LMFDB - Embedded newform 3528.2.s.i.361.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3528,2,Mod(361,3528)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3528.361"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3528, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 2, names="a")

Level:	$N$	$=$	$3528 = 2^{3} \cdot 3^{2} \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3528.s (of order $3$ , degree $2$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-2,0,0,0,0,0,2,0,-4,0,0,0,-6,0,-4,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	no
Analytic conductor:	$28.1712218331$
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:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	3528.361
Dual form			3528.2.s.i.3313.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 1.73205i) q^{5} +(1.00000 - 1.73205i) q^{11} -2.00000 q^{13} +(-3.00000 + 5.19615i) q^{17} +(-2.00000 - 3.46410i) q^{19} +(3.00000 + 5.19615i) q^{23} +(0.500000 - 0.866025i) q^{25} +(-2.00000 + 3.46410i) q^{31} +(-5.00000 - 8.66025i) q^{37} +2.00000 q^{41} -4.00000 q^{43} +(-2.00000 - 3.46410i) q^{47} +(-6.00000 + 10.3923i) q^{53} -4.00000 q^{55} +(-6.00000 + 10.3923i) q^{59} +(3.00000 + 5.19615i) q^{61} +(2.00000 + 3.46410i) q^{65} +(2.00000 - 3.46410i) q^{67} +14.0000 q^{71} +(-1.00000 + 1.73205i) q^{73} +(4.00000 + 6.92820i) q^{79} -16.0000 q^{83} +12.0000 q^{85} +(3.00000 + 5.19615i) q^{89} +(-4.00000 + 6.92820i) q^{95} +18.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{5} + 2 q^{11} - 4 q^{13} - 6 q^{17} - 4 q^{19} + 6 q^{23} + q^{25} - 4 q^{31} - 10 q^{37} + 4 q^{41} - 8 q^{43} - 4 q^{47} - 12 q^{53} - 8 q^{55} - 12 q^{59} + 6 q^{61} + 4 q^{65} + 4 q^{67}+ \cdots + 36 q^{97}+O(q^{100})$ 2 * q - 2 * q^5 + 2 * q^11 - 4 * q^13 - 6 * q^17 - 4 * q^19 + 6 * q^23 + q^25 - 4 * q^31 - 10 * q^37 + 4 * q^41 - 8 * q^43 - 4 * q^47 - 12 * q^53 - 8 * q^55 - 12 * q^59 + 6 * q^61 + 4 * q^65 + 4 * q^67 + 28 * q^71 - 2 * q^73 + 8 * q^79 - 32 * q^83 + 24 * q^85 + 6 * q^89 - 8 * q^95 + 36 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times$ .

$n$	$785$	$1081$	$1765$	$2647$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.00000	−	1.73205i	−0.447214	−	0.774597i	0.550990	−	0.834512i	$-0.314250\pi$
$5$	−1.00000	−	1.73205i	−0.447214	−	0.774597i	−0.998203	+	0.0599153i	$0.980917\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.00000	−	1.73205i	0.301511	−	0.522233i	−0.674967	−	0.737848i	$-0.735842\pi$
$11$	1.00000	−	1.73205i	0.301511	−	0.522233i	0.976478	+	0.215615i	$0.0691756\pi$
$12$		0			0
$13$	−2.00000			−0.554700			−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000			−0.554700			−0.277350	+	0.960769i	$0.589456\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.00000	+	5.19615i	−0.727607	+	1.26025i	0.230285	+	0.973123i	$0.426034\pi$
$17$	−3.00000	+	5.19615i	−0.727607	+	1.26025i	−0.957892	+	0.287129i	$0.907299\pi$
$18$		0			0
$19$	−2.00000	−	3.46410i	−0.458831	−	0.794719i	0.540068	−	0.841621i	$-0.318398\pi$
$19$	−2.00000	−	3.46410i	−0.458831	−	0.794719i	−0.998899	+	0.0469020i	$0.985065\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.00000	+	5.19615i	0.625543	+	1.08347i	0.988436	+	0.151642i	$0.0484560\pi$
$23$	3.00000	+	5.19615i	0.625543	+	1.08347i	−0.362892	+	0.931831i	$0.618211\pi$
$24$		0			0
$25$	0.500000	−	0.866025i	0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−2.00000	+	3.46410i	−0.359211	+	0.622171i	−0.987829	−	0.155543i	$-0.950287\pi$
$31$	−2.00000	+	3.46410i	−0.359211	+	0.622171i	0.628619	+	0.777714i	$0.283621\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−5.00000	−	8.66025i	−0.821995	−	1.42374i	−0.904194	−	0.427121i	$-0.859528\pi$
$37$	−5.00000	−	8.66025i	−0.821995	−	1.42374i	0.0821995	−	0.996616i	$-0.473806\pi$
$38$		0			0
$39$		0			0
$40$		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
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.00000	−	3.46410i	−0.291730	−	0.505291i	0.682489	−	0.730896i	$-0.260898\pi$
$47$	−2.00000	−	3.46410i	−0.291730	−	0.505291i	−0.974219	+	0.225605i	$0.927564\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−6.00000	+	10.3923i	−0.824163	+	1.42749i	0.0783936	+	0.996922i	$0.475021\pi$
$53$	−6.00000	+	10.3923i	−0.824163	+	1.42749i	−0.902557	+	0.430570i	$0.858312\pi$
$54$		0			0
$55$	−4.00000			−0.539360
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.00000	+	10.3923i	−0.781133	+	1.35296i	0.150148	+	0.988663i	$0.452025\pi$
$59$	−6.00000	+	10.3923i	−0.781133	+	1.35296i	−0.931282	+	0.364299i	$0.881308\pi$
$60$		0			0
$61$	3.00000	+	5.19615i	0.384111	+	0.665299i	0.991645	−	0.128994i	$-0.0411748\pi$
$61$	3.00000	+	5.19615i	0.384111	+	0.665299i	−0.607535	+	0.794293i	$0.707841\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.00000	+	3.46410i	0.248069	+	0.429669i
$66$		0			0
$67$	2.00000	−	3.46410i	0.244339	−	0.423207i	−0.717607	−	0.696449i	$-0.754762\pi$
$67$	2.00000	−	3.46410i	0.244339	−	0.423207i	0.961946	+	0.273241i	$0.0880957\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	14.0000			1.66149			0.830747	−	0.556650i	$-0.187914\pi$
$71$	14.0000			1.66149			0.830747	+	0.556650i	$0.187914\pi$
$72$		0			0
$73$	−1.00000	+	1.73205i	−0.117041	+	0.202721i	−0.918594	−	0.395203i	$-0.870674\pi$
$73$	−1.00000	+	1.73205i	−0.117041	+	0.202721i	0.801553	+	0.597924i	$0.204008\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	4.00000	+	6.92820i	0.450035	+	0.779484i	0.998388	−	0.0567635i	$-0.0180781\pi$
$79$	4.00000	+	6.92820i	0.450035	+	0.779484i	−0.548352	+	0.836247i	$0.684745\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−16.0000			−1.75623			−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000			−1.75623			−0.878114	+	0.478451i	$0.841198\pi$
$84$		0			0
$85$	12.0000			1.30158
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.00000	+	5.19615i	0.317999	+	0.550791i	0.980071	−	0.198650i	$-0.0636557\pi$
$89$	3.00000	+	5.19615i	0.317999	+	0.550791i	−0.662071	+	0.749441i	$0.730322\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.00000	+	6.92820i	−0.410391	+	0.710819i
$96$		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
$99$		0			0
$100$		0			0
$101$	−7.00000	+	12.1244i	−0.696526	+	1.20642i	0.273138	+	0.961975i	$0.411939\pi$
$101$	−7.00000	+	12.1244i	−0.696526	+	1.20642i	−0.969664	+	0.244443i	$0.921395\pi$
$102$		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
$107$	−3.00000	−	5.19615i	−0.290021	−	0.502331i	0.683793	−	0.729676i	$-0.260329\pi$
$107$	−3.00000	−	5.19615i	−0.290021	−	0.502331i	−0.973814	+	0.227345i	$0.926996\pi$
$108$		0			0
$109$	−1.00000	+	1.73205i	−0.0957826	+	0.165900i	−0.909935	−	0.414751i	$-0.863869\pi$
$109$	−1.00000	+	1.73205i	−0.0957826	+	0.165900i	0.814152	+	0.580651i	$0.197202\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	16.0000			1.50515			0.752577	−	0.658505i	$-0.228811\pi$
$113$	16.0000			1.50515			0.752577	+	0.658505i	$0.228811\pi$
$114$		0			0
$115$	6.00000	−	10.3923i	0.559503	−	0.969087i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	3.50000	+	6.06218i	0.318182	+	0.551107i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−12.0000			−1.07331
$126$		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
$129$		0			0
$130$		0			0
$131$	−4.00000	−	6.92820i	−0.349482	−	0.605320i	0.636676	−	0.771132i	$-0.280309\pi$
$131$	−4.00000	−	6.92820i	−0.349482	−	0.605320i	−0.986157	+	0.165812i	$0.946976\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.00000	+	10.3923i	−0.512615	+	0.887875i	0.487278	+	0.873247i	$0.337990\pi$
$137$	−6.00000	+	10.3923i	−0.512615	+	0.887875i	−0.999893	+	0.0146279i	$0.995344\pi$
$138$		0			0
$139$	−16.0000			−1.35710			−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000			−1.35710			−0.678551	+	0.734553i	$0.737392\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.00000	+	3.46410i	−0.167248	+	0.289683i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.00000	+	10.3923i	0.491539	+	0.851371i	0.999953	−	0.00974235i	$-0.00310113\pi$
$149$	6.00000	+	10.3923i	0.491539	+	0.851371i	−0.508413	+	0.861113i	$0.669768\pi$
$150$		0			0
$151$	−12.0000	+	20.7846i	−0.976546	+	1.69143i	−0.301811	+	0.953368i	$0.597591\pi$
$151$	−12.0000	+	20.7846i	−0.976546	+	1.69143i	−0.674735	+	0.738060i	$0.735742\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	8.00000			0.642575
$156$		0			0
$157$	3.00000	−	5.19615i	0.239426	−	0.414698i	−0.721124	−	0.692806i	$-0.756374\pi$
$157$	3.00000	−	5.19615i	0.239426	−	0.414698i	0.960550	+	0.278108i	$0.0897074\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	6.00000	+	10.3923i	0.469956	+	0.813988i	0.999410	−	0.0343508i	$-0.0109363\pi$
$163$	6.00000	+	10.3923i	0.469956	+	0.813988i	−0.529454	+	0.848339i	$0.677603\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	4.00000			0.309529			0.154765	−	0.987951i	$-0.450538\pi$
$167$	4.00000			0.309529			0.154765	+	0.987951i	$0.450538\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$		0			0
$172$		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
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−5.00000	+	8.66025i	−0.373718	+	0.647298i	−0.990134	−	0.140122i	$-0.955250\pi$
$179$	−5.00000	+	8.66025i	−0.373718	+	0.647298i	0.616417	+	0.787420i	$0.288584\pi$
$180$		0			0
$181$	14.0000			1.04061			0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000			1.04061			0.520306	+	0.853980i	$0.325818\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−10.0000	+	17.3205i	−0.735215	+	1.27343i
$186$		0			0
$187$	6.00000	+	10.3923i	0.438763	+	0.759961i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	13.0000	+	22.5167i	0.940647	+	1.62925i	0.764241	+	0.644931i	$0.223114\pi$
$191$	13.0000	+	22.5167i	0.940647	+	1.62925i	0.176406	+	0.984317i	$0.443553\pi$
$192$		0			0
$193$	13.0000	−	22.5167i	0.935760	−	1.62078i	0.162488	−	0.986710i	$-0.448048\pi$
$193$	13.0000	−	22.5167i	0.935760	−	1.62078i	0.773272	−	0.634074i	$-0.218619\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	12.0000			0.854965			0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000			0.854965			0.427482	+	0.904024i	$0.359401\pi$
$198$		0			0
$199$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$199$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−2.00000	−	3.46410i	−0.139686	−	0.241943i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−8.00000			−0.553372
$210$		0			0
$211$	−12.0000			−0.826114			−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000			−0.826114			−0.413057	+	0.910705i	$0.635539\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.00000	+	6.92820i	0.272798	+	0.472500i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	6.00000	−	10.3923i	0.403604	−	0.699062i
$222$		0			0
$223$	−24.0000			−1.60716			−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000			−1.60716			−0.803579	+	0.595198i	$0.797074\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−14.0000	+	24.2487i	−0.929213	+	1.60944i	−0.144571	+	0.989494i	$0.546180\pi$
$227$	−14.0000	+	24.2487i	−0.929213	+	1.60944i	−0.784642	+	0.619949i	$0.787153\pi$
$228$		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
$231$		0			0
$232$		0			0
$233$	−2.00000	−	3.46410i	−0.131024	−	0.226941i	0.793047	−	0.609160i	$-0.208493\pi$
$233$	−2.00000	−	3.46410i	−0.131024	−	0.226941i	−0.924072	+	0.382219i	$0.875160\pi$
$234$		0			0
$235$	−4.00000	+	6.92820i	−0.260931	+	0.451946i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−14.0000			−0.905585			−0.452792	−	0.891616i	$-0.649572\pi$
$239$	−14.0000			−0.905585			−0.452792	+	0.891616i	$0.649572\pi$
$240$		0			0
$241$	−13.0000	+	22.5167i	−0.837404	+	1.45043i	0.0546547	+	0.998505i	$0.482594\pi$
$241$	−13.0000	+	22.5167i	−0.837404	+	1.45043i	−0.892058	+	0.451920i	$0.850739\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	4.00000	+	6.92820i	0.254514	+	0.440831i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		0			0
$253$	12.0000			0.754434
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−13.0000	−	22.5167i	−0.810918	−	1.40455i	−0.912222	−	0.409695i	$-0.865635\pi$
$257$	−13.0000	−	22.5167i	−0.810918	−	1.40455i	0.101305	−	0.994855i	$-0.467698\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−9.00000	+	15.5885i	−0.554964	+	0.961225i	0.442943	+	0.896550i	$0.353935\pi$
$263$	−9.00000	+	15.5885i	−0.554964	+	0.961225i	−0.997906	+	0.0646755i	$0.979399\pi$
$264$		0			0
$265$	24.0000			1.47431
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−5.00000	+	8.66025i	−0.304855	+	0.528025i	−0.977229	−	0.212187i	$-0.931941\pi$
$269$	−5.00000	+	8.66025i	−0.304855	+	0.528025i	0.672374	+	0.740212i	$0.265275\pi$
$270$		0			0
$271$	−6.00000	−	10.3923i	−0.364474	−	0.631288i	0.624218	−	0.781251i	$-0.285418\pi$
$271$	−6.00000	−	10.3923i	−0.364474	−	0.631288i	−0.988692	+	0.149963i	$0.952085\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.00000	−	1.73205i	−0.0603023	−	0.104447i
$276$		0			0
$277$	1.00000	−	1.73205i	0.0600842	−	0.104069i	−0.834419	−	0.551131i	$-0.814196\pi$
$277$	1.00000	−	1.73205i	0.0600842	−	0.104069i	0.894503	+	0.447062i	$0.147530\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−12.0000			−0.715860			−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000			−0.715860			−0.357930	+	0.933748i	$0.616517\pi$
$282$		0			0
$283$	14.0000	−	24.2487i	0.832214	−	1.44144i	−0.0640654	−	0.997946i	$-0.520407\pi$
$283$	14.0000	−	24.2487i	0.832214	−	1.44144i	0.896279	−	0.443491i	$-0.146260\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−9.50000	−	16.4545i	−0.558824	−	0.967911i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	6.00000			0.350524			0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000			0.350524			0.175262	+	0.984522i	$0.443923\pi$
$294$		0			0
$295$	24.0000			1.39733
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6.00000	−	10.3923i	−0.346989	−	0.601003i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	6.00000	−	10.3923i	0.343559	−	0.595062i
$306$		0			0
$307$	−4.00000			−0.228292			−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000			−0.228292			−0.114146	+	0.993464i	$0.536413\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−10.0000	+	17.3205i	−0.567048	+	0.982156i	0.429808	+	0.902920i	$0.358581\pi$
$311$	−10.0000	+	17.3205i	−0.567048	+	0.982156i	−0.996856	+	0.0792356i	$0.974752\pi$
$312$		0			0
$313$	−7.00000	−	12.1244i	−0.395663	−	0.685309i	0.597522	−	0.801852i	$-0.296152\pi$
$313$	−7.00000	−	12.1244i	−0.395663	−	0.685309i	−0.993186	+	0.116543i	$0.962819\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	6.00000	+	10.3923i	0.336994	+	0.583690i	0.983866	−	0.178908i	$-0.0572566\pi$
$317$	6.00000	+	10.3923i	0.336994	+	0.583690i	−0.646872	+	0.762598i	$0.723923\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	24.0000			1.33540
$324$		0			0
$325$	−1.00000	+	1.73205i	−0.0554700	+	0.0960769i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−10.0000	−	17.3205i	−0.549650	−	0.952021i	−0.998298	−	0.0583130i	$-0.981428\pi$
$331$	−10.0000	−	17.3205i	−0.549650	−	0.952021i	0.448649	−	0.893708i	$-0.351905\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−8.00000			−0.437087
$336$		0			0
$337$	14.0000			0.762629			0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000			0.762629			0.381314	+	0.924445i	$0.375472\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	4.00000	+	6.92820i	0.216612	+	0.375183i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3.00000	−	5.19615i	0.161048	−	0.278944i	−0.774197	−	0.632945i	$-0.781846\pi$
$347$	3.00000	−	5.19615i	0.161048	−	0.278944i	0.935245	+	0.354001i	$0.115179\pi$
$348$		0			0
$349$	18.0000			0.963518			0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000			0.963518			0.481759	+	0.876304i	$0.339998\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	13.0000	−	22.5167i	0.691920	−	1.19844i	−0.279288	−	0.960207i	$-0.590098\pi$
$353$	13.0000	−	22.5167i	0.691920	−	1.19844i	0.971208	−	0.238233i	$-0.0765683\pi$
$354$		0			0
$355$	−14.0000	−	24.2487i	−0.743043	−	1.28699i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−3.00000	−	5.19615i	−0.158334	−	0.274242i	0.775934	−	0.630814i	$-0.217279\pi$
$359$	−3.00000	−	5.19615i	−0.158334	−	0.274242i	−0.934268	+	0.356572i	$0.883946\pi$
$360$		0			0
$361$	1.50000	−	2.59808i	0.0789474	−	0.136741i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	4.00000			0.209370
$366$		0			0
$367$	−16.0000	+	27.7128i	−0.835193	+	1.44660i	0.0586798	+	0.998277i	$0.481311\pi$
$367$	−16.0000	+	27.7128i	−0.835193	+	1.44660i	−0.893873	+	0.448320i	$0.852022\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−11.0000	−	19.0526i	−0.569558	−	0.986504i	−0.996610	−	0.0822766i	$-0.973781\pi$
$373$	−11.0000	−	19.0526i	−0.569558	−	0.986504i	0.427051	−	0.904227i	$-0.359552\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−28.0000			−1.43826			−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000			−1.43826			−0.719132	+	0.694874i	$0.755460\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	12.0000	+	20.7846i	0.613171	+	1.06204i	0.990702	+	0.136047i	$0.0434398\pi$
$383$	12.0000	+	20.7846i	0.613171	+	1.06204i	−0.377531	+	0.925997i	$0.623227\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−4.00000	+	6.92820i	−0.202808	+	0.351274i	−0.949432	−	0.313972i	$-0.898340\pi$
$389$	−4.00000	+	6.92820i	−0.202808	+	0.351274i	0.746624	+	0.665246i	$0.231673\pi$
$390$		0			0
$391$	−36.0000			−1.82060
$392$		0			0
$393$		0			0
$394$		0			0
$395$	8.00000	−	13.8564i	0.402524	−	0.697191i
$396$		0			0
$397$	11.0000	+	19.0526i	0.552074	+	0.956221i	0.998125	+	0.0612128i	$0.0194968\pi$
$397$	11.0000	+	19.0526i	0.552074	+	0.956221i	−0.446051	+	0.895008i	$0.647170\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	6.00000	+	10.3923i	0.299626	+	0.518967i	0.976050	−	0.217545i	$-0.0698049\pi$
$401$	6.00000	+	10.3923i	0.299626	+	0.518967i	−0.676425	+	0.736512i	$0.736472\pi$
$402$		0			0
$403$	4.00000	−	6.92820i	0.199254	−	0.345118i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−20.0000			−0.991363
$408$		0			0
$409$	−1.00000	+	1.73205i	−0.0494468	+	0.0856444i	−0.889689	−	0.456566i	$-0.849079\pi$
$409$	−1.00000	+	1.73205i	−0.0494468	+	0.0856444i	0.840243	+	0.542211i	$0.182412\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	16.0000	+	27.7128i	0.785409	+	1.36037i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−4.00000			−0.195413			−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000			−0.195413			−0.0977064	+	0.995215i	$0.531151\pi$
$420$		0			0
$421$	−2.00000			−0.0974740			−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000			−0.0974740			−0.0487370	+	0.998812i	$0.515520\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	3.00000	+	5.19615i	0.145521	+	0.252050i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	5.00000	−	8.66025i	0.240842	−	0.417150i	−0.720113	−	0.693857i	$-0.755910\pi$
$431$	5.00000	−	8.66025i	0.240842	−	0.417150i	0.960954	+	0.276707i	$0.0892433\pi$
$432$		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
$435$		0			0
$436$		0			0
$437$	12.0000	−	20.7846i	0.574038	−	0.994263i
$438$		0			0
$439$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$439$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−9.00000	−	15.5885i	−0.427603	−	0.740630i	0.569057	−	0.822298i	$-0.307309\pi$
$443$	−9.00000	−	15.5885i	−0.427603	−	0.740630i	−0.996660	+	0.0816684i	$0.973975\pi$
$444$		0			0
$445$	6.00000	−	10.3923i	0.284427	−	0.492642i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	8.00000			0.377543			0.188772	−	0.982021i	$-0.439549\pi$
$449$	8.00000			0.377543			0.188772	+	0.982021i	$0.439549\pi$
$450$		0			0
$451$	2.00000	−	3.46410i	0.0941763	−	0.163118i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−11.0000	−	19.0526i	−0.514558	−	0.891241i	−0.999857	−	0.0168929i	$-0.994623\pi$
$457$	−11.0000	−	19.0526i	−0.514558	−	0.891241i	0.485299	−	0.874348i	$-0.338711\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−18.0000			−0.838344			−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000			−0.838344			−0.419172	+	0.907907i	$0.637680\pi$
$462$		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
$467$	−2.00000	−	3.46410i	−0.0925490	−	0.160300i	0.816034	−	0.578004i	$-0.196168\pi$
$467$	−2.00000	−	3.46410i	−0.0925490	−	0.160300i	−0.908583	+	0.417704i	$0.862835\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−4.00000	+	6.92820i	−0.183920	+	0.318559i
$474$		0			0
$475$	−4.00000			−0.183533
$476$		0			0
$477$		0			0
$478$		0			0
$479$	18.0000	−	31.1769i	0.822441	−	1.42451i	−0.0814184	−	0.996680i	$-0.525945\pi$
$479$	18.0000	−	31.1769i	0.822441	−	1.42451i	0.903859	−	0.427830i	$-0.140722\pi$
$480$		0			0
$481$	10.0000	+	17.3205i	0.455961	+	0.789747i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−18.0000	−	31.1769i	−0.817338	−	1.41567i
$486$		0			0
$487$	−4.00000	+	6.92820i	−0.181257	+	0.313947i	−0.942309	−	0.334744i	$-0.891350\pi$
$487$	−4.00000	+	6.92820i	−0.181257	+	0.313947i	0.761052	+	0.648691i	$0.224683\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−6.00000			−0.270776			−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000			−0.270776			−0.135388	+	0.990793i	$0.543228\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	2.00000	+	3.46410i	0.0895323	+	0.155074i	0.907314	−	0.420455i	$-0.138129\pi$
$499$	2.00000	+	3.46410i	0.0895323	+	0.155074i	−0.817781	+	0.575529i	$0.804796\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	16.0000			0.713405			0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000			0.713405			0.356702	+	0.934218i	$0.383901\pi$
$504$		0			0
$505$	28.0000			1.24598
$506$		0			0
$507$		0			0
$508$		0			0
$509$	7.00000	+	12.1244i	0.310270	+	0.537403i	0.978421	−	0.206623i	$-0.0662474\pi$
$509$	7.00000	+	12.1244i	0.310270	+	0.537403i	−0.668151	+	0.744026i	$0.732914\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−4.00000	+	6.92820i	−0.176261	+	0.305293i
$516$		0			0
$517$	−8.00000			−0.351840
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−3.00000	+	5.19615i	−0.131432	+	0.227648i	−0.924229	−	0.381839i	$-0.875291\pi$
$521$	−3.00000	+	5.19615i	−0.131432	+	0.227648i	0.792797	+	0.609486i	$0.208624\pi$
$522$		0			0
$523$	8.00000	+	13.8564i	0.349816	+	0.605898i	0.986216	−	0.165460i	$-0.0529109\pi$
$523$	8.00000	+	13.8564i	0.349816	+	0.605898i	−0.636401	+	0.771358i	$0.719578\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−12.0000	−	20.7846i	−0.522728	−	0.905392i
$528$		0			0
$529$	−6.50000	+	11.2583i	−0.282609	+	0.489493i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−4.00000			−0.173259
$534$		0			0
$535$	−6.00000	+	10.3923i	−0.259403	+	0.449299i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	5.00000	+	8.66025i	0.214967	+	0.372333i	0.953262	−	0.302144i	$-0.0977023\pi$
$541$	5.00000	+	8.66025i	0.214967	+	0.372333i	−0.738296	+	0.674477i	$0.764369\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	4.00000			0.171341
$546$		0			0
$547$	−36.0000			−1.53925			−0.769624	−	0.638497i	$-0.779557\pi$
$547$	−36.0000			−1.53925			−0.769624	+	0.638497i	$0.779557\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	10.0000	−	17.3205i	0.423714	−	0.733893i	−0.572586	−	0.819845i	$-0.694060\pi$
$557$	10.0000	−	17.3205i	0.423714	−	0.733893i	0.996299	+	0.0859514i	$0.0273930\pi$
$558$		0			0
$559$	8.00000			0.338364
$560$		0			0
$561$		0			0
$562$		0			0
$563$	14.0000	−	24.2487i	0.590030	−	1.02196i	−0.404198	−	0.914671i	$-0.632449\pi$
$563$	14.0000	−	24.2487i	0.590030	−	1.02196i	0.994228	−	0.107290i	$-0.0342173\pi$
$564$		0			0
$565$	−16.0000	−	27.7128i	−0.673125	−	1.16589i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−2.00000	−	3.46410i	−0.0838444	−	0.145223i	0.821054	−	0.570851i	$-0.193387\pi$
$569$	−2.00000	−	3.46410i	−0.0838444	−	0.145223i	−0.904898	+	0.425628i	$0.860053\pi$
$570$		0			0
$571$	−2.00000	+	3.46410i	−0.0836974	+	0.144968i	−0.904835	−	0.425762i	$-0.860006\pi$
$571$	−2.00000	+	3.46410i	−0.0836974	+	0.144968i	0.821138	+	0.570730i	$0.193340\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	6.00000			0.250217
$576$		0			0
$577$	17.0000	−	29.4449i	0.707719	−	1.22581i	−0.257982	−	0.966150i	$-0.583058\pi$
$577$	17.0000	−	29.4449i	0.707719	−	1.22581i	0.965701	−	0.259656i	$-0.0836092\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	12.0000	+	20.7846i	0.496989	+	0.860811i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−12.0000			−0.495293			−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000			−0.495293			−0.247647	+	0.968850i	$0.579657\pi$
$588$		0			0
$589$	16.0000			0.659269
$590$		0			0
$591$		0			0
$592$		0			0
$593$	13.0000	+	22.5167i	0.533846	+	0.924648i	0.999218	+	0.0395334i	$0.0125871\pi$
$593$	13.0000	+	22.5167i	0.533846	+	0.924648i	−0.465372	+	0.885115i	$0.654080\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	9.00000	−	15.5885i	0.367730	−	0.636927i	−0.621480	−	0.783430i	$-0.713468\pi$
$599$	9.00000	−	15.5885i	0.367730	−	0.636927i	0.989210	+	0.146503i	$0.0468017\pi$
$600$		0			0
$601$	22.0000			0.897399			0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000			0.897399			0.448699	+	0.893683i	$0.351887\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	7.00000	−	12.1244i	0.284590	−	0.492925i
$606$		0			0
$607$	−20.0000	−	34.6410i	−0.811775	−	1.40604i	−0.911621	−	0.411033i	$-0.865168\pi$
$607$	−20.0000	−	34.6410i	−0.811775	−	1.40604i	0.0998457	−	0.995003i	$-0.468165\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	4.00000	+	6.92820i	0.161823	+	0.280285i
$612$		0			0
$613$	−3.00000	+	5.19615i	−0.121169	+	0.209871i	−0.920229	−	0.391381i	$-0.871998\pi$
$613$	−3.00000	+	5.19615i	−0.121169	+	0.209871i	0.799060	+	0.601251i	$0.205331\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−28.0000			−1.12724			−0.563619	−	0.826035i	$-0.690591\pi$
$617$	−28.0000			−1.12724			−0.563619	+	0.826035i	$0.690591\pi$
$618$		0			0
$619$	20.0000	−	34.6410i	0.803868	−	1.39234i	−0.113185	−	0.993574i	$-0.536105\pi$
$619$	20.0000	−	34.6410i	0.803868	−	1.39234i	0.917053	−	0.398766i	$-0.130561\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	9.50000	+	16.4545i	0.380000	+	0.658179i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	60.0000			2.39236
$630$		0			0
$631$	24.0000			0.955425			0.477712	−	0.878516i	$-0.341466\pi$
$631$	24.0000			0.955425			0.477712	+	0.878516i	$0.341466\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	8.00000	+	13.8564i	0.317470	+	0.549875i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	6.00000	−	10.3923i	0.236986	−	0.410471i	−0.722862	−	0.690992i	$-0.757174\pi$
$641$	6.00000	−	10.3923i	0.236986	−	0.410471i	0.959848	+	0.280521i	$0.0905072\pi$
$642$		0			0
$643$	44.0000			1.73519			0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000			1.73519			0.867595	+	0.497271i	$0.165665\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	10.0000	−	17.3205i	0.393141	−	0.680939i	−0.599721	−	0.800209i	$-0.704722\pi$
$647$	10.0000	−	17.3205i	0.393141	−	0.680939i	0.992862	+	0.119269i	$0.0380552\pi$
$648$		0			0
$649$	12.0000	+	20.7846i	0.471041	+	0.815867i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−16.0000	−	27.7128i	−0.626128	−	1.08449i	−0.988322	−	0.152383i	$-0.951305\pi$
$653$	−16.0000	−	27.7128i	−0.626128	−	1.08449i	0.362193	−	0.932103i	$-0.382028\pi$
$654$		0			0
$655$	−8.00000	+	13.8564i	−0.312586	+	0.541415i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−42.0000			−1.63609			−0.818044	−	0.575156i	$-0.804941\pi$
$659$	−42.0000			−1.63609			−0.818044	+	0.575156i	$0.804941\pi$
$660$		0			0
$661$	−5.00000	+	8.66025i	−0.194477	+	0.336845i	−0.946729	−	0.322031i	$-0.895634\pi$
$661$	−5.00000	+	8.66025i	−0.194477	+	0.336845i	0.752252	+	0.658876i	$0.228968\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	12.0000			0.463255
$672$		0			0
$673$	−2.00000			−0.0770943			−0.0385472	−	0.999257i	$-0.512273\pi$
$673$	−2.00000			−0.0770943			−0.0385472	+	0.999257i	$0.512273\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−15.0000	−	25.9808i	−0.576497	−	0.998522i	−0.995877	−	0.0907112i	$-0.971086\pi$
$677$	−15.0000	−	25.9808i	−0.576497	−	0.998522i	0.419380	−	0.907811i	$-0.362247\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	15.0000	−	25.9808i	0.573959	−	0.994126i	−0.422195	−	0.906505i	$-0.638740\pi$
$683$	15.0000	−	25.9808i	0.573959	−	0.994126i	0.996154	−	0.0876211i	$-0.0279265\pi$
$684$		0			0
$685$	24.0000			0.916993
$686$		0			0
$687$		0			0
$688$		0			0
$689$	12.0000	−	20.7846i	0.457164	−	0.791831i
$690$		0			0
$691$	20.0000	+	34.6410i	0.760836	+	1.31781i	0.942420	+	0.334431i	$0.108544\pi$
$691$	20.0000	+	34.6410i	0.760836	+	1.31781i	−0.181584	+	0.983375i	$0.558123\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	16.0000	+	27.7128i	0.606915	+	1.05121i
$696$		0			0
$697$	−6.00000	+	10.3923i	−0.227266	+	0.393637i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−48.0000			−1.81293			−0.906467	−	0.422276i	$-0.861231\pi$
$701$	−48.0000			−1.81293			−0.906467	+	0.422276i	$0.861231\pi$
$702$		0			0
$703$	−20.0000	+	34.6410i	−0.754314	+	1.30651i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	3.00000	+	5.19615i	0.112667	+	0.195146i	0.916845	−	0.399244i	$-0.130727\pi$
$709$	3.00000	+	5.19615i	0.112667	+	0.195146i	−0.804178	+	0.594389i	$0.797394\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−24.0000			−0.898807
$714$		0			0
$715$	8.00000			0.299183
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−16.0000	−	27.7128i	−0.596699	−	1.03351i	−0.993305	−	0.115524i	$-0.963145\pi$
$719$	−16.0000	−	27.7128i	−0.596699	−	1.03351i	0.396605	−	0.917989i	$-0.370188\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−28.0000			−1.03846			−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000			−1.03846			−0.519231	+	0.854634i	$0.673782\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	12.0000	−	20.7846i	0.443836	−	0.768747i
$732$		0			0
$733$	13.0000	+	22.5167i	0.480166	+	0.831672i	0.999741	−	0.0227529i	$-0.00724310\pi$
$733$	13.0000	+	22.5167i	0.480166	+	0.831672i	−0.519575	+	0.854425i	$0.673910\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−4.00000	−	6.92820i	−0.147342	−	0.255204i
$738$		0			0
$739$	22.0000	−	38.1051i	0.809283	−	1.40172i	−0.104078	−	0.994569i	$-0.533189\pi$
$739$	22.0000	−	38.1051i	0.809283	−	1.40172i	0.913361	−	0.407150i	$-0.133477\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−2.00000			−0.0733729			−0.0366864	−	0.999327i	$-0.511680\pi$
$743$	−2.00000			−0.0733729			−0.0366864	+	0.999327i	$0.511680\pi$
$744$		0			0
$745$	12.0000	−	20.7846i	0.439646	−	0.761489i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	4.00000	+	6.92820i	0.145962	+	0.252814i	0.929731	−	0.368238i	$-0.120039\pi$
$751$	4.00000	+	6.92820i	0.145962	+	0.252814i	−0.783769	+	0.621052i	$0.786706\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	48.0000			1.74690
$756$		0			0
$757$	−38.0000			−1.38113			−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000			−1.38113			−0.690567	+	0.723269i	$0.742639\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	15.0000	+	25.9808i	0.543750	+	0.941802i	0.998684	+	0.0512772i	$0.0163292\pi$
$761$	15.0000	+	25.9808i	0.543750	+	0.941802i	−0.454935	+	0.890525i	$0.650337\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	12.0000	−	20.7846i	0.433295	−	0.750489i
$768$		0			0
$769$	−26.0000			−0.937584			−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000			−0.937584			−0.468792	+	0.883309i	$0.655311\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	5.00000	−	8.66025i	0.179838	−	0.311488i	−0.761987	−	0.647592i	$-0.775776\pi$
$773$	5.00000	−	8.66025i	0.179838	−	0.311488i	0.941825	+	0.336104i	$0.109109\pi$
$774$		0			0
$775$	2.00000	+	3.46410i	0.0718421	+	0.124434i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4.00000	−	6.92820i	−0.143315	−	0.248229i
$780$		0			0
$781$	14.0000	−	24.2487i	0.500959	−	0.867687i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−12.0000			−0.428298
$786$		0			0
$787$	16.0000	−	27.7128i	0.570338	−	0.987855i	−0.426193	−	0.904632i	$-0.640145\pi$
$787$	16.0000	−	27.7128i	0.570338	−	0.987855i	0.996531	−	0.0832226i	$-0.0265213\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−6.00000	−	10.3923i	−0.213066	−	0.369042i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	30.0000			1.06265			0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000			1.06265			0.531327	+	0.847167i	$0.321693\pi$
$798$		0			0
$799$	24.0000			0.849059
$800$		0			0
$801$		0			0
$802$		0			0
$803$	2.00000	+	3.46410i	0.0705785	+	0.122245i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	16.0000	−	27.7128i	0.562530	−	0.974331i	−0.434745	−	0.900554i	$-0.643161\pi$
$809$	16.0000	−	27.7128i	0.562530	−	0.974331i	0.997275	−	0.0737769i	$-0.0235053\pi$
$810$		0			0
$811$	−16.0000			−0.561836			−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000			−0.561836			−0.280918	+	0.959732i	$0.590639\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	12.0000	−	20.7846i	0.420342	−	0.728053i
$816$		0			0
$817$	8.00000	+	13.8564i	0.279885	+	0.484774i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	22.0000	+	38.1051i	0.767805	+	1.32988i	0.938751	+	0.344597i	$0.111985\pi$
$821$	22.0000	+	38.1051i	0.767805	+	1.32988i	−0.170945	+	0.985281i	$0.554682\pi$
$822$		0			0
$823$	−24.0000	+	41.5692i	−0.836587	+	1.44901i	0.0561440	+	0.998423i	$0.482119\pi$
$823$	−24.0000	+	41.5692i	−0.836587	+	1.44901i	−0.892731	+	0.450589i	$0.851214\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−26.0000			−0.904109			−0.452054	−	0.891990i	$-0.649309\pi$
$827$	−26.0000			−0.904109			−0.452054	+	0.891990i	$0.649309\pi$
$828$		0			0
$829$	−23.0000	+	39.8372i	−0.798823	+	1.38360i	0.121560	+	0.992584i	$0.461210\pi$
$829$	−23.0000	+	39.8372i	−0.798823	+	1.38360i	−0.920383	+	0.391018i	$0.872123\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−4.00000	−	6.92820i	−0.138426	−	0.239760i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−36.0000			−1.24286			−0.621429	−	0.783470i	$-0.713448\pi$
$839$	−36.0000			−1.24286			−0.621429	+	0.783470i	$0.713448\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$	9.00000	+	15.5885i	0.309609	+	0.536259i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	30.0000	−	51.9615i	1.02839	−	1.78122i
$852$		0			0
$853$	10.0000			0.342393			0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000			0.342393			0.171197	+	0.985237i	$0.445237\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	15.0000	−	25.9808i	0.512390	−	0.887486i	−0.487507	−	0.873119i	$-0.662093\pi$
$857$	15.0000	−	25.9808i	0.512390	−	0.887486i	0.999897	−	0.0143666i	$-0.00457319\pi$
$858$		0			0
$859$	−2.00000	−	3.46410i	−0.0682391	−	0.118194i	0.829887	−	0.557931i	$-0.188405\pi$
$859$	−2.00000	−	3.46410i	−0.0682391	−	0.118194i	−0.898126	+	0.439738i	$0.855071\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−25.0000	−	43.3013i	−0.851010	−	1.47399i	−0.880298	−	0.474421i	$-0.842657\pi$
$863$	−25.0000	−	43.3013i	−0.851010	−	1.47399i	0.0292880	−	0.999571i	$-0.490676\pi$
$864$		0			0
$865$	−6.00000	+	10.3923i	−0.204006	+	0.353349i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	16.0000			0.542763
$870$		0			0
$871$	−4.00000	+	6.92820i	−0.135535	+	0.234753i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	9.00000	+	15.5885i	0.303908	+	0.526385i	0.977018	−	0.213158i	$-0.0683750\pi$
$877$	9.00000	+	15.5885i	0.303908	+	0.526385i	−0.673109	+	0.739543i	$0.735042\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	54.0000			1.81931			0.909653	−	0.415369i	$-0.136347\pi$
$881$	54.0000			1.81931			0.909653	+	0.415369i	$0.136347\pi$
$882$		0			0
$883$	−20.0000			−0.673054			−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000			−0.673054			−0.336527	+	0.941674i	$0.609252\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−10.0000	−	17.3205i	−0.335767	−	0.581566i	0.647865	−	0.761755i	$-0.275662\pi$
$887$	−10.0000	−	17.3205i	−0.335767	−	0.581566i	−0.983632	+	0.180190i	$0.942329\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−8.00000	+	13.8564i	−0.267710	+	0.463687i
$894$		0			0
$895$	20.0000			0.668526
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−36.0000	−	62.3538i	−1.19933	−	2.07731i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−14.0000	−	24.2487i	−0.465376	−	0.806054i
$906$		0			0
$907$	14.0000	−	24.2487i	0.464862	−	0.805165i	−0.534333	−	0.845274i	$-0.679437\pi$
$907$	14.0000	−	24.2487i	0.464862	−	0.805165i	0.999195	+	0.0401089i	$0.0127705\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	6.00000			0.198789			0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000			0.198789			0.0993944	+	0.995048i	$0.468309\pi$
$912$		0			0
$913$	−16.0000	+	27.7128i	−0.529523	+	0.917160i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−16.0000	−	27.7128i	−0.527791	−	0.914161i	−0.999475	−	0.0323936i	$-0.989687\pi$
$919$	−16.0000	−	27.7128i	−0.527791	−	0.914161i	0.471684	−	0.881768i	$-0.343646\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−28.0000			−0.921631
$924$		0			0
$925$	−10.0000			−0.328798
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−17.0000	−	29.4449i	−0.557752	−	0.966055i	−0.997684	−	0.0680235i	$-0.978331\pi$
$929$	−17.0000	−	29.4449i	−0.557752	−	0.966055i	0.439932	−	0.898031i	$-0.355003\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	12.0000	−	20.7846i	0.392442	−	0.679729i
$936$		0			0
$937$	−34.0000			−1.11073			−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000			−1.11073			−0.555366	+	0.831606i	$0.687422\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	3.00000	−	5.19615i	0.0977972	−	0.169390i	−0.812975	−	0.582298i	$-0.802154\pi$
$941$	3.00000	−	5.19615i	0.0977972	−	0.169390i	0.910773	+	0.412908i	$0.135487\pi$
$942$		0			0
$943$	6.00000	+	10.3923i	0.195387	+	0.338420i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−7.00000	−	12.1244i	−0.227469	−	0.393989i	0.729588	−	0.683887i	$-0.239712\pi$
$947$	−7.00000	−	12.1244i	−0.227469	−	0.393989i	−0.957057	+	0.289898i	$0.906379\pi$
$948$		0			0
$949$	2.00000	−	3.46410i	0.0649227	−	0.112449i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−24.0000			−0.777436			−0.388718	−	0.921357i	$-0.627082\pi$
$953$	−24.0000			−0.777436			−0.388718	+	0.921357i	$0.627082\pi$
$954$		0			0
$955$	26.0000	−	45.0333i	0.841340	−	1.45724i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	7.50000	+	12.9904i	0.241935	+	0.419045i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−52.0000			−1.67394
$966$		0			0
$967$	32.0000			1.02905			0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000			1.02905			0.514525	+	0.857475i	$0.327968\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	24.0000	+	41.5692i	0.770197	+	1.33402i	0.937455	+	0.348107i	$0.113175\pi$
$971$	24.0000	+	41.5692i	0.770197	+	1.33402i	−0.167258	+	0.985913i	$0.553491\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−26.0000	+	45.0333i	−0.831814	+	1.44074i	0.0647848	+	0.997899i	$0.479364\pi$
$977$	−26.0000	+	45.0333i	−0.831814	+	1.44074i	−0.896599	+	0.442844i	$0.853969\pi$
$978$		0			0
$979$	12.0000			0.383522
$980$		0			0
$981$		0			0
$982$		0			0
$983$	12.0000	−	20.7846i	0.382741	−	0.662926i	−0.608712	−	0.793391i	$-0.708314\pi$
$983$	12.0000	−	20.7846i	0.382741	−	0.662926i	0.991453	+	0.130465i	$0.0416470\pi$
$984$		0			0
$985$	−12.0000	−	20.7846i	−0.382352	−	0.662253i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−12.0000	−	20.7846i	−0.381578	−	0.660912i
$990$		0			0
$991$	−4.00000	+	6.92820i	−0.127064	+	0.220082i	−0.922538	−	0.385906i	$-0.873889\pi$
$991$	−4.00000	+	6.92820i	−0.127064	+	0.220082i	0.795474	+	0.605988i	$0.207222\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	11.0000	−	19.0526i	0.348373	−	0.603401i	−0.637587	−	0.770378i	$-0.720067\pi$
$997$	11.0000	−	19.0526i	0.348373	−	0.603401i	0.985961	+	0.166978i	$0.0534008\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3528.2.s.i.361.1		2
3.2	odd	2		3528.2.s.u.361.1		2
7.2	even	3	inner	3528.2.s.i.3313.1		2
7.3	odd	6		504.2.a.a.1.1	✓	1
7.4	even	3		3528.2.a.u.1.1		1
7.5	odd	6		3528.2.s.x.3313.1		2
7.6	odd	2		3528.2.s.x.361.1		2
21.2	odd	6		3528.2.s.u.3313.1		2
21.5	even	6		3528.2.s.f.3313.1		2
21.11	odd	6		3528.2.a.e.1.1		1
21.17	even	6		504.2.a.f.1.1	yes	1
21.20	even	2		3528.2.s.f.361.1		2
28.3	even	6		1008.2.a.f.1.1		1
28.11	odd	6		7056.2.a.bt.1.1		1
56.3	even	6		4032.2.a.bg.1.1		1
56.45	odd	6		4032.2.a.bf.1.1		1
84.11	even	6		7056.2.a.l.1.1		1
84.59	odd	6		1008.2.a.k.1.1		1
168.59	odd	6		4032.2.a.l.1.1		1
168.101	even	6		4032.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.a.a.1.1	✓	1	7.3	odd	6
504.2.a.f.1.1	yes	1	21.17	even	6
1008.2.a.f.1.1		1	28.3	even	6
1008.2.a.k.1.1		1	84.59	odd	6
3528.2.a.e.1.1		1	21.11	odd	6
3528.2.a.u.1.1		1	7.4	even	3
3528.2.s.f.361.1		2	21.20	even	2
3528.2.s.f.3313.1		2	21.5	even	6
3528.2.s.i.361.1		2	1.1	even	1	trivial
3528.2.s.i.3313.1		2	7.2	even	3	inner
3528.2.s.u.361.1		2	3.2	odd	2
3528.2.s.u.3313.1		2	21.2	odd	6
3528.2.s.x.361.1		2	7.6	odd	2
3528.2.s.x.3313.1		2	7.5	odd	6
4032.2.a.g.1.1		1	168.101	even	6
4032.2.a.l.1.1		1	168.59	odd	6
4032.2.a.bf.1.1		1	56.45	odd	6
4032.2.a.bg.1.1		1	56.3	even	6
7056.2.a.l.1.1		1	84.11	even	6
7056.2.a.bt.1.1		1	28.11	odd	6

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	3528.2.s.i.361.1

Level	$3528$
Weight	$2$
Character	3528.361
Analytic conductor	$28.171$
Analytic rank	$0$
Dimension	$2$
Inner twists	$2$