LMFDB - Embedded newform 1344.2.w.b.1009.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,2,Mod(337,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 0, 0]))

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

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

chi := DirichletCharacter("1344.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1344.w (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:	$10.7318940317$
Analytic rank:	$0$
Dimension:	$28$
Relative dimension:	$14$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 336)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1009.2
Character	$\chi$	$=$	1344.1009
Dual form			1344.2.w.b.337.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 + 0.707107i) q^{3} +(-1.84737 - 1.84737i) q^{5} +1.00000i q^{7} -1.00000i q^{9} +O(q^{10})$	\(q+(-0.707107 + 0.707107i) q^{3} +(-1.84737 - 1.84737i) q^{5} +1.00000i q^{7} -1.00000i q^{9} +(1.28297 + 1.28297i) q^{11} +(0.573383 - 0.573383i) q^{13} +2.61258 q^{15} -3.76640 q^{17} +(5.58859 - 5.58859i) q^{19} +(-0.707107 - 0.707107i) q^{21} +8.77599i q^{23} +1.82558i q^{25} +(0.707107 + 0.707107i) q^{27} +(-2.45512 + 2.45512i) q^{29} -7.44993 q^{31} -1.81440 q^{33} +(1.84737 - 1.84737i) q^{35} +(-8.07793 - 8.07793i) q^{37} +0.810886i q^{39} +1.18711i q^{41} +(-2.55667 - 2.55667i) q^{43} +(-1.84737 + 1.84737i) q^{45} -8.68108 q^{47} -1.00000 q^{49} +(2.66325 - 2.66325i) q^{51} +(-1.03296 - 1.03296i) q^{53} -4.74026i q^{55} +7.90346i q^{57} +(-4.49096 - 4.49096i) q^{59} +(1.77606 - 1.77606i) q^{61} +1.00000 q^{63} -2.11851 q^{65} +(-9.10323 + 9.10323i) q^{67} +(-6.20556 - 6.20556i) q^{69} +10.6094i q^{71} -1.67971i q^{73} +(-1.29088 - 1.29088i) q^{75} +(-1.28297 + 1.28297i) q^{77} -8.98738 q^{79} -1.00000 q^{81} +(-5.03598 + 5.03598i) q^{83} +(6.95796 + 6.95796i) q^{85} -3.47207i q^{87} -3.53141i q^{89} +(0.573383 + 0.573383i) q^{91} +(5.26789 - 5.26789i) q^{93} -20.6484 q^{95} +7.92244 q^{97} +(1.28297 - 1.28297i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q+O(q^{10}) $ 28 * q	$ 28 q + 4 q^{11} - 8 q^{15} - 8 q^{19} + 4 q^{29} + 24 q^{33} + 4 q^{37} - 20 q^{43} - 28 q^{49} + 8 q^{51} + 20 q^{53} - 40 q^{61} + 28 q^{63} + 16 q^{65} - 4 q^{67} - 16 q^{69} + 16 q^{75} - 4 q^{77} - 24 q^{79} - 28 q^{81} - 40 q^{83} + 48 q^{85} + 72 q^{97} + 4 q^{99}+O(q^{100}) $ 28 * q + 4 * q^11 - 8 * q^15 - 8 * q^19 + 4 * q^29 + 24 * q^33 + 4 * q^37 - 20 * q^43 - 28 * q^49 + 8 * q^51 + 20 * q^53 - 40 * q^61 + 28 * q^63 + 16 * q^65 - 4 * q^67 - 16 * q^69 + 16 * q^75 - 4 * q^77 - 24 * q^79 - 28 * q^81 - 40 * q^83 + 48 * q^85 + 72 * q^97 + 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{1}{4}\right)$

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
$3$	−0.707107	+	0.707107i	−0.408248	+	0.408248i
$3$	−0.707107	+	0.707107i	−0.408248	+	0.408248i
$4$		0			0
$5$	−1.84737	−	1.84737i	−0.826171	−	0.826171i	0.160814	−	0.986985i	$-0.448588\pi$
$5$	−1.84737	−	1.84737i	−0.826171	−	0.826171i	−0.986985	+	0.160814i	$0.948588\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$		−	1.00000i		−	0.333333i
$10$		0			0
$11$	1.28297	+	1.28297i	0.386831	+	0.386831i	0.873556	−	0.486725i	$-0.161808\pi$
$11$	1.28297	+	1.28297i	0.386831	+	0.386831i	−0.486725	+	0.873556i	$0.661808\pi$
$12$		0			0
$13$	0.573383	−	0.573383i	0.159028	−	0.159028i	−0.623108	−	0.782136i	$-0.714130\pi$
$13$	0.573383	−	0.573383i	0.159028	−	0.159028i	0.782136	+	0.623108i	$0.214130\pi$
$14$		0			0
$15$	2.61258			0.674566
$16$		0			0
$17$	−3.76640			−0.913487			−0.456743	−	0.889598i	$-0.650984\pi$
$17$	−3.76640			−0.913487			−0.456743	+	0.889598i	$0.650984\pi$
$18$		0			0
$19$	5.58859	−	5.58859i	1.28211	−	1.28211i	0.342646	−	0.939464i	$-0.388677\pi$
$19$	5.58859	−	5.58859i	1.28211	−	1.28211i	0.939464	−	0.342646i	$-0.111323\pi$
$20$		0			0
$21$	−0.707107	−	0.707107i	−0.154303	−	0.154303i
$22$		0			0
$23$			8.77599i			1.82992i	0.403544	+	0.914960i	$0.367778\pi$
$23$			8.77599i			1.82992i	−0.403544	+	0.914960i	$0.632222\pi$
$24$		0			0
$25$			1.82558i			0.365117i
$26$		0			0
$27$	0.707107	+	0.707107i	0.136083	+	0.136083i
$28$		0			0
$29$	−2.45512	+	2.45512i	−0.455905	+	0.455905i	−0.897309	−	0.441403i	$-0.854481\pi$
$29$	−2.45512	+	2.45512i	−0.455905	+	0.455905i	0.441403	+	0.897309i	$0.354481\pi$
$30$		0			0
$31$	−7.44993			−1.33805			−0.669023	−	0.743242i	$-0.733287\pi$
$31$	−7.44993			−1.33805			−0.669023	+	0.743242i	$0.733287\pi$
$32$		0			0
$33$	−1.81440			−0.315846
$34$		0			0
$35$	1.84737	−	1.84737i	0.312263	−	0.312263i
$36$		0			0
$37$	−8.07793	−	8.07793i	−1.32800	−	1.32800i	−0.907110	−	0.420894i	$-0.861716\pi$
$37$	−8.07793	−	8.07793i	−1.32800	−	1.32800i	−0.420894	−	0.907110i	$-0.638284\pi$
$38$		0			0
$39$			0.810886i			0.129846i
$40$		0			0
$41$			1.18711i			0.185395i	0.995694	+	0.0926974i	$0.0295489\pi$
$41$			1.18711i			0.185395i	−0.995694	+	0.0926974i	$0.970451\pi$
$42$		0			0
$43$	−2.55667	−	2.55667i	−0.389888	−	0.389888i	0.484759	−	0.874648i	$-0.338907\pi$
$43$	−2.55667	−	2.55667i	−0.389888	−	0.389888i	−0.874648	+	0.484759i	$0.838907\pi$
$44$		0			0
$45$	−1.84737	+	1.84737i	−0.275390	+	0.275390i
$46$		0			0
$47$	−8.68108			−1.26627			−0.633133	−	0.774043i	$-0.718231\pi$
$47$	−8.68108			−1.26627			−0.633133	+	0.774043i	$0.718231\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	2.66325	−	2.66325i	0.372929	−	0.372929i
$52$		0			0
$53$	−1.03296	−	1.03296i	−0.141888	−	0.141888i	0.632595	−	0.774483i	$-0.281990\pi$
$53$	−1.03296	−	1.03296i	−0.141888	−	0.141888i	−0.774483	+	0.632595i	$0.781990\pi$
$54$		0			0
$55$		−	4.74026i		−	0.639177i
$56$		0			0
$57$			7.90346i			1.04684i
$58$		0			0
$59$	−4.49096	−	4.49096i	−0.584673	−	0.584673i	0.351511	−	0.936184i	$-0.385668\pi$
$59$	−4.49096	−	4.49096i	−0.584673	−	0.584673i	−0.936184	+	0.351511i	$0.885668\pi$
$60$		0			0
$61$	1.77606	−	1.77606i	0.227401	−	0.227401i	−0.584205	−	0.811606i	$-0.698594\pi$
$61$	1.77606	−	1.77606i	0.227401	−	0.227401i	0.811606	+	0.584205i	$0.198594\pi$
$62$		0			0
$63$	1.00000			0.125988
$64$		0			0
$65$	−2.11851			−0.262768
$66$		0			0
$67$	−9.10323	+	9.10323i	−1.11214	+	1.11214i	−0.119276	+	0.992861i	$0.538057\pi$
$67$	−9.10323	+	9.10323i	−1.11214	+	1.11214i	−0.992861	+	0.119276i	$0.961943\pi$
$68$		0			0
$69$	−6.20556	−	6.20556i	−0.747062	−	0.747062i
$70$		0			0
$71$			10.6094i			1.25911i	0.776957	+	0.629554i	$0.216762\pi$
$71$			10.6094i			1.25911i	−0.776957	+	0.629554i	$0.783238\pi$
$72$		0			0
$73$		−	1.67971i		−	0.196596i	−0.995157	−	0.0982978i	$-0.968660\pi$
$73$		−	1.67971i		−	0.196596i	0.995157	−	0.0982978i	$-0.0313398\pi$
$74$		0			0
$75$	−1.29088	−	1.29088i	−0.149058	−	0.149058i
$76$		0			0
$77$	−1.28297	+	1.28297i	−0.146208	+	0.146208i
$78$		0			0
$79$	−8.98738			−1.01116			−0.505580	−	0.862780i	$-0.668721\pi$
$79$	−8.98738			−1.01116			−0.505580	+	0.862780i	$0.668721\pi$
$80$		0			0
$81$	−1.00000			−0.111111
$82$		0			0
$83$	−5.03598	+	5.03598i	−0.552770	+	0.552770i	−0.927239	−	0.374469i	$-0.877825\pi$
$83$	−5.03598	+	5.03598i	−0.552770	+	0.552770i	0.374469	+	0.927239i	$0.377825\pi$
$84$		0			0
$85$	6.95796	+	6.95796i	0.754696	+	0.754696i
$86$		0			0
$87$		−	3.47207i		−	0.372245i
$88$		0			0
$89$		−	3.53141i		−	0.374329i	−0.982329	−	0.187164i	$-0.940070\pi$
$89$		−	3.53141i		−	0.374329i	0.982329	−	0.187164i	$-0.0599297\pi$
$90$		0			0
$91$	0.573383	+	0.573383i	0.0601069	+	0.0601069i
$92$		0			0
$93$	5.26789	−	5.26789i	0.546255	−	0.546255i
$94$		0			0
$95$	−20.6484			−2.11849
$96$		0			0
$97$	7.92244			0.804402			0.402201	−	0.915551i	$-0.368245\pi$
$97$	7.92244			0.804402			0.402201	+	0.915551i	$0.368245\pi$
$98$		0			0
$99$	1.28297	−	1.28297i	0.128944	−	0.128944i
$100$		0			0
$101$	−3.02324	−	3.02324i	−0.300823	−	0.300823i	0.540513	−	0.841336i	$-0.318230\pi$
$101$	−3.02324	−	3.02324i	−0.300823	−	0.300823i	−0.841336	+	0.540513i	$0.818230\pi$
$102$		0			0
$103$		−	5.82296i		−	0.573753i	−0.957968	−	0.286877i	$-0.907383\pi$
$103$		−	5.82296i		−	0.573753i	0.957968	−	0.286877i	$-0.0926170\pi$
$104$		0			0
$105$			2.61258i			0.254962i
$106$		0			0
$107$	−4.48343	−	4.48343i	−0.433429	−	0.433429i	0.456364	−	0.889793i	$-0.349152\pi$
$107$	−4.48343	−	4.48343i	−0.433429	−	0.433429i	−0.889793	+	0.456364i	$0.849152\pi$
$108$		0			0
$109$	6.57337	−	6.57337i	0.629614	−	0.629614i	−0.318357	−	0.947971i	$-0.603131\pi$
$109$	6.57337	−	6.57337i	0.629614	−	0.629614i	0.947971	+	0.318357i	$0.103131\pi$
$110$		0			0
$111$	11.4239			1.08431
$112$		0			0
$113$	−12.6537			−1.19036			−0.595180	−	0.803592i	$-0.702919\pi$
$113$	−12.6537			−1.19036			−0.595180	+	0.803592i	$0.702919\pi$
$114$		0			0
$115$	16.2125	−	16.2125i	1.51183	−	1.51183i
$116$		0			0
$117$	−0.573383	−	0.573383i	−0.0530093	−	0.0530093i
$118$		0			0
$119$		−	3.76640i		−	0.345266i
$120$		0			0
$121$		−	7.70796i		−	0.700724i
$122$		0			0
$123$	−0.839410	−	0.839410i	−0.0756871	−	0.0756871i
$124$		0			0
$125$	−5.86433	+	5.86433i	−0.524522	+	0.524522i
$126$		0			0
$127$	−8.72606			−0.774313			−0.387157	−	0.922014i	$-0.626543\pi$
$127$	−8.72606			−0.774313			−0.387157	+	0.922014i	$0.626543\pi$
$128$		0			0
$129$	3.61568			0.318343
$130$		0			0
$131$	6.31850	−	6.31850i	0.552050	−	0.552050i	−0.374982	−	0.927032i	$-0.622351\pi$
$131$	6.31850	−	6.31850i	0.552050	−	0.552050i	0.927032	+	0.374982i	$0.122351\pi$
$132$		0			0
$133$	5.58859	+	5.58859i	0.484592	+	0.484592i
$134$		0			0
$135$		−	2.61258i		−	0.224855i
$136$		0			0
$137$			11.0307i			0.942416i	0.882022	+	0.471208i	$0.156182\pi$
$137$			11.0307i			0.942416i	−0.882022	+	0.471208i	$0.843818\pi$
$138$		0			0
$139$	9.38542	+	9.38542i	0.796061	+	0.796061i	0.982472	−	0.186411i	$-0.0596856\pi$
$139$	9.38542	+	9.38542i	0.796061	+	0.796061i	−0.186411	+	0.982472i	$0.559686\pi$
$140$		0			0
$141$	6.13845	−	6.13845i	0.516951	−	0.516951i
$142$		0			0
$143$	1.47127			0.123034
$144$		0			0
$145$	9.07107			0.753311
$146$		0			0
$147$	0.707107	−	0.707107i	0.0583212	−	0.0583212i
$148$		0			0
$149$	−6.04303	−	6.04303i	−0.495064	−	0.495064i	0.414833	−	0.909897i	$-0.363840\pi$
$149$	−6.04303	−	6.04303i	−0.495064	−	0.495064i	−0.909897	+	0.414833i	$0.863840\pi$
$150$		0			0
$151$			2.76077i			0.224668i	0.993671	+	0.112334i	$0.0358327\pi$
$151$			2.76077i			0.224668i	−0.993671	+	0.112334i	$0.964167\pi$
$152$		0			0
$153$			3.76640i			0.304496i
$154$		0			0
$155$	13.7628	+	13.7628i	1.10545	+	1.10545i
$156$		0			0
$157$	−12.9246	+	12.9246i	−1.03149	+	1.03149i	−0.0320071	+	0.999488i	$0.510190\pi$
$157$	−12.9246	+	12.9246i	−1.03149	+	1.03149i	−0.999488	+	0.0320071i	$0.989810\pi$
$158$		0			0
$159$	1.46083			0.115851
$160$		0			0
$161$	−8.77599			−0.691645
$162$		0			0
$163$	−7.16309	+	7.16309i	−0.561056	+	0.561056i	−0.929608	−	0.368551i	$-0.879854\pi$
$163$	−7.16309	+	7.16309i	−0.561056	+	0.561056i	0.368551	+	0.929608i	$0.379854\pi$
$164$		0			0
$165$	3.35187	+	3.35187i	0.260943	+	0.260943i
$166$		0			0
$167$		−	25.5300i		−	1.97557i	−0.155809	−	0.987787i	$-0.549799\pi$
$167$		−	25.5300i		−	1.97557i	0.155809	−	0.987787i	$-0.450201\pi$
$168$		0			0
$169$			12.3425i			0.949420i
$170$		0			0
$171$	−5.58859	−	5.58859i	−0.427370	−	0.427370i
$172$		0			0
$173$	9.22288	−	9.22288i	0.701202	−	0.701202i	−0.263466	−	0.964669i	$-0.584866\pi$
$173$	9.22288	−	9.22288i	0.701202	−	0.701202i	0.964669	+	0.263466i	$0.0848657\pi$
$174$		0			0
$175$	−1.82558			−0.138001
$176$		0			0
$177$	6.35117			0.477383
$178$		0			0
$179$	3.27756	−	3.27756i	0.244976	−	0.244976i	−0.573929	−	0.818905i	$-0.694581\pi$
$179$	3.27756	−	3.27756i	0.244976	−	0.244976i	0.818905	+	0.573929i	$0.194581\pi$
$180$		0			0
$181$	−12.7336	−	12.7336i	−0.946484	−	0.946484i	0.0521551	−	0.998639i	$-0.483391\pi$
$181$	−12.7336	−	12.7336i	−0.946484	−	0.946484i	−0.998639	+	0.0521551i	$0.983391\pi$
$182$		0			0
$183$			2.51173i			0.185673i
$184$		0			0
$185$			29.8459i			2.19432i
$186$		0			0
$187$	−4.83219	−	4.83219i	−0.353365	−	0.353365i
$188$		0			0
$189$	−0.707107	+	0.707107i	−0.0514344	+	0.0514344i
$190$		0			0
$191$	13.1825			0.953854			0.476927	−	0.878943i	$-0.341751\pi$
$191$	13.1825			0.953854			0.476927	+	0.878943i	$0.341751\pi$
$192$		0			0
$193$	24.2115			1.74278			0.871391	−	0.490589i	$-0.163219\pi$
$193$	24.2115			1.74278			0.871391	+	0.490589i	$0.163219\pi$
$194$		0			0
$195$	1.49801	−	1.49801i	0.107275	−	0.107275i
$196$		0			0
$197$	−4.20130	−	4.20130i	−0.299330	−	0.299330i	0.541421	−	0.840751i	$-0.317886\pi$
$197$	−4.20130	−	4.20130i	−0.299330	−	0.299330i	−0.840751	+	0.541421i	$0.817886\pi$
$198$		0			0
$199$			8.71100i			0.617507i	0.951142	+	0.308753i	$0.0999118\pi$
$199$			8.71100i			0.617507i	−0.951142	+	0.308753i	$0.900088\pi$
$200$		0			0
$201$		−	12.8739i		−	0.908056i
$202$		0			0
$203$	−2.45512	−	2.45512i	−0.172316	−	0.172316i
$204$		0			0
$205$	2.19303	−	2.19303i	0.153168	−	0.153168i
$206$		0			0
$207$	8.77599			0.609974
$208$		0			0
$209$	14.3400			0.991921
$210$		0			0
$211$	2.31830	−	2.31830i	0.159598	−	0.159598i	−0.622791	−	0.782389i	$-0.714001\pi$
$211$	2.31830	−	2.31830i	0.159598	−	0.159598i	0.782389	+	0.622791i	$0.214001\pi$
$212$		0			0
$213$	−7.50200	−	7.50200i	−0.514029	−	0.514029i
$214$		0			0
$215$			9.44625i			0.644229i
$216$		0			0
$217$		−	7.44993i		−	0.505734i
$218$		0			0
$219$	1.18774	+	1.18774i	0.0802598	+	0.0802598i
$220$		0			0
$221$	−2.15959	+	2.15959i	−0.145270	+	0.145270i
$222$		0			0
$223$	−9.37037			−0.627486			−0.313743	−	0.949508i	$-0.601583\pi$
$223$	−9.37037			−0.627486			−0.313743	+	0.949508i	$0.601583\pi$
$224$		0			0
$225$	1.82558			0.121706
$226$		0			0
$227$	−5.57766	+	5.57766i	−0.370202	+	0.370202i	−0.867551	−	0.497349i	$-0.834307\pi$
$227$	−5.57766	+	5.57766i	−0.370202	+	0.370202i	0.497349	+	0.867551i	$0.334307\pi$
$228$		0			0
$229$	13.8479	+	13.8479i	0.915097	+	0.915097i	0.996668	−	0.0815703i	$-0.0259935\pi$
$229$	13.8479	+	13.8479i	0.915097	+	0.915097i	−0.0815703	+	0.996668i	$0.525994\pi$
$230$		0			0
$231$		−	1.81440i		−	0.119379i
$232$		0			0
$233$			18.4216i			1.20684i	0.797424	+	0.603419i	$0.206195\pi$
$233$			18.4216i			1.20684i	−0.797424	+	0.603419i	$0.793805\pi$
$234$		0			0
$235$	16.0372	+	16.0372i	1.04615	+	1.04615i
$236$		0			0
$237$	6.35504	−	6.35504i	0.412804	−	0.412804i
$238$		0			0
$239$	14.3108			0.925691			0.462845	−	0.886439i	$-0.346828\pi$
$239$	14.3108			0.925691			0.462845	+	0.886439i	$0.346828\pi$
$240$		0			0
$241$	6.29237			0.405327			0.202664	−	0.979248i	$-0.435040\pi$
$241$	6.29237			0.405327			0.202664	+	0.979248i	$0.435040\pi$
$242$		0			0
$243$	0.707107	−	0.707107i	0.0453609	−	0.0453609i
$244$		0			0
$245$	1.84737	+	1.84737i	0.118024	+	0.118024i
$246$		0			0
$247$		−	6.40881i		−	0.407783i
$248$		0			0
$249$		−	7.12195i		−	0.451335i
$250$		0			0
$251$	−16.3705	−	16.3705i	−1.03329	−	1.03329i	−0.999426	−	0.0338675i	$-0.989218\pi$
$251$	−16.3705	−	16.3705i	−1.03329	−	1.03329i	−0.0338675	−	0.999426i	$-0.510782\pi$
$252$		0			0
$253$	−11.2594	+	11.2594i	−0.707870	+	0.707870i
$254$		0			0
$255$	−9.84004			−0.616207
$256$		0			0
$257$	5.94472			0.370821			0.185411	−	0.982661i	$-0.440638\pi$
$257$	5.94472			0.370821			0.185411	+	0.982661i	$0.440638\pi$
$258$		0			0
$259$	8.07793	−	8.07793i	0.501938	−	0.501938i
$260$		0			0
$261$	2.45512	+	2.45512i	0.151968	+	0.151968i
$262$		0			0
$263$			17.5991i			1.08521i	0.839989	+	0.542603i	$0.182561\pi$
$263$			17.5991i			1.08521i	−0.839989	+	0.542603i	$0.817439\pi$
$264$		0			0
$265$			3.81653i			0.234447i
$266$		0			0
$267$	2.49708	+	2.49708i	0.152819	+	0.152819i
$268$		0			0
$269$	−14.6340	+	14.6340i	−0.892251	+	0.892251i	−0.994735	−	0.102484i	$-0.967321\pi$
$269$	−14.6340	+	14.6340i	−0.892251	+	0.892251i	0.102484	+	0.994735i	$0.467321\pi$
$270$		0			0
$271$	1.48973			0.0904945			0.0452473	−	0.998976i	$-0.485592\pi$
$271$	1.48973			0.0904945			0.0452473	+	0.998976i	$0.485592\pi$
$272$		0			0
$273$	−0.810886			−0.0490771
$274$		0			0
$275$	−2.34218	+	2.34218i	−0.141239	+	0.141239i
$276$		0			0
$277$	6.87380	+	6.87380i	0.413006	+	0.413006i	0.882785	−	0.469778i	$-0.155666\pi$
$277$	6.87380	+	6.87380i	0.413006	+	0.413006i	−0.469778	+	0.882785i	$0.655666\pi$
$278$		0			0
$279$			7.44993i			0.446015i
$280$		0			0
$281$			1.78681i			0.106592i	0.998579	+	0.0532962i	$0.0169727\pi$
$281$			1.78681i			0.106592i	−0.998579	+	0.0532962i	$0.983027\pi$
$282$		0			0
$283$	22.5902	+	22.5902i	1.34285	+	1.34285i	0.893213	+	0.449634i	$0.148445\pi$
$283$	22.5902	+	22.5902i	1.34285	+	1.34285i	0.449634	+	0.893213i	$0.351555\pi$
$284$		0			0
$285$	14.6007	−	14.6007i	0.864868	−	0.864868i
$286$		0			0
$287$	−1.18711			−0.0700726
$288$		0			0
$289$	−2.81421			−0.165542
$290$		0			0
$291$	−5.60201	+	5.60201i	−0.328396	+	0.328396i
$292$		0			0
$293$	−5.95937	−	5.95937i	−0.348150	−	0.348150i	0.511270	−	0.859420i	$-0.329175\pi$
$293$	−5.95937	−	5.95937i	−0.348150	−	0.348150i	−0.859420	+	0.511270i	$0.829175\pi$
$294$		0			0
$295$			16.5930i			0.966080i
$296$		0			0
$297$			1.81440i			0.105282i
$298$		0			0
$299$	5.03201	+	5.03201i	0.291008	+	0.291008i
$300$		0			0
$301$	2.55667	−	2.55667i	0.147364	−	0.147364i
$302$		0			0
$303$	4.27550			0.245621
$304$		0			0
$305$	−6.56210			−0.375745
$306$		0			0
$307$	19.2863	−	19.2863i	1.10073	−	1.10073i	0.106402	−	0.994323i	$-0.466067\pi$
$307$	19.2863	−	19.2863i	1.10073	−	1.10073i	0.994323	−	0.106402i	$-0.0339331\pi$
$308$		0			0
$309$	4.11745	+	4.11745i	0.234234	+	0.234234i
$310$		0			0
$311$			30.4880i			1.72882i	0.502791	+	0.864408i	$0.332306\pi$
$311$			30.4880i			1.72882i	−0.502791	+	0.864408i	$0.667694\pi$
$312$		0			0
$313$		−	24.6544i		−	1.39355i	−0.717291	−	0.696774i	$-0.754618\pi$
$313$		−	24.6544i		−	1.39355i	0.717291	−	0.696774i	$-0.245382\pi$
$314$		0			0
$315$	−1.84737	−	1.84737i	−0.104088	−	0.104088i
$316$		0			0
$317$	18.9933	−	18.9933i	1.06677	−	1.06677i	0.0691669	−	0.997605i	$-0.477966\pi$
$317$	18.9933	−	18.9933i	1.06677	−	1.06677i	0.997605	−	0.0691669i	$-0.0220341\pi$
$318$		0			0
$319$	−6.29972			−0.352717
$320$		0			0
$321$	6.34053			0.353894
$322$		0			0
$323$	−21.0489	+	21.0489i	−1.17119	+	1.17119i
$324$		0			0
$325$	1.04676	+	1.04676i	0.0580638	+	0.0580638i
$326$		0			0
$327$			9.29614i			0.514078i
$328$		0			0
$329$		−	8.68108i		−	0.478604i
$330$		0			0
$331$	−13.2656	−	13.2656i	−0.729142	−	0.729142i	0.241307	−	0.970449i	$-0.422424\pi$
$331$	−13.2656	−	13.2656i	−0.729142	−	0.729142i	−0.970449	+	0.241307i	$0.922424\pi$
$332$		0			0
$333$	−8.07793	+	8.07793i	−0.442668	+	0.442668i
$334$		0			0
$335$	33.6342			1.83763
$336$		0			0
$337$	−5.18645			−0.282524			−0.141262	−	0.989972i	$-0.545116\pi$
$337$	−5.18645			−0.282524			−0.141262	+	0.989972i	$0.545116\pi$
$338$		0			0
$339$	8.94752	−	8.94752i	0.485963	−	0.485963i
$340$		0			0
$341$	−9.55806	−	9.55806i	−0.517598	−	0.517598i
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$			22.9280i			1.23440i
$346$		0			0
$347$	−12.7128	−	12.7128i	−0.682457	−	0.682457i	0.278096	−	0.960553i	$-0.410297\pi$
$347$	−12.7128	−	12.7128i	−0.682457	−	0.682457i	−0.960553	+	0.278096i	$0.910297\pi$
$348$		0			0
$349$	0.143369	−	0.143369i	0.00767437	−	0.00767437i	−0.703259	−	0.710934i	$-0.748273\pi$
$349$	0.143369	−	0.143369i	0.00767437	−	0.00767437i	0.710934	+	0.703259i	$0.248273\pi$
$350$		0			0
$351$	0.810886			0.0432819
$352$		0			0
$353$	7.90240			0.420602			0.210301	−	0.977637i	$-0.432556\pi$
$353$	7.90240			0.420602			0.210301	+	0.977637i	$0.432556\pi$
$354$		0			0
$355$	19.5996	−	19.5996i	1.04024	−	1.04024i
$356$		0			0
$357$	2.66325	+	2.66325i	0.140954	+	0.140954i
$358$		0			0
$359$		−	4.92045i		−	0.259691i	−0.991534	−	0.129846i	$-0.958552\pi$
$359$		−	4.92045i		−	0.259691i	0.991534	−	0.129846i	$-0.0414482\pi$
$360$		0			0
$361$		−	43.4647i		−	2.28762i
$362$		0			0
$363$	5.45035	+	5.45035i	0.286069	+	0.286069i
$364$		0			0
$365$	−3.10306	+	3.10306i	−0.162421	+	0.162421i
$366$		0			0
$367$	12.1171			0.632505			0.316253	−	0.948675i	$-0.397575\pi$
$367$	12.1171			0.632505			0.316253	+	0.948675i	$0.397575\pi$
$368$		0			0
$369$	1.18711			0.0617983
$370$		0			0
$371$	1.03296	−	1.03296i	0.0536286	−	0.0536286i
$372$		0			0
$373$	−4.24229	−	4.24229i	−0.219657	−	0.219657i	0.588697	−	0.808354i	$-0.299641\pi$
$373$	−4.24229	−	4.24229i	−0.219657	−	0.219657i	−0.808354	+	0.588697i	$0.799641\pi$
$374$		0			0
$375$		−	8.29342i		−	0.428270i
$376$		0			0
$377$			2.81545i			0.145003i
$378$		0			0
$379$	12.5478	+	12.5478i	0.644538	+	0.644538i	0.951668	−	0.307129i	$-0.0993684\pi$
$379$	12.5478	+	12.5478i	0.644538	+	0.644538i	−0.307129	+	0.951668i	$0.599368\pi$
$380$		0			0
$381$	6.17026	−	6.17026i	0.316112	−	0.316112i
$382$		0			0
$383$	−10.6477			−0.544073			−0.272036	−	0.962287i	$-0.587697\pi$
$383$	−10.6477			−0.544073			−0.272036	+	0.962287i	$0.587697\pi$
$384$		0			0
$385$	4.74026			0.241586
$386$		0			0
$387$	−2.55667	+	2.55667i	−0.129963	+	0.129963i
$388$		0			0
$389$	−7.72898	−	7.72898i	−0.391875	−	0.391875i	0.483480	−	0.875355i	$-0.339372\pi$
$389$	−7.72898	−	7.72898i	−0.391875	−	0.391875i	−0.875355	+	0.483480i	$0.839372\pi$
$390$		0			0
$391$		−	33.0539i		−	1.67161i
$392$		0			0
$393$			8.93571i			0.450747i
$394$		0			0
$395$	16.6031	+	16.6031i	0.835391	+	0.835391i
$396$		0			0
$397$	−7.15797	+	7.15797i	−0.359248	+	0.359248i	−0.863536	−	0.504288i	$-0.831755\pi$
$397$	−7.15797	+	7.15797i	−0.359248	+	0.359248i	0.504288	+	0.863536i	$0.331755\pi$
$398$		0			0
$399$	−7.90346			−0.395668
$400$		0			0
$401$	34.6204			1.72886			0.864431	−	0.502751i	$-0.167679\pi$
$401$	34.6204			1.72886			0.864431	+	0.502751i	$0.167679\pi$
$402$		0			0
$403$	−4.27166	+	4.27166i	−0.212787	+	0.212787i
$404$		0			0
$405$	1.84737	+	1.84737i	0.0917968	+	0.0917968i
$406$		0			0
$407$		−	20.7275i		−	1.02743i
$408$		0			0
$409$		−	22.6531i		−	1.12013i	−0.828450	−	0.560063i	$-0.810777\pi$
$409$		−	22.6531i		−	1.12013i	0.828450	−	0.560063i	$-0.189223\pi$
$410$		0			0
$411$	−7.79988	−	7.79988i	−0.384740	−	0.384740i
$412$		0			0
$413$	4.49096	−	4.49096i	0.220986	−	0.220986i
$414$		0			0
$415$	18.6067			0.913366
$416$		0			0
$417$	−13.2730			−0.649981
$418$		0			0
$419$	−16.0963	+	16.0963i	−0.786357	+	0.786357i	−0.980895	−	0.194538i	$-0.937679\pi$
$419$	−16.0963	+	16.0963i	−0.786357	+	0.786357i	0.194538	+	0.980895i	$0.437679\pi$
$420$		0			0
$421$	22.1632	+	22.1632i	1.08017	+	1.08017i	0.996493	+	0.0836736i	$0.0266653\pi$
$421$	22.1632	+	22.1632i	1.08017	+	1.08017i	0.0836736	+	0.996493i	$0.473335\pi$
$422$		0			0
$423$			8.68108i			0.422089i
$424$		0			0
$425$		−	6.87589i		−	0.333529i
$426$		0			0
$427$	1.77606	+	1.77606i	0.0859497	+	0.0859497i
$428$		0			0
$429$	−1.04035	+	1.04035i	−0.0502284	+	0.0502284i
$430$		0			0
$431$	−3.65156			−0.175889			−0.0879447	−	0.996125i	$-0.528030\pi$
$431$	−3.65156			−0.175889			−0.0879447	+	0.996125i	$0.528030\pi$
$432$		0			0
$433$	−37.1208			−1.78391			−0.891955	−	0.452123i	$-0.850667\pi$
$433$	−37.1208			−1.78391			−0.891955	+	0.452123i	$0.850667\pi$
$434$		0			0
$435$	−6.41422	+	6.41422i	−0.307538	+	0.307538i
$436$		0			0
$437$	49.0454	+	49.0454i	2.34616	+	2.34616i
$438$		0			0
$439$		−	23.5707i		−	1.12497i	−0.826807	−	0.562485i	$-0.809845\pi$
$439$		−	23.5707i		−	1.12497i	0.826807	−	0.562485i	$-0.190155\pi$
$440$		0			0
$441$			1.00000i			0.0476190i
$442$		0			0
$443$	10.6245	+	10.6245i	0.504786	+	0.504786i	0.912921	−	0.408136i	$-0.133821\pi$
$443$	10.6245	+	10.6245i	0.504786	+	0.504786i	−0.408136	+	0.912921i	$0.633821\pi$
$444$		0			0
$445$	−6.52383	+	6.52383i	−0.309259	+	0.309259i
$446$		0			0
$447$	8.54613			0.404218
$448$		0			0
$449$	−23.9090			−1.12833			−0.564167	−	0.825661i	$-0.690802\pi$
$449$	−23.9090			−1.12833			−0.564167	+	0.825661i	$0.690802\pi$
$450$		0			0
$451$	−1.52302	+	1.52302i	−0.0717164	+	0.0717164i
$452$		0			0
$453$	−1.95216	−	1.95216i	−0.0917203	−	0.0917203i
$454$		0			0
$455$		−	2.11851i		−	0.0993171i
$456$		0			0
$457$		−	16.0427i		−	0.750445i	−0.926935	−	0.375222i	$-0.877566\pi$
$457$		−	16.0427i		−	0.750445i	0.926935	−	0.375222i	$-0.122434\pi$
$458$		0			0
$459$	−2.66325	−	2.66325i	−0.124310	−	0.124310i
$460$		0			0
$461$	14.6403	−	14.6403i	0.681866	−	0.681866i	−0.278554	−	0.960421i	$-0.589855\pi$
$461$	14.6403	−	14.6403i	0.681866	−	0.681866i	0.960421	+	0.278554i	$0.0898551\pi$
$462$		0			0
$463$	11.5751			0.537939			0.268969	−	0.963149i	$-0.413317\pi$
$463$	11.5751			0.537939			0.268969	+	0.963149i	$0.413317\pi$
$464$		0			0
$465$	−19.4635			−0.902600
$466$		0			0
$467$	3.29239	−	3.29239i	0.152354	−	0.152354i	−0.626815	−	0.779168i	$-0.715642\pi$
$467$	3.29239	−	3.29239i	0.152354	−	0.152354i	0.779168	+	0.626815i	$0.215642\pi$
$468$		0			0
$469$	−9.10323	−	9.10323i	−0.420348	−	0.420348i
$470$		0			0
$471$		−	18.2781i		−	0.842212i
$472$		0			0
$473$		−	6.56028i		−	0.301642i
$474$		0			0
$475$	10.2024	+	10.2024i	0.468120	+	0.468120i
$476$		0			0
$477$	−1.03296	+	1.03296i	−0.0472960	+	0.0472960i
$478$		0			0
$479$	−1.56900			−0.0716893			−0.0358447	−	0.999357i	$-0.511412\pi$
$479$	−1.56900			−0.0716893			−0.0358447	+	0.999357i	$0.511412\pi$
$480$		0			0
$481$	−9.26350			−0.422379
$482$		0			0
$483$	6.20556	−	6.20556i	0.282363	−	0.282363i
$484$		0			0
$485$	−14.6357	−	14.6357i	−0.664574	−	0.664574i
$486$		0			0
$487$			13.9189i			0.630726i	0.948971	+	0.315363i	$0.102126\pi$
$487$			13.9189i			0.630726i	−0.948971	+	0.315363i	$0.897874\pi$
$488$		0			0
$489$		−	10.1301i		−	0.458101i
$490$		0			0
$491$	−30.6828	−	30.6828i	−1.38469	−	1.38469i	−0.836068	−	0.548626i	$-0.815151\pi$
$491$	−30.6828	−	30.6828i	−1.38469	−	1.38469i	−0.548626	−	0.836068i	$-0.684849\pi$
$492$		0			0
$493$	9.24699	−	9.24699i	0.416463	−	0.416463i
$494$		0			0
$495$	−4.74026			−0.213059
$496$		0			0
$497$	−10.6094			−0.475898
$498$		0			0
$499$	−7.21452	+	7.21452i	−0.322966	+	0.322966i	−0.849904	−	0.526938i	$-0.823340\pi$
$499$	−7.21452	+	7.21452i	−0.322966	+	0.322966i	0.526938	+	0.849904i	$0.323340\pi$
$500$		0			0
$501$	18.0525	+	18.0525i	0.806525	+	0.806525i
$502$		0			0
$503$		−	19.8432i		−	0.884763i	−0.896827	−	0.442381i	$-0.854134\pi$
$503$		−	19.8432i		−	0.884763i	0.896827	−	0.442381i	$-0.145866\pi$
$504$		0			0
$505$			11.1701i			0.497063i
$506$		0			0
$507$	−8.72744	−	8.72744i	−0.387599	−	0.387599i
$508$		0			0
$509$	−4.05195	+	4.05195i	−0.179599	+	0.179599i	−0.791181	−	0.611582i	$-0.790534\pi$
$509$	−4.05195	+	4.05195i	−0.179599	+	0.179599i	0.611582	+	0.791181i	$0.290534\pi$
$510$		0			0
$511$	1.67971			0.0743061
$512$		0			0
$513$	7.90346			0.348946
$514$		0			0
$515$	−10.7572	+	10.7572i	−0.474018	+	0.474018i
$516$		0			0
$517$	−11.1376	−	11.1376i	−0.489831	−	0.489831i
$518$		0			0
$519$			13.0431i			0.572529i
$520$		0			0
$521$		−	38.3311i		−	1.67932i	−0.543115	−	0.839658i	$-0.682755\pi$
$521$		−	38.3311i		−	1.67932i	0.543115	−	0.839658i	$-0.317245\pi$
$522$		0			0
$523$	−11.3713	−	11.3713i	−0.497231	−	0.497231i	0.413344	−	0.910575i	$-0.364361\pi$
$523$	−11.3713	−	11.3713i	−0.497231	−	0.497231i	−0.910575	+	0.413344i	$0.864361\pi$
$524$		0			0
$525$	1.29088	−	1.29088i	0.0563388	−	0.0563388i
$526$		0			0
$527$	28.0594			1.22229
$528$		0			0
$529$	−54.0180			−2.34861
$530$		0			0
$531$	−4.49096	+	4.49096i	−0.194891	+	0.194891i
$532$		0			0
$533$	0.680666	+	0.680666i	0.0294829	+	0.0294829i
$534$		0			0
$535$			16.5651i			0.716174i
$536$		0			0
$537$			4.63517i			0.200022i
$538$		0			0
$539$	−1.28297	−	1.28297i	−0.0552616	−	0.0552616i
$540$		0			0
$541$	−13.0681	+	13.0681i	−0.561844	+	0.561844i	−0.929831	−	0.367987i	$-0.880047\pi$
$541$	−13.0681	+	13.0681i	−0.561844	+	0.561844i	0.367987	+	0.929831i	$0.380047\pi$
$542$		0			0
$543$	18.0081			0.772801
$544$		0			0
$545$	−24.2869			−1.04034
$546$		0			0
$547$	−16.6333	+	16.6333i	−0.711189	+	0.711189i	−0.966784	−	0.255595i	$-0.917729\pi$
$547$	−16.6333	+	16.6333i	−0.711189	+	0.711189i	0.255595	+	0.966784i	$0.417729\pi$
$548$		0			0
$549$	−1.77606	−	1.77606i	−0.0758005	−	0.0758005i
$550$		0			0
$551$			27.4414i			1.16904i
$552$		0			0
$553$		−	8.98738i		−	0.382182i
$554$		0			0
$555$	−21.1043	−	21.1043i	−0.895826	−	0.895826i
$556$		0			0
$557$	27.1381	−	27.1381i	1.14988	−	1.14988i	0.163300	−	0.986576i	$-0.447786\pi$
$557$	27.1381	−	27.1381i	1.14988	−	1.14988i	0.986576	−	0.163300i	$-0.0522138\pi$
$558$		0			0
$559$	−2.93190			−0.124006
$560$		0			0
$561$	6.83375			0.288521
$562$		0			0
$563$	3.54589	−	3.54589i	0.149441	−	0.149441i	−0.628427	−	0.777869i	$-0.716301\pi$
$563$	3.54589	−	3.54589i	0.149441	−	0.149441i	0.777869	+	0.628427i	$0.216301\pi$
$564$		0			0
$565$	23.3761	+	23.3761i	0.983441	+	0.983441i
$566$		0			0
$567$		−	1.00000i		−	0.0419961i
$568$		0			0
$569$			33.1999i			1.39181i	0.718134	+	0.695905i	$0.244997\pi$
$569$			33.1999i			1.39181i	−0.718134	+	0.695905i	$0.755003\pi$
$570$		0			0
$571$	25.3515	+	25.3515i	1.06093	+	1.06093i	0.998019	+	0.0629072i	$0.0200372\pi$
$571$	25.3515	+	25.3515i	1.06093	+	1.06093i	0.0629072	+	0.998019i	$0.479963\pi$
$572$		0			0
$573$	−9.32145	+	9.32145i	−0.389409	+	0.389409i
$574$		0			0
$575$	−16.0213			−0.668135
$576$		0			0
$577$	−12.8093			−0.533258			−0.266629	−	0.963799i	$-0.585910\pi$
$577$	−12.8093			−0.533258			−0.266629	+	0.963799i	$0.585910\pi$
$578$		0			0
$579$	−17.1201	+	17.1201i	−0.711488	+	0.711488i
$580$		0			0
$581$	−5.03598	−	5.03598i	−0.208928	−	0.208928i
$582$		0			0
$583$		−	2.65052i		−	0.109773i
$584$		0			0
$585$			2.11851i			0.0875895i
$586$		0			0
$587$	−27.9293	−	27.9293i	−1.15276	−	1.15276i	−0.985996	−	0.166768i	$-0.946667\pi$
$587$	−27.9293	−	27.9293i	−1.15276	−	1.15276i	−0.166768	−	0.985996i	$-0.553333\pi$
$588$		0			0
$589$	−41.6346	+	41.6346i	−1.71552	+	1.71552i
$590$		0			0
$591$	5.94154			0.244402
$592$		0			0
$593$	−14.6589			−0.601971			−0.300985	−	0.953629i	$-0.597316\pi$
$593$	−14.6589			−0.601971			−0.300985	+	0.953629i	$0.597316\pi$
$594$		0			0
$595$	−6.95796	+	6.95796i	−0.285248	+	0.285248i
$596$		0			0
$597$	−6.15961	−	6.15961i	−0.252096	−	0.252096i
$598$		0			0
$599$		−	38.6963i		−	1.58109i	−0.612405	−	0.790545i	$-0.709798\pi$
$599$		−	38.6963i		−	1.58109i	0.612405	−	0.790545i	$-0.290202\pi$
$600$		0			0
$601$			33.2012i			1.35430i	0.735843	+	0.677152i	$0.236786\pi$
$601$			33.2012i			1.35430i	−0.735843	+	0.677152i	$0.763214\pi$
$602$		0			0
$603$	9.10323	+	9.10323i	0.370712	+	0.370712i
$604$		0			0
$605$	−14.2395	+	14.2395i	−0.578917	+	0.578917i
$606$		0			0
$607$	−11.8393			−0.480541			−0.240271	−	0.970706i	$-0.577236\pi$
$607$	−11.8393			−0.480541			−0.240271	+	0.970706i	$0.577236\pi$
$608$		0			0
$609$	3.47207			0.140695
$610$		0			0
$611$	−4.97759	+	4.97759i	−0.201372	+	0.201372i
$612$		0			0
$613$	24.5904	+	24.5904i	0.993196	+	0.993196i	0.999977	−	0.00678056i	$-0.00215834\pi$
$613$	24.5904	+	24.5904i	0.993196	+	0.993196i	−0.00678056	+	0.999977i	$0.502158\pi$
$614$		0			0
$615$			3.10141i			0.125061i
$616$		0			0
$617$			30.9399i			1.24559i	0.782383	+	0.622797i	$0.214004\pi$
$617$			30.9399i			1.24559i	−0.782383	+	0.622797i	$0.785996\pi$
$618$		0			0
$619$	−8.35827	−	8.35827i	−0.335947	−	0.335947i	0.518892	−	0.854840i	$-0.326345\pi$
$619$	−8.35827	−	8.35827i	−0.335947	−	0.335947i	−0.854840	+	0.518892i	$0.826345\pi$
$620$		0			0
$621$	−6.20556	+	6.20556i	−0.249021	+	0.249021i
$622$		0			0
$623$	3.53141			0.141483
$624$		0			0
$625$	30.7952			1.23181
$626$		0			0
$627$	−10.1399	+	10.1399i	−0.404950	+	0.404950i
$628$		0			0
$629$	30.4247	+	30.4247i	1.21311	+	1.21311i
$630$		0			0
$631$		−	32.7668i		−	1.30443i	−0.758036	−	0.652213i	$-0.773841\pi$
$631$		−	32.7668i		−	1.30443i	0.758036	−	0.652213i	$-0.226159\pi$
$632$		0			0
$633$			3.27857i			0.130311i
$634$		0			0
$635$	16.1203	+	16.1203i	0.639715	+	0.639715i
$636$		0			0
$637$	−0.573383	+	0.573383i	−0.0227183	+	0.0227183i
$638$		0			0
$639$	10.6094			0.419703
$640$		0			0
$641$	24.2376			0.957327			0.478663	−	0.877998i	$-0.341121\pi$
$641$	24.2376			0.957327			0.478663	+	0.877998i	$0.341121\pi$
$642$		0			0
$643$	−16.8282	+	16.8282i	−0.663639	+	0.663639i	−0.956236	−	0.292597i	$-0.905481\pi$
$643$	−16.8282	+	16.8282i	−0.663639	+	0.663639i	0.292597	+	0.956236i	$0.405481\pi$
$644$		0			0
$645$	−6.67951	−	6.67951i	−0.263005	−	0.263005i
$646$		0			0
$647$			8.62130i			0.338938i	0.985535	+	0.169469i	$0.0542053\pi$
$647$			8.62130i			0.338938i	−0.985535	+	0.169469i	$0.945795\pi$
$648$		0			0
$649$		−	11.5236i		−	0.452339i
$650$		0			0
$651$	5.26789	+	5.26789i	0.206465	+	0.206465i
$652$		0			0
$653$	−15.8020	+	15.8020i	−0.618380	+	0.618380i	−0.945116	−	0.326736i	$-0.894051\pi$
$653$	−15.8020	+	15.8020i	−0.618380	+	0.618380i	0.326736	+	0.945116i	$0.394051\pi$
$654$		0			0
$655$	−23.3453			−0.912175
$656$		0			0
$657$	−1.67971			−0.0655318
$658$		0			0
$659$	−19.8851	+	19.8851i	−0.774614	+	0.774614i	−0.978909	−	0.204295i	$-0.934510\pi$
$659$	−19.8851	+	19.8851i	−0.774614	+	0.774614i	0.204295	+	0.978909i	$0.434510\pi$
$660$		0			0
$661$	−9.06899	−	9.06899i	−0.352743	−	0.352743i	0.508386	−	0.861129i	$-0.330242\pi$
$661$	−9.06899	−	9.06899i	−0.352743	−	0.352743i	−0.861129	+	0.508386i	$0.830242\pi$
$662$		0			0
$663$		−	3.05412i		−	0.118612i
$664$		0			0
$665$		−	20.6484i		−	0.800712i
$666$		0			0
$667$	−21.5462	−	21.5462i	−0.834270	−	0.834270i
$668$		0			0
$669$	6.62585	−	6.62585i	0.256170	−	0.256170i
$670$		0			0
$671$	4.55728			0.175932
$672$		0			0
$673$	26.5522			1.02351			0.511756	−	0.859131i	$-0.328995\pi$
$673$	26.5522			1.02351			0.511756	+	0.859131i	$0.328995\pi$
$674$		0			0
$675$	−1.29088	+	1.29088i	−0.0496861	+	0.0496861i
$676$		0			0
$677$	−6.20763	−	6.20763i	−0.238579	−	0.238579i	0.577683	−	0.816261i	$-0.303957\pi$
$677$	−6.20763	−	6.20763i	−0.238579	−	0.238579i	−0.816261	+	0.577683i	$0.803957\pi$
$678$		0			0
$679$			7.92244i			0.304036i
$680$		0			0
$681$		−	7.88800i		−	0.302269i
$682$		0			0
$683$	−29.4938	−	29.4938i	−1.12855	−	1.12855i	−0.990414	−	0.138134i	$-0.955890\pi$
$683$	−29.4938	−	29.4938i	−1.12855	−	1.12855i	−0.138134	−	0.990414i	$-0.544110\pi$
$684$		0			0
$685$	20.3778	−	20.3778i	0.778597	−	0.778597i
$686$		0			0
$687$	−19.5839			−0.747174
$688$		0			0
$689$	−1.18456			−0.0451283
$690$		0			0
$691$	5.55775	−	5.55775i	0.211427	−	0.211427i	−0.593447	−	0.804873i	$-0.702233\pi$
$691$	5.55775	−	5.55775i	0.211427	−	0.211427i	0.804873	+	0.593447i	$0.202233\pi$
$692$		0			0
$693$	1.28297	+	1.28297i	0.0487361	+	0.0487361i
$694$		0			0
$695$		−	34.6768i		−	1.31536i
$696$		0			0
$697$		−	4.47112i		−	0.169356i
$698$		0			0
$699$	−13.0260	−	13.0260i	−0.492689	−	0.492689i
$700$		0			0
$701$	13.9287	−	13.9287i	0.526079	−	0.526079i	−0.393322	−	0.919401i	$-0.628674\pi$
$701$	13.9287	−	13.9287i	0.526079	−	0.526079i	0.919401	+	0.393322i	$0.128674\pi$
$702$		0			0
$703$	−90.2885			−3.40530
$704$		0			0
$705$	−22.6800			−0.854180
$706$		0			0
$707$	3.02324	−	3.02324i	0.113700	−	0.113700i
$708$		0			0
$709$	−16.3323	−	16.3323i	−0.613372	−	0.613372i	0.330451	−	0.943823i	$-0.392799\pi$
$709$	−16.3323	−	16.3323i	−0.613372	−	0.613372i	−0.943823	+	0.330451i	$0.892799\pi$
$710$		0			0
$711$			8.98738i			0.337053i
$712$		0			0
$713$		−	65.3805i		−	2.44852i
$714$		0			0
$715$	−2.71799	−	2.71799i	−0.101647	−	0.101647i
$716$		0			0
$717$	−10.1193	+	10.1193i	−0.377912	+	0.377912i
$718$		0			0
$719$	11.0810			0.413253			0.206627	−	0.978420i	$-0.433751\pi$
$719$	11.0810			0.413253			0.206627	+	0.978420i	$0.433751\pi$
$720$		0			0
$721$	5.82296			0.216858
$722$		0			0
$723$	−4.44938	+	4.44938i	−0.165474	+	0.165474i
$724$		0			0
$725$	−4.48204	−	4.48204i	−0.166459	−	0.166459i
$726$		0			0
$727$			6.33535i			0.234965i	0.993075	+	0.117483i	$0.0374824\pi$
$727$			6.33535i			0.234965i	−0.993075	+	0.117483i	$0.962518\pi$
$728$		0			0
$729$			1.00000i			0.0370370i
$730$		0			0
$731$	9.62945	+	9.62945i	0.356158	+	0.356158i
$732$		0			0
$733$	−32.0445	+	32.0445i	−1.18359	+	1.18359i	−0.204783	+	0.978807i	$0.565649\pi$
$733$	−32.0445	+	32.0445i	−1.18359	+	1.18359i	−0.978807	+	0.204783i	$0.934351\pi$
$734$		0			0
$735$	−2.61258			−0.0963665
$736$		0			0
$737$	−23.3584			−0.860418
$738$		0			0
$739$	25.1654	−	25.1654i	0.925724	−	0.925724i	−0.0717024	−	0.997426i	$-0.522843\pi$
$739$	25.1654	−	25.1654i	0.925724	−	0.925724i	0.997426	+	0.0717024i	$0.0228432\pi$
$740$		0			0
$741$	4.53171	+	4.53171i	0.166477	+	0.166477i
$742$		0			0
$743$			11.3500i			0.416393i	0.978087	+	0.208196i	$0.0667593\pi$
$743$			11.3500i			0.416393i	−0.978087	+	0.208196i	$0.933241\pi$
$744$		0			0
$745$			22.3275i			0.818015i
$746$		0			0
$747$	5.03598	+	5.03598i	0.184257	+	0.184257i
$748$		0			0
$749$	4.48343	−	4.48343i	0.163821	−	0.163821i
$750$		0			0
$751$	37.7729			1.37835			0.689176	−	0.724594i	$-0.257973\pi$
$751$	37.7729			1.37835			0.689176	+	0.724594i	$0.257973\pi$
$752$		0			0
$753$	23.1513			0.843681
$754$		0			0
$755$	5.10017	−	5.10017i	0.185614	−	0.185614i
$756$		0			0
$757$	−24.2029	−	24.2029i	−0.879669	−	0.879669i	0.113831	−	0.993500i	$-0.463688\pi$
$757$	−24.2029	−	24.2029i	−0.879669	−	0.879669i	−0.993500	+	0.113831i	$0.963688\pi$
$758$		0			0
$759$		−	15.9231i		−	0.577973i
$760$		0			0
$761$		−	50.5222i		−	1.83143i	−0.401831	−	0.915714i	$-0.631626\pi$
$761$		−	50.5222i		−	1.83143i	0.401831	−	0.915714i	$-0.368374\pi$
$762$		0			0
$763$	6.57337	+	6.57337i	0.237972	+	0.237972i
$764$		0			0
$765$	6.95796	−	6.95796i	0.251565	−	0.251565i
$766$		0			0
$767$	−5.15008			−0.185959
$768$		0			0
$769$	43.7545			1.57783			0.788913	−	0.614505i	$-0.210644\pi$
$769$	43.7545			1.57783			0.788913	+	0.614505i	$0.210644\pi$
$770$		0			0
$771$	−4.20355	+	4.20355i	−0.151387	+	0.151387i
$772$		0			0
$773$	35.1908	+	35.1908i	1.26572	+	1.26572i	0.948276	+	0.317448i	$0.102826\pi$
$773$	35.1908	+	35.1908i	1.26572	+	1.26572i	0.317448	+	0.948276i	$0.397174\pi$
$774$		0			0
$775$		−	13.6005i		−	0.488543i
$776$		0			0
$777$			11.4239i			0.409831i
$778$		0			0
$779$	6.63425	+	6.63425i	0.237697	+	0.237697i
$780$		0			0
$781$	−13.6116	+	13.6116i	−0.487062	+	0.487062i
$782$		0			0
$783$	−3.47207			−0.124082
$784$		0			0
$785$	47.7531			1.70438
$786$		0			0
$787$	−10.7002	+	10.7002i	−0.381423	+	0.381423i	−0.871615	−	0.490192i	$-0.836927\pi$
$787$	−10.7002	+	10.7002i	−0.381423	+	0.381423i	0.490192	+	0.871615i	$0.336927\pi$
$788$		0			0
$789$	−12.4444	−	12.4444i	−0.443034	−	0.443034i
$790$		0			0
$791$		−	12.6537i		−	0.449914i
$792$		0			0
$793$		−	2.03673i		−	0.0723264i
$794$		0			0
$795$	−2.69869	−	2.69869i	−0.0957128	−	0.0957128i
$796$		0			0
$797$	1.80566	−	1.80566i	0.0639597	−	0.0639597i	−0.674403	−	0.738363i	$-0.735599\pi$
$797$	1.80566	−	1.80566i	0.0639597	−	0.0639597i	0.738363	+	0.674403i	$0.235599\pi$
$798$		0			0
$799$	32.6965			1.15672
$800$		0			0
$801$	−3.53141			−0.124776
$802$		0			0
$803$	2.15503	−	2.15503i	0.0760492	−	0.0760492i
$804$		0			0
$805$	16.2125	+	16.2125i	0.571417	+	0.571417i
$806$		0			0
$807$		−	20.6956i		−	0.728520i
$808$		0			0
$809$			27.4525i			0.965177i	0.875847	+	0.482588i	$0.160303\pi$
$809$			27.4525i			0.965177i	−0.875847	+	0.482588i	$0.839697\pi$
$810$		0			0
$811$	−26.4118	−	26.4118i	−0.927444	−	0.927444i	0.0700961	−	0.997540i	$-0.477669\pi$
$811$	−26.4118	−	26.4118i	−0.927444	−	0.927444i	−0.997540	+	0.0700961i	$0.977669\pi$
$812$		0			0
$813$	−1.05340	+	1.05340i	−0.0369442	+	0.0369442i
$814$		0			0
$815$	26.4658			0.927057
$816$		0			0
$817$	−28.5764			−0.999760
$818$		0			0
$819$	0.573383	−	0.573383i	0.0200356	−	0.0200356i
$820$		0			0
$821$	−19.7705	−	19.7705i	−0.689997	−	0.689997i	0.272234	−	0.962231i	$-0.412237\pi$
$821$	−19.7705	−	19.7705i	−0.689997	−	0.689997i	−0.962231	+	0.272234i	$0.912237\pi$
$822$		0			0
$823$			33.0460i			1.15191i	0.817481	+	0.575955i	$0.195370\pi$
$823$			33.0460i			1.15191i	−0.817481	+	0.575955i	$0.804630\pi$
$824$		0			0
$825$		−	3.31234i		−	0.115321i
$826$		0			0
$827$	18.8609	+	18.8609i	0.655857	+	0.655857i	0.954397	−	0.298540i	$-0.0964996\pi$
$827$	18.8609	+	18.8609i	0.655857	+	0.655857i	−0.298540	+	0.954397i	$0.596500\pi$
$828$		0			0
$829$	15.6529	−	15.6529i	0.543646	−	0.543646i	−0.380950	−	0.924596i	$-0.624403\pi$
$829$	15.6529	−	15.6529i	0.543646	−	0.543646i	0.924596	+	0.380950i	$0.124403\pi$
$830$		0			0
$831$	−9.72102			−0.337218
$832$		0			0
$833$	3.76640			0.130498
$834$		0			0
$835$	−47.1636	+	47.1636i	−1.63216	+	1.63216i
$836$		0			0
$837$	−5.26789	−	5.26789i	−0.182085	−	0.182085i
$838$		0			0
$839$			43.6927i			1.50844i	0.656623	+	0.754219i	$0.271984\pi$
$839$			43.6927i			1.50844i	−0.656623	+	0.754219i	$0.728016\pi$
$840$		0			0
$841$			16.9447i			0.584301i
$842$		0			0
$843$	−1.26347	−	1.26347i	−0.0435162	−	0.0435162i
$844$		0			0
$845$	22.8012	−	22.8012i	0.784383	−	0.784383i
$846$		0			0
$847$	7.70796			0.264849
$848$		0			0
$849$	−31.9473			−1.09643
$850$		0			0
$851$	70.8919	−	70.8919i	2.43014	−	2.43014i
$852$		0			0
$853$	2.25625	+	2.25625i	0.0772526	+	0.0772526i	0.744677	−	0.667425i	$-0.232603\pi$
$853$	2.25625	+	2.25625i	0.0772526	+	0.0772526i	−0.667425	+	0.744677i	$0.732603\pi$
$854$		0			0
$855$			20.6484i			0.706162i
$856$		0			0
$857$		−	22.8421i		−	0.780271i	−0.920757	−	0.390135i	$-0.872428\pi$
$857$		−	22.8421i		−	0.780271i	0.920757	−	0.390135i	$-0.127572\pi$
$858$		0			0
$859$	−1.94160	−	1.94160i	−0.0662465	−	0.0662465i	0.673207	−	0.739454i	$-0.264916\pi$
$859$	−1.94160	−	1.94160i	−0.0662465	−	0.0662465i	−0.739454	+	0.673207i	$0.764916\pi$
$860$		0			0
$861$	0.839410	−	0.839410i	0.0286070	−	0.0286070i
$862$		0			0
$863$	7.72439			0.262941			0.131471	−	0.991320i	$-0.458030\pi$
$863$	7.72439			0.262941			0.131471	+	0.991320i	$0.458030\pi$
$864$		0			0
$865$	−34.0762			−1.15863
$866$		0			0
$867$	1.98995	−	1.98995i	0.0675822	−	0.0675822i
$868$		0			0
$869$	−11.5306	−	11.5306i	−0.391148	−	0.391148i
$870$		0			0
$871$			10.4393i			0.353722i
$872$		0			0
$873$		−	7.92244i		−	0.268134i
$874$		0			0
$875$	−5.86433	−	5.86433i	−0.198251	−	0.198251i
$876$		0			0
$877$	4.53373	−	4.53373i	0.153093	−	0.153093i	−0.626405	−	0.779498i	$-0.715474\pi$
$877$	4.53373	−	4.53373i	0.153093	−	0.153093i	0.779498	+	0.626405i	$0.215474\pi$
$878$		0			0
$879$	8.42782			0.284263
$880$		0			0
$881$	4.47594			0.150798			0.0753992	−	0.997153i	$-0.475977\pi$
$881$	4.47594			0.150798			0.0753992	+	0.997153i	$0.475977\pi$
$882$		0			0
$883$	−30.3237	+	30.3237i	−1.02047	+	1.02047i	−0.0206866	+	0.999786i	$0.506585\pi$
$883$	−30.3237	+	30.3237i	−1.02047	+	1.02047i	−0.999786	+	0.0206866i	$0.993415\pi$
$884$		0			0
$885$	−11.7330	−	11.7330i	−0.394400	−	0.394400i
$886$		0			0
$887$		−	30.3025i		−	1.01746i	−0.860926	−	0.508730i	$-0.830115\pi$
$887$		−	30.3025i		−	1.01746i	0.860926	−	0.508730i	$-0.169885\pi$
$888$		0			0
$889$		−	8.72606i		−	0.292663i
$890$		0			0
$891$	−1.28297	−	1.28297i	−0.0429812	−	0.0429812i
$892$		0			0
$893$	−48.5150	+	48.5150i	−1.62349	+	1.62349i
$894$		0			0
$895$	−12.1098			−0.404785
$896$		0			0
$897$	−7.11633			−0.237607
$898$		0			0
$899$	18.2905	−	18.2905i	0.610022	−	0.610022i
$900$		0			0
$901$	3.89054	+	3.89054i	0.129613	+	0.129613i
$902$		0			0
$903$			3.61568i			0.120322i
$904$		0			0
$905$			47.0476i			1.56392i
$906$		0			0
$907$	4.75784	+	4.75784i	0.157981	+	0.157981i	0.781672	−	0.623690i	$-0.214367\pi$
$907$	4.75784	+	4.75784i	0.157981	+	0.157981i	−0.623690	+	0.781672i	$0.714367\pi$
$908$		0			0
$909$	−3.02324	+	3.02324i	−0.100274	+	0.100274i
$910$		0			0
$911$	−11.9935			−0.397364			−0.198682	−	0.980064i	$-0.563666\pi$
$911$	−11.9935			−0.397364			−0.198682	+	0.980064i	$0.563666\pi$
$912$		0			0
$913$	−12.9221			−0.427657
$914$		0			0
$915$	4.64011	−	4.64011i	0.153397	−	0.153397i
$916$		0			0
$917$	6.31850	+	6.31850i	0.208655	+	0.208655i
$918$		0			0
$919$		−	57.8037i		−	1.90677i	−0.301759	−	0.953384i	$-0.597574\pi$
$919$		−	57.8037i		−	1.90677i	0.301759	−	0.953384i	$-0.402426\pi$
$920$		0			0
$921$			27.2749i			0.898738i
$922$		0			0
$923$	6.08327	+	6.08327i	0.200233	+	0.200233i
$924$		0			0
$925$	14.7469	−	14.7469i	0.484877	−	0.484877i
$926$		0			0
$927$	−5.82296			−0.191251
$928$		0			0
$929$	−5.01856			−0.164654			−0.0823268	−	0.996605i	$-0.526235\pi$
$929$	−5.01856			−0.164654			−0.0823268	+	0.996605i	$0.526235\pi$
$930$		0			0
$931$	−5.58859	+	5.58859i	−0.183159	+	0.183159i
$932$		0			0
$933$	−21.5583	−	21.5583i	−0.705786	−	0.705786i
$934$		0			0
$935$			17.8537i			0.583880i
$936$		0			0
$937$			14.0317i			0.458395i	0.973380	+	0.229198i	$0.0736102\pi$
$937$			14.0317i			0.458395i	−0.973380	+	0.229198i	$0.926390\pi$
$938$		0			0
$939$	17.4333	+	17.4333i	0.568913	+	0.568913i
$940$		0			0
$941$	3.87293	−	3.87293i	0.126254	−	0.126254i	−0.641156	−	0.767410i	$-0.721545\pi$
$941$	3.87293	−	3.87293i	0.126254	−	0.126254i	0.767410	+	0.641156i	$0.221545\pi$
$942$		0			0
$943$	−10.4180			−0.339258
$944$		0			0
$945$	2.61258			0.0849873
$946$		0			0
$947$	−8.14973	+	8.14973i	−0.264831	+	0.264831i	−0.827013	−	0.562183i	$-0.809962\pi$
$947$	−8.14973	+	8.14973i	−0.264831	+	0.264831i	0.562183	+	0.827013i	$0.309962\pi$
$948$		0			0
$949$	−0.963119	−	0.963119i	−0.0312642	−	0.0312642i
$950$		0			0
$951$			26.8606i			0.871016i
$952$		0			0
$953$		−	41.1170i		−	1.33191i	−0.745992	−	0.665955i	$-0.768024\pi$
$953$		−	41.1170i		−	1.33191i	0.745992	−	0.665955i	$-0.231976\pi$
$954$		0			0
$955$	−24.3531	−	24.3531i	−0.788046	−	0.788046i
$956$		0			0
$957$	4.45457	−	4.45457i	0.143996	−	0.143996i
$958$		0			0
$959$	−11.0307			−0.356200
$960$		0			0
$961$	24.5014			0.790368
$962$		0			0
$963$	−4.48343	+	4.48343i	−0.144476	+	0.144476i
$964$		0			0
$965$	−44.7277	−	44.7277i	−1.43984	−	1.43984i
$966$		0			0
$967$			31.6861i			1.01896i	0.860484	+	0.509478i	$0.170161\pi$
$967$			31.6861i			1.01896i	−0.860484	+	0.509478i	$0.829839\pi$
$968$		0			0
$969$		−	29.7676i		−	0.956274i
$970$		0			0
$971$	3.12339	+	3.12339i	0.100234	+	0.100234i	0.755446	−	0.655211i	$-0.227420\pi$
$971$	3.12339	+	3.12339i	0.100234	+	0.100234i	−0.655211	+	0.755446i	$0.727420\pi$
$972$		0			0
$973$	−9.38542	+	9.38542i	−0.300883	+	0.300883i
$974$		0			0
$975$	−1.48034			−0.0474089
$976$		0			0
$977$	−35.3476			−1.13087			−0.565435	−	0.824793i	$-0.691292\pi$
$977$	−35.3476			−1.13087			−0.565435	+	0.824793i	$0.691292\pi$
$978$		0			0
$979$	4.53070	−	4.53070i	0.144802	−	0.144802i
$980$		0			0
$981$	−6.57337	−	6.57337i	−0.209871	−	0.209871i
$982$		0			0
$983$		−	5.27253i		−	0.168168i	−0.996459	−	0.0840838i	$-0.973204\pi$
$983$		−	5.27253i		−	0.168168i	0.996459	−	0.0840838i	$-0.0267963\pi$
$984$		0			0
$985$			15.5227i			0.494596i
$986$		0			0
$987$	6.13845	+	6.13845i	0.195389	+	0.195389i
$988$		0			0
$989$	22.4373	−	22.4373i	0.713465	−	0.713465i
$990$		0			0
$991$	15.8030			0.501999			0.250999	−	0.967987i	$-0.419241\pi$
$991$	15.8030			0.501999			0.250999	+	0.967987i	$0.419241\pi$
$992$		0			0
$993$	18.7604			0.595342
$994$		0			0
$995$	16.0925	−	16.0925i	0.510166	−	0.510166i
$996$		0			0
$997$	−10.8744	−	10.8744i	−0.344397	−	0.344397i	0.513620	−	0.858018i	$-0.328304\pi$
$997$	−10.8744	−	10.8744i	−0.344397	−	0.344397i	−0.858018	+	0.513620i	$0.828304\pi$
$998$		0			0
$999$		−	11.4239i		−	0.361437i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.2.w.b.1009.2		28
4.3	odd	2		336.2.w.b.85.4	✓	28
8.3	odd	2		2688.2.w.d.2017.2		28
8.5	even	2		2688.2.w.c.2017.13		28
16.3	odd	4		336.2.w.b.253.4	yes	28
16.5	even	4		2688.2.w.c.673.13		28
16.11	odd	4		2688.2.w.d.673.2		28
16.13	even	4	inner	1344.2.w.b.337.2		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.w.b.85.4	✓	28	4.3	odd	2
336.2.w.b.253.4	yes	28	16.3	odd	4
1344.2.w.b.337.2		28	16.13	even	4	inner
1344.2.w.b.1009.2		28	1.1	even	1	trivial
2688.2.w.c.673.13		28	16.5	even	4
2688.2.w.c.2017.13		28	8.5	even	2
2688.2.w.d.673.2		28	16.11	odd	4
2688.2.w.d.2017.2		28	8.3	odd	2

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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1344.w (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 + 0.707107i) q^{3} +(-1.84737 - 1.84737i) q^{5} +1.00000i q^{7} -1.00000i q^{9} +O(q^{10})\)	\(q+(-0.707107 + 0.707107i) q^{3} +(-1.84737 - 1.84737i) q^{5} +1.00000i q^{7} -1.00000i q^{9} +(1.28297 + 1.28297i) q^{11} +(0.573383 - 0.573383i) q^{13} +2.61258 q^{15} -3.76640 q^{17} +(5.58859 - 5.58859i) q^{19} +(-0.707107 - 0.707107i) q^{21} +8.77599i q^{23} +1.82558i q^{25} +(0.707107 + 0.707107i) q^{27} +(-2.45512 + 2.45512i) q^{29} -7.44993 q^{31} -1.81440 q^{33} +(1.84737 - 1.84737i) q^{35} +(-8.07793 - 8.07793i) q^{37} +0.810886i q^{39} +1.18711i q^{41} +(-2.55667 - 2.55667i) q^{43} +(-1.84737 + 1.84737i) q^{45} -8.68108 q^{47} -1.00000 q^{49} +(2.66325 - 2.66325i) q^{51} +(-1.03296 - 1.03296i) q^{53} -4.74026i q^{55} +7.90346i q^{57} +(-4.49096 - 4.49096i) q^{59} +(1.77606 - 1.77606i) q^{61} +1.00000 q^{63} -2.11851 q^{65} +(-9.10323 + 9.10323i) q^{67} +(-6.20556 - 6.20556i) q^{69} +10.6094i q^{71} -1.67971i q^{73} +(-1.29088 - 1.29088i) q^{75} +(-1.28297 + 1.28297i) q^{77} -8.98738 q^{79} -1.00000 q^{81} +(-5.03598 + 5.03598i) q^{83} +(6.95796 + 6.95796i) q^{85} -3.47207i q^{87} -3.53141i q^{89} +(0.573383 + 0.573383i) q^{91} +(5.26789 - 5.26789i) q^{93} -20.6484 q^{95} +7.92244 q^{97} +(1.28297 - 1.28297i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q+O(q^{10}) \) 28 * q	\( 28 q + 4 q^{11} - 8 q^{15} - 8 q^{19} + 4 q^{29} + 24 q^{33} + 4 q^{37} - 20 q^{43} - 28 q^{49} + 8 q^{51} + 20 q^{53} - 40 q^{61} + 28 q^{63} + 16 q^{65} - 4 q^{67} - 16 q^{69} + 16 q^{75} - 4 q^{77} - 24 q^{79} - 28 q^{81} - 40 q^{83} + 48 q^{85} + 72 q^{97} + 4 q^{99}+O(q^{100}) \) 28 * q + 4 * q^11 - 8 * q^15 - 8 * q^19 + 4 * q^29 + 24 * q^33 + 4 * q^37 - 20 * q^43 - 28 * q^49 + 8 * q^51 + 20 * q^53 - 40 * q^61 + 28 * q^63 + 16 * q^65 - 4 * q^67 - 16 * q^69 + 16 * q^75 - 4 * q^77 - 24 * q^79 - 28 * q^81 - 40 * q^83 + 48 * q^85 + 72 * q^97 + 4 * q^99

Introduction

L-functions

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