LMFDB - Embedded newform 2160.2.q.d.721.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2160,2,Mod(721,2160)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2160, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2160.721"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$2160 = 2^{4} \cdot 3^{3} \cdot 5$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2160.q (of order $3$ , degree $2$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,1,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$17.2476868366$
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 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			721.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2160.721
Dual form			2160.2.q.d.1441.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{5} +(-2.00000 + 3.46410i) q^{7} +(-1.50000 + 2.59808i) q^{11} +(2.00000 + 3.46410i) q^{13} -3.00000 q^{17} -5.00000 q^{19} +(-3.00000 - 5.19615i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(3.00000 - 5.19615i) q^{29} +(1.00000 + 1.73205i) q^{31} -4.00000 q^{35} -4.00000 q^{37} +(-1.50000 - 2.59808i) q^{41} +(5.50000 - 9.52628i) q^{43} +(-4.50000 - 7.79423i) q^{49} -6.00000 q^{53} -3.00000 q^{55} +(1.50000 + 2.59808i) q^{59} +(5.00000 - 8.66025i) q^{61} +(-2.00000 + 3.46410i) q^{65} +(2.50000 + 4.33013i) q^{67} +6.00000 q^{71} -7.00000 q^{73} +(-6.00000 - 10.3923i) q^{77} +(7.00000 - 12.1244i) q^{79} +(-6.00000 + 10.3923i) q^{83} +(-1.50000 - 2.59808i) q^{85} -6.00000 q^{89} -16.0000 q^{91} +(-2.50000 - 4.33013i) q^{95} +(-5.50000 + 9.52628i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + q^{5} - 4 q^{7} - 3 q^{11} + 4 q^{13} - 6 q^{17} - 10 q^{19} - 6 q^{23} - q^{25} + 6 q^{29} + 2 q^{31} - 8 q^{35} - 8 q^{37} - 3 q^{41} + 11 q^{43} - 9 q^{49} - 12 q^{53} - 6 q^{55} + 3 q^{59} + 10 q^{61}+ \cdots - 11 q^{97}+O(q^{100})$ 2 * q + q^5 - 4 * q^7 - 3 * q^11 + 4 * q^13 - 6 * q^17 - 10 * q^19 - 6 * q^23 - q^25 + 6 * q^29 + 2 * q^31 - 8 * q^35 - 8 * q^37 - 3 * q^41 + 11 * q^43 - 9 * q^49 - 12 * q^53 - 6 * q^55 + 3 * q^59 + 10 * q^61 - 4 * q^65 + 5 * q^67 + 12 * q^71 - 14 * q^73 - 12 * q^77 + 14 * q^79 - 12 * q^83 - 3 * q^85 - 12 * q^89 - 32 * q^91 - 5 * q^95 - 11 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$ .

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$		0			0
$2$		0			0
$3$		0			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	0.188982	+	0.981981i	$0.439481\pi$
$7$	−2.00000	+	3.46410i	−0.755929	+	1.30931i	−0.944911	+	0.327327i	$0.893852\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.50000	+	2.59808i	−0.452267	+	0.783349i	−0.998526	−	0.0542666i	$-0.982718\pi$
$11$	−1.50000	+	2.59808i	−0.452267	+	0.783349i	0.546259	+	0.837616i	$0.316051\pi$
$12$		0			0
$13$	2.00000	+	3.46410i	0.554700	+	0.960769i	0.997927	+	0.0643593i	$0.0205004\pi$
$13$	2.00000	+	3.46410i	0.554700	+	0.960769i	−0.443227	+	0.896410i	$0.646166\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.00000			−0.727607			−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000			−0.727607			−0.363803	+	0.931476i	$0.618522\pi$
$18$		0			0
$19$	−5.00000			−1.14708			−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000			−1.14708			−0.573539	+	0.819178i	$0.694430\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.00000	−	5.19615i	−0.625543	−	1.08347i	−0.988436	−	0.151642i	$-0.951544\pi$
$23$	−3.00000	−	5.19615i	−0.625543	−	1.08347i	0.362892	−	0.931831i	$-0.381789\pi$
$24$		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	−0.440652	−	0.897678i	$-0.645253\pi$
$29$	3.00000	−	5.19615i	0.557086	−	0.964901i	0.997738	−	0.0672232i	$-0.0214140\pi$
$30$		0			0
$31$	1.00000	+	1.73205i	0.179605	+	0.311086i	0.941745	−	0.336327i	$-0.109185\pi$
$31$	1.00000	+	1.73205i	0.179605	+	0.311086i	−0.762140	+	0.647412i	$0.775851\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.00000			−0.676123
$36$		0			0
$37$	−4.00000			−0.657596			−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000			−0.657596			−0.328798	+	0.944400i	$0.606644\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.50000	−	2.59808i	−0.234261	−	0.405751i	0.724797	−	0.688963i	$-0.241934\pi$
$41$	−1.50000	−	2.59808i	−0.234261	−	0.405751i	−0.959058	+	0.283211i	$0.908600\pi$
$42$		0			0
$43$	5.50000	−	9.52628i	0.838742	−	1.45274i	−0.0522047	−	0.998636i	$-0.516625\pi$
$43$	5.50000	−	9.52628i	0.838742	−	1.45274i	0.890947	−	0.454108i	$-0.150042\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		0			0
$49$	−4.50000	−	7.79423i	−0.642857	−	1.11346i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$	−3.00000			−0.404520
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.50000	+	2.59808i	0.195283	+	0.338241i	0.946993	−	0.321253i	$-0.104104\pi$
$59$	1.50000	+	2.59808i	0.195283	+	0.338241i	−0.751710	+	0.659494i	$0.770771\pi$
$60$		0			0
$61$	5.00000	−	8.66025i	0.640184	−	1.10883i	−0.345207	−	0.938527i	$-0.612191\pi$
$61$	5.00000	−	8.66025i	0.640184	−	1.10883i	0.985391	−	0.170305i	$-0.0544754\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.00000	+	3.46410i	−0.248069	+	0.429669i
$66$		0			0
$67$	2.50000	+	4.33013i	0.305424	+	0.529009i	0.977356	−	0.211604i	$-0.0678686\pi$
$67$	2.50000	+	4.33013i	0.305424	+	0.529009i	−0.671932	+	0.740613i	$0.734535\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	6.00000			0.712069			0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000			0.712069			0.356034	+	0.934473i	$0.384129\pi$
$72$		0			0
$73$	−7.00000			−0.819288			−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000			−0.819288			−0.409644	+	0.912245i	$0.634347\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−6.00000	−	10.3923i	−0.683763	−	1.18431i
$78$		0			0
$79$	7.00000	−	12.1244i	0.787562	−	1.36410i	−0.139895	−	0.990166i	$-0.544677\pi$
$79$	7.00000	−	12.1244i	0.787562	−	1.36410i	0.927457	−	0.373930i	$-0.121990\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.00000	+	10.3923i	−0.658586	+	1.14070i	0.322396	+	0.946605i	$0.395512\pi$
$83$	−6.00000	+	10.3923i	−0.658586	+	1.14070i	−0.980982	+	0.194099i	$0.937822\pi$
$84$		0			0
$85$	−1.50000	−	2.59808i	−0.162698	−	0.281801i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$	−16.0000			−1.67726
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.50000	−	4.33013i	−0.256495	−	0.444262i
$96$		0			0
$97$	−5.50000	+	9.52628i	−0.558440	+	0.967247i	0.439187	+	0.898396i	$0.355267\pi$
$97$	−5.50000	+	9.52628i	−0.558440	+	0.967247i	−0.997627	+	0.0688512i	$0.978067\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.00000	+	10.3923i	−0.597022	+	1.03407i	0.396236	+	0.918149i	$0.370316\pi$
$101$	−6.00000	+	10.3923i	−0.597022	+	1.03407i	−0.993258	+	0.115924i	$0.963017\pi$
$102$		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
$107$	−9.00000			−0.870063			−0.435031	−	0.900415i	$-0.643263\pi$
$107$	−9.00000			−0.870063			−0.435031	+	0.900415i	$0.643263\pi$
$108$		0			0
$109$	−4.00000			−0.383131			−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000			−0.383131			−0.191565	+	0.981480i	$0.561356\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	9.00000	+	15.5885i	0.846649	+	1.46644i	0.884182	+	0.467143i	$0.154717\pi$
$113$	9.00000	+	15.5885i	0.846649	+	1.46644i	−0.0375328	+	0.999295i	$0.511950\pi$
$114$		0			0
$115$	3.00000	−	5.19615i	0.279751	−	0.484544i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	6.00000	−	10.3923i	0.550019	−	0.952661i
$120$		0			0
$121$	1.00000	+	1.73205i	0.0909091	+	0.157459i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−2.00000			−0.177471			−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000			−0.177471			−0.0887357	+	0.996055i	$0.528283\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6.00000	−	10.3923i	−0.524222	−	0.907980i	−0.999602	−	0.0281993i	$-0.991023\pi$
$131$	−6.00000	−	10.3923i	−0.524222	−	0.907980i	0.475380	−	0.879781i	$-0.342311\pi$
$132$		0			0
$133$	10.0000	−	17.3205i	0.867110	−	1.50188i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.50000	+	7.79423i	−0.384461	+	0.665906i	−0.991694	−	0.128618i	$-0.958946\pi$
$137$	−4.50000	+	7.79423i	−0.384461	+	0.665906i	0.607233	+	0.794524i	$0.292279\pi$
$138$		0			0
$139$	−0.500000	−	0.866025i	−0.0424094	−	0.0734553i	0.844042	−	0.536278i	$-0.180170\pi$
$139$	−0.500000	−	0.866025i	−0.0424094	−	0.0734553i	−0.886451	+	0.462822i	$0.846837\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−12.0000			−1.00349
$144$		0			0
$145$	6.00000			0.498273
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$149$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$150$		0			0
$151$	−5.00000	+	8.66025i	−0.406894	+	0.704761i	−0.994540	−	0.104357i	$-0.966722\pi$
$151$	−5.00000	+	8.66025i	−0.406894	+	0.704761i	0.587646	+	0.809118i	$0.300055\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.00000	+	1.73205i	−0.0803219	+	0.139122i
$156$		0			0
$157$	−4.00000	−	6.92820i	−0.319235	−	0.552931i	0.661094	−	0.750303i	$-0.270093\pi$
$157$	−4.00000	−	6.92820i	−0.319235	−	0.552931i	−0.980329	+	0.197372i	$0.936759\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	24.0000			1.89146
$162$		0			0
$163$	16.0000			1.25322			0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000			1.25322			0.626608	+	0.779334i	$0.284443\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	9.00000	+	15.5885i	0.696441	+	1.20627i	0.969693	+	0.244328i	$0.0785675\pi$
$167$	9.00000	+	15.5885i	0.696441	+	1.20627i	−0.273252	+	0.961943i	$0.588099\pi$
$168$		0			0
$169$	−1.50000	+	2.59808i	−0.115385	+	0.199852i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−9.00000	+	15.5885i	−0.684257	+	1.18517i	0.289412	+	0.957205i	$0.406540\pi$
$173$	−9.00000	+	15.5885i	−0.684257	+	1.18517i	−0.973670	+	0.227964i	$0.926793\pi$
$174$		0			0
$175$	−2.00000	−	3.46410i	−0.151186	−	0.261861i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	−16.0000			−1.18927			−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000			−1.18927			−0.594635	+	0.803996i	$0.702704\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2.00000	−	3.46410i	−0.147043	−	0.254686i
$186$		0			0
$187$	4.50000	−	7.79423i	0.329073	−	0.569970i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$191$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$192$		0			0
$193$	6.50000	+	11.2583i	0.467880	+	0.810392i	0.999326	−	0.0366998i	$-0.0116845\pi$
$193$	6.50000	+	11.2583i	0.467880	+	0.810392i	−0.531446	+	0.847092i	$0.678351\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	−20.0000			−1.41776			−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000			−1.41776			−0.708881	+	0.705328i	$0.750800\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	12.0000	+	20.7846i	0.842235	+	1.45879i
$204$		0			0
$205$	1.50000	−	2.59808i	0.104765	−	0.181458i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	7.50000	−	12.9904i	0.518786	−	0.898563i
$210$		0			0
$211$	−2.00000	−	3.46410i	−0.137686	−	0.238479i	0.788935	−	0.614477i	$-0.210633\pi$
$211$	−2.00000	−	3.46410i	−0.137686	−	0.238479i	−0.926620	+	0.375999i	$0.877300\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	11.0000			0.750194
$216$		0			0
$217$	−8.00000			−0.543075
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−6.00000	−	10.3923i	−0.403604	−	0.699062i
$222$		0			0
$223$	−11.0000	+	19.0526i	−0.736614	+	1.27585i	0.217397	+	0.976083i	$0.430243\pi$
$223$	−11.0000	+	19.0526i	−0.736614	+	1.27585i	−0.954011	+	0.299770i	$0.903090\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1.50000	−	2.59808i	0.0995585	−	0.172440i	−0.811943	−	0.583736i	$-0.801590\pi$
$227$	1.50000	−	2.59808i	0.0995585	−	0.172440i	0.911502	+	0.411296i	$0.134924\pi$
$228$		0			0
$229$	−10.0000	−	17.3205i	−0.660819	−	1.14457i	−0.980401	−	0.197013i	$-0.936876\pi$
$229$	−10.0000	−	17.3205i	−0.660819	−	1.14457i	0.319582	−	0.947559i	$-0.396457\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−21.0000			−1.37576			−0.687878	−	0.725826i	$-0.741458\pi$
$233$	−21.0000			−1.37576			−0.687878	+	0.725826i	$0.741458\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−3.00000	−	5.19615i	−0.194054	−	0.336111i	0.752536	−	0.658551i	$-0.228830\pi$
$239$	−3.00000	−	5.19615i	−0.194054	−	0.336111i	−0.946590	+	0.322440i	$0.895497\pi$
$240$		0			0
$241$	−8.50000	+	14.7224i	−0.547533	+	0.948355i	0.450910	+	0.892570i	$0.351100\pi$
$241$	−8.50000	+	14.7224i	−0.547533	+	0.948355i	−0.998443	+	0.0557856i	$0.982234\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	4.50000	−	7.79423i	0.287494	−	0.497955i
$246$		0			0
$247$	−10.0000	−	17.3205i	−0.636285	−	1.10208i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−21.0000			−1.32551			−0.662754	−	0.748837i	$-0.730613\pi$
$251$	−21.0000			−1.32551			−0.662754	+	0.748837i	$0.730613\pi$
$252$		0			0
$253$	18.0000			1.13165
$254$		0			0
$255$		0			0
$256$		0			0
$257$	4.50000	+	7.79423i	0.280702	+	0.486191i	0.971558	−	0.236802i	$-0.0760993\pi$
$257$	4.50000	+	7.79423i	0.280702	+	0.486191i	−0.690856	+	0.722993i	$0.742766\pi$
$258$		0			0
$259$	8.00000	−	13.8564i	0.497096	−	0.860995i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−12.0000	+	20.7846i	−0.739952	+	1.28163i	0.212565	+	0.977147i	$0.431818\pi$
$263$	−12.0000	+	20.7846i	−0.739952	+	1.28163i	−0.952517	+	0.304487i	$0.901515\pi$
$264$		0			0
$265$	−3.00000	−	5.19615i	−0.184289	−	0.319197i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−6.00000			−0.365826			−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000			−0.365826			−0.182913	+	0.983129i	$0.558553\pi$
$270$		0			0
$271$	16.0000			0.971931			0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000			0.971931			0.485965	+	0.873978i	$0.338468\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.50000	−	2.59808i	−0.0904534	−	0.156670i
$276$		0			0
$277$	11.0000	−	19.0526i	0.660926	−	1.14476i	−0.319447	−	0.947604i	$-0.603497\pi$
$277$	11.0000	−	19.0526i	0.660926	−	1.14476i	0.980373	−	0.197153i	$-0.0631696\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3.00000	+	5.19615i	−0.178965	+	0.309976i	−0.941526	−	0.336939i	$-0.890608\pi$
$281$	−3.00000	+	5.19615i	−0.178965	+	0.309976i	0.762561	+	0.646916i	$0.223942\pi$
$282$		0			0
$283$	10.0000	+	17.3205i	0.594438	+	1.02960i	0.993626	+	0.112728i	$0.0359589\pi$
$283$	10.0000	+	17.3205i	0.594438	+	1.02960i	−0.399188	+	0.916869i	$0.630708\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	12.0000			0.708338
$288$		0			0
$289$	−8.00000			−0.470588
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−9.00000	−	15.5885i	−0.525786	−	0.910687i	−0.999549	−	0.0300351i	$-0.990438\pi$
$293$	−9.00000	−	15.5885i	−0.525786	−	0.910687i	0.473763	−	0.880652i	$-0.342895\pi$
$294$		0			0
$295$	−1.50000	+	2.59808i	−0.0873334	+	0.151266i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	12.0000	−	20.7846i	0.693978	−	1.20201i
$300$		0			0
$301$	22.0000	+	38.1051i	1.26806	+	2.19634i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	10.0000			0.572598
$306$		0			0
$307$	7.00000			0.399511			0.199756	−	0.979846i	$-0.435985\pi$
$307$	7.00000			0.399511			0.199756	+	0.979846i	$0.435985\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3.00000	+	5.19615i	0.170114	+	0.294647i	0.938460	−	0.345389i	$-0.112253\pi$
$311$	3.00000	+	5.19615i	0.170114	+	0.294647i	−0.768345	+	0.640036i	$0.778920\pi$
$312$		0			0
$313$	0.500000	−	0.866025i	0.0282617	−	0.0489506i	−0.851549	−	0.524276i	$-0.824336\pi$
$313$	0.500000	−	0.866025i	0.0282617	−	0.0489506i	0.879810	+	0.475325i	$0.157669\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	12.0000	−	20.7846i	0.673987	−	1.16738i	−0.302777	−	0.953062i	$-0.597914\pi$
$317$	12.0000	−	20.7846i	0.673987	−	1.16738i	0.976764	−	0.214318i	$-0.0687530\pi$
$318$		0			0
$319$	9.00000	+	15.5885i	0.503903	+	0.872786i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	15.0000			0.834622
$324$		0			0
$325$	−4.00000			−0.221880
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−2.00000	+	3.46410i	−0.109930	+	0.190404i	−0.915742	−	0.401768i	$-0.868396\pi$
$331$	−2.00000	+	3.46410i	−0.109930	+	0.190404i	0.805812	+	0.592172i	$0.201729\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2.50000	+	4.33013i	−0.136590	+	0.236580i
$336$		0			0
$337$	15.5000	+	26.8468i	0.844339	+	1.46244i	0.886194	+	0.463314i	$0.153340\pi$
$337$	15.5000	+	26.8468i	0.844339	+	1.46244i	−0.0418554	+	0.999124i	$0.513327\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−6.00000			−0.324918
$342$		0			0
$343$	8.00000			0.431959
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−10.5000	−	18.1865i	−0.563670	−	0.976304i	−0.997172	−	0.0751519i	$-0.976056\pi$
$347$	−10.5000	−	18.1865i	−0.563670	−	0.976304i	0.433503	−	0.901152i	$-0.357278\pi$
$348$		0			0
$349$	8.00000	−	13.8564i	0.428230	−	0.741716i	−0.568486	−	0.822693i	$-0.692471\pi$
$349$	8.00000	−	13.8564i	0.428230	−	0.741716i	0.996716	+	0.0809766i	$0.0258039\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	4.50000	−	7.79423i	0.239511	−	0.414845i	−0.721063	−	0.692869i	$-0.756346\pi$
$353$	4.50000	−	7.79423i	0.239511	−	0.414845i	0.960574	+	0.278024i	$0.0896796\pi$
$354$		0			0
$355$	3.00000	+	5.19615i	0.159223	+	0.275783i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−24.0000			−1.26667			−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$		0			0
$361$	6.00000			0.315789
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−3.50000	−	6.06218i	−0.183198	−	0.317309i
$366$		0			0
$367$	4.00000	−	6.92820i	0.208798	−	0.361649i	−0.742538	−	0.669804i	$-0.766378\pi$
$367$	4.00000	−	6.92820i	0.208798	−	0.361649i	0.951336	+	0.308155i	$0.0997115\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	12.0000	−	20.7846i	0.623009	−	1.07908i
$372$		0			0
$373$	5.00000	+	8.66025i	0.258890	+	0.448411i	0.965945	−	0.258748i	$-0.0833099\pi$
$373$	5.00000	+	8.66025i	0.258890	+	0.448411i	−0.707055	+	0.707159i	$0.749977\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	24.0000			1.23606
$378$		0			0
$379$	−29.0000			−1.48963			−0.744815	−	0.667271i	$-0.767462\pi$
$379$	−29.0000			−1.48963			−0.744815	+	0.667271i	$0.767462\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−6.00000	−	10.3923i	−0.306586	−	0.531022i	0.671027	−	0.741433i	$-0.265853\pi$
$383$	−6.00000	−	10.3923i	−0.306586	−	0.531022i	−0.977613	+	0.210411i	$0.932520\pi$
$384$		0			0
$385$	6.00000	−	10.3923i	0.305788	−	0.529641i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	18.0000	−	31.1769i	0.912636	−	1.58073i	0.102311	−	0.994753i	$-0.467376\pi$
$389$	18.0000	−	31.1769i	0.912636	−	1.58073i	0.810326	−	0.585980i	$-0.199290\pi$
$390$		0			0
$391$	9.00000	+	15.5885i	0.455150	+	0.788342i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	14.0000			0.704416
$396$		0			0
$397$	8.00000			0.401508			0.200754	−	0.979642i	$-0.435661\pi$
$397$	8.00000			0.401508			0.200754	+	0.979642i	$0.435661\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	16.5000	+	28.5788i	0.823971	+	1.42716i	0.902703	+	0.430263i	$0.141579\pi$
$401$	16.5000	+	28.5788i	0.823971	+	1.42716i	−0.0787327	+	0.996896i	$0.525087\pi$
$402$		0			0
$403$	−4.00000	+	6.92820i	−0.199254	+	0.345118i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	6.00000	−	10.3923i	0.297409	−	0.515127i
$408$		0			0
$409$	15.5000	+	26.8468i	0.766426	+	1.32749i	0.939490	+	0.342578i	$0.111300\pi$
$409$	15.5000	+	26.8468i	0.766426	+	1.32749i	−0.173064	+	0.984911i	$0.555367\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−12.0000			−0.590481
$414$		0			0
$415$	−12.0000			−0.589057
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$419$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$420$		0			0
$421$	−1.00000	+	1.73205i	−0.0487370	+	0.0844150i	−0.889365	−	0.457198i	$-0.848853\pi$
$421$	−1.00000	+	1.73205i	−0.0487370	+	0.0844150i	0.840628	+	0.541613i	$0.182186\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.50000	−	2.59808i	0.0727607	−	0.126025i
$426$		0			0
$427$	20.0000	+	34.6410i	0.967868	+	1.67640i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	24.0000			1.15604			0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000			1.15604			0.578020	+	0.816023i	$0.303826\pi$
$432$		0			0
$433$	−13.0000			−0.624740			−0.312370	−	0.949960i	$-0.601123\pi$
$433$	−13.0000			−0.624740			−0.312370	+	0.949960i	$0.601123\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	15.0000	+	25.9808i	0.717547	+	1.24283i
$438$		0			0
$439$	−5.00000	+	8.66025i	−0.238637	+	0.413331i	−0.960323	−	0.278889i	$-0.910034\pi$
$439$	−5.00000	+	8.66025i	−0.238637	+	0.413331i	0.721686	+	0.692220i	$0.243367\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1.50000	+	2.59808i	−0.0712672	+	0.123438i	−0.899457	−	0.437009i	$-0.856038\pi$
$443$	−1.50000	+	2.59808i	−0.0712672	+	0.123438i	0.828190	+	0.560448i	$0.189371\pi$
$444$		0			0
$445$	−3.00000	−	5.19615i	−0.142214	−	0.246321i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	15.0000			0.707894			0.353947	−	0.935266i	$-0.384839\pi$
$449$	15.0000			0.707894			0.353947	+	0.935266i	$0.384839\pi$
$450$		0			0
$451$	9.00000			0.423793
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−8.00000	−	13.8564i	−0.375046	−	0.649598i
$456$		0			0
$457$	0.500000	−	0.866025i	0.0233890	−	0.0405110i	−0.854094	−	0.520119i	$-0.825888\pi$
$457$	0.500000	−	0.866025i	0.0233890	−	0.0405110i	0.877483	+	0.479608i	$0.159221\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$461$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$462$		0			0
$463$	10.0000	+	17.3205i	0.464739	+	0.804952i	0.999190	−	0.0402476i	$-0.0128147\pi$
$463$	10.0000	+	17.3205i	0.464739	+	0.804952i	−0.534450	+	0.845200i	$0.679481\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	21.0000			0.971764			0.485882	−	0.874024i	$-0.338498\pi$
$467$	21.0000			0.971764			0.485882	+	0.874024i	$0.338498\pi$
$468$		0			0
$469$	−20.0000			−0.923514
$470$		0			0
$471$		0			0
$472$		0			0
$473$	16.5000	+	28.5788i	0.758671	+	1.31406i
$474$		0			0
$475$	2.50000	−	4.33013i	0.114708	−	0.198680i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	3.00000	−	5.19615i	0.137073	−	0.237418i	−0.789314	−	0.613990i	$-0.789564\pi$
$479$	3.00000	−	5.19615i	0.137073	−	0.237418i	0.926388	+	0.376571i	$0.122897\pi$
$480$		0			0
$481$	−8.00000	−	13.8564i	−0.364769	−	0.631798i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−11.0000			−0.499484
$486$		0			0
$487$	−2.00000			−0.0906287			−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000			−0.0906287			−0.0453143	+	0.998973i	$0.514429\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	16.5000	+	28.5788i	0.744635	+	1.28974i	0.950365	+	0.311136i	$0.100710\pi$
$491$	16.5000	+	28.5788i	0.744635	+	1.28974i	−0.205731	+	0.978609i	$0.565957\pi$
$492$		0			0
$493$	−9.00000	+	15.5885i	−0.405340	+	0.702069i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−12.0000	+	20.7846i	−0.538274	+	0.932317i
$498$		0			0
$499$	−15.5000	−	26.8468i	−0.693875	−	1.20183i	−0.970558	−	0.240866i	$-0.922569\pi$
$499$	−15.5000	−	26.8468i	−0.693875	−	1.20183i	0.276683	−	0.960961i	$-0.410765\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	6.00000			0.267527			0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000			0.267527			0.133763	+	0.991013i	$0.457294\pi$
$504$		0			0
$505$	−12.0000			−0.533993
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−6.00000	−	10.3923i	−0.265945	−	0.460631i	0.701866	−	0.712309i	$-0.252351\pi$
$509$	−6.00000	−	10.3923i	−0.265945	−	0.460631i	−0.967811	+	0.251679i	$0.919017\pi$
$510$		0			0
$511$	14.0000	−	24.2487i	0.619324	−	1.07270i
$512$		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
$521$	3.00000			0.131432			0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000			0.131432			0.0657162	+	0.997838i	$0.479067\pi$
$522$		0			0
$523$	−20.0000			−0.874539			−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000			−0.874539			−0.437269	+	0.899331i	$0.644054\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3.00000	−	5.19615i	−0.130682	−	0.226348i
$528$		0			0
$529$	−6.50000	+	11.2583i	−0.282609	+	0.489493i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	6.00000	−	10.3923i	0.259889	−	0.450141i
$534$		0			0
$535$	−4.50000	−	7.79423i	−0.194552	−	0.336974i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	27.0000			1.16297
$540$		0			0
$541$	8.00000			0.343947			0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000			0.343947			0.171973	+	0.985102i	$0.444986\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2.00000	−	3.46410i	−0.0856706	−	0.148386i
$546$		0			0
$547$	−0.500000	+	0.866025i	−0.0213785	+	0.0370286i	−0.876517	−	0.481371i	$-0.840139\pi$
$547$	−0.500000	+	0.866025i	−0.0213785	+	0.0370286i	0.855138	+	0.518400i	$0.173472\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−15.0000	+	25.9808i	−0.639021	+	1.10682i
$552$		0			0
$553$	28.0000	+	48.4974i	1.19068	+	2.06232i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	12.0000			0.508456			0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000			0.508456			0.254228	+	0.967144i	$0.418179\pi$
$558$		0			0
$559$	44.0000			1.86100
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.50000	+	2.59808i	0.0632175	+	0.109496i	0.895902	−	0.444252i	$-0.146530\pi$
$563$	1.50000	+	2.59808i	0.0632175	+	0.109496i	−0.832684	+	0.553748i	$0.813197\pi$
$564$		0			0
$565$	−9.00000	+	15.5885i	−0.378633	+	0.655811i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−19.5000	+	33.7750i	−0.817483	+	1.41592i	0.0900490	+	0.995937i	$0.471298\pi$
$569$	−19.5000	+	33.7750i	−0.817483	+	1.41592i	−0.907532	+	0.419984i	$0.862036\pi$
$570$		0			0
$571$	14.5000	+	25.1147i	0.606806	+	1.05102i	0.991763	+	0.128085i	$0.0408829\pi$
$571$	14.5000	+	25.1147i	0.606806	+	1.05102i	−0.384957	+	0.922934i	$0.625784\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	6.00000			0.250217
$576$		0			0
$577$	−7.00000			−0.291414			−0.145707	−	0.989328i	$-0.546546\pi$
$577$	−7.00000			−0.291414			−0.145707	+	0.989328i	$0.546546\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−24.0000	−	41.5692i	−0.995688	−	1.72458i
$582$		0			0
$583$	9.00000	−	15.5885i	0.372742	−	0.645608i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−19.5000	+	33.7750i	−0.804851	+	1.39404i	0.111540	+	0.993760i	$0.464422\pi$
$587$	−19.5000	+	33.7750i	−0.804851	+	1.39404i	−0.916392	+	0.400283i	$0.868912\pi$
$588$		0			0
$589$	−5.00000	−	8.66025i	−0.206021	−	0.356840i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	18.0000			0.739171			0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000			0.739171			0.369586	+	0.929197i	$0.379500\pi$
$594$		0			0
$595$	12.0000			0.491952
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$		0			0
$601$	12.5000	−	21.6506i	0.509886	−	0.883148i	−0.490049	−	0.871695i	$-0.663021\pi$
$601$	12.5000	−	21.6506i	0.509886	−	0.883148i	0.999934	−	0.0114528i	$-0.00364562\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.00000	+	1.73205i	−0.0406558	+	0.0704179i
$606$		0			0
$607$	4.00000	+	6.92820i	0.162355	+	0.281207i	0.935713	−	0.352763i	$-0.114758\pi$
$607$	4.00000	+	6.92820i	0.162355	+	0.281207i	−0.773358	+	0.633970i	$0.781424\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	2.00000			0.0807792			0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000			0.0807792			0.0403896	+	0.999184i	$0.487140\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−19.5000	−	33.7750i	−0.785040	−	1.35973i	−0.928975	−	0.370143i	$-0.879309\pi$
$617$	−19.5000	−	33.7750i	−0.785040	−	1.35973i	0.143934	−	0.989587i	$-0.454025\pi$
$618$		0			0
$619$	−9.50000	+	16.4545i	−0.381837	+	0.661361i	−0.991325	−	0.131434i	$-0.958042\pi$
$619$	−9.50000	+	16.4545i	−0.381837	+	0.661361i	0.609488	+	0.792796i	$0.291375\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	12.0000	−	20.7846i	0.480770	−	0.832718i
$624$		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
$631$	34.0000			1.35352			0.676759	−	0.736204i	$-0.263384\pi$
$631$	34.0000			1.35352			0.676759	+	0.736204i	$0.263384\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1.00000	−	1.73205i	−0.0396838	−	0.0687343i
$636$		0			0
$637$	18.0000	−	31.1769i	0.713186	−	1.23527i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	4.50000	−	7.79423i	0.177739	−	0.307854i	−0.763367	−	0.645966i	$-0.776455\pi$
$641$	4.50000	−	7.79423i	0.177739	−	0.307854i	0.941106	+	0.338112i	$0.109788\pi$
$642$		0			0
$643$	11.5000	+	19.9186i	0.453516	+	0.785512i	0.998602	−	0.0528680i	$-0.0168363\pi$
$643$	11.5000	+	19.9186i	0.453516	+	0.785512i	−0.545086	+	0.838380i	$0.683503\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	6.00000			0.235884			0.117942	−	0.993020i	$-0.462370\pi$
$647$	6.00000			0.235884			0.117942	+	0.993020i	$0.462370\pi$
$648$		0			0
$649$	−9.00000			−0.353281
$650$		0			0
$651$		0			0
$652$		0			0
$653$	21.0000	+	36.3731i	0.821794	+	1.42339i	0.904345	+	0.426801i	$0.140360\pi$
$653$	21.0000	+	36.3731i	0.821794	+	1.42339i	−0.0825519	+	0.996587i	$0.526307\pi$
$654$		0			0
$655$	6.00000	−	10.3923i	0.234439	−	0.406061i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	18.0000	−	31.1769i	0.701180	−	1.21448i	−0.266872	−	0.963732i	$-0.585990\pi$
$659$	18.0000	−	31.1769i	0.701180	−	1.21448i	0.968052	−	0.250748i	$-0.0806766\pi$
$660$		0			0
$661$	−16.0000	−	27.7128i	−0.622328	−	1.07790i	−0.989051	−	0.147573i	$-0.952854\pi$
$661$	−16.0000	−	27.7128i	−0.622328	−	1.07790i	0.366723	−	0.930330i	$-0.380480\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	20.0000			0.775567
$666$		0			0
$667$	−36.0000			−1.39393
$668$		0			0
$669$		0			0
$670$		0			0
$671$	15.0000	+	25.9808i	0.579069	+	1.00298i
$672$		0			0
$673$	−7.00000	+	12.1244i	−0.269830	+	0.467360i	−0.968818	−	0.247774i	$-0.920301\pi$
$673$	−7.00000	+	12.1244i	−0.269830	+	0.467360i	0.698988	+	0.715134i	$0.253634\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−18.0000	+	31.1769i	−0.691796	+	1.19823i	0.279453	+	0.960159i	$0.409847\pi$
$677$	−18.0000	+	31.1769i	−0.691796	+	1.19823i	−0.971249	+	0.238067i	$0.923486\pi$
$678$		0			0
$679$	−22.0000	−	38.1051i	−0.844283	−	1.46234i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−21.0000			−0.803543			−0.401771	−	0.915740i	$-0.631605\pi$
$683$	−21.0000			−0.803543			−0.401771	+	0.915740i	$0.631605\pi$
$684$		0			0
$685$	−9.00000			−0.343872
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−12.0000	−	20.7846i	−0.457164	−	0.791831i
$690$		0			0
$691$	4.00000	−	6.92820i	0.152167	−	0.263561i	−0.779857	−	0.625958i	$-0.784708\pi$
$691$	4.00000	−	6.92820i	0.152167	−	0.263561i	0.932024	+	0.362397i	$0.118041\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	0.500000	−	0.866025i	0.0189661	−	0.0328502i
$696$		0			0
$697$	4.50000	+	7.79423i	0.170450	+	0.295227i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−42.0000			−1.58632			−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000			−1.58632			−0.793159	+	0.609015i	$0.791565\pi$
$702$		0			0
$703$	20.0000			0.754314
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−24.0000	−	41.5692i	−0.902613	−	1.56337i
$708$		0			0
$709$	17.0000	−	29.4449i	0.638448	−	1.10583i	−0.347325	−	0.937745i	$-0.612910\pi$
$709$	17.0000	−	29.4449i	0.638448	−	1.10583i	0.985773	−	0.168080i	$-0.0537568\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	6.00000	−	10.3923i	0.224702	−	0.389195i
$714$		0			0
$715$	−6.00000	−	10.3923i	−0.224387	−	0.388650i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		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
$727$	−14.0000	+	24.2487i	−0.519231	+	0.899335i	0.480519	+	0.876984i	$0.340448\pi$
$727$	−14.0000	+	24.2487i	−0.519231	+	0.899335i	−0.999750	+	0.0223506i	$0.992885\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−16.5000	+	28.5788i	−0.610275	+	1.05703i
$732$		0			0
$733$	−16.0000	−	27.7128i	−0.590973	−	1.02360i	−0.994102	−	0.108453i	$-0.965410\pi$
$733$	−16.0000	−	27.7128i	−0.590973	−	1.02360i	0.403128	−	0.915144i	$-0.367923\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−15.0000			−0.552532
$738$		0			0
$739$	−29.0000			−1.06678			−0.533391	−	0.845869i	$-0.679083\pi$
$739$	−29.0000			−1.06678			−0.533391	+	0.845869i	$0.679083\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	3.00000	+	5.19615i	0.110059	+	0.190628i	0.915794	−	0.401648i	$-0.131563\pi$
$743$	3.00000	+	5.19615i	0.110059	+	0.190628i	−0.805735	+	0.592277i	$0.798229\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	18.0000	−	31.1769i	0.657706	−	1.13918i
$750$		0			0
$751$	−14.0000	−	24.2487i	−0.510867	−	0.884848i	−0.999921	−	0.0125942i	$-0.995991\pi$
$751$	−14.0000	−	24.2487i	−0.510867	−	0.884848i	0.489053	−	0.872254i	$-0.337342\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−10.0000			−0.363937
$756$		0			0
$757$	38.0000			1.38113			0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000			1.38113			0.690567	+	0.723269i	$0.257361\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	9.00000	+	15.5885i	0.326250	+	0.565081i	0.981764	−	0.190101i	$-0.0608816\pi$
$761$	9.00000	+	15.5885i	0.326250	+	0.565081i	−0.655515	+	0.755182i	$0.727548\pi$
$762$		0			0
$763$	8.00000	−	13.8564i	0.289619	−	0.501636i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−6.00000	+	10.3923i	−0.216647	+	0.375244i
$768$		0			0
$769$	−25.0000	−	43.3013i	−0.901523	−	1.56148i	−0.825518	−	0.564376i	$-0.809117\pi$
$769$	−25.0000	−	43.3013i	−0.901523	−	1.56148i	−0.0760054	−	0.997107i	$-0.524217\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	30.0000			1.07903			0.539513	−	0.841978i	$-0.318609\pi$
$773$	30.0000			1.07903			0.539513	+	0.841978i	$0.318609\pi$
$774$		0			0
$775$	−2.00000			−0.0718421
$776$		0			0
$777$		0			0
$778$		0			0
$779$	7.50000	+	12.9904i	0.268715	+	0.465429i
$780$		0			0
$781$	−9.00000	+	15.5885i	−0.322045	+	0.557799i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	4.00000	−	6.92820i	0.142766	−	0.247278i
$786$		0			0
$787$	−2.00000	−	3.46410i	−0.0712923	−	0.123482i	0.828176	−	0.560469i	$-0.189379\pi$
$787$	−2.00000	−	3.46410i	−0.0712923	−	0.123482i	−0.899468	+	0.436987i	$0.856046\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−72.0000			−2.56003
$792$		0			0
$793$	40.0000			1.42044
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−24.0000	−	41.5692i	−0.850124	−	1.47246i	−0.881096	−	0.472937i	$-0.843194\pi$
$797$	−24.0000	−	41.5692i	−0.850124	−	1.47246i	0.0309726	−	0.999520i	$-0.490140\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	10.5000	−	18.1865i	0.370537	−	0.641789i
$804$		0			0
$805$	12.0000	+	20.7846i	0.422944	+	0.732561i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	39.0000			1.37117			0.685583	−	0.727994i	$-0.259547\pi$
$809$	39.0000			1.37117			0.685583	+	0.727994i	$0.259547\pi$
$810$		0			0
$811$	−35.0000			−1.22902			−0.614508	−	0.788911i	$-0.710645\pi$
$811$	−35.0000			−1.22902			−0.614508	+	0.788911i	$0.710645\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	8.00000	+	13.8564i	0.280228	+	0.485369i
$816$		0			0
$817$	−27.5000	+	47.6314i	−0.962103	+	1.66641i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−24.0000	+	41.5692i	−0.837606	+	1.45078i	0.0542853	+	0.998525i	$0.482712\pi$
$821$	−24.0000	+	41.5692i	−0.837606	+	1.45078i	−0.891891	+	0.452250i	$0.850621\pi$
$822$		0			0
$823$	−2.00000	−	3.46410i	−0.0697156	−	0.120751i	0.829060	−	0.559159i	$-0.188876\pi$
$823$	−2.00000	−	3.46410i	−0.0697156	−	0.120751i	−0.898776	+	0.438408i	$0.855543\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−36.0000			−1.25184			−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000			−1.25184			−0.625921	+	0.779886i	$0.715277\pi$
$828$		0			0
$829$	44.0000			1.52818			0.764092	−	0.645108i	$-0.223188\pi$
$829$	44.0000			1.52818			0.764092	+	0.645108i	$0.223188\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	13.5000	+	23.3827i	0.467747	+	0.810162i
$834$		0			0
$835$	−9.00000	+	15.5885i	−0.311458	+	0.539461i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	15.0000	−	25.9808i	0.517858	−	0.896956i	−0.481927	−	0.876211i	$-0.660063\pi$
$839$	15.0000	−	25.9808i	0.517858	−	0.896956i	0.999785	−	0.0207443i	$-0.00660359\pi$
$840$		0			0
$841$	−3.50000	−	6.06218i	−0.120690	−	0.209041i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3.00000			−0.103203
$846$		0			0
$847$	−8.00000			−0.274883
$848$		0			0
$849$		0			0
$850$		0			0
$851$	12.0000	+	20.7846i	0.411355	+	0.712487i
$852$		0			0
$853$	−22.0000	+	38.1051i	−0.753266	+	1.30469i	0.192966	+	0.981205i	$0.438189\pi$
$853$	−22.0000	+	38.1051i	−0.753266	+	1.30469i	−0.946232	+	0.323489i	$0.895144\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−9.00000	+	15.5885i	−0.307434	+	0.532492i	−0.977800	−	0.209539i	$-0.932804\pi$
$857$	−9.00000	+	15.5885i	−0.307434	+	0.532492i	0.670366	+	0.742030i	$0.266137\pi$
$858$		0			0
$859$	−9.50000	−	16.4545i	−0.324136	−	0.561420i	0.657201	−	0.753715i	$-0.271740\pi$
$859$	−9.50000	−	16.4545i	−0.324136	−	0.561420i	−0.981337	+	0.192295i	$0.938407\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−6.00000			−0.204242			−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000			−0.204242			−0.102121	+	0.994772i	$0.532563\pi$
$864$		0			0
$865$	−18.0000			−0.612018
$866$		0			0
$867$		0			0
$868$		0			0
$869$	21.0000	+	36.3731i	0.712376	+	1.23387i
$870$		0			0
$871$	−10.0000	+	17.3205i	−0.338837	+	0.586883i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2.00000	−	3.46410i	0.0676123	−	0.117108i
$876$		0			0
$877$	−1.00000	−	1.73205i	−0.0337676	−	0.0584872i	0.848648	−	0.528958i	$-0.177417\pi$
$877$	−1.00000	−	1.73205i	−0.0337676	−	0.0584872i	−0.882415	+	0.470471i	$0.844084\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−30.0000			−1.01073			−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000			−1.01073			−0.505363	+	0.862907i	$0.668641\pi$
$882$		0			0
$883$	−41.0000			−1.37976			−0.689880	−	0.723924i	$-0.742337\pi$
$883$	−41.0000			−1.37976			−0.689880	+	0.723924i	$0.742337\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	21.0000	+	36.3731i	0.705111	+	1.22129i	0.966651	+	0.256096i	$0.0824362\pi$
$887$	21.0000	+	36.3731i	0.705111	+	1.22129i	−0.261540	+	0.965193i	$0.584230\pi$
$888$		0			0
$889$	4.00000	−	6.92820i	0.134156	−	0.232364i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	12.0000			0.400222
$900$		0			0
$901$	18.0000			0.599667
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−8.00000	−	13.8564i	−0.265929	−	0.460603i
$906$		0			0
$907$	2.50000	−	4.33013i	0.0830111	−	0.143780i	−0.821531	−	0.570164i	$-0.806880\pi$
$907$	2.50000	−	4.33013i	0.0830111	−	0.143780i	0.904542	+	0.426385i	$0.140213\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	15.0000	−	25.9808i	0.496972	−	0.860781i	−0.503022	−	0.864274i	$-0.667778\pi$
$911$	15.0000	−	25.9808i	0.496972	−	0.860781i	0.999994	+	0.00349271i	$0.00111177\pi$
$912$		0			0
$913$	−18.0000	−	31.1769i	−0.595713	−	1.03181i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	48.0000			1.58510
$918$		0			0
$919$	34.0000			1.12156			0.560778	−	0.827966i	$-0.310502\pi$
$919$	34.0000			1.12156			0.560778	+	0.827966i	$0.310502\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	12.0000	+	20.7846i	0.394985	+	0.684134i
$924$		0			0
$925$	2.00000	−	3.46410i	0.0657596	−	0.113899i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	15.0000	−	25.9808i	0.492134	−	0.852401i	−0.507825	−	0.861460i	$-0.669550\pi$
$929$	15.0000	−	25.9808i	0.492134	−	0.852401i	0.999959	+	0.00905914i	$0.00288365\pi$
$930$		0			0
$931$	22.5000	+	38.9711i	0.737408	+	1.27723i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	9.00000			0.294331
$936$		0			0
$937$	−10.0000			−0.326686			−0.163343	−	0.986569i	$-0.552228\pi$
$937$	−10.0000			−0.326686			−0.163343	+	0.986569i	$0.552228\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	12.0000	+	20.7846i	0.391189	+	0.677559i	0.992607	−	0.121376i	$-0.0387306\pi$
$941$	12.0000	+	20.7846i	0.391189	+	0.677559i	−0.601418	+	0.798935i	$0.705397\pi$
$942$		0			0
$943$	−9.00000	+	15.5885i	−0.293080	+	0.507630i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−13.5000	+	23.3827i	−0.438691	+	0.759835i	−0.997589	−	0.0694014i	$-0.977891\pi$
$947$	−13.5000	+	23.3827i	−0.438691	+	0.759835i	0.558898	+	0.829237i	$0.311224\pi$
$948$		0			0
$949$	−14.0000	−	24.2487i	−0.454459	−	0.787146i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−51.0000			−1.65205			−0.826026	−	0.563632i	$-0.809404\pi$
$953$	−51.0000			−1.65205			−0.826026	+	0.563632i	$0.809404\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−18.0000	−	31.1769i	−0.581250	−	1.00676i
$960$		0			0
$961$	13.5000	−	23.3827i	0.435484	−	0.754280i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−6.50000	+	11.2583i	−0.209242	+	0.362418i
$966$		0			0
$967$	−11.0000	−	19.0526i	−0.353736	−	0.612689i	0.633165	−	0.774017i	$-0.281756\pi$
$967$	−11.0000	−	19.0526i	−0.353736	−	0.612689i	−0.986901	+	0.161328i	$0.948422\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	60.0000			1.92549			0.962746	−	0.270408i	$-0.0871586\pi$
$971$	60.0000			1.92549			0.962746	+	0.270408i	$0.0871586\pi$
$972$		0			0
$973$	4.00000			0.128234
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−4.50000	−	7.79423i	−0.143968	−	0.249359i	0.785020	−	0.619471i	$-0.212653\pi$
$977$	−4.50000	−	7.79423i	−0.143968	−	0.249359i	−0.928987	+	0.370111i	$0.879319\pi$
$978$		0			0
$979$	9.00000	−	15.5885i	0.287641	−	0.498209i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−18.0000	+	31.1769i	−0.574111	+	0.994389i	0.422027	+	0.906583i	$0.361319\pi$
$983$	−18.0000	+	31.1769i	−0.574111	+	0.994389i	−0.996138	+	0.0878058i	$0.972015\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−66.0000			−2.09868
$990$		0			0
$991$	−20.0000			−0.635321			−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000			−0.635321			−0.317660	+	0.948205i	$0.602897\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−10.0000	−	17.3205i	−0.317021	−	0.549097i
$996$		0			0
$997$	−13.0000	+	22.5167i	−0.411714	+	0.713110i	−0.995077	−	0.0991016i	$-0.968403\pi$
$997$	−13.0000	+	22.5167i	−0.411714	+	0.713110i	0.583363	+	0.812211i	$0.301736\pi$
$998$		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	2160.2.q.d.721.1		2
3.2	odd	2		720.2.q.c.241.1		2
4.3	odd	2		270.2.e.a.181.1		2
9.2	odd	6		6480.2.a.z.1.1		1
9.4	even	3	inner	2160.2.q.d.1441.1		2
9.5	odd	6		720.2.q.c.481.1		2
9.7	even	3		6480.2.a.l.1.1		1
12.11	even	2		90.2.e.b.61.1	yes	2
20.3	even	4		1350.2.j.c.1099.2		4
20.7	even	4		1350.2.j.c.1099.1		4
20.19	odd	2		1350.2.e.g.451.1		2
36.7	odd	6		810.2.a.e.1.1		1
36.11	even	6		810.2.a.a.1.1		1
36.23	even	6		90.2.e.b.31.1	✓	2
36.31	odd	6		270.2.e.a.91.1		2
60.23	odd	4		450.2.j.a.349.1		4
60.47	odd	4		450.2.j.a.349.2		4
60.59	even	2		450.2.e.d.151.1		2
180.7	even	12		4050.2.c.d.649.2		2
180.23	odd	12		450.2.j.a.49.2		4
180.43	even	12		4050.2.c.d.649.1		2
180.47	odd	12		4050.2.c.p.649.1		2
180.59	even	6		450.2.e.d.301.1		2
180.67	even	12		1350.2.j.c.199.2		4
180.79	odd	6		4050.2.a.q.1.1		1
180.83	odd	12		4050.2.c.p.649.2		2
180.103	even	12		1350.2.j.c.199.1		4
180.119	even	6		4050.2.a.bi.1.1		1
180.139	odd	6		1350.2.e.g.901.1		2
180.167	odd	12		450.2.j.a.49.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.2.e.b.31.1	✓	2	36.23	even	6
90.2.e.b.61.1	yes	2	12.11	even	2
270.2.e.a.91.1		2	36.31	odd	6
270.2.e.a.181.1		2	4.3	odd	2
450.2.e.d.151.1		2	60.59	even	2
450.2.e.d.301.1		2	180.59	even	6
450.2.j.a.49.1		4	180.167	odd	12
450.2.j.a.49.2		4	180.23	odd	12
450.2.j.a.349.1		4	60.23	odd	4
450.2.j.a.349.2		4	60.47	odd	4
720.2.q.c.241.1		2	3.2	odd	2
720.2.q.c.481.1		2	9.5	odd	6
810.2.a.a.1.1		1	36.11	even	6
810.2.a.e.1.1		1	36.7	odd	6
1350.2.e.g.451.1		2	20.19	odd	2
1350.2.e.g.901.1		2	180.139	odd	6
1350.2.j.c.199.1		4	180.103	even	12
1350.2.j.c.199.2		4	180.67	even	12
1350.2.j.c.1099.1		4	20.7	even	4
1350.2.j.c.1099.2		4	20.3	even	4
2160.2.q.d.721.1		2	1.1	even	1	trivial
2160.2.q.d.1441.1		2	9.4	even	3	inner
4050.2.a.q.1.1		1	180.79	odd	6
4050.2.a.bi.1.1		1	180.119	even	6
4050.2.c.d.649.1		2	180.43	even	12
4050.2.c.d.649.2		2	180.7	even	12
4050.2.c.p.649.1		2	180.47	odd	12
4050.2.c.p.649.2		2	180.83	odd	12
6480.2.a.l.1.1		1	9.7	even	3
6480.2.a.z.1.1		1	9.2	odd	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	2160.2.q.d.721.1

Level	$2160$
Weight	$2$
Character	2160.721
Analytic conductor	$17.248$
Analytic rank	$0$
Dimension	$2$
Inner twists	$2$