LMFDB - Embedded newform 140.15.d.a.41.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [140,15,Mod(41,140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(140, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 15, names="a")

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

chi := DirichletCharacter("140.41");

S:= CuspForms(chi, 15);

N := Newforms(S);

Level:	$ N $	$=$	$ 140 = 2^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 15 $
Character orbit:	$[\chi]$	$=$	140.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:	$174.060555413$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			41.10
Character	$\chi$	$=$	140.41
Dual form			140.15.d.a.41.27

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2045.44i q^{3} +34938.6i q^{5} +(-672278. - 475673. i) q^{7} +599139. q^{9} +O(q^{10})$	\(q-2045.44i q^{3} +34938.6i q^{5} +(-672278. - 475673. i) q^{7} +599139. q^{9} +2.05817e7 q^{11} -6.03298e7i q^{13} +7.14648e7 q^{15} +3.82704e8i q^{17} +8.94399e8i q^{19} +(-9.72962e8 + 1.37511e9i) q^{21} -1.78816e9 q^{23} -1.22070e9 q^{25} -1.10088e10i q^{27} +8.73289e9 q^{29} +1.20737e10i q^{31} -4.20986e10i q^{33} +(1.66193e10 - 2.34884e10i) q^{35} -1.08749e11 q^{37} -1.23401e11 q^{39} +2.46883e10i q^{41} -1.01758e10 q^{43} +2.09330e10i q^{45} -4.26103e11i q^{47} +(2.25693e11 + 6.39570e11i) q^{49} +7.82798e11 q^{51} -4.19709e11 q^{53} +7.19094e11i q^{55} +1.82944e12 q^{57} +7.09310e11i q^{59} -3.13282e11i q^{61} +(-4.02788e11 - 2.84994e11i) q^{63} +2.10784e12 q^{65} -8.39060e12 q^{67} +3.65758e12i q^{69} -1.41721e13 q^{71} +1.11185e13i q^{73} +2.49688e12i q^{75} +(-1.38366e13 - 9.79016e12i) q^{77} +3.24586e12 q^{79} -1.96522e13 q^{81} +1.83951e13i q^{83} -1.33711e13 q^{85} -1.78626e13i q^{87} +4.09351e13i q^{89} +(-2.86973e13 + 4.05584e13i) q^{91} +2.46961e13 q^{93} -3.12490e13 q^{95} -6.16639e12i q^{97} +1.23313e13 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 1364266 q^{7} - 54790830 q^{9}+O(q^{10}) $ 36 * q + 1364266 * q^7 - 54790830 * q^9	$ 36 q + 1364266 q^{7} - 54790830 q^{9} - 26192606 q^{11} + 44843750 q^{15} + 1512952694 q^{21} - 8670648636 q^{23} - 43945312500 q^{25} - 43956395706 q^{29} + 44839531250 q^{35} - 169523027308 q^{37} + 805671747486 q^{39} + 554691319560 q^{43} + 1095688125176 q^{49} + 1032170625826 q^{51} - 4262050556480 q^{53} - 3162001614828 q^{57} - 15828953775898 q^{63} - 3014492656250 q^{65} - 23495876471600 q^{67} + 22887953193352 q^{71} + 56411959501488 q^{77} + 8995204220854 q^{79} + 132868621377344 q^{81} - 2034215156250 q^{85} - 53912825209186 q^{91} + 101093199187348 q^{93} + 3862990000000 q^{95} - 416078903388420 q^{99}+O(q^{100}) $ 36 * q + 1364266 * q^7 - 54790830 * q^9 - 26192606 * q^11 + 44843750 * q^15 + 1512952694 * q^21 - 8670648636 * q^23 - 43945312500 * q^25 - 43956395706 * q^29 + 44839531250 * q^35 - 169523027308 * q^37 + 805671747486 * q^39 + 554691319560 * q^43 + 1095688125176 * q^49 + 1032170625826 * q^51 - 4262050556480 * q^53 - 3162001614828 * q^57 - 15828953775898 * q^63 - 3014492656250 * q^65 - 23495876471600 * q^67 + 22887953193352 * q^71 + 56411959501488 * q^77 + 8995204220854 * q^79 + 132868621377344 * q^81 - 2034215156250 * q^85 - 53912825209186 * q^91 + 101093199187348 * q^93 + 3862990000000 * q^95 - 416078903388420 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/140\mathbb{Z}\right)^\times$.

$n$	$57$	$71$	$101$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$		−	2045.44i		−	0.935273i	−0.883921	−	0.467636i	$-0.845106\pi$
$3$		−	2045.44i		−	0.935273i	0.883921	−	0.467636i	$-0.154894\pi$
$4$		0			0
$5$			34938.6i			0.447214i
$5$			34938.6i			0.447214i
$6$		0			0
$7$	−672278.	−	475673.i	−0.816324	−	0.577594i
$7$	−672278.	−	475673.i	−0.816324	−	0.577594i
$8$		0			0
$9$	599139.			0.125265
$10$		0			0
$11$	2.05817e7			1.05617			0.528083	−	0.849193i	$-0.322911\pi$
$11$	2.05817e7			1.05617			0.528083	+	0.849193i	$0.322911\pi$
$12$		0			0
$13$		−	6.03298e7i		−	0.961454i	−0.876870	−	0.480727i	$-0.840373\pi$
$13$		−	6.03298e7i		−	0.961454i	0.876870	−	0.480727i	$-0.159627\pi$
$14$		0			0
$15$	7.14648e7			0.418267
$16$		0			0
$17$			3.82704e8i			0.932653i	0.884613	+	0.466327i	$0.154423\pi$
$17$			3.82704e8i			0.932653i	−0.884613	+	0.466327i	$0.845577\pi$
$18$		0			0
$19$			8.94399e8i			1.00059i	0.865855	+	0.500295i	$0.166775\pi$
$19$			8.94399e8i			1.00059i	−0.865855	+	0.500295i	$0.833225\pi$
$20$		0			0
$21$	−9.72962e8	+	1.37511e9i	−0.540208	+	0.763486i
$22$		0			0
$23$	−1.78816e9			−0.525185			−0.262592	−	0.964907i	$-0.584577\pi$
$23$	−1.78816e9			−0.525185			−0.262592	+	0.964907i	$0.584577\pi$
$24$		0			0
$25$	−1.22070e9			−0.200000
$26$		0			0
$27$		−	1.10088e10i		−	1.05243i
$28$		0			0
$29$	8.73289e9			0.506258			0.253129	−	0.967432i	$-0.418540\pi$
$29$	8.73289e9			0.506258			0.253129	+	0.967432i	$0.418540\pi$
$30$		0			0
$31$			1.20737e10i			0.438843i	0.975630	+	0.219421i	$0.0704169\pi$
$31$			1.20737e10i			0.438843i	−0.975630	+	0.219421i	$0.929583\pi$
$32$		0			0
$33$		−	4.20986e10i		−	0.987803i
$34$		0			0
$35$	1.66193e10	−	2.34884e10i	0.258308	−	0.365071i
$36$		0			0
$37$	−1.08749e11			−1.14555			−0.572775	−	0.819713i	$-0.694133\pi$
$37$	−1.08749e11			−1.14555			−0.572775	+	0.819713i	$0.694133\pi$
$38$		0			0
$39$	−1.23401e11			−0.899222
$40$		0			0
$41$			2.46883e10i			0.126766i	0.997989	+	0.0633832i	$0.0201890\pi$
$41$			2.46883e10i			0.126766i	−0.997989	+	0.0633832i	$0.979811\pi$
$42$		0			0
$43$	−1.01758e10			−0.0374359			−0.0187179	−	0.999825i	$-0.505958\pi$
$43$	−1.01758e10			−0.0374359			−0.0187179	+	0.999825i	$0.505958\pi$
$44$		0			0
$45$			2.09330e10i			0.0560202i
$46$		0			0
$47$		−	4.26103e11i		−	0.841065i	−0.907277	−	0.420533i	$-0.861843\pi$
$47$		−	4.26103e11i		−	0.841065i	0.907277	−	0.420533i	$-0.138157\pi$
$48$		0			0
$49$	2.25693e11	+	6.39570e11i	0.332771	+	0.943008i
$50$		0			0
$51$	7.82798e11			0.872285
$52$		0			0
$53$	−4.19709e11			−0.357287			−0.178644	−	0.983914i	$-0.557171\pi$
$53$	−4.19709e11			−0.357287			−0.178644	+	0.983914i	$0.557171\pi$
$54$		0			0
$55$			7.19094e11i			0.472332i
$56$		0			0
$57$	1.82944e12			0.935824
$58$		0			0
$59$			7.09310e11i			0.285018i	0.989794	+	0.142509i	$0.0455170\pi$
$59$			7.09310e11i			0.285018i	−0.989794	+	0.142509i	$0.954483\pi$
$60$		0			0
$61$		−	3.13282e11i		−	0.0996842i	−0.998757	−	0.0498421i	$-0.984128\pi$
$61$		−	3.13282e11i		−	0.0996842i	0.998757	−	0.0498421i	$-0.0158718\pi$
$62$		0			0
$63$	−4.02788e11	−	2.84994e11i	−0.102257	−	0.0723523i
$64$		0			0
$65$	2.10784e12			0.429975
$66$		0			0
$67$	−8.39060e12			−1.38442			−0.692212	−	0.721694i	$-0.743364\pi$
$67$	−8.39060e12			−1.38442			−0.692212	+	0.721694i	$0.743364\pi$
$68$		0			0
$69$			3.65758e12i			0.491191i
$70$		0			0
$71$	−1.41721e13			−1.55821			−0.779106	−	0.626892i	$-0.784327\pi$
$71$	−1.41721e13			−1.55821			−0.779106	+	0.626892i	$0.784327\pi$
$72$		0			0
$73$			1.11185e13i			1.00644i	0.864158	+	0.503220i	$0.167851\pi$
$73$			1.11185e13i			1.00644i	−0.864158	+	0.503220i	$0.832149\pi$
$74$		0			0
$75$			2.49688e12i			0.187055i
$76$		0			0
$77$	−1.38366e13	−	9.79016e12i	−0.862174	−	0.610035i
$78$		0			0
$79$	3.24586e12			0.169021			0.0845103	−	0.996423i	$-0.473067\pi$
$79$	3.24586e12			0.169021			0.0845103	+	0.996423i	$0.473067\pi$
$80$		0			0
$81$	−1.96522e13			−0.859044
$82$		0			0
$83$			1.83951e13i			0.677885i	0.940807	+	0.338943i	$0.110069\pi$
$83$			1.83951e13i			0.677885i	−0.940807	+	0.338943i	$0.889931\pi$
$84$		0			0
$85$	−1.33711e13			−0.417095
$86$		0			0
$87$		−	1.78626e13i		−	0.473490i
$88$		0			0
$89$			4.09351e13i			0.925477i	0.886495	+	0.462739i	$0.153133\pi$
$89$			4.09351e13i			0.925477i	−0.886495	+	0.462739i	$0.846867\pi$
$90$		0			0
$91$	−2.86973e13	+	4.05584e13i	−0.555330	+	0.784858i
$92$		0			0
$93$	2.46961e13			0.410438
$94$		0			0
$95$	−3.12490e13			−0.447477
$96$		0			0
$97$		−	6.16639e12i		−	0.0763183i	−0.999272	−	0.0381592i	$-0.987851\pi$
$97$		−	6.16639e12i		−	0.0763183i	0.999272	−	0.0381592i	$-0.0121494\pi$
$98$		0			0
$99$	1.23313e13			0.132301
$100$		0			0
$101$			3.97962e13i			0.371186i	0.982627	+	0.185593i	$0.0594207\pi$
$101$			3.97962e13i			0.371186i	−0.982627	+	0.185593i	$0.940579\pi$
$102$		0			0
$103$		−	1.01029e14i		−	0.821456i	−0.911758	−	0.410728i	$-0.865275\pi$
$103$		−	1.01029e14i		−	0.821456i	0.911758	−	0.410728i	$-0.134725\pi$
$104$		0			0
$105$	−4.80442e13	−	3.39939e13i	−0.341441	−	0.241588i
$106$		0			0
$107$	1.60852e14			1.00171			0.500854	−	0.865532i	$-0.333019\pi$
$107$	1.60852e14			1.00171			0.500854	+	0.865532i	$0.333019\pi$
$108$		0			0
$109$	1.48692e13			0.0813396			0.0406698	−	0.999173i	$-0.487051\pi$
$109$	1.48692e13			0.0813396			0.0406698	+	0.999173i	$0.487051\pi$
$110$		0			0
$111$			2.22440e14i			1.07140i
$112$		0			0
$113$	5.43228e13			0.230905			0.115452	−	0.993313i	$-0.463168\pi$
$113$	5.43228e13			0.230905			0.115452	+	0.993313i	$0.463168\pi$
$114$		0			0
$115$		−	6.24758e13i		−	0.234870i
$116$		0			0
$117$		−	3.61459e13i		−	0.120437i
$118$		0			0
$119$	1.82042e14	−	2.57283e14i	0.538695	−	0.761348i
$120$		0			0
$121$	4.38559e13			0.115486
$122$		0			0
$123$	5.04985e13			0.118561
$124$		0			0
$125$		−	4.26496e13i		−	0.0894427i
$126$		0			0
$127$	8.85446e14			1.66164			0.830818	−	0.556544i	$-0.187873\pi$
$127$	8.85446e14			1.66164			0.830818	+	0.556544i	$0.187873\pi$
$128$		0			0
$129$			2.08139e13i			0.0350127i
$130$		0			0
$131$			5.27580e14i			0.796873i	0.917196	+	0.398436i	$0.130447\pi$
$131$			5.27580e14i			0.796873i	−0.917196	+	0.398436i	$0.869553\pi$
$132$		0			0
$133$	4.25442e14	−	6.01285e14i	0.577934	−	0.816806i
$134$		0			0
$135$	3.84631e14			0.470661
$136$		0			0
$137$	−1.68192e15			−1.85678			−0.928392	−	0.371602i	$-0.878809\pi$
$137$	−1.68192e15			−1.85678			−0.928392	+	0.371602i	$0.878809\pi$
$138$		0			0
$139$			6.00912e14i			0.599387i	0.954036	+	0.299694i	$0.0968844\pi$
$139$			6.00912e14i			0.599387i	−0.954036	+	0.299694i	$0.903116\pi$
$140$		0			0
$141$	−8.71569e14			−0.786625
$142$		0			0
$143$		−	1.24169e15i		−	1.01546i
$144$		0			0
$145$			3.05115e14i			0.226406i
$146$		0			0
$147$	1.30820e15	−	4.61642e14i	0.881969	−	0.311231i
$148$		0			0
$149$	−1.42669e13			−0.00875035			−0.00437517	−	0.999990i	$-0.501393\pi$
$149$	−1.42669e13			−0.00875035			−0.00437517	+	0.999990i	$0.501393\pi$
$150$		0			0
$151$	1.39131e15			0.777295			0.388647	−	0.921387i	$-0.372942\pi$
$151$	1.39131e15			0.777295			0.388647	+	0.921387i	$0.372942\pi$
$152$		0			0
$153$			2.29293e14i			0.116829i
$154$		0			0
$155$	−4.21838e14			−0.196256
$156$		0			0
$157$			1.69296e15i			0.720027i	0.932947	+	0.360014i	$0.117228\pi$
$157$			1.69296e15i			0.720027i	−0.932947	+	0.360014i	$0.882772\pi$
$158$		0			0
$159$			8.58490e14i			0.334161i
$160$		0			0
$161$	1.20214e15	+	8.50582e14i	0.428721	+	0.303344i
$162$		0			0
$163$	2.34131e15			0.765852			0.382926	−	0.923779i	$-0.374916\pi$
$163$	2.34131e15			0.765852			0.382926	+	0.923779i	$0.374916\pi$
$164$		0			0
$165$	1.47087e15			0.441759
$166$		0			0
$167$			6.13546e15i			1.69368i	0.531846	+	0.846841i	$0.321498\pi$
$167$			6.13546e15i			1.69368i	−0.531846	+	0.846841i	$0.678502\pi$
$168$		0			0
$169$	2.97689e14			0.0756059
$170$		0			0
$171$			5.35869e14i			0.125339i
$172$		0			0
$173$		−	3.50350e14i		−	0.0755405i	−0.999286	−	0.0377702i	$-0.987975\pi$
$173$		−	3.50350e14i		−	0.0755405i	0.999286	−	0.0377702i	$-0.0120255\pi$
$174$		0			0
$175$	8.20652e14	+	5.80656e14i	0.163265	+	0.115519i
$176$		0			0
$177$	1.45085e15			0.266569
$178$		0			0
$179$	1.81193e15			0.307730			0.153865	−	0.988092i	$-0.450828\pi$
$179$	1.81193e15			0.307730			0.153865	+	0.988092i	$0.450828\pi$
$180$		0			0
$181$			4.58659e15i			0.720675i	0.932822	+	0.360338i	$0.117339\pi$
$181$			4.58659e15i			0.720675i	−0.932822	+	0.360338i	$0.882661\pi$
$182$		0			0
$183$	−6.40800e14			−0.0932320
$184$		0			0
$185$		−	3.79954e15i		−	0.512306i
$186$		0			0
$187$			7.87669e15i			0.985037i
$188$		0			0
$189$	−5.23659e15	+	7.40097e15i	−0.607877	+	0.859124i
$190$		0			0
$191$	−9.75155e15			−1.05157			−0.525787	−	0.850616i	$-0.676229\pi$
$191$	−9.75155e15			−1.05157			−0.525787	+	0.850616i	$0.676229\pi$
$192$		0			0
$193$	1.01342e16			1.01599			0.507993	−	0.861361i	$-0.330388\pi$
$193$	1.01342e16			1.01599			0.507993	+	0.861361i	$0.330388\pi$
$194$		0			0
$195$		−	4.31146e15i		−	0.402144i
$196$		0			0
$197$	−1.14900e16			−0.997834			−0.498917	−	0.866650i	$-0.666269\pi$
$197$	−1.14900e16			−0.997834			−0.498917	+	0.866650i	$0.666269\pi$
$198$		0			0
$199$		−	3.79855e14i		−	0.0307359i	−0.999882	−	0.0153680i	$-0.995108\pi$
$199$		−	3.79855e14i		−	0.0307359i	0.999882	−	0.0153680i	$-0.00489196\pi$
$200$		0			0
$201$			1.71625e16i			1.29481i
$202$		0			0
$203$	−5.87093e15	−	4.15400e15i	−0.413271	−	0.292412i
$204$		0			0
$205$	−8.62574e14			−0.0566917
$206$		0			0
$207$	−1.07136e15			−0.0657873
$208$		0			0
$209$			1.84082e16i			1.05679i
$210$		0			0
$211$	3.47998e16			1.86896			0.934478	−	0.356020i	$-0.115866\pi$
$211$	3.47998e16			1.86896			0.934478	+	0.356020i	$0.115866\pi$
$212$		0			0
$213$			2.89882e16i			1.45735i
$214$		0			0
$215$		−	3.55527e14i		−	0.0167418i
$216$		0			0
$217$	5.74314e15	−	8.11689e15i	0.253473	−	0.358238i
$218$		0			0
$219$	2.27423e16			0.941296
$220$		0			0
$221$	2.30884e16			0.896703
$222$		0			0
$223$			1.11179e16i			0.405405i	0.979240	+	0.202702i	$0.0649724\pi$
$223$			1.11179e16i			0.405405i	−0.979240	+	0.202702i	$0.935028\pi$
$224$		0			0
$225$	−7.31371e14			−0.0250530
$226$		0			0
$227$			8.72827e15i			0.281026i	0.990079	+	0.140513i	$0.0448753\pi$
$227$			8.72827e15i			0.281026i	−0.990079	+	0.140513i	$0.955125\pi$
$228$		0			0
$229$			3.52099e16i			1.06615i	0.846070	+	0.533073i	$0.178963\pi$
$229$			3.52099e16i			1.06615i	−0.846070	+	0.533073i	$0.821037\pi$
$230$		0			0
$231$	−2.00252e16	+	2.83020e16i	−0.570549	+	0.806368i
$232$		0			0
$233$	4.92865e16			1.32201			0.661007	−	0.750379i	$-0.270129\pi$
$233$	4.92865e16			1.32201			0.661007	+	0.750379i	$0.270129\pi$
$234$		0			0
$235$	1.48874e16			0.376136
$236$		0			0
$237$		−	6.63921e15i		−	0.158080i
$238$		0			0
$239$	3.85078e16			0.864497			0.432249	−	0.901754i	$-0.357720\pi$
$239$	3.85078e16			0.864497			0.432249	+	0.901754i	$0.357720\pi$
$240$		0			0
$241$			6.51601e16i			1.37995i	0.723835	+	0.689974i	$0.242378\pi$
$241$			6.51601e16i			1.37995i	−0.723835	+	0.689974i	$0.757622\pi$
$242$		0			0
$243$		−	1.24573e16i		−	0.248990i
$244$		0			0
$245$	−2.23456e16	+	7.88538e15i	−0.421726	+	0.148820i
$246$		0			0
$247$	5.39589e16			0.962021
$248$		0			0
$249$	3.76262e16			0.634008
$250$		0			0
$251$		−	4.86880e16i		−	0.775722i	−0.921718	−	0.387861i	$-0.873214\pi$
$251$		−	4.86880e16i		−	0.775722i	0.921718	−	0.387861i	$-0.126786\pi$
$252$		0			0
$253$	−3.68034e16			−0.554682
$254$		0			0
$255$			2.73498e16i			0.390098i
$256$		0			0
$257$			3.39748e15i			0.0458802i	0.999737	+	0.0229401i	$0.00730269\pi$
$257$			3.39748e15i			0.0458802i	−0.999737	+	0.0229401i	$0.992697\pi$
$258$		0			0
$259$	7.31097e16	+	5.17291e16i	0.935141	+	0.661663i
$260$		0			0
$261$	5.23222e15			0.0634165
$262$		0			0
$263$	2.44998e16			0.281496			0.140748	−	0.990045i	$-0.455049\pi$
$263$	2.44998e16			0.281496			0.140748	+	0.990045i	$0.455049\pi$
$264$		0			0
$265$		−	1.46640e16i		−	0.159784i
$266$		0			0
$267$	8.37303e16			0.865574
$268$		0			0
$269$		−	9.68988e15i		−	0.0950719i	−0.998870	−	0.0475360i	$-0.984863\pi$
$269$		−	9.68988e15i		−	0.0950719i	0.998870	−	0.0475360i	$-0.0151369\pi$
$270$		0			0
$271$		−	5.73657e16i		−	0.534401i	−0.963641	−	0.267200i	$-0.913901\pi$
$271$		−	5.73657e16i		−	0.534401i	0.963641	−	0.267200i	$-0.0860986\pi$
$272$		0			0
$273$	8.29599e16	+	5.86986e16i	0.734057	+	0.519385i
$274$		0			0
$275$	−2.51241e16			−0.211233
$276$		0			0
$277$	6.67911e15			0.0533778			0.0266889	−	0.999644i	$-0.491504\pi$
$277$	6.67911e15			0.0533778			0.0266889	+	0.999644i	$0.491504\pi$
$278$		0			0
$279$			7.23383e15i			0.0549716i
$280$		0			0
$281$	−2.53604e14			−0.00183321			−0.000916605	−	1.00000i	$-0.500292\pi$
$281$	−2.53604e14			−0.00183321			−0.000916605		1.00000i	$0.500292\pi$
$282$		0			0
$283$			2.22870e17i			1.53302i	0.642234	+	0.766509i	$0.278008\pi$
$283$			2.22870e17i			1.53302i	−0.642234	+	0.766509i	$0.721992\pi$
$284$		0			0
$285$			6.39180e16i			0.418513i
$286$		0			0
$287$	1.17436e16	−	1.65974e16i	0.0732195	−	0.103483i
$288$		0			0
$289$	2.19157e16			0.130158
$290$		0			0
$291$	−1.26130e16			−0.0713784
$292$		0			0
$293$			3.09519e17i			1.66961i	0.550547	+	0.834804i	$0.314419\pi$
$293$			3.09519e17i			1.66961i	−0.550547	+	0.834804i	$0.685581\pi$
$294$		0			0
$295$	−2.47823e16			−0.127464
$296$		0			0
$297$		−	2.26579e17i		−	1.11154i
$298$		0			0
$299$			1.07880e17i			0.504941i
$300$		0			0
$301$	6.84094e15	+	4.84034e15i	0.0305598	+	0.0216227i
$302$		0			0
$303$	8.14008e16			0.347160
$304$		0			0
$305$	1.09456e16			0.0445802
$306$		0			0
$307$			1.13690e17i			0.442337i	0.975236	+	0.221169i	$0.0709872\pi$
$307$			1.13690e17i			0.442337i	−0.975236	+	0.221169i	$0.929013\pi$
$308$		0			0
$309$	−2.06648e17			−0.768286
$310$		0			0
$311$			1.86233e17i			0.661812i	0.943664	+	0.330906i	$0.107354\pi$
$311$			1.86233e17i			0.661812i	−0.943664	+	0.330906i	$0.892646\pi$
$312$		0			0
$313$			7.66586e16i			0.260466i	0.991483	+	0.130233i	$0.0415725\pi$
$313$			7.66586e16i			0.260466i	−0.991483	+	0.130233i	$0.958427\pi$
$314$		0			0
$315$	9.95729e15	−	1.40728e16i	0.0323569	−	0.0457307i
$316$		0			0
$317$	−2.77583e17			−0.862936			−0.431468	−	0.902128i	$-0.642004\pi$
$317$	−2.77583e17			−0.862936			−0.431468	+	0.902128i	$0.642004\pi$
$318$		0			0
$319$	1.79738e17			0.534693
$320$		0			0
$321$		−	3.29014e17i		−	0.936870i
$322$		0			0
$323$	−3.42290e17			−0.933203
$324$		0			0
$325$			7.36448e16i			0.192291i
$326$		0			0
$327$		−	3.04141e16i		−	0.0760747i
$328$		0			0
$329$	−2.02686e17	+	2.86460e17i	−0.485794	+	0.686582i
$330$		0			0
$331$	8.39753e17			1.92910			0.964552	−	0.263892i	$-0.0850064\pi$
$331$	8.39753e17			1.92910			0.964552	+	0.263892i	$0.0850064\pi$
$332$		0			0
$333$	−6.51559e16			−0.143497
$334$		0			0
$335$		−	2.93155e17i		−	0.619134i
$336$		0			0
$337$	−6.14109e17			−1.24404			−0.622022	−	0.782999i	$-0.713689\pi$
$337$	−6.14109e17			−1.24404			−0.622022	+	0.782999i	$0.713689\pi$
$338$		0			0
$339$		−	1.11114e17i		−	0.215959i
$340$		0			0
$341$			2.48497e17i			0.463491i
$342$		0			0
$343$	1.52498e17	−	5.37325e17i	0.273026	−	0.962007i
$344$		0			0
$345$	−1.27791e17			−0.219667
$346$		0			0
$347$	−4.21439e17			−0.695710			−0.347855	−	0.937548i	$-0.613090\pi$
$347$	−4.21439e17			−0.695710			−0.347855	+	0.937548i	$0.613090\pi$
$348$		0			0
$349$			9.68382e17i			1.53557i	0.640709	+	0.767784i	$0.278641\pi$
$349$			9.68382e17i			1.53557i	−0.640709	+	0.767784i	$0.721359\pi$
$350$		0			0
$351$	−6.64158e17			−1.01186
$352$		0			0
$353$		−	5.37072e16i		−	0.0786338i	−0.999227	−	0.0393169i	$-0.987482\pi$
$353$		−	5.37072e16i		−	0.0786338i	0.999227	−	0.0393169i	$-0.0125182\pi$
$354$		0			0
$355$		−	4.95154e17i		−	0.696853i
$356$		0			0
$357$	−5.26258e17	−	3.72356e17i	−0.712068	−	0.503827i
$358$		0			0
$359$	5.68525e17			0.739756			0.369878	−	0.929080i	$-0.379399\pi$
$359$	5.68525e17			0.739756			0.369878	+	0.929080i	$0.379399\pi$
$360$		0			0
$361$	−9.42554e14			−0.00117966
$362$		0			0
$363$		−	8.97046e16i		−	0.108011i
$364$		0			0
$365$	−3.88466e17			−0.450094
$366$		0			0
$367$		−	1.71491e18i		−	1.91241i	−0.292706	−	0.956203i	$-0.594556\pi$
$367$		−	1.71491e18i		−	1.91241i	0.292706	−	0.956203i	$-0.405444\pi$
$368$		0			0
$369$			1.47917e16i			0.0158794i
$370$		0			0
$371$	2.82161e17	+	1.99644e17i	0.291662	+	0.206367i
$372$		0			0
$373$	−1.22148e17			−0.121598			−0.0607989	−	0.998150i	$-0.519365\pi$
$373$	−1.22148e17			−0.121598			−0.0607989	+	0.998150i	$0.519365\pi$
$374$		0			0
$375$	−8.72373e16			−0.0836533
$376$		0			0
$377$		−	5.26854e17i		−	0.486744i
$378$		0			0
$379$	9.05706e17			0.806330			0.403165	−	0.915127i	$-0.367910\pi$
$379$	9.05706e17			0.806330			0.403165	+	0.915127i	$0.367910\pi$
$380$		0			0
$381$		−	1.81113e18i		−	1.55408i
$382$		0			0
$383$		−	8.13965e17i		−	0.673309i	−0.941628	−	0.336655i	$-0.890705\pi$
$383$		−	8.13965e17i		−	0.673309i	0.941628	−	0.336655i	$-0.109295\pi$
$384$		0			0
$385$	3.42054e17	−	4.83432e17i	0.272816	−	0.385576i
$386$		0			0
$387$	−6.09669e15			−0.00468940
$388$		0			0
$389$	1.84422e18			1.36825			0.684124	−	0.729365i	$-0.260185\pi$
$389$	1.84422e18			1.36825			0.684124	+	0.729365i	$0.260185\pi$
$390$		0			0
$391$		−	6.84337e17i		−	0.489816i
$392$		0			0
$393$	1.07913e18			0.745293
$394$		0			0
$395$			1.13406e17i			0.0755883i
$396$		0			0
$397$		−	2.65450e18i		−	1.70785i	−0.520399	−	0.853923i	$-0.674217\pi$
$397$		−	2.65450e18i		−	1.70785i	0.520399	−	0.853923i	$-0.325783\pi$
$398$		0			0
$399$	−1.22989e18	−	8.70216e17i	−0.763936	−	0.540526i
$400$		0			0
$401$	−2.12677e18			−1.27559			−0.637793	−	0.770208i	$-0.720152\pi$
$401$	−2.12677e18			−1.27559			−0.637793	+	0.770208i	$0.720152\pi$
$402$		0			0
$403$	7.28405e17			0.421927
$404$		0			0
$405$		−	6.86618e17i		−	0.384176i
$406$		0			0
$407$	−2.23824e18			−1.20989
$408$		0			0
$409$			1.61418e18i			0.843117i	0.906801	+	0.421558i	$0.138517\pi$
$409$			1.61418e18i			0.843117i	−0.906801	+	0.421558i	$0.861483\pi$
$410$		0			0
$411$			3.44027e18i			1.73660i
$412$		0			0
$413$	3.37400e17	−	4.76854e17i	0.164625	−	0.232667i
$414$		0			0
$415$	−6.42699e17			−0.303160
$416$		0			0
$417$	1.22913e18			0.560590
$418$		0			0
$419$		−	1.49090e18i		−	0.657581i	−0.944403	−	0.328790i	$-0.893359\pi$
$419$		−	1.49090e18i		−	0.657581i	0.944403	−	0.328790i	$-0.106641\pi$
$420$		0			0
$421$	−1.23497e18			−0.526843			−0.263422	−	0.964681i	$-0.584851\pi$
$421$	−1.23497e18			−0.526843			−0.263422	+	0.964681i	$0.584851\pi$
$422$		0			0
$423$		−	2.55295e17i		−	0.105356i
$424$		0			0
$425$		−	4.67168e17i		−	0.186531i
$426$		0			0
$427$	−1.49020e17	+	2.10613e17i	−0.0575770	+	0.0813747i
$428$		0			0
$429$	−2.53980e18			−0.949727
$430$		0			0
$431$	−2.45417e18			−0.888308			−0.444154	−	0.895950i	$-0.646496\pi$
$431$	−2.45417e18			−0.888308			−0.444154	+	0.895950i	$0.646496\pi$
$432$		0			0
$433$			6.21109e16i			0.0217647i	0.999941	+	0.0108823i	$0.00346402\pi$
$433$			6.21109e16i			0.0217647i	−0.999941	+	0.0108823i	$0.996536\pi$
$434$		0			0
$435$	6.24094e17			0.211751
$436$		0			0
$437$		−	1.59933e18i		−	0.525495i
$438$		0			0
$439$		−	2.25134e18i		−	0.716457i	−0.933634	−	0.358229i	$-0.883381\pi$
$439$		−	2.25134e18i		−	0.716457i	0.933634	−	0.358229i	$-0.116619\pi$
$440$		0			0
$441$	1.35221e17	+	3.83191e17i	0.0416845	+	0.118126i
$442$		0			0
$443$	−7.60393e16			−0.0227097			−0.0113549	−	0.999936i	$-0.503614\pi$
$443$	−7.60393e16			−0.0227097			−0.0113549	+	0.999936i	$0.503614\pi$
$444$		0			0
$445$	−1.43021e18			−0.413886
$446$		0			0
$447$			2.91821e16i			0.00818396i
$448$		0			0
$449$	−5.70784e18			−1.55149			−0.775743	−	0.631049i	$-0.782625\pi$
$449$	−5.70784e18			−1.55149			−0.775743	+	0.631049i	$0.782625\pi$
$450$		0			0
$451$			5.08127e17i			0.133886i
$452$		0			0
$453$		−	2.84585e18i		−	0.726983i
$454$		0			0
$455$	−1.41705e18	−	1.00264e18i	−0.350999	−	0.248351i
$456$		0			0
$457$	−6.58605e18			−1.58202			−0.791010	−	0.611803i	$-0.790444\pi$
$457$	−6.58605e18			−1.58202			−0.791010	+	0.611803i	$0.790444\pi$
$458$		0			0
$459$	4.21310e18			0.981552
$460$		0			0
$461$		−	1.69959e18i		−	0.384094i	−0.981386	−	0.192047i	$-0.938487\pi$
$461$		−	1.69959e18i		−	0.384094i	0.981386	−	0.192047i	$-0.0615126\pi$
$462$		0			0
$463$	−7.62096e18			−1.67087			−0.835435	−	0.549590i	$-0.814784\pi$
$463$	−7.62096e18			−1.67087			−0.835435	+	0.549590i	$0.814784\pi$
$464$		0			0
$465$			8.62845e17i			0.183553i
$466$		0			0
$467$		−	4.94342e18i		−	1.02049i	−0.860029	−	0.510245i	$-0.829555\pi$
$467$		−	4.94342e18i		−	1.02049i	0.860029	−	0.510245i	$-0.170445\pi$
$468$		0			0
$469$	5.64082e18	+	3.99118e18i	1.13014	+	0.799635i
$470$		0			0
$471$	3.46285e18			0.673422
$472$		0			0
$473$	−2.09434e17			−0.0395385
$474$		0			0
$475$		−	1.09180e18i		−	0.200118i
$476$		0			0
$477$	−2.51464e17			−0.0447556
$478$		0			0
$479$			7.71223e18i			1.33300i	0.745504	+	0.666502i	$0.232209\pi$
$479$			7.71223e18i			1.33300i	−0.745504	+	0.666502i	$0.767791\pi$
$480$		0			0
$481$			6.56082e18i			1.10139i
$482$		0			0
$483$	1.73981e18	−	2.45891e18i	0.283709	−	0.400971i
$484$		0			0
$485$	2.15445e17			0.0341306
$486$		0			0
$487$	−3.72892e18			−0.573958			−0.286979	−	0.957937i	$-0.592651\pi$
$487$	−3.72892e18			−0.573958			−0.286979	+	0.957937i	$0.592651\pi$
$488$		0			0
$489$		−	4.78900e18i		−	0.716280i
$490$		0			0
$491$	−7.79707e18			−1.13334			−0.566671	−	0.823944i	$-0.691769\pi$
$491$	−7.79707e18			−1.13334			−0.566671	+	0.823944i	$0.691769\pi$
$492$		0			0
$493$			3.34211e18i			0.472164i
$494$		0			0
$495$			4.30837e17i			0.0591666i
$496$		0			0
$497$	9.52761e18	+	6.74130e18i	1.27201	+	0.900013i
$498$		0			0
$499$	8.77180e18			1.13864			0.569318	−	0.822117i	$-0.307207\pi$
$499$	8.77180e18			1.13864			0.569318	+	0.822117i	$0.307207\pi$
$500$		0			0
$501$	1.25497e19			1.58405
$502$		0			0
$503$			9.30368e18i			1.14203i	0.820939	+	0.571017i	$0.193451\pi$
$503$			9.30368e18i			1.14203i	−0.820939	+	0.571017i	$0.806549\pi$
$504$		0			0
$505$	−1.39042e18			−0.166000
$506$		0			0
$507$		−	6.08905e17i		−	0.0707121i
$508$		0			0
$509$			5.41874e18i			0.612173i	0.952004	+	0.306086i	$0.0990197\pi$
$509$			5.41874e18i			0.612173i	−0.952004	+	0.306086i	$0.900980\pi$
$510$		0			0
$511$	5.28879e18	−	7.47475e18i	0.581314	−	0.821581i
$512$		0			0
$513$	9.84624e18			1.05305
$514$		0			0
$515$	3.52980e18			0.367366
$516$		0			0
$517$		−	8.76992e18i		−	0.888304i
$518$		0			0
$519$	−7.16621e17			−0.0706509
$520$		0			0
$521$		−	1.53977e19i		−	1.47772i	−0.673859	−	0.738860i	$-0.735364\pi$
$521$		−	1.53977e19i		−	1.47772i	0.673859	−	0.738860i	$-0.264636\pi$
$522$		0			0
$523$			9.24825e18i			0.864068i	0.901857	+	0.432034i	$0.142204\pi$
$523$			9.24825e18i			0.864068i	−0.901857	+	0.432034i	$0.857796\pi$
$524$		0			0
$525$	1.18770e18	−	1.67860e18i	0.108042	−	0.152697i
$526$		0			0
$527$	−4.62065e18			−0.409288
$528$		0			0
$529$	−8.39531e18			−0.724181
$530$		0			0
$531$			4.24975e17i			0.0357028i
$532$		0			0
$533$	1.48944e18			0.121880
$534$		0			0
$535$			5.61995e18i			0.447977i
$536$		0			0
$537$		−	3.70620e18i		−	0.287812i
$538$		0			0
$539$	4.64514e18	+	1.31634e19i	0.351461	+	0.995973i
$540$		0			0
$541$	−1.26511e19			−0.932714			−0.466357	−	0.884597i	$-0.654434\pi$
$541$	−1.26511e19			−0.932714			−0.466357	+	0.884597i	$0.654434\pi$
$542$		0			0
$543$	9.38159e18			0.674028
$544$		0			0
$545$			5.19508e17i			0.0363762i
$546$		0			0
$547$	−1.75098e18			−0.119500			−0.0597502	−	0.998213i	$-0.519030\pi$
$547$	−1.75098e18			−0.119500			−0.0597502	+	0.998213i	$0.519030\pi$
$548$		0			0
$549$		−	1.87699e17i		−	0.0124869i
$550$		0			0
$551$			7.81069e18i			0.506557i
$552$		0			0
$553$	−2.18212e18	−	1.54397e18i	−0.137976	−	0.0976252i
$554$		0			0
$555$	−7.77174e18			−0.479145
$556$		0			0
$557$	1.10249e19			0.662808			0.331404	−	0.943489i	$-0.392478\pi$
$557$	1.10249e19			0.662808			0.331404	+	0.943489i	$0.392478\pi$
$558$		0			0
$559$			6.13902e17i			0.0359929i
$560$		0			0
$561$	1.61113e19			0.921278
$562$		0			0
$563$			2.16087e19i			1.20523i	0.798032	+	0.602614i	$0.205874\pi$
$563$			2.16087e19i			1.20523i	−0.798032	+	0.602614i	$0.794126\pi$
$564$		0			0
$565$			1.89796e18i			0.103264i
$566$		0			0
$567$	1.32117e19	+	9.34801e18i	0.701258	+	0.496178i
$568$		0			0
$569$	9.55673e18			0.494907			0.247453	−	0.968900i	$-0.420406\pi$
$569$	9.55673e18			0.494907			0.247453	+	0.968900i	$0.420406\pi$
$570$		0			0
$571$	2.14456e19			1.08364			0.541821	−	0.840494i	$-0.317735\pi$
$571$	2.14456e19			1.08364			0.541821	+	0.840494i	$0.317735\pi$
$572$		0			0
$573$			1.99462e19i			0.983509i
$574$		0			0
$575$	2.18282e18			0.105037
$576$		0			0
$577$		−	1.20291e19i		−	0.564942i	−0.959276	−	0.282471i	$-0.908846\pi$
$577$		−	1.20291e19i		−	0.564942i	0.959276	−	0.282471i	$-0.0911541\pi$
$578$		0			0
$579$		−	2.07289e19i		−	0.950224i
$580$		0			0
$581$	8.75007e18	−	1.23666e19i	0.391542	−	0.553374i
$582$		0			0
$583$	−8.63832e18			−0.377354
$584$		0			0
$585$	1.26289e18			0.0538609
$586$		0			0
$587$			3.90443e19i			1.62589i	0.582342	+	0.812944i	$0.302137\pi$
$587$			3.90443e19i			1.62589i	−0.582342	+	0.812944i	$0.697863\pi$
$588$		0			0
$589$	−1.07987e19			−0.439101
$590$		0			0
$591$			2.35022e19i			0.933247i
$592$		0			0
$593$		−	2.35566e18i		−	0.0913544i	−0.998956	−	0.0456772i	$-0.985455\pi$
$593$		−	2.35566e18i		−	0.0913544i	0.998956	−	0.0456772i	$-0.0145446\pi$
$594$		0			0
$595$	8.98911e18	+	6.36028e18i	0.340485	+	0.240912i
$596$		0			0
$597$	−7.76971e17			−0.0287465
$598$		0			0
$599$	2.26835e19			0.819827			0.409914	−	0.912124i	$-0.365559\pi$
$599$	2.26835e19			0.819827			0.409914	+	0.912124i	$0.365559\pi$
$600$		0			0
$601$			4.52874e19i			1.59903i	0.600647	+	0.799515i	$0.294910\pi$
$601$			4.52874e19i			1.59903i	−0.600647	+	0.799515i	$0.705090\pi$
$602$		0			0
$603$	−5.02713e18			−0.173420
$604$		0			0
$605$			1.53226e18i			0.0516470i
$606$		0			0
$607$			2.51707e19i			0.829039i	0.910040	+	0.414519i	$0.136050\pi$
$607$			2.51707e19i			0.829039i	−0.910040	+	0.414519i	$0.863950\pi$
$608$		0			0
$609$	−8.49677e18	+	1.20087e19i	−0.273485	+	0.386521i
$610$		0			0
$611$	−2.57067e19			−0.808646
$612$		0			0
$613$	1.18472e19			0.364243			0.182122	−	0.983276i	$-0.441704\pi$
$613$	1.18472e19			0.364243			0.182122	+	0.983276i	$0.441704\pi$
$614$		0			0
$615$			1.76434e18i			0.0530222i
$616$		0			0
$617$	2.66864e19			0.783960			0.391980	−	0.919974i	$-0.371790\pi$
$617$	2.66864e19			0.783960			0.391980	+	0.919974i	$0.371790\pi$
$618$		0			0
$619$		−	1.64890e19i		−	0.473544i	−0.971565	−	0.236772i	$-0.923911\pi$
$619$		−	1.64890e19i		−	0.473544i	0.971565	−	0.236772i	$-0.0760894\pi$
$620$		0			0
$621$			1.96855e19i			0.552720i
$622$		0			0
$623$	1.94717e19	−	2.75198e19i	0.534550	−	0.755490i
$624$		0			0
$625$	1.49012e18			0.0400000
$626$		0			0
$627$	3.76530e19			0.988385
$628$		0			0
$629$		−	4.16187e19i		−	1.06840i
$630$		0			0
$631$	−6.26198e19			−1.57219			−0.786097	−	0.618103i	$-0.787902\pi$
$631$	−6.26198e19			−1.57219			−0.786097	+	0.618103i	$0.787902\pi$
$632$		0			0
$633$		−	7.11809e19i		−	1.74798i
$634$		0			0
$635$			3.09362e19i			0.743106i
$636$		0			0
$637$	3.85851e19	−	1.36160e19i	0.906659	−	0.319944i
$638$		0			0
$639$	−8.49107e18			−0.195189
$640$		0			0
$641$	−3.31837e19			−0.746309			−0.373154	−	0.927769i	$-0.621724\pi$
$641$	−3.31837e19			−0.746309			−0.373154	+	0.927769i	$0.621724\pi$
$642$		0			0
$643$			1.77873e19i			0.391411i	0.980663	+	0.195705i	$0.0626996\pi$
$643$			1.77873e19i			0.391411i	−0.980663	+	0.195705i	$0.937300\pi$
$644$		0			0
$645$	−7.27209e17			−0.0156582
$646$		0			0
$647$			9.06881e19i			1.91082i	0.295277	+	0.955412i	$0.404588\pi$
$647$			9.06881e19i			1.91082i	−0.295277	+	0.955412i	$0.595412\pi$
$648$		0			0
$649$			1.45988e19i			0.301026i
$650$		0			0
$651$	−1.66026e19	−	1.17473e19i	−0.335050	−	0.237066i
$652$		0			0
$653$	−5.83509e19			−1.15254			−0.576269	−	0.817260i	$-0.695492\pi$
$653$	−5.83509e19			−1.15254			−0.576269	+	0.817260i	$0.695492\pi$
$654$		0			0
$655$	−1.84329e19			−0.356372
$656$		0			0
$657$			6.66155e18i			0.126072i
$658$		0			0
$659$	−8.28719e19			−1.53536			−0.767678	−	0.640836i	$-0.778588\pi$
$659$	−8.28719e19			−1.53536			−0.767678	+	0.640836i	$0.778588\pi$
$660$		0			0
$661$			3.39989e19i			0.616672i	0.951278	+	0.308336i	$0.0997721\pi$
$661$			3.39989e19i			0.616672i	−0.951278	+	0.308336i	$0.900228\pi$
$662$		0			0
$663$		−	4.72261e19i		−	0.838662i
$664$		0			0
$665$	2.10080e19	+	1.48643e19i	0.365287	+	0.258460i
$666$		0			0
$667$	−1.56158e19			−0.265879
$668$		0			0
$669$	2.27410e19			0.379164
$670$		0			0
$671$		−	6.44787e18i		−	0.105283i
$672$		0			0
$673$	−7.98504e19			−1.27694			−0.638471	−	0.769645i	$-0.720433\pi$
$673$	−7.98504e19			−1.27694			−0.638471	+	0.769645i	$0.720433\pi$
$674$		0			0
$675$			1.34385e19i			0.210486i
$676$		0			0
$677$		−	6.78335e18i		−	0.104070i	−0.998645	−	0.0520348i	$-0.983429\pi$
$677$		−	6.78335e18i		−	0.104070i	0.998645	−	0.0520348i	$-0.0165707\pi$
$678$		0			0
$679$	−2.93319e18	+	4.14553e18i	−0.0440810	+	0.0623005i
$680$		0			0
$681$	1.78532e19			0.262836
$682$		0			0
$683$	7.26214e19			1.04742			0.523709	−	0.851898i	$-0.324548\pi$
$683$	7.26214e19			1.04742			0.523709	+	0.851898i	$0.324548\pi$
$684$		0			0
$685$		−	5.87639e19i		−	0.830379i
$686$		0			0
$687$	7.20198e19			0.997136
$688$		0			0
$689$			2.53210e19i			0.343515i
$690$		0			0
$691$		−	1.54929e19i		−	0.205962i	−0.994683	−	0.102981i	$-0.967162\pi$
$691$		−	1.54929e19i		−	0.205962i	0.994683	−	0.102981i	$-0.0328380\pi$
$692$		0			0
$693$	−8.29005e18	−	5.86566e18i	−0.108000	−	0.0764160i
$694$		0			0
$695$	−2.09950e19			−0.268054
$696$		0			0
$697$	−9.44831e18			−0.118229
$698$		0			0
$699$		−	1.00813e20i		−	1.23644i
$700$		0			0
$701$	−4.52504e19			−0.543997			−0.271998	−	0.962298i	$-0.587685\pi$
$701$	−4.52504e19			−0.543997			−0.271998	+	0.962298i	$0.587685\pi$
$702$		0			0
$703$		−	9.72652e19i		−	1.14623i
$704$		0			0
$705$		−	3.04514e19i		−	0.351790i
$706$		0			0
$707$	1.89300e19	−	2.67541e19i	0.214395	−	0.303008i
$708$		0			0
$709$	9.92762e19			1.10236			0.551178	−	0.834388i	$-0.314179\pi$
$709$	9.92762e19			1.10236			0.551178	+	0.834388i	$0.314179\pi$
$710$		0			0
$711$	1.94472e18			0.0211724
$712$		0			0
$713$		−	2.15898e19i		−	0.230474i
$714$		0			0
$715$	4.33828e19			0.454125
$716$		0			0
$717$		−	7.87655e19i		−	0.808541i
$718$		0			0
$719$		−	1.60884e20i		−	1.61961i	−0.586700	−	0.809804i	$-0.699573\pi$
$719$		−	1.60884e20i		−	1.61961i	0.586700	−	0.809804i	$-0.300427\pi$
$720$		0			0
$721$	−4.80567e19	+	6.79194e19i	−0.474468	+	0.670575i
$722$		0			0
$723$	1.33281e20			1.29063
$724$		0			0
$725$	−1.06603e19			−0.101252
$726$		0			0
$727$			4.92912e18i			0.0459228i	0.999736	+	0.0229614i	$0.00730949\pi$
$727$			4.92912e18i			0.0459228i	−0.999736	+	0.0229614i	$0.992691\pi$
$728$		0			0
$729$	−1.19476e20			−1.09192
$730$		0			0
$731$		−	3.89430e18i		−	0.0349147i
$732$		0			0
$733$			5.69563e19i			0.500973i	0.968120	+	0.250486i	$0.0805905\pi$
$733$			5.69563e19i			0.500973i	−0.968120	+	0.250486i	$0.919409\pi$
$734$		0			0
$735$	1.61291e19	+	4.57067e19i	0.139187	+	0.394429i
$736$		0			0
$737$	−1.72693e20			−1.46218
$738$		0			0
$739$	−1.37596e19			−0.114312			−0.0571562	−	0.998365i	$-0.518203\pi$
$739$	−1.37596e19			−0.114312			−0.0571562	+	0.998365i	$0.518203\pi$
$740$		0			0
$741$		−	1.10370e20i		−	0.899752i
$742$		0			0
$743$	−1.53985e19			−0.123185			−0.0615924	−	0.998101i	$-0.519618\pi$
$743$	−1.53985e19			−0.123185			−0.0615924	+	0.998101i	$0.519618\pi$
$744$		0			0
$745$		−	4.98464e17i		−	0.00391327i
$746$		0			0
$747$			1.10212e19i			0.0849153i
$748$		0			0
$749$	−1.08138e20	−	7.65132e19i	−0.817719	−	0.578580i
$750$		0			0
$751$	1.77106e20			1.31448			0.657240	−	0.753681i	$-0.271724\pi$
$751$	1.77106e20			1.31448			0.657240	+	0.753681i	$0.271724\pi$
$752$		0			0
$753$	−9.95885e19			−0.725512
$754$		0			0
$755$			4.86104e19i			0.347617i
$756$		0			0
$757$	−8.94840e17			−0.00628165			−0.00314083	−	0.999995i	$-0.501000\pi$
$757$	−8.94840e17			−0.00628165			−0.00314083	+	0.999995i	$0.501000\pi$
$758$		0			0
$759$			7.52792e19i			0.518779i
$760$		0			0
$761$		−	2.08166e20i		−	1.40837i	−0.710016	−	0.704185i	$-0.751312\pi$
$761$		−	2.08166e20i		−	1.40837i	0.710016	−	0.704185i	$-0.248688\pi$
$762$		0			0
$763$	−9.99624e18	−	7.07288e18i	−0.0663995	−	0.0469812i
$764$		0			0
$765$	−8.01115e18			−0.0522474
$766$		0			0
$767$	4.27925e19			0.274032
$768$		0			0
$769$		−	1.21564e20i		−	0.764397i	−0.924080	−	0.382199i	$-0.875167\pi$
$769$		−	1.21564e20i		−	0.764397i	0.924080	−	0.382199i	$-0.124833\pi$
$770$		0			0
$771$	6.94934e18			0.0429105
$772$		0			0
$773$		−	1.36410e20i		−	0.827163i	−0.910467	−	0.413582i	$-0.864278\pi$
$773$		−	1.36410e20i		−	0.827163i	0.910467	−	0.413582i	$-0.135722\pi$
$774$		0			0
$775$		−	1.47384e19i		−	0.0877685i
$776$		0			0
$777$	1.05809e20	−	1.49542e20i	0.618835	−	0.874611i
$778$		0			0
$779$	−2.20812e19			−0.126841
$780$		0			0
$781$	−2.91686e20			−1.64573
$782$		0			0
$783$		−	9.61386e19i		−	0.532801i
$784$		0			0
$785$	−5.91496e19			−0.322006
$786$		0			0
$787$			2.45739e19i			0.131417i	0.997839	+	0.0657085i	$0.0209308\pi$
$787$			2.45739e19i			0.131417i	−0.997839	+	0.0657085i	$0.979069\pi$
$788$		0			0
$789$		−	5.01129e19i		−	0.263275i
$790$		0			0
$791$	−3.65200e19	−	2.58399e19i	−0.188493	−	0.133369i
$792$		0			0
$793$	−1.89002e19			−0.0958418
$794$		0			0
$795$	−2.99944e19			−0.149441
$796$		0			0
$797$		−	8.76799e19i		−	0.429232i	−0.976699	−	0.214616i	$-0.931150\pi$
$797$		−	8.76799e19i		−	0.429232i	0.976699	−	0.214616i	$-0.0688500\pi$
$798$		0			0
$799$	1.63071e20			0.784422
$800$		0			0
$801$			2.45258e19i			0.115930i
$802$		0			0
$803$			2.28838e20i			1.06297i
$804$		0			0
$805$	−2.97181e19	+	4.20012e19i	−0.135659	+	0.191730i
$806$		0			0
$807$	−1.98201e19			−0.0889182
$808$		0			0
$809$	−1.63947e20			−0.722875			−0.361438	−	0.932396i	$-0.617714\pi$
$809$	−1.63947e20			−0.722875			−0.361438	+	0.932396i	$0.617714\pi$
$810$		0			0
$811$			2.00488e20i			0.868846i	0.900709	+	0.434423i	$0.143048\pi$
$811$			2.00488e20i			0.868846i	−0.900709	+	0.434423i	$0.856952\pi$
$812$		0			0
$813$	−1.17338e20			−0.499810
$814$		0			0
$815$			8.18018e19i			0.342499i
$816$		0			0
$817$		−	9.10119e18i		−	0.0374579i
$818$		0			0
$819$	−1.71937e19	+	2.43001e19i	−0.0695634	+	0.0983153i
$820$		0			0
$821$	−4.98878e20			−1.98423			−0.992115	−	0.125333i	$-0.960000\pi$
$821$	−4.98878e20			−1.98423			−0.992115	+	0.125333i	$0.960000\pi$
$822$		0			0
$823$	−3.06666e20			−1.19913			−0.599566	−	0.800326i	$-0.704660\pi$
$823$	−3.06666e20			−1.19913			−0.599566	+	0.800326i	$0.704660\pi$
$824$		0			0
$825$			5.13899e19i			0.197561i
$826$		0			0
$827$	4.34470e20			1.64218			0.821091	−	0.570797i	$-0.193366\pi$
$827$	4.34470e20			1.64218			0.821091	+	0.570797i	$0.193366\pi$
$828$		0			0
$829$			8.87812e18i			0.0329943i	0.999864	+	0.0164972i	$0.00525145\pi$
$829$			8.87812e18i			0.0329943i	−0.999864	+	0.0164972i	$0.994749\pi$
$830$		0			0
$831$		−	1.36617e19i		−	0.0499228i
$832$		0			0
$833$	−2.44766e20	+	8.63735e19i	−0.879499	+	0.310360i
$834$		0			0
$835$	−2.14364e20			−0.757437
$836$		0			0
$837$	1.32917e20			0.461851
$838$		0			0
$839$			8.24702e19i			0.281814i	0.990023	+	0.140907i	$0.0450019\pi$
$839$			8.24702e19i			0.281814i	−0.990023	+	0.140907i	$0.954998\pi$
$840$		0			0
$841$	−2.21295e20			−0.743702
$842$		0			0
$843$			5.18732e17i			0.00171455i
$844$		0			0
$845$			1.04008e19i			0.0338120i
$846$		0			0
$847$	−2.94833e19	−	2.08611e19i	−0.0942742	−	0.0667041i
$848$		0			0
$849$	4.55868e20			1.43379
$850$		0			0
$851$	1.94461e20			0.601626
$852$		0			0
$853$			9.69146e19i			0.294948i	0.989066	+	0.147474i	$0.0471144\pi$
$853$			9.69146e19i			0.294948i	−0.989066	+	0.147474i	$0.952886\pi$
$854$		0			0
$855$	−1.87225e19			−0.0560533
$856$		0			0
$857$		−	2.80107e20i		−	0.825008i	−0.910956	−	0.412504i	$-0.864654\pi$
$857$		−	2.80107e20i		−	0.825008i	0.910956	−	0.412504i	$-0.135346\pi$
$858$		0			0
$859$			2.50894e20i			0.727007i	0.931593	+	0.363504i	$0.118420\pi$
$859$			2.50894e20i			0.727007i	−0.931593	+	0.363504i	$0.881580\pi$
$860$		0			0
$861$	−3.39490e19	−	2.40208e19i	−0.0967844	−	0.0684802i
$862$		0			0
$863$	5.64844e20			1.58436			0.792178	−	0.610290i	$-0.208947\pi$
$863$	5.64844e20			1.58436			0.792178	+	0.610290i	$0.208947\pi$
$864$		0			0
$865$	1.22407e19			0.0337827
$866$		0			0
$867$		−	4.48272e19i		−	0.121733i
$868$		0			0
$869$	6.68052e19			0.178514
$870$		0			0
$871$			5.06203e20i			1.33106i
$872$		0			0
$873$		−	3.69452e18i		−	0.00956002i
$874$		0			0
$875$	−2.02873e19	+	2.86724e19i	−0.0516616	+	0.0730143i
$876$		0			0
$877$	3.45182e20			0.865070			0.432535	−	0.901617i	$-0.357619\pi$
$877$	3.45182e20			0.865070			0.432535	+	0.901617i	$0.357619\pi$
$878$		0			0
$879$	6.33104e20			1.56154
$880$		0			0
$881$			6.78968e20i			1.64823i	0.566422	+	0.824115i	$0.308327\pi$
$881$			6.78968e20i			1.64823i	−0.566422	+	0.824115i	$0.691673\pi$
$882$		0			0
$883$	1.69440e20			0.404848			0.202424	−	0.979298i	$-0.435118\pi$
$883$	1.69440e20			0.404848			0.202424	+	0.979298i	$0.435118\pi$
$884$		0			0
$885$			5.06907e19i			0.119213i
$886$		0			0
$887$		−	2.87678e20i		−	0.665949i	−0.942936	−	0.332975i	$-0.891948\pi$
$887$		−	2.87678e20i		−	0.665949i	0.942936	−	0.332975i	$-0.108052\pi$
$888$		0			0
$889$	−5.95266e20	−	4.21183e20i	−1.35643	−	0.959751i
$890$		0			0
$891$	−4.04475e20			−0.907293
$892$		0			0
$893$	3.81106e20			0.841561
$894$		0			0
$895$			6.33062e19i			0.137621i
$896$		0			0
$897$	2.20661e20			0.472258
$898$		0			0
$899$			1.05438e20i			0.222168i
$900$		0			0
$901$		−	1.60624e20i		−	0.333225i
$902$		0			0
$903$	9.90063e18	−	1.39928e19i	0.0202231	−	0.0285818i
$904$		0			0
$905$	−1.60249e20			−0.322296
$906$		0			0
$907$	−4.70688e20			−0.932143			−0.466071	−	0.884747i	$-0.654331\pi$
$907$	−4.70688e20			−0.932143			−0.466071	+	0.884747i	$0.654331\pi$
$908$		0			0
$909$			2.38434e19i			0.0464967i
$910$		0			0
$911$	2.07372e20			0.398218			0.199109	−	0.979977i	$-0.436195\pi$
$911$	2.07372e20			0.398218			0.199109	+	0.979977i	$0.436195\pi$
$912$		0			0
$913$			3.78603e20i			0.715959i
$914$		0			0
$915$		−	2.23886e19i		−	0.0416946i
$916$		0			0
$917$	2.50956e20	−	3.54680e20i	0.460269	−	0.650507i
$918$		0			0
$919$	6.38444e20			1.15323			0.576613	−	0.817018i	$-0.304374\pi$
$919$	6.38444e20			1.15323			0.576613	+	0.817018i	$0.304374\pi$
$920$		0			0
$921$	2.32546e20			0.413706
$922$		0			0
$923$			8.55002e20i			1.49815i
$924$		0			0
$925$	1.32751e20			0.229110
$926$		0			0
$927$		−	6.05302e19i		−	0.102900i
$928$		0			0
$929$			6.38097e19i			0.106851i	0.998572	+	0.0534253i	$0.0170139\pi$
$929$			6.38097e19i			0.106851i	−0.998572	+	0.0534253i	$0.982986\pi$
$930$		0			0
$931$	−5.72030e20	+	2.01859e20i	−0.943564	+	0.332967i
$932$		0			0
$933$	3.80929e20			0.618974
$934$		0			0
$935$	−2.75200e20			−0.440522
$936$		0			0
$937$		−	9.67428e20i		−	1.52560i	−0.646633	−	0.762801i	$-0.723824\pi$
$937$		−	9.67428e20i		−	1.52560i	0.646633	−	0.762801i	$-0.276176\pi$
$938$		0			0
$939$	1.56801e20			0.243607
$940$		0			0
$941$		−	2.97704e20i		−	0.455677i	−0.973699	−	0.227839i	$-0.926834\pi$
$941$		−	2.97704e20i		−	0.455677i	0.973699	−	0.227839i	$-0.0731659\pi$
$942$		0			0
$943$		−	4.41467e19i		−	0.0665758i
$944$		0			0
$945$	−2.58579e20	−	1.82959e20i	−0.384212	−	0.271851i
$946$		0			0
$947$	−3.63787e20			−0.532595			−0.266298	−	0.963891i	$-0.585800\pi$
$947$	−3.63787e20			−0.532595			−0.266298	+	0.963891i	$0.585800\pi$
$948$		0			0
$949$	6.70780e20			0.967646
$950$		0			0
$951$			5.67780e20i			0.807080i
$952$		0			0
$953$	1.01951e21			1.42804			0.714021	−	0.700124i	$-0.246872\pi$
$953$	1.01951e21			1.42804			0.714021	+	0.700124i	$0.246872\pi$
$954$		0			0
$955$		−	3.40705e20i		−	0.470279i
$956$		0			0
$957$		−	3.67643e20i		−	0.500084i
$958$		0			0
$959$	1.13072e21	+	8.00045e20i	1.51574	+	1.07247i
$960$		0			0
$961$	6.11169e20			0.807417
$962$		0			0
$963$	9.63729e19			0.125479
$964$		0			0
$965$			3.54074e20i			0.454363i
$966$		0			0
$967$	1.10471e21			1.39721			0.698607	−	0.715505i	$-0.253803\pi$
$967$	1.10471e21			1.39721			0.698607	+	0.715505i	$0.253803\pi$
$968$		0			0
$969$			7.00134e20i			0.872800i
$970$		0			0
$971$			1.36620e21i			1.67873i	0.543569	+	0.839365i	$0.317073\pi$
$971$			1.36620e21i			1.67873i	−0.543569	+	0.839365i	$0.682927\pi$
$972$		0			0
$973$	2.85838e20	−	4.03980e20i	0.346202	−	0.489294i
$974$		0			0
$975$	1.50636e20			0.179844
$976$		0			0
$977$	−5.97956e20			−0.703732			−0.351866	−	0.936050i	$-0.614453\pi$
$977$	−5.97956e20			−0.703732			−0.351866	+	0.936050i	$0.614453\pi$
$978$		0			0
$979$			8.42513e20i			0.977458i
$980$		0			0
$981$	8.90871e18			0.0101890
$982$		0			0
$983$			1.37525e21i			1.55062i	0.631580	+	0.775311i	$0.282407\pi$
$983$			1.37525e21i			1.55062i	−0.631580	+	0.775311i	$0.717593\pi$
$984$		0			0
$985$		−	4.01446e20i		−	0.446245i
$986$		0			0
$987$	5.85937e20	+	4.14582e20i	0.642141	+	0.454350i
$988$		0			0
$989$	1.81959e19			0.0196608
$990$		0			0
$991$	−1.71472e21			−1.82674			−0.913371	−	0.407128i	$-0.866530\pi$
$991$	−1.71472e21			−1.82674			−0.913371	+	0.407128i	$0.866530\pi$
$992$		0			0
$993$		−	1.71767e21i		−	1.80424i
$994$		0			0
$995$	1.32716e19			0.0137455
$996$		0			0
$997$			5.51758e20i			0.563485i	0.959490	+	0.281743i	$0.0909124\pi$
$997$			5.51758e20i			0.563485i	−0.959490	+	0.281743i	$0.909088\pi$
$998$		0			0
$999$			1.19720e21i			1.20561i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.15.d.a.41.10	✓	36
7.6	odd	2	inner	140.15.d.a.41.27	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.15.d.a.41.10	✓	36	1.1	even	1	trivial
140.15.d.a.41.27	yes	36	7.6	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	140.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2045.44i q^{3} +34938.6i q^{5} +(-672278. - 475673. i) q^{7} +599139. q^{9} +O(q^{10})\)	\(q-2045.44i q^{3} +34938.6i q^{5} +(-672278. - 475673. i) q^{7} +599139. q^{9} +2.05817e7 q^{11} -6.03298e7i q^{13} +7.14648e7 q^{15} +3.82704e8i q^{17} +8.94399e8i q^{19} +(-9.72962e8 + 1.37511e9i) q^{21} -1.78816e9 q^{23} -1.22070e9 q^{25} -1.10088e10i q^{27} +8.73289e9 q^{29} +1.20737e10i q^{31} -4.20986e10i q^{33} +(1.66193e10 - 2.34884e10i) q^{35} -1.08749e11 q^{37} -1.23401e11 q^{39} +2.46883e10i q^{41} -1.01758e10 q^{43} +2.09330e10i q^{45} -4.26103e11i q^{47} +(2.25693e11 + 6.39570e11i) q^{49} +7.82798e11 q^{51} -4.19709e11 q^{53} +7.19094e11i q^{55} +1.82944e12 q^{57} +7.09310e11i q^{59} -3.13282e11i q^{61} +(-4.02788e11 - 2.84994e11i) q^{63} +2.10784e12 q^{65} -8.39060e12 q^{67} +3.65758e12i q^{69} -1.41721e13 q^{71} +1.11185e13i q^{73} +2.49688e12i q^{75} +(-1.38366e13 - 9.79016e12i) q^{77} +3.24586e12 q^{79} -1.96522e13 q^{81} +1.83951e13i q^{83} -1.33711e13 q^{85} -1.78626e13i q^{87} +4.09351e13i q^{89} +(-2.86973e13 + 4.05584e13i) q^{91} +2.46961e13 q^{93} -3.12490e13 q^{95} -6.16639e12i q^{97} +1.23313e13 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 1364266 q^{7} - 54790830 q^{9}+O(q^{10}) \) 36 * q + 1364266 * q^7 - 54790830 * q^9	\( 36 q + 1364266 q^{7} - 54790830 q^{9} - 26192606 q^{11} + 44843750 q^{15} + 1512952694 q^{21} - 8670648636 q^{23} - 43945312500 q^{25} - 43956395706 q^{29} + 44839531250 q^{35} - 169523027308 q^{37} + 805671747486 q^{39} + 554691319560 q^{43} + 1095688125176 q^{49} + 1032170625826 q^{51} - 4262050556480 q^{53} - 3162001614828 q^{57} - 15828953775898 q^{63} - 3014492656250 q^{65} - 23495876471600 q^{67} + 22887953193352 q^{71} + 56411959501488 q^{77} + 8995204220854 q^{79} + 132868621377344 q^{81} - 2034215156250 q^{85} - 53912825209186 q^{91} + 101093199187348 q^{93} + 3862990000000 q^{95} - 416078903388420 q^{99}+O(q^{100}) \) 36 * q + 1364266 * q^7 - 54790830 * q^9 - 26192606 * q^11 + 44843750 * q^15 + 1512952694 * q^21 - 8670648636 * q^23 - 43945312500 * q^25 - 43956395706 * q^29 + 44839531250 * q^35 - 169523027308 * q^37 + 805671747486 * q^39 + 554691319560 * q^43 + 1095688125176 * q^49 + 1032170625826 * q^51 - 4262050556480 * q^53 - 3162001614828 * q^57 - 15828953775898 * q^63 - 3014492656250 * q^65 - 23495876471600 * q^67 + 22887953193352 * q^71 + 56411959501488 * q^77 + 8995204220854 * q^79 + 132868621377344 * q^81 - 2034215156250 * q^85 - 53912825209186 * q^91 + 101093199187348 * q^93 + 3862990000000 * q^95 - 416078903388420 * q^99

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