LMFDB - Embedded newform 3240.2.q.a.1081.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3240,2,Mod(1081,3240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3240, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("3240.1081");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$3240 = 2^{3} \cdot 3^{4} \cdot 5$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3240.q (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:	$25.8715302549$
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 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1081.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	3240.1081
Dual form			3240.2.q.a.2161.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{5} +(-2.00000 - 3.46410i) q^{7} +(3.00000 - 5.19615i) q^{13} +2.00000 q^{17} +4.00000 q^{19} +(-4.00000 + 6.92820i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(-3.00000 - 5.19615i) q^{29} +4.00000 q^{35} -6.00000 q^{37} +(5.00000 - 8.66025i) q^{41} +(2.00000 + 3.46410i) q^{43} +(4.00000 + 6.92820i) q^{47} +(-4.50000 + 7.79423i) q^{49} -10.0000 q^{53} +(-3.00000 - 5.19615i) q^{61} +(3.00000 + 5.19615i) q^{65} +(2.00000 - 3.46410i) q^{67} -14.0000 q^{73} +(-8.00000 - 13.8564i) q^{79} +(6.00000 + 10.3923i) q^{83} +(-1.00000 + 1.73205i) q^{85} -2.00000 q^{89} -24.0000 q^{91} +(-2.00000 + 3.46410i) q^{95} +(-1.00000 - 1.73205i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - q^{5} - 4 q^{7} + 6 q^{13} + 4 q^{17} + 8 q^{19} - 8 q^{23} - q^{25} - 6 q^{29} + 8 q^{35} - 12 q^{37} + 10 q^{41} + 4 q^{43} + 8 q^{47} - 9 q^{49} - 20 q^{53} - 6 q^{61} + 6 q^{65} + 4 q^{67} - 28 q^{73}+ \cdots - 2 q^{97}+O(q^{100})$ 2 * q - q^5 - 4 * q^7 + 6 * q^13 + 4 * q^17 + 8 * q^19 - 8 * q^23 - q^25 - 6 * q^29 + 8 * q^35 - 12 * q^37 + 10 * q^41 + 4 * q^43 + 8 * q^47 - 9 * q^49 - 20 * q^53 - 6 * q^61 + 6 * q^65 + 4 * q^67 - 28 * q^73 - 16 * q^79 + 12 * q^83 - 2 * q^85 - 4 * q^89 - 48 * q^91 - 4 * q^95 - 2 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times$ .

$n$	$1297$	$1621$	$2431$	$3161$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$\alpha_n$			$\theta_n$
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$6$		0			0
$7$	−2.00000	−	3.46410i	−0.755929	−	1.30931i	−0.944911	−	0.327327i	$-0.893852\pi$
$7$	−2.00000	−	3.46410i	−0.755929	−	1.30931i	0.188982	−	0.981981i	$-0.439481\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$11$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$12$		0			0
$13$	3.00000	−	5.19615i	0.832050	−	1.44115i	−0.0643593	−	0.997927i	$-0.520500\pi$
$13$	3.00000	−	5.19615i	0.832050	−	1.44115i	0.896410	−	0.443227i	$-0.146166\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.00000			0.485071			0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000			0.485071			0.242536	+	0.970143i	$0.422021\pi$
$18$		0			0
$19$	4.00000			0.917663			0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000			0.917663			0.458831	+	0.888523i	$0.348268\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.00000	+	6.92820i	−0.834058	+	1.44463i	0.0607377	+	0.998154i	$0.480655\pi$
$23$	−4.00000	+	6.92820i	−0.834058	+	1.44463i	−0.894795	+	0.446476i	$0.852679\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.00000	−	5.19615i	−0.557086	−	0.964901i	−0.997738	−	0.0672232i	$-0.978586\pi$
$29$	−3.00000	−	5.19615i	−0.557086	−	0.964901i	0.440652	−	0.897678i	$-0.354747\pi$
$30$		0			0
$31$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$31$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.00000			0.676123
$36$		0			0
$37$	−6.00000			−0.986394			−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000			−0.986394			−0.493197	+	0.869918i	$0.664172\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	5.00000	−	8.66025i	0.780869	−	1.35250i	−0.150567	−	0.988600i	$-0.548110\pi$
$41$	5.00000	−	8.66025i	0.780869	−	1.35250i	0.931436	−	0.363905i	$-0.118557\pi$
$42$		0			0
$43$	2.00000	+	3.46410i	0.304997	+	0.528271i	0.977261	−	0.212041i	$-0.0680112\pi$
$43$	2.00000	+	3.46410i	0.304997	+	0.528271i	−0.672264	+	0.740312i	$0.734678\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.00000	+	6.92820i	0.583460	+	1.01058i	0.995066	+	0.0992202i	$0.0316348\pi$
$47$	4.00000	+	6.92820i	0.583460	+	1.01058i	−0.411606	+	0.911362i	$0.635032\pi$
$48$		0			0
$49$	−4.50000	+	7.79423i	−0.642857	+	1.11346i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−10.0000			−1.37361			−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000			−1.37361			−0.686803	+	0.726844i	$0.740986\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		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
$61$	−3.00000	−	5.19615i	−0.384111	−	0.665299i	0.607535	−	0.794293i	$-0.292159\pi$
$61$	−3.00000	−	5.19615i	−0.384111	−	0.665299i	−0.991645	+	0.128994i	$0.958825\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.00000	+	5.19615i	0.372104	+	0.644503i
$66$		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
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	−14.0000			−1.63858			−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000			−1.63858			−0.819288	+	0.573382i	$0.805631\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−8.00000	−	13.8564i	−0.900070	−	1.55897i	−0.827401	−	0.561611i	$-0.810182\pi$
$79$	−8.00000	−	13.8564i	−0.900070	−	1.55897i	−0.0726692	−	0.997356i	$-0.523152\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	6.00000	+	10.3923i	0.658586	+	1.14070i	0.980982	+	0.194099i	$0.0621783\pi$
$83$	6.00000	+	10.3923i	0.658586	+	1.14070i	−0.322396	+	0.946605i	$0.604488\pi$
$84$		0			0
$85$	−1.00000	+	1.73205i	−0.108465	+	0.187867i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.00000			−0.212000			−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000			−0.212000			−0.106000	+	0.994366i	$0.533804\pi$
$90$		0			0
$91$	−24.0000			−2.51588
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.00000	+	3.46410i	−0.205196	+	0.355409i
$96$		0			0
$97$	−1.00000	−	1.73205i	−0.101535	−	0.175863i	0.810782	−	0.585348i	$-0.199042\pi$
$97$	−1.00000	−	1.73205i	−0.101535	−	0.175863i	−0.912317	+	0.409484i	$0.865709\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−7.00000	−	12.1244i	−0.696526	−	1.20642i	−0.969664	−	0.244443i	$-0.921395\pi$
$101$	−7.00000	−	12.1244i	−0.696526	−	1.20642i	0.273138	−	0.961975i	$-0.411939\pi$
$102$		0			0
$103$	−2.00000	+	3.46410i	−0.197066	+	0.341328i	−0.947576	−	0.319531i	$-0.896475\pi$
$103$	−2.00000	+	3.46410i	−0.197066	+	0.341328i	0.750510	+	0.660859i	$0.229808\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4.00000			−0.386695			−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000			−0.386695			−0.193347	+	0.981130i	$0.561934\pi$
$108$		0			0
$109$	−10.0000			−0.957826			−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000			−0.957826			−0.478913	+	0.877862i	$0.658969\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.00000	−	5.19615i	0.282216	−	0.488813i	−0.689714	−	0.724082i	$-0.742264\pi$
$113$	3.00000	−	5.19615i	0.282216	−	0.488813i	0.971930	+	0.235269i	$0.0755971\pi$
$114$		0			0
$115$	−4.00000	−	6.92820i	−0.373002	−	0.646058i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−4.00000	−	6.92820i	−0.366679	−	0.635107i
$120$		0			0
$121$	5.50000	−	9.52628i	0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−4.00000			−0.354943			−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000			−0.354943			−0.177471	+	0.984126i	$0.556792\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	8.00000	−	13.8564i	0.698963	−	1.21064i	−0.269863	−	0.962899i	$-0.586978\pi$
$131$	8.00000	−	13.8564i	0.698963	−	1.21064i	0.968826	−	0.247741i	$-0.0796882\pi$
$132$		0			0
$133$	−8.00000	−	13.8564i	−0.693688	−	1.20150i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−9.00000	−	15.5885i	−0.768922	−	1.33181i	−0.938148	−	0.346235i	$-0.887460\pi$
$137$	−9.00000	−	15.5885i	−0.768922	−	1.33181i	0.169226	−	0.985577i	$-0.445873\pi$
$138$		0			0
$139$	6.00000	−	10.3923i	0.508913	−	0.881464i	−0.491033	−	0.871141i	$-0.663381\pi$
$139$	6.00000	−	10.3923i	0.508913	−	0.881464i	0.999947	−	0.0103230i	$-0.00328598\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	6.00000			0.498273
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.00000	+	5.19615i	−0.245770	+	0.425685i	−0.962348	−	0.271821i	$-0.912374\pi$
$149$	−3.00000	+	5.19615i	−0.245770	+	0.425685i	0.716578	+	0.697507i	$0.245707\pi$
$150$		0			0
$151$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$151$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		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
$159$		0			0
$160$		0			0
$161$	32.0000			2.52195
$162$		0			0
$163$	−4.00000			−0.313304			−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000			−0.313304			−0.156652	+	0.987654i	$0.550070\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.00000	+	6.92820i	−0.309529	+	0.536120i	−0.978259	−	0.207385i	$-0.933505\pi$
$167$	−4.00000	+	6.92820i	−0.309529	+	0.536120i	0.668730	+	0.743505i	$0.266838\pi$
$168$		0			0
$169$	−11.5000	−	19.9186i	−0.884615	−	1.53220i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	9.00000	+	15.5885i	0.684257	+	1.18517i	0.973670	+	0.227964i	$0.0732068\pi$
$173$	9.00000	+	15.5885i	0.684257	+	1.18517i	−0.289412	+	0.957205i	$0.593460\pi$
$174$		0			0
$175$	−2.00000	+	3.46410i	−0.151186	+	0.261861i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−8.00000			−0.597948			−0.298974	−	0.954261i	$-0.596644\pi$
$179$	−8.00000			−0.597948			−0.298974	+	0.954261i	$0.596644\pi$
$180$		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
$185$	3.00000	−	5.19615i	0.220564	−	0.382029i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	4.00000	+	6.92820i	0.289430	+	0.501307i	0.973674	−	0.227946i	$-0.0732010\pi$
$191$	4.00000	+	6.92820i	0.289430	+	0.501307i	−0.684244	+	0.729253i	$0.739868\pi$
$192$		0			0
$193$	3.00000	−	5.19615i	0.215945	−	0.374027i	−0.737620	−	0.675216i	$-0.764050\pi$
$193$	3.00000	−	5.19615i	0.215945	−	0.374027i	0.953564	+	0.301189i	$0.0973836\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−10.0000			−0.712470			−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000			−0.712470			−0.356235	+	0.934396i	$0.615940\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−12.0000	+	20.7846i	−0.842235	+	1.45879i
$204$		0			0
$205$	5.00000	+	8.66025i	0.349215	+	0.604858i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−6.00000	+	10.3923i	−0.413057	+	0.715436i	−0.995222	−	0.0976347i	$-0.968872\pi$
$211$	−6.00000	+	10.3923i	−0.413057	+	0.715436i	0.582165	+	0.813070i	$0.302206\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−4.00000			−0.272798
$216$		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
$223$	−6.00000	−	10.3923i	−0.401790	−	0.695920i	0.592152	−	0.805826i	$-0.298278\pi$
$223$	−6.00000	−	10.3923i	−0.401790	−	0.695920i	−0.993942	+	0.109906i	$0.964945\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−6.00000	−	10.3923i	−0.398234	−	0.689761i	0.595274	−	0.803523i	$-0.297043\pi$
$227$	−6.00000	−	10.3923i	−0.398234	−	0.689761i	−0.993508	+	0.113761i	$0.963710\pi$
$228$		0			0
$229$	−3.00000	+	5.19615i	−0.198246	+	0.343371i	−0.947960	−	0.318390i	$-0.896858\pi$
$229$	−3.00000	+	5.19615i	−0.198246	+	0.343371i	0.749714	+	0.661762i	$0.230191\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	10.0000			0.655122			0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000			0.655122			0.327561	+	0.944830i	$0.393773\pi$
$234$		0			0
$235$	−8.00000			−0.521862
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−12.0000	+	20.7846i	−0.776215	+	1.34444i	0.157893	+	0.987456i	$0.449530\pi$
$239$	−12.0000	+	20.7846i	−0.776215	+	1.34444i	−0.934109	+	0.356988i	$0.883804\pi$
$240$		0			0
$241$	−1.00000	−	1.73205i	−0.0644157	−	0.111571i	0.832019	−	0.554747i	$-0.187185\pi$
$241$	−1.00000	−	1.73205i	−0.0644157	−	0.111571i	−0.896435	+	0.443176i	$0.853852\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4.50000	−	7.79423i	−0.287494	−	0.497955i
$246$		0			0
$247$	12.0000	−	20.7846i	0.763542	−	1.32249i
$248$		0			0
$249$		0			0
$250$		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
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3.00000	−	5.19615i	0.187135	−	0.324127i	−0.757159	−	0.653231i	$-0.773413\pi$
$257$	3.00000	−	5.19615i	0.187135	−	0.324127i	0.944294	+	0.329104i	$0.106747\pi$
$258$		0			0
$259$	12.0000	+	20.7846i	0.745644	+	1.29149i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−12.0000	−	20.7846i	−0.739952	−	1.28163i	−0.952517	−	0.304487i	$-0.901515\pi$
$263$	−12.0000	−	20.7846i	−0.739952	−	1.28163i	0.212565	−	0.977147i	$-0.431818\pi$
$264$		0			0
$265$	5.00000	−	8.66025i	0.307148	−	0.531995i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	6.00000			0.365826			0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000			0.365826			0.182913	+	0.983129i	$0.441447\pi$
$270$		0			0
$271$	8.00000			0.485965			0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000			0.485965			0.242983	+	0.970031i	$0.421874\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−5.00000	−	8.66025i	−0.300421	−	0.520344i	0.675810	−	0.737075i	$-0.263794\pi$
$277$	−5.00000	−	8.66025i	−0.300421	−	0.520344i	−0.976231	+	0.216731i	$0.930460\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.00000	+	1.73205i	0.0596550	+	0.103325i	0.894311	−	0.447447i	$-0.147667\pi$
$281$	1.00000	+	1.73205i	0.0596550	+	0.103325i	−0.834656	+	0.550772i	$0.814333\pi$
$282$		0			0
$283$	−2.00000	+	3.46410i	−0.118888	+	0.205919i	−0.919327	−	0.393494i	$-0.871266\pi$
$283$	−2.00000	+	3.46410i	−0.118888	+	0.205919i	0.800439	+	0.599414i	$0.204600\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−40.0000			−2.36113
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$		0			0
$292$		0			0
$293$	5.00000	−	8.66025i	0.292103	−	0.505937i	−0.682204	−	0.731162i	$-0.738978\pi$
$293$	5.00000	−	8.66025i	0.292103	−	0.505937i	0.974307	+	0.225225i	$0.0723116\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	24.0000	+	41.5692i	1.38796	+	2.40401i
$300$		0			0
$301$	8.00000	−	13.8564i	0.461112	−	0.798670i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	6.00000			0.343559
$306$		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
$309$		0			0
$310$		0			0
$311$	8.00000	−	13.8564i	0.453638	−	0.785725i	−0.544970	−	0.838455i	$-0.683459\pi$
$311$	8.00000	−	13.8564i	0.453638	−	0.785725i	0.998609	+	0.0527306i	$0.0167924\pi$
$312$		0			0
$313$	7.00000	+	12.1244i	0.395663	+	0.685309i	0.993186	−	0.116543i	$-0.0371814\pi$
$313$	7.00000	+	12.1244i	0.395663	+	0.685309i	−0.597522	+	0.801852i	$0.703848\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.00000	+	1.73205i	0.0561656	+	0.0972817i	0.892741	−	0.450570i	$-0.148779\pi$
$317$	1.00000	+	1.73205i	0.0561656	+	0.0972817i	−0.836576	+	0.547852i	$0.815446\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	8.00000			0.445132
$324$		0			0
$325$	−6.00000			−0.332820
$326$		0			0
$327$		0			0
$328$		0			0
$329$	16.0000	−	27.7128i	0.882109	−	1.52786i
$330$		0			0
$331$	−14.0000	−	24.2487i	−0.769510	−	1.33283i	−0.937829	−	0.347097i	$-0.887167\pi$
$331$	−14.0000	−	24.2487i	−0.769510	−	1.33283i	0.168320	−	0.985732i	$-0.446166\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2.00000	+	3.46410i	0.109272	+	0.189264i
$336$		0			0
$337$	−5.00000	+	8.66025i	−0.272367	+	0.471754i	−0.969468	−	0.245220i	$-0.921140\pi$
$337$	−5.00000	+	8.66025i	−0.272367	+	0.471754i	0.697100	+	0.716974i	$0.254473\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	8.00000			0.431959
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−18.0000	+	31.1769i	−0.966291	+	1.67366i	−0.260184	+	0.965559i	$0.583783\pi$
$347$	−18.0000	+	31.1769i	−0.966291	+	1.67366i	−0.706107	+	0.708105i	$0.749550\pi$
$348$		0			0
$349$	13.0000	+	22.5167i	0.695874	+	1.20529i	0.969885	+	0.243563i	$0.0783162\pi$
$349$	13.0000	+	22.5167i	0.695874	+	1.20529i	−0.274011	+	0.961727i	$0.588351\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9.00000	−	15.5885i	−0.479022	−	0.829690i	0.520689	−	0.853746i	$-0.325675\pi$
$353$	−9.00000	−	15.5885i	−0.479022	−	0.829690i	−0.999711	+	0.0240566i	$0.992342\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−8.00000			−0.422224			−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000			−0.422224			−0.211112	+	0.977462i	$0.567708\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$		0			0
$364$		0			0
$365$	7.00000	−	12.1244i	0.366397	−	0.634618i
$366$		0			0
$367$	−14.0000	−	24.2487i	−0.730794	−	1.26577i	−0.956544	−	0.291587i	$-0.905817\pi$
$367$	−14.0000	−	24.2487i	−0.730794	−	1.26577i	0.225750	−	0.974185i	$-0.427517\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	20.0000	+	34.6410i	1.03835	+	1.79847i
$372$		0			0
$373$	−1.00000	+	1.73205i	−0.0517780	+	0.0896822i	−0.890753	−	0.454488i	$-0.849822\pi$
$373$	−1.00000	+	1.73205i	−0.0517780	+	0.0896822i	0.838975	+	0.544170i	$0.183156\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−36.0000			−1.85409
$378$		0			0
$379$	4.00000			0.205466			0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000			0.205466			0.102733	+	0.994709i	$0.467241\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	4.00000	−	6.92820i	0.204390	−	0.354015i	−0.745548	−	0.666452i	$-0.767812\pi$
$383$	4.00000	−	6.92820i	0.204390	−	0.354015i	0.949938	+	0.312437i	$0.101145\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−3.00000	−	5.19615i	−0.152106	−	0.263455i	0.779895	−	0.625910i	$-0.215272\pi$
$389$	−3.00000	−	5.19615i	−0.152106	−	0.263455i	−0.932002	+	0.362454i	$0.881939\pi$
$390$		0			0
$391$	−8.00000	+	13.8564i	−0.404577	+	0.700749i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	16.0000			0.805047
$396$		0			0
$397$	2.00000			0.100377			0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000			0.100377			0.0501886	+	0.998740i	$0.484018\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3.00000	+	5.19615i	−0.149813	+	0.259483i	−0.931158	−	0.364615i	$-0.881200\pi$
$401$	−3.00000	+	5.19615i	−0.149813	+	0.259483i	0.781345	+	0.624099i	$0.214534\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	19.0000	−	32.9090i	0.939490	−	1.62724i	0.173064	−	0.984911i	$-0.444633\pi$
$409$	19.0000	−	32.9090i	0.939490	−	1.62724i	0.766426	−	0.642333i	$-0.222033\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−12.0000			−0.589057
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−8.00000	+	13.8564i	−0.390826	+	0.676930i	−0.992559	−	0.121768i	$-0.961144\pi$
$419$	−8.00000	+	13.8564i	−0.390826	+	0.676930i	0.601733	+	0.798697i	$0.294477\pi$
$420$		0			0
$421$	−3.00000	−	5.19615i	−0.146211	−	0.253245i	0.783613	−	0.621249i	$-0.213375\pi$
$421$	−3.00000	−	5.19615i	−0.146211	−	0.253245i	−0.929824	+	0.368004i	$0.880041\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.00000	−	1.73205i	−0.0485071	−	0.0840168i
$426$		0			0
$427$	−12.0000	+	20.7846i	−0.580721	+	1.00584i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	32.0000			1.54139			0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000			1.54139			0.770693	+	0.637207i	$0.219910\pi$
$432$		0			0
$433$	10.0000			0.480569			0.240285	−	0.970702i	$-0.422759\pi$
$433$	10.0000			0.480569			0.240285	+	0.970702i	$0.422759\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−16.0000	+	27.7128i	−0.765384	+	1.32568i
$438$		0			0
$439$	16.0000	+	27.7128i	0.763638	+	1.32266i	0.940963	+	0.338508i	$0.109922\pi$
$439$	16.0000	+	27.7128i	0.763638	+	1.32266i	−0.177325	+	0.984152i	$0.556744\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	6.00000	+	10.3923i	0.285069	+	0.493753i	0.972626	−	0.232377i	$-0.0746503\pi$
$443$	6.00000	+	10.3923i	0.285069	+	0.493753i	−0.687557	+	0.726130i	$0.741317\pi$
$444$		0			0
$445$	1.00000	−	1.73205i	0.0474045	−	0.0821071i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−26.0000			−1.22702			−0.613508	−	0.789689i	$-0.710242\pi$
$449$	−26.0000			−1.22702			−0.613508	+	0.789689i	$0.710242\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	12.0000	−	20.7846i	0.562569	−	0.974398i
$456$		0			0
$457$	19.0000	+	32.9090i	0.888783	+	1.53942i	0.841316	+	0.540544i	$0.181781\pi$
$457$	19.0000	+	32.9090i	0.888783	+	1.53942i	0.0474665	+	0.998873i	$0.484885\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1.00000	+	1.73205i	0.0465746	+	0.0806696i	0.888373	−	0.459123i	$-0.151836\pi$
$461$	1.00000	+	1.73205i	0.0465746	+	0.0806696i	−0.841798	+	0.539792i	$0.818503\pi$
$462$		0			0
$463$	−2.00000	+	3.46410i	−0.0929479	+	0.160990i	−0.908750	−	0.417340i	$-0.862962\pi$
$463$	−2.00000	+	3.46410i	−0.0929479	+	0.160990i	0.815802	+	0.578331i	$0.196296\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	20.0000			0.925490			0.462745	−	0.886492i	$-0.346865\pi$
$467$	20.0000			0.925490			0.462745	+	0.886492i	$0.346865\pi$
$468$		0			0
$469$	−16.0000			−0.738811
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−2.00000	−	3.46410i	−0.0917663	−	0.158944i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−12.0000	−	20.7846i	−0.548294	−	0.949673i	−0.998392	−	0.0566937i	$-0.981944\pi$
$479$	−12.0000	−	20.7846i	−0.548294	−	0.949673i	0.450098	−	0.892979i	$-0.351389\pi$
$480$		0			0
$481$	−18.0000	+	31.1769i	−0.820729	+	1.42154i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.00000			0.0908153
$486$		0			0
$487$	−36.0000			−1.63132			−0.815658	−	0.578535i	$-0.803625\pi$
$487$	−36.0000			−1.63132			−0.815658	+	0.578535i	$0.803625\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−20.0000	+	34.6410i	−0.902587	+	1.56333i	−0.0784639	+	0.996917i	$0.525002\pi$
$491$	−20.0000	+	34.6410i	−0.902587	+	1.56333i	−0.824123	+	0.566410i	$0.808332\pi$
$492$		0			0
$493$	−6.00000	−	10.3923i	−0.270226	−	0.468046i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−10.0000	+	17.3205i	−0.447661	+	0.775372i	−0.998233	−	0.0594153i	$-0.981076\pi$
$499$	−10.0000	+	17.3205i	−0.447661	+	0.775372i	0.550572	+	0.834788i	$0.314410\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$	14.0000			0.622992
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−3.00000	+	5.19615i	−0.132973	+	0.230315i	−0.924821	−	0.380402i	$-0.875786\pi$
$509$	−3.00000	+	5.19615i	−0.132973	+	0.230315i	0.791849	+	0.610718i	$0.209119\pi$
$510$		0			0
$511$	28.0000	+	48.4974i	1.23865	+	2.14540i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.00000	−	3.46410i	−0.0881305	−	0.152647i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	14.0000			0.613351			0.306676	−	0.951814i	$-0.400783\pi$
$521$	14.0000			0.613351			0.306676	+	0.951814i	$0.400783\pi$
$522$		0			0
$523$	36.0000			1.57417			0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000			1.57417			0.787085	+	0.616844i	$0.211589\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−20.5000	−	35.5070i	−0.891304	−	1.54378i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−30.0000	−	51.9615i	−1.29944	−	2.25070i
$534$		0			0
$535$	2.00000	−	3.46410i	0.0864675	−	0.149766i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−10.0000			−0.429934			−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000			−0.429934			−0.214967	+	0.976621i	$0.568964\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	5.00000	−	8.66025i	0.214176	−	0.370965i
$546$		0			0
$547$	−2.00000	−	3.46410i	−0.0855138	−	0.148114i	0.820096	−	0.572226i	$-0.193920\pi$
$547$	−2.00000	−	3.46410i	−0.0855138	−	0.148114i	−0.905610	+	0.424111i	$0.860587\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−12.0000	−	20.7846i	−0.511217	−	0.885454i
$552$		0			0
$553$	−32.0000	+	55.4256i	−1.36078	+	2.35694i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	14.0000			0.593199			0.296600	−	0.955002i	$-0.404147\pi$
$557$	14.0000			0.593199			0.296600	+	0.955002i	$0.404147\pi$
$558$		0			0
$559$	24.0000			1.01509
$560$		0			0
$561$		0			0
$562$		0			0
$563$	18.0000	−	31.1769i	0.758610	−	1.31395i	−0.184950	−	0.982748i	$-0.559212\pi$
$563$	18.0000	−	31.1769i	0.758610	−	1.31395i	0.943560	−	0.331202i	$-0.107454\pi$
$564$		0			0
$565$	3.00000	+	5.19615i	0.126211	+	0.218604i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−7.00000	−	12.1244i	−0.293455	−	0.508279i	0.681169	−	0.732126i	$-0.261472\pi$
$569$	−7.00000	−	12.1244i	−0.293455	−	0.508279i	−0.974624	+	0.223847i	$0.928139\pi$
$570$		0			0
$571$	10.0000	−	17.3205i	0.418487	−	0.724841i	−0.577301	−	0.816532i	$-0.695894\pi$
$571$	10.0000	−	17.3205i	0.418487	−	0.724841i	0.995788	+	0.0916910i	$0.0292272\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	8.00000			0.333623
$576$		0			0
$577$	34.0000			1.41544			0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000			1.41544			0.707719	+	0.706494i	$0.249724\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	24.0000	−	41.5692i	0.995688	−	1.72458i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	18.0000	+	31.1769i	0.742940	+	1.28681i	0.951151	+	0.308725i	$0.0999023\pi$
$587$	18.0000	+	31.1769i	0.742940	+	1.28681i	−0.208212	+	0.978084i	$0.566764\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−22.0000			−0.903432			−0.451716	−	0.892162i	$-0.649188\pi$
$593$	−22.0000			−0.903432			−0.451716	+	0.892162i	$0.649188\pi$
$594$		0			0
$595$	8.00000			0.327968
$596$		0			0
$597$		0			0
$598$		0			0
$599$	4.00000	−	6.92820i	0.163436	−	0.283079i	−0.772663	−	0.634816i	$-0.781076\pi$
$599$	4.00000	−	6.92820i	0.163436	−	0.283079i	0.936099	+	0.351738i	$0.114409\pi$
$600$		0			0
$601$	11.0000	+	19.0526i	0.448699	+	0.777170i	0.998302	−	0.0582563i	$-0.0185541\pi$
$601$	11.0000	+	19.0526i	0.448699	+	0.777170i	−0.549602	+	0.835426i	$0.685221\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	5.50000	+	9.52628i	0.223607	+	0.387298i
$606$		0			0
$607$	6.00000	−	10.3923i	0.243532	−	0.421811i	−0.718186	−	0.695852i	$-0.755027\pi$
$607$	6.00000	−	10.3923i	0.243532	−	0.421811i	0.961718	+	0.274041i	$0.0883604\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	48.0000			1.94187
$612$		0			0
$613$	26.0000			1.05013			0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000			1.05013			0.525065	+	0.851062i	$0.324041\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−13.0000	+	22.5167i	−0.523360	+	0.906487i	0.476270	+	0.879299i	$0.341988\pi$
$617$	−13.0000	+	22.5167i	−0.523360	+	0.906487i	−0.999630	+	0.0271876i	$0.991345\pi$
$618$		0			0
$619$	14.0000	+	24.2487i	0.562708	+	0.974638i	0.997259	+	0.0739910i	$0.0235736\pi$
$619$	14.0000	+	24.2487i	0.562708	+	0.974638i	−0.434551	+	0.900647i	$0.643093\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	4.00000	+	6.92820i	0.160257	+	0.277573i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−12.0000			−0.478471
$630$		0			0
$631$	40.0000			1.59237			0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000			1.59237			0.796187	+	0.605050i	$0.206847\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2.00000	−	3.46410i	0.0793676	−	0.137469i
$636$		0			0
$637$	27.0000	+	46.7654i	1.06978	+	1.85291i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	9.00000	+	15.5885i	0.355479	+	0.615707i	0.987200	−	0.159489i	$-0.0509845\pi$
$641$	9.00000	+	15.5885i	0.355479	+	0.615707i	−0.631721	+	0.775196i	$0.717651\pi$
$642$		0			0
$643$	18.0000	−	31.1769i	0.709851	−	1.22950i	−0.255062	−	0.966925i	$-0.582096\pi$
$643$	18.0000	−	31.1769i	0.709851	−	1.22950i	0.964912	−	0.262573i	$-0.0845709\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−48.0000			−1.88707			−0.943537	−	0.331266i	$-0.892524\pi$
$647$	−48.0000			−1.88707			−0.943537	+	0.331266i	$0.892524\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	9.00000	−	15.5885i	0.352197	−	0.610023i	−0.634437	−	0.772975i	$-0.718768\pi$
$653$	9.00000	−	15.5885i	0.352197	−	0.610023i	0.986634	+	0.162951i	$0.0521013\pi$
$654$		0			0
$655$	8.00000	+	13.8564i	0.312586	+	0.541415i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−8.00000	−	13.8564i	−0.311636	−	0.539769i	0.667081	−	0.744985i	$-0.267544\pi$
$659$	−8.00000	−	13.8564i	−0.311636	−	0.539769i	−0.978717	+	0.205216i	$0.934210\pi$
$660$		0			0
$661$	1.00000	−	1.73205i	0.0388955	−	0.0673690i	−0.845922	−	0.533306i	$-0.820949\pi$
$661$	1.00000	−	1.73205i	0.0388955	−	0.0673690i	0.884818	+	0.465937i	$0.154283\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	16.0000			0.620453
$666$		0			0
$667$	48.0000			1.85857
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	11.0000	+	19.0526i	0.424019	+	0.734422i	0.996328	−	0.0856156i	$-0.0272857\pi$
$673$	11.0000	+	19.0526i	0.424019	+	0.734422i	−0.572309	+	0.820038i	$0.693952\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−3.00000	−	5.19615i	−0.115299	−	0.199704i	0.802600	−	0.596518i	$-0.203449\pi$
$677$	−3.00000	−	5.19615i	−0.115299	−	0.199704i	−0.917899	+	0.396813i	$0.870116\pi$
$678$		0			0
$679$	−4.00000	+	6.92820i	−0.153506	+	0.265880i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−36.0000			−1.37750			−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000			−1.37750			−0.688751	+	0.724998i	$0.741841\pi$
$684$		0			0
$685$	18.0000			0.687745
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−30.0000	+	51.9615i	−1.14291	+	1.97958i
$690$		0			0
$691$	−2.00000	−	3.46410i	−0.0760836	−	0.131781i	0.825473	−	0.564441i	$-0.190908\pi$
$691$	−2.00000	−	3.46410i	−0.0760836	−	0.131781i	−0.901557	+	0.432660i	$0.857575\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	6.00000	+	10.3923i	0.227593	+	0.394203i
$696$		0			0
$697$	10.0000	−	17.3205i	0.378777	−	0.656061i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	38.0000			1.43524			0.717620	−	0.696435i	$-0.245231\pi$
$701$	38.0000			1.43524			0.717620	+	0.696435i	$0.245231\pi$
$702$		0			0
$703$	−24.0000			−0.905177
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−28.0000	+	48.4974i	−1.05305	+	1.82393i
$708$		0			0
$709$	−11.0000	−	19.0526i	−0.413114	−	0.715534i	0.582115	−	0.813107i	$-0.302225\pi$
$709$	−11.0000	−	19.0526i	−0.413114	−	0.715534i	−0.995228	+	0.0975728i	$0.968892\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−48.0000			−1.79010			−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000			−1.79010			−0.895049	+	0.445968i	$0.852860\pi$
$720$		0			0
$721$	16.0000			0.595871
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3.00000	+	5.19615i	−0.111417	+	0.192980i
$726$		0			0
$727$	2.00000	+	3.46410i	0.0741759	+	0.128476i	0.900728	−	0.434384i	$-0.143034\pi$
$727$	2.00000	+	3.46410i	0.0741759	+	0.128476i	−0.826552	+	0.562861i	$0.809701\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	4.00000	+	6.92820i	0.147945	+	0.256249i
$732$		0			0
$733$	23.0000	−	39.8372i	0.849524	−	1.47142i	−0.0321090	−	0.999484i	$-0.510222\pi$
$733$	23.0000	−	39.8372i	0.849524	−	1.47142i	0.881633	−	0.471935i	$-0.156444\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	4.00000			0.147142			0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000			0.147142			0.0735712	+	0.997290i	$0.476560\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$743$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$744$		0			0
$745$	−3.00000	−	5.19615i	−0.109911	−	0.190372i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	8.00000	+	13.8564i	0.292314	+	0.506302i
$750$		0			0
$751$	24.0000	−	41.5692i	0.875772	−	1.51688i	0.0198348	−	0.999803i	$-0.493686\pi$
$751$	24.0000	−	41.5692i	0.875772	−	1.51688i	0.855938	−	0.517079i	$-0.172981\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	42.0000			1.52652			0.763258	−	0.646094i	$-0.223599\pi$
$757$	42.0000			1.52652			0.763258	+	0.646094i	$0.223599\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	25.0000	−	43.3013i	0.906249	−	1.56967i	0.0870179	−	0.996207i	$-0.472266\pi$
$761$	25.0000	−	43.3013i	0.906249	−	1.56967i	0.819231	−	0.573463i	$-0.194400\pi$
$762$		0			0
$763$	20.0000	+	34.6410i	0.724049	+	1.25409i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	15.0000	−	25.9808i	0.540914	−	0.936890i	−0.457938	−	0.888984i	$-0.651412\pi$
$769$	15.0000	−	25.9808i	0.540914	−	0.936890i	0.998852	−	0.0479061i	$-0.0152548\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	38.0000			1.36677			0.683383	−	0.730061i	$-0.260508\pi$
$773$	38.0000			1.36677			0.683383	+	0.730061i	$0.260508\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	20.0000	−	34.6410i	0.716574	−	1.24114i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−5.00000	−	8.66025i	−0.178458	−	0.309098i
$786$		0			0
$787$	−6.00000	+	10.3923i	−0.213877	+	0.370446i	−0.952925	−	0.303207i	$-0.901942\pi$
$787$	−6.00000	+	10.3923i	−0.213877	+	0.370446i	0.739048	+	0.673653i	$0.235276\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−24.0000			−0.853342
$792$		0			0
$793$	−36.0000			−1.27840
$794$		0			0
$795$		0			0
$796$		0			0
$797$	17.0000	−	29.4449i	0.602171	−	1.04299i	−0.390321	−	0.920679i	$-0.627636\pi$
$797$	17.0000	−	29.4449i	0.602171	−	1.04299i	0.992492	−	0.122312i	$-0.0390308\pi$
$798$		0			0
$799$	8.00000	+	13.8564i	0.283020	+	0.490204i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−16.0000	+	27.7128i	−0.563926	+	0.976748i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	6.00000			0.210949			0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000			0.210949			0.105474	+	0.994422i	$0.466364\pi$
$810$		0			0
$811$	−12.0000			−0.421377			−0.210688	−	0.977553i	$-0.567571\pi$
$811$	−12.0000			−0.421377			−0.210688	+	0.977553i	$0.567571\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2.00000	−	3.46410i	0.0700569	−	0.121342i
$816$		0			0
$817$	8.00000	+	13.8564i	0.279885	+	0.484774i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	25.0000	+	43.3013i	0.872506	+	1.51122i	0.859396	+	0.511311i	$0.170840\pi$
$821$	25.0000	+	43.3013i	0.872506	+	1.51122i	0.0131101	+	0.999914i	$0.495827\pi$
$822$		0			0
$823$	2.00000	−	3.46410i	0.0697156	−	0.120751i	−0.829060	−	0.559159i	$-0.811124\pi$
$823$	2.00000	−	3.46410i	0.0697156	−	0.120751i	0.898776	+	0.438408i	$0.144457\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	20.0000			0.695468			0.347734	−	0.937593i	$-0.386951\pi$
$827$	20.0000			0.695468			0.347734	+	0.937593i	$0.386951\pi$
$828$		0			0
$829$	38.0000			1.31979			0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000			1.31979			0.659897	+	0.751356i	$0.270600\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−9.00000	+	15.5885i	−0.311832	+	0.540108i
$834$		0			0
$835$	−4.00000	−	6.92820i	−0.138426	−	0.239760i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−20.0000	−	34.6410i	−0.690477	−	1.19594i	−0.971682	−	0.236293i	$-0.924067\pi$
$839$	−20.0000	−	34.6410i	−0.690477	−	1.19594i	0.281205	−	0.959648i	$-0.409266\pi$
$840$		0			0
$841$	−3.50000	+	6.06218i	−0.120690	+	0.209041i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	23.0000			0.791224
$846$		0			0
$847$	−44.0000			−1.51186
$848$		0			0
$849$		0			0
$850$		0			0
$851$	24.0000	−	41.5692i	0.822709	−	1.42497i
$852$		0			0
$853$	−5.00000	−	8.66025i	−0.171197	−	0.296521i	0.767642	−	0.640879i	$-0.221430\pi$
$853$	−5.00000	−	8.66025i	−0.171197	−	0.296521i	−0.938839	+	0.344358i	$0.888097\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	19.0000	+	32.9090i	0.649028	+	1.12415i	0.983355	+	0.181692i	$0.0581574\pi$
$857$	19.0000	+	32.9090i	0.649028	+	1.12415i	−0.334328	+	0.942457i	$0.608509\pi$
$858$		0			0
$859$	−18.0000	+	31.1769i	−0.614152	+	1.06374i	0.376381	+	0.926465i	$0.377169\pi$
$859$	−18.0000	+	31.1769i	−0.614152	+	1.06374i	−0.990533	+	0.137277i	$0.956165\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−16.0000			−0.544646			−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000			−0.544646			−0.272323	+	0.962206i	$0.587792\pi$
$864$		0			0
$865$	−18.0000			−0.612018
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−12.0000	−	20.7846i	−0.406604	−	0.704260i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.00000	−	3.46410i	−0.0676123	−	0.117108i
$876$		0			0
$877$	−5.00000	+	8.66025i	−0.168838	+	0.292436i	−0.938012	−	0.346604i	$-0.887335\pi$
$877$	−5.00000	+	8.66025i	−0.168838	+	0.292436i	0.769174	+	0.639040i	$0.220668\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−26.0000			−0.875962			−0.437981	−	0.898984i	$-0.644306\pi$
$881$	−26.0000			−0.875962			−0.437981	+	0.898984i	$0.644306\pi$
$882$		0			0
$883$	28.0000			0.942275			0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000			0.942275			0.471138	+	0.882060i	$0.343844\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	12.0000	−	20.7846i	0.402921	−	0.697879i	−0.591156	−	0.806557i	$-0.701328\pi$
$887$	12.0000	−	20.7846i	0.402921	−	0.697879i	0.994077	+	0.108678i	$0.0346618\pi$
$888$		0			0
$889$	8.00000	+	13.8564i	0.268311	+	0.464729i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	16.0000	+	27.7128i	0.535420	+	0.927374i
$894$		0			0
$895$	4.00000	−	6.92820i	0.133705	−	0.231584i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−20.0000			−0.666297
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−7.00000	+	12.1244i	−0.232688	+	0.403027i
$906$		0			0
$907$	−14.0000	−	24.2487i	−0.464862	−	0.805165i	0.534333	−	0.845274i	$-0.320563\pi$
$907$	−14.0000	−	24.2487i	−0.464862	−	0.805165i	−0.999195	+	0.0401089i	$0.987230\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−16.0000	−	27.7128i	−0.530104	−	0.918166i	−0.999383	−	0.0351168i	$-0.988820\pi$
$911$	−16.0000	−	27.7128i	−0.530104	−	0.918166i	0.469280	−	0.883050i	$-0.344514\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−64.0000			−2.11347
$918$		0			0
$919$	40.0000			1.31948			0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000			1.31948			0.659739	+	0.751495i	$0.270667\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	3.00000	+	5.19615i	0.0986394	+	0.170848i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−3.00000	−	5.19615i	−0.0984268	−	0.170480i	0.812607	−	0.582812i	$-0.198048\pi$
$929$	−3.00000	−	5.19615i	−0.0984268	−	0.170480i	−0.911034	+	0.412332i	$0.864714\pi$
$930$		0			0
$931$	−18.0000	+	31.1769i	−0.589926	+	1.02178i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	26.0000			0.849383			0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000			0.849383			0.424691	+	0.905338i	$0.360383\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−7.00000	+	12.1244i	−0.228193	+	0.395243i	−0.957273	−	0.289187i	$-0.906615\pi$
$941$	−7.00000	+	12.1244i	−0.228193	+	0.395243i	0.729079	+	0.684429i	$0.239949\pi$
$942$		0			0
$943$	40.0000	+	69.2820i	1.30258	+	2.25613i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−22.0000	−	38.1051i	−0.714904	−	1.23825i	−0.962997	−	0.269514i	$-0.913137\pi$
$947$	−22.0000	−	38.1051i	−0.714904	−	1.23825i	0.248093	−	0.968736i	$-0.420196\pi$
$948$		0			0
$949$	−42.0000	+	72.7461i	−1.36338	+	2.36144i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	18.0000			0.583077			0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000			0.583077			0.291539	+	0.956559i	$0.405833\pi$
$954$		0			0
$955$	−8.00000			−0.258874
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−36.0000	+	62.3538i	−1.16250	+	2.01351i
$960$		0			0
$961$	15.5000	+	26.8468i	0.500000	+	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	3.00000	+	5.19615i	0.0965734	+	0.167270i
$966$		0			0
$967$	−10.0000	+	17.3205i	−0.321578	+	0.556990i	−0.980814	−	0.194946i	$-0.937547\pi$
$967$	−10.0000	+	17.3205i	−0.321578	+	0.556990i	0.659236	+	0.751936i	$0.270880\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	56.0000			1.79713			0.898563	−	0.438845i	$-0.144612\pi$
$971$	56.0000			1.79713			0.898563	+	0.438845i	$0.144612\pi$
$972$		0			0
$973$	−48.0000			−1.53881
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−25.0000	+	43.3013i	−0.799821	+	1.38533i	0.119912	+	0.992785i	$0.461739\pi$
$977$	−25.0000	+	43.3013i	−0.799821	+	1.38533i	−0.919732	+	0.392546i	$0.871594\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$983$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$984$		0			0
$985$	5.00000	−	8.66025i	0.159313	−	0.275939i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−32.0000			−1.01754
$990$		0			0
$991$	56.0000			1.77890			0.889449	−	0.457034i	$-0.151088\pi$
$991$	56.0000			1.77890			0.889449	+	0.457034i	$0.151088\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	15.0000	+	25.9808i	0.475055	+	0.822819i	0.999592	−	0.0285686i	$-0.00909491\pi$
$997$	15.0000	+	25.9808i	0.475055	+	0.822819i	−0.524537	+	0.851388i	$0.675762\pi$
$998$		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	3240.2.q.a.1081.1		2
3.2	odd	2		3240.2.q.m.1081.1		2
9.2	odd	6		3240.2.q.m.2161.1		2
9.4	even	3		360.2.a.e.1.1		1
9.5	odd	6		120.2.a.a.1.1	✓	1
9.7	even	3	inner	3240.2.q.a.2161.1		2
36.23	even	6		240.2.a.a.1.1		1
36.31	odd	6		720.2.a.f.1.1		1
45.4	even	6		1800.2.a.c.1.1		1
45.13	odd	12		1800.2.f.g.649.1		2
45.14	odd	6		600.2.a.a.1.1		1
45.22	odd	12		1800.2.f.g.649.2		2
45.23	even	12		600.2.f.c.49.2		2
45.32	even	12		600.2.f.c.49.1		2
63.41	even	6		5880.2.a.p.1.1		1
72.5	odd	6		960.2.a.g.1.1		1
72.13	even	6		2880.2.a.r.1.1		1
72.59	even	6		960.2.a.n.1.1		1
72.67	odd	6		2880.2.a.b.1.1		1
144.5	odd	12		3840.2.k.a.1921.1		2
144.59	even	12		3840.2.k.z.1921.2		2
144.77	odd	12		3840.2.k.a.1921.2		2
144.131	even	12		3840.2.k.z.1921.1		2
180.23	odd	12		1200.2.f.f.49.1		2
180.59	even	6		1200.2.a.r.1.1		1
180.67	even	12		3600.2.f.l.2449.1		2
180.103	even	12		3600.2.f.l.2449.2		2
180.139	odd	6		3600.2.a.bo.1.1		1
180.167	odd	12		1200.2.f.f.49.2		2
360.59	even	6		4800.2.a.bh.1.1		1
360.77	even	12		4800.2.f.u.3649.2		2
360.149	odd	6		4800.2.a.bl.1.1		1
360.203	odd	12		4800.2.f.n.3649.2		2
360.293	even	12		4800.2.f.u.3649.1		2
360.347	odd	12		4800.2.f.n.3649.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.a.a.1.1	✓	1	9.5	odd	6
240.2.a.a.1.1		1	36.23	even	6
360.2.a.e.1.1		1	9.4	even	3
600.2.a.a.1.1		1	45.14	odd	6
600.2.f.c.49.1		2	45.32	even	12
600.2.f.c.49.2		2	45.23	even	12
720.2.a.f.1.1		1	36.31	odd	6
960.2.a.g.1.1		1	72.5	odd	6
960.2.a.n.1.1		1	72.59	even	6
1200.2.a.r.1.1		1	180.59	even	6
1200.2.f.f.49.1		2	180.23	odd	12
1200.2.f.f.49.2		2	180.167	odd	12
1800.2.a.c.1.1		1	45.4	even	6
1800.2.f.g.649.1		2	45.13	odd	12
1800.2.f.g.649.2		2	45.22	odd	12
2880.2.a.b.1.1		1	72.67	odd	6
2880.2.a.r.1.1		1	72.13	even	6
3240.2.q.a.1081.1		2	1.1	even	1	trivial
3240.2.q.a.2161.1		2	9.7	even	3	inner
3240.2.q.m.1081.1		2	3.2	odd	2
3240.2.q.m.2161.1		2	9.2	odd	6
3600.2.a.bo.1.1		1	180.139	odd	6
3600.2.f.l.2449.1		2	180.67	even	12
3600.2.f.l.2449.2		2	180.103	even	12
3840.2.k.a.1921.1		2	144.5	odd	12
3840.2.k.a.1921.2		2	144.77	odd	12
3840.2.k.z.1921.1		2	144.131	even	12
3840.2.k.z.1921.2		2	144.59	even	12
4800.2.a.bh.1.1		1	360.59	even	6
4800.2.a.bl.1.1		1	360.149	odd	6
4800.2.f.n.3649.1		2	360.347	odd	12
4800.2.f.n.3649.2		2	360.203	odd	12
4800.2.f.u.3649.1		2	360.293	even	12
4800.2.f.u.3649.2		2	360.77	even	12
5880.2.a.p.1.1		1	63.41	even	6

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 $\F_{q}$

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	3240.2.q.a.1081.1

Level	$3240$
Weight	$2$
Character	3240.1081
Analytic conductor	$25.872$
Analytic rank	$0$
Dimension	$2$
Inner twists	$2$