LMFDB - Embedded newform 2880.2.bl.b.431.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(431,2880)]

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([2, 3, 2, 0]))

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

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

chi := DirichletCharacter("2880.431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.bl (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:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 720)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			431.12
Character	$\chi$	$=$	2880.431
Dual form			2880.2.bl.b.1871.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 + 0.707107i) q^{5} +2.69352 q^{7} +O(q^{10})$	$q+(0.707107 + 0.707107i) q^{5} +2.69352 q^{7} +(1.63072 - 1.63072i) q^{11} +(0.257477 + 0.257477i) q^{13} +3.62556i q^{17} +(3.93371 - 3.93371i) q^{19} +3.14141i q^{23} +1.00000i q^{25} +(4.00785 - 4.00785i) q^{29} -8.73107i q^{31} +(1.90461 + 1.90461i) q^{35} +(-2.71695 + 2.71695i) q^{37} +5.22343 q^{41} +(3.76768 + 3.76768i) q^{43} -8.54245 q^{47} +0.255048 q^{49} +(2.25619 + 2.25619i) q^{53} +2.30618 q^{55} +(-3.43351 + 3.43351i) q^{59} +(8.79557 + 8.79557i) q^{61} +0.364128i q^{65} +(5.07416 - 5.07416i) q^{67} +4.48856i q^{71} -12.9512i q^{73} +(4.39237 - 4.39237i) q^{77} -12.5985i q^{79} +(-1.25502 - 1.25502i) q^{83} +(-2.56366 + 2.56366i) q^{85} -15.8800 q^{89} +(0.693520 + 0.693520i) q^{91} +5.56310 q^{95} +11.7308 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 8 q^{7}+O(q^{10}) $ 24 * q + 8 * q^7	$ 24 q + 8 q^{7} + 24 q^{19} + 8 q^{37} - 48 q^{43} + 24 q^{49} - 24 q^{55} + 40 q^{61} + 40 q^{67} + 24 q^{85} - 40 q^{91} - 16 q^{97}+O(q^{100}) $ 24 * q + 8 * q^7 + 24 * q^19 + 8 * q^37 - 48 * q^43 + 24 * q^49 - 24 * q^55 + 40 * q^61 + 40 * q^67 + 24 * q^85 - 40 * q^91 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

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
$7$	2.69352			1.01805			0.509027	−	0.860750i	$-0.330005\pi$
$7$	2.69352			1.01805			0.509027	+	0.860750i	$0.330005\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.63072	−	1.63072i	0.491680	−	0.491680i	−0.417156	−	0.908835i	$-0.636973\pi$
$11$	1.63072	−	1.63072i	0.491680	−	0.491680i	0.908835	+	0.417156i	$0.136973\pi$
$12$		0			0
$13$	0.257477	+	0.257477i	0.0714113	+	0.0714113i	0.741910	−	0.670499i	$-0.233920\pi$
$13$	0.257477	+	0.257477i	0.0714113	+	0.0714113i	−0.670499	+	0.741910i	$0.733920\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			3.62556i			0.879328i	0.898162	+	0.439664i	$0.144902\pi$
$17$			3.62556i			0.879328i	−0.898162	+	0.439664i	$0.855098\pi$
$18$		0			0
$19$	3.93371	−	3.93371i	0.902454	−	0.902454i	−0.0931940	−	0.995648i	$-0.529708\pi$
$19$	3.93371	−	3.93371i	0.902454	−	0.902454i	0.995648	+	0.0931940i	$0.0297077\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			3.14141i			0.655030i	0.944846	+	0.327515i	$0.106211\pi$
$23$			3.14141i			0.655030i	−0.944846	+	0.327515i	$0.893789\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.00785	−	4.00785i	0.744238	−	0.744238i	−0.229152	−	0.973391i	$-0.573595\pi$
$29$	4.00785	−	4.00785i	0.744238	−	0.744238i	0.973391	+	0.229152i	$0.0735954\pi$
$30$		0			0
$31$		−	8.73107i		−	1.56815i	−0.620668	−	0.784074i	$-0.713138\pi$
$31$		−	8.73107i		−	1.56815i	0.620668	−	0.784074i	$-0.286862\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.90461	+	1.90461i	0.321937	+	0.321937i
$36$		0			0
$37$	−2.71695	+	2.71695i	−0.446664	+	0.446664i	−0.894244	−	0.447580i	$-0.852286\pi$
$37$	−2.71695	+	2.71695i	−0.446664	+	0.446664i	0.447580	+	0.894244i	$0.352286\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	5.22343			0.815762			0.407881	−	0.913035i	$-0.366268\pi$
$41$	5.22343			0.815762			0.407881	+	0.913035i	$0.366268\pi$
$42$		0			0
$43$	3.76768	+	3.76768i	0.574566	+	0.574566i	0.933401	−	0.358835i	$-0.116826\pi$
$43$	3.76768	+	3.76768i	0.574566	+	0.574566i	−0.358835	+	0.933401i	$0.616826\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−8.54245			−1.24604			−0.623022	−	0.782204i	$-0.714095\pi$
$47$	−8.54245			−1.24604			−0.623022	+	0.782204i	$0.714095\pi$
$48$		0			0
$49$	0.255048			0.0364355
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.25619	+	2.25619i	0.309912	+	0.309912i	0.844875	−	0.534963i	$-0.179675\pi$
$53$	2.25619	+	2.25619i	0.309912	+	0.309912i	−0.534963	+	0.844875i	$0.679675\pi$
$54$		0			0
$55$	2.30618			0.310965
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−3.43351	+	3.43351i	−0.447005	+	0.447005i	−0.894358	−	0.447353i	$-0.852367\pi$
$59$	−3.43351	+	3.43351i	−0.447005	+	0.447005i	0.447353	+	0.894358i	$0.352367\pi$
$60$		0			0
$61$	8.79557	+	8.79557i	1.12616	+	1.12616i	0.990796	+	0.135360i	$0.0432192\pi$
$61$	8.79557	+	8.79557i	1.12616	+	1.12616i	0.135360	+	0.990796i	$0.456781\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			0.364128i			0.0451645i
$66$		0			0
$67$	5.07416	−	5.07416i	0.619907	−	0.619907i	−0.325600	−	0.945508i	$-0.605566\pi$
$67$	5.07416	−	5.07416i	0.619907	−	0.619907i	0.945508	+	0.325600i	$0.105566\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			4.48856i			0.532694i	0.963877	+	0.266347i	$0.0858167\pi$
$71$			4.48856i			0.532694i	−0.963877	+	0.266347i	$0.914183\pi$
$72$		0			0
$73$		−	12.9512i		−	1.51583i	−0.652354	−	0.757914i	$-0.726219\pi$
$73$		−	12.9512i		−	1.51583i	0.652354	−	0.757914i	$-0.273781\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	4.39237	−	4.39237i	0.500557	−	0.500557i
$78$		0			0
$79$		−	12.5985i		−	1.41744i	−0.705490	−	0.708720i	$-0.749273\pi$
$79$		−	12.5985i		−	1.41744i	0.705490	−	0.708720i	$-0.250727\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.25502	−	1.25502i	−0.137756	−	0.137756i	0.634866	−	0.772622i	$-0.281055\pi$
$83$	−1.25502	−	1.25502i	−0.137756	−	0.137756i	−0.772622	+	0.634866i	$0.781055\pi$
$84$		0			0
$85$	−2.56366	+	2.56366i	−0.278068	+	0.278068i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−15.8800			−1.68328			−0.841640	−	0.540039i	$-0.818410\pi$
$89$	−15.8800			−1.68328			−0.841640	+	0.540039i	$0.818410\pi$
$90$		0			0
$91$	0.693520	+	0.693520i	0.0727006	+	0.0727006i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	5.56310			0.570762
$96$		0			0
$97$	11.7308			1.19108			0.595540	−	0.803326i	$-0.296938\pi$
$97$	11.7308			1.19108			0.595540	+	0.803326i	$0.296938\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	12.3238	+	12.3238i	1.22626	+	1.22626i	0.965367	+	0.260894i	$0.0840173\pi$
$101$	12.3238	+	12.3238i	1.22626	+	1.22626i	0.260894	+	0.965367i	$0.415983\pi$
$102$		0			0
$103$	−7.47988			−0.737015			−0.368507	−	0.929625i	$-0.620131\pi$
$103$	−7.47988			−0.737015			−0.368507	+	0.929625i	$0.620131\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.81647	−	1.81647i	0.175605	−	0.175605i	−0.613832	−	0.789437i	$-0.710373\pi$
$107$	1.81647	−	1.81647i	0.175605	−	0.175605i	0.789437	+	0.613832i	$0.210373\pi$
$108$		0			0
$109$	−2.06312	−	2.06312i	−0.197612	−	0.197612i	0.601364	−	0.798975i	$-0.294624\pi$
$109$	−2.06312	−	2.06312i	−0.197612	−	0.197612i	−0.798975	+	0.601364i	$0.794624\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	0.833448i		−	0.0784042i	−0.999231	−	0.0392021i	$-0.987518\pi$
$113$		−	0.833448i		−	0.0784042i	0.999231	−	0.0392021i	$-0.0124816\pi$
$114$		0			0
$115$	−2.22131	+	2.22131i	−0.207139	+	0.207139i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			9.76552i			0.895204i
$120$		0			0
$121$			5.68153i			0.516503i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.707107	+	0.707107i	−0.0632456	+	0.0632456i
$126$		0			0
$127$			9.84202i			0.873338i	0.899622	+	0.436669i	$0.143842\pi$
$127$			9.84202i			0.873338i	−0.899622	+	0.436669i	$0.856158\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.55655	−	3.55655i	−0.310737	−	0.310737i	0.534458	−	0.845195i	$-0.320516\pi$
$131$	−3.55655	−	3.55655i	−0.310737	−	0.310737i	−0.845195	+	0.534458i	$0.820516\pi$
$132$		0			0
$133$	10.5955	−	10.5955i	0.918748	−	0.918748i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	16.1237			1.37754			0.688769	−	0.724981i	$-0.258151\pi$
$137$	16.1237			1.37754			0.688769	+	0.724981i	$0.258151\pi$
$138$		0			0
$139$	4.68300	+	4.68300i	0.397206	+	0.397206i	0.877247	−	0.480040i	$-0.159378\pi$
$139$	4.68300	+	4.68300i	0.397206	+	0.397206i	−0.480040	+	0.877247i	$0.659378\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.839744			0.0702230
$144$		0			0
$145$	5.66795			0.470698
$146$		0			0
$147$		0			0
$148$		0			0
$149$	13.0936	+	13.0936i	1.07267	+	1.07267i	0.997144	+	0.0755251i	$0.0240633\pi$
$149$	13.0936	+	13.0936i	1.07267	+	1.07267i	0.0755251	+	0.997144i	$0.475937\pi$
$150$		0			0
$151$	−14.0676			−1.14480			−0.572401	−	0.819974i	$-0.693988\pi$
$151$	−14.0676			−1.14480			−0.572401	+	0.819974i	$0.693988\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	6.17380	−	6.17380i	0.495892	−	0.495892i
$156$		0			0
$157$	−5.46190	−	5.46190i	−0.435907	−	0.435907i	0.454725	−	0.890632i	$-0.349738\pi$
$157$	−5.46190	−	5.46190i	−0.435907	−	0.435907i	−0.890632	+	0.454725i	$0.849738\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			8.46145i			0.666856i
$162$		0			0
$163$	11.3222	−	11.3222i	0.886824	−	0.886824i	−0.107393	−	0.994217i	$-0.534250\pi$
$163$	11.3222	−	11.3222i	0.886824	−	0.886824i	0.994217	+	0.107393i	$0.0342502\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			22.5032i			1.74135i	0.491858	+	0.870675i	$0.336318\pi$
$167$			22.5032i			1.74135i	−0.491858	+	0.870675i	$0.663682\pi$
$168$		0			0
$169$		−	12.8674i		−	0.989801i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−8.29491	+	8.29491i	−0.630650	+	0.630650i	−0.948231	−	0.317581i	$-0.897130\pi$
$173$	−8.29491	+	8.29491i	−0.630650	+	0.630650i	0.317581	+	0.948231i	$0.397130\pi$
$174$		0			0
$175$			2.69352i			0.203611i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−13.8824	−	13.8824i	−1.03762	−	1.03762i	−0.999264	−	0.0383581i	$-0.987787\pi$
$179$	−13.8824	−	13.8824i	−1.03762	−	1.03762i	−0.0383581	−	0.999264i	$-0.512213\pi$
$180$		0			0
$181$	−3.25070	+	3.25070i	−0.241623	+	0.241623i	−0.817521	−	0.575898i	$-0.804652\pi$
$181$	−3.25070	+	3.25070i	−0.241623	+	0.241623i	0.575898	+	0.817521i	$0.304652\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.84235			−0.282495
$186$		0			0
$187$	5.91226	+	5.91226i	0.432347	+	0.432347i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	3.41263			0.246929			0.123465	−	0.992349i	$-0.460599\pi$
$191$	3.41263			0.246929			0.123465	+	0.992349i	$0.460599\pi$
$192$		0			0
$193$	5.03380			0.362341			0.181170	−	0.983452i	$-0.442011\pi$
$193$	5.03380			0.362341			0.181170	+	0.983452i	$0.442011\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	14.7210	+	14.7210i	1.04883	+	1.04883i	0.998745	+	0.0500835i	$0.0159487\pi$
$197$	14.7210	+	14.7210i	1.04883	+	1.04883i	0.0500835	+	0.998745i	$0.484051\pi$
$198$		0			0
$199$	18.4722			1.30946			0.654731	−	0.755862i	$-0.272782\pi$
$199$	18.4722			1.30946			0.654731	+	0.755862i	$0.272782\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	10.7952	−	10.7952i	0.757675	−	0.757675i
$204$		0			0
$205$	3.69352	+	3.69352i	0.257967	+	0.257967i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	12.8295i		−	0.887436i
$210$		0			0
$211$	15.3879	−	15.3879i	1.05935	−	1.05935i	0.0612232	−	0.998124i	$-0.480500\pi$
$211$	15.3879	−	15.3879i	1.05935	−	1.05935i	0.998124	−	0.0612232i	$-0.0195002\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			5.32830i			0.363387i
$216$		0			0
$217$		−	23.5173i		−	1.59646i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.933499	+	0.933499i	−0.0627939	+	0.0627939i
$222$		0			0
$223$			1.32855i			0.0889664i	0.999010	+	0.0444832i	$0.0141641\pi$
$223$			1.32855i			0.0889664i	−0.999010	+	0.0444832i	$0.985836\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	11.2200	+	11.2200i	0.744694	+	0.744694i	0.973477	−	0.228783i	$-0.0734746\pi$
$227$	11.2200	+	11.2200i	0.744694	+	0.744694i	−0.228783	+	0.973477i	$0.573475\pi$
$228$		0			0
$229$	12.5775	−	12.5775i	0.831143	−	0.831143i	−0.156530	−	0.987673i	$-0.550031\pi$
$229$	12.5775	−	12.5775i	0.831143	−	0.831143i	0.987673	+	0.156530i	$0.0500308\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−24.2114			−1.58614			−0.793070	−	0.609131i	$-0.791518\pi$
$233$	−24.2114			−1.58614			−0.793070	+	0.609131i	$0.791518\pi$
$234$		0			0
$235$	−6.04043	−	6.04043i	−0.394034	−	0.394034i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−7.78017			−0.503258			−0.251629	−	0.967824i	$-0.580966\pi$
$239$	−7.78017			−0.503258			−0.251629	+	0.967824i	$0.580966\pi$
$240$		0			0
$241$	15.3993			0.991958			0.495979	−	0.868335i	$-0.334809\pi$
$241$	15.3993			0.991958			0.495979	+	0.868335i	$0.334809\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.180347	+	0.180347i	0.0115219	+	0.0115219i
$246$		0			0
$247$	2.02568			0.128891
$248$		0			0
$249$		0			0
$250$		0			0
$251$	8.35565	−	8.35565i	0.527404	−	0.527404i	−0.392393	−	0.919798i	$-0.628353\pi$
$251$	8.35565	−	8.35565i	0.527404	−	0.527404i	0.919798	+	0.392393i	$0.128353\pi$
$252$		0			0
$253$	5.12275	+	5.12275i	0.322065	+	0.322065i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	9.49727i		−	0.592424i	−0.955122	−	0.296212i	$-0.904277\pi$
$257$		−	9.49727i		−	0.592424i	0.955122	−	0.296212i	$-0.0957234\pi$
$258$		0			0
$259$	−7.31817	+	7.31817i	−0.454729	+	0.454729i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			5.01532i			0.309258i	0.987973	+	0.154629i	$0.0494182\pi$
$263$			5.01532i			0.309258i	−0.987973	+	0.154629i	$0.950582\pi$
$264$		0			0
$265$			3.19074i			0.196006i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4.00171	+	4.00171i	−0.243989	+	0.243989i	−0.818498	−	0.574509i	$-0.805193\pi$
$269$	−4.00171	+	4.00171i	−0.243989	+	0.243989i	0.574509	+	0.818498i	$0.305193\pi$
$270$		0			0
$271$		−	0.301843i		−	0.0183357i	−0.999958	−	0.00916783i	$-0.997082\pi$
$271$		−	0.301843i		−	0.0183357i	0.999958	−	0.00916783i	$-0.00291825\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.63072	+	1.63072i	0.0983359	+	0.0983359i
$276$		0			0
$277$	−15.8734	+	15.8734i	−0.953740	+	0.953740i	−0.998976	−	0.0452363i	$-0.985596\pi$
$277$	−15.8734	+	15.8734i	−0.953740	+	0.953740i	0.0452363	+	0.998976i	$0.485596\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−24.7501			−1.47647			−0.738233	−	0.674545i	$-0.764340\pi$
$281$	−24.7501			−1.47647			−0.738233	+	0.674545i	$0.764340\pi$
$282$		0			0
$283$	18.6055	+	18.6055i	1.10598	+	1.10598i	0.993674	+	0.112307i	$0.0358240\pi$
$283$	18.6055	+	18.6055i	1.10598	+	1.10598i	0.112307	+	0.993674i	$0.464176\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	14.0694			0.830490
$288$		0			0
$289$	3.85531			0.226783
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−7.28099	−	7.28099i	−0.425360	−	0.425360i	0.461684	−	0.887044i	$-0.347245\pi$
$293$	−7.28099	−	7.28099i	−0.425360	−	0.425360i	−0.887044	+	0.461684i	$0.847245\pi$
$294$		0			0
$295$	−4.85572			−0.282711
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.808842	+	0.808842i	−0.0467765	+	0.0467765i
$300$		0			0
$301$	10.1483	+	10.1483i	0.584940	+	0.584940i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			12.4388i			0.712244i
$306$		0			0
$307$	−8.30577	+	8.30577i	−0.474036	+	0.474036i	−0.903218	−	0.429182i	$-0.858802\pi$
$307$	−8.30577	+	8.30577i	−0.474036	+	0.474036i	0.429182	+	0.903218i	$0.358802\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	0.662553i		−	0.0375699i	−0.999824	−	0.0187850i	$-0.994020\pi$
$311$		−	0.662553i		−	0.0375699i	0.999824	−	0.0187850i	$-0.00597979\pi$
$312$		0			0
$313$		−	14.0883i		−	0.796316i	−0.917317	−	0.398158i	$-0.869649\pi$
$313$		−	14.0883i		−	0.796316i	0.917317	−	0.398158i	$-0.130351\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	24.1212	−	24.1212i	1.35478	−	1.35478i	0.474558	−	0.880224i	$-0.342608\pi$
$317$	24.1212	−	24.1212i	1.35478	−	1.35478i	0.880224	−	0.474558i	$-0.157392\pi$
$318$		0			0
$319$		−	13.0713i		−	0.731853i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	14.2619	+	14.2619i	0.793553	+	0.793553i
$324$		0			0
$325$	−0.257477	+	0.257477i	−0.0142823	+	0.0142823i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−23.0093			−1.26854
$330$		0			0
$331$	−15.3212	−	15.3212i	−0.842131	−	0.842131i	0.147005	−	0.989136i	$-0.453037\pi$
$331$	−15.3212	−	15.3212i	−0.842131	−	0.842131i	−0.989136	+	0.147005i	$0.953037\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	7.17595			0.392064
$336$		0			0
$337$	−25.6177			−1.39548			−0.697742	−	0.716349i	$-0.745812\pi$
$337$	−25.6177			−1.39548			−0.697742	+	0.716349i	$0.745812\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−14.2379	−	14.2379i	−0.771026	−	0.771026i
$342$		0			0
$343$	−18.1677			−0.980961
$344$		0			0
$345$		0			0
$346$		0			0
$347$	21.7672	−	21.7672i	1.16852	−	1.16852i	0.185969	−	0.982556i	$-0.440458\pi$
$347$	21.7672	−	21.7672i	1.16852	−	1.16852i	0.982556	−	0.185969i	$-0.0595424\pi$
$348$		0			0
$349$	−17.8368	−	17.8368i	−0.954782	−	0.954782i	0.0442393	−	0.999021i	$-0.485914\pi$
$349$	−17.8368	−	17.8368i	−0.954782	−	0.954782i	−0.999021	+	0.0442393i	$0.985914\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	17.8389i		−	0.949466i	−0.880130	−	0.474733i	$-0.842545\pi$
$353$		−	17.8389i		−	0.949466i	0.880130	−	0.474733i	$-0.157455\pi$
$354$		0			0
$355$	−3.17389	+	3.17389i	−0.168453	+	0.168453i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			14.6449i			0.772930i	0.922304	+	0.386465i	$0.126304\pi$
$359$			14.6449i			0.772930i	−0.922304	+	0.386465i	$0.873696\pi$
$360$		0			0
$361$		−	11.9481i		−	0.628846i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	9.15791	−	9.15791i	0.479347	−	0.479347i
$366$		0			0
$367$			31.1733i			1.62723i	0.581402	+	0.813616i	$0.302504\pi$
$367$			31.1733i			1.62723i	−0.581402	+	0.813616i	$0.697496\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	6.07710	+	6.07710i	0.315508	+	0.315508i
$372$		0			0
$373$	−17.4475	+	17.4475i	−0.903400	+	0.903400i	−0.995729	−	0.0923289i	$-0.970569\pi$
$373$	−17.4475	+	17.4475i	−0.903400	+	0.903400i	0.0923289	+	0.995729i	$0.470569\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2.06386			0.106294
$378$		0			0
$379$	9.36464	+	9.36464i	0.481029	+	0.481029i	0.905460	−	0.424431i	$-0.139526\pi$
$379$	9.36464	+	9.36464i	0.481029	+	0.481029i	−0.424431	+	0.905460i	$0.639526\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−7.43674			−0.380000			−0.190000	−	0.981784i	$-0.560849\pi$
$383$	−7.43674			−0.380000			−0.190000	+	0.981784i	$0.560849\pi$
$384$		0			0
$385$	6.21174			0.316580
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−15.0266	−	15.0266i	−0.761880	−	0.761880i	0.214782	−	0.976662i	$-0.431096\pi$
$389$	−15.0266	−	15.0266i	−0.761880	−	0.761880i	−0.976662	+	0.214782i	$0.931096\pi$
$390$		0			0
$391$	−11.3894			−0.575986
$392$		0			0
$393$		0			0
$394$		0			0
$395$	8.90847	−	8.90847i	0.448234	−	0.448234i
$396$		0			0
$397$	14.7912	+	14.7912i	0.742348	+	0.742348i	0.973029	−	0.230682i	$-0.0740956\pi$
$397$	14.7912	+	14.7912i	0.742348	+	0.742348i	−0.230682	+	0.973029i	$0.574096\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	20.6934i		−	1.03338i	−0.856172	−	0.516691i	$-0.827164\pi$
$401$		−	20.6934i		−	1.03338i	0.856172	−	0.516691i	$-0.172836\pi$
$402$		0			0
$403$	2.24805	−	2.24805i	0.111983	−	0.111983i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			8.86116i			0.439231i
$408$		0			0
$409$			13.3867i			0.661928i	0.943643	+	0.330964i	$0.107374\pi$
$409$			13.3867i			0.661928i	−0.943643	+	0.330964i	$0.892626\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−9.24823	+	9.24823i	−0.455076	+	0.455076i
$414$		0			0
$415$		−	1.77486i		−	0.0871245i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−16.8207	−	16.8207i	−0.821747	−	0.821747i	0.164612	−	0.986358i	$-0.447363\pi$
$419$	−16.8207	−	16.8207i	−0.821747	−	0.821747i	−0.986358	+	0.164612i	$0.947363\pi$
$420$		0			0
$421$	−10.5083	+	10.5083i	−0.512144	+	0.512144i	−0.915183	−	0.403039i	$-0.867954\pi$
$421$	−10.5083	+	10.5083i	−0.512144	+	0.512144i	0.403039	+	0.915183i	$0.367954\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.62556			−0.175866
$426$		0			0
$427$	23.6910	+	23.6910i	1.14649	+	1.14649i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−23.8537			−1.14899			−0.574496	−	0.818507i	$-0.694802\pi$
$431$	−23.8537			−1.14899			−0.574496	+	0.818507i	$0.694802\pi$
$432$		0			0
$433$	−39.5380			−1.90007			−0.950037	−	0.312137i	$-0.898955\pi$
$433$	−39.5380			−1.90007			−0.950037	+	0.312137i	$0.898955\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	12.3574	+	12.3574i	0.591134	+	0.591134i
$438$		0			0
$439$	−14.1242			−0.674113			−0.337057	−	0.941484i	$-0.609431\pi$
$439$	−14.1242			−0.674113			−0.337057	+	0.941484i	$0.609431\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−13.3827	+	13.3827i	−0.635832	+	0.635832i	−0.949525	−	0.313693i	$-0.898434\pi$
$443$	−13.3827	+	13.3827i	−0.635832	+	0.635832i	0.313693	+	0.949525i	$0.398434\pi$
$444$		0			0
$445$	−11.2289	−	11.2289i	−0.532300	−	0.532300i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			13.0799i			0.617279i	0.951179	+	0.308640i	$0.0998737\pi$
$449$			13.0799i			0.617279i	−0.951179	+	0.308640i	$0.900126\pi$
$450$		0			0
$451$	8.51793	−	8.51793i	0.401093	−	0.401093i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			0.980785i			0.0459799i
$456$		0			0
$457$			24.4868i			1.14545i	0.819749	+	0.572723i	$0.194113\pi$
$457$			24.4868i			1.14545i	−0.819749	+	0.572723i	$0.805887\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−17.0710	+	17.0710i	−0.795074	+	0.795074i	−0.982314	−	0.187240i	$-0.940046\pi$
$461$	−17.0710	+	17.0710i	−0.795074	+	0.795074i	0.187240	+	0.982314i	$0.440046\pi$
$462$		0			0
$463$		−	7.49328i		−	0.348242i	−0.984724	−	0.174121i	$-0.944292\pi$
$463$		−	7.49328i		−	0.348242i	0.984724	−	0.174121i	$-0.0557084\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−6.52043	−	6.52043i	−0.301729	−	0.301729i	0.539961	−	0.841690i	$-0.318439\pi$
$467$	−6.52043	−	6.52043i	−0.301729	−	0.301729i	−0.841690	+	0.539961i	$0.818439\pi$
$468$		0			0
$469$	13.6674	−	13.6674i	0.631100	−	0.631100i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	12.2880			0.565005
$474$		0			0
$475$	3.93371	+	3.93371i	0.180491	+	0.180491i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−15.3063			−0.699365			−0.349682	−	0.936868i	$-0.613710\pi$
$479$	−15.3063			−0.699365			−0.349682	+	0.936868i	$0.613710\pi$
$480$		0			0
$481$	−1.39911			−0.0637938
$482$		0			0
$483$		0			0
$484$		0			0
$485$	8.29491	+	8.29491i	0.376652	+	0.376652i
$486$		0			0
$487$	−7.64565			−0.346457			−0.173229	−	0.984882i	$-0.555420\pi$
$487$	−7.64565			−0.346457			−0.173229	+	0.984882i	$0.555420\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	11.3221	−	11.3221i	0.510959	−	0.510959i	−0.403861	−	0.914820i	$-0.632332\pi$
$491$	11.3221	−	11.3221i	0.510959	−	0.510959i	0.914820	+	0.403861i	$0.132332\pi$
$492$		0			0
$493$	14.5307	+	14.5307i	0.654429	+	0.654429i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			12.0900i			0.542312i
$498$		0			0
$499$	−26.8759	+	26.8759i	−1.20313	+	1.20313i	−0.229923	+	0.973209i	$0.573847\pi$
$499$	−26.8759	+	26.8759i	−1.20313	+	1.20313i	−0.973209	+	0.229923i	$0.926153\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	13.0561i		−	0.582142i	−0.956701	−	0.291071i	$-0.905988\pi$
$503$		−	13.0561i		−	0.582142i	0.956701	−	0.291071i	$-0.0940116\pi$
$504$		0			0
$505$			17.4284i			0.775556i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	20.8152	−	20.8152i	0.922617	−	0.922617i	−0.0745966	−	0.997214i	$-0.523767\pi$
$509$	20.8152	−	20.8152i	0.922617	−	0.922617i	0.997214	+	0.0745966i	$0.0237669\pi$
$510$		0			0
$511$		−	34.8844i		−	1.54320i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−5.28908	−	5.28908i	−0.233064	−	0.233064i
$516$		0			0
$517$	−13.9303	+	13.9303i	−0.612655	+	0.612655i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−30.3236			−1.32850			−0.664251	−	0.747510i	$-0.731249\pi$
$521$	−30.3236			−1.32850			−0.664251	+	0.747510i	$0.731249\pi$
$522$		0			0
$523$	−20.5573	−	20.5573i	−0.898909	−	0.898909i	0.0964307	−	0.995340i	$-0.469257\pi$
$523$	−20.5573	−	20.5573i	−0.898909	−	0.898909i	−0.995340	+	0.0964307i	$0.969257\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	31.6550			1.37892
$528$		0			0
$529$	13.1315			0.570936
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.34491	+	1.34491i	0.0582546	+	0.0582546i
$534$		0			0
$535$	2.56888			0.111062
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.415912	−	0.415912i	0.0179146	−	0.0179146i
$540$		0			0
$541$	6.99841	+	6.99841i	0.300885	+	0.300885i	0.841360	−	0.540475i	$-0.181755\pi$
$541$	6.99841	+	6.99841i	0.300885	+	0.300885i	−0.540475	+	0.841360i	$0.681755\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	2.91770i		−	0.124980i
$546$		0			0
$547$	−16.1411	+	16.1411i	−0.690145	+	0.690145i	−0.962264	−	0.272118i	$-0.912276\pi$
$547$	−16.1411	+	16.1411i	−0.690145	+	0.690145i	0.272118	+	0.962264i	$0.412276\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	31.5314i		−	1.34328i
$552$		0			0
$553$		−	33.9343i		−	1.44303i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−0.917780	+	0.917780i	−0.0388876	+	0.0388876i	−0.726283	−	0.687396i	$-0.758754\pi$
$557$	−0.917780	+	0.917780i	−0.0388876	+	0.0388876i	0.687396	+	0.726283i	$0.258754\pi$
$558$		0			0
$559$			1.94018i			0.0820610i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−14.1653	−	14.1653i	−0.596995	−	0.596995i	0.342516	−	0.939512i	$-0.388721\pi$
$563$	−14.1653	−	14.1653i	−0.596995	−	0.596995i	−0.939512	+	0.342516i	$0.888721\pi$
$564$		0			0
$565$	0.589337	−	0.589337i	0.0247936	−	0.0247936i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−27.3654			−1.14722			−0.573609	−	0.819129i	$-0.694457\pi$
$569$	−27.3654			−1.14722			−0.573609	+	0.819129i	$0.694457\pi$
$570$		0			0
$571$	5.03458	+	5.03458i	0.210690	+	0.210690i	0.804561	−	0.593870i	$-0.202401\pi$
$571$	5.03458	+	5.03458i	0.210690	+	0.210690i	−0.593870	+	0.804561i	$0.702401\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−3.14141			−0.131006
$576$		0			0
$577$	−23.6881			−0.986148			−0.493074	−	0.869987i	$-0.664127\pi$
$577$	−23.6881			−0.986148			−0.493074	+	0.869987i	$0.664127\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−3.38041	−	3.38041i	−0.140243	−	0.140243i
$582$		0			0
$583$	7.35843			0.304755
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−9.70371	+	9.70371i	−0.400515	+	0.400515i	−0.878415	−	0.477899i	$-0.841398\pi$
$587$	−9.70371	+	9.70371i	−0.400515	+	0.400515i	0.477899	+	0.878415i	$0.341398\pi$
$588$		0			0
$589$	−34.3455	−	34.3455i	−1.41518	−	1.41518i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	13.9772i		−	0.573975i	−0.957934	−	0.286988i	$-0.907346\pi$
$593$		−	13.9772i		−	0.573975i	0.957934	−	0.286988i	$-0.0926539\pi$
$594$		0			0
$595$	−6.90526	+	6.90526i	−0.283088	+	0.283088i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			1.84737i			0.0754816i	0.999288	+	0.0377408i	$0.0120161\pi$
$599$			1.84737i			0.0754816i	−0.999288	+	0.0377408i	$0.987984\pi$
$600$		0			0
$601$		−	4.94507i		−	0.201714i	−0.994901	−	0.100857i	$-0.967842\pi$
$601$		−	4.94507i		−	0.201714i	0.994901	−	0.100857i	$-0.0321584\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−4.01745	+	4.01745i	−0.163332	+	0.163332i
$606$		0			0
$607$			14.9158i			0.605412i	0.953084	+	0.302706i	$0.0978901\pi$
$607$			14.9158i			0.605412i	−0.953084	+	0.302706i	$0.902110\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.19949	−	2.19949i	−0.0889817	−	0.0889817i
$612$		0			0
$613$	−17.0558	+	17.0558i	−0.688875	+	0.688875i	−0.961983	−	0.273108i	$-0.911948\pi$
$613$	−17.0558	+	17.0558i	−0.688875	+	0.688875i	0.273108	+	0.961983i	$0.411948\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	38.2383			1.53942			0.769708	−	0.638396i	$-0.220402\pi$
$617$	38.2383			1.53942			0.769708	+	0.638396i	$0.220402\pi$
$618$		0			0
$619$	−22.5791	−	22.5791i	−0.907530	−	0.907530i	0.0885421	−	0.996072i	$-0.471779\pi$
$619$	−22.5791	−	22.5791i	−0.907530	−	0.907530i	−0.996072	+	0.0885421i	$0.971779\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−42.7732			−1.71367
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−9.85048	−	9.85048i	−0.392764	−	0.392764i
$630$		0			0
$631$	−26.8116			−1.06735			−0.533676	−	0.845689i	$-0.679190\pi$
$631$	−26.8116			−1.06735			−0.533676	+	0.845689i	$0.679190\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−6.95936	+	6.95936i	−0.276174	+	0.276174i
$636$		0			0
$637$	0.0656691	+	0.0656691i	0.00260191	+	0.00260191i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			23.4014i			0.924299i	0.886802	+	0.462149i	$0.152922\pi$
$641$			23.4014i			0.924299i	−0.886802	+	0.462149i	$0.847078\pi$
$642$		0			0
$643$	21.1408	−	21.1408i	0.833711	−	0.833711i	−0.154311	−	0.988022i	$-0.549316\pi$
$643$	21.1408	−	21.1408i	0.833711	−	0.833711i	0.988022	+	0.154311i	$0.0493159\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	2.60027i		−	0.102227i	−0.998693	−	0.0511137i	$-0.983723\pi$
$647$		−	2.60027i		−	0.102227i	0.998693	−	0.0511137i	$-0.0162771\pi$
$648$		0			0
$649$			11.1982i			0.439566i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	18.6343	−	18.6343i	0.729218	−	0.729218i	−0.241246	−	0.970464i	$-0.577556\pi$
$653$	18.6343	−	18.6343i	0.729218	−	0.729218i	0.970464	+	0.241246i	$0.0775561\pi$
$654$		0			0
$655$		−	5.02972i		−	0.196527i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−28.0447	−	28.0447i	−1.09247	−	1.09247i	−0.995265	−	0.0972037i	$-0.969010\pi$
$659$	−28.0447	−	28.0447i	−1.09247	−	1.09247i	−0.0972037	−	0.995265i	$-0.530990\pi$
$660$		0			0
$661$	6.25942	−	6.25942i	0.243463	−	0.243463i	−0.574818	−	0.818281i	$-0.694927\pi$
$661$	6.25942	−	6.25942i	0.243463	−	0.243463i	0.818281	+	0.574818i	$0.194927\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	14.9843			0.581067
$666$		0			0
$667$	12.5903	+	12.5903i	0.487498	+	0.487498i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	28.6861			1.10742
$672$		0			0
$673$	28.8581			1.11240			0.556199	−	0.831049i	$-0.312259\pi$
$673$	28.8581			1.11240			0.556199	+	0.831049i	$0.312259\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	11.7021	+	11.7021i	0.449749	+	0.449749i	0.895271	−	0.445522i	$-0.146982\pi$
$677$	11.7021	+	11.7021i	0.449749	+	0.449749i	−0.445522	+	0.895271i	$0.646982\pi$
$678$		0			0
$679$	31.5971			1.21258
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−26.3064	+	26.3064i	−1.00659	+	1.00659i	−0.00660952	+	0.999978i	$0.502104\pi$
$683$	−26.3064	+	26.3064i	−1.00659	+	1.00659i	−0.999978	+	0.00660952i	$0.997896\pi$
$684$		0			0
$685$	11.4012	+	11.4012i	0.435616	+	0.435616i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			1.16184i			0.0442625i
$690$		0			0
$691$	−23.7199	+	23.7199i	−0.902347	+	0.902347i	−0.995639	−	0.0932919i	$-0.970261\pi$
$691$	−23.7199	+	23.7199i	−0.902347	+	0.902347i	0.0932919	+	0.995639i	$0.470261\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			6.62276i			0.251215i
$696$		0			0
$697$			18.9378i			0.717322i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−18.7977	+	18.7977i	−0.709981	+	0.709981i	−0.966531	−	0.256550i	$-0.917414\pi$
$701$	−18.7977	+	18.7977i	−0.709981	+	0.709981i	0.256550	+	0.966531i	$0.417414\pi$
$702$		0			0
$703$			21.3754i			0.806188i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	33.1943	+	33.1943i	1.24840	+	1.24840i
$708$		0			0
$709$	−16.5441	+	16.5441i	−0.621329	+	0.621329i	−0.945871	−	0.324543i	$-0.894790\pi$
$709$	−16.5441	+	16.5441i	−0.621329	+	0.621329i	0.324543	+	0.945871i	$0.394790\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	27.4279			1.02718
$714$		0			0
$715$	0.593789	+	0.593789i	0.0222064	+	0.0222064i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−35.4446			−1.32186			−0.660930	−	0.750447i	$-0.729838\pi$
$719$	−35.4446			−1.32186			−0.660930	+	0.750447i	$0.729838\pi$
$720$		0			0
$721$	−20.1472			−0.750321
$722$		0			0
$723$		0			0
$724$		0			0
$725$	4.00785	+	4.00785i	0.148848	+	0.148848i
$726$		0			0
$727$	−47.5983			−1.76532			−0.882662	−	0.470008i	$-0.844251\pi$
$727$	−47.5983			−1.76532			−0.882662	+	0.470008i	$0.844251\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−13.6600	+	13.6600i	−0.505232	+	0.505232i
$732$		0			0
$733$	6.41058	+	6.41058i	0.236780	+	0.236780i	0.815515	−	0.578735i	$-0.196454\pi$
$733$	6.41058	+	6.41058i	0.236780	+	0.236780i	−0.578735	+	0.815515i	$0.696454\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	16.5490i		−	0.609592i
$738$		0			0
$739$	21.1683	−	21.1683i	0.778687	−	0.778687i	−0.200920	−	0.979608i	$-0.564393\pi$
$739$	21.1683	−	21.1683i	0.778687	−	0.778687i	0.979608	+	0.200920i	$0.0643933\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	50.7869i		−	1.86319i	−0.363498	−	0.931595i	$-0.618418\pi$
$743$		−	50.7869i		−	1.86319i	0.363498	−	0.931595i	$-0.381582\pi$
$744$		0			0
$745$			18.5171i			0.678415i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	4.89270	−	4.89270i	0.178775	−	0.178775i
$750$		0			0
$751$			46.9508i			1.71326i	0.515932	+	0.856630i	$0.327446\pi$
$751$			46.9508i			1.71326i	−0.515932	+	0.856630i	$0.672554\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−9.94727	−	9.94727i	−0.362018	−	0.362018i
$756$		0			0
$757$	−2.52177	+	2.52177i	−0.0916551	+	0.0916551i	−0.751448	−	0.659793i	$-0.770644\pi$
$757$	−2.52177	+	2.52177i	−0.0916551	+	0.0916551i	0.659793	+	0.751448i	$0.270644\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	48.0326			1.74118			0.870590	−	0.492009i	$-0.163737\pi$
$761$	48.0326			1.74118			0.870590	+	0.492009i	$0.163737\pi$
$762$		0			0
$763$	−5.55707	−	5.55707i	−0.201179	−	0.201179i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.76810			−0.0638424
$768$		0			0
$769$	18.3681			0.662372			0.331186	−	0.943565i	$-0.392551\pi$
$769$	18.3681			0.662372			0.331186	+	0.943565i	$0.392551\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−9.93101	−	9.93101i	−0.357194	−	0.357194i	0.505584	−	0.862778i	$-0.331277\pi$
$773$	−9.93101	−	9.93101i	−0.357194	−	0.357194i	−0.862778	+	0.505584i	$0.831277\pi$
$774$		0			0
$775$	8.73107			0.313629
$776$		0			0
$777$		0			0
$778$		0			0
$779$	20.5474	−	20.5474i	0.736188	−	0.736188i
$780$		0			0
$781$	7.31957	+	7.31957i	0.261915	+	0.261915i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	7.72430i		−	0.275692i
$786$		0			0
$787$	−13.5531	+	13.5531i	−0.483116	+	0.483116i	−0.906125	−	0.423010i	$-0.860974\pi$
$787$	−13.5531	+	13.5531i	−0.483116	+	0.483116i	0.423010	+	0.906125i	$0.360974\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	2.24491i		−	0.0798198i
$792$		0			0
$793$			4.52931i			0.160841i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	4.93931	−	4.93931i	0.174959	−	0.174959i	−0.614195	−	0.789154i	$-0.710519\pi$
$797$	4.93931	−	4.93931i	0.174959	−	0.174959i	0.789154	+	0.614195i	$0.210519\pi$
$798$		0			0
$799$		−	30.9712i		−	1.09568i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−21.1198	−	21.1198i	−0.745302	−	0.745302i
$804$		0			0
$805$	−5.98315	+	5.98315i	−0.210878	+	0.210878i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	4.90181			0.172339			0.0861693	−	0.996281i	$-0.472537\pi$
$809$	4.90181			0.172339			0.0861693	+	0.996281i	$0.472537\pi$
$810$		0			0
$811$	−3.18129	−	3.18129i	−0.111710	−	0.111710i	0.649042	−	0.760752i	$-0.275170\pi$
$811$	−3.18129	−	3.18129i	−0.111710	−	0.111710i	−0.760752	+	0.649042i	$0.775170\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	16.0120			0.560877
$816$		0			0
$817$	29.6419			1.03704
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−13.9062	−	13.9062i	−0.485331	−	0.485331i	0.421498	−	0.906829i	$-0.361504\pi$
$821$	−13.9062	−	13.9062i	−0.485331	−	0.485331i	−0.906829	+	0.421498i	$0.861504\pi$
$822$		0			0
$823$	−46.8606			−1.63346			−0.816729	−	0.577022i	$-0.804215\pi$
$823$	−46.8606			−1.63346			−0.816729	+	0.577022i	$0.804215\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−15.8928	+	15.8928i	−0.552646	+	0.552646i	−0.927204	−	0.374558i	$-0.877795\pi$
$827$	−15.8928	+	15.8928i	−0.552646	+	0.552646i	0.374558	+	0.927204i	$0.377795\pi$
$828$		0			0
$829$	−12.4155	−	12.4155i	−0.431209	−	0.431209i	0.457830	−	0.889040i	$-0.348627\pi$
$829$	−12.4155	−	12.4155i	−0.431209	−	0.431209i	−0.889040	+	0.457830i	$0.848627\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			0.924694i			0.0320387i
$834$		0			0
$835$	−15.9122	+	15.9122i	−0.550664	+	0.550664i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			34.3222i			1.18493i	0.805595	+	0.592467i	$0.201846\pi$
$839$			34.3222i			1.18493i	−0.805595	+	0.592467i	$0.798154\pi$
$840$		0			0
$841$		−	3.12565i		−	0.107781i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	9.09863	−	9.09863i	0.313003	−	0.313003i
$846$		0			0
$847$			15.3033i			0.525828i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−8.53507	−	8.53507i	−0.292578	−	0.292578i
$852$		0			0
$853$	−1.81852	+	1.81852i	−0.0622648	+	0.0622648i	−0.737554	−	0.675289i	$-0.764019\pi$
$853$	−1.81852	+	1.81852i	−0.0622648	+	0.0622648i	0.675289	+	0.737554i	$0.264019\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	9.53296			0.325640			0.162820	−	0.986656i	$-0.447941\pi$
$857$	9.53296			0.325640			0.162820	+	0.986656i	$0.447941\pi$
$858$		0			0
$859$	−37.7308	−	37.7308i	−1.28736	−	1.28736i	−0.936386	−	0.350971i	$-0.885851\pi$
$859$	−37.7308	−	37.7308i	−1.28736	−	1.28736i	−0.350971	−	0.936386i	$-0.614149\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−20.0878			−0.683795			−0.341898	−	0.939737i	$-0.611070\pi$
$863$	−20.0878			−0.683795			−0.341898	+	0.939737i	$0.611070\pi$
$864$		0			0
$865$	−11.7308			−0.398858
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−20.5446	−	20.5446i	−0.696926	−	0.696926i
$870$		0			0
$871$	2.61296			0.0885368
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.90461	+	1.90461i	−0.0643874	+	0.0643874i
$876$		0			0
$877$	−12.7083	−	12.7083i	−0.429130	−	0.429130i	0.459202	−	0.888332i	$-0.348135\pi$
$877$	−12.7083	−	12.7083i	−0.429130	−	0.429130i	−0.888332	+	0.459202i	$0.848135\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	45.4943i		−	1.53274i	−0.642399	−	0.766370i	$-0.722061\pi$
$881$		−	45.4943i		−	1.53274i	0.642399	−	0.766370i	$-0.277939\pi$
$882$		0			0
$883$	22.8514	−	22.8514i	0.769013	−	0.769013i	−0.208920	−	0.977933i	$-0.566995\pi$
$883$	22.8514	−	22.8514i	0.769013	−	0.769013i	0.977933	+	0.208920i	$0.0669949\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	57.5409i		−	1.93203i	−0.258477	−	0.966017i	$-0.583221\pi$
$887$		−	57.5409i		−	1.93203i	0.258477	−	0.966017i	$-0.416779\pi$
$888$		0			0
$889$			26.5097i			0.889106i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−33.6035	+	33.6035i	−1.12450	+	1.12450i
$894$		0			0
$895$		−	19.6327i		−	0.656250i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−34.9928	−	34.9928i	−1.16708	−	1.16708i
$900$		0			0
$901$	−8.17997	+	8.17997i	−0.272514	+	0.272514i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−4.59719			−0.152816
$906$		0			0
$907$	−21.1176	−	21.1176i	−0.701197	−	0.701197i	0.263470	−	0.964668i	$-0.415133\pi$
$907$	−21.1176	−	21.1176i	−0.701197	−	0.701197i	−0.964668	+	0.263470i	$0.915133\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−9.64649			−0.319602			−0.159801	−	0.987149i	$-0.551085\pi$
$911$	−9.64649			−0.319602			−0.159801	+	0.987149i	$0.551085\pi$
$912$		0			0
$913$	−4.09315			−0.135463
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−9.57963	−	9.57963i	−0.316347	−	0.316347i
$918$		0			0
$919$	56.2078			1.85412			0.927062	−	0.374907i	$-0.122325\pi$
$919$	56.2078			1.85412			0.927062	+	0.374907i	$0.122325\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1.15570	+	1.15570i	−0.0380404	+	0.0380404i
$924$		0			0
$925$	−2.71695	−	2.71695i	−0.0893329	−	0.0893329i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			9.72363i			0.319022i	0.987196	+	0.159511i	$0.0509918\pi$
$929$			9.72363i			0.319022i	−0.987196	+	0.159511i	$0.949008\pi$
$930$		0			0
$931$	1.00329	−	1.00329i	0.0328814	−	0.0328814i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			8.36120i			0.273440i
$936$		0			0
$937$			40.5164i			1.32361i	0.749675	+	0.661806i	$0.230210\pi$
$937$			40.5164i			1.32361i	−0.749675	+	0.661806i	$0.769790\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	20.4541	−	20.4541i	0.666786	−	0.666786i	−0.290185	−	0.956971i	$-0.593717\pi$
$941$	20.4541	−	20.4541i	0.666786	−	0.666786i	0.956971	+	0.290185i	$0.0937168\pi$
$942$		0			0
$943$			16.4089i			0.534348i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	8.93200	+	8.93200i	0.290251	+	0.290251i	0.837179	−	0.546928i	$-0.184203\pi$
$947$	8.93200	+	8.93200i	0.290251	+	0.290251i	−0.546928	+	0.837179i	$0.684203\pi$
$948$		0			0
$949$	3.33465	−	3.33465i	0.108247	−	0.108247i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	18.2335			0.590641			0.295321	−	0.955398i	$-0.404574\pi$
$953$	18.2335			0.590641			0.295321	+	0.955398i	$0.404574\pi$
$954$		0			0
$955$	2.41309	+	2.41309i	0.0780858	+	0.0780858i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	43.4294			1.40241
$960$		0			0
$961$	−45.2317			−1.45909
$962$		0			0
$963$		0			0
$964$		0			0
$965$	3.55944	+	3.55944i	0.114582	+	0.114582i
$966$		0			0
$967$	54.2931			1.74595			0.872974	−	0.487767i	$-0.162188\pi$
$967$	54.2931			1.74595			0.872974	+	0.487767i	$0.162188\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−11.9461	+	11.9461i	−0.383370	+	0.383370i	−0.872315	−	0.488945i	$-0.837382\pi$
$971$	−11.9461	+	11.9461i	−0.383370	+	0.383370i	0.488945	+	0.872315i	$0.337382\pi$
$972$		0			0
$973$	12.6137	+	12.6137i	0.404378	+	0.404378i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	9.61818i		−	0.307713i	−0.988093	−	0.153856i	$-0.950831\pi$
$977$		−	9.61818i		−	0.307713i	0.988093	−	0.153856i	$-0.0491693\pi$
$978$		0			0
$979$	−25.8958	+	25.8958i	−0.827634	+	0.827634i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	28.5641i		−	0.911055i	−0.890222	−	0.455527i	$-0.849451\pi$
$983$		−	28.5641i		−	0.911055i	0.890222	−	0.455527i	$-0.150549\pi$
$984$		0			0
$985$			20.8187i			0.663337i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−11.8358	+	11.8358i	−0.376358	+	0.376358i
$990$		0			0
$991$			44.7917i			1.42285i	0.702760	+	0.711427i	$0.251951\pi$
$991$			44.7917i			1.42285i	−0.702760	+	0.711427i	$0.748049\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	13.0618	+	13.0618i	0.414088	+	0.414088i
$996$		0			0
$997$	34.3446	−	34.3446i	1.08770	−	1.08770i	0.0919384	−	0.995765i	$-0.470694\pi$
$997$	34.3446	−	34.3446i	1.08770	−	1.08770i	0.995765	−	0.0919384i	$-0.0293063\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.bl.b.431.12		24
3.2	odd	2	inner	2880.2.bl.b.431.5		24
4.3	odd	2		720.2.bl.b.611.10	yes	24
12.11	even	2		720.2.bl.b.611.3	yes	24
16.5	even	4		720.2.bl.b.251.3	✓	24
16.11	odd	4	inner	2880.2.bl.b.1871.5		24
48.5	odd	4		720.2.bl.b.251.10	yes	24
48.11	even	4	inner	2880.2.bl.b.1871.12		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.bl.b.251.3	✓	24	16.5	even	4
720.2.bl.b.251.10	yes	24	48.5	odd	4
720.2.bl.b.611.3	yes	24	12.11	even	2
720.2.bl.b.611.10	yes	24	4.3	odd	2
2880.2.bl.b.431.5		24	3.2	odd	2	inner
2880.2.bl.b.431.12		24	1.1	even	1	trivial
2880.2.bl.b.1871.5		24	16.11	odd	4	inner
2880.2.bl.b.1871.12		24	48.11	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.bl (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 + 0.707107i) q^{5} +2.69352 q^{7} +O(q^{10})\)	\(q+(0.707107 + 0.707107i) q^{5} +2.69352 q^{7} +(1.63072 - 1.63072i) q^{11} +(0.257477 + 0.257477i) q^{13} +3.62556i q^{17} +(3.93371 - 3.93371i) q^{19} +3.14141i q^{23} +1.00000i q^{25} +(4.00785 - 4.00785i) q^{29} -8.73107i q^{31} +(1.90461 + 1.90461i) q^{35} +(-2.71695 + 2.71695i) q^{37} +5.22343 q^{41} +(3.76768 + 3.76768i) q^{43} -8.54245 q^{47} +0.255048 q^{49} +(2.25619 + 2.25619i) q^{53} +2.30618 q^{55} +(-3.43351 + 3.43351i) q^{59} +(8.79557 + 8.79557i) q^{61} +0.364128i q^{65} +(5.07416 - 5.07416i) q^{67} +4.48856i q^{71} -12.9512i q^{73} +(4.39237 - 4.39237i) q^{77} -12.5985i q^{79} +(-1.25502 - 1.25502i) q^{83} +(-2.56366 + 2.56366i) q^{85} -15.8800 q^{89} +(0.693520 + 0.693520i) q^{91} +5.56310 q^{95} +11.7308 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 8 q^{7}+O(q^{10}) \) 24 * q + 8 * q^7	\( 24 q + 8 q^{7} + 24 q^{19} + 8 q^{37} - 48 q^{43} + 24 q^{49} - 24 q^{55} + 40 q^{61} + 40 q^{67} + 24 q^{85} - 40 q^{91} - 16 q^{97}+O(q^{100}) \) 24 * q + 8 * q^7 + 24 * q^19 + 8 * q^37 - 48 * q^43 + 24 * q^49 - 24 * q^55 + 40 * q^61 + 40 * q^67 + 24 * q^85 - 40 * q^91 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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