LMFDB - Embedded newform 448.3.l.b.433.18

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			433.18
Character	$\chi$	$=$	448.433
Dual form			448.3.l.b.209.18

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.11083 + 1.11083i) q^{3} +(-5.47687 + 5.47687i) q^{5} +(-3.11774 - 6.26735i) q^{7} -6.53210i q^{9} +(-2.57334 - 2.57334i) q^{11} +(11.1543 + 11.1543i) q^{13} -12.1678 q^{15} -24.6360i q^{17} +(-10.9902 - 10.9902i) q^{19} +(3.49869 - 10.4253i) q^{21} +10.3758i q^{23} -34.9921i q^{25} +(17.2536 - 17.2536i) q^{27} +(24.5610 - 24.5610i) q^{29} -14.5313i q^{31} -5.71711i q^{33} +(51.4009 + 17.2500i) q^{35} +(-2.55303 - 2.55303i) q^{37} +24.7811i q^{39} +48.3838 q^{41} +(-46.0243 - 46.0243i) q^{43} +(35.7754 + 35.7754i) q^{45} -19.3308i q^{47} +(-29.5593 + 39.0800i) q^{49} +(27.3665 - 27.3665i) q^{51} +(8.08934 + 8.08934i) q^{53} +28.1877 q^{55} -24.4166i q^{57} +(-61.0667 + 61.0667i) q^{59} +(-75.6742 - 75.6742i) q^{61} +(-40.9389 + 20.3654i) q^{63} -122.181 q^{65} +(81.1264 - 81.1264i) q^{67} +(-11.5258 + 11.5258i) q^{69} -9.46404i q^{71} -36.7372 q^{73} +(38.8704 - 38.8704i) q^{75} +(-8.10501 + 24.1510i) q^{77} -55.0332 q^{79} -20.4572 q^{81} +(42.2965 + 42.2965i) q^{83} +(134.928 + 134.928i) q^{85} +54.5665 q^{87} +8.11436 q^{89} +(35.1315 - 104.684i) q^{91} +(16.1418 - 16.1418i) q^{93} +120.384 q^{95} -141.282i q^{97} +(-16.8093 + 16.8093i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	1.11083	+	1.11083i	0.370278	+	0.370278i	0.867578	−	0.497300i	$-0.165675\pi$
$3$	1.11083	+	1.11083i	0.370278	+	0.370278i	−0.497300	+	0.867578i	$0.665675\pi$
$4$		0			0
$5$	−5.47687	+	5.47687i	−1.09537	+	1.09537i	−0.100429	+	0.994944i	$0.532022\pi$
$5$	−5.47687	+	5.47687i	−1.09537	+	1.09537i	−0.994944	+	0.100429i	$0.967978\pi$
$6$		0			0
$7$	−3.11774	−	6.26735i	−0.445392	−	0.895336i
$7$	−3.11774	−	6.26735i	−0.445392	−	0.895336i
$8$		0			0
$9$		−	6.53210i		−	0.725789i
$10$		0			0
$11$	−2.57334	−	2.57334i	−0.233940	−	0.233940i	0.580395	−	0.814335i	$-0.302898\pi$
$11$	−2.57334	−	2.57334i	−0.233940	−	0.233940i	−0.814335	+	0.580395i	$0.802898\pi$
$12$		0			0
$13$	11.1543	+	11.1543i	0.858021	+	0.858021i	0.991105	−	0.133084i	$-0.0424881\pi$
$13$	11.1543	+	11.1543i	0.858021	+	0.858021i	−0.133084	+	0.991105i	$0.542488\pi$
$14$		0			0
$15$	−12.1678			−0.811185
$16$		0			0
$17$		−	24.6360i		−	1.44918i	−0.689181	−	0.724589i	$-0.742029\pi$
$17$		−	24.6360i		−	1.44918i	0.689181	−	0.724589i	$-0.257971\pi$
$18$		0			0
$19$	−10.9902	−	10.9902i	−0.578432	−	0.578432i	0.356039	−	0.934471i	$-0.384127\pi$
$19$	−10.9902	−	10.9902i	−0.578432	−	0.578432i	−0.934471	+	0.356039i	$0.884127\pi$
$20$		0			0
$21$	3.49869	−	10.4253i	0.166604	−	0.496442i
$22$		0			0
$23$			10.3758i			0.451122i	0.974229	+	0.225561i	$0.0724216\pi$
$23$			10.3758i			0.451122i	−0.974229	+	0.225561i	$0.927578\pi$
$24$		0			0
$25$		−	34.9921i		−	1.39969i
$26$		0			0
$27$	17.2536	−	17.2536i	0.639021	−	0.639021i
$28$		0			0
$29$	24.5610	−	24.5610i	0.846933	−	0.846933i	−0.142817	−	0.989749i	$-0.545616\pi$
$29$	24.5610	−	24.5610i	0.846933	−	0.846933i	0.989749	+	0.142817i	$0.0456159\pi$
$30$		0			0
$31$		−	14.5313i		−	0.468751i	−0.972146	−	0.234375i	$-0.924696\pi$
$31$		−	14.5313i		−	0.468751i	0.972146	−	0.234375i	$-0.0753045\pi$
$32$		0			0
$33$		−	5.71711i		−	0.173246i
$34$		0			0
$35$	51.4009	+	17.2500i	1.46860	+	0.492856i
$36$		0			0
$37$	−2.55303	−	2.55303i	−0.0690008	−	0.0690008i	0.671764	−	0.740765i	$-0.265537\pi$
$37$	−2.55303	−	2.55303i	−0.0690008	−	0.0690008i	−0.740765	+	0.671764i	$0.765537\pi$
$38$		0			0
$39$			24.7811i			0.635412i
$40$		0			0
$41$	48.3838			1.18009			0.590046	−	0.807369i	$-0.299110\pi$
$41$	48.3838			1.18009			0.590046	+	0.807369i	$0.299110\pi$
$42$		0			0
$43$	−46.0243	−	46.0243i	−1.07033	−	1.07033i	−0.997332	−	0.0729998i	$-0.976743\pi$
$43$	−46.0243	−	46.0243i	−1.07033	−	1.07033i	−0.0729998	−	0.997332i	$-0.523257\pi$
$44$		0			0
$45$	35.7754	+	35.7754i	0.795010	+	0.795010i
$46$		0			0
$47$		−	19.3308i		−	0.411294i	−0.978626	−	0.205647i	$-0.934070\pi$
$47$		−	19.3308i		−	0.411294i	0.978626	−	0.205647i	$-0.0659298\pi$
$48$		0			0
$49$	−29.5593	+	39.0800i	−0.603252	+	0.797551i
$50$		0			0
$51$	27.3665	−	27.3665i	0.536599	−	0.536599i
$52$		0			0
$53$	8.08934	+	8.08934i	0.152629	+	0.152629i	0.779291	−	0.626662i	$-0.215579\pi$
$53$	8.08934	+	8.08934i	0.152629	+	0.152629i	−0.626662	+	0.779291i	$0.715579\pi$
$54$		0			0
$55$	28.1877			0.512503
$56$		0			0
$57$		−	24.4166i		−	0.428361i
$58$		0			0
$59$	−61.0667	+	61.0667i	−1.03503	+	1.03503i	−0.0356642	+	0.999364i	$0.511355\pi$
$59$	−61.0667	+	61.0667i	−1.03503	+	1.03503i	−0.999364	+	0.0356642i	$0.988645\pi$
$60$		0			0
$61$	−75.6742	−	75.6742i	−1.24056	−	1.24056i	−0.959768	−	0.280793i	$-0.909403\pi$
$61$	−75.6742	−	75.6742i	−1.24056	−	1.24056i	−0.280793	−	0.959768i	$-0.590597\pi$
$62$		0			0
$63$	−40.9389	+	20.3654i	−0.649824	+	0.323260i
$64$		0			0
$65$	−122.181			−1.87971
$66$		0			0
$67$	81.1264	−	81.1264i	1.21084	−	1.21084i	0.240091	−	0.970750i	$-0.422823\pi$
$67$	81.1264	−	81.1264i	1.21084	−	1.21084i	0.970750	−	0.240091i	$-0.0771774\pi$
$68$		0			0
$69$	−11.5258	+	11.5258i	−0.167041	+	0.167041i
$70$		0			0
$71$		−	9.46404i		−	0.133296i	−0.997777	−	0.0666481i	$-0.978769\pi$
$71$		−	9.46404i		−	0.133296i	0.997777	−	0.0666481i	$-0.0212305\pi$
$72$		0			0
$73$	−36.7372			−0.503249			−0.251625	−	0.967825i	$-0.580965\pi$
$73$	−36.7372			−0.503249			−0.251625	+	0.967825i	$0.580965\pi$
$74$		0			0
$75$	38.8704	−	38.8704i	0.518273	−	0.518273i
$76$		0			0
$77$	−8.10501	+	24.1510i	−0.105260	+	0.313650i
$78$		0			0
$79$	−55.0332			−0.696622			−0.348311	−	0.937379i	$-0.613245\pi$
$79$	−55.0332			−0.696622			−0.348311	+	0.937379i	$0.613245\pi$
$80$		0			0
$81$	−20.4572			−0.252558
$82$		0			0
$83$	42.2965	+	42.2965i	0.509597	+	0.509597i	0.914403	−	0.404806i	$-0.132661\pi$
$83$	42.2965	+	42.2965i	0.509597	+	0.509597i	−0.404806	+	0.914403i	$0.632661\pi$
$84$		0			0
$85$	134.928	+	134.928i	1.58739	+	1.58739i
$86$		0			0
$87$	54.5665			0.627201
$88$		0			0
$89$	8.11436			0.0911726			0.0455863	−	0.998960i	$-0.485484\pi$
$89$	8.11436			0.0911726			0.0455863	+	0.998960i	$0.485484\pi$
$90$		0			0
$91$	35.1315	−	104.684i	0.386061	−	1.15037i
$92$		0			0
$93$	16.1418	−	16.1418i	0.173568	−	0.173568i
$94$		0			0
$95$	120.384			1.26720
$96$		0			0
$97$		−	141.282i		−	1.45651i	−0.685306	−	0.728255i	$-0.740331\pi$
$97$		−	141.282i		−	1.45651i	0.685306	−	0.728255i	$-0.259669\pi$
$98$		0			0
$99$	−16.8093	+	16.8093i	−0.169791	+	0.169791i
$100$		0			0
$101$	−91.1516	+	91.1516i	−0.902491	+	0.902491i	−0.995651	−	0.0931601i	$-0.970303\pi$
$101$	−91.1516	+	91.1516i	−0.902491	+	0.902491i	0.0931601	+	0.995651i	$0.470303\pi$
$102$		0			0
$103$	−168.367			−1.63463			−0.817313	−	0.576193i	$-0.804538\pi$
$103$	−168.367			−1.63463			−0.817313	+	0.576193i	$0.804538\pi$
$104$		0			0
$105$	37.9360	+	76.2597i	0.361295	+	0.726283i
$106$		0			0
$107$	60.2166	+	60.2166i	0.562772	+	0.562772i	0.930094	−	0.367322i	$-0.119725\pi$
$107$	60.2166	+	60.2166i	0.562772	+	0.562772i	−0.367322	+	0.930094i	$0.619725\pi$
$108$		0			0
$109$	3.71167	−	3.71167i	0.0340520	−	0.0340520i	−0.689876	−	0.723928i	$-0.742335\pi$
$109$	3.71167	−	3.71167i	0.0340520	−	0.0340520i	0.723928	+	0.689876i	$0.242335\pi$
$110$		0			0
$111$		−	5.67198i		−	0.0510989i
$112$		0			0
$113$	44.1451			0.390664			0.195332	−	0.980737i	$-0.437422\pi$
$113$	44.1451			0.390664			0.195332	+	0.980737i	$0.437422\pi$
$114$		0			0
$115$	−56.8270	−	56.8270i	−0.494147	−	0.494147i
$116$		0			0
$117$	72.8608	−	72.8608i	0.622742	−	0.622742i
$118$		0			0
$119$	−154.403	+	76.8088i	−1.29750	+	0.645452i
$120$		0			0
$121$		−	107.756i		−	0.890544i
$122$		0			0
$123$	53.7463	+	53.7463i	0.436962	+	0.436962i
$124$		0			0
$125$	54.7256	+	54.7256i	0.437805	+	0.437805i
$126$		0			0
$127$	−101.945			−0.802715			−0.401358	−	0.915921i	$-0.631462\pi$
$127$	−101.945			−0.802715			−0.401358	+	0.915921i	$0.631462\pi$
$128$		0			0
$129$		−	102.251i		−	0.792640i
$130$		0			0
$131$	96.2909	+	96.2909i	0.735045	+	0.735045i	0.971615	−	0.236569i	$-0.0760231\pi$
$131$	96.2909	+	96.2909i	0.735045	+	0.735045i	−0.236569	+	0.971615i	$0.576023\pi$
$132$		0			0
$133$	−34.6148	+	103.144i	−0.260262	+	0.775520i
$134$		0			0
$135$			188.991i			1.39993i
$136$		0			0
$137$			44.6517i			0.325925i	0.986632	+	0.162963i	$0.0521050\pi$
$137$			44.6517i			0.325925i	−0.986632	+	0.162963i	$0.947895\pi$
$138$		0			0
$139$	−6.91394	+	6.91394i	−0.0497406	+	0.0497406i	−0.731540	−	0.681799i	$-0.761198\pi$
$139$	−6.91394	+	6.91394i	−0.0497406	+	0.0497406i	0.681799	+	0.731540i	$0.261198\pi$
$140$		0			0
$141$	21.4733	−	21.4733i	0.152293	−	0.152293i
$142$		0			0
$143$		−	57.4075i		−	0.401451i
$144$		0			0
$145$			269.035i			1.85541i
$146$		0			0
$147$	−76.2469	+	10.5758i	−0.518686	+	0.0719445i
$148$		0			0
$149$	−86.9376	−	86.9376i	−0.583474	−	0.583474i	0.352382	−	0.935856i	$-0.385372\pi$
$149$	−86.9376	−	86.9376i	−0.583474	−	0.583474i	−0.935856	+	0.352382i	$0.885372\pi$
$150$		0			0
$151$			136.089i			0.901251i	0.892713	+	0.450626i	$0.148799\pi$
$151$			136.089i			0.901251i	−0.892713	+	0.450626i	$0.851201\pi$
$152$		0			0
$153$	−160.925			−1.05180
$154$		0			0
$155$	79.5858	+	79.5858i	0.513457	+	0.513457i
$156$		0			0
$157$	61.2224	+	61.2224i	0.389952	+	0.389952i	0.874670	−	0.484719i	$-0.161078\pi$
$157$	61.2224	+	61.2224i	0.389952	+	0.389952i	−0.484719	+	0.874670i	$0.661078\pi$
$158$		0			0
$159$			17.9718i			0.113030i
$160$		0			0
$161$	65.0289	−	32.3491i	0.403906	−	0.200926i
$162$		0			0
$163$	−168.015	+	168.015i	−1.03077	+	1.03077i	−0.0312561	+	0.999511i	$0.509951\pi$
$163$	−168.015	+	168.015i	−1.03077	+	1.03077i	−0.999511	+	0.0312561i	$0.990049\pi$
$164$		0			0
$165$	31.3118	+	31.3118i	0.189769	+	0.189769i
$166$		0			0
$167$	−118.309			−0.708434			−0.354217	−	0.935163i	$-0.615253\pi$
$167$	−118.309			−0.708434			−0.354217	+	0.935163i	$0.615253\pi$
$168$		0			0
$169$			79.8354i			0.472399i
$170$		0			0
$171$	−71.7891	+	71.7891i	−0.419820	+	0.419820i
$172$		0			0
$173$	−8.88388	−	8.88388i	−0.0513519	−	0.0513519i	0.680965	−	0.732316i	$-0.261561\pi$
$173$	−8.88388	−	8.88388i	−0.0513519	−	0.0513519i	−0.732316	+	0.680965i	$0.761561\pi$
$174$		0			0
$175$	−219.308	+	109.097i	−1.25319	+	0.623409i
$176$		0			0
$177$	−135.670			−0.766496
$178$		0			0
$179$	134.233	−	134.233i	0.749904	−	0.749904i	−0.224557	−	0.974461i	$-0.572093\pi$
$179$	134.233	−	134.233i	0.749904	−	0.749904i	0.974461	+	0.224557i	$0.0720935\pi$
$180$		0			0
$181$	9.12602	−	9.12602i	0.0504200	−	0.0504200i	−0.681447	−	0.731867i	$-0.738649\pi$
$181$	9.12602	−	9.12602i	0.0504200	−	0.0504200i	0.731867	+	0.681447i	$0.238649\pi$
$182$		0			0
$183$		−	168.123i		−	0.918705i
$184$		0			0
$185$	27.9652			0.151163
$186$		0			0
$187$	−63.3969	+	63.3969i	−0.339021	+	0.339021i
$188$		0			0
$189$	−161.926	−	54.3420i	−0.856754	−	0.287524i
$190$		0			0
$191$	182.482			0.955405			0.477702	−	0.878522i	$-0.341470\pi$
$191$	182.482			0.955405			0.477702	+	0.878522i	$0.341470\pi$
$192$		0			0
$193$	378.766			1.96252			0.981258	−	0.192699i	$-0.0617240\pi$
$193$	378.766			1.96252			0.981258	+	0.192699i	$0.0617240\pi$
$194$		0			0
$195$	−135.723	−	135.723i	−0.696013	−	0.696013i
$196$		0			0
$197$	128.998	+	128.998i	0.654811	+	0.654811i	0.954148	−	0.299337i	$-0.0967653\pi$
$197$	128.998	+	128.998i	0.654811	+	0.654811i	−0.299337	+	0.954148i	$0.596765\pi$
$198$		0			0
$199$	274.793			1.38087			0.690436	−	0.723394i	$-0.257419\pi$
$199$	274.793			1.38087			0.690436	+	0.723394i	$0.257419\pi$
$200$		0			0
$201$	180.236			0.896696
$202$		0			0
$203$	−230.508	−	77.3576i	−1.13551	−	0.381072i
$204$		0			0
$205$	−264.992	+	264.992i	−1.29264	+	1.29264i
$206$		0			0
$207$	67.7758			0.327419
$208$		0			0
$209$			56.5631i			0.270637i
$210$		0			0
$211$	−81.1914	+	81.1914i	−0.384793	+	0.384793i	−0.872826	−	0.488032i	$-0.837715\pi$
$211$	−81.1914	+	81.1914i	−0.384793	+	0.384793i	0.488032	+	0.872826i	$0.337715\pi$
$212$		0			0
$213$	10.5130	−	10.5130i	0.0493567	−	0.0493567i
$214$		0			0
$215$	504.138			2.34483
$216$		0			0
$217$	−91.0725	+	45.3048i	−0.419689	+	0.208778i
$218$		0			0
$219$	−40.8089	−	40.8089i	−0.186342	−	0.186342i
$220$		0			0
$221$	274.797	−	274.797i	1.24343	−	1.24343i
$222$		0			0
$223$		−	265.113i		−	1.18885i	−0.804152	−	0.594424i	$-0.797380\pi$
$223$		−	265.113i		−	1.18885i	0.804152	−	0.594424i	$-0.202620\pi$
$224$		0			0
$225$	−228.572			−1.01588
$226$		0			0
$227$	35.0233	+	35.0233i	0.154288	+	0.154288i	0.780030	−	0.625742i	$-0.215204\pi$
$227$	35.0233	+	35.0233i	0.154288	+	0.154288i	−0.625742	+	0.780030i	$0.715204\pi$
$228$		0			0
$229$	16.7509	−	16.7509i	0.0731481	−	0.0731481i	−0.669586	−	0.742734i	$-0.733529\pi$
$229$	16.7509	−	16.7509i	0.0731481	−	0.0731481i	0.742734	+	0.669586i	$0.233529\pi$
$230$		0			0
$231$	−35.8311	+	17.8245i	−0.155113	+	0.0771622i
$232$		0			0
$233$			162.893i			0.699113i	0.936915	+	0.349556i	$0.113668\pi$
$233$			162.893i			0.699113i	−0.936915	+	0.349556i	$0.886332\pi$
$234$		0			0
$235$	105.872	+	105.872i	0.450520	+	0.450520i
$236$		0			0
$237$	−61.1327	−	61.1327i	−0.257944	−	0.257944i
$238$		0			0
$239$	458.835			1.91981			0.959906	−	0.280321i	$-0.0904408\pi$
$239$	458.835			1.91981			0.959906	+	0.280321i	$0.0904408\pi$
$240$		0			0
$241$			6.17751i			0.0256328i	0.999918	+	0.0128164i	$0.00407970\pi$
$241$			6.17751i			0.0256328i	−0.999918	+	0.0128164i	$0.995920\pi$
$242$		0			0
$243$	−178.007	−	178.007i	−0.732538	−	0.732538i
$244$		0			0
$245$	−52.1432	−	375.928i	−0.212830	−	1.53440i
$246$		0			0
$247$		−	245.176i		−	0.992614i
$248$		0			0
$249$			93.9688i			0.377385i
$250$		0			0
$251$	19.6790	−	19.6790i	0.0784025	−	0.0784025i	−0.666818	−	0.745221i	$-0.732344\pi$
$251$	19.6790	−	19.6790i	0.0784025	−	0.0784025i	0.745221	+	0.666818i	$0.232344\pi$
$252$		0			0
$253$	26.7005	−	26.7005i	0.105536	−	0.105536i
$254$		0			0
$255$			299.766i			1.17555i
$256$		0			0
$257$		−	46.6133i		−	0.181375i	−0.995879	−	0.0906873i	$-0.971094\pi$
$257$		−	46.6133i		−	0.181375i	0.995879	−	0.0906873i	$-0.0289064\pi$
$258$		0			0
$259$	−8.04103	+	23.9604i	−0.0310465	+	0.0925112i
$260$		0			0
$261$	−160.435	−	160.435i	−0.614694	−	0.614694i
$262$		0			0
$263$		−	233.528i		−	0.887940i	−0.896042	−	0.443970i	$-0.853570\pi$
$263$		−	233.528i		−	0.887940i	0.896042	−	0.443970i	$-0.146430\pi$
$264$		0			0
$265$	−88.6085			−0.334372
$266$		0			0
$267$	9.01371	+	9.01371i	0.0337592	+	0.0337592i
$268$		0			0
$269$	−118.706	−	118.706i	−0.441286	−	0.441286i	0.451158	−	0.892444i	$-0.351011\pi$
$269$	−118.706	−	118.706i	−0.441286	−	0.441286i	−0.892444	+	0.451158i	$0.851011\pi$
$270$		0			0
$271$			131.992i			0.487055i	0.969894	+	0.243527i	$0.0783046\pi$
$271$			131.992i			0.487055i	−0.969894	+	0.243527i	$0.921695\pi$
$272$		0			0
$273$	155.312	−	77.2610i	0.568907	−	0.283007i
$274$		0			0
$275$	−90.0467	+	90.0467i	−0.327443	+	0.327443i
$276$		0			0
$277$	−249.565	−	249.565i	−0.900958	−	0.900958i	0.0945613	−	0.995519i	$-0.469855\pi$
$277$	−249.565	−	249.565i	−0.900958	−	0.900958i	−0.995519	+	0.0945613i	$0.969855\pi$
$278$		0			0
$279$	−94.9197			−0.340214
$280$		0			0
$281$		−	344.044i		−	1.22436i	−0.790720	−	0.612178i	$-0.790294\pi$
$281$		−	344.044i		−	1.22436i	0.790720	−	0.612178i	$-0.209706\pi$
$282$		0			0
$283$	141.638	−	141.638i	0.500489	−	0.500489i	−0.411101	−	0.911590i	$-0.634856\pi$
$283$	141.638	−	141.638i	0.500489	−	0.500489i	0.911590	+	0.411101i	$0.134856\pi$
$284$		0			0
$285$	133.726	+	133.726i	0.469216	+	0.469216i
$286$		0			0
$287$	−150.848	−	303.238i	−0.525604	−	1.05658i
$288$		0			0
$289$	−317.934			−1.10012
$290$		0			0
$291$	156.940	−	156.940i	0.539314	−	0.539314i
$292$		0			0
$293$	−82.6004	+	82.6004i	−0.281913	+	0.281913i	−0.833871	−	0.551959i	$-0.813881\pi$
$293$	−82.6004	+	82.6004i	−0.281913	+	0.281913i	0.551959	+	0.833871i	$0.313881\pi$
$294$		0			0
$295$		−	668.908i		−	2.26748i
$296$		0			0
$297$	−88.7986			−0.298985
$298$		0			0
$299$	−115.735	+	115.735i	−0.387072	+	0.387072i
$300$		0			0
$301$	−144.958	+	431.942i	−0.481589	+	1.43502i
$302$		0			0
$303$	−202.509			−0.668345
$304$		0			0
$305$	828.916			2.71776
$306$		0			0
$307$	200.953	+	200.953i	0.654570	+	0.654570i	0.954090	−	0.299520i	$-0.0968265\pi$
$307$	200.953	+	200.953i	0.654570	+	0.654570i	−0.299520	+	0.954090i	$0.596826\pi$
$308$		0			0
$309$	−187.027	−	187.027i	−0.605266	−	0.605266i
$310$		0			0
$311$	−463.074			−1.48898			−0.744491	−	0.667632i	$-0.767308\pi$
$311$	−463.074			−1.48898			−0.744491	+	0.667632i	$0.767308\pi$
$312$		0			0
$313$	1.09088			0.00348523			0.00174262	−	0.999998i	$-0.499445\pi$
$313$	1.09088			0.00348523			0.00174262	+	0.999998i	$0.499445\pi$
$314$		0			0
$315$	112.679	−	335.756i	0.357710	−	1.06589i
$316$		0			0
$317$	51.1667	−	51.1667i	0.161409	−	0.161409i	−0.621782	−	0.783191i	$-0.713591\pi$
$317$	51.1667	−	51.1667i	0.161409	−	0.161409i	0.783191	+	0.621782i	$0.213591\pi$
$318$		0			0
$319$	−126.408			−0.396263
$320$		0			0
$321$			133.781i			0.416764i
$322$		0			0
$323$	−270.755	+	270.755i	−0.838252	+	0.838252i
$324$		0			0
$325$	390.312	−	390.312i	1.20096	−	1.20096i
$326$		0			0
$327$	8.24610			0.0252174
$328$		0			0
$329$	−121.153	+	60.2685i	−0.368246	+	0.183187i
$330$		0			0
$331$	−28.8219	−	28.8219i	−0.0870753	−	0.0870753i	0.662228	−	0.749303i	$-0.269611\pi$
$331$	−28.8219	−	28.8219i	−0.0870753	−	0.0870753i	−0.749303	+	0.662228i	$0.769611\pi$
$332$		0			0
$333$	−16.6766	+	16.6766i	−0.0500800	+	0.0500800i
$334$		0			0
$335$			888.637i			2.65265i
$336$		0			0
$337$	−84.7524			−0.251491			−0.125745	−	0.992063i	$-0.540132\pi$
$337$	−84.7524			−0.251491			−0.125745	+	0.992063i	$0.540132\pi$
$338$		0			0
$339$	49.0378	+	49.0378i	0.144654	+	0.144654i
$340$		0			0
$341$	−37.3939	+	37.3939i	−0.109660	+	0.109660i
$342$		0			0
$343$	337.086	+	63.4174i	0.982759	+	0.184890i
$344$		0			0
$345$		−	126.251i		−	0.365944i
$346$		0			0
$347$	38.7326	+	38.7326i	0.111621	+	0.111621i	0.760711	−	0.649090i	$-0.224850\pi$
$347$	38.7326	+	38.7326i	0.111621	+	0.111621i	−0.649090	+	0.760711i	$0.724850\pi$
$348$		0			0
$349$	3.91040	+	3.91040i	0.0112046	+	0.0112046i	0.712687	−	0.701482i	$-0.247478\pi$
$349$	3.91040	+	3.91040i	0.0112046	+	0.0112046i	−0.701482	+	0.712687i	$0.747478\pi$
$350$		0			0
$351$	384.902			1.09659
$352$		0			0
$353$			620.142i			1.75678i	0.477949	+	0.878388i	$0.341380\pi$
$353$			620.142i			1.75678i	−0.477949	+	0.878388i	$0.658620\pi$
$354$		0			0
$355$	51.8333	+	51.8333i	0.146009	+	0.146009i
$356$		0			0
$357$	−256.838	−	86.1938i	−0.719433	−	0.241439i
$358$		0			0
$359$		−	343.001i		−	0.955435i	−0.878514	−	0.477717i	$-0.841464\pi$
$359$		−	343.001i		−	0.955435i	0.878514	−	0.477717i	$-0.158536\pi$
$360$		0			0
$361$		−	119.430i		−	0.330832i
$362$		0			0
$363$	119.699	−	119.699i	0.329749	−	0.329749i
$364$		0			0
$365$	201.205	−	201.205i	0.551246	−	0.551246i
$366$		0			0
$367$		−	680.088i		−	1.85310i	−0.376171	−	0.926550i	$-0.622760\pi$
$367$		−	680.088i		−	1.85310i	0.376171	−	0.926550i	$-0.377240\pi$
$368$		0			0
$369$		−	316.048i		−	0.856498i
$370$		0			0
$371$	25.4782	−	75.9193i	0.0686745	−	0.204634i
$372$		0			0
$373$	−32.9618	−	32.9618i	−0.0883694	−	0.0883694i	0.661540	−	0.749910i	$-0.269903\pi$
$373$	−32.9618	−	32.9618i	−0.0883694	−	0.0883694i	−0.749910	+	0.661540i	$0.769903\pi$
$374$		0			0
$375$			121.582i			0.324219i
$376$		0			0
$377$	547.921			1.45337
$378$		0			0
$379$	217.828	+	217.828i	0.574745	+	0.574745i	0.933451	−	0.358706i	$-0.116782\pi$
$379$	217.828	+	217.828i	0.574745	+	0.574745i	−0.358706	+	0.933451i	$0.616782\pi$
$380$		0			0
$381$	−113.244	−	113.244i	−0.297228	−	0.297228i
$382$		0			0
$383$		−	263.286i		−	0.687430i	−0.939074	−	0.343715i	$-0.888315\pi$
$383$		−	263.286i		−	0.687430i	0.939074	−	0.343715i	$-0.111685\pi$
$384$		0			0
$385$	−87.8820	−	176.662i	−0.228265	−	0.458863i
$386$		0			0
$387$	−300.635	+	300.635i	−0.776835	+	0.776835i
$388$		0			0
$389$	28.5590	+	28.5590i	0.0734164	+	0.0734164i	0.742862	−	0.669445i	$-0.233468\pi$
$389$	28.5590	+	28.5590i	0.0734164	+	0.0734164i	−0.669445	+	0.742862i	$0.733468\pi$
$390$		0			0
$391$	255.619			0.653757
$392$		0			0
$393$			213.926i			0.544342i
$394$		0			0
$395$	301.409	−	301.409i	0.763062	−	0.763062i
$396$		0			0
$397$	−263.058	−	263.058i	−0.662615	−	0.662615i	0.293380	−	0.955996i	$-0.405220\pi$
$397$	−263.058	−	263.058i	−0.662615	−	0.662615i	−0.955996	+	0.293380i	$0.905220\pi$
$398$		0			0
$399$	−153.027	+	76.1247i	−0.383527	+	0.190789i
$400$		0			0
$401$	372.578			0.929123			0.464562	−	0.885541i	$-0.346212\pi$
$401$	372.578			0.929123			0.464562	+	0.885541i	$0.346212\pi$
$402$		0			0
$403$	162.086	−	162.086i	0.402198	−	0.402198i
$404$		0			0
$405$	112.041	−	112.041i	0.276645	−	0.276645i
$406$		0			0
$407$			13.1396i			0.0322841i
$408$		0			0
$409$	561.999			1.37408			0.687040	−	0.726620i	$-0.258910\pi$
$409$	561.999			1.37408			0.687040	+	0.726620i	$0.258910\pi$
$410$		0			0
$411$	−49.6006	+	49.6006i	−0.120683	+	0.120683i
$412$		0			0
$413$	573.116	+	192.336i	1.38769	+	0.465704i
$414$		0			0
$415$	−463.305			−1.11640
$416$		0			0
$417$	−15.3605			−0.0368357
$418$		0			0
$419$	−238.635	−	238.635i	−0.569534	−	0.569534i	0.362464	−	0.931998i	$-0.381936\pi$
$419$	−238.635	−	238.635i	−0.569534	−	0.569534i	−0.931998	+	0.362464i	$0.881936\pi$
$420$		0			0
$421$	503.838	+	503.838i	1.19676	+	1.19676i	0.975130	+	0.221635i	$0.0711393\pi$
$421$	503.838	+	503.838i	1.19676	+	1.19676i	0.221635	+	0.975130i	$0.428861\pi$
$422$		0			0
$423$	−126.271			−0.298512
$424$		0			0
$425$	−862.068			−2.02839
$426$		0			0
$427$	−238.344	+	710.210i	−0.558183	+	1.66326i
$428$		0			0
$429$	63.7701	−	63.7701i	0.148648	−	0.148648i
$430$		0			0
$431$	580.398			1.34663			0.673315	−	0.739355i	$-0.264870\pi$
$431$	580.398			1.34663			0.673315	+	0.739355i	$0.264870\pi$
$432$		0			0
$433$			161.231i			0.372359i	0.982516	+	0.186179i	$0.0596106\pi$
$433$			161.231i			0.372359i	−0.982516	+	0.186179i	$0.940389\pi$
$434$		0			0
$435$	−298.853	+	298.853i	−0.687019	+	0.687019i
$436$		0			0
$437$	114.032	−	114.032i	0.260944	−	0.260944i
$438$		0			0
$439$	−171.458			−0.390564			−0.195282	−	0.980747i	$-0.562562\pi$
$439$	−171.458			−0.390564			−0.195282	+	0.980747i	$0.562562\pi$
$440$		0			0
$441$	255.274	+	193.085i	0.578853	+	0.437833i
$442$		0			0
$443$	−362.506	−	362.506i	−0.818298	−	0.818298i	0.167563	−	0.985861i	$-0.446410\pi$
$443$	−362.506	−	362.506i	−0.818298	−	0.818298i	−0.985861	+	0.167563i	$0.946410\pi$
$444$		0			0
$445$	−44.4413	+	44.4413i	−0.0998681	+	0.0998681i
$446$		0			0
$447$		−	193.147i		−	0.432095i
$448$		0			0
$449$	−680.601			−1.51582			−0.757908	−	0.652362i	$-0.773778\pi$
$449$	−680.601			−1.51582			−0.757908	+	0.652362i	$0.773778\pi$
$450$		0			0
$451$	−124.508	−	124.508i	−0.276071	−	0.276071i
$452$		0			0
$453$	−151.172	+	151.172i	−0.333713	+	0.333713i
$454$		0			0
$455$	380.929	+	765.750i	0.837206	+	1.68297i
$456$		0			0
$457$			76.4527i			0.167293i	0.996496	+	0.0836463i	$0.0266566\pi$
$457$			76.4527i			0.167293i	−0.996496	+	0.0836463i	$0.973343\pi$
$458$		0			0
$459$	−425.060	−	425.060i	−0.926056	−	0.926056i
$460$		0			0
$461$	−9.52689	−	9.52689i	−0.0206657	−	0.0206657i	0.696698	−	0.717364i	$-0.254652\pi$
$461$	−9.52689	−	9.52689i	−0.0206657	−	0.0206657i	−0.717364	+	0.696698i	$0.754652\pi$
$462$		0			0
$463$	−108.583			−0.234521			−0.117260	−	0.993101i	$-0.537411\pi$
$463$	−108.583			−0.234521			−0.117260	+	0.993101i	$0.537411\pi$
$464$		0			0
$465$			176.813i			0.380243i
$466$		0			0
$467$	−428.836	−	428.836i	−0.918278	−	0.918278i	0.0786261	−	0.996904i	$-0.474947\pi$
$467$	−428.836	−	428.836i	−0.918278	−	0.918278i	−0.996904	+	0.0786261i	$0.974947\pi$
$468$		0			0
$469$	−761.379	−	255.516i	−1.62341	−	0.544811i
$470$		0			0
$471$			136.016i			0.288781i
$472$		0			0
$473$			236.872i			0.500787i
$474$		0			0
$475$	−384.571	+	384.571i	−0.809623	+	0.809623i
$476$		0			0
$477$	52.8404	−	52.8404i	0.110776	−	0.110776i
$478$		0			0
$479$			678.346i			1.41617i	0.706127	+	0.708085i	$0.250441\pi$
$479$			678.346i			1.41617i	−0.706127	+	0.708085i	$0.749559\pi$
$480$		0			0
$481$		−	56.9543i		−	0.118408i
$482$		0			0
$483$	108.171	+	36.3017i	0.223956	+	0.0751589i
$484$		0			0
$485$	773.780	+	773.780i	1.59542	+	1.59542i
$486$		0			0
$487$			737.584i			1.51455i	0.653098	+	0.757273i	$0.273469\pi$
$487$			737.584i			1.51455i	−0.653098	+	0.757273i	$0.726531\pi$
$488$		0			0
$489$	−373.274			−0.763341
$490$		0			0
$491$	524.855	+	524.855i	1.06895	+	1.06895i	0.997440	+	0.0715115i	$0.0227823\pi$
$491$	524.855	+	524.855i	1.06895	+	1.06895i	0.0715115	+	0.997440i	$0.477218\pi$
$492$		0			0
$493$	−605.087	−	605.087i	−1.22736	−	1.22736i
$494$		0			0
$495$		−	184.125i		−	0.371969i
$496$		0			0
$497$	−59.3144	+	29.5064i	−0.119345	+	0.0593691i
$498$		0			0
$499$	−61.5115	+	61.5115i	−0.123270	+	0.123270i	−0.766050	−	0.642781i	$-0.777781\pi$
$499$	−61.5115	+	61.5115i	−0.123270	+	0.123270i	0.642781	+	0.766050i	$0.277781\pi$
$500$		0			0
$501$	−131.421	−	131.421i	−0.262318	−	0.262318i
$502$		0			0
$503$	64.6784			0.128585			0.0642927	−	0.997931i	$-0.479521\pi$
$503$	64.6784			0.128585			0.0642927	+	0.997931i	$0.479521\pi$
$504$		0			0
$505$		−	998.450i		−	1.97713i
$506$		0			0
$507$	−88.6838	+	88.6838i	−0.174919	+	0.174919i
$508$		0			0
$509$	392.205	+	392.205i	0.770541	+	0.770541i	0.978201	−	0.207660i	$-0.0665847\pi$
$509$	392.205	+	392.205i	0.770541	+	0.770541i	−0.207660	+	0.978201i	$0.566585\pi$
$510$		0			0
$511$	114.537	+	230.245i	0.224143	+	0.450577i
$512$		0			0
$513$	−379.241			−0.739261
$514$		0			0
$515$	922.121	−	922.121i	1.79053	−	1.79053i
$516$		0			0
$517$	−49.7447	+	49.7447i	−0.0962180	+	0.0962180i
$518$		0			0
$519$		−	19.7370i		−	0.0380289i
$520$		0			0
$521$	−537.133			−1.03097			−0.515483	−	0.856900i	$-0.672387\pi$
$521$	−537.133			−1.03097			−0.515483	+	0.856900i	$0.672387\pi$
$522$		0			0
$523$	−282.626	+	282.626i	−0.540394	+	0.540394i	−0.923644	−	0.383251i	$-0.874805\pi$
$523$	−282.626	+	282.626i	−0.540394	+	0.540394i	0.383251	+	0.923644i	$0.374805\pi$
$524$		0			0
$525$	−364.803	−	122.427i	−0.694862	−	0.233194i
$526$		0			0
$527$	−357.993			−0.679303
$528$		0			0
$529$	421.342			0.796489
$530$		0			0
$531$	398.893	+	398.893i	0.751212	+	0.751212i
$532$		0			0
$533$	539.686	+	539.686i	1.01254	+	1.01254i
$534$		0			0
$535$	−659.596			−1.23289
$536$		0			0
$537$	298.221			0.555346
$538$		0			0
$539$	176.632	−	24.4998i	0.327704	−	0.0454542i
$540$		0			0
$541$	−53.3102	+	53.3102i	−0.0985401	+	0.0985401i	−0.754658	−	0.656118i	$-0.772197\pi$
$541$	−53.3102	+	53.3102i	−0.0985401	+	0.0985401i	0.656118	+	0.754658i	$0.272197\pi$
$542$		0			0
$543$	20.2750			0.0373388
$544$		0			0
$545$			40.6567i			0.0745994i
$546$		0			0
$547$	−301.530	+	301.530i	−0.551244	+	0.551244i	−0.926800	−	0.375556i	$-0.877452\pi$
$547$	−301.530	+	301.530i	−0.551244	+	0.551244i	0.375556	+	0.926800i	$0.377452\pi$
$548$		0			0
$549$	−494.312	+	494.312i	−0.900385	+	0.900385i
$550$		0			0
$551$	−539.862			−0.979786
$552$		0			0
$553$	171.579	+	344.912i	0.310270	+	0.623711i
$554$		0			0
$555$	31.0647	+	31.0647i	0.0559724	+	0.0559724i
$556$		0			0
$557$	−524.220	+	524.220i	−0.941148	+	0.941148i	−0.998362	−	0.0572135i	$-0.981778\pi$
$557$	−524.220	+	524.220i	−0.941148	+	0.941148i	0.0572135	+	0.998362i	$0.481778\pi$
$558$		0			0
$559$		−	1026.73i		−	1.83673i
$560$		0			0
$561$	−140.847			−0.251064
$562$		0			0
$563$	608.657	+	608.657i	1.08110	+	1.08110i	0.996407	+	0.0846888i	$0.0269896\pi$
$563$	608.657	+	608.657i	1.08110	+	1.08110i	0.0846888	+	0.996407i	$0.473010\pi$
$564$		0			0
$565$	−241.777	+	241.777i	−0.427923	+	0.427923i
$566$		0			0
$567$	63.7802	+	128.212i	0.112487	+	0.226124i
$568$		0			0
$569$			576.735i			1.01359i	0.862066	+	0.506797i	$0.169170\pi$
$569$			576.735i			1.01359i	−0.862066	+	0.506797i	$0.830830\pi$
$570$		0			0
$571$	401.402	+	401.402i	0.702980	+	0.702980i	0.965049	−	0.262069i	$-0.0844048\pi$
$571$	401.402	+	401.402i	0.702980	+	0.702980i	−0.262069	+	0.965049i	$0.584405\pi$
$572$		0			0
$573$	202.708	+	202.708i	0.353765	+	0.353765i
$574$		0			0
$575$	363.072			0.631430
$576$		0			0
$577$		−	933.829i		−	1.61842i	−0.587519	−	0.809210i	$-0.699895\pi$
$577$		−	933.829i		−	1.61842i	0.587519	−	0.809210i	$-0.300105\pi$
$578$		0			0
$579$	420.746	+	420.746i	0.726676	+	0.726676i
$580$		0			0
$581$	133.217	−	396.957i	0.229290	−	0.683231i
$582$		0			0
$583$		−	41.6333i		−	0.0714121i
$584$		0			0
$585$			798.097i			1.36427i
$586$		0			0
$587$	415.106	−	415.106i	0.707165	−	0.707165i	−0.258773	−	0.965938i	$-0.583318\pi$
$587$	415.106	−	415.106i	0.707165	−	0.707165i	0.965938	+	0.258773i	$0.0833181\pi$
$588$		0			0
$589$	−159.702	+	159.702i	−0.271140	+	0.271140i
$590$		0			0
$591$			286.590i			0.484924i
$592$		0			0
$593$		−	818.826i		−	1.38082i	−0.723419	−	0.690409i	$-0.757430\pi$
$593$		−	818.826i		−	1.38082i	0.723419	−	0.690409i	$-0.242570\pi$
$594$		0			0
$595$	424.971	−	1266.31i	0.714237	−	2.12826i
$596$		0			0
$597$	305.250	+	305.250i	0.511306	+	0.511306i
$598$		0			0
$599$			503.564i			0.840674i	0.907368	+	0.420337i	$0.138088\pi$
$599$			503.564i			0.840674i	−0.907368	+	0.420337i	$0.861912\pi$
$600$		0			0
$601$	−110.475			−0.183819			−0.0919093	−	0.995767i	$-0.529297\pi$
$601$	−110.475			−0.183819			−0.0919093	+	0.995767i	$0.529297\pi$
$602$		0			0
$603$	−529.925	−	529.925i	−0.878815	−	0.878815i
$604$		0			0
$605$	590.164	+	590.164i	0.975478	+	0.975478i
$606$		0			0
$607$		−	169.212i		−	0.278769i	−0.990238	−	0.139384i	$-0.955488\pi$
$607$		−	169.212i		−	0.278769i	0.990238	−	0.139384i	$-0.0445123\pi$
$608$		0			0
$609$	−170.124	−	341.987i	−0.279350	−	0.561555i
$610$		0			0
$611$	215.621	−	215.621i	0.352898	−	0.352898i
$612$		0			0
$613$	199.765	+	199.765i	0.325881	+	0.325881i	0.851018	−	0.525137i	$-0.175986\pi$
$613$	199.765	+	199.765i	0.325881	+	0.325881i	−0.525137	+	0.851018i	$0.675986\pi$
$614$		0			0
$615$	−588.723			−0.957273
$616$		0			0
$617$		−	166.208i		−	0.269381i	−0.990888	−	0.134690i	$-0.956996\pi$
$617$		−	166.208i		−	0.269381i	0.990888	−	0.134690i	$-0.0430040\pi$
$618$		0			0
$619$	474.519	−	474.519i	0.766589	−	0.766589i	−0.210915	−	0.977504i	$-0.567644\pi$
$619$	474.519	−	474.519i	0.766589	−	0.766589i	0.977504	+	0.210915i	$0.0676444\pi$
$620$		0			0
$621$	179.020	+	179.020i	0.288277	+	0.288277i
$622$		0			0
$623$	−25.2985	−	50.8555i	−0.0406075	−	0.0816301i
$624$		0			0
$625$	275.354			0.440566
$626$		0			0
$627$	−62.8322	+	62.8322i	−0.100211	+	0.100211i
$628$		0			0
$629$	−62.8965	+	62.8965i	−0.0999944	+	0.0999944i
$630$		0			0
$631$		−	326.927i		−	0.518110i	−0.965863	−	0.259055i	$-0.916589\pi$
$631$		−	326.927i		−	0.518110i	0.965863	−	0.259055i	$-0.0834111\pi$
$632$		0			0
$633$	−180.380			−0.284961
$634$		0			0
$635$	558.338	−	558.338i	0.879273	−	0.879273i
$636$		0			0
$637$	−765.621	+	106.196i	−1.20192	+	0.166712i
$638$		0			0
$639$	−61.8200			−0.0967449
$640$		0			0
$641$	−468.475			−0.730850			−0.365425	−	0.930841i	$-0.619076\pi$
$641$	−468.475			−0.730850			−0.365425	+	0.930841i	$0.619076\pi$
$642$		0			0
$643$	162.961	+	162.961i	0.253438	+	0.253438i	0.822379	−	0.568941i	$-0.192647\pi$
$643$	162.961	+	162.961i	0.253438	+	0.253438i	−0.568941	+	0.822379i	$0.692647\pi$
$644$		0			0
$645$	560.013	+	560.013i	0.868237	+	0.868237i
$646$		0			0
$647$	648.535			1.00237			0.501186	−	0.865340i	$-0.332897\pi$
$647$	648.535			1.00237			0.501186	+	0.865340i	$0.332897\pi$
$648$		0			0
$649$	314.291			0.484269
$650$		0			0
$651$	−151.493	−	50.8404i	−0.232707	−	0.0780958i
$652$		0			0
$653$	198.998	−	198.998i	0.304744	−	0.304744i	−0.538122	−	0.842867i	$-0.680866\pi$
$653$	198.998	−	198.998i	0.304744	−	0.304744i	0.842867	+	0.538122i	$0.180866\pi$
$654$		0			0
$655$	−1054.75			−1.61030
$656$		0			0
$657$			239.971i			0.365253i
$658$		0			0
$659$	−273.506	+	273.506i	−0.415032	+	0.415032i	−0.883487	−	0.468455i	$-0.844811\pi$
$659$	−273.506	+	273.506i	−0.415032	+	0.415032i	0.468455	+	0.883487i	$0.344811\pi$
$660$		0			0
$661$	−418.775	+	418.775i	−0.633547	+	0.633547i	−0.948956	−	0.315409i	$-0.897858\pi$
$661$	−418.775	+	418.775i	−0.633547	+	0.633547i	0.315409	+	0.948956i	$0.397858\pi$
$662$		0			0
$663$	610.507			0.920826
$664$		0			0
$665$	−375.326	−	754.488i	−0.564400	−	1.13457i
$666$		0			0
$667$	254.841	+	254.841i	0.382070	+	0.382070i
$668$		0			0
$669$	294.496	−	294.496i	0.440204	−	0.440204i
$670$		0			0
$671$			389.471i			0.580434i
$672$		0			0
$673$	−1171.54			−1.74077			−0.870383	−	0.492375i	$-0.836129\pi$
$673$	−1171.54			−1.74077			−0.870383	+	0.492375i	$0.836129\pi$
$674$		0			0
$675$	−603.740	−	603.740i	−0.894429	−	0.894429i
$676$		0			0
$677$	104.959	−	104.959i	0.155035	−	0.155035i	−0.625328	−	0.780362i	$-0.715035\pi$
$677$	104.959	−	104.959i	0.155035	−	0.155035i	0.780362	+	0.625328i	$0.215035\pi$
$678$		0			0
$679$	−885.461	+	440.480i	−1.30407	+	0.648718i
$680$		0			0
$681$			77.8100i			0.114259i
$682$		0			0
$683$	−598.937	−	598.937i	−0.876921	−	0.876921i	0.116294	−	0.993215i	$-0.462898\pi$
$683$	−598.937	−	598.937i	−0.876921	−	0.876921i	−0.993215	+	0.116294i	$0.962898\pi$
$684$		0			0
$685$	−244.552	−	244.552i	−0.357010	−	0.357010i
$686$		0			0
$687$	37.2150			0.0541703
$688$		0			0
$689$			180.461i			0.261918i
$690$		0			0
$691$	397.739	+	397.739i	0.575599	+	0.575599i	0.933688	−	0.358089i	$-0.116571\pi$
$691$	397.739	+	397.739i	0.575599	+	0.575599i	−0.358089	+	0.933688i	$0.616571\pi$
$692$		0			0
$693$	157.757	+	52.9427i	0.227643	+	0.0763964i
$694$		0			0
$695$		−	75.7335i		−	0.108969i
$696$		0			0
$697$		−	1191.98i		−	1.71016i
$698$		0			0
$699$	−180.947	+	180.947i	−0.258866	+	0.258866i
$700$		0			0
$701$	848.967	−	848.967i	1.21108	−	1.21108i	0.240409	−	0.970672i	$-0.422718\pi$
$701$	848.967	−	848.967i	1.21108	−	1.21108i	0.970672	−	0.240409i	$-0.0772815\pi$
$702$		0			0
$703$			56.1166i			0.0798245i
$704$		0			0
$705$			235.213i			0.333635i
$706$		0			0
$707$	855.466	+	287.092i	1.20999	+	0.406070i
$708$		0			0
$709$	250.885	+	250.885i	0.353857	+	0.353857i	0.861542	−	0.507685i	$-0.169499\pi$
$709$	250.885	+	250.885i	0.353857	+	0.353857i	−0.507685	+	0.861542i	$0.669499\pi$
$710$		0			0
$711$			359.482i			0.505601i
$712$		0			0
$713$	150.774			0.211464
$714$		0			0
$715$	314.413	+	314.413i	0.439738	+	0.439738i
$716$		0			0
$717$	509.690	+	509.690i	0.710864	+	0.710864i
$718$		0			0
$719$			1326.17i			1.84446i	0.386642	+	0.922230i	$0.373635\pi$
$719$			1326.17i			1.84446i	−0.386642	+	0.922230i	$0.626365\pi$
$720$		0			0
$721$	524.924	+	1055.21i	0.728050	+	1.46354i
$722$		0			0
$723$	−6.86219	+	6.86219i	−0.00949127	+	0.00949127i
$724$		0			0
$725$	−859.444	−	859.444i	−1.18544	−	1.18544i
$726$		0			0
$727$	135.744			0.186719			0.0933593	−	0.995632i	$-0.470239\pi$
$727$	135.744			0.186719			0.0933593	+	0.995632i	$0.470239\pi$
$728$		0			0
$729$		−	211.357i		−	0.289927i
$730$		0			0
$731$	−1133.86	+	1133.86i	−1.55110	+	1.55110i
$732$		0			0
$733$	340.606	+	340.606i	0.464674	+	0.464674i	0.900184	−	0.435510i	$-0.143432\pi$
$733$	340.606	+	340.606i	0.464674	+	0.464674i	−0.435510	+	0.900184i	$0.643432\pi$
$734$		0			0
$735$	359.672	−	475.516i	0.489349	−	0.646961i
$736$		0			0
$737$	−417.532			−0.566529
$738$		0			0
$739$	−422.281	+	422.281i	−0.571423	+	0.571423i	−0.932526	−	0.361103i	$-0.882400\pi$
$739$	−422.281	+	422.281i	−0.571423	+	0.571423i	0.361103	+	0.932526i	$0.382400\pi$
$740$		0			0
$741$	272.349	−	272.349i	0.367543	−	0.367543i
$742$		0			0
$743$		−	125.612i		−	0.169061i	−0.996421	−	0.0845303i	$-0.973061\pi$
$743$		−	125.612i		−	0.169061i	0.996421	−	0.0845303i	$-0.0269390\pi$
$744$		0			0
$745$	952.292			1.27824
$746$		0			0
$747$	276.285	−	276.285i	0.369860	−	0.369860i
$748$		0			0
$749$	189.658	−	565.138i	0.253216	−	0.754524i
$750$		0			0
$751$	964.776			1.28465			0.642327	−	0.766430i	$-0.277969\pi$
$751$	964.776			1.28465			0.642327	+	0.766430i	$0.277969\pi$
$752$		0			0
$753$	43.7202			0.0580614
$754$		0			0
$755$	−745.341	−	745.341i	−0.987207	−	0.987207i
$756$		0			0
$757$	−313.133	−	313.133i	−0.413650	−	0.413650i	0.469358	−	0.883008i	$-0.344485\pi$
$757$	−313.133	−	313.133i	−0.413650	−	0.413650i	−0.883008	+	0.469358i	$0.844485\pi$
$758$		0			0
$759$	59.3196			0.0781550
$760$		0			0
$761$	1428.00			1.87648			0.938242	−	0.345980i	$-0.112453\pi$
$761$	1428.00			1.87648			0.938242	+	0.345980i	$0.112453\pi$
$762$		0			0
$763$	−34.8344	−	11.6903i	−0.0456545	−	0.0153215i
$764$		0			0
$765$	881.365	−	881.365i	1.15211	−	1.15211i
$766$		0			0
$767$	−1362.31			−1.77615
$768$		0			0
$769$		−	870.738i		−	1.13230i	−0.824303	−	0.566149i	$-0.808433\pi$
$769$		−	870.738i		−	1.13230i	0.824303	−	0.566149i	$-0.191567\pi$
$770$		0			0
$771$	51.7796	−	51.7796i	0.0671590	−	0.0671590i
$772$		0			0
$773$	3.78212	−	3.78212i	0.00489279	−	0.00489279i	−0.704656	−	0.709549i	$-0.748899\pi$
$773$	3.78212	−	3.78212i	0.00489279	−	0.00489279i	0.709549	+	0.704656i	$0.248899\pi$
$774$		0			0
$775$	−508.480			−0.656104
$776$		0			0
$777$	−35.5483	+	17.6838i	−0.0457507	+	0.0227590i
$778$		0			0
$779$	−531.748	−	531.748i	−0.682603	−	0.682603i
$780$		0			0
$781$	−24.3542	+	24.3542i	−0.0311833	+	0.0311833i
$782$		0			0
$783$		−	847.532i		−	1.08242i
$784$		0			0
$785$	−670.614			−0.854285
$786$		0			0
$787$	376.678	+	376.678i	0.478626	+	0.478626i	0.904692	−	0.426066i	$-0.140101\pi$
$787$	376.678	+	376.678i	0.478626	+	0.478626i	−0.426066	+	0.904692i	$0.640101\pi$
$788$		0			0
$789$	259.411	−	259.411i	0.328784	−	0.328784i
$790$		0			0
$791$	−137.633	−	276.672i	−0.173999	−	0.349776i
$792$		0			0
$793$		−	1688.18i		−	2.12885i
$794$		0			0
$795$	−98.4293	−	98.4293i	−0.123810	−	0.123810i
$796$		0			0
$797$	−1054.97	−	1054.97i	−1.32367	−	1.32367i	−0.910778	−	0.412896i	$-0.864517\pi$
$797$	−1054.97	−	1054.97i	−1.32367	−	1.32367i	−0.412896	−	0.910778i	$-0.635483\pi$
$798$		0			0
$799$	−476.234			−0.596038
$800$		0			0
$801$		−	53.0038i		−	0.0661720i
$802$		0			0
$803$	94.5373	+	94.5373i	0.117730	+	0.117730i
$804$		0			0
$805$	−178.983	+	533.326i	−0.222339	+	0.662517i
$806$		0			0
$807$		−	263.725i		−	0.326797i
$808$		0			0
$809$			826.800i			1.02200i	0.859580	+	0.511001i	$0.170725\pi$
$809$			826.800i			1.02200i	−0.859580	+	0.511001i	$0.829275\pi$
$810$		0			0
$811$	−185.728	+	185.728i	−0.229012	+	0.229012i	−0.812280	−	0.583268i	$-0.801774\pi$
$811$	−185.728	+	185.728i	−0.229012	+	0.229012i	0.583268	+	0.812280i	$0.301774\pi$
$812$		0			0
$813$	−146.621	+	146.621i	−0.180346	+	0.180346i
$814$		0			0
$815$		−	1840.39i		−	2.25815i
$816$		0			0
$817$			1011.63i			1.23823i
$818$		0			0
$819$	−683.805	−	229.483i	−0.834927	−	0.280199i
$820$		0			0
$821$	−293.780	−	293.780i	−0.357832	−	0.357832i	0.505181	−	0.863013i	$-0.331426\pi$
$821$	−293.780	−	293.780i	−0.357832	−	0.357832i	−0.863013	+	0.505181i	$0.831426\pi$
$822$		0			0
$823$			118.012i			0.143393i	0.997427	+	0.0716964i	$0.0228413\pi$
$823$			118.012i			0.143393i	−0.997427	+	0.0716964i	$0.977159\pi$
$824$		0			0
$825$	−200.054			−0.242489
$826$		0			0
$827$	−36.0081	−	36.0081i	−0.0435406	−	0.0435406i	0.685001	−	0.728542i	$-0.259802\pi$
$827$	−36.0081	−	36.0081i	−0.0435406	−	0.0435406i	−0.728542	+	0.685001i	$0.759802\pi$
$828$		0			0
$829$	52.8642	+	52.8642i	0.0637686	+	0.0637686i	0.738272	−	0.674503i	$-0.235642\pi$
$829$	52.8642	+	52.8642i	0.0637686	+	0.0637686i	−0.674503	+	0.738272i	$0.735642\pi$
$830$		0			0
$831$		−	554.451i		−	0.667209i
$832$		0			0
$833$	962.776	+	728.225i	1.15579	+	0.874220i
$834$		0			0
$835$	647.960	−	647.960i	0.776000	−	0.776000i
$836$		0			0
$837$	−250.716	−	250.716i	−0.299542	−	0.299542i
$838$		0			0
$839$	1257.63			1.49897			0.749484	−	0.662023i	$-0.230302\pi$
$839$	1257.63			1.49897			0.749484	+	0.662023i	$0.230302\pi$
$840$		0			0
$841$		−	365.490i		−	0.434589i
$842$		0			0
$843$	382.176	−	382.176i	0.453352	−	0.453352i
$844$		0			0
$845$	−437.248	−	437.248i	−0.517453	−	0.517453i
$846$		0			0
$847$	−675.344	+	335.955i	−0.797336	+	0.396641i
$848$		0			0
$849$	314.674			0.370640
$850$		0			0
$851$	26.4897	−	26.4897i	0.0311278	−	0.0311278i
$852$		0			0
$853$	906.916	−	906.916i	1.06321	−	1.06321i	0.0653442	−	0.997863i	$-0.479185\pi$
$853$	906.916	−	906.916i	1.06321	−	1.06321i	0.997863	−	0.0653442i	$-0.0208145\pi$
$854$		0			0
$855$		−	786.359i		−	0.919718i
$856$		0			0
$857$	−1015.16			−1.18455			−0.592273	−	0.805737i	$-0.701769\pi$
$857$	−1015.16			−1.18455			−0.592273	+	0.805737i	$0.701769\pi$
$858$		0			0
$859$	851.483	−	851.483i	0.991249	−	0.991249i	−0.00871336	−	0.999962i	$-0.502774\pi$
$859$	851.483	−	851.483i	0.991249	−	0.991249i	0.999962	+	0.00871336i	$0.00277359\pi$
$860$		0			0
$861$	169.280	−	504.414i	0.196608	−	0.585847i
$862$		0			0
$863$	873.705			1.01240			0.506202	−	0.862415i	$-0.331049\pi$
$863$	873.705			1.01240			0.506202	+	0.862415i	$0.331049\pi$
$864$		0			0
$865$	97.3116			0.112499
$866$		0			0
$867$	−353.172	−	353.172i	−0.407350	−	0.407350i
$868$		0			0
$869$	141.619	+	141.619i	0.162968	+	0.162968i
$870$		0			0
$871$	1809.81			2.07785
$872$		0			0
$873$	−922.865			−1.05712
$874$		0			0
$875$	172.364	−	513.605i	0.196988	−	0.586977i
$876$		0			0
$877$	−1140.58	+	1140.58i	−1.30055	+	1.30055i	−0.372525	+	0.928022i	$0.621508\pi$
$877$	−1140.58	+	1140.58i	−1.30055	+	1.30055i	−0.928022	+	0.372525i	$0.878492\pi$
$878$		0			0
$879$	−183.511			−0.208772
$880$		0			0
$881$		−	626.678i		−	0.711325i	−0.934614	−	0.355663i	$-0.884255\pi$
$881$		−	626.678i		−	0.711325i	0.934614	−	0.355663i	$-0.115745\pi$
$882$		0			0
$883$	−101.385	+	101.385i	−0.114819	+	0.114819i	−0.762182	−	0.647363i	$-0.775872\pi$
$883$	−101.385	+	101.385i	−0.114819	+	0.114819i	0.647363	+	0.762182i	$0.275872\pi$
$884$		0			0
$885$	743.045	−	743.045i	0.839599	−	0.839599i
$886$		0			0
$887$	−966.644			−1.08979			−0.544895	−	0.838504i	$-0.683431\pi$
$887$	−966.644			−1.08979			−0.544895	+	0.838504i	$0.683431\pi$
$888$		0			0
$889$	317.838	+	638.924i	0.357523	+	0.718700i
$890$		0			0
$891$	52.6433	+	52.6433i	0.0590834	+	0.0590834i
$892$		0			0
$893$	−212.450	+	212.450i	−0.237905	+	0.237905i
$894$		0			0
$895$			1470.35i			1.64285i
$896$		0			0
$897$	−257.124			−0.286649
$898$		0			0
$899$	−356.903	−	356.903i	−0.397000	−	0.397000i
$900$		0			0
$901$	199.289	−	199.289i	0.221187	−	0.221187i
$902$		0			0
$903$	−640.840	+	318.791i	−0.709679	+	0.353036i
$904$		0			0
$905$			99.9640i			0.110457i
$906$		0			0
$907$	−5.22275	−	5.22275i	−0.00575826	−	0.00575826i	0.704222	−	0.709980i	$-0.251296\pi$
$907$	−5.22275	−	5.22275i	−0.00575826	−	0.00575826i	−0.709980	+	0.704222i	$0.751296\pi$
$908$		0			0
$909$	595.411	+	595.411i	0.655018	+	0.655018i
$910$		0			0
$911$	1180.89			1.29626			0.648130	−	0.761530i	$-0.275551\pi$
$911$	1180.89			1.29626			0.648130	+	0.761530i	$0.275551\pi$
$912$		0			0
$913$		−	217.687i		−	0.238430i
$914$		0			0
$915$	920.787	+	920.787i	1.00632	+	1.00632i
$916$		0			0
$917$	303.278	−	903.699i	0.330729	−	0.985495i
$918$		0			0
$919$		−	541.429i		−	0.589150i	−0.955628	−	0.294575i	$-0.904822\pi$
$919$		−	541.429i		−	0.589150i	0.955628	−	0.294575i	$-0.0951781\pi$
$920$		0			0
$921$			446.451i			0.484746i
$922$		0			0
$923$	105.564	−	105.564i	0.114371	−	0.114371i
$924$		0			0
$925$	−89.3359	+	89.3359i	−0.0965794	+	0.0965794i
$926$		0			0
$927$			1099.79i			1.18639i
$928$		0			0
$929$			362.906i			0.390641i	0.980739	+	0.195321i	$0.0625748\pi$
$929$			362.906i			0.390641i	−0.980739	+	0.195321i	$0.937425\pi$
$930$		0			0
$931$	754.361	−	104.634i	0.810269	−	0.112389i
$932$		0			0
$933$	−514.398	−	514.398i	−0.551337	−	0.551337i
$934$		0			0
$935$		−	694.433i		−	0.742709i
$936$		0			0
$937$	1172.13			1.25094			0.625469	−	0.780249i	$-0.284908\pi$
$937$	1172.13			1.25094			0.625469	+	0.780249i	$0.284908\pi$
$938$		0			0
$939$	1.21178	+	1.21178i	0.00129050	+	0.00129050i
$940$		0			0
$941$	930.677	+	930.677i	0.989030	+	0.989030i	0.999940	−	0.0109108i	$-0.00347309\pi$
$941$	930.677	+	930.677i	0.989030	+	0.989030i	−0.0109108	+	0.999940i	$0.503473\pi$
$942$		0			0
$943$			502.021i			0.532366i
$944$		0			0
$945$	1184.47	−	589.226i	1.25341	−	0.623519i
$946$		0			0
$947$	−265.946	+	265.946i	−0.280830	+	0.280830i	−0.833440	−	0.552610i	$-0.813632\pi$
$947$	−265.946	+	265.946i	−0.280830	+	0.280830i	0.552610	+	0.833440i	$0.313632\pi$
$948$		0			0
$949$	−409.777	−	409.777i	−0.431798	−	0.431798i
$950$		0			0
$951$	113.675			0.119532
$952$		0			0
$953$			1378.60i			1.44659i	0.690540	+	0.723294i	$0.257373\pi$
$953$			1378.60i			1.44659i	−0.690540	+	0.723294i	$0.742627\pi$
$954$		0			0
$955$	−999.431	+	999.431i	−1.04653	+	1.04653i
$956$		0			0
$957$	−140.418	−	140.418i	−0.146727	−	0.146727i
$958$		0			0
$959$	279.848	−	139.213i	0.291812	−	0.145164i
$960$		0			0
$961$	749.842			0.780273
$962$		0			0
$963$	393.340	−	393.340i	0.408453	−	0.408453i
$964$		0			0
$965$	−2074.45	+	2074.45i	−2.14969	+	2.14969i
$966$		0			0
$967$		−	1533.12i		−	1.58544i	−0.609584	−	0.792722i	$-0.708663\pi$
$967$		−	1533.12i		−	1.58544i	0.609584	−	0.792722i	$-0.291337\pi$
$968$		0			0
$969$	−601.528			−0.620772
$970$		0			0
$971$	−44.2230	+	44.2230i	−0.0455437	+	0.0455437i	−0.729512	−	0.683968i	$-0.760253\pi$
$971$	−44.2230	+	44.2230i	−0.0455437	+	0.0455437i	0.683968	+	0.729512i	$0.260253\pi$
$972$		0			0
$973$	64.8880	+	21.7762i	0.0666886	+	0.0223805i
$974$		0			0
$975$	867.143			0.889377
$976$		0			0
$977$	−623.770			−0.638454			−0.319227	−	0.947678i	$-0.603423\pi$
$977$	−623.770			−0.638454			−0.319227	+	0.947678i	$0.603423\pi$
$978$		0			0
$979$	−20.8810	−	20.8810i	−0.0213289	−	0.0213289i
$980$		0			0
$981$	−24.2450	−	24.2450i	−0.0247146	−	0.0247146i
$982$		0			0
$983$	−1124.01			−1.14345			−0.571724	−	0.820446i	$-0.693725\pi$
$983$	−1124.01			−1.14345			−0.571724	+	0.820446i	$0.693725\pi$
$984$		0			0
$985$	−1413.01			−1.43453
$986$		0			0
$987$	−201.529	−	67.6324i	−0.204183	−	0.0685232i
$988$		0			0
$989$	477.539	−	477.539i	0.482851	−	0.482851i
$990$		0			0
$991$	−392.328			−0.395891			−0.197946	−	0.980213i	$-0.563427\pi$
$991$	−392.328			−0.395891			−0.197946	+	0.980213i	$0.563427\pi$
$992$		0			0
$993$		−	64.0327i		−	0.0644841i
$994$		0			0
$995$	−1505.01	+	1505.01i	−1.51257	+	1.51257i
$996$		0			0
$997$	−744.756	+	744.756i	−0.746997	+	0.746997i	−0.973914	−	0.226917i	$-0.927135\pi$
$997$	−744.756	+	744.756i	−0.746997	+	0.746997i	0.226917	+	0.973914i	$0.427135\pi$
$998$		0			0
$999$	−88.0977			−0.0881859

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.3.l.b.433.18		56
4.3	odd	2		112.3.l.b.69.23	yes	56
7.6	odd	2	inner	448.3.l.b.433.11		56
16.3	odd	4		112.3.l.b.13.24	yes	56
16.13	even	4	inner	448.3.l.b.209.11		56
28.27	even	2		112.3.l.b.69.24	yes	56
112.13	odd	4	inner	448.3.l.b.209.18		56
112.83	even	4		112.3.l.b.13.23	✓	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.3.l.b.13.23	✓	56	112.83	even	4
112.3.l.b.13.24	yes	56	16.3	odd	4
112.3.l.b.69.23	yes	56	4.3	odd	2
112.3.l.b.69.24	yes	56	28.27	even	2
448.3.l.b.209.11		56	16.13	even	4	inner
448.3.l.b.209.18		56	112.13	odd	4	inner
448.3.l.b.433.11		56	7.6	odd	2	inner
448.3.l.b.433.18		56	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+(1.11083 + 1.11083i) q^{3} +(-5.47687 + 5.47687i) q^{5} +(-3.11774 - 6.26735i) q^{7} -6.53210i q^{9} +(-2.57334 - 2.57334i) q^{11} +(11.1543 + 11.1543i) q^{13} -12.1678 q^{15} -24.6360i q^{17} +(-10.9902 - 10.9902i) q^{19} +(3.49869 - 10.4253i) q^{21} +10.3758i q^{23} -34.9921i q^{25} +(17.2536 - 17.2536i) q^{27} +(24.5610 - 24.5610i) q^{29} -14.5313i q^{31} -5.71711i q^{33} +(51.4009 + 17.2500i) q^{35} +(-2.55303 - 2.55303i) q^{37} +24.7811i q^{39} +48.3838 q^{41} +(-46.0243 - 46.0243i) q^{43} +(35.7754 + 35.7754i) q^{45} -19.3308i q^{47} +(-29.5593 + 39.0800i) q^{49} +(27.3665 - 27.3665i) q^{51} +(8.08934 + 8.08934i) q^{53} +28.1877 q^{55} -24.4166i q^{57} +(-61.0667 + 61.0667i) q^{59} +(-75.6742 - 75.6742i) q^{61} +(-40.9389 + 20.3654i) q^{63} -122.181 q^{65} +(81.1264 - 81.1264i) q^{67} +(-11.5258 + 11.5258i) q^{69} -9.46404i q^{71} -36.7372 q^{73} +(38.8704 - 38.8704i) q^{75} +(-8.10501 + 24.1510i) q^{77} -55.0332 q^{79} -20.4572 q^{81} +(42.2965 + 42.2965i) q^{83} +(134.928 + 134.928i) q^{85} +54.5665 q^{87} +8.11436 q^{89} +(35.1315 - 104.684i) q^{91} +(16.1418 - 16.1418i) q^{93} +120.384 q^{95} -141.282i q^{97} +(-16.8093 + 16.8093i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 8 q^{15} - 20 q^{21} - 96 q^{29} + 100 q^{35} - 128 q^{37} + 72 q^{43} + 192 q^{49} + 128 q^{51} + 88 q^{53} - 444 q^{63} - 8 q^{65} - 440 q^{67} + 12 q^{77} + 8 q^{79} + 64 q^{81} + 96 q^{85} + 388 q^{91}+ \cdots - 8 q^{99}+O(q^{100}) \) 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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