.5000	−	23.3827i	0.684477	−	1.18555i	−0.289124	−	0.957292i	$-0.593364\pi$
$389$	13.5000	−	23.3827i	0.684477	−	1.18555i	0.973601	−	0.228257i	$-0.0733028\pi$
$390$		0			0
$391$	4.00000			0.202289
$392$		0			0
$393$	−4.00000			−0.201773
$394$		0			0
$395$	30.0000	−	51.9615i	1.50946	−	2.61447i
$396$		0			0
$397$	−13.0000	−	22.5167i	−0.652451	−	1.13008i	−0.982526	−	0.186124i	$-0.940407\pi$
$397$	−13.0000	−	22.5167i	−0.652451	−	1.13008i	0.330075	−	0.943955i	$-0.392926\pi$
$398$		0			0
$399$	−12.0000	+	10.3923i	−0.600751	+	0.520266i
$400$		0			0
$401$	9.00000	+	15.5885i	0.449439	+	0.778450i	0.998350	−	0.0574304i	$-0.0182907\pi$
$401$	9.00000	+	15.5885i	0.449439	+	0.778450i	−0.548911	+	0.835881i	$0.684957\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−4.00000			−0.198762
$406$		0			0
$407$	24.0000			1.18964
$408$		0			0
$409$	−5.50000	+	9.52628i	−0.271957	+	0.471044i	−0.969363	−	0.245633i	$-0.921004\pi$
$409$	−5.50000	+	9.52628i	−0.271957	+	0.471044i	0.697406	+	0.716677i	$0.254338\pi$
$410$		0			0
$411$	3.50000	+	6.06218i	0.172642	+	0.299025i
$412$		0			0
$413$	−1.00000	−	5.19615i	−0.0492068	−	0.255686i
$414$		0			0
$415$	24.0000	+	41.5692i	1.17811	+	2.04055i
$416$		0			0
$417$	0.500000	−	0.866025i	0.0244851	−	0.0424094i
$418$		0			0
$419$	37.0000			1.80757			0.903784	−	0.427989i	$-0.140778\pi$
$419$	37.0000			1.80757			0.903784	+	0.427989i	$0.140778\pi$
$420$		0			0
$421$	6.00000			0.292422			0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000			0.292422			0.146211	+	0.989253i	$0.453292\pi$
$422$		0			0
$423$	−2.00000	+	3.46410i	−0.0972433	+	0.168430i
$424$		0			0
$425$	5.50000	+	9.52628i	0.266789	+	0.462092i
$426$		0			0
$427$	−20.0000	−	6.92820i	−0.967868	−	0.335279i
$428$		0			0
$429$	−7.50000	−	12.9904i	−0.362103	−	0.627182i
$430$		0			0
$431$	7.50000	−	12.9904i	0.361262	−	0.625725i	−0.626907	−	0.779094i	$-0.715679\pi$
$431$	7.50000	−	12.9904i	0.361262	−	0.625725i	0.988169	+	0.153370i	$0.0490126\pi$
$432$		0			0
$433$	−10.0000			−0.480569			−0.240285	−	0.970702i	$-0.577241\pi$
$433$	−10.0000			−0.480569			−0.240285	+	0.970702i	$0.577241\pi$
$434$		0			0
$435$	−32.0000			−1.53428
$436$		0			0
$437$	12.0000	−	20.7846i	0.574038	−	0.994263i
$438$		0			0
$439$	0.500000	+	0.866025i	0.0238637	+	0.0413331i	0.877711	−	0.479191i	$-0.159070\pi$
$439$	0.500000	+	0.866025i	0.0238637	+	0.0413331i	−0.853847	+	0.520524i	$0.825737\pi$
$440$		0			0
$441$	13.0000	−	5.19615i	0.619048	−	0.247436i
$442$		0			0
$443$	−13.0000	−	22.5167i	−0.617649	−	1.06980i	−0.989914	−	0.141672i	$-0.954752\pi$
$443$	−13.0000	−	22.5167i	−0.617649	−	1.06980i	0.372265	−	0.928126i	$-0.378581\pi$
$444$		0			0
$445$	2.00000	−	3.46410i	0.0948091	−	0.164214i
$446$		0			0
$447$	−11.0000			−0.520282
$448$		0			0
$449$	−18.0000			−0.849473			−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000			−0.849473			−0.424736	+	0.905317i	$0.639633\pi$
$450$		0			0
$451$	−12.0000	+	20.7846i	−0.565058	+	0.978709i
$452$		0			0
$453$	−5.00000	−	8.66025i	−0.234920	−	0.406894i
$454$		0			0
$455$	50.0000	+	17.3205i	2.34404	+	0.811998i
$456$		0			0
$457$	13.0000	+	22.5167i	0.608114	+	1.05328i	0.991551	+	0.129718i	$0.0414071\pi$
$457$	13.0000	+	22.5167i	0.608114	+	1.05328i	−0.383437	+	0.923567i	$0.625260\pi$
$458$		0			0
$459$	−2.50000	+	4.33013i	−0.116690	+	0.202113i
$460$		0			0
$461$	15.0000			0.698620			0.349310	−	0.937007i	$-0.386416\pi$
$461$	15.0000			0.698620			0.349310	+	0.937007i	$0.386416\pi$
$462$		0			0
$463$	−16.0000			−0.743583			−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000			−0.743583			−0.371792	+	0.928316i	$0.621256\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	14.0000	+	24.2487i	0.647843	+	1.12210i	0.983637	+	0.180161i	$0.0576619\pi$
$467$	14.0000	+	24.2487i	0.647843	+	1.12210i	−0.335794	+	0.941935i	$0.609005\pi$
$468$		0			0
$469$	−7.00000	−	36.3731i	−0.323230	−	1.67955i
$470$		0			0
$471$	−0.500000	−	0.866025i	−0.0230388	−	0.0399043i
$472$		0			0
$473$	15.0000	−	25.9808i	0.689701	−	1.19460i
$474$		0			0
$475$	66.0000			3.02829
$476$		0			0
$477$	−6.00000			−0.274721
$478$		0			0
$479$	−16.0000	+	27.7128i	−0.731059	+	1.26623i	0.225372	+	0.974273i	$0.427640\pi$
$479$	−16.0000	+	27.7128i	−0.731059	+	1.26623i	−0.956431	+	0.291958i	$0.905693\pi$
$480$		0			0
$481$	−20.0000	−	34.6410i	−0.911922	−	1.57949i
$482$		0			0
$483$	8.00000	−	6.92820i	0.364013	−	0.315244i
$484$		0			0
$485$	36.0000	+	62.3538i	1.63468	+	2.83134i
$486$		0			0
$487$	2.50000	−	4.33013i	0.113286	−	0.196217i	−0.803807	−	0.594890i	$-0.797196\pi$
$487$	2.50000	−	4.33013i	0.113286	−	0.196217i	0.917093	+	0.398673i	$0.130529\pi$
$488$		0			0
$489$	4.00000			0.180886
$490$		0			0
$491$	−18.0000			−0.812329			−0.406164	−	0.913800i	$-0.633134\pi$
$491$	−18.0000			−0.812329			−0.406164	+	0.913800i	$0.633134\pi$
$492$		0			0
$493$	4.00000	−	6.92820i	0.180151	−	0.312031i
$494$		0			0
$495$	12.0000	+	20.7846i	0.539360	+	0.934199i
$496$		0			0
$497$	−14.0000	+	12.1244i	−0.627986	+	0.543852i
$498$		0			0
$499$	−19.5000	−	33.7750i	−0.872940	−	1.51198i	−0.858941	−	0.512074i	$-0.828877\pi$
$499$	−19.5000	−	33.7750i	−0.872940	−	1.51198i	−0.0139987	−	0.999902i	$-0.504456\pi$
$500$		0			0
$501$	9.50000	−	16.4545i	0.424429	−	0.735132i
$502$		0			0
$503$	21.0000			0.936344			0.468172	−	0.883637i	$-0.344913\pi$
$503$	21.0000			0.936344			0.468172	+	0.883637i	$0.344913\pi$
$504$		0			0
$505$	24.0000			1.06799
$506$		0			0
$507$	−6.00000	+	10.3923i	−0.266469	+	0.461538i
$508$		0			0
$509$	−19.0000	−	32.9090i	−0.842160	−	1.45866i	−0.888065	−	0.459718i	$-0.847950\pi$
$509$	−19.0000	−	32.9090i	−0.842160	−	1.45866i	0.0459045	−	0.998946i	$-0.485383\pi$
$510$		0			0
$511$	4.00000	+	20.7846i	0.176950	+	0.919457i
$512$		0			0
$513$	15.0000	+	25.9808i	0.662266	+	1.14708i
$514$		0			0
$515$	8.00000	−	13.8564i	0.352522	−	0.610586i
$516$		0			0
$517$	6.00000			0.263880
$518$		0			0
$519$	−2.00000			−0.0877903
$520$		0			0
$521$	12.0000	−	20.7846i	0.525730	−	0.910590i	−0.473821	−	0.880621i	$-0.657126\pi$
$521$	12.0000	−	20.7846i	0.525730	−	0.910590i	0.999551	−	0.0299693i	$-0.00954094\pi$
$522$		0			0
$523$	−3.00000	−	5.19615i	−0.131181	−	0.227212i	0.792951	−	0.609285i	$-0.208544\pi$
$523$	−3.00000	−	5.19615i	−0.131181	−	0.227212i	−0.924132	+	0.382073i	$0.875210\pi$
$524$		0			0
$525$	27.5000	+	9.52628i	1.20020	+	0.415761i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	3.50000	−	6.06218i	0.152174	−	0.263573i
$530$		0			0
$531$	−4.00000			−0.173585
$532$		0			0
$533$	40.0000			1.73259
$534$		0			0
$535$	2.00000	−	3.46410i	0.0864675	−	0.149766i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−16.5000	−	12.9904i	−0.710705	−	0.559535i
$540$		0			0
$541$	−7.00000	−	12.1244i	−0.300954	−	0.521267i	0.675399	−	0.737453i	$-0.263972\pi$
$541$	−7.00000	−	12.1244i	−0.300954	−	0.521267i	−0.976352	+	0.216186i	$0.930638\pi$
$542$		0			0
$543$	−9.00000	+	15.5885i	−0.386227	+	0.668965i
$544$		0			0
$545$	32.0000			1.37073
$546$		0			0
$547$	−13.0000			−0.555840			−0.277920	−	0.960604i	$-0.589645\pi$
$547$	−13.0000			−0.555840			−0.277920	+	0.960604i	$0.589645\pi$
$548$		0			0
$549$	−8.00000	+	13.8564i	−0.341432	+	0.591377i
$550$		0			0
$551$	−24.0000	−	41.5692i	−1.02243	−	1.77091i
$552$		0			0
$553$	37.5000	+	12.9904i	1.59466	+	0.552407i
$554$		0			0
$555$	−16.0000	−	27.7128i	−0.679162	−	1.17634i
$556$		0			0
$557$	−11.5000	+	19.9186i	−0.487271	+	0.843978i	−0.999893	−	0.0146368i	$-0.995341\pi$
$557$	−11.5000	+	19.9186i	−0.487271	+	0.843978i	0.512622	+	0.858614i	$0.328674\pi$
$558$		0			0
$559$	−50.0000			−2.11477
$560$		0			0
$561$	3.00000			0.126660
$562$		0			0
$563$	−15.0000	+	25.9808i	−0.632175	+	1.09496i	0.354932	+	0.934892i	$0.384504\pi$
$563$	−15.0000	+	25.9808i	−0.632175	+	1.09496i	−0.987106	+	0.160066i	$0.948829\pi$
$564$		0			0
$565$	−16.0000	−	27.7128i	−0.673125	−	1.16589i
$566$		0			0
$567$	−0.500000	−	2.59808i	−0.0209980	−	0.109109i
$568$		0			0
$569$	−16.5000	−	28.5788i	−0.691716	−	1.19809i	−0.971275	−	0.237959i	$-0.923522\pi$
$569$	−16.5000	−	28.5788i	−0.691716	−	1.19809i	0.279559	−	0.960128i	$-0.409812\pi$
$570$		0			0
$571$	−18.0000	+	31.1769i	−0.753277	+	1.30471i	0.192950	+	0.981209i	$0.438194\pi$
$571$	−18.0000	+	31.1769i	−0.753277	+	1.30471i	−0.946227	+	0.323505i	$0.895139\pi$
$572$		0			0
$573$	12.0000			0.501307
$574$		0			0
$575$	−44.0000			−1.83493
$576$		0			0
$577$	−6.50000	+	11.2583i	−0.270599	+	0.468690i	−0.969015	−	0.247001i	$-0.920555\pi$
$577$	−6.50000	+	11.2583i	−0.270599	+	0.468690i	0.698417	+	0.715691i	$0.253888\pi$
$578$		0			0
$579$	5.00000	+	8.66025i	0.207793	+	0.359908i
$580$		0			0
$581$	−24.0000	+	20.7846i	−0.995688	+	0.862291i
$582$		0			0
$583$	4.50000	+	7.79423i	0.186371	+	0.322804i
$584$		0			0
$585$	20.0000	−	34.6410i	0.826898	−	1.43223i
$586$		0			0
$587$	−2.00000			−0.0825488			−0.0412744	−	0.999148i	$-0.513142\pi$
$587$	−2.00000			−0.0825488			−0.0412744	+	0.999148i	$0.513142\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	−9.00000	+	15.5885i	−0.370211	+	0.641223i
$592$		0			0
$593$	−10.5000	−	18.1865i	−0.431183	−	0.746831i	0.565792	−	0.824548i	$-0.308570\pi$
$593$	−10.5000	−	18.1865i	−0.431183	−	0.746831i	−0.996976	+	0.0777165i	$0.975237\pi$
$594$		0			0
$595$	−8.00000	+	6.92820i	−0.327968	+	0.284029i
$596$		0			0
$597$	3.50000	+	6.06218i	0.143245	+	0.248108i
$598$		0			0
$599$	17.0000	−	29.4449i	0.694601	−	1.20308i	−0.275714	−	0.961240i	$-0.588914\pi$
$599$	17.0000	−	29.4449i	0.694601	−	1.20308i	0.970315	−	0.241845i	$-0.0777525\pi$
$600$		0			0
$601$	−14.0000			−0.571072			−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000			−0.571072			−0.285536	+	0.958368i	$0.592172\pi$
$602$		0			0
$603$	−28.0000			−1.14025
$604$		0			0
$605$	−4.00000	+	6.92820i	−0.162623	+	0.281672i
$606$		0			0
$607$	11.5000	+	19.9186i	0.466771	+	0.808470i	0.999279	−	0.0379540i	$-0.0120840\pi$
$607$	11.5000	+	19.9186i	0.466771	+	0.808470i	−0.532509	+	0.846424i	$0.678751\pi$
$608$		0			0
$609$	−4.00000	−	20.7846i	−0.162088	−	0.842235i
$610$		0			0
$611$	−5.00000	−	8.66025i	−0.202278	−	0.350356i
$612$		0			0
$613$	−8.50000	+	14.7224i	−0.343312	+	0.594633i	−0.985046	−	0.172294i	$-0.944882\pi$
$613$	−8.50000	+	14.7224i	−0.343312	+	0.594633i	0.641734	+	0.766927i	$0.278215\pi$
$614$		0			0
$615$	32.0000			1.29036
$616$		0			0
$617$	−36.0000			−1.44931			−0.724653	−	0.689114i	$-0.758000\pi$
$617$	−36.0000			−1.44931			−0.724653	+	0.689114i	$0.758000\pi$
$618$		0			0
$619$	8.50000	−	14.7224i	0.341644	−	0.591744i	−0.643094	−	0.765787i	$-0.722350\pi$
$619$	8.50000	−	14.7224i	0.341644	−	0.591744i	0.984738	+	0.174042i	$0.0556830\pi$
$620$		0			0
$621$	−10.0000	−	17.3205i	−0.401286	−	0.695048i
$622$		0			0
$623$	2.50000	+	0.866025i	0.100160	+	0.0346966i
$624$		0			0
$625$	−20.5000	−	35.5070i	−0.820000	−	1.42028i
$626$		0			0
$627$	9.00000	−	15.5885i	0.359425	−	0.622543i
$628$		0			0
$629$	8.00000			0.318981
$630$		0			0
$631$	−30.0000			−1.19428			−0.597141	−	0.802137i	$-0.703697\pi$
$631$	−30.0000			−1.19428			−0.597141	+	0.802137i	$0.703697\pi$
$632$		0			0
$633$	2.00000	−	3.46410i	0.0794929	−	0.137686i
$634$		0			0
$635$	16.0000	+	27.7128i	0.634941	+	1.09975i
$636$		0			0
$637$	−5.00000	+	34.6410i	−0.198107	+	1.37253i
$638$		0			0
$639$	7.00000	+	12.1244i	0.276916	+	0.479632i
$640$		0			0
$641$	−11.0000	+	19.0526i	−0.434474	+	0.752531i	−0.997253	−	0.0740768i	$-0.976399\pi$
$641$	−11.0000	+	19.0526i	−0.434474	+	0.752531i	0.562779	+	0.826608i	$0.309732\pi$
$642$		0			0
$643$	−1.00000			−0.0394362			−0.0197181	−	0.999806i	$-0.506277\pi$
$643$	−1.00000			−0.0394362			−0.0197181	+	0.999806i	$0.506277\pi$
$644$		0			0
$645$	−40.0000			−1.57500
$646$		0			0
$647$	−14.0000	+	24.2487i	−0.550397	+	0.953315i	0.447849	+	0.894109i	$0.352190\pi$
$647$	−14.0000	+	24.2487i	−0.550397	+	0.953315i	−0.998246	+	0.0592060i	$0.981143\pi$
$648$		0			0
$649$	3.00000	+	5.19615i	0.117760	+	0.203967i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	5.00000	+	8.66025i	0.195665	+	0.338902i	0.947118	−	0.320884i	$-0.103980\pi$
$653$	5.00000	+	8.66025i	0.195665	+	0.338902i	−0.751453	+	0.659786i	$0.770647\pi$
$654$		0			0
$655$	−8.00000	+	13.8564i	−0.312586	+	0.541415i
$656$		0			0
$657$	16.0000			0.624219
$658$		0			0
$659$	−30.0000			−1.16863			−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000			−1.16863			−0.584317	+	0.811525i	$0.698638\pi$
$660$		0			0
$661$	−9.00000	+	15.5885i	−0.350059	+	0.606321i	−0.986260	−	0.165203i	$-0.947172\pi$
$661$	−9.00000	+	15.5885i	−0.350059	+	0.606321i	0.636200	+	0.771524i	$0.280505\pi$
$662$		0			0
$663$	−2.50000	−	4.33013i	−0.0970920	−	0.168168i
$664$		0			0
$665$	12.0000	+	62.3538i	0.465340	+	2.41798i
$666$		0			0
$667$	16.0000	+	27.7128i	0.619522	+	1.07304i
$668$		0			0
$669$	3.00000	−	5.19615i	0.115987	−	0.200895i
$670$		0			0
$671$	24.0000			0.926510
$672$		0			0
$673$	28.0000			1.07932			0.539660	−	0.841883i	$-0.318553\pi$
$673$	28.0000			1.07932			0.539660	+	0.841883i	$0.318553\pi$
$674$		0			0
$675$	27.5000	−	47.6314i	1.05848	−	1.83333i
$676$		0			0
$677$	4.00000	+	6.92820i	0.153732	+	0.266272i	0.932597	−	0.360920i	$-0.117537\pi$
$677$	4.00000	+	6.92820i	0.153732	+	0.266272i	−0.778864	+	0.627192i	$0.784204\pi$
$678$		0			0
$679$	−36.0000	+	31.1769i	−1.38155	+	1.19646i
$680$		0			0
$681$	−12.5000	−	21.6506i	−0.479001	−	0.829654i
$682$		0			0
$683$	10.5000	−	18.1865i	0.401771	−	0.695888i	−0.592168	−	0.805814i	$-0.701728\pi$
$683$	10.5000	−	18.1865i	0.401771	−	0.695888i	0.993940	+	0.109926i	$0.0350613\pi$
$684$		0			0
$685$	28.0000			1.06983
$686$		0			0
$687$	−10.0000			−0.381524
$688$		0			0
$689$	7.50000	−	12.9904i	0.285727	−	0.494894i
$690$		0			0
$691$	−4.00000	−	6.92820i	−0.152167	−	0.263561i	0.779857	−	0.625958i	$-0.215292\pi$
$691$	−4.00000	−	6.92820i	−0.152167	−	0.263561i	−0.932024	+	0.362397i	$0.881959\pi$
$692$		0			0
$693$	−12.0000	+	10.3923i	−0.455842	+	0.394771i
$694$		0			0
$695$	−2.00000	−	3.46410i	−0.0758643	−	0.131401i
$696$		0			0
$697$	−4.00000	+	6.92820i	−0.151511	+	0.262424i
$698$		0			0
$699$	28.0000			1.05906
$700$		0			0
$701$	31.0000			1.17085			0.585427	−	0.810725i	$-0.300927\pi$
$701$	31.0000			1.17085			0.585427	+	0.810725i	$0.300927\pi$
$702$		0			0
$703$	24.0000	−	41.5692i	0.905177	−	1.56781i
$704$		0			0
$705$	−4.00000	−	6.92820i	−0.150649	−	0.260931i
$706$		0			0
$707$	3.00000	+	15.5885i	0.112827	+	0.586264i
$708$		0			0
$709$	25.0000	+	43.3013i	0.938895	+	1.62621i	0.767537	+	0.641004i	$0.221482\pi$
$709$	25.0000	+	43.3013i	0.938895	+	1.62621i	0.171358	+	0.985209i	$0.445185\pi$
$710$		0			0
$711$	15.0000	−	25.9808i	0.562544	−	0.974355i
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−60.0000			−2.24387
$716$		0			0
$717$	−1.00000	+	1.73205i	−0.0373457	+	0.0646846i
$718$		0			0
$719$	17.5000	+	30.3109i	0.652640	+	1.13041i	0.982480	+	0.186369i	$0.0596719\pi$
$719$	17.5000	+	30.3109i	0.652640	+	1.13041i	−0.329840	+	0.944037i	$0.606995\pi$
$720$		0			0
$721$	10.0000	+	3.46410i	0.372419	+	0.129010i
$722$		0			0
$723$	4.00000	+	6.92820i	0.148762	+	0.257663i
$724$		0			0
$725$	−44.0000	+	76.2102i	−1.63412	+	2.83038i
$726$		0			0
$727$	18.0000			0.667583			0.333792	−	0.942647i	$-0.391672\pi$
$727$	18.0000			0.667583			0.333792	+	0.942647i	$0.391672\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		0			0
$731$	5.00000	−	8.66025i	0.184932	−	0.320311i
$732$		0			0
$733$	−1.50000	−	2.59808i	−0.0554038	−	0.0959621i	0.836993	−	0.547213i	$-0.184311\pi$
$733$	−1.50000	−	2.59808i	−0.0554038	−	0.0959621i	−0.892397	+	0.451251i	$0.850978\pi$
$734$		0			0
$735$	−4.00000	+	27.7128i	−0.147542	+	1.02220i
$736$		0			0
$737$	21.0000	+	36.3731i	0.773545	+	1.33982i
$738$		0			0
$739$	−15.0000	+	25.9808i	−0.551784	+	0.955718i	0.446362	+	0.894852i	$0.352719\pi$
$739$	−15.0000	+	25.9808i	−0.551784	+	0.955718i	−0.998146	+	0.0608653i	$0.980614\pi$
$740$		0			0
$741$	−30.0000			−1.10208
$742$		0			0
$743$	1.00000			0.0366864			0.0183432	−	0.999832i	$-0.494161\pi$
$743$	1.00000			0.0366864			0.0183432	+	0.999832i	$0.494161\pi$
$744$		0			0
$745$	−22.0000	+	38.1051i	−0.806018	+	1.39606i
$746$		0			0
$747$	12.0000	+	20.7846i	0.439057	+	0.760469i
$748$		0			0
$749$	2.50000	+	0.866025i	0.0913480	+	0.0316439i
$750$		0			0
$751$	−1.50000	−	2.59808i	−0.0547358	−	0.0948051i	0.837359	−	0.546653i	$-0.184098\pi$
$751$	−1.50000	−	2.59808i	−0.0547358	−	0.0948051i	−0.892095	+	0.451848i	$0.850765\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−40.0000			−1.45575
$756$		0			0
$757$	9.00000			0.327111			0.163555	−	0.986534i	$-0.447704\pi$
$757$	9.00000			0.327111			0.163555	+	0.986534i	$0.447704\pi$
$758$		0			0
$759$	−6.00000	+	10.3923i	−0.217786	+	0.377217i
$760$		0			0
$761$	−22.5000	−	38.9711i	−0.815624	−	1.41270i	−0.908879	−	0.417061i	$-0.863060\pi$
$761$	−22.5000	−	38.9711i	−0.815624	−	1.41270i	0.0932544	−	0.995642i	$-0.470273\pi$
$762$		0			0
$763$	4.00000	+	20.7846i	0.144810	+	0.752453i
$764$		0			0
$765$	4.00000	+	6.92820i	0.144620	+	0.250490i
$766$		0			0
$767$	5.00000	−	8.66025i	0.180540	−	0.312704i
$768$		0			0
$769$	1.00000			0.0360609			0.0180305	−	0.999837i	$-0.494260\pi$
$769$	1.00000			0.0360609			0.0180305	+	0.999837i	$0.494260\pi$
$770$		0			0
$771$	3.00000			0.108042
$772$		0			0
$773$	−4.50000	+	7.79423i	−0.161854	+	0.280339i	−0.935534	−	0.353238i	$-0.885081\pi$
$773$	−4.50000	+	7.79423i	−0.161854	+	0.280339i	0.773680	+	0.633577i	$0.218414\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	16.0000	−	13.8564i	0.573997	−	0.497096i
$778$		0			0
$779$	24.0000	+	41.5692i	0.859889	+	1.48937i
$780$		0			0
$781$	10.5000	−	18.1865i	0.375720	−	0.650765i
$782$		0			0
$783$	−40.0000			−1.42948
$784$		0			0
$785$	−4.00000			−0.142766
$786$		0			0
$787$	−2.00000	+	3.46410i	−0.0712923	+	0.123482i	−0.899468	−	0.436987i	$-0.856046\pi$
$787$	−2.00000	+	3.46410i	−0.0712923	+	0.123482i	0.828176	+	0.560469i	$0.189379\pi$
$788$		0			0
$789$	−2.00000	−	3.46410i	−0.0712019	−	0.123325i

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:	\( N \)	\(=\)	\( 476 = 2^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	476.i (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{3} +(2.00000 + 3.46410i) q^{5} +(-2.00000 + 1.73205i) q^{7} +(1.00000 + 1.73205i) q^{9} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{3} +(2.00000 + 3.46410i) q^{5} +(-2.00000 + 1.73205i) q^{7} +(1.00000 + 1.73205i) q^{9} +(1.50000 - 2.59808i) q^{11} -5.00000 q^{13} -4.00000 q^{15} +(0.500000 - 0.866025i) q^{17} +(-3.00000 - 5.19615i) q^{19} +(-0.500000 - 2.59808i) q^{21} +(2.00000 + 3.46410i) q^{23} +(-5.50000 + 9.52628i) q^{25} -5.00000 q^{27} +8.00000 q^{29} +(1.50000 + 2.59808i) q^{33} +(-10.0000 - 3.46410i) q^{35} +(4.00000 + 6.92820i) q^{37} +(2.50000 - 4.33013i) q^{39} -8.00000 q^{41} +10.0000 q^{43} +(-4.00000 + 6.92820i) q^{45} +(1.00000 + 1.73205i) q^{47} +(1.00000 - 6.92820i) q^{49} +(0.500000 + 0.866025i) q^{51} +(-1.50000 + 2.59808i) q^{53} +12.0000 q^{55} +6.00000 q^{57} +(-1.00000 + 1.73205i) q^{59} +(4.00000 + 6.92820i) q^{61} +(-5.00000 - 1.73205i) q^{63} +(-10.0000 - 17.3205i) q^{65} +(-7.00000 + 12.1244i) q^{67} -4.00000 q^{69} +7.00000 q^{71} +(4.00000 - 6.92820i) q^{73} +(-5.50000 - 9.52628i) q^{75} +(1.50000 + 7.79423i) q^{77} +(-7.50000 - 12.9904i) q^{79} +(-0.500000 + 0.866025i) q^{81} +12.0000 q^{83} +4.00000 q^{85} +(-4.00000 + 6.92820i) q^{87} +(-0.500000 - 0.866025i) q^{89} +(10.0000 - 8.66025i) q^{91} +(12.0000 - 20.7846i) q^{95} +18.0000 q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + 4 q^{5} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - q^3 + 4 * q^5 - 4 * q^7 + 2 * q^9	\( 2 q - q^{3} + 4 q^{5} - 4 q^{7} + 2 q^{9} + 3 q^{11} - 10 q^{13} - 8 q^{15} + q^{17} - 6 q^{19} - q^{21} + 4 q^{23} - 11 q^{25} - 10 q^{27} + 16 q^{29} + 3 q^{33} - 20 q^{35} + 8 q^{37} + 5 q^{39} - 16 q^{41} + 20 q^{43} - 8 q^{45} + 2 q^{47} + 2 q^{49} + q^{51} - 3 q^{53} + 24 q^{55} + 12 q^{57} - 2 q^{59} + 8 q^{61} - 10 q^{63} - 20 q^{65} - 14 q^{67} - 8 q^{69} + 14 q^{71} + 8 q^{73} - 11 q^{75} + 3 q^{77} - 15 q^{79} - q^{81} + 24 q^{83} + 8 q^{85} - 8 q^{87} - q^{89} + 20 q^{91} + 24 q^{95} + 36 q^{97} + 12 q^{99}+O(q^{100}) \) 2 * q - q^3 + 4 * q^5 - 4 * q^7 + 2 * q^9 + 3 * q^11 - 10 * q^13 - 8 * q^15 + q^17 - 6 * q^19 - q^21 + 4 * q^23 - 11 * q^25 - 10 * q^27 + 16 * q^29 + 3 * q^33 - 20 * q^35 + 8 * q^37 + 5 * q^39 - 16 * q^41 + 20 * q^43 - 8 * q^45 + 2 * q^47 + 2 * q^49 + q^51 - 3 * q^53 + 24 * q^55 + 12 * q^57 - 2 * q^59 + 8 * q^61 - 10 * q^63 - 20 * q^65 - 14 * q^67 - 8 * q^69 + 14 * q^71 + 8 * q^73 - 11 * q^75 + 3 * q^77 - 15 * q^79 - q^81 + 24 * q^83 + 8 * q^85 - 8 * q^87 - q^89 + 20 * q^91 + 24 * q^95 + 36 * q^97 + 12 * q^99

Introduction

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