LMFDB - Embedded newform 448.3.l.b.209.23

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

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

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

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

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

Newform invariants

comment:select newform

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

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

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

Embedding invariants

Embedding label			209.23
Character	$\chi$	$=$	448.209
Dual form			448.3.l.b.433.23

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(2.42317 - 2.42317i) q^{3} +(4.90814 + 4.90814i) q^{5} +(-5.19813 + 4.68822i) q^{7} -2.74353i q^{9} +(11.4374 - 11.4374i) q^{11} +(-0.233094 + 0.233094i) q^{13} +23.7865 q^{15} +13.2735i q^{17} +(16.1828 - 16.1828i) q^{19} +(-1.23559 + 23.9563i) q^{21} +37.7293i q^{23} +23.1796i q^{25} +(15.1605 + 15.1605i) q^{27} +(23.3499 + 23.3499i) q^{29} -38.2862i q^{31} -55.4298i q^{33} +(-48.5236 - 2.50269i) q^{35} +(5.37755 - 5.37755i) q^{37} +1.12965i q^{39} +13.3775 q^{41} +(-27.7609 + 27.7609i) q^{43} +(13.4656 - 13.4656i) q^{45} -33.0328i q^{47} +(5.04112 - 48.7400i) q^{49} +(32.1640 + 32.1640i) q^{51} +(-9.52094 + 9.52094i) q^{53} +112.273 q^{55} -78.4274i q^{57} +(-46.0839 - 46.0839i) q^{59} +(58.0502 - 58.0502i) q^{61} +(12.8623 + 14.2612i) q^{63} -2.28811 q^{65} +(-30.9706 - 30.9706i) q^{67} +(91.4246 + 91.4246i) q^{69} +4.71778i q^{71} -74.6723 q^{73} +(56.1683 + 56.1683i) q^{75} +(-5.83203 + 113.075i) q^{77} -32.5841 q^{79} +98.1648 q^{81} +(-66.2230 + 66.2230i) q^{83} +(-65.1481 + 65.1481i) q^{85} +113.162 q^{87} +15.6044 q^{89} +(0.118856 - 2.30445i) q^{91} +(-92.7740 - 92.7740i) q^{93} +158.855 q^{95} -4.30378i q^{97} +(-31.3790 - 31.3790i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q + 8 q^{15} - 20 q^{21} - 96 q^{29} + 100 q^{35} - 128 q^{37} + 72 q^{43} + 192 q^{49} + 128 q^{51} + 88 q^{53} - 444 q^{63} - 8 q^{65} - 440 q^{67} + 12 q^{77} + 8 q^{79} + 64 q^{81} + 96 q^{85} + 388 q^{91}+ \cdots - 8 q^{99}+O(q^{100}) $ 56 * q + 8 * q^15 - 20 * q^21 - 96 * q^29 + 100 * q^35 - 128 * q^37 + 72 * q^43 + 192 * q^49 + 128 * q^51 + 88 * q^53 - 444 * q^63 - 8 * q^65 - 440 * q^67 + 12 * q^77 + 8 * q^79 + 64 * q^81 + 96 * q^85 + 388 * q^91 + 32 * q^93 + 776 * q^95 - 8 * q^99

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	2.42317	−	2.42317i	0.807724	−	0.807724i	−0.176565	−	0.984289i	$-0.556498\pi$
$3$	2.42317	−	2.42317i	0.807724	−	0.807724i	0.984289	+	0.176565i	$0.0564985\pi$
$4$		0			0
$5$	4.90814	+	4.90814i	0.981628	+	0.981628i	0.999834	−	0.0182065i	$-0.00579564\pi$
$5$	4.90814	+	4.90814i	0.981628	+	0.981628i	−0.0182065	+	0.999834i	$0.505796\pi$
$6$		0			0
$7$	−5.19813	+	4.68822i	−0.742590	+	0.669746i
$7$	−5.19813	+	4.68822i	−0.742590	+	0.669746i
$8$		0			0
$9$		−	2.74353i		−	0.304837i
$10$		0			0
$11$	11.4374	−	11.4374i	1.03977	−	1.03977i	0.0405918	−	0.999176i	$-0.487076\pi$
$11$	11.4374	−	11.4374i	1.03977	−	1.03977i	0.999176	−	0.0405918i	$-0.0129243\pi$
$12$		0			0
$13$	−0.233094	+	0.233094i	−0.0179303	+	0.0179303i	−0.716015	−	0.698085i	$-0.754036\pi$
$13$	−0.233094	+	0.233094i	−0.0179303	+	0.0179303i	0.698085	+	0.716015i	$0.254036\pi$
$14$		0			0
$15$	23.7865			1.58577
$16$		0			0
$17$			13.2735i			0.780794i	0.920647	+	0.390397i	$0.127662\pi$
$17$			13.2735i			0.780794i	−0.920647	+	0.390397i	$0.872338\pi$
$18$		0			0
$19$	16.1828	−	16.1828i	0.851726	−	0.851726i	−0.138620	−	0.990346i	$-0.544267\pi$
$19$	16.1828	−	16.1828i	0.851726	−	0.851726i	0.990346	+	0.138620i	$0.0442666\pi$
$20$		0			0
$21$	−1.23559	+	23.9563i	−0.0588377	+	1.14078i
$22$		0			0
$23$			37.7293i			1.64040i	0.572074	+	0.820202i	$0.306139\pi$
$23$			37.7293i			1.64040i	−0.572074	+	0.820202i	$0.693861\pi$
$24$		0			0
$25$			23.1796i			0.927186i
$26$		0			0
$27$	15.1605	+	15.1605i	0.561500	+	0.561500i
$28$		0			0
$29$	23.3499	+	23.3499i	0.805169	+	0.805169i	0.983898	−	0.178729i	$-0.0571987\pi$
$29$	23.3499	+	23.3499i	0.805169	+	0.805169i	−0.178729	+	0.983898i	$0.557199\pi$
$30$		0			0
$31$		−	38.2862i		−	1.23504i	−0.786556	−	0.617519i	$-0.788138\pi$
$31$		−	38.2862i		−	1.23504i	0.786556	−	0.617519i	$-0.211862\pi$
$32$		0			0
$33$		−	55.4298i		−	1.67969i
$34$		0			0
$35$	−48.5236	−	2.50269i	−1.38639	−	0.0715055i
$36$		0			0
$37$	5.37755	−	5.37755i	0.145339	−	0.145339i	−0.630693	−	0.776032i	$-0.717229\pi$
$37$	5.37755	−	5.37755i	0.145339	−	0.145339i	0.776032	+	0.630693i	$0.217229\pi$
$38$		0			0
$39$			1.12965i			0.0289655i
$40$		0			0
$41$	13.3775			0.326280			0.163140	−	0.986603i	$-0.447838\pi$
$41$	13.3775			0.326280			0.163140	+	0.986603i	$0.447838\pi$
$42$		0			0
$43$	−27.7609	+	27.7609i	−0.645603	+	0.645603i	−0.951927	−	0.306324i	$-0.900901\pi$
$43$	−27.7609	+	27.7609i	−0.645603	+	0.645603i	0.306324	+	0.951927i	$0.400901\pi$
$44$		0			0
$45$	13.4656	−	13.4656i	0.299236	−	0.299236i
$46$		0			0
$47$		−	33.0328i		−	0.702826i	−0.936221	−	0.351413i	$-0.885701\pi$
$47$		−	33.0328i		−	0.702826i	0.936221	−	0.351413i	$-0.114299\pi$
$48$		0			0
$49$	5.04112	−	48.7400i	0.102880	−	0.994694i
$50$		0			0
$51$	32.1640	+	32.1640i	0.630666	+	0.630666i
$52$		0			0
$53$	−9.52094	+	9.52094i	−0.179640	+	0.179640i	−0.791199	−	0.611559i	$-0.790543\pi$
$53$	−9.52094	+	9.52094i	−0.179640	+	0.179640i	0.611559	+	0.791199i	$0.290543\pi$
$54$		0			0
$55$	112.273			2.04133
$56$		0			0
$57$		−	78.4274i		−	1.37592i
$58$		0			0
$59$	−46.0839	−	46.0839i	−0.781083	−	0.781083i	0.198930	−	0.980014i	$-0.436253\pi$
$59$	−46.0839	−	46.0839i	−0.781083	−	0.781083i	−0.980014	+	0.198930i	$0.936253\pi$
$60$		0			0
$61$	58.0502	−	58.0502i	0.951643	−	0.951643i	−0.0472410	−	0.998884i	$-0.515043\pi$
$61$	58.0502	−	58.0502i	0.951643	−	0.951643i	0.998884	+	0.0472410i	$0.0150429\pi$
$62$		0			0
$63$	12.8623	+	14.2612i	0.204163	+	0.226369i
$64$		0			0
$65$	−2.28811			−0.0352018
$66$		0			0
$67$	−30.9706	−	30.9706i	−0.462248	−	0.462248i	0.437143	−	0.899392i	$-0.355990\pi$
$67$	−30.9706	−	30.9706i	−0.462248	−	0.462248i	−0.899392	+	0.437143i	$0.855990\pi$
$68$		0			0
$69$	91.4246	+	91.4246i	1.32499	+	1.32499i
$70$		0			0
$71$			4.71778i			0.0664475i	0.999448	+	0.0332238i	$0.0105774\pi$
$71$			4.71778i			0.0664475i	−0.999448	+	0.0332238i	$0.989423\pi$
$72$		0			0
$73$	−74.6723			−1.02291			−0.511454	−	0.859311i	$-0.670893\pi$
$73$	−74.6723			−1.02291			−0.511454	+	0.859311i	$0.670893\pi$
$74$		0			0
$75$	56.1683	+	56.1683i	0.748911	+	0.748911i
$76$		0			0
$77$	−5.83203	+	113.075i	−0.0757406	+	1.46850i
$78$		0			0
$79$	−32.5841			−0.412457			−0.206229	−	0.978504i	$-0.566119\pi$
$79$	−32.5841			−0.412457			−0.206229	+	0.978504i	$0.566119\pi$
$80$		0			0
$81$	98.1648			1.21191
$82$		0			0
$83$	−66.2230	+	66.2230i	−0.797867	+	0.797867i	−0.982759	−	0.184892i	$-0.940807\pi$
$83$	−66.2230	+	66.2230i	−0.797867	+	0.797867i	0.184892	+	0.982759i	$0.440807\pi$
$84$		0			0
$85$	−65.1481	+	65.1481i	−0.766449	+	0.766449i
$86$		0			0
$87$	113.162			1.30071
$88$		0			0
$89$	15.6044			0.175330			0.0876652	−	0.996150i	$-0.472059\pi$
$89$	15.6044			0.175330			0.0876652	+	0.996150i	$0.472059\pi$
$90$		0			0
$91$	0.118856	−	2.30445i	0.00130611	−	0.0253236i
$92$		0			0
$93$	−92.7740	−	92.7740i	−0.997570	−	0.997570i
$94$		0			0
$95$	158.855			1.67216
$96$		0			0
$97$		−	4.30378i		−	0.0443689i	−0.999754	−	0.0221844i	$-0.992938\pi$
$97$		−	4.30378i		−	0.0443689i	0.999754	−	0.0221844i	$-0.00706210\pi$
$98$		0			0
$99$	−31.3790	−	31.3790i	−0.316960	−	0.316960i
$100$		0			0
$101$	−103.129	−	103.129i	−1.02108	−	1.02108i	−0.999773	−	0.0213065i	$-0.993217\pi$
$101$	−103.129	−	103.129i	−1.02108	−	1.02108i	−0.0213065	−	0.999773i	$-0.506783\pi$
$102$		0			0
$103$	−152.891			−1.48438			−0.742189	−	0.670190i	$-0.766212\pi$
$103$	−152.891			−1.48438			−0.742189	+	0.670190i	$0.766212\pi$
$104$		0			0
$105$	−123.646	+	111.517i	−1.17758	+	1.06206i
$106$		0			0
$107$	−72.2804	+	72.2804i	−0.675518	+	0.675518i	−0.958983	−	0.283465i	$-0.908516\pi$
$107$	−72.2804	+	72.2804i	−0.675518	+	0.675518i	0.283465	+	0.958983i	$0.408516\pi$
$108$		0			0
$109$	5.71433	+	5.71433i	0.0524250	+	0.0524250i	0.732833	−	0.680408i	$-0.238198\pi$
$109$	5.71433	+	5.71433i	0.0524250	+	0.0524250i	−0.680408	+	0.732833i	$0.738198\pi$
$110$		0			0
$111$		−	26.0615i		−	0.234788i
$112$		0			0
$113$	150.105			1.32837			0.664184	−	0.747570i	$-0.268779\pi$
$113$	150.105			1.32837			0.664184	+	0.747570i	$0.268779\pi$
$114$		0			0
$115$	−185.181	+	185.181i	−1.61027	+	1.61027i
$116$		0			0
$117$	0.639501	+	0.639501i	0.00546582	+	0.00546582i
$118$		0			0
$119$	−62.2291	−	68.9973i	−0.522934	−	0.579810i
$120$		0			0
$121$		−	140.630i		−	1.16223i
$122$		0			0
$123$	32.4160	−	32.4160i	0.263544	−	0.263544i
$124$		0			0
$125$	8.93454	−	8.93454i	0.0714764	−	0.0714764i
$126$		0			0
$127$	−103.137			−0.812103			−0.406051	−	0.913850i	$-0.633095\pi$
$127$	−103.137			−0.812103			−0.406051	+	0.913850i	$0.633095\pi$
$128$		0			0
$129$			134.539i			1.04294i
$130$		0			0
$131$	101.501	−	101.501i	0.774818	−	0.774818i	−0.204127	−	0.978944i	$-0.565436\pi$
$131$	101.501	−	101.501i	0.774818	−	0.774818i	0.978944	+	0.204127i	$0.0654355\pi$
$132$		0			0
$133$	−8.25171	+	159.989i	−0.0620430	+	1.20292i
$134$		0			0
$135$			148.820i			1.10237i
$136$		0			0
$137$		−	233.344i		−	1.70324i	−0.524158	−	0.851621i	$-0.675620\pi$
$137$		−	233.344i		−	1.70324i	0.524158	−	0.851621i	$-0.324380\pi$
$138$		0			0
$139$	−62.9228	−	62.9228i	−0.452682	−	0.452682i	0.443562	−	0.896244i	$-0.353715\pi$
$139$	−62.9228	−	62.9228i	−0.452682	−	0.452682i	−0.896244	+	0.443562i	$0.853715\pi$
$140$		0			0
$141$	−80.0442	−	80.0442i	−0.567689	−	0.567689i
$142$		0			0
$143$			5.33200i			0.0372867i
$144$		0			0
$145$			229.209i			1.58075i
$146$		0			0
$147$	−105.890	−	130.321i	−0.720340	−	0.886537i
$148$		0			0
$149$	−147.549	+	147.549i	−0.990264	+	0.990264i	−0.999953	−	0.00968935i	$-0.996916\pi$
$149$	−147.549	+	147.549i	−0.990264	+	0.990264i	0.00968935	+	0.999953i	$0.496916\pi$
$150$		0			0
$151$		−	47.8397i		−	0.316819i	−0.987373	−	0.158410i	$-0.949363\pi$
$151$		−	47.8397i		−	0.316819i	0.987373	−	0.158410i	$-0.0506367\pi$
$152$		0			0
$153$	36.4163			0.238015
$154$		0			0
$155$	187.914	−	187.914i	1.21235	−	1.21235i
$156$		0			0
$157$	62.4830	−	62.4830i	0.397981	−	0.397981i	−0.479539	−	0.877520i	$-0.659196\pi$
$157$	62.4830	−	62.4830i	0.397981	−	0.397981i	0.877520	+	0.479539i	$0.159196\pi$
$158$		0			0
$159$			46.1418i			0.290200i
$160$		0			0
$161$	−176.883	−	196.122i	−1.09865	−	1.21815i
$162$		0			0
$163$	−43.0047	−	43.0047i	−0.263832	−	0.263832i	0.562777	−	0.826609i	$-0.309733\pi$
$163$	−43.0047	−	43.0047i	−0.263832	−	0.263832i	−0.826609	+	0.562777i	$0.809733\pi$
$164$		0			0
$165$	272.057	−	272.057i	1.64883	−	1.64883i
$166$		0			0
$167$	146.222			0.875580			0.437790	−	0.899077i	$-0.355761\pi$
$167$	146.222			0.875580			0.437790	+	0.899077i	$0.355761\pi$
$168$		0			0
$169$			168.891i			0.999357i
$170$		0			0
$171$	−44.3980	−	44.3980i	−0.259638	−	0.259638i
$172$		0			0
$173$	−180.530	+	180.530i	−1.04353	+	1.04353i	−0.0445174	+	0.999009i	$0.514175\pi$
$173$	−180.530	+	180.530i	−1.04353	+	1.04353i	−0.999009	+	0.0445174i	$0.985825\pi$
$174$		0			0
$175$	−108.671	−	120.491i	−0.620979	−	0.688519i
$176$		0			0
$177$	−223.339			−1.26180
$178$		0			0
$179$	−5.67020	−	5.67020i	−0.0316771	−	0.0316771i	0.691091	−	0.722768i	$-0.257130\pi$
$179$	−5.67020	−	5.67020i	−0.0316771	−	0.0316771i	−0.722768	+	0.691091i	$0.757130\pi$
$180$		0			0
$181$	15.3127	+	15.3127i	0.0846008	+	0.0846008i	0.748141	−	0.663540i	$-0.230947\pi$
$181$	15.3127	+	15.3127i	0.0846008	+	0.0846008i	−0.663540	+	0.748141i	$0.730947\pi$
$182$		0			0
$183$		−	281.331i		−	1.53733i
$184$		0			0
$185$	52.7875			0.285338
$186$		0			0
$187$	151.815	+	151.815i	0.811844	+	0.811844i
$188$		0			0
$189$	−149.882	−	7.73044i	−0.793027	−	0.0409018i
$190$		0			0
$191$	64.2617			0.336449			0.168224	−	0.985749i	$-0.446197\pi$
$191$	64.2617			0.336449			0.168224	+	0.985749i	$0.446197\pi$
$192$		0			0
$193$	−210.607			−1.09123			−0.545614	−	0.838037i	$-0.683704\pi$
$193$	−210.607			−1.09123			−0.545614	+	0.838037i	$0.683704\pi$
$194$		0			0
$195$	−5.54450	+	5.54450i	−0.0284333	+	0.0284333i
$196$		0			0
$197$	17.6884	−	17.6884i	0.0897890	−	0.0897890i	−0.660786	−	0.750575i	$-0.729777\pi$
$197$	17.6884	−	17.6884i	0.0897890	−	0.0897890i	0.750575	+	0.660786i	$0.229777\pi$
$198$		0			0
$199$	−300.902			−1.51207			−0.756036	−	0.654530i	$-0.772867\pi$
$199$	−300.902			−1.51207			−0.756036	+	0.654530i	$0.772867\pi$
$200$		0			0
$201$	−150.094			−0.746739
$202$		0			0
$203$	−230.845	−	11.9063i	−1.13717	−	0.0586516i
$204$		0			0
$205$	65.6586	+	65.6586i	0.320286	+	0.320286i
$206$		0			0
$207$	103.512			0.500056
$208$		0			0
$209$		−	370.180i		−	1.77119i
$210$		0			0
$211$	179.684	+	179.684i	0.851583	+	0.851583i	0.990328	−	0.138745i	$-0.0443068\pi$
$211$	179.684	+	179.684i	0.851583	+	0.851583i	−0.138745	+	0.990328i	$0.544307\pi$
$212$		0			0
$213$	11.4320	+	11.4320i	0.0536713	+	0.0536713i
$214$		0			0
$215$	−272.509			−1.26748
$216$		0			0
$217$	179.494	+	199.017i	0.827162	+	0.917127i
$218$		0			0
$219$	−180.944	+	180.944i	−0.826227	+	0.826227i
$220$		0			0
$221$	−3.09397	−	3.09397i	−0.0139999	−	0.0139999i
$222$		0			0
$223$			57.9078i			0.259676i	0.991535	+	0.129838i	$0.0414458\pi$
$223$			57.9078i			0.259676i	−0.991535	+	0.129838i	$0.958554\pi$
$224$		0			0
$225$	63.5941			0.282641
$226$		0			0
$227$	234.014	−	234.014i	1.03090	−	1.03090i	0.0313934	−	0.999507i	$-0.490006\pi$
$227$	234.014	−	234.014i	1.03090	−	1.03090i	0.999507	−	0.0313934i	$-0.00999448\pi$
$228$		0			0
$229$	−58.2181	−	58.2181i	−0.254228	−	0.254228i	0.568474	−	0.822701i	$-0.307534\pi$
$229$	−58.2181	−	58.2181i	−0.254228	−	0.254228i	−0.822701	+	0.568474i	$0.807534\pi$
$230$		0			0
$231$	259.867	+	288.131i	1.12497	+	1.24732i
$232$		0			0
$233$		−	85.3285i		−	0.366217i	−0.983093	−	0.183108i	$-0.941384\pi$
$233$		−	85.3285i		−	0.366217i	0.983093	−	0.183108i	$-0.0586159\pi$
$234$		0			0
$235$	162.130	−	162.130i	0.689913	−	0.689913i
$236$		0			0
$237$	−78.9570	+	78.9570i	−0.333152	+	0.333152i
$238$		0			0
$239$	−247.623			−1.03608			−0.518040	−	0.855356i	$-0.673338\pi$
$239$	−247.623			−1.03608			−0.518040	+	0.855356i	$0.673338\pi$
$240$		0			0
$241$			32.0526i			0.132998i	0.997786	+	0.0664991i	$0.0211830\pi$
$241$			32.0526i			0.132998i	−0.997786	+	0.0664991i	$0.978817\pi$
$242$		0			0
$243$	101.426	−	101.426i	0.417390	−	0.417390i
$244$		0			0
$245$	263.965	−	214.480i	1.07741	−	0.875429i
$246$		0			0
$247$			7.54422i			0.0305434i
$248$		0			0
$249$			320.939i			1.28891i
$250$		0			0
$251$	−30.9176	−	30.9176i	−0.123178	−	0.123178i	0.642831	−	0.766008i	$-0.277760\pi$
$251$	−30.9176	−	30.9176i	−0.123178	−	0.123178i	−0.766008	+	0.642831i	$0.777760\pi$
$252$		0			0
$253$	431.527	+	431.527i	1.70564	+	1.70564i
$254$		0			0
$255$			315.730i			1.23816i
$256$		0			0
$257$			248.945i			0.968656i	0.874886	+	0.484328i	$0.160936\pi$
$257$			248.945i			0.968656i	−0.874886	+	0.484328i	$0.839064\pi$
$258$		0			0
$259$	−2.74205	+	53.1643i	−0.0105871	+	0.205268i
$260$		0			0
$261$	64.0612	−	64.0612i	0.245445	−	0.245445i
$262$		0			0
$263$			157.837i			0.600142i	0.953917	+	0.300071i	$0.0970104\pi$
$263$			157.837i			0.600142i	−0.953917	+	0.300071i	$0.902990\pi$
$264$		0			0
$265$	−93.4602			−0.352680
$266$		0			0
$267$	37.8122	−	37.8122i	0.141619	−	0.141619i
$268$		0			0
$269$	49.6844	−	49.6844i	0.184700	−	0.184700i	−0.608700	−	0.793400i	$-0.708309\pi$
$269$	49.6844	−	49.6844i	0.184700	−	0.184700i	0.793400	+	0.608700i	$0.208309\pi$
$270$		0			0
$271$			343.687i			1.26822i	0.773244	+	0.634108i	$0.218633\pi$
$271$			343.687i			1.26822i	−0.773244	+	0.634108i	$0.781367\pi$
$272$		0			0
$273$	−5.29607	−	5.87209i	−0.0193995	−	0.0215095i
$274$		0			0
$275$	265.116	+	265.116i	0.964058	+	0.964058i
$276$		0			0
$277$	294.560	−	294.560i	1.06339	−	1.06339i	0.0655437	−	0.997850i	$-0.479122\pi$
$277$	294.560	−	294.560i	1.06339	−	1.06339i	0.997850	−	0.0655437i	$-0.0208782\pi$
$278$		0			0
$279$	−105.039			−0.376485
$280$		0			0
$281$		−	261.174i		−	0.929446i	−0.885456	−	0.464723i	$-0.846154\pi$
$281$		−	261.174i		−	0.929446i	0.885456	−	0.464723i	$-0.153846\pi$
$282$		0			0
$283$	314.008	+	314.008i	1.10957	+	1.10957i	0.993207	+	0.116361i	$0.0371228\pi$
$283$	314.008	+	314.008i	1.10957	+	1.10957i	0.116361	+	0.993207i	$0.462877\pi$
$284$		0			0
$285$	384.933	−	384.933i	1.35064	−	1.35064i
$286$		0			0
$287$	−69.5379	+	62.7167i	−0.242292	+	0.218525i
$288$		0			0
$289$	112.814			0.390361
$290$		0			0
$291$	−10.4288	−	10.4288i	−0.0358378	−	0.0358378i
$292$		0			0
$293$	−338.825	−	338.825i	−1.15640	−	1.15640i	−0.985244	−	0.171156i	$-0.945250\pi$
$293$	−338.825	−	338.825i	−1.15640	−	1.15640i	−0.171156	−	0.985244i	$-0.554750\pi$
$294$		0			0
$295$		−	452.372i		−	1.53347i
$296$		0			0
$297$	346.795			1.16766
$298$		0			0
$299$	−8.79447	−	8.79447i	−0.0294129	−	0.0294129i
$300$		0			0
$301$	14.1555	−	274.454i	0.0470282	−	0.911809i
$302$		0			0
$303$	−499.799			−1.64950
$304$		0			0
$305$	569.837			1.86832
$306$		0			0
$307$	65.2409	−	65.2409i	0.212511	−	0.212511i	−0.592822	−	0.805333i	$-0.701986\pi$
$307$	65.2409	−	65.2409i	0.212511	−	0.212511i	0.805333	+	0.592822i	$0.201986\pi$
$308$		0			0
$309$	−370.481	+	370.481i	−1.19897	+	1.19897i
$310$		0			0
$311$	251.679			0.809258			0.404629	−	0.914481i	$-0.367401\pi$
$311$	251.679			0.809258			0.404629	+	0.914481i	$0.367401\pi$
$312$		0			0
$313$	502.424			1.60519			0.802594	−	0.596526i	$-0.203453\pi$
$313$	502.424			1.60519			0.802594	+	0.596526i	$0.203453\pi$
$314$		0			0
$315$	−6.86622	+	133.126i	−0.0217975	+	0.422623i
$316$		0			0
$317$	−108.655	−	108.655i	−0.342760	−	0.342760i	0.514644	−	0.857404i	$-0.327924\pi$
$317$	−108.655	−	108.655i	−0.342760	−	0.342760i	−0.857404	+	0.514644i	$0.827924\pi$
$318$		0			0
$319$	534.126			1.67438
$320$		0			0
$321$			350.296i			1.09126i
$322$		0			0
$323$	214.802	+	214.802i	0.665022	+	0.665022i
$324$		0			0
$325$	−5.40304	−	5.40304i	−0.0166247	−	0.0166247i
$326$		0			0
$327$	27.6936			0.0846900
$328$		0			0
$329$	154.865	+	171.709i	0.470715	+	0.521911i
$330$		0			0
$331$	−135.473	+	135.473i	−0.409284	+	0.409284i	−0.881489	−	0.472205i	$-0.843458\pi$
$331$	−135.473	+	135.473i	−0.409284	+	0.409284i	0.472205	+	0.881489i	$0.343458\pi$
$332$		0			0
$333$	−14.7535	−	14.7535i	−0.0443047	−	0.0443047i
$334$		0			0
$335$		−	304.016i		−	0.907512i
$336$		0			0
$337$	96.5075			0.286372			0.143186	−	0.989696i	$-0.454265\pi$
$337$	96.5075			0.286372			0.143186	+	0.989696i	$0.454265\pi$
$338$		0			0
$339$	363.732	−	363.732i	1.07295	−	1.07295i
$340$		0			0
$341$	−437.896	−	437.896i	−1.28415	−	1.28415i
$342$		0			0
$343$	202.300	+	276.991i	0.589795	+	0.807553i
$344$		0			0
$345$			897.449i			2.60130i
$346$		0			0
$347$	310.963	−	310.963i	0.896146	−	0.896146i	−0.0989470	−	0.995093i	$-0.531547\pi$
$347$	310.963	−	310.963i	0.896146	−	0.896146i	0.995093	+	0.0989470i	$0.0315474\pi$
$348$		0			0
$349$	−92.9768	+	92.9768i	−0.266409	+	0.266409i	−0.827651	−	0.561242i	$-0.810324\pi$
$349$	−92.9768	+	92.9768i	−0.266409	+	0.266409i	0.561242	+	0.827651i	$0.310324\pi$
$350$		0			0
$351$	−7.06764			−0.0201357
$352$		0			0
$353$		−	9.13336i		−	0.0258735i	−0.999916	−	0.0129368i	$-0.995882\pi$
$353$		−	9.13336i		−	0.0258735i	0.999916	−	0.0129368i	$-0.00411802\pi$
$354$		0			0
$355$	−23.1555	+	23.1555i	−0.0652267	+	0.0652267i
$356$		0			0
$357$	−317.984	−	16.4006i	−0.890712	−	0.0459401i
$358$		0			0
$359$		−	209.563i		−	0.583741i	−0.956458	−	0.291871i	$-0.905722\pi$
$359$		−	209.563i		−	0.583741i	0.956458	−	0.291871i	$-0.0942777\pi$
$360$		0			0
$361$		−	162.765i		−	0.450874i
$362$		0			0
$363$	−340.771	−	340.771i	−0.938764	−	0.938764i
$364$		0			0
$365$	−366.502	−	366.502i	−1.00411	−	1.00411i
$366$		0			0
$367$		−	538.514i		−	1.46734i	−0.679506	−	0.733670i	$-0.737806\pi$
$367$		−	538.514i		−	1.46734i	0.679506	−	0.733670i	$-0.262194\pi$
$368$		0			0
$369$		−	36.7016i		−	0.0994623i
$370$		0			0
$371$	4.85479	−	94.1274i	0.0130857	−	0.253713i
$372$		0			0
$373$	−357.081	+	357.081i	−0.957323	+	0.957323i	−0.999126	−	0.0418032i	$-0.986690\pi$
$373$	−357.081	+	357.081i	−0.957323	+	0.957323i	0.0418032	+	0.999126i	$0.486690\pi$
$374$		0			0
$375$		−	43.2999i		−	0.115466i
$376$		0			0
$377$	−10.8854			−0.0288738
$378$		0			0
$379$	−251.266	+	251.266i	−0.662972	+	0.662972i	−0.956079	−	0.293108i	$-0.905311\pi$
$379$	−251.266	+	251.266i	−0.662972	+	0.662972i	0.293108	+	0.956079i	$0.405311\pi$
$380$		0			0
$381$	−249.919	+	249.919i	−0.655955	+	0.655955i
$382$		0			0
$383$		−	438.868i		−	1.14587i	−0.819601	−	0.572935i	$-0.805805\pi$
$383$		−	438.868i		−	1.14587i	0.819601	−	0.572935i	$-0.194195\pi$
$384$		0			0
$385$	−583.610	+	526.361i	−1.51587	+	1.36717i
$386$		0			0
$387$	76.1630	+	76.1630i	0.196804	+	0.196804i
$388$		0			0
$389$	391.861	−	391.861i	1.00735	−	1.00735i	0.00738172	−	0.999973i	$-0.497650\pi$
$389$	391.861	−	391.861i	1.00735	−	1.00735i	0.999973	−	0.00738172i	$-0.00234970\pi$
$390$		0			0
$391$	−500.799			−1.28082
$392$		0			0
$393$		−	491.909i		−	1.25168i
$394$		0			0
$395$	−159.927	−	159.927i	−0.404880	−	0.404880i
$396$		0			0
$397$	134.883	−	134.883i	0.339755	−	0.339755i	−0.516520	−	0.856275i	$-0.672773\pi$
$397$	134.883	−	134.883i	0.339755	−	0.339755i	0.856275	+	0.516520i	$0.172773\pi$
$398$		0			0
$399$	367.685	+	407.676i	0.921517	+	1.02174i
$400$		0			0
$401$	−390.589			−0.974038			−0.487019	−	0.873391i	$-0.661916\pi$
$401$	−390.589			−0.974038			−0.487019	+	0.873391i	$0.661916\pi$
$402$		0			0
$403$	8.92428	+	8.92428i	0.0221446	+	0.0221446i
$404$		0			0
$405$	481.807	+	481.807i	1.18965	+	1.18965i
$406$		0			0
$407$		−	123.011i		−	0.302238i
$408$		0			0
$409$	−111.876			−0.273536			−0.136768	−	0.990603i	$-0.543672\pi$
$409$	−111.876			−0.273536			−0.136768	+	0.990603i	$0.543672\pi$
$410$		0			0
$411$	−565.433	−	565.433i	−1.37575	−	1.37575i
$412$		0			0
$413$	455.602	+	23.4985i	1.10315	+	0.0568971i
$414$		0			0
$415$	−650.063			−1.56642
$416$		0			0
$417$	−304.946			−0.731285
$418$		0			0
$419$	−106.956	+	106.956i	−0.255264	+	0.255264i	−0.823125	−	0.567861i	$-0.807771\pi$
$419$	−106.956	+	106.956i	−0.255264	+	0.255264i	0.567861	+	0.823125i	$0.307771\pi$
$420$		0			0
$421$	−102.773	+	102.773i	−0.244115	+	0.244115i	−0.818550	−	0.574435i	$-0.805222\pi$
$421$	−102.773	+	102.773i	−0.244115	+	0.244115i	0.574435	+	0.818550i	$0.305222\pi$
$422$		0			0
$423$	−90.6266			−0.214247
$424$		0			0
$425$	−307.675			−0.723941
$426$		0			0
$427$	−29.6002	+	573.905i	−0.0693213	+	1.34404i
$428$		0			0
$429$	12.9204	+	12.9204i	0.0301174	+	0.0301174i
$430$		0			0
$431$	154.126			0.357601			0.178800	−	0.983885i	$-0.442778\pi$
$431$	154.126			0.357601			0.178800	+	0.983885i	$0.442778\pi$
$432$		0			0
$433$		−	93.0931i		−	0.214996i	−0.994205	−	0.107498i	$-0.965716\pi$
$433$		−	93.0931i		−	0.214996i	0.994205	−	0.107498i	$-0.0342838\pi$
$434$		0			0
$435$	555.413	+	555.413i	1.27681	+	1.27681i
$436$		0			0
$437$	610.565	+	610.565i	1.39717	+	1.39717i
$438$		0			0
$439$	460.835			1.04974			0.524869	−	0.851183i	$-0.324114\pi$
$439$	460.835			1.04974			0.524869	+	0.851183i	$0.324114\pi$
$440$		0			0
$441$	−133.720	−	13.8305i	−0.303219	−	0.0313616i
$442$		0			0
$443$	499.302	−	499.302i	1.12709	−	1.12709i	0.136446	−	0.990648i	$-0.456432\pi$
$443$	499.302	−	499.302i	1.12709	−	1.12709i	0.990648	−	0.136446i	$-0.0435680\pi$
$444$		0			0
$445$	76.5885	+	76.5885i	0.172109	+	0.172109i
$446$		0			0
$447$			715.075i			1.59972i
$448$		0			0
$449$	504.282			1.12312			0.561562	−	0.827435i	$-0.310201\pi$
$449$	504.282			1.12312			0.561562	+	0.827435i	$0.310201\pi$
$450$		0			0
$451$	153.004	−	153.004i	0.339256	−	0.339256i
$452$		0			0
$453$	−115.924	−	115.924i	−0.255903	−	0.255903i
$454$		0			0
$455$	11.8939	−	10.7272i	0.0261405	−	0.0235762i
$456$		0			0
$457$		−	392.566i		−	0.859006i	−0.903065	−	0.429503i	$-0.858689\pi$
$457$		−	392.566i		−	0.859006i	0.903065	−	0.429503i	$-0.141311\pi$
$458$		0			0
$459$	−201.233	+	201.233i	−0.438416	+	0.438416i
$460$		0			0
$461$	−249.810	+	249.810i	−0.541888	+	0.541888i	−0.924082	−	0.382194i	$-0.875168\pi$
$461$	−249.810	+	249.810i	−0.541888	+	0.541888i	0.382194	+	0.924082i	$0.375168\pi$
$462$		0			0
$463$	−123.976			−0.267767			−0.133884	−	0.990997i	$-0.542745\pi$
$463$	−123.976			−0.267767			−0.133884	+	0.990997i	$0.542745\pi$
$464$		0			0
$465$		−	910.696i		−	1.95849i
$466$		0			0
$467$	−254.905	+	254.905i	−0.545836	+	0.545836i	−0.925234	−	0.379398i	$-0.876131\pi$
$467$	−254.905	+	254.905i	−0.545836	+	0.545836i	0.379398	+	0.925234i	$0.376131\pi$
$468$		0			0
$469$	306.187	+	15.7921i	0.652850	+	0.0336719i
$470$		0			0
$471$		−	302.814i		−	0.642918i
$472$		0			0
$473$			635.028i			1.34255i
$474$		0			0
$475$	375.111	+	375.111i	0.789708	+	0.789708i
$476$		0			0
$477$	26.1210	+	26.1210i	0.0547610	+	0.0547610i
$478$		0			0
$479$			437.996i			0.914397i	0.889365	+	0.457199i	$0.151147\pi$
$479$			437.996i			0.914397i	−0.889365	+	0.457199i	$0.848853\pi$
$480$		0			0
$481$			2.50695i			0.00521195i
$482$		0			0
$483$	−903.856	−	46.6180i	−1.87134	−	0.0965176i
$484$		0			0
$485$	21.1235	−	21.1235i	0.0435537	−	0.0435537i
$486$		0			0
$487$		−	543.976i		−	1.11699i	−0.829507	−	0.558497i	$-0.811378\pi$
$487$		−	543.976i		−	1.11699i	0.829507	−	0.558497i	$-0.188622\pi$
$488$		0			0
$489$	−208.415			−0.426207
$490$		0			0
$491$	257.435	−	257.435i	0.524308	−	0.524308i	−0.394562	−	0.918870i	$-0.629104\pi$
$491$	257.435	−	257.435i	0.524308	−	0.524308i	0.918870	+	0.394562i	$0.129104\pi$
$492$		0			0
$493$	−309.935	+	309.935i	−0.628671	+	0.628671i
$494$		0			0
$495$		−	308.025i		−	0.622273i
$496$		0			0
$497$	−22.1180	−	24.5236i	−0.0445030	−	0.0493433i
$498$		0			0
$499$	−666.076	−	666.076i	−1.33482	−	1.33482i	−0.900997	−	0.433826i	$-0.857163\pi$
$499$	−666.076	−	666.076i	−1.33482	−	1.33482i	−0.433826	−	0.900997i	$-0.642837\pi$
$500$		0			0
$501$	354.321	−	354.321i	0.707227	−	0.707227i
$502$		0			0
$503$	−206.235			−0.410009			−0.205005	−	0.978761i	$-0.565721\pi$
$503$	−206.235			−0.410009			−0.205005	+	0.978761i	$0.565721\pi$
$504$		0			0
$505$		−	1012.34i		−	2.00464i
$506$		0			0
$507$	409.253	+	409.253i	0.807205	+	0.807205i
$508$		0			0
$509$	−239.572	+	239.572i	−0.470671	+	0.470671i	−0.902132	−	0.431461i	$-0.857998\pi$
$509$	−239.572	+	239.572i	−0.470671	+	0.470671i	0.431461	+	0.902132i	$0.357998\pi$
$510$		0			0
$511$	388.156	−	350.080i	0.759601	−	0.685088i
$512$		0			0
$513$	490.678			0.956488
$514$		0			0
$515$	−750.410	−	750.410i	−1.45711	−	1.45711i
$516$		0			0
$517$	−377.811	−	377.811i	−0.730775	−	0.730775i
$518$		0			0
$519$			874.911i			1.68576i
$520$		0			0
$521$	−630.759			−1.21067			−0.605335	−	0.795971i	$-0.706961\pi$
$521$	−630.759			−1.21067			−0.605335	+	0.795971i	$0.706961\pi$
$522$		0			0
$523$	344.460	+	344.460i	0.658624	+	0.658624i	0.955054	−	0.296431i	$-0.0957963\pi$
$523$	344.460	+	344.460i	0.658624	+	0.658624i	−0.296431	+	0.955054i	$0.595796\pi$
$524$		0			0
$525$	−555.300	−	28.6406i	−1.05771	−	0.0545535i
$526$		0			0
$527$	508.191			0.964310
$528$		0			0
$529$	−894.499			−1.69092
$530$		0			0
$531$	−126.433	+	126.433i	−0.238103	+	0.238103i
$532$		0			0
$533$	−3.11821	+	3.11821i	−0.00585030	+	0.00585030i
$534$		0			0
$535$	−709.524			−1.32621
$536$		0			0
$537$	−27.4798			−0.0511728
$538$		0			0
$539$	−499.803	−	615.118i	−0.927279	−	1.14122i
$540$		0			0
$541$	569.533	+	569.533i	1.05274	+	1.05274i	0.998529	+	0.0542113i	$0.0172645\pi$
$541$	569.533	+	569.533i	1.05274	+	1.05274i	0.0542113	+	0.998529i	$0.482736\pi$
$542$		0			0
$543$	74.2109			0.136668
$544$		0			0
$545$			56.0934i			0.102924i
$546$		0			0
$547$	476.145	+	476.145i	0.870467	+	0.870467i	0.992523	−	0.122057i	$-0.0389489\pi$
$547$	476.145	+	476.145i	0.870467	+	0.870467i	−0.122057	+	0.992523i	$0.538949\pi$
$548$		0			0
$549$	−159.263	−	159.263i	−0.290096	−	0.290096i
$550$		0			0
$551$	755.733			1.37157
$552$		0			0
$553$	169.377	−	152.762i	0.306287	−	0.276242i
$554$		0			0
$555$	127.913	−	127.913i	0.230474	−	0.230474i
$556$		0			0
$557$	−2.03506	−	2.03506i	−0.00365361	−	0.00365361i	0.705278	−	0.708931i	$-0.250822\pi$
$557$	−2.03506	−	2.03506i	−0.00365361	−	0.00365361i	−0.708931	+	0.705278i	$0.750822\pi$
$558$		0			0
$559$		−	12.9418i		−	0.0231517i
$560$		0			0
$561$	735.747			1.31149
$562$		0			0
$563$	−640.792	+	640.792i	−1.13817	+	1.13817i	−0.149397	+	0.988777i	$0.547733\pi$
$563$	−640.792	+	640.792i	−1.13817	+	1.13817i	−0.988777	+	0.149397i	$0.952267\pi$
$564$		0			0
$565$	736.739	+	736.739i	1.30396	+	1.30396i
$566$		0			0
$567$	−510.274	+	460.219i	−0.899953	+	0.811673i
$568$		0			0
$569$			148.984i			0.261835i	0.991393	+	0.130917i	$0.0417922\pi$
$569$			148.984i			0.261835i	−0.991393	+	0.130917i	$0.958208\pi$
$570$		0			0
$571$	−201.831	+	201.831i	−0.353469	+	0.353469i	−0.861399	−	0.507930i	$-0.830411\pi$
$571$	−201.831	+	201.831i	−0.353469	+	0.353469i	0.507930	+	0.861399i	$0.330411\pi$
$572$		0			0
$573$	155.717	−	155.717i	0.271758	−	0.271758i
$574$		0			0
$575$	−874.551			−1.52096
$576$		0			0
$577$			231.147i			0.400601i	0.979735	+	0.200300i	$0.0641918\pi$
$577$			231.147i			0.400601i	−0.979735	+	0.200300i	$0.935808\pi$
$578$		0			0
$579$	−510.337	+	510.337i	−0.881411	+	0.881411i
$580$		0			0
$581$	33.7675	−	654.704i	0.0581197	−	1.12686i
$582$		0			0
$583$			217.790i			0.373569i
$584$		0			0
$585$			6.27752i			0.0107308i
$586$		0			0
$587$	644.898	+	644.898i	1.09863	+	1.09863i	0.994571	+	0.104063i	$0.0331844\pi$
$587$	644.898	+	644.898i	1.09863	+	1.09863i	0.104063	+	0.994571i	$0.466816\pi$
$588$		0			0
$589$	−619.577	−	619.577i	−1.05191	−	1.05191i
$590$		0			0
$591$		−	85.7243i		−	0.145050i
$592$		0			0
$593$		−	883.947i		−	1.49064i	−0.666709	−	0.745318i	$-0.732298\pi$
$593$		−	883.947i		−	1.49064i	0.666709	−	0.745318i	$-0.267702\pi$
$594$		0			0
$595$	33.2195	−	644.077i	0.0558310	−	1.08248i
$596$		0			0
$597$	−729.139	+	729.139i	−1.22134	+	1.22134i
$598$		0			0
$599$			978.614i			1.63375i	0.576817	+	0.816873i	$0.304295\pi$
$599$			978.614i			1.63375i	−0.576817	+	0.816873i	$0.695705\pi$
$600$		0			0
$601$	−569.750			−0.948004			−0.474002	−	0.880524i	$-0.657191\pi$
$601$	−569.750			−0.948004			−0.474002	+	0.880524i	$0.657191\pi$
$602$		0			0
$603$	−84.9690	+	84.9690i	−0.140910	+	0.140910i
$604$		0			0
$605$	690.233	−	690.233i	1.14088	−	1.14088i
$606$		0			0
$607$		−	119.852i		−	0.197450i	−0.995115	−	0.0987249i	$-0.968524\pi$
$607$		−	119.852i		−	0.197450i	0.995115	−	0.0987249i	$-0.0314764\pi$
$608$		0			0
$609$	−588.229	+	530.527i	−0.965893	+	0.871145i
$610$		0			0
$611$	7.69975	+	7.69975i	0.0126019	+	0.0126019i
$612$		0			0
$613$	−442.123	+	442.123i	−0.721245	+	0.721245i	−0.968859	−	0.247614i	$-0.920353\pi$
$613$	−442.123	+	442.123i	−0.721245	+	0.721245i	0.247614	+	0.968859i	$0.420353\pi$
$614$		0			0
$615$	318.204			0.517405
$616$		0			0
$617$		−	870.225i		−	1.41041i	−0.709002	−	0.705206i	$-0.750854\pi$
$617$		−	870.225i		−	1.41041i	0.709002	−	0.705206i	$-0.249146\pi$
$618$		0			0
$619$	−392.513	−	392.513i	−0.634109	−	0.634109i	0.314987	−	0.949096i	$-0.398000\pi$
$619$	−392.513	−	392.513i	−0.634109	−	0.634109i	−0.949096	+	0.314987i	$0.898000\pi$
$620$		0			0
$621$	−571.995	+	571.995i	−0.921087	+	0.921087i
$622$		0			0
$623$	−81.1137	+	73.1569i	−0.130199	+	0.117427i
$624$		0			0
$625$	667.195			1.06751
$626$		0			0
$627$	−897.009	−	897.009i	−1.43064	−	1.43064i
$628$		0			0
$629$	71.3788	+	71.3788i	0.113480	+	0.113480i
$630$		0			0
$631$		−	252.226i		−	0.399724i	−0.979824	−	0.199862i	$-0.935951\pi$
$631$		−	252.226i		−	0.399724i	0.979824	−	0.199862i	$-0.0640494\pi$
$632$		0			0
$633$	870.811			1.37569
$634$		0			0
$635$	−506.211	−	506.211i	−0.797183	−	0.797183i
$636$		0			0
$637$	10.1859	+	12.5361i	0.0159905	+	0.0196798i
$638$		0			0
$639$	12.9434			0.0202557
$640$		0			0
$641$	874.134			1.36370			0.681852	−	0.731490i	$-0.261175\pi$
$641$	874.134			1.36370			0.681852	+	0.731490i	$0.261175\pi$
$642$		0			0
$643$	471.331	−	471.331i	0.733019	−	0.733019i	−0.238198	−	0.971217i	$-0.576557\pi$
$643$	471.331	−	471.331i	0.733019	−	0.733019i	0.971217	+	0.238198i	$0.0765566\pi$
$644$		0			0
$645$	−660.337	+	660.337i	−1.02378	+	1.02378i
$646$		0			0
$647$	−160.444			−0.247982			−0.123991	−	0.992283i	$-0.539569\pi$
$647$	−160.444			−0.247982			−0.123991	+	0.992283i	$0.539569\pi$
$648$		0			0
$649$	−1054.16			−1.62429
$650$		0			0
$651$	917.197	+	47.3061i	1.40890	+	0.0726668i
$652$		0			0
$653$	−180.797	−	180.797i	−0.276871	−	0.276871i	0.554988	−	0.831859i	$-0.312723\pi$
$653$	−180.797	−	180.797i	−0.276871	−	0.276871i	−0.831859	+	0.554988i	$0.812723\pi$
$654$		0			0
$655$	996.363			1.52116
$656$		0			0
$657$			204.866i			0.311820i
$658$		0			0
$659$	−61.5911	−	61.5911i	−0.0934615	−	0.0934615i	0.658830	−	0.752292i	$-0.271052\pi$
$659$	−61.5911	−	61.5911i	−0.0934615	−	0.0934615i	−0.752292	+	0.658830i	$0.771052\pi$
$660$		0			0
$661$	−380.712	−	380.712i	−0.575963	−	0.575963i	0.357825	−	0.933789i	$-0.383518\pi$
$661$	−380.712	−	380.712i	−0.575963	−	0.575963i	−0.933789	+	0.357825i	$0.883518\pi$
$662$		0			0
$663$	−14.9944			−0.0226161
$664$		0			0
$665$	−825.748	+	744.747i	−1.24173	+	1.11992i
$666$		0			0
$667$	−880.975	+	880.975i	−1.32080	+	1.32080i
$668$		0			0
$669$	140.321	+	140.321i	0.209747	+	0.209747i
$670$		0			0
$671$		−	1327.89i		−	1.97897i
$672$		0			0
$673$	−658.648			−0.978675			−0.489337	−	0.872095i	$-0.662761\pi$
$673$	−658.648			−0.978675			−0.489337	+	0.872095i	$0.662761\pi$
$674$		0			0
$675$	−351.415	+	351.415i	−0.520615	+	0.520615i
$676$		0			0
$677$	850.741	+	850.741i	1.25663	+	1.25663i	0.952690	+	0.303944i	$0.0983035\pi$
$677$	850.741	+	850.741i	1.25663	+	1.25663i	0.303944	+	0.952690i	$0.401697\pi$
$678$		0			0
$679$	20.1771	+	22.3716i	0.0297159	+	0.0329479i
$680$		0			0
$681$		−	1134.11i		−	1.66537i
$682$		0			0
$683$	391.331	−	391.331i	0.572958	−	0.572958i	−0.359996	−	0.932954i	$-0.617222\pi$
$683$	391.331	−	391.331i	0.572958	−	0.572958i	0.932954	+	0.359996i	$0.117222\pi$
$684$		0			0
$685$	1145.29	−	1145.29i	1.67195	−	1.67195i
$686$		0			0
$687$	−282.145			−0.410692
$688$		0			0
$689$		−	4.43855i		−	0.00644201i
$690$		0			0
$691$	144.012	−	144.012i	0.208410	−	0.208410i	−0.595181	−	0.803592i	$-0.702920\pi$
$691$	144.012	−	144.012i	0.208410	−	0.208410i	0.803592	+	0.595181i	$0.202920\pi$
$692$		0			0
$693$	310.224	+	16.0004i	0.447654	+	0.0230885i
$694$		0			0
$695$		−	617.668i		−	0.888731i
$696$		0			0
$697$			177.566i			0.254758i
$698$		0			0
$699$	−206.766	−	206.766i	−0.295802	−	0.295802i
$700$		0			0
$701$	196.926	+	196.926i	0.280922	+	0.280922i	0.833477	−	0.552555i	$-0.186347\pi$
$701$	196.926	+	196.926i	0.280922	+	0.280922i	−0.552555	+	0.833477i	$0.686347\pi$
$702$		0			0
$703$		−	174.047i		−	0.247578i
$704$		0			0
$705$		−	785.736i		−	1.11452i
$706$		0			0
$707$	1019.57	+	52.5862i	1.44211	+	0.0743793i
$708$		0			0
$709$	456.400	−	456.400i	0.643724	−	0.643724i	−0.307745	−	0.951469i	$-0.599574\pi$
$709$	456.400	−	456.400i	0.643724	−	0.643724i	0.951469	+	0.307745i	$0.0995744\pi$
$710$		0			0
$711$			89.3956i			0.125732i
$712$		0			0
$713$	1444.51			2.02596
$714$		0			0
$715$	−26.1702	+	26.1702i	−0.0366017	+	0.0366017i
$716$		0			0
$717$	−600.034	+	600.034i	−0.836867	+	0.836867i
$718$		0			0
$719$			1283.27i			1.78479i	0.451251	+	0.892397i	$0.350978\pi$
$719$			1283.27i			1.78479i	−0.451251	+	0.892397i	$0.649022\pi$
$720$		0			0
$721$	794.747	−	716.787i	1.10228	−	0.994157i
$722$		0			0
$723$	77.6689	+	77.6689i	0.107426	+	0.107426i
$724$		0			0
$725$	−541.242	+	541.242i	−0.746541	+	0.746541i
$726$		0			0
$727$	573.848			0.789337			0.394669	−	0.918823i	$-0.370859\pi$
$727$	573.848			0.789337			0.394669	+	0.918823i	$0.370859\pi$
$728$		0			0
$729$			391.939i			0.537639i
$730$		0			0
$731$	−368.485	−	368.485i	−0.504083	−	0.504083i
$732$		0			0
$733$	−21.8202	+	21.8202i	−0.0297684	+	0.0297684i	−0.721834	−	0.692066i	$-0.756701\pi$
$733$	−21.8202	+	21.8202i	−0.0297684	+	0.0297684i	0.692066	+	0.721834i	$0.256701\pi$
$734$		0			0
$735$	119.911	−	1159.36i	0.163144	−	1.57735i
$736$		0			0
$737$	−708.450			−0.961262
$738$		0			0
$739$	374.759	+	374.759i	0.507117	+	0.507117i	0.913640	−	0.406523i	$-0.133259\pi$
$739$	374.759	+	374.759i	0.507117	+	0.507117i	−0.406523	+	0.913640i	$0.633259\pi$
$740$		0			0
$741$	18.2810	+	18.2810i	0.0246707	+	0.0246707i
$742$		0			0
$743$		−	1226.17i		−	1.65029i	−0.564919	−	0.825146i	$-0.691093\pi$
$743$		−	1226.17i		−	1.65029i	0.564919	−	0.825146i	$-0.308907\pi$
$744$		0			0
$745$	−1448.38			−1.94414
$746$		0			0
$747$	181.685	+	181.685i	0.243219	+	0.243219i
$748$		0			0
$749$	36.8563	−	714.590i	0.0492073	−	0.954058i
$750$		0			0
$751$	671.894			0.894666			0.447333	−	0.894367i	$-0.352374\pi$
$751$	671.894			0.894666			0.447333	+	0.894367i	$0.352374\pi$
$752$		0			0
$753$	−149.837			−0.198987
$754$		0			0
$755$	234.804	−	234.804i	0.310999	−	0.310999i
$756$		0			0
$757$	−637.342	+	637.342i	−0.841932	+	0.841932i	−0.989110	−	0.147178i	$-0.952981\pi$
$757$	−637.342	+	637.342i	−0.841932	+	0.841932i	0.147178	+	0.989110i	$0.452981\pi$
$758$		0			0
$759$	2091.33			2.75537
$760$		0			0
$761$	1184.55			1.55657			0.778285	−	0.627911i	$-0.216090\pi$
$761$	1184.55			1.55657			0.778285	+	0.627911i	$0.216090\pi$
$762$		0			0
$763$	−56.4939	−	2.91377i	−0.0740418	−	0.00381884i
$764$		0			0
$765$	178.736	+	178.736i	0.233642	+	0.233642i
$766$		0			0
$767$	21.4838			0.0280101
$768$		0			0
$769$			880.837i			1.14543i	0.819754	+	0.572716i	$0.194110\pi$
$769$			880.837i			1.14543i	−0.819754	+	0.572716i	$0.805890\pi$
$770$		0			0
$771$	603.236	+	603.236i	0.782407	+	0.782407i
$772$		0			0
$773$	422.394	+	422.394i	0.546434	+	0.546434i	0.925408	−	0.378973i	$-0.123723\pi$
$773$	422.394	+	422.394i	0.546434	+	0.546434i	−0.378973	+	0.925408i	$0.623723\pi$
$774$		0			0
$775$	887.460			1.14511
$776$		0			0
$777$	122.182	+	135.471i	0.157248	+	0.174351i
$778$		0			0
$779$	216.485	−	216.485i	0.277901	−	0.277901i
$780$		0			0
$781$	53.9593	+	53.9593i	0.0690900	+	0.0690900i
$782$		0			0
$783$			707.992i			0.904205i
$784$		0			0
$785$	613.350			0.781338
$786$		0			0
$787$	−869.911	+	869.911i	−1.10535	+	1.10535i	−0.111598	+	0.993753i	$0.535597\pi$
$787$	−869.911	+	869.911i	−1.10535	+	1.10535i	−0.993753	+	0.111598i	$0.964403\pi$
$788$		0			0
$789$	382.467	+	382.467i	0.484749	+	0.484749i
$790$		0			0
$791$	−780.268	+	703.728i	−0.986432	+	0.889669i
$792$		0			0
$793$			27.0623i			0.0341265i
$794$		0			0
$795$	−226.470	+	226.470i	−0.284868	+	0.284868i
$796$		0			0
$797$	77.7866	−	77.7866i	0.0975992	−	0.0975992i	−0.656621	−	0.754220i	$-0.728015\pi$
$797$	77.7866	−	77.7866i	0.0975992	−	0.0975992i	0.754220	+	0.656621i	$0.228015\pi$
$798$		0			0
$799$	438.461			0.548762
$800$		0			0
$801$		−	42.8112i		−	0.0534472i
$802$		0			0
$803$	−854.060	+	854.060i	−1.06359	+	1.06359i
$804$		0			0
$805$	94.4248	−	1830.76i	0.117298	−	2.27424i
$806$		0			0
$807$		−	240.788i		−	0.298374i
$808$		0			0
$809$			1278.47i			1.58031i	0.612910	+	0.790153i	$0.289999\pi$
$809$			1278.47i			1.58031i	−0.612910	+	0.790153i	$0.710001\pi$
$810$		0			0
$811$	−135.617	−	135.617i	−0.167222	−	0.167222i	0.618535	−	0.785757i	$-0.287726\pi$
$811$	−135.617	−	135.617i	−0.167222	−	0.167222i	−0.785757	+	0.618535i	$0.787726\pi$
$812$		0			0
$813$	832.812	+	832.812i	1.02437	+	1.02437i
$814$		0			0
$815$		−	422.146i		−	0.517970i
$816$		0			0
$817$			898.499i			1.09975i
$818$		0			0
$819$	−6.32233	−	0.326086i	−0.00771958	−	0.000398151i
$820$		0			0
$821$	−225.467	+	225.467i	−0.274625	+	0.274625i	−0.830959	−	0.556334i	$-0.812208\pi$
$821$	−225.467	+	225.467i	−0.274625	+	0.274625i	0.556334	+	0.830959i	$0.312208\pi$
$822$		0			0
$823$			719.591i			0.874352i	0.899376	+	0.437176i	$0.144021\pi$
$823$			719.591i			0.874352i	−0.899376	+	0.437176i	$0.855979\pi$
$824$		0			0
$825$	1284.84			1.55739
$826$		0			0
$827$	−824.582	+	824.582i	−0.997076	+	0.997076i	−0.999996	−	0.00291969i	$-0.999071\pi$
$827$	−824.582	+	824.582i	−0.997076	+	0.997076i	0.00291969	+	0.999996i	$0.499071\pi$
$828$		0			0
$829$	559.570	−	559.570i	0.674993	−	0.674993i	−0.283869	−	0.958863i	$-0.591618\pi$
$829$	559.570	−	559.570i	0.674993	−	0.674993i	0.958863	+	0.283869i	$0.0916182\pi$
$830$		0			0
$831$		−	1427.54i		−	1.71786i
$832$		0			0
$833$	646.950	+	66.9132i	0.776651	+	0.0803280i
$834$		0			0
$835$	717.677	+	717.677i	0.859494	+	0.859494i
$836$		0			0
$837$	580.438	−	580.438i	0.693474	−	0.693474i
$838$		0			0
$839$	164.817			0.196445			0.0982224	−	0.995164i	$-0.468684\pi$
$839$	164.817			0.196445			0.0982224	+	0.995164i	$0.468684\pi$
$840$		0			0
$841$			249.435i			0.296594i
$842$		0			0
$843$	−632.871	−	632.871i	−0.750736	−	0.750736i
$844$		0			0
$845$	−828.942	+	828.942i	−0.980997	+	0.980997i
$846$		0			0
$847$	659.306	+	731.014i	0.778401	+	0.863063i
$848$		0			0
$849$	1521.79			1.79245
$850$		0			0
$851$	202.891	+	202.891i	0.238415	+	0.238415i
$852$		0			0
$853$	470.436	+	470.436i	0.551508	+	0.551508i	0.926876	−	0.375368i	$-0.122484\pi$
$853$	470.436	+	470.436i	0.551508	+	0.551508i	−0.375368	+	0.926876i	$0.622484\pi$
$854$		0			0
$855$		−	435.823i		−	0.509735i
$856$		0			0
$857$	−6.26986			−0.00731606			−0.00365803	−	0.999993i	$-0.501164\pi$
$857$	−6.26986			−0.00731606			−0.00365803	+	0.999993i	$0.501164\pi$
$858$		0			0
$859$	457.662	+	457.662i	0.532784	+	0.532784i	0.921400	−	0.388616i	$-0.127047\pi$
$859$	457.662	+	457.662i	0.532784	+	0.532784i	−0.388616	+	0.921400i	$0.627047\pi$
$860$		0			0
$861$	−16.5291	+	320.476i	−0.0191976	+	0.372213i
$862$		0			0
$863$	319.733			0.370490			0.185245	−	0.982692i	$-0.440692\pi$
$863$	319.733			0.370490			0.185245	+	0.982692i	$0.440692\pi$
$864$		0			0
$865$	−1772.13			−2.04871
$866$		0			0
$867$	273.369	−	273.369i	0.315304	−	0.315304i
$868$		0			0
$869$	−372.679	+	372.679i	−0.428860	+	0.428860i
$870$		0			0
$871$	14.4381			0.0165765
$872$		0			0
$873$	−11.8076			−0.0135253
$874$		0			0
$875$	−4.55578	+	88.3301i	−0.00520661	+	0.100949i
$876$		0			0
$877$	380.302	+	380.302i	0.433640	+	0.433640i	0.889865	−	0.456225i	$-0.150799\pi$
$877$	380.302	+	380.302i	0.433640	+	0.433640i	−0.456225	+	0.889865i	$0.650799\pi$
$878$		0			0
$879$	−1642.06			−1.86810
$880$		0			0
$881$		−	695.336i		−	0.789258i	−0.918841	−	0.394629i	$-0.870873\pi$
$881$		−	695.336i		−	0.789258i	0.918841	−	0.394629i	$-0.129127\pi$
$882$		0			0
$883$	541.268	+	541.268i	0.612987	+	0.612987i	0.943723	−	0.330736i	$-0.107297\pi$
$883$	541.268	+	541.268i	0.612987	+	0.612987i	−0.330736	+	0.943723i	$0.607297\pi$
$884$		0			0
$885$	−1096.18	−	1096.18i	−1.23862	−	1.23862i
$886$		0			0
$887$	−1253.15			−1.41279			−0.706397	−	0.707816i	$-0.749681\pi$
$887$	−1253.15			−1.41279			−0.706397	+	0.707816i	$0.749681\pi$
$888$		0			0
$889$	536.120	−	483.530i	0.603059	−	0.543903i
$890$		0			0
$891$	1122.75	−	1122.75i	1.26011	−	1.26011i
$892$		0			0
$893$	−534.563	−	534.563i	−0.598615	−	0.598615i
$894$		0			0
$895$		−	55.6603i		−	0.0621903i
$896$		0			0
$897$	−42.6210			−0.0475151
$898$		0			0
$899$	893.978	−	893.978i	0.994414	−	0.994414i
$900$		0			0
$901$	−126.376	−	126.376i	−0.140262	−	0.140262i
$902$		0			0
$903$	−630.749	−	699.352i	−0.698504	−	0.774476i
$904$		0			0
$905$			150.314i			0.166093i
$906$		0			0
$907$	−1074.26	+	1074.26i	−1.18441	+	1.18441i	−0.205817	+	0.978591i	$0.565985\pi$
$907$	−1074.26	+	1074.26i	−1.18441	+	1.18441i	−0.978591	+	0.205817i	$0.934015\pi$
$908$		0			0
$909$	−282.938	+	282.938i	−0.311263	+	0.311263i
$910$		0			0
$911$	−675.939			−0.741975			−0.370988	−	0.928638i	$-0.620981\pi$
$911$	−675.939			−0.741975			−0.370988	+	0.928638i	$0.620981\pi$
$912$		0			0
$913$			1514.84i			1.65919i
$914$		0			0
$915$	1380.81	−	1380.81i	1.50909	−	1.50909i
$916$		0			0
$917$	−51.7561	+	1003.48i	−0.0564407	+	1.09430i
$918$		0			0
$919$			545.667i			0.593762i	0.954915	+	0.296881i	$0.0959464\pi$
$919$			545.667i			0.593762i	−0.954915	+	0.296881i	$0.904054\pi$
$920$		0			0
$921$		−	316.180i		−	0.343301i
$922$		0			0
$923$	−1.09968	−	1.09968i	−0.00119142	−	0.00119142i
$924$		0			0
$925$	124.650	+	124.650i	0.134756	+	0.134756i
$926$		0			0
$927$			419.462i			0.452494i
$928$		0			0
$929$		−	803.498i		−	0.864907i	−0.901656	−	0.432453i	$-0.857648\pi$
$929$		−	803.498i		−	0.864907i	0.901656	−	0.432453i	$-0.142352\pi$
$930$		0			0
$931$	−707.170	−	870.329i	−0.759581	−	0.934832i
$932$		0			0
$933$	609.862	−	609.862i	0.653657	−	0.653657i
$934$		0			0
$935$			1490.26i			1.59386i
$936$		0			0
$937$	−1778.02			−1.89757			−0.948784	−	0.315927i	$-0.897685\pi$
$937$	−1778.02			−1.89757			−0.948784	+	0.315927i	$0.897685\pi$
$938$		0			0
$939$	1217.46	−	1217.46i	1.29655	−	1.29655i
$940$		0			0
$941$	−302.133	+	302.133i	−0.321076	+	0.321076i	−0.849180	−	0.528104i	$-0.822903\pi$
$941$	−302.133	+	302.133i	−0.321076	+	0.321076i	0.528104	+	0.849180i	$0.322903\pi$
$942$		0			0
$943$			504.723i			0.535231i
$944$		0			0
$945$	−697.700	−	773.584i	−0.738307	−	0.818607i
$946$		0			0
$947$	361.945	+	361.945i	0.382202	+	0.382202i	0.871895	−	0.489693i	$-0.162891\pi$
$947$	361.945	+	361.945i	0.382202	+	0.382202i	−0.489693	+	0.871895i	$0.662891\pi$
$948$		0			0
$949$	17.4056	−	17.4056i	0.0183410	−	0.0183410i
$950$		0			0
$951$	−526.579			−0.553711
$952$		0			0
$953$			709.369i			0.744354i	0.928162	+	0.372177i	$0.121389\pi$
$953$			709.369i			0.744354i	−0.928162	+	0.372177i	$0.878611\pi$
$954$		0			0
$955$	315.405	+	315.405i	0.330267	+	0.330267i
$956$		0			0
$957$	1294.28	−	1294.28i	1.35243	−	1.35243i
$958$		0			0
$959$	1093.97	+	1212.95i	1.14074	+	1.26481i
$960$		0			0
$961$	−504.832			−0.525320
$962$		0			0
$963$	198.304	+	198.304i	0.205923	+	0.205923i
$964$		0			0
$965$	−1033.69	−	1033.69i	−1.07118	−	1.07118i
$966$		0			0
$967$		−	78.8115i		−	0.0815011i	−0.999169	−	0.0407505i	$-0.987025\pi$
$967$		−	78.8115i		−	0.0815011i	0.999169	−	0.0407505i	$-0.0129749\pi$
$968$		0			0
$969$	1041.01			1.07431
$970$		0			0
$971$	354.033	+	354.033i	0.364606	+	0.364606i	0.865506	−	0.500899i	$-0.166997\pi$
$971$	354.033	+	354.033i	0.364606	+	0.364606i	−0.500899	+	0.865506i	$0.666997\pi$
$972$		0			0
$973$	622.077	+	32.0848i	0.639340	+	0.0329751i
$974$		0			0
$975$	−26.1850			−0.0268564
$976$		0			0
$977$	−1297.64			−1.32819			−0.664094	−	0.747649i	$-0.731182\pi$
$977$	−1297.64			−1.32819			−0.664094	+	0.747649i	$0.731182\pi$
$978$		0			0
$979$	178.474	−	178.474i	0.182303	−	0.182303i
$980$		0			0
$981$	15.6775	−	15.6775i	0.0159811	−	0.0159811i
$982$		0			0
$983$	523.198			0.532246			0.266123	−	0.963939i	$-0.414257\pi$
$983$	523.198			0.532246			0.266123	+	0.963939i	$0.414257\pi$
$984$		0			0
$985$	173.635			0.176279
$986$		0			0
$987$	791.345	+	40.8151i	0.801768	+	0.0413527i
$988$		0			0
$989$	−1047.40	−	1047.40i	−1.05905	−	1.05905i
$990$		0			0
$991$	−917.642			−0.925976			−0.462988	−	0.886365i	$-0.653223\pi$
$991$	−917.642			−0.925976			−0.462988	+	0.886365i	$0.653223\pi$
$992$		0			0
$993$			656.550i			0.661178i
$994$		0			0
$995$	−1476.87	−	1476.87i	−1.48429	−	1.48429i
$996$		0			0
$997$	−129.558	−	129.558i	−0.129948	−	0.129948i	0.639141	−	0.769089i	$-0.279290\pi$
$997$	−129.558	−	129.558i	−0.129948	−	0.129948i	−0.769089	+	0.639141i	$0.779290\pi$
$998$		0			0
$999$	163.053			0.163216

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.3.l.b.209.23		56
4.3	odd	2		112.3.l.b.13.3	✓	56
7.6	odd	2	inner	448.3.l.b.209.6		56
16.5	even	4	inner	448.3.l.b.433.6		56
16.11	odd	4		112.3.l.b.69.4	yes	56
28.27	even	2		112.3.l.b.13.4	yes	56
112.27	even	4		112.3.l.b.69.3	yes	56
112.69	odd	4	inner	448.3.l.b.433.23		56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.3.l.b.13.3	✓	56	4.3	odd	2
112.3.l.b.13.4	yes	56	28.27	even	2
112.3.l.b.69.3	yes	56	112.27	even	4
112.3.l.b.69.4	yes	56	16.11	odd	4
448.3.l.b.209.6		56	7.6	odd	2	inner
448.3.l.b.209.23		56	1.1	even	1	trivial
448.3.l.b.433.6		56	16.5	even	4	inner
448.3.l.b.433.23		56	112.69	odd	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(2.42317 - 2.42317i) q^{3} +(4.90814 + 4.90814i) q^{5} +(-5.19813 + 4.68822i) q^{7} -2.74353i q^{9} +(11.4374 - 11.4374i) q^{11} +(-0.233094 + 0.233094i) q^{13} +23.7865 q^{15} +13.2735i q^{17} +(16.1828 - 16.1828i) q^{19} +(-1.23559 + 23.9563i) q^{21} +37.7293i q^{23} +23.1796i q^{25} +(15.1605 + 15.1605i) q^{27} +(23.3499 + 23.3499i) q^{29} -38.2862i q^{31} -55.4298i q^{33} +(-48.5236 - 2.50269i) q^{35} +(5.37755 - 5.37755i) q^{37} +1.12965i q^{39} +13.3775 q^{41} +(-27.7609 + 27.7609i) q^{43} +(13.4656 - 13.4656i) q^{45} -33.0328i q^{47} +(5.04112 - 48.7400i) q^{49} +(32.1640 + 32.1640i) q^{51} +(-9.52094 + 9.52094i) q^{53} +112.273 q^{55} -78.4274i q^{57} +(-46.0839 - 46.0839i) q^{59} +(58.0502 - 58.0502i) q^{61} +(12.8623 + 14.2612i) q^{63} -2.28811 q^{65} +(-30.9706 - 30.9706i) q^{67} +(91.4246 + 91.4246i) q^{69} +4.71778i q^{71} -74.6723 q^{73} +(56.1683 + 56.1683i) q^{75} +(-5.83203 + 113.075i) q^{77} -32.5841 q^{79} +98.1648 q^{81} +(-66.2230 + 66.2230i) q^{83} +(-65.1481 + 65.1481i) q^{85} +113.162 q^{87} +15.6044 q^{89} +(0.118856 - 2.30445i) q^{91} +(-92.7740 - 92.7740i) q^{93} +158.855 q^{95} -4.30378i q^{97} +(-31.3790 - 31.3790i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 8 q^{15} - 20 q^{21} - 96 q^{29} + 100 q^{35} - 128 q^{37} + 72 q^{43} + 192 q^{49} + 128 q^{51} + 88 q^{53} - 444 q^{63} - 8 q^{65} - 440 q^{67} + 12 q^{77} + 8 q^{79} + 64 q^{81} + 96 q^{85} + 388 q^{91}+ \cdots - 8 q^{99}+O(q^{100}) \) 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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