LMFDB - Embedded newform 448.3.l.b.209.13

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)

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);

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")

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			209.13
Character	$\chi$	$=$	448.209
Dual form			448.3.l.b.433.13

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.752012 + 0.752012i) q^{3} +(6.01979 + 6.01979i) q^{5} +(4.46479 - 5.39126i) q^{7} +7.86896i q^{9} +(6.94142 - 6.94142i) q^{11} +(12.7620 - 12.7620i) q^{13} -9.05390 q^{15} -5.91610i q^{17} +(-5.24006 + 5.24006i) q^{19} +(0.696719 + 7.41186i) q^{21} +25.3271i q^{23} +47.4756i q^{25} +(-12.6857 - 12.6857i) q^{27} +(-5.79909 - 5.79909i) q^{29} -26.4139i q^{31} +10.4401i q^{33} +(59.3313 - 5.57717i) q^{35} +(-3.51860 + 3.51860i) q^{37} +19.1943i q^{39} +59.2859 q^{41} +(-1.69937 + 1.69937i) q^{43} +(-47.3694 + 47.3694i) q^{45} +81.1427i q^{47} +(-9.13136 - 48.1416i) q^{49} +(4.44897 + 4.44897i) q^{51} +(-13.0400 + 13.0400i) q^{53} +83.5717 q^{55} -7.88118i q^{57} +(20.7941 + 20.7941i) q^{59} +(-21.5617 + 21.5617i) q^{61} +(42.4236 + 35.1332i) q^{63} +153.648 q^{65} +(-40.2663 - 40.2663i) q^{67} +(-19.0462 - 19.0462i) q^{69} +99.6009i q^{71} +130.729 q^{73} +(-35.7022 - 35.7022i) q^{75} +(-6.43104 - 68.4149i) q^{77} -116.485 q^{79} -51.7411 q^{81} +(31.6396 - 31.6396i) q^{83} +(35.6136 - 35.6136i) q^{85} +8.72196 q^{87} -133.128 q^{89} +(-11.8236 - 125.782i) q^{91} +(19.8635 + 19.8635i) q^{93} -63.0881 q^{95} -27.7360i q^{97} +(54.6217 + 54.6217i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.752012	+	0.752012i	−0.250671	+	0.250671i	−0.821246	−	0.570575i	$-0.806720\pi$
$3$	−0.752012	+	0.752012i	−0.250671	+	0.250671i	0.570575	+	0.821246i	$0.306720\pi$
$4$		0			0
$5$	6.01979	+	6.01979i	1.20396	+	1.20396i	0.972953	+	0.231004i	$0.0742012\pi$
$5$	6.01979	+	6.01979i	1.20396	+	1.20396i	0.231004	+	0.972953i	$0.425799\pi$
$6$		0			0
$7$	4.46479	−	5.39126i	0.637827	−	0.770180i
$7$	4.46479	−	5.39126i	0.637827	−	0.770180i
$8$		0			0
$9$			7.86896i			0.874329i
$10$		0			0
$11$	6.94142	−	6.94142i	0.631038	−	0.631038i	−0.317290	−	0.948328i	$-0.602773\pi$
$11$	6.94142	−	6.94142i	0.631038	−	0.631038i	0.948328	+	0.317290i	$0.102773\pi$
$12$		0			0
$13$	12.7620	−	12.7620i	0.981689	−	0.981689i	−0.0181465	−	0.999835i	$-0.505777\pi$
$13$	12.7620	−	12.7620i	0.981689	−	0.981689i	0.999835	+	0.0181465i	$0.00577653\pi$
$14$		0			0
$15$	−9.05390			−0.603593
$16$		0			0
$17$		−	5.91610i		−	0.348006i	−0.984745	−	0.174003i	$-0.944330\pi$
$17$		−	5.91610i		−	0.348006i	0.984745	−	0.174003i	$-0.0556702\pi$
$18$		0			0
$19$	−5.24006	+	5.24006i	−0.275793	+	0.275793i	−0.831427	−	0.555634i	$-0.812476\pi$
$19$	−5.24006	+	5.24006i	−0.275793	+	0.275793i	0.555634	+	0.831427i	$0.312476\pi$
$20$		0			0
$21$	0.696719	+	7.41186i	0.0331771	+	0.352946i
$22$		0			0
$23$			25.3271i			1.10118i	0.834777	+	0.550588i	$0.185597\pi$
$23$			25.3271i			1.10118i	−0.834777	+	0.550588i	$0.814403\pi$
$24$		0			0
$25$			47.4756i			1.89903i
$26$		0			0
$27$	−12.6857	−	12.6857i	−0.469839	−	0.469839i
$28$		0			0
$29$	−5.79909	−	5.79909i	−0.199968	−	0.199968i	0.600018	−	0.799986i	$-0.295160\pi$
$29$	−5.79909	−	5.79909i	−0.199968	−	0.199968i	−0.799986	+	0.600018i	$0.795160\pi$
$30$		0			0
$31$		−	26.4139i		−	0.852061i	−0.904709	−	0.426030i	$-0.859912\pi$
$31$		−	26.4139i		−	0.852061i	0.904709	−	0.426030i	$-0.140088\pi$
$32$		0			0
$33$			10.4401i			0.316365i
$34$		0			0
$35$	59.3313	−	5.57717i	1.69518	−	0.159348i
$36$		0			0
$37$	−3.51860	+	3.51860i	−0.0950974	+	0.0950974i	−0.753055	−	0.657958i	$-0.771421\pi$
$37$	−3.51860	+	3.51860i	−0.0950974	+	0.0950974i	0.657958	+	0.753055i	$0.271421\pi$
$38$		0			0
$39$			19.1943i			0.492161i
$40$		0			0
$41$	59.2859			1.44600			0.722999	−	0.690849i	$-0.242763\pi$
$41$	59.2859			1.44600			0.722999	+	0.690849i	$0.242763\pi$
$42$		0			0
$43$	−1.69937	+	1.69937i	−0.0395202	+	0.0395202i	−0.726591	−	0.687071i	$-0.758896\pi$
$43$	−1.69937	+	1.69937i	−0.0395202	+	0.0395202i	0.687071	+	0.726591i	$0.258896\pi$
$44$		0			0
$45$	−47.3694	+	47.3694i	−1.05265	+	1.05265i
$46$		0			0
$47$			81.1427i			1.72644i	0.504828	+	0.863220i	$0.331556\pi$
$47$			81.1427i			1.72644i	−0.504828	+	0.863220i	$0.668444\pi$
$48$		0			0
$49$	−9.13136	−	48.1416i	−0.186354	−	0.982483i
$50$		0			0
$51$	4.44897	+	4.44897i	0.0872348	+	0.0872348i
$52$		0			0
$53$	−13.0400	+	13.0400i	−0.246037	+	0.246037i	−0.819342	−	0.573305i	$-0.805661\pi$
$53$	−13.0400	+	13.0400i	−0.246037	+	0.246037i	0.573305	+	0.819342i	$0.305661\pi$
$54$		0			0
$55$	83.5717			1.51949
$56$		0			0
$57$		−	7.88118i		−	0.138266i
$58$		0			0
$59$	20.7941	+	20.7941i	0.352442	+	0.352442i	0.861017	−	0.508575i	$-0.169828\pi$
$59$	20.7941	+	20.7941i	0.352442	+	0.352442i	−0.508575	+	0.861017i	$0.669828\pi$
$60$		0			0
$61$	−21.5617	+	21.5617i	−0.353471	+	0.353471i	−0.861399	−	0.507928i	$-0.830411\pi$
$61$	−21.5617	+	21.5617i	−0.353471	+	0.353471i	0.507928	+	0.861399i	$0.330411\pi$
$62$		0			0
$63$	42.4236	+	35.1332i	0.673390	+	0.557670i
$64$		0			0
$65$	153.648			2.36382
$66$		0			0
$67$	−40.2663	−	40.2663i	−0.600989	−	0.600989i	0.339586	−	0.940575i	$-0.389713\pi$
$67$	−40.2663	−	40.2663i	−0.600989	−	0.600989i	−0.940575	+	0.339586i	$0.889713\pi$
$68$		0			0
$69$	−19.0462	−	19.0462i	−0.276033	−	0.276033i
$70$		0			0
$71$			99.6009i			1.40283i	0.712753	+	0.701415i	$0.247448\pi$
$71$			99.6009i			1.40283i	−0.712753	+	0.701415i	$0.752552\pi$
$72$		0			0
$73$	130.729			1.79081			0.895403	−	0.445258i	$-0.146888\pi$
$73$	130.729			1.79081			0.895403	+	0.445258i	$0.146888\pi$
$74$		0			0
$75$	−35.7022	−	35.7022i	−0.476030	−	0.476030i
$76$		0			0
$77$	−6.43104	−	68.4149i	−0.0835200	−	0.888506i
$78$		0			0
$79$	−116.485			−1.47450			−0.737249	−	0.675621i	$-0.763876\pi$
$79$	−116.485			−1.47450			−0.737249	+	0.675621i	$0.763876\pi$
$80$		0			0
$81$	−51.7411			−0.638779
$82$		0			0
$83$	31.6396	−	31.6396i	0.381200	−	0.381200i	−0.490334	−	0.871534i	$-0.663125\pi$
$83$	31.6396	−	31.6396i	0.381200	−	0.381200i	0.871534	+	0.490334i	$0.163125\pi$
$84$		0			0
$85$	35.6136	−	35.6136i	0.418984	−	0.418984i
$86$		0			0
$87$	8.72196			0.100252
$88$		0			0
$89$	−133.128			−1.49583			−0.747913	−	0.663797i	$-0.768944\pi$
$89$	−133.128			−1.49583			−0.747913	+	0.663797i	$0.768944\pi$
$90$		0			0
$91$	−11.8236	−	125.782i	−0.129930	−	1.38222i
$92$		0			0
$93$	19.8635	+	19.8635i	0.213586	+	0.213586i
$94$		0			0
$95$	−63.0881			−0.664085
$96$		0			0
$97$		−	27.7360i		−	0.285938i	−0.989727	−	0.142969i	$-0.954335\pi$
$97$		−	27.7360i		−	0.285938i	0.989727	−	0.142969i	$-0.0456649\pi$
$98$		0			0
$99$	54.6217	+	54.6217i	0.551735	+	0.551735i
$100$		0			0
$101$	−100.746	−	100.746i	−0.997488	−	0.997488i	0.00250837	−	0.999997i	$-0.499202\pi$
$101$	−100.746	−	100.746i	−0.997488	−	0.997488i	−0.999997	+	0.00250837i	$0.999202\pi$
$102$		0			0
$103$	7.53632			0.0731681			0.0365841	−	0.999331i	$-0.488352\pi$
$103$	7.53632			0.0731681			0.0365841	+	0.999331i	$0.488352\pi$
$104$		0			0
$105$	−40.4237	+	48.8119i	−0.384988	+	0.464875i
$106$		0			0
$107$	82.4715	−	82.4715i	0.770762	−	0.770762i	−0.207478	−	0.978240i	$-0.566525\pi$
$107$	82.4715	−	82.4715i	0.770762	−	0.770762i	0.978240	+	0.207478i	$0.0665254\pi$
$108$		0			0
$109$	−118.767	−	118.767i	−1.08961	−	1.08961i	−0.995569	−	0.0940368i	$-0.970023\pi$
$109$	−118.767	−	118.767i	−1.08961	−	1.08961i	−0.0940368	−	0.995569i	$-0.529977\pi$
$110$		0			0
$111$		−	5.29206i		−	0.0476762i
$112$		0			0
$113$	−76.9425			−0.680907			−0.340454	−	0.940261i	$-0.610581\pi$
$113$	−76.9425			−0.680907			−0.340454	+	0.940261i	$0.610581\pi$
$114$		0			0
$115$	−152.464	+	152.464i	−1.32577	+	1.32577i
$116$		0			0
$117$	100.423	+	100.423i	0.858319	+	0.858319i
$118$		0			0
$119$	−31.8952	−	26.4141i	−0.268027	−	0.221967i
$120$		0			0
$121$			24.6334i			0.203582i
$122$		0			0
$123$	−44.5837	+	44.5837i	−0.362469	+	0.362469i
$124$		0			0
$125$	−135.299	+	135.299i	−1.08239	+	1.08239i
$126$		0			0
$127$	−102.900			−0.810240			−0.405120	−	0.914264i	$-0.632770\pi$
$127$	−102.900			−0.810240			−0.405120	+	0.914264i	$0.632770\pi$
$128$		0			0
$129$		−	2.55589i		−	0.0198131i
$130$		0			0
$131$	41.4626	−	41.4626i	0.316509	−	0.316509i	−0.530916	−	0.847425i	$-0.678152\pi$
$131$	41.4626	−	41.4626i	0.316509	−	0.316509i	0.847425	+	0.530916i	$0.178152\pi$
$132$		0			0
$133$	4.85478	+	51.6463i	0.0365021	+	0.388318i
$134$		0			0
$135$		−	152.730i		−	1.13133i
$136$		0			0
$137$			31.9294i			0.233062i	0.993187	+	0.116531i	$0.0371774\pi$
$137$			31.9294i			0.233062i	−0.993187	+	0.116531i	$0.962823\pi$
$138$		0			0
$139$	48.4192	+	48.4192i	0.348340	+	0.348340i	0.859491	−	0.511151i	$-0.170781\pi$
$139$	48.4192	+	48.4192i	0.348340	+	0.348340i	−0.511151	+	0.859491i	$0.670781\pi$
$140$		0			0
$141$	−61.0202	−	61.0202i	−0.432768	−	0.432768i
$142$		0			0
$143$		−	177.172i		−	1.23897i
$144$		0			0
$145$		−	69.8185i		−	0.481507i
$146$		0			0
$147$	43.0700	+	29.3362i	0.292993	+	0.199566i
$148$		0			0
$149$	174.695	−	174.695i	1.17245	−	1.17245i	0.190826	−	0.981624i	$-0.438883\pi$
$149$	174.695	−	174.695i	1.17245	−	1.17245i	0.981624	−	0.190826i	$-0.0611165\pi$
$150$		0			0
$151$		−	53.1868i		−	0.352230i	−0.984370	−	0.176115i	$-0.943647\pi$
$151$		−	53.1868i		−	0.352230i	0.984370	−	0.176115i	$-0.0563531\pi$
$152$		0			0
$153$	46.5535			0.304271
$154$		0			0
$155$	159.006	−	159.006i	1.02584	−	1.02584i
$156$		0			0
$157$	−52.0135	+	52.0135i	−0.331296	+	0.331296i	−0.853079	−	0.521782i	$-0.825267\pi$
$157$	−52.0135	+	52.0135i	−0.331296	+	0.331296i	0.521782	+	0.853079i	$0.325267\pi$
$158$		0			0
$159$		−	19.6124i		−	0.123349i
$160$		0			0
$161$	136.545	+	113.080i	0.848104	+	0.702360i
$162$		0			0
$163$	−135.555	−	135.555i	−0.831625	−	0.831625i	0.156114	−	0.987739i	$-0.450103\pi$
$163$	−135.555	−	135.555i	−0.831625	−	0.831625i	−0.987739	+	0.156114i	$0.950103\pi$
$164$		0			0
$165$	−62.8469	+	62.8469i	−0.380890	+	0.380890i
$166$		0			0
$167$	−268.875			−1.61003			−0.805015	−	0.593254i	$-0.797843\pi$
$167$	−268.875			−1.61003			−0.805015	+	0.593254i	$0.797843\pi$
$168$		0			0
$169$		−	156.735i		−	0.927426i
$170$		0			0
$171$	−41.2338	−	41.2338i	−0.241133	−	0.241133i
$172$		0			0
$173$	175.570	−	175.570i	1.01486	−	1.01486i	0.0149704	−	0.999888i	$-0.495235\pi$
$173$	175.570	−	175.570i	1.01486	−	1.01486i	0.999888	−	0.0149704i	$-0.00476541\pi$
$174$		0			0
$175$	255.953	+	211.969i	1.46259	+	1.21125i
$176$		0			0
$177$	−31.2748			−0.176694
$178$		0			0
$179$	−89.5549	−	89.5549i	−0.500307	−	0.500307i	0.411227	−	0.911533i	$-0.365100\pi$
$179$	−89.5549	−	89.5549i	−0.500307	−	0.500307i	−0.911533	+	0.411227i	$0.865100\pi$
$180$		0			0
$181$	−2.23766	−	2.23766i	−0.0123627	−	0.0123627i	0.700898	−	0.713261i	$-0.252783\pi$
$181$	−2.23766	−	2.23766i	−0.0123627	−	0.0123627i	−0.713261	+	0.700898i	$0.752783\pi$
$182$		0			0
$183$		−	32.4293i		−	0.177209i
$184$		0			0
$185$	−42.3625			−0.228986
$186$		0			0
$187$	−41.0661	−	41.0661i	−0.219605	−	0.219605i
$188$		0			0
$189$	−125.030	+	11.7529i	−0.661536	+	0.0621847i
$190$		0			0
$191$	202.309			1.05921			0.529606	−	0.848244i	$-0.322340\pi$
$191$	202.309			1.05921			0.529606	+	0.848244i	$0.322340\pi$
$192$		0			0
$193$	25.7587			0.133465			0.0667324	−	0.997771i	$-0.478743\pi$
$193$	25.7587			0.133465			0.0667324	+	0.997771i	$0.478743\pi$
$194$		0			0
$195$	−115.545	+	115.545i	−0.592541	+	0.592541i
$196$		0			0
$197$	−238.209	+	238.209i	−1.20918	+	1.20918i	−0.237888	+	0.971293i	$0.576455\pi$
$197$	−238.209	+	238.209i	−1.20918	+	1.20918i	−0.971293	+	0.237888i	$0.923545\pi$
$198$		0			0
$199$	70.8263			0.355911			0.177955	−	0.984039i	$-0.443052\pi$
$199$	70.8263			0.355911			0.177955	+	0.984039i	$0.443052\pi$
$200$		0			0
$201$	60.5614			0.301301
$202$		0			0
$203$	−57.1561	+	5.37270i	−0.281557	+	0.0264665i
$204$		0			0
$205$	356.888	+	356.888i	1.74092	+	1.74092i
$206$		0			0
$207$	−199.298			−0.962790
$208$		0			0
$209$			72.7469i			0.348071i
$210$		0			0
$211$	−38.5433	−	38.5433i	−0.182670	−	0.182670i	0.609848	−	0.792518i	$-0.291230\pi$
$211$	−38.5433	−	38.5433i	−0.182670	−	0.182670i	−0.792518	+	0.609848i	$0.791230\pi$
$212$		0			0
$213$	−74.9011	−	74.9011i	−0.351648	−	0.351648i
$214$		0			0
$215$	−20.4597			−0.0951613
$216$		0			0
$217$	−142.404	−	117.932i	−0.656240	−	0.543467i
$218$		0			0
$219$	−98.3096	+	98.3096i	−0.448902	+	0.448902i
$220$		0			0
$221$	−75.5010	−	75.5010i	−0.341633	−	0.341633i
$222$		0			0
$223$			231.515i			1.03818i	0.854719	+	0.519091i	$0.173729\pi$
$223$			231.515i			1.03818i	−0.854719	+	0.519091i	$0.826271\pi$
$224$		0			0
$225$	−373.584			−1.66037
$226$		0			0
$227$	164.507	−	164.507i	0.724701	−	0.724701i	−0.244858	−	0.969559i	$-0.578741\pi$
$227$	164.507	−	164.507i	0.724701	−	0.724701i	0.969559	+	0.244858i	$0.0787413\pi$
$228$		0			0
$229$	−18.0794	−	18.0794i	−0.0789495	−	0.0789495i	0.666529	−	0.745479i	$-0.267779\pi$
$229$	−18.0794	−	18.0794i	−0.0789495	−	0.0789495i	−0.745479	+	0.666529i	$0.767779\pi$
$230$		0			0
$231$	56.2850	+	46.6126i	0.243658	+	0.201786i
$232$		0			0
$233$		−	60.0557i		−	0.257750i	−0.991661	−	0.128875i	$-0.958863\pi$
$233$		−	60.0557i		−	0.257750i	0.991661	−	0.128875i	$-0.0411365\pi$
$234$		0			0
$235$	−488.461	+	488.461i	−2.07856	+	2.07856i
$236$		0			0
$237$	87.5983	−	87.5983i	0.369613	−	0.369613i
$238$		0			0
$239$	64.5787			0.270204			0.135102	−	0.990832i	$-0.456864\pi$
$239$	64.5787			0.270204			0.135102	+	0.990832i	$0.456864\pi$
$240$		0			0
$241$			77.8974i			0.323226i	0.986854	+	0.161613i	$0.0516696\pi$
$241$			77.8974i			0.323226i	−0.986854	+	0.161613i	$0.948330\pi$
$242$		0			0
$243$	153.081	−	153.081i	0.629962	−	0.629962i
$244$		0			0
$245$	234.834	−	344.771i	0.958504	−	1.40723i
$246$		0			0
$247$			133.747i			0.541485i
$248$		0			0
$249$			47.5867i			0.191111i
$250$		0			0
$251$	−41.9084	−	41.9084i	−0.166966	−	0.166966i	0.618678	−	0.785644i	$-0.287668\pi$
$251$	−41.9084	−	41.9084i	−0.166966	−	0.166966i	−0.785644	+	0.618678i	$0.787668\pi$
$252$		0			0
$253$	175.806	+	175.806i	0.694884	+	0.694884i
$254$		0			0
$255$			53.5637i			0.210054i
$256$		0			0
$257$		−	112.615i		−	0.438192i	−0.975703	−	0.219096i	$-0.929689\pi$
$257$		−	112.615i		−	0.438192i	0.975703	−	0.219096i	$-0.0703108\pi$
$258$		0			0
$259$	3.25989	+	34.6795i	0.0125865	+	0.133898i
$260$		0			0
$261$	45.6328	−	45.6328i	0.174838	−	0.174838i
$262$		0			0
$263$		−	186.564i		−	0.709367i	−0.934986	−	0.354684i	$-0.884589\pi$
$263$		−	186.564i		−	0.709367i	0.934986	−	0.354684i	$-0.115411\pi$
$264$		0			0
$265$	−156.996			−0.592437
$266$		0			0
$267$	100.114	−	100.114i	0.374959	−	0.374959i
$268$		0			0
$269$	244.189	−	244.189i	0.907768	−	0.907768i	−0.0883242	−	0.996092i	$-0.528151\pi$
$269$	244.189	−	244.189i	0.907768	−	0.907768i	0.996092	+	0.0883242i	$0.0281512\pi$
$270$		0			0
$271$		−	433.179i		−	1.59845i	−0.601034	−	0.799223i	$-0.705245\pi$
$271$		−	433.179i		−	1.59845i	0.601034	−	0.799223i	$-0.294755\pi$
$272$		0			0
$273$	103.481	+	85.6983i	0.379052	+	0.313913i
$274$		0			0
$275$	329.548	+	329.548i	1.19836	+	1.19836i
$276$		0			0
$277$	−208.653	+	208.653i	−0.753261	+	0.753261i	−0.975086	−	0.221826i	$-0.928798\pi$
$277$	−208.653	+	208.653i	−0.753261	+	0.753261i	0.221826	+	0.975086i	$0.428798\pi$
$278$		0			0
$279$	207.850			0.744981
$280$		0			0
$281$		−	244.401i		−	0.869754i	−0.900490	−	0.434877i	$-0.856792\pi$
$281$		−	244.401i		−	0.869754i	0.900490	−	0.434877i	$-0.143208\pi$
$282$		0			0
$283$	378.601	+	378.601i	1.33781	+	1.33781i	0.898176	+	0.439637i	$0.144893\pi$
$283$	378.601	+	378.601i	1.33781	+	1.33781i	0.439637	+	0.898176i	$0.355107\pi$
$284$		0			0
$285$	47.4430	−	47.4430i	0.166467	−	0.166467i
$286$		0			0
$287$	264.699	−	319.626i	0.922296	−	1.11368i
$288$		0			0
$289$	254.000			0.878892
$290$		0			0
$291$	20.8578	+	20.8578i	0.0716763	+	0.0716763i
$292$		0			0
$293$	33.2350	+	33.2350i	0.113430	+	0.113430i	0.761544	−	0.648114i	$-0.224442\pi$
$293$	33.2350	+	33.2350i	0.113430	+	0.113430i	−0.648114	+	0.761544i	$0.724442\pi$
$294$		0			0
$295$			250.352i			0.848650i
$296$		0			0
$297$	−176.113			−0.592972
$298$		0			0
$299$	323.223	+	323.223i	1.08101	+	1.08101i
$300$		0			0
$301$	1.57442	+	16.7491i	0.00523063	+	0.0556447i
$302$		0			0
$303$	151.525			0.500082
$304$		0			0
$305$	−259.594			−0.851127
$306$		0			0
$307$	68.8567	−	68.8567i	0.224289	−	0.224289i	−0.586013	−	0.810302i	$-0.699303\pi$
$307$	68.8567	−	68.8567i	0.224289	−	0.224289i	0.810302	+	0.586013i	$0.199303\pi$
$308$		0			0
$309$	−5.66740	+	5.66740i	−0.0183411	+	0.0183411i
$310$		0			0
$311$	−54.7005			−0.175886			−0.0879430	−	0.996126i	$-0.528029\pi$
$311$	−54.7005			−0.175886			−0.0879430	+	0.996126i	$0.528029\pi$
$312$		0			0
$313$	−74.2099			−0.237092			−0.118546	−	0.992949i	$-0.537823\pi$
$313$	−74.2099			−0.237092			−0.118546	+	0.992949i	$0.537823\pi$
$314$		0			0
$315$	43.8865	+	466.875i	0.139322	+	1.48214i
$316$		0			0
$317$	−100.164	−	100.164i	−0.315975	−	0.315975i	0.531244	−	0.847219i	$-0.321725\pi$
$317$	−100.164	−	100.164i	−0.315975	−	0.315975i	−0.847219	+	0.531244i	$0.821725\pi$
$318$		0			0
$319$	−80.5078			−0.252375
$320$		0			0
$321$			124.039i			0.386415i
$322$		0			0
$323$	31.0007	+	31.0007i	0.0959775	+	0.0959775i
$324$		0			0
$325$	605.882	+	605.882i	1.86425	+	1.86425i
$326$		0			0
$327$	178.628			0.546264
$328$		0			0
$329$	437.461	+	362.285i	1.32967	+	1.10117i
$330$		0			0
$331$	287.922	−	287.922i	0.869854	−	0.869854i	−0.122602	−	0.992456i	$-0.539124\pi$
$331$	287.922	−	287.922i	0.869854	−	0.869854i	0.992456	+	0.122602i	$0.0391239\pi$
$332$		0			0
$333$	−27.6877	−	27.6877i	−0.0831464	−	0.0831464i
$334$		0			0
$335$		−	484.789i		−	1.44713i
$336$		0			0
$337$	−247.389			−0.734093			−0.367047	−	0.930203i	$-0.619631\pi$
$337$	−247.389			−0.734093			−0.367047	+	0.930203i	$0.619631\pi$
$338$		0			0
$339$	57.8617	−	57.8617i	0.170683	−	0.170683i
$340$		0			0
$341$	−183.350	−	183.350i	−0.537683	−	0.537683i
$342$		0			0
$343$	−300.314	−	165.713i	−0.875550	−	0.483127i
$344$		0			0
$345$		−	229.309i		−	0.664663i
$346$		0			0
$347$	−313.519	+	313.519i	−0.903512	+	0.903512i	−0.995738	−	0.0922262i	$-0.970602\pi$
$347$	−313.519	+	313.519i	−0.903512	+	0.903512i	0.0922262	+	0.995738i	$0.470602\pi$
$348$		0			0
$349$	268.900	−	268.900i	0.770487	−	0.770487i	−0.207705	−	0.978192i	$-0.566599\pi$
$349$	268.900	−	268.900i	0.770487	−	0.770487i	0.978192	+	0.207705i	$0.0665993\pi$
$350$		0			0
$351$	−323.787			−0.922471
$352$		0			0
$353$		−	152.317i		−	0.431492i	−0.976450	−	0.215746i	$-0.930782\pi$
$353$		−	152.317i		−	0.431492i	0.976450	−	0.215746i	$-0.0692183\pi$
$354$		0			0
$355$	−599.576	+	599.576i	−1.68895	+	1.68895i
$356$		0			0
$357$	43.8493	−	4.12186i	0.122827	−	0.0115458i
$358$		0			0
$359$			99.2551i			0.276477i	0.990399	+	0.138238i	$0.0441440\pi$
$359$			99.2551i			0.276477i	−0.990399	+	0.138238i	$0.955856\pi$
$360$		0			0
$361$			306.083i			0.847877i
$362$		0			0
$363$	−18.5246	−	18.5246i	−0.0510320	−	0.0510320i
$364$		0			0
$365$	786.959	+	786.959i	2.15605	+	2.15605i
$366$		0			0
$367$			313.201i			0.853408i	0.904391	+	0.426704i	$0.140325\pi$
$367$			313.201i			0.853408i	−0.904391	+	0.426704i	$0.859675\pi$
$368$		0			0
$369$			466.518i			1.26428i
$370$		0			0
$371$	12.0812	+	128.523i	0.0325638	+	0.346422i
$372$		0			0
$373$	−192.122	+	192.122i	−0.515072	+	0.515072i	−0.916076	−	0.401004i	$-0.868661\pi$
$373$	−192.122	+	192.122i	−0.515072	+	0.515072i	0.401004	+	0.916076i	$0.368661\pi$
$374$		0			0
$375$		−	203.492i		−	0.542646i
$376$		0			0
$377$	−148.015			−0.392614
$378$		0			0
$379$	55.0996	−	55.0996i	0.145382	−	0.145382i	−0.630670	−	0.776051i	$-0.717220\pi$
$379$	55.0996	−	55.0996i	0.145382	−	0.145382i	0.776051	+	0.630670i	$0.217220\pi$
$380$		0			0
$381$	77.3823	−	77.3823i	0.203103	−	0.203103i
$382$		0			0
$383$		−	44.5676i		−	0.116365i	−0.998306	−	0.0581823i	$-0.981470\pi$
$383$		−	44.5676i		−	0.116365i	0.998306	−	0.0581823i	$-0.0185305\pi$
$384$		0			0
$385$	373.130	−	450.557i	0.969168	−	1.17028i
$386$		0			0
$387$	−13.3723	−	13.3723i	−0.0345537	−	0.0345537i
$388$		0			0
$389$	−50.2482	+	50.2482i	−0.129173	+	0.129173i	−0.768737	−	0.639565i	$-0.779115\pi$
$389$	−50.2482	+	50.2482i	−0.129173	+	0.129173i	0.639565	+	0.768737i	$0.279115\pi$
$390$		0			0
$391$	149.837			0.383216
$392$		0			0
$393$			62.3608i			0.158679i
$394$		0			0
$395$	−701.217	−	701.217i	−1.77523	−	1.77523i
$396$		0			0
$397$	−265.435	+	265.435i	−0.668602	+	0.668602i	−0.957392	−	0.288790i	$-0.906747\pi$
$397$	−265.435	+	265.435i	−0.668602	+	0.668602i	0.288790	+	0.957392i	$0.406747\pi$
$398$		0			0
$399$	−42.4895	−	35.1878i	−0.106490	−	0.0881899i
$400$		0			0
$401$	−312.566			−0.779467			−0.389734	−	0.920928i	$-0.627433\pi$
$401$	−312.566			−0.779467			−0.389734	+	0.920928i	$0.627433\pi$
$402$		0			0
$403$	−337.093	−	337.093i	−0.836458	−	0.836458i
$404$		0			0
$405$	−311.470	−	311.470i	−0.769063	−	0.769063i
$406$		0			0
$407$			48.8482i			0.120020i
$408$		0			0
$409$	−281.100			−0.687286			−0.343643	−	0.939100i	$-0.611661\pi$
$409$	−281.100			−0.687286			−0.343643	+	0.939100i	$0.611661\pi$
$410$		0			0
$411$	−24.0113	−	24.0113i	−0.0584217	−	0.0584217i
$412$		0			0
$413$	204.947	−	19.2651i	0.496241	−	0.0466469i
$414$		0			0
$415$	380.927			0.917898
$416$		0			0
$417$	−72.8236			−0.174637
$418$		0			0
$419$	348.371	−	348.371i	0.831435	−	0.831435i	−0.156278	−	0.987713i	$-0.549950\pi$
$419$	348.371	−	348.371i	0.831435	−	0.831435i	0.987713	+	0.156278i	$0.0499496\pi$
$420$		0			0
$421$	−126.230	+	126.230i	−0.299835	+	0.299835i	−0.840949	−	0.541114i	$-0.818003\pi$
$421$	−126.230	+	126.230i	−0.299835	+	0.299835i	0.541114	+	0.840949i	$0.318003\pi$
$422$		0			0
$423$	−638.508			−1.50948
$424$		0			0
$425$	280.871			0.660872
$426$		0			0
$427$	19.9763	+	212.513i	0.0467830	+	0.497689i
$428$		0			0
$429$	133.235	+	133.235i	0.310572	+	0.310572i
$430$		0			0
$431$	642.234			1.49010			0.745051	−	0.667007i	$-0.232425\pi$
$431$	642.234			1.49010			0.745051	+	0.667007i	$0.232425\pi$
$432$		0			0
$433$			92.9286i			0.214616i	0.994226	+	0.107308i	$0.0342230\pi$
$433$			92.9286i			0.214616i	−0.994226	+	0.107308i	$0.965777\pi$
$434$		0			0
$435$	52.5043	+	52.5043i	0.120700	+	0.120700i
$436$		0			0
$437$	−132.715	−	132.715i	−0.303697	−	0.303697i
$438$		0			0
$439$	−435.700			−0.992482			−0.496241	−	0.868185i	$-0.665287\pi$
$439$	−435.700			−0.992482			−0.496241	+	0.868185i	$0.665287\pi$
$440$		0			0
$441$	378.825	−	71.8543i	0.859013	−	0.162935i
$442$		0			0
$443$	355.684	−	355.684i	0.802898	−	0.802898i	−0.180649	−	0.983548i	$-0.557820\pi$
$443$	355.684	−	355.684i	0.802898	−	0.802898i	0.983548	+	0.180649i	$0.0578199\pi$
$444$		0			0
$445$	−801.405	−	801.405i	−1.80091	−	1.80091i
$446$		0			0
$447$			262.745i			0.587797i
$448$		0			0
$449$	−117.227			−0.261085			−0.130543	−	0.991443i	$-0.541672\pi$
$449$	−117.227			−0.261085			−0.130543	+	0.991443i	$0.541672\pi$
$450$		0			0
$451$	411.528	−	411.528i	0.912479	−	0.912479i
$452$		0			0
$453$	39.9971	+	39.9971i	0.0882938	+	0.0882938i
$454$		0			0
$455$	686.008	−	828.359i	1.50771	−	1.82057i
$456$		0			0
$457$		−	179.002i		−	0.391689i	−0.980635	−	0.195844i	$-0.937255\pi$
$457$		−	179.002i		−	0.391689i	0.980635	−	0.195844i	$-0.0627447\pi$
$458$		0			0
$459$	−75.0496	+	75.0496i	−0.163507	+	0.163507i
$460$		0			0
$461$	−86.8740	+	86.8740i	−0.188447	+	0.188447i	−0.795024	−	0.606578i	$-0.792542\pi$
$461$	−86.8740	+	86.8740i	−0.188447	+	0.188447i	0.606578	+	0.795024i	$0.292542\pi$
$462$		0			0
$463$	−920.087			−1.98723			−0.993614	−	0.112831i	$-0.964008\pi$
$463$	−920.087			−1.98723			−0.993614	+	0.112831i	$0.964008\pi$
$464$		0			0
$465$			239.149i			0.514298i
$466$		0			0
$467$	29.5742	−	29.5742i	0.0633281	−	0.0633281i	−0.674733	−	0.738062i	$-0.735741\pi$
$467$	29.5742	−	29.5742i	0.0633281	−	0.0633281i	0.738062	+	0.674733i	$0.235741\pi$
$468$		0			0
$469$	−396.866	+	37.3056i	−0.846197	+	0.0795429i
$470$		0			0
$471$		−	78.2296i		−	0.166092i
$472$		0			0
$473$			23.5921i			0.0498775i
$474$		0			0
$475$	−248.775	−	248.775i	−0.523737	−	0.523737i
$476$		0			0
$477$	−102.611	−	102.611i	−0.215117	−	0.215117i
$478$		0			0
$479$			773.399i			1.61461i	0.590133	+	0.807306i	$0.299075\pi$
$479$			773.399i			1.61461i	−0.590133	+	0.807306i	$0.700925\pi$
$480$		0			0
$481$			89.8085i			0.186712i
$482$		0			0
$483$	−187.721	+	17.6458i	−0.388656	+	0.0365338i
$484$		0			0
$485$	166.965	−	166.965i	0.344257	−	0.344257i
$486$		0			0
$487$			639.484i			1.31311i	0.754278	+	0.656555i	$0.227987\pi$
$487$			639.484i			1.31311i	−0.754278	+	0.656555i	$0.772013\pi$
$488$		0			0
$489$	203.878			0.416928
$490$		0			0
$491$	280.573	−	280.573i	0.571431	−	0.571431i	−0.361097	−	0.932528i	$-0.617598\pi$
$491$	280.573	−	280.573i	0.571431	−	0.571431i	0.932528	+	0.361097i	$0.117598\pi$
$492$		0			0
$493$	−34.3080	+	34.3080i	−0.0695902	+	0.0695902i
$494$		0			0
$495$			657.622i			1.32853i
$496$		0			0
$497$	536.975	+	444.697i	1.08043	+	0.894763i
$498$		0			0
$499$	−107.950	−	107.950i	−0.216332	−	0.216332i	0.590619	−	0.806951i	$-0.298884\pi$
$499$	−107.950	−	107.950i	−0.216332	−	0.216332i	−0.806951	+	0.590619i	$0.798884\pi$
$500$		0			0
$501$	202.197	−	202.197i	0.403587	−	0.403587i
$502$		0			0
$503$	−290.234			−0.577006			−0.288503	−	0.957479i	$-0.593158\pi$
$503$	−290.234			−0.577006			−0.288503	+	0.957479i	$0.593158\pi$
$504$		0			0
$505$		−	1212.94i		−	2.40187i
$506$		0			0
$507$	117.867	+	117.867i	0.232478	+	0.232478i
$508$		0			0
$509$	−605.494	+	605.494i	−1.18958	+	1.18958i	−0.212391	+	0.977185i	$0.568125\pi$
$509$	−605.494	+	605.494i	−1.18958	+	1.18958i	−0.977185	+	0.212391i	$0.931875\pi$
$510$		0			0
$511$	583.676	−	704.793i	1.14222	−	1.37924i
$512$		0			0
$513$	132.947			0.259156
$514$		0			0
$515$	45.3670	+	45.3670i	0.0880913	+	0.0880913i
$516$		0			0
$517$	563.245	+	563.245i	1.08945	+	1.08945i
$518$		0			0
$519$			264.062i			0.508790i
$520$		0			0
$521$	−514.345			−0.987227			−0.493613	−	0.869681i	$-0.664324\pi$
$521$	−514.345			−0.987227			−0.493613	+	0.869681i	$0.664324\pi$
$522$		0			0
$523$	−11.1948	−	11.1948i	−0.0214049	−	0.0214049i	0.696323	−	0.717728i	$-0.254818\pi$
$523$	−11.1948	−	11.1948i	−0.0214049	−	0.0214049i	−0.717728	+	0.696323i	$0.754818\pi$
$524$		0			0
$525$	−351.883	+	33.0772i	−0.670253	+	0.0630041i
$526$		0			0
$527$	−156.267			−0.296522
$528$		0			0
$529$	−112.460			−0.212590
$530$		0			0
$531$	−163.628	+	163.628i	−0.308150	+	0.308150i
$532$		0			0
$533$	756.604	−	756.604i	1.41952	−	1.41952i
$534$		0			0
$535$	992.922			1.85593
$536$		0			0
$537$	134.693			0.250824
$538$		0			0
$539$	−397.556	−	270.787i	−0.737581	−	0.502387i
$540$		0			0
$541$	−547.935	−	547.935i	−1.01282	−	1.01282i	−0.999917	−	0.0129021i	$-0.995893\pi$
$541$	−547.935	−	547.935i	−1.01282	−	1.01282i	−0.0129021	−	0.999917i	$-0.504107\pi$
$542$		0			0
$543$	3.36549			0.00619795
$544$		0			0
$545$		−	1429.90i		−	2.62368i
$546$		0			0
$547$	−470.095	−	470.095i	−0.859405	−	0.859405i	0.131863	−	0.991268i	$-0.457904\pi$
$547$	−470.095	−	470.095i	−0.859405	−	0.859405i	−0.991268	+	0.131863i	$0.957904\pi$
$548$		0			0
$549$	−169.668	−	169.668i	−0.309050	−	0.309050i
$550$		0			0
$551$	60.7751			0.110300
$552$		0			0
$553$	−520.082	+	628.003i	−0.940474	+	1.13563i
$554$		0			0
$555$	31.8571	−	31.8571i	0.0574001	−	0.0574001i
$556$		0			0
$557$	−299.030	−	299.030i	−0.536859	−	0.536859i	0.385746	−	0.922605i	$-0.373944\pi$
$557$	−299.030	−	299.030i	−0.536859	−	0.536859i	−0.922605	+	0.385746i	$0.873944\pi$
$558$		0			0
$559$			43.3746i			0.0775931i
$560$		0			0
$561$	61.7644			0.110097
$562$		0			0
$563$	−576.267	+	576.267i	−1.02356	+	1.02356i	−0.0238492	+	0.999716i	$0.507592\pi$
$563$	−576.267	+	576.267i	−1.02356	+	1.02356i	−0.999716	+	0.0238492i	$0.992408\pi$
$564$		0			0
$565$	−463.177	−	463.177i	−0.819783	−	0.819783i
$566$		0			0
$567$	−231.013	+	278.950i	−0.407430	+	0.491975i
$568$		0			0
$569$		−	333.539i		−	0.586185i	−0.956084	−	0.293093i	$-0.905316\pi$
$569$		−	333.539i		−	0.586185i	0.956084	−	0.293093i	$-0.0946844\pi$
$570$		0			0
$571$	287.272	−	287.272i	0.503104	−	0.503104i	−0.409297	−	0.912401i	$-0.634226\pi$
$571$	287.272	−	287.272i	0.503104	−	0.503104i	0.912401	+	0.409297i	$0.134226\pi$
$572$		0			0
$573$	−152.139	+	152.139i	−0.265513	+	0.265513i
$574$		0			0
$575$	−1202.42			−2.09116
$576$		0			0
$577$		−	254.228i		−	0.440603i	−0.975432	−	0.220302i	$-0.929296\pi$
$577$		−	254.228i		−	0.440603i	0.975432	−	0.220302i	$-0.0707042\pi$
$578$		0			0
$579$	−19.3709	+	19.3709i	−0.0334557	+	0.0334557i
$580$		0			0
$581$	−29.3133	−	311.842i	−0.0504531	−	0.536732i
$582$		0			0
$583$			181.032i			0.310518i
$584$		0			0
$585$			1209.05i			2.06676i
$586$		0			0
$587$	325.537	+	325.537i	0.554578	+	0.554578i	0.927759	−	0.373181i	$-0.121733\pi$
$587$	325.537	+	325.537i	0.554578	+	0.554578i	−0.373181	+	0.927759i	$0.621733\pi$
$588$		0			0
$589$	138.410	+	138.410i	0.234992	+	0.234992i
$590$		0			0
$591$		−	358.271i		−	0.606212i
$592$		0			0
$593$		−	365.199i		−	0.615850i	−0.951411	−	0.307925i	$-0.900365\pi$
$593$		−	365.199i		−	0.615850i	0.951411	−	0.307925i	$-0.0996346\pi$
$594$		0			0
$595$	−32.9951	−	351.010i	−0.0554539	−	0.589932i
$596$		0			0
$597$	−53.2622	+	53.2622i	−0.0892164	+	0.0892164i
$598$		0			0
$599$		−	846.590i		−	1.41334i	−0.707544	−	0.706669i	$-0.750197\pi$
$599$		−	846.590i		−	1.41334i	0.707544	−	0.706669i	$-0.249803\pi$
$600$		0			0
$601$	−264.664			−0.440372			−0.220186	−	0.975458i	$-0.570666\pi$
$601$	−264.664			−0.440372			−0.220186	+	0.975458i	$0.570666\pi$
$602$		0			0
$603$	316.854	−	316.854i	0.525462	−	0.525462i
$604$		0			0
$605$	−148.288	+	148.288i	−0.245104	+	0.245104i
$606$		0			0
$607$		−	631.165i		−	1.03981i	−0.854224	−	0.519906i	$-0.825967\pi$
$607$		−	631.165i		−	1.03981i	0.854224	−	0.519906i	$-0.174033\pi$
$608$		0			0
$609$	38.9417	−	47.0224i	0.0639437	−	0.0772124i
$610$		0			0
$611$	1035.54	+	1035.54i	1.69483	+	1.69483i
$612$		0			0
$613$	−156.240	+	156.240i	−0.254878	+	0.254878i	−0.822967	−	0.568089i	$-0.807683\pi$
$613$	−156.240	+	156.240i	−0.254878	+	0.254878i	0.568089	+	0.822967i	$0.307683\pi$
$614$		0			0
$615$	−536.768			−0.872794
$616$		0			0
$617$			523.193i			0.847963i	0.905671	+	0.423982i	$0.139368\pi$
$617$			523.193i			0.847963i	−0.905671	+	0.423982i	$0.860632\pi$
$618$		0			0
$619$	−93.7666	−	93.7666i	−0.151481	−	0.151481i	0.627298	−	0.778779i	$-0.284161\pi$
$619$	−93.7666	−	93.7666i	−0.151481	−	0.151481i	−0.778779	+	0.627298i	$0.784161\pi$
$620$		0			0
$621$	321.290	−	321.290i	0.517376	−	0.517376i
$622$		0			0
$623$	−594.390	+	717.730i	−0.954077	+	1.15205i
$624$		0			0
$625$	−442.045			−0.707273
$626$		0			0
$627$	−54.7065	−	54.7065i	−0.0872513	−	0.0872513i
$628$		0			0
$629$	20.8164	+	20.8164i	0.0330944	+	0.0330944i
$630$		0			0
$631$			997.642i			1.58105i	0.612430	+	0.790524i	$0.290192\pi$
$631$			997.642i			1.58105i	−0.612430	+	0.790524i	$0.709808\pi$
$632$		0			0
$633$	57.9700			0.0915797
$634$		0			0
$635$	−619.439	−	619.439i	−0.975494	−	0.975494i
$636$		0			0
$637$	−730.916	−	497.848i	−1.14743	−	0.781550i
$638$		0			0
$639$	−783.756			−1.22653
$640$		0			0
$641$	895.604			1.39720			0.698599	−	0.715513i	$-0.253807\pi$
$641$	895.604			1.39720			0.698599	+	0.715513i	$0.253807\pi$
$642$		0			0
$643$	−497.619	+	497.619i	−0.773902	+	0.773902i	−0.978786	−	0.204884i	$-0.934318\pi$
$643$	−497.619	+	497.619i	−0.773902	+	0.773902i	0.204884	+	0.978786i	$0.434318\pi$
$644$		0			0
$645$	15.3859	−	15.3859i	0.0238541	−	0.0238541i
$646$		0			0
$647$	58.9196			0.0910658			0.0455329	−	0.998963i	$-0.485501\pi$
$647$	58.9196			0.0910658			0.0455329	+	0.998963i	$0.485501\pi$
$648$		0			0
$649$	288.681			0.444809
$650$		0			0
$651$	195.776	−	18.4030i	0.300731	−	0.0282689i
$652$		0			0
$653$	−58.1148	−	58.1148i	−0.0889967	−	0.0889967i	0.661207	−	0.750204i	$-0.270045\pi$
$653$	−58.1148	−	58.1148i	−0.0889967	−	0.0889967i	−0.750204	+	0.661207i	$0.770045\pi$
$654$		0			0
$655$	499.192			0.762126
$656$		0			0
$657$			1028.70i			1.56575i
$658$		0			0
$659$	74.7256	+	74.7256i	0.113392	+	0.113392i	0.761526	−	0.648134i	$-0.224450\pi$
$659$	74.7256	+	74.7256i	0.113392	+	0.113392i	−0.648134	+	0.761526i	$0.724450\pi$
$660$		0			0
$661$	113.175	+	113.175i	0.171218	+	0.171218i	0.787514	−	0.616296i	$-0.211368\pi$
$661$	113.175	+	113.175i	0.171218	+	0.171218i	−0.616296	+	0.787514i	$0.711368\pi$
$662$		0			0
$663$	113.555			0.171275
$664$		0			0
$665$	−281.675	+	340.124i	−0.423571	+	0.511465i
$666$		0			0
$667$	146.874	−	146.874i	0.220201	−	0.220201i
$668$		0			0
$669$	−174.102	−	174.102i	−0.260242	−	0.260242i
$670$		0			0
$671$			299.338i			0.446107i
$672$		0			0
$673$	633.501			0.941309			0.470654	−	0.882318i	$-0.344018\pi$
$673$	633.501			0.941309			0.470654	+	0.882318i	$0.344018\pi$
$674$		0			0
$675$	602.259	−	602.259i	0.892236	−	0.892236i
$676$		0			0
$677$	439.372	+	439.372i	0.648998	+	0.648998i	0.952751	−	0.303753i	$-0.0982396\pi$
$677$	439.372	+	439.372i	0.648998	+	0.648998i	−0.303753	+	0.952751i	$0.598240\pi$
$678$		0			0
$679$	−149.532	−	123.835i	−0.220224	−	0.182379i
$680$		0			0
$681$			247.423i			0.363322i
$682$		0			0
$683$	−273.784	+	273.784i	−0.400855	+	0.400855i	−0.878534	−	0.477679i	$-0.841478\pi$
$683$	−273.784	+	273.784i	−0.400855	+	0.400855i	0.477679	+	0.878534i	$0.341478\pi$
$684$		0			0
$685$	−192.208	+	192.208i	−0.280596	+	0.280596i
$686$		0			0
$687$	27.1919			0.0395806
$688$		0			0
$689$			332.831i			0.483064i
$690$		0			0
$691$	702.619	−	702.619i	1.01681	−	1.01681i	0.0169580	−	0.999856i	$-0.494602\pi$
$691$	702.619	−	702.619i	1.01681	−	1.01681i	0.999856	−	0.0169580i	$-0.00539816\pi$
$692$		0			0
$693$	538.354	−	50.6056i	0.776846	−	0.0730239i
$694$		0			0
$695$			582.946i			0.838772i
$696$		0			0
$697$		−	350.741i		−	0.503215i
$698$		0			0
$699$	45.1626	+	45.1626i	0.0646103	+	0.0646103i
$700$		0			0
$701$	746.925	+	746.925i	1.06551	+	1.06551i	0.997698	+	0.0678159i	$0.0216031\pi$
$701$	746.925	+	746.925i	1.06551	+	1.06551i	0.0678159	+	0.997698i	$0.478397\pi$
$702$		0			0
$703$		−	36.8754i		−	0.0524543i
$704$		0			0
$705$		−	734.657i		−	1.04207i
$706$		0			0
$707$	−992.961	+	93.3388i	−1.40447	+	0.132021i
$708$		0			0
$709$	−464.990	+	464.990i	−0.655839	+	0.655839i	−0.954393	−	0.298554i	$-0.903496\pi$
$709$	−464.990	+	464.990i	−0.655839	+	0.655839i	0.298554	+	0.954393i	$0.403496\pi$
$710$		0			0
$711$		−	916.618i		−	1.28920i
$712$		0			0
$713$	668.986			0.938269
$714$		0			0
$715$	1066.54	−	1066.54i	1.49166	−	1.49166i
$716$		0			0
$717$	−48.5639	+	48.5639i	−0.0677321	+	0.0677321i
$718$		0			0
$719$			846.592i			1.17746i	0.808331	+	0.588729i	$0.200371\pi$
$719$			846.592i			1.17746i	−0.808331	+	0.588729i	$0.799629\pi$
$720$		0			0
$721$	33.6480	−	40.6302i	0.0466686	−	0.0563526i
$722$		0			0
$723$	−58.5798	−	58.5798i	−0.0810232	−	0.0810232i
$724$		0			0
$725$	275.315	−	275.315i	0.379745	−	0.379745i
$726$		0			0
$727$	1333.27			1.83393			0.916966	−	0.398965i	$-0.130630\pi$
$727$	1333.27			1.83393			0.916966	+	0.398965i	$0.130630\pi$
$728$		0			0
$729$		−	235.433i		−	0.322953i
$730$		0			0
$731$	10.0536	+	10.0536i	0.0137533	+	0.0137533i
$732$		0			0
$733$	164.938	−	164.938i	0.225018	−	0.225018i	−0.585590	−	0.810608i	$-0.699137\pi$
$733$	164.938	−	164.938i	0.225018	−	0.225018i	0.810608	+	0.585590i	$0.199137\pi$
$734$		0			0
$735$	82.6744	+	435.870i	0.112482	+	0.593020i
$736$		0			0
$737$	−559.010			−0.758494
$738$		0			0
$739$	714.408	+	714.408i	0.966722	+	0.966722i	0.999464	−	0.0327418i	$-0.0104239\pi$
$739$	714.408	+	714.408i	0.966722	+	0.966722i	−0.0327418	+	0.999464i	$0.510424\pi$
$740$		0			0
$741$	−100.579	−	100.579i	−0.135734	−	0.135734i
$742$		0			0
$743$			374.557i			0.504114i	0.967712	+	0.252057i	$0.0811071\pi$
$743$			374.557i			0.504114i	−0.967712	+	0.252057i	$0.918893\pi$
$744$		0			0
$745$	2103.25			2.82316
$746$		0			0
$747$	248.971	+	248.971i	0.333294	+	0.333294i
$748$		0			0
$749$	−76.4077	−	812.843i	−0.102013	−	1.08524i
$750$		0			0
$751$	571.874			0.761483			0.380742	−	0.924681i	$-0.375669\pi$
$751$	571.874			0.761483			0.380742	+	0.924681i	$0.375669\pi$
$752$		0			0
$753$	63.0313			0.0837068
$754$		0			0
$755$	320.173	−	320.173i	0.424070	−	0.424070i
$756$		0			0
$757$	−847.678	+	847.678i	−1.11979	+	1.11979i	−0.128014	+	0.991772i	$0.540860\pi$
$757$	−847.678	+	847.678i	−1.11979	+	1.11979i	−0.991772	+	0.128014i	$0.959140\pi$
$758$		0			0
$759$	−264.416			−0.348374
$760$		0			0
$761$	320.868			0.421639			0.210820	−	0.977525i	$-0.432387\pi$
$761$	320.868			0.421639			0.210820	+	0.977525i	$0.432387\pi$
$762$		0			0
$763$	−1170.57	+	110.034i	−1.53417	+	0.144213i
$764$		0			0
$765$	280.242	+	280.242i	0.366330	+	0.366330i
$766$		0			0
$767$	530.746			0.691977
$768$		0			0
$769$			1064.20i			1.38387i	0.721957	+	0.691937i	$0.243243\pi$
$769$			1064.20i			1.38387i	−0.721957	+	0.691937i	$0.756757\pi$
$770$		0			0
$771$	84.6881	+	84.6881i	0.109842	+	0.109842i
$772$		0			0
$773$	−903.810	−	903.810i	−1.16922	−	1.16922i	−0.982392	−	0.186832i	$-0.940178\pi$
$773$	−903.810	−	903.810i	−1.16922	−	1.16922i	−0.186832	−	0.982392i	$-0.559822\pi$
$774$		0			0
$775$	1254.02			1.61808
$776$		0			0
$777$	−28.5309	−	23.6279i	−0.0367193	−	0.0304092i
$778$		0			0
$779$	−310.662	+	310.662i	−0.398796	+	0.398796i
$780$		0			0
$781$	691.372	+	691.372i	0.885239	+	0.885239i
$782$		0			0
$783$			147.130i			0.187906i
$784$		0			0
$785$	−626.221			−0.797733
$786$		0			0
$787$	−887.233	+	887.233i	−1.12736	+	1.12736i	−0.136756	+	0.990605i	$0.543668\pi$
$787$	−887.233	+	887.233i	−1.12736	+	1.12736i	−0.990605	+	0.136756i	$0.956332\pi$
$788$		0			0
$789$	140.298	+	140.298i	0.177818	+	0.177818i
$790$		0			0
$791$	−343.532	+	414.817i	−0.434301	+	0.524421i
$792$		0			0
$793$			550.339i			0.693997i
$794$		0			0
$795$	118.063	−	118.063i	0.148506	−	0.148506i
$796$		0			0
$797$	−505.200	+	505.200i	−0.633877	+	0.633877i	−0.949038	−	0.315161i	$-0.897941\pi$
$797$	−505.200	+	505.200i	−0.633877	+	0.633877i	0.315161	+	0.949038i	$0.397941\pi$
$798$		0			0
$799$	480.048			0.600811
$800$		0			0
$801$		−	1047.58i		−	1.30784i
$802$		0			0
$803$	907.443	−	907.443i	1.13007	−	1.13007i
$804$		0			0
$805$	141.253	+	1502.69i	0.175470	+	1.86669i
$806$		0			0
$807$			367.267i			0.455101i
$808$		0			0
$809$			295.844i			0.365691i	0.983142	+	0.182846i	$0.0585308\pi$
$809$			295.844i			0.365691i	−0.983142	+	0.182846i	$0.941469\pi$
$810$		0			0
$811$	−296.074	−	296.074i	−0.365073	−	0.365073i	0.500604	−	0.865677i	$-0.333111\pi$
$811$	−296.074	−	296.074i	−0.365073	−	0.365073i	−0.865677	+	0.500604i	$0.833111\pi$
$812$		0			0
$813$	325.756	+	325.756i	0.400683	+	0.400683i
$814$		0			0
$815$		−	1632.02i		−	2.00248i
$816$		0			0
$817$		−	17.8096i		−	0.0217988i
$818$		0			0
$819$	989.776	−	93.0395i	1.20852	−	0.113601i
$820$		0			0
$821$	357.491	−	357.491i	0.435433	−	0.435433i	−0.455038	−	0.890472i	$-0.650375\pi$
$821$	357.491	−	357.491i	0.435433	−	0.435433i	0.890472	+	0.455038i	$0.150375\pi$
$822$		0			0
$823$			730.158i			0.887191i	0.896227	+	0.443596i	$0.146297\pi$
$823$			730.158i			0.887191i	−0.896227	+	0.443596i	$0.853703\pi$
$824$		0			0
$825$	−495.648			−0.600786
$826$		0			0
$827$	35.9526	−	35.9526i	0.0434735	−	0.0434735i	−0.685036	−	0.728509i	$-0.740213\pi$
$827$	35.9526	−	35.9526i	0.0434735	−	0.0434735i	0.728509	+	0.685036i	$0.240213\pi$
$828$		0			0
$829$	−652.419	+	652.419i	−0.786996	+	0.786996i	−0.981001	−	0.194005i	$-0.937852\pi$
$829$	−652.419	+	652.419i	−0.786996	+	0.786996i	0.194005	+	0.981001i	$0.437852\pi$
$830$		0			0
$831$		−	313.819i		−	0.377641i
$832$		0			0
$833$	−284.811	+	54.0220i	−0.341910	+	0.0648524i
$834$		0			0
$835$	−1618.57	−	1618.57i	−1.93841	−	1.93841i
$836$		0			0
$837$	−335.077	+	335.077i	−0.400331	+	0.400331i
$838$		0			0
$839$	−247.208			−0.294646			−0.147323	−	0.989088i	$-0.547066\pi$
$839$	−247.208			−0.294646			−0.147323	+	0.989088i	$0.547066\pi$
$840$		0			0
$841$		−	773.741i		−	0.920025i
$842$		0			0
$843$	183.792	+	183.792i	0.218022	+	0.218022i
$844$		0			0
$845$	943.511	−	943.511i	1.11658	−	1.11658i
$846$		0			0
$847$	132.805	+	109.983i	0.156795	+	0.129850i
$848$		0			0
$849$	−569.425			−0.670700
$850$		0			0
$851$	−89.1159	−	89.1159i	−0.104719	−	0.104719i
$852$		0			0
$853$	−277.858	−	277.858i	−0.325742	−	0.325742i	0.525223	−	0.850965i	$-0.323982\pi$
$853$	−277.858	−	277.858i	−0.325742	−	0.325742i	−0.850965	+	0.525223i	$0.823982\pi$
$854$		0			0
$855$		−	496.438i		−	0.580629i
$856$		0			0
$857$	419.096			0.489027			0.244513	−	0.969646i	$-0.421372\pi$
$857$	419.096			0.489027			0.244513	+	0.969646i	$0.421372\pi$
$858$		0			0
$859$	−551.198	−	551.198i	−0.641674	−	0.641674i	0.309293	−	0.950967i	$-0.399908\pi$
$859$	−551.198	−	551.198i	−0.641674	−	0.641674i	−0.950967	+	0.309293i	$0.899908\pi$
$860$		0			0
$861$	41.3056	+	439.419i	0.0479740	+	0.510359i
$862$		0			0
$863$	−814.232			−0.943491			−0.471745	−	0.881735i	$-0.656376\pi$
$863$	−814.232			−0.943491			−0.471745	+	0.881735i	$0.656376\pi$
$864$		0			0
$865$	2113.79			2.44369
$866$		0			0
$867$	−191.011	+	191.011i	−0.220312	+	0.220312i
$868$		0			0
$869$	−808.573	+	808.573i	−0.930464	+	0.930464i
$870$		0			0
$871$	−1027.75			−1.17997
$872$		0			0
$873$	218.253			0.250004
$874$		0			0
$875$	125.350	+	1333.51i	0.143258	+	1.52401i
$876$		0			0
$877$	434.182	+	434.182i	0.495077	+	0.495077i	0.909901	−	0.414824i	$-0.136157\pi$
$877$	434.182	+	434.182i	0.495077	+	0.495077i	−0.414824	+	0.909901i	$0.636157\pi$
$878$		0			0
$879$	−49.9863			−0.0568672
$880$		0			0
$881$		−	592.016i		−	0.671982i	−0.941865	−	0.335991i	$-0.890929\pi$
$881$		−	592.016i		−	0.671982i	0.941865	−	0.335991i	$-0.109071\pi$
$882$		0			0
$883$	1020.66	+	1020.66i	1.15590	+	1.15590i	0.985349	+	0.170549i	$0.0545540\pi$
$883$	1020.66	+	1020.66i	1.15590	+	1.15590i	0.170549	+	0.985349i	$0.445446\pi$
$884$		0			0
$885$	−188.267	−	188.267i	−0.212732	−	0.212732i
$886$		0			0
$887$	−958.356			−1.08045			−0.540223	−	0.841522i	$-0.681660\pi$
$887$	−958.356			−1.08045			−0.540223	+	0.841522i	$0.681660\pi$
$888$		0			0
$889$	−459.428	+	554.763i	−0.516792	+	0.624030i
$890$		0			0
$891$	−359.157	+	359.157i	−0.403094	+	0.403094i
$892$		0			0
$893$	−425.193	−	425.193i	−0.476140	−	0.476140i
$894$		0			0
$895$		−	1078.20i		−	1.20470i
$896$		0			0
$897$	−486.135			−0.541956
$898$		0			0
$899$	−153.176	+	153.176i	−0.170385	+	0.170385i
$900$		0			0
$901$	77.1458	+	77.1458i	0.0856224	+	0.0856224i
$902$		0			0
$903$	−13.7795	−	11.4115i	−0.0152597	−	0.0126373i
$904$		0			0
$905$		−	26.9404i		−	0.0297684i
$906$		0			0
$907$	−651.884	+	651.884i	−0.718725	+	0.718725i	−0.968344	−	0.249619i	$-0.919695\pi$
$907$	−651.884	+	651.884i	−0.718725	+	0.718725i	0.249619	+	0.968344i	$0.419695\pi$
$908$		0			0
$909$	792.769	−	792.769i	0.872133	−	0.872133i
$910$		0			0
$911$	270.629			0.297068			0.148534	−	0.988907i	$-0.452545\pi$
$911$	270.629			0.297068			0.148534	+	0.988907i	$0.452545\pi$
$912$		0			0
$913$		−	439.248i		−	0.481104i
$914$		0			0
$915$	195.218	−	195.218i	0.213353	−	0.213353i
$916$		0			0
$917$	−38.4140	−	408.658i	−0.0418910	−	0.445646i
$918$		0			0
$919$		−	672.828i		−	0.732131i	−0.930589	−	0.366065i	$-0.880705\pi$
$919$		−	672.828i		−	0.732131i	0.930589	−	0.366065i	$-0.119295\pi$
$920$		0			0
$921$			103.562i			0.112445i
$922$		0			0
$923$	1271.10	+	1271.10i	1.37714	+	1.37714i
$924$		0			0
$925$	−167.048	−	167.048i	−0.180592	−	0.180592i
$926$		0			0
$927$			59.3029i			0.0639730i
$928$		0			0
$929$			1329.64i			1.43125i	0.698482	+	0.715627i	$0.253859\pi$
$929$			1329.64i			1.43125i	−0.698482	+	0.715627i	$0.746141\pi$
$930$		0			0
$931$	300.114	+	204.416i	0.322357	+	0.219566i
$932$		0			0
$933$	41.1354	−	41.1354i	0.0440894	−	0.0440894i
$934$		0			0
$935$		−	494.418i		−	0.528790i
$936$		0			0
$937$	−291.259			−0.310842			−0.155421	−	0.987848i	$-0.549673\pi$
$937$	−291.259			−0.310842			−0.155421	+	0.987848i	$0.549673\pi$
$938$		0			0
$939$	55.8067	−	55.8067i	0.0594321	−	0.0594321i
$940$		0			0
$941$	419.867	−	419.867i	0.446192	−	0.446192i	−0.447894	−	0.894087i	$-0.647826\pi$
$941$	419.867	−	419.867i	0.446192	−	0.446192i	0.894087	+	0.447894i	$0.147826\pi$
$942$		0			0
$943$			1501.54i			1.59230i
$944$		0			0
$945$	−823.406	−	681.906i	−0.871329	−	0.721594i
$946$		0			0
$947$	−372.399	−	372.399i	−0.393241	−	0.393241i	0.482600	−	0.875841i	$-0.339693\pi$
$947$	−372.399	−	372.399i	−0.393241	−	0.393241i	−0.875841	+	0.482600i	$0.839693\pi$
$948$		0			0
$949$	1668.35	−	1668.35i	1.75801	−	1.75801i
$950$		0			0
$951$	150.649			0.158411
$952$		0			0
$953$			495.950i			0.520409i	0.965554	+	0.260204i	$0.0837899\pi$
$953$			495.950i			0.520409i	−0.965554	+	0.260204i	$0.916210\pi$
$954$		0			0
$955$	1217.86	+	1217.86i	1.27525	+	1.27525i
$956$		0			0
$957$	60.5428	−	60.5428i	0.0632631	−	0.0632631i
$958$		0			0
$959$	172.140	+	142.558i	0.179499	+	0.148653i
$960$		0			0
$961$	263.307			0.273993
$962$		0			0
$963$	648.965	+	648.965i	0.673899	+	0.673899i
$964$		0			0
$965$	155.062	+	155.062i	0.160686	+	0.160686i
$966$		0			0
$967$			51.0248i			0.0527660i	0.999652	+	0.0263830i	$0.00839895\pi$
$967$			51.0248i			0.0527660i	−0.999652	+	0.0263830i	$0.991601\pi$
$968$		0			0
$969$	−46.6258			−0.0481174
$970$		0			0
$971$	492.933	+	492.933i	0.507655	+	0.507655i	0.913806	−	0.406151i	$-0.133129\pi$
$971$	492.933	+	492.933i	0.507655	+	0.507655i	−0.406151	+	0.913806i	$0.633129\pi$
$972$		0			0
$973$	477.222	−	44.8591i	0.490464	−	0.0461039i
$974$		0			0
$975$	−911.261			−0.934626
$976$		0			0
$977$	520.611			0.532867			0.266433	−	0.963853i	$-0.414155\pi$
$977$	520.611			0.532867			0.266433	+	0.963853i	$0.414155\pi$
$978$		0			0
$979$	−924.100	+	924.100i	−0.943923	+	0.943923i
$980$		0			0
$981$	934.572	−	934.572i	0.952673	−	0.952673i
$982$		0			0
$983$	1507.24			1.53330			0.766651	−	0.642064i	$-0.221922\pi$
$983$	1507.24			1.53330			0.766651	+	0.642064i	$0.221922\pi$
$984$		0			0
$985$	−2867.93			−2.91160
$986$		0			0
$987$	−601.418	+	56.5336i	−0.609340	+	0.0572782i
$988$		0			0
$989$	−43.0400	−	43.0400i	−0.0435187	−	0.0435187i
$990$		0			0
$991$	988.153			0.997127			0.498563	−	0.866853i	$-0.333861\pi$
$991$	988.153			0.997127			0.498563	+	0.866853i	$0.333861\pi$
$992$		0			0
$993$			433.041i			0.436093i
$994$		0			0
$995$	426.359	+	426.359i	0.428501	+	0.428501i
$996$		0			0
$997$	613.198	+	613.198i	0.615044	+	0.615044i	0.944256	−	0.329212i	$-0.106783\pi$
$997$	613.198	+	613.198i	0.615044	+	0.615044i	−0.329212	+	0.944256i	$0.606783\pi$
$998$		0			0
$999$	89.2715			0.0893609

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.3.l.b.209.13		56
4.3	odd	2		112.3.l.b.13.12	yes	56
7.6	odd	2	inner	448.3.l.b.209.16		56
16.5	even	4	inner	448.3.l.b.433.16		56
16.11	odd	4		112.3.l.b.69.11	yes	56
28.27	even	2		112.3.l.b.13.11	✓	56
112.27	even	4		112.3.l.b.69.12	yes	56
112.69	odd	4	inner	448.3.l.b.433.13		56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.3.l.b.13.11	✓	56	28.27	even	2
112.3.l.b.13.12	yes	56	4.3	odd	2
112.3.l.b.69.11	yes	56	16.11	odd	4
112.3.l.b.69.12	yes	56	112.27	even	4
448.3.l.b.209.13		56	1.1	even	1	trivial
448.3.l.b.209.16		56	7.6	odd	2	inner
448.3.l.b.433.13		56	112.69	odd	4	inner
448.3.l.b.433.16		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)

\(f(q)\)	\(=\)	\(q+(-0.752012 + 0.752012i) q^{3} +(6.01979 + 6.01979i) q^{5} +(4.46479 - 5.39126i) q^{7} +7.86896i q^{9} +(6.94142 - 6.94142i) q^{11} +(12.7620 - 12.7620i) q^{13} -9.05390 q^{15} -5.91610i q^{17} +(-5.24006 + 5.24006i) q^{19} +(0.696719 + 7.41186i) q^{21} +25.3271i q^{23} +47.4756i q^{25} +(-12.6857 - 12.6857i) q^{27} +(-5.79909 - 5.79909i) q^{29} -26.4139i q^{31} +10.4401i q^{33} +(59.3313 - 5.57717i) q^{35} +(-3.51860 + 3.51860i) q^{37} +19.1943i q^{39} +59.2859 q^{41} +(-1.69937 + 1.69937i) q^{43} +(-47.3694 + 47.3694i) q^{45} +81.1427i q^{47} +(-9.13136 - 48.1416i) q^{49} +(4.44897 + 4.44897i) q^{51} +(-13.0400 + 13.0400i) q^{53} +83.5717 q^{55} -7.88118i q^{57} +(20.7941 + 20.7941i) q^{59} +(-21.5617 + 21.5617i) q^{61} +(42.4236 + 35.1332i) q^{63} +153.648 q^{65} +(-40.2663 - 40.2663i) q^{67} +(-19.0462 - 19.0462i) q^{69} +99.6009i q^{71} +130.729 q^{73} +(-35.7022 - 35.7022i) q^{75} +(-6.43104 - 68.4149i) q^{77} -116.485 q^{79} -51.7411 q^{81} +(31.6396 - 31.6396i) q^{83} +(35.6136 - 35.6136i) q^{85} +8.72196 q^{87} -133.128 q^{89} +(-11.8236 - 125.782i) q^{91} +(19.8635 + 19.8635i) q^{93} -63.0881 q^{95} -27.7360i q^{97} +(54.6217 + 54.6217i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 8 q^{15} - 20 q^{21} - 96 q^{29} + 100 q^{35} - 128 q^{37} + 72 q^{43} + 192 q^{49} + 128 q^{51} + 88 q^{53} - 444 q^{63} - 8 q^{65} - 440 q^{67} + 12 q^{77} + 8 q^{79} + 64 q^{81} + 96 q^{85} + 388 q^{91}+ \cdots - 8 q^{99}+O(q^{100}) \) 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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