LMFDB - Embedded newform 2880.2.t.d.2161.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(721,2880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2880, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("2880.721");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.t (of order $4$, 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:	$22.9969157821$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{18} - 4 x^{17} + 7 x^{16} + 16 x^{15} + 6 x^{14} - 36 x^{13} - 42 x^{12} + 40 x^{11} + \cdots + 1024 $ x^20 - 2x^18 - 4x^17 + 7x^16 + 16x^15 + 6x^14 - 36x^13 - 42x^12 + 40x^11 + 136x^10 + 80x^9 - 168x^8 - 288x^7 + 96x^6 + 512x^5 + 448x^4 - 512x^3 - 512*x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	no (minimal twist has level 240)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			2161.9
Root			$-0.0861743 - 1.41159i$ of defining polynomial
Character	$\chi$	$=$	2880.2161
Dual form			2880.2.t.d.721.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 + 0.707107i) q^{5} +2.76462i q^{7} +O(q^{10})$	$q+(0.707107 + 0.707107i) q^{5} +2.76462i q^{7} +(-3.51009 - 3.51009i) q^{11} +(-4.55960 + 4.55960i) q^{13} +5.00550 q^{17} +(0.812949 - 0.812949i) q^{19} -7.48205i q^{23} +1.00000i q^{25} +(-6.03354 + 6.03354i) q^{29} -7.58233 q^{31} +(-1.95488 + 1.95488i) q^{35} +(1.08674 + 1.08674i) q^{37} -3.15671i q^{41} +(3.10932 + 3.10932i) q^{43} -2.76008 q^{47} -0.643123 q^{49} +(-6.41096 - 6.41096i) q^{53} -4.96402i q^{55} +(-5.13756 - 5.13756i) q^{59} +(-2.49234 + 2.49234i) q^{61} -6.44825 q^{65} +(3.14625 - 3.14625i) q^{67} -3.50237i q^{71} -14.6145i q^{73} +(9.70408 - 9.70408i) q^{77} -8.95325 q^{79} +(-2.86293 + 2.86293i) q^{83} +(3.53942 + 3.53942i) q^{85} +7.23560i q^{89} +(-12.6056 - 12.6056i) q^{91} +1.14968 q^{95} -8.24056 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q+O(q^{10}) $ 20 * q	$ 20 q + 8 q^{11} + 24 q^{17} + 4 q^{19} - 16 q^{29} + 16 q^{37} + 8 q^{43} - 52 q^{49} + 16 q^{53} - 16 q^{59} - 4 q^{61} + 8 q^{67} + 40 q^{77} - 56 q^{79} - 48 q^{83} + 4 q^{85} + 8 q^{91} + 56 q^{97}+O(q^{100}) $ 20 * q + 8 * q^11 + 24 * q^17 + 4 * q^19 - 16 * q^29 + 16 * q^37 + 8 * q^43 - 52 * q^49 + 16 * q^53 - 16 * q^59 - 4 * q^61 + 8 * q^67 + 40 * q^77 - 56 * q^79 - 48 * q^83 + 4 * q^85 + 8 * q^91 + 56 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.707107	+	0.707107i	0.316228	+	0.316228i
$5$	0.707107	+	0.707107i	0.316228	+	0.316228i
$6$		0			0
$7$			2.76462i			1.04493i	0.852661	+	0.522464i	$0.174987\pi$
$7$			2.76462i			1.04493i	−0.852661	+	0.522464i	$0.825013\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.51009	−	3.51009i	−1.05833	−	1.05833i	−0.998190	−	0.0601437i	$-0.980844\pi$
$11$	−3.51009	−	3.51009i	−1.05833	−	1.05833i	−0.0601437	−	0.998190i	$-0.519156\pi$
$12$		0			0
$13$	−4.55960	+	4.55960i	−1.26461	+	1.26461i	−0.315771	+	0.948835i	$0.602263\pi$
$13$	−4.55960	+	4.55960i	−1.26461	+	1.26461i	−0.948835	+	0.315771i	$0.897737\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	5.00550			1.21401			0.607006	−	0.794697i	$-0.292370\pi$
$17$	5.00550			1.21401			0.607006	+	0.794697i	$0.292370\pi$
$18$		0			0
$19$	0.812949	−	0.812949i	0.186503	−	0.186503i	−0.607679	−	0.794183i	$-0.707899\pi$
$19$	0.812949	−	0.812949i	0.186503	−	0.186503i	0.794183	+	0.607679i	$0.207899\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	7.48205i		−	1.56011i	−0.625708	−	0.780057i	$-0.715190\pi$
$23$		−	7.48205i		−	1.56011i	0.625708	−	0.780057i	$-0.284810\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−6.03354	+	6.03354i	−1.12040	+	1.12040i	−0.128719	+	0.991681i	$0.541087\pi$
$29$	−6.03354	+	6.03354i	−1.12040	+	1.12040i	−0.991681	+	0.128719i	$0.958913\pi$
$30$		0			0
$31$	−7.58233			−1.36183			−0.680913	−	0.732364i	$-0.738417\pi$
$31$	−7.58233			−1.36183			−0.680913	+	0.732364i	$0.738417\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.95488	+	1.95488i	−0.330435	+	0.330435i
$36$		0			0
$37$	1.08674	+	1.08674i	0.178659	+	0.178659i	0.790771	−	0.612112i	$-0.209680\pi$
$37$	1.08674	+	1.08674i	0.178659	+	0.178659i	−0.612112	+	0.790771i	$0.709680\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	3.15671i		−	0.492995i	−0.969143	−	0.246497i	$-0.920720\pi$
$41$		−	3.15671i		−	0.492995i	0.969143	−	0.246497i	$-0.0792797\pi$
$42$		0			0
$43$	3.10932	+	3.10932i	0.474166	+	0.474166i	0.903260	−	0.429094i	$-0.141167\pi$
$43$	3.10932	+	3.10932i	0.474166	+	0.474166i	−0.429094	+	0.903260i	$0.641167\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.76008			−0.402599			−0.201299	−	0.979530i	$-0.564516\pi$
$47$	−2.76008			−0.402599			−0.201299	+	0.979530i	$0.564516\pi$
$48$		0			0
$49$	−0.643123			−0.0918747
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−6.41096	−	6.41096i	−0.880613	−	0.880613i	0.112983	−	0.993597i	$-0.463959\pi$
$53$	−6.41096	−	6.41096i	−0.880613	−	0.880613i	−0.993597	+	0.112983i	$0.963959\pi$
$54$		0			0
$55$		−	4.96402i		−	0.669349i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.13756	−	5.13756i	−0.668854	−	0.668854i	0.288597	−	0.957451i	$-0.406811\pi$
$59$	−5.13756	−	5.13756i	−0.668854	−	0.668854i	−0.957451	+	0.288597i	$0.906811\pi$
$60$		0			0
$61$	−2.49234	+	2.49234i	−0.319111	+	0.319111i	−0.848426	−	0.529314i	$-0.822449\pi$
$61$	−2.49234	+	2.49234i	−0.319111	+	0.319111i	0.529314	+	0.848426i	$0.322449\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−6.44825			−0.799807
$66$		0			0
$67$	3.14625	−	3.14625i	0.384376	−	0.384376i	−0.488300	−	0.872676i	$-0.662383\pi$
$67$	3.14625	−	3.14625i	0.384376	−	0.384376i	0.872676	+	0.488300i	$0.162383\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	3.50237i		−	0.415655i	−0.978166	−	0.207827i	$-0.933361\pi$
$71$		−	3.50237i		−	0.415655i	0.978166	−	0.207827i	$-0.0666392\pi$
$72$		0			0
$73$		−	14.6145i		−	1.71050i	−0.518217	−	0.855249i	$-0.673404\pi$
$73$		−	14.6145i		−	1.71050i	0.518217	−	0.855249i	$-0.326596\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	9.70408	−	9.70408i	1.10588	−	1.10588i
$78$		0			0
$79$	−8.95325			−1.00732			−0.503660	−	0.863902i	$-0.668013\pi$
$79$	−8.95325			−1.00732			−0.503660	+	0.863902i	$0.668013\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−2.86293	+	2.86293i	−0.314247	+	0.314247i	−0.846553	−	0.532305i	$-0.821326\pi$
$83$	−2.86293	+	2.86293i	−0.314247	+	0.314247i	0.532305	+	0.846553i	$0.321326\pi$
$84$		0			0
$85$	3.53942	+	3.53942i	0.383904	+	0.383904i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			7.23560i			0.766972i	0.923547	+	0.383486i	$0.125277\pi$
$89$			7.23560i			0.766972i	−0.923547	+	0.383486i	$0.874723\pi$
$90$		0			0
$91$	−12.6056	−	12.6056i	−1.32142	−	1.32142i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.14968			0.117955
$96$		0			0
$97$	−8.24056			−0.836702			−0.418351	−	0.908285i	$-0.637392\pi$
$97$	−8.24056			−0.836702			−0.418351	+	0.908285i	$0.637392\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.07326	−	1.07326i	−0.106794	−	0.106794i	0.651691	−	0.758485i	$-0.274060\pi$
$101$	−1.07326	−	1.07326i	−0.106794	−	0.106794i	−0.758485	+	0.651691i	$0.774060\pi$
$102$		0			0
$103$		−	1.66763i		−	0.164316i	−0.996619	−	0.0821582i	$-0.973819\pi$
$103$		−	1.66763i		−	0.164316i	0.996619	−	0.0821582i	$-0.0261813\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.08860	−	6.08860i	−0.588607	−	0.588607i	0.348647	−	0.937254i	$-0.386641\pi$
$107$	−6.08860	−	6.08860i	−0.588607	−	0.588607i	−0.937254	+	0.348647i	$0.886641\pi$
$108$		0			0
$109$	2.92136	−	2.92136i	0.279816	−	0.279816i	−0.553220	−	0.833035i	$-0.686601\pi$
$109$	2.92136	−	2.92136i	0.279816	−	0.279816i	0.833035	+	0.553220i	$0.186601\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	2.84395			0.267536			0.133768	−	0.991013i	$-0.457292\pi$
$113$	2.84395			0.267536			0.133768	+	0.991013i	$0.457292\pi$
$114$		0			0
$115$	5.29061	−	5.29061i	0.493352	−	0.493352i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			13.8383i			1.26856i
$120$		0			0
$121$			13.6415i			1.24014i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.707107	+	0.707107i	−0.0632456	+	0.0632456i
$126$		0			0
$127$	6.61073			0.586607			0.293304	−	0.956019i	$-0.405245\pi$
$127$	6.61073			0.586607			0.293304	+	0.956019i	$0.405245\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6.70125	+	6.70125i	−0.585491	+	0.585491i	−0.936407	−	0.350916i	$-0.885870\pi$
$131$	−6.70125	+	6.70125i	−0.585491	+	0.585491i	0.350916	+	0.936407i	$0.385870\pi$
$132$		0			0
$133$	2.24749	+	2.24749i	0.194883	+	0.194883i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	14.3908i		−	1.22949i	−0.788725	−	0.614746i	$-0.789258\pi$
$137$		−	14.3908i		−	1.22949i	0.788725	−	0.614746i	$-0.210742\pi$
$138$		0			0
$139$	−0.292743	−	0.292743i	−0.0248302	−	0.0248302i	0.694583	−	0.719413i	$-0.255589\pi$
$139$	−0.292743	−	0.292743i	−0.0248302	−	0.0248302i	−0.719413	+	0.694583i	$0.755589\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	32.0093			2.67675
$144$		0			0
$145$	−8.53271			−0.708603
$146$		0			0
$147$		0			0
$148$		0			0
$149$	10.9363	+	10.9363i	0.895934	+	0.895934i	0.995074	−	0.0991393i	$-0.0316089\pi$
$149$	10.9363	+	10.9363i	0.895934	+	0.895934i	−0.0991393	+	0.995074i	$0.531609\pi$
$150$		0			0
$151$			13.5225i			1.10045i	0.835017	+	0.550223i	$0.185457\pi$
$151$			13.5225i			1.10045i	−0.835017	+	0.550223i	$0.814543\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−5.36151	−	5.36151i	−0.430647	−	0.430647i
$156$		0			0
$157$	2.76487	−	2.76487i	0.220661	−	0.220661i	−0.588116	−	0.808777i	$-0.700130\pi$
$157$	2.76487	−	2.76487i	0.220661	−	0.220661i	0.808777	+	0.588116i	$0.200130\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	20.6850			1.63021
$162$		0			0
$163$	−15.4942	+	15.4942i	−1.21360	+	1.21360i	−0.243767	+	0.969834i	$0.578383\pi$
$163$	−15.4942	+	15.4942i	−1.21360	+	1.21360i	−0.969834	+	0.243767i	$0.921617\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			1.76898i			0.136888i	0.997655	+	0.0684439i	$0.0218034\pi$
$167$			1.76898i			0.136888i	−0.997655	+	0.0684439i	$0.978197\pi$
$168$		0			0
$169$		−	28.5800i		−	2.19846i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4.01351	+	4.01351i	−0.305142	+	0.305142i	−0.843022	−	0.537880i	$-0.819225\pi$
$173$	−4.01351	+	4.01351i	−0.305142	+	0.305142i	0.537880	+	0.843022i	$0.319225\pi$
$174$		0			0
$175$	−2.76462			−0.208986
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−5.31340	+	5.31340i	−0.397142	+	0.397142i	−0.877224	−	0.480082i	$-0.840607\pi$
$179$	−5.31340	+	5.31340i	−0.397142	+	0.397142i	0.480082	+	0.877224i	$0.340607\pi$
$180$		0			0
$181$	−8.74918	−	8.74918i	−0.650321	−	0.650321i	0.302749	−	0.953070i	$-0.402096\pi$
$181$	−8.74918	−	8.74918i	−0.650321	−	0.650321i	−0.953070	+	0.302749i	$0.902096\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			1.53688i			0.112994i
$186$		0			0
$187$	−17.5698	−	17.5698i	−1.28483	−	1.28483i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4.03787			−0.292170			−0.146085	−	0.989272i	$-0.546667\pi$
$191$	−4.03787			−0.292170			−0.146085	+	0.989272i	$0.546667\pi$
$192$		0			0
$193$	0.437111			0.0314639			0.0157320	−	0.999876i	$-0.494992\pi$
$193$	0.437111			0.0314639			0.0157320	+	0.999876i	$0.494992\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−11.6836	−	11.6836i	−0.832424	−	0.832424i	0.155424	−	0.987848i	$-0.450326\pi$
$197$	−11.6836	−	11.6836i	−0.832424	−	0.832424i	−0.987848	+	0.155424i	$0.950326\pi$
$198$		0			0
$199$		−	15.7412i		−	1.11587i	−0.829886	−	0.557933i	$-0.811595\pi$
$199$		−	15.7412i		−	1.11587i	0.829886	−	0.557933i	$-0.188405\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−16.6804	−	16.6804i	−1.17074	−	1.17074i
$204$		0			0
$205$	2.23213	−	2.23213i	0.155899	−	0.155899i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−5.70706			−0.394765
$210$		0			0
$211$	2.13765	−	2.13765i	0.147162	−	0.147162i	−0.629687	−	0.776849i	$-0.716817\pi$
$211$	2.13765	−	2.13765i	0.147162	−	0.147162i	0.776849	+	0.629687i	$0.216817\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			4.39724i			0.299889i
$216$		0			0
$217$		−	20.9622i		−	1.42301i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−22.8231	+	22.8231i	−1.53525	+	1.53525i
$222$		0			0
$223$	−13.4768			−0.902476			−0.451238	−	0.892404i	$-0.649017\pi$
$223$	−13.4768			−0.902476			−0.451238	+	0.892404i	$0.649017\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2.70567	+	2.70567i	−0.179582	+	0.179582i	−0.791174	−	0.611592i	$-0.790530\pi$
$227$	−2.70567	+	2.70567i	−0.179582	+	0.179582i	0.611592	+	0.791174i	$0.290530\pi$
$228$		0			0
$229$	0.507051	+	0.507051i	0.0335069	+	0.0335069i	0.723662	−	0.690155i	$-0.242458\pi$
$229$	0.507051	+	0.507051i	0.0335069	+	0.0335069i	−0.690155	+	0.723662i	$0.742458\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			8.16109i			0.534651i	0.963606	+	0.267325i	$0.0861399\pi$
$233$			8.16109i			0.534651i	−0.963606	+	0.267325i	$0.913860\pi$
$234$		0			0
$235$	−1.95167	−	1.95167i	−0.127313	−	0.127313i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−2.46397			−0.159381			−0.0796905	−	0.996820i	$-0.525393\pi$
$239$	−2.46397			−0.159381			−0.0796905	+	0.996820i	$0.525393\pi$
$240$		0			0
$241$	7.38073			0.475434			0.237717	−	0.971334i	$-0.423601\pi$
$241$	7.38073			0.475434			0.237717	+	0.971334i	$0.423601\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.454757	−	0.454757i	−0.0290533	−	0.0290533i
$246$		0			0
$247$			7.41345i			0.471707i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−5.15322	−	5.15322i	−0.325268	−	0.325268i	0.525516	−	0.850784i	$-0.323872\pi$
$251$	−5.15322	−	5.15322i	−0.325268	−	0.325268i	−0.850784	+	0.525516i	$0.823872\pi$
$252$		0			0
$253$	−26.2627	+	26.2627i	−1.65112	+	1.65112i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−9.84129			−0.613883			−0.306942	−	0.951728i	$-0.599306\pi$
$257$	−9.84129			−0.613883			−0.306942	+	0.951728i	$0.599306\pi$
$258$		0			0
$259$	−3.00442	+	3.00442i	−0.186686	+	0.186686i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			27.5027i			1.69589i	0.530086	+	0.847944i	$0.322160\pi$
$263$			27.5027i			1.69589i	−0.530086	+	0.847944i	$0.677840\pi$
$264$		0			0
$265$		−	9.06647i		−	0.556949i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−7.35280	+	7.35280i	−0.448308	+	0.448308i	−0.894792	−	0.446484i	$-0.852676\pi$
$269$	−7.35280	+	7.35280i	−0.448308	+	0.448308i	0.446484	+	0.894792i	$0.352676\pi$
$270$		0			0
$271$	16.1826			0.983023			0.491511	−	0.870871i	$-0.336445\pi$
$271$	16.1826			0.983023			0.491511	+	0.870871i	$0.336445\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.51009	−	3.51009i	0.211667	−	0.211667i
$276$		0			0
$277$	−19.3725	−	19.3725i	−1.16398	−	1.16398i	−0.983597	−	0.180381i	$-0.942267\pi$
$277$	−19.3725	−	19.3725i	−1.16398	−	1.16398i	−0.180381	−	0.983597i	$-0.557733\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	18.1786i		−	1.08445i	−0.840235	−	0.542223i	$-0.817583\pi$
$281$		−	18.1786i		−	1.08445i	0.840235	−	0.542223i	$-0.182417\pi$
$282$		0			0
$283$	7.74925	+	7.74925i	0.460645	+	0.460645i	0.898867	−	0.438222i	$-0.144391\pi$
$283$	7.74925	+	7.74925i	0.460645	+	0.460645i	−0.438222	+	0.898867i	$0.644391\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	8.72709			0.515144
$288$		0			0
$289$	8.05503			0.473825
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.24572	+	1.24572i	0.0727760	+	0.0727760i	0.742558	−	0.669782i	$-0.233612\pi$
$293$	1.24572	+	1.24572i	0.0727760	+	0.0727760i	−0.669782	+	0.742558i	$0.733612\pi$
$294$		0			0
$295$		−	7.26561i		−	0.423020i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	34.1152	+	34.1152i	1.97293	+	1.97293i
$300$		0			0
$301$	−8.59608	+	8.59608i	−0.495470	+	0.495470i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−3.52470			−0.201824
$306$		0			0
$307$	−13.5042	+	13.5042i	−0.770727	+	0.770727i	−0.978234	−	0.207506i	$-0.933465\pi$
$307$	−13.5042	+	13.5042i	−0.770727	+	0.770727i	0.207506	+	0.978234i	$0.433465\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	11.9549i		−	0.677901i	−0.940804	−	0.338951i	$-0.889928\pi$
$311$		−	11.9549i		−	0.677901i	0.940804	−	0.338951i	$-0.110072\pi$
$312$		0			0
$313$			15.2385i			0.861333i	0.902511	+	0.430667i	$0.141721\pi$
$313$			15.2385i			0.861333i	−0.902511	+	0.430667i	$0.858279\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−22.4062	+	22.4062i	−1.25846	+	1.25846i	−0.306625	+	0.951830i	$0.599200\pi$
$317$	−22.4062	+	22.4062i	−1.25846	+	1.25846i	−0.951830	+	0.306625i	$0.900800\pi$
$318$		0			0
$319$	42.3566			2.37151
$320$		0			0
$321$		0			0
$322$		0			0
$323$	4.06922	−	4.06922i	0.226417	−	0.226417i
$324$		0			0
$325$	−4.55960	−	4.55960i	−0.252921	−	0.252921i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	7.63057i		−	0.420687i
$330$		0			0
$331$	−8.11650	−	8.11650i	−0.446123	−	0.446123i	0.447940	−	0.894063i	$-0.352158\pi$
$331$	−8.11650	−	8.11650i	−0.446123	−	0.446123i	−0.894063	+	0.447940i	$0.852158\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	4.44947			0.243101
$336$		0			0
$337$	−19.1995			−1.04587			−0.522933	−	0.852374i	$-0.675162\pi$
$337$	−19.1995			−1.04587			−0.522933	+	0.852374i	$0.675162\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	26.6147	+	26.6147i	1.44127	+	1.44127i
$342$		0			0
$343$			17.5743i			0.948926i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3.64968	−	3.64968i	−0.195925	−	0.195925i	0.602325	−	0.798251i	$-0.294241\pi$
$347$	−3.64968	−	3.64968i	−0.195925	−	0.195925i	−0.798251	+	0.602325i	$0.794241\pi$
$348$		0			0
$349$	16.7180	−	16.7180i	0.894896	−	0.894896i	−0.100083	−	0.994979i	$-0.531911\pi$
$349$	16.7180	−	16.7180i	0.894896	−	0.894896i	0.994979	+	0.100083i	$0.0319107\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9.42887			−0.501848			−0.250924	−	0.968007i	$-0.580734\pi$
$353$	−9.42887			−0.501848			−0.250924	+	0.968007i	$0.580734\pi$
$354$		0			0
$355$	2.47655	−	2.47655i	0.131442	−	0.131442i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			1.12135i			0.0591826i	0.999562	+	0.0295913i	$0.00942059\pi$
$359$			1.12135i			0.0591826i	−0.999562	+	0.0295913i	$0.990579\pi$
$360$		0			0
$361$			17.6782i			0.930433i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	10.3340	−	10.3340i	0.540907	−	0.540907i
$366$		0			0
$367$	7.22666			0.377228			0.188614	−	0.982051i	$-0.439600\pi$
$367$	7.22666			0.377228			0.188614	+	0.982051i	$0.439600\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	17.7239	−	17.7239i	0.920178	−	0.920178i
$372$		0			0
$373$	−8.83590	−	8.83590i	−0.457506	−	0.457506i	0.440330	−	0.897836i	$-0.354861\pi$
$373$	−8.83590	−	8.83590i	−0.457506	−	0.457506i	−0.897836	+	0.440330i	$0.854861\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	55.0211i		−	2.83373i
$378$		0			0
$379$	26.2339	+	26.2339i	1.34754	+	1.34754i	0.888321	+	0.459223i	$0.151872\pi$
$379$	26.2339	+	26.2339i	1.34754	+	1.34754i	0.459223	+	0.888321i	$0.348128\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	14.3557			0.733544			0.366772	−	0.930311i	$-0.380463\pi$
$383$	14.3557			0.733544			0.366772	+	0.930311i	$0.380463\pi$
$384$		0			0
$385$	13.7236			0.699421
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.07721	+	1.07721i	0.0546168	+	0.0546168i	0.733888	−	0.679271i	$-0.237704\pi$
$389$	1.07721	+	1.07721i	0.0546168	+	0.0546168i	−0.679271	+	0.733888i	$0.737704\pi$
$390$		0			0
$391$		−	37.4514i		−	1.89400i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−6.33090	−	6.33090i	−0.318542	−	0.318542i
$396$		0			0
$397$	16.8838	−	16.8838i	0.847376	−	0.847376i	−0.142429	−	0.989805i	$-0.545491\pi$
$397$	16.8838	−	16.8838i	0.847376	−	0.847376i	0.989805	+	0.142429i	$0.0454913\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	0.406306			0.0202900			0.0101450	−	0.999949i	$-0.496771\pi$
$401$	0.406306			0.0202900			0.0101450	+	0.999949i	$0.496771\pi$
$402$		0			0
$403$	34.5724	−	34.5724i	1.72217	−	1.72217i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	7.62911i		−	0.378161i
$408$		0			0
$409$			38.0171i			1.87983i	0.341413	+	0.939913i	$0.389094\pi$
$409$			38.0171i			1.87983i	−0.341413	+	0.939913i	$0.610906\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	14.2034	−	14.2034i	0.698904	−	0.698904i
$414$		0			0
$415$	−4.04880			−0.198748
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−23.2302	+	23.2302i	−1.13487	+	1.13487i	−0.145512	+	0.989357i	$0.546483\pi$
$419$	−23.2302	+	23.2302i	−1.13487	+	1.13487i	−0.989357	+	0.145512i	$0.953517\pi$
$420$		0			0
$421$	26.7883	+	26.7883i	1.30558	+	1.30558i	0.924572	+	0.381009i	$0.124423\pi$
$421$	26.7883	+	26.7883i	1.30558	+	1.30558i	0.381009	+	0.924572i	$0.375577\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			5.00550i			0.242802i
$426$		0			0
$427$	−6.89037	−	6.89037i	−0.333448	−	0.333448i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−9.17509			−0.441949			−0.220974	−	0.975280i	$-0.570924\pi$
$431$	−9.17509			−0.441949			−0.220974	+	0.975280i	$0.570924\pi$
$432$		0			0
$433$	36.5762			1.75774			0.878869	−	0.477063i	$-0.158299\pi$
$433$	36.5762			1.75774			0.878869	+	0.477063i	$0.158299\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−6.08252	−	6.08252i	−0.290967	−	0.290967i
$438$		0			0
$439$			6.96346i			0.332348i	0.986096	+	0.166174i	$0.0531413\pi$
$439$			6.96346i			0.332348i	−0.986096	+	0.166174i	$0.946859\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2.15777	+	2.15777i	0.102519	+	0.102519i	0.756506	−	0.653987i	$-0.226905\pi$
$443$	2.15777	+	2.15777i	0.102519	+	0.102519i	−0.653987	+	0.756506i	$0.726905\pi$
$444$		0			0
$445$	−5.11634	+	5.11634i	−0.242538	+	0.242538i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−13.1720			−0.621627			−0.310814	−	0.950471i	$-0.600602\pi$
$449$	−13.1720			−0.621627			−0.310814	+	0.950471i	$0.600602\pi$
$450$		0			0
$451$	−11.0803	+	11.0803i	−0.521753	+	0.521753i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	17.8270i		−	0.835741i
$456$		0			0
$457$			10.0012i			0.467834i	0.972257	+	0.233917i	$0.0751545\pi$
$457$			10.0012i			0.467834i	−0.972257	+	0.233917i	$0.924846\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1.79443	−	1.79443i	0.0835750	−	0.0835750i	−0.664083	−	0.747658i	$-0.731178\pi$
$461$	1.79443	−	1.79443i	0.0835750	−	0.0835750i	0.747658	+	0.664083i	$0.231178\pi$
$462$		0			0
$463$	2.58325			0.120054			0.0600269	−	0.998197i	$-0.480881\pi$
$463$	2.58325			0.120054			0.0600269	+	0.998197i	$0.480881\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	24.6809	−	24.6809i	1.14210	−	1.14210i	0.154029	−	0.988066i	$-0.450775\pi$
$467$	24.6809	−	24.6809i	1.14210	−	1.14210i	0.988066	−	0.154029i	$-0.0492250\pi$
$468$		0			0
$469$	8.69819	+	8.69819i	0.401645	+	0.401645i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	21.8280i		−	1.00365i
$474$		0			0
$475$	0.812949	+	0.812949i	0.0373007	+	0.0373007i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1.39344			0.0636679			0.0318339	−	0.999493i	$-0.489865\pi$
$479$	1.39344			0.0636679			0.0318339	+	0.999493i	$0.489865\pi$
$480$		0			0
$481$	−9.91020			−0.451866
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−5.82695	−	5.82695i	−0.264588	−	0.264588i
$486$		0			0
$487$			26.7044i			1.21009i	0.796191	+	0.605045i	$0.206845\pi$
$487$			26.7044i			1.21009i	−0.796191	+	0.605045i	$0.793155\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	16.3259	+	16.3259i	0.736775	+	0.736775i	0.971953	−	0.235177i	$-0.0755670\pi$
$491$	16.3259	+	16.3259i	0.736775	+	0.736775i	−0.235177	+	0.971953i	$0.575567\pi$
$492$		0			0
$493$	−30.2009	+	30.2009i	−1.36018	+	1.36018i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	9.68272			0.434329
$498$		0			0
$499$	−26.4923	+	26.4923i	−1.18596	+	1.18596i	−0.207783	+	0.978175i	$0.566625\pi$
$499$	−26.4923	+	26.4923i	−1.18596	+	1.18596i	−0.978175	+	0.207783i	$0.933375\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			14.2132i			0.633737i	0.948470	+	0.316868i	$0.102631\pi$
$503$			14.2132i			0.633737i	−0.948470	+	0.316868i	$0.897369\pi$
$504$		0			0
$505$		−	1.51782i		−	0.0675422i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−4.26128	+	4.26128i	−0.188878	+	0.188878i	−0.795211	−	0.606333i	$-0.792640\pi$
$509$	−4.26128	+	4.26128i	−0.188878	+	0.188878i	0.606333	+	0.795211i	$0.292640\pi$
$510$		0			0
$511$	40.4036			1.78735
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.17919	−	1.17919i	0.0519614	−	0.0519614i
$516$		0			0
$517$	9.68814	+	9.68814i	0.426084	+	0.426084i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	31.9522i		−	1.39985i	−0.714215	−	0.699926i	$-0.753216\pi$
$521$		−	31.9522i		−	1.39985i	0.714215	−	0.699926i	$-0.246784\pi$
$522$		0			0
$523$	−12.3608	−	12.3608i	−0.540498	−	0.540498i	0.383177	−	0.923675i	$-0.374830\pi$
$523$	−12.3608	−	12.3608i	−0.540498	−	0.540498i	−0.923675	+	0.383177i	$0.874830\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−37.9533			−1.65327
$528$		0			0
$529$	−32.9810			−1.43396
$530$		0			0
$531$		0			0
$532$		0			0
$533$	14.3933	+	14.3933i	0.623444	+	0.623444i
$534$		0			0
$535$		−	8.61058i		−	0.372268i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.25742	+	2.25742i	0.0972341	+	0.0972341i
$540$		0			0
$541$	25.6126	−	25.6126i	1.10117	−	1.10117i	0.106902	−	0.994270i	$-0.465907\pi$
$541$	25.6126	−	25.6126i	1.10117	−	1.10117i	0.994270	−	0.106902i	$-0.0340932\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	4.13143			0.176971
$546$		0			0
$547$	−1.56643	+	1.56643i	−0.0669759	+	0.0669759i	−0.739801	−	0.672825i	$-0.765081\pi$
$547$	−1.56643	+	1.56643i	−0.0669759	+	0.0669759i	0.672825	+	0.739801i	$0.265081\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			9.80992i			0.417917i
$552$		0			0
$553$		−	24.7523i		−	1.05258i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−3.96707	+	3.96707i	−0.168090	+	0.168090i	−0.786139	−	0.618049i	$-0.787923\pi$
$557$	−3.96707	+	3.96707i	−0.168090	+	0.168090i	0.618049	+	0.786139i	$0.287923\pi$
$558$		0			0
$559$	−28.3545			−1.19927
$560$		0			0
$561$		0			0
$562$		0			0
$563$	23.0565	−	23.0565i	0.971715	−	0.971715i	−0.0278956	−	0.999611i	$-0.508881\pi$
$563$	23.0565	−	23.0565i	0.971715	−	0.971715i	0.999611	+	0.0278956i	$0.00888059\pi$
$564$		0			0
$565$	2.01097	+	2.01097i	0.0846023	+	0.0846023i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			32.6228i			1.36762i	0.729660	+	0.683810i	$0.239678\pi$
$569$			32.6228i			1.36762i	−0.729660	+	0.683810i	$0.760322\pi$
$570$		0			0
$571$	−13.3559	−	13.3559i	−0.558926	−	0.558926i	0.370076	−	0.929002i	$-0.379332\pi$
$571$	−13.3559	−	13.3559i	−0.558926	−	0.558926i	−0.929002	+	0.370076i	$0.879332\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	7.48205			0.312023
$576$		0			0
$577$	46.5999			1.93998			0.969989	−	0.243147i	$-0.0781798\pi$
$577$	46.5999			1.93998			0.969989	+	0.243147i	$0.0781798\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−7.91492	−	7.91492i	−0.328366	−	0.328366i
$582$		0			0
$583$			45.0062i			1.86397i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−29.5590	−	29.5590i	−1.22003	−	1.22003i	−0.967621	−	0.252408i	$-0.918778\pi$
$587$	−29.5590	−	29.5590i	−1.22003	−	1.22003i	−0.252408	−	0.967621i	$-0.581222\pi$
$588$		0			0
$589$	−6.16404	+	6.16404i	−0.253985	+	0.253985i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	35.4555			1.45598			0.727991	−	0.685587i	$-0.240454\pi$
$593$	35.4555			1.45598			0.727991	+	0.685587i	$0.240454\pi$
$594$		0			0
$595$	−9.78516	+	9.78516i	−0.401152	+	0.401152i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			22.8571i			0.933915i	0.884280	+	0.466958i	$0.154650\pi$
$599$			22.8571i			0.933915i	−0.884280	+	0.466958i	$0.845350\pi$
$600$		0			0
$601$			19.4253i			0.792374i	0.918170	+	0.396187i	$0.129667\pi$
$601$			19.4253i			0.792374i	−0.918170	+	0.396187i	$0.870333\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−9.64602	+	9.64602i	−0.392166	+	0.392166i
$606$		0			0
$607$	19.2455			0.781149			0.390574	−	0.920571i	$-0.372276\pi$
$607$	19.2455			0.781149			0.390574	+	0.920571i	$0.372276\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	12.5849	−	12.5849i	0.509129	−	0.509129i
$612$		0			0
$613$	17.3922	+	17.3922i	0.702465	+	0.702465i	0.964939	−	0.262474i	$-0.0845383\pi$
$613$	17.3922	+	17.3922i	0.702465	+	0.702465i	−0.262474	+	0.964939i	$0.584538\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			29.7863i			1.19915i	0.800318	+	0.599576i	$0.204664\pi$
$617$			29.7863i			1.19915i	−0.800318	+	0.599576i	$0.795336\pi$
$618$		0			0
$619$	10.6404	+	10.6404i	0.427673	+	0.427673i	0.887835	−	0.460162i	$-0.152209\pi$
$619$	10.6404	+	10.6404i	0.427673	+	0.427673i	−0.460162	+	0.887835i	$0.652209\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−20.0037			−0.801430
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	5.43967	+	5.43967i	0.216894	+	0.216894i
$630$		0			0
$631$			38.7660i			1.54325i	0.636078	+	0.771625i	$0.280556\pi$
$631$			38.7660i			1.54325i	−0.636078	+	0.771625i	$0.719444\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4.67449	+	4.67449i	0.185502	+	0.185502i
$636$		0			0
$637$	2.93239	−	2.93239i	0.116185	−	0.116185i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−3.84117			−0.151717			−0.0758585	−	0.997119i	$-0.524170\pi$
$641$	−3.84117			−0.151717			−0.0758585	+	0.997119i	$0.524170\pi$
$642$		0			0
$643$	4.00729	−	4.00729i	0.158032	−	0.158032i	−0.623662	−	0.781694i	$-0.714356\pi$
$643$	4.00729	−	4.00729i	0.158032	−	0.158032i	0.781694	+	0.623662i	$0.214356\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	25.3385i		−	0.996158i	−0.867132	−	0.498079i	$-0.834039\pi$
$647$		−	25.3385i		−	0.996158i	0.867132	−	0.498079i	$-0.165961\pi$
$648$		0			0
$649$			36.0667i			1.41574i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−0.690132	+	0.690132i	−0.0270069	+	0.0270069i	−0.720481	−	0.693474i	$-0.756079\pi$
$653$	−0.690132	+	0.690132i	−0.0270069	+	0.0270069i	0.693474	+	0.720481i	$0.256079\pi$
$654$		0			0
$655$	−9.47700			−0.370297
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−27.4031	+	27.4031i	−1.06747	+	1.06747i	−0.0699222	+	0.997552i	$0.522275\pi$
$659$	−27.4031	+	27.4031i	−1.06747	+	1.06747i	−0.997552	+	0.0699222i	$0.977725\pi$
$660$		0			0
$661$	14.4586	+	14.4586i	0.562373	+	0.562373i	0.929981	−	0.367608i	$-0.119823\pi$
$661$	14.4586	+	14.4586i	0.562373	+	0.562373i	−0.367608	+	0.929981i	$0.619823\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			3.17844i			0.123255i
$666$		0			0
$667$	45.1432	+	45.1432i	1.74795	+	1.74795i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	17.4967			0.675452
$672$		0			0
$673$	38.4496			1.48212			0.741061	−	0.671438i	$-0.234323\pi$
$673$	38.4496			1.48212			0.741061	+	0.671438i	$0.234323\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	15.2625	+	15.2625i	0.586586	+	0.586586i	0.936705	−	0.350119i	$-0.113859\pi$
$677$	15.2625	+	15.2625i	0.586586	+	0.586586i	−0.350119	+	0.936705i	$0.613859\pi$
$678$		0			0
$679$		−	22.7820i		−	0.874293i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−31.6516	−	31.6516i	−1.21111	−	1.21111i	−0.970661	−	0.240453i	$-0.922704\pi$
$683$	−31.6516	−	31.6516i	−1.21111	−	1.21111i	−0.240453	−	0.970661i	$-0.577296\pi$
$684$		0			0
$685$	10.1759	−	10.1759i	0.388800	−	0.388800i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	58.4629			2.22726
$690$		0			0
$691$	12.0010	−	12.0010i	0.456539	−	0.456539i	−0.440979	−	0.897518i	$-0.645369\pi$
$691$	12.0010	−	12.0010i	0.456539	−	0.456539i	0.897518	+	0.440979i	$0.145369\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	0.414002i		−	0.0157040i
$696$		0			0
$697$		−	15.8009i		−	0.598502i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	0.298277	−	0.298277i	0.0112658	−	0.0112658i	−0.701451	−	0.712717i	$-0.747464\pi$
$701$	0.298277	−	0.298277i	0.0112658	−	0.0112658i	0.712717	+	0.701451i	$0.247464\pi$
$702$		0			0
$703$	1.76693			0.0666409
$704$		0			0
$705$		0			0
$706$		0			0
$707$	2.96716	−	2.96716i	0.111592	−	0.111592i
$708$		0			0
$709$	−11.6489	−	11.6489i	−0.437485	−	0.437485i	0.453680	−	0.891165i	$-0.350111\pi$
$709$	−11.6489	−	11.6489i	−0.437485	−	0.437485i	−0.891165	+	0.453680i	$0.850111\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			56.7313i			2.12460i
$714$		0			0
$715$	22.6340	+	22.6340i	0.846463	+	0.846463i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1.69024			0.0630353			0.0315177	−	0.999503i	$-0.489966\pi$
$719$	1.69024			0.0630353			0.0315177	+	0.999503i	$0.489966\pi$
$720$		0			0
$721$	4.61036			0.171699
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−6.03354	−	6.03354i	−0.224080	−	0.224080i
$726$		0			0
$727$		−	27.2221i		−	1.00961i	−0.863233	−	0.504805i	$-0.831564\pi$
$727$		−	27.2221i		−	1.00961i	0.863233	−	0.504805i	$-0.168436\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	15.5637	+	15.5637i	0.575644	+	0.575644i
$732$		0			0
$733$	20.9172	−	20.9172i	0.772596	−	0.772596i	−0.205963	−	0.978560i	$-0.566033\pi$
$733$	20.9172	−	20.9172i	0.772596	−	0.772596i	0.978560	+	0.205963i	$0.0660328\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−22.0873			−0.813596
$738$		0			0
$739$	3.53634	−	3.53634i	0.130086	−	0.130086i	−0.639066	−	0.769152i	$-0.720679\pi$
$739$	3.53634	−	3.53634i	0.130086	−	0.130086i	0.769152	+	0.639066i	$0.220679\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			38.5438i			1.41403i	0.707196	+	0.707017i	$0.249960\pi$
$743$			38.5438i			1.41403i	−0.707196	+	0.707017i	$0.750040\pi$
$744$		0			0
$745$			15.4662i			0.566639i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	16.8327	−	16.8327i	0.615052	−	0.615052i
$750$		0			0
$751$	7.07646			0.258224			0.129112	−	0.991630i	$-0.458787\pi$
$751$	7.07646			0.258224			0.129112	+	0.991630i	$0.458787\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−9.56186	+	9.56186i	−0.347992	+	0.347992i
$756$		0			0
$757$	−32.5233	−	32.5233i	−1.18208	−	1.18208i	−0.979205	−	0.202873i	$-0.934972\pi$
$757$	−32.5233	−	32.5233i	−1.18208	−	1.18208i	−0.202873	−	0.979205i	$-0.565028\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			43.1903i			1.56565i	0.622243	+	0.782824i	$0.286222\pi$
$761$			43.1903i			1.56565i	−0.622243	+	0.782824i	$0.713778\pi$
$762$		0			0
$763$	8.07645	+	8.07645i	0.292387	+	0.292387i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	46.8505			1.69167
$768$		0			0
$769$	−31.1675			−1.12393			−0.561965	−	0.827161i	$-0.689955\pi$
$769$	−31.1675			−1.12393			−0.561965	+	0.827161i	$0.689955\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−11.7904	−	11.7904i	−0.424072	−	0.424072i	0.462531	−	0.886603i	$-0.346941\pi$
$773$	−11.7904	−	11.7904i	−0.424072	−	0.424072i	−0.886603	+	0.462531i	$0.846941\pi$
$774$		0			0
$775$		−	7.58233i		−	0.272365i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−2.56624	−	2.56624i	−0.0919452	−	0.0919452i
$780$		0			0
$781$	−12.2936	+	12.2936i	−0.439901	+	0.439901i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	3.91012			0.139558
$786$		0			0
$787$	3.50469	−	3.50469i	0.124929	−	0.124929i	−0.641878	−	0.766807i	$-0.721844\pi$
$787$	3.50469	−	3.50469i	0.124929	−	0.124929i	0.766807	+	0.641878i	$0.221844\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			7.86243i			0.279556i
$792$		0			0
$793$		−	22.7282i		−	0.807100i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	34.1882	−	34.1882i	1.21101	−	1.21101i	0.240312	−	0.970696i	$-0.422750\pi$
$797$	34.1882	−	34.1882i	1.21101	−	1.21101i	0.970696	−	0.240312i	$-0.0772497\pi$
$798$		0			0
$799$	−13.8156			−0.488760
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−51.2983	+	51.2983i	−1.81028	+	1.81028i
$804$		0			0
$805$	14.6265	+	14.6265i	0.515517	+	0.515517i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	0.637755i		−	0.0224223i	−0.999937	−	0.0112111i	$-0.996431\pi$
$809$		−	0.637755i		−	0.0224223i	0.999937	−	0.0112111i	$-0.00356869\pi$
$810$		0			0
$811$	−30.6724	−	30.6724i	−1.07705	−	1.07705i	−0.996772	−	0.0802798i	$-0.974419\pi$
$811$	−30.6724	−	30.6724i	−1.07705	−	1.07705i	−0.0802798	−	0.996772i	$-0.525581\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−21.9121			−0.767549
$816$		0			0
$817$	5.05543			0.176867
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−2.40778	−	2.40778i	−0.0840320	−	0.0840320i	0.663841	−	0.747873i	$-0.268925\pi$
$821$	−2.40778	−	2.40778i	−0.0840320	−	0.0840320i	−0.747873	+	0.663841i	$0.768925\pi$
$822$		0			0
$823$		−	32.4455i		−	1.13098i	−0.824755	−	0.565490i	$-0.808687\pi$
$823$		−	32.4455i		−	1.13098i	0.824755	−	0.565490i	$-0.191313\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	32.2278	+	32.2278i	1.12067	+	1.12067i	0.991641	+	0.129028i	$0.0411857\pi$
$827$	32.2278	+	32.2278i	1.12067	+	1.12067i	0.129028	+	0.991641i	$0.458814\pi$
$828$		0			0
$829$	29.6767	−	29.6767i	1.03072	−	1.03072i	0.0312027	−	0.999513i	$-0.490066\pi$
$829$	29.6767	−	29.6767i	1.03072	−	1.03072i	0.999513	−	0.0312027i	$-0.00993375\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−3.21915			−0.111537
$834$		0			0
$835$	−1.25086	+	1.25086i	−0.0432877	+	0.0432877i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	42.4417i		−	1.46525i	−0.680632	−	0.732625i	$-0.738295\pi$
$839$		−	42.4417i		−	1.46525i	0.680632	−	0.732625i	$-0.261705\pi$
$840$		0			0
$841$		−	43.8072i		−	1.51059i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	20.2091	−	20.2091i	0.695214	−	0.695214i
$846$		0			0
$847$	−37.7136			−1.29586
$848$		0			0
$849$		0			0
$850$		0			0
$851$	8.13103	−	8.13103i	0.278728	−	0.278728i
$852$		0			0
$853$	−16.7844	−	16.7844i	−0.574686	−	0.574686i	0.358748	−	0.933434i	$-0.383204\pi$
$853$	−16.7844	−	16.7844i	−0.574686	−	0.574686i	−0.933434	+	0.358748i	$0.883204\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	31.0924i		−	1.06210i	−0.847341	−	0.531049i	$-0.821798\pi$
$857$		−	31.0924i		−	1.06210i	0.847341	−	0.531049i	$-0.178202\pi$
$858$		0			0
$859$	16.2508	+	16.2508i	0.554469	+	0.554469i	0.927727	−	0.373258i	$-0.121759\pi$
$859$	16.2508	+	16.2508i	0.554469	+	0.554469i	−0.373258	+	0.927727i	$0.621759\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−22.4656			−0.764739			−0.382369	−	0.924010i	$-0.624892\pi$
$863$	−22.4656			−0.764739			−0.382369	+	0.924010i	$0.624892\pi$
$864$		0			0
$865$	−5.67597			−0.192989
$866$		0			0
$867$		0			0
$868$		0			0
$869$	31.4268	+	31.4268i	1.06608	+	1.06608i
$870$		0			0
$871$			28.6913i			0.972169i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.95488	−	1.95488i	−0.0660871	−	0.0660871i
$876$		0			0
$877$	1.46527	−	1.46527i	0.0494785	−	0.0494785i	−0.681935	−	0.731413i	$-0.738861\pi$
$877$	1.46527	−	1.46527i	0.0494785	−	0.0494785i	0.731413	+	0.681935i	$0.238861\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−0.578022			−0.0194740			−0.00973702	−	0.999953i	$-0.503099\pi$
$881$	−0.578022			−0.0194740			−0.00973702	+	0.999953i	$0.503099\pi$
$882$		0			0
$883$	10.1372	−	10.1372i	0.341144	−	0.341144i	−0.515653	−	0.856797i	$-0.672451\pi$
$883$	10.1372	−	10.1372i	0.341144	−	0.341144i	0.856797	+	0.515653i	$0.172451\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	56.0202i		−	1.88097i	−0.339831	−	0.940487i	$-0.610370\pi$
$887$		−	56.0202i		−	1.88097i	0.339831	−	0.940487i	$-0.389630\pi$
$888$		0			0
$889$			18.2762i			0.612962i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−2.24380	+	2.24380i	−0.0750860	+	0.0750860i
$894$		0			0
$895$	−7.51428			−0.251175
$896$		0			0
$897$		0			0
$898$		0			0
$899$	45.7483	−	45.7483i	1.52579	−	1.52579i
$900$		0			0
$901$	−32.0901	−	32.0901i	−1.06908	−	1.06908i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	12.3732i		−	0.411299i
$906$		0			0
$907$	−31.3292	−	31.3292i	−1.04027	−	1.04027i	−0.999154	−	0.0411133i	$-0.986910\pi$
$907$	−31.3292	−	31.3292i	−1.04027	−	1.04027i	−0.0411133	−	0.999154i	$-0.513090\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−24.7701			−0.820670			−0.410335	−	0.911935i	$-0.634588\pi$
$911$	−24.7701			−0.820670			−0.410335	+	0.911935i	$0.634588\pi$
$912$		0			0
$913$	20.0983			0.665157
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−18.5264	−	18.5264i	−0.611796	−	0.611796i
$918$		0			0
$919$			7.82495i			0.258121i	0.991637	+	0.129061i	$0.0411962\pi$
$919$			7.82495i			0.258121i	−0.991637	+	0.129061i	$0.958804\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	15.9694	+	15.9694i	0.525640	+	0.525640i
$924$		0			0
$925$	−1.08674	+	1.08674i	−0.0357318	+	0.0357318i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−54.8650			−1.80006			−0.900031	−	0.435826i	$-0.856456\pi$
$929$	−54.8650			−1.80006			−0.900031	+	0.435826i	$0.856456\pi$
$930$		0			0
$931$	−0.522826	+	0.522826i	−0.0171349	+	0.0171349i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	24.8474i		−	0.812598i
$936$		0			0
$937$		−	19.3306i		−	0.631504i	−0.948842	−	0.315752i	$-0.897743\pi$
$937$		−	19.3306i		−	0.631504i	0.948842	−	0.315752i	$-0.102257\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−11.4226	+	11.4226i	−0.372365	+	0.372365i	−0.868338	−	0.495973i	$-0.834812\pi$
$941$	−11.4226	+	11.4226i	−0.372365	+	0.372365i	0.495973	+	0.868338i	$0.334812\pi$
$942$		0			0
$943$	−23.6186			−0.769129
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−26.8761	+	26.8761i	−0.873357	+	0.873357i	−0.992837	−	0.119480i	$-0.961877\pi$
$947$	−26.8761	+	26.8761i	−0.873357	+	0.873357i	0.119480	+	0.992837i	$0.461877\pi$
$948$		0			0
$949$	66.6364	+	66.6364i	2.16311	+	2.16311i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			36.6661i			1.18773i	0.804564	+	0.593866i	$0.202399\pi$
$953$			36.6661i			1.18773i	−0.804564	+	0.593866i	$0.797601\pi$
$954$		0			0
$955$	−2.85521	−	2.85521i	−0.0923923	−	0.0923923i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	39.7852			1.28473
$960$		0			0
$961$	26.4917			0.854570
$962$		0			0
$963$		0			0
$964$		0			0
$965$	0.309084	+	0.309084i	0.00994976	+	0.00994976i
$966$		0			0
$967$		−	43.0462i		−	1.38427i	−0.721767	−	0.692136i	$-0.756670\pi$
$967$		−	43.0462i		−	1.38427i	0.721767	−	0.692136i	$-0.243330\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−3.26831	−	3.26831i	−0.104885	−	0.104885i	0.652717	−	0.757602i	$-0.273629\pi$
$971$	−3.26831	−	3.26831i	−0.104885	−	0.104885i	−0.757602	+	0.652717i	$0.773629\pi$
$972$		0			0
$973$	0.809324	−	0.809324i	0.0259457	−	0.0259457i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.62437			−0.0519681			−0.0259840	−	0.999662i	$-0.508272\pi$
$977$	−1.62437			−0.0519681			−0.0259840	+	0.999662i	$0.508272\pi$
$978$		0			0
$979$	25.3976	−	25.3976i	0.811712	−	0.811712i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			45.7756i			1.46001i	0.683439	+	0.730007i	$0.260483\pi$
$983$			45.7756i			1.46001i	−0.683439	+	0.730007i	$0.739517\pi$
$984$		0			0
$985$		−	16.5231i		−	0.526471i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	23.2641	−	23.2641i	0.739754	−	0.739754i
$990$		0			0
$991$	−11.0608			−0.351357			−0.175679	−	0.984448i	$-0.556212\pi$
$991$	−11.0608			−0.351357			−0.175679	+	0.984448i	$0.556212\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	11.1307	−	11.1307i	0.352868	−	0.352868i
$996$		0			0
$997$	21.9716	+	21.9716i	0.695848	+	0.695848i	0.963512	−	0.267664i	$-0.0862518\pi$
$997$	21.9716	+	21.9716i	0.695848	+	0.695848i	−0.267664	+	0.963512i	$0.586252\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.t.d.2161.9		20
3.2	odd	2		960.2.s.c.241.9		20
4.3	odd	2		720.2.t.d.181.5		20
12.11	even	2		240.2.s.c.181.6	yes	20
16.3	odd	4		720.2.t.d.541.5		20
16.13	even	4	inner	2880.2.t.d.721.7		20
24.5	odd	2		1920.2.s.f.481.4		20
24.11	even	2		1920.2.s.e.481.7		20
48.5	odd	4		1920.2.s.f.1441.2		20
48.11	even	4		1920.2.s.e.1441.9		20
48.29	odd	4		960.2.s.c.721.7		20
48.35	even	4		240.2.s.c.61.6	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.2.s.c.61.6	✓	20	48.35	even	4
240.2.s.c.181.6	yes	20	12.11	even	2
720.2.t.d.181.5		20	4.3	odd	2
720.2.t.d.541.5		20	16.3	odd	4
960.2.s.c.241.9		20	3.2	odd	2
960.2.s.c.721.7		20	48.29	odd	4
1920.2.s.e.481.7		20	24.11	even	2
1920.2.s.e.1441.9		20	48.11	even	4
1920.2.s.f.481.4		20	24.5	odd	2
1920.2.s.f.1441.2		20	48.5	odd	4
2880.2.t.d.721.7		20	16.13	even	4	inner
2880.2.t.d.2161.9		20	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2880.t (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 + 0.707107i) q^{5} +2.76462i q^{7} +O(q^{10})\)	\(q+(0.707107 + 0.707107i) q^{5} +2.76462i q^{7} +(-3.51009 - 3.51009i) q^{11} +(-4.55960 + 4.55960i) q^{13} +5.00550 q^{17} +(0.812949 - 0.812949i) q^{19} -7.48205i q^{23} +1.00000i q^{25} +(-6.03354 + 6.03354i) q^{29} -7.58233 q^{31} +(-1.95488 + 1.95488i) q^{35} +(1.08674 + 1.08674i) q^{37} -3.15671i q^{41} +(3.10932 + 3.10932i) q^{43} -2.76008 q^{47} -0.643123 q^{49} +(-6.41096 - 6.41096i) q^{53} -4.96402i q^{55} +(-5.13756 - 5.13756i) q^{59} +(-2.49234 + 2.49234i) q^{61} -6.44825 q^{65} +(3.14625 - 3.14625i) q^{67} -3.50237i q^{71} -14.6145i q^{73} +(9.70408 - 9.70408i) q^{77} -8.95325 q^{79} +(-2.86293 + 2.86293i) q^{83} +(3.53942 + 3.53942i) q^{85} +7.23560i q^{89} +(-12.6056 - 12.6056i) q^{91} +1.14968 q^{95} -8.24056 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q+O(q^{10}) \) 20 * q	\( 20 q + 8 q^{11} + 24 q^{17} + 4 q^{19} - 16 q^{29} + 16 q^{37} + 8 q^{43} - 52 q^{49} + 16 q^{53} - 16 q^{59} - 4 q^{61} + 8 q^{67} + 40 q^{77} - 56 q^{79} - 48 q^{83} + 4 q^{85} + 8 q^{91} + 56 q^{97}+O(q^{100}) \) 20 * q + 8 * q^11 + 24 * q^17 + 4 * q^19 - 16 * q^29 + 16 * q^37 + 8 * q^43 - 52 * q^49 + 16 * q^53 - 16 * q^59 - 4 * q^61 + 8 * q^67 + 40 * q^77 - 56 * q^79 - 48 * q^83 + 4 * q^85 + 8 * q^91 + 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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