LMFDB - Embedded newform 476.2.i.c.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:	$\Z[a_1, \ldots, a_{5}]$
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.c.205.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.50000 - 2.59808i) q^{3} +(-2.00000 - 3.46410i) q^{5} +(2.00000 + 1.73205i) q^{7} +(-3.00000 - 5.19615i) q^{9} +(-0.500000 + 0.866025i) q^{11} +3.00000 q^{13} -12.0000 q^{15} +(0.500000 - 0.866025i) q^{17} +(1.00000 + 1.73205i) q^{19} +(7.50000 - 2.59808i) q^{21} +(-2.00000 - 3.46410i) q^{23} +(-5.50000 + 9.52628i) q^{25} -9.00000 q^{27} +(-4.00000 + 6.92820i) q^{31} +(1.50000 + 2.59808i) q^{33} +(2.00000 - 10.3923i) q^{35} +(-4.00000 - 6.92820i) q^{37} +(4.50000 - 7.79423i) q^{39} +10.0000 q^{43} +(-12.0000 + 20.7846i) q^{45} +(5.00000 + 8.66025i) q^{47} +(1.00000 + 6.92820i) q^{49} +(-1.50000 - 2.59808i) q^{51} +(-1.50000 + 2.59808i) q^{53} +4.00000 q^{55} +6.00000 q^{57} +(7.00000 - 12.1244i) q^{59} +(4.00000 + 6.92820i) q^{61} +(3.00000 - 15.5885i) q^{63} +(-6.00000 - 10.3923i) q^{65} +(5.00000 - 8.66025i) q^{67} -12.0000 q^{69} -5.00000 q^{71} +(8.00000 - 13.8564i) q^{73} +(16.5000 + 28.5788i) q^{75} +(-2.50000 + 0.866025i) q^{77} +(-5.50000 - 9.52628i) q^{79} +(-4.50000 + 7.79423i) q^{81} +12.0000 q^{83} -4.00000 q^{85} +(-4.50000 - 7.79423i) q^{89} +(6.00000 + 5.19615i) q^{91} +(12.0000 + 20.7846i) q^{93} +(4.00000 - 6.92820i) q^{95} +10.0000 q^{97} +6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{3} - 4 q^{5} + 4 q^{7} - 6 q^{9} - q^{11} + 6 q^{13} - 24 q^{15} + q^{17} + 2 q^{19} + 15 q^{21} - 4 q^{23} - 11 q^{25} - 18 q^{27} - 8 q^{31} + 3 q^{33} + 4 q^{35} - 8 q^{37} + 9 q^{39} + 20 q^{43}+ \cdots + 12 q^{99}+O(q^{100}) $ 2 * q + 3 * q^3 - 4 * q^5 + 4 * q^7 - 6 * q^9 - q^11 + 6 * q^13 - 24 * q^15 + q^17 + 2 * q^19 + 15 * q^21 - 4 * q^23 - 11 * q^25 - 18 * q^27 - 8 * q^31 + 3 * q^33 + 4 * q^35 - 8 * q^37 + 9 * q^39 + 20 * q^43 - 24 * q^45 + 10 * q^47 + 2 * q^49 - 3 * q^51 - 3 * q^53 + 8 * q^55 + 12 * q^57 + 14 * q^59 + 8 * q^61 + 6 * q^63 - 12 * q^65 + 10 * q^67 - 24 * q^69 - 10 * q^71 + 16 * q^73 + 33 * q^75 - 5 * q^77 - 11 * q^79 - 9 * q^81 + 24 * q^83 - 8 * q^85 - 9 * q^89 + 12 * q^91 + 24 * q^93 + 8 * q^95 + 20 * q^97 + 12 * q^99

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
$3$	1.50000	−	2.59808i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$3$	1.50000	−	2.59808i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$4$		0			0
$5$	−2.00000	−	3.46410i	−0.894427	−	1.54919i	−0.834512	−	0.550990i	$-0.814250\pi$
$5$	−2.00000	−	3.46410i	−0.894427	−	1.54919i	−0.0599153	−	0.998203i	$-0.519083\pi$
$6$		0			0
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$8$		0			0
$9$	−3.00000	−	5.19615i	−1.00000	−	1.73205i
$10$		0			0
$11$	−0.500000	+	0.866025i	−0.150756	+	0.261116i	−0.931505	−	0.363727i	$-0.881504\pi$
$11$	−0.500000	+	0.866025i	−0.150756	+	0.261116i	0.780750	+	0.624844i	$0.214837\pi$
$12$		0			0
$13$	3.00000			0.832050			0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000			0.832050			0.416025	+	0.909353i	$0.363423\pi$
$14$		0			0
$15$	−12.0000			−3.09839
$16$		0			0
$17$	0.500000	−	0.866025i	0.121268	−	0.210042i
$17$	0.500000	−	0.866025i	0.121268	−	0.210042i
$18$		0			0
$19$	1.00000	+	1.73205i	0.229416	+	0.397360i	0.957635	−	0.287984i	$-0.0929851\pi$
$19$	1.00000	+	1.73205i	0.229416	+	0.397360i	−0.728219	+	0.685344i	$0.759652\pi$
$20$		0			0
$21$	7.50000	−	2.59808i	1.63663	−	0.566947i
$22$		0			0
$23$	−2.00000	−	3.46410i	−0.417029	−	0.722315i	0.578610	−	0.815604i	$-0.303595\pi$
$23$	−2.00000	−	3.46410i	−0.417029	−	0.722315i	−0.995639	+	0.0932891i	$0.970262\pi$
$24$		0			0
$25$	−5.50000	+	9.52628i	−1.10000	+	1.90526i
$26$		0			0
$27$	−9.00000			−1.73205
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−4.00000	+	6.92820i	−0.718421	+	1.24434i	0.243204	+	0.969975i	$0.421802\pi$
$31$	−4.00000	+	6.92820i	−0.718421	+	1.24434i	−0.961625	+	0.274367i	$0.911532\pi$
$32$		0			0
$33$	1.50000	+	2.59808i	0.261116	+	0.452267i
$34$		0			0
$35$	2.00000	−	10.3923i	0.338062	−	1.75662i
$36$		0			0
$37$	−4.00000	−	6.92820i	−0.657596	−	1.13899i	−0.981236	−	0.192809i	$-0.938240\pi$
$37$	−4.00000	−	6.92820i	−0.657596	−	1.13899i	0.323640	−	0.946180i	$-0.395093\pi$
$38$		0			0
$39$	4.50000	−	7.79423i	0.720577	−	1.24808i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		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
$45$	−12.0000	+	20.7846i	−1.78885	+	3.09839i
$46$		0			0
$47$	5.00000	+	8.66025i	0.729325	+	1.26323i	0.957169	+	0.289530i	$0.0934991\pi$
$47$	5.00000	+	8.66025i	0.729325	+	1.26323i	−0.227844	+	0.973698i	$0.573168\pi$
$48$		0			0
$49$	1.00000	+	6.92820i	0.142857	+	0.989743i
$50$		0			0
$51$	−1.50000	−	2.59808i	−0.210042	−	0.363803i
$52$		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
$55$	4.00000			0.539360
$56$		0			0
$57$	6.00000			0.794719
$58$		0			0
$59$	7.00000	−	12.1244i	0.911322	−	1.57846i	0.0991242	−	0.995075i	$-0.468396\pi$
$59$	7.00000	−	12.1244i	0.911322	−	1.57846i	0.812198	−	0.583382i	$-0.198271\pi$
$60$		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
$63$	3.00000	−	15.5885i	0.377964	−	1.96396i
$64$		0			0
$65$	−6.00000	−	10.3923i	−0.744208	−	1.28901i
$66$		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	$-0.0425044\pi$
$68$		0			0
$69$	−12.0000			−1.44463
$70$		0			0
$71$	−5.00000			−0.593391			−0.296695	−	0.954972i	$-0.595885\pi$
$71$	−5.00000			−0.593391			−0.296695	+	0.954972i	$0.595885\pi$
$72$		0			0
$73$	8.00000	−	13.8564i	0.936329	−	1.62177i	0.164083	−	0.986447i	$-0.447534\pi$
$73$	8.00000	−	13.8564i	0.936329	−	1.62177i	0.772246	−	0.635323i	$-0.219133\pi$
$74$		0			0
$75$	16.5000	+	28.5788i	1.90526	+	3.30000i
$76$		0			0
$77$	−2.50000	+	0.866025i	−0.284901	+	0.0986928i
$78$		0			0
$79$	−5.50000	−	9.52628i	−0.618798	−	1.07179i	−0.989705	−	0.143120i	$-0.954286\pi$
$79$	−5.50000	−	9.52628i	−0.618798	−	1.07179i	0.370907	−	0.928670i	$-0.379047\pi$
$80$		0			0
$81$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$82$		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
$87$		0			0
$88$		0			0
$89$	−4.50000	−	7.79423i	−0.476999	−	0.826187i	0.522654	−	0.852545i	$-0.324942\pi$
$89$	−4.50000	−	7.79423i	−0.476999	−	0.826187i	−0.999653	+	0.0263586i	$0.991609\pi$
$90$		0			0
$91$	6.00000	+	5.19615i	0.628971	+	0.544705i
$92$		0			0
$93$	12.0000	+	20.7846i	1.24434	+	2.15526i
$94$		0			0
$95$	4.00000	−	6.92820i	0.410391	−	0.710819i
$96$		0			0
$97$	10.0000			1.01535			0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000			1.01535			0.507673	+	0.861550i	$0.330506\pi$
$98$		0			0
$99$	6.00000			0.603023
$100$		0			0
$101$	−5.00000	+	8.66025i	−0.497519	+	0.861727i	−0.999996	−	0.00286291i	$-0.999089\pi$
$101$	−5.00000	+	8.66025i	−0.497519	+	0.861727i	0.502477	+	0.864590i	$0.332422\pi$
$102$		0			0
$103$	2.00000	+	3.46410i	0.197066	+	0.341328i	0.947576	−	0.319531i	$-0.103525\pi$
$103$	2.00000	+	3.46410i	0.197066	+	0.341328i	−0.750510	+	0.660859i	$0.770192\pi$
$104$		0			0
$105$	−24.0000	−	20.7846i	−2.34216	−	2.02837i
$106$		0			0
$107$	1.50000	+	2.59808i	0.145010	+	0.251166i	0.929377	−	0.369132i	$-0.120345\pi$
$107$	1.50000	+	2.59808i	0.145010	+	0.251166i	−0.784366	+	0.620298i	$0.787012\pi$
$108$		0			0
$109$	−4.00000	+	6.92820i	−0.383131	+	0.663602i	−0.991508	−	0.130046i	$-0.958487\pi$
$109$	−4.00000	+	6.92820i	−0.383131	+	0.663602i	0.608377	+	0.793648i	$0.291821\pi$
$110$		0			0
$111$	−24.0000			−2.27798
$112$		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
$117$	−9.00000	−	15.5885i	−0.832050	−	1.44115i
$118$		0			0
$119$	2.50000	−	0.866025i	0.229175	−	0.0793884i
$120$		0			0
$121$	5.00000	+	8.66025i	0.454545	+	0.787296i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	24.0000			2.14663
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		0			0
$129$	15.0000	−	25.9808i	1.32068	−	2.28748i
$130$		0			0
$131$	6.00000	+	10.3923i	0.524222	+	0.907980i	0.999602	+	0.0281993i	$0.00897729\pi$
$131$	6.00000	+	10.3923i	0.524222	+	0.907980i	−0.475380	+	0.879781i	$0.657689\pi$
$132$		0			0
$133$	−1.00000	+	5.19615i	−0.0867110	+	0.450564i
$134$		0			0
$135$	18.0000	+	31.1769i	1.54919	+	2.68328i
$136$		0			0
$137$	−8.50000	+	14.7224i	−0.726204	+	1.25782i	0.232273	+	0.972651i	$0.425384\pi$
$137$	−8.50000	+	14.7224i	−0.726204	+	1.25782i	−0.958477	+	0.285171i	$0.907949\pi$
$138$		0			0
$139$	−21.0000			−1.78120			−0.890598	−	0.454791i	$-0.849714\pi$
$139$	−21.0000			−1.78120			−0.890598	+	0.454791i	$0.849714\pi$
$140$		0			0
$141$	30.0000			2.52646
$142$		0			0
$143$	−1.50000	+	2.59808i	−0.125436	+	0.217262i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	19.5000	+	7.79423i	1.60833	+	0.642857i
$148$		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
$151$	−1.00000	+	1.73205i	−0.0813788	+	0.140952i	−0.903842	−	0.427865i	$-0.859266\pi$
$151$	−1.00000	+	1.73205i	−0.0813788	+	0.140952i	0.822464	+	0.568818i	$0.192599\pi$
$152$		0			0
$153$	−6.00000			−0.485071
$154$		0			0
$155$	32.0000			2.57030
$156$		0			0
$157$	−8.50000	+	14.7224i	−0.678374	+	1.17498i	0.297097	+	0.954847i	$0.403982\pi$
$157$	−8.50000	+	14.7224i	−0.678374	+	1.17498i	−0.975470	+	0.220131i	$0.929352\pi$
$158$		0			0
$159$	4.50000	+	7.79423i	0.356873	+	0.618123i
$160$		0			0
$161$	2.00000	−	10.3923i	0.157622	−	0.819028i
$162$		0			0
$163$	−6.00000	−	10.3923i	−0.469956	−	0.813988i	0.529454	−	0.848339i	$-0.322397\pi$
$163$	−6.00000	−	10.3923i	−0.469956	−	0.813988i	−0.999410	+	0.0343508i	$0.989064\pi$
$164$		0			0
$165$	6.00000	−	10.3923i	0.467099	−	0.809040i
$166$		0			0
$167$	17.0000			1.31550			0.657750	−	0.753237i	$-0.271508\pi$
$167$	17.0000			1.31550			0.657750	+	0.753237i	$0.271508\pi$
$168$		0			0
$169$	−4.00000			−0.307692
$170$		0			0
$171$	6.00000	−	10.3923i	0.458831	−	0.794719i
$172$		0			0
$173$	−3.00000	−	5.19615i	−0.228086	−	0.395056i	0.729155	−	0.684349i	$-0.239913\pi$
$173$	−3.00000	−	5.19615i	−0.228086	−	0.395056i	−0.957241	+	0.289292i	$0.906580\pi$
$174$		0			0
$175$	−27.5000	+	9.52628i	−2.07880	+	0.720119i
$176$		0			0
$177$	−21.0000	−	36.3731i	−1.57846	−	2.73397i
$178$		0			0
$179$	−8.00000	+	13.8564i	−0.597948	+	1.03568i	0.395175	+	0.918606i	$0.370684\pi$
$179$	−8.00000	+	13.8564i	−0.597948	+	1.03568i	−0.993124	+	0.117071i	$0.962650\pi$
$180$		0			0
$181$	−6.00000			−0.445976			−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000			−0.445976			−0.222988	+	0.974821i	$0.571581\pi$
$182$		0			0
$183$	24.0000			1.77413
$184$		0			0
$185$	−16.0000	+	27.7128i	−1.17634	+	2.03749i
$186$		0			0
$187$	0.500000	+	0.866025i	0.0365636	+	0.0633300i
$188$		0			0
$189$	−18.0000	−	15.5885i	−1.30931	−	1.13389i
$190$		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
$193$	−7.00000	+	12.1244i	−0.503871	+	0.872730i	0.496119	+	0.868255i	$0.334758\pi$
$193$	−7.00000	+	12.1244i	−0.503871	+	0.872730i	−0.999990	+	0.00447566i	$0.998575\pi$
$194$		0			0
$195$	−36.0000			−2.57801
$196$		0			0
$197$	−14.0000			−0.997459			−0.498729	−	0.866758i	$-0.666200\pi$
$197$	−14.0000			−0.997459			−0.498729	+	0.866758i	$0.666200\pi$
$198$		0			0
$199$	1.50000	−	2.59808i	0.106332	−	0.184173i	−0.807950	−	0.589252i	$-0.799423\pi$
$199$	1.50000	−	2.59808i	0.106332	−	0.184173i	0.914282	+	0.405079i	$0.132756\pi$
$200$		0			0
$201$	−15.0000	−	25.9808i	−1.05802	−	1.83254i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−12.0000	+	20.7846i	−0.834058	+	1.44463i
$208$		0			0
$209$	−2.00000			−0.138343
$210$		0			0
$211$	20.0000			1.37686			0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000			1.37686			0.688428	+	0.725304i	$0.258301\pi$
$212$		0			0
$213$	−7.50000	+	12.9904i	−0.513892	+	0.890086i
$214$		0			0
$215$	−20.0000	−	34.6410i	−1.36399	−	2.36250i
$216$		0			0
$217$	−20.0000	+	6.92820i	−1.35769	+	0.470317i
$218$		0			0
$219$	−24.0000	−	41.5692i	−1.62177	−	2.80899i
$220$		0			0
$221$	1.50000	−	2.59808i	0.100901	−	0.174766i
$222$		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
$225$	66.0000			4.40000
$226$		0			0
$227$	1.50000	−	2.59808i	0.0995585	−	0.172440i	−0.811943	−	0.583736i	$-0.801590\pi$
$227$	1.50000	−	2.59808i	0.0995585	−	0.172440i	0.911502	+	0.411296i	$0.134924\pi$
$228$		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
$231$	−1.50000	+	7.79423i	−0.0986928	+	0.512823i
$232$		0			0
$233$	10.0000	+	17.3205i	0.655122	+	1.13470i	0.981863	+	0.189590i	$0.0607160\pi$
$233$	10.0000	+	17.3205i	0.655122	+	1.13470i	−0.326741	+	0.945114i	$0.605951\pi$
$234$		0			0
$235$	20.0000	−	34.6410i	1.30466	−	2.25973i
$236$		0			0
$237$	−33.0000			−2.14358
$238$		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
$241$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$241$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	22.0000	−	17.3205i	1.40553	−	1.10657i
$246$		0			0
$247$	3.00000	+	5.19615i	0.190885	+	0.330623i
$248$		0			0
$249$	18.0000	−	31.1769i	1.14070	−	1.97576i
$250$		0			0
$251$	8.00000			0.504956			0.252478	−	0.967603i	$-0.418755\pi$
$251$	8.00000			0.504956			0.252478	+	0.967603i	$0.418755\pi$
$252$		0			0
$253$	4.00000			0.251478
$254$		0			0
$255$	−6.00000	+	10.3923i	−0.375735	+	0.650791i
$256$		0			0
$257$	−5.50000	−	9.52628i	−0.343081	−	0.594233i	0.641923	−	0.766769i	$-0.278137\pi$
$257$	−5.50000	−	9.52628i	−0.343081	−	0.594233i	−0.985003	+	0.172536i	$0.944804\pi$
$258$		0			0
$259$	4.00000	−	20.7846i	0.248548	−	1.29149i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6.00000	+	10.3923i	−0.369976	+	0.640817i	−0.989561	−	0.144112i	$-0.953967\pi$
$263$	−6.00000	+	10.3923i	−0.369976	+	0.640817i	0.619586	+	0.784929i	$0.287301\pi$
$264$		0			0
$265$	12.0000			0.737154
$266$		0			0
$267$	−27.0000			−1.65237
$268$		0			0
$269$	1.00000	−	1.73205i	0.0609711	−	0.105605i	−0.833929	−	0.551872i	$-0.813914\pi$
$269$	1.00000	−	1.73205i	0.0609711	−	0.105605i	0.894900	+	0.446267i	$0.147247\pi$
$270$		0			0
$271$	−4.00000	−	6.92820i	−0.242983	−	0.420858i	0.718580	−	0.695444i	$-0.244792\pi$
$271$	−4.00000	−	6.92820i	−0.242983	−	0.420858i	−0.961563	+	0.274586i	$0.911459\pi$
$272$		0			0
$273$	22.5000	−	7.79423i	1.36176	−	0.471728i
$274$		0			0
$275$	−5.50000	−	9.52628i	−0.331662	−	0.574456i
$276$		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
$279$	48.0000			2.87368
$280$		0			0
$281$	−3.00000			−0.178965			−0.0894825	−	0.995988i	$-0.528521\pi$
$281$	−3.00000			−0.178965			−0.0894825	+	0.995988i	$0.528521\pi$
$282$		0			0
$283$	−9.50000	+	16.4545i	−0.564716	+	0.978117i	0.432360	+	0.901701i	$0.357681\pi$
$283$	−9.50000	+	16.4545i	−0.564716	+	0.978117i	−0.997076	+	0.0764162i	$0.975652\pi$
$284$		0			0
$285$	−12.0000	−	20.7846i	−0.710819	−	1.23117i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.500000	−	0.866025i	−0.0294118	−	0.0509427i
$290$		0			0
$291$	15.0000	−	25.9808i	0.879316	−	1.52302i
$292$		0			0
$293$	1.00000			0.0584206			0.0292103	−	0.999573i	$-0.490701\pi$
$293$	1.00000			0.0584206			0.0292103	+	0.999573i	$0.490701\pi$
$294$		0			0
$295$	−56.0000			−3.26045
$296$		0			0
$297$	4.50000	−	7.79423i	0.261116	−	0.452267i
$298$		0			0
$299$	−6.00000	−	10.3923i	−0.346989	−	0.601003i
$300$		0			0
$301$	20.0000	+	17.3205i	1.15278	+	0.998337i
$302$		0			0
$303$	15.0000	+	25.9808i	0.861727	+	1.49256i
$304$		0			0
$305$	16.0000	−	27.7128i	0.916157	−	1.58683i
$306$		0			0
$307$	4.00000			0.228292			0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000			0.228292			0.114146	+	0.993464i	$0.463587\pi$
$308$		0			0
$309$	12.0000			0.682656
$310$		0			0
$311$	4.50000	−	7.79423i	0.255172	−	0.441970i	−0.709771	−	0.704433i	$-0.751201\pi$
$311$	4.50000	−	7.79423i	0.255172	−	0.441970i	0.964942	+	0.262463i	$0.0845347\pi$
$312$		0			0
$313$	9.00000	+	15.5885i	0.508710	+	0.881112i	0.999949	+	0.0100869i	$0.00321082\pi$
$313$	9.00000	+	15.5885i	0.508710	+	0.881112i	−0.491239	+	0.871025i	$0.663456\pi$
$314$		0			0
$315$	−60.0000	+	20.7846i	−3.38062	+	1.17108i
$316$		0			0
$317$	−3.00000	−	5.19615i	−0.168497	−	0.291845i	0.769395	−	0.638774i	$-0.220558\pi$
$317$	−3.00000	−	5.19615i	−0.168497	−	0.291845i	−0.937892	+	0.346929i	$0.887225\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	9.00000			0.502331
$322$		0			0
$323$	2.00000			0.111283
$324$		0			0
$325$	−16.5000	+	28.5788i	−0.915255	+	1.58527i
$326$		0			0
$327$	12.0000	+	20.7846i	0.663602	+	1.14939i
$328$		0			0
$329$	−5.00000	+	25.9808i	−0.275659	+	1.43237i
$330$		0			0
$331$	−5.00000	−	8.66025i	−0.274825	−	0.476011i	0.695266	−	0.718752i	$-0.255287\pi$
$331$	−5.00000	−	8.66025i	−0.274825	−	0.476011i	−0.970091	+	0.242742i	$0.921953\pi$
$332$		0			0
$333$	−24.0000	+	41.5692i	−1.31519	+	2.27798i
$334$		0			0
$335$	−40.0000			−2.18543
$336$		0			0
$337$	−8.00000			−0.435788			−0.217894	−	0.975972i	$-0.569919\pi$
$337$	−8.00000			−0.435788			−0.217894	+	0.975972i	$0.569919\pi$
$338$		0			0
$339$	−12.0000	+	20.7846i	−0.651751	+	1.12887i
$340$		0			0
$341$	−4.00000	−	6.92820i	−0.216612	−	0.375183i
$342$		0			0
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$		0			0
$345$	24.0000	+	41.5692i	1.29212	+	2.23801i
$346$		0			0
$347$	2.00000	−	3.46410i	0.107366	−	0.185963i	−0.807337	−	0.590091i	$-0.799092\pi$
$347$	2.00000	−	3.46410i	0.107366	−	0.185963i	0.914702	+	0.404128i	$0.132425\pi$
$348$		0			0
$349$	30.0000			1.60586			0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000			1.60586			0.802932	+	0.596071i	$0.203272\pi$
$350$		0			0
$351$	−27.0000			−1.44115
$352$		0			0
$353$	−5.50000	+	9.52628i	−0.292735	+	0.507033i	−0.974456	−	0.224580i	$-0.927899\pi$
$353$	−5.50000	+	9.52628i	−0.292735	+	0.507033i	0.681720	+	0.731613i	$0.261232\pi$
$354$		0			0
$355$	10.0000	+	17.3205i	0.530745	+	0.919277i
$356$		0			0
$357$	1.50000	−	7.79423i	0.0793884	−	0.412514i
$358$		0			0
$359$	5.00000	+	8.66025i	0.263890	+	0.457071i	0.967272	−	0.253741i	$-0.0816611\pi$
$359$	5.00000	+	8.66025i	0.263890	+	0.457071i	−0.703382	+	0.710812i	$0.748328\pi$
$360$		0			0
$361$	7.50000	−	12.9904i	0.394737	−	0.683704i
$362$		0			0
$363$	30.0000			1.57459
$364$		0			0
$365$	−64.0000			−3.34991
$366$		0			0
$367$	−1.50000	+	2.59808i	−0.0782994	+	0.135618i	−0.902516	−	0.430656i	$-0.858282\pi$
$367$	−1.50000	+	2.59808i	−0.0782994	+	0.135618i	0.824217	+	0.566274i	$0.191616\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−7.50000	+	2.59808i	−0.389381	+	0.134885i
$372$		0			0
$373$	−3.50000	−	6.06218i	−0.181223	−	0.313888i	0.761074	−	0.648665i	$-0.224672\pi$
$373$	−3.50000	−	6.06218i	−0.181223	−	0.313888i	−0.942297	+	0.334777i	$0.891339\pi$
$374$		0			0
$375$	36.0000	−	62.3538i	1.85903	−	3.21994i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	13.0000			0.667765			0.333883	−	0.942615i	$-0.391641\pi$
$379$	13.0000			0.667765			0.333883	+	0.942615i	$0.391641\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−17.0000	−	29.4449i	−0.868659	−	1.50456i	−0.863367	−	0.504576i	$-0.831649\pi$
$383$	−17.0000	−	29.4449i	−0.868659	−	1.50456i	−0.00529229	−	0.999986i	$-0.501685\pi$
$384$		0			0
$385$	8.00000	+	6.92820i	0.407718	+	0.353094i
$386$		0			0
$387$	−30.0000	−	51.9615i	−1.52499	−	2.64135i
$388$		0			0
$389$	1.50000	−	2.59808i	0.0760530	−	0.131728i	−0.825491	−	0.564416i	$-0.809102\pi$
$389$	1.50000	−	2.59808i	0.0760530	−	0.131728i	0.901544	+	0.432688i	$0.142435\pi$
$390$		0			0
$391$	−4.00000			−0.202289
$392$		0			0
$393$	36.0000			1.81596
$394$		0			0
$395$	−22.0000	+	38.1051i	−1.10694	+	1.91728i
$396$		0			0
$397$	−1.00000	−	1.73205i	−0.0501886	−	0.0869291i	0.839840	−	0.542834i	$-0.182649\pi$
$397$	−1.00000	−	1.73205i	−0.0501886	−	0.0869291i	−0.890028	+	0.455905i	$0.849316\pi$
$398$		0			0
$399$	12.0000	+	10.3923i	0.600751	+	0.520266i
$400$		0			0
$401$	13.0000	+	22.5167i	0.649189	+	1.12443i	0.983317	+	0.181901i	$0.0582249\pi$
$401$	13.0000	+	22.5167i	0.649189	+	1.12443i	−0.334128	+	0.942528i	$0.608442\pi$
$402$		0			0
$403$	−12.0000	+	20.7846i	−0.597763	+	1.03536i
$404$		0			0
$405$	36.0000			1.78885
$406$		0			0
$407$	8.00000			0.396545
$408$		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
$411$	25.5000	+	44.1673i	1.25782	+	2.17861i
$412$		0			0
$413$	35.0000	−	12.1244i	1.72224	−	0.596601i
$414$		0			0
$415$	−24.0000	−	41.5692i	−1.17811	−	2.04055i
$416$		0			0
$417$	−31.5000	+	54.5596i	−1.54256	+	2.67180i
$418$		0			0
$419$	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
$421$	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
$423$	30.0000	−	51.9615i	1.45865	−	2.52646i
$424$		0			0
$425$	5.50000	+	9.52628i	0.266789	+	0.462092i
$426$		0			0
$427$	−4.00000	+	20.7846i	−0.193574	+	1.00584i
$428$		0			0
$429$	4.50000	+	7.79423i	0.217262	+	0.376309i
$430$		0			0
$431$	9.50000	−	16.4545i	0.457599	−	0.792585i	−0.541235	−	0.840872i	$-0.682043\pi$
$431$	9.50000	−	16.4545i	0.457599	−	0.792585i	0.998833	+	0.0482871i	$0.0153762\pi$
$432$		0			0
$433$	6.00000			0.288342			0.144171	−	0.989553i	$-0.453949\pi$
$433$	6.00000			0.288342			0.144171	+	0.989553i	$0.453949\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	4.00000	−	6.92820i	0.191346	−	0.331421i
$438$		0			0
$439$	14.5000	+	25.1147i	0.692047	+	1.19866i	0.971166	+	0.238404i	$0.0766244\pi$
$439$	14.5000	+	25.1147i	0.692047	+	1.19866i	−0.279119	+	0.960257i	$0.590042\pi$
$440$		0			0
$441$	33.0000	−	25.9808i	1.57143	−	1.23718i
$442$		0			0
$443$	3.00000	+	5.19615i	0.142534	+	0.246877i	0.928450	−	0.371457i	$-0.121142\pi$
$443$	3.00000	+	5.19615i	0.142534	+	0.246877i	−0.785916	+	0.618333i	$0.787808\pi$
$444$		0			0
$445$	−18.0000	+	31.1769i	−0.853282	+	1.47793i
$446$		0			0
$447$	33.0000			1.56085
$448$		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
$451$		0			0
$452$		0			0
$453$	3.00000	+	5.19615i	0.140952	+	0.244137i
$454$		0			0
$455$	6.00000	−	31.1769i	0.281284	−	1.46160i
$456$		0			0
$457$	5.00000	+	8.66025i	0.233890	+	0.405110i	0.958950	−	0.283577i	$-0.0915211\pi$
$457$	5.00000	+	8.66025i	0.233890	+	0.405110i	−0.725059	+	0.688686i	$0.758188\pi$
$458$		0			0
$459$	−4.50000	+	7.79423i	−0.210042	+	0.363803i
$460$		0			0
$461$	−17.0000			−0.791769			−0.395884	−	0.918300i	$-0.629562\pi$
$461$	−17.0000			−0.791769			−0.395884	+	0.918300i	$0.629562\pi$
$462$		0			0
$463$	8.00000			0.371792			0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000			0.371792			0.185896	+	0.982569i	$0.440481\pi$
$464$		0			0
$465$	48.0000	−	83.1384i	2.22595	−	3.85545i
$466$		0			0
$467$	18.0000	+	31.1769i	0.832941	+	1.44270i	0.895696	+	0.444667i	$0.146678\pi$
$467$	18.0000	+	31.1769i	0.832941	+	1.44270i	−0.0627555	+	0.998029i	$0.519989\pi$
$468$		0			0
$469$	25.0000	−	8.66025i	1.15439	−	0.399893i
$470$		0			0
$471$	25.5000	+	44.1673i	1.17498	+	2.03512i
$472$		0			0
$473$	−5.00000	+	8.66025i	−0.229900	+	0.398199i
$474$		0			0
$475$	−22.0000			−1.00943
$476$		0			0
$477$	18.0000			0.824163
$478$		0			0
$479$	−12.0000	+	20.7846i	−0.548294	+	0.949673i	0.450098	+	0.892979i	$0.351389\pi$
$479$	−12.0000	+	20.7846i	−0.548294	+	0.949673i	−0.998392	+	0.0566937i	$0.981944\pi$
$480$		0			0
$481$	−12.0000	−	20.7846i	−0.547153	−	0.947697i
$482$		0			0
$483$	−24.0000	−	20.7846i	−1.09204	−	0.945732i
$484$		0			0
$485$	−20.0000	−	34.6410i	−0.908153	−	1.57297i
$486$		0			0
$487$	−11.5000	+	19.9186i	−0.521115	+	0.902597i	0.478584	+	0.878042i	$0.341150\pi$
$487$	−11.5000	+	19.9186i	−0.521115	+	0.902597i	−0.999698	+	0.0245553i	$0.992183\pi$
$488$		0			0
$489$	−36.0000			−1.62798
$490$		0			0
$491$	−26.0000			−1.17336			−0.586682	−	0.809818i	$-0.699566\pi$
$491$	−26.0000			−1.17336			−0.586682	+	0.809818i	$0.699566\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−12.0000	−	20.7846i	−0.539360	−	0.934199i
$496$		0			0
$497$	−10.0000	−	8.66025i	−0.448561	−	0.388465i
$498$		0			0
$499$	2.50000	+	4.33013i	0.111915	+	0.193843i	0.916542	−	0.399937i	$-0.130968\pi$
$499$	2.50000	+	4.33013i	0.111915	+	0.193843i	−0.804627	+	0.593780i	$0.797635\pi$
$500$		0			0
$501$	25.5000	−	44.1673i	1.13926	−	1.97325i
$502$		0			0
$503$	33.0000			1.47140			0.735699	−	0.677309i	$-0.236854\pi$
$503$	33.0000			1.47140			0.735699	+	0.677309i	$0.236854\pi$
$504$		0			0
$505$	40.0000			1.77998
$506$		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
$511$	40.0000	−	13.8564i	1.76950	−	0.612971i
$512$		0			0
$513$	−9.00000	−	15.5885i	−0.397360	−	0.688247i
$514$		0			0
$515$	8.00000	−	13.8564i	0.352522	−	0.610586i
$516$		0			0
$517$	−10.0000			−0.439799
$518$		0			0
$519$	−18.0000			−0.790112
$520$		0			0
$521$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$521$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$522$		0			0
$523$	−11.0000	−	19.0526i	−0.480996	−	0.833110i	0.518766	−	0.854916i	$-0.326392\pi$
$523$	−11.0000	−	19.0526i	−0.480996	−	0.833110i	−0.999762	+	0.0218062i	$0.993058\pi$
$524$		0			0
$525$	−16.5000	+	85.7365i	−0.720119	+	3.74185i
$526$		0			0
$527$	4.00000	+	6.92820i	0.174243	+	0.301797i
$528$		0			0
$529$	3.50000	−	6.06218i	0.152174	−	0.263573i
$530$		0			0
$531$	−84.0000			−3.64529
$532$		0			0
$533$		0			0
$534$		0			0
$535$	6.00000	−	10.3923i	0.259403	−	0.449299i
$536$		0			0
$537$	24.0000	+	41.5692i	1.03568	+	1.79384i
$538$		0			0
$539$	−6.50000	−	2.59808i	−0.279975	−	0.111907i
$540$		0			0
$541$	−11.0000	−	19.0526i	−0.472927	−	0.819133i	0.526593	−	0.850118i	$-0.323469\pi$
$541$	−11.0000	−	19.0526i	−0.472927	−	0.819133i	−0.999520	+	0.0309841i	$0.990136\pi$
$542$		0			0
$543$	−9.00000	+	15.5885i	−0.386227	+	0.668965i
$544$		0			0
$545$	32.0000			1.37073
$546$		0			0
$547$	−1.00000			−0.0427569			−0.0213785	−	0.999771i	$-0.506805\pi$
$547$	−1.00000			−0.0427569			−0.0213785	+	0.999771i	$0.506805\pi$
$548$		0			0
$549$	24.0000	−	41.5692i	1.02430	−	1.77413i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	5.50000	−	28.5788i	0.233884	−	1.21530i
$554$		0			0
$555$	48.0000	+	83.1384i	2.03749	+	3.52903i
$556$		0			0
$557$	−7.50000	+	12.9904i	−0.317785	+	0.550420i	−0.980026	−	0.198871i	$-0.936272\pi$
$557$	−7.50000	+	12.9904i	−0.317785	+	0.550420i	0.662240	+	0.749291i	$0.269606\pi$
$558$		0			0
$559$	30.0000			1.26886
$560$		0			0
$561$	3.00000			0.126660
$562$		0			0
$563$	1.00000	−	1.73205i	0.0421450	−	0.0729972i	−0.844183	−	0.536054i	$-0.819914\pi$
$563$	1.00000	−	1.73205i	0.0421450	−	0.0729972i	0.886328	+	0.463057i	$0.153248\pi$
$564$		0			0
$565$	16.0000	+	27.7128i	0.673125	+	1.16589i
$566$		0			0
$567$	−22.5000	+	7.79423i	−0.944911	+	0.327327i
$568$		0			0
$569$	7.50000	+	12.9904i	0.314416	+	0.544585i	0.979313	−	0.202350i	$-0.0648579\pi$
$569$	7.50000	+	12.9904i	0.314416	+	0.544585i	−0.664897	+	0.746935i	$0.731525\pi$
$570$		0			0
$571$	2.00000	−	3.46410i	0.0836974	−	0.144968i	−0.821138	−	0.570730i	$-0.806660\pi$
$571$	2.00000	−	3.46410i	0.0836974	−	0.144968i	0.904835	+	0.425762i	$0.139994\pi$
$572$		0			0
$573$	−36.0000			−1.50392
$574$		0			0
$575$	44.0000			1.83493
$576$		0			0
$577$	21.5000	−	37.2391i	0.895057	−	1.55028i	0.0613223	−	0.998118i	$-0.480468\pi$
$577$	21.5000	−	37.2391i	0.895057	−	1.55028i	0.833734	−	0.552166i	$-0.186198\pi$
$578$		0			0
$579$	21.0000	+	36.3731i	0.872730	+	1.51161i
$580$		0			0
$581$	24.0000	+	20.7846i	0.995688	+	0.862291i
$582$		0			0
$583$	−1.50000	−	2.59808i	−0.0621237	−	0.107601i
$584$		0			0
$585$	−36.0000	+	62.3538i	−1.48842	+	2.57801i
$586$		0			0
$587$	−10.0000			−0.412744			−0.206372	−	0.978474i	$-0.566166\pi$
$587$	−10.0000			−0.412744			−0.206372	+	0.978474i	$0.566166\pi$
$588$		0			0
$589$	−16.0000			−0.659269
$590$		0			0
$591$	−21.0000	+	36.3731i	−0.863825	+	1.49619i
$592$		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
$597$	−4.50000	−	7.79423i	−0.184173	−	0.318997i
$598$		0			0
$599$	−15.0000	+	25.9808i	−0.612883	+	1.06155i	0.377869	+	0.925859i	$0.376657\pi$
$599$	−15.0000	+	25.9808i	−0.612883	+	1.06155i	−0.990752	+	0.135686i	$0.956676\pi$
$600$		0			0
$601$	−46.0000			−1.87638			−0.938190	−	0.346122i	$-0.887498\pi$
$601$	−46.0000			−1.87638			−0.938190	+	0.346122i	$0.887498\pi$
$602$		0			0
$603$	−60.0000			−2.44339
$604$		0			0
$605$	20.0000	−	34.6410i	0.813116	−	1.40836i
$606$		0			0
$607$	−14.5000	−	25.1147i	−0.588537	−	1.01938i	−0.994424	−	0.105453i	$-0.966371\pi$
$607$	−14.5000	−	25.1147i	−0.588537	−	1.01938i	0.405887	−	0.913923i	$-0.366962\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	15.0000	+	25.9808i	0.606835	+	1.05107i
$612$		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
$615$		0			0
$616$		0			0
$617$	−44.0000			−1.77137			−0.885687	−	0.464283i	$-0.846312\pi$
$617$	−44.0000			−1.77137			−0.885687	+	0.464283i	$0.846312\pi$
$618$		0			0
$619$	14.5000	−	25.1147i	0.582804	−	1.00945i	−0.412341	−	0.911030i	$-0.635289\pi$
$619$	14.5000	−	25.1147i	0.582804	−	1.00945i	0.995145	−	0.0984169i	$-0.0313779\pi$
$620$		0			0
$621$	18.0000	+	31.1769i	0.722315	+	1.25109i
$622$		0			0
$623$	4.50000	−	23.3827i	0.180289	−	0.936808i
$624$		0			0
$625$	−20.5000	−	35.5070i	−0.820000	−	1.42028i
$626$		0			0
$627$	−3.00000	+	5.19615i	−0.119808	+	0.207514i
$628$		0			0
$629$	−8.00000			−0.318981
$630$		0			0
$631$	−46.0000			−1.83123			−0.915616	−	0.402055i	$-0.868296\pi$
$631$	−46.0000			−1.83123			−0.915616	+	0.402055i	$0.868296\pi$
$632$		0			0
$633$	30.0000	−	51.9615i	1.19239	−	2.06529i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	3.00000	+	20.7846i	0.118864	+	0.823516i
$638$		0			0
$639$	15.0000	+	25.9808i	0.593391	+	1.02778i
$640$		0			0
$641$	13.0000	−	22.5167i	0.513469	−	0.889355i	−0.486409	−	0.873731i	$-0.661693\pi$
$641$	13.0000	−	22.5167i	0.513469	−	0.889355i	0.999878	−	0.0156233i	$-0.00497325\pi$
$642$		0			0
$643$	27.0000			1.06478			0.532388	−	0.846500i	$-0.321295\pi$
$643$	27.0000			1.06478			0.532388	+	0.846500i	$0.321295\pi$
$644$		0			0
$645$	−120.000			−4.72500
$646$		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
$649$	7.00000	+	12.1244i	0.274774	+	0.475923i
$650$		0			0
$651$	−12.0000	+	62.3538i	−0.470317	+	2.44384i
$652$		0			0
$653$	−23.0000	−	39.8372i	−0.900060	−	1.55895i	−0.827415	−	0.561591i	$-0.810189\pi$
$653$	−23.0000	−	39.8372i	−0.900060	−	1.55895i	−0.0726446	−	0.997358i	$-0.523144\pi$
$654$		0			0
$655$	24.0000	−	41.5692i	0.937758	−	1.62424i
$656$		0			0
$657$	−96.0000			−3.74532
$658$		0			0
$659$	2.00000			0.0779089			0.0389545	−	0.999241i	$-0.487597\pi$
$659$	2.00000			0.0779089			0.0389545	+	0.999241i	$0.487597\pi$
$660$		0			0
$661$	15.0000	−	25.9808i	0.583432	−	1.01053i	−0.411636	−	0.911348i	$-0.635043\pi$
$661$	15.0000	−	25.9808i	0.583432	−	1.01053i	0.995069	−	0.0991864i	$-0.0316240\pi$
$662$		0			0
$663$	−4.50000	−	7.79423i	−0.174766	−	0.302703i
$664$		0			0
$665$	20.0000	−	6.92820i	0.775567	−	0.268664i
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−9.00000	+	15.5885i	−0.347960	+	0.602685i
$670$		0			0
$671$	−8.00000			−0.308837
$672$		0			0
$673$	−44.0000			−1.69608			−0.848038	−	0.529936i	$-0.822216\pi$
$673$	−44.0000			−1.69608			−0.848038	+	0.529936i	$0.822216\pi$
$674$		0			0
$675$	49.5000	−	85.7365i	1.90526	−	3.30000i
$676$		0			0
$677$	24.0000	+	41.5692i	0.922395	+	1.59763i	0.795698	+	0.605693i	$0.207104\pi$
$677$	24.0000	+	41.5692i	0.922395	+	1.59763i	0.126697	+	0.991941i	$0.459562\pi$
$678$		0			0
$679$	20.0000	+	17.3205i	0.767530	+	0.664700i
$680$		0			0
$681$	−4.50000	−	7.79423i	−0.172440	−	0.298675i
$682$		0			0
$683$	4.50000	−	7.79423i	0.172188	−	0.298238i	−0.766997	−	0.641651i	$-0.778250\pi$
$683$	4.50000	−	7.79423i	0.172188	−	0.298238i	0.939184	+	0.343413i	$0.111583\pi$
$684$		0			0
$685$	68.0000			2.59815
$686$		0			0
$687$	30.0000			1.14457
$688$		0			0
$689$	−4.50000	+	7.79423i	−0.171436	+	0.296936i
$690$		0			0
$691$	−8.00000	−	13.8564i	−0.304334	−	0.527123i	0.672779	−	0.739844i	$-0.265101\pi$
$691$	−8.00000	−	13.8564i	−0.304334	−	0.527123i	−0.977113	+	0.212721i	$0.931767\pi$
$692$		0			0
$693$	12.0000	+	10.3923i	0.455842	+	0.394771i
$694$		0			0
$695$	42.0000	+	72.7461i	1.59315	+	2.75942i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	60.0000			2.26941
$700$		0			0
$701$	39.0000			1.47301			0.736505	−	0.676432i	$-0.236475\pi$
$701$	39.0000			1.47301			0.736505	+	0.676432i	$0.236475\pi$
$702$		0			0
$703$	8.00000	−	13.8564i	0.301726	−	0.522604i
$704$		0			0
$705$	−60.0000	−	103.923i	−2.25973	−	3.91397i
$706$		0			0
$707$	−25.0000	+	8.66025i	−0.940222	+	0.325702i
$708$		0			0
$709$	1.00000	+	1.73205i	0.0375558	+	0.0650485i	0.884192	−	0.467123i	$-0.154709\pi$
$709$	1.00000	+	1.73205i	0.0375558	+	0.0650485i	−0.846637	+	0.532172i	$0.821376\pi$
$710$		0			0
$711$	−33.0000	+	57.1577i	−1.23760	+	2.14358i
$712$		0			0
$713$	32.0000			1.19841
$714$		0			0
$715$	12.0000			0.448775
$716$		0			0
$717$	3.00000	−	5.19615i	0.112037	−	0.194054i
$718$		0			0
$719$	7.50000	+	12.9904i	0.279703	+	0.484459i	0.971311	−	0.237814i	$-0.0764307\pi$
$719$	7.50000	+	12.9904i	0.279703	+	0.484459i	−0.691608	+	0.722273i	$0.743097\pi$
$720$		0			0
$721$	−2.00000	+	10.3923i	−0.0744839	+	0.387030i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−14.0000			−0.519231			−0.259616	−	0.965712i	$-0.583596\pi$
$727$	−14.0000			−0.519231			−0.259616	+	0.965712i	$0.583596\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$	5.00000	−	8.66025i	0.184932	−	0.320311i
$732$		0			0
$733$	6.50000	+	11.2583i	0.240083	+	0.415836i	0.960738	−	0.277458i	$-0.0894920\pi$
$733$	6.50000	+	11.2583i	0.240083	+	0.415836i	−0.720655	+	0.693294i	$0.756159\pi$
$734$		0			0
$735$	−12.0000	−	83.1384i	−0.442627	−	3.06661i
$736$		0			0
$737$	5.00000	+	8.66025i	0.184177	+	0.319005i
$738$		0			0
$739$	17.0000	−	29.4449i	0.625355	−	1.08315i	−0.363117	−	0.931744i	$-0.618287\pi$
$739$	17.0000	−	29.4449i	0.625355	−	1.08315i	0.988472	−	0.151403i	$-0.0483792\pi$
$740$		0			0
$741$	18.0000			0.661247
$742$		0			0
$743$	−27.0000			−0.990534			−0.495267	−	0.868741i	$-0.664930\pi$
$743$	−27.0000			−0.990534			−0.495267	+	0.868741i	$0.664930\pi$
$744$		0			0
$745$	22.0000	−	38.1051i	0.806018	−	1.39606i
$746$		0			0
$747$	−36.0000	−	62.3538i	−1.31717	−	2.28141i
$748$		0			0
$749$	−1.50000	+	7.79423i	−0.0548088	+	0.284795i
$750$		0			0
$751$	−11.5000	−	19.9186i	−0.419641	−	0.726839i	0.576262	−	0.817265i	$-0.304511\pi$
$751$	−11.5000	−	19.9186i	−0.419641	−	0.726839i	−0.995903	+	0.0904254i	$0.971177\pi$
$752$		0			0
$753$	12.0000	−	20.7846i	0.437304	−	0.757433i
$754$		0			0
$755$	8.00000			0.291150
$756$		0			0
$757$	17.0000			0.617876			0.308938	−	0.951082i	$-0.400027\pi$
$757$	17.0000			0.617876			0.308938	+	0.951082i	$0.400027\pi$
$758$		0			0
$759$	6.00000	−	10.3923i	0.217786	−	0.377217i
$760$		0			0
$761$	13.5000	+	23.3827i	0.489375	+	0.847622i	0.999925	−	0.0122260i	$-0.00389175\pi$
$761$	13.5000	+	23.3827i	0.489375	+	0.847622i	−0.510551	+	0.859848i	$0.670558\pi$
$762$		0			0
$763$	−20.0000	+	6.92820i	−0.724049	+	0.250818i
$764$		0			0
$765$	12.0000	+	20.7846i	0.433861	+	0.751469i
$766$		0			0
$767$	21.0000	−	36.3731i	0.758266	−	1.31336i
$768$		0			0
$769$	25.0000			0.901523			0.450762	−	0.892644i	$-0.351152\pi$
$769$	25.0000			0.901523			0.450762	+	0.892644i	$0.351152\pi$
$770$		0			0
$771$	−33.0000			−1.18847
$772$		0			0
$773$	−16.5000	+	28.5788i	−0.593464	+	1.02791i	0.400298	+	0.916385i	$0.368907\pi$
$773$	−16.5000	+	28.5788i	−0.593464	+	1.02791i	−0.993762	+	0.111524i	$0.964427\pi$
$774$		0			0
$775$	−44.0000	−	76.2102i	−1.58053	−	2.73755i
$776$		0			0
$777$	−48.0000	−	41.5692i	−1.72199	−	1.49129i
$778$		0			0
$779$		0			0
$780$		0			0
$781$	2.50000	−	4.33013i	0.0894570	−	0.154944i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	68.0000			2.42702
$786$		0			0
$787$	2.00000	−	3.46410i	0.0712923	−	0.123482i	−0.828176	−	0.560469i	$-0.810621\pi$
$787$	2.00000	−	3.46410i	0.0712923	−	0.123482i	0.899468	+	0.436987i	$0.143954\pi$
$788$		0			0
$789$	18.0000	+	31.1769i	0.640817	+	1.10993i
$790$		0			0
$791$	−16.0000	−	13.8564i	−0.568895	−	0.492677i
$792$		0			0

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(1.50000 - 2.59808i) q^{3} +(-2.00000 - 3.46410i) q^{5} +(2.00000 + 1.73205i) q^{7} +(-3.00000 - 5.19615i) q^{9} +(-0.500000 + 0.866025i) q^{11} +3.00000 q^{13} -12.0000 q^{15} +(0.500000 - 0.866025i) q^{17} +(1.00000 + 1.73205i) q^{19} +(7.50000 - 2.59808i) q^{21} +(-2.00000 - 3.46410i) q^{23} +(-5.50000 + 9.52628i) q^{25} -9.00000 q^{27} +(-4.00000 + 6.92820i) q^{31} +(1.50000 + 2.59808i) q^{33} +(2.00000 - 10.3923i) q^{35} +(-4.00000 - 6.92820i) q^{37} +(4.50000 - 7.79423i) q^{39} +10.0000 q^{43} +(-12.0000 + 20.7846i) q^{45} +(5.00000 + 8.66025i) q^{47} +(1.00000 + 6.92820i) q^{49} +(-1.50000 - 2.59808i) q^{51} +(-1.50000 + 2.59808i) q^{53} +4.00000 q^{55} +6.00000 q^{57} +(7.00000 - 12.1244i) q^{59} +(4.00000 + 6.92820i) q^{61} +(3.00000 - 15.5885i) q^{63} +(-6.00000 - 10.3923i) q^{65} +(5.00000 - 8.66025i) q^{67} -12.0000 q^{69} -5.00000 q^{71} +(8.00000 - 13.8564i) q^{73} +(16.5000 + 28.5788i) q^{75} +(-2.50000 + 0.866025i) q^{77} +(-5.50000 - 9.52628i) q^{79} +(-4.50000 + 7.79423i) q^{81} +12.0000 q^{83} -4.00000 q^{85} +(-4.50000 - 7.79423i) q^{89} +(6.00000 + 5.19615i) q^{91} +(12.0000 + 20.7846i) q^{93} +(4.00000 - 6.92820i) q^{95} +10.0000 q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} - 4 q^{5} + 4 q^{7} - 6 q^{9} - q^{11} + 6 q^{13} - 24 q^{15} + q^{17} + 2 q^{19} + 15 q^{21} - 4 q^{23} - 11 q^{25} - 18 q^{27} - 8 q^{31} + 3 q^{33} + 4 q^{35} - 8 q^{37} + 9 q^{39} + 20 q^{43}+ \cdots + 12 q^{99}+O(q^{100}) \) 2 * q + 3 * q^3 - 4 * q^5 + 4 * q^7 - 6 * q^9 - q^11 + 6 * q^13 - 24 * q^15 + q^17 + 2 * q^19 + 15 * q^21 - 4 * q^23 - 11 * q^25 - 18 * q^27 - 8 * q^31 + 3 * q^33 + 4 * q^35 - 8 * q^37 + 9 * q^39 + 20 * q^43 - 24 * q^45 + 10 * q^47 + 2 * q^49 - 3 * q^51 - 3 * q^53 + 8 * q^55 + 12 * q^57 + 14 * q^59 + 8 * q^61 + 6 * q^63 - 12 * q^65 + 10 * q^67 - 24 * q^69 - 10 * q^71 + 16 * q^73 + 33 * q^75 - 5 * q^77 - 11 * q^79 - 9 * q^81 + 24 * q^83 - 8 * q^85 - 9 * q^89 + 12 * q^91 + 24 * q^93 + 8 * q^95 + 20 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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