LMFDB - Embedded newform 448.4.j.b.335.21

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,4,Mod(111,448)]

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([2, 3, 2]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("448.111");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	448.j (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:	$26.4328556826$
Analytic rank:	$0$
Dimension:	$88$
Relative dimension:	$44$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			335.21
Character	$\chi$	$=$	448.335
Dual form			448.4.j.b.111.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.342891 + 0.342891i) q^{3} +(10.0615 - 10.0615i) q^{5} +(11.3901 + 14.6036i) q^{7} +26.7649i q^{9} +O(q^{10})$	\(q+(-0.342891 + 0.342891i) q^{3} +(10.0615 - 10.0615i) q^{5} +(11.3901 + 14.6036i) q^{7} +26.7649i q^{9} +(-23.5470 + 23.5470i) q^{11} +(-27.7952 - 27.7952i) q^{13} +6.90000i q^{15} +120.290i q^{17} +(-7.21660 + 7.21660i) q^{19} +(-8.91302 - 1.10188i) q^{21} -50.4833 q^{23} -77.4676i q^{25} +(-18.4355 - 18.4355i) q^{27} +(-108.139 + 108.139i) q^{29} +129.020 q^{31} -16.1481i q^{33} +(261.536 + 32.3325i) q^{35} +(-219.446 - 219.446i) q^{37} +19.0614 q^{39} -109.587 q^{41} +(-35.8480 + 35.8480i) q^{43} +(269.295 + 269.295i) q^{45} +566.754 q^{47} +(-83.5304 + 332.674i) q^{49} +(-41.2464 - 41.2464i) q^{51} +(212.984 + 212.984i) q^{53} +473.837i q^{55} -4.94901i q^{57} +(166.740 + 166.740i) q^{59} +(474.293 + 474.293i) q^{61} +(-390.863 + 304.855i) q^{63} -559.323 q^{65} +(6.09734 + 6.09734i) q^{67} +(17.3103 - 17.3103i) q^{69} -992.670 q^{71} -504.285 q^{73} +(26.5630 + 26.5630i) q^{75} +(-612.074 - 75.6680i) q^{77} +777.732i q^{79} -710.008 q^{81} +(-283.766 + 283.766i) q^{83} +(1210.30 + 1210.30i) q^{85} -74.1596i q^{87} +784.187 q^{89} +(89.3194 - 722.500i) q^{91} +(-44.2397 + 44.2397i) q^{93} +145.220i q^{95} +189.493i q^{97} +(-630.232 - 630.232i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 88 q + 4 q^{7}+O(q^{10}) $ 88 * q + 4 * q^7	$ 88 q + 4 q^{7} - 112 q^{11} + 52 q^{21} - 160 q^{23} + 528 q^{29} - 476 q^{35} - 896 q^{37} + 8 q^{39} - 40 q^{43} - 1376 q^{49} - 1504 q^{51} - 1560 q^{53} - 8 q^{65} - 648 q^{67} + 456 q^{71} + 292 q^{77} - 1952 q^{81} + 496 q^{85} + 1964 q^{91} - 112 q^{93} - 7592 q^{99}+O(q^{100}) $ 88 * q + 4 * q^7 - 112 * q^11 + 52 * q^21 - 160 * q^23 + 528 * q^29 - 476 * q^35 - 896 * q^37 + 8 * q^39 - 40 * q^43 - 1376 * q^49 - 1504 * q^51 - 1560 * q^53 - 8 * q^65 - 648 * q^67 + 456 * q^71 + 292 * q^77 - 1952 * q^81 + 496 * q^85 + 1964 * q^91 - 112 * q^93 - 7592 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−0.342891	+	0.342891i	−0.0659894	+	0.0659894i	−0.739331	−	0.673342i	$-0.764858\pi$
$3$	−0.342891	+	0.342891i	−0.0659894	+	0.0659894i	0.673342	+	0.739331i	$0.264858\pi$
$4$		0			0
$5$	10.0615	−	10.0615i	0.899928	−	0.899928i	−0.0955012	−	0.995429i	$-0.530445\pi$
$5$	10.0615	−	10.0615i	0.899928	−	0.899928i	0.995429	+	0.0955012i	$0.0304454\pi$
$6$		0			0
$7$	11.3901	+	14.6036i	0.615009	+	0.788520i
$7$	11.3901	+	14.6036i	0.615009	+	0.788520i
$8$		0			0
$9$			26.7649i			0.991291i
$10$		0			0
$11$	−23.5470	+	23.5470i	−0.645426	+	0.645426i	−0.951884	−	0.306458i	$-0.900856\pi$
$11$	−23.5470	+	23.5470i	−0.645426	+	0.645426i	0.306458	+	0.951884i	$0.400856\pi$
$12$		0			0
$13$	−27.7952	−	27.7952i	−0.593000	−	0.593000i	0.345441	−	0.938441i	$-0.387729\pi$
$13$	−27.7952	−	27.7952i	−0.593000	−	0.593000i	−0.938441	+	0.345441i	$0.887729\pi$
$14$		0			0
$15$			6.90000i			0.118771i
$16$		0			0
$17$			120.290i			1.71615i	0.513522	+	0.858077i	$0.328341\pi$
$17$			120.290i			1.71615i	−0.513522	+	0.858077i	$0.671659\pi$
$18$		0			0
$19$	−7.21660	+	7.21660i	−0.0871369	+	0.0871369i	−0.749332	−	0.662195i	$-0.769625\pi$
$19$	−7.21660	+	7.21660i	−0.0871369	+	0.0871369i	0.662195	+	0.749332i	$0.269625\pi$
$20$		0			0
$21$	−8.91302	−	1.10188i	−0.0926181	−	0.0114499i
$22$		0			0
$23$	−50.4833			−0.457674			−0.228837	−	0.973465i	$-0.573492\pi$
$23$	−50.4833			−0.457674			−0.228837	+	0.973465i	$0.573492\pi$
$24$		0			0
$25$		−	77.4676i		−	0.619741i
$26$		0			0
$27$	−18.4355	−	18.4355i	−0.131404	−	0.131404i
$28$		0			0
$29$	−108.139	+	108.139i	−0.692443	+	0.692443i	−0.962769	−	0.270326i	$-0.912869\pi$
$29$	−108.139	+	108.139i	−0.692443	+	0.692443i	0.270326	+	0.962769i	$0.412869\pi$
$30$		0			0
$31$	129.020			0.747503			0.373752	−	0.927529i	$-0.378071\pi$
$31$	129.020			0.747503			0.373752	+	0.927529i	$0.378071\pi$
$32$		0			0
$33$		−	16.1481i		−	0.0851826i
$34$		0			0
$35$	261.536	+	32.3325i	1.26308	+	0.156148i
$36$		0			0
$37$	−219.446	−	219.446i	−0.975046	−	0.975046i	0.0246503	−	0.999696i	$-0.492153\pi$
$37$	−219.446	−	219.446i	−0.975046	−	0.975046i	−0.999696	+	0.0246503i	$0.992153\pi$
$38$		0			0
$39$	19.0614			0.0782634
$40$		0			0
$41$	−109.587			−0.417430			−0.208715	−	0.977977i	$-0.566928\pi$
$41$	−109.587			−0.417430			−0.208715	+	0.977977i	$0.566928\pi$
$42$		0			0
$43$	−35.8480	+	35.8480i	−0.127134	+	0.127134i	−0.767811	−	0.640677i	$-0.778654\pi$
$43$	−35.8480	+	35.8480i	−0.127134	+	0.127134i	0.640677	+	0.767811i	$0.278654\pi$
$44$		0			0
$45$	269.295	+	269.295i	0.892090	+	0.892090i
$46$		0			0
$47$	566.754			1.75893			0.879464	−	0.475965i	$-0.157901\pi$
$47$	566.754			1.75893			0.879464	+	0.475965i	$0.157901\pi$
$48$		0			0
$49$	−83.5304	+	332.674i	−0.243529	+	0.969894i
$50$		0			0
$51$	−41.2464	−	41.2464i	−0.113248	−	0.113248i
$52$		0			0
$53$	212.984	+	212.984i	0.551991	+	0.551991i	0.927015	−	0.375024i	$-0.122366\pi$
$53$	212.984	+	212.984i	0.551991	+	0.551991i	−0.375024	+	0.927015i	$0.622366\pi$
$54$		0			0
$55$			473.837i			1.16167i
$56$		0			0
$57$		−	4.94901i		−	0.0115002i
$58$		0			0
$59$	166.740	+	166.740i	0.367927	+	0.367927i	0.866721	−	0.498794i	$-0.166223\pi$
$59$	166.740	+	166.740i	0.367927	+	0.367927i	−0.498794	+	0.866721i	$0.666223\pi$
$60$		0			0
$61$	474.293	+	474.293i	0.995524	+	0.995524i	0.999990	−	0.00446644i	$-0.00142172\pi$
$61$	474.293	+	474.293i	0.995524	+	0.995524i	−0.00446644	+	0.999990i	$0.501422\pi$
$62$		0			0
$63$	−390.863	+	304.855i	−0.781653	+	0.609652i
$64$		0			0
$65$	−559.323			−1.06731
$66$		0			0
$67$	6.09734	+	6.09734i	0.0111180	+	0.0111180i	0.712644	−	0.701526i	$-0.247498\pi$
$67$	6.09734	+	6.09734i	0.0111180	+	0.0111180i	−0.701526	+	0.712644i	$0.747498\pi$
$68$		0			0
$69$	17.3103	−	17.3103i	0.0302016	−	0.0302016i
$70$		0			0
$71$	−992.670			−1.65927			−0.829636	−	0.558305i	$-0.811452\pi$
$71$	−992.670			−1.65927			−0.829636	+	0.558305i	$0.811452\pi$
$72$		0			0
$73$	−504.285			−0.808522			−0.404261	−	0.914644i	$-0.632471\pi$
$73$	−504.285			−0.808522			−0.404261	+	0.914644i	$0.632471\pi$
$74$		0			0
$75$	26.5630	+	26.5630i	0.0408964	+	0.0408964i
$76$		0			0
$77$	−612.074	−	75.6680i	−0.905875	−	0.111989i
$78$		0			0
$79$			777.732i			1.10762i	0.832644	+	0.553808i	$0.186826\pi$
$79$			777.732i			1.10762i	−0.832644	+	0.553808i	$0.813174\pi$
$80$		0			0
$81$	−710.008			−0.973948
$82$		0			0
$83$	−283.766	+	283.766i	−0.375270	+	0.375270i	−0.869392	−	0.494122i	$-0.835489\pi$
$83$	−283.766	+	283.766i	−0.375270	+	0.375270i	0.494122	+	0.869392i	$0.335489\pi$
$84$		0			0
$85$	1210.30	+	1210.30i	1.54441	+	1.54441i
$86$		0			0
$87$		−	74.1596i		−	0.0913879i
$88$		0			0
$89$	784.187			0.933974			0.466987	−	0.884264i	$-0.345340\pi$
$89$	784.187			0.933974			0.466987	+	0.884264i	$0.345340\pi$
$90$		0			0
$91$	89.3194	−	722.500i	0.102893	−	0.832292i
$92$		0			0
$93$	−44.2397	+	44.2397i	−0.0493273	+	0.0493273i
$94$		0			0
$95$			145.220i			0.156834i
$96$		0			0
$97$			189.493i			0.198352i	0.995070	+	0.0991758i	$0.0316206\pi$
$97$			189.493i			0.198352i	−0.995070	+	0.0991758i	$0.968379\pi$
$98$		0			0
$99$	−630.232	−	630.232i	−0.639805	−	0.639805i
$100$		0			0
$101$	1217.41	−	1217.41i	1.19937	−	1.19937i	0.225015	−	0.974355i	$-0.427757\pi$
$101$	1217.41	−	1217.41i	1.19937	−	1.19937i	0.974355	−	0.225015i	$-0.0722433\pi$
$102$		0			0
$103$			1749.98i			1.67408i	0.547141	+	0.837041i	$0.315716\pi$
$103$			1749.98i			1.67408i	−0.547141	+	0.837041i	$0.684284\pi$
$104$		0			0
$105$	−100.765	+	78.5918i	−0.0936537	+	0.0730455i
$106$		0			0
$107$	262.668	−	262.668i	0.237318	−	0.237318i	−0.578421	−	0.815739i	$-0.696331\pi$
$107$	262.668	−	262.668i	0.237318	−	0.237318i	0.815739	+	0.578421i	$0.196331\pi$
$108$		0			0
$109$	−50.7414	+	50.7414i	−0.0445885	+	0.0445885i	−0.729049	−	0.684461i	$-0.760038\pi$
$109$	−50.7414	+	50.7414i	−0.0445885	+	0.0445885i	0.684461	+	0.729049i	$0.260038\pi$
$110$		0			0
$111$	150.492			0.128685
$112$		0			0
$113$	1202.25			1.00087			0.500435	−	0.865774i	$-0.333173\pi$
$113$	1202.25			1.00087			0.500435	+	0.865774i	$0.333173\pi$
$114$		0			0
$115$	−507.938	+	507.938i	−0.411874	+	0.411874i
$116$		0			0
$117$	743.934	−	743.934i	0.587835	−	0.587835i
$118$		0			0
$119$	−1756.67	+	1370.12i	−1.35322	+	1.05545i
$120$		0			0
$121$			222.077i			0.166849i
$122$		0			0
$123$	37.5764	−	37.5764i	0.0275460	−	0.0275460i
$124$		0			0
$125$	478.247	+	478.247i	0.342206	+	0.342206i
$126$		0			0
$127$		−	1068.61i		−	0.746645i	−0.927702	−	0.373323i	$-0.878218\pi$
$127$		−	1068.61i		−	0.746645i	0.927702	−	0.373323i	$-0.121782\pi$
$128$		0			0
$129$		−	24.5839i		−	0.0167790i
$130$		0			0
$131$	1533.19	−	1533.19i	1.02256	−	1.02256i	0.0228184	−	0.999740i	$-0.492736\pi$
$131$	1533.19	−	1533.19i	1.02256	−	1.02256i	0.999740	−	0.0228184i	$-0.00726395\pi$
$132$		0			0
$133$	−187.586	−	23.1904i	−0.122299	−	0.0151193i
$134$		0			0
$135$	−370.977			−0.236509
$136$		0			0
$137$		−	2743.91i		−	1.71115i	−0.517677	−	0.855576i	$-0.673203\pi$
$137$		−	2743.91i		−	1.71115i	0.517677	−	0.855576i	$-0.326797\pi$
$138$		0			0
$139$	−1737.73	−	1737.73i	−1.06037	−	1.06037i	−0.998056	−	0.0623173i	$-0.980151\pi$
$139$	−1737.73	−	1737.73i	−1.06037	−	1.06037i	−0.0623173	−	0.998056i	$-0.519849\pi$
$140$		0			0
$141$	−194.335	+	194.335i	−0.116071	+	0.116071i
$142$		0			0
$143$	1308.99			0.765475
$144$		0			0
$145$			2176.08i			1.24630i
$146$		0			0
$147$	−85.4289	−	142.713i	−0.0479324	−	0.0800731i
$148$		0			0
$149$	2.90649	+	2.90649i	0.00159805	+	0.00159805i	0.707905	−	0.706307i	$-0.249640\pi$
$149$	2.90649	+	2.90649i	0.00159805	+	0.00159805i	−0.706307	+	0.707905i	$0.749640\pi$
$150$		0			0
$151$	2478.29			1.33563			0.667817	−	0.744325i	$-0.267229\pi$
$151$	2478.29			1.33563			0.667817	+	0.744325i	$0.267229\pi$
$152$		0			0
$153$	−3219.54			−1.70121
$154$		0			0
$155$	1298.13	−	1298.13i	0.672699	−	0.672699i
$156$		0			0
$157$	1224.66	+	1224.66i	0.622541	+	0.622541i	0.946180	−	0.323640i	$-0.104907\pi$
$157$	1224.66	+	1224.66i	0.622541	+	0.622541i	−0.323640	+	0.946180i	$0.604907\pi$
$158$		0			0
$159$	−146.060			−0.0728512
$160$		0			0
$161$	−575.011	−	737.238i	−0.281473	−	0.360885i
$162$		0			0
$163$	1573.82	+	1573.82i	0.756265	+	0.756265i	0.975640	−	0.219376i	$-0.0704020\pi$
$163$	1573.82	+	1573.82i	0.756265	+	0.756265i	−0.219376	+	0.975640i	$0.570402\pi$
$164$		0			0
$165$	−162.474	−	162.474i	−0.0766582	−	0.0766582i
$166$		0			0
$167$		−	1951.37i		−	0.904203i	−0.891967	−	0.452101i	$-0.850674\pi$
$167$		−	1951.37i		−	0.904203i	0.891967	−	0.452101i	$-0.149326\pi$
$168$		0			0
$169$		−	651.855i		−	0.296703i
$170$		0			0
$171$	−193.151	−	193.151i	−0.0863780	−	0.0863780i
$172$		0			0
$173$	−3013.12	−	3013.12i	−1.32418	−	1.32418i	−0.910359	−	0.413820i	$-0.864194\pi$
$173$	−3013.12	−	3013.12i	−1.32418	−	1.32418i	−0.413820	−	0.910359i	$-0.635806\pi$
$174$		0			0
$175$	1131.31	−	882.366i	0.488679	−	0.381146i
$176$		0			0
$177$	−114.347			−0.0485586
$178$		0			0
$179$	2593.46	+	2593.46i	1.08293	+	1.08293i	0.996235	+	0.0866958i	$0.0276308\pi$
$179$	2593.46	+	2593.46i	1.08293	+	1.08293i	0.0866958	+	0.996235i	$0.472369\pi$
$180$		0			0
$181$	859.073	−	859.073i	0.352787	−	0.352787i	−0.508359	−	0.861145i	$-0.669748\pi$
$181$	859.073	−	859.073i	0.352787	−	0.352787i	0.861145	+	0.508359i	$0.169748\pi$
$182$		0			0
$183$	−325.261			−0.131388
$184$		0			0
$185$	−4415.91			−1.75494
$186$		0			0
$187$	−2832.47	−	2832.47i	−1.10765	−	1.10765i
$188$		0			0
$189$	59.2422	−	479.207i	0.0228002	−	0.184429i
$190$		0			0
$191$			2917.88i			1.10540i	0.833382	+	0.552698i	$0.186402\pi$
$191$			2917.88i			1.10540i	−0.833382	+	0.552698i	$0.813598\pi$
$192$		0			0
$193$	−802.630			−0.299350			−0.149675	−	0.988735i	$-0.547823\pi$
$193$	−802.630			−0.299350			−0.149675	+	0.988735i	$0.547823\pi$
$194$		0			0
$195$	191.787	−	191.787i	0.0704315	−	0.0704315i
$196$		0			0
$197$	621.538	+	621.538i	0.224785	+	0.224785i	0.810510	−	0.585725i	$-0.199190\pi$
$197$	621.538	+	621.538i	0.224785	+	0.224785i	−0.585725	+	0.810510i	$0.699190\pi$
$198$		0			0
$199$		−	1991.53i		−	0.709425i	−0.934975	−	0.354713i	$-0.884579\pi$
$199$		−	1991.53i		−	0.709425i	0.934975	−	0.354713i	$-0.115421\pi$
$200$		0			0
$201$	−4.18145			−0.00146735
$202$		0			0
$203$	−2810.93	−	347.502i	−0.971864	−	0.120147i
$204$		0			0
$205$	−1102.61	+	1102.61i	−0.375657	+	0.375657i
$206$		0			0
$207$		−	1351.18i		−	0.453688i
$208$		0			0
$209$		−	339.859i		−	0.112481i
$210$		0			0
$211$	−2921.65	−	2921.65i	−0.953246	−	0.953246i	0.0457089	−	0.998955i	$-0.485445\pi$
$211$	−2921.65	−	2921.65i	−0.953246	−	0.953246i	−0.998955	+	0.0457089i	$0.985445\pi$
$212$		0			0
$213$	340.378	−	340.378i	0.109494	−	0.109494i
$214$		0			0
$215$			721.369i			0.228823i
$216$		0			0
$217$	1469.55	+	1884.15i	0.459721	+	0.589422i
$218$		0			0
$219$	172.915	−	172.915i	0.0533539	−	0.0533539i
$220$		0			0
$221$	3343.48	−	3343.48i	1.01768	−	1.01768i
$222$		0			0
$223$	−214.075			−0.0642850			−0.0321425	−	0.999483i	$-0.510233\pi$
$223$	−214.075			−0.0642850			−0.0321425	+	0.999483i	$0.510233\pi$
$224$		0			0
$225$	2073.41			0.614344
$226$		0			0
$227$	−1977.97	+	1977.97i	−0.578336	+	0.578336i	−0.934445	−	0.356108i	$-0.884103\pi$
$227$	−1977.97	+	1977.97i	−0.578336	+	0.578336i	0.356108	+	0.934445i	$0.384103\pi$
$228$		0			0
$229$	−1231.97	+	1231.97i	−0.355506	+	0.355506i	−0.862153	−	0.506647i	$-0.830885\pi$
$229$	−1231.97	+	1231.97i	−0.355506	+	0.355506i	0.506647	+	0.862153i	$0.330885\pi$
$230$		0			0
$231$	235.821	−	183.929i	0.0671682	−	0.0523880i
$232$		0			0
$233$		−	4736.44i		−	1.33174i	−0.746070	−	0.665868i	$-0.768061\pi$
$233$		−	4736.44i		−	1.33174i	0.746070	−	0.665868i	$-0.231939\pi$
$234$		0			0
$235$	5702.40	−	5702.40i	1.58291	−	1.58291i
$236$		0			0
$237$	−266.678	−	266.678i	−0.0730910	−	0.0730910i
$238$		0			0
$239$			1373.12i			0.371630i	0.982585	+	0.185815i	$0.0594926\pi$
$239$			1373.12i			0.371630i	−0.982585	+	0.185815i	$0.940507\pi$
$240$		0			0
$241$		−	1877.13i		−	0.501728i	−0.968022	−	0.250864i	$-0.919285\pi$
$241$		−	1877.13i		−	0.501728i	0.968022	−	0.250864i	$-0.0807148\pi$
$242$		0			0
$243$	741.214	−	741.214i	0.195674	−	0.195674i
$244$		0			0
$245$	2506.75	+	4187.64i	0.653676	+	1.09199i
$246$		0			0
$247$	401.173			0.103344
$248$		0			0
$249$		−	194.602i		−	0.0495277i
$250$		0			0
$251$	4593.31	+	4593.31i	1.15509	+	1.15509i	0.985518	+	0.169571i	$0.0542381\pi$
$251$	4593.31	+	4593.31i	1.15509	+	1.15509i	0.169571	+	0.985518i	$0.445762\pi$
$252$		0			0
$253$	1188.73	−	1188.73i	0.295395	−	0.295395i
$254$		0			0
$255$	−830.001			−0.203830
$256$		0			0
$257$		−	2757.30i		−	0.669244i	−0.942352	−	0.334622i	$-0.891391\pi$
$257$		−	2757.30i		−	0.669244i	0.942352	−	0.334622i	$-0.108609\pi$
$258$		0			0
$259$	705.186	−	5704.22i	0.169182	−	1.36851i
$260$		0			0
$261$	−2894.32	−	2894.32i	−0.686413	−	0.686413i
$262$		0			0
$263$	−1406.10			−0.329672			−0.164836	−	0.986321i	$-0.552710\pi$
$263$	−1406.10			−0.329672			−0.164836	+	0.986321i	$0.552710\pi$
$264$		0			0
$265$	4285.87			0.993505
$266$		0			0
$267$	−268.891	+	268.891i	−0.0616324	+	0.0616324i
$268$		0			0
$269$	864.430	+	864.430i	0.195930	+	0.195930i	0.798253	−	0.602323i	$-0.205758\pi$
$269$	864.430	+	864.430i	0.195930	+	0.195930i	−0.602323	+	0.798253i	$0.705758\pi$
$270$		0			0
$271$	−6018.21			−1.34901			−0.674503	−	0.738272i	$-0.735642\pi$
$271$	−6018.21			−1.34901			−0.674503	+	0.738272i	$0.735642\pi$
$272$		0			0
$273$	217.112	+	278.366i	0.0481327	+	0.0617123i
$274$		0			0
$275$	1824.13	+	1824.13i	0.399997	+	0.399997i
$276$		0			0
$277$	−68.6407	−	68.6407i	−0.0148889	−	0.0148889i	0.699623	−	0.714512i	$-0.253351\pi$
$277$	−68.6407	−	68.6407i	−0.0148889	−	0.0148889i	−0.714512	+	0.699623i	$0.753351\pi$
$278$		0			0
$279$			3453.19i			0.740993i
$280$		0			0
$281$			1826.33i			0.387722i	0.981029	+	0.193861i	$0.0621010\pi$
$281$			1826.33i			0.387722i	−0.981029	+	0.193861i	$0.937899\pi$
$282$		0			0
$283$	−1116.25	−	1116.25i	−0.234467	−	0.234467i	0.580087	−	0.814554i	$-0.303018\pi$
$283$	−1116.25	−	1116.25i	−0.234467	−	0.234467i	−0.814554	+	0.580087i	$0.803018\pi$
$284$		0			0
$285$	−49.7945	−	49.7945i	−0.0103494	−	0.0103494i
$286$		0			0
$287$	−1248.21	−	1600.37i	−0.256723	−	0.329152i
$288$		0			0
$289$	−9556.68			−1.94518
$290$		0			0
$291$	−64.9755	−	64.9755i	−0.0130891	−	0.0130891i
$292$		0			0
$293$	2108.76	−	2108.76i	0.420462	−	0.420462i	−0.464901	−	0.885363i	$-0.653910\pi$
$293$	2108.76	−	2108.76i	0.420462	−	0.420462i	0.885363	+	0.464901i	$0.153910\pi$
$294$		0			0
$295$	3355.31			0.662216
$296$		0			0
$297$	868.201			0.169623
$298$		0			0
$299$	1403.19	+	1403.19i	0.271400	+	0.271400i
$300$		0			0
$301$	−931.822	−	115.197i	−0.178436	−	0.0220593i
$302$		0			0
$303$			834.875i			0.158292i
$304$		0			0
$305$	9544.19			1.79180
$306$		0			0
$307$	455.427	−	455.427i	0.0846665	−	0.0846665i	−0.663505	−	0.748172i	$-0.730932\pi$
$307$	455.427	−	455.427i	0.0846665	−	0.0846665i	0.748172	+	0.663505i	$0.230932\pi$
$308$		0			0
$309$	−600.052	−	600.052i	−0.110472	−	0.110472i
$310$		0			0
$311$		−	6873.90i		−	1.25332i	−0.779292	−	0.626661i	$-0.784421\pi$
$311$		−	6873.90i		−	1.25332i	0.779292	−	0.626661i	$-0.215579\pi$
$312$		0			0
$313$	−3305.64			−0.596952			−0.298476	−	0.954417i	$-0.596478\pi$
$313$	−3305.64			−0.596952			−0.298476	+	0.954417i	$0.596478\pi$
$314$		0			0
$315$	−865.374	+	6999.97i	−0.154788	+	1.25207i
$316$		0			0
$317$	−6336.25	+	6336.25i	−1.12265	+	1.12265i	−0.131306	+	0.991342i	$0.541917\pi$
$317$	−6336.25	+	6336.25i	−1.12265	+	1.12265i	−0.991342	+	0.131306i	$0.958083\pi$
$318$		0			0
$319$		−	5092.69i		−	0.893842i
$320$		0			0
$321$			180.133i			0.0313210i
$322$		0			0
$323$	−868.084	−	868.084i	−0.149540	−	0.149540i
$324$		0			0
$325$	−2153.23	+	2153.23i	−0.367506	+	0.367506i
$326$		0			0
$327$		−	34.7975i		−	0.00588473i
$328$		0			0
$329$	6455.40	+	8276.66i	1.08176	+	1.38695i
$330$		0			0
$331$	−6859.04	+	6859.04i	−1.13899	+	1.13899i	−0.150363	+	0.988631i	$0.548044\pi$
$331$	−6859.04	+	6859.04i	−1.13899	+	1.13899i	−0.988631	+	0.150363i	$0.951956\pi$
$332$		0			0
$333$	5873.44	−	5873.44i	0.966554	−	0.966554i
$334$		0			0
$335$	122.697			0.0200109
$336$		0			0
$337$	5072.19			0.819881			0.409941	−	0.912112i	$-0.365549\pi$
$337$	5072.19			0.819881			0.409941	+	0.912112i	$0.365549\pi$
$338$		0			0
$339$	−412.242	+	412.242i	−0.0660469	+	0.0660469i
$340$		0			0
$341$	−3038.03	+	3038.03i	−0.482458	+	0.482458i
$342$		0			0
$343$	−5809.65	+	2569.35i	−0.914553	+	0.404465i
$344$		0			0
$345$		−	348.335i		−	0.0543586i
$346$		0			0
$347$	−2075.94	+	2075.94i	−0.321159	+	0.321159i	−0.849212	−	0.528053i	$-0.822922\pi$
$347$	−2075.94	+	2075.94i	−0.321159	+	0.321159i	0.528053	+	0.849212i	$0.322922\pi$
$348$		0			0
$349$	251.547	+	251.547i	0.0385816	+	0.0385816i	0.726134	−	0.687553i	$-0.241315\pi$
$349$	251.547	+	251.547i	0.0385816	+	0.0385816i	−0.687553	+	0.726134i	$0.741315\pi$
$350$		0			0
$351$			1024.84i			0.155845i
$352$		0			0
$353$			1260.83i			0.190106i	0.995472	+	0.0950529i	$0.0303020\pi$
$353$			1260.83i			0.190106i	−0.995472	+	0.0950529i	$0.969698\pi$
$354$		0			0
$355$	−9987.75	+	9987.75i	−1.49322	+	1.49322i
$356$		0			0
$357$	132.545	−	1072.15i	0.0196499	−	0.158947i
$358$		0			0
$359$	3069.01			0.451186			0.225593	−	0.974222i	$-0.427568\pi$
$359$	3069.01			0.451186			0.225593	+	0.974222i	$0.427568\pi$
$360$		0			0
$361$			6754.84i			0.984814i
$362$		0			0
$363$	−76.1481	−	76.1481i	−0.0110103	−	0.0110103i
$364$		0			0
$365$	−5073.86	+	5073.86i	−0.727611	+	0.727611i
$366$		0			0
$367$	286.940			0.0408123			0.0204062	−	0.999792i	$-0.493504\pi$
$367$	286.940			0.0408123			0.0204062	+	0.999792i	$0.493504\pi$
$368$		0			0
$369$		−	2933.08i		−	0.413794i
$370$		0			0
$371$	−684.419	+	5536.23i	−0.0957770	+	0.774736i
$372$		0			0
$373$	−4100.53	−	4100.53i	−0.569216	−	0.569216i	0.362693	−	0.931909i	$-0.381857\pi$
$373$	−4100.53	−	4100.53i	−0.569216	−	0.569216i	−0.931909	+	0.362693i	$0.881857\pi$
$374$		0			0
$375$	−327.973			−0.0451639
$376$		0			0
$377$	6011.47			0.821237
$378$		0			0
$379$	457.310	−	457.310i	0.0619800	−	0.0619800i	−0.675437	−	0.737417i	$-0.736045\pi$
$379$	457.310	−	457.310i	0.0619800	−	0.0619800i	0.737417	+	0.675437i	$0.236045\pi$
$380$		0			0
$381$	366.417	+	366.417i	0.0492707	+	0.0492707i
$382$		0			0
$383$	12171.1			1.62380			0.811899	−	0.583798i	$-0.198434\pi$
$383$	12171.1			1.62380			0.811899	+	0.583798i	$0.198434\pi$
$384$		0			0
$385$	−6919.72	+	5397.05i	−0.916004	+	0.714440i
$386$		0			0
$387$	−959.466	−	959.466i	−0.126027	−	0.126027i
$388$		0			0
$389$	−618.718	−	618.718i	−0.0806433	−	0.0806433i	0.665635	−	0.746278i	$-0.268161\pi$
$389$	−618.718	−	618.718i	−0.0806433	−	0.0806433i	−0.746278	+	0.665635i	$0.768161\pi$
$390$		0			0
$391$		−	6072.64i		−	0.785438i
$392$		0			0
$393$			1051.43i			0.134956i
$394$		0			0
$395$	7825.16	+	7825.16i	0.996776	+	0.996776i
$396$		0			0
$397$	3326.00	+	3326.00i	0.420471	+	0.420471i	0.885366	−	0.464895i	$-0.153908\pi$
$397$	3326.00	+	3326.00i	0.420471	+	0.420471i	−0.464895	+	0.885366i	$0.653908\pi$
$398$		0			0
$399$	72.2734	−	56.3698i	0.00906816	−	0.00707274i
$400$		0			0
$401$	1177.84			0.146679			0.0733396	−	0.997307i	$-0.476634\pi$
$401$	1177.84			0.146679			0.0733396	+	0.997307i	$0.476634\pi$
$402$		0			0
$403$	−3586.12	−	3586.12i	−0.443269	−	0.443269i
$404$		0			0
$405$	−7143.75	+	7143.75i	−0.876483	+	0.876483i
$406$		0			0
$407$	10334.6			1.25864
$408$		0			0
$409$	8970.23			1.08447			0.542236	−	0.840226i	$-0.317578\pi$
$409$	8970.23			1.08447			0.542236	+	0.840226i	$0.317578\pi$
$410$		0			0
$411$	940.862	+	940.862i	0.112918	+	0.112918i
$412$		0			0
$413$	−535.816	+	4334.19i	−0.0638397	+	0.516396i
$414$		0			0
$415$			5710.23i			0.675432i
$416$		0			0
$417$	1191.70			0.139947
$418$		0			0
$419$	1295.20	−	1295.20i	0.151013	−	0.151013i	−0.627557	−	0.778570i	$-0.715945\pi$
$419$	1295.20	−	1295.20i	0.151013	−	0.151013i	0.778570	+	0.627557i	$0.215945\pi$
$420$		0			0
$421$	−9589.76	−	9589.76i	−1.11016	−	1.11016i	−0.993128	−	0.117030i	$-0.962663\pi$
$421$	−9589.76	−	9589.76i	−1.11016	−	1.11016i	−0.117030	−	0.993128i	$-0.537337\pi$
$422$		0			0
$423$			15169.1i			1.74361i
$424$		0			0
$425$	9318.58			1.06357
$426$		0			0
$427$	−1524.13	+	12328.6i	−0.172735	+	1.39725i
$428$		0			0
$429$	−448.840	+	448.840i	−0.0505133	+	0.0505133i
$430$		0			0
$431$		−	16019.8i		−	1.79037i	−0.445698	−	0.895183i	$-0.647044\pi$
$431$		−	16019.8i		−	1.79037i	0.445698	−	0.895183i	$-0.352956\pi$
$432$		0			0
$433$		−	2932.82i		−	0.325502i	−0.986667	−	0.162751i	$-0.947963\pi$
$433$		−	2932.82i		−	0.325502i	0.986667	−	0.162751i	$-0.0520367\pi$
$434$		0			0
$435$	−746.157	−	746.157i	−0.0822425	−	0.0822425i
$436$		0			0
$437$	364.318	−	364.318i	0.0398803	−	0.0398803i
$438$		0			0
$439$		−	9898.86i		−	1.07619i	−0.842885	−	0.538095i	$-0.819144\pi$
$439$		−	9898.86i		−	1.07619i	0.842885	−	0.538095i	$-0.180856\pi$
$440$		0			0
$441$	−8903.96	−	2235.68i	−0.961447	−	0.241408i
$442$		0			0
$443$	−7430.43	+	7430.43i	−0.796908	+	0.796908i	−0.982607	−	0.185698i	$-0.940545\pi$
$443$	−7430.43	+	7430.43i	−0.796908	+	0.796908i	0.185698	+	0.982607i	$0.440545\pi$
$444$		0			0
$445$	7890.10	−	7890.10i	0.840509	−	0.840509i
$446$		0			0
$447$	−1.99322			−0.000210909
$448$		0			0
$449$	8935.86			0.939219			0.469609	−	0.882874i	$-0.344395\pi$
$449$	8935.86			0.939219			0.469609	+	0.882874i	$0.344395\pi$
$450$		0			0
$451$	2580.45	−	2580.45i	0.269420	−	0.269420i
$452$		0			0
$453$	−849.785	+	849.785i	−0.0881377	+	0.0881377i
$454$		0			0
$455$	−6370.75	−	8168.13i	−0.656407	−	0.841599i
$456$		0			0
$457$			15455.4i			1.58200i	0.611819	+	0.790998i	$0.290438\pi$
$457$			15455.4i			1.58200i	−0.611819	+	0.790998i	$0.709562\pi$
$458$		0			0
$459$	2217.60	−	2217.60i	0.225510	−	0.225510i
$460$		0			0
$461$	2961.04	+	2961.04i	0.299153	+	0.299153i	0.840682	−	0.541529i	$-0.182154\pi$
$461$	2961.04	+	2961.04i	0.299153	+	0.299153i	−0.541529	+	0.840682i	$0.682154\pi$
$462$		0			0
$463$			11030.6i			1.10720i	0.832783	+	0.553600i	$0.186746\pi$
$463$			11030.6i			1.10720i	−0.832783	+	0.553600i	$0.813254\pi$
$464$		0			0
$465$			890.235i			0.0887820i
$466$		0			0
$467$	11720.5	−	11720.5i	1.16137	−	1.16137i	0.177192	−	0.984176i	$-0.443299\pi$
$467$	11720.5	−	11720.5i	1.16137	−	1.16137i	0.984176	−	0.177192i	$-0.0567013\pi$
$468$		0			0
$469$	−19.5937	+	158.492i	−0.00192911	+	0.0156045i
$470$		0			0
$471$	−839.853			−0.0821622
$472$		0			0
$473$		−	1688.23i		−	0.164111i
$474$		0			0
$475$	559.053	+	559.053i	0.0540023	+	0.0540023i
$476$		0			0
$477$	−5700.47	+	5700.47i	−0.547184	+	0.547184i
$478$		0			0
$479$	−7614.60			−0.726346			−0.363173	−	0.931722i	$-0.618307\pi$
$479$	−7614.60			−0.726346			−0.363173	+	0.931722i	$0.618307\pi$
$480$		0			0
$481$			12199.1i			1.15640i
$482$		0			0
$483$	449.959	+	55.6263i	0.0423889	+	0.00524034i
$484$		0			0
$485$	1906.58	+	1906.58i	0.178502	+	0.178502i
$486$		0			0
$487$	10320.2			0.960277			0.480139	−	0.877193i	$-0.340586\pi$
$487$	10320.2			0.960277			0.480139	+	0.877193i	$0.340586\pi$
$488$		0			0
$489$	−1079.30			−0.0998110
$490$		0			0
$491$	4552.94	−	4552.94i	0.418476	−	0.418476i	−0.466202	−	0.884678i	$-0.654378\pi$
$491$	4552.94	−	4552.94i	0.418476	−	0.418476i	0.884678	+	0.466202i	$0.154378\pi$
$492$		0			0
$493$	−13008.0	−	13008.0i	−1.18834	−	1.18834i
$494$		0			0
$495$	−12682.2			−1.15156
$496$		0			0
$497$	−11306.6	−	14496.6i	−1.02047	−	1.30837i
$498$		0			0
$499$	−6229.73	−	6229.73i	−0.558880	−	0.558880i	0.370109	−	0.928989i	$-0.379320\pi$
$499$	−6229.73	−	6229.73i	−0.558880	−	0.558880i	−0.928989	+	0.370109i	$0.879320\pi$
$500$		0			0
$501$	669.109	+	669.109i	0.0596678	+	0.0596678i
$502$		0			0
$503$			432.677i			0.0383541i	0.999816	+	0.0191770i	$0.00610461\pi$
$503$			432.677i			0.0383541i	−0.999816	+	0.0191770i	$0.993895\pi$
$504$		0			0
$505$		−	24497.9i		−	2.15869i
$506$		0			0
$507$	223.515	+	223.515i	0.0195792	+	0.0195792i
$508$		0			0
$509$	2643.47	+	2643.47i	0.230196	+	0.230196i	0.812774	−	0.582578i	$-0.197956\pi$
$509$	2643.47	+	2643.47i	0.230196	+	0.230196i	−0.582578	+	0.812774i	$0.697956\pi$
$510$		0			0
$511$	−5743.86	−	7364.38i	−0.497248	−	0.637536i
$512$		0			0
$513$	266.083			0.0229003
$514$		0			0
$515$	17607.4	+	17607.4i	1.50655	+	1.50655i
$516$		0			0
$517$	−13345.4	+	13345.4i	−1.13526	+	1.13526i
$518$		0			0
$519$	2066.34			0.174764
$520$		0			0
$521$	9858.61			0.829008			0.414504	−	0.910047i	$-0.363955\pi$
$521$	9858.61			0.829008			0.414504	+	0.910047i	$0.363955\pi$
$522$		0			0
$523$	15527.5	+	15527.5i	1.29822	+	1.29822i	0.929565	+	0.368658i	$0.120183\pi$
$523$	15527.5	+	15527.5i	1.29822	+	1.29822i	0.368658	+	0.929565i	$0.379817\pi$
$524$		0			0
$525$	−85.3597	+	690.470i	−0.00709600	+	0.0573992i
$526$		0			0
$527$			15519.8i			1.28283i
$528$		0			0
$529$	−9618.44			−0.790535
$530$		0			0
$531$	−4462.77	+	4462.77i	−0.364723	+	0.364723i
$532$		0			0
$533$	3045.99	+	3045.99i	0.247536	+	0.247536i
$534$		0			0
$535$		−	5285.66i		−	0.427138i
$536$		0			0
$537$	−1778.55			−0.142924
$538$		0			0
$539$	−5866.57	−	9800.36i	−0.468815	−	0.783175i
$540$		0			0
$541$	−7786.22	+	7786.22i	−0.618772	+	0.618772i	−0.945216	−	0.326444i	$-0.894149\pi$
$541$	−7786.22	+	7786.22i	−0.618772	+	0.618772i	0.326444	+	0.945216i	$0.394149\pi$
$542$		0			0
$543$			589.137i			0.0465604i
$544$		0			0
$545$			1021.07i			0.0802528i
$546$		0			0
$547$	−9962.02	−	9962.02i	−0.778693	−	0.778693i	0.200915	−	0.979609i	$-0.435608\pi$
$547$	−9962.02	−	9962.02i	−0.778693	−	0.778693i	−0.979609	+	0.200915i	$0.935608\pi$
$548$		0			0
$549$	−12694.4	+	12694.4i	−0.986853	+	0.986853i
$550$		0			0
$551$		−	1560.79i		−	0.120675i
$552$		0			0
$553$	−11357.7	+	8858.46i	−0.873379	+	0.681194i
$554$		0			0
$555$	1514.18	−	1514.18i	0.115808	−	0.115808i
$556$		0			0
$557$	16847.7	−	16847.7i	1.28162	−	1.28162i	0.341867	−	0.939748i	$-0.388941\pi$
$557$	16847.7	−	16847.7i	1.28162	−	1.28162i	0.939748	−	0.341867i	$-0.111059\pi$
$558$		0			0
$559$	1992.80			0.150781
$560$		0			0
$561$	1942.46			0.146186
$562$		0			0
$563$	12084.8	−	12084.8i	0.904642	−	0.904642i	−0.0911911	−	0.995833i	$-0.529067\pi$
$563$	12084.8	−	12084.8i	0.904642	−	0.904642i	0.995833	+	0.0911911i	$0.0290674\pi$
$564$		0			0
$565$	12096.5	−	12096.5i	0.900712	−	0.900712i
$566$		0			0
$567$	−8087.08	−	10368.7i	−0.598986	−	0.767978i
$568$		0			0
$569$			15786.6i			1.16311i	0.813506	+	0.581556i	$0.197556\pi$
$569$			15786.6i			1.16311i	−0.813506	+	0.581556i	$0.802444\pi$
$570$		0			0
$571$	10067.9	−	10067.9i	0.737880	−	0.737880i	−0.234288	−	0.972167i	$-0.575276\pi$
$571$	10067.9	−	10067.9i	0.737880	−	0.737880i	0.972167	+	0.234288i	$0.0752757\pi$
$572$		0			0
$573$	−1000.52	−	1000.52i	−0.0729444	−	0.0729444i
$574$		0			0
$575$			3910.82i			0.283639i
$576$		0			0
$577$			4908.81i			0.354171i	0.984196	+	0.177085i	$0.0566668\pi$
$577$			4908.81i			0.354171i	−0.984196	+	0.177085i	$0.943333\pi$
$578$		0			0
$579$	275.215	−	275.215i	0.0197539	−	0.0197539i
$580$		0			0
$581$	−7376.14	−	911.879i	−0.526702	−	0.0651138i
$582$		0			0
$583$	−10030.2			−0.712539
$584$		0			0
$585$		−	14970.2i		−	1.05802i
$586$		0			0
$587$	−7677.59	−	7677.59i	−0.539843	−	0.539843i	0.383640	−	0.923483i	$-0.374670\pi$
$587$	−7677.59	−	7677.59i	−0.539843	−	0.539843i	−0.923483	+	0.383640i	$0.874670\pi$
$588$		0			0
$589$	−931.082	+	931.082i	−0.0651351	+	0.0651351i
$590$		0			0
$591$	−426.240			−0.0296669
$592$		0			0
$593$		−	16863.3i		−	1.16778i	−0.811832	−	0.583890i	$-0.801530\pi$
$593$		−	16863.3i		−	1.16778i	0.811832	−	0.583890i	$-0.198470\pi$
$594$		0			0
$595$	−3889.27	+	31460.1i	−0.267974	+	2.16763i
$596$		0			0
$597$	682.877	+	682.877i	0.0468146	+	0.0468146i
$598$		0			0
$599$	23012.2			1.56970			0.784852	−	0.619683i	$-0.212739\pi$
$599$	23012.2			1.56970			0.784852	+	0.619683i	$0.212739\pi$
$600$		0			0
$601$	−8460.85			−0.574251			−0.287126	−	0.957893i	$-0.592700\pi$
$601$	−8460.85			−0.574251			−0.287126	+	0.957893i	$0.592700\pi$
$602$		0			0
$603$	−163.194	+	163.194i	−0.0110212	+	0.0110212i
$604$		0			0
$605$	2234.42	+	2234.42i	0.150153	+	0.150153i
$606$		0			0
$607$	17227.4			1.15196			0.575978	−	0.817465i	$-0.304621\pi$
$607$	17227.4			1.15196			0.575978	+	0.817465i	$0.304621\pi$
$608$		0			0
$609$	1083.00	−	844.687i	0.0720612	−	0.0562043i
$610$		0			0
$611$	−15753.0	−	15753.0i	−1.04304	−	1.04304i
$612$		0			0
$613$	9657.01	+	9657.01i	0.636285	+	0.636285i	0.949637	−	0.313352i	$-0.101452\pi$
$613$	9657.01	+	9657.01i	0.636285	+	0.636285i	−0.313352	+	0.949637i	$0.601452\pi$
$614$		0			0
$615$		−	756.151i		−	0.0495788i
$616$		0			0
$617$			521.695i			0.0340399i	0.999855	+	0.0170200i	$0.00541788\pi$
$617$			521.695i			0.0340399i	−0.999855	+	0.0170200i	$0.994582\pi$
$618$		0			0
$619$	−20718.5	−	20718.5i	−1.34531	−	1.34531i	−0.890683	−	0.454625i	$-0.849773\pi$
$619$	−20718.5	−	20718.5i	−1.34531	−	1.34531i	−0.454625	−	0.890683i	$-0.650227\pi$
$620$		0			0
$621$	930.685	+	930.685i	0.0601402	+	0.0601402i
$622$		0			0
$623$	8931.98	+	11452.0i	0.574402	+	0.736457i
$624$		0			0
$625$	19307.2			1.23566
$626$		0			0
$627$	116.534	+	116.534i	0.00742255	+	0.00742255i
$628$		0			0
$629$	26397.2	−	26397.2i	1.67333	−	1.67333i
$630$		0			0
$631$	5819.96			0.367178			0.183589	−	0.983003i	$-0.441229\pi$
$631$	5819.96			0.367178			0.183589	+	0.983003i	$0.441229\pi$
$632$		0			0
$633$	2003.62			0.125808
$634$		0			0
$635$	−10751.8	−	10751.8i	−0.671927	−	0.671927i
$636$		0			0
$637$	11568.5	−	6924.98i	0.719559	−	0.430734i
$638$		0			0
$639$		−	26568.7i		−	1.64482i
$640$		0			0
$641$	3403.15			0.209698			0.104849	−	0.994488i	$-0.466564\pi$
$641$	3403.15			0.209698			0.104849	+	0.994488i	$0.466564\pi$
$642$		0			0
$643$	−6001.94	+	6001.94i	−0.368108	+	0.368108i	−0.866787	−	0.498679i	$-0.833819\pi$
$643$	−6001.94	+	6001.94i	−0.368108	+	0.368108i	0.498679	+	0.866787i	$0.333819\pi$
$644$		0			0
$645$	−247.351	−	247.351i	−0.0150999	−	0.0150999i
$646$		0			0
$647$			16816.2i			1.02181i	0.859637	+	0.510905i	$0.170690\pi$
$647$			16816.2i			1.02181i	−0.859637	+	0.510905i	$0.829310\pi$
$648$		0			0
$649$	−7852.46			−0.474940
$650$		0			0
$651$	−1149.95	−	142.164i	−0.0692323	−	0.00855887i
$652$		0			0
$653$	10942.9	−	10942.9i	0.655787	−	0.655787i	−0.298594	−	0.954380i	$-0.596517\pi$
$653$	10942.9	−	10942.9i	0.655787	−	0.655787i	0.954380	+	0.298594i	$0.0965175\pi$
$654$		0			0
$655$		−	30852.3i		−	1.84046i
$656$		0			0
$657$		−	13497.1i		−	0.801480i
$658$		0			0
$659$	−13080.2	−	13080.2i	−0.773191	−	0.773191i	0.205472	−	0.978663i	$-0.434127\pi$
$659$	−13080.2	−	13080.2i	−0.773191	−	0.773191i	−0.978663	+	0.205472i	$0.934127\pi$
$660$		0			0
$661$	13385.3	−	13385.3i	0.787636	−	0.787636i	−0.193470	−	0.981106i	$-0.561974\pi$
$661$	13385.3	−	13385.3i	0.787636	−	0.787636i	0.981106	+	0.193470i	$0.0619742\pi$
$662$		0			0
$663$			2292.90i			0.134312i
$664$		0			0
$665$	−2120.73	+	1654.07i	−0.123667	+	0.0964542i
$666$		0			0
$667$	5459.20	−	5459.20i	0.316913	−	0.316913i
$668$		0			0
$669$	73.4046	−	73.4046i	0.00424213	−	0.00424213i
$670$		0			0
$671$	−22336.3			−1.28507
$672$		0			0
$673$	22224.0			1.27291			0.636457	−	0.771312i	$-0.280399\pi$
$673$	22224.0			1.27291			0.636457	+	0.771312i	$0.280399\pi$
$674$		0			0
$675$	−1428.15	+	1428.15i	−0.0814365	+	0.0814365i
$676$		0			0
$677$	4955.51	−	4955.51i	0.281323	−	0.281323i	−0.552314	−	0.833636i	$-0.686255\pi$
$677$	4955.51	−	4955.51i	0.281323	−	0.281323i	0.833636	+	0.552314i	$0.186255\pi$
$678$		0			0
$679$	−2767.28	+	2158.35i	−0.156404	+	0.121988i
$680$		0			0
$681$		−	1356.46i		−	0.0763282i
$682$		0			0
$683$	−1229.73	+	1229.73i	−0.0688933	+	0.0688933i	−0.740714	−	0.671821i	$-0.765513\pi$
$683$	−1229.73	+	1229.73i	−0.0688933	+	0.0688933i	0.671821	+	0.740714i	$0.265513\pi$
$684$		0			0
$685$	−27607.8	−	27607.8i	−1.53991	−	1.53991i
$686$		0			0
$687$		−	844.864i		−	0.0469193i
$688$		0			0
$689$		−	11839.8i		−	0.654661i
$690$		0			0
$691$	−21512.7	+	21512.7i	−1.18434	+	1.18434i	−0.205736	+	0.978608i	$0.565959\pi$
$691$	−21512.7	+	21512.7i	−1.18434	+	1.18434i	−0.978608	+	0.205736i	$0.934041\pi$
$692$		0			0
$693$	2025.24	−	16382.1i	0.111014	−	0.897985i
$694$		0			0
$695$	−34968.3			−1.90852
$696$		0			0
$697$		−	13182.2i		−	0.716374i
$698$		0			0
$699$	1624.08	+	1624.08i	0.0878805	+	0.0878805i
$700$		0			0
$701$	−20884.4	+	20884.4i	−1.12524	+	1.12524i	−0.134296	+	0.990941i	$0.542877\pi$
$701$	−20884.4	+	20884.4i	−1.12524	+	1.12524i	−0.990941	+	0.134296i	$0.957123\pi$
$702$		0			0
$703$	3167.31			0.169925
$704$		0			0
$705$			3910.60i			0.208910i
$706$		0			0
$707$	31644.9	+	3912.12i	1.68335	+	0.208105i
$708$		0			0
$709$	14090.4	+	14090.4i	0.746372	+	0.746372i	0.973796	−	0.227424i	$-0.0730303\pi$
$709$	14090.4	+	14090.4i	0.746372	+	0.746372i	−0.227424	+	0.973796i	$0.573030\pi$
$710$		0			0
$711$	−20815.9			−1.09797
$712$		0			0
$713$	−6513.34			−0.342113
$714$		0			0
$715$	13170.4	−	13170.4i	0.688873	−	0.688873i
$716$		0			0
$717$	−470.830	−	470.830i	−0.0245237	−	0.0245237i
$718$		0			0
$719$	−4216.09			−0.218684			−0.109342	−	0.994004i	$-0.534874\pi$
$719$	−4216.09			−0.218684			−0.109342	+	0.994004i	$0.534874\pi$
$720$		0			0
$721$	−25556.0	+	19932.4i	−1.32005	+	1.02957i
$722$		0			0
$723$	643.651	+	643.651i	0.0331088	+	0.0331088i
$724$		0			0
$725$	8377.25	+	8377.25i	0.429136	+	0.429136i
$726$		0			0
$727$			21701.0i			1.10708i	0.832824	+	0.553538i	$0.186723\pi$
$727$			21701.0i			1.10708i	−0.832824	+	0.553538i	$0.813277\pi$
$728$		0			0
$729$		−	18661.9i		−	0.948123i
$730$		0			0
$731$	−4312.15	−	4312.15i	−0.218182	−	0.218182i
$732$		0			0
$733$	9256.12	+	9256.12i	0.466416	+	0.466416i	0.900751	−	0.434336i	$-0.143017\pi$
$733$	9256.12	+	9256.12i	0.466416	+	0.466416i	−0.434336	+	0.900751i	$0.643017\pi$
$734$		0			0
$735$	−2295.45	−	576.360i	−0.115196	−	0.0289243i
$736$		0			0
$737$	−287.148			−0.0143517
$738$		0			0
$739$	10966.7	+	10966.7i	0.545894	+	0.545894i	0.925251	−	0.379357i	$-0.123855\pi$
$739$	10966.7	+	10966.7i	0.545894	+	0.545894i	−0.379357	+	0.925251i	$0.623855\pi$
$740$		0			0
$741$	−137.559	+	137.559i	−0.00681963	+	0.00681963i
$742$		0			0
$743$	26265.5			1.29689			0.648444	−	0.761262i	$-0.275420\pi$
$743$	26265.5			1.29689			0.648444	+	0.761262i	$0.275420\pi$
$744$		0			0
$745$	58.4874			0.00287626
$746$		0			0
$747$	−7594.96	−	7594.96i	−0.372002	−	0.372002i
$748$		0			0
$749$	6827.71	+	844.078i	0.333083	+	0.0411775i
$750$		0			0
$751$			9601.26i			0.466518i	0.972415	+	0.233259i	$0.0749390\pi$
$751$			9601.26i			0.466518i	−0.972415	+	0.233259i	$0.925061\pi$
$752$		0			0
$753$	−3150.01			−0.152447
$754$		0			0
$755$	24935.4	−	24935.4i	1.20197	−	1.20197i
$756$		0			0
$757$	5546.05	+	5546.05i	0.266281	+	0.266281i	0.827600	−	0.561319i	$-0.189706\pi$
$757$	5546.05	+	5546.05i	0.266281	+	0.266281i	−0.561319	+	0.827600i	$0.689706\pi$
$758$		0			0
$759$			815.211i			0.0389859i
$760$		0			0
$761$	−27393.4			−1.30487			−0.652437	−	0.757843i	$-0.726253\pi$
$761$	−27393.4			−1.30487			−0.652437	+	0.757843i	$0.726253\pi$
$762$		0			0
$763$	−1318.96	−	163.057i	−0.0625812	−	0.00773663i
$764$		0			0
$765$	−32393.4	+	32393.4i	−1.53096	+	1.53096i
$766$		0			0
$767$		−	9269.14i		−	0.436361i
$768$		0			0
$769$			30407.0i			1.42589i	0.701222	+	0.712943i	$0.252638\pi$
$769$			30407.0i			1.42589i	−0.701222	+	0.712943i	$0.747362\pi$
$770$		0			0
$771$	945.454	+	945.454i	0.0441630	+	0.0441630i
$772$		0			0
$773$	−2476.14	+	2476.14i	−0.115214	+	0.115214i	−0.762363	−	0.647149i	$-0.775961\pi$
$773$	−2476.14	+	2476.14i	−0.115214	+	0.115214i	0.647149	+	0.762363i	$0.275961\pi$
$774$		0			0
$775$		−	9994.84i		−	0.463258i
$776$		0			0
$777$	1714.12	+	2197.73i	0.0791426	+	0.101471i
$778$		0			0
$779$	790.846	−	790.846i	0.0363735	−	0.0363735i
$780$		0			0
$781$	23374.4	−	23374.4i	1.07094	−	1.07094i
$782$		0			0
$783$	3987.18			0.181980
$784$		0			0
$785$	24643.9			1.12048
$786$		0			0
$787$	2253.18	−	2253.18i	0.102055	−	0.102055i	−0.654236	−	0.756291i	$-0.727010\pi$
$787$	2253.18	−	2253.18i	0.102055	−	0.102055i	0.756291	+	0.654236i	$0.227010\pi$
$788$		0			0
$789$	482.139	−	482.139i	0.0217549	−	0.0217549i
$790$		0			0
$791$	13693.8	+	17557.2i	0.615544	+	0.789207i
$792$		0			0
$793$		−	26366.1i		−	1.18069i
$794$		0			0
$795$	−1469.59	+	1469.59i	−0.0655608	+	0.0655608i
$796$		0			0
$797$	16000.4	+	16000.4i	0.711121	+	0.711121i	0.966770	−	0.255649i	$-0.0822892\pi$
$797$	16000.4	+	16000.4i	0.711121	+	0.711121i	−0.255649	+	0.966770i	$0.582289\pi$
$798$		0			0
$799$			68174.9i			3.01859i
$800$		0			0
$801$			20988.6i			0.925839i
$802$		0			0
$803$	11874.4	−	11874.4i	0.521841	−	0.521841i
$804$		0			0
$805$	−13203.2	−	1632.25i	−0.578076	−	0.0714650i
$806$		0			0
$807$	−592.811			−0.0258587
$808$		0			0
$809$			27106.7i			1.17802i	0.808125	+	0.589011i	$0.200483\pi$
$809$			27106.7i			1.17802i	−0.808125	+	0.589011i	$0.799517\pi$
$810$		0			0
$811$	−13252.6	−	13252.6i	−0.573812	−	0.573812i	0.359379	−	0.933192i	$-0.382988\pi$
$811$	−13252.6	−	13252.6i	−0.573812	−	0.573812i	−0.933192	+	0.359379i	$0.882988\pi$
$812$		0			0
$813$	2063.59	−	2063.59i	0.0890201	−	0.0890201i
$814$		0			0
$815$	31670.0			1.36117
$816$		0			0
$817$		−	517.401i		−	0.0221561i
$818$		0			0
$819$	19337.6	+	2390.62i	0.825044	+	0.101996i
$820$		0			0
$821$	12774.8	+	12774.8i	0.543048	+	0.543048i	0.924421	−	0.381373i	$-0.124549\pi$
$821$	12774.8	+	12774.8i	0.543048	+	0.543048i	−0.381373	+	0.924421i	$0.624549\pi$
$822$		0			0
$823$	−17526.6			−0.742331			−0.371166	−	0.928567i	$-0.621042\pi$
$823$	−17526.6			−0.742331			−0.371166	+	0.928567i	$0.621042\pi$
$824$		0			0
$825$	−1250.96			−0.0527912
$826$		0			0
$827$	6285.90	−	6285.90i	0.264307	−	0.264307i	−0.562494	−	0.826801i	$-0.690158\pi$
$827$	6285.90	−	6285.90i	0.264307	−	0.264307i	0.826801	+	0.562494i	$0.190158\pi$
$828$		0			0
$829$	−5698.29	−	5698.29i	−0.238733	−	0.238733i	0.577592	−	0.816325i	$-0.303992\pi$
$829$	−5698.29	−	5698.29i	−0.238733	−	0.238733i	−0.816325	+	0.577592i	$0.803992\pi$
$830$		0			0
$831$	47.0725			0.00196502
$832$		0			0
$833$	−40017.3	−	10047.9i	−1.66449	−	0.417933i
$834$		0			0
$835$	−19633.8	−	19633.8i	−0.813718	−	0.813718i
$836$		0			0
$837$	−2378.54	−	2378.54i	−0.0982250	−	0.0982250i
$838$		0			0
$839$		−	3202.64i		−	0.131785i	−0.997827	−	0.0658924i	$-0.979011\pi$
$839$		−	3202.64i		−	0.131785i	0.997827	−	0.0658924i	$-0.0209894\pi$
$840$		0			0
$841$			1001.04i			0.0410447i
$842$		0			0
$843$	−626.233	−	626.233i	−0.0255855	−	0.0255855i
$844$		0			0
$845$	−6558.64	−	6558.64i	−0.267011	−	0.267011i
$846$		0			0
$847$	−3243.12	+	2529.48i	−0.131564	+	0.102614i
$848$		0			0
$849$	765.505			0.0309447
$850$		0			0
$851$	11078.4	+	11078.4i	0.446253	+	0.446253i
$852$		0			0
$853$	2553.19	−	2553.19i	0.102485	−	0.102485i	−0.654005	−	0.756490i	$-0.726913\pi$
$853$	2553.19	−	2553.19i	0.102485	−	0.102485i	0.756490	+	0.654005i	$0.226913\pi$
$854$		0			0
$855$	−3886.78			−0.155468
$856$		0			0
$857$	−33164.1			−1.32189			−0.660947	−	0.750432i	$-0.729845\pi$
$857$	−33164.1			−1.32189			−0.660947	+	0.750432i	$0.729845\pi$
$858$		0			0
$859$	−7590.88	−	7590.88i	−0.301510	−	0.301510i	0.540094	−	0.841605i	$-0.318389\pi$
$859$	−7590.88	−	7590.88i	−0.301510	−	0.301510i	−0.841605	+	0.540094i	$0.818389\pi$
$860$		0			0
$861$	976.751	+	120.751i	0.0386615	+	0.00477955i
$862$		0			0
$863$			17015.2i			0.671152i	0.942013	+	0.335576i	$0.108931\pi$
$863$			17015.2i			0.671152i	−0.942013	+	0.335576i	$0.891069\pi$
$864$		0			0
$865$	−60632.9			−2.38333
$866$		0			0
$867$	3276.90	−	3276.90i	0.128361	−	0.128361i
$868$		0			0
$869$	−18313.3	−	18313.3i	−0.714885	−	0.714885i
$870$		0			0
$871$		−	338.953i		−	0.0131860i
$872$		0			0
$873$	−5071.75			−0.196624
$874$		0			0
$875$	−1536.84	+	12431.4i	−0.0593767	+	0.480296i
$876$		0			0
$877$	18.4854	−	18.4854i	0.000711752	−	0.000711752i	−0.706751	−	0.707463i	$-0.749840\pi$
$877$	18.4854	−	18.4854i	0.000711752	−	0.000711752i	0.707463	+	0.706751i	$0.249840\pi$
$878$		0			0
$879$			1446.15i			0.0554921i
$880$		0			0
$881$			19084.9i			0.729838i	0.931039	+	0.364919i	$0.118903\pi$
$881$			19084.9i			0.729838i	−0.931039	+	0.364919i	$0.881097\pi$
$882$		0			0
$883$	−24207.0	−	24207.0i	−0.922571	−	0.922571i	0.0746392	−	0.997211i	$-0.476219\pi$
$883$	−24207.0	−	24207.0i	−0.922571	−	0.922571i	−0.997211	+	0.0746392i	$0.976219\pi$
$884$		0			0
$885$	−1150.51	+	1150.51i	−0.0436992	+	0.0436992i
$886$		0			0
$887$			23895.7i			0.904554i	0.891878	+	0.452277i	$0.149388\pi$
$887$			23895.7i			0.904554i	−0.891878	+	0.452277i	$0.850612\pi$
$888$		0			0
$889$	15605.6	−	12171.6i	0.588745	−	0.459193i
$890$		0			0
$891$	16718.6	−	16718.6i	0.628612	−	0.628612i
$892$		0			0
$893$	−4090.04	+	4090.04i	−0.153267	+	0.153267i
$894$		0			0
$895$	52188.3			1.94912
$896$		0			0
$897$	−962.285			−0.0358191
$898$		0			0
$899$	−13952.0	+	13952.0i	−0.517604	+	0.517604i
$900$		0			0
$901$	−25619.8	+	25619.8i	−0.947302	+	0.947302i
$902$		0			0
$903$	359.014	−	280.014i	0.0132306	−	0.0103192i
$904$		0			0
$905$		−	17287.1i		−	0.634966i
$906$		0			0
$907$	13458.2	−	13458.2i	0.492694	−	0.492694i	−0.416460	−	0.909154i	$-0.636729\pi$
$907$	13458.2	−	13458.2i	0.492694	−	0.492694i	0.909154	+	0.416460i	$0.136729\pi$
$908$		0			0
$909$	32583.7	+	32583.7i	1.18893	+	1.18893i
$910$		0			0
$911$			12028.6i			0.437458i	0.975786	+	0.218729i	$0.0701910\pi$
$911$			12028.6i			0.437458i	−0.975786	+	0.218729i	$0.929809\pi$
$912$		0			0
$913$		−	13363.7i		−	0.484418i
$914$		0			0
$915$	−3272.62	+	3272.62i	−0.118240	+	0.118240i
$916$		0			0
$917$	39853.2	+	4926.87i	1.43519	+	0.177426i
$918$		0			0
$919$	6224.99			0.223442			0.111721	−	0.993740i	$-0.464364\pi$
$919$	6224.99			0.223442			0.111721	+	0.993740i	$0.464364\pi$
$920$		0			0
$921$			312.324i			0.0111742i
$922$		0			0
$923$	27591.4	+	27591.4i	0.983947	+	0.983947i
$924$		0			0
$925$	−17000.0	+	17000.0i	−0.604276	+	0.604276i
$926$		0			0
$927$	−46837.9			−1.65950
$928$		0			0
$929$			10863.7i			0.383666i	0.981428	+	0.191833i	$0.0614432\pi$
$929$			10863.7i			0.383666i	−0.981428	+	0.191833i	$0.938557\pi$
$930$		0			0
$931$	−1797.96	−	3003.58i	−0.0632931	−	0.105734i
$932$		0			0
$933$	2357.00	+	2357.00i	0.0827060	+	0.0827060i
$934$		0			0
$935$	−56997.8			−1.99361
$936$		0			0
$937$	36380.3			1.26840			0.634201	−	0.773168i	$-0.281329\pi$
$937$	36380.3			1.26840			0.634201	+	0.773168i	$0.281329\pi$
$938$		0			0
$939$	1133.48	−	1133.48i	0.0393925	−	0.0393925i
$940$		0			0
$941$	29521.1	+	29521.1i	1.02270	+	1.02270i	0.999736	+	0.0229626i	$0.00730987\pi$
$941$	29521.1	+	29521.1i	1.02270	+	1.02270i	0.0229626	+	0.999736i	$0.492690\pi$
$942$		0			0
$943$	5532.32			0.191047
$944$		0			0
$945$	−4225.48	−	5417.61i	−0.145455	−	0.186492i
$946$		0			0
$947$	26494.7	+	26494.7i	0.909147	+	0.909147i	0.996203	−	0.0870568i	$-0.0277461\pi$
$947$	26494.7	+	26494.7i	0.909147	+	0.909147i	−0.0870568	+	0.996203i	$0.527746\pi$
$948$		0			0
$949$	14016.7	+	14016.7i	0.479453	+	0.479453i
$950$		0			0
$951$		−	4345.29i		−	0.148166i
$952$		0			0
$953$		−	34129.8i		−	1.16010i	−0.814582	−	0.580049i	$-0.803033\pi$
$953$		−	34129.8i		−	1.16010i	0.814582	−	0.580049i	$-0.196967\pi$
$954$		0			0
$955$	29358.3	+	29358.3i	0.994777	+	0.994777i
$956$		0			0
$957$	1746.24	+	1746.24i	0.0589841	+	0.0589841i
$958$		0			0
$959$	40070.9	−	31253.4i	1.34928	−	1.05237i
$960$		0			0
$961$	−13145.0			−0.441239
$962$		0			0
$963$	7030.26	+	7030.26i	0.235251	+	0.235251i
$964$		0			0
$965$	−8075.66	+	8075.66i	−0.269393	+	0.269393i
$966$		0			0
$967$	8378.57			0.278632			0.139316	−	0.990248i	$-0.455510\pi$
$967$	8378.57			0.278632			0.139316	+	0.990248i	$0.455510\pi$
$968$		0			0
$969$	595.317			0.0197361
$970$		0			0
$971$	−4397.60	−	4397.60i	−0.145340	−	0.145340i	0.630692	−	0.776033i	$-0.282771\pi$
$971$	−4397.60	−	4397.60i	−0.145340	−	0.145340i	−0.776033	+	0.630692i	$0.782771\pi$
$972$		0			0
$973$	5584.16	−	45169.9i	0.183987	−	1.48827i
$974$		0			0
$975$		−	1476.64i		−	0.0485031i
$976$		0			0
$977$	30785.5			1.00810			0.504051	−	0.863674i	$-0.331842\pi$
$977$	30785.5			1.00810			0.504051	+	0.863674i	$0.331842\pi$
$978$		0			0
$979$	−18465.3	+	18465.3i	−0.602811	+	0.602811i
$980$		0			0
$981$	−1358.09	−	1358.09i	−0.0442001	−	0.0442001i
$982$		0			0
$983$		−	4454.97i		−	0.144549i	−0.997385	−	0.0722744i	$-0.976974\pi$
$983$		−	4454.97i		−	0.144549i	0.997385	−	0.0722744i	$-0.0230257\pi$
$984$		0			0
$985$	12507.2			0.404581
$986$		0			0
$987$	−5051.49	−	624.493i	−0.162909	−	0.0201396i
$988$		0			0
$989$	1809.72	−	1809.72i	0.0581859	−	0.0581859i
$990$		0			0
$991$			6910.57i			0.221515i	0.993847	+	0.110758i	$0.0353277\pi$
$991$			6910.57i			0.221515i	−0.993847	+	0.110758i	$0.964672\pi$
$992$		0			0
$993$		−	4703.81i		−	0.150323i
$994$		0			0
$995$	−20037.8	−	20037.8i	−0.638432	−	0.638432i
$996$		0			0
$997$	23672.4	−	23672.4i	0.751969	−	0.751969i	−0.222878	−	0.974846i	$-0.571545\pi$
$997$	23672.4	−	23672.4i	0.751969	−	0.751969i	0.974846	+	0.222878i	$0.0715450\pi$
$998$		0			0
$999$			8091.19i			0.256250i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.4.j.b.335.21		88
4.3	odd	2		112.4.j.b.27.42	yes	88
7.6	odd	2	inner	448.4.j.b.335.24		88
16.3	odd	4	inner	448.4.j.b.111.24		88
16.13	even	4		112.4.j.b.83.41	yes	88
28.27	even	2		112.4.j.b.27.41	✓	88
112.13	odd	4		112.4.j.b.83.42	yes	88
112.83	even	4	inner	448.4.j.b.111.21		88

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.4.j.b.27.41	✓	88	28.27	even	2
112.4.j.b.27.42	yes	88	4.3	odd	2
112.4.j.b.83.41	yes	88	16.13	even	4
112.4.j.b.83.42	yes	88	112.13	odd	4
448.4.j.b.111.21		88	112.83	even	4	inner
448.4.j.b.111.24		88	16.3	odd	4	inner
448.4.j.b.335.21		88	1.1	even	1	trivial
448.4.j.b.335.24		88	7.6	odd	2	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	448.j (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.342891 + 0.342891i) q^{3} +(10.0615 - 10.0615i) q^{5} +(11.3901 + 14.6036i) q^{7} +26.7649i q^{9} +O(q^{10})\)	\(q+(-0.342891 + 0.342891i) q^{3} +(10.0615 - 10.0615i) q^{5} +(11.3901 + 14.6036i) q^{7} +26.7649i q^{9} +(-23.5470 + 23.5470i) q^{11} +(-27.7952 - 27.7952i) q^{13} +6.90000i q^{15} +120.290i q^{17} +(-7.21660 + 7.21660i) q^{19} +(-8.91302 - 1.10188i) q^{21} -50.4833 q^{23} -77.4676i q^{25} +(-18.4355 - 18.4355i) q^{27} +(-108.139 + 108.139i) q^{29} +129.020 q^{31} -16.1481i q^{33} +(261.536 + 32.3325i) q^{35} +(-219.446 - 219.446i) q^{37} +19.0614 q^{39} -109.587 q^{41} +(-35.8480 + 35.8480i) q^{43} +(269.295 + 269.295i) q^{45} +566.754 q^{47} +(-83.5304 + 332.674i) q^{49} +(-41.2464 - 41.2464i) q^{51} +(212.984 + 212.984i) q^{53} +473.837i q^{55} -4.94901i q^{57} +(166.740 + 166.740i) q^{59} +(474.293 + 474.293i) q^{61} +(-390.863 + 304.855i) q^{63} -559.323 q^{65} +(6.09734 + 6.09734i) q^{67} +(17.3103 - 17.3103i) q^{69} -992.670 q^{71} -504.285 q^{73} +(26.5630 + 26.5630i) q^{75} +(-612.074 - 75.6680i) q^{77} +777.732i q^{79} -710.008 q^{81} +(-283.766 + 283.766i) q^{83} +(1210.30 + 1210.30i) q^{85} -74.1596i q^{87} +784.187 q^{89} +(89.3194 - 722.500i) q^{91} +(-44.2397 + 44.2397i) q^{93} +145.220i q^{95} +189.493i q^{97} +(-630.232 - 630.232i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 4 q^{7}+O(q^{10}) \) 88 * q + 4 * q^7	\( 88 q + 4 q^{7} - 112 q^{11} + 52 q^{21} - 160 q^{23} + 528 q^{29} - 476 q^{35} - 896 q^{37} + 8 q^{39} - 40 q^{43} - 1376 q^{49} - 1504 q^{51} - 1560 q^{53} - 8 q^{65} - 648 q^{67} + 456 q^{71} + 292 q^{77} - 1952 q^{81} + 496 q^{85} + 1964 q^{91} - 112 q^{93} - 7592 q^{99}+O(q^{100}) \) 88 * q + 4 * q^7 - 112 * q^11 + 52 * q^21 - 160 * q^23 + 528 * q^29 - 476 * q^35 - 896 * q^37 + 8 * q^39 - 40 * q^43 - 1376 * q^49 - 1504 * q^51 - 1560 * q^53 - 8 * q^65 - 648 * q^67 + 456 * q^71 + 292 * q^77 - 1952 * q^81 + 496 * q^85 + 1964 * q^91 - 112 * q^93 - 7592 * q^99

Introduction

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