LMFDB - Embedded newform 2028.2.i.a.2005.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2028,2,Mod(529,2028)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2028.529"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2028, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2028 = 2^{2} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2028.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,-1,0,-4,0,-4,0,-1,0,6,0,0,0,2,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$16.1936615299$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 156)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			2005.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2028.2005
Dual form			2028.2.i.a.529.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.500000 - 0.866025i) q^{3} -2.00000 q^{5} +(-2.00000 + 3.46410i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(3.00000 + 5.19615i) q^{11} +(1.00000 + 1.73205i) q^{15} +(-1.00000 + 1.73205i) q^{17} +4.00000 q^{21} +(-4.00000 - 6.92820i) q^{23} -1.00000 q^{25} +1.00000 q^{27} +(-1.00000 - 1.73205i) q^{29} +8.00000 q^{31} +(3.00000 - 5.19615i) q^{33} +(4.00000 - 6.92820i) q^{35} +(-4.00000 - 6.92820i) q^{37} +(-1.00000 - 1.73205i) q^{41} +(-4.00000 + 6.92820i) q^{43} +(1.00000 - 1.73205i) q^{45} -6.00000 q^{47} +(-4.50000 - 7.79423i) q^{49} +2.00000 q^{51} +6.00000 q^{53} +(-6.00000 - 10.3923i) q^{55} +(1.00000 - 1.73205i) q^{59} +(-1.00000 + 1.73205i) q^{61} +(-2.00000 - 3.46410i) q^{63} +(2.00000 + 3.46410i) q^{67} +(-4.00000 + 6.92820i) q^{69} +(3.00000 - 5.19615i) q^{71} -4.00000 q^{73} +(0.500000 + 0.866025i) q^{75} -24.0000 q^{77} +(-0.500000 - 0.866025i) q^{81} -14.0000 q^{83} +(2.00000 - 3.46410i) q^{85} +(-1.00000 + 1.73205i) q^{87} +(-3.00000 - 5.19615i) q^{89} +(-4.00000 - 6.92820i) q^{93} +(-6.00000 + 10.3923i) q^{97} -6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 4 q^{5} - 4 q^{7} - q^{9} + 6 q^{11} + 2 q^{15} - 2 q^{17} + 8 q^{21} - 8 q^{23} - 2 q^{25} + 2 q^{27} - 2 q^{29} + 16 q^{31} + 6 q^{33} + 8 q^{35} - 8 q^{37} - 2 q^{41} - 8 q^{43} + 2 q^{45}+ \cdots - 12 q^{99}+O(q^{100}) $ 2 * q - q^3 - 4 * q^5 - 4 * q^7 - q^9 + 6 * q^11 + 2 * q^15 - 2 * q^17 + 8 * q^21 - 8 * q^23 - 2 * q^25 + 2 * q^27 - 2 * q^29 + 16 * q^31 + 6 * q^33 + 8 * q^35 - 8 * q^37 - 2 * q^41 - 8 * q^43 + 2 * q^45 - 12 * q^47 - 9 * q^49 + 4 * q^51 + 12 * q^53 - 12 * q^55 + 2 * q^59 - 2 * q^61 - 4 * q^63 + 4 * q^67 - 8 * q^69 + 6 * q^71 - 8 * q^73 + q^75 - 48 * q^77 - q^81 - 28 * q^83 + 4 * q^85 - 2 * q^87 - 6 * q^89 - 8 * q^93 - 12 * q^97 - 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2028\mathbb{Z}\right)^\times$.

$n$	$677$	$1015$	$1861$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$3$	−0.500000	−	0.866025i	−0.288675	−	0.500000i
$3$	−0.500000	−	0.866025i	−0.288675	−	0.500000i
$4$		0			0
$5$	−2.00000			−0.894427			−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000			−0.894427			−0.447214	+	0.894427i	$0.647584\pi$
$6$		0			0
$7$	−2.00000	+	3.46410i	−0.755929	+	1.30931i	0.188982	+	0.981981i	$0.439481\pi$
$7$	−2.00000	+	3.46410i	−0.755929	+	1.30931i	−0.944911	+	0.327327i	$0.893852\pi$
$8$		0			0
$9$	−0.500000	+	0.866025i	−0.166667	+	0.288675i
$10$		0			0
$11$	3.00000	+	5.19615i	0.904534	+	1.56670i	0.821541	+	0.570149i	$0.193114\pi$
$11$	3.00000	+	5.19615i	0.904534	+	1.56670i	0.0829925	+	0.996550i	$0.473552\pi$
$12$		0			0
$13$		0			0
$13$		0			0
$14$		0			0
$15$	1.00000	+	1.73205i	0.258199	+	0.447214i
$16$		0			0
$17$	−1.00000	+	1.73205i	−0.242536	+	0.420084i	−0.961436	−	0.275029i	$-0.911312\pi$
$17$	−1.00000	+	1.73205i	−0.242536	+	0.420084i	0.718900	+	0.695113i	$0.244646\pi$
$18$		0			0
$19$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$19$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$20$		0			0
$21$	4.00000			0.872872
$22$		0			0
$23$	−4.00000	−	6.92820i	−0.834058	−	1.44463i	−0.894795	−	0.446476i	$-0.852679\pi$
$23$	−4.00000	−	6.92820i	−0.834058	−	1.44463i	0.0607377	−	0.998154i	$-0.480655\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	−1.00000	−	1.73205i	−0.185695	−	0.321634i	0.758115	−	0.652121i	$-0.226120\pi$
$29$	−1.00000	−	1.73205i	−0.185695	−	0.321634i	−0.943811	+	0.330487i	$0.892787\pi$
$30$		0			0
$31$	8.00000			1.43684			0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000			1.43684			0.718421	+	0.695608i	$0.244865\pi$
$32$		0			0
$33$	3.00000	−	5.19615i	0.522233	−	0.904534i
$34$		0			0
$35$	4.00000	−	6.92820i	0.676123	−	1.17108i
$36$		0			0
$37$	−4.00000	−	6.92820i	−0.657596	−	1.13899i	−0.981236	−	0.192809i	$-0.938240\pi$
$37$	−4.00000	−	6.92820i	−0.657596	−	1.13899i	0.323640	−	0.946180i	$-0.395093\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.00000	−	1.73205i	−0.156174	−	0.270501i	0.777312	−	0.629115i	$-0.216583\pi$
$41$	−1.00000	−	1.73205i	−0.156174	−	0.270501i	−0.933486	+	0.358614i	$0.883249\pi$
$42$		0			0
$43$	−4.00000	+	6.92820i	−0.609994	+	1.05654i	0.381246	+	0.924473i	$0.375495\pi$
$43$	−4.00000	+	6.92820i	−0.609994	+	1.05654i	−0.991241	+	0.132068i	$0.957838\pi$
$44$		0			0
$45$	1.00000	−	1.73205i	0.149071	−	0.258199i
$46$		0			0
$47$	−6.00000			−0.875190			−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000			−0.875190			−0.437595	+	0.899172i	$0.644170\pi$
$48$		0			0
$49$	−4.50000	−	7.79423i	−0.642857	−	1.11346i
$50$		0			0
$51$	2.00000			0.280056
$52$		0			0
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$	−6.00000	−	10.3923i	−0.809040	−	1.40130i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.00000	−	1.73205i	0.130189	−	0.225494i	−0.793560	−	0.608492i	$-0.791775\pi$
$59$	1.00000	−	1.73205i	0.130189	−	0.225494i	0.923749	+	0.382998i	$0.125108\pi$
$60$		0			0
$61$	−1.00000	+	1.73205i	−0.128037	+	0.221766i	−0.922916	−	0.385002i	$-0.874201\pi$
$61$	−1.00000	+	1.73205i	−0.128037	+	0.221766i	0.794879	+	0.606768i	$0.207534\pi$
$62$		0			0
$63$	−2.00000	−	3.46410i	−0.251976	−	0.436436i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	2.00000	+	3.46410i	0.244339	+	0.423207i	0.961946	−	0.273241i	$-0.0880957\pi$
$67$	2.00000	+	3.46410i	0.244339	+	0.423207i	−0.717607	+	0.696449i	$0.754762\pi$
$68$		0			0
$69$	−4.00000	+	6.92820i	−0.481543	+	0.834058i
$70$		0			0
$71$	3.00000	−	5.19615i	0.356034	−	0.616670i	−0.631260	−	0.775571i	$-0.717462\pi$
$71$	3.00000	−	5.19615i	0.356034	−	0.616670i	0.987294	+	0.158901i	$0.0507952\pi$
$72$		0			0
$73$	−4.00000			−0.468165			−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000			−0.468165			−0.234082	+	0.972217i	$0.575209\pi$
$74$		0			0
$75$	0.500000	+	0.866025i	0.0577350	+	0.100000i
$76$		0			0
$77$	−24.0000			−2.73505
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	−14.0000			−1.53670			−0.768350	−	0.640030i	$-0.778922\pi$
$83$	−14.0000			−1.53670			−0.768350	+	0.640030i	$0.778922\pi$
$84$		0			0
$85$	2.00000	−	3.46410i	0.216930	−	0.375735i
$86$		0			0
$87$	−1.00000	+	1.73205i	−0.107211	+	0.185695i
$88$		0			0
$89$	−3.00000	−	5.19615i	−0.317999	−	0.550791i	0.662071	−	0.749441i	$-0.269678\pi$
$89$	−3.00000	−	5.19615i	−0.317999	−	0.550791i	−0.980071	+	0.198650i	$0.936344\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−4.00000	−	6.92820i	−0.414781	−	0.718421i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−6.00000	+	10.3923i	−0.609208	+	1.05518i	0.382164	+	0.924095i	$0.375179\pi$
$97$	−6.00000	+	10.3923i	−0.609208	+	1.05518i	−0.991371	+	0.131084i	$0.958154\pi$
$98$		0			0
$99$	−6.00000			−0.603023
$100$		0			0
$101$	−5.00000	−	8.66025i	−0.497519	−	0.861727i	0.502477	−	0.864590i	$-0.332422\pi$
$101$	−5.00000	−	8.66025i	−0.497519	−	0.861727i	−0.999996	+	0.00286291i	$0.999089\pi$
$102$		0			0
$103$	4.00000			0.394132			0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000			0.394132			0.197066	+	0.980390i	$0.436859\pi$
$104$		0			0
$105$	−8.00000			−0.780720
$106$		0			0
$107$	−10.0000	−	17.3205i	−0.966736	−	1.67444i	−0.704875	−	0.709331i	$-0.748997\pi$
$107$	−10.0000	−	17.3205i	−0.966736	−	1.67444i	−0.261861	−	0.965106i	$-0.584336\pi$
$108$		0			0
$109$	−4.00000			−0.383131			−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000			−0.383131			−0.191565	+	0.981480i	$0.561356\pi$
$110$		0			0
$111$	−4.00000	+	6.92820i	−0.379663	+	0.657596i
$112$		0			0
$113$	5.00000	−	8.66025i	0.470360	−	0.814688i	−0.529065	−	0.848581i	$-0.677457\pi$
$113$	5.00000	−	8.66025i	0.470360	−	0.814688i	0.999425	+	0.0338931i	$0.0107906\pi$
$114$		0			0
$115$	8.00000	+	13.8564i	0.746004	+	1.29212i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−4.00000	−	6.92820i	−0.366679	−	0.635107i
$120$		0			0
$121$	−12.5000	+	21.6506i	−1.13636	+	1.96824i
$122$		0			0
$123$	−1.00000	+	1.73205i	−0.0901670	+	0.156174i
$124$		0			0
$125$	12.0000			1.07331
$126$		0			0
$127$	4.00000	+	6.92820i	0.354943	+	0.614779i	0.987108	−	0.160055i	$-0.0511671\pi$
$127$	4.00000	+	6.92820i	0.354943	+	0.614779i	−0.632166	+	0.774833i	$0.717834\pi$
$128$		0			0
$129$	8.00000			0.704361
$130$		0			0
$131$	4.00000			0.349482			0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000			0.349482			0.174741	+	0.984614i	$0.444091\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−2.00000			−0.172133
$136$		0			0
$137$	−5.00000	+	8.66025i	−0.427179	+	0.739895i	−0.996621	−	0.0821359i	$-0.973826\pi$
$137$	−5.00000	+	8.66025i	−0.427179	+	0.739895i	0.569442	+	0.822031i	$0.307159\pi$
$138$		0			0
$139$	4.00000	−	6.92820i	0.339276	−	0.587643i	−0.645021	−	0.764165i	$-0.723151\pi$
$139$	4.00000	−	6.92820i	0.339276	−	0.587643i	0.984297	+	0.176522i	$0.0564848\pi$
$140$		0			0
$141$	3.00000	+	5.19615i	0.252646	+	0.437595i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	2.00000	+	3.46410i	0.166091	+	0.287678i
$146$		0			0
$147$	−4.50000	+	7.79423i	−0.371154	+	0.642857i
$148$		0			0
$149$	−5.00000	+	8.66025i	−0.409616	+	0.709476i	−0.994847	−	0.101391i	$-0.967671\pi$
$149$	−5.00000	+	8.66025i	−0.409616	+	0.709476i	0.585231	+	0.810867i	$0.301004\pi$
$150$		0			0
$151$	16.0000			1.30206			0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000			1.30206			0.651031	+	0.759051i	$0.274337\pi$
$152$		0			0
$153$	−1.00000	−	1.73205i	−0.0808452	−	0.140028i
$154$		0			0
$155$	−16.0000			−1.28515
$156$		0			0
$157$	−14.0000			−1.11732			−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000			−1.11732			−0.558661	+	0.829396i	$0.688685\pi$
$158$		0			0
$159$	−3.00000	−	5.19615i	−0.237915	−	0.412082i
$160$		0			0
$161$	32.0000			2.52195
$162$		0			0
$163$	4.00000	−	6.92820i	0.313304	−	0.542659i	−0.665771	−	0.746156i	$-0.731897\pi$
$163$	4.00000	−	6.92820i	0.313304	−	0.542659i	0.979076	+	0.203497i	$0.0652307\pi$
$164$		0			0
$165$	−6.00000	+	10.3923i	−0.467099	+	0.809040i
$166$		0			0
$167$	−9.00000	−	15.5885i	−0.696441	−	1.20627i	−0.969693	−	0.244328i	$-0.921432\pi$
$167$	−9.00000	−	15.5885i	−0.696441	−	1.20627i	0.273252	−	0.961943i	$-0.411901\pi$
$168$		0			0
$169$		0			0
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.00000	−	5.19615i	0.228086	−	0.395056i	−0.729155	−	0.684349i	$-0.760087\pi$
$173$	3.00000	−	5.19615i	0.228086	−	0.395056i	0.957241	+	0.289292i	$0.0934200\pi$
$174$		0			0
$175$	2.00000	−	3.46410i	0.151186	−	0.261861i
$176$		0			0
$177$	−2.00000			−0.150329
$178$		0			0
$179$	−6.00000	−	10.3923i	−0.448461	−	0.776757i	0.549825	−	0.835280i	$-0.314694\pi$
$179$	−6.00000	−	10.3923i	−0.448461	−	0.776757i	−0.998286	+	0.0585225i	$0.981361\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$		0			0
$183$	2.00000			0.147844
$184$		0			0
$185$	8.00000	+	13.8564i	0.588172	+	1.01874i
$186$		0			0
$187$	−12.0000			−0.877527
$188$		0			0
$189$	−2.00000	+	3.46410i	−0.145479	+	0.251976i
$190$		0			0
$191$	12.0000	−	20.7846i	0.868290	−	1.50392i	0.00454614	−	0.999990i	$-0.498553\pi$
$191$	12.0000	−	20.7846i	0.868290	−	1.50392i	0.863743	−	0.503932i	$-0.168114\pi$
$192$		0			0
$193$	4.00000	+	6.92820i	0.287926	+	0.498703i	0.973315	−	0.229475i	$-0.0737008\pi$
$193$	4.00000	+	6.92820i	0.287926	+	0.498703i	−0.685388	+	0.728178i	$0.740368\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	13.0000	+	22.5167i	0.926212	+	1.60425i	0.789601	+	0.613621i	$0.210288\pi$
$197$	13.0000	+	22.5167i	0.926212	+	1.60425i	0.136611	+	0.990625i	$0.456379\pi$
$198$		0			0
$199$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$199$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$200$		0			0
$201$	2.00000	−	3.46410i	0.141069	−	0.244339i
$202$		0			0
$203$	8.00000			0.561490
$204$		0			0
$205$	2.00000	+	3.46410i	0.139686	+	0.241943i
$206$		0			0
$207$	8.00000			0.556038
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−6.00000	−	10.3923i	−0.413057	−	0.715436i	0.582165	−	0.813070i	$-0.302206\pi$
$211$	−6.00000	−	10.3923i	−0.413057	−	0.715436i	−0.995222	+	0.0976347i	$0.968872\pi$
$212$		0			0
$213$	−6.00000			−0.411113
$214$		0			0
$215$	8.00000	−	13.8564i	0.545595	−	0.944999i
$216$		0			0
$217$	−16.0000	+	27.7128i	−1.08615	+	1.88127i
$218$		0			0
$219$	2.00000	+	3.46410i	0.135147	+	0.234082i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−4.00000	−	6.92820i	−0.267860	−	0.463947i	0.700449	−	0.713702i	$-0.252983\pi$
$223$	−4.00000	−	6.92820i	−0.267860	−	0.463947i	−0.968309	+	0.249756i	$0.919650\pi$
$224$		0			0
$225$	0.500000	−	0.866025i	0.0333333	−	0.0577350i
$226$		0			0
$227$	1.00000	−	1.73205i	0.0663723	−	0.114960i	−0.830930	−	0.556378i	$-0.812191\pi$
$227$	1.00000	−	1.73205i	0.0663723	−	0.114960i	0.897302	+	0.441417i	$0.145524\pi$
$228$		0			0
$229$	−20.0000			−1.32164			−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000			−1.32164			−0.660819	+	0.750546i	$0.729791\pi$
$230$		0			0
$231$	12.0000	+	20.7846i	0.789542	+	1.36753i
$232$		0			0
$233$	−18.0000			−1.17922			−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000			−1.17922			−0.589610	+	0.807688i	$0.700718\pi$
$234$		0			0
$235$	12.0000			0.782794
$236$		0			0
$237$		0			0
$238$		0			0
$239$	10.0000			0.646846			0.323423	−	0.946254i	$-0.395166\pi$
$239$	10.0000			0.646846			0.323423	+	0.946254i	$0.395166\pi$
$240$		0			0
$241$	−10.0000	+	17.3205i	−0.644157	+	1.11571i	0.340339	+	0.940303i	$0.389458\pi$
$241$	−10.0000	+	17.3205i	−0.644157	+	1.11571i	−0.984496	+	0.175409i	$0.943875\pi$
$242$		0			0
$243$	−0.500000	+	0.866025i	−0.0320750	+	0.0555556i
$244$		0			0
$245$	9.00000	+	15.5885i	0.574989	+	0.995910i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	7.00000	+	12.1244i	0.443607	+	0.768350i
$250$		0			0
$251$	10.0000	−	17.3205i	0.631194	−	1.09326i	−0.356113	−	0.934443i	$-0.615898\pi$
$251$	10.0000	−	17.3205i	0.631194	−	1.09326i	0.987308	−	0.158818i	$-0.0507683\pi$
$252$		0			0
$253$	24.0000	−	41.5692i	1.50887	−	2.61343i
$254$		0			0
$255$	−4.00000			−0.250490
$256$		0			0
$257$	−3.00000	−	5.19615i	−0.187135	−	0.324127i	0.757159	−	0.653231i	$-0.226587\pi$
$257$	−3.00000	−	5.19615i	−0.187135	−	0.324127i	−0.944294	+	0.329104i	$0.893253\pi$
$258$		0			0
$259$	32.0000			1.98838
$260$		0			0
$261$	2.00000			0.123797
$262$		0			0
$263$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$263$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$264$		0			0
$265$	−12.0000			−0.737154
$266$		0			0
$267$	−3.00000	+	5.19615i	−0.183597	+	0.317999i
$268$		0			0
$269$	11.0000	−	19.0526i	0.670682	−	1.16166i	−0.307029	−	0.951700i	$-0.599335\pi$
$269$	11.0000	−	19.0526i	0.670682	−	1.16166i	0.977711	−	0.209955i	$-0.0673317\pi$
$270$		0			0
$271$	−10.0000	−	17.3205i	−0.607457	−	1.05215i	−0.991658	−	0.128897i	$-0.958856\pi$
$271$	−10.0000	−	17.3205i	−0.607457	−	1.05215i	0.384201	−	0.923249i	$-0.374477\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3.00000	−	5.19615i	−0.180907	−	0.313340i
$276$		0			0
$277$	11.0000	−	19.0526i	0.660926	−	1.14476i	−0.319447	−	0.947604i	$-0.603497\pi$
$277$	11.0000	−	19.0526i	0.660926	−	1.14476i	0.980373	−	0.197153i	$-0.0631696\pi$
$278$		0			0
$279$	−4.00000	+	6.92820i	−0.239474	+	0.414781i
$280$		0			0
$281$	−22.0000			−1.31241			−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000			−1.31241			−0.656205	+	0.754583i	$0.727839\pi$
$282$		0			0
$283$	2.00000	+	3.46410i	0.118888	+	0.205919i	0.919327	−	0.393494i	$-0.128734\pi$
$283$	2.00000	+	3.46410i	0.118888	+	0.205919i	−0.800439	+	0.599414i	$0.795400\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	8.00000			0.472225
$288$		0			0
$289$	6.50000	+	11.2583i	0.382353	+	0.662255i
$290$		0			0
$291$	12.0000			0.703452
$292$		0			0
$293$	7.00000	−	12.1244i	0.408944	−	0.708312i	−0.585827	−	0.810436i	$-0.699230\pi$
$293$	7.00000	−	12.1244i	0.408944	−	0.708312i	0.994772	+	0.102123i	$0.0325637\pi$
$294$		0			0
$295$	−2.00000	+	3.46410i	−0.116445	+	0.201688i
$296$		0			0
$297$	3.00000	+	5.19615i	0.174078	+	0.301511i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−16.0000	−	27.7128i	−0.922225	−	1.59734i
$302$		0			0
$303$	−5.00000	+	8.66025i	−0.287242	+	0.497519i
$304$		0			0
$305$	2.00000	−	3.46410i	0.114520	−	0.198354i
$306$		0			0
$307$	−20.0000			−1.14146			−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000			−1.14146			−0.570730	+	0.821138i	$0.693340\pi$
$308$		0			0
$309$	−2.00000	−	3.46410i	−0.113776	−	0.197066i
$310$		0			0
$311$	16.0000			0.907277			0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000			0.907277			0.453638	+	0.891186i	$0.350126\pi$
$312$		0			0
$313$	−14.0000			−0.791327			−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000			−0.791327			−0.395663	+	0.918396i	$0.629485\pi$
$314$		0			0
$315$	4.00000	+	6.92820i	0.225374	+	0.390360i
$316$		0			0
$317$	−18.0000			−1.01098			−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000			−1.01098			−0.505490	+	0.862832i	$0.668688\pi$
$318$		0			0
$319$	6.00000	−	10.3923i	0.335936	−	0.581857i
$320$		0			0
$321$	−10.0000	+	17.3205i	−0.558146	+	0.966736i
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$	2.00000	+	3.46410i	0.110600	+	0.191565i
$328$		0			0
$329$	12.0000	−	20.7846i	0.661581	−	1.14589i
$330$		0			0
$331$	−14.0000	+	24.2487i	−0.769510	+	1.33283i	0.168320	+	0.985732i	$0.446166\pi$
$331$	−14.0000	+	24.2487i	−0.769510	+	1.33283i	−0.937829	+	0.347097i	$0.887167\pi$
$332$		0			0
$333$	8.00000			0.438397
$334$		0			0
$335$	−4.00000	−	6.92820i	−0.218543	−	0.378528i
$336$		0			0
$337$	14.0000			0.762629			0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000			0.762629			0.381314	+	0.924445i	$0.375472\pi$
$338$		0			0
$339$	−10.0000			−0.543125
$340$		0			0
$341$	24.0000	+	41.5692i	1.29967	+	2.25110i
$342$		0			0
$343$	8.00000			0.431959
$344$		0			0
$345$	8.00000	−	13.8564i	0.430706	−	0.746004i
$346$		0			0
$347$	−6.00000	+	10.3923i	−0.322097	+	0.557888i	−0.980921	−	0.194409i	$-0.937721\pi$
$347$	−6.00000	+	10.3923i	−0.322097	+	0.557888i	0.658824	+	0.752297i	$0.271054\pi$
$348$		0			0
$349$	−4.00000	−	6.92820i	−0.214115	−	0.370858i	0.738883	−	0.673833i	$-0.235353\pi$
$349$	−4.00000	−	6.92820i	−0.214115	−	0.370858i	−0.952998	+	0.302975i	$0.902020\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−5.00000	−	8.66025i	−0.266123	−	0.460939i	0.701734	−	0.712439i	$-0.252409\pi$
$353$	−5.00000	−	8.66025i	−0.266123	−	0.460939i	−0.967857	+	0.251500i	$0.919076\pi$
$354$		0			0
$355$	−6.00000	+	10.3923i	−0.318447	+	0.551566i
$356$		0			0
$357$	−4.00000	+	6.92820i	−0.211702	+	0.366679i
$358$		0			0
$359$	10.0000			0.527780			0.263890	−	0.964553i	$-0.414994\pi$
$359$	10.0000			0.527780			0.263890	+	0.964553i	$0.414994\pi$
$360$		0			0
$361$	9.50000	+	16.4545i	0.500000	+	0.866025i
$362$		0			0
$363$	25.0000			1.31216
$364$		0			0
$365$	8.00000			0.418739
$366$		0			0
$367$	14.0000	+	24.2487i	0.730794	+	1.26577i	0.956544	+	0.291587i	$0.0941834\pi$
$367$	14.0000	+	24.2487i	0.730794	+	1.26577i	−0.225750	+	0.974185i	$0.572483\pi$
$368$		0			0
$369$	2.00000			0.104116
$370$		0			0
$371$	−12.0000	+	20.7846i	−0.623009	+	1.07908i
$372$		0			0
$373$	−3.00000	+	5.19615i	−0.155334	+	0.269047i	−0.933181	−	0.359408i	$-0.882979\pi$
$373$	−3.00000	+	5.19615i	−0.155334	+	0.269047i	0.777847	+	0.628454i	$0.216312\pi$
$374$		0			0
$375$	−6.00000	−	10.3923i	−0.309839	−	0.536656i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	8.00000	+	13.8564i	0.410932	+	0.711756i	0.994992	−	0.0999550i	$-0.0318699\pi$
$379$	8.00000	+	13.8564i	0.410932	+	0.711756i	−0.584060	+	0.811711i	$0.698537\pi$
$380$		0			0
$381$	4.00000	−	6.92820i	0.204926	−	0.354943i
$382$		0			0
$383$	−11.0000	+	19.0526i	−0.562074	+	0.973540i	0.435242	+	0.900314i	$0.356663\pi$
$383$	−11.0000	+	19.0526i	−0.562074	+	0.973540i	−0.997315	+	0.0732266i	$0.976670\pi$
$384$		0			0
$385$	48.0000			2.44631
$386$		0			0
$387$	−4.00000	−	6.92820i	−0.203331	−	0.352180i
$388$		0			0
$389$	−18.0000			−0.912636			−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000			−0.912636			−0.456318	+	0.889817i	$0.650832\pi$
$390$		0			0
$391$	16.0000			0.809155
$392$		0			0
$393$	−2.00000	−	3.46410i	−0.100887	−	0.174741i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	8.00000	−	13.8564i	0.401508	−	0.695433i	−0.592400	−	0.805644i	$-0.701819\pi$
$397$	8.00000	−	13.8564i	0.401508	−	0.695433i	0.993908	+	0.110211i	$0.0351527\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3.00000	−	5.19615i	−0.149813	−	0.259483i	0.781345	−	0.624099i	$-0.214534\pi$
$401$	−3.00000	−	5.19615i	−0.149813	−	0.259483i	−0.931158	+	0.364615i	$0.881200\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	1.00000	+	1.73205i	0.0496904	+	0.0860663i
$406$		0			0
$407$	24.0000	−	41.5692i	1.18964	−	2.06051i
$408$		0			0
$409$	−10.0000	+	17.3205i	−0.494468	+	0.856444i	−0.999980	−	0.00637586i	$-0.997970\pi$
$409$	−10.0000	+	17.3205i	−0.494468	+	0.856444i	0.505511	+	0.862820i	$0.331304\pi$
$410$		0			0
$411$	10.0000			0.493264
$412$		0			0
$413$	4.00000	+	6.92820i	0.196827	+	0.340915i
$414$		0			0
$415$	28.0000			1.37447
$416$		0			0
$417$	−8.00000			−0.391762
$418$		0			0
$419$	−14.0000	−	24.2487i	−0.683945	−	1.18463i	−0.973767	−	0.227547i	$-0.926930\pi$
$419$	−14.0000	−	24.2487i	−0.683945	−	1.18463i	0.289822	−	0.957080i	$-0.406404\pi$
$420$		0			0
$421$	4.00000			0.194948			0.0974740	−	0.995238i	$-0.468924\pi$
$421$	4.00000			0.194948			0.0974740	+	0.995238i	$0.468924\pi$
$422$		0			0
$423$	3.00000	−	5.19615i	0.145865	−	0.252646i
$424$		0			0
$425$	1.00000	−	1.73205i	0.0485071	−	0.0840168i
$426$		0			0
$427$	−4.00000	−	6.92820i	−0.193574	−	0.335279i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	5.00000	+	8.66025i	0.240842	+	0.417150i	0.960954	−	0.276707i	$-0.0892433\pi$
$431$	5.00000	+	8.66025i	0.240842	+	0.417150i	−0.720113	+	0.693857i	$0.755910\pi$
$432$		0			0
$433$	11.0000	−	19.0526i	0.528626	−	0.915608i	−0.470817	−	0.882231i	$-0.656041\pi$
$433$	11.0000	−	19.0526i	0.528626	−	0.915608i	0.999443	−	0.0333764i	$-0.0106260\pi$
$434$		0			0
$435$	2.00000	−	3.46410i	0.0958927	−	0.166091i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	18.0000	+	31.1769i	0.859093	+	1.48799i	0.872795	+	0.488087i	$0.162305\pi$
$439$	18.0000	+	31.1769i	0.859093	+	1.48799i	−0.0137020	+	0.999906i	$0.504362\pi$
$440$		0			0
$441$	9.00000			0.428571
$442$		0			0
$443$	20.0000			0.950229			0.475114	−	0.879924i	$-0.342407\pi$
$443$	20.0000			0.950229			0.475114	+	0.879924i	$0.342407\pi$
$444$		0			0
$445$	6.00000	+	10.3923i	0.284427	+	0.492642i
$446$		0			0
$447$	10.0000			0.472984
$448$		0			0
$449$	−15.0000	+	25.9808i	−0.707894	+	1.22611i	0.257743	+	0.966213i	$0.417021\pi$
$449$	−15.0000	+	25.9808i	−0.707894	+	1.22611i	−0.965637	+	0.259895i	$0.916312\pi$
$450$		0			0
$451$	6.00000	−	10.3923i	0.282529	−	0.489355i
$452$		0			0
$453$	−8.00000	−	13.8564i	−0.375873	−	0.651031i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	2.00000	+	3.46410i	0.0935561	+	0.162044i	0.909005	−	0.416785i	$-0.136843\pi$
$457$	2.00000	+	3.46410i	0.0935561	+	0.162044i	−0.815449	+	0.578829i	$0.803510\pi$
$458$		0			0
$459$	−1.00000	+	1.73205i	−0.0466760	+	0.0808452i
$460$		0			0
$461$	−3.00000	+	5.19615i	−0.139724	+	0.242009i	−0.927392	−	0.374091i	$-0.877955\pi$
$461$	−3.00000	+	5.19615i	−0.139724	+	0.242009i	0.787668	+	0.616100i	$0.211288\pi$
$462$		0			0
$463$	12.0000			0.557687			0.278844	−	0.960337i	$-0.410049\pi$
$463$	12.0000			0.557687			0.278844	+	0.960337i	$0.410049\pi$
$464$		0			0
$465$	8.00000	+	13.8564i	0.370991	+	0.642575i
$466$		0			0
$467$	−12.0000			−0.555294			−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000			−0.555294			−0.277647	+	0.960683i	$0.589555\pi$
$468$		0			0
$469$	−16.0000			−0.738811
$470$		0			0
$471$	7.00000	+	12.1244i	0.322543	+	0.558661i
$472$		0			0
$473$	−48.0000			−2.20704
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−3.00000	+	5.19615i	−0.137361	+	0.237915i
$478$		0			0
$479$	9.00000	+	15.5885i	0.411220	+	0.712255i	0.995023	−	0.0996406i	$-0.0317693\pi$
$479$	9.00000	+	15.5885i	0.411220	+	0.712255i	−0.583803	+	0.811895i	$0.698436\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	−16.0000	−	27.7128i	−0.728025	−	1.26098i
$484$		0			0
$485$	12.0000	−	20.7846i	0.544892	−	0.943781i
$486$		0			0
$487$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$487$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$488$		0			0
$489$	−8.00000			−0.361773
$490$		0			0
$491$	10.0000	+	17.3205i	0.451294	+	0.781664i	0.998467	−	0.0553560i	$-0.0176294\pi$
$491$	10.0000	+	17.3205i	0.451294	+	0.781664i	−0.547173	+	0.837020i	$0.684296\pi$
$492$		0			0
$493$	4.00000			0.180151
$494$		0			0
$495$	12.0000			0.539360
$496$		0			0
$497$	12.0000	+	20.7846i	0.538274	+	0.932317i
$498$		0			0
$499$	4.00000			0.179065			0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000			0.179065			0.0895323	+	0.995984i	$0.471463\pi$
$500$		0			0
$501$	−9.00000	+	15.5885i	−0.402090	+	0.696441i
$502$		0			0
$503$	−12.0000	+	20.7846i	−0.535054	+	0.926740i	0.464107	+	0.885779i	$0.346375\pi$
$503$	−12.0000	+	20.7846i	−0.535054	+	0.926740i	−0.999161	+	0.0409609i	$0.986958\pi$
$504$		0			0
$505$	10.0000	+	17.3205i	0.444994	+	0.770752i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−15.0000	−	25.9808i	−0.664863	−	1.15158i	−0.979322	−	0.202306i	$-0.935156\pi$
$509$	−15.0000	−	25.9808i	−0.664863	−	1.15158i	0.314459	−	0.949271i	$-0.398177\pi$
$510$		0			0
$511$	8.00000	−	13.8564i	0.353899	−	0.612971i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−8.00000			−0.352522
$516$		0			0
$517$	−18.0000	−	31.1769i	−0.791639	−	1.37116i
$518$		0			0
$519$	−6.00000			−0.263371
$520$		0			0
$521$	−18.0000			−0.788594			−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000			−0.788594			−0.394297	+	0.918983i	$0.629012\pi$
$522$		0			0
$523$	4.00000	+	6.92820i	0.174908	+	0.302949i	0.940129	−	0.340818i	$-0.110704\pi$
$523$	4.00000	+	6.92820i	0.174908	+	0.302949i	−0.765222	+	0.643767i	$0.777371\pi$
$524$		0			0
$525$	−4.00000			−0.174574
$526$		0			0
$527$	−8.00000	+	13.8564i	−0.348485	+	0.603595i
$528$		0			0
$529$	−20.5000	+	35.5070i	−0.891304	+	1.54378i
$530$		0			0
$531$	1.00000	+	1.73205i	0.0433963	+	0.0751646i
$532$		0			0
$533$		0			0
$534$		0			0
$535$	20.0000	+	34.6410i	0.864675	+	1.49766i
$536$		0			0
$537$	−6.00000	+	10.3923i	−0.258919	+	0.448461i
$538$		0			0
$539$	27.0000	−	46.7654i	1.16297	−	2.01433i
$540$		0			0
$541$	−20.0000			−0.859867			−0.429934	−	0.902861i	$-0.641463\pi$
$541$	−20.0000			−0.859867			−0.429934	+	0.902861i	$0.641463\pi$
$542$		0			0
$543$	5.00000	+	8.66025i	0.214571	+	0.371647i
$544$		0			0
$545$	8.00000			0.342682
$546$		0			0
$547$	−8.00000			−0.342055			−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000			−0.342055			−0.171028	+	0.985266i	$0.554709\pi$
$548$		0			0
$549$	−1.00000	−	1.73205i	−0.0426790	−	0.0739221i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	8.00000	−	13.8564i	0.339581	−	0.588172i
$556$		0			0
$557$	15.0000	+	25.9808i	0.635570	+	1.10084i	0.986394	+	0.164399i	$0.0525683\pi$
$557$	15.0000	+	25.9808i	0.635570	+	1.10084i	−0.350824	+	0.936442i	$0.614098\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	6.00000	+	10.3923i	0.253320	+	0.438763i
$562$		0			0
$563$	−2.00000	+	3.46410i	−0.0842900	+	0.145994i	−0.905088	−	0.425223i	$-0.860196\pi$
$563$	−2.00000	+	3.46410i	−0.0842900	+	0.145994i	0.820798	+	0.571218i	$0.193529\pi$
$564$		0			0
$565$	−10.0000	+	17.3205i	−0.420703	+	0.728679i
$566$		0			0
$567$	4.00000			0.167984
$568$		0			0
$569$	17.0000	+	29.4449i	0.712677	+	1.23439i	0.963849	+	0.266450i	$0.0858508\pi$
$569$	17.0000	+	29.4449i	0.712677	+	1.23439i	−0.251172	+	0.967943i	$0.580816\pi$
$570$		0			0
$571$	−20.0000			−0.836974			−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000			−0.836974			−0.418487	+	0.908223i	$0.637439\pi$
$572$		0			0
$573$	−24.0000			−1.00261
$574$		0			0
$575$	4.00000	+	6.92820i	0.166812	+	0.288926i
$576$		0			0
$577$	40.0000			1.66522			0.832611	−	0.553858i	$-0.186845\pi$
$577$	40.0000			1.66522			0.832611	+	0.553858i	$0.186845\pi$
$578$		0			0
$579$	4.00000	−	6.92820i	0.166234	−	0.287926i
$580$		0			0
$581$	28.0000	−	48.4974i	1.16164	−	2.01201i
$582$		0			0
$583$	18.0000	+	31.1769i	0.745484	+	1.29122i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.00000	−	1.73205i	−0.0412744	−	0.0714894i	0.844650	−	0.535319i	$-0.179808\pi$
$587$	−1.00000	−	1.73205i	−0.0412744	−	0.0714894i	−0.885925	+	0.463829i	$0.846475\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	13.0000	−	22.5167i	0.534749	−	0.926212i
$592$		0			0
$593$	−34.0000			−1.39621			−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000			−1.39621			−0.698106	+	0.715994i	$0.745974\pi$
$594$		0			0
$595$	8.00000	+	13.8564i	0.327968	+	0.568057i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	7.00000	+	12.1244i	0.285536	+	0.494563i	0.972739	−	0.231903i	$-0.0744951\pi$
$601$	7.00000	+	12.1244i	0.285536	+	0.494563i	−0.687203	+	0.726465i	$0.741162\pi$
$602$		0			0
$603$	−4.00000			−0.162893
$604$		0			0
$605$	25.0000	−	43.3013i	1.01639	−	1.76045i
$606$		0			0
$607$	−2.00000	+	3.46410i	−0.0811775	+	0.140604i	−0.903756	−	0.428048i	$-0.859201\pi$
$607$	−2.00000	+	3.46410i	−0.0811775	+	0.140604i	0.822578	+	0.568652i	$0.192535\pi$
$608$		0			0
$609$	−4.00000	−	6.92820i	−0.162088	−	0.280745i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	8.00000	+	13.8564i	0.323117	+	0.559655i	0.981129	−	0.193352i	$-0.0619359\pi$
$613$	8.00000	+	13.8564i	0.323117	+	0.559655i	−0.658012	+	0.753007i	$0.728603\pi$
$614$		0			0
$615$	2.00000	−	3.46410i	0.0806478	−	0.139686i
$616$		0			0
$617$	7.00000	−	12.1244i	0.281809	−	0.488108i	−0.690021	−	0.723789i	$-0.742399\pi$
$617$	7.00000	−	12.1244i	0.281809	−	0.488108i	0.971830	+	0.235681i	$0.0757321\pi$
$618$		0			0
$619$	−20.0000			−0.803868			−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000			−0.803868			−0.401934	+	0.915669i	$0.631662\pi$
$620$		0			0
$621$	−4.00000	−	6.92820i	−0.160514	−	0.278019i
$622$		0			0
$623$	24.0000			0.961540
$624$		0			0
$625$	−19.0000			−0.760000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	16.0000			0.637962
$630$		0			0
$631$	4.00000	−	6.92820i	0.159237	−	0.275807i	−0.775356	−	0.631524i	$-0.782430\pi$
$631$	4.00000	−	6.92820i	0.159237	−	0.275807i	0.934594	+	0.355716i	$0.115763\pi$
$632$		0			0
$633$	−6.00000	+	10.3923i	−0.238479	+	0.413057i
$634$		0			0
$635$	−8.00000	−	13.8564i	−0.317470	−	0.549875i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	3.00000	+	5.19615i	0.118678	+	0.205557i
$640$		0			0
$641$	−21.0000	+	36.3731i	−0.829450	+	1.43665i	0.0690201	+	0.997615i	$0.478013\pi$
$641$	−21.0000	+	36.3731i	−0.829450	+	1.43665i	−0.898470	+	0.439034i	$0.855321\pi$
$642$		0			0
$643$	18.0000	−	31.1769i	0.709851	−	1.22950i	−0.255062	−	0.966925i	$-0.582096\pi$
$643$	18.0000	−	31.1769i	0.709851	−	1.22950i	0.964912	−	0.262573i	$-0.0845709\pi$
$644$		0			0
$645$	−16.0000			−0.629999
$646$		0			0
$647$	−16.0000	−	27.7128i	−0.629025	−	1.08950i	−0.987748	−	0.156059i	$-0.950121\pi$
$647$	−16.0000	−	27.7128i	−0.629025	−	1.08950i	0.358723	−	0.933444i	$-0.383212\pi$
$648$		0			0
$649$	12.0000			0.471041
$650$		0			0
$651$	32.0000			1.25418
$652$		0			0
$653$	7.00000	+	12.1244i	0.273931	+	0.474463i	0.969865	−	0.243643i	$-0.0783426\pi$
$653$	7.00000	+	12.1244i	0.273931	+	0.474463i	−0.695934	+	0.718106i	$0.745009\pi$
$654$		0			0
$655$	−8.00000			−0.312586
$656$		0			0
$657$	2.00000	−	3.46410i	0.0780274	−	0.135147i
$658$		0			0
$659$	−10.0000	+	17.3205i	−0.389545	+	0.674711i	−0.992388	−	0.123148i	$-0.960701\pi$
$659$	−10.0000	+	17.3205i	−0.389545	+	0.674711i	0.602844	+	0.797859i	$0.294034\pi$
$660$		0			0
$661$	−16.0000	−	27.7128i	−0.622328	−	1.07790i	−0.989051	−	0.147573i	$-0.952854\pi$
$661$	−16.0000	−	27.7128i	−0.622328	−	1.07790i	0.366723	−	0.930330i	$-0.380480\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−8.00000	+	13.8564i	−0.309761	+	0.536522i
$668$		0			0
$669$	−4.00000	+	6.92820i	−0.154649	+	0.267860i
$670$		0			0
$671$	−12.0000			−0.463255
$672$		0			0
$673$	15.0000	+	25.9808i	0.578208	+	1.00148i	0.995685	+	0.0927975i	$0.0295809\pi$
$673$	15.0000	+	25.9808i	0.578208	+	1.00148i	−0.417477	+	0.908687i	$0.637086\pi$
$674$		0			0
$675$	−1.00000			−0.0384900
$676$		0			0
$677$	−18.0000			−0.691796			−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000			−0.691796			−0.345898	+	0.938272i	$0.612426\pi$
$678$		0			0
$679$	−24.0000	−	41.5692i	−0.921035	−	1.59528i
$680$		0			0
$681$	−2.00000			−0.0766402
$682$		0			0
$683$	3.00000	−	5.19615i	0.114792	−	0.198825i	−0.802905	−	0.596107i	$-0.796713\pi$
$683$	3.00000	−	5.19615i	0.114792	−	0.198825i	0.917697	+	0.397282i	$0.130047\pi$
$684$		0			0
$685$	10.0000	−	17.3205i	0.382080	−	0.661783i
$686$		0			0
$687$	10.0000	+	17.3205i	0.381524	+	0.660819i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−2.00000	−	3.46410i	−0.0760836	−	0.131781i	0.825473	−	0.564441i	$-0.190908\pi$
$691$	−2.00000	−	3.46410i	−0.0760836	−	0.131781i	−0.901557	+	0.432660i	$0.857575\pi$
$692$		0			0
$693$	12.0000	−	20.7846i	0.455842	−	0.789542i
$694$		0			0
$695$	−8.00000	+	13.8564i	−0.303457	+	0.525603i
$696$		0			0
$697$	4.00000			0.151511
$698$		0			0
$699$	9.00000	+	15.5885i	0.340411	+	0.589610i
$700$		0			0
$701$	−10.0000			−0.377695			−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000			−0.377695			−0.188847	+	0.982006i	$0.560475\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−6.00000	−	10.3923i	−0.225973	−	0.391397i
$706$		0			0
$707$	40.0000			1.50435
$708$		0			0
$709$	−18.0000	+	31.1769i	−0.676004	+	1.17087i	0.300170	+	0.953886i	$0.402957\pi$
$709$	−18.0000	+	31.1769i	−0.676004	+	1.17087i	−0.976174	+	0.216988i	$0.930377\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−32.0000	−	55.4256i	−1.19841	−	2.07571i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−5.00000	−	8.66025i	−0.186728	−	0.323423i
$718$		0			0
$719$	−8.00000	+	13.8564i	−0.298350	+	0.516757i	−0.975759	−	0.218850i	$-0.929769\pi$
$719$	−8.00000	+	13.8564i	−0.298350	+	0.516757i	0.677409	+	0.735607i	$0.263103\pi$
$720$		0			0
$721$	−8.00000	+	13.8564i	−0.297936	+	0.516040i
$722$		0			0
$723$	20.0000			0.743808
$724$		0			0
$725$	1.00000	+	1.73205i	0.0371391	+	0.0643268i
$726$		0			0
$727$	−8.00000			−0.296704			−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000			−0.296704			−0.148352	+	0.988935i	$0.547397\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−8.00000	−	13.8564i	−0.295891	−	0.512498i
$732$		0			0
$733$	20.0000			0.738717			0.369358	−	0.929287i	$-0.379577\pi$
$733$	20.0000			0.738717			0.369358	+	0.929287i	$0.379577\pi$
$734$		0			0
$735$	9.00000	−	15.5885i	0.331970	−	0.574989i
$736$		0			0
$737$	−12.0000	+	20.7846i	−0.442026	+	0.765611i
$738$		0			0
$739$	−14.0000	−	24.2487i	−0.514998	−	0.892003i	−0.999849	−	0.0174060i	$-0.994459\pi$
$739$	−14.0000	−	24.2487i	−0.514998	−	0.892003i	0.484850	−	0.874597i	$-0.338874\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	11.0000	+	19.0526i	0.403551	+	0.698971i	0.994152	−	0.107993i	$-0.0344425\pi$
$743$	11.0000	+	19.0526i	0.403551	+	0.698971i	−0.590601	+	0.806964i	$0.701109\pi$
$744$		0			0
$745$	10.0000	−	17.3205i	0.366372	−	0.634574i
$746$		0			0
$747$	7.00000	−	12.1244i	0.256117	−	0.443607i
$748$		0			0
$749$	80.0000			2.92314
$750$		0			0
$751$	10.0000	+	17.3205i	0.364905	+	0.632034i	0.988761	−	0.149505i	$-0.0477681\pi$
$751$	10.0000	+	17.3205i	0.364905	+	0.632034i	−0.623856	+	0.781540i	$0.714435\pi$
$752$		0			0
$753$	−20.0000			−0.728841
$754$		0			0
$755$	−32.0000			−1.16460
$756$		0			0
$757$	−1.00000	−	1.73205i	−0.0363456	−	0.0629525i	0.847280	−	0.531146i	$-0.178238\pi$
$757$	−1.00000	−	1.73205i	−0.0363456	−	0.0629525i	−0.883626	+	0.468193i	$0.844905\pi$
$758$		0			0
$759$	−48.0000			−1.74229
$760$		0			0
$761$	15.0000	−	25.9808i	0.543750	−	0.941802i	−0.454935	−	0.890525i	$-0.650337\pi$
$761$	15.0000	−	25.9808i	0.543750	−	0.941802i	0.998684	−	0.0512772i	$-0.0163292\pi$
$762$		0			0
$763$	8.00000	−	13.8564i	0.289619	−	0.501636i
$764$		0			0
$765$	2.00000	+	3.46410i	0.0723102	+	0.125245i
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$769$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$770$		0			0
$771$	−3.00000	+	5.19615i	−0.108042	+	0.187135i
$772$		0			0
$773$	−9.00000	+	15.5885i	−0.323708	+	0.560678i	−0.981250	−	0.192740i	$-0.938263\pi$
$773$	−9.00000	+	15.5885i	−0.323708	+	0.560678i	0.657542	+	0.753418i	$0.271596\pi$
$774$		0			0
$775$	−8.00000			−0.287368
$776$		0			0
$777$	−16.0000	−	27.7128i	−0.573997	−	0.994192i
$778$		0			0
$779$		0			0
$780$		0			0
$781$	36.0000			1.28818
$782$		0			0
$783$	−1.00000	−	1.73205i	−0.0357371	−	0.0618984i
$784$		0			0
$785$	28.0000			0.999363
$786$		0			0
$787$	−22.0000	+	38.1051i	−0.784215	+	1.35830i	0.145251	+	0.989395i	$0.453601\pi$
$787$	−22.0000	+	38.1051i	−0.784215	+	1.35830i	−0.929467	+	0.368906i	$0.879732\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	20.0000	+	34.6410i	0.711118	+	1.23169i
$792$		0			0
$793$		0			0
$794$		0			0
$795$	6.00000	+	10.3923i	0.212798	+	0.368577i
$796$		0			0
$797$	5.00000	−	8.66025i	0.177109	−	0.306762i	−0.763780	−	0.645477i	$-0.776659\pi$
$797$	5.00000	−	8.66025i	0.177109	−	0.306762i	0.940889	+	0.338715i	$0.109992\pi$
$798$		0			0
$799$	6.00000	−	10.3923i	0.212265	−	0.367653i
$800$		0			0
$801$	6.00000			0.212000
$802$		0			0
$803$	−12.0000	−	20.7846i	−0.423471	−	0.733473i
$804$		0			0
$805$	−64.0000			−2.25570
$806$		0			0
$807$	−22.0000			−0.774437
$808$		0			0
$809$	−11.0000	−	19.0526i	−0.386739	−	0.669852i	0.605269	−	0.796021i	$-0.293065\pi$
$809$	−11.0000	−	19.0526i	−0.386739	−	0.669852i	−0.992009	+	0.126168i	$0.959732\pi$
$810$		0			0
$811$	−40.0000			−1.40459			−0.702295	−	0.711886i	$-0.747841\pi$
$811$	−40.0000			−1.40459			−0.702295	+	0.711886i	$0.747841\pi$
$812$		0			0
$813$	−10.0000	+	17.3205i	−0.350715	+	0.607457i
$814$		0			0
$815$	−8.00000	+	13.8564i	−0.280228	+	0.485369i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	15.0000	+	25.9808i	0.523504	+	0.906735i	0.999626	+	0.0273557i	$0.00870868\pi$
$821$	15.0000	+	25.9808i	0.523504	+	0.906735i	−0.476122	+	0.879379i	$0.657958\pi$
$822$		0			0
$823$	−18.0000	+	31.1769i	−0.627441	+	1.08676i	0.360623	+	0.932712i	$0.382564\pi$
$823$	−18.0000	+	31.1769i	−0.627441	+	1.08676i	−0.988063	+	0.154047i	$0.950769\pi$
$824$		0			0
$825$	−3.00000	+	5.19615i	−0.104447	+	0.180907i
$826$		0			0
$827$	18.0000			0.625921			0.312961	−	0.949766i	$-0.398679\pi$
$827$	18.0000			0.625921			0.312961	+	0.949766i	$0.398679\pi$
$828$		0			0
$829$	−27.0000	−	46.7654i	−0.937749	−	1.62423i	−0.769657	−	0.638457i	$-0.779573\pi$
$829$	−27.0000	−	46.7654i	−0.937749	−	1.62423i	−0.168091	−	0.985771i	$-0.553760\pi$
$830$		0			0
$831$	−22.0000			−0.763172
$832$		0			0
$833$	18.0000			0.623663
$834$		0			0
$835$	18.0000	+	31.1769i	0.622916	+	1.07892i
$836$		0			0
$837$	8.00000			0.276520
$838$		0			0
$839$	5.00000	−	8.66025i	0.172619	−	0.298985i	−0.766716	−	0.641987i	$-0.778110\pi$
$839$	5.00000	−	8.66025i	0.172619	−	0.298985i	0.939335	+	0.343002i	$0.111444\pi$
$840$		0			0
$841$	12.5000	−	21.6506i	0.431034	−	0.746574i
$842$		0			0
$843$	11.0000	+	19.0526i	0.378860	+	0.656205i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−50.0000	−	86.6025i	−1.71802	−	2.97570i
$848$		0			0
$849$	2.00000	−	3.46410i	0.0686398	−	0.118888i
$850$		0			0
$851$	−32.0000	+	55.4256i	−1.09695	+	1.89997i
$852$		0			0
$853$	−16.0000			−0.547830			−0.273915	−	0.961754i	$-0.588319\pi$
$853$	−16.0000			−0.547830			−0.273915	+	0.961754i	$0.588319\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	22.0000			0.751506			0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000			0.751506			0.375753	+	0.926720i	$0.377384\pi$
$858$		0			0
$859$	52.0000			1.77422			0.887109	−	0.461561i	$-0.152710\pi$
$859$	52.0000			1.77422			0.887109	+	0.461561i	$0.152710\pi$
$860$		0			0
$861$	−4.00000	−	6.92820i	−0.136320	−	0.236113i
$862$		0			0
$863$	−34.0000			−1.15737			−0.578687	−	0.815550i	$-0.696435\pi$
$863$	−34.0000			−1.15737			−0.578687	+	0.815550i	$0.696435\pi$
$864$		0			0
$865$	−6.00000	+	10.3923i	−0.204006	+	0.353349i
$866$		0			0
$867$	6.50000	−	11.2583i	0.220752	−	0.382353i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−6.00000	−	10.3923i	−0.203069	−	0.351726i
$874$		0			0
$875$	−24.0000	+	41.5692i	−0.811348	+	1.40530i
$876$		0			0
$877$	−8.00000	+	13.8564i	−0.270141	+	0.467898i	−0.968898	−	0.247462i	$-0.920404\pi$
$877$	−8.00000	+	13.8564i	−0.270141	+	0.467898i	0.698757	+	0.715359i	$0.253737\pi$
$878$		0			0
$879$	−14.0000			−0.472208
$880$		0			0
$881$	21.0000	+	36.3731i	0.707508	+	1.22544i	0.965779	+	0.259367i	$0.0835140\pi$
$881$	21.0000	+	36.3731i	0.707508	+	1.22544i	−0.258271	+	0.966073i	$0.583153\pi$
$882$		0			0
$883$	12.0000			0.403832			0.201916	−	0.979403i	$-0.435283\pi$
$883$	12.0000			0.403832			0.201916	+	0.979403i	$0.435283\pi$
$884$		0			0
$885$	4.00000			0.134459
$886$		0			0
$887$	−12.0000	−	20.7846i	−0.402921	−	0.697879i	0.591156	−	0.806557i	$-0.298672\pi$
$887$	−12.0000	−	20.7846i	−0.402921	−	0.697879i	−0.994077	+	0.108678i	$0.965338\pi$
$888$		0			0
$889$	−32.0000			−1.07325
$890$		0			0
$891$	3.00000	−	5.19615i	0.100504	−	0.174078i
$892$		0			0
$893$		0			0
$894$		0			0
$895$	12.0000	+	20.7846i	0.401116	+	0.694753i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−8.00000	−	13.8564i	−0.266815	−	0.462137i
$900$		0			0
$901$	−6.00000	+	10.3923i	−0.199889	+	0.346218i
$902$		0			0
$903$	−16.0000	+	27.7128i	−0.532447	+	0.922225i
$904$		0			0
$905$	20.0000			0.664822
$906$		0			0
$907$	−22.0000	−	38.1051i	−0.730498	−	1.26526i	−0.956671	−	0.291172i	$-0.905955\pi$
$907$	−22.0000	−	38.1051i	−0.730498	−	1.26526i	0.226173	−	0.974087i	$-0.427379\pi$
$908$		0			0
$909$	10.0000			0.331679
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$	−42.0000	−	72.7461i	−1.39000	−	2.40755i
$914$		0			0
$915$	−4.00000			−0.132236
$916$		0			0
$917$	−8.00000	+	13.8564i	−0.264183	+	0.457579i
$918$		0			0
$919$	12.0000	−	20.7846i	0.395843	−	0.685621i	−0.597365	−	0.801970i	$-0.703786\pi$
$919$	12.0000	−	20.7846i	0.395843	−	0.685621i	0.993208	+	0.116348i	$0.0371189\pi$
$920$		0			0
$921$	10.0000	+	17.3205i	0.329511	+	0.570730i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	4.00000	+	6.92820i	0.131519	+	0.227798i
$926$		0			0
$927$	−2.00000	+	3.46410i	−0.0656886	+	0.113776i
$928$		0			0
$929$	15.0000	−	25.9808i	0.492134	−	0.852401i	−0.507825	−	0.861460i	$-0.669550\pi$
$929$	15.0000	−	25.9808i	0.492134	−	0.852401i	0.999959	+	0.00905914i	$0.00288365\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−8.00000	−	13.8564i	−0.261908	−	0.453638i
$934$		0			0
$935$	24.0000			0.784884
$936$		0			0
$937$	−50.0000			−1.63343			−0.816714	−	0.577042i	$-0.804207\pi$
$937$	−50.0000			−1.63343			−0.816714	+	0.577042i	$0.804207\pi$
$938$		0			0
$939$	7.00000	+	12.1244i	0.228436	+	0.395663i
$940$		0			0
$941$	−38.0000			−1.23876			−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000			−1.23876			−0.619382	+	0.785090i	$0.712617\pi$
$942$		0			0
$943$	−8.00000	+	13.8564i	−0.260516	+	0.451227i
$944$		0			0
$945$	4.00000	−	6.92820i	0.130120	−	0.225374i
$946$		0			0
$947$	−19.0000	−	32.9090i	−0.617417	−	1.06940i	−0.989955	−	0.141381i	$-0.954846\pi$
$947$	−19.0000	−	32.9090i	−0.617417	−	1.06940i	0.372538	−	0.928017i	$-0.378488\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	9.00000	+	15.5885i	0.291845	+	0.505490i
$952$		0			0
$953$	−27.0000	+	46.7654i	−0.874616	+	1.51488i	−0.0174443	+	0.999848i	$0.505553\pi$
$953$	−27.0000	+	46.7654i	−0.874616	+	1.51488i	−0.857171	+	0.515031i	$0.827780\pi$
$954$		0			0
$955$	−24.0000	+	41.5692i	−0.776622	+	1.34515i
$956$		0			0
$957$	−12.0000			−0.387905
$958$		0			0
$959$	−20.0000	−	34.6410i	−0.645834	−	1.11862i
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$	20.0000			0.644491
$964$		0			0
$965$	−8.00000	−	13.8564i	−0.257529	−	0.446054i
$966$		0			0
$967$	−8.00000			−0.257263			−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000			−0.257263			−0.128631	+	0.991692i	$0.541058\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	6.00000	−	10.3923i	0.192549	−	0.333505i	−0.753545	−	0.657396i	$-0.771658\pi$
$971$	6.00000	−	10.3923i	0.192549	−	0.333505i	0.946094	+	0.323891i	$0.104991\pi$
$972$		0			0
$973$	16.0000	+	27.7128i	0.512936	+	0.888432i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−15.0000	−	25.9808i	−0.479893	−	0.831198i	0.519841	−	0.854263i	$-0.325991\pi$
$977$	−15.0000	−	25.9808i	−0.479893	−	0.831198i	−0.999734	+	0.0230645i	$0.992658\pi$
$978$		0			0
$979$	18.0000	−	31.1769i	0.575282	−	0.996419i
$980$		0			0
$981$	2.00000	−	3.46410i	0.0638551	−	0.110600i
$982$		0			0
$983$	−18.0000			−0.574111			−0.287055	−	0.957914i	$-0.592676\pi$
$983$	−18.0000			−0.574111			−0.287055	+	0.957914i	$0.592676\pi$
$984$		0			0
$985$	−26.0000	−	45.0333i	−0.828429	−	1.43488i
$986$		0			0
$987$	−24.0000			−0.763928
$988$		0			0
$989$	64.0000			2.03508
$990$		0			0
$991$	22.0000	+	38.1051i	0.698853	+	1.21045i	0.968864	+	0.247592i	$0.0796392\pi$
$991$	22.0000	+	38.1051i	0.698853	+	1.21045i	−0.270011	+	0.962857i	$0.587027\pi$
$992$		0			0
$993$	28.0000			0.888553
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−3.00000	+	5.19615i	−0.0950110	+	0.164564i	−0.909613	−	0.415456i	$-0.863622\pi$
$997$	−3.00000	+	5.19615i	−0.0950110	+	0.164564i	0.814602	+	0.580020i	$0.196955\pi$
$998$		0			0
$999$	−4.00000	−	6.92820i	−0.126554	−	0.219199i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2028.2.i.a.2005.1		2
13.2	odd	12		156.2.b.b.25.2	yes	2
13.3	even	3		2028.2.a.d.1.1		1
13.4	even	6		2028.2.i.d.529.1		2
13.5	odd	4		2028.2.q.e.361.2		4
13.6	odd	12		2028.2.q.e.1837.2		4
13.7	odd	12		2028.2.q.e.1837.1		4
13.8	odd	4		2028.2.q.e.361.1		4
13.9	even	3	inner	2028.2.i.a.529.1		2
13.10	even	6		2028.2.a.f.1.1		1
13.11	odd	12		156.2.b.b.25.1	✓	2
13.12	even	2		2028.2.i.d.2005.1		2
39.2	even	12		468.2.b.c.181.1		2
39.11	even	12		468.2.b.c.181.2		2
39.23	odd	6		6084.2.a.d.1.1		1
39.29	odd	6		6084.2.a.n.1.1		1
52.3	odd	6		8112.2.a.d.1.1		1
52.11	even	12		624.2.c.d.337.1		2
52.15	even	12		624.2.c.d.337.2		2
52.23	odd	6		8112.2.a.l.1.1		1
65.2	even	12		3900.2.j.b.649.1		2
65.24	odd	12		3900.2.c.a.3301.2		2
65.28	even	12		3900.2.j.e.649.2		2
65.37	even	12		3900.2.j.e.649.1		2
65.54	odd	12		3900.2.c.a.3301.1		2
65.63	even	12		3900.2.j.b.649.2		2
91.41	even	12		7644.2.e.b.4705.1		2
91.76	even	12		7644.2.e.b.4705.2		2
104.11	even	12		2496.2.c.i.961.2		2
104.37	odd	12		2496.2.c.b.961.2		2
104.67	even	12		2496.2.c.i.961.1		2
104.93	odd	12		2496.2.c.b.961.1		2
156.11	odd	12		1872.2.c.h.1585.2		2
156.119	odd	12		1872.2.c.h.1585.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.2.b.b.25.1	✓	2	13.11	odd	12
156.2.b.b.25.2	yes	2	13.2	odd	12
468.2.b.c.181.1		2	39.2	even	12
468.2.b.c.181.2		2	39.11	even	12
624.2.c.d.337.1		2	52.11	even	12
624.2.c.d.337.2		2	52.15	even	12
1872.2.c.h.1585.1		2	156.119	odd	12
1872.2.c.h.1585.2		2	156.11	odd	12
2028.2.a.d.1.1		1	13.3	even	3
2028.2.a.f.1.1		1	13.10	even	6
2028.2.i.a.529.1		2	13.9	even	3	inner
2028.2.i.a.2005.1		2	1.1	even	1	trivial
2028.2.i.d.529.1		2	13.4	even	6
2028.2.i.d.2005.1		2	13.12	even	2
2028.2.q.e.361.1		4	13.8	odd	4
2028.2.q.e.361.2		4	13.5	odd	4
2028.2.q.e.1837.1		4	13.7	odd	12
2028.2.q.e.1837.2		4	13.6	odd	12
2496.2.c.b.961.1		2	104.93	odd	12
2496.2.c.b.961.2		2	104.37	odd	12
2496.2.c.i.961.1		2	104.67	even	12
2496.2.c.i.961.2		2	104.11	even	12
3900.2.c.a.3301.1		2	65.54	odd	12
3900.2.c.a.3301.2		2	65.24	odd	12
3900.2.j.b.649.1		2	65.2	even	12
3900.2.j.b.649.2		2	65.63	even	12
3900.2.j.e.649.1		2	65.37	even	12
3900.2.j.e.649.2		2	65.28	even	12
6084.2.a.d.1.1		1	39.23	odd	6
6084.2.a.n.1.1		1	39.29	odd	6
7644.2.e.b.4705.1		2	91.41	even	12
7644.2.e.b.4705.2		2	91.76	even	12
8112.2.a.d.1.1		1	52.3	odd	6
8112.2.a.l.1.1		1	52.23	odd	6

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2028.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{3} -2.00000 q^{5} +(-2.00000 + 3.46410i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(3.00000 + 5.19615i) q^{11} +(1.00000 + 1.73205i) q^{15} +(-1.00000 + 1.73205i) q^{17} +4.00000 q^{21} +(-4.00000 - 6.92820i) q^{23} -1.00000 q^{25} +1.00000 q^{27} +(-1.00000 - 1.73205i) q^{29} +8.00000 q^{31} +(3.00000 - 5.19615i) q^{33} +(4.00000 - 6.92820i) q^{35} +(-4.00000 - 6.92820i) q^{37} +(-1.00000 - 1.73205i) q^{41} +(-4.00000 + 6.92820i) q^{43} +(1.00000 - 1.73205i) q^{45} -6.00000 q^{47} +(-4.50000 - 7.79423i) q^{49} +2.00000 q^{51} +6.00000 q^{53} +(-6.00000 - 10.3923i) q^{55} +(1.00000 - 1.73205i) q^{59} +(-1.00000 + 1.73205i) q^{61} +(-2.00000 - 3.46410i) q^{63} +(2.00000 + 3.46410i) q^{67} +(-4.00000 + 6.92820i) q^{69} +(3.00000 - 5.19615i) q^{71} -4.00000 q^{73} +(0.500000 + 0.866025i) q^{75} -24.0000 q^{77} +(-0.500000 - 0.866025i) q^{81} -14.0000 q^{83} +(2.00000 - 3.46410i) q^{85} +(-1.00000 + 1.73205i) q^{87} +(-3.00000 - 5.19615i) q^{89} +(-4.00000 - 6.92820i) q^{93} +(-6.00000 + 10.3923i) q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 4 q^{5} - 4 q^{7} - q^{9} + 6 q^{11} + 2 q^{15} - 2 q^{17} + 8 q^{21} - 8 q^{23} - 2 q^{25} + 2 q^{27} - 2 q^{29} + 16 q^{31} + 6 q^{33} + 8 q^{35} - 8 q^{37} - 2 q^{41} - 8 q^{43} + 2 q^{45}+ \cdots - 12 q^{99}+O(q^{100}) \) 2 * q - q^3 - 4 * q^5 - 4 * q^7 - q^9 + 6 * q^11 + 2 * q^15 - 2 * q^17 + 8 * q^21 - 8 * q^23 - 2 * q^25 + 2 * q^27 - 2 * q^29 + 16 * q^31 + 6 * q^33 + 8 * q^35 - 8 * q^37 - 2 * q^41 - 8 * q^43 + 2 * q^45 - 12 * q^47 - 9 * q^49 + 4 * q^51 + 12 * q^53 - 12 * q^55 + 2 * q^59 - 2 * q^61 - 4 * q^63 + 4 * q^67 - 8 * q^69 + 6 * q^71 - 8 * q^73 + q^75 - 48 * q^77 - q^81 - 28 * q^83 + 4 * q^85 - 2 * q^87 - 6 * q^89 - 8 * q^93 - 12 * q^97 - 12 * q^99

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