LMFDB - Embedded newform 448.3.l.b.209.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("448.209");

S:= CuspForms(chi, 3);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

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

Embedding invariants

Embedding label			209.5
Character	$\chi$	$=$	448.209
Dual form			448.3.l.b.433.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.99374 + 2.99374i) q^{3} +(0.657693 + 0.657693i) q^{5} +(-6.11030 - 3.41529i) q^{7} -8.92498i q^{9} +O(q^{10})$	\(q+(-2.99374 + 2.99374i) q^{3} +(0.657693 + 0.657693i) q^{5} +(-6.11030 - 3.41529i) q^{7} -8.92498i q^{9} +(10.3321 - 10.3321i) q^{11} +(6.46869 - 6.46869i) q^{13} -3.93793 q^{15} +12.4267i q^{17} +(9.56839 - 9.56839i) q^{19} +(28.5172 - 8.06819i) q^{21} +19.8839i q^{23} -24.1349i q^{25} +(-0.224592 - 0.224592i) q^{27} +(19.8270 + 19.8270i) q^{29} +46.5731i q^{31} +61.8630i q^{33} +(-1.77249 - 6.26492i) q^{35} +(-14.3809 + 14.3809i) q^{37} +38.7312i q^{39} +32.8728 q^{41} +(35.2241 - 35.2241i) q^{43} +(5.86990 - 5.86990i) q^{45} -36.8393i q^{47} +(25.6716 + 41.7369i) q^{49} +(-37.2023 - 37.2023i) q^{51} +(-27.7586 + 27.7586i) q^{53} +13.5907 q^{55} +57.2906i q^{57} +(76.0220 + 76.0220i) q^{59} +(66.0243 - 66.0243i) q^{61} +(-30.4814 + 54.5343i) q^{63} +8.50883 q^{65} +(59.7469 + 59.7469i) q^{67} +(-59.5273 - 59.5273i) q^{69} -39.5053i q^{71} -22.3061 q^{73} +(72.2536 + 72.2536i) q^{75} +(-98.4190 + 27.8451i) q^{77} +125.343 q^{79} +81.6696 q^{81} +(82.4016 - 82.4016i) q^{83} +(-8.17295 + 8.17295i) q^{85} -118.714 q^{87} -41.7209 q^{89} +(-61.6181 + 17.4332i) q^{91} +(-139.428 - 139.428i) q^{93} +12.5861 q^{95} -115.081i q^{97} +(-92.2134 - 92.2134i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q+O(q^{10}) $ 56 * q	$ 56 q + 8 q^{15} - 20 q^{21} - 96 q^{29} + 100 q^{35} - 128 q^{37} + 72 q^{43} + 192 q^{49} + 128 q^{51} + 88 q^{53} - 444 q^{63} - 8 q^{65} - 440 q^{67} + 12 q^{77} + 8 q^{79} + 64 q^{81} + 96 q^{85} + 388 q^{91} + 32 q^{93} + 776 q^{95} - 8 q^{99}+O(q^{100}) $ 56 * q + 8 * q^15 - 20 * q^21 - 96 * q^29 + 100 * q^35 - 128 * q^37 + 72 * q^43 + 192 * q^49 + 128 * q^51 + 88 * q^53 - 444 * q^63 - 8 * q^65 - 440 * q^67 + 12 * q^77 + 8 * q^79 + 64 * q^81 + 96 * q^85 + 388 * q^91 + 32 * q^93 + 776 * q^95 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−2.99374	+	2.99374i	−0.997914	+	0.997914i	−0.999998	−	0.00208391i	$-0.999337\pi$
$3$	−2.99374	+	2.99374i	−0.997914	+	0.997914i	0.00208391	+	0.999998i	$0.499337\pi$
$4$		0			0
$5$	0.657693	+	0.657693i	0.131539	+	0.131539i	0.769811	−	0.638272i	$-0.220351\pi$
$5$	0.657693	+	0.657693i	0.131539	+	0.131539i	−0.638272	+	0.769811i	$0.720351\pi$
$6$		0			0
$7$	−6.11030	−	3.41529i	−0.872901	−	0.487898i
$7$	−6.11030	−	3.41529i	−0.872901	−	0.487898i
$8$		0			0
$9$		−	8.92498i		−	0.991664i
$10$		0			0
$11$	10.3321	−	10.3321i	0.939278	−	0.939278i	−0.0589808	−	0.998259i	$-0.518785\pi$
$11$	10.3321	−	10.3321i	0.939278	−	0.939278i	0.998259	+	0.0589808i	$0.0187851\pi$
$12$		0			0
$13$	6.46869	−	6.46869i	0.497592	−	0.497592i	−0.413096	−	0.910688i	$-0.635553\pi$
$13$	6.46869	−	6.46869i	0.497592	−	0.497592i	0.910688	+	0.413096i	$0.135553\pi$
$14$		0			0
$15$	−3.93793			−0.262528
$16$		0			0
$17$			12.4267i			0.730981i	0.930815	+	0.365491i	$0.119099\pi$
$17$			12.4267i			0.730981i	−0.930815	+	0.365491i	$0.880901\pi$
$18$		0			0
$19$	9.56839	−	9.56839i	0.503599	−	0.503599i	−0.408955	−	0.912555i	$-0.634107\pi$
$19$	9.56839	−	9.56839i	0.503599	−	0.503599i	0.912555	+	0.408955i	$0.134107\pi$
$20$		0			0
$21$	28.5172	−	8.06819i	1.35796	−	0.384199i
$22$		0			0
$23$			19.8839i			0.864518i	0.901750	+	0.432259i	$0.142283\pi$
$23$			19.8839i			0.864518i	−0.901750	+	0.432259i	$0.857717\pi$
$24$		0			0
$25$		−	24.1349i		−	0.965395i
$26$		0			0
$27$	−0.224592	−	0.224592i	−0.00831823	−	0.00831823i
$28$		0			0
$29$	19.8270	+	19.8270i	0.683689	+	0.683689i	0.960829	−	0.277140i	$-0.0893867\pi$
$29$	19.8270	+	19.8270i	0.683689	+	0.683689i	−0.277140	+	0.960829i	$0.589387\pi$
$30$		0			0
$31$			46.5731i			1.50236i	0.660099	+	0.751179i	$0.270514\pi$
$31$			46.5731i			1.50236i	−0.660099	+	0.751179i	$0.729486\pi$
$32$		0			0
$33$			61.8630i			1.87464i
$34$		0			0
$35$	−1.77249	−	6.26492i	−0.0506427	−	0.178998i
$36$		0			0
$37$	−14.3809	+	14.3809i	−0.388674	+	0.388674i	−0.874214	−	0.485540i	$-0.838623\pi$
$37$	−14.3809	+	14.3809i	−0.388674	+	0.388674i	0.485540	+	0.874214i	$0.338623\pi$
$38$		0			0
$39$			38.7312i			0.993108i
$40$		0			0
$41$	32.8728			0.801776			0.400888	−	0.916127i	$-0.368702\pi$
$41$	32.8728			0.801776			0.400888	+	0.916127i	$0.368702\pi$
$42$		0			0
$43$	35.2241	−	35.2241i	0.819166	−	0.819166i	−0.166821	−	0.985987i	$-0.553350\pi$
$43$	35.2241	−	35.2241i	0.819166	−	0.819166i	0.985987	+	0.166821i	$0.0533503\pi$
$44$		0			0
$45$	5.86990	−	5.86990i	0.130442	−	0.130442i
$46$		0			0
$47$		−	36.8393i		−	0.783814i	−0.920005	−	0.391907i	$-0.871816\pi$
$47$		−	36.8393i		−	0.783814i	0.920005	−	0.391907i	$-0.128184\pi$
$48$		0			0
$49$	25.6716	+	41.7369i	0.523911	+	0.851773i
$50$		0			0
$51$	−37.2023	−	37.2023i	−0.729457	−	0.729457i
$52$		0			0
$53$	−27.7586	+	27.7586i	−0.523748	+	0.523748i	−0.918701	−	0.394954i	$-0.870761\pi$
$53$	−27.7586	+	27.7586i	−0.523748	+	0.523748i	0.394954	+	0.918701i	$0.370761\pi$
$54$		0			0
$55$	13.5907			0.247103
$56$		0			0
$57$			57.2906i			1.00510i
$58$		0			0
$59$	76.0220	+	76.0220i	1.28851	+	1.28851i	0.935692	+	0.352817i	$0.114776\pi$
$59$	76.0220	+	76.0220i	1.28851	+	1.28851i	0.352817	+	0.935692i	$0.385224\pi$
$60$		0			0
$61$	66.0243	−	66.0243i	1.08237	−	1.08237i	0.0860768	−	0.996289i	$-0.472567\pi$
$61$	66.0243	−	66.0243i	1.08237	−	1.08237i	0.996289	−	0.0860768i	$-0.0274331\pi$
$62$		0			0
$63$	−30.4814	+	54.5343i	−0.483831	+	0.865624i
$64$		0			0
$65$	8.50883			0.130905
$66$		0			0
$67$	59.7469	+	59.7469i	0.891745	+	0.891745i	0.994687	−	0.102942i	$-0.0328257\pi$
$67$	59.7469	+	59.7469i	0.891745	+	0.891745i	−0.102942	+	0.994687i	$0.532826\pi$
$68$		0			0
$69$	−59.5273	−	59.5273i	−0.862714	−	0.862714i
$70$		0			0
$71$		−	39.5053i		−	0.556412i	−0.960521	−	0.278206i	$-0.910260\pi$
$71$		−	39.5053i		−	0.556412i	0.960521	−	0.278206i	$-0.0897398\pi$
$72$		0			0
$73$	−22.3061			−0.305563			−0.152782	−	0.988260i	$-0.548823\pi$
$73$	−22.3061			−0.305563			−0.152782	+	0.988260i	$0.548823\pi$
$74$		0			0
$75$	72.2536	+	72.2536i	0.963381	+	0.963381i
$76$		0			0
$77$	−98.4190	+	27.8451i	−1.27817	+	0.361625i
$78$		0			0
$79$	125.343			1.58662			0.793311	−	0.608817i	$-0.208356\pi$
$79$	125.343			1.58662			0.793311	+	0.608817i	$0.208356\pi$
$80$		0			0
$81$	81.6696			1.00827
$82$		0			0
$83$	82.4016	−	82.4016i	0.992790	−	0.992790i	−0.00718397	−	0.999974i	$-0.502287\pi$
$83$	82.4016	−	82.4016i	0.992790	−	0.992790i	0.999974	+	0.00718397i	$0.00228675\pi$
$84$		0			0
$85$	−8.17295	+	8.17295i	−0.0961523	+	0.0961523i
$86$		0			0
$87$	−118.714			−1.36453
$88$		0			0
$89$	−41.7209			−0.468774			−0.234387	−	0.972143i	$-0.575308\pi$
$89$	−41.7209			−0.468774			−0.234387	+	0.972143i	$0.575308\pi$
$90$		0			0
$91$	−61.6181	+	17.4332i	−0.677122	+	0.191574i
$92$		0			0
$93$	−139.428	−	139.428i	−1.49922	−	1.49922i
$94$		0			0
$95$	12.5861			0.132486
$96$		0			0
$97$		−	115.081i		−	1.18640i	−0.805054	−	0.593201i	$-0.797864\pi$
$97$		−	115.081i		−	1.18640i	0.805054	−	0.593201i	$-0.202136\pi$
$98$		0			0
$99$	−92.2134	−	92.2134i	−0.931449	−	0.931449i
$100$		0			0
$101$	−25.3448	−	25.3448i	−0.250939	−	0.250939i	0.570417	−	0.821355i	$-0.306782\pi$
$101$	−25.3448	−	25.3448i	−0.250939	−	0.250939i	−0.821355	+	0.570417i	$0.806782\pi$
$102$		0			0
$103$	−191.118			−1.85552			−0.927759	−	0.373179i	$-0.878268\pi$
$103$	−191.118			−1.85552			−0.927759	+	0.373179i	$0.878268\pi$
$104$		0			0
$105$	24.0619	+	13.4491i	0.229161	+	0.128087i
$106$		0			0
$107$	49.5841	−	49.5841i	0.463403	−	0.463403i	−0.436366	−	0.899769i	$-0.643735\pi$
$107$	49.5841	−	49.5841i	0.463403	−	0.463403i	0.899769	+	0.436366i	$0.143735\pi$
$108$		0			0
$109$	13.1098	+	13.1098i	0.120273	+	0.120273i	0.764682	−	0.644408i	$-0.222896\pi$
$109$	13.1098	+	13.1098i	0.120273	+	0.120273i	−0.644408	+	0.764682i	$0.722896\pi$
$110$		0			0
$111$		−	86.1056i		−	0.775727i
$112$		0			0
$113$	47.1505			0.417261			0.208630	−	0.977995i	$-0.433099\pi$
$113$	47.1505			0.417261			0.208630	+	0.977995i	$0.433099\pi$
$114$		0			0
$115$	−13.0775	+	13.0775i	−0.113718	+	0.113718i
$116$		0			0
$117$	−57.7330	−	57.7330i	−0.493444	−	0.493444i
$118$		0			0
$119$	42.4407	−	75.9308i	0.356644	−	0.638074i
$120$		0			0
$121$		−	92.5030i		−	0.764487i
$122$		0			0
$123$	−98.4127	+	98.4127i	−0.800103	+	0.800103i
$124$		0			0
$125$	32.3157	−	32.3157i	0.258525	−	0.258525i
$126$		0			0
$127$	−91.8303			−0.723073			−0.361537	−	0.932358i	$-0.617748\pi$
$127$	−91.8303			−0.723073			−0.361537	+	0.932358i	$0.617748\pi$
$128$		0			0
$129$			210.904i			1.63491i
$130$		0			0
$131$	83.1791	−	83.1791i	0.634955	−	0.634955i	−0.314351	−	0.949307i	$-0.601787\pi$
$131$	83.1791	−	83.1791i	0.634955	−	0.634955i	0.949307	+	0.314351i	$0.101787\pi$
$132$		0			0
$133$	−91.1446	+	25.7870i	−0.685297	+	0.193887i
$134$		0			0
$135$		−	0.295426i		−	0.00218834i
$136$		0			0
$137$			87.9696i			0.642114i	0.947060	+	0.321057i	$0.104038\pi$
$137$			87.9696i			0.642114i	−0.947060	+	0.321057i	$0.895962\pi$
$138$		0			0
$139$	−170.053	−	170.053i	−1.22340	−	1.22340i	−0.966414	−	0.256991i	$-0.917269\pi$
$139$	−170.053	−	170.053i	−1.22340	−	1.22340i	−0.256991	−	0.966414i	$-0.582731\pi$
$140$		0			0
$141$	110.287	+	110.287i	0.782179	+	0.782179i
$142$		0			0
$143$		−	133.670i		−	0.934754i
$144$		0			0
$145$			26.0801i			0.179863i
$146$		0			0
$147$	−201.804	−	48.0952i	−1.37281	−	0.327178i
$148$		0			0
$149$	26.4600	−	26.4600i	0.177584	−	0.177584i	−0.612718	−	0.790302i	$-0.709924\pi$
$149$	26.4600	−	26.4600i	0.177584	−	0.177584i	0.790302	+	0.612718i	$0.209924\pi$
$150$		0			0
$151$			142.320i			0.942519i	0.881994	+	0.471260i	$0.156201\pi$
$151$			142.320i			0.942519i	−0.881994	+	0.471260i	$0.843799\pi$
$152$		0			0
$153$	110.908			0.724888
$154$		0			0
$155$	−30.6308	+	30.6308i	−0.197618	+	0.197618i
$156$		0			0
$157$	−35.0827	+	35.0827i	−0.223457	+	0.223457i	−0.809952	−	0.586495i	$-0.800507\pi$
$157$	−35.0827	+	35.0827i	−0.223457	+	0.223457i	0.586495	+	0.809952i	$0.300507\pi$
$158$		0			0
$159$		−	166.204i		−	1.04531i
$160$		0			0
$161$	67.9093	−	121.497i	0.421797	−	0.754638i
$162$		0			0
$163$	146.517	+	146.517i	0.898876	+	0.898876i	0.995337	−	0.0964607i	$-0.0307522\pi$
$163$	146.517	+	146.517i	0.898876	+	0.898876i	−0.0964607	+	0.995337i	$0.530752\pi$
$164$		0			0
$165$	−40.6869	+	40.6869i	−0.246587	+	0.246587i
$166$		0			0
$167$	60.5631			0.362654			0.181327	−	0.983423i	$-0.441961\pi$
$167$	60.5631			0.362654			0.181327	+	0.983423i	$0.441961\pi$
$168$		0			0
$169$			85.3120i			0.504805i
$170$		0			0
$171$	−85.3977	−	85.3977i	−0.499402	−	0.499402i
$172$		0			0
$173$	219.202	−	219.202i	1.26707	−	1.26707i	0.319470	−	0.947597i	$-0.396495\pi$
$173$	219.202	−	219.202i	1.26707	−	1.26707i	0.947597	−	0.319470i	$-0.103505\pi$
$174$		0			0
$175$	−82.4275	+	147.471i	−0.471014	+	0.842694i
$176$		0			0
$177$	−455.181			−2.57164
$178$		0			0
$179$	133.334	+	133.334i	0.744885	+	0.744885i	0.973514	−	0.228629i	$-0.0734242\pi$
$179$	133.334	+	133.334i	0.744885	+	0.744885i	−0.228629	+	0.973514i	$0.573424\pi$
$180$		0			0
$181$	63.6266	+	63.6266i	0.351528	+	0.351528i	0.860678	−	0.509150i	$-0.170040\pi$
$181$	63.6266	+	63.6266i	0.351528	+	0.351528i	−0.509150	+	0.860678i	$0.670040\pi$
$182$		0			0
$183$			395.319i			2.16021i
$184$		0			0
$185$	−18.9165			−0.102251
$186$		0			0
$187$	128.393	+	128.393i	0.686595	+	0.686595i
$188$		0			0
$189$	0.605280	+	2.13937i	0.00320254	+	0.0113194i
$190$		0			0
$191$	−327.403			−1.71415			−0.857076	−	0.515191i	$-0.827721\pi$
$191$	−327.403			−1.71415			−0.857076	+	0.515191i	$0.827721\pi$
$192$		0			0
$193$	130.387			0.675582			0.337791	−	0.941221i	$-0.390320\pi$
$193$	130.387			0.675582			0.337791	+	0.941221i	$0.390320\pi$
$194$		0			0
$195$	−25.4732	+	25.4732i	−0.130632	+	0.130632i
$196$		0			0
$197$	53.0684	−	53.0684i	0.269383	−	0.269383i	−0.559469	−	0.828851i	$-0.688995\pi$
$197$	53.0684	−	53.0684i	0.269383	−	0.269383i	0.828851	+	0.559469i	$0.188995\pi$
$198$		0			0
$199$	39.1841			0.196905			0.0984525	−	0.995142i	$-0.468611\pi$
$199$	39.1841			0.196905			0.0984525	+	0.995142i	$0.468611\pi$
$200$		0			0
$201$	−357.734			−1.77977
$202$		0			0
$203$	−53.4341	−	188.864i	−0.263222	−	0.930363i
$204$		0			0
$205$	21.6202	+	21.6202i	0.105464	+	0.105464i
$206$		0			0
$207$	177.464			0.857312
$208$		0			0
$209$		−	197.722i		−	0.946040i
$210$		0			0
$211$	−98.4909	−	98.4909i	−0.466781	−	0.466781i	0.434089	−	0.900870i	$-0.357070\pi$
$211$	−98.4909	−	98.4909i	−0.466781	−	0.466781i	−0.900870	+	0.434089i	$0.857070\pi$
$212$		0			0
$213$	118.269	+	118.269i	0.555251	+	0.555251i
$214$		0			0
$215$	46.3333			0.215504
$216$		0			0
$217$	159.060	−	284.576i	0.732998	−	1.31141i
$218$		0			0
$219$	66.7787	−	66.7787i	0.304926	−	0.304926i
$220$		0			0
$221$	80.3844	+	80.3844i	0.363730	+	0.363730i
$222$		0			0
$223$			373.131i			1.67324i	0.547787	+	0.836618i	$0.315470\pi$
$223$			373.131i			1.67324i	−0.547787	+	0.836618i	$0.684530\pi$
$224$		0			0
$225$	−215.403			−0.957348
$226$		0			0
$227$	45.1003	−	45.1003i	0.198680	−	0.198680i	−0.600754	−	0.799434i	$-0.705133\pi$
$227$	45.1003	−	45.1003i	0.198680	−	0.198680i	0.799434	+	0.600754i	$0.205133\pi$
$228$		0			0
$229$	−32.1518	−	32.1518i	−0.140401	−	0.140401i	0.633413	−	0.773814i	$-0.281653\pi$
$229$	−32.1518	−	32.1518i	−0.140401	−	0.140401i	−0.773814	+	0.633413i	$0.781653\pi$
$230$		0			0
$231$	211.280	−	378.002i	0.914632	−	1.63637i
$232$		0			0
$233$		−	254.578i		−	1.09261i	−0.837587	−	0.546304i	$-0.816034\pi$
$233$		−	254.578i		−	1.09261i	0.837587	−	0.546304i	$-0.183966\pi$
$234$		0			0
$235$	24.2289	−	24.2289i	0.103102	−	0.103102i
$236$		0			0
$237$	−375.245	+	375.245i	−1.58331	+	1.58331i
$238$		0			0
$239$	200.937			0.840740			0.420370	−	0.907353i	$-0.361900\pi$
$239$	200.937			0.840740			0.420370	+	0.907353i	$0.361900\pi$
$240$		0			0
$241$		−	300.467i		−	1.24675i	−0.781922	−	0.623376i	$-0.785761\pi$
$241$		−	300.467i		−	1.24675i	0.781922	−	0.623376i	$-0.214239\pi$
$242$		0			0
$243$	−242.476	+	242.476i	−0.997845	+	0.997845i
$244$		0			0
$245$	−10.5660	+	44.3341i	−0.0431265	+	0.180956i
$246$		0			0
$247$		−	123.790i		−	0.501174i
$248$		0			0
$249$			493.378i			1.98144i
$250$		0			0
$251$	−51.4543	−	51.4543i	−0.204997	−	0.204997i	0.597140	−	0.802137i	$-0.296304\pi$
$251$	−51.4543	−	51.4543i	−0.204997	−	0.204997i	−0.802137	+	0.597140i	$0.796304\pi$
$252$		0			0
$253$	205.442	+	205.442i	0.812023	+	0.812023i
$254$		0			0
$255$		−	48.9354i		−	0.191903i
$256$		0			0
$257$			305.136i			1.18730i	0.804724	+	0.593649i	$0.202313\pi$
$257$			305.136i			1.18730i	−0.804724	+	0.593649i	$0.797687\pi$
$258$		0			0
$259$	136.987	−	38.7569i	0.528907	−	0.149641i
$260$		0			0
$261$	176.955	−	176.955i	0.677990	−	0.677990i
$262$		0			0
$263$		−	392.985i		−	1.49424i	−0.664690	−	0.747119i	$-0.731437\pi$
$263$		−	392.985i		−	1.49424i	0.664690	−	0.747119i	$-0.268563\pi$
$264$		0			0
$265$	−36.5133			−0.137786
$266$		0			0
$267$	124.902	−	124.902i	0.467796	−	0.467796i
$268$		0			0
$269$	70.7183	−	70.7183i	0.262893	−	0.262893i	−0.563335	−	0.826228i	$-0.690482\pi$
$269$	70.7183	−	70.7183i	0.262893	−	0.262893i	0.826228	+	0.563335i	$0.190482\pi$
$270$		0			0
$271$			18.6577i			0.0688476i	0.999407	+	0.0344238i	$0.0109596\pi$
$271$			18.6577i			0.0688476i	−0.999407	+	0.0344238i	$0.989040\pi$
$272$		0			0
$273$	132.278	−	236.659i	0.484535	−	0.866884i
$274$		0			0
$275$	−249.363	−	249.363i	−0.906775	−	0.906775i
$276$		0			0
$277$	−181.808	+	181.808i	−0.656347	+	0.656347i	−0.954514	−	0.298167i	$-0.903625\pi$
$277$	−181.808	+	181.808i	−0.656347	+	0.656347i	0.298167	+	0.954514i	$0.403625\pi$
$278$		0			0
$279$	415.664			1.48983
$280$		0			0
$281$		−	292.196i		−	1.03984i	−0.854213	−	0.519922i	$-0.825961\pi$
$281$		−	292.196i		−	1.03984i	0.854213	−	0.519922i	$-0.174039\pi$
$282$		0			0
$283$	24.3621	+	24.3621i	0.0860851	+	0.0860851i	0.748838	−	0.662753i	$-0.230612\pi$
$283$	24.3621	+	24.3621i	0.0860851	+	0.0860851i	−0.662753	+	0.748838i	$0.730612\pi$
$284$		0			0
$285$	−37.6796	+	37.6796i	−0.132209	+	0.132209i
$286$		0			0
$287$	−200.863	−	112.270i	−0.699870	−	0.391185i
$288$		0			0
$289$	134.577			0.465666
$290$		0			0
$291$	344.523	+	344.523i	1.18393	+	1.18393i
$292$		0			0
$293$	176.954	+	176.954i	0.603939	+	0.603939i	0.941356	−	0.337416i	$-0.109553\pi$
$293$	176.954	+	176.954i	0.603939	+	0.603939i	−0.337416	+	0.941356i	$0.609553\pi$
$294$		0			0
$295$			99.9984i			0.338977i
$296$		0			0
$297$	−4.64100			−0.0156263
$298$		0			0
$299$	128.623	+	128.623i	0.430177	+	0.430177i
$300$		0			0
$301$	−335.531	+	94.9297i	−1.11472	+	0.315381i
$302$		0			0
$303$	151.752			0.500830
$304$		0			0
$305$	86.8474			0.284746
$306$		0			0
$307$	−319.326	+	319.326i	−1.04015	+	1.04015i	−0.0409888	+	0.999160i	$0.513051\pi$
$307$	−319.326	+	319.326i	−1.04015	+	1.04015i	−0.999160	+	0.0409888i	$0.986949\pi$
$308$		0			0
$309$	572.159	−	572.159i	1.85165	−	1.85165i
$310$		0			0
$311$	383.985			1.23468			0.617339	−	0.786697i	$-0.288211\pi$
$311$	383.985			1.23468			0.617339	+	0.786697i	$0.288211\pi$
$312$		0			0
$313$	259.627			0.829479			0.414739	−	0.909940i	$-0.363873\pi$
$313$	259.627			0.829479			0.414739	+	0.909940i	$0.363873\pi$
$314$		0			0
$315$	−55.9142	+	15.8195i	−0.177506	+	0.0502206i
$316$		0			0
$317$	−208.220	−	208.220i	−0.656846	−	0.656846i	0.297786	−	0.954633i	$-0.403752\pi$
$317$	−208.220	−	208.220i	−0.656846	−	0.656846i	−0.954633	+	0.297786i	$0.903752\pi$
$318$		0			0
$319$	409.707			1.28435
$320$		0			0
$321$			296.884i			0.924873i
$322$		0			0
$323$	118.903	+	118.903i	0.368122	+	0.368122i
$324$		0			0
$325$	−156.121	−	156.121i	−0.480373	−	0.480373i
$326$		0			0
$327$	−78.4946			−0.240045
$328$		0			0
$329$	−125.817	+	225.099i	−0.382421	+	0.684192i
$330$		0			0
$331$	−108.375	+	108.375i	−0.327418	+	0.327418i	−0.851604	−	0.524186i	$-0.824370\pi$
$331$	−108.375	+	108.375i	−0.327418	+	0.327418i	0.524186	+	0.851604i	$0.324370\pi$
$332$		0			0
$333$	128.350	+	128.350i	0.385434	+	0.385434i
$334$		0			0
$335$			78.5903i			0.234598i
$336$		0			0
$337$	−326.397			−0.968536			−0.484268	−	0.874920i	$-0.660914\pi$
$337$	−326.397			−0.968536			−0.484268	+	0.874920i	$0.660914\pi$
$338$		0			0
$339$	−141.156	+	141.156i	−0.416390	+	0.416390i
$340$		0			0
$341$	481.196	+	481.196i	1.41113	+	1.41113i
$342$		0			0
$343$	−14.3181	−	342.701i	−0.0417437	−	0.999128i
$344$		0			0
$345$		−	78.3014i		−	0.226961i
$346$		0			0
$347$	−429.230	+	429.230i	−1.23697	+	1.23697i	−0.275742	+	0.961232i	$0.588924\pi$
$347$	−429.230	+	429.230i	−1.23697	+	1.23697i	−0.961232	+	0.275742i	$0.911076\pi$
$348$		0			0
$349$	−125.553	+	125.553i	−0.359750	+	0.359750i	−0.863721	−	0.503970i	$-0.831872\pi$
$349$	−125.553	+	125.553i	−0.359750	+	0.359750i	0.503970	+	0.863721i	$0.331872\pi$
$350$		0			0
$351$	−2.90564			−0.00827817
$352$		0			0
$353$		−	275.094i		−	0.779303i	−0.920962	−	0.389652i	$-0.872595\pi$
$353$		−	275.094i		−	0.779303i	0.920962	−	0.389652i	$-0.127405\pi$
$354$		0			0
$355$	25.9823	−	25.9823i	0.0731897	−	0.0731897i
$356$		0			0
$357$	100.261	+	354.374i	0.280843	+	0.992644i
$358$		0			0
$359$		−	118.673i		−	0.330564i	−0.986246	−	0.165282i	$-0.947147\pi$
$359$		−	118.673i		−	0.330564i	0.986246	−	0.165282i	$-0.0528535\pi$
$360$		0			0
$361$			177.892i			0.492775i
$362$		0			0
$363$	276.930	+	276.930i	0.762893	+	0.762893i
$364$		0			0
$365$	−14.6706	−	14.6706i	−0.0401934	−	0.0401934i
$366$		0			0
$367$			232.590i			0.633759i	0.948466	+	0.316879i	$0.102635\pi$
$367$			232.590i			0.633759i	−0.948466	+	0.316879i	$0.897365\pi$
$368$		0			0
$369$		−	293.389i		−	0.795092i
$370$		0			0
$371$	264.417	−	74.8100i	0.712715	−	0.201644i
$372$		0			0
$373$	−24.9792	+	24.9792i	−0.0669683	+	0.0669683i	−0.739798	−	0.672829i	$-0.765079\pi$
$373$	−24.9792	+	24.9792i	−0.0669683	+	0.0669683i	0.672829	+	0.739798i	$0.265079\pi$
$374$		0			0
$375$			193.490i			0.515972i
$376$		0			0
$377$	256.509			0.680396
$378$		0			0
$379$	−14.8957	+	14.8957i	−0.0393025	+	0.0393025i	−0.726485	−	0.687182i	$-0.758847\pi$
$379$	−14.8957	+	14.8957i	−0.0393025	+	0.0393025i	0.687182	+	0.726485i	$0.258847\pi$
$380$		0			0
$381$	274.916	−	274.916i	0.721565	−	0.721565i
$382$		0			0
$383$		−	27.6638i		−	0.0722291i	−0.999348	−	0.0361146i	$-0.988502\pi$
$383$		−	27.6638i		−	0.0722291i	0.999348	−	0.0361146i	$-0.0114981\pi$
$384$		0			0
$385$	−83.0430	−	46.4160i	−0.215696	−	0.120561i
$386$		0			0
$387$	−314.375	−	314.375i	−0.812338	−	0.812338i
$388$		0			0
$389$	−128.491	+	128.491i	−0.330311	+	0.330311i	−0.852705	−	0.522394i	$-0.825039\pi$
$389$	−128.491	+	128.491i	−0.330311	+	0.330311i	0.522394	+	0.852705i	$0.325039\pi$
$390$		0			0
$391$	−247.091			−0.631947
$392$		0			0
$393$			498.034i			1.26726i
$394$		0			0
$395$	82.4373	+	82.4373i	0.208702	+	0.208702i
$396$		0			0
$397$	−451.607	+	451.607i	−1.13755	+	1.13755i	−0.148660	+	0.988888i	$0.547496\pi$
$397$	−451.607	+	451.607i	−1.13755	+	1.13755i	−0.988888	+	0.148660i	$0.952504\pi$
$398$		0			0
$399$	195.664	−	350.063i	0.490385	−	0.877350i
$400$		0			0
$401$	127.385			0.317668			0.158834	−	0.987305i	$-0.449226\pi$
$401$	127.385			0.317668			0.158834	+	0.987305i	$0.449226\pi$
$402$		0			0
$403$	301.267	+	301.267i	0.747561	+	0.747561i
$404$		0			0
$405$	53.7135	+	53.7135i	0.132626	+	0.132626i
$406$		0			0
$407$			297.170i			0.730146i
$408$		0			0
$409$	52.3077			0.127892			0.0639458	−	0.997953i	$-0.479632\pi$
$409$	52.3077			0.127892			0.0639458	+	0.997953i	$0.479632\pi$
$410$		0			0
$411$	−263.358	−	263.358i	−0.640774	−	0.640774i
$412$		0			0
$413$	−204.881	−	724.155i	−0.496079	−	1.75340i
$414$		0			0
$415$	108.390			0.261181
$416$		0			0
$417$	1018.19			2.44171
$418$		0			0
$419$	−309.486	+	309.486i	−0.738629	+	0.738629i	−0.972313	−	0.233684i	$-0.924922\pi$
$419$	−309.486	+	309.486i	−0.738629	+	0.738629i	0.233684	+	0.972313i	$0.424922\pi$
$420$		0			0
$421$	−223.706	+	223.706i	−0.531368	+	0.531368i	−0.920979	−	0.389612i	$-0.872609\pi$
$421$	−223.706	+	223.706i	−0.531368	+	0.531368i	0.389612	+	0.920979i	$0.372609\pi$
$422$		0			0
$423$	−328.790			−0.777280
$424$		0			0
$425$	299.917			0.705686
$426$		0			0
$427$	−628.920	+	177.937i	−1.47288	+	0.416713i
$428$		0			0
$429$	400.173	+	400.173i	0.932804	+	0.932804i
$430$		0			0
$431$	−598.931			−1.38963			−0.694815	−	0.719189i	$-0.744514\pi$
$431$	−598.931			−1.38963			−0.694815	+	0.719189i	$0.744514\pi$
$432$		0			0
$433$			197.759i			0.456718i	0.973577	+	0.228359i	$0.0733360\pi$
$433$			197.759i			0.456718i	−0.973577	+	0.228359i	$0.926664\pi$
$434$		0			0
$435$	−78.0772	−	78.0772i	−0.179488	−	0.179488i
$436$		0			0
$437$	190.257	+	190.257i	0.435371	+	0.435371i
$438$		0			0
$439$	116.437			0.265232			0.132616	−	0.991167i	$-0.457662\pi$
$439$	116.437			0.265232			0.132616	+	0.991167i	$0.457662\pi$
$440$		0			0
$441$	372.501	−	229.119i	0.844673	−	0.519544i
$442$		0			0
$443$	250.712	−	250.712i	0.565941	−	0.565941i	−0.365048	−	0.930989i	$-0.618947\pi$
$443$	250.712	−	250.712i	0.565941	−	0.565941i	0.930989	+	0.365048i	$0.118947\pi$
$444$		0			0
$445$	−27.4396	−	27.4396i	−0.0616619	−	0.0616619i
$446$		0			0
$447$			158.429i			0.354426i
$448$		0			0
$449$	−869.681			−1.93693			−0.968464	−	0.249153i	$-0.919848\pi$
$449$	−869.681			−1.93693			−0.968464	+	0.249153i	$0.919848\pi$
$450$		0			0
$451$	339.644	−	339.644i	0.753090	−	0.753090i
$452$		0			0
$453$	−426.071	−	426.071i	−0.940553	−	0.940553i
$454$		0			0
$455$	−51.9915	−	29.0601i	−0.114267	−	0.0638683i
$456$		0			0
$457$		−	81.7790i		−	0.178947i	−0.995989	−	0.0894737i	$-0.971481\pi$
$457$		−	81.7790i		−	0.178947i	0.995989	−	0.0894737i	$-0.0285185\pi$
$458$		0			0
$459$	2.79094	−	2.79094i	0.00608047	−	0.00608047i
$460$		0			0
$461$	117.326	−	117.326i	0.254504	−	0.254504i	−0.568310	−	0.822814i	$-0.692403\pi$
$461$	117.326	−	117.326i	0.254504	−	0.254504i	0.822814	+	0.568310i	$0.192403\pi$
$462$		0			0
$463$	−129.550			−0.279806			−0.139903	−	0.990165i	$-0.544679\pi$
$463$	−129.550			−0.279806			−0.139903	+	0.990165i	$0.544679\pi$
$464$		0			0
$465$		−	183.401i		−	0.394412i
$466$		0			0
$467$	339.918	−	339.918i	0.727875	−	0.727875i	−0.242321	−	0.970196i	$-0.577909\pi$
$467$	339.918	−	339.918i	0.727875	−	0.727875i	0.970196	+	0.242321i	$0.0779087\pi$
$468$		0			0
$469$	−161.019	−	569.125i	−0.343324	−	1.21349i
$470$		0			0
$471$		−	210.057i		−	0.445982i
$472$		0			0
$473$		−	727.876i		−	1.53885i
$474$		0			0
$475$	−230.932	−	230.932i	−0.486172	−	0.486172i
$476$		0			0
$477$	247.745	+	247.745i	0.519382	+	0.519382i
$478$		0			0
$479$		−	462.374i		−	0.965290i	−0.875816	−	0.482645i	$-0.839676\pi$
$479$		−	462.374i		−	0.965290i	0.875816	−	0.482645i	$-0.160324\pi$
$480$		0			0
$481$			186.052i			0.386802i
$482$		0			0
$483$	160.427	+	567.033i	0.332147	+	1.17398i
$484$		0			0
$485$	75.6880	−	75.6880i	0.156058	−	0.156058i
$486$		0			0
$487$		−	232.597i		−	0.477611i	−0.971067	−	0.238806i	$-0.923244\pi$
$487$		−	232.597i		−	0.477611i	0.971067	−	0.238806i	$-0.0767559\pi$
$488$		0			0
$489$	−877.267			−1.79400
$490$		0			0
$491$	603.262	−	603.262i	1.22864	−	1.22864i	0.264161	−	0.964479i	$-0.414905\pi$
$491$	603.262	−	603.262i	1.22864	−	1.22864i	0.964479	−	0.264161i	$-0.0850951\pi$
$492$		0			0
$493$	−246.384	+	246.384i	−0.499764	+	0.499764i
$494$		0			0
$495$		−	121.296i		−	0.245043i
$496$		0			0
$497$	−134.922	+	241.389i	−0.271472	+	0.485692i
$498$		0			0
$499$	−522.572	−	522.572i	−1.04724	−	1.04724i	−0.998827	−	0.0484112i	$-0.984584\pi$
$499$	−522.572	−	522.572i	−1.04724	−	1.04724i	−0.0484112	−	0.998827i	$-0.515416\pi$
$500$		0			0
$501$	−181.310	+	181.310i	−0.361897	+	0.361897i
$502$		0			0
$503$	−580.953			−1.15498			−0.577488	−	0.816399i	$-0.695967\pi$
$503$	−580.953			−1.15498			−0.577488	+	0.816399i	$0.695967\pi$
$504$		0			0
$505$		−	33.3382i		−	0.0660162i
$506$		0			0
$507$	−255.402	−	255.402i	−0.503752	−	0.503752i
$508$		0			0
$509$	198.424	−	198.424i	0.389832	−	0.389832i	−0.484796	−	0.874627i	$-0.661106\pi$
$509$	198.424	−	198.424i	0.389832	−	0.389832i	0.874627	+	0.484796i	$0.161106\pi$
$510$		0			0
$511$	136.297	+	76.1818i	0.266726	+	0.149084i
$512$		0			0
$513$	−4.29797			−0.00837811
$514$		0			0
$515$	−125.697	−	125.697i	−0.244072	−	0.244072i
$516$		0			0
$517$	−380.625	−	380.625i	−0.736219	−	0.736219i
$518$		0			0
$519$			1312.47i			2.52885i
$520$		0			0
$521$	−247.183			−0.474439			−0.237220	−	0.971456i	$-0.576236\pi$
$521$	−247.183			−0.474439			−0.237220	+	0.971456i	$0.576236\pi$
$522$		0			0
$523$	−144.472	−	144.472i	−0.276237	−	0.276237i	0.555368	−	0.831605i	$-0.312577\pi$
$523$	−144.472	−	144.472i	−0.276237	−	0.276237i	−0.831605	+	0.555368i	$0.812577\pi$
$524$		0			0
$525$	−194.725	−	688.258i	−0.370904	−	1.31097i
$526$		0			0
$527$	−578.749			−1.09820
$528$		0			0
$529$	133.630			0.252609
$530$		0			0
$531$	678.495	−	678.495i	1.27777	−	1.27777i
$532$		0			0
$533$	212.644	−	212.644i	0.398957	−	0.398957i
$534$		0			0
$535$	65.2223			0.121911
$536$		0			0
$537$	−798.338			−1.48666
$538$		0			0
$539$	696.469	+	165.987i	1.29215	+	0.307954i
$540$		0			0
$541$	27.5816	+	27.5816i	0.0509825	+	0.0509825i	0.732138	−	0.681156i	$-0.238522\pi$
$541$	27.5816	+	27.5816i	0.0509825	+	0.0509825i	−0.681156	+	0.732138i	$0.738522\pi$
$542$		0			0
$543$	−380.963			−0.701589
$544$		0			0
$545$			17.2444i			0.0316412i
$546$		0			0
$547$	−515.129	−	515.129i	−0.941734	−	0.941734i	0.0566591	−	0.998394i	$-0.481955\pi$
$547$	−515.129	−	515.129i	−0.941734	−	0.941734i	−0.998394	+	0.0566591i	$0.981955\pi$
$548$		0			0
$549$	−589.265	−	589.265i	−1.07334	−	1.07334i
$550$		0			0
$551$	379.425			0.688611
$552$		0			0
$553$	−765.885	−	428.083i	−1.38496	−	0.774110i
$554$		0			0
$555$	56.6311	−	56.6311i	0.102038	−	0.102038i
$556$		0			0
$557$	152.354	+	152.354i	0.273526	+	0.273526i	0.830518	−	0.556992i	$-0.188045\pi$
$557$	152.354	+	152.354i	0.273526	+	0.273526i	−0.556992	+	0.830518i	$0.688045\pi$
$558$		0			0
$559$		−	455.708i		−	0.815221i
$560$		0			0
$561$	−768.753			−1.37033
$562$		0			0
$563$	−155.608	+	155.608i	−0.276392	+	0.276392i	−0.831667	−	0.555275i	$-0.812613\pi$
$563$	−155.608	+	155.608i	−0.276392	+	0.276392i	0.555275	+	0.831667i	$0.312613\pi$
$564$		0			0
$565$	31.0105	+	31.0105i	0.0548859	+	0.0548859i
$566$		0			0
$567$	−499.026	−	278.925i	−0.880116	−	0.491931i
$568$		0			0
$569$			1047.03i			1.84012i	0.391779	+	0.920059i	$0.371860\pi$
$569$			1047.03i			1.84012i	−0.391779	+	0.920059i	$0.628140\pi$
$570$		0			0
$571$	145.325	−	145.325i	0.254510	−	0.254510i	−0.568307	−	0.822817i	$-0.692401\pi$
$571$	145.325	−	145.325i	0.254510	−	0.254510i	0.822817	+	0.568307i	$0.192401\pi$
$572$		0			0
$573$	980.160	−	980.160i	1.71058	−	1.71058i
$574$		0			0
$575$	479.896			0.834601
$576$		0			0
$577$			995.546i			1.72538i	0.505730	+	0.862692i	$0.331223\pi$
$577$			995.546i			1.72538i	−0.505730	+	0.862692i	$0.668777\pi$
$578$		0			0
$579$	−390.346	+	390.346i	−0.674173	+	0.674173i
$580$		0			0
$581$	−784.924	+	222.074i	−1.35099	+	0.382227i
$582$		0			0
$583$			573.608i			0.983889i
$584$		0			0
$585$		−	75.9411i		−	0.129814i
$586$		0			0
$587$	271.642	+	271.642i	0.462763	+	0.462763i	0.899560	−	0.436797i	$-0.143887\pi$
$587$	271.642	+	271.642i	0.462763	+	0.462763i	−0.436797	+	0.899560i	$0.643887\pi$
$588$		0			0
$589$	445.630	+	445.630i	0.756587	+	0.756587i
$590$		0			0
$591$			317.746i			0.537641i
$592$		0			0
$593$		−	980.919i		−	1.65416i	−0.562081	−	0.827082i	$-0.689999\pi$
$593$		−	980.919i		−	1.65416i	0.562081	−	0.827082i	$-0.310001\pi$
$594$		0			0
$595$	77.8521	−	22.0262i	0.130844	−	0.0370189i
$596$		0			0
$597$	−117.307	+	117.307i	−0.196494	+	0.196494i
$598$		0			0
$599$			164.810i			0.275141i	0.990492	+	0.137571i	$0.0439294\pi$
$599$			164.810i			0.275141i	−0.990492	+	0.137571i	$0.956071\pi$
$600$		0			0
$601$	445.621			0.741465			0.370733	−	0.928740i	$-0.379107\pi$
$601$	445.621			0.741465			0.370733	+	0.928740i	$0.379107\pi$
$602$		0			0
$603$	533.240	−	533.240i	0.884312	−	0.884312i
$604$		0			0
$605$	60.8386	−	60.8386i	0.100560	−	0.100560i
$606$		0			0
$607$			1068.50i			1.76029i	0.474704	+	0.880146i	$0.342555\pi$
$607$			1068.50i			1.76029i	−0.474704	+	0.880146i	$0.657445\pi$
$608$		0			0
$609$	725.377	+	405.441i	1.19110	+	0.665749i
$610$		0			0
$611$	−238.302	−	238.302i	−0.390019	−	0.390019i
$612$		0			0
$613$	−557.460	+	557.460i	−0.909396	+	0.909396i	−0.996223	−	0.0868277i	$-0.972327\pi$
$613$	−557.460	+	557.460i	−0.909396	+	0.909396i	0.0868277	+	0.996223i	$0.472327\pi$
$614$		0			0
$615$	−129.451			−0.210489
$616$		0			0
$617$		−	41.8092i		−	0.0677621i	−0.999426	−	0.0338810i	$-0.989213\pi$
$617$		−	41.8092i		−	0.0677621i	0.999426	−	0.0338810i	$-0.0107867\pi$
$618$		0			0
$619$	−330.047	−	330.047i	−0.533194	−	0.533194i	0.388328	−	0.921521i	$-0.373053\pi$
$619$	−330.047	−	330.047i	−0.533194	−	0.533194i	−0.921521	+	0.388328i	$0.873053\pi$
$620$		0			0
$621$	4.46577	−	4.46577i	0.00719126	−	0.00719126i
$622$		0			0
$623$	254.927	+	142.489i	0.409193	+	0.228714i
$624$		0			0
$625$	−560.864			−0.897383
$626$		0			0
$627$	591.930	+	591.930i	0.944066	+	0.944066i
$628$		0			0
$629$	−178.707	−	178.707i	−0.284114	−	0.284114i
$630$		0			0
$631$		−	37.5011i		−	0.0594312i	−0.999558	−	0.0297156i	$-0.990540\pi$
$631$		−	37.5011i		−	0.0594312i	0.999558	−	0.0297156i	$-0.00946016\pi$
$632$		0			0
$633$	589.712			0.931615
$634$		0			0
$635$	−60.3962	−	60.3962i	−0.0951121	−	0.0951121i
$636$		0			0
$637$	436.045	+	103.921i	0.684529	+	0.163142i
$638$		0			0
$639$	−352.584			−0.551774
$640$		0			0
$641$	36.1839			0.0564491			0.0282245	−	0.999602i	$-0.491015\pi$
$641$	36.1839			0.0564491			0.0282245	+	0.999602i	$0.491015\pi$
$642$		0			0
$643$	674.115	−	674.115i	1.04839	−	1.04839i	0.0496218	−	0.998768i	$-0.484198\pi$
$643$	674.115	−	674.115i	1.04839	−	1.04839i	0.998768	−	0.0496218i	$-0.0158016\pi$
$644$		0			0
$645$	−138.710	+	138.710i	−0.215054	+	0.215054i
$646$		0			0
$647$	185.531			0.286755			0.143378	−	0.989668i	$-0.454204\pi$
$647$	185.531			0.286755			0.143378	+	0.989668i	$0.454204\pi$
$648$		0			0
$649$	1570.93			2.42054
$650$		0			0
$651$	375.760	+	1328.13i	0.577205	+	2.04014i
$652$		0			0
$653$	228.859	+	228.859i	0.350473	+	0.350473i	0.860286	−	0.509812i	$-0.170285\pi$
$653$	228.859	+	228.859i	0.350473	+	0.350473i	−0.509812	+	0.860286i	$0.670285\pi$
$654$		0			0
$655$	109.413			0.167042
$656$		0			0
$657$			199.082i			0.303016i
$658$		0			0
$659$	−175.358	−	175.358i	−0.266097	−	0.266097i	0.561428	−	0.827525i	$-0.310252\pi$
$659$	−175.358	−	175.358i	−0.266097	−	0.266097i	−0.827525	+	0.561428i	$0.810252\pi$
$660$		0			0
$661$	371.415	+	371.415i	0.561899	+	0.561899i	0.929847	−	0.367947i	$-0.119939\pi$
$661$	371.415	+	371.415i	0.561899	+	0.561899i	−0.367947	+	0.929847i	$0.619939\pi$
$662$		0			0
$663$	−481.300			−0.725943
$664$		0			0
$665$	−76.9051	−	42.9852i	−0.115647	−	0.0646395i
$666$		0			0
$667$	−394.238	+	394.238i	−0.591061	+	0.591061i
$668$		0			0
$669$	−1117.06	−	1117.06i	−1.66974	−	1.66974i
$670$		0			0
$671$		−	1364.33i		−	2.03328i
$672$		0			0
$673$	342.937			0.509564			0.254782	−	0.966998i	$-0.417996\pi$
$673$	342.937			0.509564			0.254782	+	0.966998i	$0.417996\pi$
$674$		0			0
$675$	−5.42051	+	5.42051i	−0.00803038	+	0.00803038i
$676$		0			0
$677$	−882.092	−	882.092i	−1.30294	−	1.30294i	−0.926399	−	0.376543i	$-0.877113\pi$
$677$	−882.092	−	882.092i	−1.30294	−	1.30294i	−0.376543	−	0.926399i	$-0.622887\pi$
$678$		0			0
$679$	−393.035	+	703.180i	−0.578844	+	1.03561i
$680$		0			0
$681$			270.038i			0.396531i
$682$		0			0
$683$	−705.361	+	705.361i	−1.03274	+	1.03274i	−0.0332932	+	0.999446i	$0.510600\pi$
$683$	−705.361	+	705.361i	−1.03274	+	1.03274i	−0.999446	+	0.0332932i	$0.989400\pi$
$684$		0			0
$685$	−57.8570	+	57.8570i	−0.0844627	+	0.0844627i
$686$		0			0
$687$	192.508			0.280216
$688$		0			0
$689$			359.124i			0.521225i
$690$		0			0
$691$	471.203	−	471.203i	0.681915	−	0.681915i	−0.278517	−	0.960431i	$-0.589843\pi$
$691$	471.203	−	471.203i	0.681915	−	0.681915i	0.960431	+	0.278517i	$0.0898428\pi$
$692$		0			0
$693$	248.517	+	878.387i	0.358610	+	1.26751i
$694$		0			0
$695$		−	223.686i		−	0.321850i
$696$		0			0
$697$			408.500i			0.586083i
$698$		0			0
$699$	762.140	+	762.140i	1.09033	+	1.09033i
$700$		0			0
$701$	−397.070	−	397.070i	−0.566434	−	0.566434i	0.364694	−	0.931127i	$-0.381174\pi$
$701$	−397.070	−	397.070i	−0.566434	−	0.566434i	−0.931127	+	0.364694i	$0.881174\pi$
$702$		0			0
$703$			275.205i			0.391472i
$704$		0			0
$705$			145.070i			0.205773i
$706$		0			0
$707$	68.3047	+	241.424i	0.0966120	+	0.341477i
$708$		0			0
$709$	431.219	−	431.219i	0.608207	−	0.608207i	−0.334270	−	0.942477i	$-0.608490\pi$
$709$	431.219	−	431.219i	0.608207	−	0.608207i	0.942477	+	0.334270i	$0.108490\pi$
$710$		0			0
$711$		−	1118.68i		−	1.57340i
$712$		0			0
$713$	−926.055			−1.29882
$714$		0			0
$715$	87.9138	−	87.9138i	0.122956	−	0.122956i
$716$		0			0
$717$	−601.553	+	601.553i	−0.838986	+	0.838986i
$718$		0			0
$719$			553.599i			0.769956i	0.922926	+	0.384978i	$0.125791\pi$
$719$			553.599i			0.769956i	−0.922926	+	0.384978i	$0.874209\pi$
$720$		0			0
$721$	1167.79	+	652.724i	1.61968	+	0.905304i
$722$		0			0
$723$	899.521	+	899.521i	1.24415	+	1.24415i
$724$		0			0
$725$	478.522	−	478.522i	0.660030	−	0.660030i
$726$		0			0
$727$	−536.095			−0.737407			−0.368703	−	0.929547i	$-0.620198\pi$
$727$	−536.095			−0.737407			−0.368703	+	0.929547i	$0.620198\pi$
$728$		0			0
$729$		−	716.796i		−	0.983260i
$730$		0			0
$731$	437.719	+	437.719i	0.598795	+	0.598795i
$732$		0			0
$733$	−630.257	+	630.257i	−0.859833	+	0.859833i	−0.991318	−	0.131486i	$-0.958025\pi$
$733$	−630.257	+	630.257i	−0.859833	+	0.859833i	0.131486	+	0.991318i	$0.458025\pi$
$734$		0			0
$735$	−101.093	−	164.357i	−0.137542	−	0.223615i
$736$		0			0
$737$	1234.62			1.67519
$738$		0			0
$739$	−596.381	−	596.381i	−0.807010	−	0.807010i	0.177170	−	0.984180i	$-0.443306\pi$
$739$	−596.381	−	596.381i	−0.807010	−	0.807010i	−0.984180	+	0.177170i	$0.943306\pi$
$740$		0			0
$741$	370.595	+	370.595i	0.500128	+	0.500128i
$742$		0			0
$743$			436.927i			0.588057i	0.955797	+	0.294029i	$0.0949961\pi$
$743$			436.927i			0.588057i	−0.955797	+	0.294029i	$0.905004\pi$
$744$		0			0
$745$	34.8051			0.0467182
$746$		0			0
$747$	−735.432	−	735.432i	−0.984515	−	0.984515i
$748$		0			0
$749$	−472.318	+	133.630i	−0.630598	+	0.178411i
$750$		0			0
$751$	134.317			0.178851			0.0894254	−	0.995994i	$-0.471497\pi$
$751$	134.317			0.178851			0.0894254	+	0.995994i	$0.471497\pi$
$752$		0			0
$753$	308.082			0.409139
$754$		0			0
$755$	−93.6032	+	93.6032i	−0.123978	+	0.123978i
$756$		0			0
$757$	222.243	−	222.243i	0.293584	−	0.293584i	−0.544910	−	0.838494i	$-0.683436\pi$
$757$	222.243	−	222.243i	0.293584	−	0.293584i	0.838494	+	0.544910i	$0.183436\pi$
$758$		0			0
$759$	−1230.08			−1.62066
$760$		0			0
$761$	611.801			0.803943			0.401972	−	0.915652i	$-0.368325\pi$
$761$	611.801			0.803943			0.401972	+	0.915652i	$0.368325\pi$
$762$		0			0
$763$	−35.3311	−	124.878i	−0.0463055	−	0.163668i
$764$		0			0
$765$	72.9434	+	72.9434i	0.0953508	+	0.0953508i
$766$		0			0
$767$	983.526			1.28230
$768$		0			0
$769$			126.534i			0.164544i	0.996610	+	0.0822718i	$0.0262176\pi$
$769$			126.534i			0.164544i	−0.996610	+	0.0822718i	$0.973782\pi$
$770$		0			0
$771$	−913.497	−	913.497i	−1.18482	−	1.18482i
$772$		0			0
$773$	651.858	+	651.858i	0.843283	+	0.843283i	0.989284	−	0.146002i	$-0.0466405\pi$
$773$	651.858	+	651.858i	0.843283	+	0.843283i	−0.146002	+	0.989284i	$0.546640\pi$
$774$		0			0
$775$	1124.04			1.45037
$776$		0			0
$777$	−294.075	+	526.132i	−0.378476	+	0.677132i
$778$		0			0
$779$	314.540	−	314.540i	0.403774	−	0.403774i
$780$		0			0
$781$	−408.171	−	408.171i	−0.522626	−	0.522626i
$782$		0			0
$783$		−	8.90597i		−	0.0113742i
$784$		0			0
$785$	−46.1474			−0.0587865
$786$		0			0
$787$	15.8719	−	15.8719i	0.0201676	−	0.0201676i	−0.696951	−	0.717119i	$-0.745461\pi$
$787$	15.8719	−	15.8719i	0.0201676	−	0.0201676i	0.717119	+	0.696951i	$0.245461\pi$
$788$		0			0
$789$	1176.49	+	1176.49i	1.49112	+	1.49112i
$790$		0			0
$791$	−288.104	−	161.032i	−0.364227	−	0.203581i
$792$		0			0
$793$		−	854.182i		−	1.07715i
$794$		0			0
$795$	109.311	−	109.311i	0.137499	−	0.137499i
$796$		0			0
$797$	825.437	−	825.437i	1.03568	−	1.03568i	0.0363401	−	0.999339i	$-0.488430\pi$
$797$	825.437	−	825.437i	1.03568	−	1.03568i	0.999339	−	0.0363401i	$-0.0115700\pi$
$798$		0			0
$799$	457.790			0.572954
$800$		0			0
$801$			372.358i			0.464867i
$802$		0			0
$803$	−230.468	+	230.468i	−0.287009	+	0.287009i
$804$		0			0
$805$	124.571	−	35.2441i	0.154747	−	0.0437815i
$806$		0			0
$807$			423.424i			0.524689i
$808$		0			0
$809$			568.791i			0.703079i	0.936173	+	0.351539i	$0.114342\pi$
$809$			568.791i			0.703079i	−0.936173	+	0.351539i	$0.885658\pi$
$810$		0			0
$811$	−147.502	−	147.502i	−0.181877	−	0.181877i	0.610296	−	0.792173i	$-0.291050\pi$
$811$	−147.502	−	147.502i	−0.181877	−	0.181877i	−0.792173	+	0.610296i	$0.791050\pi$
$812$		0			0
$813$	−55.8564	−	55.8564i	−0.0687040	−	0.0687040i
$814$		0			0
$815$			192.726i			0.236474i
$816$		0			0
$817$		−	674.076i		−	0.825063i
$818$		0			0
$819$	155.591	+	549.940i	0.189977	+	0.671478i
$820$		0			0
$821$	332.180	−	332.180i	0.404604	−	0.404604i	−0.475248	−	0.879852i	$-0.657642\pi$
$821$	332.180	−	332.180i	0.404604	−	0.404604i	0.879852	+	0.475248i	$0.157642\pi$
$822$		0			0
$823$		−	1591.77i		−	1.93410i	−0.254584	−	0.967051i	$-0.581939\pi$
$823$		−	1591.77i		−	1.93410i	0.254584	−	0.967051i	$-0.418061\pi$
$824$		0			0
$825$	1493.06			1.80977
$826$		0			0
$827$	−12.4095	+	12.4095i	−0.0150055	+	0.0150055i	−0.714570	−	0.699564i	$-0.753377\pi$
$827$	−12.4095	+	12.4095i	−0.0150055	+	0.0150055i	0.699564	+	0.714570i	$0.253377\pi$
$828$		0			0
$829$	−375.156	+	375.156i	−0.452541	+	0.452541i	−0.896197	−	0.443656i	$-0.853681\pi$
$829$	−375.156	+	375.156i	−0.452541	+	0.452541i	0.443656	+	0.896197i	$0.353681\pi$
$830$		0			0
$831$		−	1088.57i		−	1.30996i
$832$		0			0
$833$	−518.651	+	319.013i	−0.622630	+	0.382969i
$834$		0			0
$835$	39.8320	+	39.8320i	0.0477030	+	0.0477030i
$836$		0			0
$837$	10.4600	−	10.4600i	0.0124970	−	0.0124970i
$838$		0			0
$839$	−132.372			−0.157774			−0.0788868	−	0.996884i	$-0.525137\pi$
$839$	−132.372			−0.157774			−0.0788868	+	0.996884i	$0.525137\pi$
$840$		0			0
$841$		−	54.7816i		−	0.0651387i
$842$		0			0
$843$	874.761	+	874.761i	1.03768	+	1.03768i
$844$		0			0
$845$	−56.1091	+	56.1091i	−0.0664013	+	0.0664013i
$846$		0			0
$847$	−315.924	+	565.221i	−0.372992	+	0.667322i
$848$		0			0
$849$	−145.868			−0.171811
$850$		0			0
$851$	−285.949	−	285.949i	−0.336016	−	0.336016i
$852$		0			0
$853$	−565.441	−	565.441i	−0.662885	−	0.662885i	0.293174	−	0.956059i	$-0.405289\pi$
$853$	−565.441	−	565.441i	−0.662885	−	0.662885i	−0.956059	+	0.293174i	$0.905289\pi$
$854$		0			0
$855$		−	112.331i		−	0.131381i
$856$		0			0
$857$	−222.141			−0.259207			−0.129604	−	0.991566i	$-0.541371\pi$
$857$	−222.141			−0.259207			−0.129604	+	0.991566i	$0.541371\pi$
$858$		0			0
$859$	−345.234	−	345.234i	−0.401903	−	0.401903i	0.477001	−	0.878903i	$-0.341724\pi$
$859$	−345.234	−	345.234i	−0.401903	−	0.401903i	−0.878903	+	0.477001i	$0.841724\pi$
$860$		0			0
$861$	937.439	−	265.224i	1.08878	−	0.308042i
$862$		0			0
$863$	600.557			0.695895			0.347947	−	0.937514i	$-0.386879\pi$
$863$	600.557			0.695895			0.347947	+	0.937514i	$0.386879\pi$
$864$		0			0
$865$	288.336			0.333336
$866$		0			0
$867$	−402.890	+	402.890i	−0.464695	+	0.464695i
$868$		0			0
$869$	1295.05	−	1295.05i	1.49028	−	1.49028i
$870$		0			0
$871$	772.969			0.887450
$872$		0			0
$873$	−1027.10			−1.17651
$874$		0			0
$875$	−307.826	+	87.0913i	−0.351801	+	0.0995329i
$876$		0			0
$877$	320.870	+	320.870i	0.365873	+	0.365873i	0.865970	−	0.500097i	$-0.166702\pi$
$877$	320.870	+	320.870i	0.365873	+	0.365873i	−0.500097	+	0.865970i	$0.666702\pi$
$878$		0			0
$879$	−1059.51			−1.20536
$880$		0			0
$881$		−	15.7893i		−	0.0179220i	−0.999960	−	0.00896102i	$-0.997148\pi$
$881$		−	15.7893i		−	0.0179220i	0.999960	−	0.00896102i	$-0.00285242\pi$
$882$		0			0
$883$	382.109	+	382.109i	0.432739	+	0.432739i	0.889559	−	0.456820i	$-0.151012\pi$
$883$	382.109	+	382.109i	0.432739	+	0.432739i	−0.456820	+	0.889559i	$0.651012\pi$
$884$		0			0
$885$	−299.369	−	299.369i	−0.338270	−	0.338270i
$886$		0			0
$887$	616.000			0.694476			0.347238	−	0.937777i	$-0.387120\pi$
$887$	616.000			0.694476			0.347238	+	0.937777i	$0.387120\pi$
$888$		0			0
$889$	561.111	+	313.627i	0.631171	+	0.352786i
$890$		0			0
$891$	843.815	−	843.815i	0.947042	−	0.947042i
$892$		0			0
$893$	−352.492	−	352.492i	−0.394728	−	0.394728i
$894$		0			0
$895$			175.386i			0.195962i
$896$		0			0
$897$	−770.128			−0.858559
$898$		0			0
$899$	−923.404	+	923.404i	−1.02715	+	1.02715i
$900$		0			0
$901$	−344.948	−	344.948i	−0.382850	−	0.382850i
$902$		0			0
$903$	720.297	−	1288.69i	0.797672	−	1.42712i
$904$		0			0
$905$			83.6935i			0.0924790i
$906$		0			0
$907$	542.025	−	542.025i	0.597602	−	0.597602i	−0.342072	−	0.939674i	$-0.611129\pi$
$907$	542.025	−	542.025i	0.597602	−	0.597602i	0.939674	+	0.342072i	$0.111129\pi$
$908$		0			0
$909$	−226.202	+	226.202i	−0.248847	+	0.248847i
$910$		0			0
$911$	−307.047			−0.337044			−0.168522	−	0.985698i	$-0.553899\pi$
$911$	−307.047			−0.337044			−0.168522	+	0.985698i	$0.553899\pi$
$912$		0			0
$913$		−	1702.76i		−	1.86501i
$914$		0			0
$915$	−259.999	+	259.999i	−0.284152	+	0.284152i
$916$		0			0
$917$	−792.330	+	224.169i	−0.864046	+	0.244459i
$918$		0			0
$919$			739.177i			0.804327i	0.915568	+	0.402164i	$0.131742\pi$
$919$			739.177i			0.804327i	−0.915568	+	0.402164i	$0.868258\pi$
$920$		0			0
$921$		−	1911.96i		−	2.07596i
$922$		0			0
$923$	−255.547	−	255.547i	−0.276866	−	0.276866i
$924$		0			0
$925$	347.082	+	347.082i	0.375224	+	0.375224i
$926$		0			0
$927$			1705.73i			1.84005i
$928$		0			0
$929$			1473.87i			1.58651i	0.608888	+	0.793256i	$0.291616\pi$
$929$			1473.87i			1.58651i	−0.608888	+	0.793256i	$0.708384\pi$
$930$		0			0
$931$	644.991	+	153.719i	0.692794	+	0.165111i
$932$		0			0
$933$	−1149.55	+	1149.55i	−1.23210	+	1.23210i
$934$		0			0
$935$			168.887i			0.180628i
$936$		0			0
$937$	−234.391			−0.250151			−0.125075	−	0.992147i	$-0.539917\pi$
$937$	−234.391			−0.250151			−0.125075	+	0.992147i	$0.539917\pi$
$938$		0			0
$939$	−777.256	+	777.256i	−0.827748	+	0.827748i
$940$		0			0
$941$	324.740	−	324.740i	0.345101	−	0.345101i	−0.513180	−	0.858281i	$-0.671533\pi$
$941$	324.740	−	324.740i	0.345101	−	0.345101i	0.858281	+	0.513180i	$0.171533\pi$
$942$		0			0
$943$			653.640i			0.693149i
$944$		0			0
$945$	−1.00896	+	1.80514i	−0.00106769	+	0.00191020i
$946$		0			0
$947$	764.773	+	764.773i	0.807574	+	0.807574i	0.984266	−	0.176692i	$-0.0565397\pi$
$947$	764.773	+	764.773i	0.807574	+	0.807574i	−0.176692	+	0.984266i	$0.556540\pi$
$948$		0			0
$949$	−144.291	+	144.291i	−0.152046	+	0.152046i
$950$		0			0
$951$	1246.72			1.31095
$952$		0			0
$953$			811.387i			0.851403i	0.904864	+	0.425701i	$0.139973\pi$
$953$			811.387i			0.851403i	−0.904864	+	0.425701i	$0.860027\pi$
$954$		0			0
$955$	−215.331	−	215.331i	−0.225477	−	0.225477i
$956$		0			0
$957$	−1226.56	+	1226.56i	−1.28167	+	1.28167i
$958$		0			0
$959$	300.441	−	537.521i	0.313286	−	0.560501i
$960$		0			0
$961$	−1208.05			−1.25708
$962$		0			0
$963$	−442.537	−	442.537i	−0.459540	−	0.459540i
$964$		0			0
$965$	85.7549	+	85.7549i	0.0888652	+	0.0888652i
$966$		0			0
$967$			1449.05i			1.49850i	0.662288	+	0.749249i	$0.269585\pi$
$967$			1449.05i			1.49850i	−0.662288	+	0.749249i	$0.730415\pi$
$968$		0			0
$969$	−711.932			−0.734708
$970$		0			0
$971$	412.622	+	412.622i	0.424945	+	0.424945i	0.886902	−	0.461957i	$-0.152853\pi$
$971$	412.622	+	412.622i	0.424945	+	0.424945i	−0.461957	+	0.886902i	$0.652853\pi$
$972$		0			0
$973$	458.297	+	1619.86i	0.471014	+	1.66481i
$974$		0			0
$975$	934.773			0.958741
$976$		0			0
$977$	−610.912			−0.625294			−0.312647	−	0.949869i	$-0.601216\pi$
$977$	−610.912			−0.625294			−0.312647	+	0.949869i	$0.601216\pi$
$978$		0			0
$979$	−431.063	+	431.063i	−0.440310	+	0.440310i
$980$		0			0
$981$	117.005	−	117.005i	0.119271	−	0.119271i
$982$		0			0
$983$	1050.92			1.06910			0.534549	−	0.845137i	$-0.320481\pi$
$983$	1050.92			1.06910			0.534549	+	0.845137i	$0.320481\pi$
$984$		0			0
$985$	69.8054			0.0708685
$986$		0			0
$987$	−297.226	−	1050.55i	−0.301141	−	1.06439i
$988$		0			0
$989$	700.394	+	700.394i	0.708184	+	0.708184i
$990$		0			0
$991$	712.709			0.719181			0.359591	−	0.933110i	$-0.382916\pi$
$991$	712.709			0.719181			0.359591	+	0.933110i	$0.382916\pi$
$992$		0			0
$993$		−	648.895i		−	0.653469i
$994$		0			0
$995$	25.7711	+	25.7711i	0.0259006	+	0.0259006i
$996$		0			0
$997$	−338.064	−	338.064i	−0.339081	−	0.339081i	0.516940	−	0.856021i	$-0.327071\pi$
$997$	−338.064	−	338.064i	−0.339081	−	0.339081i	−0.856021	+	0.516940i	$0.827071\pi$
$998$		0			0
$999$	6.45969			0.00646616

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.3.l.b.209.5		56
4.3	odd	2		112.3.l.b.13.20	yes	56
7.6	odd	2	inner	448.3.l.b.209.24		56
16.5	even	4	inner	448.3.l.b.433.24		56
16.11	odd	4		112.3.l.b.69.19	yes	56
28.27	even	2		112.3.l.b.13.19	✓	56
112.27	even	4		112.3.l.b.69.20	yes	56
112.69	odd	4	inner	448.3.l.b.433.5		56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.3.l.b.13.19	✓	56	28.27	even	2
112.3.l.b.13.20	yes	56	4.3	odd	2
112.3.l.b.69.19	yes	56	16.11	odd	4
112.3.l.b.69.20	yes	56	112.27	even	4
448.3.l.b.209.5		56	1.1	even	1	trivial
448.3.l.b.209.24		56	7.6	odd	2	inner
448.3.l.b.433.5		56	112.69	odd	4	inner
448.3.l.b.433.24		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(-2.99374 + 2.99374i) q^{3} +(0.657693 + 0.657693i) q^{5} +(-6.11030 - 3.41529i) q^{7} -8.92498i q^{9} +O(q^{10})\)	\(q+(-2.99374 + 2.99374i) q^{3} +(0.657693 + 0.657693i) q^{5} +(-6.11030 - 3.41529i) q^{7} -8.92498i q^{9} +(10.3321 - 10.3321i) q^{11} +(6.46869 - 6.46869i) q^{13} -3.93793 q^{15} +12.4267i q^{17} +(9.56839 - 9.56839i) q^{19} +(28.5172 - 8.06819i) q^{21} +19.8839i q^{23} -24.1349i q^{25} +(-0.224592 - 0.224592i) q^{27} +(19.8270 + 19.8270i) q^{29} +46.5731i q^{31} +61.8630i q^{33} +(-1.77249 - 6.26492i) q^{35} +(-14.3809 + 14.3809i) q^{37} +38.7312i q^{39} +32.8728 q^{41} +(35.2241 - 35.2241i) q^{43} +(5.86990 - 5.86990i) q^{45} -36.8393i q^{47} +(25.6716 + 41.7369i) q^{49} +(-37.2023 - 37.2023i) q^{51} +(-27.7586 + 27.7586i) q^{53} +13.5907 q^{55} +57.2906i q^{57} +(76.0220 + 76.0220i) q^{59} +(66.0243 - 66.0243i) q^{61} +(-30.4814 + 54.5343i) q^{63} +8.50883 q^{65} +(59.7469 + 59.7469i) q^{67} +(-59.5273 - 59.5273i) q^{69} -39.5053i q^{71} -22.3061 q^{73} +(72.2536 + 72.2536i) q^{75} +(-98.4190 + 27.8451i) q^{77} +125.343 q^{79} +81.6696 q^{81} +(82.4016 - 82.4016i) q^{83} +(-8.17295 + 8.17295i) q^{85} -118.714 q^{87} -41.7209 q^{89} +(-61.6181 + 17.4332i) q^{91} +(-139.428 - 139.428i) q^{93} +12.5861 q^{95} -115.081i q^{97} +(-92.2134 - 92.2134i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q+O(q^{10}) \) 56 * q	\( 56 q + 8 q^{15} - 20 q^{21} - 96 q^{29} + 100 q^{35} - 128 q^{37} + 72 q^{43} + 192 q^{49} + 128 q^{51} + 88 q^{53} - 444 q^{63} - 8 q^{65} - 440 q^{67} + 12 q^{77} + 8 q^{79} + 64 q^{81} + 96 q^{85} + 388 q^{91} + 32 q^{93} + 776 q^{95} - 8 q^{99}+O(q^{100}) \) 56 * q + 8 * q^15 - 20 * q^21 - 96 * q^29 + 100 * q^35 - 128 * q^37 + 72 * q^43 + 192 * q^49 + 128 * q^51 + 88 * q^53 - 444 * q^63 - 8 * q^65 - 440 * q^67 + 12 * q^77 + 8 * q^79 + 64 * q^81 + 96 * q^85 + 388 * q^91 + 32 * q^93 + 776 * q^95 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more