LMFDB - Embedded newform 576.2.l.a.143.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,2,Mod(143,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("576.143");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	576.l (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:	$4.59938315643$
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} - 4x^{14} + 6x^{12} - 12x^{10} + 33x^{8} - 48x^{6} + 96x^{4} - 256x^{2} + 256 $ x^16 - 4x^14 + 6x^12 - 12x^10 + 33x^8 - 48x^6 + 96x^4 - 256*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			143.5
Root			$-1.40927 + 0.118126i$ of defining polynomial
Character	$\chi$	$=$	576.143
Dual form			576.2.l.a.431.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.236253 - 0.236253i) q^{5} -3.27830 q^{7} +O(q^{10})$	$q+(0.236253 - 0.236253i) q^{5} -3.27830 q^{7} +(-2.58229 - 2.58229i) q^{11} +(-1.70773 + 1.70773i) q^{13} -7.05130i q^{17} +(-3.04184 - 3.04184i) q^{19} -1.47338i q^{23} +4.88837i q^{25} +(-2.98575 - 2.98575i) q^{29} +8.02552i q^{31} +(-0.774506 + 0.774506i) q^{35} +(-7.93021 - 7.93021i) q^{37} +2.22112 q^{41} +(4.61007 - 4.61007i) q^{43} +7.13023 q^{47} +3.74723 q^{49} +(-5.81417 + 5.81417i) q^{53} -1.22015 q^{55} +(-7.46464 - 7.46464i) q^{59} +(-4.04184 + 4.04184i) q^{61} +0.806909i q^{65} +(2.90468 + 2.90468i) q^{67} -1.02064i q^{71} -4.08367i q^{73} +(8.46551 + 8.46551i) q^{77} +5.36197i q^{79} +(3.93734 - 3.93734i) q^{83} +(-1.66589 - 1.66589i) q^{85} -2.35922 q^{89} +(5.59843 - 5.59843i) q^{91} -1.43728 q^{95} +9.97204 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 16 q^{19} + 32 q^{43} + 16 q^{49} + 64 q^{55} - 32 q^{61} + 16 q^{67} - 32 q^{85} - 48 q^{91}+O(q^{100}) $ 16 * q - 16 * q^19 + 32 * q^43 + 16 * q^49 + 64 * q^55 - 32 * q^61 + 16 * q^67 - 32 * q^85 - 48 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{4}\right)$

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
$5$	0.236253	−	0.236253i	0.105655	−	0.105655i	−0.652303	−	0.757958i	$-0.726197\pi$
$5$	0.236253	−	0.236253i	0.105655	−	0.105655i	0.757958	+	0.652303i	$0.226197\pi$
$6$		0			0
$7$	−3.27830			−1.23908			−0.619540	−	0.784965i	$-0.712681\pi$
$7$	−3.27830			−1.23908			−0.619540	+	0.784965i	$0.712681\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.58229	−	2.58229i	−0.778590	−	0.778590i	0.201001	−	0.979591i	$-0.435580\pi$
$11$	−2.58229	−	2.58229i	−0.778590	−	0.778590i	−0.979591	+	0.201001i	$0.935580\pi$
$12$		0			0
$13$	−1.70773	+	1.70773i	−0.473638	+	0.473638i	−0.903090	−	0.429452i	$-0.858707\pi$
$13$	−1.70773	+	1.70773i	−0.473638	+	0.473638i	0.429452	+	0.903090i	$0.358707\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	7.05130i		−	1.71019i	−0.518470	−	0.855096i	$-0.673498\pi$
$17$		−	7.05130i		−	1.71019i	0.518470	−	0.855096i	$-0.326502\pi$
$18$		0			0
$19$	−3.04184	−	3.04184i	−0.697845	−	0.697845i	0.266100	−	0.963945i	$-0.414265\pi$
$19$	−3.04184	−	3.04184i	−0.697845	−	0.697845i	−0.963945	+	0.266100i	$0.914265\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	1.47338i		−	0.307221i	−0.988131	−	0.153610i	$-0.950910\pi$
$23$		−	1.47338i		−	0.307221i	0.988131	−	0.153610i	$-0.0490901\pi$
$24$		0			0
$25$			4.88837i			0.977674i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.98575	−	2.98575i	−0.554439	−	0.554439i	0.373280	−	0.927719i	$-0.378233\pi$
$29$	−2.98575	−	2.98575i	−0.554439	−	0.554439i	−0.927719	+	0.373280i	$0.878233\pi$
$30$		0			0
$31$			8.02552i			1.44143i	0.693233	+	0.720713i	$0.256186\pi$
$31$			8.02552i			1.44143i	−0.693233	+	0.720713i	$0.743814\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.774506	+	0.774506i	−0.130915	+	0.130915i
$36$		0			0
$37$	−7.93021	−	7.93021i	−1.30372	−	1.30372i	−0.925866	−	0.377852i	$-0.876663\pi$
$37$	−7.93021	−	7.93021i	−1.30372	−	1.30372i	−0.377852	−	0.925866i	$-0.623337\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.22112			0.346881			0.173441	−	0.984844i	$-0.444512\pi$
$41$	2.22112			0.346881			0.173441	+	0.984844i	$0.444512\pi$
$42$		0			0
$43$	4.61007	−	4.61007i	0.703030	−	0.703030i	−0.262030	−	0.965060i	$-0.584392\pi$
$43$	4.61007	−	4.61007i	0.703030	−	0.703030i	0.965060	+	0.262030i	$0.0843920\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	7.13023			1.04005			0.520026	−	0.854151i	$-0.325922\pi$
$47$	7.13023			1.04005			0.520026	+	0.854151i	$0.325922\pi$
$48$		0			0
$49$	3.74723			0.535318
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−5.81417	+	5.81417i	−0.798638	+	0.798638i	−0.982881	−	0.184243i	$-0.941017\pi$
$53$	−5.81417	+	5.81417i	−0.798638	+	0.798638i	0.184243	+	0.982881i	$0.441017\pi$
$54$		0			0
$55$	−1.22015			−0.164524
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−7.46464	−	7.46464i	−0.971813	−	0.971813i	0.0278004	−	0.999613i	$-0.491150\pi$
$59$	−7.46464	−	7.46464i	−0.971813	−	0.971813i	−0.999613	+	0.0278004i	$0.991150\pi$
$60$		0			0
$61$	−4.04184	+	4.04184i	−0.517504	+	0.517504i	−0.916815	−	0.399311i	$-0.869249\pi$
$61$	−4.04184	+	4.04184i	−0.517504	+	0.517504i	0.399311	+	0.916815i	$0.369249\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			0.806909i			0.100085i
$66$		0			0
$67$	2.90468	+	2.90468i	0.354863	+	0.354863i	0.861915	−	0.507052i	$-0.169265\pi$
$67$	2.90468	+	2.90468i	0.354863	+	0.354863i	−0.507052	+	0.861915i	$0.669265\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	1.02064i		−	0.121128i	−0.998164	−	0.0605640i	$-0.980710\pi$
$71$		−	1.02064i		−	0.121128i	0.998164	−	0.0605640i	$-0.0192899\pi$
$72$		0			0
$73$		−	4.08367i		−	0.477958i	−0.971025	−	0.238979i	$-0.923187\pi$
$73$		−	4.08367i		−	0.477958i	0.971025	−	0.238979i	$-0.0768127\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	8.46551	+	8.46551i	0.964735	+	0.964735i
$78$		0			0
$79$			5.36197i			0.603269i	0.953424	+	0.301634i	$0.0975322\pi$
$79$			5.36197i			0.603269i	−0.953424	+	0.301634i	$0.902468\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3.93734	−	3.93734i	0.432179	−	0.432179i	−0.457190	−	0.889369i	$-0.651144\pi$
$83$	3.93734	−	3.93734i	0.432179	−	0.432179i	0.889369	+	0.457190i	$0.151144\pi$
$84$		0			0
$85$	−1.66589	−	1.66589i	−0.180691	−	0.180691i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.35922			−0.250077			−0.125039	−	0.992152i	$-0.539905\pi$
$89$	−2.35922			−0.250077			−0.125039	+	0.992152i	$0.539905\pi$
$90$		0			0
$91$	5.59843	−	5.59843i	0.586875	−	0.586875i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.43728			−0.147462
$96$		0			0
$97$	9.97204			1.01251			0.506254	−	0.862385i	$-0.331030\pi$
$97$	9.97204			1.01251			0.506254	+	0.862385i	$0.331030\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−2.65134	+	2.65134i	−0.263818	+	0.263818i	−0.826603	−	0.562785i	$-0.809730\pi$
$101$	−2.65134	+	2.65134i	−0.263818	+	0.263818i	0.562785	+	0.826603i	$0.309730\pi$
$102$		0			0
$103$	0.0255237			0.00251492			0.00125746	−	0.999999i	$-0.499600\pi$
$103$	0.0255237			0.00251492			0.00125746	+	0.999999i	$0.499600\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	11.7664	+	11.7664i	1.13751	+	1.13751i	0.988896	+	0.148609i	$0.0474797\pi$
$107$	11.7664	+	11.7664i	1.13751	+	1.13751i	0.148609	+	0.988896i	$0.452520\pi$
$108$		0			0
$109$	6.26432	−	6.26432i	0.600013	−	0.600013i	−0.340303	−	0.940316i	$-0.610530\pi$
$109$	6.26432	−	6.26432i	0.600013	−	0.600013i	0.940316	+	0.340303i	$0.110530\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			4.36097i			0.410246i	0.978736	+	0.205123i	$0.0657594\pi$
$113$			4.36097i			0.410246i	−0.978736	+	0.205123i	$0.934241\pi$
$114$		0			0
$115$	−0.348090	−	0.348090i	−0.0324596	−	0.0324596i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			23.1162i			2.11906i
$120$		0			0
$121$			2.33645i			0.212404i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	2.33615	+	2.33615i	0.208952	+	0.208952i
$126$		0			0
$127$		−	8.66579i		−	0.768965i	−0.923132	−	0.384482i	$-0.874380\pi$
$127$		−	8.66579i		−	0.768965i	0.923132	−	0.384482i	$-0.125620\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−14.9293	+	14.9293i	−1.30438	+	1.30438i	−0.378967	+	0.925410i	$0.623720\pi$
$131$	−14.9293	+	14.9293i	−1.30438	+	1.30438i	−0.925410	+	0.378967i	$0.876280\pi$
$132$		0			0
$133$	9.97204	+	9.97204i	0.864686	+	0.864686i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	10.9365			0.934365			0.467183	−	0.884161i	$-0.345269\pi$
$137$	10.9365			0.934365			0.467183	+	0.884161i	$0.345269\pi$
$138$		0			0
$139$	1.09532	−	1.09532i	0.0929036	−	0.0929036i	−0.659128	−	0.752031i	$-0.729074\pi$
$139$	1.09532	−	1.09532i	0.0929036	−	0.0929036i	0.752031	+	0.659128i	$0.229074\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	8.81969			0.737539
$144$		0			0
$145$	−1.41078			−0.117159
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.42060	−	5.42060i	0.444073	−	0.444073i	−0.449305	−	0.893378i	$-0.648328\pi$
$149$	5.42060	−	5.42060i	0.444073	−	0.444073i	0.893378	+	0.449305i	$0.148328\pi$
$150$		0			0
$151$	14.5821			1.18668			0.593338	−	0.804953i	$-0.297810\pi$
$151$	14.5821			1.18668			0.593338	+	0.804953i	$0.297810\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.89605	+	1.89605i	0.152295	+	0.152295i
$156$		0			0
$157$	6.04184	−	6.04184i	0.482191	−	0.482191i	−0.423640	−	0.905831i	$-0.639248\pi$
$157$	6.04184	−	6.04184i	0.482191	−	0.482191i	0.905831	+	0.423640i	$0.139248\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			4.83018i			0.380671i
$162$		0			0
$163$	−3.16667	−	3.16667i	−0.248032	−	0.248032i	0.572130	−	0.820163i	$-0.306117\pi$
$163$	−3.16667	−	3.16667i	−0.248032	−	0.248032i	−0.820163	+	0.572130i	$0.806117\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	15.5333i		−	1.20200i	−0.799249	−	0.601001i	$-0.794769\pi$
$167$		−	15.5333i		−	1.20200i	0.799249	−	0.601001i	$-0.205231\pi$
$168$		0			0
$169$			7.16735i			0.551334i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−11.0577	−	11.0577i	−0.840700	−	0.840700i	0.148250	−	0.988950i	$-0.452636\pi$
$173$	−11.0577	−	11.0577i	−0.840700	−	0.840700i	−0.988950	+	0.148250i	$0.952636\pi$
$174$		0			0
$175$		−	16.0255i		−	1.21142i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3.84907	−	3.84907i	0.287693	−	0.287693i	−0.548474	−	0.836167i	$-0.684791\pi$
$179$	3.84907	−	3.84907i	0.287693	−	0.287693i	0.836167	+	0.548474i	$0.184791\pi$
$180$		0			0
$181$	4.29227	+	4.29227i	0.319042	+	0.319042i	0.848399	−	0.529357i	$-0.177567\pi$
$181$	4.29227	+	4.29227i	0.319042	+	0.319042i	−0.529357	+	0.848399i	$0.677567\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.74706			−0.275490
$186$		0			0
$187$	−18.2085	+	18.2085i	−1.33154	+	1.33154i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−24.8057			−1.79488			−0.897439	−	0.441139i	$-0.854575\pi$
$191$	−24.8057			−1.79488			−0.897439	+	0.441139i	$0.854575\pi$
$192$		0			0
$193$	−8.08367			−0.581876			−0.290938	−	0.956742i	$-0.593967\pi$
$193$	−8.08367			−0.581876			−0.290938	+	0.956742i	$0.593967\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	7.34342	−	7.34342i	0.523197	−	0.523197i	−0.395339	−	0.918535i	$-0.629373\pi$
$197$	7.34342	−	7.34342i	0.523197	−	0.523197i	0.918535	+	0.395339i	$0.129373\pi$
$198$		0			0
$199$	−11.5526			−0.818942			−0.409471	−	0.912323i	$-0.634287\pi$
$199$	−11.5526			−0.818942			−0.409471	+	0.912323i	$0.634287\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	9.78816	+	9.78816i	0.686994	+	0.686994i
$204$		0			0
$205$	0.524746	−	0.524746i	0.0366499	−	0.0366499i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			15.7098i			1.08667i
$210$		0			0
$211$	−0.821009	−	0.821009i	−0.0565206	−	0.0565206i	0.678282	−	0.734802i	$-0.262725\pi$
$211$	−0.821009	−	0.821009i	−0.0565206	−	0.0565206i	−0.734802	+	0.678282i	$0.762725\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	2.17828i		−	0.148558i
$216$		0			0
$217$		−	26.3100i		−	1.78604i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	12.0417	+	12.0417i	0.810011	+	0.810011i
$222$		0			0
$223$			9.83489i			0.658593i	0.944227	+	0.329296i	$0.106812\pi$
$223$			9.83489i			0.658593i	−0.944227	+	0.329296i	$0.893188\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	15.6642	−	15.6642i	1.03967	−	1.03967i	0.0404927	−	0.999180i	$-0.487107\pi$
$227$	15.6642	−	15.6642i	1.03967	−	1.03967i	0.999180	−	0.0404927i	$-0.0128928\pi$
$228$		0			0
$229$	11.4845	+	11.4845i	0.758915	+	0.758915i	0.976125	−	0.217210i	$-0.0696957\pi$
$229$	11.4845	+	11.4845i	0.758915	+	0.758915i	−0.217210	+	0.976125i	$0.569696\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−22.8992			−1.50018			−0.750089	−	0.661337i	$-0.769989\pi$
$233$	−22.8992			−1.50018			−0.750089	+	0.661337i	$0.769989\pi$
$234$		0			0
$235$	1.68454	−	1.68454i	0.109887	−	0.109887i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.50948			−0.0976399			−0.0488199	−	0.998808i	$-0.515546\pi$
$239$	−1.50948			−0.0976399			−0.0488199	+	0.998808i	$0.515546\pi$
$240$		0			0
$241$	−19.3922			−1.24916			−0.624580	−	0.780961i	$-0.714730\pi$
$241$	−19.3922			−1.24916			−0.624580	+	0.780961i	$0.714730\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.885292	−	0.885292i	0.0565593	−	0.0565593i
$246$		0			0
$247$	10.3892			0.661052
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−12.3470	−	12.3470i	−0.779335	−	0.779335i	0.200383	−	0.979718i	$-0.435781\pi$
$251$	−12.3470	−	12.3470i	−0.779335	−	0.779335i	−0.979718	+	0.200383i	$0.935781\pi$
$252$		0			0
$253$	−3.80470	+	3.80470i	−0.239199	+	0.239199i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	10.5288i		−	0.656766i	−0.944545	−	0.328383i	$-0.893496\pi$
$257$		−	10.5288i		−	0.656766i	0.944545	−	0.328383i	$-0.106504\pi$
$258$		0			0
$259$	25.9976	+	25.9976i	1.61541	+	1.61541i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	13.8725i		−	0.855417i	−0.903917	−	0.427709i	$-0.859321\pi$
$263$		−	13.8725i		−	0.855417i	0.903917	−	0.427709i	$-0.140679\pi$
$264$		0			0
$265$			2.74723i			0.168761i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−11.0971	−	11.0971i	−0.676601	−	0.676601i	0.282628	−	0.959230i	$-0.408794\pi$
$269$	−11.0971	−	11.0971i	−0.676601	−	0.676601i	−0.959230	+	0.282628i	$0.908794\pi$
$270$		0			0
$271$		−	30.0022i		−	1.82251i	−0.411847	−	0.911253i	$-0.635116\pi$
$271$		−	30.0022i		−	1.82251i	0.411847	−	0.911253i	$-0.364884\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	12.6232	−	12.6232i	0.761207	−	0.761207i
$276$		0			0
$277$	−5.62872	−	5.62872i	−0.338197	−	0.338197i	0.517491	−	0.855688i	$-0.326866\pi$
$277$	−5.62872	−	5.62872i	−0.338197	−	0.338197i	−0.855688	+	0.517491i	$0.826866\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	11.9012			0.709969			0.354984	−	0.934872i	$-0.384486\pi$
$281$	11.9012			0.709969			0.354984	+	0.934872i	$0.384486\pi$
$282$		0			0
$283$	−17.3875	+	17.3875i	−1.03358	+	1.03358i	−0.0341630	+	0.999416i	$0.510877\pi$
$283$	−17.3875	+	17.3875i	−1.03358	+	1.03358i	−0.999416	+	0.0341630i	$0.989123\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−7.28150			−0.429813
$288$		0			0
$289$	−32.7208			−1.92475
$290$		0			0
$291$		0			0
$292$		0			0
$293$	5.14536	−	5.14536i	0.300595	−	0.300595i	−0.540651	−	0.841247i	$-0.681822\pi$
$293$	5.14536	−	5.14536i	0.300595	−	0.300595i	0.841247	+	0.540651i	$0.181822\pi$
$294$		0			0
$295$	−3.52708			−0.205355
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.51613	+	2.51613i	0.145511	+	0.145511i
$300$		0			0
$301$	−15.1132	+	15.1132i	−0.871110	+	0.871110i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			1.90979i			0.109354i
$306$		0			0
$307$	−14.9557	−	14.9557i	−0.853569	−	0.853569i	0.137002	−	0.990571i	$-0.456253\pi$
$307$	−14.9557	−	14.9557i	−0.853569	−	0.853569i	−0.990571	+	0.137002i	$0.956253\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	16.8915i		−	0.957829i	−0.877862	−	0.478914i	$-0.841030\pi$
$311$		−	16.8915i		−	0.957829i	0.877862	−	0.478914i	$-0.158970\pi$
$312$		0			0
$313$			1.42012i			0.0802700i	0.999194	+	0.0401350i	$0.0127788\pi$
$313$			1.42012i			0.0802700i	−0.999194	+	0.0401350i	$0.987221\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−23.3492	−	23.3492i	−1.31142	−	1.31142i	−0.920368	−	0.391054i	$-0.872111\pi$
$317$	−23.3492	−	23.3492i	−1.31142	−	1.31142i	−0.391054	−	0.920368i	$-0.627889\pi$
$318$		0			0
$319$			15.4201i			0.863361i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−21.4489	+	21.4489i	−1.19345	+	1.19345i
$324$		0			0
$325$	−8.34799	−	8.34799i	−0.463063	−	0.463063i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−23.3750			−1.28871
$330$		0			0
$331$	−2.95573	+	2.95573i	−0.162462	+	0.162462i	−0.783656	−	0.621195i	$-0.786648\pi$
$331$	−2.95573	+	2.95573i	−0.162462	+	0.162462i	0.621195	+	0.783656i	$0.286648\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.37248			0.0749865
$336$		0			0
$337$	−1.41078			−0.0768501			−0.0384251	−	0.999261i	$-0.512234\pi$
$337$	−1.41078			−0.0768501			−0.0384251	+	0.999261i	$0.512234\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	20.7242	−	20.7242i	1.12228	−	1.12228i
$342$		0			0
$343$	10.6636			0.575778
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3.11726	+	3.11726i	0.167344	+	0.167344i	0.785811	−	0.618467i	$-0.212246\pi$
$347$	3.11726	+	3.11726i	0.167344	+	0.167344i	−0.618467	+	0.785811i	$0.712246\pi$
$348$		0			0
$349$	11.8465	−	11.8465i	0.634130	−	0.634130i	−0.314971	−	0.949101i	$-0.601995\pi$
$349$	11.8465	−	11.8465i	0.634130	−	0.634130i	0.949101	+	0.314971i	$0.101995\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	5.30598i		−	0.282409i	−0.989980	−	0.141205i	$-0.954902\pi$
$353$		−	5.30598i		−	0.282409i	0.989980	−	0.141205i	$-0.0450975\pi$
$354$		0			0
$355$	−0.241130	−	0.241130i	−0.0127978	−	0.0127978i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	17.8399i		−	0.941556i	−0.882252	−	0.470778i	$-0.843973\pi$
$359$		−	17.8399i		−	0.941556i	0.882252	−	0.470778i	$-0.156027\pi$
$360$		0			0
$361$		−	0.494455i		−	0.0260239i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−0.964779	−	0.964779i	−0.0504988	−	0.0504988i
$366$		0			0
$367$		−	2.05815i		−	0.107435i	−0.998556	−	0.0537173i	$-0.982893\pi$
$367$		−	2.05815i		−	0.107435i	0.998556	−	0.0537173i	$-0.0171070\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	19.0606	−	19.0606i	0.989576	−	0.989576i
$372$		0			0
$373$	12.7891	+	12.7891i	0.662193	+	0.662193i	0.955896	−	0.293704i	$-0.0948879\pi$
$373$	12.7891	+	12.7891i	0.662193	+	0.662193i	−0.293704	+	0.955896i	$0.594888\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	10.1977			0.525206
$378$		0			0
$379$	−6.76753	+	6.76753i	−0.347625	+	0.347625i	−0.859224	−	0.511599i	$-0.829053\pi$
$379$	−6.76753	+	6.76753i	−0.347625	+	0.347625i	0.511599	+	0.859224i	$0.329053\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−7.48207			−0.382316			−0.191158	−	0.981559i	$-0.561224\pi$
$383$	−7.48207			−0.382316			−0.191158	+	0.981559i	$0.561224\pi$
$384$		0			0
$385$	4.00000			0.203859
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−20.7180	+	20.7180i	−1.05045	+	1.05045i	−0.0517883	+	0.998658i	$0.516492\pi$
$389$	−20.7180	+	20.7180i	−1.05045	+	1.05045i	−0.998658	+	0.0517883i	$0.983508\pi$
$390$		0			0
$391$	−10.3892			−0.525407
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.26678	+	1.26678i	0.0637386	+	0.0637386i
$396$		0			0
$397$	12.5194	−	12.5194i	0.628332	−	0.628332i	−0.319316	−	0.947648i	$-0.603453\pi$
$397$	12.5194	−	12.5194i	0.628332	−	0.628332i	0.947648	+	0.319316i	$0.103453\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	11.4935i		−	0.573960i	−0.957937	−	0.286980i	$-0.907349\pi$
$401$		−	11.4935i		−	0.573960i	0.957937	−	0.286980i	$-0.0926514\pi$
$402$		0			0
$403$	−13.7054	−	13.7054i	−0.682714	−	0.682714i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			40.9562i			2.03012i
$408$		0			0
$409$			23.2432i			1.14930i	0.818398	+	0.574652i	$0.194863\pi$
$409$			23.2432i			1.14930i	−0.818398	+	0.574652i	$0.805137\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	24.4713	+	24.4713i	1.20415	+	1.20415i
$414$		0			0
$415$		−	1.86041i		−	0.0913241i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−21.1446	+	21.1446i	−1.03298	+	1.03298i	−0.0335424	+	0.999437i	$0.510679\pi$
$419$	−21.1446	+	21.1446i	−1.03298	+	1.03298i	−0.999437	+	0.0335424i	$0.989321\pi$
$420$		0			0
$421$	9.60077	+	9.60077i	0.467913	+	0.467913i	0.901238	−	0.433325i	$-0.142660\pi$
$421$	9.60077	+	9.60077i	0.467913	+	0.467913i	−0.433325	+	0.901238i	$0.642660\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	34.4694			1.67201
$426$		0			0
$427$	13.2503	−	13.2503i	0.641229	−	0.641229i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	18.2795			0.880491			0.440246	−	0.897877i	$-0.354891\pi$
$431$	18.2795			0.880491			0.440246	+	0.897877i	$0.354891\pi$
$432$		0			0
$433$	34.3844			1.65241			0.826204	−	0.563371i	$-0.190496\pi$
$433$	34.3844			1.65241			0.826204	+	0.563371i	$0.190496\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−4.48178	+	4.48178i	−0.214393	+	0.214393i
$438$		0			0
$439$	−15.1713			−0.724088			−0.362044	−	0.932161i	$-0.617921\pi$
$439$	−15.1713			−0.724088			−0.362044	+	0.932161i	$0.617921\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−18.1978	−	18.1978i	−0.864604	−	0.864604i	0.127265	−	0.991869i	$-0.459380\pi$
$443$	−18.1978	−	18.1978i	−0.864604	−	0.864604i	−0.991869	+	0.127265i	$0.959380\pi$
$444$		0			0
$445$	−0.557373	+	0.557373i	−0.0264220	+	0.0264220i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			8.89518i			0.419789i	0.977724	+	0.209895i	$0.0673121\pi$
$449$			8.89518i			0.419789i	−0.977724	+	0.209895i	$0.932688\pi$
$450$		0			0
$451$	−5.73558	−	5.73558i	−0.270078	−	0.270078i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	2.64529i		−	0.124013i
$456$		0			0
$457$		−	11.3085i		−	0.528989i	−0.964387	−	0.264494i	$-0.914795\pi$
$457$		−	11.3085i		−	0.528989i	0.964387	−	0.264494i	$-0.0852051\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	10.8614	+	10.8614i	0.505865	+	0.505865i	0.913255	−	0.407389i	$-0.133561\pi$
$461$	10.8614	+	10.8614i	0.505865	+	0.505865i	−0.407389	+	0.913255i	$0.633561\pi$
$462$		0			0
$463$			3.05504i			0.141980i	0.997477	+	0.0709898i	$0.0226158\pi$
$463$			3.05504i			0.141980i	−0.997477	+	0.0709898i	$0.977384\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	3.93734	−	3.93734i	0.182198	−	0.182198i	−0.610115	−	0.792313i	$-0.708877\pi$
$467$	3.93734	−	3.93734i	0.182198	−	0.182198i	0.792313	+	0.610115i	$0.208877\pi$
$468$		0			0
$469$	−9.52241	−	9.52241i	−0.439704	−	0.439704i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−23.8091			−1.09474
$474$		0			0
$475$	14.8696	−	14.8696i	0.682265	−	0.682265i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−27.6167			−1.26184			−0.630920	−	0.775848i	$-0.717322\pi$
$479$	−27.6167			−1.26184			−0.630920	+	0.775848i	$0.717322\pi$
$480$		0			0
$481$	27.0852			1.23498
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.35592	−	2.35592i	0.106977	−	0.106977i
$486$		0			0
$487$	−13.0783			−0.592635			−0.296318	−	0.955089i	$-0.595759\pi$
$487$	−13.0783			−0.592635			−0.296318	+	0.955089i	$0.595759\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	21.8368	+	21.8368i	0.985483	+	0.985483i	0.999896	−	0.0144135i	$-0.00458813\pi$
$491$	21.8368	+	21.8368i	0.985483	+	0.985483i	−0.0144135	+	0.999896i	$0.504588\pi$
$492$		0			0
$493$	−21.0534	+	21.0534i	−0.948197	+	0.948197i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			3.34597i			0.150087i
$498$		0			0
$499$	16.4170	+	16.4170i	0.734926	+	0.734926i	0.971591	−	0.236665i	$-0.0760544\pi$
$499$	16.4170	+	16.4170i	0.734926	+	0.734926i	−0.236665	+	0.971591i	$0.576054\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			22.1243i			0.986475i	0.869895	+	0.493237i	$0.164187\pi$
$503$			22.1243i			0.986475i	−0.869895	+	0.493237i	$0.835813\pi$
$504$		0			0
$505$			1.25277i			0.0557477i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−13.7152	−	13.7152i	−0.607916	−	0.607916i	0.334485	−	0.942401i	$-0.391438\pi$
$509$	−13.7152	−	13.7152i	−0.607916	−	0.607916i	−0.942401	+	0.334485i	$0.891438\pi$
$510$		0			0
$511$			13.3875i			0.592228i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	0.00603003	−	0.00603003i	0.000265715	−	0.000265715i
$516$		0			0
$517$	−18.4123	−	18.4123i	−0.809774	−	0.809774i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	0.888181			0.0389119			0.0194560	−	0.999811i	$-0.493807\pi$
$521$	0.888181			0.0389119			0.0194560	+	0.999811i	$0.493807\pi$
$522$		0			0
$523$	14.8186	−	14.8186i	0.647971	−	0.647971i	−0.304531	−	0.952502i	$-0.598500\pi$
$523$	14.8186	−	14.8186i	0.647971	−	0.647971i	0.952502	+	0.304531i	$0.0984998\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	56.5904			2.46512
$528$		0			0
$529$	20.8292			0.905615
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−3.79307	+	3.79307i	−0.164296	+	0.164296i
$534$		0			0
$535$	5.55971			0.240367
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−9.67643	−	9.67643i	−0.416793	−	0.416793i
$540$		0			0
$541$	−4.26432	+	4.26432i	−0.183337	+	0.183337i	−0.792808	−	0.609471i	$-0.791382\pi$
$541$	−4.26432	+	4.26432i	−0.183337	+	0.183337i	0.609471	+	0.792808i	$0.291382\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	2.95992i		−	0.126789i
$546$		0			0
$547$	0.559026	+	0.559026i	0.0239022	+	0.0239022i	0.718957	−	0.695055i	$-0.244620\pi$
$547$	0.559026	+	0.559026i	0.0239022	+	0.0239022i	−0.695055	+	0.718957i	$0.744620\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			18.1643i			0.773825i
$552$		0			0
$553$		−	17.5781i		−	0.747498i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	20.4287	+	20.4287i	0.865591	+	0.865591i	0.991981	−	0.126390i	$-0.0403390\pi$
$557$	20.4287	+	20.4287i	0.865591	+	0.865591i	−0.126390	+	0.991981i	$0.540339\pi$
$558$		0			0
$559$			15.7455i			0.665963i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−13.9682	+	13.9682i	−0.588689	+	0.588689i	−0.937276	−	0.348587i	$-0.886661\pi$
$563$	−13.9682	+	13.9682i	−0.588689	+	0.588689i	0.348587	+	0.937276i	$0.386661\pi$
$564$		0			0
$565$	1.03029	+	1.03029i	0.0433447	+	0.0433447i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	21.3052			0.893159			0.446579	−	0.894744i	$-0.352642\pi$
$569$	21.3052			0.893159			0.446579	+	0.894744i	$0.352642\pi$
$570$		0			0
$571$	18.3333	−	18.3333i	0.767226	−	0.767226i	−0.210391	−	0.977617i	$-0.567474\pi$
$571$	18.3333	−	18.3333i	0.767226	−	0.767226i	0.977617	+	0.210391i	$0.0674738\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	7.20243			0.300362
$576$		0			0
$577$	7.57813			0.315482			0.157741	−	0.987481i	$-0.449579\pi$
$577$	7.57813			0.315482			0.157741	+	0.987481i	$0.449579\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−12.9078	+	12.9078i	−0.535504	+	0.535504i
$582$		0			0
$583$	30.0278			1.24362
$584$		0			0
$585$		0			0
$586$		0			0
$587$	2.75279	+	2.75279i	0.113620	+	0.113620i	0.761631	−	0.648011i	$-0.224399\pi$
$587$	2.75279	+	2.75279i	0.113620	+	0.113620i	−0.648011	+	0.761631i	$0.724399\pi$
$588$		0			0
$589$	24.4123	−	24.4123i	1.00589	−	1.00589i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	11.3554i		−	0.466312i	−0.972439	−	0.233156i	$-0.925095\pi$
$593$		−	11.3554i		−	0.466312i	0.972439	−	0.233156i	$-0.0749053\pi$
$594$		0			0
$595$	5.46128	+	5.46128i	0.223890	+	0.223890i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			25.8985i			1.05819i	0.848564	+	0.529093i	$0.177468\pi$
$599$			25.8985i			1.05819i	−0.848564	+	0.529093i	$0.822532\pi$
$600$		0			0
$601$			15.9753i			0.651648i	0.945430	+	0.325824i	$0.105642\pi$
$601$			15.9753i			0.651648i	−0.945430	+	0.325824i	$0.894358\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.551992	+	0.551992i	0.0224417	+	0.0224417i
$606$		0			0
$607$			15.9745i			0.648384i	0.945991	+	0.324192i	$0.105092\pi$
$607$			15.9745i			0.648384i	−0.945991	+	0.324192i	$0.894908\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−12.1765	+	12.1765i	−0.492608	+	0.492608i
$612$		0			0
$613$	−20.2125	−	20.2125i	−0.816375	−	0.816375i	0.169206	−	0.985581i	$-0.445880\pi$
$613$	−20.2125	−	20.2125i	−0.816375	−	0.816375i	−0.985581	+	0.169206i	$0.945880\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	4.04523			0.162855			0.0814275	−	0.996679i	$-0.474052\pi$
$617$	4.04523			0.162855			0.0814275	+	0.996679i	$0.474052\pi$
$618$		0			0
$619$	20.7472	−	20.7472i	0.833901	−	0.833901i	−0.154147	−	0.988048i	$-0.549263\pi$
$619$	20.7472	−	20.7472i	0.833901	−	0.833901i	0.988048	+	0.154147i	$0.0492628\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	7.73424			0.309866
$624$		0			0
$625$	−23.3380			−0.933520
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−55.9183	+	55.9183i	−2.22961	+	2.22961i
$630$		0			0
$631$	38.0533			1.51488			0.757439	−	0.652906i	$-0.226450\pi$
$631$	38.0533			1.51488			0.757439	+	0.652906i	$0.226450\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2.04732	−	2.04732i	−0.0812453	−	0.0812453i
$636$		0			0
$637$	−6.39923	+	6.39923i	−0.253547	+	0.253547i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			27.3678i			1.08096i	0.841355	+	0.540482i	$0.181758\pi$
$641$			27.3678i			1.08096i	−0.841355	+	0.540482i	$0.818242\pi$
$642$		0			0
$643$	−8.88438	−	8.88438i	−0.350366	−	0.350366i	0.509880	−	0.860246i	$-0.329690\pi$
$643$	−8.88438	−	8.88438i	−0.350366	−	0.350366i	−0.860246	+	0.509880i	$0.829690\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	40.9923i		−	1.61157i	−0.592206	−	0.805787i	$-0.701743\pi$
$647$		−	40.9923i		−	1.61157i	0.592206	−	0.805787i	$-0.298257\pi$
$648$		0			0
$649$			38.5517i			1.51329i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	22.0952	+	22.0952i	0.864652	+	0.864652i	0.991874	−	0.127222i	$-0.0406062\pi$
$653$	22.0952	+	22.0952i	0.864652	+	0.864652i	−0.127222	+	0.991874i	$0.540606\pi$
$654$		0			0
$655$			7.05416i			0.275629i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	14.3943	−	14.3943i	0.560722	−	0.560722i	−0.368790	−	0.929513i	$-0.620228\pi$
$659$	14.3943	−	14.3943i	0.560722	−	0.560722i	0.929513	+	0.368790i	$0.120228\pi$
$660$		0			0
$661$	−27.1550	−	27.1550i	−1.05621	−	1.05621i	−0.998323	−	0.0578847i	$-0.981564\pi$
$661$	−27.1550	−	27.1550i	−1.05621	−	1.05621i	−0.0578847	−	0.998323i	$-0.518436\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	4.71184			0.182717
$666$		0			0
$667$	−4.39914	+	4.39914i	−0.170335	+	0.170335i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	20.8744			0.805847
$672$		0			0
$673$	20.8899			0.805247			0.402624	−	0.915366i	$-0.368098\pi$
$673$	20.8899			0.805247			0.402624	+	0.915366i	$0.368098\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	19.2536	−	19.2536i	0.739976	−	0.739976i	−0.232597	−	0.972573i	$-0.574722\pi$
$677$	19.2536	−	19.2536i	0.739976	−	0.739976i	0.972573	+	0.232597i	$0.0747223\pi$
$678$		0			0
$679$	−32.6913			−1.25458
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−4.11387	−	4.11387i	−0.157413	−	0.157413i	0.624006	−	0.781419i	$-0.285504\pi$
$683$	−4.11387	−	4.11387i	−0.157413	−	0.157413i	−0.781419	+	0.624006i	$0.785504\pi$
$684$		0			0
$685$	2.58377	−	2.58377i	0.0987207	−	0.0987207i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	19.8580i		−	0.756530i
$690$		0			0
$691$	−25.2503	−	25.2503i	−0.960568	−	0.960568i	0.0386833	−	0.999252i	$-0.487684\pi$
$691$	−25.2503	−	25.2503i	−0.960568	−	0.960568i	−0.999252	+	0.0386833i	$0.987684\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	0.517543i		−	0.0196315i
$696$		0			0
$697$		−	15.6618i		−	0.593233i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−5.51930	−	5.51930i	−0.208461	−	0.208461i	0.595152	−	0.803613i	$-0.297092\pi$
$701$	−5.51930	−	5.51930i	−0.208461	−	0.208461i	−0.803613	+	0.595152i	$0.797092\pi$
$702$		0			0
$703$			48.2448i			1.81959i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	8.69188	−	8.69188i	0.326892	−	0.326892i
$708$		0			0
$709$	−5.23948	−	5.23948i	−0.196773	−	0.196773i	0.601842	−	0.798615i	$-0.294434\pi$
$709$	−5.23948	−	5.23948i	−0.196773	−	0.196773i	−0.798615	+	0.601842i	$0.794434\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	11.8246			0.442837
$714$		0			0
$715$	2.08367	−	2.08367i	0.0779250	−	0.0779250i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−36.4570			−1.35962			−0.679808	−	0.733390i	$-0.737937\pi$
$719$	−36.4570			−1.35962			−0.679808	+	0.733390i	$0.737937\pi$
$720$		0			0
$721$	−0.0836741			−0.00311619
$722$		0			0
$723$		0			0
$724$		0			0
$725$	14.5954	−	14.5954i	0.542060	−	0.542060i
$726$		0			0
$727$	−31.5790			−1.17120			−0.585600	−	0.810600i	$-0.699141\pi$
$727$	−31.5790			−1.17120			−0.585600	+	0.810600i	$0.699141\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−32.5070	−	32.5070i	−1.20232	−	1.20232i
$732$		0			0
$733$	−10.8720	+	10.8720i	−0.401565	+	0.401565i	−0.878784	−	0.477219i	$-0.841645\pi$
$733$	−10.8720	+	10.8720i	−0.401565	+	0.401565i	0.477219	+	0.878784i	$0.341645\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	15.0015i		−	0.552586i
$738$		0			0
$739$	34.2774	+	34.2774i	1.26092	+	1.26092i	0.950650	+	0.310265i	$0.100418\pi$
$739$	34.2774	+	34.2774i	1.26092	+	1.26092i	0.310265	+	0.950650i	$0.399582\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	16.5900i		−	0.608629i	−0.952572	−	0.304314i	$-0.901573\pi$
$743$		−	16.5900i		−	0.608629i	0.952572	−	0.304314i	$-0.0984273\pi$
$744$		0			0
$745$		−	2.56126i		−	0.0938374i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−38.5739	−	38.5739i	−1.40946	−	1.40946i
$750$		0			0
$751$		−	14.8054i		−	0.540256i	−0.962824	−	0.270128i	$-0.912934\pi$
$751$		−	14.8054i		−	0.540256i	0.962824	−	0.270128i	$-0.0870660\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	3.44506	−	3.44506i	0.125379	−	0.125379i
$756$		0			0
$757$	32.8209	+	32.8209i	1.19290	+	1.19290i	0.976250	+	0.216646i	$0.0695118\pi$
$757$	32.8209	+	32.8209i	1.19290	+	1.19290i	0.216646	+	0.976250i	$0.430488\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−42.5206			−1.54137			−0.770685	−	0.637217i	$-0.780086\pi$
$761$	−42.5206			−1.54137			−0.770685	+	0.637217i	$0.780086\pi$
$762$		0			0
$763$	−20.5363	+	20.5363i	−0.743464	+	0.743464i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	25.4951			0.920575
$768$		0			0
$769$	22.9146			0.826321			0.413160	−	0.910658i	$-0.364425\pi$
$769$	22.9146			0.826321			0.413160	+	0.910658i	$0.364425\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	15.8155	−	15.8155i	0.568845	−	0.568845i	−0.362960	−	0.931805i	$-0.618234\pi$
$773$	15.8155	−	15.8155i	0.568845	−	0.568845i	0.931805	+	0.362960i	$0.118234\pi$
$774$		0			0
$775$	−39.2317			−1.40925
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−6.75629	−	6.75629i	−0.242069	−	0.242069i
$780$		0			0
$781$	−2.63560	+	2.63560i	−0.0943091	+	0.0943091i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	2.85480i		−	0.101892i
$786$		0			0
$787$	−18.8790	−	18.8790i	−0.672962	−	0.672962i	0.285436	−	0.958398i	$-0.407862\pi$
$787$	−18.8790	−	18.8790i	−0.672962	−	0.672962i	−0.958398	+	0.285436i	$0.907862\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	14.2966i		−	0.508327i
$792$		0			0
$793$		−	13.8047i		−	0.490219i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	15.5930	+	15.5930i	0.552332	+	0.552332i	0.927113	−	0.374781i	$-0.122282\pi$
$797$	15.5930	+	15.5930i	0.552332	+	0.552332i	−0.374781	+	0.927113i	$0.622282\pi$
$798$		0			0
$799$		−	50.2774i		−	1.77869i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−10.5452	+	10.5452i	−0.372133	+	0.372133i
$804$		0			0
$805$	1.14114	+	1.14114i	0.0402200	+	0.0402200i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	14.7494			0.518563			0.259281	−	0.965802i	$-0.416514\pi$
$809$	14.7494			0.518563			0.259281	+	0.965802i	$0.416514\pi$
$810$		0			0
$811$	−9.71473	+	9.71473i	−0.341130	+	0.341130i	−0.856792	−	0.515662i	$-0.827546\pi$
$811$	−9.71473	+	9.71473i	−0.341130	+	0.341130i	0.515662	+	0.856792i	$0.327546\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1.49627			−0.0524119
$816$		0			0
$817$	−28.0462			−0.981212
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−10.2311	+	10.2311i	−0.357070	+	0.357070i	−0.862732	−	0.505662i	$-0.831248\pi$
$821$	−10.2311	+	10.2311i	−0.357070	+	0.357070i	0.505662	+	0.862732i	$0.331248\pi$
$822$		0			0
$823$	22.8564			0.796725			0.398362	−	0.917228i	$-0.369579\pi$
$823$	22.8564			0.796725			0.398362	+	0.917228i	$0.369579\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	19.1488	+	19.1488i	0.665871	+	0.665871i	0.956757	−	0.290887i	$-0.0939503\pi$
$827$	19.1488	+	19.1488i	0.665871	+	0.665871i	−0.290887	+	0.956757i	$0.593950\pi$
$828$		0			0
$829$	−32.8813	+	32.8813i	−1.14201	+	1.14201i	−0.153934	+	0.988081i	$0.549194\pi$
$829$	−32.8813	+	32.8813i	−1.14201	+	1.14201i	−0.988081	+	0.153934i	$0.950806\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	26.4228i		−	0.915497i
$834$		0			0
$835$	−3.66978	−	3.66978i	−0.126998	−	0.126998i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			41.2084i			1.42267i	0.702853	+	0.711335i	$0.251909\pi$
$839$			41.2084i			1.42267i	−0.702853	+	0.711335i	$0.748091\pi$
$840$		0			0
$841$		−	11.1707i		−	0.385195i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.69331	+	1.69331i	0.0582515	+	0.0582515i
$846$		0			0
$847$		−	7.65957i		−	0.263186i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−11.6842	+	11.6842i	−0.400530	+	0.400530i
$852$		0			0
$853$	−16.3985	−	16.3985i	−0.561472	−	0.561472i	0.368253	−	0.929726i	$-0.379956\pi$
$853$	−16.3985	−	16.3985i	−0.561472	−	0.561472i	−0.929726	+	0.368253i	$0.879956\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−3.15292			−0.107702			−0.0538509	−	0.998549i	$-0.517150\pi$
$857$	−3.15292			−0.107702			−0.0538509	+	0.998549i	$0.517150\pi$
$858$		0			0
$859$	−14.6691	+	14.6691i	−0.500503	+	0.500503i	−0.911594	−	0.411091i	$-0.865148\pi$
$859$	−14.6691	+	14.6691i	−0.500503	+	0.500503i	0.411091	+	0.911594i	$0.365148\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−42.0851			−1.43259			−0.716297	−	0.697795i	$-0.754164\pi$
$863$	−42.0851			−1.43259			−0.716297	+	0.697795i	$0.754164\pi$
$864$		0			0
$865$	−5.22482			−0.177649
$866$		0			0
$867$		0			0
$868$		0			0
$869$	13.8462	−	13.8462i	0.469699	−	0.469699i
$870$		0			0
$871$	−9.92080			−0.336154
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−7.65860	−	7.65860i	−0.258908	−	0.258908i
$876$		0			0
$877$	−2.42641	+	2.42641i	−0.0819341	+	0.0819341i	−0.746886	−	0.664952i	$-0.768452\pi$
$877$	−2.42641	+	2.42641i	−0.0819341	+	0.0819341i	0.664952	+	0.746886i	$0.268452\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	32.2178i		−	1.08544i	−0.839912	−	0.542722i	$-0.817394\pi$
$881$		−	32.2178i		−	1.08544i	0.839912	−	0.542722i	$-0.182606\pi$
$882$		0			0
$883$	−0.924984	−	0.924984i	−0.0311282	−	0.0311282i	0.691371	−	0.722500i	$-0.257007\pi$
$883$	−0.924984	−	0.924984i	−0.0311282	−	0.0311282i	−0.722500	+	0.691371i	$0.757007\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	17.9482i		−	0.602643i	−0.953523	−	0.301321i	$-0.902572\pi$
$887$		−	17.9482i		−	0.602643i	0.953523	−	0.301321i	$-0.0974277\pi$
$888$		0			0
$889$			28.4090i			0.952808i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−21.6890	−	21.6890i	−0.725795	−	0.725795i
$894$		0			0
$895$		−	1.81871i		−	0.0607926i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	23.9622	−	23.9622i	0.799183	−	0.799183i
$900$		0			0
$901$	40.9975	+	40.9975i	1.36582	+	1.36582i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2.02812			0.0674171
$906$		0			0
$907$	17.2503	−	17.2503i	0.572788	−	0.572788i	−0.360118	−	0.932907i	$-0.617264\pi$
$907$	17.2503	−	17.2503i	0.572788	−	0.572788i	0.932907	+	0.360118i	$0.117264\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−19.5368			−0.647282			−0.323641	−	0.946180i	$-0.604907\pi$
$911$	−19.5368			−0.647282			−0.323641	+	0.946180i	$0.604907\pi$
$912$		0			0
$913$	−20.3347			−0.672980
$914$		0			0
$915$		0			0
$916$		0			0
$917$	48.9426	−	48.9426i	1.61623	−	1.61623i
$918$		0			0
$919$	−15.9943			−0.527602			−0.263801	−	0.964577i	$-0.584976\pi$
$919$	−15.9943			−0.527602			−0.263801	+	0.964577i	$0.584976\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.74298	+	1.74298i	0.0573708	+	0.0573708i
$924$		0			0
$925$	38.7658	−	38.7658i	1.27461	−	1.27461i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			1.11164i			0.0364717i	0.999834	+	0.0182358i	$0.00580497\pi$
$929$			1.11164i			0.0364717i	−0.999834	+	0.0182358i	$0.994195\pi$
$930$		0			0
$931$	−11.3985	−	11.3985i	−0.373569	−	0.373569i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			8.60361i			0.281368i
$936$		0			0
$937$			17.7208i			0.578914i	0.957191	+	0.289457i	$0.0934747\pi$
$937$			17.7208i			0.578914i	−0.957191	+	0.289457i	$0.906525\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−30.8171	−	30.8171i	−1.00461	−	1.00461i	−0.999989	−	0.00462076i	$-0.998529\pi$
$941$	−30.8171	−	30.8171i	−1.00461	−	1.00461i	−0.00462076	−	0.999989i	$-0.501471\pi$
$942$		0			0
$943$		−	3.27256i		−	0.106569i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	16.6327	−	16.6327i	0.540491	−	0.540491i	−0.383182	−	0.923673i	$-0.625172\pi$
$947$	16.6327	−	16.6327i	0.540491	−	0.540491i	0.923673	+	0.383182i	$0.125172\pi$
$948$		0			0
$949$	6.97379	+	6.97379i	0.226379	+	0.226379i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−14.5520			−0.471387			−0.235693	−	0.971827i	$-0.575736\pi$
$953$	−14.5520			−0.471387			−0.235693	+	0.971827i	$0.575736\pi$
$954$		0			0
$955$	−5.86041	+	5.86041i	−0.189639	+	0.189639i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−35.8530			−1.15775
$960$		0			0
$961$	−33.4090			−1.07771
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1.90979	+	1.90979i	−0.0614783	+	0.0614783i
$966$		0			0
$967$	17.3884			0.559172			0.279586	−	0.960121i	$-0.409803\pi$
$967$	17.3884			0.559172			0.279586	+	0.960121i	$0.409803\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	20.2644	+	20.2644i	0.650314	+	0.650314i	0.953069	−	0.302754i	$-0.0979062\pi$
$971$	20.2644	+	20.2644i	0.650314	+	0.650314i	−0.302754	+	0.953069i	$0.597906\pi$
$972$		0			0
$973$	−3.59077	+	3.59077i	−0.115115	+	0.115115i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			35.3026i			1.12943i	0.825285	+	0.564716i	$0.191014\pi$
$977$			35.3026i			1.12943i	−0.825285	+	0.564716i	$0.808986\pi$
$978$		0			0
$979$	6.09220	+	6.09220i	0.194708	+	0.194708i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	51.7381i		−	1.65019i	−0.564995	−	0.825094i	$-0.691122\pi$
$983$		−	51.7381i		−	1.65019i	0.564995	−	0.825094i	$-0.308878\pi$
$984$		0			0
$985$		−	3.46980i		−	0.110557i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−6.79239	−	6.79239i	−0.215985	−	0.215985i
$990$		0			0
$991$			25.0317i			0.795160i	0.917568	+	0.397580i	$0.130150\pi$
$991$			25.0317i			0.795160i	−0.917568	+	0.397580i	$0.869850\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−2.72933	+	2.72933i	−0.0865257	+	0.0865257i
$996$		0			0
$997$	−17.3224	−	17.3224i	−0.548605	−	0.548605i	0.377432	−	0.926037i	$-0.376807\pi$
$997$	−17.3224	−	17.3224i	−0.548605	−	0.548605i	−0.926037	+	0.377432i	$0.876807\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.l.a.143.5		16
3.2	odd	2	inner	576.2.l.a.143.4		16
4.3	odd	2		144.2.l.a.107.3	yes	16
8.3	odd	2		1152.2.l.a.287.4		16
8.5	even	2		1152.2.l.b.287.4		16
12.11	even	2		144.2.l.a.107.6	yes	16
16.3	odd	4	inner	576.2.l.a.431.4		16
16.5	even	4		1152.2.l.a.863.5		16
16.11	odd	4		1152.2.l.b.863.5		16
16.13	even	4		144.2.l.a.35.6	yes	16
24.5	odd	2		1152.2.l.b.287.5		16
24.11	even	2		1152.2.l.a.287.5		16
48.5	odd	4		1152.2.l.a.863.4		16
48.11	even	4		1152.2.l.b.863.4		16
48.29	odd	4		144.2.l.a.35.3	✓	16
48.35	even	4	inner	576.2.l.a.431.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.l.a.35.3	✓	16	48.29	odd	4
144.2.l.a.35.6	yes	16	16.13	even	4
144.2.l.a.107.3	yes	16	4.3	odd	2
144.2.l.a.107.6	yes	16	12.11	even	2
576.2.l.a.143.4		16	3.2	odd	2	inner
576.2.l.a.143.5		16	1.1	even	1	trivial
576.2.l.a.431.4		16	16.3	odd	4	inner
576.2.l.a.431.5		16	48.35	even	4	inner
1152.2.l.a.287.4		16	8.3	odd	2
1152.2.l.a.287.5		16	24.11	even	2
1152.2.l.a.863.4		16	48.5	odd	4
1152.2.l.a.863.5		16	16.5	even	4
1152.2.l.b.287.4		16	8.5	even	2
1152.2.l.b.287.5		16	24.5	odd	2
1152.2.l.b.863.4		16	48.11	even	4
1152.2.l.b.863.5		16	16.11	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:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	576.l (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.236253 - 0.236253i) q^{5} -3.27830 q^{7} +O(q^{10})\)	\(q+(0.236253 - 0.236253i) q^{5} -3.27830 q^{7} +(-2.58229 - 2.58229i) q^{11} +(-1.70773 + 1.70773i) q^{13} -7.05130i q^{17} +(-3.04184 - 3.04184i) q^{19} -1.47338i q^{23} +4.88837i q^{25} +(-2.98575 - 2.98575i) q^{29} +8.02552i q^{31} +(-0.774506 + 0.774506i) q^{35} +(-7.93021 - 7.93021i) q^{37} +2.22112 q^{41} +(4.61007 - 4.61007i) q^{43} +7.13023 q^{47} +3.74723 q^{49} +(-5.81417 + 5.81417i) q^{53} -1.22015 q^{55} +(-7.46464 - 7.46464i) q^{59} +(-4.04184 + 4.04184i) q^{61} +0.806909i q^{65} +(2.90468 + 2.90468i) q^{67} -1.02064i q^{71} -4.08367i q^{73} +(8.46551 + 8.46551i) q^{77} +5.36197i q^{79} +(3.93734 - 3.93734i) q^{83} +(-1.66589 - 1.66589i) q^{85} -2.35922 q^{89} +(5.59843 - 5.59843i) q^{91} -1.43728 q^{95} +9.97204 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 16 q^{19} + 32 q^{43} + 16 q^{49} + 64 q^{55} - 32 q^{61} + 16 q^{67} - 32 q^{85} - 48 q^{91}+O(q^{100}) \) 16 * q - 16 * q^19 + 32 * q^43 + 16 * q^49 + 64 * q^55 - 32 * q^61 + 16 * q^67 - 32 * q^85 - 48 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more