LMFDB - Embedded newform 1152.3.m.f.991.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,3,Mod(415,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1152.415");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1152.m (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:	$31.3897264543$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6 x^{14} - 4 x^{13} + 10 x^{12} + 56 x^{11} + 88 x^{10} - 128 x^{9} - 496 x^{8} - 512 x^{7} + \cdots + 65536 $ x^16 - 6x^14 - 4x^13 + 10x^12 + 56x^11 + 88x^10 - 128x^9 - 496x^8 - 512x^7 + 1408x^6 + 3584x^5 + 2560x^4 - 4096x^3 - 24576*x^2 + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{28} $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			991.2
Root			$1.84258 + 0.777752i$ of defining polynomial
Character	$\chi$	$=$	1152.991
Dual form			1152.3.m.f.415.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.78830 + 4.78830i) q^{5} -10.3302 q^{7} +O(q^{10})$	$q+(-4.78830 + 4.78830i) q^{5} -10.3302 q^{7} +(-0.526169 - 0.526169i) q^{11} +(-17.2840 - 17.2840i) q^{13} -4.71650 q^{17} +(2.53604 - 2.53604i) q^{19} +12.5864 q^{23} -20.8557i q^{25} +(-2.19683 - 2.19683i) q^{29} +28.0521i q^{31} +(49.4644 - 49.4644i) q^{35} +(32.1128 - 32.1128i) q^{37} +23.1145i q^{41} +(-4.79441 - 4.79441i) q^{43} +39.0095i q^{47} +57.7141 q^{49} +(-27.9768 + 27.9768i) q^{53} +5.03891 q^{55} +(79.8538 + 79.8538i) q^{59} +(36.7762 + 36.7762i) q^{61} +165.522 q^{65} +(10.9869 - 10.9869i) q^{67} -52.6605 q^{71} -67.8061i q^{73} +(5.43545 + 5.43545i) q^{77} +56.4602i q^{79} +(-58.3697 + 58.3697i) q^{83} +(22.5840 - 22.5840i) q^{85} -131.566i q^{89} +(178.548 + 178.548i) q^{91} +24.2866i q^{95} +60.9413 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 32 q^{11} + 32 q^{19} + 128 q^{23} + 32 q^{29} + 96 q^{35} + 96 q^{37} - 160 q^{43} + 112 q^{49} - 160 q^{53} - 256 q^{55} - 128 q^{59} + 32 q^{61} + 32 q^{65} - 320 q^{67} - 512 q^{71} + 224 q^{77} - 160 q^{83} - 160 q^{85} + 480 q^{91}+O(q^{100}) $ 16 * q + 32 * q^11 + 32 * q^19 + 128 * q^23 + 32 * q^29 + 96 * q^35 + 96 * q^37 - 160 * q^43 + 112 * q^49 - 160 * q^53 - 256 * q^55 - 128 * q^59 + 32 * q^61 + 32 * q^65 - 320 * q^67 - 512 * q^71 + 224 * q^77 - 160 * q^83 - 160 * q^85 + 480 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$-1$	$1$	$e\left(\frac{3}{4}\right)$

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
$5$	−4.78830	+	4.78830i	−0.957661	+	0.957661i	−0.999139	−	0.0414785i	$-0.986793\pi$
$5$	−4.78830	+	4.78830i	−0.957661	+	0.957661i	0.0414785	+	0.999139i	$0.486793\pi$
$6$		0			0
$7$	−10.3302			−1.47575			−0.737875	−	0.674937i	$-0.764171\pi$
$7$	−10.3302			−1.47575			−0.737875	+	0.674937i	$0.764171\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.526169	−	0.526169i	−0.0478335	−	0.0478335i	0.682785	−	0.730619i	$-0.260768\pi$
$11$	−0.526169	−	0.526169i	−0.0478335	−	0.0478335i	−0.730619	+	0.682785i	$0.760768\pi$
$12$		0			0
$13$	−17.2840	−	17.2840i	−1.32953	−	1.32953i	−0.905774	−	0.423761i	$-0.860710\pi$
$13$	−17.2840	−	17.2840i	−1.32953	−	1.32953i	−0.423761	−	0.905774i	$-0.639290\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.71650			−0.277441			−0.138721	−	0.990332i	$-0.544299\pi$
$17$	−4.71650			−0.277441			−0.138721	+	0.990332i	$0.544299\pi$
$18$		0			0
$19$	2.53604	−	2.53604i	0.133476	−	0.133476i	−0.637213	−	0.770688i	$-0.719913\pi$
$19$	2.53604	−	2.53604i	0.133476	−	0.133476i	0.770688	+	0.637213i	$0.219913\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	12.5864			0.547236			0.273618	−	0.961838i	$-0.411780\pi$
$23$	12.5864			0.547236			0.273618	+	0.961838i	$0.411780\pi$
$24$		0			0
$25$		−	20.8557i		−	0.834229i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.19683	−	2.19683i	−0.0757526	−	0.0757526i	0.668215	−	0.743968i	$-0.267058\pi$
$29$	−2.19683	−	2.19683i	−0.0757526	−	0.0757526i	−0.743968	+	0.668215i	$0.767058\pi$
$30$		0			0
$31$			28.0521i			0.904908i	0.891788	+	0.452454i	$0.149451\pi$
$31$			28.0521i			0.904908i	−0.891788	+	0.452454i	$0.850549\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	49.4644	−	49.4644i	1.41327	−	1.41327i
$36$		0			0
$37$	32.1128	−	32.1128i	0.867914	−	0.867914i	−0.124327	−	0.992241i	$-0.539677\pi$
$37$	32.1128	−	32.1128i	0.867914	−	0.867914i	0.992241	+	0.124327i	$0.0396773\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			23.1145i			0.563768i	0.959449	+	0.281884i	$0.0909593\pi$
$41$			23.1145i			0.563768i	−0.959449	+	0.281884i	$0.909041\pi$
$42$		0			0
$43$	−4.79441	−	4.79441i	−0.111498	−	0.111498i	0.649157	−	0.760655i	$-0.275122\pi$
$43$	−4.79441	−	4.79441i	−0.111498	−	0.111498i	−0.760655	+	0.649157i	$0.775122\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			39.0095i			0.829989i	0.909824	+	0.414994i	$0.136216\pi$
$47$			39.0095i			0.829989i	−0.909824	+	0.414994i	$0.863784\pi$
$48$		0			0
$49$	57.7141			1.17784
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−27.9768	+	27.9768i	−0.527864	+	0.527864i	−0.919935	−	0.392071i	$-0.871759\pi$
$53$	−27.9768	+	27.9768i	−0.527864	+	0.527864i	0.392071	+	0.919935i	$0.371759\pi$
$54$		0			0
$55$	5.03891			0.0916166
$56$		0			0
$57$		0			0
$58$		0			0
$59$	79.8538	+	79.8538i	1.35345	+	1.35345i	0.881764	+	0.471691i	$0.156356\pi$
$59$	79.8538	+	79.8538i	1.35345	+	1.35345i	0.471691	+	0.881764i	$0.343644\pi$
$60$		0			0
$61$	36.7762	+	36.7762i	0.602888	+	0.602888i	0.941078	−	0.338190i	$-0.109815\pi$
$61$	36.7762	+	36.7762i	0.602888	+	0.602888i	−0.338190	+	0.941078i	$0.609815\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	165.522			2.54649
$66$		0			0
$67$	10.9869	−	10.9869i	0.163984	−	0.163984i	−0.620345	−	0.784329i	$-0.713008\pi$
$67$	10.9869	−	10.9869i	0.163984	−	0.163984i	0.784329	+	0.620345i	$0.213008\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−52.6605			−0.741697			−0.370849	−	0.928693i	$-0.620933\pi$
$71$	−52.6605			−0.741697			−0.370849	+	0.928693i	$0.620933\pi$
$72$		0			0
$73$		−	67.8061i		−	0.928850i	−0.885612	−	0.464425i	$-0.846261\pi$
$73$		−	67.8061i		−	0.928850i	0.885612	−	0.464425i	$-0.153739\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	5.43545	+	5.43545i	0.0705903	+	0.0705903i
$78$		0			0
$79$			56.4602i			0.714686i	0.933973	+	0.357343i	$0.116317\pi$
$79$			56.4602i			0.714686i	−0.933973	+	0.357343i	$0.883683\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−58.3697	+	58.3697i	−0.703249	+	0.703249i	−0.965107	−	0.261857i	$-0.915665\pi$
$83$	−58.3697	+	58.3697i	−0.703249	+	0.703249i	0.261857	+	0.965107i	$0.415665\pi$
$84$		0			0
$85$	22.5840	−	22.5840i	0.265694	−	0.265694i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	131.566i		−	1.47827i	−0.673558	−	0.739135i	$-0.735235\pi$
$89$		−	131.566i		−	1.47827i	0.673558	−	0.739135i	$-0.264765\pi$
$90$		0			0
$91$	178.548	+	178.548i	1.96206	+	1.96206i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			24.2866i			0.255649i
$96$		0			0
$97$	60.9413			0.628261			0.314131	−	0.949380i	$-0.398287\pi$
$97$	60.9413			0.628261			0.314131	+	0.949380i	$0.398287\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	109.986	−	109.986i	1.08897	−	1.08897i	0.0933326	−	0.995635i	$-0.470248\pi$
$101$	109.986	−	109.986i	1.08897	−	1.08897i	0.995635	−	0.0933326i	$-0.0297520\pi$
$102$		0			0
$103$	173.295			1.68248			0.841239	−	0.540663i	$-0.181826\pi$
$103$	173.295			1.68248			0.841239	+	0.540663i	$0.181826\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−25.4747	−	25.4747i	−0.238081	−	0.238081i	0.577974	−	0.816055i	$-0.303844\pi$
$107$	−25.4747	−	25.4747i	−0.238081	−	0.238081i	−0.816055	+	0.577974i	$0.803844\pi$
$108$		0			0
$109$	−33.0605	−	33.0605i	−0.303307	−	0.303307i	0.538999	−	0.842306i	$-0.318803\pi$
$109$	−33.0605	−	33.0605i	−0.303307	−	0.303307i	−0.842306	+	0.538999i	$0.818803\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−140.159			−1.24034			−0.620171	−	0.784466i	$-0.712937\pi$
$113$	−140.159			−1.24034			−0.620171	+	0.784466i	$0.712937\pi$
$114$		0			0
$115$	−60.2677	+	60.2677i	−0.524067	+	0.524067i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	48.7226			0.409434
$120$		0			0
$121$		−	120.446i		−	0.995424i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−19.8441	−	19.8441i	−0.158752	−	0.158752i
$126$		0			0
$127$		−	40.8458i		−	0.321620i	−0.986985	−	0.160810i	$-0.948589\pi$
$127$		−	40.8458i		−	0.321620i	0.986985	−	0.160810i	$-0.0514107\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−75.0168	+	75.0168i	−0.572647	+	0.572647i	−0.932867	−	0.360220i	$-0.882702\pi$
$131$	−75.0168	+	75.0168i	−0.572647	+	0.572647i	0.360220	+	0.932867i	$0.382702\pi$
$132$		0			0
$133$	−26.1979	+	26.1979i	−0.196977	+	0.196977i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	134.028i		−	0.978308i	−0.872197	−	0.489154i	$-0.837306\pi$
$137$		−	134.028i		−	0.978308i	0.872197	−	0.489154i	$-0.162694\pi$
$138$		0			0
$139$	−22.8798	−	22.8798i	−0.164603	−	0.164603i	0.619999	−	0.784602i	$-0.287133\pi$
$139$	−22.8798	−	22.8798i	−0.164603	−	0.164603i	−0.784602	+	0.619999i	$0.787133\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			18.1885i			0.127193i
$144$		0			0
$145$	21.0381			0.145091
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−9.32124	+	9.32124i	−0.0625587	+	0.0625587i	−0.737694	−	0.675135i	$-0.764085\pi$
$149$	−9.32124	+	9.32124i	−0.0625587	+	0.0625587i	0.675135	+	0.737694i	$0.264085\pi$
$150$		0			0
$151$	−50.5403			−0.334704			−0.167352	−	0.985897i	$-0.553522\pi$
$151$	−50.5403			−0.334704			−0.167352	+	0.985897i	$0.553522\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−134.322	−	134.322i	−0.866595	−	0.866595i
$156$		0			0
$157$	95.8844	+	95.8844i	0.610729	+	0.610729i	0.943136	−	0.332407i	$-0.107861\pi$
$157$	95.8844	+	95.8844i	0.610729	+	0.610729i	−0.332407	+	0.943136i	$0.607861\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−130.021			−0.807584
$162$		0			0
$163$	140.885	−	140.885i	0.864324	−	0.864324i	−0.127513	−	0.991837i	$-0.540699\pi$
$163$	140.885	−	140.885i	0.864324	−	0.864324i	0.991837	+	0.127513i	$0.0406994\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	107.849			0.645800			0.322900	−	0.946433i	$-0.395342\pi$
$167$	107.849			0.645800			0.322900	+	0.946433i	$0.395342\pi$
$168$		0			0
$169$			428.470i			2.53533i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−53.8845	−	53.8845i	−0.311471	−	0.311471i	0.534008	−	0.845479i	$-0.320685\pi$
$173$	−53.8845	−	53.8845i	−0.311471	−	0.311471i	−0.845479	+	0.534008i	$0.820685\pi$
$174$		0			0
$175$			215.445i			1.23111i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	104.178	−	104.178i	0.582002	−	0.582002i	−0.353451	−	0.935453i	$-0.614992\pi$
$179$	104.178	−	104.178i	0.582002	−	0.582002i	0.935453	+	0.353451i	$0.114992\pi$
$180$		0			0
$181$	205.498	−	205.498i	1.13535	−	1.13535i	0.146073	−	0.989274i	$-0.453336\pi$
$181$	205.498	−	205.498i	1.13535	−	1.13535i	0.989274	−	0.146073i	$-0.0466635\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			307.532i			1.66233i
$186$		0			0
$187$	2.48167	+	2.48167i	0.0132710	+	0.0132710i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	248.255i		−	1.29977i	−0.760034	−	0.649883i	$-0.774818\pi$
$191$		−	248.255i		−	1.29977i	0.760034	−	0.649883i	$-0.225182\pi$
$192$		0			0
$193$	−129.921			−0.673166			−0.336583	−	0.941654i	$-0.609271\pi$
$193$	−129.921			−0.673166			−0.336583	+	0.941654i	$0.609271\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	237.001	−	237.001i	1.20305	−	1.20305i	0.229816	−	0.973234i	$-0.426188\pi$
$197$	237.001	−	237.001i	1.20305	−	1.20305i	0.973234	−	0.229816i	$-0.0738123\pi$
$198$		0			0
$199$	246.508			1.23873			0.619366	−	0.785102i	$-0.287390\pi$
$199$	246.508			1.23873			0.619366	+	0.785102i	$0.287390\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	22.6938	+	22.6938i	0.111792	+	0.111792i
$204$		0			0
$205$	−110.679	−	110.679i	−0.539898	−	0.539898i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2.66877			−0.0127692
$210$		0			0
$211$	13.4139	−	13.4139i	0.0635728	−	0.0635728i	−0.674606	−	0.738178i	$-0.735686\pi$
$211$	13.4139	−	13.4139i	0.0635728	−	0.0635728i	0.738178	+	0.674606i	$0.235686\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	45.9142			0.213554
$216$		0			0
$217$		−	289.786i		−	1.33542i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	81.5197	+	81.5197i	0.368867	+	0.368867i
$222$		0			0
$223$		−	295.580i		−	1.32547i	−0.748854	−	0.662735i	$-0.769396\pi$
$223$		−	295.580i		−	1.32547i	0.748854	−	0.662735i	$-0.230604\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−97.0742	+	97.0742i	−0.427640	+	0.427640i	−0.887824	−	0.460184i	$-0.847783\pi$
$227$	−97.0742	+	97.0742i	−0.427640	+	0.427640i	0.460184	+	0.887824i	$0.347783\pi$
$228$		0			0
$229$	−34.2565	+	34.2565i	−0.149592	+	0.149592i	−0.777936	−	0.628344i	$-0.783733\pi$
$229$	−34.2565	+	34.2565i	−0.149592	+	0.149592i	0.628344	+	0.777936i	$0.283733\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			62.8176i			0.269604i	0.990873	+	0.134802i	$0.0430398\pi$
$233$			62.8176i			0.269604i	−0.990873	+	0.134802i	$0.956960\pi$
$234$		0			0
$235$	−186.789	−	186.789i	−0.794848	−	0.794848i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			355.910i			1.48916i	0.667532	+	0.744581i	$0.267351\pi$
$239$			355.910i			1.48916i	−0.667532	+	0.744581i	$0.732649\pi$
$240$		0			0
$241$	66.2545			0.274915			0.137458	−	0.990508i	$-0.456107\pi$
$241$	66.2545			0.274915			0.137458	+	0.990508i	$0.456107\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−276.352	+	276.352i	−1.12797	+	1.12797i
$246$		0			0
$247$	−87.6655			−0.354921
$248$		0			0
$249$		0			0
$250$		0			0
$251$	325.395	+	325.395i	1.29640	+	1.29640i	0.930757	+	0.365638i	$0.119149\pi$
$251$	325.395	+	325.395i	1.29640	+	1.29640i	0.365638	+	0.930757i	$0.380851\pi$
$252$		0			0
$253$	−6.62259	−	6.62259i	−0.0261762	−	0.0261762i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	312.011			1.21405			0.607026	−	0.794682i	$-0.292362\pi$
$257$	312.011			1.21405			0.607026	+	0.794682i	$0.292362\pi$
$258$		0			0
$259$	−331.733	+	331.733i	−1.28082	+	1.28082i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	168.163			0.639403			0.319702	−	0.947518i	$-0.396417\pi$
$263$	168.163			0.639403			0.319702	+	0.947518i	$0.396417\pi$
$264$		0			0
$265$		−	267.923i		−	1.01103i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	212.116	+	212.116i	0.788535	+	0.788535i	0.981254	−	0.192719i	$-0.0617306\pi$
$269$	212.116	+	212.116i	0.788535	+	0.788535i	−0.192719	+	0.981254i	$0.561731\pi$
$270$		0			0
$271$			173.450i			0.640037i	0.947411	+	0.320019i	$0.103689\pi$
$271$			173.450i			0.640037i	−0.947411	+	0.320019i	$0.896311\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−10.9736	+	10.9736i	−0.0399041	+	0.0399041i
$276$		0			0
$277$	38.4049	−	38.4049i	0.138646	−	0.138646i	−0.634377	−	0.773023i	$-0.718743\pi$
$277$	38.4049	−	38.4049i	0.138646	−	0.138646i	0.773023	+	0.634377i	$0.218743\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			223.573i			0.795632i	0.917465	+	0.397816i	$0.130232\pi$
$281$			223.573i			0.795632i	−0.917465	+	0.397816i	$0.869768\pi$
$282$		0			0
$283$	−247.755	−	247.755i	−0.875459	−	0.875459i	0.117602	−	0.993061i	$-0.462479\pi$
$283$	−247.755	−	247.755i	−0.875459	−	0.875459i	−0.993061	+	0.117602i	$0.962479\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	238.778i		−	0.831980i
$288$		0			0
$289$	−266.755			−0.923026
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−102.262	+	102.262i	−0.349016	+	0.349016i	−0.859743	−	0.510727i	$-0.829376\pi$
$293$	−102.262	+	102.262i	−0.349016	+	0.349016i	0.510727	+	0.859743i	$0.329376\pi$
$294$		0			0
$295$	−764.729			−2.59230
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−217.543	−	217.543i	−0.727570	−	0.727570i
$300$		0			0
$301$	49.5275	+	49.5275i	0.164543	+	0.164543i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−352.191			−1.15472
$306$		0			0
$307$	−138.292	+	138.292i	−0.450463	+	0.450463i	−0.895508	−	0.445045i	$-0.853188\pi$
$307$	−138.292	+	138.292i	−0.450463	+	0.450463i	0.445045	+	0.895508i	$0.353188\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−205.789			−0.661702			−0.330851	−	0.943683i	$-0.607336\pi$
$311$	−205.789			−0.661702			−0.330851	+	0.943683i	$0.607336\pi$
$312$		0			0
$313$			223.861i			0.715209i	0.933873	+	0.357605i	$0.116406\pi$
$313$			223.861i			0.715209i	−0.933873	+	0.357605i	$0.883594\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−176.488	−	176.488i	−0.556744	−	0.556744i	0.371635	−	0.928379i	$-0.378797\pi$
$317$	−176.488	−	176.488i	−0.556744	−	0.556744i	−0.928379	+	0.371635i	$0.878797\pi$
$318$		0			0
$319$			2.31180i			0.00724703i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−11.9612	+	11.9612i	−0.0370316	+	0.0370316i
$324$		0			0
$325$	−360.469	+	360.469i	−1.10914	+	1.10914i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	402.978i		−	1.22486i
$330$		0			0
$331$	183.939	+	183.939i	0.555706	+	0.555706i	0.928082	−	0.372376i	$-0.121457\pi$
$331$	183.939	+	183.939i	0.555706	+	0.555706i	−0.372376	+	0.928082i	$0.621457\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			105.217i			0.314081i
$336$		0			0
$337$	12.7162			0.0377336			0.0188668	−	0.999822i	$-0.493994\pi$
$337$	12.7162			0.0377336			0.0188668	+	0.999822i	$0.493994\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	14.7602	−	14.7602i	0.0432849	−	0.0432849i
$342$		0			0
$343$	−90.0184			−0.262444
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−113.546	−	113.546i	−0.327221	−	0.327221i	0.524308	−	0.851529i	$-0.324324\pi$
$347$	−113.546	−	113.546i	−0.327221	−	0.327221i	−0.851529	+	0.524308i	$0.824324\pi$
$348$		0			0
$349$	−90.9653	−	90.9653i	−0.260645	−	0.260645i	0.564671	−	0.825316i	$-0.309003\pi$
$349$	−90.9653	−	90.9653i	−0.260645	−	0.260645i	−0.825316	+	0.564671i	$0.809003\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−36.2208			−0.102609			−0.0513043	−	0.998683i	$-0.516338\pi$
$353$	−36.2208			−0.102609			−0.0513043	+	0.998683i	$0.516338\pi$
$354$		0			0
$355$	252.155	−	252.155i	0.710294	−	0.710294i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−142.121			−0.395880			−0.197940	−	0.980214i	$-0.563425\pi$
$359$	−142.121			−0.395880			−0.197940	+	0.980214i	$0.563425\pi$
$360$		0			0
$361$			348.137i			0.964369i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	324.676	+	324.676i	0.889523	+	0.889523i
$366$		0			0
$367$		−	654.218i		−	1.78261i	−0.453404	−	0.891305i	$-0.649791\pi$
$367$		−	654.218i		−	1.78261i	0.453404	−	0.891305i	$-0.350209\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	289.007	−	289.007i	0.778995	−	0.778995i
$372$		0			0
$373$	−335.277	+	335.277i	−0.898867	+	0.898867i	−0.995336	−	0.0964690i	$-0.969245\pi$
$373$	−335.277	+	335.277i	−0.898867	+	0.898867i	0.0964690	+	0.995336i	$0.469245\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			75.9397i			0.201432i
$378$		0			0
$379$	98.7497	+	98.7497i	0.260553	+	0.260553i	0.825279	−	0.564725i	$-0.191018\pi$
$379$	98.7497	+	98.7497i	0.260553	+	0.260553i	−0.564725	+	0.825279i	$0.691018\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	156.144i		−	0.407687i	−0.979003	−	0.203844i	$-0.934657\pi$
$383$		−	156.144i		−	0.407687i	0.979003	−	0.203844i	$-0.0653434\pi$
$384$		0			0
$385$	−52.0532			−0.135203
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−391.047	+	391.047i	−1.00526	+	1.00526i	−0.00527486	+	0.999986i	$0.501679\pi$
$389$	−391.047	+	391.047i	−1.00526	+	1.00526i	−0.999986	+	0.00527486i	$0.998321\pi$
$390$		0			0
$391$	−59.3639			−0.151826
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−270.349	−	270.349i	−0.684427	−	0.684427i
$396$		0			0
$397$	−243.862	−	243.862i	−0.614262	−	0.614262i	0.329791	−	0.944054i	$-0.393022\pi$
$397$	−243.862	−	243.862i	−0.614262	−	0.614262i	−0.944054	+	0.329791i	$0.893022\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	175.261			0.437059			0.218529	−	0.975830i	$-0.429874\pi$
$401$	175.261			0.437059			0.218529	+	0.975830i	$0.429874\pi$
$402$		0			0
$403$	484.852	−	484.852i	1.20311	−	1.20311i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−33.7935			−0.0830307
$408$		0			0
$409$		−	44.4504i		−	0.108681i	−0.998522	−	0.0543404i	$-0.982694\pi$
$409$		−	44.4504i		−	0.108681i	0.998522	−	0.0543404i	$-0.0173056\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−824.910	−	824.910i	−1.99736	−	1.99736i
$414$		0			0
$415$		−	558.984i		−	1.34695i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	14.9985	−	14.9985i	0.0357959	−	0.0357959i	−0.688982	−	0.724778i	$-0.741942\pi$
$419$	14.9985	−	14.9985i	0.0357959	−	0.0357959i	0.724778	+	0.688982i	$0.241942\pi$
$420$		0			0
$421$	−312.907	+	312.907i	−0.743247	+	0.743247i	−0.973201	−	0.229954i	$-0.926142\pi$
$421$	−312.907	+	312.907i	−0.743247	+	0.743247i	0.229954	+	0.973201i	$0.426142\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			98.3660i			0.231449i
$426$		0			0
$427$	−379.907	−	379.907i	−0.889712	−	0.889712i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	532.400i		−	1.23527i	−0.786466	−	0.617633i	$-0.788092\pi$
$431$		−	532.400i		−	1.23527i	0.786466	−	0.617633i	$-0.211908\pi$
$432$		0			0
$433$	553.451			1.27818			0.639089	−	0.769133i	$-0.279312\pi$
$433$	553.451			1.27818			0.639089	+	0.769133i	$0.279312\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	31.9197	−	31.9197i	0.0730427	−	0.0730427i
$438$		0			0
$439$	645.291			1.46991			0.734956	−	0.678115i	$-0.237203\pi$
$439$	645.291			1.46991			0.734956	+	0.678115i	$0.237203\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	315.833	+	315.833i	0.712941	+	0.712941i	0.967149	−	0.254208i	$-0.0818149\pi$
$443$	315.833	+	315.833i	0.712941	+	0.712941i	−0.254208	+	0.967149i	$0.581815\pi$
$444$		0			0
$445$	629.978	+	629.978i	1.41568	+	1.41568i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−218.589			−0.486835			−0.243417	−	0.969922i	$-0.578268\pi$
$449$	−218.589			−0.486835			−0.243417	+	0.969922i	$0.578268\pi$
$450$		0			0
$451$	12.1621	−	12.1621i	0.0269670	−	0.0269670i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1709.88			−3.75798
$456$		0			0
$457$			296.561i			0.648930i	0.945898	+	0.324465i	$0.105184\pi$
$457$			296.561i			0.648930i	−0.945898	+	0.324465i	$0.894816\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	118.061	+	118.061i	0.256097	+	0.256097i	0.823465	−	0.567368i	$-0.192038\pi$
$461$	118.061	+	118.061i	0.256097	+	0.256097i	−0.567368	+	0.823465i	$0.692038\pi$
$462$		0			0
$463$			409.453i			0.884348i	0.896929	+	0.442174i	$0.145793\pi$
$463$			409.453i			0.884348i	−0.896929	+	0.442174i	$0.854207\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	494.764	−	494.764i	1.05945	−	1.05945i	0.0613343	−	0.998117i	$-0.480464\pi$
$467$	494.764	−	494.764i	1.05945	−	1.05945i	0.998117	−	0.0613343i	$-0.0195356\pi$
$468$		0			0
$469$	−113.497	+	113.497i	−0.241999	+	0.241999i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			5.04534i			0.0106667i
$474$		0			0
$475$	−52.8909	−	52.8909i	−0.111349	−	0.111349i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			558.806i			1.16661i	0.812254	+	0.583305i	$0.198241\pi$
$479$			558.806i			1.16661i	−0.812254	+	0.583305i	$0.801759\pi$
$480$		0			0
$481$	−1110.07			−2.30784
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−291.806	+	291.806i	−0.601661	+	0.601661i
$486$		0			0
$487$	−361.328			−0.741946			−0.370973	−	0.928644i	$-0.620976\pi$
$487$	−361.328			−0.741946			−0.370973	+	0.928644i	$0.620976\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−488.975	−	488.975i	−0.995876	−	0.995876i	0.00411514	−	0.999992i	$-0.498690\pi$
$491$	−488.975	−	488.975i	−0.995876	−	0.995876i	−0.999992	+	0.00411514i	$0.998690\pi$
$492$		0			0
$493$	10.3613	+	10.3613i	0.0210169	+	0.0210169i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	543.996			1.09456
$498$		0			0
$499$	102.895	−	102.895i	0.206203	−	0.206203i	−0.596448	−	0.802652i	$-0.703422\pi$
$499$	102.895	−	102.895i	0.206203	−	0.206203i	0.802652	+	0.596448i	$0.203422\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	881.975			1.75343			0.876715	−	0.481011i	$-0.159730\pi$
$503$	881.975			1.75343			0.876715	+	0.481011i	$0.159730\pi$
$504$		0			0
$505$			1053.29i			2.08572i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	161.639	+	161.639i	0.317563	+	0.317563i	0.847830	−	0.530268i	$-0.177909\pi$
$509$	161.639	+	161.639i	0.317563	+	0.317563i	−0.530268	+	0.847830i	$0.677909\pi$
$510$		0			0
$511$			700.454i			1.37075i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−829.791	+	829.791i	−1.61124	+	1.61124i
$516$		0			0
$517$	20.5256	−	20.5256i	0.0397013	−	0.0397013i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	763.931i		−	1.46628i	−0.680078	−	0.733140i	$-0.738054\pi$
$521$		−	763.931i		−	1.46628i	0.680078	−	0.733140i	$-0.261946\pi$
$522$		0			0
$523$	−295.573	−	295.573i	−0.565150	−	0.565150i	0.365616	−	0.930766i	$-0.380858\pi$
$523$	−295.573	−	295.573i	−0.565150	−	0.565150i	−0.930766	+	0.365616i	$0.880858\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	132.308i		−	0.251059i
$528$		0			0
$529$	−370.582			−0.700532
$530$		0			0
$531$		0			0
$532$		0			0
$533$	399.509	−	399.509i	0.749549	−	0.749549i
$534$		0			0
$535$	243.961			0.456002
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−30.3673	−	30.3673i	−0.0563401	−	0.0563401i
$540$		0			0
$541$	−243.037	−	243.037i	−0.449236	−	0.449236i	0.445865	−	0.895100i	$-0.352896\pi$
$541$	−243.037	−	243.037i	−0.449236	−	0.449236i	−0.895100	+	0.445865i	$0.852896\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	316.607			0.580931
$546$		0			0
$547$	−424.574	+	424.574i	−0.776187	+	0.776187i	−0.979180	−	0.202993i	$-0.934933\pi$
$547$	−424.574	+	424.574i	−0.776187	+	0.776187i	0.202993	+	0.979180i	$0.434933\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−11.1425			−0.0202223
$552$		0			0
$553$		−	583.248i		−	1.05470i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−445.773	−	445.773i	−0.800311	−	0.800311i	0.182833	−	0.983144i	$-0.441473\pi$
$557$	−445.773	−	445.773i	−0.800311	−	0.800311i	−0.983144	+	0.182833i	$0.941473\pi$
$558$		0			0
$559$			165.733i			0.296481i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−529.295	+	529.295i	−0.940133	+	0.940133i	−0.998307	−	0.0581732i	$-0.981472\pi$
$563$	−529.295	+	529.295i	−0.940133	+	0.940133i	0.0581732	+	0.998307i	$0.481472\pi$
$564$		0			0
$565$	671.123	−	671.123i	1.18783	−	1.18783i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			346.814i			0.609516i	0.952430	+	0.304758i	$0.0985755\pi$
$569$			346.814i			0.609516i	−0.952430	+	0.304758i	$0.901424\pi$
$570$		0			0
$571$	155.711	+	155.711i	0.272699	+	0.272699i	0.830186	−	0.557487i	$-0.188234\pi$
$571$	155.711	+	155.711i	0.272699	+	0.272699i	−0.557487	+	0.830186i	$0.688234\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	262.499i		−	0.456520i
$576$		0			0
$577$	620.510			1.07541			0.537704	−	0.843134i	$-0.319292\pi$
$577$	620.510			1.07541			0.537704	+	0.843134i	$0.319292\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	602.974	−	602.974i	1.03782	−	1.03782i
$582$		0			0
$583$	29.4410			0.0504992
$584$		0			0
$585$		0			0
$586$		0			0
$587$	561.656	+	561.656i	0.956825	+	0.956825i	0.999106	−	0.0422810i	$-0.0134625\pi$
$587$	561.656	+	561.656i	0.956825	+	0.956825i	−0.0422810	+	0.999106i	$0.513462\pi$
$588$		0			0
$589$	71.1413	+	71.1413i	0.120783	+	0.120783i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−851.739			−1.43632			−0.718161	−	0.695877i	$-0.755016\pi$
$593$	−851.739			−1.43632			−0.718161	+	0.695877i	$0.755016\pi$
$594$		0			0
$595$	−233.299	+	233.299i	−0.392099	+	0.392099i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1001.69			1.67228			0.836138	−	0.548519i	$-0.184808\pi$
$599$	1001.69			1.67228			0.836138	+	0.548519i	$0.184808\pi$
$600$		0			0
$601$		−	955.182i		−	1.58932i	−0.607054	−	0.794661i	$-0.707649\pi$
$601$		−	955.182i		−	1.58932i	0.607054	−	0.794661i	$-0.292351\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	576.734	+	576.734i	0.953279	+	0.953279i
$606$		0			0
$607$			291.885i			0.480865i	0.970666	+	0.240432i	$0.0772892\pi$
$607$			291.885i			0.480865i	−0.970666	+	0.240432i	$0.922711\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	674.238	−	674.238i	1.10350	−	1.10350i
$612$		0			0
$613$	332.933	−	332.933i	0.543121	−	0.543121i	−0.381322	−	0.924442i	$-0.624531\pi$
$613$	332.933	−	332.933i	0.543121	−	0.543121i	0.924442	+	0.381322i	$0.124531\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			970.864i			1.57352i	0.617257	+	0.786762i	$0.288244\pi$
$617$			970.864i			1.57352i	−0.617257	+	0.786762i	$0.711756\pi$
$618$		0			0
$619$	696.761	+	696.761i	1.12562	+	1.12562i	0.990881	+	0.134744i	$0.0430210\pi$
$619$	696.761	+	696.761i	1.12562	+	1.12562i	0.134744	+	0.990881i	$0.456979\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			1359.11i			2.18156i
$624$		0			0
$625$	711.432			1.13829
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−151.460	+	151.460i	−0.240795	+	0.240795i
$630$		0			0
$631$	−377.591			−0.598401			−0.299200	−	0.954190i	$-0.596720\pi$
$631$	−377.591			−0.598401			−0.299200	+	0.954190i	$0.596720\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	195.582	+	195.582i	0.308003	+	0.308003i
$636$		0			0
$637$	−997.527	−	997.527i	−1.56598	−	1.56598i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−729.200			−1.13760			−0.568799	−	0.822477i	$-0.692592\pi$
$641$	−729.200			−1.13760			−0.568799	+	0.822477i	$0.692592\pi$
$642$		0			0
$643$	−243.958	+	243.958i	−0.379406	+	0.379406i	−0.870888	−	0.491482i	$-0.836455\pi$
$643$	−243.958	+	243.958i	−0.379406	+	0.379406i	0.491482	+	0.870888i	$0.336455\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	281.594			0.435230			0.217615	−	0.976035i	$-0.430172\pi$
$647$	281.594			0.435230			0.217615	+	0.976035i	$0.430172\pi$
$648$		0			0
$649$		−	84.0331i		−	0.129481i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	323.704	+	323.704i	0.495718	+	0.495718i	0.910102	−	0.414384i	$-0.136003\pi$
$653$	323.704	+	323.704i	0.495718	+	0.495718i	−0.414384	+	0.910102i	$0.636003\pi$
$654$		0			0
$655$		−	718.407i		−	1.09680i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	507.811	−	507.811i	0.770578	−	0.770578i	−0.207629	−	0.978208i	$-0.566575\pi$
$659$	507.811	−	507.811i	0.770578	−	0.770578i	0.978208	+	0.207629i	$0.0665748\pi$
$660$		0			0
$661$	−57.1593	+	57.1593i	−0.0864741	+	0.0864741i	−0.749021	−	0.662547i	$-0.769476\pi$
$661$	−57.1593	+	57.1593i	−0.0864741	+	0.0864741i	0.662547	+	0.749021i	$0.269476\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	250.887i		−	0.377274i
$666$		0			0
$667$	−27.6502	−	27.6502i	−0.0414546	−	0.0414546i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	38.7009i		−	0.0576765i
$672$		0			0
$673$	1110.84			1.65059			0.825293	−	0.564705i	$-0.191010\pi$
$673$	1110.84			1.65059			0.825293	+	0.564705i	$0.191010\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	397.465	−	397.465i	0.587097	−	0.587097i	−0.349747	−	0.936844i	$-0.613732\pi$
$677$	397.465	−	397.465i	0.587097	−	0.587097i	0.936844	+	0.349747i	$0.113732\pi$
$678$		0			0
$679$	−629.539			−0.927156
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−238.015	−	238.015i	−0.348485	−	0.348485i	0.511060	−	0.859545i	$-0.329253\pi$
$683$	−238.015	−	238.015i	−0.348485	−	0.348485i	−0.859545	+	0.511060i	$0.829253\pi$
$684$		0			0
$685$	641.768	+	641.768i	0.936887	+	0.936887i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	967.099			1.40363
$690$		0			0
$691$	685.172	−	685.172i	0.991565	−	0.991565i	−0.00839951	−	0.999965i	$-0.502674\pi$
$691$	685.172	−	685.172i	0.991565	−	0.991565i	0.999965	+	0.00839951i	$0.00267368\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	219.111			0.315267
$696$		0			0
$697$		−	109.019i		−	0.156412i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	543.074	+	543.074i	0.774713	+	0.774713i	0.978926	−	0.204214i	$-0.0654637\pi$
$701$	543.074	+	543.074i	0.774713	+	0.774713i	−0.204214	+	0.978926i	$0.565464\pi$
$702$		0			0
$703$		−	162.879i		−	0.231691i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1136.18	+	1136.18i	−1.60704	+	1.60704i
$708$		0			0
$709$	488.019	−	488.019i	0.688320	−	0.688320i	−0.273541	−	0.961860i	$-0.588195\pi$
$709$	488.019	−	488.019i	0.688320	−	0.688320i	0.961860	+	0.273541i	$0.0881948\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			353.076i			0.495198i
$714$		0			0
$715$	−87.0923	−	87.0923i	−0.121807	−	0.121807i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	297.369i		−	0.413587i	−0.978385	−	0.206793i	$-0.933697\pi$
$719$		−	297.369i		−	0.413587i	0.978385	−	0.206793i	$-0.0663028\pi$
$720$		0			0
$721$	−1790.18			−2.48292
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−45.8164	+	45.8164i	−0.0631950	+	0.0631950i
$726$		0			0
$727$	1158.85			1.59402			0.797009	−	0.603967i	$-0.206414\pi$
$727$	1158.85			1.59402			0.797009	+	0.603967i	$0.206414\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	22.6128	+	22.6128i	0.0309341	+	0.0309341i
$732$		0			0
$733$	348.835	+	348.835i	0.475901	+	0.475901i	0.903818	−	0.427917i	$-0.140752\pi$
$733$	348.835	+	348.835i	0.475901	+	0.475901i	−0.427917	+	0.903818i	$0.640752\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−11.5619			−0.0156878
$738$		0			0
$739$	−825.489	+	825.489i	−1.11703	+	1.11703i	−0.124860	+	0.992174i	$0.539848\pi$
$739$	−825.489	+	825.489i	−1.11703	+	1.11703i	−0.992174	+	0.124860i	$0.960152\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	899.725			1.21094			0.605468	−	0.795870i	$-0.292986\pi$
$743$	899.725			1.21094			0.605468	+	0.795870i	$0.292986\pi$
$744$		0			0
$745$		−	89.2659i		−	0.119820i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	263.160	+	263.160i	0.351348	+	0.351348i
$750$		0			0
$751$		−	80.4386i		−	0.107109i	−0.998565	−	0.0535543i	$-0.982945\pi$
$751$		−	80.4386i		−	0.107109i	0.998565	−	0.0535543i	$-0.0170550\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	242.003	−	242.003i	0.320533	−	0.320533i
$756$		0			0
$757$	−233.298	+	233.298i	−0.308187	+	0.308187i	−0.844206	−	0.536019i	$-0.819928\pi$
$757$	−233.298	+	233.298i	−0.308187	+	0.308187i	0.536019	+	0.844206i	$0.319928\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			56.1906i			0.0738378i	0.999318	+	0.0369189i	$0.0117543\pi$
$761$			56.1906i			0.0738378i	−0.999318	+	0.0369189i	$0.988246\pi$
$762$		0			0
$763$	341.523	+	341.523i	0.447606	+	0.447606i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	2760.38i		−	3.59893i
$768$		0			0
$769$	517.343			0.672748			0.336374	−	0.941728i	$-0.390799\pi$
$769$	517.343			0.672748			0.336374	+	0.941728i	$0.390799\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−523.925	+	523.925i	−0.677781	+	0.677781i	−0.959498	−	0.281716i	$-0.909096\pi$
$773$	−523.925	+	523.925i	−0.677781	+	0.677781i	0.281716	+	0.959498i	$0.409096\pi$
$774$		0			0
$775$	585.048			0.754900
$776$		0			0
$777$		0			0
$778$		0			0
$779$	58.6192	+	58.6192i	0.0752492	+	0.0752492i
$780$		0			0
$781$	27.7083	+	27.7083i	0.0354780	+	0.0354780i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−918.248			−1.16974
$786$		0			0
$787$	46.6965	−	46.6965i	0.0593348	−	0.0593348i	−0.676817	−	0.736152i	$-0.736641\pi$
$787$	46.6965	−	46.6965i	0.0593348	−	0.0593348i	0.736152	+	0.676817i	$0.236641\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1447.87			1.83044
$792$		0			0
$793$		−	1271.27i		−	1.60312i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	127.126	+	127.126i	0.159505	+	0.159505i	0.782348	−	0.622842i	$-0.214022\pi$
$797$	127.126	+	127.126i	0.159505	+	0.159505i	−0.622842	+	0.782348i	$0.714022\pi$
$798$		0			0
$799$		−	183.988i		−	0.230273i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−35.6774	+	35.6774i	−0.0444302	+	0.0444302i
$804$		0			0
$805$	622.580	−	622.580i	0.773392	−	0.773392i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	1047.16i		−	1.29439i	−0.762325	−	0.647194i	$-0.775942\pi$
$809$		−	1047.16i		−	1.29439i	0.762325	−	0.647194i	$-0.224058\pi$
$810$		0			0
$811$	112.206	+	112.206i	0.138356	+	0.138356i	0.772893	−	0.634537i	$-0.218809\pi$
$811$	112.206	+	112.206i	0.138356	+	0.138356i	−0.634537	+	0.772893i	$0.718809\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			1349.20i			1.65546i
$816$		0			0
$817$	−24.3176			−0.0297645
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−7.63080	+	7.63080i	−0.00929452	+	0.00929452i	−0.711739	−	0.702444i	$-0.752092\pi$
$821$	−7.63080	+	7.63080i	−0.00929452	+	0.00929452i	0.702444	+	0.711739i	$0.252092\pi$
$822$		0			0
$823$	1316.28			1.59937			0.799687	−	0.600417i	$-0.204999\pi$
$823$	1316.28			1.59937			0.799687	+	0.600417i	$0.204999\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	341.515	+	341.515i	0.412957	+	0.412957i	0.882767	−	0.469810i	$-0.155678\pi$
$827$	341.515	+	341.515i	0.412957	+	0.412957i	−0.469810	+	0.882767i	$0.655678\pi$
$828$		0			0
$829$	621.672	+	621.672i	0.749905	+	0.749905i	0.974461	−	0.224556i	$-0.0720931\pi$
$829$	621.672	+	621.672i	0.749905	+	0.749905i	−0.224556	+	0.974461i	$0.572093\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−272.208			−0.326781
$834$		0			0
$835$	−516.412	+	516.412i	−0.618457	+	0.618457i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1440.49			−1.71692			−0.858459	−	0.512883i	$-0.828578\pi$
$839$	−1440.49			−1.71692			−0.858459	+	0.512883i	$0.828578\pi$
$840$		0			0
$841$		−	831.348i		−	0.988523i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−2051.65	−	2051.65i	−2.42798	−	2.42798i
$846$		0			0
$847$			1244.24i			1.46900i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	404.186	−	404.186i	0.474954	−	0.474954i
$852$		0			0
$853$	625.193	−	625.193i	0.732934	−	0.732934i	−0.238266	−	0.971200i	$-0.576579\pi$
$853$	625.193	−	625.193i	0.732934	−	0.732934i	0.971200	+	0.238266i	$0.0765789\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	1105.18i		−	1.28959i	−0.764356	−	0.644794i	$-0.776943\pi$
$857$		−	1105.18i		−	1.28959i	0.764356	−	0.644794i	$-0.223057\pi$
$858$		0			0
$859$	−379.841	−	379.841i	−0.442190	−	0.442190i	0.450558	−	0.892747i	$-0.351225\pi$
$859$	−379.841	−	379.841i	−0.442190	−	0.442190i	−0.892747	+	0.450558i	$0.851225\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			381.969i			0.442606i	0.975205	+	0.221303i	$0.0710311\pi$
$863$			381.969i			0.442606i	−0.975205	+	0.221303i	$0.928969\pi$
$864$		0			0
$865$	516.031			0.596567
$866$		0			0
$867$		0			0
$868$		0			0
$869$	29.7076	−	29.7076i	0.0341859	−	0.0341859i
$870$		0			0
$871$	−379.794			−0.436044
$872$		0			0
$873$		0			0
$874$		0			0
$875$	204.994	+	204.994i	0.234279	+	0.234279i
$876$		0			0
$877$	−638.602	−	638.602i	−0.728166	−	0.728166i	0.242088	−	0.970254i	$-0.422168\pi$
$877$	−638.602	−	638.602i	−0.728166	−	0.728166i	−0.970254	+	0.242088i	$0.922168\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1362.97			1.54707			0.773533	−	0.633756i	$-0.218488\pi$
$881$	1362.97			1.54707			0.773533	+	0.633756i	$0.218488\pi$
$882$		0			0
$883$	897.988	−	897.988i	1.01697	−	1.01697i	0.0171209	−	0.999853i	$-0.494550\pi$
$883$	897.988	−	897.988i	1.01697	−	1.01697i	0.999853	−	0.0171209i	$-0.00545001\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1343.56			−1.51472			−0.757359	−	0.652998i	$-0.773511\pi$
$887$	−1343.56			−1.51472			−0.757359	+	0.652998i	$0.773511\pi$
$888$		0			0
$889$			421.947i			0.474631i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	98.9295	+	98.9295i	0.110783	+	0.110783i
$894$		0			0
$895$			997.676i			1.11472i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	61.6257	−	61.6257i	0.0685491	−	0.0685491i
$900$		0			0
$901$	131.952	−	131.952i	0.146451	−	0.146451i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			1967.97i			2.17456i
$906$		0			0
$907$	671.651	+	671.651i	0.740519	+	0.740519i	0.972678	−	0.232159i	$-0.0745789\pi$
$907$	671.651	+	671.651i	0.740519	+	0.740519i	−0.232159	+	0.972678i	$0.574579\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			770.729i			0.846025i	0.906124	+	0.423012i	$0.139027\pi$
$911$			770.729i			0.846025i	−0.906124	+	0.423012i	$0.860973\pi$
$912$		0			0
$913$	61.4246			0.0672778
$914$		0			0
$915$		0			0
$916$		0			0
$917$	774.942	−	774.942i	0.845084	−	0.845084i
$918$		0			0
$919$	−1153.98			−1.25569			−0.627843	−	0.778340i	$-0.716062\pi$
$919$	−1153.98			−1.25569			−0.627843	+	0.778340i	$0.716062\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	910.182	+	910.182i	0.986112	+	0.986112i
$924$		0			0
$925$	−669.736	−	669.736i	−0.724039	−	0.724039i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	652.736			0.702622			0.351311	−	0.936259i	$-0.385736\pi$
$929$	652.736			0.702622			0.351311	+	0.936259i	$0.385736\pi$
$930$		0			0
$931$	146.365	−	146.365i	0.157213	−	0.157213i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−23.7660			−0.0254182
$936$		0			0
$937$			644.074i			0.687378i	0.939083	+	0.343689i	$0.111677\pi$
$937$			644.074i			0.687378i	−0.939083	+	0.343689i	$0.888323\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	171.348	+	171.348i	0.182092	+	0.182092i	0.792267	−	0.610175i	$-0.208901\pi$
$941$	171.348	+	171.348i	0.182092	+	0.182092i	−0.610175	+	0.792267i	$0.708901\pi$
$942$		0			0
$943$			290.929i			0.308514i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	731.249	−	731.249i	0.772174	−	0.772174i	−0.206312	−	0.978486i	$-0.566146\pi$
$947$	731.249	−	731.249i	0.772174	−	0.772174i	0.978486	+	0.206312i	$0.0661461\pi$
$948$		0			0
$949$	−1171.96	+	1171.96i	−1.23494	+	1.23494i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1745.08i		−	1.83115i	−0.402152	−	0.915573i	$-0.631738\pi$
$953$		−	1745.08i		−	1.83115i	0.402152	−	0.915573i	$-0.368262\pi$
$954$		0			0
$955$	1188.72	+	1188.72i	1.24473	+	1.24473i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			1384.54i			1.44374i
$960$		0			0
$961$	174.077			0.181142
$962$		0			0
$963$		0			0
$964$		0			0
$965$	622.102	−	622.102i	0.644665	−	0.644665i
$966$		0			0
$967$	904.237			0.935095			0.467548	−	0.883968i	$-0.345138\pi$
$967$	904.237			0.935095			0.467548	+	0.883968i	$0.345138\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1010.37	−	1010.37i	−1.04055	−	1.04055i	−0.999143	−	0.0414029i	$-0.986817\pi$
$971$	−1010.37	−	1010.37i	−1.04055	−	1.04055i	−0.0414029	−	0.999143i	$-0.513183\pi$
$972$		0			0
$973$	236.354	+	236.354i	0.242913	+	0.242913i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	396.922			0.406266			0.203133	−	0.979151i	$-0.434888\pi$
$977$	396.922			0.406266			0.203133	+	0.979151i	$0.434888\pi$
$978$		0			0
$979$	−69.2259	+	69.2259i	−0.0707108	+	0.0707108i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1672.52			−1.70145			−0.850724	−	0.525612i	$-0.823836\pi$
$983$	−1672.52			−1.70145			−0.850724	+	0.525612i	$0.823836\pi$
$984$		0			0
$985$			2269.66i			2.30423i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−60.3446	−	60.3446i	−0.0610157	−	0.0610157i
$990$		0			0
$991$		−	775.801i		−	0.782847i	−0.920211	−	0.391423i	$-0.871983\pi$
$991$		−	775.801i		−	0.782847i	0.920211	−	0.391423i	$-0.128017\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1180.35	+	1180.35i	−1.18629	+	1.18629i
$996$		0			0
$997$	−201.495	+	201.495i	−0.202101	+	0.202101i	−0.800900	−	0.598799i	$-0.795645\pi$
$997$	−201.495	+	201.495i	−0.202101	+	0.202101i	0.598799	+	0.800900i	$0.295645\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.3.m.f.991.2		16
3.2	odd	2		384.3.l.a.223.8		16
4.3	odd	2		1152.3.m.c.991.2		16
8.3	odd	2		576.3.m.c.559.7		16
8.5	even	2		144.3.m.c.19.8		16
12.11	even	2		384.3.l.b.223.4		16
16.3	odd	4		144.3.m.c.91.8		16
16.5	even	4		1152.3.m.c.415.2		16
16.11	odd	4	inner	1152.3.m.f.415.2		16
16.13	even	4		576.3.m.c.271.7		16
24.5	odd	2		48.3.l.a.19.1	✓	16
24.11	even	2		192.3.l.a.175.5		16
48.5	odd	4		384.3.l.b.31.4		16
48.11	even	4		384.3.l.a.31.8		16
48.29	odd	4		192.3.l.a.79.5		16
48.35	even	4		48.3.l.a.43.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.3.l.a.19.1	✓	16	24.5	odd	2
48.3.l.a.43.1	yes	16	48.35	even	4
144.3.m.c.19.8		16	8.5	even	2
144.3.m.c.91.8		16	16.3	odd	4
192.3.l.a.79.5		16	48.29	odd	4
192.3.l.a.175.5		16	24.11	even	2
384.3.l.a.31.8		16	48.11	even	4
384.3.l.a.223.8		16	3.2	odd	2
384.3.l.b.31.4		16	48.5	odd	4
384.3.l.b.223.4		16	12.11	even	2
576.3.m.c.271.7		16	16.13	even	4
576.3.m.c.559.7		16	8.3	odd	2
1152.3.m.c.415.2		16	16.5	even	4
1152.3.m.c.991.2		16	4.3	odd	2
1152.3.m.f.415.2		16	16.11	odd	4	inner
1152.3.m.f.991.2		16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1152.m (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.78830 + 4.78830i) q^{5} -10.3302 q^{7} +O(q^{10})\)	\(q+(-4.78830 + 4.78830i) q^{5} -10.3302 q^{7} +(-0.526169 - 0.526169i) q^{11} +(-17.2840 - 17.2840i) q^{13} -4.71650 q^{17} +(2.53604 - 2.53604i) q^{19} +12.5864 q^{23} -20.8557i q^{25} +(-2.19683 - 2.19683i) q^{29} +28.0521i q^{31} +(49.4644 - 49.4644i) q^{35} +(32.1128 - 32.1128i) q^{37} +23.1145i q^{41} +(-4.79441 - 4.79441i) q^{43} +39.0095i q^{47} +57.7141 q^{49} +(-27.9768 + 27.9768i) q^{53} +5.03891 q^{55} +(79.8538 + 79.8538i) q^{59} +(36.7762 + 36.7762i) q^{61} +165.522 q^{65} +(10.9869 - 10.9869i) q^{67} -52.6605 q^{71} -67.8061i q^{73} +(5.43545 + 5.43545i) q^{77} +56.4602i q^{79} +(-58.3697 + 58.3697i) q^{83} +(22.5840 - 22.5840i) q^{85} -131.566i q^{89} +(178.548 + 178.548i) q^{91} +24.2866i q^{95} +60.9413 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 32 q^{11} + 32 q^{19} + 128 q^{23} + 32 q^{29} + 96 q^{35} + 96 q^{37} - 160 q^{43} + 112 q^{49} - 160 q^{53} - 256 q^{55} - 128 q^{59} + 32 q^{61} + 32 q^{65} - 320 q^{67} - 512 q^{71} + 224 q^{77} - 160 q^{83} - 160 q^{85} + 480 q^{91}+O(q^{100}) \) 16 * q + 32 * q^11 + 32 * q^19 + 128 * q^23 + 32 * q^29 + 96 * q^35 + 96 * q^37 - 160 * q^43 + 112 * q^49 - 160 * q^53 - 256 * q^55 - 128 * q^59 + 32 * q^61 + 32 * q^65 - 320 * q^67 - 512 * q^71 + 224 * q^77 - 160 * q^83 - 160 * q^85 + 480 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more