LMFDB - Embedded newform 448.4.j.b.111.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [448,4,Mod(111,448)] 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([2, 3, 2])) N = Newforms(chi, 4, names="a")

magma://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:traces = [88] 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:	$26.4328556826$
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			111.8
Character	$\chi$	$=$	448.111
Dual form			448.4.j.b.335.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-4.79702 - 4.79702i) q^{3} +(-2.04624 - 2.04624i) q^{5} +(-13.3740 + 12.8115i) q^{7} +19.0229i q^{9} +(-34.5515 - 34.5515i) q^{11} +(7.32563 - 7.32563i) q^{13} +19.6317i q^{15} +133.707i q^{17} +(-13.6735 - 13.6735i) q^{19} +(125.613 + 2.69841i) q^{21} -129.773 q^{23} -116.626i q^{25} +(-38.2664 + 38.2664i) q^{27} +(85.7909 + 85.7909i) q^{29} +239.073 q^{31} +331.489i q^{33} +(53.5820 + 1.15105i) q^{35} +(-50.7975 + 50.7975i) q^{37} -70.2824 q^{39} -118.276 q^{41} +(75.4750 + 75.4750i) q^{43} +(38.9254 - 38.9254i) q^{45} +169.043 q^{47} +(14.7299 - 342.684i) q^{49} +(641.398 - 641.398i) q^{51} +(482.264 - 482.264i) q^{53} +141.401i q^{55} +131.184i q^{57} +(285.231 - 285.231i) q^{59} +(313.070 - 313.070i) q^{61} +(-243.712 - 254.413i) q^{63} -29.9800 q^{65} +(58.2711 - 58.2711i) q^{67} +(622.523 + 622.523i) q^{69} +374.109 q^{71} -610.003 q^{73} +(-559.457 + 559.457i) q^{75} +(904.751 + 19.4359i) q^{77} +460.899i q^{79} +880.748 q^{81} +(875.405 + 875.405i) q^{83} +(273.598 - 273.598i) q^{85} -823.082i q^{87} +633.231 q^{89} +(-4.12080 + 191.826i) q^{91} +(-1146.84 - 1146.84i) q^{93} +55.9584i q^{95} +946.138i q^{97} +(657.269 - 657.269i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots - 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

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
$3$	−4.79702	−	4.79702i	−0.923188	−	0.923188i	0.0740657	−	0.997253i	$-0.476403\pi$
$3$	−4.79702	−	4.79702i	−0.923188	−	0.923188i	−0.997253	+	0.0740657i	$0.976403\pi$
$4$		0			0
$5$	−2.04624	−	2.04624i	−0.183021	−	0.183021i	0.609650	−	0.792671i	$-0.291310\pi$
$5$	−2.04624	−	2.04624i	−0.183021	−	0.183021i	−0.792671	+	0.609650i	$0.791310\pi$
$6$		0			0
$7$	−13.3740	+	12.8115i	−0.722130	+	0.691757i
$7$	−13.3740	+	12.8115i	−0.722130	+	0.691757i
$8$		0			0
$9$			19.0229i			0.704551i
$10$		0			0
$11$	−34.5515	−	34.5515i	−0.947061	−	0.947061i	0.0516064	−	0.998668i	$-0.483566\pi$
$11$	−34.5515	−	34.5515i	−0.947061	−	0.947061i	−0.998668	+	0.0516064i	$0.983566\pi$
$12$		0			0
$13$	7.32563	−	7.32563i	0.156290	−	0.156290i	−0.624631	−	0.780920i	$-0.714750\pi$
$13$	7.32563	−	7.32563i	0.156290	−	0.156290i	0.780920	+	0.624631i	$0.214750\pi$
$14$		0			0
$15$			19.6317i			0.337926i
$16$		0			0
$17$			133.707i			1.90758i	0.300477	+	0.953789i	$0.402854\pi$
$17$			133.707i			1.90758i	−0.300477	+	0.953789i	$0.597146\pi$
$18$		0			0
$19$	−13.6735	−	13.6735i	−0.165100	−	0.165100i	0.619722	−	0.784822i	$-0.287246\pi$
$19$	−13.6735	−	13.6735i	−0.165100	−	0.165100i	−0.784822	+	0.619722i	$0.787246\pi$
$20$		0			0
$21$	125.613	+	2.69841i	1.30528	+	0.0280401i
$22$		0			0
$23$	−129.773			−1.17650			−0.588250	−	0.808679i	$-0.700183\pi$
$23$	−129.773			−1.17650			−0.588250	+	0.808679i	$0.700183\pi$
$24$		0			0
$25$		−	116.626i		−	0.933006i
$26$		0			0
$27$	−38.2664	+	38.2664i	−0.272755	+	0.272755i
$28$		0			0
$29$	85.7909	+	85.7909i	0.549344	+	0.549344i	0.926251	−	0.376907i	$-0.123012\pi$
$29$	85.7909	+	85.7909i	0.549344	+	0.549344i	−0.376907	+	0.926251i	$0.623012\pi$
$30$		0			0
$31$	239.073			1.38512			0.692560	−	0.721360i	$-0.256483\pi$
$31$	239.073			1.38512			0.692560	+	0.721360i	$0.256483\pi$
$32$		0			0
$33$			331.489i			1.74863i
$34$		0			0
$35$	53.5820	+	1.15105i	0.258772	+	0.00555893i
$36$		0			0
$37$	−50.7975	+	50.7975i	−0.225704	+	0.225704i	−0.810895	−	0.585191i	$-0.801019\pi$
$37$	−50.7975	+	50.7975i	−0.225704	+	0.225704i	0.585191	+	0.810895i	$0.301019\pi$
$38$		0			0
$39$	−70.2824			−0.288569
$40$		0			0
$41$	−118.276			−0.450527			−0.225264	−	0.974298i	$-0.572324\pi$
$41$	−118.276			−0.450527			−0.225264	+	0.974298i	$0.572324\pi$
$42$		0			0
$43$	75.4750	+	75.4750i	0.267671	+	0.267671i	0.828161	−	0.560490i	$-0.189387\pi$
$43$	75.4750	+	75.4750i	0.267671	+	0.267671i	−0.560490	+	0.828161i	$0.689387\pi$
$44$		0			0
$45$	38.9254	−	38.9254i	0.128948	−	0.128948i
$46$		0			0
$47$	169.043			0.524625			0.262313	−	0.964983i	$-0.415515\pi$
$47$	169.043			0.524625			0.262313	+	0.964983i	$0.415515\pi$
$48$		0			0
$49$	14.7299	−	342.684i	0.0429442	−	0.999077i
$50$		0			0
$51$	641.398	−	641.398i	1.76105	−	1.76105i
$52$		0			0
$53$	482.264	−	482.264i	1.24989	−	1.24989i	0.294118	−	0.955769i	$-0.404974\pi$
$53$	482.264	−	482.264i	1.24989	−	1.24989i	0.955769	−	0.294118i	$-0.0950257\pi$
$54$		0			0
$55$			141.401i			0.346665i
$56$		0			0
$57$			131.184i			0.304837i
$58$		0			0
$59$	285.231	−	285.231i	0.629389	−	0.629389i	−0.318525	−	0.947914i	$-0.603188\pi$
$59$	285.231	−	285.231i	0.629389	−	0.629389i	0.947914	+	0.318525i	$0.103188\pi$
$60$		0			0
$61$	313.070	−	313.070i	0.657124	−	0.657124i	−0.297575	−	0.954699i	$-0.596178\pi$
$61$	313.070	−	313.070i	0.657124	−	0.657124i	0.954699	+	0.297575i	$0.0961777\pi$
$62$		0			0
$63$	−243.712	−	254.413i	−0.487378	−	0.508778i
$64$		0			0
$65$	−29.9800			−0.0572086
$66$		0			0
$67$	58.2711	−	58.2711i	0.106253	−	0.106253i	−0.651982	−	0.758235i	$-0.726062\pi$
$67$	58.2711	−	58.2711i	0.106253	−	0.106253i	0.758235	+	0.651982i	$0.226062\pi$
$68$		0			0
$69$	622.523	+	622.523i	1.08613	+	1.08613i
$70$		0			0
$71$	374.109			0.625331			0.312666	−	0.949863i	$-0.398778\pi$
$71$	374.109			0.625331			0.312666	+	0.949863i	$0.398778\pi$
$72$		0			0
$73$	−610.003			−0.978020			−0.489010	−	0.872278i	$-0.662642\pi$
$73$	−610.003			−0.978020			−0.489010	+	0.872278i	$0.662642\pi$
$74$		0			0
$75$	−559.457	+	559.457i	−0.861340	+	0.861340i
$76$		0			0
$77$	904.751	+	19.4359i	1.33904	+	0.0287652i
$78$		0			0
$79$			460.899i			0.656395i	0.944609	+	0.328198i	$0.106441\pi$
$79$			460.899i			0.656395i	−0.944609	+	0.328198i	$0.893559\pi$
$80$		0			0
$81$	880.748			1.20816
$82$		0			0
$83$	875.405	+	875.405i	1.15769	+	1.15769i	0.984971	+	0.172717i	$0.0552548\pi$
$83$	875.405	+	875.405i	1.15769	+	1.15769i	0.172717	+	0.984971i	$0.444745\pi$
$84$		0			0
$85$	273.598	−	273.598i	0.349128	−	0.349128i
$86$		0			0
$87$		−	823.082i		−	1.01430i
$88$		0			0
$89$	633.231			0.754184			0.377092	−	0.926176i	$-0.376924\pi$
$89$	633.231			0.754184			0.377092	+	0.926176i	$0.376924\pi$
$90$		0			0
$91$	−4.12080	+	191.826i	−0.00474700	+	0.220976i
$92$		0			0
$93$	−1146.84	−	1146.84i	−1.27873	−	1.27873i
$94$		0			0
$95$			55.9584i			0.0604338i
$96$		0			0
$97$			946.138i			0.990368i	0.868788	+	0.495184i	$0.164899\pi$
$97$			946.138i			0.990368i	−0.868788	+	0.495184i	$0.835101\pi$
$98$		0			0
$99$	657.269	−	657.269i	0.667253	−	0.667253i
$100$		0			0
$101$	−141.100	−	141.100i	−0.139010	−	0.139010i	0.634178	−	0.773187i	$-0.281339\pi$
$101$	−141.100	−	141.100i	−0.139010	−	0.139010i	−0.773187	+	0.634178i	$0.781339\pi$
$102$		0			0
$103$			891.445i			0.852784i	0.904539	+	0.426392i	$0.140216\pi$
$103$			891.445i			0.852784i	−0.904539	+	0.426392i	$0.859784\pi$
$104$		0			0
$105$	−251.512	−	262.556i	−0.233763	−	0.244027i
$106$		0			0
$107$	727.560	+	727.560i	0.657345	+	0.657345i	0.954751	−	0.297406i	$-0.0961215\pi$
$107$	727.560	+	727.560i	0.657345	+	0.657345i	−0.297406	+	0.954751i	$0.596122\pi$
$108$		0			0
$109$	−1136.70	−	1136.70i	−0.998867	−	0.998867i	0.00113245	−	0.999999i	$-0.499640\pi$
$109$	−1136.70	−	1136.70i	−0.998867	−	0.998867i	−0.999999	+	0.00113245i	$0.999640\pi$
$110$		0			0
$111$	487.353			0.416735
$112$		0			0
$113$	−556.606			−0.463373			−0.231686	−	0.972791i	$-0.574424\pi$
$113$	−556.606			−0.463373			−0.231686	+	0.972791i	$0.574424\pi$
$114$		0			0
$115$	265.546	+	265.546i	0.215325	+	0.215325i
$116$		0			0
$117$	139.355	+	139.355i	0.110114	+	0.110114i
$118$		0			0
$119$	−1713.00	−	1788.21i	−1.31958	−	1.37752i
$120$		0			0
$121$			1056.61i			0.793849i
$122$		0			0
$123$	567.373	+	567.373i	0.415921	+	0.415921i
$124$		0			0
$125$	−494.425	+	494.425i	−0.353781	+	0.353781i
$126$		0			0
$127$		−	1588.68i		−	1.11002i	−0.831843	−	0.555011i	$-0.812714\pi$
$127$		−	1588.68i		−	1.11002i	0.831843	−	0.555011i	$-0.187286\pi$
$128$		0			0
$129$		−	724.111i		−	0.494220i
$130$		0			0
$131$	1500.21	+	1500.21i	1.00057	+	1.00057i	1.00000		0.000565274i	$0.000179932\pi$
$131$	1500.21	+	1500.21i	1.00057	+	1.00057i	0.000565274		1.00000i	$0.499820\pi$
$132$		0			0
$133$	358.047	+	7.69157i	0.233433	+	0.00501461i
$134$		0			0
$135$	156.605			0.0998399
$136$		0			0
$137$			2259.10i			1.40882i	0.709795	+	0.704408i	$0.248788\pi$
$137$			2259.10i			1.40882i	−0.709795	+	0.704408i	$0.751212\pi$
$138$		0			0
$139$	−2145.82	+	2145.82i	−1.30940	+	1.30940i	−0.387547	+	0.921850i	$0.626677\pi$
$139$	−2145.82	+	2145.82i	−1.30940	+	1.30940i	−0.921850	+	0.387547i	$0.873323\pi$
$140$		0			0
$141$	−810.901	−	810.901i	−0.484328	−	0.484328i
$142$		0			0
$143$	−506.223			−0.296031
$144$		0			0
$145$		−	351.098i		−	0.201083i
$146$		0			0
$147$	−1714.52	+	1573.20i	−0.961982	+	0.882690i
$148$		0			0
$149$	−1085.99	+	1085.99i	−0.597100	+	0.597100i	−0.939540	−	0.342440i	$-0.888747\pi$
$149$	−1085.99	+	1085.99i	−0.597100	+	0.597100i	0.342440	+	0.939540i	$0.388747\pi$
$150$		0			0
$151$	−257.765			−0.138918			−0.0694590	−	0.997585i	$-0.522127\pi$
$151$	−257.765			−0.138918			−0.0694590	+	0.997585i	$0.522127\pi$
$152$		0			0
$153$	−2543.50			−1.34399
$154$		0			0
$155$	−489.200	−	489.200i	−0.253507	−	0.253507i
$156$		0			0
$157$	1532.56	−	1532.56i	0.779053	−	0.779053i	−0.200617	−	0.979670i	$-0.564295\pi$
$157$	1532.56	−	1532.56i	0.779053	−	0.779053i	0.979670	+	0.200617i	$0.0642946\pi$
$158$		0			0
$159$	−4626.86			−2.30776
$160$		0			0
$161$	1735.59	−	1662.59i	0.849586	−	0.813852i
$162$		0			0
$163$	177.457	−	177.457i	0.0852729	−	0.0852729i	−0.663184	−	0.748457i	$-0.730795\pi$
$163$	177.457	−	177.457i	0.0852729	−	0.0852729i	0.748457	+	0.663184i	$0.230795\pi$
$164$		0			0
$165$	678.306	−	678.306i	0.320037	−	0.320037i
$166$		0			0
$167$		−	1722.29i		−	0.798054i	−0.916939	−	0.399027i	$-0.869348\pi$
$167$		−	1722.29i		−	0.798054i	0.916939	−	0.399027i	$-0.130652\pi$
$168$		0			0
$169$			2089.67i			0.951147i
$170$		0			0
$171$	260.108	−	260.108i	0.116322	−	0.116322i
$172$		0			0
$173$	−1795.20	+	1795.20i	−0.788939	+	0.788939i	−0.981320	−	0.192381i	$-0.938379\pi$
$173$	−1795.20	+	1795.20i	−0.788939	+	0.788939i	0.192381	+	0.981320i	$0.438379\pi$
$174$		0			0
$175$	1494.15	+	1559.76i	0.645414	+	0.673752i
$176$		0			0
$177$	−2736.52			−1.16209
$178$		0			0
$179$	953.910	−	953.910i	0.398316	−	0.398316i	−0.479323	−	0.877639i	$-0.659118\pi$
$179$	953.910	−	953.910i	0.398316	−	0.398316i	0.877639	+	0.479323i	$0.159118\pi$
$180$		0			0
$181$	1780.34	+	1780.34i	0.731114	+	0.731114i	0.970840	−	0.239727i	$-0.0770578\pi$
$181$	1780.34	+	1780.34i	0.731114	+	0.731114i	−0.239727	+	0.970840i	$0.577058\pi$
$182$		0			0
$183$	−3003.61			−1.21330
$184$		0			0
$185$	207.888			0.0826173
$186$		0			0
$187$	4619.79	−	4619.79i	1.80659	−	1.80659i
$188$		0			0
$189$	21.5256	−	1002.03i	0.00828442	−	0.385645i
$190$		0			0
$191$			2064.00i			0.781916i	0.920409	+	0.390958i	$0.127856\pi$
$191$			2064.00i			0.781916i	−0.920409	+	0.390958i	$0.872144\pi$
$192$		0			0
$193$	1142.67			0.426172			0.213086	−	0.977033i	$-0.431649\pi$
$193$	1142.67			0.426172			0.213086	+	0.977033i	$0.431649\pi$
$194$		0			0
$195$	143.815	+	143.815i	0.0528143	+	0.0528143i
$196$		0			0
$197$	837.511	−	837.511i	0.302894	−	0.302894i	−0.539251	−	0.842145i	$-0.681292\pi$
$197$	837.511	−	837.511i	0.302894	−	0.302894i	0.842145	+	0.539251i	$0.181292\pi$
$198$		0			0
$199$		−	2165.33i		−	0.771338i	−0.922637	−	0.385669i	$-0.873971\pi$
$199$		−	2165.33i		−	0.771338i	0.922637	−	0.385669i	$-0.126029\pi$
$200$		0			0
$201$	−559.056			−0.196183
$202$		0			0
$203$	−2246.48	−	48.2590i	−0.776710	−	0.0166853i
$204$		0			0
$205$	242.021	+	242.021i	0.0824561	+	0.0824561i
$206$		0			0
$207$		−	2468.65i		−	0.828904i
$208$		0			0
$209$			944.877i			0.312720i
$210$		0			0
$211$	−130.423	+	130.423i	−0.0425531	+	0.0425531i	−0.728063	−	0.685510i	$-0.759579\pi$
$211$	−130.423	+	130.423i	−0.0425531	+	0.0425531i	0.685510	+	0.728063i	$0.259579\pi$
$212$		0			0
$213$	−1794.61	−	1794.61i	−0.577298	−	0.577298i
$214$		0			0
$215$		−	308.880i		−	0.0979789i
$216$		0			0
$217$	−3197.37	+	3062.89i	−1.00024	+	0.958167i
$218$		0			0
$219$	2926.20	+	2926.20i	0.902896	+	0.902896i
$220$		0			0
$221$	979.491	+	979.491i	0.298134	+	0.298134i
$222$		0			0
$223$	1221.92			0.366933			0.183466	−	0.983026i	$-0.441268\pi$
$223$	1221.92			0.366933			0.183466	+	0.983026i	$0.441268\pi$
$224$		0			0
$225$	2218.56			0.657351
$226$		0			0
$227$	−3515.16	−	3515.16i	−1.02780	−	1.02780i	−0.999602	−	0.0281934i	$-0.991025\pi$
$227$	−3515.16	−	3515.16i	−1.02780	−	1.02780i	−0.0281934	−	0.999602i	$-0.508975\pi$
$228$		0			0
$229$	−3345.39	−	3345.39i	−0.965369	−	0.965369i	0.0340515	−	0.999420i	$-0.489159\pi$
$229$	−3345.39	−	3345.39i	−0.965369	−	0.965369i	−0.999420	+	0.0340515i	$0.989159\pi$
$230$		0			0
$231$	−4246.88	−	4433.34i	−1.20963	−	1.26274i
$232$		0			0
$233$			537.847i			0.151225i	0.997137	+	0.0756127i	$0.0240913\pi$
$233$			537.847i			0.151225i	−0.997137	+	0.0756127i	$0.975909\pi$
$234$		0			0
$235$	−345.902	−	345.902i	−0.0960177	−	0.0960177i
$236$		0			0
$237$	2210.95	−	2210.95i	0.605976	−	0.605976i
$238$		0			0
$239$			6403.30i			1.73303i	0.499149	+	0.866516i	$0.333646\pi$
$239$			6403.30i			1.73303i	−0.499149	+	0.866516i	$0.666354\pi$
$240$		0			0
$241$		−	1648.67i		−	0.440664i	−0.975425	−	0.220332i	$-0.929286\pi$
$241$		−	1648.67i		−	0.440664i	0.975425	−	0.220332i	$-0.0707140\pi$
$242$		0			0
$243$	−3191.77	−	3191.77i	−0.842603	−	0.842603i
$244$		0			0
$245$	−731.354	+	671.072i	−0.190712	+	0.174993i
$246$		0			0
$247$	−200.333			−0.0516069
$248$		0			0
$249$		−	8398.68i		−	2.13753i
$250$		0			0
$251$	−3775.90	+	3775.90i	−0.949534	+	0.949534i	−0.998786	−	0.0492528i	$-0.984316\pi$
$251$	−3775.90	+	3775.90i	−0.949534	+	0.949534i	0.0492528	+	0.998786i	$0.484316\pi$
$252$		0			0
$253$	4483.84	+	4483.84i	1.11422	+	1.11422i
$254$		0			0
$255$	−2624.91			−0.644620
$256$		0			0
$257$			6959.86i			1.68928i	0.535337	+	0.844638i	$0.320185\pi$
$257$			6959.86i			1.68928i	−0.535337	+	0.844638i	$0.679815\pi$
$258$		0			0
$259$	28.5745	−	1330.16i	0.00685534	−	0.319120i
$260$		0			0
$261$	−1631.99	+	1631.99i	−0.387041	+	0.387041i
$262$		0			0
$263$	4461.40			1.04602			0.523008	−	0.852328i	$-0.324810\pi$
$263$	4461.40			1.04602			0.523008	+	0.852328i	$0.324810\pi$
$264$		0			0
$265$	−1973.66			−0.457512
$266$		0			0
$267$	−3037.62	−	3037.62i	−0.696253	−	0.696253i
$268$		0			0
$269$	2252.15	−	2252.15i	0.510469	−	0.510469i	−0.404201	−	0.914670i	$-0.632450\pi$
$269$	2252.15	−	2252.15i	0.510469	−	0.510469i	0.914670	+	0.404201i	$0.132450\pi$
$270$		0			0
$271$	6670.05			1.49512			0.747559	−	0.664196i	$-0.231226\pi$
$271$	6670.05			1.49512			0.747559	+	0.664196i	$0.231226\pi$
$272$		0			0
$273$	939.960	−	900.425i	0.208384	−	0.199620i
$274$		0			0
$275$	−4029.60	+	4029.60i	−0.883614	+	0.883614i
$276$		0			0
$277$	−1517.50	+	1517.50i	−0.329161	+	0.329161i	−0.852267	−	0.523106i	$-0.824773\pi$
$277$	−1517.50	+	1517.50i	−0.329161	+	0.329161i	0.523106	+	0.852267i	$0.324773\pi$
$278$		0			0
$279$			4547.85i			0.975888i
$280$		0			0
$281$			559.278i			0.118732i	0.998236	+	0.0593661i	$0.0189079\pi$
$281$			559.278i			0.118732i	−0.998236	+	0.0593661i	$0.981092\pi$
$282$		0			0
$283$	25.8379	−	25.8379i	0.00542721	−	0.00542721i	−0.704388	−	0.709815i	$-0.748778\pi$
$283$	25.8379	−	25.8379i	0.00542721	−	0.00542721i	0.709815	+	0.704388i	$0.248778\pi$
$284$		0			0
$285$	268.434	−	268.434i	0.0557917	−	0.0557917i
$286$		0			0
$287$	1581.83	−	1515.30i	0.325339	−	0.311655i
$288$		0			0
$289$	−12964.7			−2.63885
$290$		0			0
$291$	4538.65	−	4538.65i	0.914296	−	0.914296i
$292$		0			0
$293$	3862.53	+	3862.53i	0.770142	+	0.770142i	0.978131	−	0.207989i	$-0.0666918\pi$
$293$	3862.53	+	3862.53i	0.770142	+	0.770142i	−0.207989	+	0.978131i	$0.566692\pi$
$294$		0			0
$295$	−1167.30			−0.230383
$296$		0			0
$297$	2644.33			0.516631
$298$		0			0
$299$	−950.667	+	950.667i	−0.183875	+	0.183875i
$300$		0			0
$301$	−1976.36	−	42.4561i	−0.378456	−	0.00812999i
$302$		0			0
$303$			1353.72i			0.256664i
$304$		0			0
$305$	−1281.23			−0.240535
$306$		0			0
$307$	2044.42	+	2044.42i	0.380069	+	0.380069i	0.871127	−	0.491058i	$-0.163390\pi$
$307$	2044.42	+	2044.42i	0.380069	+	0.380069i	−0.491058	+	0.871127i	$0.663390\pi$
$308$		0			0
$309$	4276.28	−	4276.28i	0.787279	−	0.787279i
$310$		0			0
$311$			9371.35i			1.70868i	0.519711	+	0.854342i	$0.326040\pi$
$311$			9371.35i			1.70868i	−0.519711	+	0.854342i	$0.673960\pi$
$312$		0			0
$313$	8978.95			1.62147			0.810735	−	0.585413i	$-0.199068\pi$
$313$	8978.95			1.62147			0.810735	+	0.585413i	$0.199068\pi$
$314$		0			0
$315$	−21.8962	+	1019.28i	−0.00391655	+	0.182318i
$316$		0			0
$317$	−1767.75	−	1767.75i	−0.313207	−	0.313207i	0.532944	−	0.846151i	$-0.321086\pi$
$317$	−1767.75	−	1767.75i	−0.313207	−	0.313207i	−0.846151	+	0.532944i	$0.821086\pi$
$318$		0			0
$319$		−	5928.41i		−	1.04052i
$320$		0			0
$321$		−	6980.25i		−	1.21371i
$322$		0			0
$323$	1828.24	−	1828.24i	0.314942	−	0.314942i
$324$		0			0
$325$	−854.357	−	854.357i	−0.145819	−	0.145819i
$326$		0			0
$327$			10905.6i			1.84428i
$328$		0			0
$329$	−2260.78	+	2165.69i	−0.378848	+	0.362913i
$330$		0			0
$331$	−4730.32	−	4730.32i	−0.785504	−	0.785504i	0.195250	−	0.980754i	$-0.437448\pi$
$331$	−4730.32	−	4730.32i	−0.785504	−	0.785504i	−0.980754	+	0.195250i	$0.937448\pi$
$332$		0			0
$333$	−966.314	−	966.314i	−0.159020	−	0.159020i
$334$		0			0
$335$	−238.473			−0.0388931
$336$		0			0
$337$	1717.91			0.277688			0.138844	−	0.990314i	$-0.455661\pi$
$337$	1717.91			0.277688			0.138844	+	0.990314i	$0.455661\pi$
$338$		0			0
$339$	2670.05	+	2670.05i	0.427780	+	0.427780i
$340$		0			0
$341$	−8260.32	−	8260.32i	−1.31179	−	1.31179i
$342$		0			0
$343$	4193.30	+	4771.78i	0.660108	+	0.751171i
$344$		0			0
$345$		−	2547.66i		−	0.397570i
$346$		0			0
$347$	7793.80	+	7793.80i	1.20574	+	1.20574i	0.972392	+	0.233352i	$0.0749695\pi$
$347$	7793.80	+	7793.80i	1.20574	+	1.20574i	0.233352	+	0.972392i	$0.425031\pi$
$348$		0			0
$349$	−4211.57	+	4211.57i	−0.645960	+	0.645960i	−0.952014	−	0.306054i	$-0.900991\pi$
$349$	−4211.57	+	4211.57i	−0.645960	+	0.645960i	0.306054	+	0.952014i	$0.400991\pi$
$350$		0			0
$351$			560.652i			0.0852574i
$352$		0			0
$353$			7172.87i			1.08151i	0.841180	+	0.540755i	$0.181862\pi$
$353$			7172.87i			1.08151i	−0.841180	+	0.540755i	$0.818138\pi$
$354$		0			0
$355$	−765.516	−	765.516i	−0.114449	−	0.114449i
$356$		0			0
$357$	−360.798	+	16795.4i	−0.0534887	+	2.48993i
$358$		0			0
$359$	12810.1			1.88326			0.941630	−	0.336650i	$-0.109294\pi$
$359$	12810.1			1.88326			0.941630	+	0.336650i	$0.109294\pi$
$360$		0			0
$361$		−	6485.07i		−	0.945484i
$362$		0			0
$363$	5068.60	−	5068.60i	0.732872	−	0.732872i
$364$		0			0
$365$	1248.21	+	1248.21i	0.178998	+	0.178998i
$366$		0			0
$367$	5502.08			0.782578			0.391289	−	0.920268i	$-0.372029\pi$
$367$	5502.08			0.782578			0.391289	+	0.920268i	$0.372029\pi$
$368$		0			0
$369$		−	2249.95i		−	0.317419i
$370$		0			0
$371$	−271.282	+	12628.3i	−0.0379630	+	1.76720i
$372$		0			0
$373$	−5026.12	+	5026.12i	−0.697702	+	0.697702i	−0.963914	−	0.266213i	$-0.914228\pi$
$373$	−5026.12	+	5026.12i	−0.697702	+	0.697702i	0.266213	+	0.963914i	$0.414228\pi$
$374$		0			0
$375$	4743.53			0.653213
$376$		0			0
$377$	1256.94			0.171713
$378$		0			0
$379$	−6001.11	−	6001.11i	−0.813342	−	0.813342i	0.171792	−	0.985133i	$-0.445044\pi$
$379$	−6001.11	−	6001.11i	−0.813342	−	0.813342i	−0.985133	+	0.171792i	$0.945044\pi$
$380$		0			0
$381$	−7620.96	+	7620.96i	−1.02476	+	1.02476i
$382$		0			0
$383$	−12452.4			−1.66133			−0.830665	−	0.556773i	$-0.812039\pi$
$383$	−12452.4			−1.66133			−0.830665	+	0.556773i	$0.812039\pi$
$384$		0			0
$385$	−1811.57	−	1891.11i	−0.239808	−	0.250337i
$386$		0			0
$387$	−1435.75	+	1435.75i	−0.188588	+	0.188588i
$388$		0			0
$389$	7568.58	−	7568.58i	0.986483	−	0.986483i	−0.0134265	−	0.999910i	$-0.504274\pi$
$389$	7568.58	−	7568.58i	0.986483	−	0.986483i	0.999910	+	0.0134265i	$0.00427393\pi$
$390$		0			0
$391$		−	17351.6i		−	2.24426i
$392$		0			0
$393$		−	14393.1i		−	1.84742i
$394$		0			0
$395$	943.111	−	943.111i	0.120134	−	0.120134i
$396$		0			0
$397$	746.476	−	746.476i	0.0943692	−	0.0943692i	−0.658346	−	0.752715i	$-0.728744\pi$
$397$	746.476	−	746.476i	0.0943692	−	0.0943692i	0.752715	+	0.658346i	$0.228744\pi$
$398$		0			0
$399$	−1680.66	−	1754.46i	−0.210873	−	0.220132i
$400$		0			0
$401$	8402.79			1.04642			0.523211	−	0.852203i	$-0.324734\pi$
$401$	8402.79			1.04642			0.523211	+	0.852203i	$0.324734\pi$
$402$		0			0
$403$	1751.36	−	1751.36i	0.216480	−	0.216480i
$404$		0			0
$405$	−1802.22	−	1802.22i	−0.221119	−	0.221119i
$406$		0			0
$407$	3510.26			0.427511
$408$		0			0
$409$	8430.15			1.01918			0.509589	−	0.860418i	$-0.329797\pi$
$409$	8430.15			1.01918			0.509589	+	0.860418i	$0.329797\pi$
$410$		0			0
$411$	10837.0	−	10837.0i	1.30060	−	1.30060i
$412$		0			0
$413$	−160.448	+	7468.95i	−0.0191165	+	0.889886i
$414$		0			0
$415$		−	3582.58i		−	0.423764i
$416$		0			0
$417$	20587.1			2.41764
$418$		0			0
$419$	−8672.14	−	8672.14i	−1.01113	−	1.01113i	−0.999937	−	0.0111887i	$-0.996438\pi$
$419$	−8672.14	−	8672.14i	−1.01113	−	1.01113i	−0.0111887	−	0.999937i	$-0.503562\pi$
$420$		0			0
$421$	10398.8	−	10398.8i	1.20382	−	1.20382i	0.230818	−	0.972997i	$-0.425860\pi$
$421$	10398.8	−	10398.8i	1.20382	−	1.20382i	0.972997	−	0.230818i	$-0.0741403\pi$
$422$		0			0
$423$			3215.68i			0.369625i
$424$		0			0
$425$	15593.7			1.77978
$426$		0			0
$427$	−176.108	+	8197.92i	−0.0199589	+	0.929099i
$428$		0			0
$429$	2428.36	+	2428.36i	0.273293	+	0.273293i
$430$		0			0
$431$		−	12243.6i		−	1.36834i	−0.729322	−	0.684170i	$-0.760165\pi$
$431$		−	12243.6i		−	1.36834i	0.729322	−	0.684170i	$-0.239835\pi$
$432$		0			0
$433$		−	8787.77i		−	0.975320i	−0.873034	−	0.487660i	$-0.837851\pi$
$433$		−	8787.77i		−	0.975320i	0.873034	−	0.487660i	$-0.162149\pi$
$434$		0			0
$435$	−1684.22	+	1684.22i	−0.185638	+	0.185638i
$436$		0			0
$437$	1774.44	+	1774.44i	0.194240	+	0.194240i
$438$		0			0
$439$			6163.76i			0.670115i	0.942198	+	0.335057i	$0.108756\pi$
$439$			6163.76i			0.670115i	−0.942198	+	0.335057i	$0.891244\pi$
$440$		0			0
$441$	6518.83	+	280.204i	0.703901	+	0.0302564i
$442$		0			0
$443$	−705.371	−	705.371i	−0.0756505	−	0.0756505i	0.668269	−	0.743920i	$-0.267035\pi$
$443$	−705.371	−	705.371i	−0.0756505	−	0.0756505i	−0.743920	+	0.668269i	$0.767035\pi$
$444$		0			0
$445$	−1295.74	−	1295.74i	−0.138032	−	0.138032i
$446$		0			0
$447$	10419.1			1.10247
$448$		0			0
$449$	1808.14			0.190048			0.0950240	−	0.995475i	$-0.469707\pi$
$449$	1808.14			0.190048			0.0950240	+	0.995475i	$0.469707\pi$
$450$		0			0
$451$	4086.62	+	4086.62i	0.426677	+	0.426677i
$452$		0			0
$453$	1236.51	+	1236.51i	0.128247	+	0.128247i
$454$		0			0
$455$	400.954	−	384.089i	0.0413121	−	0.0395745i
$456$		0			0
$457$			13173.0i			1.34838i	0.738560	+	0.674188i	$0.235506\pi$
$457$			13173.0i			1.34838i	−0.738560	+	0.674188i	$0.764494\pi$
$458$		0			0
$459$	−5116.51	−	5116.51i	−0.520301	−	0.520301i
$460$		0			0
$461$	2780.05	−	2780.05i	0.280867	−	0.280867i	−0.552588	−	0.833455i	$-0.686360\pi$
$461$	2780.05	−	2780.05i	0.280867	−	0.280867i	0.833455	+	0.552588i	$0.186360\pi$
$462$		0			0
$463$		−	3630.01i		−	0.364365i	−0.983265	−	0.182182i	$-0.941684\pi$
$463$		−	3630.01i		−	0.364365i	0.983265	−	0.182182i	$-0.0583161\pi$
$464$		0			0
$465$			4693.41i			0.468068i
$466$		0			0
$467$	−39.0209	−	39.0209i	−0.00386654	−	0.00386654i	0.705171	−	0.709037i	$-0.250870\pi$
$467$	−39.0209	−	39.0209i	−0.00386654	−	0.00386654i	−0.709037	+	0.705171i	$0.750870\pi$
$468$		0			0
$469$	−32.7786	+	1525.86i	−0.00322723	+	0.150230i
$470$		0			0
$471$	−14703.4			−1.43842
$472$		0			0
$473$		−	5215.55i		−	0.507001i
$474$		0			0
$475$	−1594.68	+	1594.68i	−0.154040	+	0.154040i
$476$		0			0
$477$	9174.04	+	9174.04i	0.880609	+	0.880609i
$478$		0			0
$479$	−1088.16			−0.103798			−0.0518990	−	0.998652i	$-0.516527\pi$
$479$	−1088.16			−0.103798			−0.0518990	+	0.998652i	$0.516527\pi$
$480$		0			0
$481$			744.247i			0.0705504i
$482$		0			0
$483$	−16301.1	−	350.181i	−1.53567	−	0.0329892i
$484$		0			0
$485$	1936.03	−	1936.03i	0.181259	−	0.181259i
$486$		0			0
$487$	17740.1			1.65068			0.825339	−	0.564638i	$-0.190984\pi$
$487$	17740.1			1.65068			0.825339	+	0.564638i	$0.190984\pi$
$488$		0			0
$489$	−1702.53			−0.157446
$490$		0			0
$491$	−5821.10	−	5821.10i	−0.535036	−	0.535036i	0.387031	−	0.922067i	$-0.373501\pi$
$491$	−5821.10	−	5821.10i	−0.535036	−	0.535036i	−0.922067	+	0.387031i	$0.873501\pi$
$492$		0			0
$493$	−11470.9	+	11470.9i	−1.04792	+	1.04792i
$494$		0			0
$495$	−2689.86			−0.244243
$496$		0			0
$497$	−5003.34	+	4792.90i	−0.451571	+	0.432577i
$498$		0			0
$499$	−4283.85	+	4283.85i	−0.384312	+	0.384312i	−0.872653	−	0.488341i	$-0.837602\pi$
$499$	−4283.85	+	4283.85i	−0.384312	+	0.384312i	0.488341	+	0.872653i	$0.337602\pi$
$500$		0			0
$501$	−8261.88	+	8261.88i	−0.736753	+	0.736753i
$502$		0			0
$503$			743.585i			0.0659142i	0.999457	+	0.0329571i	$0.0104925\pi$
$503$			743.585i			0.0659142i	−0.999457	+	0.0329571i	$0.989508\pi$
$504$		0			0
$505$			577.449i			0.0508834i
$506$		0			0
$507$	10024.2	−	10024.2i	0.878087	−	0.878087i
$508$		0			0
$509$	−6189.71	+	6189.71i	−0.539006	+	0.539006i	−0.923237	−	0.384231i	$-0.874467\pi$
$509$	−6189.71	+	6189.71i	−0.539006	+	0.539006i	0.384231	+	0.923237i	$0.374467\pi$
$510$		0			0
$511$	8158.20	−	7815.06i	0.706258	−	0.676552i
$512$		0			0
$513$	1046.47			0.0900638
$514$		0			0
$515$	1824.11	−	1824.11i	0.156078	−	0.156078i
$516$		0			0
$517$	−5840.68	−	5840.68i	−0.496852	−	0.496852i
$518$		0			0
$519$	17223.2			1.45668
$520$		0			0
$521$	−11779.3			−0.990521			−0.495260	−	0.868745i	$-0.664927\pi$
$521$	−11779.3			−0.990521			−0.495260	+	0.868745i	$0.664927\pi$
$522$		0			0
$523$	−1261.69	+	1261.69i	−0.105488	+	0.105488i	−0.757881	−	0.652393i	$-0.773765\pi$
$523$	−1261.69	+	1261.69i	−0.105488	+	0.105488i	0.652393	+	0.757881i	$0.273765\pi$
$524$		0			0
$525$	314.705	−	14649.7i	0.0261616	−	1.21784i
$526$		0			0
$527$			31965.8i			2.64222i
$528$		0			0
$529$	4673.97			0.384151
$530$		0			0
$531$	5425.92	+	5425.92i	0.443437	+	0.443437i
$532$		0			0
$533$	−866.447	+	866.447i	−0.0704127	+	0.0704127i
$534$		0			0
$535$		−	2977.53i		−	0.240616i
$536$		0			0
$537$	−9151.86			−0.735441
$538$		0			0
$539$	−12349.2	+	11331.3i	−0.986858	+	0.905517i
$540$		0			0
$541$	3180.80	+	3180.80i	0.252778	+	0.252778i	0.822109	−	0.569330i	$-0.192797\pi$
$541$	3180.80	+	3180.80i	0.252778	+	0.252778i	−0.569330	+	0.822109i	$0.692797\pi$
$542$		0			0
$543$		−	17080.7i		−	1.34991i
$544$		0			0
$545$			4651.94i			0.365628i
$546$		0			0
$547$	2706.48	−	2706.48i	0.211555	−	0.211555i	−0.593373	−	0.804928i	$-0.702204\pi$
$547$	2706.48	−	2706.48i	0.211555	−	0.211555i	0.804928	+	0.593373i	$0.202204\pi$
$548$		0			0
$549$	5955.50	+	5955.50i	0.462977	+	0.462977i
$550$		0			0
$551$		−	2346.12i		−	0.181394i
$552$		0			0
$553$	−5904.82	−	6164.09i	−0.454066	−	0.474003i
$554$		0			0
$555$	−997.242	−	997.242i	−0.0762713	−	0.0762713i
$556$		0			0
$557$	−1976.55	−	1976.55i	−0.150358	−	0.150358i	0.627920	−	0.778278i	$-0.283906\pi$
$557$	−1976.55	−	1976.55i	−0.150358	−	0.150358i	−0.778278	+	0.627920i	$0.783906\pi$
$558$		0			0
$559$	1105.80			0.0836682
$560$		0			0
$561$	−44322.5			−3.33565
$562$		0			0
$563$	−1969.01	−	1969.01i	−0.147396	−	0.147396i	0.629558	−	0.776954i	$-0.283236\pi$
$563$	−1969.01	−	1969.01i	−0.147396	−	0.147396i	−0.776954	+	0.629558i	$0.783236\pi$
$564$		0			0
$565$	1138.95	+	1138.95i	0.0848071	+	0.0848071i
$566$		0			0
$567$	−11779.2	+	11283.7i	−0.872448	+	0.835752i
$568$		0			0
$569$		−	12025.5i		−	0.886004i	−0.896521	−	0.443002i	$-0.853913\pi$
$569$		−	12025.5i		−	0.886004i	0.896521	−	0.443002i	$-0.146087\pi$
$570$		0			0
$571$	3067.62	+	3067.62i	0.224826	+	0.224826i	0.810527	−	0.585701i	$-0.199181\pi$
$571$	3067.62	+	3067.62i	0.224826	+	0.224826i	−0.585701	+	0.810527i	$0.699181\pi$
$572$		0			0
$573$	9901.06	−	9901.06i	0.721855	−	0.721855i
$574$		0			0
$575$			15134.8i			1.09768i
$576$		0			0
$577$		−	4574.91i		−	0.330080i	−0.986287	−	0.165040i	$-0.947225\pi$
$577$		−	4574.91i		−	0.330080i	0.986287	−	0.165040i	$-0.0527753\pi$
$578$		0			0
$579$	−5481.42	−	5481.42i	−0.393437	−	0.393437i
$580$		0			0
$581$	−22923.0	−	492.431i	−1.63684	−	0.0351626i
$582$		0			0
$583$	−33325.9			−2.36744
$584$		0			0
$585$		−	570.306i		−	0.0403064i
$586$		0			0
$587$	1826.09	−	1826.09i	0.128400	−	0.128400i	−0.639986	−	0.768386i	$-0.721060\pi$
$587$	1826.09	−	1826.09i	0.128400	−	0.128400i	0.768386	+	0.639986i	$0.221060\pi$
$588$		0			0
$589$	−3268.95	−	3268.95i	−0.228684	−	0.228684i
$590$		0			0
$591$	−8035.12			−0.559257
$592$		0			0
$593$		−	5705.55i		−	0.395107i	−0.980292	−	0.197554i	$-0.936700\pi$
$593$		−	5705.55i		−	0.395107i	0.980292	−	0.197554i	$-0.0632997\pi$
$594$		0			0
$595$	−153.904	+	7164.31i	−0.0106041	+	0.493627i
$596$		0			0
$597$	−10387.1	+	10387.1i	−0.712090	+	0.712090i
$598$		0			0
$599$	−10497.7			−0.716070			−0.358035	−	0.933708i	$-0.616553\pi$
$599$	−10497.7			−0.716070			−0.358035	+	0.933708i	$0.616553\pi$
$600$		0			0
$601$	9375.74			0.636347			0.318173	−	0.948033i	$-0.396931\pi$
$601$	9375.74			0.636347			0.318173	+	0.948033i	$0.396931\pi$
$602$		0			0
$603$	1108.48	+	1108.48i	0.0748606	+	0.0748606i
$604$		0			0
$605$	2162.09	−	2162.09i	0.145291	−	0.145291i
$606$		0			0
$607$	5702.60			0.381320			0.190660	−	0.981656i	$-0.438937\pi$
$607$	5702.60			0.381320			0.190660	+	0.981656i	$0.438937\pi$
$608$		0			0
$609$	10544.9	+	11007.9i	0.701646	+	0.732453i
$610$		0			0
$611$	1238.34	−	1238.34i	0.0819934	−	0.0819934i
$612$		0			0
$613$	7424.25	−	7424.25i	0.489172	−	0.489172i	−0.418873	−	0.908045i	$-0.637575\pi$
$613$	7424.25	−	7424.25i	0.489172	−	0.489172i	0.908045	+	0.418873i	$0.137575\pi$
$614$		0			0
$615$		−	2321.96i		−	0.152245i
$616$		0			0
$617$			22808.8i			1.48825i	0.668043	+	0.744123i	$0.267132\pi$
$617$			22808.8i			1.48825i	−0.668043	+	0.744123i	$0.732868\pi$
$618$		0			0
$619$	−15749.6	+	15749.6i	−1.02267	+	1.02267i	−0.0229291	+	0.999737i	$0.507299\pi$
$619$	−15749.6	+	15749.6i	−1.02267	+	1.02267i	−0.999737	+	0.0229291i	$0.992701\pi$
$620$		0			0
$621$	4965.94	−	4965.94i	0.320896	−	0.320896i
$622$		0			0
$623$	−8468.86	+	8112.65i	−0.544619	+	0.521712i
$624$		0			0
$625$	−12554.8			−0.803507
$626$		0			0
$627$	4532.60	−	4532.60i	0.288699	−	0.288699i
$628$		0			0
$629$	−6792.00	−	6792.00i	−0.430548	−	0.430548i
$630$		0			0
$631$	7632.20			0.481510			0.240755	−	0.970586i	$-0.422605\pi$
$631$	7632.20			0.481510			0.240755	+	0.970586i	$0.422605\pi$
$632$		0			0
$633$	1251.29			0.0785689
$634$		0			0
$635$	−3250.83	+	3250.83i	−0.203158	+	0.203158i
$636$		0			0
$637$	−2402.47	−	2618.28i	−0.149434	−	0.162857i
$638$		0			0
$639$			7116.62i			0.440578i
$640$		0			0
$641$	18811.7			1.15915			0.579576	−	0.814918i	$-0.303218\pi$
$641$	18811.7			1.15915			0.579576	+	0.814918i	$0.303218\pi$
$642$		0			0
$643$	−14360.3	−	14360.3i	−0.880740	−	0.880740i	0.112869	−	0.993610i	$-0.463996\pi$
$643$	−14360.3	−	14360.3i	−0.880740	−	0.880740i	−0.993610	+	0.112869i	$0.963996\pi$
$644$		0			0
$645$	−1481.71	+	1481.71i	−0.0904529	+	0.0904529i
$646$		0			0
$647$			13880.5i			0.843430i	0.906728	+	0.421715i	$0.138572\pi$
$647$			13880.5i			0.843430i	−0.906728	+	0.421715i	$0.861428\pi$
$648$		0			0
$649$	−19710.4			−1.19214
$650$		0			0
$651$	30030.6	+	645.117i	1.80797	+	0.0388389i
$652$		0			0
$653$	6926.86	+	6926.86i	0.415113	+	0.415113i	0.883515	−	0.468402i	$-0.155170\pi$
$653$	6926.86	+	6926.86i	0.415113	+	0.415113i	−0.468402	+	0.883515i	$0.655170\pi$
$654$		0			0
$655$		−	6139.59i		−	0.366250i
$656$		0			0
$657$		−	11604.0i		−	0.689065i
$658$		0			0
$659$	21306.6	−	21306.6i	1.25947	−	1.25947i	0.308118	−	0.951348i	$-0.400301\pi$
$659$	21306.6	−	21306.6i	1.25947	−	1.25947i	0.951348	−	0.308118i	$-0.0996992\pi$
$660$		0			0
$661$	−9192.89	−	9192.89i	−0.540941	−	0.540941i	0.382864	−	0.923805i	$-0.374938\pi$
$661$	−9192.89	−	9192.89i	−0.540941	−	0.540941i	−0.923805	+	0.382864i	$0.874938\pi$
$662$		0			0
$663$		−	9397.29i		−	0.550468i
$664$		0			0
$665$	−716.912	−	748.389i	−0.0418055	−	0.0436410i
$666$		0			0
$667$	−11133.3	−	11133.3i	−0.646303	−	0.646303i
$668$		0			0
$669$	−5861.59	−	5861.59i	−0.338748	−	0.338748i
$670$		0			0
$671$	−21634.1			−1.24467
$672$		0			0
$673$	−24859.6			−1.42388			−0.711938	−	0.702242i	$-0.752182\pi$
$673$	−24859.6			−1.42388			−0.711938	+	0.702242i	$0.752182\pi$
$674$		0			0
$675$	4462.85	+	4462.85i	0.254482	+	0.254482i
$676$		0			0
$677$	11098.7	+	11098.7i	0.630070	+	0.630070i	0.948086	−	0.318016i	$-0.103016\pi$
$677$	11098.7	+	11098.7i	0.630070	+	0.630070i	−0.318016	+	0.948086i	$0.603016\pi$
$678$		0			0
$679$	−12121.5	−	12653.7i	−0.685094	−	0.715175i
$680$		0			0
$681$			33724.7i			1.89770i
$682$		0			0
$683$	5129.88	+	5129.88i	0.287393	+	0.287393i	0.836048	−	0.548656i	$-0.184860\pi$
$683$	5129.88	+	5129.88i	0.287393	+	0.287393i	−0.548656	+	0.836048i	$0.684860\pi$
$684$		0			0
$685$	4622.66	−	4622.66i	0.257844	−	0.257844i
$686$		0			0
$687$			32095.8i			1.78243i
$688$		0			0
$689$		−	7065.77i		−	0.390688i
$690$		0			0
$691$	−1243.53	−	1243.53i	−0.0684606	−	0.0684606i	0.672047	−	0.740508i	$-0.265415\pi$
$691$	−1243.53	−	1243.53i	−0.0684606	−	0.0684606i	−0.740508	+	0.672047i	$0.765415\pi$
$692$		0			0
$693$	−369.726	+	17211.0i	−0.0202666	+	0.943420i
$694$		0			0
$695$	8781.73			0.479295
$696$		0			0
$697$		−	15814.4i		−	0.859416i
$698$		0			0
$699$	2580.07	−	2580.07i	0.139609	−	0.139609i
$700$		0			0
$701$	8513.11	+	8513.11i	0.458681	+	0.458681i	0.898223	−	0.439541i	$-0.144859\pi$
$701$	8513.11	+	8513.11i	0.458681	+	0.458681i	−0.439541	+	0.898223i	$0.644859\pi$
$702$		0			0
$703$	1389.15			0.0745276
$704$		0			0
$705$			3318.60i			0.177285i
$706$		0			0
$707$	3694.78	+	79.3713i	0.196544	+	0.00422216i
$708$		0			0
$709$	20499.4	−	20499.4i	1.08585	−	1.08585i	0.0899043	−	0.995950i	$-0.471344\pi$
$709$	20499.4	−	20499.4i	1.08585	−	1.08585i	0.995950	−	0.0899043i	$-0.0286561\pi$
$710$		0			0
$711$	−8767.63			−0.462464
$712$		0			0
$713$	−31025.1			−1.62959
$714$		0			0
$715$	1035.85	+	1035.85i	0.0541801	+	0.0541801i
$716$		0			0
$717$	30716.8	−	30716.8i	1.59991	−	1.59991i
$718$		0			0
$719$	14074.7			0.730037			0.365018	−	0.931000i	$-0.381063\pi$
$719$	14074.7			0.730037			0.365018	+	0.931000i	$0.381063\pi$
$720$		0			0
$721$	−11420.8	−	11922.2i	−0.589919	−	0.615821i
$722$		0			0
$723$	−7908.69	+	7908.69i	−0.406815	+	0.406815i
$724$		0			0
$725$	10005.4	−	10005.4i	0.512541	−	0.512541i
$726$		0			0
$727$			95.2264i			0.00485798i	0.999997	+	0.00242899i	$0.000773172\pi$
$727$			95.2264i			0.00485798i	−0.999997	+	0.00242899i	$0.999227\pi$
$728$		0			0
$729$			6841.84i			0.347602i
$730$		0			0
$731$	−10091.6	+	10091.6i	−0.510602	+	0.510602i
$732$		0			0
$733$	13958.3	−	13958.3i	0.703357	−	0.703357i	−0.261772	−	0.965130i	$-0.584307\pi$
$733$	13958.3	−	13958.3i	0.703357	−	0.703357i	0.965130	+	0.261772i	$0.0843070\pi$
$734$		0			0
$735$	6727.47	+	289.173i	0.337614	+	0.0145120i
$736$		0			0
$737$	−4026.71			−0.201256
$738$		0			0
$739$	−22132.3	+	22132.3i	−1.10169	+	1.10169i	−0.107485	+	0.994207i	$0.534280\pi$
$739$	−22132.3	+	22132.3i	−1.10169	+	1.10169i	−0.994207	+	0.107485i	$0.965720\pi$
$740$		0			0
$741$	961.004	+	961.004i	0.0476428	+	0.0476428i
$742$		0			0
$743$	5532.55			0.273176			0.136588	−	0.990628i	$-0.456386\pi$
$743$	5532.55			0.273176			0.136588	+	0.990628i	$0.456386\pi$
$744$		0			0
$745$	4444.40			0.218564
$746$		0			0
$747$	−16652.7	+	16652.7i	−0.815651	+	0.815651i
$748$		0			0
$749$	−19051.6	−	409.266i	−0.929412	−	0.0199656i
$750$		0			0
$751$		−	36330.7i		−	1.76528i	−0.470048	−	0.882641i	$-0.655764\pi$
$751$		−	36330.7i		−	1.76528i	0.470048	−	0.882641i	$-0.344236\pi$
$752$		0			0
$753$	36226.2			1.75320
$754$		0			0
$755$	527.449	+	527.449i	0.0254250	+	0.0254250i
$756$		0			0
$757$	15674.3	−	15674.3i	0.752564	−	0.752564i	−0.222393	−	0.974957i	$-0.571387\pi$
$757$	15674.3	−	15674.3i	0.752564	−	0.752564i	0.974957	+	0.222393i	$0.0713869\pi$
$758$		0			0
$759$		−	43018.2i		−	2.05726i
$760$		0			0
$761$	−16656.7			−0.793437			−0.396718	−	0.917940i	$-0.629851\pi$
$761$	−16656.7			−0.793437			−0.396718	+	0.917940i	$0.629851\pi$
$762$		0			0
$763$	29765.2	+	639.417i	1.41229	+	0.0303387i
$764$		0			0
$765$	5204.62	+	5204.62i	0.245978	+	0.245978i
$766$		0			0
$767$		−	4179.00i		−	0.196734i
$768$		0			0
$769$			9190.37i			0.430966i	0.976508	+	0.215483i	$0.0691327\pi$
$769$			9190.37i			0.430966i	−0.976508	+	0.215483i	$0.930867\pi$
$770$		0			0
$771$	33386.6	−	33386.6i	1.55952	−	1.55952i
$772$		0			0
$773$	3285.14	+	3285.14i	0.152857	+	0.152857i	0.779393	−	0.626536i	$-0.215528\pi$
$773$	3285.14	+	3285.14i	0.152857	+	0.152857i	−0.626536	+	0.779393i	$0.715528\pi$
$774$		0			0
$775$		−	27882.0i		−	1.29233i
$776$		0			0
$777$	−6517.88	+	6243.74i	−0.300937	+	0.288279i
$778$		0			0
$779$	1617.24	+	1617.24i	0.0743822	+	0.0743822i
$780$		0			0
$781$	−12926.0	−	12926.0i	−0.592227	−	0.592227i
$782$		0			0
$783$	−6565.83			−0.299672
$784$		0			0
$785$	−6271.96			−0.285167
$786$		0			0
$787$	−11875.1	−	11875.1i	−0.537867	−	0.537867i	0.385035	−	0.922902i	$-0.374189\pi$
$787$	−11875.1	−	11875.1i	−0.537867	−	0.537867i	−0.922902	+	0.385035i	$0.874189\pi$
$788$		0			0
$789$	−21401.5	−	21401.5i	−0.965668	−	0.965668i
$790$		0			0
$791$	7444.08	−	7130.98i	0.334616	−	0.320541i
$792$		0			0
$793$		−	4586.87i		−	0.205403i
$794$		0			0
$795$	9467.67	+	9467.67i	0.422369	+	0.422369i
$796$		0			0
$797$	−27267.4	+	27267.4i	−1.21187	+	1.21187i	−0.241460	+	0.970411i	$0.577626\pi$
$797$	−27267.4	+	27267.4i	−1.21187	+	1.21187i	−0.970411	+	0.241460i	$0.922374\pi$
$798$		0			0
$799$			22602.3i			1.00076i
$800$		0			0
$801$			12045.9i			0.531361i
$802$		0			0
$803$	21076.5	+	21076.5i	0.926244	+	0.926244i
$804$		0			0
$805$	−6953.48	−	149.375i	−0.304445	−	0.00654008i
$806$		0			0
$807$	−21607.3			−0.942518
$808$		0			0
$809$			37899.6i			1.64707i	0.567265	+	0.823535i	$0.308001\pi$
$809$			37899.6i			1.64707i	−0.567265	+	0.823535i	$0.691999\pi$
$810$		0			0
$811$	2453.46	−	2453.46i	0.106230	−	0.106230i	−0.651994	−	0.758224i	$-0.726067\pi$
$811$	2453.46	−	2453.46i	0.106230	−	0.106230i	0.758224	+	0.651994i	$0.226067\pi$
$812$		0			0
$813$	−31996.4	−	31996.4i	−1.38027	−	1.38027i
$814$		0			0
$815$	−726.238			−0.0312135
$816$		0			0
$817$		−	2064.01i		−	0.0883850i
$818$		0			0
$819$	−3649.08	−	78.3895i	−0.155689	−	0.00334451i
$820$		0			0
$821$	−16123.1	+	16123.1i	−0.685385	+	0.685385i	−0.961208	−	0.275823i	$-0.911050\pi$
$821$	−16123.1	+	16123.1i	−0.685385	+	0.685385i	0.275823	+	0.961208i	$0.411050\pi$
$822$		0			0
$823$	−24632.9			−1.04332			−0.521658	−	0.853155i	$-0.674686\pi$
$823$	−24632.9			−1.04332			−0.521658	+	0.853155i	$0.674686\pi$
$824$		0			0
$825$	38660.1			1.63148
$826$		0			0
$827$	−1089.80	−	1089.80i	−0.0458236	−	0.0458236i	0.683824	−	0.729647i	$-0.260316\pi$
$827$	−1089.80	−	1089.80i	−0.0458236	−	0.0458236i	−0.729647	+	0.683824i	$0.760316\pi$
$828$		0			0
$829$	−4055.04	+	4055.04i	−0.169888	+	0.169888i	−0.786930	−	0.617042i	$-0.788331\pi$
$829$	−4055.04	+	4055.04i	−0.169888	+	0.169888i	0.617042	+	0.786930i	$0.288331\pi$
$830$		0			0
$831$	14559.0			0.607755
$832$		0			0
$833$	45819.4	+	1969.49i	1.90582	+	0.0819194i
$834$		0			0
$835$	−3524.22	+	3524.22i	−0.146061	+	0.146061i
$836$		0			0
$837$	−9148.46	+	9148.46i	−0.377798	+	0.377798i
$838$		0			0
$839$			35349.1i			1.45457i	0.686335	+	0.727285i	$0.259218\pi$
$839$			35349.1i			1.45457i	−0.686335	+	0.727285i	$0.740782\pi$
$840$		0			0
$841$		−	9668.83i		−	0.396442i
$842$		0			0
$843$	2682.87	−	2682.87i	0.109612	−	0.109612i
$844$		0			0
$845$	4275.97	−	4275.97i	0.174080	−	0.174080i
$846$		0			0
$847$	−13536.8	−	14131.2i	−0.549151	−	0.573263i
$848$		0			0
$849$	−247.890			−0.0100207
$850$		0			0
$851$	6592.13	−	6592.13i	0.265541	−	0.265541i
$852$		0			0
$853$	28356.9	+	28356.9i	1.13824	+	1.13824i	0.988765	+	0.149477i	$0.0477591\pi$
$853$	28356.9	+	28356.9i	1.13824	+	1.13824i	0.149477	+	0.988765i	$0.452241\pi$
$854$		0			0
$855$	−1064.49			−0.0425787
$856$		0			0
$857$	−5617.28			−0.223900			−0.111950	−	0.993714i	$-0.535710\pi$
$857$	−5617.28			−0.223900			−0.111950	+	0.993714i	$0.535710\pi$
$858$		0			0
$859$	−26923.7	+	26923.7i	−1.06941	+	1.06941i	−0.0720059	+	0.997404i	$0.522940\pi$
$859$	−26923.7	+	26923.7i	−1.06941	+	1.06941i	−0.997404	+	0.0720059i	$0.977060\pi$
$860$		0			0
$861$	−14857.0	−	319.158i	−0.588066	−	0.0126328i
$862$		0			0
$863$			7764.09i			0.306249i	0.988207	+	0.153124i	$0.0489335\pi$
$863$			7764.09i			0.306249i	−0.988207	+	0.153124i	$0.951067\pi$
$864$		0			0
$865$	7346.82			0.288785
$866$		0			0
$867$	62191.9	+	62191.9i	2.43616	+	2.43616i
$868$		0			0
$869$	15924.8	−	15924.8i	0.621646	−	0.621646i
$870$		0			0
$871$		−	853.745i		−	0.0332124i
$872$		0			0
$873$	−17998.3			−0.697765
$874$		0			0
$875$	278.123	−	12946.8i	0.0107455	−	0.500207i
$876$		0			0
$877$	15619.2	+	15619.2i	0.601396	+	0.601396i	0.940683	−	0.339287i	$-0.110186\pi$
$877$	15619.2	+	15619.2i	0.601396	+	0.601396i	−0.339287	+	0.940683i	$0.610186\pi$
$878$		0			0
$879$		−	37057.3i		−	1.42197i
$880$		0			0
$881$			45320.9i			1.73314i	0.499053	+	0.866571i	$0.333681\pi$
$881$			45320.9i			1.73314i	−0.499053	+	0.866571i	$0.666319\pi$
$882$		0			0
$883$	5998.33	−	5998.33i	0.228607	−	0.228607i	−0.583504	−	0.812110i	$-0.698319\pi$
$883$	5998.33	−	5998.33i	0.228607	−	0.228607i	0.812110	+	0.583504i	$0.198319\pi$
$884$		0			0
$885$	5599.59	+	5599.59i	0.212687	+	0.212687i
$886$		0			0
$887$		−	25507.8i		−	0.965578i	−0.875737	−	0.482789i	$-0.839624\pi$
$887$		−	25507.8i		−	0.965578i	0.875737	−	0.482789i	$-0.160376\pi$
$888$		0			0
$889$	20353.5	+	21247.1i	0.767866	+	0.801581i
$890$		0			0
$891$	−30431.2	−	30431.2i	−1.14420	−	1.14420i
$892$		0			0
$893$	−2311.40	−	2311.40i	−0.0866158	−	0.0866158i
$894$		0			0
$895$	−3903.86			−0.145801
$896$		0			0
$897$	9120.74			0.339501
$898$		0			0
$899$	20510.3	+	20510.3i	0.760907	+	0.760907i
$900$		0			0
$901$	64482.2	+	64482.2i	2.38426	+	2.38426i
$902$		0			0
$903$	9276.96	+	9684.29i	0.341880	+	0.356891i
$904$		0			0
$905$		−	7286.01i		−	0.267619i
$906$		0			0
$907$	26334.9	+	26334.9i	0.964096	+	0.964096i	0.999377	−	0.0352818i	$-0.0112329\pi$
$907$	26334.9	+	26334.9i	0.964096	+	0.964096i	−0.0352818	+	0.999377i	$0.511233\pi$
$908$		0			0
$909$	2684.13	−	2684.13i	0.0979393	−	0.0979393i
$910$		0			0
$911$		−	16395.8i		−	0.596285i	−0.954521	−	0.298142i	$-0.903633\pi$
$911$		−	16395.8i		−	0.596285i	0.954521	−	0.298142i	$-0.0963671\pi$
$912$		0			0
$913$		−	60493.1i		−	2.19280i
$914$		0			0
$915$	6146.11	+	6146.11i	0.222059	+	0.222059i
$916$		0			0
$917$	−39283.9	−	843.896i	−1.41469	−	0.0303903i
$918$		0			0
$919$	26851.4			0.963815			0.481908	−	0.876222i	$-0.339944\pi$
$919$	26851.4			0.963815			0.481908	+	0.876222i	$0.339944\pi$
$920$		0			0
$921$		−	19614.3i		−	0.701750i
$922$		0			0
$923$	2740.58	−	2740.58i	0.0977327	−	0.0977327i
$924$		0			0
$925$	5924.29	+	5924.29i	0.210583	+	0.210583i
$926$		0			0
$927$	−16957.9			−0.600830
$928$		0			0
$929$			19627.5i			0.693171i	0.938018	+	0.346585i	$0.112659\pi$
$929$			19627.5i			0.693171i	−0.938018	+	0.346585i	$0.887341\pi$
$930$		0			0
$931$	−4887.08	+	4484.26i	−0.172038	+	0.157858i
$932$		0			0
$933$	44954.6	−	44954.6i	1.57744	−	1.57744i
$934$		0			0
$935$	−18906.4			−0.661290
$936$		0			0
$937$	4230.42			0.147494			0.0737469	−	0.997277i	$-0.476504\pi$
$937$	4230.42			0.147494			0.0737469	+	0.997277i	$0.476504\pi$
$938$		0			0
$939$	−43072.2	−	43072.2i	−1.49692	−	1.49692i
$940$		0			0
$941$	−3133.94	+	3133.94i	−0.108569	+	0.108569i	−0.759305	−	0.650735i	$-0.774461\pi$
$941$	−3133.94	+	3133.94i	−0.108569	+	0.108569i	0.650735	+	0.759305i	$0.274461\pi$
$942$		0			0
$943$	15349.0			0.530045
$944$		0			0
$945$	−2094.44	+	2006.35i	−0.0720974	+	0.0690650i
$946$		0			0
$947$	−273.354	+	273.354i	−0.00937997	+	0.00937997i	−0.711781	−	0.702401i	$-0.752111\pi$
$947$	−273.354	+	273.354i	−0.00937997	+	0.00937997i	0.702401	+	0.711781i	$0.252111\pi$
$948$		0			0
$949$	−4468.65	+	4468.65i	−0.152854	+	0.152854i
$950$		0			0
$951$			16959.9i			0.578298i
$952$		0			0
$953$			20906.0i			0.710610i	0.934750	+	0.355305i	$0.115623\pi$
$953$			20906.0i			0.710610i	−0.934750	+	0.355305i	$0.884377\pi$
$954$		0			0
$955$	4223.44	−	4223.44i	0.143107	−	0.143107i
$956$		0			0
$957$	−28438.7	+	28438.7i	−0.960599	+	0.960599i
$958$		0			0
$959$	−28942.5	−	30213.3i	−0.974559	−	1.01735i
$960$		0			0
$961$	27364.8			0.918558
$962$		0			0
$963$	−13840.3	+	13840.3i	−0.463133	+	0.463133i
$964$		0			0
$965$	−2338.18	−	2338.18i	−0.0779986	−	0.0779986i
$966$		0			0
$967$	−9771.57			−0.324956			−0.162478	−	0.986712i	$-0.551949\pi$
$967$	−9771.57			−0.324956			−0.162478	+	0.986712i	$0.551949\pi$
$968$		0			0
$969$	−17540.3			−0.581501
$970$		0			0
$971$	9977.45	−	9977.45i	0.329754	−	0.329754i	−0.522739	−	0.852493i	$-0.675090\pi$
$971$	9977.45	−	9977.45i	0.329754	−	0.329754i	0.852493	+	0.522739i	$0.175090\pi$
$972$		0			0
$973$	1207.06	−	56189.5i	0.0397705	−	1.85134i
$974$		0			0
$975$			8196.74i			0.269237i
$976$		0			0
$977$	15291.4			0.500732			0.250366	−	0.968151i	$-0.419449\pi$
$977$	15291.4			0.500732			0.250366	+	0.968151i	$0.419449\pi$
$978$		0			0
$979$	−21879.1	−	21879.1i	−0.714258	−	0.714258i
$980$		0			0
$981$	21623.4	−	21623.4i	0.703753	−	0.703753i
$982$		0			0
$983$		−	14778.2i		−	0.479504i	−0.970834	−	0.239752i	$-0.922934\pi$
$983$		−	14778.2i		−	0.479504i	0.970834	−	0.239752i	$-0.0770661\pi$
$984$		0			0
$985$	−3427.50			−0.110872
$986$		0			0
$987$	21233.9	+	456.147i	0.684785	+	0.0147106i
$988$		0			0
$989$	−9794.60	−	9794.60i	−0.314914	−	0.314914i
$990$		0			0
$991$			7672.15i			0.245927i	0.992411	+	0.122964i	$0.0392398\pi$
$991$			7672.15i			0.245927i	−0.992411	+	0.122964i	$0.960760\pi$
$992$		0			0
$993$			45382.9i			1.45033i
$994$		0			0
$995$	−4430.79	+	4430.79i	−0.141171	+	0.141171i
$996$		0			0
$997$	28731.6	+	28731.6i	0.912677	+	0.912677i	0.996482	−	0.0838054i	$-0.0267074\pi$
$997$	28731.6	+	28731.6i	0.912677	+	0.912677i	−0.0838054	+	0.996482i	$0.526707\pi$
$998$		0			0
$999$		−	3887.68i		−	0.123124i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.4.j.b.111.8		88
4.3	odd	2		112.4.j.b.83.10	yes	88
7.6	odd	2	inner	448.4.j.b.111.37		88
16.5	even	4		112.4.j.b.27.9	✓	88
16.11	odd	4	inner	448.4.j.b.335.37		88
28.27	even	2		112.4.j.b.83.9	yes	88
112.27	even	4	inner	448.4.j.b.335.8		88
112.69	odd	4		112.4.j.b.27.10	yes	88

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.4.j.b.27.9	✓	88	16.5	even	4
112.4.j.b.27.10	yes	88	112.69	odd	4
112.4.j.b.83.9	yes	88	28.27	even	2
112.4.j.b.83.10	yes	88	4.3	odd	2
448.4.j.b.111.8		88	1.1	even	1	trivial
448.4.j.b.111.37		88	7.6	odd	2	inner
448.4.j.b.335.8		88	112.27	even	4	inner
448.4.j.b.335.37		88	16.11	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	448.j (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.79702 - 4.79702i) q^{3} +(-2.04624 - 2.04624i) q^{5} +(-13.3740 + 12.8115i) q^{7} +19.0229i q^{9} +(-34.5515 - 34.5515i) q^{11} +(7.32563 - 7.32563i) q^{13} +19.6317i q^{15} +133.707i q^{17} +(-13.6735 - 13.6735i) q^{19} +(125.613 + 2.69841i) q^{21} -129.773 q^{23} -116.626i q^{25} +(-38.2664 + 38.2664i) q^{27} +(85.7909 + 85.7909i) q^{29} +239.073 q^{31} +331.489i q^{33} +(53.5820 + 1.15105i) q^{35} +(-50.7975 + 50.7975i) q^{37} -70.2824 q^{39} -118.276 q^{41} +(75.4750 + 75.4750i) q^{43} +(38.9254 - 38.9254i) q^{45} +169.043 q^{47} +(14.7299 - 342.684i) q^{49} +(641.398 - 641.398i) q^{51} +(482.264 - 482.264i) q^{53} +141.401i q^{55} +131.184i q^{57} +(285.231 - 285.231i) q^{59} +(313.070 - 313.070i) q^{61} +(-243.712 - 254.413i) q^{63} -29.9800 q^{65} +(58.2711 - 58.2711i) q^{67} +(622.523 + 622.523i) q^{69} +374.109 q^{71} -610.003 q^{73} +(-559.457 + 559.457i) q^{75} +(904.751 + 19.4359i) q^{77} +460.899i q^{79} +880.748 q^{81} +(875.405 + 875.405i) q^{83} +(273.598 - 273.598i) q^{85} -823.082i q^{87} +633.231 q^{89} +(-4.12080 + 191.826i) q^{91} +(-1146.84 - 1146.84i) q^{93} +55.9584i q^{95} +946.138i q^{97} +(657.269 - 657.269i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 4 q^{7} - 112 q^{11} + 52 q^{21} - 160 q^{23} + 528 q^{29} - 476 q^{35} - 896 q^{37} + 8 q^{39} - 40 q^{43} - 1376 q^{49} - 1504 q^{51} - 1560 q^{53} - 8 q^{65} - 648 q^{67} + 456 q^{71} + 292 q^{77}+ \cdots - 7592 q^{99}+O(q^{100}) \) 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

L-functions

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