LMFDB - Embedded newform 2880.2.bc.a.593.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(593,2880)]

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, 2, 3]))

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

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

chi := DirichletCharacter("2880.593");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.bc (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:	$96$
Relative dimension:	$48$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 720)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			593.17
Character	$\chi$	$=$	2880.593
Dual form			2880.2.bc.a.1457.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.09255 + 1.95099i) q^{5} +(1.70979 + 1.70979i) q^{7} +O(q^{10})$	$q+(-1.09255 + 1.95099i) q^{5} +(1.70979 + 1.70979i) q^{7} +(0.701463 + 0.701463i) q^{11} -0.350324i q^{13} +(4.15556 + 4.15556i) q^{17} +(-5.84772 + 5.84772i) q^{19} +(3.23953 + 3.23953i) q^{23} +(-2.61269 - 4.26308i) q^{25} +(1.23660 + 1.23660i) q^{29} -5.33324 q^{31} +(-5.20380 + 1.46775i) q^{35} -11.3001i q^{37} +5.43035 q^{41} -3.39882 q^{43} +(0.157234 + 0.157234i) q^{47} -1.15324i q^{49} -4.57970i q^{53} +(-2.13492 + 0.602163i) q^{55} +(-9.97123 + 9.97123i) q^{59} +(6.17129 + 6.17129i) q^{61} +(0.683477 + 0.382745i) q^{65} +5.97529 q^{67} +9.10388 q^{71} +(-9.21077 + 9.21077i) q^{73} +2.39871i q^{77} -3.05255i q^{79} -2.92670 q^{83} +(-12.6476 + 3.56729i) q^{85} +12.0075i q^{89} +(0.598980 - 0.598980i) q^{91} +(-5.01991 - 17.7977i) q^{95} +(2.74811 - 2.74811i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 96 q+O(q^{10}) $ 96 * q	$ 96 q - 16 q^{19} - 64 q^{43} + 32 q^{61}+O(q^{100}) $ 96 * q - 16 * q^19 - 64 * q^43 + 32 * q^61

Display 100 coefficients 10 coefficients

Character values

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

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

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
$5$	−1.09255	+	1.95099i	−0.488601	+	0.872507i
$5$	−1.09255	+	1.95099i	−0.488601	+	0.872507i
$6$		0			0
$7$	1.70979	+	1.70979i	0.646240	+	0.646240i	0.952082	−	0.305843i	$-0.0989381\pi$
$7$	1.70979	+	1.70979i	0.646240	+	0.646240i	−0.305843	+	0.952082i	$0.598938\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.701463	+	0.701463i	0.211499	+	0.211499i	0.804904	−	0.593405i	$-0.202217\pi$
$11$	0.701463	+	0.701463i	0.211499	+	0.211499i	−0.593405	+	0.804904i	$0.702217\pi$
$12$		0			0
$13$		−	0.350324i		−	0.0971624i	−0.998819	−	0.0485812i	$-0.984530\pi$
$13$		−	0.350324i		−	0.0971624i	0.998819	−	0.0485812i	$-0.0154700\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.15556	+	4.15556i	1.00787	+	1.00787i	0.999969	+	0.00790150i	$0.00251515\pi$
$17$	4.15556	+	4.15556i	1.00787	+	1.00787i	0.00790150	+	0.999969i	$0.497485\pi$
$18$		0			0
$19$	−5.84772	+	5.84772i	−1.34156	+	1.34156i	−0.447050	+	0.894509i	$0.647525\pi$
$19$	−5.84772	+	5.84772i	−1.34156	+	1.34156i	−0.894509	+	0.447050i	$0.852475\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.23953	+	3.23953i	0.675488	+	0.675488i	0.958976	−	0.283488i	$-0.0914916\pi$
$23$	3.23953	+	3.23953i	0.675488	+	0.675488i	−0.283488	+	0.958976i	$0.591492\pi$
$24$		0			0
$25$	−2.61269	−	4.26308i	−0.522537	−	0.852616i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.23660	+	1.23660i	0.229631	+	0.229631i	0.812538	−	0.582908i	$-0.198085\pi$
$29$	1.23660	+	1.23660i	0.229631	+	0.229631i	−0.582908	+	0.812538i	$0.698085\pi$
$30$		0			0
$31$	−5.33324			−0.957878			−0.478939	−	0.877848i	$-0.658978\pi$
$31$	−5.33324			−0.957878			−0.478939	+	0.877848i	$0.658978\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−5.20380	+	1.46775i	−0.879602	+	0.248095i
$36$		0			0
$37$		−	11.3001i		−	1.85773i	−0.370420	−	0.928864i	$-0.620786\pi$
$37$		−	11.3001i		−	1.85773i	0.370420	−	0.928864i	$-0.379214\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	5.43035			0.848079			0.424039	−	0.905644i	$-0.360612\pi$
$41$	5.43035			0.848079			0.424039	+	0.905644i	$0.360612\pi$
$42$		0			0
$43$	−3.39882			−0.518315			−0.259158	−	0.965835i	$-0.583445\pi$
$43$	−3.39882			−0.518315			−0.259158	+	0.965835i	$0.583445\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.157234	+	0.157234i	0.0229350	+	0.0229350i	0.718481	−	0.695546i	$-0.244838\pi$
$47$	0.157234	+	0.157234i	0.0229350	+	0.0229350i	−0.695546	+	0.718481i	$0.744838\pi$
$48$		0			0
$49$		−	1.15324i		−	0.164749i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	4.57970i		−	0.629070i	−0.949246	−	0.314535i	$-0.898151\pi$
$53$		−	4.57970i		−	0.629070i	0.949246	−	0.314535i	$-0.101849\pi$
$54$		0			0
$55$	−2.13492	+	0.602163i	−0.287873	+	0.0811957i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−9.97123	+	9.97123i	−1.29814	+	1.29814i	−0.368527	+	0.929617i	$0.620138\pi$
$59$	−9.97123	+	9.97123i	−1.29814	+	1.29814i	−0.929617	+	0.368527i	$0.879862\pi$
$60$		0			0
$61$	6.17129	+	6.17129i	0.790152	+	0.790152i	0.981519	−	0.191366i	$-0.0612918\pi$
$61$	6.17129	+	6.17129i	0.790152	+	0.790152i	−0.191366	+	0.981519i	$0.561292\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.683477	+	0.382745i	0.0847749	+	0.0474737i
$66$		0			0
$67$	5.97529			0.729998			0.364999	−	0.931008i	$-0.381069\pi$
$67$	5.97529			0.729998			0.364999	+	0.931008i	$0.381069\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	9.10388			1.08043			0.540216	−	0.841526i	$-0.318343\pi$
$71$	9.10388			1.08043			0.540216	+	0.841526i	$0.318343\pi$
$72$		0			0
$73$	−9.21077	+	9.21077i	−1.07804	+	1.07804i	−0.0813543	+	0.996685i	$0.525925\pi$
$73$	−9.21077	+	9.21077i	−1.07804	+	1.07804i	−0.996685	+	0.0813543i	$0.974075\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			2.39871i			0.273358i
$78$		0			0
$79$		−	3.05255i		−	0.343438i	−0.985146	−	0.171719i	$-0.945068\pi$
$79$		−	3.05255i		−	0.343438i	0.985146	−	0.171719i	$-0.0549322\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−2.92670			−0.321247			−0.160623	−	0.987016i	$-0.551350\pi$
$83$	−2.92670			−0.321247			−0.160623	+	0.987016i	$0.551350\pi$
$84$		0			0
$85$	−12.6476	+	3.56729i	−1.37182	+	0.386927i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			12.0075i			1.27279i	0.771364	+	0.636395i	$0.219575\pi$
$89$			12.0075i			1.27279i	−0.771364	+	0.636395i	$0.780425\pi$
$90$		0			0
$91$	0.598980	−	0.598980i	0.0627902	−	0.0627902i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−5.01991	−	17.7977i	−0.515032	−	1.82601i
$96$		0			0
$97$	2.74811	−	2.74811i	0.279029	−	0.279029i	−0.553693	−	0.832721i	$-0.686782\pi$
$97$	2.74811	−	2.74811i	0.279029	−	0.279029i	0.832721	+	0.553693i	$0.186782\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.51459	−	6.51459i	−0.648226	−	0.648226i	0.304338	−	0.952564i	$-0.401565\pi$
$101$	−6.51459	−	6.51459i	−0.648226	−	0.648226i	−0.952564	+	0.304338i	$0.901565\pi$
$102$		0			0
$103$	−3.77640	+	3.77640i	−0.372100	+	0.372100i	−0.868242	−	0.496142i	$-0.834750\pi$
$103$	−3.77640	+	3.77640i	−0.372100	+	0.372100i	0.496142	+	0.868242i	$0.334750\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.45784			−0.720977			−0.360488	−	0.932764i	$-0.617390\pi$
$107$	−7.45784			−0.720977			−0.360488	+	0.932764i	$0.617390\pi$
$108$		0			0
$109$	1.73676	−	1.73676i	0.166352	−	0.166352i	−0.619022	−	0.785374i	$-0.712471\pi$
$109$	1.73676	−	1.73676i	0.166352	−	0.166352i	0.785374	+	0.619022i	$0.212471\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−6.83534	+	6.83534i	−0.643014	+	0.643014i	−0.951295	−	0.308281i	$-0.900246\pi$
$113$	−6.83534	+	6.83534i	−0.643014	+	0.643014i	0.308281	+	0.951295i	$0.400246\pi$
$114$		0			0
$115$	−9.85960	+	2.78094i	−0.919412	+	0.259324i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			14.2102i			1.30265i
$120$		0			0
$121$		−	10.0159i		−	0.910536i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.1717	−	0.439700i	0.999226	−	0.0393280i
$126$		0			0
$127$	−4.83381	+	4.83381i	−0.428931	+	0.428931i	−0.888264	−	0.459333i	$-0.848088\pi$
$127$	−4.83381	+	4.83381i	−0.428931	+	0.428931i	0.459333	+	0.888264i	$0.348088\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.26253	+	3.26253i	−0.285049	+	0.285049i	−0.835119	−	0.550070i	$-0.814601\pi$
$131$	−3.26253	+	3.26253i	−0.285049	+	0.285049i	0.550070	+	0.835119i	$0.314601\pi$
$132$		0			0
$133$	−19.9967			−1.73394
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2.01493	−	2.01493i	0.172147	−	0.172147i	−0.615775	−	0.787922i	$-0.711157\pi$
$137$	2.01493	−	2.01493i	0.172147	−	0.172147i	0.787922	+	0.615775i	$0.211157\pi$
$138$		0			0
$139$	−12.1616	−	12.1616i	−1.03153	−	1.03153i	−0.999486	−	0.0320471i	$-0.989797\pi$
$139$	−12.1616	−	12.1616i	−1.03153	−	1.03153i	−0.0320471	−	0.999486i	$-0.510203\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.245739	−	0.245739i	0.0205498	−	0.0205498i
$144$		0			0
$145$	−3.76363	+	1.06155i	−0.312552	+	0.0881566i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.50261	−	5.50261i	0.450792	−	0.450792i	−0.444826	−	0.895617i	$-0.646734\pi$
$149$	5.50261	−	5.50261i	0.450792	−	0.450792i	0.895617	+	0.444826i	$0.146734\pi$
$150$		0			0
$151$		−	12.6800i		−	1.03188i	−0.856624	−	0.515941i	$-0.827443\pi$
$151$		−	12.6800i		−	1.03188i	0.856624	−	0.515941i	$-0.172557\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	5.82681	−	10.4051i	0.468020	−	0.835755i
$156$		0			0
$157$	−15.2601			−1.21789			−0.608943	−	0.793214i	$-0.708406\pi$
$157$	−15.2601			−1.21789			−0.608943	+	0.793214i	$0.708406\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			11.0778i			0.873054i
$162$		0			0
$163$			22.3339i			1.74932i	0.484733	+	0.874662i	$0.338917\pi$
$163$			22.3339i			1.74932i	−0.484733	+	0.874662i	$0.661083\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−7.61241	+	7.61241i	−0.589066	+	0.589066i	−0.937378	−	0.348313i	$-0.886755\pi$
$167$	−7.61241	+	7.61241i	−0.589066	+	0.589066i	0.348313	+	0.937378i	$0.386755\pi$
$168$		0			0
$169$	12.8773			0.990559
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−14.2692			−1.08487			−0.542435	−	0.840098i	$-0.682498\pi$
$173$	−14.2692			−1.08487			−0.542435	+	0.840098i	$0.682498\pi$
$174$		0			0
$175$	2.82183	−	11.7561i	0.213310	−	0.888679i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1.87756	−	1.87756i	−0.140336	−	0.140336i	0.633449	−	0.773785i	$-0.281639\pi$
$179$	−1.87756	−	1.87756i	−0.140336	−	0.140336i	−0.773785	+	0.633449i	$0.781639\pi$
$180$		0			0
$181$	−11.6354	+	11.6354i	−0.864850	+	0.864850i	−0.991897	−	0.127047i	$-0.959450\pi$
$181$	−11.6354	+	11.6354i	−0.864850	+	0.864850i	0.127047	+	0.991897i	$0.459450\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	22.0464	+	12.3459i	1.62088	+	0.907689i
$186$		0			0
$187$			5.82994i			0.426327i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	6.81196i		−	0.492896i	−0.969156	−	0.246448i	$-0.920737\pi$
$191$		−	6.81196i		−	0.492896i	0.969156	−	0.246448i	$-0.0792635\pi$
$192$		0			0
$193$	7.32620	+	7.32620i	0.527352	+	0.527352i	0.919782	−	0.392430i	$-0.128366\pi$
$193$	7.32620	+	7.32620i	0.527352	+	0.527352i	−0.392430	+	0.919782i	$0.628366\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−8.49110			−0.604966			−0.302483	−	0.953155i	$-0.597816\pi$
$197$	−8.49110			−0.604966			−0.302483	+	0.953155i	$0.597816\pi$
$198$		0			0
$199$	−19.3403			−1.37100			−0.685498	−	0.728074i	$-0.740416\pi$
$199$	−19.3403			−1.37100			−0.685498	+	0.728074i	$0.740416\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			4.22865i			0.296793i
$204$		0			0
$205$	−5.93291	+	10.5945i	−0.414373	+	0.739955i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−8.20392			−0.567477
$210$		0			0
$211$	11.5189	+	11.5189i	0.792993	+	0.792993i	0.981980	−	0.188987i	$-0.0605204\pi$
$211$	11.5189	+	11.5189i	0.792993	+	0.792993i	−0.188987	+	0.981980i	$0.560520\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3.71337	−	6.63105i	0.253250	−	0.452234i
$216$		0			0
$217$	−9.11871	−	9.11871i	−0.619019	−	0.619019i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.45579	−	1.45579i	0.0979271	−	0.0979271i
$222$		0			0
$223$	−4.17970	−	4.17970i	−0.279893	−	0.279893i	0.553173	−	0.833066i	$-0.313417\pi$
$223$	−4.17970	−	4.17970i	−0.279893	−	0.279893i	−0.833066	+	0.553173i	$0.813417\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			22.0979i			1.46669i	0.679857	+	0.733345i	$0.262042\pi$
$227$			22.0979i			1.46669i	−0.679857	+	0.733345i	$0.737958\pi$
$228$		0			0
$229$	1.81023	+	1.81023i	0.119623	+	0.119623i	0.764384	−	0.644761i	$-0.223043\pi$
$229$	1.81023	+	1.81023i	0.119623	+	0.119623i	−0.644761	+	0.764384i	$0.723043\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	16.6515	+	16.6515i	1.09088	+	1.09088i	0.995435	+	0.0954409i	$0.0304261\pi$
$233$	16.6515	+	16.6515i	1.09088	+	1.09088i	0.0954409	+	0.995435i	$0.469574\pi$
$234$		0			0
$235$	−0.478548	+	0.134976i	−0.0312170	+	0.00880487i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	18.6114			1.20387			0.601935	−	0.798545i	$-0.294397\pi$
$239$	18.6114			1.20387			0.601935	+	0.798545i	$0.294397\pi$
$240$		0			0
$241$	9.42655			0.607218			0.303609	−	0.952797i	$-0.401808\pi$
$241$	9.42655			0.607218			0.303609	+	0.952797i	$0.401808\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.24996	+	1.25997i	0.143745	+	0.0804966i
$246$		0			0
$247$	2.04860	+	2.04860i	0.130349	+	0.130349i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	12.8666	+	12.8666i	0.812131	+	0.812131i	0.984953	−	0.172822i	$-0.0552885\pi$
$251$	12.8666	+	12.8666i	0.812131	+	0.812131i	−0.172822	+	0.984953i	$0.555289\pi$
$252$		0			0
$253$			4.54482i			0.285730i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−11.2530	−	11.2530i	−0.701940	−	0.701940i	0.262886	−	0.964827i	$-0.415326\pi$
$257$	−11.2530	−	11.2530i	−0.701940	−	0.701940i	−0.964827	+	0.262886i	$0.915326\pi$
$258$		0			0
$259$	19.3208	−	19.3208i	1.20054	−	1.20054i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1.12411	+	1.12411i	0.0693156	+	0.0693156i	0.740915	−	0.671599i	$-0.234392\pi$
$263$	1.12411	+	1.12411i	0.0693156	+	0.0693156i	−0.671599	+	0.740915i	$0.734392\pi$
$264$		0			0
$265$	8.93492	+	5.00353i	0.548868	+	0.307364i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	9.47153	+	9.47153i	0.577490	+	0.577490i	0.934211	−	0.356721i	$-0.116105\pi$
$269$	9.47153	+	9.47153i	0.577490	+	0.577490i	−0.356721	+	0.934211i	$0.616105\pi$
$270$		0			0
$271$	−17.0874			−1.03799			−0.518993	−	0.854779i	$-0.673693\pi$
$271$	−17.0874			−1.03799			−0.518993	+	0.854779i	$0.673693\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.15769	−	4.82310i	0.0698114	−	0.290844i
$276$		0			0
$277$			5.32430i			0.319906i	0.987125	+	0.159953i	$0.0511343\pi$
$277$			5.32430i			0.319906i	−0.987125	+	0.159953i	$0.948866\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−16.7000			−0.996239			−0.498119	−	0.867108i	$-0.665976\pi$
$281$	−16.7000			−0.996239			−0.498119	+	0.867108i	$0.665976\pi$
$282$		0			0
$283$	16.1534			0.960221			0.480111	−	0.877208i	$-0.340597\pi$
$283$	16.1534			0.960221			0.480111	+	0.877208i	$0.340597\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	9.28476	+	9.28476i	0.548062	+	0.548062i
$288$		0			0
$289$			17.5373i			1.03161i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	32.7905i		−	1.91564i	−0.287369	−	0.957820i	$-0.592781\pi$
$293$		−	32.7905i		−	1.91564i	0.287369	−	0.957820i	$-0.407219\pi$
$294$		0			0
$295$	−8.55970	−	30.3478i	−0.498365	−	1.76691i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.13488	−	1.13488i	0.0656320	−	0.0656320i
$300$		0			0
$301$	−5.81127	−	5.81127i	−0.334956	−	0.334956i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−18.7825	+	5.29768i	−1.07548	+	0.303344i
$306$		0			0
$307$	8.38256			0.478418			0.239209	−	0.970968i	$-0.423112\pi$
$307$	8.38256			0.478418			0.239209	+	0.970968i	$0.423112\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	33.8278			1.91820			0.959099	−	0.283070i	$-0.0913528\pi$
$311$	33.8278			1.91820			0.959099	+	0.283070i	$0.0913528\pi$
$312$		0			0
$313$	−15.8613	+	15.8613i	−0.896535	+	0.896535i	−0.995128	−	0.0985931i	$-0.968566\pi$
$313$	−15.8613	+	15.8613i	−0.896535	+	0.896535i	0.0985931	+	0.995128i	$0.468566\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			15.6755i			0.880422i	0.897894	+	0.440211i	$0.145096\pi$
$317$			15.6755i			0.880422i	−0.897894	+	0.440211i	$0.854904\pi$
$318$		0			0
$319$			1.73486i			0.0971334i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−48.6010			−2.70423
$324$		0			0
$325$	−1.49346	+	0.915287i	−0.0828422	+	0.0507710i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			0.537675i			0.0296430i
$330$		0			0
$331$	−2.63339	+	2.63339i	−0.144744	+	0.144744i	−0.775766	−	0.631021i	$-0.782636\pi$
$331$	−2.63339	+	2.63339i	−0.144744	+	0.144744i	0.631021	+	0.775766i	$0.282636\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−6.52828	+	11.6577i	−0.356678	+	0.636929i
$336$		0			0
$337$	19.3480	−	19.3480i	1.05395	−	1.05395i	0.0554957	−	0.998459i	$-0.482326\pi$
$337$	19.3480	−	19.3480i	1.05395	−	1.05395i	0.998459	−	0.0554957i	$-0.0176739\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−3.74107	−	3.74107i	−0.202590	−	0.202590i
$342$		0			0
$343$	13.9403	−	13.9403i	0.752707	−	0.752707i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	26.1374			1.40313			0.701564	−	0.712606i	$-0.252485\pi$
$347$	26.1374			1.40313			0.701564	+	0.712606i	$0.252485\pi$
$348$		0			0
$349$	4.50325	−	4.50325i	0.241054	−	0.241054i	−0.576232	−	0.817286i	$-0.695478\pi$
$349$	4.50325	−	4.50325i	0.241054	−	0.241054i	0.817286	+	0.576232i	$0.195478\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−18.3500	+	18.3500i	−0.976674	+	0.976674i	−0.999734	−	0.0230604i	$-0.992659\pi$
$353$	−18.3500	+	18.3500i	−0.976674	+	0.976674i	0.0230604	+	0.999734i	$0.492659\pi$
$354$		0			0
$355$	−9.94641	+	17.7615i	−0.527900	+	0.942684i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			22.2948i			1.17667i	0.808616	+	0.588337i	$0.200217\pi$
$359$			22.2948i			1.17667i	−0.808616	+	0.588337i	$0.799783\pi$
$360$		0			0
$361$		−	49.3916i		−	2.59956i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−7.90689	−	28.0333i	−0.413866	−	1.46733i
$366$		0			0
$367$	14.8081	−	14.8081i	0.772975	−	0.772975i	−0.205651	−	0.978625i	$-0.565931\pi$
$367$	14.8081	−	14.8081i	0.772975	−	0.772975i	0.978625	+	0.205651i	$0.0659311\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	7.83031	−	7.83031i	0.406530	−	0.406530i
$372$		0			0
$373$	18.6724			0.966821			0.483410	−	0.875394i	$-0.339398\pi$
$373$	18.6724			0.966821			0.483410	+	0.875394i	$0.339398\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	0.433211	−	0.433211i	0.0223115	−	0.0223115i
$378$		0			0
$379$	−18.8088	−	18.8088i	−0.966141	−	0.966141i	0.0333040	−	0.999445i	$-0.489397\pi$
$379$	−18.8088	−	18.8088i	−0.966141	−	0.966141i	−0.999445	+	0.0333040i	$0.989397\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	23.6584	−	23.6584i	1.20889	−	1.20889i	0.237502	−	0.971387i	$-0.423671\pi$
$383$	23.6584	−	23.6584i	1.20889	−	1.20889i	0.971387	−	0.237502i	$-0.0763286\pi$
$384$		0			0
$385$	−4.67984	−	2.62070i	−0.238507	−	0.133563i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−5.76236	+	5.76236i	−0.292163	+	0.292163i	−0.837934	−	0.545771i	$-0.816237\pi$
$389$	−5.76236	+	5.76236i	−0.292163	+	0.292163i	0.545771	+	0.837934i	$0.316237\pi$
$390$		0			0
$391$			26.9241i			1.36161i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	5.95548	+	3.33505i	0.299652	+	0.167805i
$396$		0			0
$397$	23.7724			1.19310			0.596552	−	0.802575i	$-0.296537\pi$
$397$	23.7724			1.19310			0.596552	+	0.802575i	$0.296537\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	28.0151i		−	1.39901i	−0.714629	−	0.699504i	$-0.753404\pi$
$401$		−	28.0151i		−	1.39901i	0.714629	−	0.699504i	$-0.246596\pi$
$402$		0			0
$403$			1.86836i			0.0930697i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	7.92662	−	7.92662i	0.392908	−	0.392908i
$408$		0			0
$409$	14.8889			0.736208			0.368104	−	0.929785i	$-0.380007\pi$
$409$	14.8889			0.736208			0.368104	+	0.929785i	$0.380007\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−34.0974			−1.67782
$414$		0			0
$415$	3.19755	−	5.70994i	0.156962	−	0.280290i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−3.83156	−	3.83156i	−0.187184	−	0.187184i	0.607294	−	0.794478i	$-0.292255\pi$
$419$	−3.83156	−	3.83156i	−0.187184	−	0.187184i	−0.794478	+	0.607294i	$0.792255\pi$
$420$		0			0
$421$	−8.11188	+	8.11188i	−0.395349	+	0.395349i	−0.876589	−	0.481240i	$-0.840187\pi$
$421$	−8.11188	+	8.11188i	−0.395349	+	0.395349i	0.481240	+	0.876589i	$0.340187\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	6.85831	−	28.5726i	0.332677	−	1.38598i
$426$		0			0
$427$			21.1032i			1.02126i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			2.57559i			0.124062i	0.998074	+	0.0620310i	$0.0197578\pi$
$431$			2.57559i			0.124062i	−0.998074	+	0.0620310i	$0.980242\pi$
$432$		0			0
$433$	3.60543	+	3.60543i	0.173266	+	0.173266i	0.788413	−	0.615147i	$-0.210903\pi$
$433$	3.60543	+	3.60543i	0.173266	+	0.173266i	−0.615147	+	0.788413i	$0.710903\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−37.8877			−1.81241
$438$		0			0
$439$	30.9865			1.47890			0.739452	−	0.673210i	$-0.235085\pi$
$439$	30.9865			1.47890			0.739452	+	0.673210i	$0.235085\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	5.00471i		−	0.237781i	−0.992907	−	0.118890i	$-0.962066\pi$
$443$		−	5.00471i		−	0.237781i	0.992907	−	0.118890i	$-0.0379337\pi$
$444$		0			0
$445$	−23.4264	−	13.1187i	−1.11052	−	0.621887i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	37.8212			1.78489			0.892446	−	0.451153i	$-0.148987\pi$
$449$	37.8212			1.78489			0.892446	+	0.451153i	$0.148987\pi$
$450$		0			0
$451$	3.80919	+	3.80919i	0.179368	+	0.179368i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	0.514188	+	1.82301i	0.0241055	+	0.0854642i
$456$		0			0
$457$	18.3136	+	18.3136i	0.856675	+	0.856675i	0.990945	−	0.134270i	$-0.0428688\pi$
$457$	18.3136	+	18.3136i	0.856675	+	0.856675i	−0.134270	+	0.990945i	$0.542869\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	26.3763	−	26.3763i	1.22847	−	1.22847i	0.263923	−	0.964544i	$-0.414984\pi$
$461$	26.3763	−	26.3763i	1.22847	−	1.22847i	0.964544	−	0.263923i	$-0.0850164\pi$
$462$		0			0
$463$	0.245818	+	0.245818i	0.0114241	+	0.0114241i	0.712796	−	0.701372i	$-0.247429\pi$
$463$	0.245818	+	0.245818i	0.0114241	+	0.0114241i	−0.701372	+	0.712796i	$0.747429\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			14.4794i			0.670025i	0.942214	+	0.335012i	$0.108740\pi$
$467$			14.4794i			0.670025i	−0.942214	+	0.335012i	$0.891260\pi$
$468$		0			0
$469$	10.2165	+	10.2165i	0.471754	+	0.471754i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2.38415	−	2.38415i	−0.109623	−	0.109623i
$474$		0			0
$475$	40.2076	+	9.65105i	1.84485	+	0.442820i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	20.7708			0.949041			0.474521	−	0.880244i	$-0.342621\pi$
$479$	20.7708			0.949041			0.474521	+	0.880244i	$0.342621\pi$
$480$		0			0
$481$	−3.95870			−0.180501
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.35909	+	8.36397i	0.107121	+	0.379788i
$486$		0			0
$487$	24.1831	+	24.1831i	1.09584	+	1.09584i	0.994891	+	0.100951i	$0.0321886\pi$
$487$	24.1831	+	24.1831i	1.09584	+	1.09584i	0.100951	+	0.994891i	$0.467811\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−23.9774	−	23.9774i	−1.08208	−	1.08208i	−0.996315	−	0.0857692i	$-0.972665\pi$
$491$	−23.9774	−	23.9774i	−1.08208	−	1.08208i	−0.0857692	−	0.996315i	$-0.527335\pi$
$492$		0			0
$493$			10.2775i			0.462876i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	15.5657	+	15.5657i	0.698218	+	0.698218i
$498$		0			0
$499$	−8.30857	+	8.30857i	−0.371943	+	0.371943i	−0.868184	−	0.496242i	$-0.834713\pi$
$499$	−8.30857	+	8.30857i	−0.371943	+	0.371943i	0.496242	+	0.868184i	$0.334713\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−25.5272	−	25.5272i	−1.13820	−	1.13820i	−0.988773	−	0.149428i	$-0.952257\pi$
$503$	−25.5272	−	25.5272i	−1.13820	−	1.13820i	−0.149428	−	0.988773i	$-0.547743\pi$
$504$		0			0
$505$	19.8274	−	5.59238i	0.882306	−	0.248858i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−4.16293	−	4.16293i	−0.184519	−	0.184519i	0.608803	−	0.793322i	$-0.291650\pi$
$509$	−4.16293	−	4.16293i	−0.184519	−	0.184519i	−0.793322	+	0.608803i	$0.791650\pi$
$510$		0			0
$511$	−31.4970			−1.39334
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−3.24181	−	11.4936i	−0.142851	−	0.506468i
$516$		0			0
$517$			0.220588i			0.00970146i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−5.01151			−0.219558			−0.109779	−	0.993956i	$-0.535014\pi$
$521$	−5.01151			−0.219558			−0.109779	+	0.993956i	$0.535014\pi$
$522$		0			0
$523$	−24.8875			−1.08825			−0.544127	−	0.839003i	$-0.683139\pi$
$523$	−24.8875			−1.08825			−0.544127	+	0.839003i	$0.683139\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−22.1626	−	22.1626i	−0.965417	−	0.965417i
$528$		0			0
$529$		−	2.01094i		−	0.0874320i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	1.90238i		−	0.0824014i
$534$		0			0
$535$	8.14803	−	14.5501i	0.352270	−	0.629057i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.808957	−	0.808957i	0.0348443	−	0.0348443i
$540$		0			0
$541$	6.92151	+	6.92151i	0.297579	+	0.297579i	0.840065	−	0.542486i	$-0.182517\pi$
$541$	6.92151	+	6.92151i	0.297579	+	0.297579i	−0.542486	+	0.840065i	$0.682517\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1.49090	+	5.28589i	0.0638633	+	0.226422i
$546$		0			0
$547$	−1.22898			−0.0525473			−0.0262737	−	0.999655i	$-0.508364\pi$
$547$	−1.22898			−0.0525473			−0.0262737	+	0.999655i	$0.508364\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−14.4626			−0.616127
$552$		0			0
$553$	5.21921	−	5.21921i	0.221943	−	0.221943i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			22.2557i			0.943003i	0.881865	+	0.471501i	$0.156288\pi$
$557$			22.2557i			0.943003i	−0.881865	+	0.471501i	$0.843712\pi$
$558$		0			0
$559$			1.19069i			0.0503607i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−23.5319			−0.991750			−0.495875	−	0.868394i	$-0.665153\pi$
$563$	−23.5319			−0.991750			−0.495875	+	0.868394i	$0.665153\pi$
$564$		0			0
$565$	−5.86772	−	20.8036i	−0.246857	−	0.875212i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	0.716133i		−	0.0300219i	−0.999887	−	0.0150109i	$-0.995222\pi$
$569$		−	0.716133i		−	0.0300219i	0.999887	−	0.0150109i	$-0.00477831\pi$
$570$		0			0
$571$	16.9873	−	16.9873i	0.710895	−	0.710895i	−0.255828	−	0.966722i	$-0.582348\pi$
$571$	16.9873	−	16.9873i	0.710895	−	0.710895i	0.966722	+	0.255828i	$0.0823480\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	5.34650	−	22.2742i	0.222964	−	0.928900i
$576$		0			0
$577$	−30.4952	+	30.4952i	−1.26953	+	1.26953i	−0.323202	+	0.946330i	$0.604759\pi$
$577$	−30.4952	+	30.4952i	−1.26953	+	1.26953i	−0.946330	+	0.323202i	$0.895241\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−5.00403	−	5.00403i	−0.207602	−	0.207602i
$582$		0			0
$583$	3.21249	−	3.21249i	0.133048	−	0.133048i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−15.6031			−0.644007			−0.322004	−	0.946738i	$-0.604356\pi$
$587$	−15.6031			−0.644007			−0.322004	+	0.946738i	$0.604356\pi$
$588$		0			0
$589$	31.1873	−	31.1873i	1.28505	−	1.28505i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	4.83979	−	4.83979i	0.198746	−	0.198746i	−0.600716	−	0.799462i	$-0.705118\pi$
$593$	4.83979	−	4.83979i	0.198746	−	0.198746i	0.799462	+	0.600716i	$0.205118\pi$
$594$		0			0
$595$	−27.7240	−	15.5253i	−1.13657	−	0.636477i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	25.0371i		−	1.02299i	−0.859287	−	0.511494i	$-0.829092\pi$
$599$		−	25.0371i		−	1.02299i	0.859287	−	0.511494i	$-0.170908\pi$
$600$		0			0
$601$		−	5.70141i		−	0.232565i	−0.993216	−	0.116283i	$-0.962902\pi$
$601$		−	5.70141i		−	0.232565i	0.993216	−	0.116283i	$-0.0370978\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	19.5409	+	10.9428i	0.794449	+	0.444889i
$606$		0			0
$607$	−11.8178	+	11.8178i	−0.479668	+	0.479668i	−0.905026	−	0.425357i	$-0.860148\pi$
$607$	−11.8178	+	11.8178i	−0.479668	+	0.479668i	0.425357	+	0.905026i	$0.360148\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	0.0550830	−	0.0550830i	0.00222842	−	0.00222842i
$612$		0			0
$613$	−5.44770			−0.220030			−0.110015	−	0.993930i	$-0.535090\pi$
$613$	−5.44770			−0.220030			−0.110015	+	0.993930i	$0.535090\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−12.9090	+	12.9090i	−0.519698	+	0.519698i	−0.917480	−	0.397782i	$-0.869780\pi$
$617$	−12.9090	+	12.9090i	−0.519698	+	0.519698i	0.397782	+	0.917480i	$0.369780\pi$
$618$		0			0
$619$	14.5979	+	14.5979i	0.586740	+	0.586740i	0.936747	−	0.350007i	$-0.113821\pi$
$619$	14.5979	+	14.5979i	0.586740	+	0.586740i	−0.350007	+	0.936747i	$0.613821\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−20.5302	+	20.5302i	−0.822527	+	0.822527i
$624$		0			0
$625$	−11.3477	+	22.2762i	−0.453909	+	0.891048i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	46.9583	−	46.9583i	1.87235	−	1.87235i
$630$		0			0
$631$		−	19.0468i		−	0.758240i	−0.925347	−	0.379120i	$-0.876227\pi$
$631$		−	19.0468i		−	0.758240i	0.925347	−	0.379120i	$-0.123773\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4.14953	−	14.7118i	−0.164669	−	0.583822i
$636$		0			0
$637$	−0.404009			−0.0160074
$638$		0			0
$639$		0			0
$640$		0			0
$641$			25.2388i			0.996874i	0.866926	+	0.498437i	$0.166092\pi$
$641$			25.2388i			0.996874i	−0.866926	+	0.498437i	$0.833908\pi$
$642$		0			0
$643$		−	24.6600i		−	0.972495i	−0.873821	−	0.486247i	$-0.838365\pi$
$643$		−	24.6600i		−	0.972495i	0.873821	−	0.486247i	$-0.161635\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	2.74892	−	2.74892i	0.108071	−	0.108071i	−0.651003	−	0.759075i	$-0.725652\pi$
$647$	2.74892	−	2.74892i	0.108071	−	0.108071i	0.759075	+	0.651003i	$0.225652\pi$
$648$		0			0
$649$	−13.9889			−0.549113
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−9.93904			−0.388945			−0.194472	−	0.980908i	$-0.562299\pi$
$653$	−9.93904			−0.388945			−0.194472	+	0.980908i	$0.562299\pi$
$654$		0			0
$655$	−2.80069	−	9.92962i	−0.109432	−	0.387982i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	22.1095	+	22.1095i	0.861263	+	0.861263i	0.991485	−	0.130222i	$-0.0415689\pi$
$659$	22.1095	+	22.1095i	0.861263	+	0.861263i	−0.130222	+	0.991485i	$0.541569\pi$
$660$		0			0
$661$	11.3870	−	11.3870i	0.442904	−	0.442904i	−0.450083	−	0.892987i	$-0.648606\pi$
$661$	11.3870	−	11.3870i	0.442904	−	0.442904i	0.892987	+	0.450083i	$0.148606\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	21.8473	−	39.0133i	0.847204	−	1.51287i
$666$		0			0
$667$			8.01200i			0.310226i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			8.65786i			0.334233i
$672$		0			0
$673$	10.1521	+	10.1521i	0.391335	+	0.391335i	0.875163	−	0.483828i	$-0.160754\pi$
$673$	10.1521	+	10.1521i	0.391335	+	0.391335i	−0.483828	+	0.875163i	$0.660754\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	36.8264			1.41535			0.707677	−	0.706536i	$-0.249743\pi$
$677$	36.8264			1.41535			0.707677	+	0.706536i	$0.249743\pi$
$678$		0			0
$679$	9.39739			0.360639
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	7.05388i		−	0.269909i	−0.990852	−	0.134955i	$-0.956911\pi$
$683$		−	7.05388i		−	0.269909i	0.990852	−	0.134955i	$-0.0430889\pi$
$684$		0			0
$685$	1.72969	+	6.13249i	0.0660881	+	0.234310i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1.60438			−0.0611219
$690$		0			0
$691$	3.26394	+	3.26394i	0.124166	+	0.124166i	0.766459	−	0.642293i	$-0.222017\pi$
$691$	3.26394	+	3.26394i	0.124166	+	0.124166i	−0.642293	+	0.766459i	$0.722017\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	37.0142	−	10.4400i	1.40403	−	0.396012i
$696$		0			0
$697$	22.5661	+	22.5661i	0.854753	+	0.854753i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−7.06532	+	7.06532i	−0.266853	+	0.266853i	−0.827831	−	0.560978i	$-0.810425\pi$
$701$	−7.06532	+	7.06532i	−0.266853	+	0.266853i	0.560978	+	0.827831i	$0.310425\pi$
$702$		0			0
$703$	66.0799	+	66.0799i	2.49225	+	2.49225i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	22.2771i		−	0.837818i
$708$		0			0
$709$	33.5742	+	33.5742i	1.26091	+	1.26091i	0.950654	+	0.310252i	$0.100413\pi$
$709$	33.5742	+	33.5742i	1.26091	+	1.26091i	0.310252	+	0.950654i	$0.399587\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−17.2772	−	17.2772i	−0.647035	−	0.647035i
$714$		0			0
$715$	0.210952	+	0.747915i	0.00788917	+	0.0279704i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−16.6274			−0.620097			−0.310049	−	0.950721i	$-0.600345\pi$
$719$	−16.6274			−0.620097			−0.310049	+	0.950721i	$0.600345\pi$
$720$		0			0
$721$	−12.9137			−0.480931
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2.04088	−	8.50258i	0.0757963	−	0.315778i
$726$		0			0
$727$	−7.54239	−	7.54239i	−0.279732	−	0.279732i	0.553270	−	0.833002i	$-0.313380\pi$
$727$	−7.54239	−	7.54239i	−0.279732	−	0.279732i	−0.833002	+	0.553270i	$0.813380\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−14.1240	−	14.1240i	−0.522394	−	0.522394i
$732$		0			0
$733$		−	3.71206i		−	0.137108i	−0.997647	−	0.0685541i	$-0.978161\pi$
$733$		−	3.71206i		−	0.137108i	0.997647	−	0.0685541i	$-0.0218386\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4.19145	+	4.19145i	0.154394	+	0.154394i
$738$		0			0
$739$	31.9791	−	31.9791i	1.17637	−	1.17637i	0.195709	−	0.980662i	$-0.437299\pi$
$739$	31.9791	−	31.9791i	1.17637	−	1.17637i	0.980662	−	0.195709i	$-0.0627007\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−15.7609	−	15.7609i	−0.578212	−	0.578212i	0.356198	−	0.934410i	$-0.384073\pi$
$743$	−15.7609	−	15.7609i	−0.578212	−	0.578212i	−0.934410	+	0.356198i	$0.884073\pi$
$744$		0			0
$745$	4.72366	+	16.7474i	0.173061	+	0.613576i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−12.7513	−	12.7513i	−0.465924	−	0.465924i
$750$		0			0
$751$	−14.6752			−0.535504			−0.267752	−	0.963488i	$-0.586281\pi$
$751$	−14.6752			−0.535504			−0.267752	+	0.963488i	$0.586281\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	24.7384	+	13.8534i	0.900324	+	0.504179i
$756$		0			0
$757$		−	6.39340i		−	0.232372i	−0.993227	−	0.116186i	$-0.962933\pi$
$757$		−	6.39340i		−	0.232372i	0.993227	−	0.116186i	$-0.0370669\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	16.2322			0.588416			0.294208	−	0.955741i	$-0.404944\pi$
$761$	16.2322			0.588416			0.294208	+	0.955741i	$0.404944\pi$
$762$		0			0
$763$	5.93899			0.215006
$764$		0			0
$765$		0			0
$766$		0			0
$767$	3.49316	+	3.49316i	0.126131	+	0.126131i
$768$		0			0
$769$		−	38.2217i		−	1.37831i	−0.724615	−	0.689154i	$-0.757982\pi$
$769$		−	38.2217i		−	1.37831i	0.724615	−	0.689154i	$-0.242018\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			16.5524i			0.595349i	0.954667	+	0.297674i	$0.0962110\pi$
$773$			16.5524i			0.595349i	−0.954667	+	0.297674i	$0.903789\pi$
$774$		0			0
$775$	13.9341	+	22.7360i	0.500527	+	0.816702i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−31.7552	+	31.7552i	−1.13775	+	1.13775i
$780$		0			0
$781$	6.38604	+	6.38604i	0.228510	+	0.228510i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	16.6723	−	29.7722i	0.595061	−	1.06261i
$786$		0			0
$787$	−9.81578			−0.349895			−0.174947	−	0.984578i	$-0.555976\pi$
$787$	−9.81578			−0.349895			−0.174947	+	0.984578i	$0.555976\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−23.3740			−0.831083
$792$		0			0
$793$	2.16195	−	2.16195i	0.0767731	−	0.0767731i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	15.5379i		−	0.550381i	−0.961390	−	0.275190i	$-0.911259\pi$
$797$		−	15.5379i		−	0.550381i	0.961390	−	0.275190i	$-0.0887409\pi$
$798$		0			0
$799$			1.30679i			0.0462310i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−12.9220			−0.456009
$804$		0			0
$805$	−21.6127	−	12.1030i	−0.761746	−	0.426575i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			34.0108i			1.19575i	0.801587	+	0.597877i	$0.203989\pi$
$809$			34.0108i			1.19575i	−0.801587	+	0.597877i	$0.796011\pi$
$810$		0			0
$811$	26.0581	−	26.0581i	0.915023	−	0.915023i	−0.0816390	−	0.996662i	$-0.526015\pi$
$811$	26.0581	−	26.0581i	0.915023	−	0.915023i	0.996662	+	0.0816390i	$0.0260155\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−43.5731	−	24.4008i	−1.52630	−	0.854723i
$816$		0			0
$817$	19.8753	−	19.8753i	0.695350	−	0.695350i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	29.5405	+	29.5405i	1.03097	+	1.03097i	0.999505	+	0.0314667i	$0.0100178\pi$
$821$	29.5405	+	29.5405i	1.03097	+	1.03097i	0.0314667	+	0.999505i	$0.489982\pi$
$822$		0			0
$823$	−25.6776	+	25.6776i	−0.895064	+	0.895064i	−0.994994	−	0.0999304i	$-0.968138\pi$
$823$	−25.6776	+	25.6776i	−0.895064	+	0.895064i	0.0999304	+	0.994994i	$0.468138\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	27.3164			0.949883			0.474941	−	0.880017i	$-0.342469\pi$
$827$	27.3164			0.949883			0.474941	+	0.880017i	$0.342469\pi$
$828$		0			0
$829$	31.8750	−	31.8750i	1.10706	−	1.10706i	0.113530	−	0.993535i	$-0.463784\pi$
$829$	31.8750	−	31.8750i	1.10706	−	1.10706i	0.993535	−	0.113530i	$-0.0362159\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	4.79237	−	4.79237i	0.166046	−	0.166046i
$834$		0			0
$835$	−6.53479	−	23.1686i	−0.226146	−	0.801782i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			23.5736i			0.813852i	0.913461	+	0.406926i	$0.133399\pi$
$839$			23.5736i			0.813852i	−0.913461	+	0.406926i	$0.866601\pi$
$840$		0			0
$841$		−	25.9416i		−	0.894539i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−14.0690	+	25.1234i	−0.483989	+	0.864270i
$846$		0			0
$847$	17.1251	−	17.1251i	0.588425	−	0.588425i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	36.6070	−	36.6070i	1.25487	−	1.25487i
$852$		0			0
$853$	15.4412			0.528697			0.264348	−	0.964427i	$-0.414843\pi$
$853$	15.4412			0.528697			0.264348	+	0.964427i	$0.414843\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−15.0075	+	15.0075i	−0.512645	+	0.512645i	−0.915336	−	0.402691i	$-0.868075\pi$
$857$	−15.0075	+	15.0075i	−0.512645	+	0.512645i	0.402691	+	0.915336i	$0.368075\pi$
$858$		0			0
$859$	29.6625	+	29.6625i	1.01207	+	1.01207i	0.999926	+	0.0121452i	$0.00386604\pi$
$859$	29.6625	+	29.6625i	1.01207	+	1.01207i	0.0121452	+	0.999926i	$0.496134\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−11.4520	+	11.4520i	−0.389830	+	0.389830i	−0.874627	−	0.484797i	$-0.838893\pi$
$863$	−11.4520	+	11.4520i	−0.389830	+	0.389830i	0.484797	+	0.874627i	$0.338893\pi$
$864$		0			0
$865$	15.5898	−	27.8391i	0.530069	−	0.946557i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	2.14125	−	2.14125i	0.0726369	−	0.0726369i
$870$		0			0
$871$		−	2.09329i		−	0.0709284i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	19.8530	+	18.3494i	0.671155	+	0.620324i
$876$		0			0
$877$	−11.2559			−0.380083			−0.190042	−	0.981776i	$-0.560862\pi$
$877$	−11.2559			−0.380083			−0.190042	+	0.981776i	$0.560862\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			33.5847i			1.13150i	0.824577	+	0.565749i	$0.191413\pi$
$881$			33.5847i			1.13150i	−0.824577	+	0.565749i	$0.808587\pi$
$882$		0			0
$883$			6.17485i			0.207800i	0.994588	+	0.103900i	$0.0331322\pi$
$883$			6.17485i			0.207800i	−0.994588	+	0.103900i	$0.966868\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−12.6490	+	12.6490i	−0.424711	+	0.424711i	−0.886822	−	0.462111i	$-0.847092\pi$
$887$	−12.6490	+	12.6490i	−0.424711	+	0.424711i	0.462111	+	0.886822i	$0.347092\pi$
$888$		0			0
$889$	−16.5296			−0.554384
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.83892			−0.0615373
$894$		0			0
$895$	5.71442	−	1.61177i	0.191012	−	0.0538756i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−6.59508	−	6.59508i	−0.219958	−	0.219958i
$900$		0			0
$901$	19.0312	−	19.0312i	0.634021	−	0.634021i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−9.98825	−	35.4126i	−0.332021	−	1.17715i
$906$		0			0
$907$		−	0.741561i		−	0.0246231i	−0.999924	−	0.0123116i	$-0.996081\pi$
$907$		−	0.741561i		−	0.0246231i	0.999924	−	0.0123116i	$-0.00391899\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			54.5704i			1.80800i	0.427533	+	0.904000i	$0.359383\pi$
$911$			54.5704i			1.80800i	−0.427533	+	0.904000i	$0.640617\pi$
$912$		0			0
$913$	−2.05297	−	2.05297i	−0.0679434	−	0.0679434i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−11.1565			−0.368420
$918$		0			0
$919$	56.6439			1.86851			0.934255	−	0.356607i	$-0.116066\pi$
$919$	56.6439			1.86851			0.934255	+	0.356607i	$0.116066\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	3.18931i		−	0.104977i
$924$		0			0
$925$	−48.1733	+	29.5237i	−1.58393	+	0.970733i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	5.73521			0.188166			0.0940831	−	0.995564i	$-0.470008\pi$
$929$	5.73521			0.188166			0.0940831	+	0.995564i	$0.470008\pi$
$930$		0			0
$931$	6.74384	+	6.74384i	0.221020	+	0.221020i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−11.3741	−	6.36947i	−0.371974	−	0.208304i
$936$		0			0
$937$	15.6669	+	15.6669i	0.511815	+	0.511815i	0.915082	−	0.403267i	$-0.132126\pi$
$937$	15.6669	+	15.6669i	0.511815	+	0.511815i	−0.403267	+	0.915082i	$0.632126\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	21.6702	−	21.6702i	0.706429	−	0.706429i	−0.259354	−	0.965782i	$-0.583510\pi$
$941$	21.6702	−	21.6702i	0.706429	−	0.706429i	0.965782	+	0.259354i	$0.0835095\pi$
$942$		0			0
$943$	17.5918	+	17.5918i	0.572867	+	0.572867i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	0.412354i		−	0.0133997i	−0.999978	−	0.00669985i	$-0.997867\pi$
$947$		−	0.412354i		−	0.0133997i	0.999978	−	0.00669985i	$-0.00213264\pi$
$948$		0			0
$949$	3.22676	+	3.22676i	0.104745	+	0.104745i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−30.5110	−	30.5110i	−0.988347	−	0.988347i	0.0115860	−	0.999933i	$-0.496312\pi$
$953$	−30.5110	−	30.5110i	−0.988347	−	0.988347i	−0.999933	+	0.0115860i	$0.996312\pi$
$954$		0			0
$955$	13.2900	+	7.44238i	0.430055	+	0.240830i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	6.89020			0.222496
$960$		0			0
$961$	−2.55657			−0.0824699
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−22.2975	+	6.28910i	−0.717783	+	0.202453i
$966$		0			0
$967$	−32.9528	−	32.9528i	−1.05969	−	1.05969i	−0.998102	−	0.0615904i	$-0.980383\pi$
$967$	−32.9528	−	32.9528i	−1.05969	−	1.05969i	−0.0615904	−	0.998102i	$-0.519617\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1.21142	+	1.21142i	0.0388763	+	0.0388763i	0.726278	−	0.687401i	$-0.241249\pi$
$971$	1.21142	+	1.21142i	0.0388763	+	0.0388763i	−0.687401	+	0.726278i	$0.741249\pi$
$972$		0			0
$973$		−	41.5875i		−	1.33324i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	7.89981	+	7.89981i	0.252737	+	0.252737i	0.822092	−	0.569355i	$-0.192807\pi$
$977$	7.89981	+	7.89981i	0.252737	+	0.252737i	−0.569355	+	0.822092i	$0.692807\pi$
$978$		0			0
$979$	−8.42280	+	8.42280i	−0.269194	+	0.269194i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−25.4490	−	25.4490i	−0.811696	−	0.811696i	0.173192	−	0.984888i	$-0.444592\pi$
$983$	−25.4490	−	25.4490i	−0.811696	−	0.811696i	−0.984888	+	0.173192i	$0.944592\pi$
$984$		0			0
$985$	9.27692	−	16.5660i	0.295587	−	0.527837i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−11.0106	−	11.0106i	−0.350116	−	0.350116i
$990$		0			0
$991$	−50.4773			−1.60346			−0.801732	−	0.597683i	$-0.796088\pi$
$991$	−50.4773			−1.60346			−0.801732	+	0.597683i	$0.796088\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	21.1301	−	37.7326i	0.669871	−	1.19620i
$996$		0			0
$997$		−	25.4699i		−	0.806641i	−0.915059	−	0.403321i	$-0.867856\pi$
$997$		−	25.4699i		−	0.806641i	0.915059	−	0.403321i	$-0.132144\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.bc.a.593.17		96
3.2	odd	2	inner	2880.2.bc.a.593.32		96
4.3	odd	2		720.2.bc.a.413.20	yes	96
5.2	odd	4		2880.2.bg.a.17.41		96
12.11	even	2		720.2.bc.a.413.29	yes	96
15.2	even	4		2880.2.bg.a.17.8		96
16.5	even	4		2880.2.bg.a.2033.8		96
16.11	odd	4		720.2.bg.a.53.5	yes	96
20.7	even	4		720.2.bg.a.557.44	yes	96
48.5	odd	4		2880.2.bg.a.2033.41		96
48.11	even	4		720.2.bg.a.53.44	yes	96
60.47	odd	4		720.2.bg.a.557.5	yes	96
80.27	even	4		720.2.bc.a.197.29	yes	96
80.37	odd	4	inner	2880.2.bc.a.1457.32		96
240.107	odd	4		720.2.bc.a.197.20	✓	96
240.197	even	4	inner	2880.2.bc.a.1457.17		96

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.bc.a.197.20	✓	96	240.107	odd	4
720.2.bc.a.197.29	yes	96	80.27	even	4
720.2.bc.a.413.20	yes	96	4.3	odd	2
720.2.bc.a.413.29	yes	96	12.11	even	2
720.2.bg.a.53.5	yes	96	16.11	odd	4
720.2.bg.a.53.44	yes	96	48.11	even	4
720.2.bg.a.557.5	yes	96	60.47	odd	4
720.2.bg.a.557.44	yes	96	20.7	even	4
2880.2.bc.a.593.17		96	1.1	even	1	trivial
2880.2.bc.a.593.32		96	3.2	odd	2	inner
2880.2.bc.a.1457.17		96	240.197	even	4	inner
2880.2.bc.a.1457.32		96	80.37	odd	4	inner
2880.2.bg.a.17.8		96	15.2	even	4
2880.2.bg.a.17.41		96	5.2	odd	4
2880.2.bg.a.2033.8		96	16.5	even	4
2880.2.bg.a.2033.41		96	48.5	odd	4

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.bc (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.09255 + 1.95099i) q^{5} +(1.70979 + 1.70979i) q^{7} +O(q^{10})\)	\(q+(-1.09255 + 1.95099i) q^{5} +(1.70979 + 1.70979i) q^{7} +(0.701463 + 0.701463i) q^{11} -0.350324i q^{13} +(4.15556 + 4.15556i) q^{17} +(-5.84772 + 5.84772i) q^{19} +(3.23953 + 3.23953i) q^{23} +(-2.61269 - 4.26308i) q^{25} +(1.23660 + 1.23660i) q^{29} -5.33324 q^{31} +(-5.20380 + 1.46775i) q^{35} -11.3001i q^{37} +5.43035 q^{41} -3.39882 q^{43} +(0.157234 + 0.157234i) q^{47} -1.15324i q^{49} -4.57970i q^{53} +(-2.13492 + 0.602163i) q^{55} +(-9.97123 + 9.97123i) q^{59} +(6.17129 + 6.17129i) q^{61} +(0.683477 + 0.382745i) q^{65} +5.97529 q^{67} +9.10388 q^{71} +(-9.21077 + 9.21077i) q^{73} +2.39871i q^{77} -3.05255i q^{79} -2.92670 q^{83} +(-12.6476 + 3.56729i) q^{85} +12.0075i q^{89} +(0.598980 - 0.598980i) q^{91} +(-5.01991 - 17.7977i) q^{95} +(2.74811 - 2.74811i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q+O(q^{10}) \) 96 * q	\( 96 q - 16 q^{19} - 64 q^{43} + 32 q^{61}+O(q^{100}) \) 96 * q - 16 * q^19 - 64 * q^43 + 32 * q^61

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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