LMFDB - Embedded newform 448.2.j.c.335.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,2,Mod(111,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 2]))

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

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

chi := DirichletCharacter("448.111");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	448.j (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:	$3.57729801055$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			335.2
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	448.335
Dual form			448.2.j.c.111.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.73205 - 1.73205i) q^{3} +(1.73205 - 1.73205i) q^{5} +(2.00000 - 1.73205i) q^{7} -3.00000i q^{9} +(-3.00000 + 3.00000i) q^{11} +(1.73205 + 1.73205i) q^{13} -6.00000i q^{15} +(-5.19615 + 5.19615i) q^{19} +(0.464102 - 6.46410i) q^{21} -4.00000 q^{23} -1.00000i q^{25} +(-1.00000 + 1.00000i) q^{29} +6.92820 q^{31} +10.3923i q^{33} +(0.464102 - 6.46410i) q^{35} +(-5.00000 - 5.00000i) q^{37} +6.00000 q^{39} -3.46410 q^{41} +(1.00000 - 1.00000i) q^{43} +(-5.19615 - 5.19615i) q^{45} +6.92820 q^{47} +(1.00000 - 6.92820i) q^{49} +(3.00000 + 3.00000i) q^{53} +10.3923i q^{55} +18.0000i q^{57} +(-8.66025 - 8.66025i) q^{59} +(-5.19615 - 5.19615i) q^{61} +(-5.19615 - 6.00000i) q^{63} +6.00000 q^{65} +(7.00000 + 7.00000i) q^{67} +(-6.92820 + 6.92820i) q^{69} -4.00000 q^{71} +10.3923 q^{73} +(-1.73205 - 1.73205i) q^{75} +(-0.803848 + 11.1962i) q^{77} -2.00000i q^{79} +9.00000 q^{81} +(1.73205 - 1.73205i) q^{83} +3.46410i q^{87} -3.46410 q^{89} +(6.46410 + 0.464102i) q^{91} +(12.0000 - 12.0000i) q^{93} +18.0000i q^{95} +(9.00000 + 9.00000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{7} - 12 q^{11} - 12 q^{21} - 16 q^{23} - 4 q^{29} - 12 q^{35} - 20 q^{37} + 24 q^{39} + 4 q^{43} + 4 q^{49} + 12 q^{53} + 24 q^{65} + 28 q^{67} - 16 q^{71} - 24 q^{77} + 36 q^{81} + 12 q^{91}+ \cdots + 36 q^{99}+O(q^{100}) $ 4 * q + 8 * q^7 - 12 * q^11 - 12 * q^21 - 16 * q^23 - 4 * q^29 - 12 * q^35 - 20 * q^37 + 24 * q^39 + 4 * q^43 + 4 * q^49 + 12 * q^53 + 24 * q^65 + 28 * q^67 - 16 * q^71 - 24 * q^77 + 36 * q^81 + 12 * q^91 + 48 * q^93 + 36 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{4}\right)$

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
$3$	1.73205	−	1.73205i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$3$	1.73205	−	1.73205i	1.00000	−	1.00000i	1.00000			$0$
$4$		0			0
$5$	1.73205	−	1.73205i	0.774597	−	0.774597i	−0.204310	−	0.978906i	$-0.565495\pi$
$5$	1.73205	−	1.73205i	0.774597	−	0.774597i	0.978906	+	0.204310i	$0.0654949\pi$
$6$		0			0
$7$	2.00000	−	1.73205i	0.755929	−	0.654654i
$7$	2.00000	−	1.73205i	0.755929	−	0.654654i
$8$		0			0
$9$		−	3.00000i		−	1.00000i
$10$		0			0
$11$	−3.00000	+	3.00000i	−0.904534	+	0.904534i	−0.995824	−	0.0912903i	$-0.970901\pi$
$11$	−3.00000	+	3.00000i	−0.904534	+	0.904534i	0.0912903	+	0.995824i	$0.470901\pi$
$12$		0			0
$13$	1.73205	+	1.73205i	0.480384	+	0.480384i	0.905254	−	0.424870i	$-0.139680\pi$
$13$	1.73205	+	1.73205i	0.480384	+	0.480384i	−0.424870	+	0.905254i	$0.639680\pi$
$14$		0			0
$15$		−	6.00000i		−	1.54919i
$16$		0			0
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$	−5.19615	+	5.19615i	−1.19208	+	1.19208i	−0.215597	+	0.976483i	$0.569170\pi$
$19$	−5.19615	+	5.19615i	−1.19208	+	1.19208i	−0.976483	+	0.215597i	$0.930830\pi$
$20$		0			0
$21$	0.464102	−	6.46410i	0.101275	−	1.41058i
$22$		0			0
$23$	−4.00000			−0.834058			−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000			−0.834058			−0.417029	+	0.908893i	$0.636929\pi$
$24$		0			0
$25$		−	1.00000i		−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.00000	+	1.00000i	−0.185695	+	0.185695i	−0.793832	−	0.608137i	$-0.791917\pi$
$29$	−1.00000	+	1.00000i	−0.185695	+	0.185695i	0.608137	+	0.793832i	$0.291917\pi$
$30$		0			0
$31$	6.92820			1.24434			0.622171	−	0.782881i	$-0.286251\pi$
$31$	6.92820			1.24434			0.622171	+	0.782881i	$0.286251\pi$
$32$		0			0
$33$			10.3923i			1.80907i
$34$		0			0
$35$	0.464102	−	6.46410i	0.0784475	−	1.09263i
$36$		0			0
$37$	−5.00000	−	5.00000i	−0.821995	−	0.821995i	0.164399	−	0.986394i	$-0.447432\pi$
$37$	−5.00000	−	5.00000i	−0.821995	−	0.821995i	−0.986394	+	0.164399i	$0.947432\pi$
$38$		0			0
$39$	6.00000			0.960769
$40$		0			0
$41$	−3.46410			−0.541002			−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410			−0.541002			−0.270501	+	0.962720i	$0.587189\pi$
$42$		0			0
$43$	1.00000	−	1.00000i	0.152499	−	0.152499i	−0.626734	−	0.779233i	$-0.715609\pi$
$43$	1.00000	−	1.00000i	0.152499	−	0.152499i	0.779233	+	0.626734i	$0.215609\pi$
$44$		0			0
$45$	−5.19615	−	5.19615i	−0.774597	−	0.774597i
$46$		0			0
$47$	6.92820			1.01058			0.505291	−	0.862949i	$-0.331385\pi$
$47$	6.92820			1.01058			0.505291	+	0.862949i	$0.331385\pi$
$48$		0			0
$49$	1.00000	−	6.92820i	0.142857	−	0.989743i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.00000	+	3.00000i	0.412082	+	0.412082i	0.882463	−	0.470381i	$-0.155884\pi$
$53$	3.00000	+	3.00000i	0.412082	+	0.412082i	−0.470381	+	0.882463i	$0.655884\pi$
$54$		0			0
$55$			10.3923i			1.40130i
$56$		0			0
$57$			18.0000i			2.38416i
$58$		0			0
$59$	−8.66025	−	8.66025i	−1.12747	−	1.12747i	−0.990587	−	0.136882i	$-0.956292\pi$
$59$	−8.66025	−	8.66025i	−1.12747	−	1.12747i	−0.136882	−	0.990587i	$-0.543708\pi$
$60$		0			0
$61$	−5.19615	−	5.19615i	−0.665299	−	0.665299i	0.291325	−	0.956624i	$-0.405904\pi$
$61$	−5.19615	−	5.19615i	−0.665299	−	0.665299i	−0.956624	+	0.291325i	$0.905904\pi$
$62$		0			0
$63$	−5.19615	−	6.00000i	−0.654654	−	0.755929i
$64$		0			0
$65$	6.00000			0.744208
$66$		0			0
$67$	7.00000	+	7.00000i	0.855186	+	0.855186i	0.990766	−	0.135580i	$-0.0432899\pi$
$67$	7.00000	+	7.00000i	0.855186	+	0.855186i	−0.135580	+	0.990766i	$0.543290\pi$
$68$		0			0
$69$	−6.92820	+	6.92820i	−0.834058	+	0.834058i
$70$		0			0
$71$	−4.00000			−0.474713			−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000			−0.474713			−0.237356	+	0.971423i	$0.576281\pi$
$72$		0			0
$73$	10.3923			1.21633			0.608164	−	0.793812i	$-0.291906\pi$
$73$	10.3923			1.21633			0.608164	+	0.793812i	$0.291906\pi$
$74$		0			0
$75$	−1.73205	−	1.73205i	−0.200000	−	0.200000i
$76$		0			0
$77$	−0.803848	+	11.1962i	−0.0916069	+	1.27592i
$78$		0			0
$79$		−	2.00000i		−	0.225018i	−0.993651	−	0.112509i	$-0.964111\pi$
$79$		−	2.00000i		−	0.225018i	0.993651	−	0.112509i	$-0.0358886\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$	1.73205	−	1.73205i	0.190117	−	0.190117i	−0.605629	−	0.795747i	$-0.707079\pi$
$83$	1.73205	−	1.73205i	0.190117	−	0.190117i	0.795747	+	0.605629i	$0.207079\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$			3.46410i			0.371391i
$88$		0			0
$89$	−3.46410			−0.367194			−0.183597	−	0.983002i	$-0.558774\pi$
$89$	−3.46410			−0.367194			−0.183597	+	0.983002i	$0.558774\pi$
$90$		0			0
$91$	6.46410	+	0.464102i	0.677622	+	0.0486511i
$92$		0			0
$93$	12.0000	−	12.0000i	1.24434	−	1.24434i
$94$		0			0
$95$			18.0000i			1.84676i
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$	9.00000	+	9.00000i	0.904534	+	0.904534i
$100$		0			0
$101$	1.73205	−	1.73205i	0.172345	−	0.172345i	−0.615664	−	0.788009i	$-0.711112\pi$
$101$	1.73205	−	1.73205i	0.172345	−	0.172345i	0.788009	+	0.615664i	$0.211112\pi$
$102$		0			0
$103$		−	10.3923i		−	1.02398i	−0.858990	−	0.511992i	$-0.828908\pi$
$103$		−	10.3923i		−	1.02398i	0.858990	−	0.511992i	$-0.171092\pi$
$104$		0			0
$105$	−10.3923	−	12.0000i	−1.01419	−	1.17108i
$106$		0			0
$107$	−7.00000	+	7.00000i	−0.676716	+	0.676716i	−0.959256	−	0.282540i	$-0.908823\pi$
$107$	−7.00000	+	7.00000i	−0.676716	+	0.676716i	0.282540	+	0.959256i	$0.408823\pi$
$108$		0			0
$109$	−13.0000	+	13.0000i	−1.24517	+	1.24517i	−0.287348	+	0.957826i	$0.592774\pi$
$109$	−13.0000	+	13.0000i	−1.24517	+	1.24517i	−0.957826	+	0.287348i	$0.907226\pi$
$110$		0			0
$111$	−17.3205			−1.64399
$112$		0			0
$113$	−10.0000			−0.940721			−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000			−0.940721			−0.470360	+	0.882474i	$0.655876\pi$
$114$		0			0
$115$	−6.92820	+	6.92820i	−0.646058	+	0.646058i
$116$		0			0
$117$	5.19615	−	5.19615i	0.480384	−	0.480384i
$118$		0			0
$119$		0			0
$120$		0			0
$121$		−	7.00000i		−	0.636364i
$122$		0			0
$123$	−6.00000	+	6.00000i	−0.541002	+	0.541002i
$124$		0			0
$125$	6.92820	+	6.92820i	0.619677	+	0.619677i
$126$		0			0
$127$			6.00000i			0.532414i	0.963916	+	0.266207i	$0.0857705\pi$
$127$			6.00000i			0.532414i	−0.963916	+	0.266207i	$0.914230\pi$
$128$		0			0
$129$		−	3.46410i		−	0.304997i
$130$		0			0
$131$	−5.19615	+	5.19615i	−0.453990	+	0.453990i	−0.896676	−	0.442687i	$-0.854025\pi$
$131$	−5.19615	+	5.19615i	−0.453990	+	0.453990i	0.442687	+	0.896676i	$0.354025\pi$
$132$		0			0
$133$	−1.39230	+	19.3923i	−0.120728	+	1.68153i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			8.00000i			0.683486i	0.939793	+	0.341743i	$0.111017\pi$
$137$			8.00000i			0.683486i	−0.939793	+	0.341743i	$0.888983\pi$
$138$		0			0
$139$	5.19615	+	5.19615i	0.440732	+	0.440732i	0.892258	−	0.451526i	$-0.149120\pi$
$139$	5.19615	+	5.19615i	0.440732	+	0.440732i	−0.451526	+	0.892258i	$0.649120\pi$
$140$		0			0
$141$	12.0000	−	12.0000i	1.01058	−	1.01058i
$142$		0			0
$143$	−10.3923			−0.869048
$144$		0			0
$145$			3.46410i			0.287678i
$146$		0			0
$147$	−10.2679	−	13.7321i	−0.846886	−	1.13260i
$148$		0			0
$149$	−5.00000	−	5.00000i	−0.409616	−	0.409616i	0.471989	−	0.881605i	$-0.343536\pi$
$149$	−5.00000	−	5.00000i	−0.409616	−	0.409616i	−0.881605	+	0.471989i	$0.843536\pi$
$150$		0			0
$151$	12.0000			0.976546			0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000			0.976546			0.488273	+	0.872691i	$0.337627\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	12.0000	−	12.0000i	0.963863	−	0.963863i
$156$		0			0
$157$	8.66025	+	8.66025i	0.691164	+	0.691164i	0.962488	−	0.271324i	$-0.0874616\pi$
$157$	8.66025	+	8.66025i	0.691164	+	0.691164i	−0.271324	+	0.962488i	$0.587462\pi$
$158$		0			0
$159$	10.3923			0.824163
$160$		0			0
$161$	−8.00000	+	6.92820i	−0.630488	+	0.546019i
$162$		0			0
$163$	−5.00000	−	5.00000i	−0.391630	−	0.391630i	0.483638	−	0.875268i	$-0.339315\pi$
$163$	−5.00000	−	5.00000i	−0.391630	−	0.391630i	−0.875268	+	0.483638i	$0.839315\pi$
$164$		0			0
$165$	18.0000	+	18.0000i	1.40130	+	1.40130i
$166$		0			0
$167$		−	10.3923i		−	0.804181i	−0.915600	−	0.402090i	$-0.868284\pi$
$167$		−	10.3923i		−	0.804181i	0.915600	−	0.402090i	$-0.131716\pi$
$168$		0			0
$169$		−	7.00000i		−	0.538462i
$170$		0			0
$171$	15.5885	+	15.5885i	1.19208	+	1.19208i
$172$		0			0
$173$	−12.1244	−	12.1244i	−0.921798	−	0.921798i	0.0753588	−	0.997156i	$-0.475990\pi$
$173$	−12.1244	−	12.1244i	−0.921798	−	0.921798i	−0.997156	+	0.0753588i	$0.975990\pi$
$174$		0			0
$175$	−1.73205	−	2.00000i	−0.130931	−	0.151186i
$176$		0			0
$177$	−30.0000			−2.25494
$178$		0			0
$179$	−17.0000	−	17.0000i	−1.27064	−	1.27064i	−0.945753	−	0.324887i	$-0.894674\pi$
$179$	−17.0000	−	17.0000i	−1.27064	−	1.27064i	−0.324887	−	0.945753i	$-0.605326\pi$
$180$		0			0
$181$	8.66025	−	8.66025i	0.643712	−	0.643712i	−0.307754	−	0.951466i	$-0.599577\pi$
$181$	8.66025	−	8.66025i	0.643712	−	0.643712i	0.951466	+	0.307754i	$0.0995775\pi$
$182$		0			0
$183$	−18.0000			−1.33060
$184$		0			0
$185$	−17.3205			−1.27343
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	10.0000i		−	0.723575i	−0.932261	−	0.361787i	$-0.882167\pi$
$191$		−	10.0000i		−	0.723575i	0.932261	−	0.361787i	$-0.117833\pi$
$192$		0			0
$193$	10.0000			0.719816			0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000			0.719816			0.359908	+	0.932988i	$0.382808\pi$
$194$		0			0
$195$	10.3923	−	10.3923i	0.744208	−	0.744208i
$196$		0			0
$197$	−1.00000	−	1.00000i	−0.0712470	−	0.0712470i	0.670585	−	0.741832i	$-0.266043\pi$
$197$	−1.00000	−	1.00000i	−0.0712470	−	0.0712470i	−0.741832	+	0.670585i	$0.766043\pi$
$198$		0			0
$199$		−	10.3923i		−	0.736691i	−0.929689	−	0.368345i	$-0.879924\pi$
$199$		−	10.3923i		−	0.736691i	0.929689	−	0.368345i	$-0.120076\pi$
$200$		0			0
$201$	24.2487			1.71037
$202$		0			0
$203$	−0.267949	+	3.73205i	−0.0188063	+	0.261939i
$204$		0			0
$205$	−6.00000	+	6.00000i	−0.419058	+	0.419058i
$206$		0			0
$207$			12.0000i			0.834058i
$208$		0			0
$209$		−	31.1769i		−	2.15655i
$210$		0			0
$211$	19.0000	+	19.0000i	1.30801	+	1.30801i	0.922847	+	0.385167i	$0.125856\pi$
$211$	19.0000	+	19.0000i	1.30801	+	1.30801i	0.385167	+	0.922847i	$0.374144\pi$
$212$		0			0
$213$	−6.92820	+	6.92820i	−0.474713	+	0.474713i
$214$		0			0
$215$		−	3.46410i		−	0.236250i
$216$		0			0
$217$	13.8564	−	12.0000i	0.940634	−	0.814613i
$218$		0			0
$219$	18.0000	−	18.0000i	1.21633	−	1.21633i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	13.8564			0.927894			0.463947	−	0.885863i	$-0.346433\pi$
$223$	13.8564			0.927894			0.463947	+	0.885863i	$0.346433\pi$
$224$		0			0
$225$	−3.00000			−0.200000
$226$		0			0
$227$	8.66025	−	8.66025i	0.574801	−	0.574801i	−0.358665	−	0.933466i	$-0.616768\pi$
$227$	8.66025	−	8.66025i	0.574801	−	0.574801i	0.933466	+	0.358665i	$0.116768\pi$
$228$		0			0
$229$	−5.19615	+	5.19615i	−0.343371	+	0.343371i	−0.857633	−	0.514262i	$-0.828066\pi$
$229$	−5.19615	+	5.19615i	−0.343371	+	0.343371i	0.514262	+	0.857633i	$0.328066\pi$
$230$		0			0
$231$	18.0000	+	20.7846i	1.18431	+	1.36753i
$232$		0			0
$233$		−	4.00000i		−	0.262049i	−0.991379	−	0.131024i	$-0.958173\pi$
$233$		−	4.00000i		−	0.262049i	0.991379	−	0.131024i	$-0.0418266\pi$
$234$		0			0
$235$	12.0000	−	12.0000i	0.782794	−	0.782794i
$236$		0			0
$237$	−3.46410	−	3.46410i	−0.225018	−	0.225018i
$238$		0			0
$239$		−	2.00000i		−	0.129369i	−0.997906	−	0.0646846i	$-0.979396\pi$
$239$		−	2.00000i		−	0.129369i	0.997906	−	0.0646846i	$-0.0206041\pi$
$240$		0			0
$241$			13.8564i			0.892570i	0.894891	+	0.446285i	$0.147253\pi$
$241$			13.8564i			0.892570i	−0.894891	+	0.446285i	$0.852747\pi$
$242$		0			0
$243$	15.5885	−	15.5885i	1.00000	−	1.00000i
$244$		0			0
$245$	−10.2679	−	13.7321i	−0.655995	−	0.877309i
$246$		0			0
$247$	−18.0000			−1.14531
$248$		0			0
$249$		−	6.00000i		−	0.380235i
$250$		0			0
$251$	−1.73205	−	1.73205i	−0.109326	−	0.109326i	0.650328	−	0.759654i	$-0.274632\pi$
$251$	−1.73205	−	1.73205i	−0.109326	−	0.109326i	−0.759654	+	0.650328i	$0.774632\pi$
$252$		0			0
$253$	12.0000	−	12.0000i	0.754434	−	0.754434i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	27.7128i		−	1.72868i	−0.502910	−	0.864339i	$-0.667737\pi$
$257$		−	27.7128i		−	1.72868i	0.502910	−	0.864339i	$-0.332263\pi$
$258$		0			0
$259$	−18.6603	−	1.33975i	−1.15949	−	0.0832478i
$260$		0			0
$261$	3.00000	+	3.00000i	0.185695	+	0.185695i
$262$		0			0
$263$	4.00000			0.246651			0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000			0.246651			0.123325	+	0.992366i	$0.460644\pi$
$264$		0			0
$265$	10.3923			0.638394
$266$		0			0
$267$	−6.00000	+	6.00000i	−0.367194	+	0.367194i
$268$		0			0
$269$	−5.19615	−	5.19615i	−0.316815	−	0.316815i	0.530728	−	0.847543i	$-0.321919\pi$
$269$	−5.19615	−	5.19615i	−0.316815	−	0.316815i	−0.847543	+	0.530728i	$0.821919\pi$
$270$		0			0
$271$	−13.8564			−0.841717			−0.420858	−	0.907126i	$-0.638271\pi$
$271$	−13.8564			−0.841717			−0.420858	+	0.907126i	$0.638271\pi$
$272$		0			0
$273$	12.0000	−	10.3923i	0.726273	−	0.628971i
$274$		0			0
$275$	3.00000	+	3.00000i	0.180907	+	0.180907i
$276$		0			0
$277$	−1.00000	−	1.00000i	−0.0600842	−	0.0600842i	0.676426	−	0.736510i	$-0.263528\pi$
$277$	−1.00000	−	1.00000i	−0.0600842	−	0.0600842i	−0.736510	+	0.676426i	$0.763528\pi$
$278$		0			0
$279$		−	20.7846i		−	1.24434i
$280$		0			0
$281$			28.0000i			1.67034i	0.549992	+	0.835170i	$0.314631\pi$
$281$			28.0000i			1.67034i	−0.549992	+	0.835170i	$0.685369\pi$
$282$		0			0
$283$	5.19615	+	5.19615i	0.308879	+	0.308879i	0.844475	−	0.535595i	$-0.179913\pi$
$283$	5.19615	+	5.19615i	0.308879	+	0.308879i	−0.535595	+	0.844475i	$0.679913\pi$
$284$		0			0
$285$	31.1769	+	31.1769i	1.84676	+	1.84676i
$286$		0			0
$287$	−6.92820	+	6.00000i	−0.408959	+	0.354169i
$288$		0			0
$289$	17.0000			1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.73205	−	1.73205i	0.101187	−	0.101187i	−0.654701	−	0.755888i	$-0.727205\pi$
$293$	1.73205	−	1.73205i	0.101187	−	0.101187i	0.755888	+	0.654701i	$0.227205\pi$
$294$		0			0
$295$	−30.0000			−1.74667
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6.92820	−	6.92820i	−0.400668	−	0.400668i
$300$		0			0
$301$	0.267949	−	3.73205i	0.0154443	−	0.215112i
$302$		0			0
$303$		−	6.00000i		−	0.344691i
$304$		0			0
$305$	−18.0000			−1.03068
$306$		0			0
$307$	1.73205	−	1.73205i	0.0988534	−	0.0988534i	−0.655951	−	0.754804i	$-0.727732\pi$
$307$	1.73205	−	1.73205i	0.0988534	−	0.0988534i	0.754804	+	0.655951i	$0.227732\pi$
$308$		0			0
$309$	−18.0000	−	18.0000i	−1.02398	−	1.02398i
$310$		0			0
$311$			10.3923i			0.589294i	0.955606	+	0.294647i	$0.0952020\pi$
$311$			10.3923i			0.589294i	−0.955606	+	0.294647i	$0.904798\pi$
$312$		0			0
$313$	−17.3205			−0.979013			−0.489506	−	0.872000i	$-0.662823\pi$
$313$	−17.3205			−0.979013			−0.489506	+	0.872000i	$0.662823\pi$
$314$		0			0
$315$	−19.3923	−	1.39230i	−1.09263	−	0.0784475i
$316$		0			0
$317$	3.00000	−	3.00000i	0.168497	−	0.168497i	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	3.00000	−	3.00000i	0.168497	−	0.168497i	0.786318	+	0.617822i	$0.211985\pi$
$318$		0			0
$319$		−	6.00000i		−	0.335936i
$320$		0			0
$321$			24.2487i			1.35343i
$322$		0			0
$323$		0			0
$324$		0			0
$325$	1.73205	−	1.73205i	0.0960769	−	0.0960769i
$326$		0			0
$327$			45.0333i			2.49035i
$328$		0			0
$329$	13.8564	−	12.0000i	0.763928	−	0.661581i
$330$		0			0
$331$	17.0000	−	17.0000i	0.934405	−	0.934405i	−0.0635727	−	0.997977i	$-0.520249\pi$
$331$	17.0000	−	17.0000i	0.934405	−	0.934405i	0.997977	+	0.0635727i	$0.0202495\pi$
$332$		0			0
$333$	−15.0000	+	15.0000i	−0.821995	+	0.821995i
$334$		0			0
$335$	24.2487			1.32485
$336$		0			0
$337$	−10.0000			−0.544735			−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000			−0.544735			−0.272367	+	0.962193i	$0.587807\pi$
$338$		0			0
$339$	−17.3205	+	17.3205i	−0.940721	+	0.940721i
$340$		0			0
$341$	−20.7846	+	20.7846i	−1.12555	+	1.12555i
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$		0			0
$345$			24.0000i			1.29212i
$346$		0			0
$347$	−23.0000	+	23.0000i	−1.23470	+	1.23470i	−0.272568	+	0.962136i	$0.587873\pi$
$347$	−23.0000	+	23.0000i	−1.23470	+	1.23470i	−0.962136	+	0.272568i	$0.912127\pi$
$348$		0			0
$349$	−19.0526	−	19.0526i	−1.01986	−	1.01986i	−0.999799	−	0.0200613i	$-0.993614\pi$
$349$	−19.0526	−	19.0526i	−1.01986	−	1.01986i	−0.0200613	−	0.999799i	$-0.506386\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			13.8564i			0.737502i	0.929528	+	0.368751i	$0.120215\pi$
$353$			13.8564i			0.737502i	−0.929528	+	0.368751i	$0.879785\pi$
$354$		0			0
$355$	−6.92820	+	6.92820i	−0.367711	+	0.367711i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	4.00000			0.211112			0.105556	−	0.994413i	$-0.466338\pi$
$359$	4.00000			0.211112			0.105556	+	0.994413i	$0.466338\pi$
$360$		0			0
$361$		−	35.0000i		−	1.84211i
$362$		0			0
$363$	−12.1244	−	12.1244i	−0.636364	−	0.636364i
$364$		0			0
$365$	18.0000	−	18.0000i	0.942163	−	0.942163i
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		0			0
$369$			10.3923i			0.541002i
$370$		0			0
$371$	11.1962	+	0.803848i	0.581275	+	0.0417337i
$372$		0			0
$373$	7.00000	+	7.00000i	0.362446	+	0.362446i	0.864713	−	0.502267i	$-0.167500\pi$
$373$	7.00000	+	7.00000i	0.362446	+	0.362446i	−0.502267	+	0.864713i	$0.667500\pi$
$374$		0			0
$375$	24.0000			1.23935
$376$		0			0
$377$	−3.46410			−0.178410
$378$		0			0
$379$	−11.0000	+	11.0000i	−0.565032	+	0.565032i	−0.930733	−	0.365701i	$-0.880829\pi$
$379$	−11.0000	+	11.0000i	−0.565032	+	0.565032i	0.365701	+	0.930733i	$0.380829\pi$
$380$		0			0
$381$	10.3923	+	10.3923i	0.532414	+	0.532414i
$382$		0			0
$383$	−34.6410			−1.77007			−0.885037	−	0.465521i	$-0.845867\pi$
$383$	−34.6410			−1.77007			−0.885037	+	0.465521i	$0.845867\pi$
$384$		0			0
$385$	18.0000	+	20.7846i	0.917365	+	1.05928i
$386$		0			0
$387$	−3.00000	−	3.00000i	−0.152499	−	0.152499i
$388$		0			0
$389$	3.00000	+	3.00000i	0.152106	+	0.152106i	0.779058	−	0.626952i	$-0.215698\pi$
$389$	3.00000	+	3.00000i	0.152106	+	0.152106i	−0.626952	+	0.779058i	$0.715698\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$			18.0000i			0.907980i
$394$		0			0
$395$	−3.46410	−	3.46410i	−0.174298	−	0.174298i
$396$		0			0
$397$	−19.0526	−	19.0526i	−0.956221	−	0.956221i	0.0428605	−	0.999081i	$-0.486353\pi$
$397$	−19.0526	−	19.0526i	−0.956221	−	0.956221i	−0.999081	+	0.0428605i	$0.986353\pi$
$398$		0			0
$399$	31.1769	+	36.0000i	1.56080	+	1.80225i
$400$		0			0
$401$	2.00000			0.0998752			0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000			0.0998752			0.0499376	+	0.998752i	$0.484098\pi$
$402$		0			0
$403$	12.0000	+	12.0000i	0.597763	+	0.597763i
$404$		0			0
$405$	15.5885	−	15.5885i	0.774597	−	0.774597i
$406$		0			0
$407$	30.0000			1.48704
$408$		0			0
$409$	−17.3205			−0.856444			−0.428222	−	0.903674i	$-0.640860\pi$
$409$	−17.3205			−0.856444			−0.428222	+	0.903674i	$0.640860\pi$
$410$		0			0
$411$	13.8564	+	13.8564i	0.683486	+	0.683486i
$412$		0			0
$413$	−32.3205	−	2.32051i	−1.59039	−	0.114185i
$414$		0			0
$415$		−	6.00000i		−	0.294528i
$416$		0			0
$417$	18.0000			0.881464
$418$		0			0
$419$	−25.9808	+	25.9808i	−1.26924	+	1.26924i	−0.322764	+	0.946480i	$0.604612\pi$
$419$	−25.9808	+	25.9808i	−1.26924	+	1.26924i	−0.946480	+	0.322764i	$0.895388\pi$
$420$		0			0
$421$	11.0000	+	11.0000i	0.536107	+	0.536107i	0.922383	−	0.386276i	$-0.126239\pi$
$421$	11.0000	+	11.0000i	0.536107	+	0.536107i	−0.386276	+	0.922383i	$0.626239\pi$
$422$		0			0
$423$		−	20.7846i		−	1.01058i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−19.3923	−	1.39230i	−0.938459	−	0.0673784i
$428$		0			0
$429$	−18.0000	+	18.0000i	−0.869048	+	0.869048i
$430$		0			0
$431$		−	10.0000i		−	0.481683i	−0.970564	−	0.240842i	$-0.922577\pi$
$431$		−	10.0000i		−	0.481683i	0.970564	−	0.240842i	$-0.0774234\pi$
$432$		0			0
$433$			27.7128i			1.33179i	0.746044	+	0.665896i	$0.231951\pi$
$433$			27.7128i			1.33179i	−0.746044	+	0.665896i	$0.768049\pi$
$434$		0			0
$435$	6.00000	+	6.00000i	0.287678	+	0.287678i
$436$		0			0
$437$	20.7846	−	20.7846i	0.994263	−	0.994263i
$438$		0			0
$439$		−	17.3205i		−	0.826663i	−0.910581	−	0.413331i	$-0.864365\pi$
$439$		−	17.3205i		−	0.826663i	0.910581	−	0.413331i	$-0.135635\pi$
$440$		0			0
$441$	−20.7846	−	3.00000i	−0.989743	−	0.142857i
$442$		0			0
$443$	−15.0000	+	15.0000i	−0.712672	+	0.712672i	−0.967093	−	0.254422i	$-0.918115\pi$
$443$	−15.0000	+	15.0000i	−0.712672	+	0.712672i	0.254422	+	0.967093i	$0.418115\pi$
$444$		0			0
$445$	−6.00000	+	6.00000i	−0.284427	+	0.284427i
$446$		0			0
$447$	−17.3205			−0.819232
$448$		0			0
$449$	22.0000			1.03824			0.519122	−	0.854700i	$-0.326259\pi$
$449$	22.0000			1.03824			0.519122	+	0.854700i	$0.326259\pi$
$450$		0			0
$451$	10.3923	−	10.3923i	0.489355	−	0.489355i
$452$		0			0
$453$	20.7846	−	20.7846i	0.976546	−	0.976546i
$454$		0			0
$455$	12.0000	−	10.3923i	0.562569	−	0.487199i
$456$		0			0
$457$			36.0000i			1.68401i	0.539471	+	0.842004i	$0.318624\pi$
$457$			36.0000i			1.68401i	−0.539471	+	0.842004i	$0.681376\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	8.66025	+	8.66025i	0.403348	+	0.403348i	0.879411	−	0.476063i	$-0.157937\pi$
$461$	8.66025	+	8.66025i	0.403348	+	0.403348i	−0.476063	+	0.879411i	$0.657937\pi$
$462$		0			0
$463$		−	34.0000i		−	1.58011i	−0.613033	−	0.790057i	$-0.710051\pi$
$463$		−	34.0000i		−	1.58011i	0.613033	−	0.790057i	$-0.289949\pi$
$464$		0			0
$465$		−	41.5692i		−	1.92773i
$466$		0			0
$467$	15.5885	−	15.5885i	0.721348	−	0.721348i	−0.247532	−	0.968880i	$-0.579620\pi$
$467$	15.5885	−	15.5885i	0.721348	−	0.721348i	0.968880	+	0.247532i	$0.0796195\pi$
$468$		0			0
$469$	26.1244	+	1.87564i	1.20631	+	0.0866092i
$470$		0			0
$471$	30.0000			1.38233
$472$		0			0
$473$			6.00000i			0.275880i
$474$		0			0
$475$	5.19615	+	5.19615i	0.238416	+	0.238416i
$476$		0			0
$477$	9.00000	−	9.00000i	0.412082	−	0.412082i
$478$		0			0
$479$	−6.92820			−0.316558			−0.158279	−	0.987394i	$-0.550594\pi$
$479$	−6.92820			−0.316558			−0.158279	+	0.987394i	$0.550594\pi$
$480$		0			0
$481$		−	17.3205i		−	0.789747i
$482$		0			0
$483$	−1.85641	+	25.8564i	−0.0844694	+	1.17651i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	12.0000			0.543772			0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000			0.543772			0.271886	+	0.962329i	$0.412353\pi$
$488$		0			0
$489$	−17.3205			−0.783260
$490$		0			0
$491$	9.00000	−	9.00000i	0.406164	−	0.406164i	−0.474234	−	0.880399i	$-0.657275\pi$
$491$	9.00000	−	9.00000i	0.406164	−	0.406164i	0.880399	+	0.474234i	$0.157275\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	31.1769			1.40130
$496$		0			0
$497$	−8.00000	+	6.92820i	−0.358849	+	0.310772i
$498$		0			0
$499$	11.0000	+	11.0000i	0.492428	+	0.492428i	0.909070	−	0.416643i	$-0.136793\pi$
$499$	11.0000	+	11.0000i	0.492428	+	0.492428i	−0.416643	+	0.909070i	$0.636793\pi$
$500$		0			0
$501$	−18.0000	−	18.0000i	−0.804181	−	0.804181i
$502$		0			0
$503$			38.1051i			1.69902i	0.527570	+	0.849512i	$0.323103\pi$
$503$			38.1051i			1.69902i	−0.527570	+	0.849512i	$0.676897\pi$
$504$		0			0
$505$		−	6.00000i		−	0.266996i
$506$		0			0
$507$	−12.1244	−	12.1244i	−0.538462	−	0.538462i
$508$		0			0
$509$	29.4449	+	29.4449i	1.30512	+	1.30512i	0.924893	+	0.380228i	$0.124154\pi$
$509$	29.4449	+	29.4449i	1.30512	+	1.30512i	0.380228	+	0.924893i	$0.375846\pi$
$510$		0			0
$511$	20.7846	−	18.0000i	0.919457	−	0.796273i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−18.0000	−	18.0000i	−0.793175	−	0.793175i
$516$		0			0
$517$	−20.7846	+	20.7846i	−0.914106	+	0.914106i
$518$		0			0
$519$	−42.0000			−1.84360
$520$		0			0
$521$	10.3923			0.455295			0.227648	−	0.973744i	$-0.426897\pi$
$521$	10.3923			0.455295			0.227648	+	0.973744i	$0.426897\pi$
$522$		0			0
$523$	−1.73205	−	1.73205i	−0.0757373	−	0.0757373i	0.668223	−	0.743961i	$-0.267055\pi$
$523$	−1.73205	−	1.73205i	−0.0757373	−	0.0757373i	−0.743961	+	0.668223i	$0.767055\pi$
$524$		0			0
$525$	−6.46410	−	0.464102i	−0.282117	−	0.0202551i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−7.00000			−0.304348
$530$		0			0
$531$	−25.9808	+	25.9808i	−1.12747	+	1.12747i
$532$		0			0
$533$	−6.00000	−	6.00000i	−0.259889	−	0.259889i
$534$		0			0
$535$			24.2487i			1.04836i
$536$		0			0
$537$	−58.8897			−2.54128
$538$		0			0
$539$	17.7846	+	23.7846i	0.766037	+	1.02448i
$540$		0			0
$541$	−5.00000	+	5.00000i	−0.214967	+	0.214967i	−0.806373	−	0.591407i	$-0.798573\pi$
$541$	−5.00000	+	5.00000i	−0.214967	+	0.214967i	0.591407	+	0.806373i	$0.298573\pi$
$542$		0			0
$543$		−	30.0000i		−	1.28742i
$544$		0			0
$545$			45.0333i			1.92902i
$546$		0			0
$547$	−1.00000	−	1.00000i	−0.0427569	−	0.0427569i	0.685405	−	0.728162i	$-0.259625\pi$
$547$	−1.00000	−	1.00000i	−0.0427569	−	0.0427569i	−0.728162	+	0.685405i	$0.759625\pi$
$548$		0			0
$549$	−15.5885	+	15.5885i	−0.665299	+	0.665299i
$550$		0			0
$551$		−	10.3923i		−	0.442727i
$552$		0			0
$553$	−3.46410	−	4.00000i	−0.147309	−	0.170097i
$554$		0			0
$555$	−30.0000	+	30.0000i	−1.27343	+	1.27343i
$556$		0			0
$557$	7.00000	−	7.00000i	0.296600	−	0.296600i	−0.543081	−	0.839680i	$-0.682742\pi$
$557$	7.00000	−	7.00000i	0.296600	−	0.296600i	0.839680	+	0.543081i	$0.182742\pi$
$558$		0			0
$559$	3.46410			0.146516
$560$		0			0
$561$		0			0
$562$		0			0
$563$	22.5167	−	22.5167i	0.948964	−	0.948964i	−0.0497953	−	0.998759i	$-0.515857\pi$
$563$	22.5167	−	22.5167i	0.948964	−	0.948964i	0.998759	+	0.0497953i	$0.0158569\pi$
$564$		0			0
$565$	−17.3205	+	17.3205i	−0.728679	+	0.728679i
$566$		0			0
$567$	18.0000	−	15.5885i	0.755929	−	0.654654i
$568$		0			0
$569$			4.00000i			0.167689i	0.996479	+	0.0838444i	$0.0267199\pi$
$569$			4.00000i			0.167689i	−0.996479	+	0.0838444i	$0.973280\pi$
$570$		0			0
$571$	5.00000	−	5.00000i	0.209243	−	0.209243i	−0.594702	−	0.803946i	$-0.702730\pi$
$571$	5.00000	−	5.00000i	0.209243	−	0.209243i	0.803946	+	0.594702i	$0.202730\pi$
$572$		0			0
$573$	−17.3205	−	17.3205i	−0.723575	−	0.723575i
$574$		0			0
$575$			4.00000i			0.166812i
$576$		0			0
$577$		−	13.8564i		−	0.576850i	−0.957503	−	0.288425i	$-0.906868\pi$
$577$		−	13.8564i		−	0.576850i	0.957503	−	0.288425i	$-0.0931316\pi$
$578$		0			0
$579$	17.3205	−	17.3205i	0.719816	−	0.719816i
$580$		0			0
$581$	0.464102	−	6.46410i	0.0192542	−	0.268176i
$582$		0			0
$583$	−18.0000			−0.745484
$584$		0			0
$585$		−	18.0000i		−	0.744208i
$586$		0			0
$587$	−1.73205	−	1.73205i	−0.0714894	−	0.0714894i	0.670458	−	0.741947i	$-0.266098\pi$
$587$	−1.73205	−	1.73205i	−0.0714894	−	0.0714894i	−0.741947	+	0.670458i	$0.766098\pi$
$588$		0			0
$589$	−36.0000	+	36.0000i	−1.48335	+	1.48335i
$590$		0			0
$591$	−3.46410			−0.142494
$592$		0			0
$593$			13.8564i			0.569014i	0.958674	+	0.284507i	$0.0918300\pi$
$593$			13.8564i			0.569014i	−0.958674	+	0.284507i	$0.908170\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−18.0000	−	18.0000i	−0.736691	−	0.736691i
$598$		0			0
$599$	−44.0000			−1.79779			−0.898896	−	0.438163i	$-0.855629\pi$
$599$	−44.0000			−1.79779			−0.898896	+	0.438163i	$0.855629\pi$
$600$		0			0
$601$	38.1051			1.55434			0.777170	−	0.629291i	$-0.216654\pi$
$601$	38.1051			1.55434			0.777170	+	0.629291i	$0.216654\pi$
$602$		0			0
$603$	21.0000	−	21.0000i	0.855186	−	0.855186i
$604$		0			0
$605$	−12.1244	−	12.1244i	−0.492925	−	0.492925i
$606$		0			0
$607$	−27.7128			−1.12483			−0.562414	−	0.826856i	$-0.690127\pi$
$607$	−27.7128			−1.12483			−0.562414	+	0.826856i	$0.690127\pi$
$608$		0			0
$609$	6.00000	+	6.92820i	0.243132	+	0.280745i
$610$		0			0
$611$	12.0000	+	12.0000i	0.485468	+	0.485468i
$612$		0			0
$613$	−29.0000	−	29.0000i	−1.17130	−	1.17130i	−0.981900	−	0.189399i	$-0.939346\pi$
$613$	−29.0000	−	29.0000i	−1.17130	−	1.17130i	−0.189399	−	0.981900i	$-0.560654\pi$
$614$		0			0
$615$			20.7846i			0.838116i
$616$		0			0
$617$		−	16.0000i		−	0.644136i	−0.946717	−	0.322068i	$-0.895622\pi$
$617$		−	16.0000i		−	0.644136i	0.946717	−	0.322068i	$-0.104378\pi$
$618$		0			0
$619$	−22.5167	−	22.5167i	−0.905021	−	0.905021i	0.0908441	−	0.995865i	$-0.471043\pi$
$619$	−22.5167	−	22.5167i	−0.905021	−	0.905021i	−0.995865	+	0.0908441i	$0.971043\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−6.92820	+	6.00000i	−0.277573	+	0.240385i
$624$		0			0
$625$	29.0000			1.16000
$626$		0			0
$627$	−54.0000	−	54.0000i	−2.15655	−	2.15655i
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−12.0000			−0.477712			−0.238856	−	0.971055i	$-0.576772\pi$
$631$	−12.0000			−0.477712			−0.238856	+	0.971055i	$0.576772\pi$
$632$		0			0
$633$	65.8179			2.61603
$634$		0			0
$635$	10.3923	+	10.3923i	0.412406	+	0.412406i
$636$		0			0
$637$	13.7321	−	10.2679i	0.544084	−	0.406831i
$638$		0			0
$639$			12.0000i			0.474713i
$640$		0			0
$641$	10.0000			0.394976			0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000			0.394976			0.197488	+	0.980305i	$0.436722\pi$
$642$		0			0
$643$	1.73205	−	1.73205i	0.0683054	−	0.0683054i	−0.672129	−	0.740434i	$-0.734620\pi$
$643$	1.73205	−	1.73205i	0.0683054	−	0.0683054i	0.740434	+	0.672129i	$0.234620\pi$
$644$		0			0
$645$	−6.00000	−	6.00000i	−0.236250	−	0.236250i
$646$		0			0
$647$			17.3205i			0.680939i	0.940255	+	0.340470i	$0.110586\pi$
$647$			17.3205i			0.680939i	−0.940255	+	0.340470i	$0.889414\pi$
$648$		0			0
$649$	51.9615			2.03967
$650$		0			0
$651$	3.21539	−	44.7846i	0.126021	−	1.75525i
$652$		0			0
$653$	31.0000	−	31.0000i	1.21312	−	1.21312i	0.243130	−	0.969994i	$-0.421826\pi$
$653$	31.0000	−	31.0000i	1.21312	−	1.21312i	0.969994	−	0.243130i	$-0.0781742\pi$
$654$		0			0
$655$			18.0000i			0.703318i
$656$		0			0
$657$		−	31.1769i		−	1.21633i
$658$		0			0
$659$	23.0000	+	23.0000i	0.895953	+	0.895953i	0.995075	−	0.0991224i	$-0.0316036\pi$
$659$	23.0000	+	23.0000i	0.895953	+	0.895953i	−0.0991224	+	0.995075i	$0.531604\pi$
$660$		0			0
$661$	8.66025	−	8.66025i	0.336845	−	0.336845i	−0.518334	−	0.855178i	$-0.673447\pi$
$661$	8.66025	−	8.66025i	0.336845	−	0.336845i	0.855178	+	0.518334i	$0.173447\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	31.1769	+	36.0000i	1.20899	+	1.39602i
$666$		0			0
$667$	4.00000	−	4.00000i	0.154881	−	0.154881i
$668$		0			0
$669$	24.0000	−	24.0000i	0.927894	−	0.927894i
$670$		0			0
$671$	31.1769			1.20357
$672$		0			0
$673$	−42.0000			−1.61898			−0.809491	−	0.587133i	$-0.800257\pi$
$673$	−42.0000			−1.61898			−0.809491	+	0.587133i	$0.800257\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	22.5167	−	22.5167i	0.865386	−	0.865386i	−0.126572	−	0.991957i	$-0.540397\pi$
$677$	22.5167	−	22.5167i	0.865386	−	0.865386i	0.991957	+	0.126572i	$0.0403974\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		−	30.0000i		−	1.14960i
$682$		0			0
$683$	21.0000	−	21.0000i	0.803543	−	0.803543i	−0.180105	−	0.983647i	$-0.557644\pi$
$683$	21.0000	−	21.0000i	0.803543	−	0.803543i	0.983647	+	0.180105i	$0.0576437\pi$
$684$		0			0
$685$	13.8564	+	13.8564i	0.529426	+	0.529426i
$686$		0			0
$687$			18.0000i			0.686743i
$688$		0			0
$689$			10.3923i			0.395915i
$690$		0			0
$691$	1.73205	−	1.73205i	0.0658903	−	0.0658903i	−0.673394	−	0.739284i	$-0.735164\pi$
$691$	1.73205	−	1.73205i	0.0658903	−	0.0658903i	0.739284	+	0.673394i	$0.235164\pi$
$692$		0			0
$693$	33.5885	+	2.41154i	1.27592	+	0.0916069i
$694$		0			0
$695$	18.0000			0.682779
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−6.92820	−	6.92820i	−0.262049	−	0.262049i
$700$		0			0
$701$	27.0000	−	27.0000i	1.01978	−	1.01978i	0.0199755	−	0.999800i	$-0.493641\pi$
$701$	27.0000	−	27.0000i	1.01978	−	1.01978i	0.999800	−	0.0199755i	$-0.00635881\pi$
$702$		0			0
$703$	51.9615			1.95977
$704$		0			0
$705$		−	41.5692i		−	1.56559i
$706$		0			0
$707$	0.464102	−	6.46410i	0.0174543	−	0.243108i
$708$		0			0
$709$	19.0000	+	19.0000i	0.713560	+	0.713560i	0.967278	−	0.253718i	$-0.0816536\pi$
$709$	19.0000	+	19.0000i	0.713560	+	0.713560i	−0.253718	+	0.967278i	$0.581654\pi$
$710$		0			0
$711$	−6.00000			−0.225018
$712$		0			0
$713$	−27.7128			−1.03785
$714$		0			0
$715$	−18.0000	+	18.0000i	−0.673162	+	0.673162i
$716$		0			0
$717$	−3.46410	−	3.46410i	−0.129369	−	0.129369i
$718$		0			0
$719$	−6.92820			−0.258378			−0.129189	−	0.991620i	$-0.541237\pi$
$719$	−6.92820			−0.258378			−0.129189	+	0.991620i	$0.541237\pi$
$720$		0			0
$721$	−18.0000	−	20.7846i	−0.670355	−	0.774059i
$722$		0			0
$723$	24.0000	+	24.0000i	0.892570	+	0.892570i
$724$		0			0
$725$	1.00000	+	1.00000i	0.0371391	+	0.0371391i
$726$		0			0
$727$		−	38.1051i		−	1.41324i	−0.707593	−	0.706620i	$-0.750219\pi$
$727$		−	38.1051i		−	1.41324i	0.707593	−	0.706620i	$-0.249781\pi$
$728$		0			0
$729$		−	27.0000i		−	1.00000i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	1.73205	+	1.73205i	0.0639748	+	0.0639748i	0.738370	−	0.674396i	$-0.235596\pi$
$733$	1.73205	+	1.73205i	0.0639748	+	0.0639748i	−0.674396	+	0.738370i	$0.735596\pi$
$734$		0			0
$735$	−41.5692	−	6.00000i	−1.53330	−	0.221313i
$736$		0			0
$737$	−42.0000			−1.54709
$738$		0			0
$739$	11.0000	+	11.0000i	0.404642	+	0.404642i	0.879865	−	0.475224i	$-0.157633\pi$
$739$	11.0000	+	11.0000i	0.404642	+	0.404642i	−0.475224	+	0.879865i	$0.657633\pi$
$740$		0			0
$741$	−31.1769	+	31.1769i	−1.14531	+	1.14531i
$742$		0			0
$743$	20.0000			0.733729			0.366864	−	0.930274i	$-0.380431\pi$
$743$	20.0000			0.733729			0.366864	+	0.930274i	$0.380431\pi$
$744$		0			0
$745$	−17.3205			−0.634574
$746$		0			0
$747$	−5.19615	−	5.19615i	−0.190117	−	0.190117i
$748$		0			0
$749$	−1.87564	+	26.1244i	−0.0685346	+	0.954563i
$750$		0			0
$751$			46.0000i			1.67856i	0.543696	+	0.839282i	$0.317024\pi$
$751$			46.0000i			1.67856i	−0.543696	+	0.839282i	$0.682976\pi$
$752$		0			0
$753$	−6.00000			−0.218652
$754$		0			0
$755$	20.7846	−	20.7846i	0.756429	−	0.756429i
$756$		0			0
$757$	−13.0000	−	13.0000i	−0.472493	−	0.472493i	0.430227	−	0.902721i	$-0.358433\pi$
$757$	−13.0000	−	13.0000i	−0.472493	−	0.472493i	−0.902721	+	0.430227i	$0.858433\pi$
$758$		0			0
$759$		−	41.5692i		−	1.50887i
$760$		0			0
$761$	−3.46410			−0.125574			−0.0627868	−	0.998027i	$-0.519999\pi$
$761$	−3.46410			−0.125574			−0.0627868	+	0.998027i	$0.519999\pi$
$762$		0			0
$763$	−3.48334	+	48.5167i	−0.126105	+	1.75642i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	30.0000i		−	1.08324i
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	−48.0000	−	48.0000i	−1.72868	−	1.72868i
$772$		0			0
$773$	−12.1244	+	12.1244i	−0.436083	+	0.436083i	−0.890691	−	0.454609i	$-0.849779\pi$
$773$	−12.1244	+	12.1244i	−0.436083	+	0.436083i	0.454609	+	0.890691i	$0.349779\pi$
$774$		0			0
$775$		−	6.92820i		−	0.248868i
$776$		0			0
$777$	−34.6410	+	30.0000i	−1.24274	+	1.07624i
$778$		0			0
$779$	18.0000	−	18.0000i	0.644917	−	0.644917i
$780$		0			0
$781$	12.0000	−	12.0000i	0.429394	−	0.429394i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	30.0000			1.07075
$786$		0			0
$787$	1.73205	−	1.73205i	0.0617409	−	0.0617409i	−0.675562	−	0.737303i	$-0.736099\pi$
$787$	1.73205	−	1.73205i	0.0617409	−	0.0617409i	0.737303	+	0.675562i	$0.236099\pi$
$788$		0			0
$789$	6.92820	−	6.92820i	0.246651	−	0.246651i
$790$		0			0
$791$	−20.0000	+	17.3205i	−0.711118	+	0.615846i
$792$		0			0
$793$		−	18.0000i		−	0.639199i
$794$		0			0
$795$	18.0000	−	18.0000i	0.638394	−	0.638394i
$796$		0			0
$797$	15.5885	+	15.5885i	0.552171	+	0.552171i	0.927067	−	0.374896i	$-0.122321\pi$
$797$	15.5885	+	15.5885i	0.552171	+	0.552171i	−0.374896	+	0.927067i	$0.622321\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$			10.3923i			0.367194i
$802$		0			0
$803$	−31.1769	+	31.1769i	−1.10021	+	1.10021i
$804$		0			0
$805$	−1.85641	+	25.8564i	−0.0654297	+	0.911319i
$806$		0			0
$807$	−18.0000			−0.633630
$808$		0			0
$809$		−	20.0000i		−	0.703163i	−0.936157	−	0.351581i	$-0.885644\pi$
$809$		−	20.0000i		−	0.703163i	0.936157	−	0.351581i	$-0.114356\pi$
$810$		0			0
$811$	19.0526	+	19.0526i	0.669026	+	0.669026i	0.957491	−	0.288465i	$-0.0931448\pi$
$811$	19.0526	+	19.0526i	0.669026	+	0.669026i	−0.288465	+	0.957491i	$0.593145\pi$
$812$		0			0
$813$	−24.0000	+	24.0000i	−0.841717	+	0.841717i
$814$		0			0
$815$	−17.3205			−0.606711
$816$		0			0
$817$			10.3923i			0.363581i
$818$		0			0
$819$	1.39230	−	19.3923i	0.0486511	−	0.677622i
$820$		0			0
$821$	39.0000	+	39.0000i	1.36111	+	1.36111i	0.872506	+	0.488603i	$0.162493\pi$
$821$	39.0000	+	39.0000i	1.36111	+	1.36111i	0.488603	+	0.872506i	$0.337507\pi$
$822$		0			0
$823$	−12.0000			−0.418294			−0.209147	−	0.977884i	$-0.567069\pi$
$823$	−12.0000			−0.418294			−0.209147	+	0.977884i	$0.567069\pi$
$824$		0			0
$825$	10.3923			0.361814
$826$		0			0
$827$	29.0000	−	29.0000i	1.00843	−	1.00843i	0.00846463	−	0.999964i	$-0.497306\pi$
$827$	29.0000	−	29.0000i	1.00843	−	1.00843i	0.999964	−	0.00846463i	$-0.00269441\pi$
$828$		0			0
$829$	−12.1244	−	12.1244i	−0.421096	−	0.421096i	0.464485	−	0.885581i	$-0.346240\pi$
$829$	−12.1244	−	12.1244i	−0.421096	−	0.421096i	−0.885581	+	0.464485i	$0.846240\pi$
$830$		0			0
$831$	−3.46410			−0.120168
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−18.0000	−	18.0000i	−0.622916	−	0.622916i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	10.3923i		−	0.358782i	−0.983778	−	0.179391i	$-0.942587\pi$
$839$		−	10.3923i		−	0.358782i	0.983778	−	0.179391i	$-0.0574128\pi$
$840$		0			0
$841$			27.0000i			0.931034i
$842$		0			0
$843$	48.4974	+	48.4974i	1.67034	+	1.67034i
$844$		0			0
$845$	−12.1244	−	12.1244i	−0.417091	−	0.417091i
$846$		0			0
$847$	−12.1244	−	14.0000i	−0.416598	−	0.481046i
$848$		0			0
$849$	18.0000			0.617758
$850$		0			0
$851$	20.0000	+	20.0000i	0.685591	+	0.685591i
$852$		0			0
$853$	29.4449	−	29.4449i	1.00817	−	1.00817i	0.00820661	−	0.999966i	$-0.497388\pi$
$853$	29.4449	−	29.4449i	1.00817	−	1.00817i	0.999966	−	0.00820661i	$-0.00261227\pi$
$854$		0			0
$855$	54.0000			1.84676
$856$		0			0
$857$	−45.0333			−1.53831			−0.769154	−	0.639063i	$-0.779322\pi$
$857$	−45.0333			−1.53831			−0.769154	+	0.639063i	$0.779322\pi$
$858$		0			0
$859$	19.0526	+	19.0526i	0.650065	+	0.650065i	0.953008	−	0.302944i	$-0.0979694\pi$
$859$	19.0526	+	19.0526i	0.650065	+	0.650065i	−0.302944	+	0.953008i	$0.597969\pi$
$860$		0			0
$861$	−1.60770	+	22.3923i	−0.0547901	+	0.763128i
$862$		0			0
$863$			38.0000i			1.29354i	0.762687	+	0.646768i	$0.223880\pi$
$863$			38.0000i			1.29354i	−0.762687	+	0.646768i	$0.776120\pi$
$864$		0			0
$865$	−42.0000			−1.42804
$866$		0			0
$867$	29.4449	−	29.4449i	1.00000	−	1.00000i
$868$		0			0
$869$	6.00000	+	6.00000i	0.203536	+	0.203536i
$870$		0			0
$871$			24.2487i			0.821636i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	25.8564	+	1.85641i	0.874106	+	0.0627580i
$876$		0			0
$877$	−5.00000	+	5.00000i	−0.168838	+	0.168838i	−0.786468	−	0.617630i	$-0.788093\pi$
$877$	−5.00000	+	5.00000i	−0.168838	+	0.168838i	0.617630	+	0.786468i	$0.288093\pi$
$878$		0			0
$879$		−	6.00000i		−	0.202375i
$880$		0			0
$881$		−	55.4256i		−	1.86734i	−0.358139	−	0.933668i	$-0.616589\pi$
$881$		−	55.4256i		−	1.86734i	0.358139	−	0.933668i	$-0.383411\pi$
$882$		0			0
$883$	−13.0000	−	13.0000i	−0.437485	−	0.437485i	0.453680	−	0.891165i	$-0.350111\pi$
$883$	−13.0000	−	13.0000i	−0.437485	−	0.437485i	−0.891165	+	0.453680i	$0.850111\pi$
$884$		0			0
$885$	−51.9615	+	51.9615i	−1.74667	+	1.74667i
$886$		0			0
$887$		−	10.3923i		−	0.348939i	−0.984663	−	0.174470i	$-0.944179\pi$
$887$		−	10.3923i		−	0.348939i	0.984663	−	0.174470i	$-0.0558211\pi$
$888$		0			0
$889$	10.3923	+	12.0000i	0.348547	+	0.402467i
$890$		0			0
$891$	−27.0000	+	27.0000i	−0.904534	+	0.904534i
$892$		0			0
$893$	−36.0000	+	36.0000i	−1.20469	+	1.20469i
$894$		0			0
$895$	−58.8897			−1.96847
$896$		0			0
$897$	−24.0000			−0.801337
$898$		0			0
$899$	−6.92820	+	6.92820i	−0.231069	+	0.231069i
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−6.00000	−	6.92820i	−0.199667	−	0.230556i
$904$		0			0
$905$		−	30.0000i		−	0.997234i
$906$		0			0
$907$	5.00000	−	5.00000i	0.166022	−	0.166022i	−0.619206	−	0.785228i	$-0.712545\pi$
$907$	5.00000	−	5.00000i	0.166022	−	0.166022i	0.785228	+	0.619206i	$0.212545\pi$
$908$		0			0
$909$	−5.19615	−	5.19615i	−0.172345	−	0.172345i
$910$		0			0
$911$		−	34.0000i		−	1.12647i	−0.826297	−	0.563235i	$-0.809557\pi$
$911$		−	34.0000i		−	1.12647i	0.826297	−	0.563235i	$-0.190443\pi$
$912$		0			0
$913$			10.3923i			0.343935i
$914$		0			0
$915$	−31.1769	+	31.1769i	−1.03068	+	1.03068i
$916$		0			0
$917$	−1.39230	+	19.3923i	−0.0459780	+	0.640390i
$918$		0			0
$919$	−28.0000			−0.923635			−0.461817	−	0.886975i	$-0.652802\pi$
$919$	−28.0000			−0.923635			−0.461817	+	0.886975i	$0.652802\pi$
$920$		0			0
$921$		−	6.00000i		−	0.197707i
$922$		0			0
$923$	−6.92820	−	6.92820i	−0.228045	−	0.228045i
$924$		0			0
$925$	−5.00000	+	5.00000i	−0.164399	+	0.164399i
$926$		0			0
$927$	−31.1769			−1.02398
$928$		0			0
$929$			41.5692i			1.36384i	0.731426	+	0.681921i	$0.238855\pi$
$929$			41.5692i			1.36384i	−0.731426	+	0.681921i	$0.761145\pi$
$930$		0			0
$931$	30.8038	+	41.1962i	1.00956	+	1.35015i
$932$		0			0
$933$	18.0000	+	18.0000i	0.589294	+	0.589294i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−17.3205			−0.565836			−0.282918	−	0.959144i	$-0.591302\pi$
$937$	−17.3205			−0.565836			−0.282918	+	0.959144i	$0.591302\pi$
$938$		0			0
$939$	−30.0000	+	30.0000i	−0.979013	+	0.979013i
$940$		0			0
$941$	22.5167	+	22.5167i	0.734022	+	0.734022i	0.971414	−	0.237392i	$-0.0762925\pi$
$941$	22.5167	+	22.5167i	0.734022	+	0.734022i	−0.237392	+	0.971414i	$0.576293\pi$
$942$		0			0
$943$	13.8564			0.451227
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−5.00000	−	5.00000i	−0.162478	−	0.162478i	0.621185	−	0.783664i	$-0.286651\pi$
$947$	−5.00000	−	5.00000i	−0.162478	−	0.162478i	−0.783664	+	0.621185i	$0.786651\pi$
$948$		0			0
$949$	18.0000	+	18.0000i	0.584305	+	0.584305i
$950$		0			0
$951$		−	10.3923i		−	0.336994i
$952$		0			0
$953$		−	32.0000i		−	1.03658i	−0.855204	−	0.518291i	$-0.826568\pi$
$953$		−	32.0000i		−	1.03658i	0.855204	−	0.518291i	$-0.173432\pi$
$954$		0			0
$955$	−17.3205	−	17.3205i	−0.560478	−	0.560478i
$956$		0			0
$957$	−10.3923	−	10.3923i	−0.335936	−	0.335936i
$958$		0			0
$959$	13.8564	+	16.0000i	0.447447	+	0.516667i
$960$		0			0
$961$	17.0000			0.548387
$962$		0			0
$963$	21.0000	+	21.0000i	0.676716	+	0.676716i
$964$		0			0
$965$	17.3205	−	17.3205i	0.557567	−	0.557567i
$966$		0			0
$967$	44.0000			1.41494			0.707472	−	0.706741i	$-0.249835\pi$
$967$	44.0000			1.41494			0.707472	+	0.706741i	$0.249835\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1.73205	−	1.73205i	−0.0555842	−	0.0555842i	0.678768	−	0.734352i	$-0.262514\pi$
$971$	−1.73205	−	1.73205i	−0.0555842	−	0.0555842i	−0.734352	+	0.678768i	$0.762514\pi$
$972$		0			0
$973$	19.3923	+	1.39230i	0.621689	+	0.0446352i
$974$		0			0
$975$		−	6.00000i		−	0.192154i
$976$		0			0
$977$	−10.0000			−0.319928			−0.159964	−	0.987123i	$-0.551138\pi$
$977$	−10.0000			−0.319928			−0.159964	+	0.987123i	$0.551138\pi$
$978$		0			0
$979$	10.3923	−	10.3923i	0.332140	−	0.332140i
$980$		0			0
$981$	39.0000	+	39.0000i	1.24517	+	1.24517i
$982$		0			0
$983$			45.0333i			1.43634i	0.695868	+	0.718170i	$0.255020\pi$
$983$			45.0333i			1.43634i	−0.695868	+	0.718170i	$0.744980\pi$
$984$		0			0
$985$	−3.46410			−0.110375
$986$		0			0
$987$	3.21539	−	44.7846i	0.102347	−	1.42551i
$988$		0			0
$989$	−4.00000	+	4.00000i	−0.127193	+	0.127193i
$990$		0			0
$991$			6.00000i			0.190596i	0.995449	+	0.0952981i	$0.0303804\pi$
$991$			6.00000i			0.190596i	−0.995449	+	0.0952981i	$0.969620\pi$
$992$		0			0
$993$		−	58.8897i		−	1.86881i
$994$		0			0
$995$	−18.0000	−	18.0000i	−0.570638	−	0.570638i
$996$		0			0
$997$	22.5167	−	22.5167i	0.713110	−	0.713110i	−0.254075	−	0.967185i	$-0.581771\pi$
$997$	22.5167	−	22.5167i	0.713110	−	0.713110i	0.967185	+	0.254075i	$0.0817709\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.2.j.c.335.2		4
4.3	odd	2		112.2.j.c.27.1	✓	4
7.6	odd	2	inner	448.2.j.c.335.1		4
8.3	odd	2		896.2.j.a.671.2		4
8.5	even	2		896.2.j.f.671.1		4
16.3	odd	4	inner	448.2.j.c.111.1		4
16.5	even	4		896.2.j.a.223.1		4
16.11	odd	4		896.2.j.f.223.2		4
16.13	even	4		112.2.j.c.83.2	yes	4
28.3	even	6		784.2.w.a.411.1		4
28.11	odd	6		784.2.w.b.411.1		4
28.19	even	6		784.2.w.b.619.1		4
28.23	odd	6		784.2.w.a.619.1		4
28.27	even	2		112.2.j.c.27.2	yes	4
56.13	odd	2		896.2.j.f.671.2		4
56.27	even	2		896.2.j.a.671.1		4
112.13	odd	4		112.2.j.c.83.1	yes	4
112.27	even	4		896.2.j.f.223.1		4
112.45	odd	12		784.2.w.a.19.1		4
112.61	odd	12		784.2.w.b.227.1		4
112.69	odd	4		896.2.j.a.223.2		4
112.83	even	4	inner	448.2.j.c.111.2		4
112.93	even	12		784.2.w.a.227.1		4
112.109	even	12		784.2.w.b.19.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.j.c.27.1	✓	4	4.3	odd	2
112.2.j.c.27.2	yes	4	28.27	even	2
112.2.j.c.83.1	yes	4	112.13	odd	4
112.2.j.c.83.2	yes	4	16.13	even	4
448.2.j.c.111.1		4	16.3	odd	4	inner
448.2.j.c.111.2		4	112.83	even	4	inner
448.2.j.c.335.1		4	7.6	odd	2	inner
448.2.j.c.335.2		4	1.1	even	1	trivial
784.2.w.a.19.1		4	112.45	odd	12
784.2.w.a.227.1		4	112.93	even	12
784.2.w.a.411.1		4	28.3	even	6
784.2.w.a.619.1		4	28.23	odd	6
784.2.w.b.19.1		4	112.109	even	12
784.2.w.b.227.1		4	112.61	odd	12
784.2.w.b.411.1		4	28.11	odd	6
784.2.w.b.619.1		4	28.19	even	6
896.2.j.a.223.1		4	16.5	even	4
896.2.j.a.223.2		4	112.69	odd	4
896.2.j.a.671.1		4	56.27	even	2
896.2.j.a.671.2		4	8.3	odd	2
896.2.j.f.223.1		4	112.27	even	4
896.2.j.f.223.2		4	16.11	odd	4
896.2.j.f.671.1		4	8.5	even	2
896.2.j.f.671.2		4	56.13	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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	448.j (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.73205 - 1.73205i) q^{3} +(1.73205 - 1.73205i) q^{5} +(2.00000 - 1.73205i) q^{7} -3.00000i q^{9} +(-3.00000 + 3.00000i) q^{11} +(1.73205 + 1.73205i) q^{13} -6.00000i q^{15} +(-5.19615 + 5.19615i) q^{19} +(0.464102 - 6.46410i) q^{21} -4.00000 q^{23} -1.00000i q^{25} +(-1.00000 + 1.00000i) q^{29} +6.92820 q^{31} +10.3923i q^{33} +(0.464102 - 6.46410i) q^{35} +(-5.00000 - 5.00000i) q^{37} +6.00000 q^{39} -3.46410 q^{41} +(1.00000 - 1.00000i) q^{43} +(-5.19615 - 5.19615i) q^{45} +6.92820 q^{47} +(1.00000 - 6.92820i) q^{49} +(3.00000 + 3.00000i) q^{53} +10.3923i q^{55} +18.0000i q^{57} +(-8.66025 - 8.66025i) q^{59} +(-5.19615 - 5.19615i) q^{61} +(-5.19615 - 6.00000i) q^{63} +6.00000 q^{65} +(7.00000 + 7.00000i) q^{67} +(-6.92820 + 6.92820i) q^{69} -4.00000 q^{71} +10.3923 q^{73} +(-1.73205 - 1.73205i) q^{75} +(-0.803848 + 11.1962i) q^{77} -2.00000i q^{79} +9.00000 q^{81} +(1.73205 - 1.73205i) q^{83} +3.46410i q^{87} -3.46410 q^{89} +(6.46410 + 0.464102i) q^{91} +(12.0000 - 12.0000i) q^{93} +18.0000i q^{95} +(9.00000 + 9.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{7} - 12 q^{11} - 12 q^{21} - 16 q^{23} - 4 q^{29} - 12 q^{35} - 20 q^{37} + 24 q^{39} + 4 q^{43} + 4 q^{49} + 12 q^{53} + 24 q^{65} + 28 q^{67} - 16 q^{71} - 24 q^{77} + 36 q^{81} + 12 q^{91}+ \cdots + 36 q^{99}+O(q^{100}) \) 4 * q + 8 * q^7 - 12 * q^11 - 12 * q^21 - 16 * q^23 - 4 * q^29 - 12 * q^35 - 20 * q^37 + 24 * q^39 + 4 * q^43 + 4 * q^49 + 12 * q^53 + 24 * q^65 + 28 * q^67 - 16 * q^71 - 24 * q^77 + 36 * q^81 + 12 * q^91 + 48 * q^93 + 36 * q^99

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