LMFDB - Embedded newform 2880.2.t.e.721.12

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:	$32$
Relative dimension:	$16$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 720)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			721.12
Character	$\chi$	$=$	2880.721
Dual form			2880.2.t.e.2161.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 - 0.707107i) q^{5} -0.635963i q^{7} +O(q^{10})$	$q+(0.707107 - 0.707107i) q^{5} -0.635963i q^{7} +(-2.35248 + 2.35248i) q^{11} +(0.577587 + 0.577587i) q^{13} -0.902391 q^{17} +(-0.663835 - 0.663835i) q^{19} +5.57937i q^{23} -1.00000i q^{25} +(3.45325 + 3.45325i) q^{29} -5.79463 q^{31} +(-0.449694 - 0.449694i) q^{35} +(-1.34051 + 1.34051i) q^{37} -1.91461i q^{41} +(-5.60151 + 5.60151i) q^{43} -0.204529 q^{47} +6.59555 q^{49} +(-3.45626 + 3.45626i) q^{53} +3.32691i q^{55} +(-4.16569 + 4.16569i) q^{59} +(3.62778 + 3.62778i) q^{61} +0.816832 q^{65} +(5.47831 + 5.47831i) q^{67} +11.3735i q^{71} -16.3717i q^{73} +(1.49609 + 1.49609i) q^{77} -11.5640 q^{79} +(5.37333 + 5.37333i) q^{83} +(-0.638087 + 0.638087i) q^{85} +16.9135i q^{89} +(0.367324 - 0.367324i) q^{91} -0.938805 q^{95} +11.5475 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q - 8 q^{19} - 32 q^{37} - 16 q^{43} - 32 q^{49} - 16 q^{61} + 16 q^{67} + 16 q^{79} + 16 q^{85} + 80 q^{91}+O(q^{100}) $ 32 * q - 8 * q^19 - 32 * q^37 - 16 * q^43 - 32 * q^49 - 16 * q^61 + 16 * q^67 + 16 * q^79 + 16 * q^85 + 80 * q^91

Display 100 coefficients 10 coefficients

Character values

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

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$	$1$

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
$7$		−	0.635963i		−	0.240372i	−0.992751	−	0.120186i	$-0.961651\pi$
$7$		−	0.635963i		−	0.240372i	0.992751	−	0.120186i	$-0.0383490\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.35248	+	2.35248i	−0.709300	+	0.709300i	−0.966388	−	0.257088i	$-0.917237\pi$
$11$	−2.35248	+	2.35248i	−0.709300	+	0.709300i	0.257088	+	0.966388i	$0.417237\pi$
$12$		0			0
$13$	0.577587	+	0.577587i	0.160194	+	0.160194i	0.782653	−	0.622459i	$-0.213866\pi$
$13$	0.577587	+	0.577587i	0.160194	+	0.160194i	−0.622459	+	0.782653i	$0.713866\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.902391			−0.218862			−0.109431	−	0.993994i	$-0.534903\pi$
$17$	−0.902391			−0.218862			−0.109431	+	0.993994i	$0.534903\pi$
$18$		0			0
$19$	−0.663835	−	0.663835i	−0.152294	−	0.152294i	0.626848	−	0.779142i	$-0.284345\pi$
$19$	−0.663835	−	0.663835i	−0.152294	−	0.152294i	−0.779142	+	0.626848i	$0.784345\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			5.57937i			1.16338i	0.813411	+	0.581689i	$0.197608\pi$
$23$			5.57937i			1.16338i	−0.813411	+	0.581689i	$0.802392\pi$
$24$		0			0
$25$		−	1.00000i		−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.45325	+	3.45325i	0.641253	+	0.641253i	0.950864	−	0.309610i	$-0.100199\pi$
$29$	3.45325	+	3.45325i	0.641253	+	0.641253i	−0.309610	+	0.950864i	$0.600199\pi$
$30$		0			0
$31$	−5.79463			−1.04075			−0.520373	−	0.853939i	$-0.674207\pi$
$31$	−5.79463			−1.04075			−0.520373	+	0.853939i	$0.674207\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.449694	−	0.449694i	−0.0760121	−	0.0760121i
$36$		0			0
$37$	−1.34051	+	1.34051i	−0.220378	+	0.220378i	−0.808658	−	0.588280i	$-0.799805\pi$
$37$	−1.34051	+	1.34051i	−0.220378	+	0.220378i	0.588280	+	0.808658i	$0.299805\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	1.91461i		−	0.299012i	−0.988761	−	0.149506i	$-0.952232\pi$
$41$		−	1.91461i		−	0.299012i	0.988761	−	0.149506i	$-0.0477683\pi$
$42$		0			0
$43$	−5.60151	+	5.60151i	−0.854222	+	0.854222i	−0.990650	−	0.136428i	$-0.956438\pi$
$43$	−5.60151	+	5.60151i	−0.854222	+	0.854222i	0.136428	+	0.990650i	$0.456438\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−0.204529			−0.0298336			−0.0149168	−	0.999889i	$-0.504748\pi$
$47$	−0.204529			−0.0298336			−0.0149168	+	0.999889i	$0.504748\pi$
$48$		0			0
$49$	6.59555			0.942222
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−3.45626	+	3.45626i	−0.474753	+	0.474753i	−0.903449	−	0.428696i	$-0.858973\pi$
$53$	−3.45626	+	3.45626i	−0.474753	+	0.474753i	0.428696	+	0.903449i	$0.358973\pi$
$54$		0			0
$55$			3.32691i			0.448601i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.16569	+	4.16569i	−0.542327	+	0.542327i	−0.924210	−	0.381884i	$-0.875275\pi$
$59$	−4.16569	+	4.16569i	−0.542327	+	0.542327i	0.381884	+	0.924210i	$0.375275\pi$
$60$		0			0
$61$	3.62778	+	3.62778i	0.464490	+	0.464490i	0.900124	−	0.435634i	$-0.143476\pi$
$61$	3.62778	+	3.62778i	0.464490	+	0.464490i	−0.435634	+	0.900124i	$0.643476\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.816832			0.101316
$66$		0			0
$67$	5.47831	+	5.47831i	0.669282	+	0.669282i	0.957550	−	0.288268i	$-0.0930793\pi$
$67$	5.47831	+	5.47831i	0.669282	+	0.669282i	−0.288268	+	0.957550i	$0.593079\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			11.3735i			1.34979i	0.737914	+	0.674895i	$0.235811\pi$
$71$			11.3735i			1.34979i	−0.737914	+	0.674895i	$0.764189\pi$
$72$		0			0
$73$		−	16.3717i		−	1.91617i	−0.286488	−	0.958084i	$-0.592488\pi$
$73$		−	16.3717i		−	1.91617i	0.286488	−	0.958084i	$-0.407512\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.49609	+	1.49609i	0.170496	+	0.170496i
$78$		0			0
$79$	−11.5640			−1.30106			−0.650528	−	0.759482i	$-0.725452\pi$
$79$	−11.5640			−1.30106			−0.650528	+	0.759482i	$0.725452\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	5.37333	+	5.37333i	0.589799	+	0.589799i	0.937577	−	0.347778i	$-0.113064\pi$
$83$	5.37333	+	5.37333i	0.589799	+	0.589799i	−0.347778	+	0.937577i	$0.613064\pi$
$84$		0			0
$85$	−0.638087	+	0.638087i	−0.0692102	+	0.0692102i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			16.9135i			1.79283i	0.443220	+	0.896413i	$0.353836\pi$
$89$			16.9135i			1.79283i	−0.443220	+	0.896413i	$0.646164\pi$
$90$		0			0
$91$	0.367324	−	0.367324i	0.0385061	−	0.0385061i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.938805			−0.0963193
$96$		0			0
$97$	11.5475			1.17247			0.586237	−	0.810139i	$-0.300609\pi$
$97$	11.5475			1.17247			0.586237	+	0.810139i	$0.300609\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	0.335283	−	0.335283i	0.0333619	−	0.0333619i	−0.690229	−	0.723591i	$-0.742490\pi$
$101$	0.335283	−	0.335283i	0.0333619	−	0.0333619i	0.723591	+	0.690229i	$0.242490\pi$
$102$		0			0
$103$		−	2.46558i		−	0.242940i	−0.992595	−	0.121470i	$-0.961239\pi$
$103$		−	2.46558i		−	0.242940i	0.992595	−	0.121470i	$-0.0387609\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	4.18207	−	4.18207i	0.404296	−	0.404296i	−0.475448	−	0.879744i	$-0.657714\pi$
$107$	4.18207	−	4.18207i	0.404296	−	0.404296i	0.879744	+	0.475448i	$0.157714\pi$
$108$		0			0
$109$	−7.90968	−	7.90968i	−0.757610	−	0.757610i	0.218277	−	0.975887i	$-0.429956\pi$
$109$	−7.90968	−	7.90968i	−0.757610	−	0.757610i	−0.975887	+	0.218277i	$0.929956\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−7.02622			−0.660971			−0.330485	−	0.943811i	$-0.607212\pi$
$113$	−7.02622			−0.660971			−0.330485	+	0.943811i	$0.607212\pi$
$114$		0			0
$115$	3.94521	+	3.94521i	0.367893	+	0.367893i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			0.573887i			0.0526082i
$120$		0			0
$121$		−	0.0683426i		−	0.00621296i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.707107	−	0.707107i	−0.0632456	−	0.0632456i
$126$		0			0
$127$	−5.35870			−0.475507			−0.237754	−	0.971325i	$-0.576411\pi$
$127$	−5.35870			−0.475507			−0.237754	+	0.971325i	$0.576411\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	9.31042	+	9.31042i	0.813455	+	0.813455i	0.985150	−	0.171695i	$-0.0549243\pi$
$131$	9.31042	+	9.31042i	0.813455	+	0.813455i	−0.171695	+	0.985150i	$0.554924\pi$
$132$		0			0
$133$	−0.422175	+	0.422175i	−0.0366072	+	0.0366072i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			18.2485i			1.55907i	0.626357	+	0.779536i	$0.284545\pi$
$137$			18.2485i			1.55907i	−0.626357	+	0.779536i	$0.715455\pi$
$138$		0			0
$139$	6.78866	−	6.78866i	0.575807	−	0.575807i	−0.357938	−	0.933745i	$-0.616520\pi$
$139$	6.78866	−	6.78866i	0.575807	−	0.575807i	0.933745	+	0.357938i	$0.116520\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.71753			−0.227251
$144$		0			0
$145$	4.88364			0.405564
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−6.87049	+	6.87049i	−0.562853	+	0.562853i	−0.930117	−	0.367264i	$-0.880295\pi$
$149$	−6.87049	+	6.87049i	−0.562853	+	0.562853i	0.367264	+	0.930117i	$0.380295\pi$
$150$		0			0
$151$		−	18.5235i		−	1.50742i	−0.657207	−	0.753710i	$-0.728262\pi$
$151$		−	18.5235i		−	1.50742i	0.657207	−	0.753710i	$-0.271738\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.09742	+	4.09742i	−0.329113	+	0.329113i
$156$		0			0
$157$	15.5997	+	15.5997i	1.24500	+	1.24500i	0.957903	+	0.287092i	$0.0926886\pi$
$157$	15.5997	+	15.5997i	1.24500	+	1.24500i	0.287092	+	0.957903i	$0.407311\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	3.54827			0.279643
$162$		0			0
$163$	13.2383	+	13.2383i	1.03690	+	1.03690i	0.999292	+	0.0376104i	$0.0119746\pi$
$163$	13.2383	+	13.2383i	1.03690	+	1.03690i	0.0376104	+	0.999292i	$0.488025\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	4.82546i		−	0.373405i	−0.982416	−	0.186703i	$-0.940220\pi$
$167$		−	4.82546i		−	0.373405i	0.982416	−	0.186703i	$-0.0597800\pi$
$168$		0			0
$169$		−	12.3328i		−	0.948676i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	11.0762	+	11.0762i	0.842106	+	0.842106i	0.989133	−	0.147026i	$-0.0469702\pi$
$173$	11.0762	+	11.0762i	0.842106	+	0.842106i	−0.147026	+	0.989133i	$0.546970\pi$
$174$		0			0
$175$	−0.635963			−0.0480743
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−8.56879	−	8.56879i	−0.640462	−	0.640462i	0.310207	−	0.950669i	$-0.399602\pi$
$179$	−8.56879	−	8.56879i	−0.640462	−	0.640462i	−0.950669	+	0.310207i	$0.899602\pi$
$180$		0			0
$181$	−14.0159	+	14.0159i	−1.04179	+	1.04179i	−0.0427052	+	0.999088i	$0.513598\pi$
$181$	−14.0159	+	14.0159i	−1.04179	+	1.04179i	−0.999088	+	0.0427052i	$0.986402\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			1.89576i			0.139379i
$186$		0			0
$187$	2.12286	−	2.12286i	0.155239	−	0.155239i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−8.69253			−0.628969			−0.314485	−	0.949263i	$-0.601832\pi$
$191$	−8.69253			−0.628969			−0.314485	+	0.949263i	$0.601832\pi$
$192$		0			0
$193$	−5.14305			−0.370205			−0.185103	−	0.982719i	$-0.559262\pi$
$193$	−5.14305			−0.370205			−0.185103	+	0.982719i	$0.559262\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−15.7527	+	15.7527i	−1.12233	+	1.12233i	−0.130941	+	0.991390i	$0.541800\pi$
$197$	−15.7527	+	15.7527i	−1.12233	+	1.12233i	−0.991390	+	0.130941i	$0.958200\pi$
$198$		0			0
$199$		−	12.6091i		−	0.893839i	−0.894574	−	0.446919i	$-0.852521\pi$
$199$		−	12.6091i		−	0.893839i	0.894574	−	0.446919i	$-0.147479\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2.19614	−	2.19614i	0.154139	−	0.154139i
$204$		0			0
$205$	−1.35383	−	1.35383i	−0.0945557	−	0.0945557i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	3.12332			0.216045
$210$		0			0
$211$	2.80503	+	2.80503i	0.193106	+	0.193106i	0.797037	−	0.603931i	$-0.206400\pi$
$211$	2.80503	+	2.80503i	0.193106	+	0.193106i	−0.603931	+	0.797037i	$0.706400\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			7.92173i			0.540257i
$216$		0			0
$217$			3.68517i			0.250166i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.521209	−	0.521209i	−0.0350603	−	0.0350603i
$222$		0			0
$223$	−10.8014			−0.723315			−0.361657	−	0.932311i	$-0.617789\pi$
$223$	−10.8014			−0.723315			−0.361657	+	0.932311i	$0.617789\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	4.25178	+	4.25178i	0.282201	+	0.282201i	0.833986	−	0.551785i	$-0.186053\pi$
$227$	4.25178	+	4.25178i	0.282201	+	0.282201i	−0.551785	+	0.833986i	$0.686053\pi$
$228$		0			0
$229$	−14.8409	+	14.8409i	−0.980712	+	0.980712i	−0.999817	−	0.0191054i	$-0.993918\pi$
$229$	−14.8409	+	14.8409i	−0.980712	+	0.980712i	0.0191054	+	0.999817i	$0.493918\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	5.52959i		−	0.362256i	−0.983460	−	0.181128i	$-0.942025\pi$
$233$		−	5.52959i		−	0.362256i	0.983460	−	0.181128i	$-0.0579748\pi$
$234$		0			0
$235$	−0.144624	+	0.144624i	−0.00943421	+	0.00943421i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	24.4860			1.58387			0.791934	−	0.610606i	$-0.209074\pi$
$239$	24.4860			1.58387			0.791934	+	0.610606i	$0.209074\pi$
$240$		0			0
$241$	6.89740			0.444300			0.222150	−	0.975012i	$-0.428692\pi$
$241$	6.89740			0.444300			0.222150	+	0.975012i	$0.428692\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	4.66376	−	4.66376i	0.297957	−	0.297957i
$246$		0			0
$247$		−	0.766846i		−	0.0487932i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−15.6417	+	15.6417i	−0.987295	+	0.987295i	−0.999920	−	0.0126252i	$-0.995981\pi$
$251$	−15.6417	+	15.6417i	−0.987295	+	0.987295i	0.0126252	+	0.999920i	$0.495981\pi$
$252$		0			0
$253$	−13.1254	−	13.1254i	−0.825184	−	0.825184i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−6.06912			−0.378582			−0.189291	−	0.981921i	$-0.560619\pi$
$257$	−6.06912			−0.378582			−0.189291	+	0.981921i	$0.560619\pi$
$258$		0			0
$259$	0.852513	+	0.852513i	0.0529726	+	0.0529726i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	6.63806i		−	0.409320i	−0.978833	−	0.204660i	$-0.934391\pi$
$263$		−	6.63806i		−	0.409320i	0.978833	−	0.204660i	$-0.0656089\pi$
$264$		0			0
$265$			4.88789i			0.300260i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−11.9789	−	11.9789i	−0.730367	−	0.730367i	0.240325	−	0.970692i	$-0.422746\pi$
$269$	−11.9789	−	11.9789i	−0.730367	−	0.730367i	−0.970692	+	0.240325i	$0.922746\pi$
$270$		0			0
$271$	−1.88686			−0.114619			−0.0573094	−	0.998356i	$-0.518252\pi$
$271$	−1.88686			−0.114619			−0.0573094	+	0.998356i	$0.518252\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.35248	+	2.35248i	0.141860	+	0.141860i
$276$		0			0
$277$	−18.4284	+	18.4284i	−1.10726	+	1.10726i	−0.113746	+	0.993510i	$0.536285\pi$
$277$	−18.4284	+	18.4284i	−1.10726	+	1.10726i	−0.993510	+	0.113746i	$0.963715\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	22.5211i		−	1.34349i	−0.740781	−	0.671747i	$-0.765544\pi$
$281$		−	22.5211i		−	1.34349i	0.740781	−	0.671747i	$-0.234456\pi$
$282$		0			0
$283$	8.67520	−	8.67520i	0.515687	−	0.515687i	−0.400576	−	0.916263i	$-0.631190\pi$
$283$	8.67520	−	8.67520i	0.515687	−	0.515687i	0.916263	+	0.400576i	$0.131190\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1.21762			−0.0718739
$288$		0			0
$289$	−16.1857			−0.952099
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−9.74466	+	9.74466i	−0.569289	+	0.569289i	−0.931929	−	0.362640i	$-0.881875\pi$
$293$	−9.74466	+	9.74466i	−0.569289	+	0.569289i	0.362640	+	0.931929i	$0.381875\pi$
$294$		0			0
$295$			5.89117i			0.342997i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.22257	+	3.22257i	−0.186366	+	0.186366i
$300$		0			0
$301$	3.56235	+	3.56235i	0.205331	+	0.205331i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	5.13045			0.293769
$306$		0			0
$307$	23.9750	+	23.9750i	1.36833	+	1.36833i	0.862827	+	0.505500i	$0.168692\pi$
$307$	23.9750	+	23.9750i	1.36833	+	1.36833i	0.505500	+	0.862827i	$0.331308\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			14.6746i			0.832120i	0.909337	+	0.416060i	$0.136589\pi$
$311$			14.6746i			0.832120i	−0.909337	+	0.416060i	$0.863411\pi$
$312$		0			0
$313$		−	10.5051i		−	0.593782i	−0.954911	−	0.296891i	$-0.904050\pi$
$313$		−	10.5051i		−	0.593782i	0.954911	−	0.296891i	$-0.0959498\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	12.4853	+	12.4853i	0.701247	+	0.701247i	0.964678	−	0.263431i	$-0.0848542\pi$
$317$	12.4853	+	12.4853i	0.701247	+	0.701247i	−0.263431	+	0.964678i	$0.584854\pi$
$318$		0			0
$319$	−16.2474			−0.909682
$320$		0			0
$321$		0			0
$322$		0			0
$323$	0.599039	+	0.599039i	0.0333314	+	0.0333314i
$324$		0			0
$325$	0.577587	−	0.577587i	0.0320388	−	0.0320388i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			0.130073i			0.00717114i
$330$		0			0
$331$	2.95247	−	2.95247i	0.162283	−	0.162283i	−0.621295	−	0.783577i	$-0.713393\pi$
$331$	2.95247	−	2.95247i	0.162283	−	0.162283i	0.783577	+	0.621295i	$0.213393\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	7.74750			0.423291
$336$		0			0
$337$	9.90826			0.539737			0.269869	−	0.962897i	$-0.413020\pi$
$337$	9.90826			0.539737			0.269869	+	0.962897i	$0.413020\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	13.6318	−	13.6318i	0.738201	−	0.738201i
$342$		0			0
$343$		−	8.64627i		−	0.466855i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	4.41559	−	4.41559i	0.237042	−	0.237042i	−0.578582	−	0.815624i	$-0.696394\pi$
$347$	4.41559	−	4.41559i	0.237042	−	0.237042i	0.815624	+	0.578582i	$0.196394\pi$
$348$		0			0
$349$	−8.25420	−	8.25420i	−0.441837	−	0.441837i	0.450792	−	0.892629i	$-0.351142\pi$
$349$	−8.25420	−	8.25420i	−0.441837	−	0.441837i	−0.892629	+	0.450792i	$0.851142\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	16.0753			0.855601			0.427801	−	0.903873i	$-0.359289\pi$
$353$	16.0753			0.855601			0.427801	+	0.903873i	$0.359289\pi$
$354$		0			0
$355$	8.04230	+	8.04230i	0.426841	+	0.426841i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	17.5075i		−	0.924008i	−0.886878	−	0.462004i	$-0.847130\pi$
$359$		−	17.5075i		−	0.924008i	0.886878	−	0.462004i	$-0.152870\pi$
$360$		0			0
$361$		−	18.1186i		−	0.953613i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−11.5766	−	11.5766i	−0.605945	−	0.605945i
$366$		0			0
$367$	25.6267			1.33770			0.668851	−	0.743397i	$-0.266787\pi$
$367$	25.6267			1.33770			0.668851	+	0.743397i	$0.266787\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2.19805	+	2.19805i	0.114117	+	0.114117i
$372$		0			0
$373$	−13.7029	+	13.7029i	−0.709512	+	0.709512i	−0.966432	−	0.256921i	$-0.917292\pi$
$373$	−13.7029	+	13.7029i	−0.709512	+	0.709512i	0.256921	+	0.966432i	$0.417292\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			3.98911i			0.205450i
$378$		0			0
$379$	1.80429	−	1.80429i	0.0926803	−	0.0926803i	−0.659247	−	0.751927i	$-0.729125\pi$
$379$	1.80429	−	1.80429i	0.0926803	−	0.0926803i	0.751927	+	0.659247i	$0.229125\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−35.0165			−1.78926			−0.894630	−	0.446807i	$-0.852561\pi$
$383$	−35.0165			−1.78926			−0.894630	+	0.446807i	$0.852561\pi$
$384$		0			0
$385$	2.11579			0.107831
$386$		0			0
$387$		0			0
$388$		0			0
$389$	21.9669	−	21.9669i	1.11376	−	1.11376i	0.121128	−	0.992637i	$-0.461349\pi$
$389$	21.9669	−	21.9669i	1.11376	−	1.11376i	0.992637	−	0.121128i	$-0.0386512\pi$
$390$		0			0
$391$		−	5.03477i		−	0.254619i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−8.17701	+	8.17701i	−0.411430	+	0.411430i
$396$		0			0
$397$	9.64651	+	9.64651i	0.484144	+	0.484144i	0.906452	−	0.422308i	$-0.138780\pi$
$397$	9.64651	+	9.64651i	0.484144	+	0.484144i	−0.422308	+	0.906452i	$0.638780\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2.37443			0.118573			0.0592866	−	0.998241i	$-0.481117\pi$
$401$	2.37443			0.118573			0.0592866	+	0.998241i	$0.481117\pi$
$402$		0			0
$403$	−3.34690	−	3.34690i	−0.166721	−	0.166721i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	6.30704i		−	0.312628i
$408$		0			0
$409$		−	27.5053i		−	1.36005i	−0.733189	−	0.680025i	$-0.761969\pi$
$409$		−	27.5053i		−	1.36005i	0.733189	−	0.680025i	$-0.238031\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.64923	+	2.64923i	0.130360	+	0.130360i
$414$		0			0
$415$	7.59903			0.373022
$416$		0			0
$417$		0			0
$418$		0			0
$419$	0.491941	+	0.491941i	0.0240329	+	0.0240329i	0.719021	−	0.694988i	$-0.244590\pi$
$419$	0.491941	+	0.491941i	0.0240329	+	0.0240329i	−0.694988	+	0.719021i	$0.744590\pi$
$420$		0			0
$421$	−4.58992	+	4.58992i	−0.223699	+	0.223699i	−0.810054	−	0.586355i	$-0.800562\pi$
$421$	−4.58992	+	4.58992i	−0.223699	+	0.223699i	0.586355	+	0.810054i	$0.300562\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			0.902391i			0.0437724i
$426$		0			0
$427$	2.30713	−	2.30713i	0.111650	−	0.111650i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−18.9558			−0.913069			−0.456535	−	0.889706i	$-0.650910\pi$
$431$	−18.9558			−0.913069			−0.456535	+	0.889706i	$0.650910\pi$
$432$		0			0
$433$	7.32444			0.351990			0.175995	−	0.984391i	$-0.443686\pi$
$433$	7.32444			0.351990			0.175995	+	0.984391i	$0.443686\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3.70378	−	3.70378i	0.177176	−	0.177176i
$438$		0			0
$439$		−	34.9314i		−	1.66718i	−0.552381	−	0.833592i	$-0.686281\pi$
$439$		−	34.9314i		−	1.66718i	0.552381	−	0.833592i	$-0.313719\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	19.9694	−	19.9694i	0.948776	−	0.948776i	−0.0499742	−	0.998751i	$-0.515914\pi$
$443$	19.9694	−	19.9694i	0.948776	−	0.948776i	0.998751	+	0.0499742i	$0.0159139\pi$
$444$		0			0
$445$	11.9596	+	11.9596i	0.566941	+	0.566941i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−9.20896			−0.434598			−0.217299	−	0.976105i	$-0.569725\pi$
$449$	−9.20896			−0.434598			−0.217299	+	0.976105i	$0.569725\pi$
$450$		0			0
$451$	4.50408	+	4.50408i	0.212089	+	0.212089i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	0.519475i		−	0.0243534i
$456$		0			0
$457$			24.9334i			1.16633i	0.812352	+	0.583167i	$0.198187\pi$
$457$			24.9334i			1.16633i	−0.812352	+	0.583167i	$0.801813\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	2.56075	+	2.56075i	0.119266	+	0.119266i	0.764221	−	0.644955i	$-0.223124\pi$
$461$	2.56075	+	2.56075i	0.119266	+	0.119266i	−0.644955	+	0.764221i	$0.723124\pi$
$462$		0			0
$463$	−20.9730			−0.974696			−0.487348	−	0.873208i	$-0.662036\pi$
$463$	−20.9730			−0.974696			−0.487348	+	0.873208i	$0.662036\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−17.2099	−	17.2099i	−0.796381	−	0.796381i	0.186142	−	0.982523i	$-0.440401\pi$
$467$	−17.2099	−	17.2099i	−0.796381	−	0.796381i	−0.982523	+	0.186142i	$0.940401\pi$
$468$		0			0
$469$	3.48400	−	3.48400i	0.160876	−	0.160876i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	26.3549i		−	1.21180i
$474$		0			0
$475$	−0.663835	+	0.663835i	−0.0304588	+	0.0304588i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	25.8313			1.18026			0.590132	−	0.807307i	$-0.299076\pi$
$479$	25.8313			1.18026			0.590132	+	0.807307i	$0.299076\pi$
$480$		0			0
$481$	−1.54852			−0.0706064
$482$		0			0
$483$		0			0
$484$		0			0
$485$	8.16534	−	8.16534i	0.370769	−	0.370769i
$486$		0			0
$487$			29.7345i			1.34740i	0.739006	+	0.673698i	$0.235295\pi$
$487$			29.7345i			1.34740i	−0.739006	+	0.673698i	$0.764705\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	21.0366	−	21.0366i	0.949367	−	0.949367i	−0.0494118	−	0.998778i	$-0.515735\pi$
$491$	21.0366	−	21.0366i	0.949367	−	0.949367i	0.998778	+	0.0494118i	$0.0157347\pi$
$492$		0			0
$493$	−3.11618	−	3.11618i	−0.140346	−	0.140346i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	7.23315			0.324451
$498$		0			0
$499$	−18.9594	−	18.9594i	−0.848741	−	0.848741i	0.141235	−	0.989976i	$-0.454893\pi$
$499$	−18.9594	−	18.9594i	−0.848741	−	0.848741i	−0.989976	+	0.141235i	$0.954893\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	0.199360i		−	0.00888902i	−0.999990	−	0.00444451i	$-0.998585\pi$
$503$		−	0.199360i		−	0.00888902i	0.999990	−	0.00444451i	$-0.00141474\pi$
$504$		0			0
$505$		−	0.474161i		−	0.0210999i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−25.0123	−	25.0123i	−1.10865	−	1.10865i	−0.993328	−	0.115321i	$-0.963210\pi$
$509$	−25.0123	−	25.0123i	−1.10865	−	1.10865i	−0.115321	−	0.993328i	$-0.536790\pi$
$510$		0			0
$511$	−10.4118			−0.460592
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.74343	−	1.74343i	−0.0768245	−	0.0768245i
$516$		0			0
$517$	0.481150	−	0.481150i	0.0211610	−	0.0211610i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	32.6531i		−	1.43056i	−0.698838	−	0.715280i	$-0.746299\pi$
$521$		−	32.6531i		−	1.43056i	0.698838	−	0.715280i	$-0.253701\pi$
$522$		0			0
$523$	6.92135	−	6.92135i	0.302650	−	0.302650i	−0.539400	−	0.842050i	$-0.681349\pi$
$523$	6.92135	−	6.92135i	0.302650	−	0.302650i	0.842050	+	0.539400i	$0.181349\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	5.22902			0.227780
$528$		0			0
$529$	−8.12933			−0.353449
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.10585	−	1.10585i	0.0478998	−	0.0478998i
$534$		0			0
$535$		−	5.91434i		−	0.255699i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−15.5159	+	15.5159i	−0.668318	+	0.668318i
$540$		0			0
$541$	−11.9660	−	11.9660i	−0.514461	−	0.514461i	0.401429	−	0.915890i	$-0.368514\pi$
$541$	−11.9660	−	11.9660i	−0.514461	−	0.514461i	−0.915890	+	0.401429i	$0.868514\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−11.1860			−0.479155
$546$		0			0
$547$	21.3285	+	21.3285i	0.911941	+	0.911941i	0.996425	−	0.0844841i	$-0.0269242\pi$
$547$	21.3285	+	21.3285i	0.911941	+	0.911941i	−0.0844841	+	0.996425i	$0.526924\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	4.58478i		−	0.195318i
$552$		0			0
$553$			7.35431i			0.312737i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−13.3326	−	13.3326i	−0.564921	−	0.564921i	0.365780	−	0.930701i	$-0.380802\pi$
$557$	−13.3326	−	13.3326i	−0.564921	−	0.564921i	−0.930701	+	0.365780i	$0.880802\pi$
$558$		0			0
$559$	−6.47072			−0.273682
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6.57088	−	6.57088i	−0.276930	−	0.276930i	0.554952	−	0.831882i	$-0.312736\pi$
$563$	−6.57088	−	6.57088i	−0.276930	−	0.276930i	−0.831882	+	0.554952i	$0.812736\pi$
$564$		0			0
$565$	−4.96829	+	4.96829i	−0.209017	+	0.209017i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			46.2800i			1.94016i	0.242792	+	0.970078i	$0.421937\pi$
$569$			46.2800i			1.94016i	−0.242792	+	0.970078i	$0.578063\pi$
$570$		0			0
$571$	15.7220	−	15.7220i	0.657943	−	0.657943i	−0.296950	−	0.954893i	$-0.595969\pi$
$571$	15.7220	−	15.7220i	0.657943	−	0.657943i	0.954893	+	0.296950i	$0.0959694\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	5.57937			0.232676
$576$		0			0
$577$	28.4494			1.18436			0.592181	−	0.805805i	$-0.298267\pi$
$577$	28.4494			1.18436			0.592181	+	0.805805i	$0.298267\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	3.41724	−	3.41724i	0.141771	−	0.141771i
$582$		0			0
$583$		−	16.2616i		−	0.673485i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	29.7737	−	29.7737i	1.22889	−	1.22889i	0.264511	−	0.964383i	$-0.414790\pi$
$587$	29.7737	−	29.7737i	1.22889	−	1.22889i	0.964383	−	0.264511i	$-0.0852105\pi$
$588$		0			0
$589$	3.84668	+	3.84668i	0.158500	+	0.158500i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−19.2483			−0.790433			−0.395217	−	0.918588i	$-0.629330\pi$
$593$	−19.2483			−0.790433			−0.395217	+	0.918588i	$0.629330\pi$
$594$		0			0
$595$	0.405800	+	0.405800i	0.0166362	+	0.0166362i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	0.563616i		−	0.0230287i	−0.999934	−	0.0115144i	$-0.996335\pi$
$599$		−	0.563616i		−	0.0230287i	0.999934	−	0.0115144i	$-0.00366522\pi$
$600$		0			0
$601$			3.86618i			0.157705i	0.996886	+	0.0788524i	$0.0251256\pi$
$601$			3.86618i			0.157705i	−0.996886	+	0.0788524i	$0.974874\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.0483255	−	0.0483255i	−0.00196471	−	0.00196471i
$606$		0			0
$607$	−1.08864			−0.0441864			−0.0220932	−	0.999756i	$-0.507033\pi$
$607$	−1.08864			−0.0441864			−0.0220932	+	0.999756i	$0.507033\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−0.118133	−	0.118133i	−0.00477916	−	0.00477916i
$612$		0			0
$613$	−0.853904	+	0.853904i	−0.0344889	+	0.0344889i	−0.724141	−	0.689652i	$-0.757763\pi$
$613$	−0.853904	+	0.853904i	−0.0344889	+	0.0344889i	0.689652	+	0.724141i	$0.257763\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	15.8444i		−	0.637872i	−0.947776	−	0.318936i	$-0.896675\pi$
$617$		−	15.8444i		−	0.637872i	0.947776	−	0.318936i	$-0.103325\pi$
$618$		0			0
$619$	−28.7182	+	28.7182i	−1.15428	+	1.15428i	−0.168598	+	0.985685i	$0.553924\pi$
$619$	−28.7182	+	28.7182i	−1.15428	+	1.15428i	−0.985685	+	0.168598i	$0.946076\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	10.7564			0.430944
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1.20966	−	1.20966i	0.0482323	−	0.0482323i
$630$		0			0
$631$			33.9701i			1.35233i	0.736751	+	0.676164i	$0.236359\pi$
$631$			33.9701i			1.35233i	−0.736751	+	0.676164i	$0.763641\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3.78917	+	3.78917i	−0.150369	+	0.150369i
$636$		0			0
$637$	3.80951	+	3.80951i	0.150938	+	0.150938i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	10.1321			0.400192			0.200096	−	0.979776i	$-0.435875\pi$
$641$	10.1321			0.400192			0.200096	+	0.979776i	$0.435875\pi$
$642$		0			0
$643$	−35.0098	−	35.0098i	−1.38065	−	1.38065i	−0.843460	−	0.537191i	$-0.819485\pi$
$643$	−35.0098	−	35.0098i	−1.38065	−	1.38065i	−0.537191	−	0.843460i	$-0.680515\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	43.3669i		−	1.70493i	−0.522786	−	0.852464i	$-0.675107\pi$
$647$		−	43.3669i		−	1.70493i	0.522786	−	0.852464i	$-0.324893\pi$
$648$		0			0
$649$		−	19.5994i		−	0.769344i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	19.2284	+	19.2284i	0.752464	+	0.752464i	0.974939	−	0.222474i	$-0.0714134\pi$
$653$	19.2284	+	19.2284i	0.752464	+	0.752464i	−0.222474	+	0.974939i	$0.571413\pi$
$654$		0			0
$655$	13.1669			0.514474
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−30.1413	−	30.1413i	−1.17414	−	1.17414i	−0.981214	−	0.192923i	$-0.938203\pi$
$659$	−30.1413	−	30.1413i	−1.17414	−	1.17414i	−0.192923	−	0.981214i	$-0.561797\pi$
$660$		0			0
$661$	27.2272	−	27.2272i	1.05902	−	1.05902i	0.0608694	−	0.998146i	$-0.480613\pi$
$661$	27.2272	−	27.2272i	1.05902	−	1.05902i	0.998146	−	0.0608694i	$-0.0193873\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			0.597045i			0.0231524i
$666$		0			0
$667$	−19.2670	+	19.2670i	−0.746020	+	0.746020i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−17.0686			−0.658925
$672$		0			0
$673$	11.1628			0.430296			0.215148	−	0.976581i	$-0.430977\pi$
$673$	11.1628			0.430296			0.215148	+	0.976581i	$0.430977\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	3.76635	−	3.76635i	0.144753	−	0.144753i	−0.631017	−	0.775769i	$-0.717362\pi$
$677$	3.76635	−	3.76635i	0.144753	−	0.144753i	0.775769	+	0.631017i	$0.217362\pi$
$678$		0			0
$679$		−	7.34381i		−	0.281829i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	23.8889	−	23.8889i	0.914083	−	0.914083i	−0.0825077	−	0.996590i	$-0.526293\pi$
$683$	23.8889	−	23.8889i	0.914083	−	0.914083i	0.996590	+	0.0825077i	$0.0262929\pi$
$684$		0			0
$685$	12.9036	+	12.9036i	0.493022	+	0.493022i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−3.99258			−0.152105
$690$		0			0
$691$	3.31980	+	3.31980i	0.126291	+	0.126291i	0.767427	−	0.641136i	$-0.221537\pi$
$691$	3.31980	+	3.31980i	0.126291	+	0.126291i	−0.641136	+	0.767427i	$0.721537\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	9.60062i		−	0.364172i
$696$		0			0
$697$			1.72772i			0.0654422i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−3.41258	−	3.41258i	−0.128891	−	0.128891i	0.639718	−	0.768609i	$-0.279051\pi$
$701$	−3.41258	−	3.41258i	−0.128891	−	0.128891i	−0.768609	+	0.639718i	$0.779051\pi$
$702$		0			0
$703$	1.77975			0.0671246
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−0.213227	−	0.213227i	−0.00801924	−	0.00801924i
$708$		0			0
$709$	17.3069	−	17.3069i	0.649973	−	0.649973i	−0.303014	−	0.952986i	$-0.597993\pi$
$709$	17.3069	−	17.3069i	0.649973	−	0.649973i	0.952986	+	0.303014i	$0.0979928\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	32.3303i		−	1.21078i
$714$		0			0
$715$	−1.92158	+	1.92158i	−0.0718631	+	0.0718631i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	27.9048			1.04067			0.520337	−	0.853961i	$-0.325806\pi$
$719$	27.9048			1.04067			0.520337	+	0.853961i	$0.325806\pi$
$720$		0			0
$721$	−1.56802			−0.0583960
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3.45325	−	3.45325i	0.128251	−	0.128251i
$726$		0			0
$727$			21.0702i			0.781449i	0.920508	+	0.390725i	$0.127776\pi$
$727$			21.0702i			0.781449i	−0.920508	+	0.390725i	$0.872224\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	5.05475	−	5.05475i	0.186957	−	0.186957i
$732$		0			0
$733$	9.27608	+	9.27608i	0.342620	+	0.342620i	0.857351	−	0.514732i	$-0.172108\pi$
$733$	9.27608	+	9.27608i	0.342620	+	0.342620i	−0.514732	+	0.857351i	$0.672108\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−25.7753			−0.949444
$738$		0			0
$739$	−0.781108	−	0.781108i	−0.0287335	−	0.0287335i	0.692594	−	0.721328i	$-0.256468\pi$
$739$	−0.781108	−	0.781108i	−0.0287335	−	0.0287335i	−0.721328	+	0.692594i	$0.756468\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	24.6032i		−	0.902603i	−0.892371	−	0.451302i	$-0.850960\pi$
$743$		−	24.6032i		−	0.902603i	0.892371	−	0.451302i	$-0.149040\pi$
$744$		0			0
$745$			9.71634i			0.355979i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−2.65964	−	2.65964i	−0.0971812	−	0.0971812i
$750$		0			0
$751$	−15.8768			−0.579351			−0.289676	−	0.957125i	$-0.593547\pi$
$751$	−15.8768			−0.579351			−0.289676	+	0.957125i	$0.593547\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−13.0981	−	13.0981i	−0.476688	−	0.476688i
$756$		0			0
$757$	29.3690	−	29.3690i	1.06744	−	1.06744i	0.0698797	−	0.997555i	$-0.477738\pi$
$757$	29.3690	−	29.3690i	1.06744	−	1.06744i	0.997555	−	0.0698797i	$-0.0222616\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	9.67224i		−	0.350619i	−0.984513	−	0.175309i	$-0.943907\pi$
$761$		−	9.67224i		−	0.350619i	0.984513	−	0.175309i	$-0.0560925\pi$
$762$		0			0
$763$	−5.03027	+	5.03027i	−0.182108	+	0.182108i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−4.81210			−0.173755
$768$		0			0
$769$	−18.3177			−0.660553			−0.330276	−	0.943884i	$-0.607142\pi$
$769$	−18.3177			−0.660553			−0.330276	+	0.943884i	$0.607142\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−11.9406	+	11.9406i	−0.429474	+	0.429474i	−0.888449	−	0.458975i	$-0.848217\pi$
$773$	−11.9406	+	11.9406i	−0.429474	+	0.429474i	0.458975	+	0.888449i	$0.348217\pi$
$774$		0			0
$775$			5.79463i			0.208149i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1.27098	+	1.27098i	−0.0455377	+	0.0455377i
$780$		0			0
$781$	−26.7560	−	26.7560i	−0.957406	−	0.957406i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	22.0614			0.787404
$786$		0			0
$787$	4.73565	+	4.73565i	0.168808	+	0.168808i	0.786455	−	0.617647i	$-0.211914\pi$
$787$	4.73565	+	4.73565i	0.168808	+	0.168808i	−0.617647	+	0.786455i	$0.711914\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			4.46842i			0.158879i
$792$		0			0
$793$			4.19072i			0.148817i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	7.96817	+	7.96817i	0.282247	+	0.282247i	0.834005	−	0.551757i	$-0.186043\pi$
$797$	7.96817	+	7.96817i	0.282247	+	0.282247i	−0.551757	+	0.834005i	$0.686043\pi$
$798$		0			0
$799$	0.184565			0.00652944
$800$		0			0
$801$		0			0
$802$		0			0
$803$	38.5142	+	38.5142i	1.35914	+	1.35914i
$804$		0			0
$805$	2.50901	−	2.50901i	0.0884309	−	0.0884309i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			24.8804i			0.874750i	0.899279	+	0.437375i	$0.144092\pi$
$809$			24.8804i			0.874750i	−0.899279	+	0.437375i	$0.855908\pi$
$810$		0			0
$811$	−27.4397	+	27.4397i	−0.963539	+	0.963539i	−0.999358	−	0.0358188i	$-0.988596\pi$
$811$	−27.4397	+	27.4397i	−0.963539	+	0.963539i	0.0358188	+	0.999358i	$0.488596\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	18.7218			0.655795
$816$		0			0
$817$	7.43696			0.260186
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−13.7145	+	13.7145i	−0.478638	+	0.478638i	−0.904696	−	0.426058i	$-0.859902\pi$
$821$	−13.7145	+	13.7145i	−0.478638	+	0.478638i	0.426058	+	0.904696i	$0.359902\pi$
$822$		0			0
$823$			5.63219i			0.196326i	0.995170	+	0.0981629i	$0.0312966\pi$
$823$			5.63219i			0.196326i	−0.995170	+	0.0981629i	$0.968703\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−10.8755	+	10.8755i	−0.378178	+	0.378178i	−0.870444	−	0.492267i	$-0.836168\pi$
$827$	−10.8755	+	10.8755i	−0.378178	+	0.378178i	0.492267	+	0.870444i	$0.336168\pi$
$828$		0			0
$829$	−28.6159	−	28.6159i	−0.993870	−	0.993870i	0.00611096	−	0.999981i	$-0.498055\pi$
$829$	−28.6159	−	28.6159i	−0.993870	−	0.993870i	−0.999981	+	0.00611096i	$0.998055\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−5.95176			−0.206216
$834$		0			0
$835$	−3.41211	−	3.41211i	−0.118081	−	0.118081i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	21.8709i		−	0.755067i	−0.925996	−	0.377533i	$-0.876772\pi$
$839$		−	21.8709i		−	0.755067i	0.925996	−	0.377533i	$-0.123228\pi$
$840$		0			0
$841$		−	5.15006i		−	0.177588i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−8.72060	−	8.72060i	−0.299998	−	0.299998i
$846$		0			0
$847$	−0.0434634			−0.00149342
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−7.47918	−	7.47918i	−0.256383	−	0.256383i
$852$		0			0
$853$	15.7986	−	15.7986i	0.540934	−	0.540934i	−0.382869	−	0.923803i	$-0.625064\pi$
$853$	15.7986	−	15.7986i	0.540934	−	0.540934i	0.923803	+	0.382869i	$0.125064\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	48.7010i		−	1.66360i	−0.555078	−	0.831798i	$-0.687312\pi$
$857$		−	48.7010i		−	1.66360i	0.555078	−	0.831798i	$-0.312688\pi$
$858$		0			0
$859$	16.1807	−	16.1807i	0.552078	−	0.552078i	−0.374962	−	0.927040i	$-0.622344\pi$
$859$	16.1807	−	16.1807i	0.552078	−	0.552078i	0.927040	+	0.374962i	$0.122344\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−48.6053			−1.65454			−0.827271	−	0.561803i	$-0.810108\pi$
$863$	−48.6053			−1.65454			−0.827271	+	0.561803i	$0.810108\pi$
$864$		0			0
$865$	15.6641			0.532595
$866$		0			0
$867$		0			0
$868$		0			0
$869$	27.2042	−	27.2042i	0.922839	−	0.922839i
$870$		0			0
$871$			6.32841i			0.214430i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−0.449694	+	0.449694i	−0.0152024	+	0.0152024i
$876$		0			0
$877$	35.9674	+	35.9674i	1.21453	+	1.21453i	0.969520	+	0.245014i	$0.0787925\pi$
$877$	35.9674	+	35.9674i	1.21453	+	1.21453i	0.245014	+	0.969520i	$0.421207\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−33.4850			−1.12814			−0.564070	−	0.825727i	$-0.690765\pi$
$881$	−33.4850			−1.12814			−0.564070	+	0.825727i	$0.690765\pi$
$882$		0			0
$883$	−15.7115	−	15.7115i	−0.528733	−	0.528733i	0.391461	−	0.920195i	$-0.371970\pi$
$883$	−15.7115	−	15.7115i	−0.528733	−	0.528733i	−0.920195	+	0.391461i	$0.871970\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			24.3892i			0.818909i	0.912331	+	0.409455i	$0.134281\pi$
$887$			24.3892i			0.818909i	−0.912331	+	0.409455i	$0.865719\pi$
$888$		0			0
$889$			3.40793i			0.114298i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	0.135773	+	0.135773i	0.00454348	+	0.00454348i
$894$		0			0
$895$	−12.1181			−0.405063
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−20.0103	−	20.0103i	−0.667382	−	0.667382i
$900$		0			0
$901$	3.11889	−	3.11889i	0.103905	−	0.103905i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			19.8215i			0.658888i
$906$		0			0
$907$	8.52873	−	8.52873i	0.283192	−	0.283192i	−0.551189	−	0.834381i	$-0.685826\pi$
$907$	8.52873	−	8.52873i	0.283192	−	0.283192i	0.834381	+	0.551189i	$0.185826\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	34.5986			1.14630			0.573152	−	0.819449i	$-0.305720\pi$
$911$	34.5986			1.14630			0.573152	+	0.819449i	$0.305720\pi$
$912$		0			0
$913$	−25.2813			−0.836689
$914$		0			0
$915$		0			0
$916$		0			0
$917$	5.92109	−	5.92109i	0.195531	−	0.195531i
$918$		0			0
$919$			15.5458i			0.512807i	0.966570	+	0.256404i	$0.0825376\pi$
$919$			15.5458i			0.512807i	−0.966570	+	0.256404i	$0.917462\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−6.56921	+	6.56921i	−0.216228	+	0.216228i
$924$		0			0
$925$	1.34051	+	1.34051i	0.0440756	+	0.0440756i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	60.3172			1.97894			0.989472	−	0.144728i	$-0.0462307\pi$
$929$	60.3172			1.97894			0.989472	+	0.144728i	$0.0462307\pi$
$930$		0			0
$931$	−4.37836	−	4.37836i	−0.143495	−	0.143495i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	3.00217i		−	0.0981816i
$936$		0			0
$937$			25.5917i			0.836043i	0.908437	+	0.418021i	$0.137276\pi$
$937$			25.5917i			0.836043i	−0.908437	+	0.418021i	$0.862724\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	15.9931	+	15.9931i	0.521360	+	0.521360i	0.917982	−	0.396622i	$-0.129818\pi$
$941$	15.9931	+	15.9931i	0.521360	+	0.521360i	−0.396622	+	0.917982i	$0.629818\pi$
$942$		0			0
$943$	10.6823			0.347864
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−0.149135	−	0.149135i	−0.00484624	−	0.00484624i	0.704680	−	0.709526i	$-0.251091\pi$
$947$	−0.149135	−	0.149135i	−0.00484624	−	0.00484624i	−0.709526	+	0.704680i	$0.751091\pi$
$948$		0			0
$949$	9.45611	−	9.45611i	0.306958	−	0.306958i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	35.1017i		−	1.13706i	−0.822664	−	0.568528i	$-0.807513\pi$
$953$		−	35.1017i		−	1.13706i	0.822664	−	0.568528i	$-0.192487\pi$
$954$		0			0
$955$	−6.14655	+	6.14655i	−0.198898	+	0.198898i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	11.6054			0.374757
$960$		0			0
$961$	2.57769			0.0831514
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−3.63669	+	3.63669i	−0.117069	+	0.117069i
$966$		0			0
$967$			46.3335i			1.48998i	0.667073	+	0.744992i	$0.267547\pi$
$967$			46.3335i			1.48998i	−0.667073	+	0.744992i	$0.732453\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	10.4510	−	10.4510i	0.335389	−	0.335389i	−0.519240	−	0.854628i	$-0.673785\pi$
$971$	10.4510	−	10.4510i	0.335389	−	0.335389i	0.854628	+	0.519240i	$0.173785\pi$
$972$		0			0
$973$	−4.31734	−	4.31734i	−0.138408	−	0.138408i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	32.2019			1.03023			0.515114	−	0.857121i	$-0.327749\pi$
$977$	32.2019			1.03023			0.515114	+	0.857121i	$0.327749\pi$
$978$		0			0
$979$	−39.7887	−	39.7887i	−1.27165	−	1.27165i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			53.0129i			1.69085i	0.534096	+	0.845424i	$0.320652\pi$
$983$			53.0129i			1.69085i	−0.534096	+	0.845424i	$0.679348\pi$
$984$		0			0
$985$			22.2776i			0.709825i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−31.2529	−	31.2529i	−0.993783	−	0.993783i
$990$		0			0
$991$	37.0151			1.17582			0.587912	−	0.808925i	$-0.299950\pi$
$991$	37.0151			1.17582			0.587912	+	0.808925i	$0.299950\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−8.91601	−	8.91601i	−0.282657	−	0.282657i
$996$		0			0
$997$	−6.24280	+	6.24280i	−0.197711	+	0.197711i	−0.799018	−	0.601307i	$-0.794647\pi$
$997$	−6.24280	+	6.24280i	−0.197711	+	0.197711i	0.601307	+	0.799018i	$0.294647\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.t.e.721.12		32
3.2	odd	2	inner	2880.2.t.e.721.6		32
4.3	odd	2		720.2.t.e.541.3	yes	32
12.11	even	2		720.2.t.e.541.14	yes	32
16.5	even	4	inner	2880.2.t.e.2161.12		32
16.11	odd	4		720.2.t.e.181.3	✓	32
48.5	odd	4	inner	2880.2.t.e.2161.6		32
48.11	even	4		720.2.t.e.181.14	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.t.e.181.3	✓	32	16.11	odd	4
720.2.t.e.181.14	yes	32	48.11	even	4
720.2.t.e.541.3	yes	32	4.3	odd	2
720.2.t.e.541.14	yes	32	12.11	even	2
2880.2.t.e.721.6		32	3.2	odd	2	inner
2880.2.t.e.721.12		32	1.1	even	1	trivial
2880.2.t.e.2161.6		32	48.5	odd	4	inner
2880.2.t.e.2161.12		32	16.5	even	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(0.707107 - 0.707107i) q^{5} -0.635963i q^{7} +O(q^{10})\)	\(q+(0.707107 - 0.707107i) q^{5} -0.635963i q^{7} +(-2.35248 + 2.35248i) q^{11} +(0.577587 + 0.577587i) q^{13} -0.902391 q^{17} +(-0.663835 - 0.663835i) q^{19} +5.57937i q^{23} -1.00000i q^{25} +(3.45325 + 3.45325i) q^{29} -5.79463 q^{31} +(-0.449694 - 0.449694i) q^{35} +(-1.34051 + 1.34051i) q^{37} -1.91461i q^{41} +(-5.60151 + 5.60151i) q^{43} -0.204529 q^{47} +6.59555 q^{49} +(-3.45626 + 3.45626i) q^{53} +3.32691i q^{55} +(-4.16569 + 4.16569i) q^{59} +(3.62778 + 3.62778i) q^{61} +0.816832 q^{65} +(5.47831 + 5.47831i) q^{67} +11.3735i q^{71} -16.3717i q^{73} +(1.49609 + 1.49609i) q^{77} -11.5640 q^{79} +(5.37333 + 5.37333i) q^{83} +(-0.638087 + 0.638087i) q^{85} +16.9135i q^{89} +(0.367324 - 0.367324i) q^{91} -0.938805 q^{95} +11.5475 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q - 8 q^{19} - 32 q^{37} - 16 q^{43} - 32 q^{49} - 16 q^{61} + 16 q^{67} + 16 q^{79} + 16 q^{85} + 80 q^{91}+O(q^{100}) \) 32 * q - 8 * q^19 - 32 * q^37 - 16 * q^43 - 32 * q^49 - 16 * q^61 + 16 * q^67 + 16 * q^79 + 16 * q^85 + 80 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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