LMFDB - Embedded newform 2548.2.j.j.1145.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2548,2,Mod(1145,2548)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2548, 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("2548.1145");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2548 = 2^{2} \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2548.j (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$20.3458824350$
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:	no (minimal twist has level 364)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1145.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2548.1145
Dual form			2548.2.j.j.1353.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.73205i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+(1.00000 - 1.73205i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(-0.500000 - 0.866025i) q^{9} +(2.00000 - 3.46410i) q^{11} +1.00000 q^{13} -2.00000 q^{15} +(1.00000 - 1.73205i) q^{17} +(0.500000 + 0.866025i) q^{19} +(3.50000 + 6.06218i) q^{23} +(2.00000 - 3.46410i) q^{25} +4.00000 q^{27} -5.00000 q^{29} +(4.50000 - 7.79423i) q^{31} +(-4.00000 - 6.92820i) q^{33} +(1.00000 + 1.73205i) q^{37} +(1.00000 - 1.73205i) q^{39} +2.00000 q^{41} +1.00000 q^{43} +(-0.500000 + 0.866025i) q^{45} +(-4.50000 - 7.79423i) q^{47} +(-2.00000 - 3.46410i) q^{51} +(-1.50000 + 2.59808i) q^{53} -4.00000 q^{55} +2.00000 q^{57} +(-7.00000 - 12.1244i) q^{61} +(-0.500000 - 0.866025i) q^{65} +(-5.00000 + 8.66025i) q^{67} +14.0000 q^{69} -14.0000 q^{71} +(-1.50000 + 2.59808i) q^{73} +(-4.00000 - 6.92820i) q^{75} +(-2.50000 - 4.33013i) q^{79} +(5.50000 - 9.52628i) q^{81} +5.00000 q^{83} -2.00000 q^{85} +(-5.00000 + 8.66025i) q^{87} +(4.50000 + 7.79423i) q^{89} +(-9.00000 - 15.5885i) q^{93} +(0.500000 - 0.866025i) q^{95} -1.00000 q^{97} -4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - q^{5} - q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - q^5 - q^9	$ 2 q + 2 q^{3} - q^{5} - q^{9} + 4 q^{11} + 2 q^{13} - 4 q^{15} + 2 q^{17} + q^{19} + 7 q^{23} + 4 q^{25} + 8 q^{27} - 10 q^{29} + 9 q^{31} - 8 q^{33} + 2 q^{37} + 2 q^{39} + 4 q^{41} + 2 q^{43} - q^{45} - 9 q^{47} - 4 q^{51} - 3 q^{53} - 8 q^{55} + 4 q^{57} - 14 q^{61} - q^{65} - 10 q^{67} + 28 q^{69} - 28 q^{71} - 3 q^{73} - 8 q^{75} - 5 q^{79} + 11 q^{81} + 10 q^{83} - 4 q^{85} - 10 q^{87} + 9 q^{89} - 18 q^{93} + q^{95} - 2 q^{97} - 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - q^5 - q^9 + 4 * q^11 + 2 * q^13 - 4 * q^15 + 2 * q^17 + q^19 + 7 * q^23 + 4 * q^25 + 8 * q^27 - 10 * q^29 + 9 * q^31 - 8 * q^33 + 2 * q^37 + 2 * q^39 + 4 * q^41 + 2 * q^43 - q^45 - 9 * q^47 - 4 * q^51 - 3 * q^53 - 8 * q^55 + 4 * q^57 - 14 * q^61 - q^65 - 10 * q^67 + 28 * q^69 - 28 * q^71 - 3 * q^73 - 8 * q^75 - 5 * q^79 + 11 * q^81 + 10 * q^83 - 4 * q^85 - 10 * q^87 + 9 * q^89 - 18 * q^93 + q^95 - 2 * q^97 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2548\mathbb{Z}\right)^\times$.

$n$	$197$	$885$	$1275$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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
$3$	1.00000	−	1.73205i	0.577350	−	1.00000i	−0.418432	−	0.908248i	$-0.637420\pi$
$3$	1.00000	−	1.73205i	0.577350	−	1.00000i	0.995782	−	0.0917517i	$-0.0292466\pi$
$4$		0			0
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i	0.732294	−	0.680989i	$-0.238450\pi$
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i	−0.955901	+	0.293691i	$0.905116\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−0.500000	−	0.866025i	−0.166667	−	0.288675i
$10$		0			0
$11$	2.00000	−	3.46410i	0.603023	−	1.04447i	−0.389338	−	0.921095i	$-0.627296\pi$
$11$	2.00000	−	3.46410i	0.603023	−	1.04447i	0.992361	−	0.123371i	$-0.0393705\pi$
$12$		0			0
$13$	1.00000			0.277350
$13$	1.00000			0.277350
$14$		0			0
$15$	−2.00000			−0.516398
$16$		0			0
$17$	1.00000	−	1.73205i	0.242536	−	0.420084i	−0.718900	−	0.695113i	$-0.755354\pi$
$17$	1.00000	−	1.73205i	0.242536	−	0.420084i	0.961436	+	0.275029i	$0.0886875\pi$
$18$		0			0
$19$	0.500000	+	0.866025i	0.114708	+	0.198680i	0.917663	−	0.397360i	$-0.130073\pi$
$19$	0.500000	+	0.866025i	0.114708	+	0.198680i	−0.802955	+	0.596040i	$0.796740\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.50000	+	6.06218i	0.729800	+	1.26405i	0.956967	+	0.290196i	$0.0937204\pi$
$23$	3.50000	+	6.06218i	0.729800	+	1.26405i	−0.227167	+	0.973856i	$0.572946\pi$
$24$		0			0
$25$	2.00000	−	3.46410i	0.400000	−	0.692820i
$26$		0			0
$27$	4.00000			0.769800
$28$		0			0
$29$	−5.00000			−0.928477			−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000			−0.928477			−0.464238	+	0.885710i	$0.653672\pi$
$30$		0			0
$31$	4.50000	−	7.79423i	0.808224	−	1.39988i	−0.105869	−	0.994380i	$-0.533762\pi$
$31$	4.50000	−	7.79423i	0.808224	−	1.39988i	0.914093	−	0.405505i	$-0.132904\pi$
$32$		0			0
$33$	−4.00000	−	6.92820i	−0.696311	−	1.20605i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	1.00000	+	1.73205i	0.164399	+	0.284747i	0.936442	−	0.350823i	$-0.114098\pi$
$37$	1.00000	+	1.73205i	0.164399	+	0.284747i	−0.772043	+	0.635571i	$0.780765\pi$
$38$		0			0
$39$	1.00000	−	1.73205i	0.160128	−	0.277350i
$40$		0			0
$41$	2.00000			0.312348			0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000			0.312348			0.156174	+	0.987730i	$0.450084\pi$
$42$		0			0
$43$	1.00000			0.152499			0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000			0.152499			0.0762493	+	0.997089i	$0.475706\pi$
$44$		0			0
$45$	−0.500000	+	0.866025i	−0.0745356	+	0.129099i
$46$		0			0
$47$	−4.50000	−	7.79423i	−0.656392	−	1.13691i	−0.981543	−	0.191243i	$-0.938748\pi$
$47$	−4.50000	−	7.79423i	−0.656392	−	1.13691i	0.325150	−	0.945662i	$-0.394585\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−2.00000	−	3.46410i	−0.280056	−	0.485071i
$52$		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
$57$	2.00000			0.264906
$58$		0			0
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$	−7.00000	−	12.1244i	−0.896258	−	1.55236i	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−7.00000	−	12.1244i	−0.896258	−	1.55236i	−0.0640184	−	0.997949i	$-0.520392\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.500000	−	0.866025i	−0.0620174	−	0.107417i
$66$		0			0
$67$	−5.00000	+	8.66025i	−0.610847	+	1.05802i	0.380251	+	0.924883i	$0.375838\pi$
$67$	−5.00000	+	8.66025i	−0.610847	+	1.05802i	−0.991098	+	0.133135i	$0.957496\pi$
$68$		0			0
$69$	14.0000			1.68540
$70$		0			0
$71$	−14.0000			−1.66149			−0.830747	−	0.556650i	$-0.812086\pi$
$71$	−14.0000			−1.66149			−0.830747	+	0.556650i	$0.812086\pi$
$72$		0			0
$73$	−1.50000	+	2.59808i	−0.175562	+	0.304082i	−0.940356	−	0.340193i	$-0.889507\pi$
$73$	−1.50000	+	2.59808i	−0.175562	+	0.304082i	0.764794	+	0.644275i	$0.222841\pi$
$74$		0			0
$75$	−4.00000	−	6.92820i	−0.461880	−	0.800000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−2.50000	−	4.33013i	−0.281272	−	0.487177i	0.690426	−	0.723403i	$-0.257423\pi$
$79$	−2.50000	−	4.33013i	−0.281272	−	0.487177i	−0.971698	+	0.236225i	$0.924090\pi$
$80$		0			0
$81$	5.50000	−	9.52628i	0.611111	−	1.05848i
$82$		0			0
$83$	5.00000			0.548821			0.274411	−	0.961613i	$-0.411517\pi$
$83$	5.00000			0.548821			0.274411	+	0.961613i	$0.411517\pi$
$84$		0			0
$85$	−2.00000			−0.216930
$86$		0			0
$87$	−5.00000	+	8.66025i	−0.536056	+	0.928477i
$88$		0			0
$89$	4.50000	+	7.79423i	0.476999	+	0.826187i	0.999653	−	0.0263586i	$-0.00839118\pi$
$89$	4.50000	+	7.79423i	0.476999	+	0.826187i	−0.522654	+	0.852545i	$0.675058\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−9.00000	−	15.5885i	−0.933257	−	1.61645i
$94$		0			0
$95$	0.500000	−	0.866025i	0.0512989	−	0.0888523i
$96$		0			0
$97$	−1.00000			−0.101535			−0.0507673	−	0.998711i	$-0.516167\pi$
$97$	−1.00000			−0.101535			−0.0507673	+	0.998711i	$0.516167\pi$
$98$		0			0
$99$	−4.00000			−0.402015
$100$		0			0
$101$	6.00000	−	10.3923i	0.597022	−	1.03407i	−0.396236	−	0.918149i	$-0.629684\pi$
$101$	6.00000	−	10.3923i	0.597022	−	1.03407i	0.993258	−	0.115924i	$-0.0369830\pi$
$102$		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
$105$		0			0
$106$		0			0
$107$	6.00000	+	10.3923i	0.580042	+	1.00466i	0.995474	+	0.0950377i	$0.0302972\pi$
$107$	6.00000	+	10.3923i	0.580042	+	1.00466i	−0.415432	+	0.909624i	$0.636370\pi$
$108$		0			0
$109$	2.00000	−	3.46410i	0.191565	−	0.331801i	−0.754204	−	0.656640i	$-0.771977\pi$
$109$	2.00000	−	3.46410i	0.191565	−	0.331801i	0.945769	+	0.324840i	$0.105310\pi$
$110$		0			0
$111$	4.00000			0.379663
$112$		0			0
$113$	1.00000			0.0940721			0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000			0.0940721			0.0470360	+	0.998893i	$0.485022\pi$
$114$		0			0
$115$	3.50000	−	6.06218i	0.326377	−	0.565301i
$116$		0			0
$117$	−0.500000	−	0.866025i	−0.0462250	−	0.0800641i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−2.50000	−	4.33013i	−0.227273	−	0.393648i
$122$		0			0
$123$	2.00000	−	3.46410i	0.180334	−	0.312348i
$124$		0			0
$125$	−9.00000			−0.804984
$126$		0			0
$127$	−8.00000			−0.709885			−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000			−0.709885			−0.354943	+	0.934888i	$0.615500\pi$
$128$		0			0
$129$	1.00000	−	1.73205i	0.0880451	−	0.152499i
$130$		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
$133$		0			0
$134$		0			0
$135$	−2.00000	−	3.46410i	−0.172133	−	0.298142i
$136$		0			0
$137$	7.00000	−	12.1244i	0.598050	−	1.03585i	−0.395058	−	0.918656i	$-0.629276\pi$
$137$	7.00000	−	12.1244i	0.598050	−	1.03585i	0.993109	−	0.117198i	$-0.0373911\pi$
$138$		0			0
$139$	16.0000			1.35710			0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000			1.35710			0.678551	+	0.734553i	$0.262608\pi$
$140$		0			0
$141$	−18.0000			−1.51587
$142$		0			0
$143$	2.00000	−	3.46410i	0.167248	−	0.289683i
$144$		0			0
$145$	2.50000	+	4.33013i	0.207614	+	0.359597i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−6.00000	−	10.3923i	−0.491539	−	0.851371i	0.508413	−	0.861113i	$-0.330232\pi$
$149$	−6.00000	−	10.3923i	−0.491539	−	0.851371i	−0.999953	+	0.00974235i	$0.996899\pi$
$150$		0			0
$151$	−7.00000	+	12.1244i	−0.569652	+	0.986666i	0.426948	+	0.904276i	$0.359589\pi$
$151$	−7.00000	+	12.1244i	−0.569652	+	0.986666i	−0.996600	+	0.0823900i	$0.973745\pi$
$152$		0			0
$153$	−2.00000			−0.161690
$154$		0			0
$155$	−9.00000			−0.722897
$156$		0			0
$157$	−5.00000	+	8.66025i	−0.399043	+	0.691164i	−0.993608	−	0.112884i	$-0.963991\pi$
$157$	−5.00000	+	8.66025i	−0.399043	+	0.691164i	0.594565	+	0.804048i	$0.297324\pi$
$158$		0			0
$159$	3.00000	+	5.19615i	0.237915	+	0.412082i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−12.0000	−	20.7846i	−0.939913	−	1.62798i	−0.765631	−	0.643280i	$-0.777573\pi$
$163$	−12.0000	−	20.7846i	−0.939913	−	1.62798i	−0.174282	−	0.984696i	$-0.555760\pi$
$164$		0			0
$165$	−4.00000	+	6.92820i	−0.311400	+	0.539360i
$166$		0			0
$167$	3.00000			0.232147			0.116073	−	0.993241i	$-0.462969\pi$
$167$	3.00000			0.232147			0.116073	+	0.993241i	$0.462969\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$	0.500000	−	0.866025i	0.0382360	−	0.0662266i
$172$		0			0
$173$	−1.00000	−	1.73205i	−0.0760286	−	0.131685i	0.825505	−	0.564396i	$-0.190891\pi$
$173$	−1.00000	−	1.73205i	−0.0760286	−	0.131685i	−0.901533	+	0.432710i	$0.857557\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.50000	+	7.79423i	−0.336346	+	0.582568i	−0.983742	−	0.179585i	$-0.942524\pi$
$179$	−4.50000	+	7.79423i	−0.336346	+	0.582568i	0.647397	+	0.762153i	$0.275858\pi$
$180$		0			0
$181$	−20.0000			−1.48659			−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−20.0000			−1.48659			−0.743294	+	0.668965i	$0.766738\pi$
$182$		0			0
$183$	−28.0000			−2.06982
$184$		0			0
$185$	1.00000	−	1.73205i	0.0735215	−	0.127343i
$186$		0			0
$187$	−4.00000	−	6.92820i	−0.292509	−	0.506640i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	12.0000	+	20.7846i	0.868290	+	1.50392i	0.863743	+	0.503932i	$0.168114\pi$
$191$	12.0000	+	20.7846i	0.868290	+	1.50392i	0.00454614	+	0.999990i	$0.498553\pi$
$192$		0			0
$193$	−1.00000	+	1.73205i	−0.0719816	+	0.124676i	−0.899770	−	0.436365i	$-0.856266\pi$
$193$	−1.00000	+	1.73205i	−0.0719816	+	0.124676i	0.827788	+	0.561041i	$0.189599\pi$
$194$		0			0
$195$	−2.00000			−0.143223
$196$		0			0
$197$	6.00000			0.427482			0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000			0.427482			0.213741	+	0.976890i	$0.431435\pi$
$198$		0			0
$199$	7.00000	−	12.1244i	0.496217	−	0.859473i	−0.503774	−	0.863836i	$-0.668055\pi$
$199$	7.00000	−	12.1244i	0.496217	−	0.859473i	0.999990	+	0.00436292i	$0.00138876\pi$
$200$		0			0
$201$	10.0000	+	17.3205i	0.705346	+	1.22169i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−1.00000	−	1.73205i	−0.0698430	−	0.120972i
$206$		0			0
$207$	3.50000	−	6.06218i	0.243267	−	0.421350i
$208$		0			0
$209$	4.00000			0.276686
$210$		0			0
$211$	−23.0000			−1.58339			−0.791693	−	0.610920i	$-0.790800\pi$
$211$	−23.0000			−1.58339			−0.791693	+	0.610920i	$0.790800\pi$
$212$		0			0
$213$	−14.0000	+	24.2487i	−0.959264	+	1.66149i
$214$		0			0
$215$	−0.500000	−	0.866025i	−0.0340997	−	0.0590624i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	3.00000	+	5.19615i	0.202721	+	0.351123i
$220$		0			0
$221$	1.00000	−	1.73205i	0.0672673	−	0.116510i
$222$		0			0
$223$	21.0000			1.40626			0.703132	−	0.711059i	$-0.251784\pi$
$223$	21.0000			1.40626			0.703132	+	0.711059i	$0.251784\pi$
$224$		0			0
$225$	−4.00000			−0.266667
$226$		0			0
$227$	4.00000	−	6.92820i	0.265489	−	0.459841i	−0.702202	−	0.711977i	$-0.747800\pi$
$227$	4.00000	−	6.92820i	0.265489	−	0.459841i	0.967692	+	0.252136i	$0.0811332\pi$
$228$		0			0
$229$	1.00000	+	1.73205i	0.0660819	+	0.114457i	0.897173	−	0.441679i	$-0.145617\pi$
$229$	1.00000	+	1.73205i	0.0660819	+	0.114457i	−0.831092	+	0.556136i	$0.812283\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	12.5000	+	21.6506i	0.818902	+	1.41838i	0.906492	+	0.422224i	$0.138750\pi$
$233$	12.5000	+	21.6506i	0.818902	+	1.41838i	−0.0875895	+	0.996157i	$0.527916\pi$
$234$		0			0
$235$	−4.50000	+	7.79423i	−0.293548	+	0.508439i
$236$		0			0
$237$	−10.0000			−0.649570
$238$		0			0
$239$	12.0000			0.776215			0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000			0.776215			0.388108	+	0.921614i	$0.373129\pi$
$240$		0			0
$241$	12.5000	−	21.6506i	0.805196	−	1.39464i	−0.110963	−	0.993825i	$-0.535394\pi$
$241$	12.5000	−	21.6506i	0.805196	−	1.39464i	0.916159	−	0.400815i	$-0.131273\pi$
$242$		0			0
$243$	−5.00000	−	8.66025i	−0.320750	−	0.555556i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	0.500000	+	0.866025i	0.0318142	+	0.0551039i
$248$		0			0
$249$	5.00000	−	8.66025i	0.316862	−	0.548821i
$250$		0			0
$251$	−16.0000			−1.00991			−0.504956	−	0.863145i	$-0.668491\pi$
$251$	−16.0000			−1.00991			−0.504956	+	0.863145i	$0.668491\pi$
$252$		0			0
$253$	28.0000			1.76034
$254$		0			0
$255$	−2.00000	+	3.46410i	−0.125245	+	0.216930i
$256$		0			0
$257$	6.00000	+	10.3923i	0.374270	+	0.648254i	0.990217	−	0.139533i	$-0.0445601\pi$
$257$	6.00000	+	10.3923i	0.374270	+	0.648254i	−0.615948	+	0.787787i	$0.711227\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	2.50000	+	4.33013i	0.154746	+	0.268028i
$262$		0			0
$263$	−9.50000	+	16.4545i	−0.585795	+	1.01463i	0.408981	+	0.912543i	$0.365884\pi$
$263$	−9.50000	+	16.4545i	−0.585795	+	1.01463i	−0.994776	+	0.102084i	$0.967449\pi$
$264$		0			0
$265$	3.00000			0.184289
$266$		0			0
$267$	18.0000			1.10158
$268$		0			0
$269$	−12.0000	+	20.7846i	−0.731653	+	1.26726i	0.224523	+	0.974469i	$0.427917\pi$
$269$	−12.0000	+	20.7846i	−0.731653	+	1.26726i	−0.956176	+	0.292791i	$0.905416\pi$
$270$		0			0
$271$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$271$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−8.00000	−	13.8564i	−0.482418	−	0.835573i
$276$		0			0
$277$	11.5000	−	19.9186i	0.690968	−	1.19679i	−0.280553	−	0.959839i	$-0.590518\pi$
$277$	11.5000	−	19.9186i	0.690968	−	1.19679i	0.971521	−	0.236953i	$-0.0761488\pi$
$278$		0			0
$279$	−9.00000			−0.538816
$280$		0			0
$281$	−8.00000			−0.477240			−0.238620	−	0.971113i	$-0.576695\pi$
$281$	−8.00000			−0.477240			−0.238620	+	0.971113i	$0.576695\pi$
$282$		0			0
$283$	2.00000	−	3.46410i	0.118888	−	0.205919i	−0.800439	−	0.599414i	$-0.795400\pi$
$283$	2.00000	−	3.46410i	0.118888	−	0.205919i	0.919327	+	0.393494i	$0.128734\pi$
$284$		0			0
$285$	−1.00000	−	1.73205i	−0.0592349	−	0.102598i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	6.50000	+	11.2583i	0.382353	+	0.662255i
$290$		0			0
$291$	−1.00000	+	1.73205i	−0.0586210	+	0.101535i
$292$		0			0
$293$	5.00000			0.292103			0.146052	−	0.989277i	$-0.453343\pi$
$293$	5.00000			0.292103			0.146052	+	0.989277i	$0.453343\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	8.00000	−	13.8564i	0.464207	−	0.804030i
$298$		0			0
$299$	3.50000	+	6.06218i	0.202410	+	0.350585i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−12.0000	−	20.7846i	−0.689382	−	1.19404i
$304$		0			0
$305$	−7.00000	+	12.1244i	−0.400819	+	0.694239i
$306$		0			0
$307$	−27.0000			−1.54097			−0.770486	−	0.637457i	$-0.779986\pi$
$307$	−27.0000			−1.54097			−0.770486	+	0.637457i	$0.779986\pi$
$308$		0			0
$309$	−8.00000			−0.455104
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	6.00000	+	10.3923i	0.339140	+	0.587408i	0.984271	−	0.176664i	$-0.0565306\pi$
$313$	6.00000	+	10.3923i	0.339140	+	0.587408i	−0.645131	+	0.764072i	$0.723197\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−10.0000	−	17.3205i	−0.561656	−	0.972817i	−0.997352	−	0.0727229i	$-0.976831\pi$
$317$	−10.0000	−	17.3205i	−0.561656	−	0.972817i	0.435696	−	0.900094i	$-0.356502\pi$
$318$		0			0
$319$	−10.0000	+	17.3205i	−0.559893	+	0.969762i
$320$		0			0
$321$	24.0000			1.33955
$322$		0			0
$323$	2.00000			0.111283
$324$		0			0
$325$	2.00000	−	3.46410i	0.110940	−	0.192154i
$326$		0			0
$327$	−4.00000	−	6.92820i	−0.221201	−	0.383131i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−15.0000	−	25.9808i	−0.824475	−	1.42803i	−0.902320	−	0.431066i	$-0.858137\pi$
$331$	−15.0000	−	25.9808i	−0.824475	−	1.42803i	0.0778456	−	0.996965i	$-0.475196\pi$
$332$		0			0
$333$	1.00000	−	1.73205i	0.0547997	−	0.0949158i
$334$		0			0
$335$	10.0000			0.546358
$336$		0			0
$337$	21.0000			1.14394			0.571971	−	0.820274i	$-0.306179\pi$
$337$	21.0000			1.14394			0.571971	+	0.820274i	$0.306179\pi$
$338$		0			0
$339$	1.00000	−	1.73205i	0.0543125	−	0.0940721i
$340$		0			0
$341$	−18.0000	−	31.1769i	−0.974755	−	1.68832i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−7.00000	−	12.1244i	−0.376867	−	0.652753i
$346$		0			0
$347$	−8.00000	+	13.8564i	−0.429463	+	0.743851i	−0.996826	−	0.0796169i	$-0.974630\pi$
$347$	−8.00000	+	13.8564i	−0.429463	+	0.743851i	0.567363	+	0.823468i	$0.307964\pi$
$348$		0			0
$349$	35.0000			1.87351			0.936754	−	0.349990i	$-0.113815\pi$
$349$	35.0000			1.87351			0.936754	+	0.349990i	$0.113815\pi$
$350$		0			0
$351$	4.00000			0.213504
$352$		0			0
$353$	1.00000	−	1.73205i	0.0532246	−	0.0921878i	−0.838186	−	0.545385i	$-0.816383\pi$
$353$	1.00000	−	1.73205i	0.0532246	−	0.0921878i	0.891410	+	0.453197i	$0.149717\pi$
$354$		0			0
$355$	7.00000	+	12.1244i	0.371521	+	0.643494i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	15.0000	+	25.9808i	0.791670	+	1.37121i	0.924932	+	0.380131i	$0.124121\pi$
$359$	15.0000	+	25.9808i	0.791670	+	1.37121i	−0.133263	+	0.991081i	$0.542545\pi$
$360$		0			0
$361$	9.00000	−	15.5885i	0.473684	−	0.820445i
$362$		0			0
$363$	−10.0000			−0.524864
$364$		0			0
$365$	3.00000			0.157027
$366$		0			0
$367$	−12.0000	+	20.7846i	−0.626395	+	1.08495i	0.361874	+	0.932227i	$0.382137\pi$
$367$	−12.0000	+	20.7846i	−0.626395	+	1.08495i	−0.988269	+	0.152721i	$0.951196\pi$
$368$		0			0
$369$	−1.00000	−	1.73205i	−0.0520579	−	0.0901670i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	13.0000	+	22.5167i	0.673114	+	1.16587i	0.977016	+	0.213165i	$0.0683772\pi$
$373$	13.0000	+	22.5167i	0.673114	+	1.16587i	−0.303902	+	0.952703i	$0.598289\pi$
$374$		0			0
$375$	−9.00000	+	15.5885i	−0.464758	+	0.804984i
$376$		0			0
$377$	−5.00000			−0.257513
$378$		0			0
$379$	28.0000			1.43826			0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000			1.43826			0.719132	+	0.694874i	$0.244540\pi$
$380$		0			0
$381$	−8.00000	+	13.8564i	−0.409852	+	0.709885i
$382$		0			0
$383$	−12.0000	−	20.7846i	−0.613171	−	1.06204i	−0.990702	−	0.136047i	$-0.956560\pi$
$383$	−12.0000	−	20.7846i	−0.613171	−	1.06204i	0.377531	−	0.925997i	$-0.376773\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−0.500000	−	0.866025i	−0.0254164	−	0.0440225i
$388$		0			0
$389$	11.0000	−	19.0526i	0.557722	−	0.966003i	−0.439964	−	0.898015i	$-0.645009\pi$
$389$	11.0000	−	19.0526i	0.557722	−	0.966003i	0.997686	−	0.0679877i	$-0.0216579\pi$
$390$		0			0
$391$	14.0000			0.708010
$392$		0			0
$393$	24.0000			1.21064
$394$		0			0
$395$	−2.50000	+	4.33013i	−0.125789	+	0.217872i
$396$		0			0
$397$	12.5000	+	21.6506i	0.627357	+	1.08661i	0.988080	+	0.153941i	$0.0491966\pi$
$397$	12.5000	+	21.6506i	0.627357	+	1.08661i	−0.360723	+	0.932673i	$0.617470\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	14.0000	+	24.2487i	0.699127	+	1.21092i	0.968770	+	0.247962i	$0.0797610\pi$
$401$	14.0000	+	24.2487i	0.699127	+	1.21092i	−0.269643	+	0.962960i	$0.586906\pi$
$402$		0			0
$403$	4.50000	−	7.79423i	0.224161	−	0.388258i
$404$		0			0
$405$	−11.0000			−0.546594
$406$		0			0
$407$	8.00000			0.396545
$408$		0			0
$409$	12.5000	−	21.6506i	0.618085	−	1.07056i	−0.371750	−	0.928333i	$-0.621242\pi$
$409$	12.5000	−	21.6506i	0.618085	−	1.07056i	0.989835	−	0.142222i	$-0.0454247\pi$
$410$		0			0
$411$	−14.0000	−	24.2487i	−0.690569	−	1.19610i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−2.50000	−	4.33013i	−0.122720	−	0.212558i
$416$		0			0
$417$	16.0000	−	27.7128i	0.783523	−	1.35710i
$418$		0			0
$419$	−16.0000			−0.781651			−0.390826	−	0.920465i	$-0.627810\pi$
$419$	−16.0000			−0.781651			−0.390826	+	0.920465i	$0.627810\pi$
$420$		0			0
$421$	10.0000			0.487370			0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000			0.487370			0.243685	+	0.969854i	$0.421644\pi$
$422$		0			0
$423$	−4.50000	+	7.79423i	−0.218797	+	0.378968i
$424$		0			0
$425$	−4.00000	−	6.92820i	−0.194029	−	0.336067i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−4.00000	−	6.92820i	−0.193122	−	0.334497i
$430$		0			0
$431$	−7.00000	+	12.1244i	−0.337178	+	0.584010i	−0.983901	−	0.178716i	$-0.942806\pi$
$431$	−7.00000	+	12.1244i	−0.337178	+	0.584010i	0.646723	+	0.762725i	$0.276139\pi$
$432$		0			0
$433$	24.0000			1.15337			0.576683	−	0.816968i	$-0.304347\pi$
$433$	24.0000			1.15337			0.576683	+	0.816968i	$0.304347\pi$
$434$		0			0
$435$	10.0000			0.479463
$436$		0			0
$437$	−3.50000	+	6.06218i	−0.167428	+	0.289993i
$438$		0			0
$439$	−5.00000	−	8.66025i	−0.238637	−	0.413331i	0.721686	−	0.692220i	$-0.243367\pi$
$439$	−5.00000	−	8.66025i	−0.238637	−	0.413331i	−0.960323	+	0.278889i	$0.910034\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	19.5000	+	33.7750i	0.926473	+	1.60470i	0.789175	+	0.614168i	$0.210508\pi$
$443$	19.5000	+	33.7750i	0.926473	+	1.60470i	0.137298	+	0.990530i	$0.456158\pi$
$444$		0			0
$445$	4.50000	−	7.79423i	0.213320	−	0.369482i
$446$		0			0
$447$	−24.0000			−1.13516
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$	4.00000	−	6.92820i	0.188353	−	0.326236i
$452$		0			0
$453$	14.0000	+	24.2487i	0.657777	+	1.13930i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	3.00000	+	5.19615i	0.140334	+	0.243066i	0.927622	−	0.373519i	$-0.121849\pi$
$457$	3.00000	+	5.19615i	0.140334	+	0.243066i	−0.787288	+	0.616585i	$0.788516\pi$
$458$		0			0
$459$	4.00000	−	6.92820i	0.186704	−	0.323381i
$460$		0			0
$461$	10.0000			0.465746			0.232873	−	0.972507i	$-0.425187\pi$
$461$	10.0000			0.465746			0.232873	+	0.972507i	$0.425187\pi$
$462$		0			0
$463$	24.0000			1.11537			0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000			1.11537			0.557687	+	0.830051i	$0.311689\pi$
$464$		0			0
$465$	−9.00000	+	15.5885i	−0.417365	+	0.722897i
$466$		0			0
$467$	4.00000	+	6.92820i	0.185098	+	0.320599i	0.943610	−	0.331061i	$-0.107406\pi$
$467$	4.00000	+	6.92820i	0.185098	+	0.320599i	−0.758512	+	0.651660i	$0.774073\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	10.0000	+	17.3205i	0.460776	+	0.798087i
$472$		0			0
$473$	2.00000	−	3.46410i	0.0919601	−	0.159280i
$474$		0			0
$475$	4.00000			0.183533
$476$		0			0
$477$	3.00000			0.137361
$478$		0			0
$479$	−13.5000	+	23.3827i	−0.616831	+	1.06838i	0.373230	+	0.927739i	$0.378250\pi$
$479$	−13.5000	+	23.3827i	−0.616831	+	1.06838i	−0.990060	+	0.140643i	$0.955083\pi$
$480$		0			0
$481$	1.00000	+	1.73205i	0.0455961	+	0.0789747i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	0.500000	+	0.866025i	0.0227038	+	0.0393242i
$486$		0			0
$487$	17.0000	−	29.4449i	0.770344	−	1.33427i	−0.167031	−	0.985952i	$-0.553418\pi$
$487$	17.0000	−	29.4449i	0.770344	−	1.33427i	0.937375	−	0.348323i	$-0.113249\pi$
$488$		0			0
$489$	−48.0000			−2.17064
$490$		0			0
$491$	−8.00000			−0.361035			−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000			−0.361035			−0.180517	+	0.983572i	$0.557777\pi$
$492$		0			0
$493$	−5.00000	+	8.66025i	−0.225189	+	0.390038i
$494$		0			0
$495$	2.00000	+	3.46410i	0.0898933	+	0.155700i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	1.00000	+	1.73205i	0.0447661	+	0.0775372i	0.887540	−	0.460730i	$-0.152412\pi$
$499$	1.00000	+	1.73205i	0.0447661	+	0.0775372i	−0.842774	+	0.538267i	$0.819079\pi$
$500$		0			0
$501$	3.00000	−	5.19615i	0.134030	−	0.232147i
$502$		0			0
$503$	−34.0000			−1.51599			−0.757993	−	0.652263i	$-0.773820\pi$
$503$	−34.0000			−1.51599			−0.757993	+	0.652263i	$0.773820\pi$
$504$		0			0
$505$	−12.0000			−0.533993
$506$		0			0
$507$	1.00000	−	1.73205i	0.0444116	−	0.0769231i
$508$		0			0
$509$	−4.50000	−	7.79423i	−0.199459	−	0.345473i	0.748894	−	0.662690i	$-0.230585\pi$
$509$	−4.50000	−	7.79423i	−0.199459	−	0.345473i	−0.948353	+	0.317217i	$0.897252\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	2.00000	+	3.46410i	0.0883022	+	0.152944i
$514$		0			0
$515$	−2.00000	+	3.46410i	−0.0881305	+	0.152647i
$516$		0			0
$517$	−36.0000			−1.58328
$518$		0			0
$519$	−4.00000			−0.175581
$520$		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
$523$	9.00000	+	15.5885i	0.393543	+	0.681636i	0.992914	−	0.118835i	$-0.0379161\pi$
$523$	9.00000	+	15.5885i	0.393543	+	0.681636i	−0.599371	+	0.800471i	$0.704583\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−9.00000	−	15.5885i	−0.392046	−	0.679044i
$528$		0			0
$529$	−13.0000	+	22.5167i	−0.565217	+	0.978985i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	2.00000			0.0866296
$534$		0			0
$535$	6.00000	−	10.3923i	0.259403	−	0.449299i
$536$		0			0
$537$	9.00000	+	15.5885i	0.388379	+	0.672692i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	19.0000	+	32.9090i	0.816874	+	1.41487i	0.907975	+	0.419025i	$0.137628\pi$
$541$	19.0000	+	32.9090i	0.816874	+	1.41487i	−0.0911008	+	0.995842i	$0.529039\pi$
$542$		0			0
$543$	−20.0000	+	34.6410i	−0.858282	+	1.48659i
$544$		0			0
$545$	−4.00000			−0.171341
$546$		0			0
$547$	23.0000			0.983409			0.491704	−	0.870762i	$-0.336374\pi$
$547$	23.0000			0.983409			0.491704	+	0.870762i	$0.336374\pi$
$548$		0			0
$549$	−7.00000	+	12.1244i	−0.298753	+	0.517455i
$550$		0			0
$551$	−2.50000	−	4.33013i	−0.106504	−	0.184470i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−2.00000	−	3.46410i	−0.0848953	−	0.147043i
$556$		0			0
$557$	−2.00000	+	3.46410i	−0.0847427	+	0.146779i	−0.905282	−	0.424812i	$-0.860340\pi$
$557$	−2.00000	+	3.46410i	−0.0847427	+	0.146779i	0.820539	+	0.571591i	$0.193674\pi$
$558$		0			0
$559$	1.00000			0.0422955
$560$		0			0
$561$	−16.0000			−0.675521
$562$		0			0
$563$	20.0000	−	34.6410i	0.842900	−	1.45994i	−0.0445334	−	0.999008i	$-0.514180\pi$
$563$	20.0000	−	34.6410i	0.842900	−	1.45994i	0.887433	−	0.460937i	$-0.152487\pi$
$564$		0			0
$565$	−0.500000	−	0.866025i	−0.0210352	−	0.0364340i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	18.5000	+	32.0429i	0.775560	+	1.34331i	0.934479	+	0.356018i	$0.115866\pi$
$569$	18.5000	+	32.0429i	0.775560	+	1.34331i	−0.158919	+	0.987292i	$0.550801\pi$
$570$		0			0
$571$	3.50000	−	6.06218i	0.146470	−	0.253694i	−0.783450	−	0.621455i	$-0.786542\pi$
$571$	3.50000	−	6.06218i	0.146470	−	0.253694i	0.929921	+	0.367760i	$0.119875\pi$
$572$		0			0
$573$	48.0000			2.00523
$574$		0			0
$575$	28.0000			1.16768
$576$		0			0
$577$	9.00000	−	15.5885i	0.374675	−	0.648956i	−0.615603	−	0.788056i	$-0.711088\pi$
$577$	9.00000	−	15.5885i	0.374675	−	0.648956i	0.990278	+	0.139100i	$0.0444210\pi$
$578$		0			0
$579$	2.00000	+	3.46410i	0.0831172	+	0.143963i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	6.00000	+	10.3923i	0.248495	+	0.430405i
$584$		0			0
$585$	−0.500000	+	0.866025i	−0.0206725	+	0.0358057i
$586$		0			0
$587$	−15.0000			−0.619116			−0.309558	−	0.950881i	$-0.600181\pi$
$587$	−15.0000			−0.619116			−0.309558	+	0.950881i	$0.600181\pi$
$588$		0			0
$589$	9.00000			0.370839
$590$		0			0
$591$	6.00000	−	10.3923i	0.246807	−	0.427482i
$592$		0			0
$593$	−20.5000	−	35.5070i	−0.841834	−	1.45810i	−0.888342	−	0.459182i	$-0.848143\pi$
$593$	−20.5000	−	35.5070i	−0.841834	−	1.45810i	0.0465084	−	0.998918i	$-0.485191\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−14.0000	−	24.2487i	−0.572982	−	0.992434i
$598$		0			0
$599$	−10.5000	+	18.1865i	−0.429018	+	0.743082i	−0.996786	−	0.0801071i	$-0.974474\pi$
$599$	−10.5000	+	18.1865i	−0.429018	+	0.743082i	0.567768	+	0.823189i	$0.307807\pi$
$600$		0			0
$601$	2.00000			0.0815817			0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000			0.0815817			0.0407909	+	0.999168i	$0.487012\pi$
$602$		0			0
$603$	10.0000			0.407231
$604$		0			0
$605$	−2.50000	+	4.33013i	−0.101639	+	0.176045i
$606$		0			0
$607$	20.0000	+	34.6410i	0.811775	+	1.40604i	0.911621	+	0.411033i	$0.134832\pi$
$607$	20.0000	+	34.6410i	0.811775	+	1.40604i	−0.0998457	+	0.995003i	$0.531835\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−4.50000	−	7.79423i	−0.182051	−	0.315321i
$612$		0			0
$613$	8.00000	−	13.8564i	0.323117	−	0.559655i	−0.658012	−	0.753007i	$-0.728603\pi$
$613$	8.00000	−	13.8564i	0.323117	−	0.559655i	0.981129	+	0.193352i	$0.0619359\pi$
$614$		0			0
$615$	−4.00000			−0.161296
$616$		0			0
$617$	34.0000			1.36879			0.684394	−	0.729112i	$-0.260067\pi$
$617$	34.0000			1.36879			0.684394	+	0.729112i	$0.260067\pi$
$618$		0			0
$619$	−10.0000	+	17.3205i	−0.401934	+	0.696170i	−0.993959	−	0.109749i	$-0.964995\pi$
$619$	−10.0000	+	17.3205i	−0.401934	+	0.696170i	0.592025	+	0.805919i	$0.298329\pi$
$620$		0			0
$621$	14.0000	+	24.2487i	0.561801	+	0.973067i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−5.50000	−	9.52628i	−0.220000	−	0.381051i
$626$		0			0
$627$	4.00000	−	6.92820i	0.159745	−	0.276686i
$628$		0			0
$629$	4.00000			0.159490
$630$		0			0
$631$	−2.00000			−0.0796187			−0.0398094	−	0.999207i	$-0.512675\pi$
$631$	−2.00000			−0.0796187			−0.0398094	+	0.999207i	$0.512675\pi$
$632$		0			0
$633$	−23.0000	+	39.8372i	−0.914168	+	1.58339i
$634$		0			0
$635$	4.00000	+	6.92820i	0.158735	+	0.274937i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	7.00000	+	12.1244i	0.276916	+	0.479632i
$640$		0			0
$641$	−20.5000	+	35.5070i	−0.809701	+	1.40244i	0.103370	+	0.994643i	$0.467038\pi$
$641$	−20.5000	+	35.5070i	−0.809701	+	1.40244i	−0.913071	+	0.407801i	$0.866296\pi$
$642$		0			0
$643$	−28.0000			−1.10421			−0.552106	−	0.833774i	$-0.686176\pi$
$643$	−28.0000			−1.10421			−0.552106	+	0.833774i	$0.686176\pi$
$644$		0			0
$645$	−2.00000			−0.0787499
$646$		0			0
$647$	−6.00000	+	10.3923i	−0.235884	+	0.408564i	−0.959529	−	0.281609i	$-0.909132\pi$
$647$	−6.00000	+	10.3923i	−0.235884	+	0.408564i	0.723645	+	0.690172i	$0.242465\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	3.00000	+	5.19615i	0.117399	+	0.203341i	0.918736	−	0.394872i	$-0.129211\pi$
$653$	3.00000	+	5.19615i	0.117399	+	0.203341i	−0.801337	+	0.598213i	$0.795878\pi$
$654$		0			0
$655$	6.00000	−	10.3923i	0.234439	−	0.406061i
$656$		0			0
$657$	3.00000			0.117041
$658$		0			0
$659$	−45.0000			−1.75295			−0.876476	−	0.481446i	$-0.840112\pi$
$659$	−45.0000			−1.75295			−0.876476	+	0.481446i	$0.840112\pi$
$660$		0			0
$661$	−15.5000	+	26.8468i	−0.602880	+	1.04422i	0.389503	+	0.921025i	$0.372647\pi$
$661$	−15.5000	+	26.8468i	−0.602880	+	1.04422i	−0.992383	+	0.123194i	$0.960686\pi$
$662$		0			0
$663$	−2.00000	−	3.46410i	−0.0776736	−	0.134535i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−17.5000	−	30.3109i	−0.677603	−	1.17364i
$668$		0			0
$669$	21.0000	−	36.3731i	0.811907	−	1.40626i
$670$		0			0
$671$	−56.0000			−2.16186
$672$		0			0
$673$	21.0000			0.809491			0.404745	−	0.914429i	$-0.367360\pi$
$673$	21.0000			0.809491			0.404745	+	0.914429i	$0.367360\pi$
$674$		0			0
$675$	8.00000	−	13.8564i	0.307920	−	0.533333i
$676$		0			0
$677$	−8.00000	−	13.8564i	−0.307465	−	0.532545i	0.670342	−	0.742052i	$-0.266147\pi$
$677$	−8.00000	−	13.8564i	−0.307465	−	0.532545i	−0.977807	+	0.209507i	$0.932814\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−8.00000	−	13.8564i	−0.306561	−	0.530979i
$682$		0			0
$683$	−2.00000	+	3.46410i	−0.0765279	+	0.132550i	−0.901750	−	0.432259i	$-0.857717\pi$
$683$	−2.00000	+	3.46410i	−0.0765279	+	0.132550i	0.825222	+	0.564809i	$0.191050\pi$
$684$		0			0
$685$	−14.0000			−0.534913
$686$		0			0
$687$	4.00000			0.152610
$688$		0			0
$689$	−1.50000	+	2.59808i	−0.0571454	+	0.0989788i
$690$		0			0
$691$	17.5000	+	30.3109i	0.665731	+	1.15308i	0.979086	+	0.203445i	$0.0652137\pi$
$691$	17.5000	+	30.3109i	0.665731	+	1.15308i	−0.313355	+	0.949636i	$0.601453\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−8.00000	−	13.8564i	−0.303457	−	0.525603i
$696$		0			0
$697$	2.00000	−	3.46410i	0.0757554	−	0.131212i
$698$		0			0
$699$	50.0000			1.89117
$700$		0			0
$701$	45.0000			1.69963			0.849813	−	0.527084i	$-0.176715\pi$
$701$	45.0000			1.69963			0.849813	+	0.527084i	$0.176715\pi$
$702$		0			0
$703$	−1.00000	+	1.73205i	−0.0377157	+	0.0653255i
$704$		0			0
$705$	9.00000	+	15.5885i	0.338960	+	0.587095i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	6.00000	+	10.3923i	0.225335	+	0.390291i	0.956420	−	0.291995i	$-0.0943191\pi$
$709$	6.00000	+	10.3923i	0.225335	+	0.390291i	−0.731085	+	0.682286i	$0.760986\pi$
$710$		0			0
$711$	−2.50000	+	4.33013i	−0.0937573	+	0.162392i
$712$		0			0
$713$	63.0000			2.35937
$714$		0			0
$715$	−4.00000			−0.149592
$716$		0			0
$717$	12.0000	−	20.7846i	0.448148	−	0.776215i
$718$		0			0
$719$	20.0000	+	34.6410i	0.745874	+	1.29189i	0.949785	+	0.312903i	$0.101301\pi$
$719$	20.0000	+	34.6410i	0.745874	+	1.29189i	−0.203911	+	0.978989i	$0.565365\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−25.0000	−	43.3013i	−0.929760	−	1.61039i
$724$		0			0
$725$	−10.0000	+	17.3205i	−0.371391	+	0.643268i
$726$		0			0
$727$	−42.0000			−1.55769			−0.778847	−	0.627214i	$-0.784195\pi$
$727$	−42.0000			−1.55769			−0.778847	+	0.627214i	$0.784195\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		0			0
$731$	1.00000	−	1.73205i	0.0369863	−	0.0640622i
$732$		0			0
$733$	10.5000	+	18.1865i	0.387826	+	0.671735i	0.992157	−	0.124999i	$-0.0398927\pi$
$733$	10.5000	+	18.1865i	0.387826	+	0.671735i	−0.604331	+	0.796734i	$0.706559\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	20.0000	+	34.6410i	0.736709	+	1.27602i
$738$		0			0
$739$	4.00000	−	6.92820i	0.147142	−	0.254858i	−0.783028	−	0.621987i	$-0.786326\pi$
$739$	4.00000	−	6.92820i	0.147142	−	0.254858i	0.930170	+	0.367129i	$0.119659\pi$
$740$		0			0
$741$	2.00000			0.0734718
$742$		0			0
$743$	−8.00000			−0.293492			−0.146746	−	0.989174i	$-0.546880\pi$
$743$	−8.00000			−0.293492			−0.146746	+	0.989174i	$0.546880\pi$
$744$		0			0
$745$	−6.00000	+	10.3923i	−0.219823	+	0.380745i
$746$		0			0
$747$	−2.50000	−	4.33013i	−0.0914702	−	0.158431i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−12.5000	−	21.6506i	−0.456131	−	0.790043i	0.542621	−	0.839978i	$-0.317432\pi$
$751$	−12.5000	−	21.6506i	−0.456131	−	0.790043i	−0.998752	+	0.0499348i	$0.984099\pi$
$752$		0			0
$753$	−16.0000	+	27.7128i	−0.583072	+	1.00991i
$754$		0			0
$755$	14.0000			0.509512
$756$		0			0
$757$	−47.0000			−1.70824			−0.854122	−	0.520073i	$-0.825905\pi$
$757$	−47.0000			−1.70824			−0.854122	+	0.520073i	$0.825905\pi$
$758$		0			0
$759$	28.0000	−	48.4974i	1.01634	−	1.76034i
$760$		0			0
$761$	17.5000	+	30.3109i	0.634375	+	1.09877i	0.986647	+	0.162872i	$0.0520756\pi$
$761$	17.5000	+	30.3109i	0.634375	+	1.09877i	−0.352273	+	0.935897i	$0.614591\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	1.00000	+	1.73205i	0.0361551	+	0.0626224i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	37.0000			1.33425			0.667127	−	0.744944i	$-0.267524\pi$
$769$	37.0000			1.33425			0.667127	+	0.744944i	$0.267524\pi$
$770$		0			0
$771$	24.0000			0.864339
$772$		0			0
$773$	−19.0000	+	32.9090i	−0.683383	+	1.18365i	0.290560	+	0.956857i	$0.406159\pi$
$773$	−19.0000	+	32.9090i	−0.683383	+	1.18365i	−0.973942	+	0.226796i	$0.927175\pi$
$774$		0			0
$775$	−18.0000	−	31.1769i	−0.646579	−	1.11991i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.00000	+	1.73205i	0.0358287	+	0.0620572i
$780$		0			0
$781$	−28.0000	+	48.4974i	−1.00192	+	1.73537i
$782$		0			0
$783$	−20.0000			−0.714742
$784$		0			0
$785$	10.0000			0.356915
$786$		0			0
$787$	18.5000	−	32.0429i	0.659454	−	1.14221i	−0.321303	−	0.946976i	$-0.604121\pi$
$787$	18.5000	−	32.0429i	0.659454	−	1.14221i	0.980757	−	0.195231i	$-0.0625457\pi$
$788$		0			0
$789$	19.0000	+	32.9090i	0.676418	+	1.17159i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−7.00000	−	12.1244i	−0.248577	−	0.430548i
$794$		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:	\( N \)	\(=\)	\( 2548 = 2^{2} \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2548.j (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.73205i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(1.00000 - 1.73205i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(-0.500000 - 0.866025i) q^{9} +(2.00000 - 3.46410i) q^{11} +1.00000 q^{13} -2.00000 q^{15} +(1.00000 - 1.73205i) q^{17} +(0.500000 + 0.866025i) q^{19} +(3.50000 + 6.06218i) q^{23} +(2.00000 - 3.46410i) q^{25} +4.00000 q^{27} -5.00000 q^{29} +(4.50000 - 7.79423i) q^{31} +(-4.00000 - 6.92820i) q^{33} +(1.00000 + 1.73205i) q^{37} +(1.00000 - 1.73205i) q^{39} +2.00000 q^{41} +1.00000 q^{43} +(-0.500000 + 0.866025i) q^{45} +(-4.50000 - 7.79423i) q^{47} +(-2.00000 - 3.46410i) q^{51} +(-1.50000 + 2.59808i) q^{53} -4.00000 q^{55} +2.00000 q^{57} +(-7.00000 - 12.1244i) q^{61} +(-0.500000 - 0.866025i) q^{65} +(-5.00000 + 8.66025i) q^{67} +14.0000 q^{69} -14.0000 q^{71} +(-1.50000 + 2.59808i) q^{73} +(-4.00000 - 6.92820i) q^{75} +(-2.50000 - 4.33013i) q^{79} +(5.50000 - 9.52628i) q^{81} +5.00000 q^{83} -2.00000 q^{85} +(-5.00000 + 8.66025i) q^{87} +(4.50000 + 7.79423i) q^{89} +(-9.00000 - 15.5885i) q^{93} +(0.500000 - 0.866025i) q^{95} -1.00000 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - q^{5} - q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - q^5 - q^9	\( 2 q + 2 q^{3} - q^{5} - q^{9} + 4 q^{11} + 2 q^{13} - 4 q^{15} + 2 q^{17} + q^{19} + 7 q^{23} + 4 q^{25} + 8 q^{27} - 10 q^{29} + 9 q^{31} - 8 q^{33} + 2 q^{37} + 2 q^{39} + 4 q^{41} + 2 q^{43} - q^{45} - 9 q^{47} - 4 q^{51} - 3 q^{53} - 8 q^{55} + 4 q^{57} - 14 q^{61} - q^{65} - 10 q^{67} + 28 q^{69} - 28 q^{71} - 3 q^{73} - 8 q^{75} - 5 q^{79} + 11 q^{81} + 10 q^{83} - 4 q^{85} - 10 q^{87} + 9 q^{89} - 18 q^{93} + q^{95} - 2 q^{97} - 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - q^5 - q^9 + 4 * q^11 + 2 * q^13 - 4 * q^15 + 2 * q^17 + q^19 + 7 * q^23 + 4 * q^25 + 8 * q^27 - 10 * q^29 + 9 * q^31 - 8 * q^33 + 2 * q^37 + 2 * q^39 + 4 * q^41 + 2 * q^43 - q^45 - 9 * q^47 - 4 * q^51 - 3 * q^53 - 8 * q^55 + 4 * q^57 - 14 * q^61 - q^65 - 10 * q^67 + 28 * q^69 - 28 * q^71 - 3 * q^73 - 8 * q^75 - 5 * q^79 + 11 * q^81 + 10 * q^83 - 4 * q^85 - 10 * q^87 + 9 * q^89 - 18 * q^93 + q^95 - 2 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more