LMFDB - Embedded newform 2160.2.x.b.1567.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,2,Mod(703,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 0, 3]))

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

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

chi := DirichletCharacter("2160.703");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2160.x (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$17.2476868366$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1567.3
Root			$1.14412 + 1.14412i$ of defining polynomial
Character	$\chi$	$=$	2160.1567
Dual form			2160.2.x.b.703.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.58114 - 1.58114i) q^{5} +O(q^{10})$	$q+(1.58114 - 1.58114i) q^{5} -4.24264i q^{11} +(-2.00000 + 2.00000i) q^{13} +(-1.58114 - 1.58114i) q^{17} -6.70820 q^{19} +(-2.12132 - 2.12132i) q^{23} -5.00000i q^{25} +6.32456i q^{29} -6.70820i q^{31} +(-1.00000 - 1.00000i) q^{37} -3.16228 q^{41} +(-6.70820 - 6.70820i) q^{43} +(-4.24264 + 4.24264i) q^{47} +7.00000i q^{49} +(1.58114 - 1.58114i) q^{53} +(-6.70820 - 6.70820i) q^{55} +8.48528 q^{59} +9.00000 q^{61} +6.32456i q^{65} +(-6.70820 + 6.70820i) q^{67} -4.24264i q^{71} +(-1.00000 + 1.00000i) q^{73} -6.70820 q^{79} +(10.6066 + 10.6066i) q^{83} -5.00000 q^{85} -15.8114i q^{89} +(-10.6066 + 10.6066i) q^{95} +(-4.00000 - 4.00000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 16 q^{13} - 8 q^{37} + 72 q^{61} - 8 q^{73} - 40 q^{85} - 32 q^{97}+O(q^{100}) $ 8 * q - 16 * q^13 - 8 * q^37 + 72 * q^61 - 8 * q^73 - 40 * q^85 - 32 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$	$1$	$1$

Coefficient data

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

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.58114	−	1.58114i	0.707107	−	0.707107i
$5$	1.58114	−	1.58114i	0.707107	−	0.707107i
$6$		0			0
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	4.24264i		−	1.27920i	−0.768706	−	0.639602i	$-0.779099\pi$
$11$		−	4.24264i		−	1.27920i	0.768706	−	0.639602i	$-0.220901\pi$
$12$		0			0
$13$	−2.00000	+	2.00000i	−0.554700	+	0.554700i	−0.927794	−	0.373094i	$-0.878297\pi$
$13$	−2.00000	+	2.00000i	−0.554700	+	0.554700i	0.373094	+	0.927794i	$0.378297\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.58114	−	1.58114i	−0.383482	−	0.383482i	0.488873	−	0.872355i	$-0.337408\pi$
$17$	−1.58114	−	1.58114i	−0.383482	−	0.383482i	−0.872355	+	0.488873i	$0.837408\pi$
$18$		0			0
$19$	−6.70820			−1.53897			−0.769484	−	0.638666i	$-0.779486\pi$
$19$	−6.70820			−1.53897			−0.769484	+	0.638666i	$0.779486\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.12132	−	2.12132i	−0.442326	−	0.442326i	0.450467	−	0.892793i	$-0.351257\pi$
$23$	−2.12132	−	2.12132i	−0.442326	−	0.442326i	−0.892793	+	0.450467i	$0.851257\pi$
$24$		0			0
$25$		−	5.00000i		−	1.00000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			6.32456i			1.17444i	0.809427	+	0.587220i	$0.199778\pi$
$29$			6.32456i			1.17444i	−0.809427	+	0.587220i	$0.800222\pi$
$30$		0			0
$31$		−	6.70820i		−	1.20483i	−0.798183	−	0.602414i	$-0.794205\pi$
$31$		−	6.70820i		−	1.20483i	0.798183	−	0.602414i	$-0.205795\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1.00000	−	1.00000i	−0.164399	−	0.164399i	0.620113	−	0.784512i	$-0.287087\pi$
$37$	−1.00000	−	1.00000i	−0.164399	−	0.164399i	−0.784512	+	0.620113i	$0.787087\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.16228			−0.493865			−0.246932	−	0.969033i	$-0.579423\pi$
$41$	−3.16228			−0.493865			−0.246932	+	0.969033i	$0.579423\pi$
$42$		0			0
$43$	−6.70820	−	6.70820i	−1.02299	−	1.02299i	−0.999729	−	0.0232621i	$-0.992595\pi$
$43$	−6.70820	−	6.70820i	−1.02299	−	1.02299i	−0.0232621	−	0.999729i	$-0.507405\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.24264	+	4.24264i	−0.618853	+	0.618853i	−0.945237	−	0.326384i	$-0.894170\pi$
$47$	−4.24264	+	4.24264i	−0.618853	+	0.618853i	0.326384	+	0.945237i	$0.394170\pi$
$48$		0			0
$49$			7.00000i			1.00000i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.58114	−	1.58114i	0.217186	−	0.217186i	−0.590125	−	0.807312i	$-0.700922\pi$
$53$	1.58114	−	1.58114i	0.217186	−	0.217186i	0.807312	+	0.590125i	$0.200922\pi$
$54$		0			0
$55$	−6.70820	−	6.70820i	−0.904534	−	0.904534i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	8.48528			1.10469			0.552345	−	0.833616i	$-0.313733\pi$
$59$	8.48528			1.10469			0.552345	+	0.833616i	$0.313733\pi$
$60$		0			0
$61$	9.00000			1.15233			0.576166	−	0.817333i	$-0.304548\pi$
$61$	9.00000			1.15233			0.576166	+	0.817333i	$0.304548\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			6.32456i			0.784465i
$66$		0			0
$67$	−6.70820	+	6.70820i	−0.819538	+	0.819538i	−0.986041	−	0.166503i	$-0.946752\pi$
$67$	−6.70820	+	6.70820i	−0.819538	+	0.819538i	0.166503	+	0.986041i	$0.446752\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	4.24264i		−	0.503509i	−0.967791	−	0.251754i	$-0.918992\pi$
$71$		−	4.24264i		−	0.503509i	0.967791	−	0.251754i	$-0.0810075\pi$
$72$		0			0
$73$	−1.00000	+	1.00000i	−0.117041	+	0.117041i	−0.763202	−	0.646160i	$-0.776374\pi$
$73$	−1.00000	+	1.00000i	−0.117041	+	0.117041i	0.646160	+	0.763202i	$0.276374\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−6.70820			−0.754732			−0.377366	−	0.926064i	$-0.623170\pi$
$79$	−6.70820			−0.754732			−0.377366	+	0.926064i	$0.623170\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	10.6066	+	10.6066i	1.16423	+	1.16423i	0.983541	+	0.180685i	$0.0578314\pi$
$83$	10.6066	+	10.6066i	1.16423	+	1.16423i	0.180685	+	0.983541i	$0.442169\pi$
$84$		0			0
$85$	−5.00000			−0.542326
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	15.8114i		−	1.67600i	−0.545667	−	0.838002i	$-0.683724\pi$
$89$		−	15.8114i		−	1.67600i	0.545667	−	0.838002i	$-0.316276\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−10.6066	+	10.6066i	−1.08821	+	1.08821i
$96$		0			0
$97$	−4.00000	−	4.00000i	−0.406138	−	0.406138i	0.474251	−	0.880390i	$-0.342719\pi$
$97$	−4.00000	−	4.00000i	−0.406138	−	0.406138i	−0.880390	+	0.474251i	$0.842719\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.32456			0.629317			0.314658	−	0.949205i	$-0.398110\pi$
$101$	6.32456			0.629317			0.314658	+	0.949205i	$0.398110\pi$
$102$		0			0
$103$	−13.4164	−	13.4164i	−1.32196	−	1.32196i	−0.912190	−	0.409768i	$-0.865610\pi$
$103$	−13.4164	−	13.4164i	−1.32196	−	1.32196i	−0.409768	−	0.912190i	$-0.634390\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4.24264	+	4.24264i	−0.410152	+	0.410152i	−0.881791	−	0.471640i	$-0.843662\pi$
$107$	−4.24264	+	4.24264i	−0.410152	+	0.410152i	0.471640	+	0.881791i	$0.343662\pi$
$108$		0			0
$109$		−	9.00000i		−	0.862044i	−0.902342	−	0.431022i	$-0.858153\pi$
$109$		−	9.00000i		−	0.862044i	0.902342	−	0.431022i	$-0.141847\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$113$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$114$		0			0
$115$	−6.70820			−0.625543
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−7.90569	−	7.90569i	−0.707107	−	0.707107i
$126$		0			0
$127$	6.70820	−	6.70820i	0.595257	−	0.595257i	−0.343790	−	0.939047i	$-0.611711\pi$
$127$	6.70820	−	6.70820i	0.595257	−	0.595257i	0.939047	+	0.343790i	$0.111711\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	4.24264i		−	0.370681i	−0.982674	−	0.185341i	$-0.940661\pi$
$131$		−	4.24264i		−	0.370681i	0.982674	−	0.185341i	$-0.0593388\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	14.2302	+	14.2302i	1.21577	+	1.21577i	0.969099	+	0.246674i	$0.0793376\pi$
$137$	14.2302	+	14.2302i	1.21577	+	1.21577i	0.246674	+	0.969099i	$0.420662\pi$
$138$		0			0
$139$	−13.4164			−1.13796			−0.568982	−	0.822350i	$-0.692663\pi$
$139$	−13.4164			−1.13796			−0.568982	+	0.822350i	$0.692663\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	8.48528	+	8.48528i	0.709575	+	0.709575i
$144$		0			0
$145$	10.0000	+	10.0000i	0.830455	+	0.830455i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	3.16228i		−	0.259064i	−0.991575	−	0.129532i	$-0.958653\pi$
$149$		−	3.16228i		−	0.259064i	0.991575	−	0.129532i	$-0.0413475\pi$
$150$		0			0
$151$		−	13.4164i		−	1.09181i	−0.837846	−	0.545906i	$-0.816186\pi$
$151$		−	13.4164i		−	1.09181i	0.837846	−	0.545906i	$-0.183814\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−10.6066	−	10.6066i	−0.851943	−	0.851943i
$156$		0			0
$157$	7.00000	+	7.00000i	0.558661	+	0.558661i	0.928926	−	0.370265i	$-0.120733\pi$
$157$	7.00000	+	7.00000i	0.558661	+	0.558661i	−0.370265	+	0.928926i	$0.620733\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−6.70820	−	6.70820i	−0.525427	−	0.525427i	0.393778	−	0.919205i	$-0.371168\pi$
$163$	−6.70820	−	6.70820i	−0.525427	−	0.525427i	−0.919205	+	0.393778i	$0.871168\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.36396	−	6.36396i	0.492458	−	0.492458i	−0.416622	−	0.909080i	$-0.636786\pi$
$167$	6.36396	−	6.36396i	0.492458	−	0.492458i	0.909080	+	0.416622i	$0.136786\pi$
$168$		0			0
$169$			5.00000i			0.384615i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−14.2302	+	14.2302i	−1.08191	+	1.08191i	−0.0855740	+	0.996332i	$0.527272\pi$
$173$	−14.2302	+	14.2302i	−1.08191	+	1.08191i	−0.996332	+	0.0855740i	$0.972728\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	8.48528			0.634220			0.317110	−	0.948389i	$-0.397288\pi$
$179$	8.48528			0.634220			0.317110	+	0.948389i	$0.397288\pi$
$180$		0			0
$181$	−9.00000			−0.668965			−0.334482	−	0.942402i	$-0.608561\pi$
$181$	−9.00000			−0.668965			−0.334482	+	0.942402i	$0.608561\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.16228			−0.232495
$186$		0			0
$187$	−6.70820	+	6.70820i	−0.490552	+	0.490552i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	21.2132i		−	1.53493i	−0.641089	−	0.767467i	$-0.721517\pi$
$191$		−	21.2132i		−	1.53493i	0.641089	−	0.767467i	$-0.278483\pi$
$192$		0			0
$193$	16.0000	−	16.0000i	1.15171	−	1.15171i	0.165494	−	0.986211i	$-0.447078\pi$
$193$	16.0000	−	16.0000i	1.15171	−	1.15171i	0.986211	−	0.165494i	$-0.0529220\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−14.2302	−	14.2302i	−1.01386	−	1.01386i	−0.999903	−	0.0139608i	$-0.995556\pi$
$197$	−14.2302	−	14.2302i	−1.01386	−	1.01386i	−0.0139608	−	0.999903i	$-0.504444\pi$
$198$		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
$203$		0			0
$204$		0			0
$205$	−5.00000	+	5.00000i	−0.349215	+	0.349215i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			28.4605i			1.96865i
$210$		0			0
$211$			6.70820i			0.461812i	0.972976	+	0.230906i	$0.0741690\pi$
$211$			6.70820i			0.461812i	−0.972976	+	0.230906i	$0.925831\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−21.2132			−1.44673
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	6.32456			0.425436
$222$		0			0
$223$	6.70820	+	6.70820i	0.449215	+	0.449215i	0.895093	−	0.445879i	$-0.147109\pi$
$223$	6.70820	+	6.70820i	0.449215	+	0.449215i	−0.445879	+	0.895093i	$0.647109\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−6.36396	+	6.36396i	−0.422391	+	0.422391i	−0.886026	−	0.463635i	$-0.846545\pi$
$227$	−6.36396	+	6.36396i	−0.422391	+	0.422391i	0.463635	+	0.886026i	$0.346545\pi$
$228$		0			0
$229$		−	13.0000i		−	0.859064i	−0.903052	−	0.429532i	$-0.858679\pi$
$229$		−	13.0000i		−	0.859064i	0.903052	−	0.429532i	$-0.141321\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	12.6491	−	12.6491i	0.828671	−	0.828671i	−0.158662	−	0.987333i	$-0.550718\pi$
$233$	12.6491	−	12.6491i	0.828671	−	0.828671i	0.987333	+	0.158662i	$0.0507181\pi$
$234$		0			0
$235$			13.4164i			0.875190i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	12.7279			0.823301			0.411650	−	0.911342i	$-0.364952\pi$
$239$	12.7279			0.823301			0.411650	+	0.911342i	$0.364952\pi$
$240$		0			0
$241$	−19.0000			−1.22390			−0.611949	−	0.790897i	$-0.709614\pi$
$241$	−19.0000			−1.22390			−0.611949	+	0.790897i	$0.709614\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	11.0680	+	11.0680i	0.707107	+	0.707107i
$246$		0			0
$247$	13.4164	−	13.4164i	0.853666	−	0.853666i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	25.4558i		−	1.60676i	−0.595468	−	0.803379i	$-0.703033\pi$
$251$		−	25.4558i		−	1.60676i	0.595468	−	0.803379i	$-0.296967\pi$
$252$		0			0
$253$	−9.00000	+	9.00000i	−0.565825	+	0.565825i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−14.2302	−	14.2302i	−0.887659	−	0.887659i	0.106639	−	0.994298i	$-0.465991\pi$
$257$	−14.2302	−	14.2302i	−0.887659	−	0.887659i	−0.994298	+	0.106639i	$0.965991\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	8.48528	+	8.48528i	0.523225	+	0.523225i	0.918544	−	0.395319i	$-0.129366\pi$
$263$	8.48528	+	8.48528i	0.523225	+	0.523225i	−0.395319	+	0.918544i	$0.629366\pi$
$264$		0			0
$265$		−	5.00000i		−	0.307148i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	15.8114i		−	0.964037i	−0.876161	−	0.482019i	$-0.839904\pi$
$269$		−	15.8114i		−	0.964037i	0.876161	−	0.482019i	$-0.160096\pi$
$270$		0			0
$271$		−	6.70820i		−	0.407494i	−0.979024	−	0.203747i	$-0.934688\pi$
$271$		−	6.70820i		−	0.407494i	0.979024	−	0.203747i	$-0.0653121\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−21.2132			−1.27920
$276$		0			0
$277$	13.0000	+	13.0000i	0.781094	+	0.781094i	0.980015	−	0.198921i	$-0.0637438\pi$
$277$	13.0000	+	13.0000i	0.781094	+	0.781094i	−0.198921	+	0.980015i	$0.563744\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−6.32456			−0.377291			−0.188646	−	0.982045i	$-0.560410\pi$
$281$	−6.32456			−0.377291			−0.188646	+	0.982045i	$0.560410\pi$
$282$		0			0
$283$	20.1246	+	20.1246i	1.19628	+	1.19628i	0.975271	+	0.221013i	$0.0709364\pi$
$283$	20.1246	+	20.1246i	1.19628	+	1.19628i	0.221013	+	0.975271i	$0.429064\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$		−	12.0000i		−	0.705882i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	14.2302	−	14.2302i	0.831340	−	0.831340i	−0.156360	−	0.987700i	$-0.549976\pi$
$293$	14.2302	−	14.2302i	0.831340	−	0.831340i	0.987700	+	0.156360i	$0.0499760\pi$
$294$		0			0
$295$	13.4164	−	13.4164i	0.781133	−	0.781133i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	8.48528			0.490716
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	14.2302	−	14.2302i	0.814822	−	0.814822i
$306$		0			0
$307$	−20.1246	+	20.1246i	−1.14857	+	1.14857i	−0.161739	+	0.986834i	$0.551710\pi$
$307$	−20.1246	+	20.1246i	−1.14857	+	1.14857i	−0.986834	+	0.161739i	$0.948290\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	25.4558i		−	1.44347i	−0.692170	−	0.721734i	$-0.743345\pi$
$311$		−	25.4558i		−	1.44347i	0.692170	−	0.721734i	$-0.256655\pi$
$312$		0			0
$313$	−4.00000	+	4.00000i	−0.226093	+	0.226093i	−0.811058	−	0.584965i	$-0.801108\pi$
$313$	−4.00000	+	4.00000i	−0.226093	+	0.226093i	0.584965	+	0.811058i	$0.301108\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	20.5548	+	20.5548i	1.15447	+	1.15447i	0.985646	+	0.168827i	$0.0539980\pi$
$317$	20.5548	+	20.5548i	1.15447	+	1.15447i	0.168827	+	0.985646i	$0.446002\pi$
$318$		0			0
$319$	26.8328			1.50235
$320$		0			0
$321$		0			0
$322$		0			0
$323$	10.6066	+	10.6066i	0.590167	+	0.590167i
$324$		0			0
$325$	10.0000	+	10.0000i	0.554700	+	0.554700i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$			26.8328i			1.47486i	0.675421	+	0.737432i	$0.263962\pi$
$331$			26.8328i			1.47486i	−0.675421	+	0.737432i	$0.736038\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			21.2132i			1.15900i
$336$		0			0
$337$	−2.00000	−	2.00000i	−0.108947	−	0.108947i	0.650532	−	0.759479i	$-0.274546\pi$
$337$	−2.00000	−	2.00000i	−0.108947	−	0.108947i	−0.759479	+	0.650532i	$0.774546\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−28.4605			−1.54122
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−21.2132	+	21.2132i	−1.13878	+	1.13878i	−0.150116	+	0.988668i	$0.547965\pi$
$347$	−21.2132	+	21.2132i	−1.13878	+	1.13878i	−0.988668	+	0.150116i	$0.952035\pi$
$348$		0			0
$349$			9.00000i			0.481759i	0.970555	+	0.240879i	$0.0774359\pi$
$349$			9.00000i			0.481759i	−0.970555	+	0.240879i	$0.922564\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6.32456	+	6.32456i	−0.336622	+	0.336622i	−0.855094	−	0.518472i	$-0.826501\pi$
$353$	−6.32456	+	6.32456i	−0.336622	+	0.336622i	0.518472	+	0.855094i	$0.326501\pi$
$354$		0			0
$355$	−6.70820	−	6.70820i	−0.356034	−	0.356034i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	12.7279			0.671754			0.335877	−	0.941906i	$-0.390967\pi$
$359$	12.7279			0.671754			0.335877	+	0.941906i	$0.390967\pi$
$360$		0			0
$361$	26.0000			1.36842
$362$		0			0
$363$		0			0
$364$		0			0
$365$			3.16228i			0.165521i
$366$		0			0
$367$	13.4164	−	13.4164i	0.700331	−	0.700331i	−0.264151	−	0.964481i	$-0.585092\pi$
$367$	13.4164	−	13.4164i	0.700331	−	0.700331i	0.964481	+	0.264151i	$0.0850916\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	16.0000	−	16.0000i	0.828449	−	0.828449i	−0.158854	−	0.987302i	$-0.550780\pi$
$373$	16.0000	−	16.0000i	0.828449	−	0.828449i	0.987302	+	0.158854i	$0.0507798\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−12.6491	−	12.6491i	−0.651462	−	0.651462i
$378$		0			0
$379$	20.1246			1.03373			0.516866	−	0.856066i	$-0.327099\pi$
$379$	20.1246			1.03373			0.516866	+	0.856066i	$0.327099\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	23.3345	+	23.3345i	1.19234	+	1.19234i	0.976409	+	0.215930i	$0.0692781\pi$
$383$	23.3345	+	23.3345i	1.19234	+	1.19234i	0.215930	+	0.976409i	$0.430722\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$			22.1359i			1.12234i	0.827702	+	0.561168i	$0.189648\pi$
$389$			22.1359i			1.12234i	−0.827702	+	0.561168i	$0.810352\pi$
$390$		0			0
$391$			6.70820i			0.339248i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−10.6066	+	10.6066i	−0.533676	+	0.533676i
$396$		0			0
$397$	−26.0000	−	26.0000i	−1.30490	−	1.30490i	−0.925045	−	0.379858i	$-0.875973\pi$
$397$	−26.0000	−	26.0000i	−1.30490	−	1.30490i	−0.379858	−	0.925045i	$-0.624027\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3.16228			−0.157917			−0.0789583	−	0.996878i	$-0.525159\pi$
$401$	−3.16228			−0.157917			−0.0789583	+	0.996878i	$0.525159\pi$
$402$		0			0
$403$	13.4164	+	13.4164i	0.668319	+	0.668319i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−4.24264	+	4.24264i	−0.210300	+	0.210300i
$408$		0			0
$409$			27.0000i			1.33506i	0.744581	+	0.667532i	$0.232649\pi$
$409$			27.0000i			1.33506i	−0.744581	+	0.667532i	$0.767351\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	33.5410			1.64646
$416$		0			0
$417$		0			0
$418$		0			0
$419$	21.2132			1.03633			0.518166	−	0.855280i	$-0.326615\pi$
$419$	21.2132			1.03633			0.518166	+	0.855280i	$0.326615\pi$
$420$		0			0
$421$	1.00000			0.0487370			0.0243685	−	0.999703i	$-0.492242\pi$
$421$	1.00000			0.0487370			0.0243685	+	0.999703i	$0.492242\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−7.90569	+	7.90569i	−0.383482	+	0.383482i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	16.9706i		−	0.817443i	−0.912659	−	0.408722i	$-0.865975\pi$
$431$		−	16.9706i		−	0.817443i	0.912659	−	0.408722i	$-0.134025\pi$
$432$		0			0
$433$	−28.0000	+	28.0000i	−1.34559	+	1.34559i	−0.455210	+	0.890384i	$0.650436\pi$
$433$	−28.0000	+	28.0000i	−1.34559	+	1.34559i	−0.890384	+	0.455210i	$0.849564\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	14.2302	+	14.2302i	0.680725	+	0.680725i
$438$		0			0
$439$	20.1246			0.960495			0.480248	−	0.877133i	$-0.340547\pi$
$439$	20.1246			0.960495			0.480248	+	0.877133i	$0.340547\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2.12132	−	2.12132i	−0.100787	−	0.100787i	0.654915	−	0.755702i	$-0.272704\pi$
$443$	−2.12132	−	2.12132i	−0.100787	−	0.100787i	−0.755702	+	0.654915i	$0.772704\pi$
$444$		0			0
$445$	−25.0000	−	25.0000i	−1.18511	−	1.18511i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	12.6491i		−	0.596948i	−0.954418	−	0.298474i	$-0.903522\pi$
$449$		−	12.6491i		−	0.596948i	0.954418	−	0.298474i	$-0.0964777\pi$
$450$		0			0
$451$			13.4164i			0.631754i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−4.00000	−	4.00000i	−0.187112	−	0.187112i	0.607334	−	0.794446i	$-0.292239\pi$
$457$	−4.00000	−	4.00000i	−0.187112	−	0.187112i	−0.794446	+	0.607334i	$0.792239\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	22.1359			1.03097			0.515487	−	0.856897i	$-0.327611\pi$
$461$	22.1359			1.03097			0.515487	+	0.856897i	$0.327611\pi$
$462$		0			0
$463$	20.1246	+	20.1246i	0.935270	+	0.935270i	0.998029	−	0.0627587i	$-0.0199899\pi$
$463$	20.1246	+	20.1246i	0.935270	+	0.935270i	−0.0627587	+	0.998029i	$0.519990\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−27.5772	+	27.5772i	−1.27612	+	1.27612i	−0.333297	+	0.942822i	$0.608161\pi$
$467$	−27.5772	+	27.5772i	−1.27612	+	1.27612i	−0.942822	+	0.333297i	$0.891839\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−28.4605	+	28.4605i	−1.30862	+	1.30862i
$474$		0			0
$475$			33.5410i			1.53897i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	21.2132			0.969256			0.484628	−	0.874720i	$-0.338955\pi$
$479$	21.2132			0.969256			0.484628	+	0.874720i	$0.338955\pi$
$480$		0			0
$481$	4.00000			0.182384
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−12.6491			−0.574367
$486$		0			0
$487$	6.70820	−	6.70820i	0.303978	−	0.303978i	−0.538590	−	0.842568i	$-0.681043\pi$
$487$	6.70820	−	6.70820i	0.303978	−	0.303978i	0.842568	+	0.538590i	$0.181043\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	21.2132i		−	0.957338i	−0.877995	−	0.478669i	$-0.841119\pi$
$491$		−	21.2132i		−	0.957338i	0.877995	−	0.478669i	$-0.158881\pi$
$492$		0			0
$493$	10.0000	−	10.0000i	0.450377	−	0.450377i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−6.70820			−0.300300			−0.150150	−	0.988663i	$-0.547976\pi$
$499$	−6.70820			−0.300300			−0.150150	+	0.988663i	$0.547976\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	10.6066	+	10.6066i	0.472925	+	0.472925i	0.902860	−	0.429935i	$-0.141463\pi$
$503$	10.6066	+	10.6066i	0.472925	+	0.472925i	−0.429935	+	0.902860i	$0.641463\pi$
$504$		0			0
$505$	10.0000	−	10.0000i	0.444994	−	0.444994i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	25.2982i		−	1.12132i	−0.828045	−	0.560662i	$-0.810547\pi$
$509$		−	25.2982i		−	1.12132i	0.828045	−	0.560662i	$-0.189453\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−42.4264			−1.86953
$516$		0			0
$517$	18.0000	+	18.0000i	0.791639	+	0.791639i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	41.1096			1.80104			0.900522	−	0.434810i	$-0.143184\pi$
$521$	41.1096			1.80104			0.900522	+	0.434810i	$0.143184\pi$
$522$		0			0
$523$	26.8328	+	26.8328i	1.17332	+	1.17332i	0.981414	+	0.191903i	$0.0614660\pi$
$523$	26.8328	+	26.8328i	1.17332	+	1.17332i	0.191903	+	0.981414i	$0.438534\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−10.6066	+	10.6066i	−0.462031	+	0.462031i
$528$		0			0
$529$		−	14.0000i		−	0.608696i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	6.32456	−	6.32456i	0.273947	−	0.273947i
$534$		0			0
$535$			13.4164i			0.580042i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	29.6985			1.27920
$540$		0			0
$541$	36.0000			1.54776			0.773880	−	0.633332i	$-0.218313\pi$
$541$	36.0000			1.54776			0.773880	+	0.633332i	$0.218313\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−14.2302	−	14.2302i	−0.609557	−	0.609557i
$546$		0			0
$547$	6.70820	−	6.70820i	0.286822	−	0.286822i	−0.549000	−	0.835822i	$-0.684991\pi$
$547$	6.70820	−	6.70820i	0.286822	−	0.286822i	0.835822	+	0.549000i	$0.184991\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	42.4264i		−	1.80743i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−12.6491	−	12.6491i	−0.535960	−	0.535960i	0.386380	−	0.922340i	$-0.373725\pi$
$557$	−12.6491	−	12.6491i	−0.535960	−	0.535960i	−0.922340	+	0.386380i	$0.873725\pi$
$558$		0			0
$559$	26.8328			1.13491
$560$		0			0
$561$		0			0
$562$		0			0
$563$	12.7279	+	12.7279i	0.536418	+	0.536418i	0.922475	−	0.386057i	$-0.126163\pi$
$563$	12.7279	+	12.7279i	0.536418	+	0.536418i	−0.386057	+	0.922475i	$0.626163\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	6.32456i		−	0.265139i	−0.991174	−	0.132570i	$-0.957677\pi$
$569$		−	6.32456i		−	0.265139i	0.991174	−	0.132570i	$-0.0423228\pi$
$570$		0			0
$571$			20.1246i			0.842189i	0.907017	+	0.421094i	$0.138354\pi$
$571$			20.1246i			0.842189i	−0.907017	+	0.421094i	$0.861646\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−10.6066	+	10.6066i	−0.442326	+	0.442326i
$576$		0			0
$577$	−1.00000	−	1.00000i	−0.0416305	−	0.0416305i	0.685985	−	0.727616i	$-0.259372\pi$
$577$	−1.00000	−	1.00000i	−0.0416305	−	0.0416305i	−0.727616	+	0.685985i	$0.759372\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−6.70820	−	6.70820i	−0.277825	−	0.277825i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	10.6066	−	10.6066i	0.437781	−	0.437781i	−0.453483	−	0.891265i	$-0.649819\pi$
$587$	10.6066	−	10.6066i	0.437781	−	0.437781i	0.891265	+	0.453483i	$0.149819\pi$
$588$		0			0
$589$			45.0000i			1.85419i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	14.2302	−	14.2302i	0.584366	−	0.584366i	−0.351734	−	0.936100i	$-0.614408\pi$
$593$	14.2302	−	14.2302i	0.584366	−	0.584366i	0.936100	+	0.351734i	$0.114408\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−42.4264			−1.73350			−0.866748	−	0.498746i	$-0.833794\pi$
$599$	−42.4264			−1.73350			−0.866748	+	0.498746i	$0.833794\pi$
$600$		0			0
$601$	9.00000			0.367118			0.183559	−	0.983009i	$-0.441238\pi$
$601$	9.00000			0.367118			0.183559	+	0.983009i	$0.441238\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−11.0680	+	11.0680i	−0.449977	+	0.449977i
$606$		0			0
$607$	6.70820	−	6.70820i	0.272278	−	0.272278i	−0.557739	−	0.830016i	$-0.688331\pi$
$607$	6.70820	−	6.70820i	0.272278	−	0.272278i	0.830016	+	0.557739i	$0.188331\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	16.9706i		−	0.686555i
$612$		0			0
$613$	−28.0000	+	28.0000i	−1.13091	+	1.13091i	−0.140883	+	0.990026i	$0.544994\pi$
$613$	−28.0000	+	28.0000i	−1.13091	+	1.13091i	−0.990026	+	0.140883i	$0.955006\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	11.0680	+	11.0680i	0.445580	+	0.445580i	0.893882	−	0.448302i	$-0.147971\pi$
$617$	11.0680	+	11.0680i	0.445580	+	0.445580i	−0.448302	+	0.893882i	$0.647971\pi$
$618$		0			0
$619$	13.4164			0.539251			0.269625	−	0.962965i	$-0.413100\pi$
$619$	13.4164			0.539251			0.269625	+	0.962965i	$0.413100\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−25.0000			−1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$			3.16228i			0.126088i
$630$		0			0
$631$		−	20.1246i		−	0.801148i	−0.916264	−	0.400574i	$-0.868811\pi$
$631$		−	20.1246i		−	0.801148i	0.916264	−	0.400574i	$-0.131189\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	21.2132i		−	0.841820i
$636$		0			0
$637$	−14.0000	−	14.0000i	−0.554700	−	0.554700i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	6.32456			0.249805			0.124902	−	0.992169i	$-0.460138\pi$
$641$	6.32456			0.249805			0.124902	+	0.992169i	$0.460138\pi$
$642$		0			0
$643$	−13.4164	−	13.4164i	−0.529091	−	0.529091i	0.391210	−	0.920301i	$-0.372057\pi$
$643$	−13.4164	−	13.4164i	−0.529091	−	0.529091i	−0.920301	+	0.391210i	$0.872057\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	27.5772	−	27.5772i	1.08417	−	1.08417i	0.0880546	−	0.996116i	$-0.471935\pi$
$647$	27.5772	−	27.5772i	1.08417	−	1.08417i	0.996116	−	0.0880546i	$-0.0280650\pi$
$648$		0			0
$649$		−	36.0000i		−	1.41312i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	17.3925	−	17.3925i	0.680622	−	0.680622i	−0.279518	−	0.960140i	$-0.590175\pi$
$653$	17.3925	−	17.3925i	0.680622	−	0.680622i	0.960140	+	0.279518i	$0.0901747\pi$
$654$		0			0
$655$	−6.70820	−	6.70820i	−0.262111	−	0.262111i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	42.4264			1.65270			0.826349	−	0.563158i	$-0.190414\pi$
$659$	42.4264			1.65270			0.826349	+	0.563158i	$0.190414\pi$
$660$		0			0
$661$	−36.0000			−1.40024			−0.700119	−	0.714026i	$-0.746870\pi$
$661$	−36.0000			−1.40024			−0.700119	+	0.714026i	$0.746870\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	13.4164	−	13.4164i	0.519485	−	0.519485i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	38.1838i		−	1.47407i
$672$		0			0
$673$	22.0000	−	22.0000i	0.848038	−	0.848038i	−0.141850	−	0.989888i	$-0.545305\pi$
$673$	22.0000	−	22.0000i	0.848038	−	0.848038i	0.989888	+	0.141850i	$0.0453052\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−6.32456	−	6.32456i	−0.243072	−	0.243072i	0.575048	−	0.818120i	$-0.304984\pi$
$677$	−6.32456	−	6.32456i	−0.243072	−	0.243072i	−0.818120	+	0.575048i	$0.804984\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−19.0919	−	19.0919i	−0.730531	−	0.730531i	0.240194	−	0.970725i	$-0.422789\pi$
$683$	−19.0919	−	19.0919i	−0.730531	−	0.730531i	−0.970725	+	0.240194i	$0.922789\pi$
$684$		0			0
$685$	45.0000			1.71936
$686$		0			0
$687$		0			0
$688$		0			0
$689$			6.32456i			0.240946i
$690$		0			0
$691$			6.70820i			0.255192i	0.991826	+	0.127596i	$0.0407261\pi$
$691$			6.70820i			0.255192i	−0.991826	+	0.127596i	$0.959274\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−21.2132	+	21.2132i	−0.804663	+	0.804663i
$696$		0			0
$697$	5.00000	+	5.00000i	0.189389	+	0.189389i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−41.1096			−1.55269			−0.776344	−	0.630309i	$-0.782928\pi$
$701$	−41.1096			−1.55269			−0.776344	+	0.630309i	$0.782928\pi$
$702$		0			0
$703$	6.70820	+	6.70820i	0.253005	+	0.253005i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	8.00000i		−	0.300446i	−0.988652	−	0.150223i	$-0.952001\pi$
$709$		−	8.00000i		−	0.300446i	0.988652	−	0.150223i	$-0.0479992\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−14.2302	+	14.2302i	−0.532927	+	0.532927i
$714$		0			0
$715$	26.8328			1.00349
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−29.6985			−1.10757			−0.553783	−	0.832661i	$-0.686816\pi$
$719$	−29.6985			−1.10757			−0.553783	+	0.832661i	$0.686816\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	31.6228			1.17444
$726$		0			0
$727$	−13.4164	+	13.4164i	−0.497587	+	0.497587i	−0.910686	−	0.413099i	$-0.864446\pi$
$727$	−13.4164	+	13.4164i	−0.497587	+	0.497587i	0.413099	+	0.910686i	$0.364446\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			21.2132i			0.784599i
$732$		0			0
$733$	−2.00000	+	2.00000i	−0.0738717	+	0.0738717i	−0.743077	−	0.669206i	$-0.766635\pi$
$733$	−2.00000	+	2.00000i	−0.0738717	+	0.0738717i	0.669206	+	0.743077i	$0.266635\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	28.4605	+	28.4605i	1.04836	+	1.04836i
$738$		0			0
$739$	−33.5410			−1.23383			−0.616913	−	0.787031i	$-0.711617\pi$
$739$	−33.5410			−1.23383			−0.616913	+	0.787031i	$0.711617\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	29.6985	+	29.6985i	1.08953	+	1.08953i	0.995576	+	0.0939553i	$0.0299511\pi$
$743$	29.6985	+	29.6985i	1.08953	+	1.08953i	0.0939553	+	0.995576i	$0.470049\pi$
$744$		0			0
$745$	−5.00000	−	5.00000i	−0.183186	−	0.183186i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		−	6.70820i		−	0.244786i	−0.992482	−	0.122393i	$-0.960943\pi$
$751$		−	6.70820i		−	0.244786i	0.992482	−	0.122393i	$-0.0390568\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−21.2132	−	21.2132i	−0.772028	−	0.772028i
$756$		0			0
$757$	−19.0000	−	19.0000i	−0.690567	−	0.690567i	0.271790	−	0.962357i	$-0.412384\pi$
$757$	−19.0000	−	19.0000i	−0.690567	−	0.690567i	−0.962357	+	0.271790i	$0.912384\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−25.2982			−0.917060			−0.458530	−	0.888679i	$-0.651624\pi$
$761$	−25.2982			−0.917060			−0.458530	+	0.888679i	$0.651624\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−16.9706	+	16.9706i	−0.612772	+	0.612772i
$768$		0			0
$769$			9.00000i			0.324548i	0.986746	+	0.162274i	$0.0518829\pi$
$769$			9.00000i			0.324548i	−0.986746	+	0.162274i	$0.948117\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−30.0416	+	30.0416i	−1.08052	+	1.08052i	−0.0840621	+	0.996461i	$0.526789\pi$
$773$	−30.0416	+	30.0416i	−1.08052	+	1.08052i	−0.996461	+	0.0840621i	$0.973211\pi$
$774$		0			0
$775$	−33.5410			−1.20483
$776$		0			0
$777$		0			0
$778$		0			0
$779$	21.2132			0.760042
$780$		0			0
$781$	−18.0000			−0.644091
$782$		0			0
$783$		0			0
$784$		0			0
$785$	22.1359			0.790066
$786$		0			0
$787$	−20.1246	+	20.1246i	−0.717365	+	0.717365i	−0.968065	−	0.250700i	$-0.919339\pi$
$787$	−20.1246	+	20.1246i	−0.717365	+	0.717365i	0.250700	+	0.968065i	$0.419339\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−18.0000	+	18.0000i	−0.639199	+	0.639199i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−14.2302	−	14.2302i	−0.504061	−	0.504061i	0.408636	−	0.912697i	$-0.366005\pi$
$797$	−14.2302	−	14.2302i	−0.504061	−	0.504061i	−0.912697	+	0.408636i	$0.866005\pi$
$798$		0			0
$799$	13.4164			0.474638
$800$		0			0
$801$		0			0
$802$		0			0
$803$	4.24264	+	4.24264i	0.149720	+	0.149720i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			15.8114i			0.555899i	0.960596	+	0.277949i	$0.0896548\pi$
$809$			15.8114i			0.555899i	−0.960596	+	0.277949i	$0.910345\pi$
$810$		0			0
$811$			40.2492i			1.41334i	0.707543	+	0.706671i	$0.249804\pi$
$811$			40.2492i			1.41334i	−0.707543	+	0.706671i	$0.750196\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−21.2132			−0.743066
$816$		0			0
$817$	45.0000	+	45.0000i	1.57435	+	1.57435i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−34.7851			−1.21401			−0.607003	−	0.794699i	$-0.707629\pi$
$821$	−34.7851			−1.21401			−0.607003	+	0.794699i	$0.707629\pi$
$822$		0			0
$823$	−20.1246	−	20.1246i	−0.701500	−	0.701500i	0.263233	−	0.964732i	$-0.415211\pi$
$823$	−20.1246	−	20.1246i	−0.701500	−	0.701500i	−0.964732	+	0.263233i	$0.915211\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−6.36396	+	6.36396i	−0.221297	+	0.221297i	−0.809044	−	0.587748i	$-0.800015\pi$
$827$	−6.36396	+	6.36396i	−0.221297	+	0.221297i	0.587748	+	0.809044i	$0.300015\pi$
$828$		0			0
$829$		−	18.0000i		−	0.625166i	−0.949890	−	0.312583i	$-0.898806\pi$
$829$		−	18.0000i		−	0.625166i	0.949890	−	0.312583i	$-0.101194\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	11.0680	−	11.0680i	0.383482	−	0.383482i
$834$		0			0
$835$		−	20.1246i		−	0.696441i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−8.48528			−0.292944			−0.146472	−	0.989215i	$-0.546792\pi$
$839$	−8.48528			−0.292944			−0.146472	+	0.989215i	$0.546792\pi$
$840$		0			0
$841$	−11.0000			−0.379310
$842$		0			0
$843$		0			0
$844$		0			0
$845$	7.90569	+	7.90569i	0.271964	+	0.271964i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$			4.24264i			0.145436i
$852$		0			0
$853$	22.0000	−	22.0000i	0.753266	−	0.753266i	−0.221822	−	0.975087i	$-0.571200\pi$
$853$	22.0000	−	22.0000i	0.753266	−	0.753266i	0.975087	+	0.221822i	$0.0712003\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−20.5548	−	20.5548i	−0.702139	−	0.702139i	0.262731	−	0.964869i	$-0.415377\pi$
$857$	−20.5548	−	20.5548i	−0.702139	−	0.702139i	−0.964869	+	0.262731i	$0.915377\pi$
$858$		0			0
$859$	20.1246			0.686643			0.343321	−	0.939218i	$-0.388448\pi$
$859$	20.1246			0.686643			0.343321	+	0.939218i	$0.388448\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	31.8198	+	31.8198i	1.08316	+	1.08316i	0.996213	+	0.0869457i	$0.0277107\pi$
$863$	31.8198	+	31.8198i	1.08316	+	1.08316i	0.0869457	+	0.996213i	$0.472289\pi$
$864$		0			0
$865$			45.0000i			1.53005i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			28.4605i			0.965456i
$870$		0			0
$871$		−	26.8328i		−	0.909195i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	29.0000	+	29.0000i	0.979260	+	0.979260i	0.999789	−	0.0205288i	$-0.00653499\pi$
$877$	29.0000	+	29.0000i	0.979260	+	0.979260i	−0.0205288	+	0.999789i	$0.506535\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	12.6491			0.426159			0.213080	−	0.977035i	$-0.431651\pi$
$881$	12.6491			0.426159			0.213080	+	0.977035i	$0.431651\pi$
$882$		0			0
$883$	−26.8328	−	26.8328i	−0.902996	−	0.902996i	0.0926981	−	0.995694i	$-0.470451\pi$
$883$	−26.8328	−	26.8328i	−0.902996	−	0.902996i	−0.995694	+	0.0926981i	$0.970451\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−6.36396	+	6.36396i	−0.213681	+	0.213681i	−0.805829	−	0.592148i	$-0.798280\pi$
$887$	−6.36396	+	6.36396i	−0.213681	+	0.213681i	0.592148	+	0.805829i	$0.298280\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	28.4605	−	28.4605i	0.952394	−	0.952394i
$894$		0			0
$895$	13.4164	−	13.4164i	0.448461	−	0.448461i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	42.4264			1.41500
$900$		0			0
$901$	−5.00000			−0.166574
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−14.2302	+	14.2302i	−0.473029	+	0.473029i
$906$		0			0
$907$	−13.4164	+	13.4164i	−0.445485	+	0.445485i	−0.893850	−	0.448366i	$-0.852006\pi$
$907$	−13.4164	+	13.4164i	−0.445485	+	0.445485i	0.448366	+	0.893850i	$0.352006\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	16.9706i		−	0.562260i	−0.959670	−	0.281130i	$-0.909291\pi$
$911$		−	16.9706i		−	0.562260i	0.959670	−	0.281130i	$-0.0907092\pi$
$912$		0			0
$913$	45.0000	−	45.0000i	1.48928	−	1.48928i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−40.2492			−1.32770			−0.663850	−	0.747866i	$-0.731079\pi$
$919$	−40.2492			−1.32770			−0.663850	+	0.747866i	$0.731079\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	8.48528	+	8.48528i	0.279296	+	0.279296i
$924$		0			0
$925$	−5.00000	+	5.00000i	−0.164399	+	0.164399i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	22.1359i		−	0.726257i	−0.931739	−	0.363128i	$-0.881709\pi$
$929$		−	22.1359i		−	0.726257i	0.931739	−	0.363128i	$-0.118291\pi$
$930$		0			0
$931$		−	46.9574i		−	1.53897i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			21.2132i			0.693746i
$936$		0			0
$937$	−26.0000	−	26.0000i	−0.849383	−	0.849383i	0.140673	−	0.990056i	$-0.455073\pi$
$937$	−26.0000	−	26.0000i	−0.849383	−	0.849383i	−0.990056	+	0.140673i	$0.955073\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	53.7587			1.75248			0.876242	−	0.481871i	$-0.160043\pi$
$941$	53.7587			1.75248			0.876242	+	0.481871i	$0.160043\pi$
$942$		0			0
$943$	6.70820	+	6.70820i	0.218449	+	0.218449i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	10.6066	−	10.6066i	0.344668	−	0.344668i	−0.513451	−	0.858119i	$-0.671633\pi$
$947$	10.6066	−	10.6066i	0.344668	−	0.344668i	0.858119	+	0.513451i	$0.171633\pi$
$948$		0			0
$949$		−	4.00000i		−	0.129845i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	28.4605	−	28.4605i	0.921926	−	0.921926i	−0.0752395	−	0.997165i	$-0.523972\pi$
$953$	28.4605	−	28.4605i	0.921926	−	0.921926i	0.997165	+	0.0752395i	$0.0239721\pi$
$954$		0			0
$955$	−33.5410	−	33.5410i	−1.08536	−	1.08536i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−14.0000			−0.451613
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	50.5964i		−	1.62876i
$966$		0			0
$967$	−6.70820	+	6.70820i	−0.215721	+	0.215721i	−0.806693	−	0.590971i	$-0.798745\pi$
$967$	−6.70820	+	6.70820i	−0.215721	+	0.215721i	0.590971	+	0.806693i	$0.298745\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			38.1838i			1.22538i	0.790325	+	0.612688i	$0.209912\pi$
$971$			38.1838i			1.22538i	−0.790325	+	0.612688i	$0.790088\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−25.2982	−	25.2982i	−0.809362	−	0.809362i	0.175175	−	0.984537i	$-0.443951\pi$
$977$	−25.2982	−	25.2982i	−0.809362	−	0.809362i	−0.984537	+	0.175175i	$0.943951\pi$
$978$		0			0
$979$	−67.0820			−2.14395
$980$		0			0
$981$		0			0
$982$		0			0
$983$	2.12132	+	2.12132i	0.0676596	+	0.0676596i	0.740127	−	0.672467i	$-0.234765\pi$
$983$	2.12132	+	2.12132i	0.0676596	+	0.0676596i	−0.672467	+	0.740127i	$0.734765\pi$
$984$		0			0
$985$	−45.0000			−1.43382
$986$		0			0
$987$		0			0
$988$		0			0
$989$			28.4605i			0.904991i
$990$		0			0
$991$			6.70820i			0.213093i	0.994308	+	0.106547i	$0.0339793\pi$
$991$			6.70820i			0.213093i	−0.994308	+	0.106547i	$0.966021\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−31.0000	−	31.0000i	−0.981780	−	0.981780i	0.0180571	−	0.999837i	$-0.494252\pi$
$997$	−31.0000	−	31.0000i	−0.981780	−	0.981780i	−0.999837	+	0.0180571i	$0.994252\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.2.x.b.1567.3	yes	8
3.2	odd	2	inner	2160.2.x.b.1567.2	yes	8
4.3	odd	2	inner	2160.2.x.b.1567.4	yes	8
5.3	odd	4	inner	2160.2.x.b.703.3	yes	8
12.11	even	2	inner	2160.2.x.b.1567.1	yes	8
15.8	even	4	inner	2160.2.x.b.703.2	yes	8
20.3	even	4	inner	2160.2.x.b.703.4	yes	8
60.23	odd	4	inner	2160.2.x.b.703.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2160.2.x.b.703.1	✓	8	60.23	odd	4	inner
2160.2.x.b.703.2	yes	8	15.8	even	4	inner
2160.2.x.b.703.3	yes	8	5.3	odd	4	inner
2160.2.x.b.703.4	yes	8	20.3	even	4	inner
2160.2.x.b.1567.1	yes	8	12.11	even	2	inner
2160.2.x.b.1567.2	yes	8	3.2	odd	2	inner
2160.2.x.b.1567.3	yes	8	1.1	even	1	trivial
2160.2.x.b.1567.4	yes	8	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2160.x (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.58114 - 1.58114i) q^{5} +O(q^{10})\)	\(q+(1.58114 - 1.58114i) q^{5} -4.24264i q^{11} +(-2.00000 + 2.00000i) q^{13} +(-1.58114 - 1.58114i) q^{17} -6.70820 q^{19} +(-2.12132 - 2.12132i) q^{23} -5.00000i q^{25} +6.32456i q^{29} -6.70820i q^{31} +(-1.00000 - 1.00000i) q^{37} -3.16228 q^{41} +(-6.70820 - 6.70820i) q^{43} +(-4.24264 + 4.24264i) q^{47} +7.00000i q^{49} +(1.58114 - 1.58114i) q^{53} +(-6.70820 - 6.70820i) q^{55} +8.48528 q^{59} +9.00000 q^{61} +6.32456i q^{65} +(-6.70820 + 6.70820i) q^{67} -4.24264i q^{71} +(-1.00000 + 1.00000i) q^{73} -6.70820 q^{79} +(10.6066 + 10.6066i) q^{83} -5.00000 q^{85} -15.8114i q^{89} +(-10.6066 + 10.6066i) q^{95} +(-4.00000 - 4.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 16 q^{13} - 8 q^{37} + 72 q^{61} - 8 q^{73} - 40 q^{85} - 32 q^{97}+O(q^{100}) \) 8 * q - 16 * q^13 - 8 * q^37 + 72 * q^61 - 8 * q^73 - 40 * q^85 - 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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