LMFDB - Embedded newform 448.3.l.b.209.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [448,3,Mod(209,448)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 2])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("448.209"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	448.l (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$12.2071158433$
Analytic rank:	$0$
Dimension:	$56$
Relative dimension:	$28$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			209.12
Character	$\chi$	$=$	448.209
Dual form			448.3.l.b.433.12

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.00109 + 1.00109i) q^{3} +(4.03888 + 4.03888i) q^{5} +(-6.50171 + 2.59380i) q^{7} +6.99566i q^{9} +(6.25649 - 6.25649i) q^{11} +(-4.97013 + 4.97013i) q^{13} -8.08654 q^{15} -5.44402i q^{17} +(-17.3201 + 17.3201i) q^{19} +(3.91215 - 9.10538i) q^{21} -8.07936i q^{23} +7.62516i q^{25} +(-16.0130 - 16.0130i) q^{27} +(-27.3552 - 27.3552i) q^{29} +53.1316i q^{31} +12.5266i q^{33} +(-36.7357 - 15.7836i) q^{35} +(13.0922 - 13.0922i) q^{37} -9.95105i q^{39} +3.54538 q^{41} +(-13.9574 + 13.9574i) q^{43} +(-28.2546 + 28.2546i) q^{45} +38.9511i q^{47} +(35.5444 - 33.7283i) q^{49} +(5.44993 + 5.44993i) q^{51} +(-51.8947 + 51.8947i) q^{53} +50.5385 q^{55} -34.6778i q^{57} +(-56.4368 - 56.4368i) q^{59} +(-84.3789 + 84.3789i) q^{61} +(-18.1453 - 45.4837i) q^{63} -40.1475 q^{65} +(-0.603225 - 0.603225i) q^{67} +(8.08813 + 8.08813i) q^{69} +78.2455i q^{71} -79.7314 q^{73} +(-7.63344 - 7.63344i) q^{75} +(-24.4498 + 56.9060i) q^{77} +85.2911 q^{79} -30.9001 q^{81} +(5.96262 - 5.96262i) q^{83} +(21.9878 - 21.9878i) q^{85} +54.7699 q^{87} +163.829 q^{89} +(19.4228 - 45.2058i) q^{91} +(-53.1892 - 53.1892i) q^{93} -139.908 q^{95} -89.6393i q^{97} +(43.7683 + 43.7683i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q + 8 q^{15} - 20 q^{21} - 96 q^{29} + 100 q^{35} - 128 q^{37} + 72 q^{43} + 192 q^{49} + 128 q^{51} + 88 q^{53} - 444 q^{63} - 8 q^{65} - 440 q^{67} + 12 q^{77} + 8 q^{79} + 64 q^{81} + 96 q^{85} + 388 q^{91}+ \cdots - 8 q^{99}+O(q^{100}) $ 56 * q + 8 * q^15 - 20 * q^21 - 96 * q^29 + 100 * q^35 - 128 * q^37 + 72 * q^43 + 192 * q^49 + 128 * q^51 + 88 * q^53 - 444 * q^63 - 8 * q^65 - 440 * q^67 + 12 * q^77 + 8 * q^79 + 64 * q^81 + 96 * q^85 + 388 * q^91 + 32 * q^93 + 776 * q^95 - 8 * q^99

Character values

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

$n$	$127$	$129$	$197$
$\chi(n)$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−1.00109	+	1.00109i	−0.333695	+	0.333695i	−0.853988	−	0.520293i	$-0.825823\pi$
$3$	−1.00109	+	1.00109i	−0.333695	+	0.333695i	0.520293	+	0.853988i	$0.325823\pi$
$4$		0			0
$5$	4.03888	+	4.03888i	0.807777	+	0.807777i	0.984297	−	0.176520i	$-0.0564841\pi$
$5$	4.03888	+	4.03888i	0.807777	+	0.807777i	−0.176520	+	0.984297i	$0.556484\pi$
$6$		0			0
$7$	−6.50171	+	2.59380i	−0.928815	+	0.370543i
$7$	−6.50171	+	2.59380i	−0.928815	+	0.370543i
$8$		0			0
$9$			6.99566i			0.777295i
$10$		0			0
$11$	6.25649	−	6.25649i	0.568772	−	0.568772i	−0.363012	−	0.931784i	$-0.618252\pi$
$11$	6.25649	−	6.25649i	0.568772	−	0.568772i	0.931784	+	0.363012i	$0.118252\pi$
$12$		0			0
$13$	−4.97013	+	4.97013i	−0.382318	+	0.382318i	−0.871936	−	0.489619i	$-0.837136\pi$
$13$	−4.97013	+	4.97013i	−0.382318	+	0.382318i	0.489619	+	0.871936i	$0.337136\pi$
$14$		0			0
$15$	−8.08654			−0.539102
$16$		0			0
$17$		−	5.44402i		−	0.320237i	−0.987098	−	0.160118i	$-0.948812\pi$
$17$		−	5.44402i		−	0.320237i	0.987098	−	0.160118i	$-0.0511876\pi$
$18$		0			0
$19$	−17.3201	+	17.3201i	−0.911585	+	0.911585i	−0.996397	−	0.0848120i	$-0.972971\pi$
$19$	−17.3201	+	17.3201i	−0.911585	+	0.911585i	0.0848120	+	0.996397i	$0.472971\pi$
$20$		0			0
$21$	3.91215	−	9.10538i	0.186293	−	0.433590i
$22$		0			0
$23$		−	8.07936i		−	0.351276i	−0.984455	−	0.175638i	$-0.943801\pi$
$23$		−	8.07936i		−	0.351276i	0.984455	−	0.175638i	$-0.0561989\pi$
$24$		0			0
$25$			7.62516i			0.305006i
$26$		0			0
$27$	−16.0130	−	16.0130i	−0.593075	−	0.593075i
$28$		0			0
$29$	−27.3552	−	27.3552i	−0.943284	−	0.943284i	0.0551914	−	0.998476i	$-0.482423\pi$
$29$	−27.3552	−	27.3552i	−0.943284	−	0.943284i	−0.998476	+	0.0551914i	$0.982423\pi$
$30$		0			0
$31$			53.1316i			1.71392i	0.515382	+	0.856961i	$0.327650\pi$
$31$			53.1316i			1.71392i	−0.515382	+	0.856961i	$0.672350\pi$
$32$		0			0
$33$			12.5266i			0.379593i
$34$		0			0
$35$	−36.7357	−	15.7836i	−1.04959	−	0.450959i
$36$		0			0
$37$	13.0922	−	13.0922i	0.353842	−	0.353842i	−0.507695	−	0.861537i	$-0.669502\pi$
$37$	13.0922	−	13.0922i	0.353842	−	0.353842i	0.861537	+	0.507695i	$0.169502\pi$
$38$		0			0
$39$		−	9.95105i		−	0.255155i
$40$		0			0
$41$	3.54538			0.0864728			0.0432364	−	0.999065i	$-0.486233\pi$
$41$	3.54538			0.0864728			0.0432364	+	0.999065i	$0.486233\pi$
$42$		0			0
$43$	−13.9574	+	13.9574i	−0.324590	+	0.324590i	−0.850525	−	0.525935i	$-0.823715\pi$
$43$	−13.9574	+	13.9574i	−0.324590	+	0.324590i	0.525935	+	0.850525i	$0.323715\pi$
$44$		0			0
$45$	−28.2546	+	28.2546i	−0.627881	+	0.627881i
$46$		0			0
$47$			38.9511i			0.828747i	0.910107	+	0.414374i	$0.135999\pi$
$47$			38.9511i			0.828747i	−0.910107	+	0.414374i	$0.864001\pi$
$48$		0			0
$49$	35.5444	−	33.7283i	0.725396	−	0.688332i
$50$		0			0
$51$	5.44993	+	5.44993i	0.106861	+	0.106861i
$52$		0			0
$53$	−51.8947	+	51.8947i	−0.979144	+	0.979144i	−0.999787	−	0.0206426i	$-0.993429\pi$
$53$	−51.8947	+	51.8947i	−0.979144	+	0.979144i	0.0206426	+	0.999787i	$0.493429\pi$
$54$		0			0
$55$	50.5385			0.918882
$56$		0			0
$57$		−	34.6778i		−	0.608383i
$58$		0			0
$59$	−56.4368	−	56.4368i	−0.956556	−	0.956556i	0.0425390	−	0.999095i	$-0.486455\pi$
$59$	−56.4368	−	56.4368i	−0.956556	−	0.956556i	−0.999095	+	0.0425390i	$0.986455\pi$
$60$		0			0
$61$	−84.3789	+	84.3789i	−1.38326	+	1.38326i	−0.544499	+	0.838762i	$0.683280\pi$
$61$	−84.3789	+	84.3789i	−1.38326	+	1.38326i	−0.838762	+	0.544499i	$0.816720\pi$
$62$		0			0
$63$	−18.1453	−	45.4837i	−0.288021	−	0.721964i
$64$		0			0
$65$	−40.1475			−0.617654
$66$		0			0
$67$	−0.603225	−	0.603225i	−0.00900335	−	0.00900335i	0.702591	−	0.711594i	$-0.252026\pi$
$67$	−0.603225	−	0.603225i	−0.00900335	−	0.00900335i	−0.711594	+	0.702591i	$0.752026\pi$
$68$		0			0
$69$	8.08813	+	8.08813i	0.117219	+	0.117219i
$70$		0			0
$71$			78.2455i			1.10205i	0.834489	+	0.551024i	$0.185763\pi$
$71$			78.2455i			1.10205i	−0.834489	+	0.551024i	$0.814237\pi$
$72$		0			0
$73$	−79.7314			−1.09221			−0.546106	−	0.837716i	$-0.683890\pi$
$73$	−79.7314			−1.09221			−0.546106	+	0.837716i	$0.683890\pi$
$74$		0			0
$75$	−7.63344	−	7.63344i	−0.101779	−	0.101779i
$76$		0			0
$77$	−24.4498	+	56.9060i	−0.317530	+	0.739039i
$78$		0			0
$79$	85.2911			1.07963			0.539817	−	0.841782i	$-0.318493\pi$
$79$	85.2911			1.07963			0.539817	+	0.841782i	$0.318493\pi$
$80$		0			0
$81$	−30.9001			−0.381483
$82$		0			0
$83$	5.96262	−	5.96262i	0.0718387	−	0.0718387i	−0.670275	−	0.742113i	$-0.733824\pi$
$83$	5.96262	−	5.96262i	0.0718387	−	0.0718387i	0.742113	+	0.670275i	$0.233824\pi$
$84$		0			0
$85$	21.9878	−	21.9878i	0.258680	−	0.258680i
$86$		0			0
$87$	54.7699			0.629539
$88$		0			0
$89$	163.829			1.84077			0.920385	−	0.391014i	$-0.127875\pi$
$89$	163.829			1.84077			0.920385	+	0.391014i	$0.127875\pi$
$90$		0			0
$91$	19.4228	−	45.2058i	0.213437	−	0.496767i
$92$		0			0
$93$	−53.1892	−	53.1892i	−0.571927	−	0.571927i
$94$		0			0
$95$	−139.908			−1.47271
$96$		0			0
$97$		−	89.6393i		−	0.924117i	−0.886850	−	0.462058i	$-0.847111\pi$
$97$		−	89.6393i		−	0.924117i	0.886850	−	0.462058i	$-0.152889\pi$
$98$		0			0
$99$	43.7683	+	43.7683i	0.442104	+	0.442104i
$100$		0			0
$101$	50.7141	+	50.7141i	0.502120	+	0.502120i	0.912096	−	0.409976i	$-0.134463\pi$
$101$	50.7141	+	50.7141i	0.502120	+	0.502120i	−0.409976	+	0.912096i	$0.634463\pi$
$102$		0			0
$103$	77.0434			0.747994			0.373997	−	0.927430i	$-0.377987\pi$
$103$	77.0434			0.747994			0.373997	+	0.927430i	$0.377987\pi$
$104$		0			0
$105$	52.5763	−	20.9749i	0.500727	−	0.199761i
$106$		0			0
$107$	140.533	−	140.533i	1.31339	−	1.31339i	0.394494	−	0.918899i	$-0.370920\pi$
$107$	140.533	−	140.533i	1.31339	−	1.31339i	0.918899	−	0.394494i	$-0.129080\pi$
$108$		0			0
$109$	46.2295	+	46.2295i	0.424124	+	0.424124i	0.886621	−	0.462497i	$-0.153046\pi$
$109$	46.2295	+	46.2295i	0.424124	+	0.424124i	−0.462497	+	0.886621i	$0.653046\pi$
$110$		0			0
$111$			26.2127i			0.236151i
$112$		0			0
$113$	163.269			1.44486			0.722431	−	0.691443i	$-0.243025\pi$
$113$	163.269			1.44486			0.722431	+	0.691443i	$0.243025\pi$
$114$		0			0
$115$	32.6316	−	32.6316i	0.283753	−	0.283753i
$116$		0			0
$117$	−34.7693	−	34.7693i	−0.297173	−	0.297173i
$118$		0			0
$119$	14.1207	+	35.3954i	0.118661	+	0.297441i
$120$		0			0
$121$			42.7126i			0.352997i
$122$		0			0
$123$	−3.54923	+	3.54923i	−0.0288556	+	0.0288556i
$124$		0			0
$125$	70.1750	−	70.1750i	0.561400	−	0.561400i
$126$		0			0
$127$	−31.3100			−0.246535			−0.123268	−	0.992373i	$-0.539337\pi$
$127$	−31.3100			−0.246535			−0.123268	+	0.992373i	$0.539337\pi$
$128$		0			0
$129$		−	27.9450i		−	0.216628i
$130$		0			0
$131$	21.8179	−	21.8179i	0.166549	−	0.166549i	−0.618912	−	0.785461i	$-0.712426\pi$
$131$	21.8179	−	21.8179i	0.166549	−	0.166549i	0.785461	+	0.618912i	$0.212426\pi$
$132$		0			0
$133$	67.6854	−	157.535i	0.508913	−	1.18448i
$134$		0			0
$135$		−	129.349i		−	0.958144i
$136$		0			0
$137$			125.561i			0.916503i	0.888823	+	0.458251i	$0.151524\pi$
$137$			125.561i			0.916503i	−0.888823	+	0.458251i	$0.848476\pi$
$138$		0			0
$139$	9.99666	+	9.99666i	0.0719184	+	0.0719184i	0.742151	−	0.670233i	$-0.233806\pi$
$139$	9.99666	+	9.99666i	0.0719184	+	0.0719184i	−0.670233	+	0.742151i	$0.733806\pi$
$140$		0			0
$141$	−38.9934	−	38.9934i	−0.276549	−	0.276549i
$142$		0			0
$143$			62.1911i			0.434903i
$144$		0			0
$145$		−	220.969i		−	1.52393i
$146$		0			0
$147$	−1.81808	+	69.3479i	−0.0123679	+	0.471754i
$148$		0			0
$149$	−109.367	+	109.367i	−0.734004	+	0.734004i	−0.971410	−	0.237406i	$-0.923703\pi$
$149$	−109.367	+	109.367i	−0.734004	+	0.734004i	0.237406	+	0.971410i	$0.423703\pi$
$150$		0			0
$151$			212.815i			1.40937i	0.709519	+	0.704686i	$0.248912\pi$
$151$			212.815i			1.40937i	−0.709519	+	0.704686i	$0.751088\pi$
$152$		0			0
$153$	38.0845			0.248918
$154$		0			0
$155$	−214.592	+	214.592i	−1.38447	+	1.38447i
$156$		0			0
$157$	159.296	−	159.296i	1.01463	−	1.01463i	0.0147347	−	0.999891i	$-0.495310\pi$
$157$	159.296	−	159.296i	1.01463	−	1.01463i	0.999891	−	0.0147347i	$-0.00469038\pi$
$158$		0			0
$159$		−	103.902i		−	0.653472i
$160$		0			0
$161$	20.9562	+	52.5296i	0.130163	+	0.326271i
$162$		0			0
$163$	12.8052	+	12.8052i	0.0785594	+	0.0785594i	0.745295	−	0.666735i	$-0.232309\pi$
$163$	12.8052	+	12.8052i	0.0785594	+	0.0785594i	−0.666735	+	0.745295i	$0.732309\pi$
$164$		0			0
$165$	−50.5934	+	50.5934i	−0.306626	+	0.306626i
$166$		0			0
$167$	29.3992			0.176043			0.0880214	−	0.996119i	$-0.471946\pi$
$167$	29.3992			0.176043			0.0880214	+	0.996119i	$0.471946\pi$
$168$		0			0
$169$			119.596i			0.707667i
$170$		0			0
$171$	−121.166	−	121.166i	−0.708570	−	0.708570i
$172$		0			0
$173$	21.6282	−	21.6282i	0.125018	−	0.125018i	−0.641829	−	0.766848i	$-0.721824\pi$
$173$	21.6282	−	21.6282i	0.125018	−	0.125018i	0.766848	+	0.641829i	$0.221824\pi$
$174$		0			0
$175$	−19.7781	−	49.5765i	−0.113018	−	0.283295i
$176$		0			0
$177$	112.996			0.638396
$178$		0			0
$179$	−42.8249	−	42.8249i	−0.239245	−	0.239245i	0.577292	−	0.816538i	$-0.304109\pi$
$179$	−42.8249	−	42.8249i	−0.239245	−	0.239245i	−0.816538	+	0.577292i	$0.804109\pi$
$180$		0			0
$181$	148.030	+	148.030i	0.817848	+	0.817848i	0.985796	−	0.167948i	$-0.0537141\pi$
$181$	148.030	+	148.030i	0.817848	+	0.817848i	−0.167948	+	0.985796i	$0.553714\pi$
$182$		0			0
$183$		−	168.941i		−	0.923175i
$184$		0			0
$185$	105.755			0.571651
$186$		0			0
$187$	−34.0605	−	34.0605i	−0.182142	−	0.182142i
$188$		0			0
$189$	145.647	+	62.5774i	0.770617	+	0.331097i
$190$		0			0
$191$	266.000			1.39267			0.696336	−	0.717716i	$-0.254813\pi$
$191$	266.000			1.39267			0.696336	+	0.717716i	$0.254813\pi$
$192$		0			0
$193$	−287.377			−1.48900			−0.744499	−	0.667624i	$-0.767312\pi$
$193$	−287.377			−1.48900			−0.744499	+	0.667624i	$0.767312\pi$
$194$		0			0
$195$	40.1911	−	40.1911i	0.206108	−	0.206108i
$196$		0			0
$197$	−7.88931	+	7.88931i	−0.0400473	+	0.0400473i	−0.726847	−	0.686800i	$-0.759015\pi$
$197$	−7.88931	+	7.88931i	−0.0400473	+	0.0400473i	0.686800	+	0.726847i	$0.259015\pi$
$198$		0			0
$199$	−55.3752			−0.278267			−0.139134	−	0.990274i	$-0.544432\pi$
$199$	−55.3752			−0.278267			−0.139134	+	0.990274i	$0.544432\pi$
$200$		0			0
$201$	1.20776			0.00600875
$202$		0			0
$203$	248.810	+	106.902i	1.22566	+	0.526609i
$204$		0			0
$205$	14.3194	+	14.3194i	0.0698507	+	0.0698507i
$206$		0			0
$207$	56.5204			0.273045
$208$		0			0
$209$			216.726i			1.03697i
$210$		0			0
$211$	−120.578	−	120.578i	−0.571461	−	0.571461i	0.361075	−	0.932537i	$-0.382410\pi$
$211$	−120.578	−	120.578i	−0.571461	−	0.571461i	−0.932537	+	0.361075i	$0.882410\pi$
$212$		0			0
$213$	−78.3304	−	78.3304i	−0.367748	−	0.367748i
$214$		0			0
$215$	−112.744			−0.524392
$216$		0			0
$217$	−137.813	−	345.446i	−0.635082	−	1.59192i
$218$		0			0
$219$	79.8180	−	79.8180i	0.364466	−	0.364466i
$220$		0			0
$221$	27.0575	+	27.0575i	0.122432	+	0.122432i
$222$		0			0
$223$		−	120.836i		−	0.541865i	−0.962598	−	0.270933i	$-0.912668\pi$
$223$		−	120.836i		−	0.541865i	0.962598	−	0.270933i	$-0.0873321\pi$
$224$		0			0
$225$	−53.3430			−0.237080
$226$		0			0
$227$	57.4724	−	57.4724i	0.253183	−	0.253183i	−0.569092	−	0.822274i	$-0.692705\pi$
$227$	57.4724	−	57.4724i	0.253183	−	0.253183i	0.822274	+	0.569092i	$0.192705\pi$
$228$		0			0
$229$	−175.590	−	175.590i	−0.766771	−	0.766771i	0.210766	−	0.977537i	$-0.432404\pi$
$229$	−175.590	−	175.590i	−0.766771	−	0.766771i	−0.977537	+	0.210766i	$0.932404\pi$
$230$		0			0
$231$	−32.4914	−	81.4441i	−0.140656	−	0.352572i
$232$		0			0
$233$		−	184.061i		−	0.789959i	−0.918690	−	0.394980i	$-0.870752\pi$
$233$		−	184.061i		−	0.789959i	0.918690	−	0.394980i	$-0.129248\pi$
$234$		0			0
$235$	−157.319	+	157.319i	−0.669443	+	0.669443i
$236$		0			0
$237$	−85.3837	+	85.3837i	−0.360269	+	0.360269i
$238$		0			0
$239$	33.1847			0.138848			0.0694240	−	0.997587i	$-0.477884\pi$
$239$	33.1847			0.138848			0.0694240	+	0.997587i	$0.477884\pi$
$240$		0			0
$241$			55.7768i			0.231439i	0.993282	+	0.115719i	$0.0369173\pi$
$241$			55.7768i			0.231439i	−0.993282	+	0.115719i	$0.963083\pi$
$242$		0			0
$243$	175.051	−	175.051i	0.720374	−	0.720374i
$244$		0			0
$245$	279.784	+	7.33506i	1.14198	+	0.0299390i
$246$		0			0
$247$		−	172.166i		−	0.697030i
$248$		0			0
$249$			11.9382i			0.0479445i
$250$		0			0
$251$	181.168	+	181.168i	0.721785	+	0.721785i	0.968969	−	0.247184i	$-0.0795051\pi$
$251$	181.168	+	181.168i	0.721785	+	0.721785i	−0.247184	+	0.968969i	$0.579505\pi$
$252$		0			0
$253$	−50.5484	−	50.5484i	−0.199796	−	0.199796i
$254$		0			0
$255$			44.0233i			0.172640i
$256$		0			0
$257$			363.112i			1.41289i	0.707770	+	0.706443i	$0.249701\pi$
$257$			363.112i			1.41289i	−0.707770	+	0.706443i	$0.750299\pi$
$258$		0			0
$259$	−51.1629	+	119.080i	−0.197540	+	0.459768i
$260$		0			0
$261$	191.368	−	191.368i	0.733210	−	0.733210i
$262$		0			0
$263$			421.236i			1.60166i	0.598892	+	0.800829i	$0.295608\pi$
$263$			421.236i			1.60166i	−0.598892	+	0.800829i	$0.704392\pi$
$264$		0			0
$265$	−419.193			−1.58186
$266$		0			0
$267$	−164.006	+	164.006i	−0.614256	+	0.614256i
$268$		0			0
$269$	−171.203	+	171.203i	−0.636441	+	0.636441i	−0.949676	−	0.313235i	$-0.898587\pi$
$269$	−171.203	+	171.203i	−0.636441	+	0.636441i	0.313235	+	0.949676i	$0.398587\pi$
$270$		0			0
$271$			15.9792i			0.0589637i	0.999565	+	0.0294819i	$0.00938573\pi$
$271$			15.9792i			0.0589637i	−0.999565	+	0.0294819i	$0.990614\pi$
$272$		0			0
$273$	25.8110	+	64.6988i	0.0945459	+	0.236992i
$274$		0			0
$275$	47.7068	+	47.7068i	0.173479	+	0.173479i
$276$		0			0
$277$	−201.173	+	201.173i	−0.726256	+	0.726256i	−0.969872	−	0.243615i	$-0.921667\pi$
$277$	−201.173	+	201.173i	−0.726256	+	0.726256i	0.243615	+	0.969872i	$0.421667\pi$
$278$		0			0
$279$	−371.690			−1.33222
$280$		0			0
$281$			364.700i			1.29786i	0.760846	+	0.648932i	$0.224784\pi$
$281$			364.700i			1.29786i	−0.760846	+	0.648932i	$0.775216\pi$
$282$		0			0
$283$	311.354	+	311.354i	1.10019	+	1.10019i	0.994387	+	0.105806i	$0.0337421\pi$
$283$	311.354	+	311.354i	1.10019	+	1.10019i	0.105806	+	0.994387i	$0.466258\pi$
$284$		0			0
$285$	140.060	−	140.060i	0.491438	−	0.491438i
$286$		0			0
$287$	−23.0511	+	9.19602i	−0.0803173	+	0.0320419i
$288$		0			0
$289$	259.363			0.897449
$290$		0			0
$291$	89.7366	+	89.7366i	0.308373	+	0.308373i
$292$		0			0
$293$	−249.211	−	249.211i	−0.850550	−	0.850550i	0.139651	−	0.990201i	$-0.455402\pi$
$293$	−249.211	−	249.211i	−0.850550	−	0.850550i	−0.990201	+	0.139651i	$0.955402\pi$
$294$		0			0
$295$		−	455.883i		−	1.54537i
$296$		0			0
$297$	−200.371			−0.674649
$298$		0			0
$299$	40.1554	+	40.1554i	0.134299	+	0.134299i
$300$		0			0
$301$	54.5440	−	126.949i	0.181209	−	0.421758i
$302$		0			0
$303$	−101.538			−0.335110
$304$		0			0
$305$	−681.593			−2.23473
$306$		0			0
$307$	−265.892	+	265.892i	−0.866098	+	0.866098i	−0.992038	−	0.125940i	$-0.959805\pi$
$307$	−265.892	+	265.892i	−0.866098	+	0.866098i	0.125940	+	0.992038i	$0.459805\pi$
$308$		0			0
$309$	−77.1270	+	77.1270i	−0.249602	+	0.249602i
$310$		0			0
$311$	−115.392			−0.371035			−0.185518	−	0.982641i	$-0.559396\pi$
$311$	−115.392			−0.371035			−0.185518	+	0.982641i	$0.559396\pi$
$312$		0			0
$313$	−334.318			−1.06811			−0.534054	−	0.845450i	$-0.679332\pi$
$313$	−334.318			−1.06811			−0.534054	+	0.845450i	$0.679332\pi$
$314$		0			0
$315$	110.416	−	256.990i	0.350528	−	0.815842i
$316$		0			0
$317$	26.5251	+	26.5251i	0.0836755	+	0.0836755i	0.747706	−	0.664030i	$-0.231155\pi$
$317$	26.5251	+	26.5251i	0.0836755	+	0.0836755i	−0.664030	+	0.747706i	$0.731155\pi$
$318$		0			0
$319$	−342.296			−1.07303
$320$		0			0
$321$			281.371i			0.876545i
$322$		0			0
$323$	94.2911	+	94.2911i	0.291923	+	0.291923i
$324$		0			0
$325$	−37.8980	−	37.8980i	−0.116609	−	0.116609i
$326$		0			0
$327$	−92.5593			−0.283056
$328$		0			0
$329$	−101.031	−	253.249i	−0.307087	−	0.769753i
$330$		0			0
$331$	−33.5611	+	33.5611i	−0.101393	+	0.101393i	−0.755984	−	0.654591i	$-0.772841\pi$
$331$	−33.5611	+	33.5611i	−0.101393	+	0.101393i	0.654591	+	0.755984i	$0.272841\pi$
$332$		0			0
$333$	91.5882	+	91.5882i	0.275040	+	0.275040i
$334$		0			0
$335$		−	4.87271i		−	0.0145454i
$336$		0			0
$337$	402.236			1.19358			0.596789	−	0.802398i	$-0.296443\pi$
$337$	402.236			1.19358			0.596789	+	0.802398i	$0.296443\pi$
$338$		0			0
$339$	−163.447	+	163.447i	−0.482144	+	0.482144i
$340$		0			0
$341$	332.417	+	332.417i	0.974830	+	0.974830i
$342$		0			0
$343$	−143.615	+	311.486i	−0.418702	+	0.908124i
$344$		0			0
$345$			65.3340i			0.189374i
$346$		0			0
$347$	66.0997	−	66.0997i	0.190489	−	0.190489i	−0.605418	−	0.795907i	$-0.706994\pi$
$347$	66.0997	−	66.0997i	0.190489	−	0.190489i	0.795907	+	0.605418i	$0.206994\pi$
$348$		0			0
$349$	224.860	−	224.860i	0.644299	−	0.644299i	−0.307310	−	0.951609i	$-0.599429\pi$
$349$	224.860	−	224.860i	0.644299	−	0.644299i	0.951609	+	0.307310i	$0.0994290\pi$
$350$		0			0
$351$	159.174			0.453486
$352$		0			0
$353$			278.155i			0.787974i	0.919116	+	0.393987i	$0.128905\pi$
$353$			278.155i			0.787974i	−0.919116	+	0.393987i	$0.871095\pi$
$354$		0			0
$355$	−316.024	+	316.024i	−0.890209	+	0.890209i
$356$		0			0
$357$	−49.5699	−	21.2978i	−0.138851	−	0.0596578i
$358$		0			0
$359$		−	473.744i		−	1.31962i	−0.751432	−	0.659810i	$-0.770637\pi$
$359$		−	473.744i		−	1.31962i	0.751432	−	0.659810i	$-0.229363\pi$
$360$		0			0
$361$		−	238.973i		−	0.661974i
$362$		0			0
$363$	−42.7590	−	42.7590i	−0.117793	−	0.117793i
$364$		0			0
$365$	−322.026	−	322.026i	−0.882263	−	0.882263i
$366$		0			0
$367$		−	717.099i		−	1.95395i	−0.213360	−	0.976974i	$-0.568441\pi$
$367$		−	717.099i		−	1.95395i	0.213360	−	0.976974i	$-0.431559\pi$
$368$		0			0
$369$			24.8023i			0.0672149i
$370$		0			0
$371$	202.799	−	472.008i	0.546629	−	1.27226i
$372$		0			0
$373$	226.840	−	226.840i	0.608150	−	0.608150i	−0.334312	−	0.942462i	$-0.608504\pi$
$373$	226.840	−	226.840i	0.608150	−	0.608150i	0.942462	+	0.334312i	$0.108504\pi$
$374$		0			0
$375$			140.502i			0.374673i
$376$		0			0
$377$	271.918			0.721268
$378$		0			0
$379$	−181.444	+	181.444i	−0.478744	+	0.478744i	−0.904730	−	0.425986i	$-0.859927\pi$
$379$	−181.444	+	181.444i	−0.478744	+	0.478744i	0.425986	+	0.904730i	$0.359927\pi$
$380$		0			0
$381$	31.3440	−	31.3440i	0.0822677	−	0.0822677i
$382$		0			0
$383$			453.443i			1.18393i	0.805965	+	0.591963i	$0.201647\pi$
$383$			453.443i			1.18393i	−0.805965	+	0.591963i	$0.798353\pi$
$384$		0			0
$385$	−328.586	+	131.087i	−0.853471	+	0.340485i
$386$		0			0
$387$	−97.6408	−	97.6408i	−0.252302	−	0.252302i
$388$		0			0
$389$	−0.485255	+	0.485255i	−0.00124744	+	0.00124744i	−0.707730	−	0.706483i	$-0.750281\pi$
$389$	−0.485255	+	0.485255i	−0.00124744	+	0.00124744i	0.706483	+	0.707730i	$0.250281\pi$
$390$		0			0
$391$	−43.9842			−0.112492
$392$		0			0
$393$			43.6832i			0.111153i
$394$		0			0
$395$	344.481	+	344.481i	0.872103	+	0.872103i
$396$		0			0
$397$	−113.596	+	113.596i	−0.286135	+	0.286135i	−0.835550	−	0.549415i	$-0.814851\pi$
$397$	−113.596	+	113.596i	−0.286135	+	0.286135i	0.549415	+	0.835550i	$0.314851\pi$
$398$		0			0
$399$	89.9474	+	225.465i	0.225432	+	0.565075i
$400$		0			0
$401$	−115.312			−0.287561			−0.143781	−	0.989610i	$-0.545926\pi$
$401$	−115.312			−0.287561			−0.143781	+	0.989610i	$0.545926\pi$
$402$		0			0
$403$	−264.071	−	264.071i	−0.655262	−	0.655262i
$404$		0			0
$405$	−124.802	−	124.802i	−0.308153	−	0.308153i
$406$		0			0
$407$		−	163.822i		−	0.402511i
$408$		0			0
$409$	498.247			1.21821			0.609104	−	0.793091i	$-0.291529\pi$
$409$	498.247			1.21821			0.609104	+	0.793091i	$0.291529\pi$
$410$		0			0
$411$	−125.697	−	125.697i	−0.305832	−	0.305832i
$412$		0			0
$413$	513.321	+	220.550i	1.24291	+	0.534019i
$414$		0			0
$415$	48.1646			0.116059
$416$		0			0
$417$	−20.0150			−0.0479977
$418$		0			0
$419$	63.7497	−	63.7497i	0.152147	−	0.152147i	−0.626929	−	0.779076i	$-0.715688\pi$
$419$	63.7497	−	63.7497i	0.152147	−	0.152147i	0.779076	+	0.626929i	$0.215688\pi$
$420$		0			0
$421$	−454.190	+	454.190i	−1.07884	+	1.07884i	−0.0822213	+	0.996614i	$0.526201\pi$
$421$	−454.190	+	454.190i	−1.07884	+	1.07884i	−0.996614	+	0.0822213i	$0.973799\pi$
$422$		0			0
$423$	−272.489			−0.644181
$424$		0			0
$425$	41.5115			0.0976742
$426$		0			0
$427$	329.745	−	767.469i	0.772236	−	1.79735i
$428$		0			0
$429$	−62.2586	−	62.2586i	−0.145125	−	0.145125i
$430$		0			0
$431$	−726.777			−1.68626			−0.843128	−	0.537712i	$-0.819289\pi$
$431$	−726.777			−1.68626			−0.843128	+	0.537712i	$0.819289\pi$
$432$		0			0
$433$		−	797.491i		−	1.84178i	−0.389822	−	0.920890i	$-0.627464\pi$
$433$		−	797.491i		−	1.84178i	0.389822	−	0.920890i	$-0.372536\pi$
$434$		0			0
$435$	221.209	+	221.209i	0.508527	+	0.508527i
$436$		0			0
$437$	139.935	+	139.935i	0.320218	+	0.320218i
$438$		0			0
$439$	−238.672			−0.543671			−0.271835	−	0.962344i	$-0.587631\pi$
$439$	−238.672			−0.543671			−0.271835	+	0.962344i	$0.587631\pi$
$440$		0			0
$441$	235.951	+	248.656i	0.535037	+	0.563846i
$442$		0			0
$443$	−340.379	+	340.379i	−0.768351	+	0.768351i	−0.977816	−	0.209466i	$-0.932828\pi$
$443$	−340.379	+	340.379i	−0.768351	+	0.768351i	0.209466	+	0.977816i	$0.432828\pi$
$444$		0			0
$445$	661.684	+	661.684i	1.48693	+	1.48693i
$446$		0			0
$447$		−	218.971i		−	0.489867i
$448$		0			0
$449$	260.491			0.580159			0.290079	−	0.957003i	$-0.406318\pi$
$449$	260.491			0.580159			0.290079	+	0.957003i	$0.406318\pi$
$450$		0			0
$451$	22.1817	−	22.1817i	0.0491833	−	0.0491833i
$452$		0			0
$453$	−213.046	−	213.046i	−0.470301	−	0.470301i
$454$		0			0
$455$	261.027	−	104.135i	0.573687	−	0.228868i
$456$		0			0
$457$		−	599.916i		−	1.31273i	−0.754445	−	0.656363i	$-0.772094\pi$
$457$		−	599.916i		−	1.31273i	0.754445	−	0.656363i	$-0.227906\pi$
$458$		0			0
$459$	−87.1752	+	87.1752i	−0.189924	+	0.189924i
$460$		0			0
$461$	−140.443	+	140.443i	−0.304648	+	0.304648i	−0.842829	−	0.538181i	$-0.819112\pi$
$461$	−140.443	+	140.443i	−0.304648	+	0.304648i	0.538181	+	0.842829i	$0.319112\pi$
$462$		0			0
$463$	830.793			1.79437			0.897185	−	0.441655i	$-0.145608\pi$
$463$	830.793			1.79437			0.897185	+	0.441655i	$0.145608\pi$
$464$		0			0
$465$		−	429.650i		−	0.923979i
$466$		0			0
$467$	198.168	−	198.168i	0.424343	−	0.424343i	−0.462353	−	0.886696i	$-0.652995\pi$
$467$	198.168	−	198.168i	0.424343	−	0.424343i	0.886696	+	0.462353i	$0.152995\pi$
$468$		0			0
$469$	5.48664	+	2.35735i	0.0116986	+	0.00502632i
$470$		0			0
$471$			318.938i			0.677152i
$472$		0			0
$473$			174.648i			0.369235i
$474$		0			0
$475$	−132.069	−	132.069i	−0.278039	−	0.278039i
$476$		0			0
$477$	−363.037	−	363.037i	−0.761084	−	0.761084i
$478$		0			0
$479$			309.380i			0.645887i	0.946418	+	0.322944i	$0.104672\pi$
$479$			309.380i			0.645887i	−0.946418	+	0.322944i	$0.895328\pi$
$480$		0			0
$481$			130.139i			0.270560i
$482$		0			0
$483$	−73.5656	−	31.6076i	−0.152310	−	0.0654402i
$484$		0			0
$485$	362.043	−	362.043i	0.746480	−	0.746480i
$486$		0			0
$487$			376.712i			0.773537i	0.922177	+	0.386768i	$0.126409\pi$
$487$			376.712i			0.773537i	−0.922177	+	0.386768i	$0.873591\pi$
$488$		0			0
$489$	−25.6382			−0.0524298
$490$		0			0
$491$	−404.340	+	404.340i	−0.823502	+	0.823502i	−0.986608	−	0.163106i	$-0.947849\pi$
$491$	−404.340	+	404.340i	−0.823502	+	0.823502i	0.163106	+	0.986608i	$0.447849\pi$
$492$		0			0
$493$	−148.923	+	148.923i	−0.302074	+	0.302074i
$494$		0			0
$495$			353.550i			0.714242i
$496$		0			0
$497$	−202.953	−	508.729i	−0.408357	−	1.02360i
$498$		0			0
$499$	134.474	+	134.474i	0.269487	+	0.269487i	0.828893	−	0.559406i	$-0.188971\pi$
$499$	134.474	+	134.474i	0.269487	+	0.269487i	−0.559406	+	0.828893i	$0.688971\pi$
$500$		0			0
$501$	−29.4311	+	29.4311i	−0.0587446	+	0.0587446i
$502$		0			0
$503$	550.735			1.09490			0.547451	−	0.836838i	$-0.315598\pi$
$503$	550.735			1.09490			0.547451	+	0.836838i	$0.315598\pi$
$504$		0			0
$505$			409.657i			0.811202i
$506$		0			0
$507$	−119.725	−	119.725i	−0.236145	−	0.236145i
$508$		0			0
$509$	244.631	−	244.631i	0.480610	−	0.480610i	−0.424716	−	0.905327i	$-0.639626\pi$
$509$	244.631	−	244.631i	0.480610	−	0.480610i	0.905327	+	0.424716i	$0.139626\pi$
$510$		0			0
$511$	518.390	−	206.807i	1.01446	−	0.404711i
$512$		0			0
$513$	554.695			1.08128
$514$		0			0
$515$	311.169	+	311.169i	0.604212	+	0.604212i
$516$		0			0
$517$	243.697	+	243.697i	0.471368	+	0.471368i
$518$		0			0
$519$			43.3034i			0.0834361i
$520$		0			0
$521$	244.953			0.470160			0.235080	−	0.971976i	$-0.424465\pi$
$521$	244.953			0.470160			0.235080	+	0.971976i	$0.424465\pi$
$522$		0			0
$523$	113.030	+	113.030i	0.216118	+	0.216118i	0.806860	−	0.590742i	$-0.201165\pi$
$523$	113.030	+	113.030i	0.216118	+	0.216118i	−0.590742	+	0.806860i	$0.701165\pi$
$524$		0			0
$525$	69.4300	+	29.8307i	0.132248	+	0.0568205i
$526$		0			0
$527$	289.249			0.548860
$528$		0			0
$529$	463.724			0.876605
$530$		0			0
$531$	394.812	−	394.812i	0.743526	−	0.743526i
$532$		0			0
$533$	−17.6210	+	17.6210i	−0.0330601	+	0.0330601i
$534$		0			0
$535$	1135.19			2.12186
$536$		0			0
$537$	85.7428			0.159670
$538$		0			0
$539$	11.3625	−	433.404i	0.0210807	−	0.804089i
$540$		0			0
$541$	−282.140	−	282.140i	−0.521515	−	0.521515i	0.396514	−	0.918029i	$-0.370220\pi$
$541$	−282.140	−	282.140i	−0.521515	−	0.521515i	−0.918029	+	0.396514i	$0.870220\pi$
$542$		0			0
$543$	−296.382			−0.545824
$544$		0			0
$545$			373.431i			0.685195i
$546$		0			0
$547$	−152.625	−	152.625i	−0.279023	−	0.279023i	0.553696	−	0.832719i	$-0.313217\pi$
$547$	−152.625	−	152.625i	−0.279023	−	0.279023i	−0.832719	+	0.553696i	$0.813217\pi$
$548$		0			0
$549$	−590.286	−	590.286i	−1.07520	−	1.07520i
$550$		0			0
$551$	947.592			1.71977
$552$		0			0
$553$	−554.538	+	221.228i	−1.00278	+	0.400051i
$554$		0			0
$555$	−105.870	+	105.870i	−0.190757	+	0.190757i
$556$		0			0
$557$	−55.7663	−	55.7663i	−0.100119	−	0.100119i	0.655273	−	0.755392i	$-0.272554\pi$
$557$	−55.7663	−	55.7663i	−0.100119	−	0.100119i	−0.755392	+	0.655273i	$0.772554\pi$
$558$		0			0
$559$		−	138.740i		−	0.248193i
$560$		0			0
$561$	68.1949			0.121560
$562$		0			0
$563$	−101.194	+	101.194i	−0.179741	+	0.179741i	−0.791243	−	0.611502i	$-0.790566\pi$
$563$	−101.194	+	101.194i	−0.179741	+	0.179741i	0.611502	+	0.791243i	$0.290566\pi$
$564$		0			0
$565$	659.426	+	659.426i	1.16713	+	1.16713i
$566$		0			0
$567$	200.903	−	80.1487i	0.354327	−	0.141356i
$568$		0			0
$569$			754.689i			1.32634i	0.748468	+	0.663171i	$0.230790\pi$
$569$			754.689i			1.32634i	−0.748468	+	0.663171i	$0.769210\pi$
$570$		0			0
$571$	595.857	−	595.857i	1.04353	−	1.04353i	0.0445235	−	0.999008i	$-0.485823\pi$
$571$	595.857	−	595.857i	1.04353	−	1.04353i	0.999008	−	0.0445235i	$-0.0141770\pi$
$572$		0			0
$573$	−266.289	+	266.289i	−0.464728	+	0.464728i
$574$		0			0
$575$	61.6064			0.107142
$576$		0			0
$577$		−	118.627i		−	0.205593i	−0.994702	−	0.102796i	$-0.967221\pi$
$577$		−	118.627i		−	0.205593i	0.994702	−	0.102796i	$-0.0327790\pi$
$578$		0			0
$579$	287.689	−	287.689i	0.496872	−	0.496872i
$580$		0			0
$581$	−23.3013	+	54.2330i	−0.0401056	+	0.0933443i
$582$		0			0
$583$			649.357i			1.11382i
$584$		0			0
$585$		−	280.858i		−	0.480100i
$586$		0			0
$587$	−386.365	−	386.365i	−0.658203	−	0.658203i	0.296751	−	0.954955i	$-0.404097\pi$
$587$	−386.365	−	386.365i	−0.658203	−	0.658203i	−0.954955	+	0.296751i	$0.904097\pi$
$588$		0			0
$589$	−920.245	−	920.245i	−1.56238	−	1.56238i
$590$		0			0
$591$		−	15.7958i		−	0.0267272i
$592$		0			0
$593$		−	475.843i		−	0.802433i	−0.915983	−	0.401216i	$-0.868588\pi$
$593$		−	475.843i		−	0.802433i	0.915983	−	0.401216i	$-0.131412\pi$
$594$		0			0
$595$	−85.9261	+	199.990i	−0.144414	+	0.336118i
$596$		0			0
$597$	55.4353	−	55.4353i	0.0928564	−	0.0928564i
$598$		0			0
$599$			38.9554i			0.0650341i	0.999471	+	0.0325171i	$0.0103523\pi$
$599$			38.9554i			0.0650341i	−0.999471	+	0.0325171i	$0.989648\pi$
$600$		0			0
$601$	646.318			1.07540			0.537702	−	0.843135i	$-0.319292\pi$
$601$	646.318			1.07540			0.537702	+	0.843135i	$0.319292\pi$
$602$		0			0
$603$	4.21995	−	4.21995i	0.00699826	−	0.00699826i
$604$		0			0
$605$	−172.511	+	172.511i	−0.285143	+	0.285143i
$606$		0			0
$607$		−	1192.00i		−	1.96376i	−0.189505	−	0.981880i	$-0.560688\pi$
$607$		−	1192.00i		−	1.96376i	0.189505	−	0.981880i	$-0.439312\pi$
$608$		0			0
$609$	−356.098	+	142.062i	−0.584725	+	0.233271i
$610$		0			0
$611$	−193.592	−	193.592i	−0.316845	−	0.316845i
$612$		0			0
$613$	48.8446	−	48.8446i	0.0796812	−	0.0796812i	−0.666143	−	0.745824i	$-0.732056\pi$
$613$	48.8446	−	48.8446i	0.0796812	−	0.0796812i	0.745824	+	0.666143i	$0.232056\pi$
$614$		0			0
$615$	−28.6699			−0.0466177
$616$		0			0
$617$			225.471i			0.365432i	0.983166	+	0.182716i	$0.0584888\pi$
$617$			225.471i			0.365432i	−0.983166	+	0.182716i	$0.941511\pi$
$618$		0			0
$619$	−801.515	−	801.515i	−1.29485	−	1.29485i	−0.931747	−	0.363108i	$-0.881716\pi$
$619$	−801.515	−	801.515i	−1.29485	−	1.29485i	−0.363108	−	0.931747i	$-0.618284\pi$
$620$		0			0
$621$	−129.375	+	129.375i	−0.208333	+	0.208333i
$622$		0			0
$623$	−1065.16	+	424.939i	−1.70974	+	0.682084i
$624$		0			0
$625$	757.486			1.21198
$626$		0			0
$627$	−216.962	−	216.962i	−0.346031	−	0.346031i
$628$		0			0
$629$	−71.2740	−	71.2740i	−0.113313	−	0.113313i
$630$		0			0
$631$		−	826.708i		−	1.31016i	−0.755561	−	0.655078i	$-0.772636\pi$
$631$		−	826.708i		−	1.31016i	0.755561	−	0.655078i	$-0.227364\pi$
$632$		0			0
$633$	241.419			0.381388
$634$		0			0
$635$	−126.457	−	126.457i	−0.199146	−	0.199146i
$636$		0			0
$637$	−9.02631	+	344.294i	−0.0141700	+	0.540493i
$638$		0			0
$639$	−547.378			−0.856617
$640$		0			0
$641$	−616.538			−0.961838			−0.480919	−	0.876765i	$-0.659697\pi$
$641$	−616.538			−0.961838			−0.480919	+	0.876765i	$0.659697\pi$
$642$		0			0
$643$	266.078	−	266.078i	0.413807	−	0.413807i	−0.469256	−	0.883062i	$-0.655478\pi$
$643$	266.078	−	266.078i	0.413807	−	0.413807i	0.883062	+	0.469256i	$0.155478\pi$
$644$		0			0
$645$	112.867	−	112.867i	0.174987	−	0.174987i
$646$		0			0
$647$	−856.520			−1.32383			−0.661917	−	0.749577i	$-0.730257\pi$
$647$	−856.520			−1.32383			−0.661917	+	0.749577i	$0.730257\pi$
$648$		0			0
$649$	−706.193			−1.08812
$650$		0			0
$651$	483.783	+	207.858i	0.743138	+	0.319291i
$652$		0			0
$653$	−36.2597	−	36.2597i	−0.0555279	−	0.0555279i	0.678798	−	0.734325i	$-0.262501\pi$
$653$	−36.2597	−	36.2597i	−0.0555279	−	0.0555279i	−0.734325	+	0.678798i	$0.762501\pi$
$654$		0			0
$655$	176.240			0.269069
$656$		0			0
$657$		−	557.773i		−	0.848970i
$658$		0			0
$659$	875.515	+	875.515i	1.32855	+	1.32855i	0.906634	+	0.421917i	$0.138643\pi$
$659$	875.515	+	875.515i	1.32855	+	1.32855i	0.421917	+	0.906634i	$0.361357\pi$
$660$		0			0
$661$	837.005	+	837.005i	1.26627	+	1.26627i	0.948000	+	0.318271i	$0.103102\pi$
$661$	837.005	+	837.005i	1.26627	+	1.26627i	0.318271	+	0.948000i	$0.396898\pi$
$662$		0			0
$663$	−54.1737			−0.0817100
$664$		0			0
$665$	909.640	−	362.893i	1.36788	−	0.545704i
$666$		0			0
$667$	−221.013	+	221.013i	−0.331354	+	0.331354i
$668$		0			0
$669$	120.967	+	120.967i	0.180818	+	0.180818i
$670$		0			0
$671$			1055.83i			1.57352i
$672$		0			0
$673$	104.129			0.154723			0.0773615	−	0.997003i	$-0.475350\pi$
$673$	104.129			0.154723			0.0773615	+	0.997003i	$0.475350\pi$
$674$		0			0
$675$	122.102	−	122.102i	0.180892	−	0.180892i
$676$		0			0
$677$	−465.462	−	465.462i	−0.687536	−	0.687536i	0.274151	−	0.961687i	$-0.411603\pi$
$677$	−465.462	−	465.462i	−0.687536	−	0.687536i	−0.961687	+	0.274151i	$0.911603\pi$
$678$		0			0
$679$	232.507	+	582.809i	0.342425	+	0.858334i
$680$		0			0
$681$			115.070i			0.168972i
$682$		0			0
$683$	616.133	−	616.133i	0.902098	−	0.902098i	−0.0935198	−	0.995617i	$-0.529812\pi$
$683$	616.133	−	616.133i	0.902098	−	0.902098i	0.995617	+	0.0935198i	$0.0298118\pi$
$684$		0			0
$685$	−507.126	+	507.126i	−0.740329	+	0.740329i
$686$		0			0
$687$	351.562			0.511735
$688$		0			0
$689$		−	515.846i		−	0.748688i
$690$		0			0
$691$	−512.637	+	512.637i	−0.741878	+	0.741878i	−0.972939	−	0.231061i	$-0.925780\pi$
$691$	−512.637	+	512.637i	−0.741878	+	0.741878i	0.231061	+	0.972939i	$0.425780\pi$
$692$		0			0
$693$	−398.095	−	171.042i	−0.574451	−	0.246814i
$694$		0			0
$695$			80.7507i			0.116188i
$696$		0			0
$697$		−	19.3012i		−	0.0276918i
$698$		0			0
$699$	184.260	+	184.260i	0.263606	+	0.263606i
$700$		0			0
$701$	−11.7578	−	11.7578i	−0.0167729	−	0.0167729i	0.698671	−	0.715443i	$-0.253775\pi$
$701$	−11.7578	−	11.7578i	−0.0167729	−	0.0167729i	−0.715443	+	0.698671i	$0.753775\pi$
$702$		0			0
$703$			453.515i			0.645114i
$704$		0			0
$705$		−	314.980i		−	0.446780i
$706$		0			0
$707$	−461.271	−	198.186i	−0.652434	−	0.280320i
$708$		0			0
$709$	−400.746	+	400.746i	−0.565228	+	0.565228i	−0.930788	−	0.365560i	$-0.880877\pi$
$709$	−400.746	+	400.746i	−0.565228	+	0.565228i	0.365560	+	0.930788i	$0.380877\pi$
$710$		0			0
$711$			596.667i			0.839194i
$712$		0			0
$713$	429.269			0.602060
$714$		0			0
$715$	−251.183	+	251.183i	−0.351305	+	0.351305i
$716$		0			0
$717$	−33.2207	+	33.2207i	−0.0463329	+	0.0463329i
$718$		0			0
$719$			125.701i			0.174827i	0.996172	+	0.0874135i	$0.0278601\pi$
$719$			125.701i			0.174827i	−0.996172	+	0.0874135i	$0.972140\pi$
$720$		0			0
$721$	−500.914	+	199.835i	−0.694748	+	0.277164i
$722$		0			0
$723$	−55.8373	−	55.8373i	−0.0772300	−	0.0772300i
$724$		0			0
$725$	208.588	−	208.588i	0.287708	−	0.287708i
$726$		0			0
$727$	216.647			0.298001			0.149001	−	0.988837i	$-0.452394\pi$
$727$	216.647			0.298001			0.149001	+	0.988837i	$0.452394\pi$
$728$		0			0
$729$			72.3809i			0.0992879i
$730$		0			0
$731$	75.9842	+	75.9842i	0.103945	+	0.103945i
$732$		0			0
$733$	71.7298	−	71.7298i	0.0978579	−	0.0978579i	−0.656483	−	0.754341i	$-0.727957\pi$
$733$	71.7298	−	71.7298i	0.0978579	−	0.0978579i	0.754341	+	0.656483i	$0.227957\pi$
$734$		0			0
$735$	−287.431	+	272.745i	−0.391063	+	0.371082i
$736$		0			0
$737$	−7.54814			−0.0102417
$738$		0			0
$739$	−532.903	−	532.903i	−0.721113	−	0.721113i	0.247719	−	0.968832i	$-0.420319\pi$
$739$	−532.903	−	532.903i	−0.721113	−	0.721113i	−0.968832	+	0.247719i	$0.920319\pi$
$740$		0			0
$741$	172.353	+	172.353i	0.232595	+	0.232595i
$742$		0			0
$743$		−	579.975i		−	0.780585i	−0.920691	−	0.390292i	$-0.872374\pi$
$743$		−	579.975i		−	0.780585i	0.920691	−	0.390292i	$-0.127626\pi$
$744$		0			0
$745$	−883.438			−1.18582
$746$		0			0
$747$	41.7124	+	41.7124i	0.0558399	+	0.0558399i
$748$		0			0
$749$	−549.190	+	1278.22i	−0.733230	+	1.70657i
$750$		0			0
$751$	−666.251			−0.887151			−0.443576	−	0.896237i	$-0.646290\pi$
$751$	−666.251			−0.887151			−0.443576	+	0.896237i	$0.646290\pi$
$752$		0			0
$753$	−362.729			−0.481712
$754$		0			0
$755$	−859.536	+	859.536i	−1.13846	+	1.13846i
$756$		0			0
$757$	−148.454	+	148.454i	−0.196108	+	0.196108i	−0.798329	−	0.602221i	$-0.794282\pi$
$757$	−148.454	+	148.454i	−0.196108	+	0.196108i	0.602221	+	0.798329i	$0.294282\pi$
$758$		0			0
$759$	101.207			0.133342
$760$		0			0
$761$	714.400			0.938765			0.469382	−	0.882995i	$-0.344477\pi$
$761$	714.400			0.938765			0.469382	+	0.882995i	$0.344477\pi$
$762$		0			0
$763$	−420.481	−	180.660i	−0.551089	−	0.236776i
$764$		0			0
$765$	153.819	+	153.819i	0.201070	+	0.201070i
$766$		0			0
$767$	560.996			0.731416
$768$		0			0
$769$			175.219i			0.227854i	0.993489	+	0.113927i	$0.0363429\pi$
$769$			175.219i			0.227854i	−0.993489	+	0.113927i	$0.963657\pi$
$770$		0			0
$771$	−363.506	−	363.506i	−0.471473	−	0.471473i
$772$		0			0
$773$	−385.461	−	385.461i	−0.498655	−	0.498655i	0.412364	−	0.911019i	$-0.364703\pi$
$773$	−385.461	−	385.461i	−0.498655	−	0.498655i	−0.911019	+	0.412364i	$0.864703\pi$
$774$		0			0
$775$	−405.137			−0.522757
$776$		0			0
$777$	−67.9906	−	170.428i	−0.0875040	−	0.219340i
$778$		0			0
$779$	−61.4065	+	61.4065i	−0.0788273	+	0.0788273i
$780$		0			0
$781$	489.542	+	489.542i	0.626814	+	0.626814i
$782$		0			0
$783$			876.080i			1.11888i
$784$		0			0
$785$	1286.76			1.63918
$786$		0			0
$787$	−661.030	+	661.030i	−0.839937	+	0.839937i	−0.988850	−	0.148913i	$-0.952422\pi$
$787$	−661.030	+	661.030i	−0.839937	+	0.839937i	0.148913	+	0.988850i	$0.452422\pi$
$788$		0			0
$789$	−421.694	−	421.694i	−0.534466	−	0.534466i
$790$		0			0
$791$	−1061.53	+	423.489i	−1.34201	+	0.535384i
$792$		0			0
$793$		−	838.748i		−	1.05769i
$794$		0			0
$795$	419.648	−	419.648i	0.527859	−	0.527859i
$796$		0			0
$797$	644.542	−	644.542i	0.808711	−	0.808711i	−0.175728	−	0.984439i	$-0.556228\pi$
$797$	644.542	−	644.542i	0.808711	−	0.808711i	0.984439	+	0.175728i	$0.0562279\pi$
$798$		0			0
$799$	212.051			0.265395
$800$		0			0
$801$			1146.09i			1.43082i
$802$		0			0
$803$	−498.839	+	498.839i	−0.621219	+	0.621219i
$804$		0			0
$805$	−127.521	+	296.801i	−0.158411	+	0.368697i
$806$		0			0
$807$		−	342.777i		−	0.424755i
$808$		0			0
$809$		−	1222.06i		−	1.51058i	−0.655388	−	0.755292i	$-0.727495\pi$
$809$		−	1222.06i		−	1.51058i	0.655388	−	0.755292i	$-0.272505\pi$
$810$		0			0
$811$	236.769	+	236.769i	0.291947	+	0.291947i	0.837849	−	0.545902i	$-0.183813\pi$
$811$	236.769	+	236.769i	0.291947	+	0.291947i	−0.545902	+	0.837849i	$0.683813\pi$
$812$		0			0
$813$	−15.9965	−	15.9965i	−0.0196759	−	0.0196759i
$814$		0			0
$815$			103.437i			0.126917i
$816$		0			0
$817$		−	483.486i		−	0.591782i
$818$		0			0
$819$	316.244	+	135.875i	0.386135	+	0.165904i
$820$		0			0
$821$	−927.111	+	927.111i	−1.12925	+	1.12925i	−0.138946	+	0.990300i	$0.544371\pi$
$821$	−927.111	+	927.111i	−1.12925	+	1.12925i	−0.990300	+	0.138946i	$0.955629\pi$
$822$		0			0
$823$		−	386.940i		−	0.470158i	−0.971976	−	0.235079i	$-0.924465\pi$
$823$		−	386.940i		−	0.470158i	0.971976	−	0.235079i	$-0.0755348\pi$
$824$		0			0
$825$	−95.5171			−0.115778
$826$		0			0
$827$	−431.979	+	431.979i	−0.522345	+	0.522345i	−0.918279	−	0.395934i	$-0.870421\pi$
$827$	−431.979	+	431.979i	−0.522345	+	0.522345i	0.395934	+	0.918279i	$0.370421\pi$
$828$		0			0
$829$	546.644	−	546.644i	0.659401	−	0.659401i	−0.295837	−	0.955238i	$-0.595599\pi$
$829$	546.644	−	546.644i	0.659401	−	0.659401i	0.955238	+	0.295837i	$0.0955986\pi$
$830$		0			0
$831$		−	402.783i		−	0.484697i
$832$		0			0
$833$	−183.617	−	193.504i	−0.220429	−	0.232298i
$834$		0			0
$835$	118.740	+	118.740i	0.142203	+	0.142203i
$836$		0			0
$837$	850.797	−	850.797i	1.01648	−	1.01648i
$838$		0			0
$839$	196.129			0.233765			0.116883	−	0.993146i	$-0.462710\pi$
$839$	196.129			0.233765			0.116883	+	0.993146i	$0.462710\pi$
$840$		0			0
$841$			655.619i			0.779571i
$842$		0			0
$843$	−365.096	−	365.096i	−0.433091	−	0.433091i
$844$		0			0
$845$	−483.033	+	483.033i	−0.571637	+	0.571637i
$846$		0			0
$847$	−110.788	−	277.705i	−0.130800	−	0.327869i
$848$		0			0
$849$	−623.385			−0.734258
$850$		0			0
$851$	−105.776	−	105.776i	−0.124296	−	0.124296i
$852$		0			0
$853$	463.772	+	463.772i	0.543695	+	0.543695i	0.924610	−	0.380915i	$-0.124391\pi$
$853$	463.772	+	463.772i	0.543695	+	0.543695i	−0.380915	+	0.924610i	$0.624391\pi$
$854$		0			0
$855$		−	978.747i		−	1.14473i
$856$		0			0
$857$	−1536.03			−1.79234			−0.896168	−	0.443714i	$-0.853661\pi$
$857$	−1536.03			−1.79234			−0.896168	+	0.443714i	$0.853661\pi$
$858$		0			0
$859$	−36.4535	−	36.4535i	−0.0424371	−	0.0424371i	0.685570	−	0.728007i	$-0.259553\pi$
$859$	−36.4535	−	36.4535i	−0.0424371	−	0.0424371i	−0.728007	+	0.685570i	$0.759553\pi$
$860$		0			0
$861$	13.8701	−	32.2821i	0.0161093	−	0.0374937i
$862$		0			0
$863$	304.834			0.353226			0.176613	−	0.984280i	$-0.443486\pi$
$863$	304.834			0.353226			0.176613	+	0.984280i	$0.443486\pi$
$864$		0			0
$865$	174.708			0.201974
$866$		0			0
$867$	−259.644	+	259.644i	−0.299474	+	0.299474i
$868$		0			0
$869$	533.623	−	533.623i	0.614066	−	0.614066i
$870$		0			0
$871$	5.99621			0.00688428
$872$		0			0
$873$	627.086			0.718311
$874$		0			0
$875$	−274.237	+	638.277i	−0.313414	+	0.729459i
$876$		0			0
$877$	−545.809	−	545.809i	−0.622359	−	0.622359i	0.323775	−	0.946134i	$-0.395048\pi$
$877$	−545.809	−	545.809i	−0.622359	−	0.622359i	−0.946134	+	0.323775i	$0.895048\pi$
$878$		0			0
$879$	498.963			0.567649
$880$		0			0
$881$			1051.42i			1.19344i	0.802449	+	0.596721i	$0.203530\pi$
$881$			1051.42i			1.19344i	−0.802449	+	0.596721i	$0.796470\pi$
$882$		0			0
$883$	−693.577	−	693.577i	−0.785478	−	0.785478i	0.195271	−	0.980749i	$-0.437441\pi$
$883$	−693.577	−	693.577i	−0.785478	−	0.785478i	−0.980749	+	0.195271i	$0.937441\pi$
$884$		0			0
$885$	456.378	+	456.378i	0.515682	+	0.515682i
$886$		0			0
$887$	−579.072			−0.652843			−0.326422	−	0.945224i	$-0.605843\pi$
$887$	−579.072			−0.652843			−0.326422	+	0.945224i	$0.605843\pi$
$888$		0			0
$889$	203.568	−	81.2119i	0.228986	−	0.0913520i
$890$		0			0
$891$	−193.326	+	193.326i	−0.216977	+	0.216977i
$892$		0			0
$893$	−674.638	−	674.638i	−0.755474	−	0.755474i
$894$		0			0
$895$		−	345.929i		−	0.386513i
$896$		0			0
$897$	−80.3981			−0.0896299
$898$		0			0
$899$	1453.43	−	1453.43i	1.61672	−	1.61672i
$900$		0			0
$901$	282.516	+	282.516i	0.313558	+	0.313558i
$902$		0			0
$903$	72.4838	+	181.690i	0.0802700	+	0.201207i
$904$		0			0
$905$			1195.76i			1.32128i
$906$		0			0
$907$	134.035	−	134.035i	0.147778	−	0.147778i	−0.629347	−	0.777125i	$-0.716677\pi$
$907$	134.035	−	134.035i	0.147778	−	0.147778i	0.777125	+	0.629347i	$0.216677\pi$
$908$		0			0
$909$	−354.779	+	354.779i	−0.390296	+	0.390296i
$910$		0			0
$911$	−41.6323			−0.0456996			−0.0228498	−	0.999739i	$-0.507274\pi$
$911$	−41.6323			−0.0456996			−0.0228498	+	0.999739i	$0.507274\pi$
$912$		0			0
$913$		−	74.6101i		−	0.0817197i
$914$		0			0
$915$	682.333	−	682.333i	0.745719	−	0.745719i
$916$		0			0
$917$	−85.2623	+	198.445i	−0.0929796	+	0.216407i
$918$		0			0
$919$		−	772.416i		−	0.840496i	−0.907409	−	0.420248i	$-0.861943\pi$
$919$		−	772.416i		−	0.840496i	0.907409	−	0.420248i	$-0.138057\pi$
$920$		0			0
$921$		−	532.361i		−	0.578025i
$922$		0			0
$923$	−388.890	−	388.890i	−0.421333	−	0.421333i
$924$		0			0
$925$	99.8298	+	99.8298i	0.107924	+	0.107924i
$926$		0			0
$927$			538.969i			0.581412i
$928$		0			0
$929$			1353.43i			1.45687i	0.685114	+	0.728436i	$0.259753\pi$
$929$			1353.43i			1.45687i	−0.685114	+	0.728436i	$0.740247\pi$
$930$		0			0
$931$	−31.4553	+	1199.81i	−0.0337865	+	1.28873i
$932$		0			0
$933$	115.517	−	115.517i	0.123813	−	0.123813i
$934$		0			0
$935$		−	275.133i		−	0.294260i
$936$		0			0
$937$	−1628.25			−1.73772			−0.868861	−	0.495056i	$-0.835148\pi$
$937$	−1628.25			−1.73772			−0.868861	+	0.495056i	$0.835148\pi$
$938$		0			0
$939$	334.681	−	334.681i	0.356423	−	0.356423i
$940$		0			0
$941$	293.378	−	293.378i	0.311772	−	0.311772i	−0.533824	−	0.845596i	$-0.679245\pi$
$941$	293.378	−	293.378i	0.311772	−	0.311772i	0.845596	+	0.533824i	$0.179245\pi$
$942$		0			0
$943$		−	28.6444i		−	0.0303759i
$944$		0			0
$945$	335.507	+	840.992i	0.355034	+	0.889939i
$946$		0			0
$947$	375.023	+	375.023i	0.396012	+	0.396012i	0.876824	−	0.480812i	$-0.159658\pi$
$947$	375.023	+	375.023i	0.396012	+	0.396012i	−0.480812	+	0.876824i	$0.659658\pi$
$948$		0			0
$949$	396.275	−	396.275i	0.417571	−	0.417571i
$950$		0			0
$951$	−53.1079			−0.0558442
$952$		0			0
$953$			263.530i			0.276527i	0.990395	+	0.138264i	$0.0441521\pi$
$953$			263.530i			0.276527i	−0.990395	+	0.138264i	$0.955848\pi$
$954$		0			0
$955$	1074.34	+	1074.34i	1.12497	+	1.12497i
$956$		0			0
$957$	342.667	−	342.667i	0.358064	−	0.358064i
$958$		0			0
$959$	−325.680	−	816.360i	−0.339604	−	0.851262i
$960$		0			0
$961$	−1861.96			−1.93753
$962$		0			0
$963$	983.120	+	983.120i	1.02089	+	1.02089i
$964$		0			0
$965$	−1160.68	−	1160.68i	−1.20278	−	1.20278i
$966$		0			0
$967$			791.495i			0.818505i	0.912421	+	0.409253i	$0.134211\pi$
$967$			791.495i			0.818505i	−0.912421	+	0.409253i	$0.865789\pi$
$968$		0			0
$969$	−188.787			−0.194827
$970$		0			0
$971$	1184.27	+	1184.27i	1.21964	+	1.21964i	0.967758	+	0.251883i	$0.0810499\pi$
$971$	1184.27	+	1184.27i	1.21964	+	1.21964i	0.251883	+	0.967758i	$0.418950\pi$
$972$		0			0
$973$	−90.9247	−	39.0660i	−0.0934478	−	0.0401501i
$974$		0			0
$975$	75.8783			0.0778239
$976$		0			0
$977$	615.856			0.630354			0.315177	−	0.949033i	$-0.397936\pi$
$977$	615.856			0.630354			0.315177	+	0.949033i	$0.397936\pi$
$978$		0			0
$979$	1024.99	−	1024.99i	1.04698	−	1.04698i
$980$		0			0
$981$	−323.406	+	323.406i	−0.329669	+	0.329669i
$982$		0			0
$983$	−534.869			−0.544119			−0.272059	−	0.962280i	$-0.587705\pi$
$983$	−534.869			−0.544119			−0.272059	+	0.962280i	$0.587705\pi$
$984$		0			0
$985$	−63.7280			−0.0646985
$986$		0			0
$987$	354.665	+	152.383i	0.359336	+	0.154390i
$988$		0			0
$989$	112.766	+	112.766i	0.114021	+	0.114021i
$990$		0			0
$991$	125.613			0.126754			0.0633768	−	0.997990i	$-0.479813\pi$
$991$	125.613			0.126754			0.0633768	+	0.997990i	$0.479813\pi$
$992$		0			0
$993$		−	67.1950i		−	0.0676687i
$994$		0			0
$995$	−223.654	−	223.654i	−0.224778	−	0.224778i
$996$		0			0
$997$	83.7554	+	83.7554i	0.0840075	+	0.0840075i	0.747862	−	0.663854i	$-0.231081\pi$
$997$	83.7554	+	83.7554i	0.0840075	+	0.0840075i	−0.663854	+	0.747862i	$0.731081\pi$
$998$		0			0
$999$	−419.290			−0.419710

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.3.l.b.209.12		56
4.3	odd	2		112.3.l.b.13.18	yes	56
7.6	odd	2	inner	448.3.l.b.209.17		56
16.5	even	4	inner	448.3.l.b.433.17		56
16.11	odd	4		112.3.l.b.69.17	yes	56
28.27	even	2		112.3.l.b.13.17	✓	56
112.27	even	4		112.3.l.b.69.18	yes	56
112.69	odd	4	inner	448.3.l.b.433.12		56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.3.l.b.13.17	✓	56	28.27	even	2
112.3.l.b.13.18	yes	56	4.3	odd	2
112.3.l.b.69.17	yes	56	16.11	odd	4
112.3.l.b.69.18	yes	56	112.27	even	4
448.3.l.b.209.12		56	1.1	even	1	trivial
448.3.l.b.209.17		56	7.6	odd	2	inner
448.3.l.b.433.12		56	112.69	odd	4	inner
448.3.l.b.433.17		56	16.5	even	4	inner

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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	448.l (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00109 + 1.00109i) q^{3} +(4.03888 + 4.03888i) q^{5} +(-6.50171 + 2.59380i) q^{7} +6.99566i q^{9} +(6.25649 - 6.25649i) q^{11} +(-4.97013 + 4.97013i) q^{13} -8.08654 q^{15} -5.44402i q^{17} +(-17.3201 + 17.3201i) q^{19} +(3.91215 - 9.10538i) q^{21} -8.07936i q^{23} +7.62516i q^{25} +(-16.0130 - 16.0130i) q^{27} +(-27.3552 - 27.3552i) q^{29} +53.1316i q^{31} +12.5266i q^{33} +(-36.7357 - 15.7836i) q^{35} +(13.0922 - 13.0922i) q^{37} -9.95105i q^{39} +3.54538 q^{41} +(-13.9574 + 13.9574i) q^{43} +(-28.2546 + 28.2546i) q^{45} +38.9511i q^{47} +(35.5444 - 33.7283i) q^{49} +(5.44993 + 5.44993i) q^{51} +(-51.8947 + 51.8947i) q^{53} +50.5385 q^{55} -34.6778i q^{57} +(-56.4368 - 56.4368i) q^{59} +(-84.3789 + 84.3789i) q^{61} +(-18.1453 - 45.4837i) q^{63} -40.1475 q^{65} +(-0.603225 - 0.603225i) q^{67} +(8.08813 + 8.08813i) q^{69} +78.2455i q^{71} -79.7314 q^{73} +(-7.63344 - 7.63344i) q^{75} +(-24.4498 + 56.9060i) q^{77} +85.2911 q^{79} -30.9001 q^{81} +(5.96262 - 5.96262i) q^{83} +(21.9878 - 21.9878i) q^{85} +54.7699 q^{87} +163.829 q^{89} +(19.4228 - 45.2058i) q^{91} +(-53.1892 - 53.1892i) q^{93} -139.908 q^{95} -89.6393i q^{97} +(43.7683 + 43.7683i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 8 q^{15} - 20 q^{21} - 96 q^{29} + 100 q^{35} - 128 q^{37} + 72 q^{43} + 192 q^{49} + 128 q^{51} + 88 q^{53} - 444 q^{63} - 8 q^{65} - 440 q^{67} + 12 q^{77} + 8 q^{79} + 64 q^{81} + 96 q^{85} + 388 q^{91}+ \cdots - 8 q^{99}+O(q^{100}) \) 56 * q + 8 * q^15 - 20 * q^21 - 96 * q^29 + 100 * q^35 - 128 * q^37 + 72 * q^43 + 192 * q^49 + 128 * q^51 + 88 * q^53 - 444 * q^63 - 8 * q^65 - 440 * q^67 + 12 * q^77 + 8 * q^79 + 64 * q^81 + 96 * q^85 + 388 * q^91 + 32 * q^93 + 776 * q^95 - 8 * q^99

Introduction

L-functions

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