LMFDB - Embedded newform 448.8.a.l.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [448,8,Mod(1,448)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("448.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	448.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-70,0,-126,0,686] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$139.948491417$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1969}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 492 $ x^2 - x - 492
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 14)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-21.6867$ of defining polynomial
Character	$\chi$	$=$	448.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+9.37342 q^{3} -462.361 q^{5} +343.000 q^{7} -2099.14 q^{9} -3881.05 q^{11} +11585.6 q^{13} -4333.90 q^{15} -15242.4 q^{17} +32461.7 q^{19} +3215.08 q^{21} +56146.1 q^{23} +135652. q^{25} -40175.8 q^{27} +26442.0 q^{29} -45448.2 q^{31} -36378.7 q^{33} -158590. q^{35} +555245. q^{37} +108596. q^{39} +306385. q^{41} -780755. q^{43} +970559. q^{45} -531243. q^{47} +117649. q^{49} -142873. q^{51} +363593. q^{53} +1.79445e6 q^{55} +304277. q^{57} +2.14095e6 q^{59} -888824. q^{61} -720005. q^{63} -5.35672e6 q^{65} -4.34541e6 q^{67} +526281. q^{69} +663207. q^{71} +1.34202e6 q^{73} +1.27153e6 q^{75} -1.33120e6 q^{77} +7.05834e6 q^{79} +4.21423e6 q^{81} -6.60874e6 q^{83} +7.04748e6 q^{85} +247851. q^{87} -5.32998e6 q^{89} +3.97385e6 q^{91} -426005. q^{93} -1.50090e7 q^{95} -2.09810e6 q^{97} +8.14686e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 70 q^{3} - 126 q^{5} + 686 q^{7} + 2014 q^{9} + 3420 q^{11} + 6398 q^{13} - 31032 q^{15} - 38472 q^{17} + 43358 q^{19} - 24010 q^{21} + 89928 q^{23} + 170666 q^{25} - 193060 q^{27} - 159576 q^{29}+ \cdots + 38177100 q^{99}+O(q^{100}) $ 2 * q - 70 * q^3 - 126 * q^5 + 686 * q^7 + 2014 * q^9 + 3420 * q^11 + 6398 * q^13 - 31032 * q^15 - 38472 * q^17 + 43358 * q^19 - 24010 * q^21 + 89928 * q^23 + 170666 * q^25 - 193060 * q^27 - 159576 * q^29 - 143612 * q^31 - 615888 * q^33 - 43218 * q^35 + 271832 * q^37 + 520352 * q^39 + 64848 * q^41 - 1527964 * q^43 + 2354058 * q^45 + 485436 * q^47 + 235298 * q^49 + 1700940 * q^51 + 145716 * q^53 + 4250232 * q^55 - 560596 * q^57 + 4183662 * q^59 + 280658 * q^61 + 690802 * q^63 - 7101612 * q^65 - 5671648 * q^67 - 2155104 * q^69 - 619272 * q^71 + 3939628 * q^73 - 1507618 * q^75 + 1173060 * q^77 + 4656616 * q^79 + 7353742 * q^81 - 1235850 * q^83 - 766044 * q^85 + 15012732 * q^87 - 17241420 * q^89 + 2194514 * q^91 + 7365592 * q^93 - 11343960 * q^95 - 740936 * q^97 + 38177100 * q^99

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
$3$	9.37342	0.200435	0.100217	−	0.994966i	$-0.468046\pi$
$3$	9.37342	0.200435	0.100217	+	0.994966i	$0.468046\pi$
$4$	0	0
$5$	−462.361	−1.65419	−0.827096	−	0.562061i	$-0.810009\pi$
$5$	−462.361	−1.65419	−0.827096	+	0.562061i	$0.810009\pi$
$6$	0	0
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	0	0
$9$	−2099.14	−0.959826
$10$	0	0
$11$	−3881.05	−0.879174	−0.439587	−	0.898200i	$-0.644875\pi$
$11$	−3881.05	−0.879174	−0.439587	+	0.898200i	$0.644875\pi$
$12$	0	0
$13$	11585.6	1.46257	0.731284	−	0.682073i	$-0.238922\pi$
$13$	11585.6	1.46257	0.731284	+	0.682073i	$0.238922\pi$
$14$	0	0
$15$	−4333.90	−0.331558
$16$	0	0
$17$	−15242.4	−0.752457	−0.376229	−	0.926527i	$-0.622779\pi$
$17$	−15242.4	−0.752457	−0.376229	+	0.926527i	$0.622779\pi$
$18$	0	0
$19$	32461.7	1.08576	0.542880	−	0.839810i	$-0.317334\pi$
$19$	32461.7	1.08576	0.542880	+	0.839810i	$0.317334\pi$
$20$	0	0
$21$	3215.08	0.0757573
$22$	0	0
$23$	56146.1	0.962215	0.481108	−	0.876662i	$-0.340235\pi$
$23$	56146.1	0.962215	0.481108	+	0.876662i	$0.340235\pi$
$24$	0	0
$25$	135652.	1.73635
$26$	0	0
$27$	−40175.8	−0.392818
$28$	0	0
$29$	26442.0	0.201326	0.100663	−	0.994921i	$-0.467904\pi$
$29$	26442.0	0.201326	0.100663	+	0.994921i	$0.467904\pi$
$30$	0	0
$31$	−45448.2	−0.274000	−0.137000	−	0.990571i	$-0.543746\pi$
$31$	−45448.2	−0.274000	−0.137000	+	0.990571i	$0.543746\pi$
$32$	0	0
$33$	−36378.7	−0.176217
$34$	0	0
$35$	−158590.	−0.625226
$36$	0	0
$37$	555245.	1.80210	0.901049	−	0.433717i	$-0.142798\pi$
$37$	555245.	1.80210	0.901049	+	0.433717i	$0.142798\pi$
$38$	0	0
$39$	108596.	0.293150
$40$	0	0
$41$	306385.	0.694264	0.347132	−	0.937816i	$-0.387156\pi$
$41$	306385.	0.694264	0.347132	+	0.937816i	$0.387156\pi$
$42$	0	0
$43$	−780755.	−1.49753	−0.748765	−	0.662836i	$-0.769353\pi$
$43$	−780755.	−1.49753	−0.748765	+	0.662836i	$0.769353\pi$
$44$	0	0
$45$	970559.	1.58774
$46$	0	0
$47$	−531243.	−0.746364	−0.373182	−	0.927758i	$-0.621733\pi$
$47$	−531243.	−0.746364	−0.373182	+	0.927758i	$0.621733\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	0	0
$51$	−142873.	−0.150819
$52$	0	0
$53$	363593.	0.335467	0.167733	−	0.985832i	$-0.446355\pi$
$53$	363593.	0.335467	0.167733	+	0.985832i	$0.446355\pi$
$54$	0	0
$55$	1.79445e6	1.45432
$56$	0	0
$57$	304277.	0.217624
$58$	0	0
$59$	2.14095e6	1.35714	0.678570	−	0.734535i	$-0.262600\pi$
$59$	2.14095e6	1.35714	0.678570	+	0.734535i	$0.262600\pi$
$60$	0	0
$61$	−888824.	−0.501373	−0.250687	−	0.968068i	$-0.580656\pi$
$61$	−888824.	−0.501373	−0.250687	+	0.968068i	$0.580656\pi$
$62$	0	0
$63$	−720005.	−0.362780
$64$	0	0
$65$	−5.35672e6	−2.41937
$66$	0	0
$67$	−4.34541e6	−1.76510	−0.882549	−	0.470221i	$-0.844174\pi$
$67$	−4.34541e6	−1.76510	−0.882549	+	0.470221i	$0.844174\pi$
$68$	0	0
$69$	526281.	0.192862
$70$	0	0
$71$	663207.	0.219910	0.109955	−	0.993937i	$-0.464929\pi$
$71$	663207.	0.219910	0.109955	+	0.993937i	$0.464929\pi$
$72$	0	0
$73$	1.34202e6	0.403765	0.201882	−	0.979410i	$-0.435294\pi$
$73$	1.34202e6	0.403765	0.201882	+	0.979410i	$0.435294\pi$
$74$	0	0
$75$	1.27153e6	0.348026
$76$	0	0
$77$	−1.33120e6	−0.332297
$78$	0	0
$79$	7.05834e6	1.61067	0.805337	−	0.592818i	$-0.201984\pi$
$79$	7.05834e6	1.61067	0.805337	+	0.592818i	$0.201984\pi$
$80$	0	0
$81$	4.21423e6	0.881091
$82$	0	0
$83$	−6.60874e6	−1.26866	−0.634330	−	0.773063i	$-0.718724\pi$
$83$	−6.60874e6	−1.26866	−0.634330	+	0.773063i	$0.718724\pi$
$84$	0	0
$85$	7.04748e6	1.24471
$86$	0	0
$87$	247851.	0.0403528
$88$	0	0
$89$	−5.32998e6	−0.801420	−0.400710	−	0.916205i	$-0.631237\pi$
$89$	−5.32998e6	−0.801420	−0.400710	+	0.916205i	$0.631237\pi$
$90$	0	0
$91$	3.97385e6	0.552799
$92$	0	0
$93$	−426005.	−0.0549192
$94$	0	0
$95$	−1.50090e7	−1.79606
$96$	0	0
$97$	−2.09810e6	−0.233413	−0.116707	−	0.993166i	$-0.537234\pi$
$97$	−2.09810e6	−0.233413	−0.116707	+	0.993166i	$0.537234\pi$
$98$	0	0
$99$	8.14686e6	0.843854
$100$	0	0
$101$	−1.95538e7	−1.88845	−0.944227	−	0.329294i	$-0.893189\pi$
$101$	−1.95538e7	−1.88845	−0.944227	+	0.329294i	$0.893189\pi$
$102$	0	0
$103$	−8.95093e6	−0.807119	−0.403560	−	0.914953i	$-0.632227\pi$
$103$	−8.95093e6	−0.807119	−0.403560	+	0.914953i	$0.632227\pi$
$104$	0	0
$105$	−1.48653e6	−0.125317
$106$	0	0
$107$	1.39509e7	1.10093	0.550466	−	0.834858i	$-0.314450\pi$
$107$	1.39509e7	1.10093	0.550466	+	0.834858i	$0.314450\pi$
$108$	0	0
$109$	−5.84295e6	−0.432155	−0.216077	−	0.976376i	$-0.569326\pi$
$109$	−5.84295e6	−0.432155	−0.216077	+	0.976376i	$0.569326\pi$
$110$	0	0
$111$	5.20454e6	0.361203
$112$	0	0
$113$	−1.20951e7	−0.788562	−0.394281	−	0.918990i	$-0.629006\pi$
$113$	−1.20951e7	−0.788562	−0.394281	+	0.918990i	$0.629006\pi$
$114$	0	0
$115$	−2.59598e7	−1.59169
$116$	0	0
$117$	−2.43197e7	−1.40381
$118$	0	0
$119$	−5.22814e6	−0.284402
$120$	0	0
$121$	−4.42462e6	−0.227053
$122$	0	0
$123$	2.87188e6	0.139155
$124$	0	0
$125$	−2.65984e7	−1.21807
$126$	0	0
$127$	−325760.	−0.0141119	−0.00705594	−	0.999975i	$-0.502246\pi$
$127$	−325760.	−0.0141119	−0.00705594	+	0.999975i	$0.502246\pi$
$128$	0	0
$129$	−7.31834e6	−0.300157
$130$	0	0
$131$	−6.38049e6	−0.247973	−0.123986	−	0.992284i	$-0.539568\pi$
$131$	−6.38049e6	−0.247973	−0.123986	+	0.992284i	$0.539568\pi$
$132$	0	0
$133$	1.11344e7	0.410379
$134$	0	0
$135$	1.85757e7	0.649796
$136$	0	0
$137$	4.62324e6	0.153612	0.0768059	−	0.997046i	$-0.475528\pi$
$137$	4.62324e6	0.153612	0.0768059	+	0.997046i	$0.475528\pi$
$138$	0	0
$139$	3.55394e7	1.12243	0.561214	−	0.827671i	$-0.310335\pi$
$139$	3.55394e7	1.12243	0.561214	+	0.827671i	$0.310335\pi$
$140$	0	0
$141$	−4.97956e6	−0.149597
$142$	0	0
$143$	−4.49642e7	−1.28585
$144$	0	0
$145$	−1.22257e7	−0.333032
$146$	0	0
$147$	1.10277e6	0.0286336
$148$	0	0
$149$	−1.01708e7	−0.251885	−0.125943	−	0.992038i	$-0.540196\pi$
$149$	−1.01708e7	−0.251885	−0.125943	+	0.992038i	$0.540196\pi$
$150$	0	0
$151$	2.45272e7	0.579735	0.289867	−	0.957067i	$-0.406389\pi$
$151$	2.45272e7	0.579735	0.289867	+	0.957067i	$0.406389\pi$
$152$	0	0
$153$	3.19959e7	0.722228
$154$	0	0
$155$	2.10135e7	0.453249
$156$	0	0
$157$	1.27379e7	0.262693	0.131346	−	0.991337i	$-0.458070\pi$
$157$	1.27379e7	0.262693	0.131346	+	0.991337i	$0.458070\pi$
$158$	0	0
$159$	3.40810e6	0.0672393
$160$	0	0
$161$	1.92581e7	0.363683
$162$	0	0
$163$	8.51531e7	1.54008	0.770041	−	0.637995i	$-0.220236\pi$
$163$	8.51531e7	1.54008	0.770041	+	0.637995i	$0.220236\pi$
$164$	0	0
$165$	1.68201e7	0.291497
$166$	0	0
$167$	−3.74260e6	−0.0621822	−0.0310911	−	0.999517i	$-0.509898\pi$
$167$	−3.74260e6	−0.0621822	−0.0310911	+	0.999517i	$0.509898\pi$
$168$	0	0
$169$	7.14770e7	1.13910
$170$	0	0
$171$	−6.81417e7	−1.04214
$172$	0	0
$173$	7.26865e7	1.06731	0.533657	−	0.845701i	$-0.320817\pi$
$173$	7.26865e7	1.06731	0.533657	+	0.845701i	$0.320817\pi$
$174$	0	0
$175$	4.65288e7	0.656279
$176$	0	0
$177$	2.00680e7	0.272018
$178$	0	0
$179$	−1.02445e8	−1.33507	−0.667536	−	0.744578i	$-0.732651\pi$
$179$	−1.02445e8	−1.33507	−0.667536	+	0.744578i	$0.732651\pi$
$180$	0	0
$181$	−4.79161e6	−0.0600630	−0.0300315	−	0.999549i	$-0.509561\pi$
$181$	−4.79161e6	−0.0600630	−0.0300315	+	0.999549i	$0.509561\pi$
$182$	0	0
$183$	−8.33131e6	−0.100493
$184$	0	0
$185$	−2.56723e8	−2.98102
$186$	0	0
$187$	5.91565e7	0.661541
$188$	0	0
$189$	−1.37803e7	−0.148471
$190$	0	0
$191$	2.60173e7	0.270175	0.135087	−	0.990834i	$-0.456868\pi$
$191$	2.60173e7	0.270175	0.135087	+	0.990834i	$0.456868\pi$
$192$	0	0
$193$	−7.58326e7	−0.759286	−0.379643	−	0.925133i	$-0.623953\pi$
$193$	−7.58326e7	−0.759286	−0.379643	+	0.925133i	$0.623953\pi$
$194$	0	0
$195$	−5.02107e7	−0.484926
$196$	0	0
$197$	−1.19192e8	−1.11075	−0.555374	−	0.831601i	$-0.687425\pi$
$197$	−1.19192e8	−1.11075	−0.555374	+	0.831601i	$0.687425\pi$
$198$	0	0
$199$	6.72662e7	0.605078	0.302539	−	0.953137i	$-0.402166\pi$
$199$	6.72662e7	0.605078	0.302539	+	0.953137i	$0.402166\pi$
$200$	0	0
$201$	−4.07313e7	−0.353787
$202$	0	0
$203$	9.06959e6	0.0760942
$204$	0	0
$205$	−1.41661e8	−1.14845
$206$	0	0
$207$	−1.17858e8	−0.923559
$208$	0	0
$209$	−1.25986e8	−0.954573
$210$	0	0
$211$	−2.32421e8	−1.70328	−0.851641	−	0.524126i	$-0.824392\pi$
$211$	−2.32421e8	−1.70328	−0.851641	+	0.524126i	$0.824392\pi$
$212$	0	0
$213$	6.21651e6	0.0440776
$214$	0	0
$215$	3.60991e8	2.47720
$216$	0	0
$217$	−1.55887e7	−0.103562
$218$	0	0
$219$	1.25793e7	0.0809286
$220$	0	0
$221$	−1.76592e8	−1.10052
$222$	0	0
$223$	−2.53319e8	−1.52968	−0.764840	−	0.644221i	$-0.777182\pi$
$223$	−2.53319e8	−1.52968	−0.764840	+	0.644221i	$0.777182\pi$
$224$	0	0
$225$	−2.84753e8	−1.66659
$226$	0	0
$227$	−4.50124e7	−0.255412	−0.127706	−	0.991812i	$-0.540761\pi$
$227$	−4.50124e7	−0.255412	−0.127706	+	0.991812i	$0.540761\pi$
$228$	0	0
$229$	−1.86973e8	−1.02886	−0.514429	−	0.857533i	$-0.671996\pi$
$229$	−1.86973e8	−1.02886	−0.514429	+	0.857533i	$0.671996\pi$
$230$	0	0
$231$	−1.24779e7	−0.0666039
$232$	0	0
$233$	−2.57588e8	−1.33408	−0.667038	−	0.745024i	$-0.732438\pi$
$233$	−2.57588e8	−1.33408	−0.667038	+	0.745024i	$0.732438\pi$
$234$	0	0
$235$	2.45626e8	1.23463
$236$	0	0
$237$	6.61607e7	0.322835
$238$	0	0
$239$	−3.15071e8	−1.49285	−0.746425	−	0.665469i	$-0.768231\pi$
$239$	−3.15071e8	−1.49285	−0.746425	+	0.665469i	$0.768231\pi$
$240$	0	0
$241$	8.39900e7	0.386516	0.193258	−	0.981148i	$-0.438095\pi$
$241$	8.39900e7	0.386516	0.193258	+	0.981148i	$0.438095\pi$
$242$	0	0
$243$	1.27366e8	0.569419
$244$	0	0
$245$	−5.43963e7	−0.236313
$246$	0	0
$247$	3.76088e8	1.58800
$248$	0	0
$249$	−6.19464e7	−0.254284
$250$	0	0
$251$	5.33130e6	0.0212802	0.0106401	−	0.999943i	$-0.496613\pi$
$251$	5.33130e6	0.0212802	0.0106401	+	0.999943i	$0.496613\pi$
$252$	0	0
$253$	−2.17906e8	−0.845955
$254$	0	0
$255$	6.60590e7	0.249483
$256$	0	0
$257$	−2.09942e8	−0.771495	−0.385747	−	0.922604i	$-0.626056\pi$
$257$	−2.09942e8	−0.771495	−0.385747	+	0.922604i	$0.626056\pi$
$258$	0	0
$259$	1.90449e8	0.681129
$260$	0	0
$261$	−5.55053e7	−0.193238
$262$	0	0
$263$	−2.50805e8	−0.850143	−0.425071	−	0.905160i	$-0.639751\pi$
$263$	−2.50805e8	−0.850143	−0.425071	+	0.905160i	$0.639751\pi$
$264$	0	0
$265$	−1.68111e8	−0.554926
$266$	0	0
$267$	−4.99601e7	−0.160633
$268$	0	0
$269$	−3.39197e8	−1.06248	−0.531238	−	0.847223i	$-0.678273\pi$
$269$	−3.39197e8	−1.06248	−0.531238	+	0.847223i	$0.678273\pi$
$270$	0	0
$271$	7.59574e7	0.231834	0.115917	−	0.993259i	$-0.463019\pi$
$271$	7.59574e7	0.231834	0.115917	+	0.993259i	$0.463019\pi$
$272$	0	0
$273$	3.72486e7	0.110800
$274$	0	0
$275$	−5.26474e8	−1.52656
$276$	0	0
$277$	5.00883e7	0.141598	0.0707990	−	0.997491i	$-0.477445\pi$
$277$	5.00883e7	0.141598	0.0707990	+	0.997491i	$0.477445\pi$
$278$	0	0
$279$	9.54021e7	0.262992
$280$	0	0
$281$	3.41917e8	0.919283	0.459641	−	0.888105i	$-0.347978\pi$
$281$	3.41917e8	0.919283	0.459641	+	0.888105i	$0.347978\pi$
$282$	0	0
$283$	−3.69077e8	−0.967977	−0.483988	−	0.875074i	$-0.660812\pi$
$283$	−3.69077e8	−0.967977	−0.483988	+	0.875074i	$0.660812\pi$
$284$	0	0
$285$	−1.40686e8	−0.359993
$286$	0	0
$287$	1.05090e8	0.262407
$288$	0	0
$289$	−1.78008e8	−0.433808
$290$	0	0
$291$	−1.96664e7	−0.0467842
$292$	0	0
$293$	−5.92780e8	−1.37676	−0.688378	−	0.725352i	$-0.741677\pi$
$293$	−5.92780e8	−1.37676	−0.688378	+	0.725352i	$0.741677\pi$
$294$	0	0
$295$	−9.89892e8	−2.24497
$296$	0	0
$297$	1.55924e8	0.345355
$298$	0	0
$299$	6.50485e8	1.40730
$300$	0	0
$301$	−2.67799e8	−0.566013
$302$	0	0
$303$	−1.83286e8	−0.378512
$304$	0	0
$305$	4.10957e8	0.829368
$306$	0	0
$307$	9.43846e8	1.86173	0.930865	−	0.365363i	$-0.119055\pi$
$307$	9.43846e8	1.86173	0.930865	+	0.365363i	$0.119055\pi$
$308$	0	0
$309$	−8.39008e7	−0.161775
$310$	0	0
$311$	−1.64065e8	−0.309282	−0.154641	−	0.987971i	$-0.549422\pi$
$311$	−1.64065e8	−0.309282	−0.154641	+	0.987971i	$0.549422\pi$
$312$	0	0
$313$	3.46012e8	0.637801	0.318901	−	0.947788i	$-0.396686\pi$
$313$	3.46012e8	0.637801	0.318901	+	0.947788i	$0.396686\pi$
$314$	0	0
$315$	3.32902e8	0.600108
$316$	0	0
$317$	9.53240e8	1.68072	0.840359	−	0.542030i	$-0.182344\pi$
$317$	9.53240e8	1.68072	0.840359	+	0.542030i	$0.182344\pi$
$318$	0	0
$319$	−1.02623e8	−0.177001
$320$	0	0
$321$	1.30768e8	0.220665
$322$	0	0
$323$	−4.94795e8	−0.816989
$324$	0	0
$325$	1.57161e9	2.53953
$326$	0	0
$327$	−5.47684e7	−0.0866189
$328$	0	0
$329$	−1.82216e8	−0.282099
$330$	0	0
$331$	−2.24263e8	−0.339906	−0.169953	−	0.985452i	$-0.554362\pi$
$331$	−2.24263e8	−0.339906	−0.169953	+	0.985452i	$0.554362\pi$
$332$	0	0
$333$	−1.16554e9	−1.72970
$334$	0	0
$335$	2.00915e9	2.91981
$336$	0	0
$337$	3.56750e8	0.507761	0.253880	−	0.967236i	$-0.418293\pi$
$337$	3.56750e8	0.507761	0.253880	+	0.967236i	$0.418293\pi$
$338$	0	0
$339$	−1.13373e8	−0.158055
$340$	0	0
$341$	1.76387e8	0.240894
$342$	0	0
$343$	4.03536e7	0.0539949
$344$	0	0
$345$	−2.43332e8	−0.319030
$346$	0	0
$347$	−1.61839e7	−0.0207936	−0.0103968	−	0.999946i	$-0.503309\pi$
$347$	−1.61839e7	−0.0207936	−0.0103968	+	0.999946i	$0.503309\pi$
$348$	0	0
$349$	−2.07780e8	−0.261647	−0.130823	−	0.991406i	$-0.541762\pi$
$349$	−2.07780e8	−0.261647	−0.130823	+	0.991406i	$0.541762\pi$
$350$	0	0
$351$	−4.65459e8	−0.574522
$352$	0	0
$353$	−1.31624e9	−1.59266	−0.796328	−	0.604865i	$-0.793227\pi$
$353$	−1.31624e9	−1.59266	−0.796328	+	0.604865i	$0.793227\pi$
$354$	0	0
$355$	−3.06641e8	−0.363773
$356$	0	0
$357$	−4.90055e7	−0.0570041
$358$	0	0
$359$	−6.19271e8	−0.706400	−0.353200	−	0.935548i	$-0.614906\pi$
$359$	−6.19271e8	−0.706400	−0.353200	+	0.935548i	$0.614906\pi$
$360$	0	0
$361$	1.59893e8	0.178877
$362$	0	0
$363$	−4.14738e7	−0.0455093
$364$	0	0
$365$	−6.20497e8	−0.667905
$366$	0	0
$367$	2.78628e8	0.294235	0.147117	−	0.989119i	$-0.453000\pi$
$367$	2.78628e8	0.294235	0.147117	+	0.989119i	$0.453000\pi$
$368$	0	0
$369$	−6.43146e8	−0.666373
$370$	0	0
$371$	1.24712e8	0.126795
$372$	0	0
$373$	1.65056e8	0.164684	0.0823420	−	0.996604i	$-0.473760\pi$
$373$	1.65056e8	0.164684	0.0823420	+	0.996604i	$0.473760\pi$
$374$	0	0
$375$	−2.49318e8	−0.244143
$376$	0	0
$377$	3.06345e8	0.294453
$378$	0	0
$379$	−4.29433e8	−0.405190	−0.202595	−	0.979263i	$-0.564937\pi$
$379$	−4.29433e8	−0.405190	−0.202595	+	0.979263i	$0.564937\pi$
$380$	0	0
$381$	−3.05348e6	−0.00282851
$382$	0	0
$383$	1.34378e9	1.22217	0.611084	−	0.791566i	$-0.290734\pi$
$383$	1.34378e9	1.22217	0.611084	+	0.791566i	$0.290734\pi$
$384$	0	0
$385$	6.15495e8	0.549682
$386$	0	0
$387$	1.63891e9	1.43737
$388$	0	0
$389$	1.07871e8	0.0929139	0.0464569	−	0.998920i	$-0.485207\pi$
$389$	1.07871e8	0.0929139	0.0464569	+	0.998920i	$0.485207\pi$
$390$	0	0
$391$	−8.55801e8	−0.724026
$392$	0	0
$393$	−5.98070e7	−0.0497024
$394$	0	0
$395$	−3.26350e9	−2.66436
$396$	0	0
$397$	1.86525e9	1.49613	0.748066	−	0.663625i	$-0.230983\pi$
$397$	1.86525e9	1.49613	0.748066	+	0.663625i	$0.230983\pi$
$398$	0	0
$399$	1.04367e8	0.0822543
$400$	0	0
$401$	9.94724e6	0.00770367	0.00385183	−	0.999993i	$-0.498774\pi$
$401$	9.94724e6	0.00770367	0.00385183	+	0.999993i	$0.498774\pi$
$402$	0	0
$403$	−5.26543e8	−0.400744
$404$	0	0
$405$	−1.94850e9	−1.45749
$406$	0	0
$407$	−2.15493e9	−1.58436
$408$	0	0
$409$	−8.76376e8	−0.633372	−0.316686	−	0.948530i	$-0.602570\pi$
$409$	−8.76376e8	−0.633372	−0.316686	+	0.948530i	$0.602570\pi$
$410$	0	0
$411$	4.33356e7	0.0307892
$412$	0	0
$413$	7.34347e8	0.512951
$414$	0	0
$415$	3.05562e9	2.09861
$416$	0	0
$417$	3.33126e8	0.224974
$418$	0	0
$419$	2.01605e9	1.33891	0.669457	−	0.742851i	$-0.266527\pi$
$419$	2.01605e9	1.33891	0.669457	+	0.742851i	$0.266527\pi$
$420$	0	0
$421$	−5.69546e8	−0.371999	−0.186000	−	0.982550i	$-0.559552\pi$
$421$	−5.69546e8	−0.371999	−0.186000	+	0.982550i	$0.559552\pi$
$422$	0	0
$423$	1.11515e9	0.716380
$424$	0	0
$425$	−2.06767e9	−1.30653
$426$	0	0
$427$	−3.04867e8	−0.189501
$428$	0	0
$429$	−4.21468e8	−0.257730
$430$	0	0
$431$	−7.71940e8	−0.464422	−0.232211	−	0.972665i	$-0.574596\pi$
$431$	−7.71940e8	−0.464422	−0.232211	+	0.972665i	$0.574596\pi$
$432$	0	0
$433$	1.24002e9	0.734043	0.367021	−	0.930213i	$-0.380378\pi$
$433$	1.24002e9	0.734043	0.367021	+	0.930213i	$0.380378\pi$
$434$	0	0
$435$	−1.14597e8	−0.0667513
$436$	0	0
$437$	1.82260e9	1.04474
$438$	0	0
$439$	1.90621e9	1.07534	0.537669	−	0.843156i	$-0.319305\pi$
$439$	1.90621e9	1.07534	0.537669	+	0.843156i	$0.319305\pi$
$440$	0	0
$441$	−2.46962e8	−0.137118
$442$	0	0
$443$	5.04738e8	0.275837	0.137919	−	0.990444i	$-0.455959\pi$
$443$	5.04738e8	0.275837	0.137919	+	0.990444i	$0.455959\pi$
$444$	0	0
$445$	2.46437e9	1.32570
$446$	0	0
$447$	−9.53351e7	−0.0504866
$448$	0	0
$449$	−2.02721e9	−1.05691	−0.528453	−	0.848962i	$-0.677228\pi$
$449$	−2.02721e9	−1.05691	−0.528453	+	0.848962i	$0.677228\pi$
$450$	0	0
$451$	−1.18910e9	−0.610379
$452$	0	0
$453$	2.29904e8	0.116199
$454$	0	0
$455$	−1.83735e9	−0.914435
$456$	0	0
$457$	2.03969e9	0.999672	0.499836	−	0.866120i	$-0.333394\pi$
$457$	2.03969e9	0.999672	0.499836	+	0.866120i	$0.333394\pi$
$458$	0	0
$459$	6.12375e8	0.295579
$460$	0	0
$461$	3.41390e9	1.62292	0.811461	−	0.584406i	$-0.198673\pi$
$461$	3.41390e9	1.62292	0.811461	+	0.584406i	$0.198673\pi$
$462$	0	0
$463$	−1.97893e9	−0.926608	−0.463304	−	0.886199i	$-0.653336\pi$
$463$	−1.97893e9	−0.926608	−0.463304	+	0.886199i	$0.653336\pi$
$464$	0	0
$465$	1.96968e8	0.0908469
$466$	0	0
$467$	−1.86422e9	−0.847007	−0.423504	−	0.905894i	$-0.639200\pi$
$467$	−1.86422e9	−0.847007	−0.423504	+	0.905894i	$0.639200\pi$
$468$	0	0
$469$	−1.49047e9	−0.667144
$470$	0	0
$471$	1.19397e8	0.0526528
$472$	0	0
$473$	3.03015e9	1.31659
$474$	0	0
$475$	4.40351e9	1.88526
$476$	0	0
$477$	−7.63231e8	−0.321990
$478$	0	0
$479$	3.71828e9	1.54585	0.772926	−	0.634496i	$-0.218792\pi$
$479$	3.71828e9	1.54585	0.772926	+	0.634496i	$0.218792\pi$
$480$	0	0
$481$	6.43283e9	2.63569
$482$	0	0
$483$	1.80514e8	0.0728948
$484$	0	0
$485$	9.70080e8	0.386110
$486$	0	0
$487$	−2.16341e9	−0.848767	−0.424383	−	0.905483i	$-0.639509\pi$
$487$	−2.16341e9	−0.848767	−0.424383	+	0.905483i	$0.639509\pi$
$488$	0	0
$489$	7.98175e8	0.308686
$490$	0	0
$491$	−2.37608e8	−0.0905890	−0.0452945	−	0.998974i	$-0.514423\pi$
$491$	−2.37608e8	−0.0905890	−0.0452945	+	0.998974i	$0.514423\pi$
$492$	0	0
$493$	−4.03039e8	−0.151489
$494$	0	0
$495$	−3.76679e9	−1.39590
$496$	0	0
$497$	2.27480e8	0.0831181
$498$	0	0
$499$	−2.65166e9	−0.955358	−0.477679	−	0.878535i	$-0.658522\pi$
$499$	−2.65166e9	−0.955358	−0.477679	+	0.878535i	$0.658522\pi$
$500$	0	0
$501$	−3.50810e7	−0.0124635
$502$	0	0
$503$	2.15865e9	0.756300	0.378150	−	0.925744i	$-0.376560\pi$
$503$	2.15865e9	0.756300	0.378150	+	0.925744i	$0.376560\pi$
$504$	0	0
$505$	9.04091e9	3.12387
$506$	0	0
$507$	6.69984e8	0.228316
$508$	0	0
$509$	−2.47831e9	−0.832998	−0.416499	−	0.909136i	$-0.636743\pi$
$509$	−2.47831e9	−0.832998	−0.416499	+	0.909136i	$0.636743\pi$
$510$	0	0
$511$	4.60312e8	0.152609
$512$	0	0
$513$	−1.30418e9	−0.426506
$514$	0	0
$515$	4.13856e9	1.33513
$516$	0	0
$517$	2.06178e9	0.656184
$518$	0	0
$519$	6.81321e8	0.213927
$520$	0	0
$521$	−3.99735e9	−1.23834	−0.619170	−	0.785257i	$-0.712531\pi$
$521$	−3.99735e9	−1.23834	−0.619170	+	0.785257i	$0.712531\pi$
$522$	0	0
$523$	3.93132e9	1.20166	0.600831	−	0.799376i	$-0.294837\pi$
$523$	3.93132e9	1.20166	0.600831	+	0.799376i	$0.294837\pi$
$524$	0	0
$525$	4.36134e8	0.131541
$526$	0	0
$527$	6.92739e8	0.206173
$528$	0	0
$529$	−2.52441e8	−0.0741421
$530$	0	0
$531$	−4.49416e9	−1.30262
$532$	0	0
$533$	3.54965e9	1.01541
$534$	0	0
$535$	−6.45037e9	−1.82115
$536$	0	0
$537$	−9.60258e8	−0.267595
$538$	0	0
$539$	−4.56602e8	−0.125596
$540$	0	0
$541$	2.74766e9	0.746057	0.373028	−	0.927820i	$-0.378319\pi$
$541$	2.74766e9	0.746057	0.373028	+	0.927820i	$0.378319\pi$
$542$	0	0
$543$	−4.49138e7	−0.0120387
$544$	0	0
$545$	2.70155e9	0.714867
$546$	0	0
$547$	1.40581e9	0.367258	0.183629	−	0.982996i	$-0.441216\pi$
$547$	1.40581e9	0.367258	0.183629	+	0.982996i	$0.441216\pi$
$548$	0	0
$549$	1.86576e9	0.481231
$550$	0	0
$551$	8.58352e8	0.218592
$552$	0	0
$553$	2.42101e9	0.608777
$554$	0	0
$555$	−2.40637e9	−0.597500
$556$	0	0
$557$	−9.87805e8	−0.242202	−0.121101	−	0.992640i	$-0.538643\pi$
$557$	−9.87805e8	−0.242202	−0.121101	+	0.992640i	$0.538643\pi$
$558$	0	0
$559$	−9.04550e9	−2.19024
$560$	0	0
$561$	5.54498e8	0.132596
$562$	0	0
$563$	−1.88115e9	−0.444268	−0.222134	−	0.975016i	$-0.571302\pi$
$563$	−1.88115e9	−0.444268	−0.222134	+	0.975016i	$0.571302\pi$
$564$	0	0
$565$	5.59231e9	1.30443
$566$	0	0
$567$	1.44548e9	0.333021
$568$	0	0
$569$	−3.88531e9	−0.884164	−0.442082	−	0.896975i	$-0.645760\pi$
$569$	−3.88531e9	−0.884164	−0.442082	+	0.896975i	$0.645760\pi$
$570$	0	0
$571$	−2.44900e9	−0.550505	−0.275253	−	0.961372i	$-0.588762\pi$
$571$	−2.44900e9	−0.550505	−0.275253	+	0.961372i	$0.588762\pi$
$572$	0	0
$573$	2.43871e8	0.0541525
$574$	0	0
$575$	7.61636e9	1.67074
$576$	0	0
$577$	2.88218e9	0.624605	0.312303	−	0.949983i	$-0.398900\pi$
$577$	2.88218e9	0.624605	0.312303	+	0.949983i	$0.398900\pi$
$578$	0	0
$579$	−7.10811e8	−0.152188
$580$	0	0
$581$	−2.26680e9	−0.479508
$582$	0	0
$583$	−1.41112e9	−0.294934
$584$	0	0
$585$	1.12445e10	2.32217
$586$	0	0
$587$	−4.72250e9	−0.963692	−0.481846	−	0.876256i	$-0.660034\pi$
$587$	−4.72250e9	−0.963692	−0.481846	+	0.876256i	$0.660034\pi$
$588$	0	0
$589$	−1.47533e9	−0.297499
$590$	0	0
$591$	−1.11724e9	−0.222633
$592$	0	0
$593$	1.23167e8	0.0242550	0.0121275	−	0.999926i	$-0.496140\pi$
$593$	1.23167e8	0.0242550	0.0121275	+	0.999926i	$0.496140\pi$
$594$	0	0
$595$	2.41729e9	0.470456
$596$	0	0
$597$	6.30514e8	0.121279
$598$	0	0
$599$	1.35497e9	0.257594	0.128797	−	0.991671i	$-0.458888\pi$
$599$	1.35497e9	0.257594	0.128797	+	0.991671i	$0.458888\pi$
$600$	0	0
$601$	−5.25107e9	−0.986704	−0.493352	−	0.869830i	$-0.664228\pi$
$601$	−5.25107e9	−0.986704	−0.493352	+	0.869830i	$0.664228\pi$
$602$	0	0
$603$	9.12161e9	1.69419
$604$	0	0
$605$	2.04577e9	0.375589
$606$	0	0
$607$	4.99012e9	0.905630	0.452815	−	0.891605i	$-0.350420\pi$
$607$	4.99012e9	0.905630	0.452815	+	0.891605i	$0.350420\pi$
$608$	0	0
$609$	8.50130e7	0.0152519
$610$	0	0
$611$	−6.15476e9	−1.09161
$612$	0	0
$613$	2.46028e9	0.431393	0.215696	−	0.976460i	$-0.430798\pi$
$613$	2.46028e9	0.431393	0.215696	+	0.976460i	$0.430798\pi$
$614$	0	0
$615$	−1.32784e9	−0.230189
$616$	0	0
$617$	−1.93223e9	−0.331178	−0.165589	−	0.986195i	$-0.552953\pi$
$617$	−1.93223e9	−0.331178	−0.165589	+	0.986195i	$0.552953\pi$
$618$	0	0
$619$	8.13699e9	1.37894	0.689472	−	0.724313i	$-0.257843\pi$
$619$	8.13699e9	1.37894	0.689472	+	0.724313i	$0.257843\pi$
$620$	0	0
$621$	−2.25571e9	−0.377975
$622$	0	0
$623$	−1.82818e9	−0.302908
$624$	0	0
$625$	1.70022e9	0.278565
$626$	0	0
$627$	−1.18092e9	−0.191330
$628$	0	0
$629$	−8.46326e9	−1.35600
$630$	0	0
$631$	8.83737e9	1.40030	0.700149	−	0.713997i	$-0.253117\pi$
$631$	8.83737e9	1.40030	0.700149	+	0.713997i	$0.253117\pi$
$632$	0	0
$633$	−2.17858e9	−0.341397
$634$	0	0
$635$	1.50619e8	0.0233438
$636$	0	0
$637$	1.36303e9	0.208938
$638$	0	0
$639$	−1.39216e9	−0.211075
$640$	0	0
$641$	−1.25905e10	−1.88817	−0.944086	−	0.329700i	$-0.893052\pi$
$641$	−1.25905e10	−1.88817	−0.944086	+	0.329700i	$0.893052\pi$
$642$	0	0
$643$	−6.47578e9	−0.960625	−0.480312	−	0.877098i	$-0.659477\pi$
$643$	−6.47578e9	−0.960625	−0.480312	+	0.877098i	$0.659477\pi$
$644$	0	0
$645$	3.38371e9	0.496518
$646$	0	0
$647$	8.69520e9	1.26216	0.631081	−	0.775717i	$-0.282612\pi$
$647$	8.69520e9	1.26216	0.631081	+	0.775717i	$0.282612\pi$
$648$	0	0
$649$	−8.30914e9	−1.19316
$650$	0	0
$651$	−1.46120e8	−0.0207575
$652$	0	0
$653$	3.87898e9	0.545156	0.272578	−	0.962134i	$-0.412124\pi$
$653$	3.87898e9	0.545156	0.272578	+	0.962134i	$0.412124\pi$
$654$	0	0
$655$	2.95009e9	0.410195
$656$	0	0
$657$	−2.81708e9	−0.387544
$658$	0	0
$659$	1.30333e9	0.177400	0.0887000	−	0.996058i	$-0.471729\pi$
$659$	1.30333e9	0.177400	0.0887000	+	0.996058i	$0.471729\pi$
$660$	0	0
$661$	−6.85888e9	−0.923736	−0.461868	−	0.886949i	$-0.652821\pi$
$661$	−6.85888e9	−0.923736	−0.461868	+	0.886949i	$0.652821\pi$
$662$	0	0
$663$	−1.65527e9	−0.220583
$664$	0	0
$665$	−5.14810e9	−0.678846
$666$	0	0
$667$	1.48461e9	0.193719
$668$	0	0
$669$	−2.37446e9	−0.306601
$670$	0	0
$671$	3.44957e9	0.440794
$672$	0	0
$673$	−6.44953e9	−0.815596	−0.407798	−	0.913072i	$-0.633703\pi$
$673$	−6.44953e9	−0.815596	−0.407798	+	0.913072i	$0.633703\pi$
$674$	0	0
$675$	−5.44994e9	−0.682069
$676$	0	0
$677$	−3.83027e9	−0.474426	−0.237213	−	0.971458i	$-0.576234\pi$
$677$	−3.83027e9	−0.474426	−0.237213	+	0.971458i	$0.576234\pi$
$678$	0	0
$679$	−7.19649e8	−0.0882219
$680$	0	0
$681$	−4.21920e8	−0.0511936
$682$	0	0
$683$	1.97836e9	0.237592	0.118796	−	0.992919i	$-0.462097\pi$
$683$	1.97836e9	0.237592	0.118796	+	0.992919i	$0.462097\pi$
$684$	0	0
$685$	−2.13761e9	−0.254103
$686$	0	0
$687$	−1.75258e9	−0.206219
$688$	0	0
$689$	4.21243e9	0.490643
$690$	0	0
$691$	−1.00255e10	−1.15593	−0.577967	−	0.816060i	$-0.696154\pi$
$691$	−1.00255e10	−1.15593	−0.577967	+	0.816060i	$0.696154\pi$
$692$	0	0
$693$	2.79437e9	0.318947
$694$	0	0
$695$	−1.64320e10	−1.85671
$696$	0	0
$697$	−4.67005e9	−0.522404
$698$	0	0
$699$	−2.41448e9	−0.267395
$700$	0	0
$701$	−1.21616e9	−0.133345	−0.0666727	−	0.997775i	$-0.521238\pi$
$701$	−1.21616e9	−0.133345	−0.0666727	+	0.997775i	$0.521238\pi$
$702$	0	0
$703$	1.80242e10	1.95665
$704$	0	0
$705$	2.30235e9	0.247463
$706$	0	0
$707$	−6.70696e9	−0.713769
$708$	0	0
$709$	−1.04706e10	−1.10334	−0.551671	−	0.834062i	$-0.686010\pi$
$709$	−1.04706e10	−1.10334	−0.551671	+	0.834062i	$0.686010\pi$
$710$	0	0
$711$	−1.48164e10	−1.54597
$712$	0	0
$713$	−2.55174e9	−0.263647
$714$	0	0
$715$	2.07897e10	2.12705
$716$	0	0
$717$	−2.95330e9	−0.299219
$718$	0	0
$719$	−2.99586e9	−0.300587	−0.150294	−	0.988641i	$-0.548022\pi$
$719$	−2.99586e9	−0.300587	−0.150294	+	0.988641i	$0.548022\pi$
$720$	0	0
$721$	−3.07017e9	−0.305062
$722$	0	0
$723$	7.87273e8	0.0774714
$724$	0	0
$725$	3.58692e9	0.349573
$726$	0	0
$727$	−6.52469e9	−0.629781	−0.314890	−	0.949128i	$-0.601968\pi$
$727$	−6.52469e9	−0.629781	−0.314890	+	0.949128i	$0.601968\pi$
$728$	0	0
$729$	−8.02267e9	−0.766960
$730$	0	0
$731$	1.19006e10	1.12683
$732$	0	0
$733$	1.19856e10	1.12408	0.562040	−	0.827110i	$-0.310017\pi$
$733$	1.19856e10	1.12408	0.562040	+	0.827110i	$0.310017\pi$
$734$	0	0
$735$	−5.09879e8	−0.0473654
$736$	0	0
$737$	1.68647e10	1.55183
$738$	0	0
$739$	−1.81843e10	−1.65746	−0.828728	−	0.559651i	$-0.810935\pi$
$739$	−1.81843e10	−1.65746	−0.828728	+	0.559651i	$0.810935\pi$
$740$	0	0
$741$	3.52523e9	0.318290
$742$	0	0
$743$	6.10111e9	0.545692	0.272846	−	0.962058i	$-0.412035\pi$
$743$	6.10111e9	0.545692	0.272846	+	0.962058i	$0.412035\pi$
$744$	0	0
$745$	4.70257e9	0.416667
$746$	0	0
$747$	1.38727e10	1.21769
$748$	0	0
$749$	4.78517e9	0.416113
$750$	0	0
$751$	−3.69384e9	−0.318228	−0.159114	−	0.987260i	$-0.550864\pi$
$751$	−3.69384e9	−0.318228	−0.159114	+	0.987260i	$0.550864\pi$
$752$	0	0
$753$	4.99725e7	0.00426529
$754$	0	0
$755$	−1.13404e10	−0.958992
$756$	0	0
$757$	−1.35894e10	−1.13858	−0.569292	−	0.822135i	$-0.692783\pi$
$757$	−1.35894e10	−1.13858	−0.569292	+	0.822135i	$0.692783\pi$
$758$	0	0
$759$	−2.04252e9	−0.169559
$760$	0	0
$761$	1.79994e10	1.48051	0.740255	−	0.672326i	$-0.234705\pi$
$761$	1.79994e10	1.48051	0.740255	+	0.672326i	$0.234705\pi$
$762$	0	0
$763$	−2.00413e9	−0.163339
$764$	0	0
$765$	−1.47936e10	−1.19470
$766$	0	0
$767$	2.48042e10	1.98491
$768$	0	0
$769$	1.41340e10	1.12079	0.560394	−	0.828226i	$-0.310650\pi$
$769$	1.41340e10	1.12079	0.560394	+	0.828226i	$0.310650\pi$
$770$	0	0
$771$	−1.96787e9	−0.154634
$772$	0	0
$773$	7.39484e9	0.575838	0.287919	−	0.957655i	$-0.407037\pi$
$773$	7.39484e9	0.575838	0.287919	+	0.957655i	$0.407037\pi$
$774$	0	0
$775$	−6.16516e9	−0.475760
$776$	0	0
$777$	1.78516e9	0.136522
$778$	0	0
$779$	9.94581e9	0.753805
$780$	0	0
$781$	−2.57394e9	−0.193339
$782$	0	0
$783$	−1.06233e9	−0.0790845
$784$	0	0
$785$	−5.88949e9	−0.434544
$786$	0	0
$787$	9.49146e9	0.694099	0.347050	−	0.937847i	$-0.387184\pi$
$787$	9.49146e9	0.694099	0.347050	+	0.937847i	$0.387184\pi$
$788$	0	0
$789$	−2.35090e9	−0.170398
$790$	0	0
$791$	−4.14863e9	−0.298049
$792$	0	0
$793$	−1.02975e10	−0.733292
$794$	0	0
$795$	−1.57577e9	−0.111227
$796$	0	0
$797$	8.83350e9	0.618057	0.309029	−	0.951053i	$-0.399996\pi$
$797$	8.83350e9	0.618057	0.309029	+	0.951053i	$0.399996\pi$
$798$	0	0
$799$	8.09742e9	0.561607
$800$	0	0
$801$	1.11884e10	0.769224
$802$	0	0
$803$	−5.20844e9	−0.354980
$804$	0	0
$805$	−8.90420e9	−0.601602
$806$	0	0
$807$	−3.17943e9	−0.212957
$808$	0	0
$809$	6.15622e9	0.408784	0.204392	−	0.978889i	$-0.434478\pi$
$809$	6.15622e9	0.408784	0.204392	+	0.978889i	$0.434478\pi$
$810$	0	0
$811$	1.53945e10	1.01343	0.506715	−	0.862114i	$-0.330860\pi$
$811$	1.53945e10	1.01343	0.506715	+	0.862114i	$0.330860\pi$
$812$	0	0
$813$	7.11980e8	0.0464677
$814$	0	0
$815$	−3.93714e10	−2.54759
$816$	0	0
$817$	−2.53447e10	−1.62596
$818$	0	0
$819$	−8.34167e9	−0.530590
$820$	0	0
$821$	2.74907e10	1.73374	0.866871	−	0.498532i	$-0.166128\pi$
$821$	2.74907e10	1.73374	0.866871	+	0.498532i	$0.166128\pi$
$822$	0	0
$823$	1.61300e9	0.100864	0.0504318	−	0.998728i	$-0.483940\pi$
$823$	1.61300e9	0.100864	0.0504318	+	0.998728i	$0.483940\pi$
$824$	0	0
$825$	−4.93486e9	−0.305975
$826$	0	0
$827$	−9.03950e9	−0.555745	−0.277872	−	0.960618i	$-0.589629\pi$
$827$	−9.03950e9	−0.555745	−0.277872	+	0.960618i	$0.589629\pi$
$828$	0	0
$829$	−2.46326e10	−1.50165	−0.750826	−	0.660501i	$-0.770344\pi$
$829$	−2.46326e10	−1.50165	−0.750826	+	0.660501i	$0.770344\pi$
$830$	0	0
$831$	4.69499e8	0.0283812
$832$	0	0
$833$	−1.79325e9	−0.107494
$834$	0	0
$835$	1.73043e9	0.102861
$836$	0	0
$837$	1.82592e9	0.107632
$838$	0	0
$839$	−6.84347e9	−0.400046	−0.200023	−	0.979791i	$-0.564102\pi$
$839$	−6.84347e9	−0.400046	−0.200023	+	0.979791i	$0.564102\pi$
$840$	0	0
$841$	−1.65507e10	−0.959468
$842$	0	0
$843$	3.20493e9	0.184256
$844$	0	0
$845$	−3.30482e10	−1.88430
$846$	0	0
$847$	−1.51764e9	−0.0858179
$848$	0	0
$849$	−3.45952e9	−0.194016
$850$	0	0
$851$	3.11748e10	1.73401
$852$	0	0
$853$	−3.54072e10	−1.95330	−0.976652	−	0.214826i	$-0.931081\pi$
$853$	−3.54072e10	−1.95330	−0.976652	+	0.214826i	$0.931081\pi$
$854$	0	0
$855$	3.15061e10	1.72390
$856$	0	0
$857$	−1.87299e10	−1.01649	−0.508245	−	0.861213i	$-0.669706\pi$
$857$	−1.87299e10	−1.01649	−0.508245	+	0.861213i	$0.669706\pi$
$858$	0	0
$859$	5.38272e9	0.289752	0.144876	−	0.989450i	$-0.453722\pi$
$859$	5.38272e9	0.289752	0.144876	+	0.989450i	$0.453722\pi$
$860$	0	0
$861$	9.85054e8	0.0525956
$862$	0	0
$863$	7.29444e9	0.386326	0.193163	−	0.981167i	$-0.438125\pi$
$863$	7.29444e9	0.386326	0.193163	+	0.981167i	$0.438125\pi$
$864$	0	0
$865$	−3.36074e10	−1.76554
$866$	0	0
$867$	−1.66854e9	−0.0869503
$868$	0	0
$869$	−2.73938e10	−1.41606
$870$	0	0
$871$	−5.03440e10	−2.58157
$872$	0	0
$873$	4.40421e9	0.224036
$874$	0	0
$875$	−9.12326e9	−0.460386
$876$	0	0
$877$	−3.05623e10	−1.52998	−0.764992	−	0.644039i	$-0.777257\pi$
$877$	−3.05623e10	−1.52998	−0.764992	+	0.644039i	$0.777257\pi$
$878$	0	0
$879$	−5.55637e9	−0.275950
$880$	0	0
$881$	−3.10736e10	−1.53100	−0.765500	−	0.643436i	$-0.777508\pi$
$881$	−3.10736e10	−1.53100	−0.765500	+	0.643436i	$0.777508\pi$
$882$	0	0
$883$	−9.10432e9	−0.445026	−0.222513	−	0.974930i	$-0.571426\pi$
$883$	−9.10432e9	−0.445026	−0.222513	+	0.974930i	$0.571426\pi$
$884$	0	0
$885$	−9.27867e9	−0.449971
$886$	0	0
$887$	−5.03417e8	−0.0242212	−0.0121106	−	0.999927i	$-0.503855\pi$
$887$	−5.03417e8	−0.0242212	−0.0121106	+	0.999927i	$0.503855\pi$
$888$	0	0
$889$	−1.11736e8	−0.00533379
$890$	0	0
$891$	−1.63557e10	−0.774633
$892$	0	0
$893$	−1.72451e10	−0.810373
$894$	0	0
$895$	4.73665e10	2.20846
$896$	0	0
$897$	6.09727e9	0.282073
$898$	0	0
$899$	−1.20174e9	−0.0551634
$900$	0	0
$901$	−5.54202e9	−0.252424
$902$	0	0
$903$	−2.51019e9	−0.113449
$904$	0	0
$905$	2.21545e9	0.0993557
$906$	0	0
$907$	−3.55392e10	−1.58155	−0.790774	−	0.612108i	$-0.790322\pi$
$907$	−3.55392e10	−1.58155	−0.790774	+	0.612108i	$0.790322\pi$
$908$	0	0
$909$	4.10462e10	1.81259
$910$	0	0
$911$	3.60841e10	1.58125	0.790627	−	0.612299i	$-0.209755\pi$
$911$	3.60841e10	1.58125	0.790627	+	0.612299i	$0.209755\pi$
$912$	0	0
$913$	2.56488e10	1.11537
$914$	0	0
$915$	3.85207e9	0.166234
$916$	0	0
$917$	−2.18851e9	−0.0937250
$918$	0	0
$919$	−7.06975e9	−0.300469	−0.150235	−	0.988650i	$-0.548003\pi$
$919$	−7.06975e9	−0.300469	−0.150235	+	0.988650i	$0.548003\pi$
$920$	0	0
$921$	8.84706e9	0.373156
$922$	0	0
$923$	7.68363e9	0.321633
$924$	0	0
$925$	7.53203e10	3.12908
$926$	0	0
$927$	1.87892e10	0.774694
$928$	0	0
$929$	−1.55102e10	−0.634691	−0.317345	−	0.948310i	$-0.602791\pi$
$929$	−1.55102e10	−0.634691	−0.317345	+	0.948310i	$0.602791\pi$
$930$	0	0
$931$	3.81909e9	0.155109
$932$	0	0
$933$	−1.53785e9	−0.0619910
$934$	0	0
$935$	−2.73516e10	−1.09432
$936$	0	0
$937$	−3.61814e10	−1.43680	−0.718400	−	0.695630i	$-0.755125\pi$
$937$	−3.61814e10	−1.43680	−0.718400	+	0.695630i	$0.755125\pi$
$938$	0	0
$939$	3.24331e9	0.127838
$940$	0	0
$941$	−1.86518e10	−0.729720	−0.364860	−	0.931062i	$-0.618883\pi$
$941$	−1.86518e10	−0.729720	−0.364860	+	0.931062i	$0.618883\pi$
$942$	0	0
$943$	1.72023e10	0.668031
$944$	0	0
$945$	6.37146e9	0.245600
$946$	0	0
$947$	−8.55630e9	−0.327387	−0.163693	−	0.986511i	$-0.552341\pi$
$947$	−8.55630e9	−0.327387	−0.163693	+	0.986511i	$0.552341\pi$
$948$	0	0
$949$	1.55481e10	0.590533
$950$	0	0
$951$	8.93512e9	0.336875
$952$	0	0
$953$	−5.92820e8	−0.0221870	−0.0110935	−	0.999938i	$-0.503531\pi$
$953$	−5.92820e8	−0.0221870	−0.0110935	+	0.999938i	$0.503531\pi$
$954$	0	0
$955$	−1.20294e10	−0.446921
$956$	0	0
$957$	−9.61924e8	−0.0354772
$958$	0	0
$959$	1.58577e9	0.0580598
$960$	0	0
$961$	−2.54471e10	−0.924924
$962$	0	0
$963$	−2.92850e10	−1.05670
$964$	0	0
$965$	3.50620e10	1.25601
$966$	0	0
$967$	−5.25393e10	−1.86849	−0.934247	−	0.356626i	$-0.883927\pi$
$967$	−5.25393e10	−1.86849	−0.934247	+	0.356626i	$0.883927\pi$
$968$	0	0
$969$	−4.63792e9	−0.163753
$970$	0	0
$971$	9.16576e9	0.321293	0.160647	−	0.987012i	$-0.448642\pi$
$971$	9.16576e9	0.321293	0.160647	+	0.987012i	$0.448642\pi$
$972$	0	0
$973$	1.21900e10	0.424238
$974$	0	0
$975$	1.47314e10	0.509011
$976$	0	0
$977$	−2.01603e10	−0.691618	−0.345809	−	0.938305i	$-0.612396\pi$
$977$	−2.01603e10	−0.691618	−0.345809	+	0.938305i	$0.612396\pi$
$978$	0	0
$979$	2.06859e10	0.704588
$980$	0	0
$981$	1.22652e10	0.414793
$982$	0	0
$983$	−1.18989e9	−0.0399549	−0.0199775	−	0.999800i	$-0.506359\pi$
$983$	−1.18989e9	−0.0399549	−0.0199775	+	0.999800i	$0.506359\pi$
$984$	0	0
$985$	5.51097e10	1.83739
$986$	0	0
$987$	−1.70799e9	−0.0565425
$988$	0	0
$989$	−4.38364e10	−1.44095
$990$	0	0
$991$	2.09986e10	0.685382	0.342691	−	0.939448i	$-0.388662\pi$
$991$	2.09986e10	0.685382	0.342691	+	0.939448i	$0.388662\pi$
$992$	0	0
$993$	−2.10211e9	−0.0681291
$994$	0	0
$995$	−3.11013e10	−1.00092
$996$	0	0
$997$	−3.21657e10	−1.02792	−0.513961	−	0.857814i	$-0.671822\pi$
$997$	−3.21657e10	−1.02792	−0.513961	+	0.857814i	$0.671822\pi$
$998$	0	0
$999$	−2.23074e10	−0.707896

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.8.a.l.1.2		2
4.3	odd	2		448.8.a.s.1.1		2
8.3	odd	2		112.8.a.g.1.2		2
8.5	even	2		14.8.a.c.1.1	✓	2
24.5	odd	2		126.8.a.i.1.1		2
40.13	odd	4		350.8.c.k.99.1		4
40.29	even	2		350.8.a.j.1.2		2
40.37	odd	4		350.8.c.k.99.4		4
56.5	odd	6		98.8.c.k.67.1		4
56.13	odd	2		98.8.a.g.1.2		2
56.37	even	6		98.8.c.g.67.2		4
56.45	odd	6		98.8.c.k.79.1		4
56.53	even	6		98.8.c.g.79.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.8.a.c.1.1	✓	2	8.5	even	2
98.8.a.g.1.2		2	56.13	odd	2
98.8.c.g.67.2		4	56.37	even	6
98.8.c.g.79.2		4	56.53	even	6
98.8.c.k.67.1		4	56.5	odd	6
98.8.c.k.79.1		4	56.45	odd	6
112.8.a.g.1.2		2	8.3	odd	2
126.8.a.i.1.1		2	24.5	odd	2
350.8.a.j.1.2		2	40.29	even	2
350.8.c.k.99.1		4	40.13	odd	4
350.8.c.k.99.4		4	40.37	odd	4
448.8.a.l.1.2		2	1.1	even	1	trivial
448.8.a.s.1.1		2	4.3	odd	2

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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	448.a (trivial)

\(f(q)\)	\(=\)	\(q+9.37342 q^{3} -462.361 q^{5} +343.000 q^{7} -2099.14 q^{9} -3881.05 q^{11} +11585.6 q^{13} -4333.90 q^{15} -15242.4 q^{17} +32461.7 q^{19} +3215.08 q^{21} +56146.1 q^{23} +135652. q^{25} -40175.8 q^{27} +26442.0 q^{29} -45448.2 q^{31} -36378.7 q^{33} -158590. q^{35} +555245. q^{37} +108596. q^{39} +306385. q^{41} -780755. q^{43} +970559. q^{45} -531243. q^{47} +117649. q^{49} -142873. q^{51} +363593. q^{53} +1.79445e6 q^{55} +304277. q^{57} +2.14095e6 q^{59} -888824. q^{61} -720005. q^{63} -5.35672e6 q^{65} -4.34541e6 q^{67} +526281. q^{69} +663207. q^{71} +1.34202e6 q^{73} +1.27153e6 q^{75} -1.33120e6 q^{77} +7.05834e6 q^{79} +4.21423e6 q^{81} -6.60874e6 q^{83} +7.04748e6 q^{85} +247851. q^{87} -5.32998e6 q^{89} +3.97385e6 q^{91} -426005. q^{93} -1.50090e7 q^{95} -2.09810e6 q^{97} +8.14686e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 70 q^{3} - 126 q^{5} + 686 q^{7} + 2014 q^{9} + 3420 q^{11} + 6398 q^{13} - 31032 q^{15} - 38472 q^{17} + 43358 q^{19} - 24010 q^{21} + 89928 q^{23} + 170666 q^{25} - 193060 q^{27} - 159576 q^{29}+ \cdots + 38177100 q^{99}+O(q^{100}) \) 2 * q - 70 * q^3 - 126 * q^5 + 686 * q^7 + 2014 * q^9 + 3420 * q^11 + 6398 * q^13 - 31032 * q^15 - 38472 * q^17 + 43358 * q^19 - 24010 * q^21 + 89928 * q^23 + 170666 * q^25 - 193060 * q^27 - 159576 * q^29 - 143612 * q^31 - 615888 * q^33 - 43218 * q^35 + 271832 * q^37 + 520352 * q^39 + 64848 * q^41 - 1527964 * q^43 + 2354058 * q^45 + 485436 * q^47 + 235298 * q^49 + 1700940 * q^51 + 145716 * q^53 + 4250232 * q^55 - 560596 * q^57 + 4183662 * q^59 + 280658 * q^61 + 690802 * q^63 - 7101612 * q^65 - 5671648 * q^67 - 2155104 * q^69 - 619272 * q^71 + 3939628 * q^73 - 1507618 * q^75 + 1173060 * q^77 + 4656616 * q^79 + 7353742 * q^81 - 1235850 * q^83 - 766044 * q^85 + 15012732 * q^87 - 17241420 * q^89 + 2194514 * q^91 + 7365592 * q^93 - 11343960 * q^95 - 740936 * q^97 + 38177100 * q^99

Introduction

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