LMFDB - Embedded newform 448.3.l.b.209.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,3,Mod(209,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 3, 2]))

N = Newforms(chi, 3, names="a")

//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: f = N[0] # Warning: the index may be different

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			209.17
Character	$\chi$	$=$	448.209
Dual form			448.3.l.b.433.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00109 - 1.00109i) q^{3} +(-4.03888 - 4.03888i) q^{5} +(6.50171 + 2.59380i) q^{7} +6.99566i q^{9} +O(q^{10})$	\(q+(1.00109 - 1.00109i) q^{3} +(-4.03888 - 4.03888i) q^{5} +(6.50171 + 2.59380i) q^{7} +6.99566i q^{9} +(6.25649 - 6.25649i) q^{11} +(4.97013 - 4.97013i) q^{13} -8.08654 q^{15} +5.44402i q^{17} +(17.3201 - 17.3201i) q^{19} +(9.10538 - 3.91215i) q^{21} -8.07936i q^{23} +7.62516i q^{25} +(16.0130 + 16.0130i) q^{27} +(-27.3552 - 27.3552i) q^{29} -53.1316i q^{31} -12.5266i q^{33} +(-15.7836 - 36.7357i) q^{35} +(13.0922 - 13.0922i) q^{37} -9.95105i q^{39} -3.54538 q^{41} +(-13.9574 + 13.9574i) q^{43} +(28.2546 - 28.2546i) q^{45} -38.9511i q^{47} +(35.5444 + 33.7283i) q^{49} +(5.44993 + 5.44993i) q^{51} +(-51.8947 + 51.8947i) q^{53} -50.5385 q^{55} -34.6778i q^{57} +(56.4368 + 56.4368i) q^{59} +(84.3789 - 84.3789i) q^{61} +(-18.1453 + 45.4837i) q^{63} -40.1475 q^{65} +(-0.603225 - 0.603225i) q^{67} +(-8.08813 - 8.08813i) q^{69} +78.2455i q^{71} +79.7314 q^{73} +(7.63344 + 7.63344i) q^{75} +(56.9060 - 24.4498i) q^{77} +85.2911 q^{79} -30.9001 q^{81} +(-5.96262 + 5.96262i) q^{83} +(21.9878 - 21.9878i) q^{85} -54.7699 q^{87} -163.829 q^{89} +(45.2058 - 19.4228i) q^{91} +(-53.1892 - 53.1892i) q^{93} -139.908 q^{95} +89.6393i q^{97} +(43.7683 + 43.7683i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q+O(q^{10}) $ 56 * 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} + 32 q^{93} + 776 q^{95} - 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

Display 100 coefficients 10 coefficients

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	−0.520293	−	0.853988i	$-0.674177\pi$
$3$	1.00109	−	1.00109i	0.333695	−	0.333695i	0.853988	+	0.520293i	$0.174177\pi$
$4$		0			0
$5$	−4.03888	−	4.03888i	−0.807777	−	0.807777i	0.176520	−	0.984297i	$-0.443516\pi$
$5$	−4.03888	−	4.03888i	−0.807777	−	0.807777i	−0.984297	+	0.176520i	$0.943516\pi$
$6$		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	−0.489619	−	0.871936i	$-0.662864\pi$
$13$	4.97013	−	4.97013i	0.382318	−	0.382318i	0.871936	+	0.489619i	$0.162864\pi$
$14$		0			0
$15$	−8.08654			−0.539102
$16$		0			0
$17$			5.44402i			0.320237i	0.987098	+	0.160118i	$0.0511876\pi$
$17$			5.44402i			0.320237i	−0.987098	+	0.160118i	$0.948812\pi$
$18$		0			0
$19$	17.3201	−	17.3201i	0.911585	−	0.911585i	−0.0848120	−	0.996397i	$-0.527029\pi$
$19$	17.3201	−	17.3201i	0.911585	−	0.911585i	0.996397	+	0.0848120i	$0.0270290\pi$
$20$		0			0
$21$	9.10538	−	3.91215i	0.433590	−	0.186293i
$22$		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	$-0.672350\pi$
$31$		−	53.1316i		−	1.71392i	0.515382	−	0.856961i	$-0.327650\pi$
$32$		0			0
$33$		−	12.5266i		−	0.379593i
$34$		0			0
$35$	−15.7836	−	36.7357i	−0.450959	−	1.04959i
$36$		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	$-0.513767\pi$
$41$	−3.54538			−0.0864728			−0.0432364	+	0.999065i	$0.513767\pi$
$42$		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	$-0.864001\pi$
$47$		−	38.9511i		−	0.828747i	0.910107	−	0.414374i	$-0.135999\pi$
$48$		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	0.999095	−	0.0425390i	$-0.0135447\pi$
$59$	56.4368	+	56.4368i	0.956556	+	0.956556i	−0.0425390	+	0.999095i	$0.513545\pi$
$60$		0			0
$61$	84.3789	−	84.3789i	1.38326	−	1.38326i	0.544499	−	0.838762i	$-0.316720\pi$
$61$	84.3789	−	84.3789i	1.38326	−	1.38326i	0.838762	−	0.544499i	$-0.183280\pi$
$62$		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	$-0.316110\pi$
$73$	79.7314			1.09221			0.546106	+	0.837716i	$0.316110\pi$
$74$		0			0
$75$	7.63344	+	7.63344i	0.101779	+	0.101779i
$76$		0			0
$77$	56.9060	−	24.4498i	0.739039	−	0.317530i
$78$		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	−0.742113	−	0.670275i	$-0.766176\pi$
$83$	−5.96262	+	5.96262i	−0.0718387	+	0.0718387i	0.670275	+	0.742113i	$0.266176\pi$
$84$		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	$-0.872125\pi$
$89$	−163.829			−1.84077			−0.920385	+	0.391014i	$0.872125\pi$
$90$		0			0
$91$	45.2058	−	19.4228i	0.496767	−	0.213437i
$92$		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	$0.152889\pi$
$97$			89.6393i			0.924117i	−0.886850	+	0.462058i	$0.847111\pi$
$98$		0			0
$99$	43.7683	+	43.7683i	0.442104	+	0.442104i
$100$		0			0
$101$	−50.7141	−	50.7141i	−0.502120	−	0.502120i	0.409976	−	0.912096i	$-0.365537\pi$
$101$	−50.7141	−	50.7141i	−0.502120	−	0.502120i	−0.912096	+	0.409976i	$0.865537\pi$
$102$		0			0
$103$	−77.0434			−0.747994			−0.373997	−	0.927430i	$-0.622013\pi$
$103$	−77.0434			−0.747994			−0.373997	+	0.927430i	$0.622013\pi$
$104$		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	−0.785461	−	0.618912i	$-0.787574\pi$
$131$	−21.8179	+	21.8179i	−0.166549	+	0.166549i	0.618912	+	0.785461i	$0.287574\pi$
$132$		0			0
$133$	157.535	−	67.6854i	1.18448	−	0.508913i
$134$		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	0.670233	−	0.742151i	$-0.266194\pi$
$139$	−9.99666	−	9.99666i	−0.0719184	−	0.0719184i	−0.742151	+	0.670233i	$0.766194\pi$
$140$		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
$147$	69.3479	−	1.81808i	0.471754	−	0.0123679i
$148$		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	$0.504690\pi$
$157$	−159.296	+	159.296i	−1.01463	+	1.01463i	−0.999891	+	0.0147347i	$0.995310\pi$
$158$		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	$-0.528054\pi$
$167$	−29.3992			−0.176043			−0.0880214	+	0.996119i	$0.528054\pi$
$168$		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	−0.766848	−	0.641829i	$-0.778176\pi$
$173$	−21.6282	+	21.6282i	−0.125018	+	0.125018i	0.641829	+	0.766848i	$0.278176\pi$
$174$		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	0.167948	−	0.985796i	$-0.446286\pi$
$181$	−148.030	−	148.030i	−0.817848	−	0.817848i	−0.985796	+	0.167948i	$0.946286\pi$
$182$		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
$189$	62.5774	+	145.647i	0.331097	+	0.770617i
$190$		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	$-0.455568\pi$
$199$	55.3752			0.278267			0.139134	+	0.990274i	$0.455568\pi$
$200$		0			0
$201$	−1.20776			−0.00600875
$202$		0			0
$203$	−106.902	−	248.810i	−0.526609	−	1.22566i
$204$		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	$0.0873321\pi$
$223$			120.836i			0.541865i	−0.962598	+	0.270933i	$0.912668\pi$
$224$		0			0
$225$	−53.3430			−0.237080
$226$		0			0
$227$	−57.4724	+	57.4724i	−0.253183	+	0.253183i	−0.822274	−	0.569092i	$-0.807295\pi$
$227$	−57.4724	+	57.4724i	−0.253183	+	0.253183i	0.569092	+	0.822274i	$0.307295\pi$
$228$		0			0
$229$	175.590	+	175.590i	0.766771	+	0.766771i	0.977537	−	0.210766i	$-0.0675958\pi$
$229$	175.590	+	175.590i	0.766771	+	0.766771i	−0.210766	+	0.977537i	$0.567596\pi$
$230$		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	$-0.963083\pi$
$241$		−	55.7768i		−	0.231439i	0.993282	−	0.115719i	$-0.0369173\pi$
$242$		0			0
$243$	−175.051	+	175.051i	−0.720374	+	0.720374i
$244$		0			0
$245$	−7.33506	−	279.784i	−0.0299390	−	1.14198i
$246$		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	0.247184	−	0.968969i	$-0.420495\pi$
$251$	−181.168	−	181.168i	−0.721785	−	0.721785i	−0.968969	+	0.247184i	$0.920495\pi$
$252$		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	$-0.750299\pi$
$257$		−	363.112i		−	1.41289i	0.707770	−	0.706443i	$-0.249701\pi$
$258$		0			0
$259$	119.080	−	51.1629i	0.459768	−	0.197540i
$260$		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	−0.313235	−	0.949676i	$-0.601413\pi$
$269$	171.203	−	171.203i	0.636441	−	0.636441i	0.949676	+	0.313235i	$0.101413\pi$
$270$		0			0
$271$		−	15.9792i		−	0.0589637i	−0.999565	−	0.0294819i	$-0.990614\pi$
$271$		−	15.9792i		−	0.0589637i	0.999565	−	0.0294819i	$-0.00938573\pi$
$272$		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	$-0.966258\pi$
$283$	−311.354	−	311.354i	−1.10019	−	1.10019i	−0.105806	−	0.994387i	$-0.533742\pi$
$284$		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	0.990201	−	0.139651i	$-0.0445980\pi$
$293$	249.211	+	249.211i	0.850550	+	0.850550i	−0.139651	+	0.990201i	$0.544598\pi$
$294$		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
$301$	−126.949	+	54.5440i	−0.421758	+	0.181209i
$302$		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	−0.125940	−	0.992038i	$-0.540195\pi$
$307$	265.892	−	265.892i	0.866098	−	0.866098i	0.992038	+	0.125940i	$0.0401947\pi$
$308$		0			0
$309$	−77.1270	+	77.1270i	−0.249602	+	0.249602i
$310$		0			0
$311$	115.392			0.371035			0.185518	−	0.982641i	$-0.440604\pi$
$311$	115.392			0.371035			0.185518	+	0.982641i	$0.440604\pi$
$312$		0			0
$313$	334.318			1.06811			0.534054	−	0.845450i	$-0.320668\pi$
$313$	334.318			1.06811			0.534054	+	0.845450i	$0.320668\pi$
$314$		0			0
$315$	256.990	−	110.416i	0.815842	−	0.350528i
$316$		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	−0.951609	−	0.307310i	$-0.900571\pi$
$349$	−224.860	+	224.860i	−0.644299	+	0.644299i	0.307310	+	0.951609i	$0.400571\pi$
$350$		0			0
$351$	159.174			0.453486
$352$		0			0
$353$		−	278.155i		−	0.787974i	−0.919116	−	0.393987i	$-0.871095\pi$
$353$		−	278.155i		−	0.787974i	0.919116	−	0.393987i	$-0.128905\pi$
$354$		0			0
$355$	316.024	−	316.024i	0.890209	−	0.890209i
$356$		0			0
$357$	21.2978	+	49.5699i	0.0596578	+	0.138851i
$358$		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	$0.431559\pi$
$367$			717.099i			1.95395i	−0.213360	+	0.976974i	$0.568441\pi$
$368$		0			0
$369$		−	24.8023i		−	0.0672149i
$370$		0			0
$371$	−472.008	+	202.799i	−1.27226	+	0.546629i
$372$		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	$-0.798353\pi$
$383$		−	453.443i		−	1.18393i	0.805965	−	0.591963i	$-0.201647\pi$
$384$		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	−0.549415	−	0.835550i	$-0.685149\pi$
$397$	113.596	−	113.596i	0.286135	−	0.286135i	0.835550	+	0.549415i	$0.185149\pi$
$398$		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	$-0.708471\pi$
$409$	−498.247			−1.21821			−0.609104	+	0.793091i	$0.708471\pi$
$410$		0			0
$411$	125.697	+	125.697i	0.305832	+	0.305832i
$412$		0			0
$413$	220.550	+	513.321i	0.534019	+	1.24291i
$414$		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	−0.779076	−	0.626929i	$-0.784312\pi$
$419$	−63.7497	+	63.7497i	−0.152147	+	0.152147i	0.626929	+	0.779076i	$0.284312\pi$
$420$		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
$427$	767.469	−	329.745i	1.79735	−	0.772236i
$428$		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	$0.372536\pi$
$433$			797.491i			1.84178i	−0.389822	+	0.920890i	$0.627464\pi$
$434$		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	$-0.412369\pi$
$439$	238.672			0.543671			0.271835	+	0.962344i	$0.412369\pi$
$440$		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	−0.538181	−	0.842829i	$-0.680888\pi$
$461$	140.443	−	140.443i	0.304648	−	0.304648i	0.842829	+	0.538181i	$0.180888\pi$
$462$		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	−0.886696	−	0.462353i	$-0.847005\pi$
$467$	−198.168	+	198.168i	−0.424343	+	0.424343i	0.462353	+	0.886696i	$0.347005\pi$
$468$		0			0
$469$	−2.35735	−	5.48664i	−0.00502632	−	0.0116986i
$470$		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	$-0.895328\pi$
$479$		−	309.380i		−	0.645887i	0.946418	−	0.322944i	$-0.104672\pi$
$480$		0			0
$481$		−	130.139i		−	0.270560i
$482$		0			0
$483$	−31.6076	−	73.5656i	−0.0654402	−	0.152310i
$484$		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	$-0.684402\pi$
$503$	−550.735			−1.09490			−0.547451	+	0.836838i	$0.684402\pi$
$504$		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	−0.905327	−	0.424716i	$-0.860374\pi$
$509$	−244.631	+	244.631i	−0.480610	+	0.480610i	0.424716	+	0.905327i	$0.360374\pi$
$510$		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	$-0.575535\pi$
$521$	−244.953			−0.470160			−0.235080	+	0.971976i	$0.575535\pi$
$522$		0			0
$523$	−113.030	−	113.030i	−0.216118	−	0.216118i	0.590742	−	0.806860i	$-0.298835\pi$
$523$	−113.030	−	113.030i	−0.216118	−	0.216118i	−0.806860	+	0.590742i	$0.798835\pi$
$524$		0			0
$525$	29.8307	+	69.4300i	0.0568205	+	0.132248i
$526$		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
$539$	433.404	−	11.3625i	0.804089	−	0.0210807i
$540$		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	−0.611502	−	0.791243i	$-0.709434\pi$
$563$	101.194	−	101.194i	0.179741	−	0.179741i	0.791243	+	0.611502i	$0.209434\pi$
$564$		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	$0.0327790\pi$
$577$			118.627i			0.205593i	−0.994702	+	0.102796i	$0.967221\pi$
$578$		0			0
$579$	−287.689	+	287.689i	−0.496872	+	0.496872i
$580$		0			0
$581$	−54.2330	+	23.3013i	−0.0933443	+	0.0401056i
$582$		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	0.954955	−	0.296751i	$-0.0959032\pi$
$587$	386.365	+	386.365i	0.658203	+	0.658203i	−0.296751	+	0.954955i	$0.595903\pi$
$588$		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	$0.131412\pi$
$593$			475.843i			0.802433i	−0.915983	+	0.401216i	$0.868588\pi$
$594$		0			0
$595$	199.990	−	85.9261i	0.336118	−	0.144414i
$596$		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	$-0.680708\pi$
$601$	−646.318			−1.07540			−0.537702	+	0.843135i	$0.680708\pi$
$602$		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	$0.439312\pi$
$607$			1192.00i			1.96376i	−0.189505	+	0.981880i	$0.560688\pi$
$608$		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	$0.118284\pi$
$619$	801.515	+	801.515i	1.29485	+	1.29485i	0.363108	+	0.931747i	$0.381716\pi$
$620$		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
$637$	344.294	−	9.02631i	0.540493	−	0.0141700i
$638$		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	−0.883062	−	0.469256i	$-0.844522\pi$
$643$	−266.078	+	266.078i	−0.413807	+	0.413807i	0.469256	+	0.883062i	$0.344522\pi$
$644$		0			0
$645$	112.867	−	112.867i	0.174987	−	0.174987i
$646$		0			0
$647$	856.520			1.32383			0.661917	−	0.749577i	$-0.269743\pi$
$647$	856.520			1.32383			0.661917	+	0.749577i	$0.269743\pi$
$648$		0			0
$649$	706.193			1.08812
$650$		0			0
$651$	−207.858	−	483.783i	−0.319291	−	0.743138i
$652$		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	$-0.896898\pi$
$661$	−837.005	−	837.005i	−1.26627	−	1.26627i	−0.318271	−	0.948000i	$-0.603102\pi$
$662$		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	0.961687	−	0.274151i	$-0.0883968\pi$
$677$	465.462	+	465.462i	0.687536	+	0.687536i	−0.274151	+	0.961687i	$0.588397\pi$
$678$		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	−0.231061	−	0.972939i	$-0.574220\pi$
$691$	512.637	−	512.637i	0.741878	−	0.741878i	0.972939	+	0.231061i	$0.0742198\pi$
$692$		0			0
$693$	171.042	+	398.095i	0.246814	+	0.574451i
$694$		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
$707$	−198.186	−	461.271i	−0.280320	−	0.652434i
$708$		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	$-0.972140\pi$
$719$		−	125.701i		−	0.174827i	0.996172	−	0.0874135i	$-0.0278601\pi$
$720$		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	$-0.547606\pi$
$727$	−216.647			−0.298001			−0.149001	+	0.988837i	$0.547606\pi$
$728$		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	−0.754341	−	0.656483i	$-0.772043\pi$
$733$	−71.7298	+	71.7298i	−0.0978579	+	0.0978579i	0.656483	+	0.754341i	$0.272043\pi$
$734$		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
$749$	1278.22	−	549.190i	1.70657	−	0.733230i
$750$		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	$-0.655523\pi$
$761$	−714.400			−0.938765			−0.469382	+	0.882995i	$0.655523\pi$
$762$		0			0
$763$	180.660	+	420.481i	0.236776	+	0.551089i
$764$		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	$-0.963657\pi$
$769$		−	175.219i		−	0.227854i	0.993489	−	0.113927i	$-0.0363429\pi$
$770$		0			0
$771$	−363.506	−	363.506i	−0.471473	−	0.471473i
$772$		0			0
$773$	385.461	+	385.461i	0.498655	+	0.498655i	0.911019	−	0.412364i	$-0.135297\pi$
$773$	385.461	+	385.461i	0.498655	+	0.498655i	−0.412364	+	0.911019i	$0.635297\pi$
$774$		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	−0.148913	−	0.988850i	$-0.547578\pi$
$787$	661.030	−	661.030i	0.839937	−	0.839937i	0.988850	+	0.148913i	$0.0475775\pi$
$788$		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	−0.984439	−	0.175728i	$-0.943772\pi$
$797$	−644.542	+	644.542i	−0.808711	+	0.808711i	0.175728	+	0.984439i	$0.443772\pi$
$798$		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
$805$	−296.801	+	127.521i	−0.368697	+	0.158411i
$806$		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	0.545902	−	0.837849i	$-0.316187\pi$
$811$	−236.769	−	236.769i	−0.291947	−	0.291947i	−0.837849	+	0.545902i	$0.816187\pi$
$812$		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
$819$	135.875	+	316.244i	0.165904	+	0.386135i
$820$		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	−0.955238	−	0.295837i	$-0.904401\pi$
$829$	−546.644	+	546.644i	−0.659401	+	0.659401i	0.295837	+	0.955238i	$0.404401\pi$
$830$		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	$-0.537290\pi$
$839$	−196.129			−0.233765			−0.116883	+	0.993146i	$0.537290\pi$
$840$		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	0.380915	−	0.924610i	$-0.375609\pi$
$853$	−463.772	−	463.772i	−0.543695	−	0.543695i	−0.924610	+	0.380915i	$0.875609\pi$
$854$		0			0
$855$		−	978.747i		−	1.14473i
$856$		0			0
$857$	1536.03			1.79234			0.896168	−	0.443714i	$-0.146339\pi$
$857$	1536.03			1.79234			0.896168	+	0.443714i	$0.146339\pi$
$858$		0			0
$859$	36.4535	+	36.4535i	0.0424371	+	0.0424371i	0.728007	−	0.685570i	$-0.240447\pi$
$859$	36.4535	+	36.4535i	0.0424371	+	0.0424371i	−0.685570	+	0.728007i	$0.740447\pi$
$860$		0			0
$861$	−32.2821	+	13.8701i	−0.0374937	+	0.0161093i
$862$		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
$875$	−638.277	+	274.237i	−0.729459	+	0.313414i
$876$		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	$-0.796470\pi$
$881$		−	1051.42i		−	1.19344i	0.802449	−	0.596721i	$-0.203530\pi$
$882$		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	$-0.394157\pi$
$887$	579.072			0.652843			0.326422	+	0.945224i	$0.394157\pi$
$888$		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
$917$	−198.445	+	85.2623i	−0.216407	+	0.0929796i
$918$		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	$-0.740247\pi$
$929$		−	1353.43i		−	1.45687i	0.685114	−	0.728436i	$-0.259753\pi$
$930$		0			0
$931$	1199.81	−	31.4553i	1.28873	−	0.0337865i
$932$		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	$-0.164852\pi$
$937$	1628.25			1.73772			0.868861	+	0.495056i	$0.164852\pi$
$938$		0			0
$939$	334.681	−	334.681i	0.356423	−	0.356423i
$940$		0			0
$941$	−293.378	+	293.378i	−0.311772	+	0.311772i	−0.845596	−	0.533824i	$-0.820755\pi$
$941$	−293.378	+	293.378i	−0.311772	+	0.311772i	0.533824	+	0.845596i	$0.320755\pi$
$942$		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	$-0.918950\pi$
$971$	−1184.27	−	1184.27i	−1.21964	−	1.21964i	−0.251883	−	0.967758i	$-0.581050\pi$
$972$		0			0
$973$	−39.0660	−	90.9247i	−0.0401501	−	0.0934478i
$974$		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	$-0.412295\pi$
$983$	534.869			0.544119			0.272059	+	0.962280i	$0.412295\pi$
$984$		0			0
$985$	63.7280			0.0646985
$986$		0			0
$987$	−152.383	−	354.665i	−0.154390	−	0.359336i
$988$		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	0.663854	−	0.747862i	$-0.268919\pi$
$997$	−83.7554	−	83.7554i	−0.0840075	−	0.0840075i	−0.747862	+	0.663854i	$0.768919\pi$
$998$		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.209.17		56
4.3	odd	2		112.3.l.b.13.17	✓	56
7.6	odd	2	inner	448.3.l.b.209.12		56
16.5	even	4	inner	448.3.l.b.433.12		56
16.11	odd	4		112.3.l.b.69.18	yes	56
28.27	even	2		112.3.l.b.13.18	yes	56
112.27	even	4		112.3.l.b.69.17	yes	56
112.69	odd	4	inner	448.3.l.b.433.17		56

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

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

\(f(q)\)	\(=\)	\(q+(1.00109 - 1.00109i) q^{3} +(-4.03888 - 4.03888i) q^{5} +(6.50171 + 2.59380i) q^{7} +6.99566i q^{9} +O(q^{10})\)	\(q+(1.00109 - 1.00109i) q^{3} +(-4.03888 - 4.03888i) q^{5} +(6.50171 + 2.59380i) q^{7} +6.99566i q^{9} +(6.25649 - 6.25649i) q^{11} +(4.97013 - 4.97013i) q^{13} -8.08654 q^{15} +5.44402i q^{17} +(17.3201 - 17.3201i) q^{19} +(9.10538 - 3.91215i) q^{21} -8.07936i q^{23} +7.62516i q^{25} +(16.0130 + 16.0130i) q^{27} +(-27.3552 - 27.3552i) q^{29} -53.1316i q^{31} -12.5266i q^{33} +(-15.7836 - 36.7357i) q^{35} +(13.0922 - 13.0922i) q^{37} -9.95105i q^{39} -3.54538 q^{41} +(-13.9574 + 13.9574i) q^{43} +(28.2546 - 28.2546i) q^{45} -38.9511i q^{47} +(35.5444 + 33.7283i) q^{49} +(5.44993 + 5.44993i) q^{51} +(-51.8947 + 51.8947i) q^{53} -50.5385 q^{55} -34.6778i q^{57} +(56.4368 + 56.4368i) q^{59} +(84.3789 - 84.3789i) q^{61} +(-18.1453 + 45.4837i) q^{63} -40.1475 q^{65} +(-0.603225 - 0.603225i) q^{67} +(-8.08813 - 8.08813i) q^{69} +78.2455i q^{71} +79.7314 q^{73} +(7.63344 + 7.63344i) q^{75} +(56.9060 - 24.4498i) q^{77} +85.2911 q^{79} -30.9001 q^{81} +(-5.96262 + 5.96262i) q^{83} +(21.9878 - 21.9878i) q^{85} -54.7699 q^{87} -163.829 q^{89} +(45.2058 - 19.4228i) q^{91} +(-53.1892 - 53.1892i) q^{93} -139.908 q^{95} +89.6393i q^{97} +(43.7683 + 43.7683i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q+O(q^{10}) \) 56 * q	\( 56 q + 8 q^{15} - 20 q^{21} - 96 q^{29} + 100 q^{35} - 128 q^{37} + 72 q^{43} + 192 q^{49} + 128 q^{51} + 88 q^{53} - 444 q^{63} - 8 q^{65} - 440 q^{67} + 12 q^{77} + 8 q^{79} + 64 q^{81} + 96 q^{85} + 388 q^{91} + 32 q^{93} + 776 q^{95} - 8 q^{99}+O(q^{100}) \) 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

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