LMFDB - Embedded newform 2880.2.t.d.2161.6

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			2161.6
Root			$1.32147 + 0.503713i$ of defining polynomial
Character	$\chi$	$=$	2880.2161
Dual form			2880.2.t.d.721.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 + 0.707107i) q^{5} -2.69529i q^{7} +O(q^{10})$	$q+(0.707107 + 0.707107i) q^{5} -2.69529i q^{7} +(2.72735 + 2.72735i) q^{11} +(1.82449 - 1.82449i) q^{13} +7.33517 q^{17} +(-3.62540 + 3.62540i) q^{19} -8.95345i q^{23} +1.00000i q^{25} +(-2.84302 + 2.84302i) q^{29} +3.37977 q^{31} +(1.90586 - 1.90586i) q^{35} +(-0.190364 - 0.190364i) q^{37} +7.67786i q^{41} +(-7.98115 - 7.98115i) q^{43} -1.31537 q^{47} -0.264584 q^{49} +(6.71014 + 6.71014i) q^{53} +3.85706i q^{55} +(1.01464 + 1.01464i) q^{59} +(2.38996 - 2.38996i) q^{61} +2.58022 q^{65} +(7.22173 - 7.22173i) q^{67} -2.28859i q^{71} -1.31098i q^{73} +(7.35101 - 7.35101i) q^{77} +2.59319 q^{79} +(-5.36584 + 5.36584i) q^{83} +(5.18675 + 5.18675i) q^{85} -14.8944i q^{89} +(-4.91753 - 4.91753i) q^{91} -5.12708 q^{95} +0.694695 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$		−	2.69529i		−	1.01872i	−0.860552	−	0.509362i	$-0.829882\pi$
$7$		−	2.69529i		−	1.01872i	0.860552	−	0.509362i	$-0.170118\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.72735	+	2.72735i	0.822328	+	0.822328i	0.986442	−	0.164113i	$-0.0524762\pi$
$11$	2.72735	+	2.72735i	0.822328	+	0.822328i	−0.164113	+	0.986442i	$0.552476\pi$
$12$		0			0
$13$	1.82449	−	1.82449i	0.506023	−	0.506023i	−0.407281	−	0.913303i	$-0.633523\pi$
$13$	1.82449	−	1.82449i	0.506023	−	0.506023i	0.913303	+	0.407281i	$0.133523\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	7.33517			1.77904			0.889520	−	0.456897i	$-0.151039\pi$
$17$	7.33517			1.77904			0.889520	+	0.456897i	$0.151039\pi$
$18$		0			0
$19$	−3.62540	+	3.62540i	−0.831723	+	0.831723i	−0.987752	−	0.156029i	$-0.950131\pi$
$19$	−3.62540	+	3.62540i	−0.831723	+	0.831723i	0.156029	+	0.987752i	$0.450131\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	8.95345i		−	1.86692i	−0.358678	−	0.933461i	$-0.616772\pi$
$23$		−	8.95345i		−	1.86692i	0.358678	−	0.933461i	$-0.383228\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.84302	+	2.84302i	−0.527935	+	0.527935i	−0.919956	−	0.392021i	$-0.871776\pi$
$29$	−2.84302	+	2.84302i	−0.527935	+	0.527935i	0.392021	+	0.919956i	$0.371776\pi$
$30$		0			0
$31$	3.37977			0.607024			0.303512	−	0.952828i	$-0.401841\pi$
$31$	3.37977			0.607024			0.303512	+	0.952828i	$0.401841\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.90586	−	1.90586i	0.322149	−	0.322149i
$36$		0			0
$37$	−0.190364	−	0.190364i	−0.0312956	−	0.0312956i	0.691286	−	0.722581i	$-0.257045\pi$
$37$	−0.190364	−	0.190364i	−0.0312956	−	0.0312956i	−0.722581	+	0.691286i	$0.757045\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			7.67786i			1.19908i	0.800345	+	0.599540i	$0.204650\pi$
$41$			7.67786i			1.19908i	−0.800345	+	0.599540i	$0.795350\pi$
$42$		0			0
$43$	−7.98115	−	7.98115i	−1.21711	−	1.21711i	−0.968638	−	0.248477i	$-0.920070\pi$
$43$	−7.98115	−	7.98115i	−1.21711	−	1.21711i	−0.248477	−	0.968638i	$-0.579930\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.31537			−0.191866			−0.0959332	−	0.995388i	$-0.530584\pi$
$47$	−1.31537			−0.191866			−0.0959332	+	0.995388i	$0.530584\pi$
$48$		0			0
$49$	−0.264584			−0.0377977
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.71014	+	6.71014i	0.921709	+	0.921709i	0.997150	−	0.0754412i	$-0.0240365\pi$
$53$	6.71014	+	6.71014i	0.921709	+	0.921709i	−0.0754412	+	0.997150i	$0.524037\pi$
$54$		0			0
$55$			3.85706i			0.520086i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.01464	+	1.01464i	0.132094	+	0.132094i	0.770063	−	0.637968i	$-0.220225\pi$
$59$	1.01464	+	1.01464i	0.132094	+	0.132094i	−0.637968	+	0.770063i	$0.720225\pi$
$60$		0			0
$61$	2.38996	−	2.38996i	0.306004	−	0.306004i	−0.537354	−	0.843357i	$-0.680576\pi$
$61$	2.38996	−	2.38996i	0.306004	−	0.306004i	0.843357	+	0.537354i	$0.180576\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.58022			0.320037
$66$		0			0
$67$	7.22173	−	7.22173i	0.882275	−	0.882275i	−0.111490	−	0.993766i	$-0.535562\pi$
$67$	7.22173	−	7.22173i	0.882275	−	0.882275i	0.993766	+	0.111490i	$0.0355624\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	2.28859i		−	0.271606i	−0.990736	−	0.135803i	$-0.956639\pi$
$71$		−	2.28859i		−	0.271606i	0.990736	−	0.135803i	$-0.0433614\pi$
$72$		0			0
$73$		−	1.31098i		−	0.153439i	−0.997053	−	0.0767196i	$-0.975555\pi$
$73$		−	1.31098i		−	0.153439i	0.997053	−	0.0767196i	$-0.0244446\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	7.35101	−	7.35101i	0.837725	−	0.837725i
$78$		0			0
$79$	2.59319			0.291757			0.145879	−	0.989303i	$-0.453399\pi$
$79$	2.59319			0.291757			0.145879	+	0.989303i	$0.453399\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5.36584	+	5.36584i	−0.588977	+	0.588977i	−0.937354	−	0.348377i	$-0.886733\pi$
$83$	−5.36584	+	5.36584i	−0.588977	+	0.588977i	0.348377	+	0.937354i	$0.386733\pi$
$84$		0			0
$85$	5.18675	+	5.18675i	0.562582	+	0.562582i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	14.8944i		−	1.57880i	−0.613877	−	0.789402i	$-0.710391\pi$
$89$		−	14.8944i		−	1.57880i	0.613877	−	0.789402i	$-0.289609\pi$
$90$		0			0
$91$	−4.91753	−	4.91753i	−0.515497	−	0.515497i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−5.12708			−0.526028
$96$		0			0
$97$	0.694695			0.0705356			0.0352678	−	0.999378i	$-0.488772\pi$
$97$	0.694695			0.0705356			0.0352678	+	0.999378i	$0.488772\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.88955	+	6.88955i	0.685536	+	0.685536i	0.961242	−	0.275706i	$-0.0889115\pi$
$101$	6.88955	+	6.88955i	0.685536	+	0.685536i	−0.275706	+	0.961242i	$0.588912\pi$
$102$		0			0
$103$		−	4.39299i		−	0.432854i	−0.976299	−	0.216427i	$-0.930560\pi$
$103$		−	4.39299i		−	0.432854i	0.976299	−	0.216427i	$-0.0694403\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.09279	+	7.09279i	0.685686	+	0.685686i	0.961275	−	0.275590i	$-0.0888730\pi$
$107$	7.09279	+	7.09279i	0.685686	+	0.685686i	−0.275590	+	0.961275i	$0.588873\pi$
$108$		0			0
$109$	7.83035	−	7.83035i	0.750012	−	0.750012i	−0.224469	−	0.974481i	$-0.572065\pi$
$109$	7.83035	−	7.83035i	0.750012	−	0.750012i	0.974481	+	0.224469i	$0.0720649\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	4.63828			0.436332			0.218166	−	0.975912i	$-0.429993\pi$
$113$	4.63828			0.436332			0.218166	+	0.975912i	$0.429993\pi$
$114$		0			0
$115$	6.33104	−	6.33104i	0.590373	−	0.590373i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	19.7704i		−	1.81235i
$120$		0			0
$121$			3.87693i			0.352448i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.707107	+	0.707107i	−0.0632456	+	0.0632456i
$126$		0			0
$127$	8.57901			0.761264			0.380632	−	0.924727i	$-0.375706\pi$
$127$	8.57901			0.761264			0.380632	+	0.924727i	$0.375706\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.33779	+	5.33779i	−0.466365	+	0.466365i	−0.900735	−	0.434370i	$-0.856971\pi$
$131$	−5.33779	+	5.33779i	−0.466365	+	0.466365i	0.434370	+	0.900735i	$0.356971\pi$
$132$		0			0
$133$	9.77149	+	9.77149i	0.847296	+	0.847296i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	12.2694i		−	1.04825i	−0.851642	−	0.524124i	$-0.824393\pi$
$137$		−	12.2694i		−	1.04825i	0.851642	−	0.524124i	$-0.175607\pi$
$138$		0			0
$139$	12.7593	+	12.7593i	1.08223	+	1.08223i	0.996301	+	0.0859285i	$0.0273857\pi$
$139$	12.7593	+	12.7593i	1.08223	+	1.08223i	0.0859285	+	0.996301i	$0.472614\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	9.95206			0.832233
$144$		0			0
$145$	−4.02063			−0.333895
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−6.47238	−	6.47238i	−0.530238	−	0.530238i	0.390405	−	0.920643i	$-0.372335\pi$
$149$	−6.47238	−	6.47238i	−0.530238	−	0.530238i	−0.920643	+	0.390405i	$0.872335\pi$
$150$		0			0
$151$		−	14.7782i		−	1.20263i	−0.799012	−	0.601316i	$-0.794643\pi$
$151$		−	14.7782i		−	1.20263i	0.799012	−	0.601316i	$-0.205357\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.38986	+	2.38986i	0.191958	+	0.191958i
$156$		0			0
$157$	14.0590	−	14.0590i	1.12203	−	1.12203i	0.130592	−	0.991436i	$-0.458312\pi$
$157$	14.0590	−	14.0590i	1.12203	−	1.12203i	0.991436	−	0.130592i	$-0.0416878\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−24.1321			−1.90188
$162$		0			0
$163$	−2.09819	+	2.09819i	−0.164343	+	0.164343i	−0.784487	−	0.620145i	$-0.787074\pi$
$163$	−2.09819	+	2.09819i	−0.164343	+	0.164343i	0.620145	+	0.784487i	$0.287074\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			15.7672i			1.22010i	0.792361	+	0.610052i	$0.208852\pi$
$167$			15.7672i			1.22010i	−0.792361	+	0.610052i	$0.791148\pi$
$168$		0			0
$169$			6.34247i			0.487882i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−8.73572	+	8.73572i	−0.664164	+	0.664164i	−0.956359	−	0.292195i	$-0.905615\pi$
$173$	−8.73572	+	8.73572i	−0.664164	+	0.664164i	0.292195	+	0.956359i	$0.405615\pi$
$174$		0			0
$175$	2.69529			0.203745
$176$		0			0
$177$		0			0
$178$		0			0
$179$	5.72942	−	5.72942i	0.428237	−	0.428237i	−0.459790	−	0.888027i	$-0.652075\pi$
$179$	5.72942	−	5.72942i	0.428237	−	0.428237i	0.888027	+	0.459790i	$0.152075\pi$
$180$		0			0
$181$	−1.48317	−	1.48317i	−0.110243	−	0.110243i	0.649833	−	0.760077i	$-0.274839\pi$
$181$	−1.48317	−	1.48317i	−0.110243	−	0.110243i	−0.760077	+	0.649833i	$0.774839\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	0.269215i		−	0.0197931i
$186$		0			0
$187$	20.0056	+	20.0056i	1.46295	+	1.46295i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.00884			0.0729970			0.0364985	−	0.999334i	$-0.488380\pi$
$191$	1.00884			0.0729970			0.0364985	+	0.999334i	$0.488380\pi$
$192$		0			0
$193$	−20.0273			−1.44160			−0.720798	−	0.693145i	$-0.756225\pi$
$193$	−20.0273			−1.44160			−0.720798	+	0.693145i	$0.756225\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	12.7615	+	12.7615i	0.909217	+	0.909217i	0.996209	−	0.0869919i	$-0.0277254\pi$
$197$	12.7615	+	12.7615i	0.909217	+	0.909217i	−0.0869919	+	0.996209i	$0.527725\pi$
$198$		0			0
$199$			11.6629i			0.826764i	0.910558	+	0.413382i	$0.135653\pi$
$199$			11.6629i			0.826764i	−0.910558	+	0.413382i	$0.864347\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	7.66275	+	7.66275i	0.537820	+	0.537820i
$204$		0			0
$205$	−5.42907	+	5.42907i	−0.379183	+	0.379183i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−19.7755			−1.36790
$210$		0			0
$211$	0.419270	−	0.419270i	0.0288637	−	0.0288637i	−0.692528	−	0.721391i	$-0.743503\pi$
$211$	0.419270	−	0.419270i	0.0288637	−	0.0288637i	0.721391	+	0.692528i	$0.243503\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	11.2871i		−	0.769771i
$216$		0			0
$217$		−	9.10945i		−	0.618389i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	13.3829	−	13.3829i	0.900234	−	0.900234i
$222$		0			0
$223$	20.3746			1.36438			0.682191	−	0.731174i	$-0.261027\pi$
$223$	20.3746			1.36438			0.682191	+	0.731174i	$0.261027\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2.87480	−	2.87480i	0.190807	−	0.190807i	−0.605238	−	0.796045i	$-0.706922\pi$
$227$	2.87480	−	2.87480i	0.190807	−	0.190807i	0.796045	+	0.605238i	$0.206922\pi$
$228$		0			0
$229$	−6.78564	−	6.78564i	−0.448408	−	0.448408i	0.446417	−	0.894825i	$-0.352700\pi$
$229$	−6.78564	−	6.78564i	−0.448408	−	0.448408i	−0.894825	+	0.446417i	$0.852700\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			27.8637i			1.82541i	0.408619	+	0.912705i	$0.366010\pi$
$233$			27.8637i			1.82541i	−0.408619	+	0.912705i	$0.633990\pi$
$234$		0			0
$235$	−0.930107	−	0.930107i	−0.0606735	−	0.0606735i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−19.7188			−1.27550			−0.637750	−	0.770243i	$-0.720135\pi$
$239$	−19.7188			−1.27550			−0.637750	+	0.770243i	$0.720135\pi$
$240$		0			0
$241$	7.74263			0.498747			0.249373	−	0.968407i	$-0.419775\pi$
$241$	7.74263			0.498747			0.249373	+	0.968407i	$0.419775\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.187089	−	0.187089i	−0.0119527	−	0.0119527i
$246$		0			0
$247$			13.2290i			0.841741i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1.46277	+	1.46277i	0.0923293	+	0.0923293i	0.751763	−	0.659434i	$-0.229204\pi$
$251$	1.46277	+	1.46277i	0.0923293	+	0.0923293i	−0.659434	+	0.751763i	$0.729204\pi$
$252$		0			0
$253$	24.4192	−	24.4192i	1.53522	−	1.53522i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−20.6684			−1.28926			−0.644630	−	0.764495i	$-0.722988\pi$
$257$	−20.6684			−1.28926			−0.644630	+	0.764495i	$0.722988\pi$
$258$		0			0
$259$	−0.513085	+	0.513085i	−0.0318815	+	0.0318815i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	26.3239i		−	1.62320i	−0.584215	−	0.811599i	$-0.698597\pi$
$263$		−	26.3239i		−	1.62320i	0.584215	−	0.811599i	$-0.301403\pi$
$264$		0			0
$265$			9.48958i			0.582940i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	17.8129	−	17.8129i	1.08607	−	1.08607i	0.0901396	−	0.995929i	$-0.471269\pi$
$269$	17.8129	−	17.8129i	1.08607	−	1.08607i	0.995929	−	0.0901396i	$-0.0287313\pi$
$270$		0			0
$271$	7.19848			0.437277			0.218638	−	0.975806i	$-0.429838\pi$
$271$	7.19848			0.437277			0.218638	+	0.975806i	$0.429838\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.72735	+	2.72735i	−0.164466	+	0.164466i
$276$		0			0
$277$	10.4666	+	10.4666i	0.628879	+	0.628879i	0.947786	−	0.318907i	$-0.103316\pi$
$277$	10.4666	+	10.4666i	0.628879	+	0.628879i	−0.318907	+	0.947786i	$0.603316\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	17.0567i		−	1.01752i	−0.860909	−	0.508759i	$-0.830104\pi$
$281$		−	17.0567i		−	1.01752i	0.860909	−	0.508759i	$-0.169896\pi$
$282$		0			0
$283$	−2.70727	−	2.70727i	−0.160931	−	0.160931i	0.622048	−	0.782979i	$-0.286301\pi$
$283$	−2.70727	−	2.70727i	−0.160931	−	0.160931i	−0.782979	+	0.622048i	$0.786301\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	20.6941			1.22153
$288$		0			0
$289$	36.8047			2.16498
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−16.6594	−	16.6594i	−0.973255	−	0.973255i	0.0263968	−	0.999652i	$-0.491597\pi$
$293$	−16.6594	−	16.6594i	−0.973255	−	0.973255i	−0.999652	+	0.0263968i	$0.991597\pi$
$294$		0			0
$295$			1.43491i			0.0835439i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−16.3355	−	16.3355i	−0.944705	−	0.944705i
$300$		0			0
$301$	−21.5115	+	21.5115i	−1.23990	+	1.23990i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	3.37992			0.193534
$306$		0			0
$307$	−18.1991	+	18.1991i	−1.03868	+	1.03868i	−0.0394558	+	0.999221i	$0.512562\pi$
$307$	−18.1991	+	18.1991i	−1.03868	+	1.03868i	−0.999221	+	0.0394558i	$0.987438\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			15.3933i			0.872874i	0.899735	+	0.436437i	$0.143760\pi$
$311$			15.3933i			0.872874i	−0.899735	+	0.436437i	$0.856240\pi$
$312$		0			0
$313$		−	23.7790i		−	1.34407i	−0.740520	−	0.672034i	$-0.765421\pi$
$313$		−	23.7790i		−	1.34407i	0.740520	−	0.672034i	$-0.234579\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	9.29662	−	9.29662i	0.522150	−	0.522150i	−0.396070	−	0.918220i	$-0.629626\pi$
$317$	9.29662	−	9.29662i	0.522150	−	0.522150i	0.918220	+	0.396070i	$0.129626\pi$
$318$		0			0
$319$	−15.5078			−0.868272
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−26.5929	+	26.5929i	−1.47967	+	1.47967i
$324$		0			0
$325$	1.82449	+	1.82449i	0.101205	+	0.101205i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			3.54530i			0.195459i
$330$		0			0
$331$	15.0564	+	15.0564i	0.827572	+	0.827572i	0.987180	−	0.159608i	$-0.0510231\pi$
$331$	15.0564	+	15.0564i	0.827572	+	0.827572i	−0.159608	+	0.987180i	$0.551023\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	10.2131			0.558000
$336$		0			0
$337$	−11.7116			−0.637969			−0.318985	−	0.947760i	$-0.603342\pi$
$337$	−11.7116			−0.637969			−0.318985	+	0.947760i	$0.603342\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	9.21782	+	9.21782i	0.499173	+	0.499173i
$342$		0			0
$343$		−	18.1539i		−	0.980218i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	12.5589	+	12.5589i	0.674197	+	0.674197i	0.958681	−	0.284484i	$-0.0918223\pi$
$347$	12.5589	+	12.5589i	0.674197	+	0.674197i	−0.284484	+	0.958681i	$0.591822\pi$
$348$		0			0
$349$	−12.3114	+	12.3114i	−0.659014	+	0.659014i	−0.955147	−	0.296133i	$-0.904303\pi$
$349$	−12.3114	+	12.3114i	−0.659014	+	0.659014i	0.296133	+	0.955147i	$0.404303\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	5.99485			0.319073			0.159537	−	0.987192i	$-0.449000\pi$
$353$	5.99485			0.319073			0.159537	+	0.987192i	$0.449000\pi$
$354$		0			0
$355$	1.61828	−	1.61828i	0.0858894	−	0.0858894i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			4.95429i			0.261477i	0.991417	+	0.130739i	$0.0417349\pi$
$359$			4.95429i			0.261477i	−0.991417	+	0.130739i	$0.958265\pi$
$360$		0			0
$361$		−	7.28700i		−	0.383526i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	0.927006	−	0.927006i	0.0485217	−	0.0485217i
$366$		0			0
$367$	−26.2297			−1.36918			−0.684591	−	0.728928i	$-0.740019\pi$
$367$	−26.2297			−1.36918			−0.684591	+	0.728928i	$0.740019\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	18.0858	−	18.0858i	0.938967	−	0.938967i
$372$		0			0
$373$	−8.58218	−	8.58218i	−0.444368	−	0.444368i	0.449109	−	0.893477i	$-0.351742\pi$
$373$	−8.58218	−	8.58218i	−0.444368	−	0.444368i	−0.893477	+	0.449109i	$0.851742\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			10.3741i			0.534294i
$378$		0			0
$379$	−4.04145	−	4.04145i	−0.207595	−	0.207595i	0.595649	−	0.803245i	$-0.296895\pi$
$379$	−4.04145	−	4.04145i	−0.207595	−	0.207595i	−0.803245	+	0.595649i	$0.796895\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.97475			−0.100905			−0.0504526	−	0.998726i	$-0.516066\pi$
$383$	−1.97475			−0.100905			−0.0504526	+	0.998726i	$0.516066\pi$
$384$		0			0
$385$	10.3959			0.529824
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−27.0045	−	27.0045i	−1.36918	−	1.36918i	−0.861606	−	0.507579i	$-0.830541\pi$
$389$	−27.0045	−	27.0045i	−1.36918	−	1.36918i	−0.507579	−	0.861606i	$-0.669459\pi$
$390$		0			0
$391$		−	65.6750i		−	3.32133i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.83367	+	1.83367i	0.0922617	+	0.0922617i
$396$		0			0
$397$	−13.2948	+	13.2948i	−0.667248	+	0.667248i	−0.957078	−	0.289830i	$-0.906401\pi$
$397$	−13.2948	+	13.2948i	−0.667248	+	0.667248i	0.289830	+	0.957078i	$0.406401\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3.97892			−0.198698			−0.0993490	−	0.995053i	$-0.531676\pi$
$401$	−3.97892			−0.198698			−0.0993490	+	0.995053i	$0.531676\pi$
$402$		0			0
$403$	6.16635	−	6.16635i	0.307168	−	0.307168i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	1.03838i		−	0.0514705i
$408$		0			0
$409$			23.6091i			1.16740i	0.811971	+	0.583698i	$0.198395\pi$
$409$			23.6091i			1.16740i	−0.811971	+	0.583698i	$0.801605\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.73474	−	2.73474i	0.134568	−	0.134568i
$414$		0			0
$415$	−7.58844			−0.372502
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−13.5761	+	13.5761i	−0.663235	+	0.663235i	−0.956141	−	0.292906i	$-0.905378\pi$
$419$	−13.5761	+	13.5761i	−0.663235	+	0.663235i	0.292906	+	0.956141i	$0.405378\pi$
$420$		0			0
$421$	9.30166	+	9.30166i	0.453335	+	0.453335i	0.896460	−	0.443125i	$-0.146130\pi$
$421$	9.30166	+	9.30166i	0.453335	+	0.453335i	−0.443125	+	0.896460i	$0.646130\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			7.33517i			0.355808i
$426$		0			0
$427$	−6.44164	−	6.44164i	−0.311733	−	0.311733i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	6.40406			0.308473			0.154236	−	0.988034i	$-0.450708\pi$
$431$	6.40406			0.308473			0.154236	+	0.988034i	$0.450708\pi$
$432$		0			0
$433$	−33.9234			−1.63025			−0.815127	−	0.579282i	$-0.803333\pi$
$433$	−33.9234			−1.63025			−0.815127	+	0.579282i	$0.803333\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	32.4598	+	32.4598i	1.55276	+	1.55276i
$438$		0			0
$439$			3.97789i			0.189855i	0.995484	+	0.0949273i	$0.0302619\pi$
$439$			3.97789i			0.189855i	−0.995484	+	0.0949273i	$0.969738\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	14.5607	+	14.5607i	0.691802	+	0.691802i	0.962628	−	0.270826i	$-0.0872969\pi$
$443$	14.5607	+	14.5607i	0.691802	+	0.691802i	−0.270826	+	0.962628i	$0.587297\pi$
$444$		0			0
$445$	10.5319	−	10.5319i	0.499261	−	0.499261i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−11.3678			−0.536481			−0.268240	−	0.963352i	$-0.586442\pi$
$449$	−11.3678			−0.536481			−0.268240	+	0.963352i	$0.586442\pi$
$450$		0			0
$451$	−20.9402	+	20.9402i	−0.986038	+	0.986038i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	6.95444i		−	0.326029i
$456$		0			0
$457$		−	39.0892i		−	1.82851i	−0.405135	−	0.914257i	$-0.632775\pi$
$457$		−	39.0892i		−	1.82851i	0.405135	−	0.914257i	$-0.367225\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	6.05492	−	6.05492i	0.282006	−	0.282006i	−0.551903	−	0.833908i	$-0.686098\pi$
$461$	6.05492	−	6.05492i	0.282006	−	0.282006i	0.833908	+	0.551903i	$0.186098\pi$
$462$		0			0
$463$	−36.1375			−1.67945			−0.839725	−	0.543012i	$-0.817284\pi$
$463$	−36.1375			−1.67945			−0.839725	+	0.543012i	$0.817284\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−13.0890	+	13.0890i	−0.605689	+	0.605689i	−0.941816	−	0.336128i	$-0.890883\pi$
$467$	−13.0890	+	13.0890i	−0.605689	+	0.605689i	0.336128	+	0.941816i	$0.390883\pi$
$468$		0			0
$469$	−19.4647	−	19.4647i	−0.898795	−	0.898795i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	43.5349i		−	2.00174i
$474$		0			0
$475$	−3.62540	−	3.62540i	−0.166345	−	0.166345i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−3.78947			−0.173145			−0.0865726	−	0.996246i	$-0.527591\pi$
$479$	−3.78947			−0.173145			−0.0865726	+	0.996246i	$0.527591\pi$
$480$		0			0
$481$	−0.694633			−0.0316725
$482$		0			0
$483$		0			0
$484$		0			0
$485$	0.491223	+	0.491223i	0.0223053	+	0.0223053i
$486$		0			0
$487$		−	3.39864i		−	0.154007i	−0.997031	−	0.0770036i	$-0.975465\pi$
$487$		−	3.39864i		−	0.154007i	0.997031	−	0.0770036i	$-0.0245353\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−8.70898	−	8.70898i	−0.393031	−	0.393031i	0.482735	−	0.875766i	$-0.339643\pi$
$491$	−8.70898	−	8.70898i	−0.393031	−	0.393031i	−0.875766	+	0.482735i	$0.839643\pi$
$492$		0			0
$493$	−20.8540	+	20.8540i	−0.939217	+	0.939217i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6.16842			−0.276691
$498$		0			0
$499$	15.9542	−	15.9542i	0.714210	−	0.714210i	−0.253203	−	0.967413i	$-0.581484\pi$
$499$	15.9542	−	15.9542i	0.714210	−	0.714210i	0.967413	+	0.253203i	$0.0814841\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			18.4604i			0.823107i	0.911386	+	0.411553i	$0.135014\pi$
$503$			18.4604i			0.823107i	−0.911386	+	0.411553i	$0.864986\pi$
$504$		0			0
$505$			9.74330i			0.433571i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−8.47641	+	8.47641i	−0.375710	+	0.375710i	−0.869552	−	0.493841i	$-0.835592\pi$
$509$	−8.47641	+	8.47641i	−0.375710	+	0.375710i	0.493841	+	0.869552i	$0.335592\pi$
$510$		0			0
$511$	−3.53348			−0.156312
$512$		0			0
$513$		0			0
$514$		0			0
$515$	3.10631	−	3.10631i	0.136880	−	0.136880i
$516$		0			0
$517$	−3.58748	−	3.58748i	−0.157777	−	0.157777i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			22.3693i			0.980017i	0.871718	+	0.490009i	$0.163006\pi$
$521$			22.3693i			0.980017i	−0.871718	+	0.490009i	$0.836994\pi$
$522$		0			0
$523$	2.39235	+	2.39235i	0.104610	+	0.104610i	0.757475	−	0.652865i	$-0.226433\pi$
$523$	2.39235	+	2.39235i	0.104610	+	0.104610i	−0.652865	+	0.757475i	$0.726433\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	24.7911			1.07992
$528$		0			0
$529$	−57.1642			−2.48540
$530$		0			0
$531$		0			0
$532$		0			0
$533$	14.0082	+	14.0082i	0.606762	+	0.606762i
$534$		0			0
$535$			10.0307i			0.433666i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−0.721614	−	0.721614i	−0.0310821	−	0.0310821i
$540$		0			0
$541$	−10.4567	+	10.4567i	−0.449570	+	0.449570i	−0.895211	−	0.445642i	$-0.852976\pi$
$541$	−10.4567	+	10.4567i	−0.449570	+	0.449570i	0.445642	+	0.895211i	$0.352976\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	11.0738			0.474349
$546$		0			0
$547$	−27.1404	+	27.1404i	−1.16044	+	1.16044i	−0.176060	+	0.984379i	$0.556335\pi$
$547$	−27.1404	+	27.1404i	−1.16044	+	1.16044i	−0.984379	+	0.176060i	$0.943665\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	20.6141i		−	0.878191i
$552$		0			0
$553$		−	6.98941i		−	0.297220i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1.29047	−	1.29047i	0.0546790	−	0.0546790i	−0.679239	−	0.733918i	$-0.737690\pi$
$557$	1.29047	−	1.29047i	0.0546790	−	0.0546790i	0.733918	+	0.679239i	$0.237690\pi$
$558$		0			0
$559$	−29.1231			−1.23177
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−3.42706	+	3.42706i	−0.144433	+	0.144433i	−0.775626	−	0.631193i	$-0.782566\pi$
$563$	−3.42706	+	3.42706i	−0.144433	+	0.144433i	0.631193	+	0.775626i	$0.282566\pi$
$564$		0			0
$565$	3.27976	+	3.27976i	0.137980	+	0.137980i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			26.4853i			1.11032i	0.831743	+	0.555161i	$0.187344\pi$
$569$			26.4853i			1.11032i	−0.831743	+	0.555161i	$0.812656\pi$
$570$		0			0
$571$	1.42398	+	1.42398i	0.0595918	+	0.0595918i	0.736275	−	0.676683i	$-0.236583\pi$
$571$	1.42398	+	1.42398i	0.0595918	+	0.0595918i	−0.676683	+	0.736275i	$0.736583\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	8.95345			0.373385
$576$		0			0
$577$	37.6681			1.56814			0.784071	−	0.620671i	$-0.213140\pi$
$577$	37.6681			1.56814			0.784071	+	0.620671i	$0.213140\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	14.4625	+	14.4625i	0.600005	+	0.600005i
$582$		0			0
$583$			36.6019i			1.51589i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1.99399	+	1.99399i	0.0823008	+	0.0823008i	0.747059	−	0.664758i	$-0.231465\pi$
$587$	1.99399	+	1.99399i	0.0823008	+	0.0823008i	−0.664758	+	0.747059i	$0.731465\pi$
$588$		0			0
$589$	−12.2530	+	12.2530i	−0.504876	+	0.504876i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−24.0023			−0.985657			−0.492828	−	0.870127i	$-0.664037\pi$
$593$	−24.0023			−0.985657			−0.492828	+	0.870127i	$0.664037\pi$
$594$		0			0
$595$	13.9798	−	13.9798i	0.573115	−	0.573115i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			34.5205i			1.41047i	0.708974	+	0.705234i	$0.249158\pi$
$599$			34.5205i			1.41047i	−0.708974	+	0.705234i	$0.750842\pi$
$600$		0			0
$601$			3.45113i			0.140775i	0.997520	+	0.0703873i	$0.0224235\pi$
$601$			3.45113i			0.140775i	−0.997520	+	0.0703873i	$0.977577\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2.74140	+	2.74140i	−0.111454	+	0.111454i
$606$		0			0
$607$	43.2408			1.75509			0.877545	−	0.479494i	$-0.159180\pi$
$607$	43.2408			1.75509			0.877545	+	0.479494i	$0.159180\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.39988	+	2.39988i	−0.0970887	+	0.0970887i
$612$		0			0
$613$	31.4157	+	31.4157i	1.26887	+	1.26887i	0.946673	+	0.322195i	$0.104421\pi$
$613$	31.4157	+	31.4157i	1.26887	+	1.26887i	0.322195	+	0.946673i	$0.395579\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			26.4920i			1.06653i	0.845949	+	0.533265i	$0.179035\pi$
$617$			26.4920i			1.06653i	−0.845949	+	0.533265i	$0.820965\pi$
$618$		0			0
$619$	−23.9945	−	23.9945i	−0.964422	−	0.964422i	0.0349667	−	0.999388i	$-0.488867\pi$
$619$	−23.9945	−	23.9945i	−0.964422	−	0.964422i	−0.999388	+	0.0349667i	$0.988867\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−40.1447			−1.60836
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1.39635	−	1.39635i	−0.0556761	−	0.0556761i
$630$		0			0
$631$			34.6813i			1.38064i	0.723504	+	0.690320i	$0.242530\pi$
$631$			34.6813i			1.38064i	−0.723504	+	0.690320i	$0.757470\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	6.06628	+	6.06628i	0.240733	+	0.240733i
$636$		0			0
$637$	−0.482731	+	0.482731i	−0.0191265	+	0.0191265i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−25.5477			−1.00908			−0.504538	−	0.863390i	$-0.668337\pi$
$641$	−25.5477			−1.00908			−0.504538	+	0.863390i	$0.668337\pi$
$642$		0			0
$643$	−12.0319	+	12.0319i	−0.474490	+	0.474490i	−0.903364	−	0.428874i	$-0.858910\pi$
$643$	−12.0319	+	12.0319i	−0.474490	+	0.474490i	0.428874	+	0.903364i	$0.358910\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			20.4148i			0.802587i	0.915950	+	0.401293i	$0.131439\pi$
$647$			20.4148i			0.802587i	−0.915950	+	0.401293i	$0.868561\pi$
$648$		0			0
$649$			5.53455i			0.217250i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−15.3217	+	15.3217i	−0.599586	+	0.599586i	−0.940202	−	0.340616i	$-0.889364\pi$
$653$	−15.3217	+	15.3217i	−0.599586	+	0.599586i	0.340616	+	0.940202i	$0.389364\pi$
$654$		0			0
$655$	−7.54877			−0.294955
$656$		0			0
$657$		0			0
$658$		0			0
$659$	3.52074	−	3.52074i	0.137148	−	0.137148i	−0.635200	−	0.772348i	$-0.719082\pi$
$659$	3.52074	−	3.52074i	0.137148	−	0.137148i	0.772348	+	0.635200i	$0.219082\pi$
$660$		0			0
$661$	−23.8945	−	23.8945i	−0.929388	−	0.929388i	0.0682781	−	0.997666i	$-0.478249\pi$
$661$	−23.8945	−	23.8945i	−0.929388	−	0.929388i	−0.997666	+	0.0682781i	$0.978249\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			13.8190i			0.535877i
$666$		0			0
$667$	25.4548	+	25.4548i	0.985613	+	0.985613i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	13.0366			0.503271
$672$		0			0
$673$	−25.4607			−0.981437			−0.490719	−	0.871318i	$-0.663266\pi$
$673$	−25.4607			−0.981437			−0.490719	+	0.871318i	$0.663266\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−21.1318	−	21.1318i	−0.812162	−	0.812162i	0.172796	−	0.984958i	$-0.444720\pi$
$677$	−21.1318	−	21.1318i	−0.812162	−	0.812162i	−0.984958	+	0.172796i	$0.944720\pi$
$678$		0			0
$679$		−	1.87240i		−	0.0718562i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−6.14096	−	6.14096i	−0.234977	−	0.234977i	0.579789	−	0.814767i	$-0.303135\pi$
$683$	−6.14096	−	6.14096i	−0.234977	−	0.234977i	−0.814767	+	0.579789i	$0.803135\pi$
$684$		0			0
$685$	8.67579	−	8.67579i	0.331485	−	0.331485i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	24.4852			0.932811
$690$		0			0
$691$	−1.37557	+	1.37557i	−0.0523291	+	0.0523291i	−0.732787	−	0.680458i	$-0.761781\pi$
$691$	−1.37557	+	1.37557i	−0.0523291	+	0.0523291i	0.680458	+	0.732787i	$0.261781\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			18.0444i			0.684462i
$696$		0			0
$697$			56.3184i			2.13321i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−27.0610	+	27.0610i	−1.02208	+	1.02208i	−0.0223289	+	0.999751i	$0.507108\pi$
$701$	−27.0610	+	27.0610i	−1.02208	+	1.02208i	−0.999751	+	0.0223289i	$0.992892\pi$
$702$		0			0
$703$	1.38029			0.0520585
$704$		0			0
$705$		0			0
$706$		0			0
$707$	18.5693	−	18.5693i	0.698372	−	0.698372i
$708$		0			0
$709$	−32.8479	−	32.8479i	−1.23363	−	1.23363i	−0.962560	−	0.271068i	$-0.912623\pi$
$709$	−32.8479	−	32.8479i	−1.23363	−	1.23363i	−0.271068	−	0.962560i	$-0.587377\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	30.2605i		−	1.13327i
$714$		0			0
$715$	7.03717	+	7.03717i	0.263175	+	0.263175i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−33.7746			−1.25958			−0.629791	−	0.776765i	$-0.716859\pi$
$719$	−33.7746			−1.25958			−0.629791	+	0.776765i	$0.716859\pi$
$720$		0			0
$721$	−11.8404			−0.440958
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.84302	−	2.84302i	−0.105587	−	0.105587i
$726$		0			0
$727$		−	7.54917i		−	0.279983i	−0.990153	−	0.139992i	$-0.955292\pi$
$727$		−	7.54917i		−	0.279983i	0.990153	−	0.139992i	$-0.0447075\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−58.5431	−	58.5431i	−2.16529	−	2.16529i
$732$		0			0
$733$	−13.2524	+	13.2524i	−0.489487	+	0.489487i	−0.908144	−	0.418657i	$-0.862501\pi$
$733$	−13.2524	+	13.2524i	−0.489487	+	0.489487i	0.418657	+	0.908144i	$0.362501\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	39.3925			1.45104
$738$		0			0
$739$	−4.42190	+	4.42190i	−0.162662	+	0.162662i	−0.783745	−	0.621083i	$-0.786693\pi$
$739$	−4.42190	+	4.42190i	−0.162662	+	0.162662i	0.621083	+	0.783745i	$0.286693\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	45.5703i		−	1.67181i	−0.548873	−	0.835906i	$-0.684943\pi$
$743$		−	45.5703i		−	1.67181i	0.548873	−	0.835906i	$-0.315057\pi$
$744$		0			0
$745$		−	9.15332i		−	0.335352i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	19.1171	−	19.1171i	0.698524	−	0.698524i
$750$		0			0
$751$	−10.6007			−0.386824			−0.193412	−	0.981118i	$-0.561955\pi$
$751$	−10.6007			−0.386824			−0.193412	+	0.981118i	$0.561955\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	10.4498	−	10.4498i	0.380305	−	0.380305i
$756$		0			0
$757$	−14.7340	−	14.7340i	−0.535517	−	0.535517i	0.386692	−	0.922209i	$-0.373618\pi$
$757$	−14.7340	−	14.7340i	−0.535517	−	0.535517i	−0.922209	+	0.386692i	$0.873618\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	8.89182i		−	0.322328i	−0.986928	−	0.161164i	$-0.948475\pi$
$761$		−	8.89182i		−	0.322328i	0.986928	−	0.161164i	$-0.0515248\pi$
$762$		0			0
$763$	−21.1051	−	21.1051i	−0.764055	−	0.764055i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	3.70239			0.133686
$768$		0			0
$769$	32.7022			1.17927			0.589636	−	0.807669i	$-0.299271\pi$
$769$	32.7022			1.17927			0.589636	+	0.807669i	$0.299271\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−12.4209	−	12.4209i	−0.446747	−	0.446747i	0.447525	−	0.894272i	$-0.352306\pi$
$773$	−12.4209	−	12.4209i	−0.446747	−	0.446747i	−0.894272	+	0.447525i	$0.852306\pi$
$774$		0			0
$775$			3.37977i			0.121405i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−27.8353	−	27.8353i	−0.997303	−	0.997303i
$780$		0			0
$781$	6.24180	−	6.24180i	0.223349	−	0.223349i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	19.8824			0.709633
$786$		0			0
$787$	22.6663	−	22.6663i	0.807965	−	0.807965i	−0.176361	−	0.984326i	$-0.556433\pi$
$787$	22.6663	−	22.6663i	0.807965	−	0.807965i	0.984326	+	0.176361i	$0.0564325\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	12.5015i		−	0.444502i
$792$		0			0
$793$		−	8.72093i		−	0.309689i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−7.47619	+	7.47619i	−0.264820	+	0.264820i	−0.827009	−	0.562189i	$-0.809959\pi$
$797$	−7.47619	+	7.47619i	−0.264820	+	0.264820i	0.562189	+	0.827009i	$0.309959\pi$
$798$		0			0
$799$	−9.64846			−0.341338
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3.57552	−	3.57552i	0.126177	−	0.126177i
$804$		0			0
$805$	−17.0640	−	17.0640i	−0.601427	−	0.601427i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	9.20349i		−	0.323577i	−0.986825	−	0.161789i	$-0.948274\pi$
$809$		−	9.20349i		−	0.323577i	0.986825	−	0.161789i	$-0.0517263\pi$
$810$		0			0
$811$	−31.0252	−	31.0252i	−1.08944	−	1.08944i	−0.995586	−	0.0938555i	$-0.970081\pi$
$811$	−31.0252	−	31.0252i	−1.08944	−	1.08944i	−0.0938555	−	0.995586i	$-0.529919\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2.96728			−0.103939
$816$		0			0
$817$	57.8697			2.02460
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−5.31973	−	5.31973i	−0.185660	−	0.185660i	0.608157	−	0.793817i	$-0.291909\pi$
$821$	−5.31973	−	5.31973i	−0.185660	−	0.185660i	−0.793817	+	0.608157i	$0.791909\pi$
$822$		0			0
$823$			24.8425i			0.865956i	0.901404	+	0.432978i	$0.142537\pi$
$823$			24.8425i			0.865956i	−0.901404	+	0.432978i	$0.857463\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	11.8871	+	11.8871i	0.413354	+	0.413354i	0.882905	−	0.469551i	$-0.155584\pi$
$827$	11.8871	+	11.8871i	0.413354	+	0.413354i	−0.469551	+	0.882905i	$0.655584\pi$
$828$		0			0
$829$	6.17740	−	6.17740i	0.214550	−	0.214550i	−0.591647	−	0.806197i	$-0.701522\pi$
$829$	6.17740	−	6.17740i	0.214550	−	0.214550i	0.806197	+	0.591647i	$0.201522\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.94077			−0.0672436
$834$		0			0
$835$	−11.1491	+	11.1491i	−0.385831	+	0.385831i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	8.14180i		−	0.281086i	−0.990075	−	0.140543i	$-0.955115\pi$
$839$		−	8.14180i		−	0.281086i	0.990075	−	0.140543i	$-0.0448848\pi$
$840$		0			0
$841$			12.8345i			0.442570i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−4.48480	+	4.48480i	−0.154282	+	0.154282i
$846$		0			0
$847$	10.4494			0.359047
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1.70441	+	1.70441i	−0.0584264	+	0.0584264i
$852$		0			0
$853$	27.9298	+	27.9298i	0.956297	+	0.956297i	0.999084	−	0.0427875i	$-0.0136239\pi$
$853$	27.9298	+	27.9298i	0.956297	+	0.956297i	−0.0427875	+	0.999084i	$0.513624\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			20.7075i			0.707356i	0.935367	+	0.353678i	$0.115069\pi$
$857$			20.7075i			0.707356i	−0.935367	+	0.353678i	$0.884931\pi$
$858$		0			0
$859$	12.8547	+	12.8547i	0.438597	+	0.438597i	0.891539	−	0.452943i	$-0.149626\pi$
$859$	12.8547	+	12.8547i	0.438597	+	0.438597i	−0.452943	+	0.891539i	$0.649626\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	11.9606			0.407143			0.203572	−	0.979060i	$-0.434745\pi$
$863$	11.9606			0.407143			0.203572	+	0.979060i	$0.434745\pi$
$864$		0			0
$865$	−12.3542			−0.420054
$866$		0			0
$867$		0			0
$868$		0			0
$869$	7.07256	+	7.07256i	0.239920	+	0.239920i
$870$		0			0
$871$		−	26.3520i		−	0.892902i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.90586	+	1.90586i	0.0644297	+	0.0644297i
$876$		0			0
$877$	6.15758	−	6.15758i	0.207927	−	0.207927i	−0.595459	−	0.803386i	$-0.703030\pi$
$877$	6.15758	−	6.15758i	0.207927	−	0.207927i	0.803386	+	0.595459i	$0.203030\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	19.5267			0.657871			0.328936	−	0.944352i	$-0.393310\pi$
$881$	19.5267			0.657871			0.328936	+	0.944352i	$0.393310\pi$
$882$		0			0
$883$	−17.6166	+	17.6166i	−0.592845	+	0.592845i	−0.938399	−	0.345554i	$-0.887691\pi$
$883$	−17.6166	+	17.6166i	−0.592845	+	0.592845i	0.345554	+	0.938399i	$0.387691\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	29.2567i		−	0.982343i	−0.871063	−	0.491172i	$-0.836569\pi$
$887$		−	29.2567i		−	0.982343i	0.871063	−	0.491172i	$-0.163431\pi$
$888$		0			0
$889$		−	23.1229i		−	0.775518i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	4.76874	−	4.76874i	0.159580	−	0.159580i
$894$		0			0
$895$	8.10262			0.270841
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−9.60873	+	9.60873i	−0.320469	+	0.320469i
$900$		0			0
$901$	49.2200	+	49.2200i	1.63976	+	1.63976i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	2.09752i		−	0.0697240i
$906$		0			0
$907$	−32.6905	−	32.6905i	−1.08547	−	1.08547i	−0.995988	−	0.0894835i	$-0.971478\pi$
$907$	−32.6905	−	32.6905i	−1.08547	−	1.08547i	−0.0894835	−	0.995988i	$-0.528522\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−44.0834			−1.46055			−0.730275	−	0.683153i	$-0.760608\pi$
$911$	−44.0834			−1.46055			−0.730275	+	0.683153i	$0.760608\pi$
$912$		0			0
$913$	−29.2691			−0.968665
$914$		0			0
$915$		0			0
$916$		0			0
$917$	14.3869	+	14.3869i	0.475097	+	0.475097i
$918$		0			0
$919$			21.4544i			0.707716i	0.935299	+	0.353858i	$0.115130\pi$
$919$			21.4544i			0.707716i	−0.935299	+	0.353858i	$0.884870\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−4.17551	−	4.17551i	−0.137439	−	0.137439i
$924$		0			0
$925$	0.190364	−	0.190364i	0.00625912	−	0.00625912i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−33.6688			−1.10464			−0.552318	−	0.833634i	$-0.686257\pi$
$929$	−33.6688			−1.10464			−0.552318	+	0.833634i	$0.686257\pi$
$930$		0			0
$931$	0.959222	−	0.959222i	0.0314372	−	0.0314372i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			28.2922i			0.925254i
$936$		0			0
$937$			10.5938i			0.346084i	0.984914	+	0.173042i	$0.0553596\pi$
$937$			10.5938i			0.346084i	−0.984914	+	0.173042i	$0.944640\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−7.72584	+	7.72584i	−0.251855	+	0.251855i	−0.821731	−	0.569876i	$-0.806991\pi$
$941$	−7.72584	+	7.72584i	−0.251855	+	0.251855i	0.569876	+	0.821731i	$0.306991\pi$
$942$		0			0
$943$	68.7433			2.23859
$944$		0			0
$945$		0			0
$946$		0			0
$947$	21.4749	−	21.4749i	0.697841	−	0.697841i	−0.266103	−	0.963945i	$-0.585736\pi$
$947$	21.4749	−	21.4749i	0.697841	−	0.697841i	0.963945	+	0.266103i	$0.0857362\pi$
$948$		0			0
$949$	−2.39188	−	2.39188i	−0.0776437	−	0.0776437i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	29.0291i		−	0.940344i	−0.882575	−	0.470172i	$-0.844192\pi$
$953$		−	29.0291i		−	0.940344i	0.882575	−	0.470172i	$-0.155808\pi$
$954$		0			0
$955$	0.713357	+	0.713357i	0.0230837	+	0.0230837i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−33.0696			−1.06787
$960$		0			0
$961$	−19.5772			−0.631522
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−14.1614	−	14.1614i	−0.455873	−	0.455873i
$966$		0			0
$967$			4.50118i			0.144748i	0.997378	+	0.0723741i	$0.0230576\pi$
$967$			4.50118i			0.144748i	−0.997378	+	0.0723741i	$0.976942\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	30.6131	+	30.6131i	0.982420	+	0.982420i	0.999848	−	0.0174280i	$-0.00554777\pi$
$971$	30.6131	+	30.6131i	0.982420	+	0.982420i	−0.0174280	+	0.999848i	$0.505548\pi$
$972$		0			0
$973$	34.3900	−	34.3900i	1.10249	−	1.10249i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	36.6473			1.17245			0.586225	−	0.810148i	$-0.300613\pi$
$977$	36.6473			1.17245			0.586225	+	0.810148i	$0.300613\pi$
$978$		0			0
$979$	40.6223	−	40.6223i	1.29829	−	1.29829i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	16.3748i		−	0.522276i	−0.965301	−	0.261138i	$-0.915902\pi$
$983$		−	16.3748i		−	0.522276i	0.965301	−	0.261138i	$-0.0840978\pi$
$984$		0			0
$985$			18.0474i			0.575039i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−71.4588	+	71.4588i	−2.27226	+	2.27226i
$990$		0			0
$991$	−13.8052			−0.438537			−0.219269	−	0.975665i	$-0.570367\pi$
$991$	−13.8052			−0.438537			−0.219269	+	0.975665i	$0.570367\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−8.24694	+	8.24694i	−0.261446	+	0.261446i
$996$		0			0
$997$	10.9547	+	10.9547i	0.346938	+	0.346938i	0.858968	−	0.512030i	$-0.171106\pi$
$997$	10.9547	+	10.9547i	0.346938	+	0.346938i	−0.512030	+	0.858968i	$0.671106\pi$
$998$		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.2161.6		20
3.2	odd	2		960.2.s.c.241.6		20
4.3	odd	2		720.2.t.d.181.1		20
12.11	even	2		240.2.s.c.181.10	yes	20
16.3	odd	4		720.2.t.d.541.1		20
16.13	even	4	inner	2880.2.t.d.721.10		20
24.5	odd	2		1920.2.s.f.481.1		20
24.11	even	2		1920.2.s.e.481.10		20
48.5	odd	4		1920.2.s.f.1441.5		20
48.11	even	4		1920.2.s.e.1441.6		20
48.29	odd	4		960.2.s.c.721.10		20
48.35	even	4		240.2.s.c.61.10	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.2.s.c.61.10	✓	20	48.35	even	4
240.2.s.c.181.10	yes	20	12.11	even	2
720.2.t.d.181.1		20	4.3	odd	2
720.2.t.d.541.1		20	16.3	odd	4
960.2.s.c.241.6		20	3.2	odd	2
960.2.s.c.721.10		20	48.29	odd	4
1920.2.s.e.481.10		20	24.11	even	2
1920.2.s.e.1441.6		20	48.11	even	4
1920.2.s.f.481.1		20	24.5	odd	2
1920.2.s.f.1441.5		20	48.5	odd	4
2880.2.t.d.721.10		20	16.13	even	4	inner
2880.2.t.d.2161.6		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)

\(f(q)\)	\(=\)	\(q+(0.707107 + 0.707107i) q^{5} -2.69529i q^{7} +O(q^{10})\)	\(q+(0.707107 + 0.707107i) q^{5} -2.69529i q^{7} +(2.72735 + 2.72735i) q^{11} +(1.82449 - 1.82449i) q^{13} +7.33517 q^{17} +(-3.62540 + 3.62540i) q^{19} -8.95345i q^{23} +1.00000i q^{25} +(-2.84302 + 2.84302i) q^{29} +3.37977 q^{31} +(1.90586 - 1.90586i) q^{35} +(-0.190364 - 0.190364i) q^{37} +7.67786i q^{41} +(-7.98115 - 7.98115i) q^{43} -1.31537 q^{47} -0.264584 q^{49} +(6.71014 + 6.71014i) q^{53} +3.85706i q^{55} +(1.01464 + 1.01464i) q^{59} +(2.38996 - 2.38996i) q^{61} +2.58022 q^{65} +(7.22173 - 7.22173i) q^{67} -2.28859i q^{71} -1.31098i q^{73} +(7.35101 - 7.35101i) q^{77} +2.59319 q^{79} +(-5.36584 + 5.36584i) q^{83} +(5.18675 + 5.18675i) q^{85} -14.8944i q^{89} +(-4.91753 - 4.91753i) q^{91} -5.12708 q^{95} +0.694695 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q+O(q^{10}) \) 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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