LMFDB - Embedded newform 126.4.d.a.125.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [126,4,Mod(125,126)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(126, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

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

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

chi := DirichletCharacter("126.125");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 126 = 2 \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	126.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$7.43424066072$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.849346560000.6
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 50625 $ x^8 + 50625
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			125.4
Root			$-3.57817 + 1.48213i$ of defining polynomial
Character	$\chi$	$=$	126.125
Dual form			126.4.d.a.125.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} -4.00000 q^{4} +14.3127 q^{5} +(9.24264 + 16.0491i) q^{7} +8.00000i q^{8} +O(q^{10})$	\(q-2.00000i q^{2} -4.00000 q^{4} +14.3127 q^{5} +(9.24264 + 16.0491i) q^{7} +8.00000i q^{8} -28.6254i q^{10} +59.6985i q^{11} -53.7779i q^{13} +(32.0982 - 18.4853i) q^{14} +16.0000 q^{16} +135.760 q^{17} -82.4032i q^{19} -57.2507 q^{20} +119.397 q^{22} -20.7868i q^{23} +79.8528 q^{25} -107.556 q^{26} +(-36.9706 - 64.1964i) q^{28} -63.0366i q^{29} +121.447i q^{31} -32.0000i q^{32} -271.520i q^{34} +(132.287 + 229.706i) q^{35} -325.470 q^{37} -164.806 q^{38} +114.501i q^{40} -206.902 q^{41} +274.250 q^{43} -238.794i q^{44} -41.5736 q^{46} +294.042 q^{47} +(-172.147 + 296.672i) q^{49} -159.706i q^{50} +215.112i q^{52} +178.919i q^{53} +854.445i q^{55} +(-128.393 + 73.9411i) q^{56} -126.073 q^{58} -170.067 q^{59} +72.8274i q^{61} +242.894 q^{62} -64.0000 q^{64} -769.706i q^{65} -1053.97 q^{67} -543.039 q^{68} +(459.411 - 264.574i) q^{70} -28.8154i q^{71} -482.316i q^{73} +650.940i q^{74} +329.613i q^{76} +(-958.107 + 551.772i) q^{77} -707.675 q^{79} +229.003 q^{80} +413.804i q^{82} +778.145 q^{83} +1943.09 q^{85} -548.500i q^{86} -477.588 q^{88} -9.89496 q^{89} +(863.087 - 497.050i) q^{91} +83.1472i q^{92} -588.084i q^{94} -1179.41i q^{95} -1561.45i q^{97} +(593.344 + 344.294i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 32 q^{4} + 40 q^{7}+O(q^{10}) $ 8 * q - 32 * q^4 + 40 * q^7	$ 8 q - 32 q^{4} + 40 q^{7} + 128 q^{16} + 480 q^{22} - 40 q^{25} - 160 q^{28} - 160 q^{37} + 1040 q^{43} - 672 q^{46} - 2056 q^{49} + 960 q^{58} - 512 q^{64} - 3680 q^{67} + 960 q^{70} + 448 q^{79} + 6720 q^{85} - 1920 q^{88} - 1920 q^{91}+O(q^{100}) $ 8 * q - 32 * q^4 + 40 * q^7 + 128 * q^16 + 480 * q^22 - 40 * q^25 - 160 * q^28 - 160 * q^37 + 1040 * q^43 - 672 * q^46 - 2056 * q^49 + 960 * q^58 - 512 * q^64 - 3680 * q^67 + 960 * q^70 + 448 * q^79 + 6720 * q^85 - 1920 * q^88 - 1920 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/126\mathbb{Z}\right)^\times$.

$n$	$29$	$73$
$\chi(n)$	$-1$	$-1$

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 $

$2$		−	2.00000i		−	0.707107i
$2$		−	2.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−4.00000			−0.500000
$5$	14.3127			1.28017			0.640083	−	0.768306i	$-0.278900\pi$
$5$	14.3127			1.28017			0.640083	+	0.768306i	$0.278900\pi$
$6$		0			0
$7$	9.24264	+	16.0491i	0.499056	+	0.866570i
$7$	9.24264	+	16.0491i	0.499056	+	0.866570i
$8$			8.00000i			0.353553i
$9$		0			0
$10$		−	28.6254i		−	0.905213i
$11$			59.6985i			1.63634i	0.574974	+	0.818171i	$0.305012\pi$
$11$			59.6985i			1.63634i	−0.574974	+	0.818171i	$0.694988\pi$
$12$		0			0
$13$		−	53.7779i		−	1.14733i	−0.819090	−	0.573665i	$-0.805521\pi$
$13$		−	53.7779i		−	1.14733i	0.819090	−	0.573665i	$-0.194479\pi$
$14$	32.0982	−	18.4853i	0.612757	−	0.352886i
$15$		0			0
$16$	16.0000			0.250000
$17$	135.760			1.93686			0.968429	−	0.249289i	$-0.0801970\pi$
$17$	135.760			1.93686			0.968429	+	0.249289i	$0.0801970\pi$
$18$		0			0
$19$		−	82.4032i		−	0.994979i	−0.867470	−	0.497489i	$-0.834255\pi$
$19$		−	82.4032i		−	0.994979i	0.867470	−	0.497489i	$-0.165745\pi$
$20$	−57.2507			−0.640083
$21$		0			0
$22$	119.397			1.15707
$23$		−	20.7868i		−	0.188450i	−0.995551	−	0.0942249i	$-0.969963\pi$
$23$		−	20.7868i		−	0.188450i	0.995551	−	0.0942249i	$-0.0300373\pi$
$24$		0			0
$25$	79.8528			0.638823
$26$	−107.556			−0.811285
$27$		0			0
$28$	−36.9706	−	64.1964i	−0.249528	−	0.433285i
$29$		−	63.0366i		−	0.403641i	−0.979422	−	0.201821i	$-0.935314\pi$
$29$		−	63.0366i		−	0.403641i	0.979422	−	0.201821i	$-0.0646858\pi$
$30$		0			0
$31$			121.447i			0.703631i	0.936069	+	0.351815i	$0.114435\pi$
$31$			121.447i			0.703631i	−0.936069	+	0.351815i	$0.885565\pi$
$32$		−	32.0000i		−	0.176777i
$33$		0			0
$34$		−	271.520i		−	1.36957i
$35$	132.287	+	229.706i	0.638874	+	1.10935i
$36$		0			0
$37$	−325.470			−1.44613			−0.723067	−	0.690778i	$-0.757268\pi$
$37$	−325.470			−1.44613			−0.723067	+	0.690778i	$0.757268\pi$
$38$	−164.806			−0.703556
$39$		0			0
$40$			114.501i			0.452607i
$41$	−206.902			−0.788113			−0.394057	−	0.919086i	$-0.628929\pi$
$41$	−206.902			−0.788113			−0.394057	+	0.919086i	$0.628929\pi$
$42$		0			0
$43$	274.250			0.972621			0.486310	−	0.873786i	$-0.338342\pi$
$43$	274.250			0.972621			0.486310	+	0.873786i	$0.338342\pi$
$44$		−	238.794i		−	0.818171i
$45$		0			0
$46$	−41.5736			−0.133254
$47$	294.042			0.912562			0.456281	−	0.889836i	$-0.349181\pi$
$47$	294.042			0.912562			0.456281	+	0.889836i	$0.349181\pi$
$48$		0			0
$49$	−172.147	+	296.672i	−0.501887	+	0.864933i
$50$		−	159.706i		−	0.451716i
$51$		0			0
$52$			215.112i			0.573665i
$53$			178.919i			0.463706i	0.972751	+	0.231853i	$0.0744788\pi$
$53$			178.919i			0.463706i	−0.972751	+	0.231853i	$0.925521\pi$
$54$		0			0
$55$			854.445i			2.09479i
$56$	−128.393	+	73.9411i	−0.306379	+	0.176443i
$57$		0			0
$58$	−126.073			−0.285418
$59$	−170.067			−0.375268			−0.187634	−	0.982239i	$-0.560082\pi$
$59$	−170.067			−0.375268			−0.187634	+	0.982239i	$0.560082\pi$
$60$		0			0
$61$			72.8274i			0.152862i	0.997075	+	0.0764311i	$0.0243525\pi$
$61$			72.8274i			0.152862i	−0.997075	+	0.0764311i	$0.975647\pi$
$62$	242.894			0.497542
$63$		0			0
$64$	−64.0000			−0.125000
$65$		−	769.706i		−	1.46877i
$66$		0			0
$67$	−1053.97			−1.92183			−0.960917	−	0.276836i	$-0.910714\pi$
$67$	−1053.97			−1.92183			−0.960917	+	0.276836i	$0.910714\pi$
$68$	−543.039			−0.968429
$69$		0			0
$70$	459.411	−	264.574i	0.784431	−	0.451752i
$71$		−	28.8154i		−	0.0481656i	−0.999710	−	0.0240828i	$-0.992333\pi$
$71$		−	28.8154i		−	0.0481656i	0.999710	−	0.0240828i	$-0.00766653\pi$
$72$		0			0
$73$		−	482.316i		−	0.773298i	−0.922227	−	0.386649i	$-0.873632\pi$
$73$		−	482.316i		−	0.773298i	0.922227	−	0.386649i	$-0.126368\pi$
$74$			650.940i			1.02257i
$75$		0			0
$76$			329.613i			0.497489i
$77$	−958.107	+	551.772i	−1.41801	+	0.816626i
$78$		0			0
$79$	−707.675			−1.00784			−0.503922	−	0.863749i	$-0.668110\pi$
$79$	−707.675			−1.00784			−0.503922	+	0.863749i	$0.668110\pi$
$80$	229.003			0.320041
$81$		0			0
$82$			413.804i			0.557280i
$83$	778.145			1.02907			0.514533	−	0.857470i	$-0.327965\pi$
$83$	778.145			1.02907			0.514533	+	0.857470i	$0.327965\pi$
$84$		0			0
$85$	1943.09			2.47950
$86$		−	548.500i		−	0.687747i
$87$		0			0
$88$	−477.588			−0.578535
$89$	−9.89496			−0.0117850			−0.00589249	−	0.999983i	$-0.501876\pi$
$89$	−9.89496			−0.0117850			−0.00589249	+	0.999983i	$0.501876\pi$
$90$		0			0
$91$	863.087	−	497.050i	0.994242	−	0.572582i
$92$			83.1472i			0.0942249i
$93$		0			0
$94$		−	588.084i		−	0.645279i
$95$		−	1179.41i		−	1.27374i
$96$		0			0
$97$		−	1561.45i		−	1.63444i	−0.576323	−	0.817222i	$-0.695513\pi$
$97$		−	1561.45i		−	1.63444i	0.576323	−	0.817222i	$-0.304487\pi$
$98$	593.344	+	344.294i	0.611600	+	0.354888i
$99$		0			0
$100$	−319.411			−0.319411
$101$	−1004.84			−0.989951			−0.494975	−	0.868907i	$-0.664823\pi$
$101$	−1004.84			−0.989951			−0.494975	+	0.868907i	$0.664823\pi$
$102$		0			0
$103$		−	560.301i		−	0.536001i	−0.963419	−	0.268001i	$-0.913637\pi$
$103$		−	560.301i		−	0.536001i	0.963419	−	0.268001i	$-0.0863629\pi$
$104$	430.223			0.405643
$105$		0			0
$106$	357.838			0.327889
$107$			331.021i			0.299074i	0.988756	+	0.149537i	$0.0477784\pi$
$107$			331.021i			0.299074i	−0.988756	+	0.149537i	$0.952222\pi$
$108$		0			0
$109$	1215.15			1.06780			0.533899	−	0.845548i	$-0.320726\pi$
$109$	1215.15			1.06780			0.533899	+	0.845548i	$0.320726\pi$
$110$	1708.89			1.48124
$111$		0			0
$112$	147.882	+	256.786i	0.124764	+	0.216642i
$113$			99.2435i			0.0826199i	0.999146	+	0.0413099i	$0.0131531\pi$
$113$			99.2435i			0.0826199i	−0.999146	+	0.0413099i	$0.986847\pi$
$114$		0			0
$115$		−	297.515i		−	0.241247i
$116$			252.146i			0.201821i
$117$		0			0
$118$			340.134i			0.265355i
$119$	1254.78	+	2178.82i	0.966600	+	1.67842i
$120$		0			0
$121$	−2232.91			−1.67762
$122$	145.655			0.108090
$123$		0			0
$124$		−	485.788i		−	0.351815i
$125$	−646.177			−0.462367
$126$		0			0
$127$	703.087			0.491251			0.245625	−	0.969365i	$-0.421007\pi$
$127$	703.087			0.491251			0.245625	+	0.969365i	$0.421007\pi$
$128$			128.000i			0.0883883i
$129$		0			0
$130$	−1539.41			−1.03858
$131$	−1119.96			−0.746960			−0.373480	−	0.927638i	$-0.621835\pi$
$131$	−1119.96			−0.746960			−0.373480	+	0.927638i	$0.621835\pi$
$132$		0			0
$133$	1322.50	−	761.624i	0.862219	−	0.496550i
$134$			2107.94i			1.35894i
$135$		0			0
$136$			1086.08i			0.684783i
$137$		−	1692.56i		−	1.05551i	−0.849395	−	0.527757i	$-0.823033\pi$
$137$		−	1692.56i		−	1.05551i	0.849395	−	0.527757i	$-0.176967\pi$
$138$		0			0
$139$			733.943i			0.447858i	0.974605	+	0.223929i	$0.0718883\pi$
$139$			733.943i			0.447858i	−0.974605	+	0.223929i	$0.928112\pi$
$140$	−529.148	−	918.823i	−0.319437	−	0.554676i
$141$		0			0
$142$	−57.6307			−0.0340582
$143$	3210.46			1.87743
$144$		0			0
$145$		−	902.222i		−	0.516728i
$146$	−964.631			−0.546804
$147$		0			0
$148$	1301.88			0.723067
$149$		−	2923.65i		−	1.60748i	−0.594978	−	0.803742i	$-0.702839\pi$
$149$		−	2923.65i		−	1.60748i	0.594978	−	0.803742i	$-0.297161\pi$
$150$		0			0
$151$	333.604			0.179790			0.0898950	−	0.995951i	$-0.471347\pi$
$151$	333.604			0.179790			0.0898950	+	0.995951i	$0.471347\pi$
$152$	659.226			0.351778
$153$		0			0
$154$	1103.54	+	1916.21i	0.577442	+	1.00268i
$155$			1738.23i			0.900763i
$156$		0			0
$157$			2140.80i			1.08824i	0.839006	+	0.544122i	$0.183137\pi$
$157$			2140.80i			1.08824i	−0.839006	+	0.544122i	$0.816863\pi$
$158$			1415.35i			0.712653i
$159$		0			0
$160$		−	458.006i		−	0.226303i
$161$	333.609	−	192.125i	0.163305	−	0.0940470i
$162$		0			0
$163$	−3860.14			−1.85491			−0.927453	−	0.373939i	$-0.878007\pi$
$163$	−3860.14			−1.85491			−0.927453	+	0.373939i	$0.878007\pi$
$164$	827.607			0.394057
$165$		0			0
$166$		−	1556.29i		−	0.727660i
$167$	−3030.28			−1.40413			−0.702066	−	0.712112i	$-0.747739\pi$
$167$	−3030.28			−1.40413			−0.702066	+	0.712112i	$0.747739\pi$
$168$		0			0
$169$	−695.061			−0.316368
$170$		−	3886.17i		−	1.75327i
$171$		0			0
$172$	−1097.00			−0.486310
$173$	−2290.45			−1.00659			−0.503294	−	0.864115i	$-0.667879\pi$
$173$	−2290.45			−1.00659			−0.503294	+	0.864115i	$0.667879\pi$
$174$		0			0
$175$	738.051	+	1281.57i	0.318808	+	0.553584i
$176$			955.176i			0.409086i
$177$		0			0
$178$			19.7899i			0.00833324i
$179$		−	1081.64i		−	0.451650i	−0.974168	−	0.225825i	$-0.927492\pi$
$179$		−	1081.64i		−	0.451650i	0.974168	−	0.225825i	$-0.0725077\pi$
$180$		0			0
$181$		−	3076.27i		−	1.26330i	−0.775254	−	0.631650i	$-0.782378\pi$
$181$		−	3076.27i		−	1.26330i	0.775254	−	0.631650i	$-0.217622\pi$
$182$	−994.099	−	1726.17i	−0.404877	−	0.703036i
$183$		0			0
$184$	166.294			0.0666271
$185$	−4658.35			−1.85129
$186$		0			0
$187$			8104.65i			3.16936i
$188$	−1176.17			−0.456281
$189$		0			0
$190$	−2358.82			−0.900668
$191$			1274.99i			0.483010i	0.970400	+	0.241505i	$0.0776410\pi$
$191$			1274.99i			0.483010i	−0.970400	+	0.241505i	$0.922359\pi$
$192$		0			0
$193$	1752.74			0.653703			0.326851	−	0.945076i	$-0.394012\pi$
$193$	1752.74			0.653703			0.326851	+	0.945076i	$0.394012\pi$
$194$	−3122.90			−1.15573
$195$		0			0
$196$	688.589	−	1186.69i	0.250943	−	0.432467i
$197$			2996.70i			1.08379i	0.840448	+	0.541893i	$0.182292\pi$
$197$			2996.70i			1.08379i	−0.840448	+	0.541893i	$0.817708\pi$
$198$		0			0
$199$		−	1191.00i		−	0.424261i	−0.977241	−	0.212131i	$-0.931960\pi$
$199$		−	1191.00i		−	0.424261i	0.977241	−	0.212131i	$-0.0680403\pi$
$200$			638.823i			0.225858i
$201$		0			0
$202$			2009.67i			0.700001i
$203$	1011.68	−	582.624i	0.349783	−	0.201440i
$204$		0			0
$205$	−2961.32			−1.00892
$206$	−1120.60			−0.379010
$207$		0			0
$208$		−	860.446i		−	0.286833i
$209$	4919.35			1.62813
$210$		0			0
$211$	1525.81			0.497824			0.248912	−	0.968526i	$-0.419927\pi$
$211$	1525.81			0.497824			0.248912	+	0.968526i	$0.419927\pi$
$212$		−	715.675i		−	0.231853i
$213$		0			0
$214$	662.041			0.211478
$215$	3925.25			1.24512
$216$		0			0
$217$	−1949.12	+	1122.49i	−0.609745	+	0.351151i
$218$		−	2430.29i		−	0.755047i
$219$		0			0
$220$		−	3417.78i		−	1.04739i
$221$		−	7300.87i		−	2.22222i
$222$		0			0
$223$			3687.92i			1.10745i	0.832699	+	0.553725i	$0.186794\pi$
$223$			3687.92i			1.10745i	−0.832699	+	0.553725i	$0.813206\pi$
$224$	513.571	−	295.765i	0.153189	−	0.0882214i
$225$		0			0
$226$	198.487			0.0584211
$227$	3440.10			1.00585			0.502924	−	0.864331i	$-0.332258\pi$
$227$	3440.10			1.00585			0.502924	+	0.864331i	$0.332258\pi$
$228$		0			0
$229$		−	5192.45i		−	1.49837i	−0.662361	−	0.749185i	$-0.730445\pi$
$229$		−	5192.45i		−	1.49837i	0.662361	−	0.749185i	$-0.269555\pi$
$230$	−595.030			−0.170587
$231$		0			0
$232$	504.293			0.142709
$233$			5676.44i			1.59603i	0.602634	+	0.798017i	$0.294118\pi$
$233$			5676.44i			1.59603i	−0.602634	+	0.798017i	$0.705882\pi$
$234$		0			0
$235$	4208.53			1.16823
$236$	680.267			0.187634
$237$		0			0
$238$	4357.65	−	2509.56i	1.18682	−	0.683489i
$239$			2649.43i			0.717061i	0.933518	+	0.358530i	$0.116722\pi$
$239$			2649.43i			0.717061i	−0.933518	+	0.358530i	$0.883278\pi$
$240$		0			0
$241$			2338.96i			0.625167i	0.949890	+	0.312584i	$0.101194\pi$
$241$			2338.96i			0.625167i	−0.949890	+	0.312584i	$0.898806\pi$
$242$			4465.82i			1.18625i
$243$		0			0
$244$		−	291.310i		−	0.0764311i
$245$	−2463.89	+	4246.17i	−0.642498	+	1.10726i
$246$		0			0
$247$	−4431.47			−1.14157
$248$	−971.577			−0.248771
$249$		0			0
$250$			1292.35i			0.326943i
$251$	4862.30			1.22273			0.611366	−	0.791348i	$-0.290620\pi$
$251$	4862.30			1.22273			0.611366	+	0.791348i	$0.290620\pi$
$252$		0			0
$253$	1240.94			0.308369
$254$		−	1406.17i		−	0.347367i
$255$		0			0
$256$	256.000			0.0625000
$257$	3778.76			0.917171			0.458585	−	0.888650i	$-0.348356\pi$
$257$	3778.76			0.917171			0.458585	+	0.888650i	$0.348356\pi$
$258$		0			0
$259$	−3008.20	−	5223.50i	−0.721701	−	1.25318i
$260$			3078.82i			0.734386i
$261$		0			0
$262$			2239.93i			0.528180i
$263$			26.4210i			0.00619463i	0.999995	+	0.00309732i	$0.000985908\pi$
$263$			26.4210i			0.00619463i	−0.999995	+	0.00309732i	$0.999014\pi$
$264$		0			0
$265$			2560.81i			0.593620i
$266$	−1523.25	−	2645.00i	−0.351114	−	0.609681i
$267$		0			0
$268$	4215.88			0.960917
$269$	348.548			0.0790013			0.0395006	−	0.999220i	$-0.487423\pi$
$269$	348.548			0.0790013			0.0395006	+	0.999220i	$0.487423\pi$
$270$		0			0
$271$		−	7598.97i		−	1.70334i	−0.524080	−	0.851669i	$-0.675591\pi$
$271$		−	7598.97i		−	1.70334i	0.524080	−	0.851669i	$-0.324409\pi$
$272$	2172.16			0.484215
$273$		0			0
$274$	−3385.13			−0.746361
$275$			4767.09i			1.04533i
$276$		0			0
$277$	−1001.61			−0.217261			−0.108630	−	0.994082i	$-0.534646\pi$
$277$	−1001.61			−0.217261			−0.108630	+	0.994082i	$0.534646\pi$
$278$	1467.89			0.316683
$279$		0			0
$280$	−1837.65	+	1058.30i	−0.392215	+	0.225876i
$281$		−	3753.32i		−	0.796812i	−0.917209	−	0.398406i	$-0.869564\pi$
$281$		−	3753.32i		−	0.796812i	0.917209	−	0.398406i	$-0.130436\pi$
$282$		0			0
$283$		−	5859.67i		−	1.23082i	−0.788208	−	0.615409i	$-0.788991\pi$
$283$		−	5859.67i		−	1.23082i	0.788208	−	0.615409i	$-0.211009\pi$
$284$			115.261i			0.0240828i
$285$		0			0
$286$		−	6420.92i		−	1.32754i
$287$	−1912.32	−	3320.59i	−0.393312	−	0.682955i
$288$		0			0
$289$	13517.7			2.75142
$290$	−1804.44			−0.365382
$291$		0			0
$292$			1929.26i			0.386649i
$293$	−1165.23			−0.232332			−0.116166	−	0.993230i	$-0.537060\pi$
$293$	−1165.23			−0.232332			−0.116166	+	0.993230i	$0.537060\pi$
$294$		0			0
$295$	−2434.11			−0.480405
$296$		−	2603.76i		−	0.511286i
$297$		0			0
$298$	−5847.31			−1.13666
$299$	−1117.87			−0.216214
$300$		0			0
$301$	2534.79	+	4401.46i	0.485392	+	0.842844i
$302$		−	667.208i		−	0.127131i
$303$		0			0
$304$		−	1318.45i		−	0.248745i
$305$			1042.35i			0.195689i
$306$		0			0
$307$			2460.63i			0.457445i	0.973492	+	0.228723i	$0.0734549\pi$
$307$			2460.63i			0.457445i	−0.973492	+	0.228723i	$0.926545\pi$
$308$	3832.43	−	2207.09i	0.709003	−	0.408313i
$309$		0			0
$310$	3476.47			0.636936
$311$	−3358.44			−0.612346			−0.306173	−	0.951976i	$-0.599049\pi$
$311$	−3358.44			−0.612346			−0.306173	+	0.951976i	$0.599049\pi$
$312$		0			0
$313$			4559.43i			0.823367i	0.911327	+	0.411684i	$0.135059\pi$
$313$			4559.43i			0.823367i	−0.911327	+	0.411684i	$0.864941\pi$
$314$	4281.60			0.769505
$315$		0			0
$316$	2830.70			0.503922
$317$		−	140.575i		−	0.0249068i	−0.999922	−	0.0124534i	$-0.996036\pi$
$317$		−	140.575i		−	0.0249068i	0.999922	−	0.0124534i	$-0.00396415\pi$
$318$		0			0
$319$	3763.19			0.660496
$320$	−916.012			−0.160021
$321$		0			0
$322$	−384.250	−	667.219i	−0.0665013	−	0.115474i
$323$		−	11187.0i		−	1.92713i
$324$		0			0
$325$		−	4294.32i		−	0.732941i
$326$			7720.29i			1.31162i
$327$		0			0
$328$		−	1655.21i		−	0.278640i
$329$	2717.72	+	4719.11i	0.455419	+	0.790799i
$330$		0			0
$331$	3203.01			0.531883			0.265941	−	0.963989i	$-0.414317\pi$
$331$	3203.01			0.531883			0.265941	+	0.963989i	$0.414317\pi$
$332$	−3112.58			−0.514533
$333$		0			0
$334$			6060.56i			0.992871i
$335$	−15085.1			−2.46026
$336$		0			0
$337$	−9458.69			−1.52892			−0.764462	−	0.644668i	$-0.776996\pi$
$337$	−9458.69			−1.52892			−0.764462	+	0.644668i	$0.776996\pi$
$338$			1390.12i			0.223706i
$339$		0			0
$340$	−7772.35			−1.23975
$341$	−7250.21			−1.15138
$342$		0			0
$343$	−6352.41	−	20.7735i	−0.999995	−	0.00327016i
$344$			2194.00i			0.343873i
$345$		0			0
$346$			4580.90i			0.711765i
$347$		−	2645.38i		−	0.409254i	−0.978840	−	0.204627i	$-0.934402\pi$
$347$		−	2645.38i		−	0.409254i	0.978840	−	0.204627i	$-0.0655982\pi$
$348$		0			0
$349$		−	6172.76i		−	0.946763i	−0.880857	−	0.473382i	$-0.843033\pi$
$349$		−	6172.76i		−	0.946763i	0.880857	−	0.473382i	$-0.156967\pi$
$350$	2563.13	−	1476.10i	0.391443	−	0.225431i
$351$		0			0
$352$	1910.35			0.289267
$353$	8527.20			1.28571			0.642857	−	0.765986i	$-0.277749\pi$
$353$	8527.20			1.28571			0.642857	+	0.765986i	$0.277749\pi$
$354$		0			0
$355$		−	412.425i		−	0.0616599i
$356$	39.5798			0.00589249
$357$		0			0
$358$	−2163.27			−0.319365
$359$			3456.78i			0.508195i	0.967179	+	0.254098i	$0.0817784\pi$
$359$			3456.78i			0.508195i	−0.967179	+	0.254098i	$0.918222\pi$
$360$		0			0
$361$	68.7056			0.0100169
$362$	−6152.54			−0.893288
$363$		0			0
$364$	−3452.35	+	1988.20i	−0.497121	+	0.286291i
$365$		−	6903.23i		−	0.989949i
$366$		0			0
$367$			12039.0i			1.71234i	0.516694	+	0.856170i	$0.327163\pi$
$367$			12039.0i			1.71234i	−0.516694	+	0.856170i	$0.672837\pi$
$368$		−	332.589i		−	0.0471125i
$369$		0			0
$370$			9316.70i			1.30906i
$371$	−2871.49	+	1653.68i	−0.401833	+	0.231415i
$372$		0			0
$373$	−2327.95			−0.323154			−0.161577	−	0.986860i	$-0.551658\pi$
$373$	−2327.95			−0.323154			−0.161577	+	0.986860i	$0.551658\pi$
$374$	16209.3			2.24108
$375$		0			0
$376$			2352.34i			0.322639i
$377$	−3389.97			−0.463110
$378$		0			0
$379$	1313.46			0.178016			0.0890079	−	0.996031i	$-0.471630\pi$
$379$	1313.46			0.178016			0.0890079	+	0.996031i	$0.471630\pi$
$380$			4717.65i			0.636869i
$381$		0			0
$382$	2549.98			0.341540
$383$	589.131			0.0785984			0.0392992	−	0.999227i	$-0.487487\pi$
$383$	589.131			0.0785984			0.0392992	+	0.999227i	$0.487487\pi$
$384$		0			0
$385$	−13713.1	+	7897.33i	−1.81528	+	1.04542i
$386$		−	3505.47i		−	0.462238i
$387$		0			0
$388$			6245.79i			0.817222i
$389$		−	5561.37i		−	0.724865i	−0.932010	−	0.362433i	$-0.881946\pi$
$389$		−	5561.37i		−	0.724865i	0.932010	−	0.362433i	$-0.118054\pi$
$390$		0			0
$391$		−	2822.01i		−	0.365001i
$392$	−2373.38	−	1377.18i	−0.305800	−	0.177444i
$393$		0			0
$394$	5993.39			0.766352
$395$	−10128.7			−1.29021
$396$		0			0
$397$			79.1597i			0.0100073i	0.999987	+	0.00500366i	$0.00159272\pi$
$397$			79.1597i			0.0100073i	−0.999987	+	0.00500366i	$0.998407\pi$
$398$	−2382.01			−0.299998
$399$		0			0
$400$	1277.65			0.159706
$401$			11297.6i			1.40693i	0.710732	+	0.703463i	$0.248364\pi$
$401$			11297.6i			1.40693i	−0.710732	+	0.703463i	$0.751636\pi$
$402$		0			0
$403$	6531.17			0.807297
$404$	4019.35			0.494975
$405$		0			0
$406$	−1165.25	−	2023.36i	−0.142439	−	0.247334i
$407$		−	19430.1i		−	2.36637i
$408$		0			0
$409$			4875.66i			0.589452i	0.955582	+	0.294726i	$0.0952283\pi$
$409$			4875.66i			0.589452i	−0.955582	+	0.294726i	$0.904772\pi$
$410$			5922.64i			0.713411i
$411$		0			0
$412$			2241.20i			0.268001i
$413$	−1571.87	−	2729.42i	−0.187280	−	0.325196i
$414$		0			0
$415$	11137.3			1.31737
$416$	−1720.89			−0.202821
$417$		0			0
$418$		−	9838.70i		−	1.15126i
$419$	3758.73			0.438248			0.219124	−	0.975697i	$-0.429680\pi$
$419$	3758.73			0.438248			0.219124	+	0.975697i	$0.429680\pi$
$420$		0			0
$421$	−1365.09			−0.158029			−0.0790145	−	0.996873i	$-0.525177\pi$
$421$	−1365.09			−0.158029			−0.0790145	+	0.996873i	$0.525177\pi$
$422$		−	3051.61i		−	0.352015i
$423$		0			0
$424$	−1431.35			−0.163945
$425$	10840.8			1.23731
$426$		0			0
$427$	−1168.81	+	673.117i	−0.132466	+	0.0762867i
$428$		−	1324.08i		−	0.149537i
$429$		0			0
$430$		−	7850.50i		−	0.880429i
$431$			8072.43i			0.902170i	0.892481	+	0.451085i	$0.148963\pi$
$431$			8072.43i			0.902170i	−0.892481	+	0.451085i	$0.851037\pi$
$432$		0			0
$433$			687.622i			0.0763164i	0.999272	+	0.0381582i	$0.0121491\pi$
$433$			687.622i			0.0763164i	−0.999272	+	0.0381582i	$0.987851\pi$
$434$	2244.98	+	3898.23i	0.248301	+	0.431155i
$435$		0			0
$436$	−4860.59			−0.533899
$437$	−1712.90			−0.187504
$438$		0			0
$439$		−	11064.0i		−	1.20286i	−0.798924	−	0.601432i	$-0.794597\pi$
$439$		−	11064.0i		−	1.20286i	0.798924	−	0.601432i	$-0.205403\pi$
$440$	−6835.56			−0.740620
$441$		0			0
$442$	−14601.7			−1.57134
$443$			12836.9i			1.37675i	0.725357	+	0.688373i	$0.241675\pi$
$443$			12836.9i			1.37675i	−0.725357	+	0.688373i	$0.758325\pi$
$444$		0			0
$445$	−141.623			−0.0150867
$446$	7375.84			0.783086
$447$		0			0
$448$	−591.529	−	1027.14i	−0.0623820	−	0.108321i
$449$			10439.7i			1.09728i	0.836059	+	0.548640i	$0.184854\pi$
$449$			10439.7i			1.09728i	−0.836059	+	0.548640i	$0.815146\pi$
$450$		0			0
$451$		−	12351.7i		−	1.28962i
$452$		−	396.974i		−	0.0413099i
$453$		0			0
$454$		−	6880.20i		−	0.711242i
$455$	12353.1	−	7114.11i	1.27279	−	0.732999i
$456$		0			0
$457$	13774.9			1.40998			0.704991	−	0.709217i	$-0.250951\pi$
$457$	13774.9			1.40998			0.704991	+	0.709217i	$0.250951\pi$
$458$	−10384.9			−1.05951
$459$		0			0
$460$			1190.06i			0.120623i
$461$	6904.60			0.697569			0.348784	−	0.937203i	$-0.386595\pi$
$461$	6904.60			0.697569			0.348784	+	0.937203i	$0.386595\pi$
$462$		0			0
$463$	14767.2			1.48227			0.741134	−	0.671357i	$-0.234288\pi$
$463$	14767.2			1.48227			0.741134	+	0.671357i	$0.234288\pi$
$464$		−	1008.59i		−	0.100910i
$465$		0			0
$466$	11352.9			1.12857
$467$	−17212.9			−1.70560			−0.852802	−	0.522234i	$-0.825099\pi$
$467$	−17212.9			−1.70560			−0.852802	+	0.522234i	$0.825099\pi$
$468$		0			0
$469$	−9741.46	−	16915.3i	−0.959102	−	1.66540i
$470$		−	8417.06i		−	0.826064i
$471$		0			0
$472$		−	1360.53i		−	0.132677i
$473$			16372.3i			1.59154i
$474$		0			0
$475$		−	6580.13i		−	0.635615i
$476$	−5019.12	−	8715.29i	−0.483300	−	0.839211i
$477$		0			0
$478$	5298.87			0.507039
$479$	−4862.12			−0.463791			−0.231896	−	0.972741i	$-0.574493\pi$
$479$	−4862.12			−0.463791			−0.231896	+	0.972741i	$0.574493\pi$
$480$		0			0
$481$			17503.1i			1.65919i
$482$	4677.91			0.442060
$483$		0			0
$484$	8931.64			0.838809
$485$		−	22348.5i		−	2.09236i
$486$		0			0
$487$	−5368.72			−0.499548			−0.249774	−	0.968304i	$-0.580356\pi$
$487$	−5368.72			−0.499548			−0.249774	+	0.968304i	$0.580356\pi$
$488$	−582.619			−0.0540449
$489$		0			0
$490$	8492.35	+	4927.78i	0.782949	+	0.454315i
$491$			5093.75i			0.468183i	0.972215	+	0.234092i	$0.0752115\pi$
$491$			5093.75i			0.468183i	−0.972215	+	0.234092i	$0.924788\pi$
$492$		0			0
$493$		−	8557.83i		−	0.781796i
$494$			8862.94i			0.807212i
$495$		0			0
$496$			1943.15i			0.175908i
$497$	462.461	−	266.330i	0.0417388	−	0.0240373i
$498$		0			0
$499$	−2047.15			−0.183653			−0.0918265	−	0.995775i	$-0.529271\pi$
$499$	−2047.15			−0.183653			−0.0918265	+	0.995775i	$0.529271\pi$
$500$	2584.71			0.231183
$501$		0			0
$502$		−	9724.60i		−	0.864602i
$503$	16598.7			1.47137			0.735685	−	0.677323i	$-0.236860\pi$
$503$	16598.7			1.47137			0.735685	+	0.677323i	$0.236860\pi$
$504$		0			0
$505$	−14381.9			−1.26730
$506$		−	2481.88i		−	0.218049i
$507$		0			0
$508$	−2812.35			−0.245625
$509$	11437.9			0.996024			0.498012	−	0.867170i	$-0.334064\pi$
$509$	11437.9			0.996024			0.498012	+	0.867170i	$0.334064\pi$
$510$		0			0
$511$	7740.73	−	4457.87i	0.670117	−	0.385919i
$512$		−	512.000i		−	0.0441942i
$513$		0			0
$514$		−	7557.53i		−	0.648538i
$515$		−	8019.41i		−	0.686170i
$516$		0			0
$517$			17553.9i			1.49326i
$518$	−10447.0	+	6016.41i	−0.886129	+	0.510320i
$519$		0			0
$520$	6157.65			0.519290
$521$	23419.6			1.96935			0.984674	−	0.174408i	$-0.0558011\pi$
$521$	23419.6			1.96935			0.984674	+	0.174408i	$0.0558011\pi$
$522$		0			0
$523$			8945.82i			0.747942i	0.927440	+	0.373971i	$0.122004\pi$
$523$			8945.82i			0.747942i	−0.927440	+	0.373971i	$0.877996\pi$
$524$	4479.86			0.373480
$525$		0			0
$526$	52.8420			0.00438027
$527$			16487.6i			1.36283i
$528$		0			0
$529$	11734.9			0.964487
$530$	5121.62			0.419752
$531$		0			0
$532$	−5289.99	+	3046.49i	−0.431109	+	0.248275i
$533$			11126.7i			0.904227i
$534$		0			0
$535$			4737.79i			0.382865i
$536$		−	8431.76i		−	0.679471i
$537$		0			0
$538$		−	697.096i		−	0.0558623i
$539$	−17710.9	−	10276.9i	−1.41533	−	0.821259i
$540$		0			0
$541$	2499.34			0.198623			0.0993114	−	0.995056i	$-0.468336\pi$
$541$	2499.34			0.198623			0.0993114	+	0.995056i	$0.468336\pi$
$542$	−15197.9			−1.20444
$543$		0			0
$544$		−	4344.31i		−	0.342391i
$545$	17392.0			1.36696
$546$		0			0
$547$	−19005.7			−1.48561			−0.742803	−	0.669510i	$-0.766504\pi$
$547$	−19005.7			−1.48561			−0.742803	+	0.669510i	$0.766504\pi$
$548$			6770.26i			0.527757i
$549$		0			0
$550$	9534.18			0.739162
$551$	−5194.42			−0.401615
$552$		0			0
$553$	−6540.79	−	11357.6i	−0.502970	−	0.873367i
$554$			2003.23i			0.153626i
$555$		0			0
$556$		−	2935.77i		−	0.223929i
$557$			8178.32i			0.622130i	0.950389	+	0.311065i	$0.100686\pi$
$557$			8178.32i			0.622130i	−0.950389	+	0.311065i	$0.899314\pi$
$558$		0			0
$559$		−	14748.6i		−	1.11592i
$560$	2116.59	+	3675.29i	0.159718	+	0.277338i
$561$		0			0
$562$	−7506.63			−0.563431
$563$	7619.17			0.570355			0.285177	−	0.958475i	$-0.407947\pi$
$563$	7619.17			0.570355			0.285177	+	0.958475i	$0.407947\pi$
$564$		0			0
$565$			1420.44i			0.105767i
$566$	−11719.3			−0.870319
$567$		0			0
$568$	230.523			0.0170291
$569$			6806.43i			0.501477i	0.968055	+	0.250738i	$0.0806734\pi$
$569$			6806.43i			0.501477i	−0.968055	+	0.250738i	$0.919327\pi$
$570$		0			0
$571$	−18259.8			−1.33826			−0.669131	−	0.743144i	$-0.733334\pi$
$571$	−18259.8			−1.33826			−0.669131	+	0.743144i	$0.733334\pi$
$572$	−12841.8			−0.938713
$573$		0			0
$574$	−6641.18	+	3824.64i	−0.482922	+	0.278114i
$575$		−	1659.88i		−	0.120386i
$576$		0			0
$577$			19239.4i			1.38812i	0.719916	+	0.694062i	$0.244180\pi$
$577$			19239.4i			1.38812i	−0.719916	+	0.694062i	$0.755820\pi$
$578$		−	27035.4i		−	1.94555i
$579$		0			0
$580$			3608.89i			0.258364i
$581$	7192.12	+	12488.5i	0.513562	+	0.891758i
$582$		0			0
$583$	−10681.2			−0.758781
$584$	3858.53			0.273402
$585$		0			0
$586$			2330.45i			0.164283i
$587$	−3612.03			−0.253977			−0.126988	−	0.991904i	$-0.540531\pi$
$587$	−3612.03			−0.253977			−0.126988	+	0.991904i	$0.540531\pi$
$588$		0			0
$589$	10007.6			0.700098
$590$			4868.23i			0.339698i
$591$		0			0
$592$	−5207.52			−0.361534
$593$	4338.25			0.300422			0.150211	−	0.988654i	$-0.452005\pi$
$593$	4338.25			0.300422			0.150211	+	0.988654i	$0.452005\pi$
$594$		0			0
$595$	17959.3	+	31184.8i	1.23741	+	2.14866i
$596$			11694.6i			0.803742i
$597$		0			0
$598$			2235.74i			0.152887i
$599$		−	17593.2i		−	1.20006i	−0.799976	−	0.600032i	$-0.795155\pi$
$599$		−	17593.2i		−	1.20006i	0.799976	−	0.600032i	$-0.204845\pi$
$600$		0			0
$601$			28189.4i			1.91326i	0.291303	+	0.956631i	$0.405911\pi$
$601$			28189.4i			1.91326i	−0.291303	+	0.956631i	$0.594089\pi$
$602$	8802.92	−	5069.58i	0.595981	−	0.343224i
$603$		0			0
$604$	−1334.42			−0.0898950
$605$	−31958.9			−2.14763
$606$		0			0
$607$			6643.76i			0.444254i	0.975018	+	0.222127i	$0.0712999\pi$
$607$			6643.76i			0.444254i	−0.975018	+	0.222127i	$0.928700\pi$
$608$	−2636.90			−0.175889
$609$		0			0
$610$	2084.71			0.138373
$611$		−	15813.0i		−	1.04701i
$612$		0			0
$613$	16410.9			1.08129			0.540644	−	0.841252i	$-0.318181\pi$
$613$	16410.9			1.08129			0.540644	+	0.841252i	$0.318181\pi$
$614$	4921.26			0.323463
$615$		0			0
$616$	−4414.17	−	7664.86i	−0.288721	−	0.501341i
$617$			8724.75i			0.569279i	0.958635	+	0.284640i	$0.0918739\pi$
$617$			8724.75i			0.569279i	−0.958635	+	0.284640i	$0.908126\pi$
$618$		0			0
$619$		−	3503.43i		−	0.227487i	−0.993510	−	0.113744i	$-0.963716\pi$
$619$		−	3503.43i		−	0.227487i	0.993510	−	0.113744i	$-0.0362843\pi$
$620$		−	6952.94i		−	0.450382i
$621$		0			0
$622$			6716.87i			0.432994i
$623$	−91.4555	−	158.805i	−0.00588136	−	0.0102125i
$624$		0			0
$625$	−19230.1			−1.23073
$626$	9118.85			0.582209
$627$		0			0
$628$		−	8563.20i		−	0.544122i
$629$	−44185.8			−2.80096
$630$		0			0
$631$	−9975.21			−0.629329			−0.314665	−	0.949203i	$-0.601892\pi$
$631$	−9975.21			−0.629329			−0.314665	+	0.949203i	$0.601892\pi$
$632$		−	5661.40i		−	0.356327i
$633$		0			0
$634$	−281.149			−0.0176118
$635$	10063.1			0.628882
$636$		0			0
$637$	15954.4	+	9257.71i	0.992365	+	0.575830i
$638$		−	7526.38i		−	0.467041i
$639$		0			0
$640$			1832.02i			0.113152i
$641$		−	14573.8i		−	0.898021i	−0.893527	−	0.449010i	$-0.851777\pi$
$641$		−	14573.8i		−	0.898021i	0.893527	−	0.449010i	$-0.148223\pi$
$642$		0			0
$643$		−	21396.3i		−	1.31227i	−0.754646	−	0.656133i	$-0.772191\pi$
$643$		−	21396.3i		−	1.31227i	0.754646	−	0.656133i	$-0.227809\pi$
$644$	−1334.44	+	768.500i	−0.0816525	+	0.0470235i
$645$		0			0
$646$	−22374.1			−1.36269
$647$	−12162.0			−0.739005			−0.369503	−	0.929230i	$-0.620472\pi$
$647$	−12162.0			−0.739005			−0.369503	+	0.929230i	$0.620472\pi$
$648$		0			0
$649$		−	10152.7i		−	0.614067i
$650$	−8588.63			−0.518267
$651$		0			0
$652$	15440.6			0.927453
$653$		−	28768.8i		−	1.72406i	−0.506860	−	0.862028i	$-0.669194\pi$
$653$		−	28768.8i		−	1.72406i	0.506860	−	0.862028i	$-0.330806\pi$
$654$		0			0
$655$	−16029.7			−0.956232
$656$	−3310.43			−0.197028
$657$		0			0
$658$	9438.22	−	5435.45i	0.559179	−	0.322030i
$659$		−	2881.81i		−	0.170348i	−0.996366	−	0.0851740i	$-0.972855\pi$
$659$		−	2881.81i		−	0.170348i	0.996366	−	0.0851740i	$-0.0271446\pi$
$660$		0			0
$661$		−	2853.04i		−	0.167882i	−0.996471	−	0.0839412i	$-0.973249\pi$
$661$		−	2853.04i		−	0.167882i	0.996471	−	0.0839412i	$-0.0267508\pi$
$662$		−	6406.01i		−	0.376098i
$663$		0			0
$664$			6225.16i			0.363830i
$665$	18928.5	−	10900.9i	1.10378	−	0.635666i
$666$		0			0
$667$	−1310.33			−0.0760662
$668$	12121.1			0.702066
$669$		0			0
$670$			30170.3i			1.73967i
$671$	−4347.68			−0.250135
$672$		0			0
$673$	−20107.9			−1.15171			−0.575855	−	0.817552i	$-0.695331\pi$
$673$	−20107.9			−1.15171			−0.575855	+	0.817552i	$0.695331\pi$
$674$			18917.4i			1.08111i
$675$		0			0
$676$	2780.24			0.158184
$677$	−28605.6			−1.62393			−0.811966	−	0.583705i	$-0.801603\pi$
$677$	−28605.6			−1.62393			−0.811966	+	0.583705i	$0.801603\pi$
$678$		0			0
$679$	25059.8	−	14431.9i	1.41636	−	0.815679i
$680$			15544.7i			0.876635i
$681$		0			0
$682$			14500.4i			0.814149i
$683$			33665.7i			1.88607i	0.332698	+	0.943034i	$0.392041\pi$
$683$			33665.7i			1.88607i	−0.332698	+	0.943034i	$0.607959\pi$
$684$		0			0
$685$		−	24225.1i		−	1.35123i
$686$	−41.5470	+	12704.8i	−0.00231235	+	0.707103i
$687$		0			0
$688$	4388.00			0.243155
$689$	9621.88			0.532024
$690$		0			0
$691$		−	20687.9i		−	1.13894i	−0.822013	−	0.569469i	$-0.807149\pi$
$691$		−	20687.9i		−	1.13894i	0.822013	−	0.569469i	$-0.192851\pi$
$692$	9161.80			0.503294
$693$		0			0
$694$	−5290.75			−0.289386
$695$			10504.7i			0.573332i
$696$		0			0
$697$	−28089.0			−1.52646
$698$	−12345.5			−0.669463
$699$		0			0
$700$	−2952.20	−	5126.26i	−0.159404	−	0.276792i
$701$			33866.4i			1.82470i	0.409410	+	0.912351i	$0.365735\pi$
$701$			33866.4i			1.82470i	−0.409410	+	0.912351i	$0.634265\pi$
$702$		0			0
$703$			26819.8i			1.43887i
$704$		−	3820.70i		−	0.204543i
$705$		0			0
$706$		−	17054.4i		−	0.909138i
$707$	−9287.35	−	16126.7i	−0.494040	−	0.857861i
$708$		0			0
$709$	16743.4			0.886899			0.443449	−	0.896299i	$-0.353755\pi$
$709$	16743.4			0.886899			0.443449	+	0.896299i	$0.353755\pi$
$710$	−824.850			−0.0436001
$711$		0			0
$712$		−	79.1597i		−	0.00416662i
$713$	2524.50			0.132599
$714$		0			0
$715$	45950.3			2.40342
$716$			4326.54i			0.225825i
$717$		0			0
$718$	6913.57			0.359348
$719$	10445.3			0.541784			0.270892	−	0.962610i	$-0.412681\pi$
$719$	10445.3			0.541784			0.270892	+	0.962610i	$0.412681\pi$
$720$		0			0
$721$	8992.33	−	5178.66i	0.464482	−	0.267494i
$722$		−	137.411i		−	0.00708299i
$723$		0			0
$724$			12305.1i			0.631650i
$725$		−	5033.65i		−	0.257855i
$726$		0			0
$727$			8798.07i			0.448834i	0.974493	+	0.224417i	$0.0720478\pi$
$727$			8798.07i			0.448834i	−0.974493	+	0.224417i	$0.927952\pi$
$728$	3976.40	+	6904.69i	0.202438	+	0.351518i
$729$		0			0
$730$	−13806.5			−0.700000
$731$	37232.1			1.88383
$732$		0			0
$733$		−	17833.1i		−	0.898607i	−0.893379	−	0.449304i	$-0.851672\pi$
$733$		−	17833.1i		−	0.898607i	0.893379	−	0.449304i	$-0.148328\pi$
$734$	24077.9			1.21081
$735$		0			0
$736$	−665.177			−0.0333135
$737$		−	62920.4i		−	3.14478i
$738$		0			0
$739$	−3942.38			−0.196242			−0.0981209	−	0.995175i	$-0.531283\pi$
$739$	−3942.38			−0.196242			−0.0981209	+	0.995175i	$0.531283\pi$
$740$	18633.4			0.925645
$741$		0			0
$742$	3307.36	+	5742.97i	0.163635	+	0.284139i
$743$			4260.50i			0.210367i	0.994453	+	0.105183i	$0.0335430\pi$
$743$			4260.50i			0.210367i	−0.994453	+	0.105183i	$0.966457\pi$
$744$		0			0
$745$		−	41845.3i		−	2.05784i
$746$			4655.90i			0.228505i
$747$		0			0
$748$		−	32418.6i		−	1.58468i
$749$	−5312.58	+	3059.50i	−0.259169	+	0.149255i
$750$		0			0
$751$	−30893.6			−1.50110			−0.750548	−	0.660816i	$-0.770210\pi$
$751$	−30893.6			−1.50110			−0.750548	+	0.660816i	$0.770210\pi$
$752$	4704.67			0.228141
$753$		0			0
$754$			6779.95i			0.327468i
$755$	4774.77			0.230161
$756$		0			0
$757$	12559.1			0.602995			0.301497	−	0.953467i	$-0.402514\pi$
$757$	12559.1			0.602995			0.301497	+	0.953467i	$0.402514\pi$
$758$		−	2626.92i		−	0.125876i
$759$		0			0
$760$	9435.29			0.450334
$761$	1308.17			0.0623144			0.0311572	−	0.999514i	$-0.490081\pi$
$761$	1308.17			0.0623144			0.0311572	+	0.999514i	$0.490081\pi$
$762$		0			0
$763$	11231.2	+	19502.0i	0.532891	+	0.925322i
$764$		−	5099.95i		−	0.241505i
$765$		0			0
$766$		−	1178.26i		−	0.0555775i
$767$			9145.84i			0.430557i
$768$		0			0
$769$			3616.24i			0.169578i	0.996399	+	0.0847888i	$0.0270215\pi$
$769$			3616.24i			0.169578i	−0.996399	+	0.0847888i	$0.972978\pi$
$770$	15794.7	+	27426.2i	0.739221	+	1.28360i
$771$		0			0
$772$	−7010.94			−0.326851
$773$	17407.9			0.809987			0.404994	−	0.914320i	$-0.367274\pi$
$773$	17407.9			0.809987			0.404994	+	0.914320i	$0.367274\pi$
$774$		0			0
$775$			9697.89i			0.449495i
$776$	12491.6			0.577863
$777$		0			0
$778$	−11122.7			−0.512557
$779$			17049.4i			0.784156i
$780$		0			0
$781$	1720.23			0.0788154
$782$	−5644.02			−0.258094
$783$		0			0
$784$	−2754.35	+	4746.75i	−0.125472	+	0.216233i
$785$			30640.6i			1.39313i
$786$		0			0
$787$		−	27090.1i		−	1.22701i	−0.789691	−	0.613505i	$-0.789759\pi$
$787$		−	27090.1i		−	1.22701i	0.789691	−	0.613505i	$-0.210241\pi$
$788$		−	11986.8i		−	0.541893i
$789$		0			0
$790$			20257.5i			0.912314i
$791$	−1592.77	+	917.272i	−0.0715959	+	0.0412319i
$792$		0			0
$793$	3916.50			0.175383
$794$	158.319			0.00707625
$795$		0			0
$796$			4764.02i			0.212131i
$797$	−25584.0			−1.13705			−0.568526	−	0.822665i	$-0.692486\pi$
$797$	−25584.0			−1.13705			−0.568526	+	0.822665i	$0.692486\pi$
$798$		0			0
$799$	39919.1			1.76750
$800$		−	2555.29i		−	0.112929i
$801$		0			0
$802$	22595.3			0.994847
$803$	28793.5			1.26538
$804$		0			0
$805$	4774.84	−	2749.82i	0.209057	−	0.120396i
$806$		−	13062.3i		−	0.570845i
$807$		0			0
$808$		−	8038.70i		−	0.350000i
$809$			10535.8i			0.457872i	0.973442	+	0.228936i	$0.0735246\pi$
$809$			10535.8i			0.457872i	−0.973442	+	0.228936i	$0.926475\pi$
$810$		0			0
$811$			30858.9i			1.33613i	0.744102	+	0.668066i	$0.232878\pi$
$811$			30858.9i			1.33613i	−0.744102	+	0.668066i	$0.767122\pi$
$812$	−4046.72	+	2330.50i	−0.174892	+	0.100720i
$813$		0			0
$814$	−38860.1			−1.67328
$815$	−55249.0			−2.37459
$816$		0			0
$817$		−	22599.1i		−	0.967737i
$818$	9751.32			0.416805
$819$		0			0
$820$	11845.3			0.504458
$821$		−	21084.8i		−	0.896304i	−0.893957	−	0.448152i	$-0.852082\pi$
$821$		−	21084.8i		−	0.896304i	0.893957	−	0.448152i	$-0.147918\pi$
$822$		0			0
$823$	8517.47			0.360754			0.180377	−	0.983598i	$-0.442268\pi$
$823$	8517.47			0.360754			0.180377	+	0.983598i	$0.442268\pi$
$824$	4482.41			0.189505
$825$		0			0
$826$	−5458.84	+	3143.73i	−0.229948	+	0.132427i
$827$			11824.9i			0.497210i	0.968605	+	0.248605i	$0.0799721\pi$
$827$			11824.9i			0.497210i	−0.968605	+	0.248605i	$0.920028\pi$
$828$		0			0
$829$		−	32293.2i		−	1.35294i	−0.736468	−	0.676472i	$-0.763508\pi$
$829$		−	32293.2i		−	1.35294i	0.736468	−	0.676472i	$-0.236492\pi$
$830$		−	22274.7i		−	0.931525i
$831$		0			0
$832$			3441.78i			0.143416i
$833$	−23370.7	+	40276.1i	−0.972084	+	1.67525i
$834$		0			0
$835$	−43371.4			−1.79752
$836$	−19677.4			−0.814063
$837$		0			0
$838$		−	7517.47i		−	0.309888i
$839$	31256.5			1.28617			0.643084	−	0.765796i	$-0.277655\pi$
$839$	31256.5			1.28617			0.643084	+	0.765796i	$0.277655\pi$
$840$		0			0
$841$	20415.4			0.837074
$842$			2730.17i			0.111743i
$843$		0			0
$844$	−6103.23			−0.248912
$845$	−9948.18			−0.405003
$846$		0			0
$847$	−20638.0	−	35836.2i	−0.837225	−	1.45377i
$848$			2862.70i			0.115926i
$849$		0			0
$850$		−	21681.6i		−	0.874909i
$851$			6765.48i			0.272524i
$852$		0			0
$853$			6514.43i			0.261488i	0.991416	+	0.130744i	$0.0417367\pi$
$853$			6514.43i			0.261488i	−0.991416	+	0.130744i	$0.958263\pi$
$854$	1346.23	+	2337.63i	0.0539429	+	0.0936674i
$855$		0			0
$856$	−2648.16			−0.105739
$857$	2962.52			0.118084			0.0590419	−	0.998256i	$-0.481195\pi$
$857$	2962.52			0.118084			0.0590419	+	0.998256i	$0.481195\pi$
$858$		0			0
$859$			9325.46i			0.370408i	0.982700	+	0.185204i	$0.0592946\pi$
$859$			9325.46i			0.370408i	−0.982700	+	0.185204i	$0.940705\pi$
$860$	−15701.0			−0.622558
$861$		0			0
$862$	16144.9			0.637931
$863$		−	29756.4i		−	1.17372i	−0.809689	−	0.586859i	$-0.800364\pi$
$863$		−	29756.4i		−	1.17372i	0.809689	−	0.586859i	$-0.199636\pi$
$864$		0			0
$865$	−32782.5			−1.28860
$866$	1375.24			0.0539639
$867$		0			0
$868$	7796.47	−	4489.97i	0.304873	−	0.175575i
$869$		−	42247.1i		−	1.64918i
$870$		0			0
$871$			56680.3i			2.20498i
$872$			9721.18i			0.377524i
$873$		0			0
$874$			3425.80i			0.132585i
$875$	−5972.38	−	10370.6i	−0.230747	−	0.400673i
$876$		0			0
$877$	32926.5			1.26779			0.633893	−	0.773421i	$-0.281456\pi$
$877$	32926.5			1.26779			0.633893	+	0.773421i	$0.281456\pi$
$878$	−22128.0			−0.850553
$879$		0			0
$880$			13671.1i			0.523697i
$881$	26780.0			1.02411			0.512056	−	0.858952i	$-0.328884\pi$
$881$	26780.0			1.02411			0.512056	+	0.858952i	$0.328884\pi$
$882$		0			0
$883$	−8561.73			−0.326302			−0.163151	−	0.986601i	$-0.552166\pi$
$883$	−8561.73			−0.326302			−0.163151	+	0.986601i	$0.552166\pi$
$884$			29203.5i			1.11111i
$885$		0			0
$886$	25673.8			0.973507
$887$	14445.5			0.546825			0.273412	−	0.961897i	$-0.411848\pi$
$887$	14445.5			0.546825			0.273412	+	0.961897i	$0.411848\pi$
$888$		0			0
$889$	6498.38	+	11283.9i	0.245161	+	0.425703i
$890$			283.247i			0.0106679i
$891$		0			0
$892$		−	14751.7i		−	0.553725i
$893$		−	24230.0i		−	0.907980i
$894$		0			0
$895$		−	15481.1i		−	0.578186i
$896$	−2054.28	+	1183.06i	−0.0765947	+	0.0441107i
$897$		0			0
$898$	20879.4			0.775894
$899$	7655.61			0.284014
$900$		0			0
$901$			24290.0i			0.898132i
$902$	−24703.5			−0.911901
$903$		0			0
$904$	−793.948			−0.0292105
$905$		−	44029.7i		−	1.61723i
$906$		0			0
$907$	−10804.2			−0.395531			−0.197766	−	0.980249i	$-0.563368\pi$
$907$	−10804.2			−0.395531			−0.197766	+	0.980249i	$0.563368\pi$
$908$	−13760.4			−0.502924
$909$		0			0
$910$	−14228.2	−	24706.2i	−0.518309	−	0.900002i
$911$		−	21876.6i		−	0.795612i	−0.917470	−	0.397806i	$-0.869772\pi$
$911$		−	21876.6i		−	0.795612i	0.917470	−	0.397806i	$-0.130228\pi$
$912$		0			0
$913$			46454.1i			1.68391i
$914$		−	27549.7i		−	0.997007i
$915$		0			0
$916$			20769.8i			0.749185i
$917$	−10351.4	−	17974.4i	−0.372774	−	0.647293i
$918$		0			0
$919$	−41888.0			−1.50354			−0.751772	−	0.659423i	$-0.770801\pi$
$919$	−41888.0			−1.50354			−0.751772	+	0.659423i	$0.770801\pi$
$920$	2380.12			0.0852937
$921$		0			0
$922$		−	13809.2i		−	0.493256i
$923$	−1549.63			−0.0552618
$924$		0			0
$925$	−25989.7			−0.923823
$926$		−	29534.4i		−	1.04812i
$927$		0			0
$928$	−2017.17			−0.0713544
$929$	−42563.6			−1.50319			−0.751596	−	0.659624i	$-0.770716\pi$
$929$	−42563.6			−1.50319			−0.751596	+	0.659624i	$0.770716\pi$
$930$		0			0
$931$	24446.7	+	14185.5i	0.860590	+	0.499367i
$932$		−	22705.8i		−	0.798017i
$933$		0			0
$934$			34425.8i			1.20604i
$935$			115999.i			4.05731i
$936$		0			0
$937$			14367.8i			0.500936i	0.968125	+	0.250468i	$0.0805844\pi$
$937$			14367.8i			0.500936i	−0.968125	+	0.250468i	$0.919416\pi$
$938$	−33830.5	+	19482.9i	−1.17762	+	0.678188i
$939$		0			0
$940$	−16834.1			−0.584115
$941$	−31175.0			−1.08000			−0.539999	−	0.841666i	$-0.681575\pi$
$941$	−31175.0			−1.08000			−0.539999	+	0.841666i	$0.681575\pi$
$942$		0			0
$943$			4300.83i			0.148520i
$944$	−2721.07			−0.0938170
$945$		0			0
$946$	32744.6			1.12539
$947$			23601.7i			0.809876i	0.914344	+	0.404938i	$0.132707\pi$
$947$			23601.7i			0.809876i	−0.914344	+	0.404938i	$0.867293\pi$
$948$		0			0
$949$	−25937.9			−0.887229
$950$	−13160.3			−0.449448
$951$		0			0
$952$	−17430.6	+	10038.2i	−0.593412	+	0.341745i
$953$			16096.1i			0.547117i	0.961855	+	0.273559i	$0.0882008\pi$
$953$			16096.1i			0.547117i	−0.961855	+	0.273559i	$0.911799\pi$
$954$		0			0
$955$			18248.5i			0.618333i
$956$		−	10597.7i		−	0.358530i
$957$		0			0
$958$			9724.25i			0.327950i
$959$	27164.1	−	15643.8i	0.914677	−	0.526761i
$960$		0			0
$961$	15041.6			0.504904
$962$	35006.2			1.17323
$963$		0			0
$964$		−	9355.82i		−	0.312584i
$965$	25086.3			0.836847
$966$		0			0
$967$	−11261.0			−0.374487			−0.187243	−	0.982314i	$-0.559955\pi$
$967$	−11261.0			−0.374487			−0.187243	+	0.982314i	$0.559955\pi$
$968$		−	17863.3i		−	0.593127i
$969$		0			0
$970$	−44697.0			−1.47952
$971$	6524.08			0.215621			0.107810	−	0.994171i	$-0.465616\pi$
$971$	6524.08			0.215621			0.107810	+	0.994171i	$0.465616\pi$
$972$		0			0
$973$	−11779.1	+	6783.57i	−0.388100	+	0.223506i
$974$			10737.4i			0.353234i
$975$		0			0
$976$			1165.24i			0.0382155i
$977$			22701.3i			0.743378i	0.928357	+	0.371689i	$0.121221\pi$
$977$			22701.3i			0.743378i	−0.928357	+	0.371689i	$0.878779\pi$
$978$		0			0
$979$		−	590.714i		−	0.0192843i
$980$	9855.55	−	16984.7i	0.321249	−	0.553629i
$981$		0			0
$982$	10187.5			0.331055
$983$	13841.1			0.449096			0.224548	−	0.974463i	$-0.427909\pi$
$983$	13841.1			0.449096			0.224548	+	0.974463i	$0.427909\pi$
$984$		0			0
$985$			42890.7i			1.38742i
$986$	−17115.7			−0.552813
$987$		0			0
$988$	17725.9			0.570785
$989$		−	5700.77i		−	0.183290i
$990$		0			0
$991$	−36303.4			−1.16369			−0.581845	−	0.813300i	$-0.697669\pi$
$991$	−36303.4			−1.16369			−0.581845	+	0.813300i	$0.697669\pi$
$992$	3886.31			0.124385
$993$		0			0
$994$	−532.660	−	924.921i	−0.0169969	−	0.0295138i
$995$		−	17046.5i		−	0.543125i
$996$		0			0
$997$		−	22810.6i		−	0.724593i	−0.932063	−	0.362296i	$-0.881993\pi$
$997$		−	22810.6i		−	0.724593i	0.932063	−	0.362296i	$-0.118007\pi$
$998$			4094.29i			0.129862i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	126.4.d.a.125.4	yes	8
3.2	odd	2	inner	126.4.d.a.125.5	yes	8
4.3	odd	2		1008.4.k.c.881.7		8
7.2	even	3		882.4.k.c.521.1		16
7.3	odd	6		882.4.k.c.215.8		16
7.4	even	3		882.4.k.c.215.5		16
7.5	odd	6		882.4.k.c.521.4		16
7.6	odd	2	inner	126.4.d.a.125.1	✓	8
12.11	even	2		1008.4.k.c.881.1		8
21.2	odd	6		882.4.k.c.521.8		16
21.5	even	6		882.4.k.c.521.5		16
21.11	odd	6		882.4.k.c.215.4		16
21.17	even	6		882.4.k.c.215.1		16
21.20	even	2	inner	126.4.d.a.125.8	yes	8
28.27	even	2		1008.4.k.c.881.2		8
84.83	odd	2		1008.4.k.c.881.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.4.d.a.125.1	✓	8	7.6	odd	2	inner
126.4.d.a.125.4	yes	8	1.1	even	1	trivial
126.4.d.a.125.5	yes	8	3.2	odd	2	inner
126.4.d.a.125.8	yes	8	21.20	even	2	inner
882.4.k.c.215.1		16	21.17	even	6
882.4.k.c.215.4		16	21.11	odd	6
882.4.k.c.215.5		16	7.4	even	3
882.4.k.c.215.8		16	7.3	odd	6
882.4.k.c.521.1		16	7.2	even	3
882.4.k.c.521.4		16	7.5	odd	6
882.4.k.c.521.5		16	21.5	even	6
882.4.k.c.521.8		16	21.2	odd	6
1008.4.k.c.881.1		8	12.11	even	2
1008.4.k.c.881.2		8	28.27	even	2
1008.4.k.c.881.7		8	4.3	odd	2
1008.4.k.c.881.8		8	84.83	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	126.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -4.00000 q^{4} +14.3127 q^{5} +(9.24264 + 16.0491i) q^{7} +8.00000i q^{8} +O(q^{10})\)	\(q-2.00000i q^{2} -4.00000 q^{4} +14.3127 q^{5} +(9.24264 + 16.0491i) q^{7} +8.00000i q^{8} -28.6254i q^{10} +59.6985i q^{11} -53.7779i q^{13} +(32.0982 - 18.4853i) q^{14} +16.0000 q^{16} +135.760 q^{17} -82.4032i q^{19} -57.2507 q^{20} +119.397 q^{22} -20.7868i q^{23} +79.8528 q^{25} -107.556 q^{26} +(-36.9706 - 64.1964i) q^{28} -63.0366i q^{29} +121.447i q^{31} -32.0000i q^{32} -271.520i q^{34} +(132.287 + 229.706i) q^{35} -325.470 q^{37} -164.806 q^{38} +114.501i q^{40} -206.902 q^{41} +274.250 q^{43} -238.794i q^{44} -41.5736 q^{46} +294.042 q^{47} +(-172.147 + 296.672i) q^{49} -159.706i q^{50} +215.112i q^{52} +178.919i q^{53} +854.445i q^{55} +(-128.393 + 73.9411i) q^{56} -126.073 q^{58} -170.067 q^{59} +72.8274i q^{61} +242.894 q^{62} -64.0000 q^{64} -769.706i q^{65} -1053.97 q^{67} -543.039 q^{68} +(459.411 - 264.574i) q^{70} -28.8154i q^{71} -482.316i q^{73} +650.940i q^{74} +329.613i q^{76} +(-958.107 + 551.772i) q^{77} -707.675 q^{79} +229.003 q^{80} +413.804i q^{82} +778.145 q^{83} +1943.09 q^{85} -548.500i q^{86} -477.588 q^{88} -9.89496 q^{89} +(863.087 - 497.050i) q^{91} +83.1472i q^{92} -588.084i q^{94} -1179.41i q^{95} -1561.45i q^{97} +(593.344 + 344.294i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 32 q^{4} + 40 q^{7}+O(q^{10}) \) 8 * q - 32 * q^4 + 40 * q^7	\( 8 q - 32 q^{4} + 40 q^{7} + 128 q^{16} + 480 q^{22} - 40 q^{25} - 160 q^{28} - 160 q^{37} + 1040 q^{43} - 672 q^{46} - 2056 q^{49} + 960 q^{58} - 512 q^{64} - 3680 q^{67} + 960 q^{70} + 448 q^{79} + 6720 q^{85} - 1920 q^{88} - 1920 q^{91}+O(q^{100}) \) 8 * q - 32 * q^4 + 40 * q^7 + 128 * q^16 + 480 * q^22 - 40 * q^25 - 160 * q^28 - 160 * q^37 + 1040 * q^43 - 672 * q^46 - 2056 * q^49 + 960 * q^58 - 512 * q^64 - 3680 * q^67 + 960 * q^70 + 448 * q^79 + 6720 * q^85 - 1920 * q^88 - 1920 * q^91

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