LMFDB - Embedded newform 140.15.d.a.41.1

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			41.1
Character	$\chi$	$=$	140.41
Dual form			140.15.d.a.41.36

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4265.32i q^{3} +34938.6i q^{5} +(584126. - 580535. i) q^{7} -1.34099e7 q^{9} +O(q^{10})$	\(q-4265.32i q^{3} +34938.6i q^{5} +(584126. - 580535. i) q^{7} -1.34099e7 q^{9} +2.12377e7 q^{11} +5.11474e7i q^{13} +1.49024e8 q^{15} -4.00063e8i q^{17} +7.97279e8i q^{19} +(-2.47616e9 - 2.49148e9i) q^{21} -3.69946e8 q^{23} -1.22070e9 q^{25} +3.67968e10i q^{27} -3.76406e9 q^{29} +3.53542e10i q^{31} -9.05853e10i q^{33} +(2.02830e10 + 2.04085e10i) q^{35} -9.12282e10 q^{37} +2.18160e11 q^{39} +5.23075e10i q^{41} +4.54451e11 q^{43} -4.68524e11i q^{45} +4.20385e9i q^{47} +(4.18218e9 - 6.78210e11i) q^{49} -1.70639e12 q^{51} +1.36796e12 q^{53} +7.42013e11i q^{55} +3.40065e12 q^{57} +2.36823e12i q^{59} +3.98635e12i q^{61} +(-7.83309e12 + 7.78494e12i) q^{63} -1.78702e12 q^{65} +7.42846e12 q^{67} +1.57794e12i q^{69} -9.43543e11 q^{71} +1.50515e13i q^{73} +5.20668e12i q^{75} +(1.24055e13 - 1.23292e13i) q^{77} +3.07309e13 q^{79} +9.28105e13 q^{81} +3.87814e13i q^{83} +1.39776e13 q^{85} +1.60549e13i q^{87} -2.82570e13i q^{89} +(2.96928e13 + 2.98765e13i) q^{91} +1.50797e14 q^{93} -2.78558e13 q^{95} +4.24375e13i q^{97} -2.84796e14 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	4265.32i		−	1.95030i	−0.221536	−	0.975152i	$-0.571107\pi$
$3$		−	4265.32i		−	1.95030i	0.221536	−	0.975152i	$-0.428893\pi$
$4$		0			0
$5$			34938.6i			0.447214i
$5$			34938.6i			0.447214i
$6$		0			0
$7$	584126.	−	580535.i	0.709284	−	0.704923i
$7$	584126.	−	580535.i	0.709284	−	0.704923i
$8$		0			0
$9$	−1.34099e7			−2.80369
$10$		0			0
$11$	2.12377e7			1.08983			0.544914	−	0.838492i	$-0.316562\pi$
$11$	2.12377e7			1.08983			0.544914	+	0.838492i	$0.316562\pi$
$12$		0			0
$13$			5.11474e7i			0.815117i	0.913179	+	0.407559i	$0.133620\pi$
$13$			5.11474e7i			0.815117i	−0.913179	+	0.407559i	$0.866380\pi$
$14$		0			0
$15$	1.49024e8			0.872203
$16$		0			0
$17$		−	4.00063e8i		−	0.974957i	−0.873135	−	0.487479i	$-0.837917\pi$
$17$		−	4.00063e8i		−	0.974957i	0.873135	−	0.487479i	$-0.162083\pi$
$18$		0			0
$19$			7.97279e8i			0.891939i	0.895048	+	0.445969i	$0.147141\pi$
$19$			7.97279e8i			0.891939i	−0.895048	+	0.445969i	$0.852859\pi$
$20$		0			0
$21$	−2.47616e9	−	2.49148e9i	−1.37481	−	1.38332i
$22$		0			0
$23$	−3.69946e8			−0.108653			−0.0543267	−	0.998523i	$-0.517301\pi$
$23$	−3.69946e8			−0.108653			−0.0543267	+	0.998523i	$0.517301\pi$
$24$		0			0
$25$	−1.22070e9			−0.200000
$26$		0			0
$27$			3.67968e10i			3.51774i
$28$		0			0
$29$	−3.76406e9			−0.218208			−0.109104	−	0.994030i	$-0.534798\pi$
$29$	−3.76406e9			−0.218208			−0.109104	+	0.994030i	$0.534798\pi$
$30$		0			0
$31$			3.53542e10i			1.28502i	0.766278	+	0.642510i	$0.222107\pi$
$31$			3.53542e10i			1.28502i	−0.766278	+	0.642510i	$0.777893\pi$
$32$		0			0
$33$		−	9.05853e10i		−	2.12550i
$34$		0			0
$35$	2.02830e10	+	2.04085e10i	0.315251	+	0.317201i
$36$		0			0
$37$	−9.12282e10			−0.960986			−0.480493	−	0.876999i	$-0.659542\pi$
$37$	−9.12282e10			−0.960986			−0.480493	+	0.876999i	$0.659542\pi$
$38$		0			0
$39$	2.18160e11			1.58973
$40$		0			0
$41$			5.23075e10i			0.268582i	0.990942	+	0.134291i	$0.0428757\pi$
$41$			5.23075e10i			0.268582i	−0.990942	+	0.134291i	$0.957124\pi$
$42$		0			0
$43$	4.54451e11			1.67189			0.835945	−	0.548814i	$-0.184920\pi$
$43$	4.54451e11			1.67189			0.835945	+	0.548814i	$0.184920\pi$
$44$		0			0
$45$		−	4.68524e11i		−	1.25385i
$46$		0			0
$47$			4.20385e9i			0.00829779i	0.999991	+	0.00414890i	$0.00132064\pi$
$47$			4.20385e9i			0.00829779i	−0.999991	+	0.00414890i	$0.998679\pi$
$48$		0			0
$49$	4.18218e9	−	6.78210e11i	0.00616637	−	0.999981i
$50$		0			0
$51$	−1.70639e12			−1.90146
$52$		0			0
$53$	1.36796e12			1.16451			0.582255	−	0.813006i	$-0.302170\pi$
$53$	1.36796e12			1.16451			0.582255	+	0.813006i	$0.302170\pi$
$54$		0			0
$55$			7.42013e11i			0.487386i
$56$		0			0
$57$	3.40065e12			1.73955
$58$		0			0
$59$			2.36823e12i			0.951611i	0.879551	+	0.475805i	$0.157843\pi$
$59$			2.36823e12i			0.951611i	−0.879551	+	0.475805i	$0.842157\pi$
$60$		0			0
$61$			3.98635e12i			1.26843i	0.773157	+	0.634215i	$0.218676\pi$
$61$			3.98635e12i			1.26843i	−0.773157	+	0.634215i	$0.781324\pi$
$62$		0			0
$63$	−7.83309e12	+	7.78494e12i	−1.98861	+	1.97638i
$64$		0			0
$65$	−1.78702e12			−0.364532
$66$		0			0
$67$	7.42846e12			1.22567			0.612837	−	0.790209i	$-0.290028\pi$
$67$	7.42846e12			1.22567			0.612837	+	0.790209i	$0.290028\pi$
$68$		0			0
$69$			1.57794e12i			0.211907i
$70$		0			0
$71$	−9.43543e11			−0.103742			−0.0518708	−	0.998654i	$-0.516518\pi$
$71$	−9.43543e11			−0.103742			−0.0518708	+	0.998654i	$0.516518\pi$
$72$		0			0
$73$			1.50515e13i			1.36244i	0.732077	+	0.681222i	$0.238551\pi$
$73$			1.50515e13i			1.36244i	−0.732077	+	0.681222i	$0.761449\pi$
$74$		0			0
$75$			5.20668e12i			0.390061i
$76$		0			0
$77$	1.24055e13	−	1.23292e13i	0.772997	−	0.768245i
$78$		0			0
$79$	3.07309e13			1.60024			0.800120	−	0.599840i	$-0.204769\pi$
$79$	3.07309e13			1.60024			0.800120	+	0.599840i	$0.204769\pi$
$80$		0			0
$81$	9.28105e13			4.05697
$82$		0			0
$83$			3.87814e13i			1.42915i	0.699561	+	0.714573i	$0.253379\pi$
$83$			3.87814e13i			1.42915i	−0.699561	+	0.714573i	$0.746621\pi$
$84$		0			0
$85$	1.39776e13			0.436014
$86$		0			0
$87$			1.60549e13i			0.425572i
$88$		0			0
$89$		−	2.82570e13i		−	0.638847i	−0.947612	−	0.319423i	$-0.896511\pi$
$89$		−	2.82570e13i		−	0.638847i	0.947612	−	0.319423i	$-0.103489\pi$
$90$		0			0
$91$	2.96928e13	+	2.98765e13i	0.574595	+	0.578149i
$92$		0			0
$93$	1.50797e14			2.50618
$94$		0			0
$95$	−2.78558e13			−0.398887
$96$		0			0
$97$			4.24375e13i			0.525228i	0.964901	+	0.262614i	$0.0845846\pi$
$97$			4.24375e13i			0.525228i	−0.964901	+	0.262614i	$0.915415\pi$
$98$		0			0
$99$	−2.84796e14			−3.05554
$100$		0			0
$101$			7.88338e13i			0.735297i	0.929965	+	0.367648i	$0.119837\pi$
$101$			7.88338e13i			0.735297i	−0.929965	+	0.367648i	$0.880163\pi$
$102$		0			0
$103$			8.52770e13i			0.693380i	0.937980	+	0.346690i	$0.112694\pi$
$103$			8.52770e13i			0.693380i	−0.937980	+	0.346690i	$0.887306\pi$
$104$		0			0
$105$	8.70487e13	−	8.65136e13i	0.618639	−	0.614836i
$106$		0			0
$107$	−2.58600e14			−1.61043			−0.805216	−	0.592982i	$-0.797950\pi$
$107$	−2.58600e14			−1.61043			−0.805216	+	0.592982i	$0.797950\pi$
$108$		0			0
$109$	−2.89325e14			−1.58271			−0.791353	−	0.611360i	$-0.790623\pi$
$109$	−2.89325e14			−1.58271			−0.791353	+	0.611360i	$0.790623\pi$
$110$		0			0
$111$			3.89117e14i			1.87421i
$112$		0			0
$113$	1.66558e14			0.707972			0.353986	−	0.935251i	$-0.384826\pi$
$113$	1.66558e14			0.707972			0.353986	+	0.935251i	$0.384826\pi$
$114$		0			0
$115$		−	1.29254e13i		−	0.0485913i
$116$		0			0
$117$		−	6.85884e14i		−	2.28533i
$118$		0			0
$119$	−2.32250e14	−	2.33687e14i	−0.687270	−	0.691521i
$120$		0			0
$121$	7.12885e13			0.187725
$122$		0			0
$123$	2.23108e14			0.523817
$124$		0			0
$125$		−	4.26496e13i		−	0.0894427i
$126$		0			0
$127$	6.46529e14			1.21328			0.606641	−	0.794976i	$-0.292516\pi$
$127$	6.46529e14			1.21328			0.606641	+	0.794976i	$0.292516\pi$
$128$		0			0
$129$		−	1.93838e15i		−	3.26069i
$130$		0			0
$131$		−	1.28806e15i		−	1.94552i	−0.231812	−	0.972761i	$-0.574465\pi$
$131$		−	1.28806e15i		−	1.94552i	0.231812	−	0.972761i	$-0.425535\pi$
$132$		0			0
$133$	4.62848e14	+	4.65711e14i	0.628748	+	0.632637i
$134$		0			0
$135$	−1.28563e15			−1.57318
$136$		0			0
$137$	6.43107e14			0.709969			0.354984	−	0.934872i	$-0.384486\pi$
$137$	6.43107e14			0.709969			0.354984	+	0.934872i	$0.384486\pi$
$138$		0			0
$139$		−	7.98598e14i		−	0.796571i	−0.917262	−	0.398285i	$-0.869605\pi$
$139$		−	7.98598e14i		−	0.796571i	0.917262	−	0.398285i	$-0.130395\pi$
$140$		0			0
$141$	1.79308e13			0.0161832
$142$		0			0
$143$			1.08625e15i			0.888338i
$144$		0			0
$145$		−	1.31511e14i		−	0.0975855i
$146$		0			0
$147$	−2.89278e15	−	1.78383e13i	−1.95027	−	0.0120263i
$148$		0			0
$149$	−2.51848e15			−1.54467			−0.772334	−	0.635217i	$-0.780911\pi$
$149$	−2.51848e15			−1.54467			−0.772334	+	0.635217i	$0.780911\pi$
$150$		0			0
$151$	1.60824e15			0.898486			0.449243	−	0.893410i	$-0.351694\pi$
$151$	1.60824e15			0.898486			0.449243	+	0.893410i	$0.351694\pi$
$152$		0			0
$153$			5.36482e15i			2.73347i
$154$		0			0
$155$	−1.23523e15			−0.574678
$156$		0			0
$157$		−	6.70726e14i		−	0.285264i	−0.989776	−	0.142632i	$-0.954443\pi$
$157$		−	6.70726e14i		−	0.285264i	0.989776	−	0.142632i	$-0.0455566\pi$
$158$		0			0
$159$		−	5.83479e15i		−	2.27115i
$160$		0			0
$161$	−2.16095e14	+	2.14767e14i	−0.0770661	+	0.0765924i
$162$		0			0
$163$	2.06199e15			0.674488			0.337244	−	0.941417i	$-0.390505\pi$
$163$	2.06199e15			0.674488			0.337244	+	0.941417i	$0.390505\pi$
$164$		0			0
$165$	3.16492e15			0.950551
$166$		0			0
$167$			3.84347e15i			1.06098i	0.847690	+	0.530491i	$0.177992\pi$
$167$			3.84347e15i			1.06098i	−0.847690	+	0.530491i	$0.822008\pi$
$168$		0			0
$169$	1.32132e15			0.335584
$170$		0			0
$171$		−	1.06915e16i		−	2.50072i
$172$		0			0
$173$		−	4.18659e15i		−	0.902688i	−0.892350	−	0.451344i	$-0.850945\pi$
$173$		−	4.18659e15i		−	0.902688i	0.892350	−	0.451344i	$-0.149055\pi$
$174$		0			0
$175$	−7.13044e14	+	7.08660e14i	−0.141857	+	0.140985i
$176$		0			0
$177$	1.01012e16			1.85593
$178$		0			0
$179$	−1.33665e15			−0.227012			−0.113506	−	0.993537i	$-0.536208\pi$
$179$	−1.33665e15			−0.227012			−0.113506	+	0.993537i	$0.536208\pi$
$180$		0			0
$181$			9.27076e15i			1.45668i	0.685214	+	0.728342i	$0.259709\pi$
$181$			9.27076e15i			1.45668i	−0.685214	+	0.728342i	$0.740291\pi$
$182$		0			0
$183$	1.70030e16			2.47382
$184$		0			0
$185$		−	3.18738e15i		−	0.429766i
$186$		0			0
$187$		−	8.49639e15i		−	1.06254i
$188$		0			0
$189$	2.13618e16	+	2.14939e16i	2.47974	+	2.49507i
$190$		0			0
$191$	1.32054e16			1.42403			0.712014	−	0.702165i	$-0.247783\pi$
$191$	1.32054e16			1.42403			0.712014	+	0.702165i	$0.247783\pi$
$192$		0			0
$193$	−1.10359e16			−1.10639			−0.553194	−	0.833053i	$-0.686591\pi$
$193$	−1.10359e16			−1.10639			−0.553194	+	0.833053i	$0.686591\pi$
$194$		0			0
$195$			7.62219e15i			0.710947i
$196$		0			0
$197$	−3.01430e15			−0.261772			−0.130886	−	0.991397i	$-0.541782\pi$
$197$	−3.01430e15			−0.261772			−0.130886	+	0.991397i	$0.541782\pi$
$198$		0			0
$199$		−	1.43908e16i		−	1.16443i	−0.813034	−	0.582216i	$-0.802186\pi$
$199$		−	1.43908e16i		−	1.16443i	0.813034	−	0.582216i	$-0.197814\pi$
$200$		0			0
$201$		−	3.16847e16i		−	2.39044i
$202$		0			0
$203$	−2.19868e15	+	2.18517e15i	−0.154771	+	0.153820i
$204$		0			0
$205$	−1.82755e15			−0.120114
$206$		0			0
$207$	4.96096e15			0.304630
$208$		0			0
$209$			1.69323e16i			0.972060i
$210$		0			0
$211$	−1.38505e16			−0.743855			−0.371927	−	0.928262i	$-0.621303\pi$
$211$	−1.38505e16			−0.743855			−0.371927	+	0.928262i	$0.621303\pi$
$212$		0			0
$213$			4.02451e15i			0.202328i
$214$		0			0
$215$			1.58779e16i			0.747692i
$216$		0			0
$217$	2.05244e16	+	2.06513e16i	0.905840	+	0.911443i
$218$		0			0
$219$	6.41992e16			2.65718
$220$		0			0
$221$	2.04622e16			0.794704
$222$		0			0
$223$		−	2.01988e16i		−	0.736532i	−0.929720	−	0.368266i	$-0.879952\pi$
$223$		−	2.01988e16i		−	0.736532i	0.929720	−	0.368266i	$-0.120048\pi$
$224$		0			0
$225$	1.63696e16			0.560737
$226$		0			0
$227$			9.90283e15i			0.318844i	0.987211	+	0.159422i	$0.0509630\pi$
$227$			9.90283e15i			0.318844i	−0.987211	+	0.159422i	$0.949037\pi$
$228$		0			0
$229$		−	2.18102e16i		−	0.660407i	−0.943910	−	0.330204i	$-0.892883\pi$
$229$		−	2.18102e16i		−	0.660407i	0.943910	−	0.330204i	$-0.107117\pi$
$230$		0			0
$231$	−5.25879e16	−	5.29132e16i	−1.49831	−	1.50758i
$232$		0			0
$233$	−3.57216e16			−0.958163			−0.479081	−	0.877770i	$-0.659030\pi$
$233$	−3.57216e16			−0.958163			−0.479081	+	0.877770i	$0.659030\pi$
$234$		0			0
$235$	−1.46877e14			−0.00371089
$236$		0			0
$237$		−	1.31077e17i		−	3.12095i
$238$		0			0
$239$	−3.36392e16			−0.755198			−0.377599	−	0.925969i	$-0.623250\pi$
$239$	−3.36392e16			−0.755198			−0.377599	+	0.925969i	$0.623250\pi$
$240$		0			0
$241$		−	3.78837e16i		−	0.802292i	−0.916014	−	0.401146i	$-0.868612\pi$
$241$		−	3.78837e16i		−	0.802292i	0.916014	−	0.401146i	$-0.131388\pi$
$242$		0			0
$243$		−	2.19868e17i		−	4.39459i
$244$		0			0
$245$	2.36957e16	+	1.46119e14i	0.447205	+	0.00275769i
$246$		0			0
$247$	−4.07787e16			−0.727034
$248$		0			0
$249$	1.65415e17			2.78727
$250$		0			0
$251$			8.31971e16i			1.32554i	0.748824	+	0.662769i	$0.230619\pi$
$251$			8.31971e16i			1.32554i	−0.748824	+	0.662769i	$0.769381\pi$
$252$		0			0
$253$	−7.85679e15			−0.118414
$254$		0			0
$255$		−	5.96189e16i		−	0.850360i
$256$		0			0
$257$			5.39802e16i			0.728958i	0.931212	+	0.364479i	$0.118753\pi$
$257$			5.39802e16i			0.728958i	−0.931212	+	0.364479i	$0.881247\pi$
$258$		0			0
$259$	−5.32887e16	+	5.29611e16i	−0.681611	+	0.677421i
$260$		0			0
$261$	5.04758e16			0.611787
$262$		0			0
$263$	2.38023e16			0.273482			0.136741	−	0.990607i	$-0.456337\pi$
$263$	2.38023e16			0.273482			0.136741	+	0.990607i	$0.456337\pi$
$264$		0			0
$265$			4.77947e16i			0.520785i
$266$		0			0
$267$	−1.20525e17			−1.24595
$268$		0			0
$269$		−	1.23686e17i		−	1.21354i	−0.794879	−	0.606768i	$-0.792466\pi$
$269$		−	1.23686e17i		−	1.21354i	0.794879	−	0.606768i	$-0.207534\pi$
$270$		0			0
$271$			9.13802e16i			0.851269i	0.904895	+	0.425635i	$0.139949\pi$
$271$			9.13802e16i			0.851269i	−0.904895	+	0.425635i	$0.860051\pi$
$272$		0			0
$273$	1.27433e17	−	1.26649e17i	1.12757	−	1.12064i
$274$		0			0
$275$	−2.59249e16			−0.217966
$276$		0			0
$277$	−1.93609e17			−1.54727			−0.773636	−	0.633630i	$-0.781564\pi$
$277$	−1.93609e17			−1.54727			−0.773636	+	0.633630i	$0.781564\pi$
$278$		0			0
$279$		−	4.74098e17i		−	3.60279i
$280$		0			0
$281$	2.07213e17			1.49786			0.748931	−	0.662648i	$-0.230568\pi$
$281$	2.07213e17			1.49786			0.748931	+	0.662648i	$0.230568\pi$
$282$		0			0
$283$			1.85155e17i			1.27359i	0.771033	+	0.636795i	$0.219740\pi$
$283$			1.85155e17i			1.27359i	−0.771033	+	0.636795i	$0.780260\pi$
$284$		0			0
$285$			1.18814e17i			0.777951i
$286$		0			0
$287$	3.03663e16	+	3.05542e16i	0.189330	+	0.190501i
$288$		0			0
$289$	8.32777e15			0.0494588
$290$		0			0
$291$	1.81009e17			1.02435
$292$		0			0
$293$		−	1.91833e17i		−	1.03479i	−0.855748	−	0.517393i	$-0.826902\pi$
$293$		−	1.91833e17i		−	1.03479i	0.855748	−	0.517393i	$-0.173098\pi$
$294$		0			0
$295$	−8.27425e16			−0.425573
$296$		0			0
$297$			7.81478e17i			3.83373i
$298$		0			0
$299$		−	1.89218e16i		−	0.0885653i
$300$		0			0
$301$	2.65456e17	−	2.63824e17i	1.18584	−	1.17855i
$302$		0			0
$303$	3.36251e17			1.43405
$304$		0			0
$305$	−1.39277e17			−0.567259
$306$		0			0
$307$		−	4.93031e17i		−	1.91825i	−0.282978	−	0.959126i	$-0.591322\pi$
$307$		−	4.93031e17i		−	1.91825i	0.282978	−	0.959126i	$-0.408678\pi$
$308$		0			0
$309$	3.63733e17			1.35230
$310$		0			0
$311$		−	4.44317e17i		−	1.57896i	−0.613779	−	0.789478i	$-0.710351\pi$
$311$		−	4.44317e17i		−	1.57896i	0.613779	−	0.789478i	$-0.289649\pi$
$312$		0			0
$313$			3.90722e17i			1.32757i	0.747923	+	0.663786i	$0.231051\pi$
$313$			3.90722e17i			1.32757i	−0.747923	+	0.663786i	$0.768949\pi$
$314$		0			0
$315$	−2.71995e17	−	2.73677e17i	−0.883866	−	0.889333i
$316$		0			0
$317$	3.96140e17			1.23150			0.615749	−	0.787942i	$-0.288853\pi$
$317$	3.96140e17			1.23150			0.615749	+	0.787942i	$0.288853\pi$
$318$		0			0
$319$	−7.99398e16			−0.237809
$320$		0			0
$321$			1.10301e18i			3.14083i
$322$		0			0
$323$	3.18961e17			0.869602
$324$		0			0
$325$		−	6.24358e16i		−	0.163023i
$326$		0			0
$327$			1.23406e18i			3.08676i
$328$		0			0
$329$	2.44048e15	+	2.45558e15i	0.00584931	+	0.00588549i
$330$		0			0
$331$	−2.77885e17			−0.638364			−0.319182	−	0.947693i	$-0.603408\pi$
$331$	−2.77885e17			−0.638364			−0.319182	+	0.947693i	$0.603408\pi$
$332$		0			0
$333$	1.22336e18			2.69430
$334$		0			0
$335$			2.59540e17i			0.548138i
$336$		0			0
$337$	−8.01555e17			−1.62377			−0.811885	−	0.583818i	$-0.801558\pi$
$337$	−8.01555e17			−1.62377			−0.811885	+	0.583818i	$0.801558\pi$
$338$		0			0
$339$		−	7.10422e17i		−	1.38076i
$340$		0			0
$341$			7.50841e17i			1.40045i
$342$		0			0
$343$	−3.91282e17	−	3.98588e17i	−0.700536	−	0.713617i
$344$		0			0
$345$	−5.51309e16			−0.0947679
$346$		0			0
$347$	3.20891e17			0.529725			0.264863	−	0.964286i	$-0.414673\pi$
$347$	3.20891e17			0.529725			0.264863	+	0.964286i	$0.414673\pi$
$348$		0			0
$349$		−	6.87137e17i		−	1.08960i	−0.838568	−	0.544798i	$-0.816606\pi$
$349$		−	6.87137e17i		−	1.08960i	0.838568	−	0.544798i	$-0.183394\pi$
$350$		0			0
$351$	−1.88206e18			−2.86737
$352$		0			0
$353$			9.16661e16i			0.134210i	0.997746	+	0.0671051i	$0.0213763\pi$
$353$			9.16661e16i			0.134210i	−0.997746	+	0.0671051i	$0.978624\pi$
$354$		0			0
$355$		−	3.29660e16i		−	0.0463947i
$356$		0			0
$357$	−9.96748e17	+	9.90620e17i	−1.34868	+	1.34039i
$358$		0			0
$359$	4.61731e17			0.600797			0.300398	−	0.953814i	$-0.402880\pi$
$359$	4.61731e17			0.600797			0.300398	+	0.953814i	$0.402880\pi$
$360$		0			0
$361$	1.63353e17			0.204446
$362$		0			0
$363$		−	3.04068e17i		−	0.366121i
$364$		0			0
$365$	−5.25876e17			−0.609303
$366$		0			0
$367$			1.20582e18i			1.34468i	0.740241	+	0.672341i	$0.234711\pi$
$367$			1.20582e18i			1.34468i	−0.740241	+	0.672341i	$0.765289\pi$
$368$		0			0
$369$		−	7.01441e17i		−	0.753021i
$370$		0			0
$371$	7.99062e17	−	7.94150e17i	0.825968	−	0.820890i
$372$		0			0
$373$	5.36763e17			0.534344			0.267172	−	0.963649i	$-0.413911\pi$
$373$	5.36763e17			0.534344			0.267172	+	0.963649i	$0.413911\pi$
$374$		0			0
$375$	−1.81914e17			−0.174441
$376$		0			0
$377$		−	1.92522e17i		−	0.177865i
$378$		0			0
$379$	1.82256e16			0.0162258			0.00811291	−	0.999967i	$-0.497418\pi$
$379$	1.82256e16			0.0162258			0.00811291	+	0.999967i	$0.497418\pi$
$380$		0			0
$381$		−	2.75765e18i		−	2.36627i
$382$		0			0
$383$		−	3.89866e17i		−	0.322495i	−0.986914	−	0.161248i	$-0.948448\pi$
$383$		−	3.89866e17i		−	0.322495i	0.986914	−	0.161248i	$-0.0515518\pi$
$384$		0			0
$385$	4.30764e17	+	4.33429e17i	0.343570	+	0.345695i
$386$		0			0
$387$	−6.09416e18			−4.68745
$388$		0			0
$389$	1.07241e18			0.795635			0.397818	−	0.917464i	$-0.369768\pi$
$389$	1.07241e18			0.795635			0.397818	+	0.917464i	$0.369768\pi$
$390$		0			0
$391$			1.48002e17i			0.105932i
$392$		0			0
$393$	−5.49397e18			−3.79436
$394$		0			0
$395$			1.07369e18i			0.715649i
$396$		0			0
$397$			5.89618e16i			0.0379348i	0.999820	+	0.0189674i	$0.00603787\pi$
$397$			5.89618e16i			0.0379348i	−0.999820	+	0.0189674i	$0.993962\pi$
$398$		0			0
$399$	1.98640e18	−	1.97419e18i	1.23384	−	1.22625i
$400$		0			0
$401$	1.35392e18			0.812047			0.406023	−	0.913863i	$-0.366915\pi$
$401$	1.35392e18			0.812047			0.406023	+	0.913863i	$0.366915\pi$
$402$		0			0
$403$	−1.80828e18			−1.04744
$404$		0			0
$405$			3.24267e18i			1.81433i
$406$		0			0
$407$	−1.93747e18			−1.04731
$408$		0			0
$409$			2.45733e18i			1.28351i	0.766908	+	0.641757i	$0.221794\pi$
$409$			2.45733e18i			1.28351i	−0.766908	+	0.641757i	$0.778206\pi$
$410$		0			0
$411$		−	2.74305e18i		−	1.38466i
$412$		0			0
$413$	1.37484e18	+	1.38334e18i	0.670813	+	0.674962i
$414$		0			0
$415$	−1.35497e18			−0.639134
$416$		0			0
$417$	−3.40627e18			−1.55356
$418$		0			0
$419$			2.60686e18i			1.14979i	0.818227	+	0.574895i	$0.194957\pi$
$419$			2.60686e18i			1.14979i	−0.818227	+	0.574895i	$0.805043\pi$
$420$		0			0
$421$	4.41517e17			0.188353			0.0941764	−	0.995556i	$-0.469978\pi$
$421$	4.41517e17			0.188353			0.0941764	+	0.995556i	$0.469978\pi$
$422$		0			0
$423$		−	5.63734e16i		−	0.0232644i
$424$		0			0
$425$			4.88358e17i			0.194991i
$426$		0			0
$427$	2.31421e18	+	2.32853e18i	0.894145	+	0.899676i
$428$		0			0
$429$	4.63320e18			1.73253
$430$		0			0
$431$	−3.11290e18			−1.12674			−0.563371	−	0.826204i	$-0.690496\pi$
$431$	−3.11290e18			−1.12674			−0.563371	+	0.826204i	$0.690496\pi$
$432$		0			0
$433$		−	6.28051e17i		−	0.220079i	−0.993927	−	0.110040i	$-0.964902\pi$
$433$		−	6.28051e17i		−	0.220079i	0.993927	−	0.110040i	$-0.0350978\pi$
$434$		0			0
$435$	−5.60935e17			−0.190321
$436$		0			0
$437$		−	2.94950e17i		−	0.0969122i
$438$		0			0
$439$		−	2.47956e18i		−	0.789084i	−0.918878	−	0.394542i	$-0.870903\pi$
$439$		−	2.47956e18i		−	0.789084i	0.918878	−	0.394542i	$-0.129097\pi$
$440$		0			0
$441$	−5.60828e16	+	9.09476e18i	−0.0172886	+	2.80363i
$442$		0			0
$443$	1.97005e18			0.588371			0.294185	−	0.955748i	$-0.404952\pi$
$443$	1.97005e18			0.588371			0.294185	+	0.955748i	$0.404952\pi$
$444$		0			0
$445$	9.87260e17			0.285701
$446$		0			0
$447$			1.07421e19i			3.01257i
$448$		0			0
$449$	−4.76619e18			−1.29553			−0.647764	−	0.761841i	$-0.724296\pi$
$449$	−4.76619e18			−1.29553			−0.647764	+	0.761841i	$0.724296\pi$
$450$		0			0
$451$			1.11089e18i			0.292708i
$452$		0			0
$453$		−	6.85963e18i		−	1.75232i
$454$		0			0
$455$	−1.04384e18	+	1.03742e18i	−0.258556	+	0.256967i
$456$		0			0
$457$	−7.70469e17			−0.185072			−0.0925362	−	0.995709i	$-0.529497\pi$
$457$	−7.70469e17			−0.185072			−0.0925362	+	0.995709i	$0.529497\pi$
$458$		0			0
$459$	1.47210e19			3.42964
$460$		0			0
$461$		−	5.53578e17i		−	0.125104i	−0.998042	−	0.0625522i	$-0.980076\pi$
$461$		−	5.53578e17i		−	0.125104i	0.998042	−	0.0625522i	$-0.0199240\pi$
$462$		0			0
$463$	7.92861e18			1.73832			0.869161	−	0.494530i	$-0.164660\pi$
$463$	7.92861e18			1.73832			0.869161	+	0.494530i	$0.164660\pi$
$464$		0			0
$465$			5.26863e18i			1.12080i
$466$		0			0
$467$			6.97421e18i			1.43972i	0.694121	+	0.719858i	$0.255793\pi$
$467$			6.97421e18i			1.43972i	−0.694121	+	0.719858i	$0.744207\pi$
$468$		0			0
$469$	4.33915e18	−	4.31248e18i	0.869351	−	0.864007i
$470$		0			0
$471$	−2.86086e18			−0.556352
$472$		0			0
$473$	9.65147e18			1.82207
$474$		0			0
$475$		−	9.73241e17i		−	0.178388i
$476$		0			0
$477$	−1.83443e19			−3.26492
$478$		0			0
$479$		−	9.78617e17i		−	0.169147i	−0.996417	−	0.0845735i	$-0.973047\pi$
$479$		−	9.78617e17i		−	0.169147i	0.996417	−	0.0845735i	$-0.0269528\pi$
$480$		0			0
$481$		−	4.66608e18i		−	0.783316i
$482$		0			0
$483$	9.16047e17	+	9.21713e17i	0.149378	+	0.150302i
$484$		0			0
$485$	−1.48271e18			−0.234889
$486$		0			0
$487$	5.77711e18			0.889217			0.444608	−	0.895725i	$-0.353343\pi$
$487$	5.77711e18			0.889217			0.444608	+	0.895725i	$0.353343\pi$
$488$		0			0
$489$		−	8.79505e18i		−	1.31546i
$490$		0			0
$491$	−4.73217e18			−0.687844			−0.343922	−	0.938998i	$-0.611756\pi$
$491$	−4.73217e18			−0.687844			−0.343922	+	0.938998i	$0.611756\pi$
$492$		0			0
$493$			1.50586e18i			0.212743i
$494$		0			0
$495$		−	9.95036e18i		−	1.36648i
$496$		0			0
$497$	−5.51147e17	+	5.47759e17i	−0.0735822	+	0.0731299i
$498$		0			0
$499$	−9.90238e18			−1.28539			−0.642696	−	0.766121i	$-0.722184\pi$
$499$	−9.90238e18			−1.28539			−0.642696	+	0.766121i	$0.722184\pi$
$500$		0			0
$501$	1.63936e19			2.06924
$502$		0			0
$503$			2.78757e18i			0.342176i	0.985256	+	0.171088i	$0.0547282\pi$
$503$			2.78757e18i			0.342176i	−0.985256	+	0.171088i	$0.945272\pi$
$504$		0			0
$505$	−2.75434e18			−0.328835
$506$		0			0
$507$		−	5.63585e18i		−	0.654491i
$508$		0			0
$509$		−	7.58239e17i		−	0.0856607i	−0.999082	−	0.0428303i	$-0.986363\pi$
$509$		−	7.58239e17i		−	0.0856607i	0.999082	−	0.0428303i	$-0.0136375\pi$
$510$		0			0
$511$	8.73789e18	+	8.79194e18i	0.960418	+	0.966359i
$512$		0			0
$513$	−2.93373e19			−3.13761
$514$		0			0
$515$	−2.97945e18			−0.310089
$516$		0			0
$517$			8.92800e16i			0.00904317i
$518$		0			0
$519$	−1.78571e19			−1.76052
$520$		0			0
$521$			1.38399e19i			1.32822i	0.747635	+	0.664110i	$0.231190\pi$
$521$			1.38399e19i			1.32822i	−0.747635	+	0.664110i	$0.768810\pi$
$522$		0			0
$523$			9.03332e18i			0.843987i	0.906599	+	0.421993i	$0.138669\pi$
$523$			9.03332e18i			0.843987i	−0.906599	+	0.421993i	$0.861331\pi$
$524$		0			0
$525$	3.02266e18	+	3.04136e18i	0.274963	+	0.276664i
$526$		0			0
$527$	1.41439e19			1.25284
$528$		0			0
$529$	−1.14560e19			−0.988194
$530$		0			0
$531$		−	3.17578e19i		−	2.66802i
$532$		0			0
$533$	−2.67539e18			−0.218926
$534$		0			0
$535$		−	9.03511e18i		−	0.720207i
$536$		0			0
$537$			5.70126e18i			0.442742i
$538$		0			0
$539$	8.88196e16	−	1.44036e19i	0.00672028	−	1.08981i
$540$		0			0
$541$	1.23140e19			0.907862			0.453931	−	0.891037i	$-0.350021\pi$
$541$	1.23140e19			0.907862			0.453931	+	0.891037i	$0.350021\pi$
$542$		0			0
$543$	3.95427e19			2.84098
$544$		0			0
$545$		−	1.01086e19i		−	0.707808i
$546$		0			0
$547$	6.64701e18			0.453644			0.226822	−	0.973936i	$-0.427166\pi$
$547$	6.64701e18			0.453644			0.226822	+	0.973936i	$0.427166\pi$
$548$		0			0
$549$		−	5.34567e19i		−	3.55628i
$550$		0			0
$551$		−	3.00100e18i		−	0.194628i
$552$		0			0
$553$	1.79507e19	−	1.78403e19i	1.13502	−	1.12805i
$554$		0			0
$555$	−1.35952e19			−0.838174
$556$		0			0
$557$	−1.50244e19			−0.903257			−0.451629	−	0.892206i	$-0.649157\pi$
$557$	−1.50244e19			−0.903257			−0.451629	+	0.892206i	$0.649157\pi$
$558$		0			0
$559$			2.32440e19i			1.36279i
$560$		0			0
$561$	−3.62398e19			−2.07227
$562$		0			0
$563$			1.31341e19i			0.732555i	0.930506	+	0.366277i	$0.119368\pi$
$563$			1.31341e19i			0.732555i	−0.930506	+	0.366277i	$0.880632\pi$
$564$		0			0
$565$			5.81929e18i			0.316615i
$566$		0			0
$567$	5.42130e19	−	5.38797e19i	2.87754	−	2.85985i
$568$		0			0
$569$	1.73796e19			0.900024			0.450012	−	0.893022i	$-0.351420\pi$
$569$	1.73796e19			0.900024			0.450012	+	0.893022i	$0.351420\pi$
$570$		0			0
$571$	−2.19326e18			−0.110825			−0.0554124	−	0.998464i	$-0.517647\pi$
$571$	−2.19326e18			−0.110825			−0.0554124	+	0.998464i	$0.517647\pi$
$572$		0			0
$573$		−	5.63253e19i		−	2.77729i
$574$		0			0
$575$	4.51594e17			0.0217307
$576$		0			0
$577$		−	1.26284e19i		−	0.593084i	−0.955020	−	0.296542i	$-0.904167\pi$
$577$		−	1.26284e19i		−	0.593084i	0.955020	−	0.296542i	$-0.0958334\pi$
$578$		0			0
$579$			4.70717e19i			2.15779i
$580$		0			0
$581$	2.25139e19	+	2.26532e19i	1.00744	+	1.01367i
$582$		0			0
$583$	2.90523e19			1.26912
$584$		0			0
$585$	2.39638e19			1.02203
$586$		0			0
$587$			4.75058e18i			0.197824i	0.995096	+	0.0989122i	$0.0315363\pi$
$587$			4.75058e18i			0.197824i	−0.995096	+	0.0989122i	$0.968464\pi$
$588$		0			0
$589$	−2.81872e19			−1.14616
$590$		0			0
$591$			1.28569e19i			0.510534i
$592$		0			0
$593$			2.50074e18i			0.0969806i	0.998824	+	0.0484903i	$0.0154410\pi$
$593$			2.50074e18i			0.0969806i	−0.998824	+	0.0484903i	$0.984559\pi$
$594$		0			0
$595$	8.16468e18	−	8.11449e18i	0.309258	−	0.307356i
$596$		0			0
$597$	−6.13814e19			−2.27100
$598$		0			0
$599$	3.54598e19			1.28159			0.640793	−	0.767713i	$-0.278606\pi$
$599$	3.54598e19			1.28159			0.640793	+	0.767713i	$0.278606\pi$
$600$		0			0
$601$			1.46556e18i			0.0517466i	0.999665	+	0.0258733i	$0.00823665\pi$
$601$			1.46556e18i			0.0517466i	−0.999665	+	0.0258733i	$0.991763\pi$
$602$		0			0
$603$	−9.96153e19			−3.43641
$604$		0			0
$605$			2.49072e18i			0.0839532i
$606$		0			0
$607$			1.37901e19i			0.454199i	0.973872	+	0.227099i	$0.0729242\pi$
$607$			1.37901e19i			0.454199i	−0.973872	+	0.227099i	$0.927076\pi$
$608$		0			0
$609$	9.32042e18	+	9.37808e18i	0.299995	+	0.301851i
$610$		0			0
$611$	−2.15016e17			−0.00676367
$612$		0			0
$613$	1.93795e19			0.595826			0.297913	−	0.954593i	$-0.403709\pi$
$613$	1.93795e19			0.595826			0.297913	+	0.954593i	$0.403709\pi$
$614$		0			0
$615$			7.79508e18i			0.234258i
$616$		0			0
$617$	4.31076e19			1.26636			0.633180	−	0.774005i	$-0.281749\pi$
$617$	4.31076e19			1.26636			0.633180	+	0.774005i	$0.281749\pi$
$618$		0			0
$619$			3.16212e19i			0.908123i	0.890970	+	0.454061i	$0.150025\pi$
$619$			3.16212e19i			0.908123i	−0.890970	+	0.454061i	$0.849975\pi$
$620$		0			0
$621$		−	1.36128e19i		−	0.382215i
$622$		0			0
$623$	−1.64042e19	−	1.65057e19i	−0.450338	−	0.453123i
$624$		0			0
$625$	1.49012e18			0.0400000
$626$		0			0
$627$	7.22218e19			1.89581
$628$		0			0
$629$			3.64970e19i			0.936920i
$630$		0			0
$631$	−2.23483e19			−0.561098			−0.280549	−	0.959840i	$-0.590517\pi$
$631$	−2.23483e19			−0.561098			−0.280549	+	0.959840i	$0.590517\pi$
$632$		0			0
$633$			5.90767e19i			1.45074i
$634$		0			0
$635$			2.25888e19i			0.542596i
$636$		0			0
$637$	3.46887e19	+	2.13907e17i	0.815102	+	0.00502632i
$638$		0			0
$639$	1.26529e19			0.290859
$640$		0			0
$641$	5.51746e19			1.24089			0.620444	−	0.784251i	$-0.286952\pi$
$641$	5.51746e19			1.24089			0.620444	+	0.784251i	$0.286952\pi$
$642$		0			0
$643$			8.64214e19i			1.90171i	0.309635	+	0.950856i	$0.399793\pi$
$643$			8.64214e19i			1.90171i	−0.309635	+	0.950856i	$0.600207\pi$
$644$		0			0
$645$	6.77240e19			1.45823
$646$		0			0
$647$		−	5.50912e19i		−	1.16079i	−0.814336	−	0.580393i	$-0.802899\pi$
$647$		−	5.50912e19i		−	1.16079i	0.814336	−	0.580393i	$-0.197101\pi$
$648$		0			0
$649$			5.02956e19i			1.03709i
$650$		0			0
$651$	8.80844e19	−	8.75429e19i	1.77759	−	1.76666i
$652$		0			0
$653$	−7.01321e19			−1.38524			−0.692619	−	0.721304i	$-0.743543\pi$
$653$	−7.01321e19			−1.38524			−0.692619	+	0.721304i	$0.743543\pi$
$654$		0			0
$655$	4.50029e19			0.870064
$656$		0			0
$657$		−	2.01839e20i		−	3.81987i
$658$		0			0
$659$	1.00009e19			0.185285			0.0926424	−	0.995699i	$-0.470469\pi$
$659$	1.00009e19			0.185285			0.0926424	+	0.995699i	$0.470469\pi$
$660$		0			0
$661$			4.09942e19i			0.743553i	0.928322	+	0.371776i	$0.121251\pi$
$661$			4.09942e19i			0.743553i	−0.928322	+	0.371776i	$0.878749\pi$
$662$		0			0
$663$		−	8.72776e19i		−	1.54992i
$664$		0			0
$665$	−1.62713e19	+	1.61712e19i	−0.282924	+	0.281185i
$666$		0			0
$667$	1.39250e18			0.0237090
$668$		0			0
$669$	−8.61543e19			−1.43646
$670$		0			0
$671$			8.46607e19i			1.38237i
$672$		0			0
$673$	6.52413e18			0.104332			0.0521659	−	0.998638i	$-0.483388\pi$
$673$	6.52413e18			0.104332			0.0521659	+	0.998638i	$0.483388\pi$
$674$		0			0
$675$		−	4.49180e19i		−	0.703548i
$676$		0			0
$677$		−	4.45002e19i		−	0.682717i	−0.939933	−	0.341359i	$-0.889113\pi$
$677$		−	4.45002e19i		−	0.682717i	0.939933	−	0.341359i	$-0.110887\pi$
$678$		0			0
$679$	2.46365e19	+	2.47888e19i	0.370246	+	0.372536i
$680$		0			0
$681$	4.22387e19			0.621843
$682$		0			0
$683$	9.45909e19			1.36428			0.682141	−	0.731220i	$-0.261049\pi$
$683$	9.45909e19			1.36428			0.682141	+	0.731220i	$0.261049\pi$
$684$		0			0
$685$			2.24692e19i			0.317508i
$686$		0			0
$687$	−9.30275e19			−1.28800
$688$		0			0
$689$			6.99678e19i			0.949212i
$690$		0			0
$691$		−	1.22370e19i		−	0.162677i	−0.996687	−	0.0813387i	$-0.974080\pi$
$691$		−	1.22370e19i		−	0.162677i	0.996687	−	0.0813387i	$-0.0259195\pi$
$692$		0			0
$693$	−1.66357e20	+	1.65334e20i	−2.16724	+	2.15392i
$694$		0			0
$695$	2.79018e19			0.356237
$696$		0			0
$697$	2.09263e19			0.261856
$698$		0			0
$699$			1.52364e20i			1.86871i
$700$		0			0
$701$	−9.79039e19			−1.17699			−0.588497	−	0.808500i	$-0.700280\pi$
$701$	−9.79039e19			−1.17699			−0.588497	+	0.808500i	$0.700280\pi$
$702$		0			0
$703$		−	7.27343e19i		−	0.857140i
$704$		0			0
$705$			6.26475e17i			0.00723736i
$706$		0			0
$707$	4.57657e19	+	4.60488e19i	0.518328	+	0.521534i
$708$		0			0
$709$	−5.74217e19			−0.637606			−0.318803	−	0.947821i	$-0.603281\pi$
$709$	−5.74217e19			−0.637606			−0.318803	+	0.947821i	$0.603281\pi$
$710$		0			0
$711$	−4.12099e20			−4.48657
$712$		0			0
$713$		−	1.30792e19i		−	0.139622i
$714$		0			0
$715$	−3.79521e19			−0.397277
$716$		0			0
$717$			1.43482e20i			1.47287i
$718$		0			0
$719$		−	1.71211e20i		−	1.72357i	−0.507276	−	0.861784i	$-0.669347\pi$
$719$		−	1.71211e20i		−	1.72357i	0.507276	−	0.861784i	$-0.330653\pi$
$720$		0			0
$721$	4.95062e19	+	4.98125e19i	0.488780	+	0.491803i
$722$		0			0
$723$	−1.61586e20			−1.56471
$724$		0			0
$725$	4.59480e18			0.0436416
$726$		0			0
$727$		−	6.12590e19i		−	0.570728i	−0.958419	−	0.285364i	$-0.907886\pi$
$727$		−	6.12590e19i		−	0.570728i	0.958419	−	0.285364i	$-0.0921145\pi$
$728$		0			0
$729$	−4.93898e20			−4.51382
$730$		0			0
$731$		−	1.81809e20i		−	1.63002i
$732$		0			0
$733$			1.48684e20i			1.30778i	0.756588	+	0.653891i	$0.226865\pi$
$733$			1.48684e20i			1.30778i	−0.756588	+	0.653891i	$0.773135\pi$
$734$		0			0
$735$	6.23244e17	−	1.01070e20i	0.00537833	−	0.872186i
$736$		0			0
$737$	1.57763e20			1.33577
$738$		0			0
$739$	−6.22571e19			−0.517223			−0.258612	−	0.965981i	$-0.583265\pi$
$739$	−6.22571e19			−0.517223			−0.258612	+	0.965981i	$0.583265\pi$
$740$		0			0
$741$			1.73934e20i			1.41794i
$742$		0			0
$743$	−2.87002e19			−0.229596			−0.114798	−	0.993389i	$-0.536622\pi$
$743$	−2.87002e19			−0.229596			−0.114798	+	0.993389i	$0.536622\pi$
$744$		0			0
$745$		−	8.79921e19i		−	0.690796i
$746$		0			0
$747$		−	5.20056e20i		−	4.00688i
$748$		0			0
$749$	−1.51055e20	+	1.50126e20i	−1.14225	+	1.13523i
$750$		0			0
$751$	1.97960e20			1.46926			0.734628	−	0.678470i	$-0.237357\pi$
$751$	1.97960e20			1.46926			0.734628	+	0.678470i	$0.237357\pi$
$752$		0			0
$753$	3.54862e20			2.58520
$754$		0			0
$755$			5.61895e19i			0.401815i
$756$		0			0
$757$	1.48584e20			1.04304			0.521519	−	0.853239i	$-0.325365\pi$
$757$	1.48584e20			1.04304			0.521519	+	0.853239i	$0.325365\pi$
$758$		0			0
$759$			3.35117e19i			0.230943i
$760$		0			0
$761$			6.20022e19i			0.419483i	0.977757	+	0.209741i	$0.0672622\pi$
$761$			6.20022e19i			0.419483i	−0.977757	+	0.209741i	$0.932738\pi$
$762$		0			0
$763$	−1.69002e20	+	1.67963e20i	−1.12259	+	1.11569i
$764$		0			0
$765$	−1.87439e20			−1.22245
$766$		0			0
$767$	−1.21129e20			−0.775674
$768$		0			0
$769$		−	1.92341e20i		−	1.20945i	−0.796435	−	0.604724i	$-0.793283\pi$
$769$		−	1.92341e20i		−	1.20945i	0.796435	−	0.604724i	$-0.206717\pi$
$770$		0			0
$771$	2.30242e20			1.42169
$772$		0			0
$773$			2.14732e20i			1.30209i	0.759040	+	0.651044i	$0.225669\pi$
$773$			2.14732e20i			1.30209i	−0.759040	+	0.651044i	$0.774331\pi$
$774$		0			0
$775$		−	4.31570e19i		−	0.257004i
$776$		0			0
$777$	2.25896e20	+	2.27293e20i	1.32118	+	1.32935i
$778$		0			0
$779$	−4.17037e19			−0.239559
$780$		0			0
$781$	−2.00386e19			−0.113061
$782$		0			0
$783$		−	1.38505e20i		−	0.767598i
$784$		0			0
$785$	2.34342e19			0.127574
$786$		0			0
$787$		−	6.44334e19i		−	0.344578i	−0.985046	−	0.172289i	$-0.944884\pi$
$787$		−	6.44334e19i		−	0.344578i	0.985046	−	0.172289i	$-0.0551163\pi$
$788$		0			0
$789$		−	1.01524e20i		−	0.533374i
$790$		0			0
$791$	9.72907e19	−	9.66926e19i	0.502153	−	0.499066i
$792$		0			0
$793$	−2.03891e20			−1.03392
$794$		0			0
$795$	2.03859e20			1.01569
$796$		0			0
$797$			2.03164e20i			0.994577i	0.867585	+	0.497288i	$0.165671\pi$
$797$			2.03164e20i			0.994577i	−0.867585	+	0.497288i	$0.834329\pi$
$798$		0			0
$799$	1.68180e18			0.00808999
$800$		0			0
$801$			3.78925e20i			1.79113i
$802$		0			0
$803$			3.19658e20i			1.48483i
$804$		0			0
$805$	−7.50363e18	−	7.55005e18i	−0.0342532	−	0.0344650i
$806$		0			0
$807$	−5.27558e20			−2.36677
$808$		0			0
$809$	3.58307e20			1.57985			0.789925	−	0.613204i	$-0.210120\pi$
$809$	3.58307e20			1.57985			0.789925	+	0.613204i	$0.210120\pi$
$810$		0			0
$811$			3.96421e20i			1.71795i	0.512017	+	0.858976i	$0.328899\pi$
$811$			3.96421e20i			1.71795i	−0.512017	+	0.858976i	$0.671101\pi$
$812$		0			0
$813$	3.89765e20			1.66023
$814$		0			0
$815$			7.20431e19i			0.301640i
$816$		0			0
$817$			3.62324e20i			1.49122i
$818$		0			0
$819$	−3.98179e20	−	4.00642e20i	−1.61098	−	1.62095i
$820$		0			0
$821$	3.54798e20			1.41117			0.705585	−	0.708625i	$-0.250684\pi$
$821$	3.54798e20			1.41117			0.705585	+	0.708625i	$0.250684\pi$
$822$		0			0
$823$	3.16402e20			1.23720			0.618600	−	0.785706i	$-0.287700\pi$
$823$	3.16402e20			1.23720			0.618600	+	0.785706i	$0.287700\pi$
$824$		0			0
$825$			1.10578e20i			0.425099i
$826$		0			0
$827$	1.80175e20			0.681014			0.340507	−	0.940242i	$-0.389401\pi$
$827$	1.80175e20			0.681014			0.340507	+	0.940242i	$0.389401\pi$
$828$		0			0
$829$			1.51491e20i			0.562996i	0.959562	+	0.281498i	$0.0908313\pi$
$829$			1.51491e20i			0.562996i	−0.959562	+	0.281498i	$0.909169\pi$
$830$		0			0
$831$			8.25803e20i			3.01765i
$832$		0			0
$833$	−2.71327e20	−	1.67313e18i	−0.974938	−	0.00601195i
$834$		0			0
$835$	−1.34285e20			−0.474486
$836$		0			0
$837$	−1.30092e21			−4.52036
$838$		0			0
$839$			3.59530e20i			1.22858i	0.789082	+	0.614288i	$0.210557\pi$
$839$			3.59530e20i			1.22858i	−0.789082	+	0.614288i	$0.789443\pi$
$840$		0			0
$841$	−2.83390e20			−0.952385
$842$		0			0
$843$		−	8.83827e20i		−	2.92129i
$844$		0			0
$845$			4.61650e19i			0.150078i
$846$		0			0
$847$	4.16415e19	−	4.13855e19i	0.133150	−	0.132332i
$848$		0			0
$849$	7.89744e20			2.48389
$850$		0			0
$851$	3.37495e19			0.104414
$852$		0			0
$853$		−	3.84656e20i		−	1.17066i	−0.810796	−	0.585329i	$-0.800966\pi$
$853$		−	3.84656e20i		−	1.17066i	0.810796	−	0.585329i	$-0.199034\pi$
$854$		0			0
$855$	3.73544e20			1.11835
$856$		0			0
$857$			2.89146e20i			0.851632i	0.904810	+	0.425816i	$0.140013\pi$
$857$			2.89146e20i			0.851632i	−0.904810	+	0.425816i	$0.859987\pi$
$858$		0			0
$859$		−	1.36931e20i		−	0.396781i	−0.980123	−	0.198390i	$-0.936429\pi$
$859$		−	1.36931e20i		−	0.396781i	0.980123	−	0.198390i	$-0.0635714\pi$
$860$		0			0
$861$	1.30323e20	−	1.29522e20i	0.371535	−	0.369251i
$862$		0			0
$863$	4.38366e19			0.122959			0.0614797	−	0.998108i	$-0.480418\pi$
$863$	4.38366e19			0.122959			0.0614797	+	0.998108i	$0.480418\pi$
$864$		0			0
$865$	1.46273e20			0.403695
$866$		0			0
$867$		−	3.55205e19i		−	0.0964597i
$868$		0			0
$869$	6.52652e20			1.74399
$870$		0			0
$871$			3.79946e20i			0.999069i
$872$		0			0
$873$		−	5.69085e20i		−	1.47258i
$874$		0			0
$875$	−2.47596e19	−	2.49127e19i	−0.0630503	−	0.0634403i
$876$		0			0
$877$	3.95501e20			0.991176			0.495588	−	0.868558i	$-0.334953\pi$
$877$	3.95501e20			0.991176			0.495588	+	0.868558i	$0.334953\pi$
$878$		0			0
$879$	−8.18230e20			−2.01815
$880$		0			0
$881$		−	7.13891e20i		−	1.73301i	−0.499170	−	0.866504i	$-0.666362\pi$
$881$		−	7.13891e20i		−	1.73301i	0.499170	−	0.866504i	$-0.333638\pi$
$882$		0			0
$883$	7.17451e20			1.71422			0.857112	−	0.515130i	$-0.172256\pi$
$883$	7.17451e20			1.71422			0.857112	+	0.515130i	$0.172256\pi$
$884$		0			0
$885$			3.52923e20i			0.829997i
$886$		0			0
$887$		−	4.51399e20i		−	1.04495i	−0.852654	−	0.522475i	$-0.825009\pi$
$887$		−	4.51399e20i		−	1.04495i	0.852654	−	0.522475i	$-0.174991\pi$
$888$		0			0
$889$	3.77654e20	−	3.75332e20i	0.860561	−	0.855271i
$890$		0			0
$891$	1.97108e21			4.42140
$892$		0			0
$893$	−3.35164e18			−0.00740112
$894$		0			0
$895$		−	4.67008e19i		−	0.101523i
$896$		0			0
$897$	−8.07074e19			−0.172729
$898$		0			0
$899$		−	1.33075e20i		−	0.280401i
$900$		0			0
$901$		−	5.47271e20i		−	1.13535i
$902$		0			0
$903$	−1.12529e21	−	1.13225e21i	−2.29854	−	2.31276i
$904$		0			0
$905$	−3.23907e20			−0.651449
$906$		0			0
$907$	−5.50598e20			−1.09039			−0.545197	−	0.838308i	$-0.683545\pi$
$907$	−5.50598e20			−1.09039			−0.545197	+	0.838308i	$0.683545\pi$
$908$		0			0
$909$		−	1.05716e21i		−	2.06154i
$910$		0			0
$911$	−1.54917e20			−0.297488			−0.148744	−	0.988876i	$-0.547523\pi$
$911$	−1.54917e20			−0.297488			−0.148744	+	0.988876i	$0.547523\pi$
$912$		0			0
$913$			8.23626e20i			1.55752i
$914$		0			0
$915$			5.94061e20i			1.10633i
$916$		0			0
$917$	−7.47762e20	−	7.52387e20i	−1.37144	−	1.37993i
$918$		0			0
$919$	8.31811e20			1.50250			0.751252	−	0.660015i	$-0.229450\pi$
$919$	8.31811e20			1.50250			0.751252	+	0.660015i	$0.229450\pi$
$920$		0			0
$921$	−2.10293e21			−3.74118
$922$		0			0
$923$		−	4.82597e19i		−	0.0845616i
$924$		0			0
$925$	1.11363e20			0.192197
$926$		0			0
$927$		−	1.14356e21i		−	1.94402i
$928$		0			0
$929$			1.07761e21i			1.80448i	0.431236	+	0.902239i	$0.358078\pi$
$929$			1.07761e21i			1.80448i	−0.431236	+	0.902239i	$0.641922\pi$
$930$		0			0
$931$	5.40723e20	+	3.33436e18i	0.891922	+	0.00550002i
$932$		0			0
$933$	−1.89515e21			−3.07944
$934$		0			0
$935$	2.96852e20			0.475180
$936$		0			0
$937$			8.52797e20i			1.34483i	0.740173	+	0.672417i	$0.234744\pi$
$937$			8.52797e20i			1.34483i	−0.740173	+	0.672417i	$0.765256\pi$
$938$		0			0
$939$	1.66655e21			2.58917
$940$		0			0
$941$		−	3.23271e20i		−	0.494811i	−0.968912	−	0.247405i	$-0.920422\pi$
$941$		−	3.23271e20i		−	0.494811i	0.968912	−	0.247405i	$-0.0795779\pi$
$942$		0			0
$943$		−	1.93510e19i		−	0.0291824i
$944$		0			0
$945$	−7.50967e20	+	7.46351e20i	−1.11583	+	1.10897i
$946$		0			0
$947$	7.62536e19			0.111638			0.0558188	−	0.998441i	$-0.482223\pi$
$947$	7.62536e19			0.111638			0.0558188	+	0.998441i	$0.482223\pi$
$948$		0			0
$949$	−7.69843e20			−1.11055
$950$		0			0
$951$		−	1.68966e21i		−	2.40180i
$952$		0			0
$953$	−5.03432e20			−0.705165			−0.352583	−	0.935781i	$-0.614696\pi$
$953$	−5.03432e20			−0.705165			−0.352583	+	0.935781i	$0.614696\pi$
$954$		0			0
$955$			4.61378e20i			0.636845i
$956$		0			0
$957$			3.40969e20i			0.463800i
$958$		0			0
$959$	3.75655e20	−	3.73346e20i	0.503569	−	0.500473i
$960$		0			0
$961$	−4.92978e20			−0.651274
$962$		0			0
$963$	3.46781e21			4.51514
$964$		0			0
$965$		−	3.85579e20i		−	0.494792i
$966$		0			0
$967$	−9.43557e20			−1.19339			−0.596695	−	0.802468i	$-0.703520\pi$
$967$	−9.43557e20			−1.19339			−0.596695	+	0.802468i	$0.703520\pi$
$968$		0			0
$969$		−	1.36047e21i		−	1.69599i
$970$		0			0
$971$			5.68512e20i			0.698562i	0.937018	+	0.349281i	$0.113574\pi$
$971$			5.68512e20i			0.698562i	−0.937018	+	0.349281i	$0.886426\pi$
$972$		0			0
$973$	−4.63614e20	−	4.66481e20i	−0.561521	−	0.564995i
$974$		0			0
$975$	−2.66308e20			−0.317945
$976$		0			0
$977$	−1.03065e21			−1.21296			−0.606481	−	0.795098i	$-0.707419\pi$
$977$	−1.03065e21			−1.21296			−0.606481	+	0.795098i	$0.707419\pi$
$978$		0			0
$979$		−	6.00113e20i		−	0.696233i
$980$		0			0
$981$	3.87983e21			4.43741
$982$		0			0
$983$		−	8.26760e20i		−	0.932191i	−0.884734	−	0.466096i	$-0.845660\pi$
$983$		−	8.26760e20i		−	0.932191i	0.884734	−	0.466096i	$-0.154340\pi$
$984$		0			0
$985$		−	1.05315e20i		−	0.117068i
$986$		0			0
$987$	1.04738e19	−	1.04094e19i	0.0114785	−	0.0114079i
$988$		0			0
$989$	−1.68122e20			−0.181657
$990$		0			0
$991$	−2.23505e20			−0.238107			−0.119053	−	0.992888i	$-0.537986\pi$
$991$	−2.23505e20			−0.238107			−0.119053	+	0.992888i	$0.537986\pi$
$992$		0			0
$993$			1.18527e21i			1.24500i
$994$		0			0
$995$	5.02795e20			0.520750
$996$		0			0
$997$		−	7.76991e19i		−	0.0793505i	−0.999213	−	0.0396753i	$-0.987368\pi$
$997$		−	7.76991e19i		−	0.0793505i	0.999213	−	0.0396753i	$-0.0126323\pi$
$998$		0			0
$999$		−	3.35690e21i		−	3.38050i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.15.d.a.41.1	✓	36
7.6	odd	2	inner	140.15.d.a.41.36	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.15.d.a.41.1	✓	36	1.1	even	1	trivial
140.15.d.a.41.36	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)

\(f(q)\)	\(=\)	\(q-4265.32i q^{3} +34938.6i q^{5} +(584126. - 580535. i) q^{7} -1.34099e7 q^{9} +O(q^{10})\)	\(q-4265.32i q^{3} +34938.6i q^{5} +(584126. - 580535. i) q^{7} -1.34099e7 q^{9} +2.12377e7 q^{11} +5.11474e7i q^{13} +1.49024e8 q^{15} -4.00063e8i q^{17} +7.97279e8i q^{19} +(-2.47616e9 - 2.49148e9i) q^{21} -3.69946e8 q^{23} -1.22070e9 q^{25} +3.67968e10i q^{27} -3.76406e9 q^{29} +3.53542e10i q^{31} -9.05853e10i q^{33} +(2.02830e10 + 2.04085e10i) q^{35} -9.12282e10 q^{37} +2.18160e11 q^{39} +5.23075e10i q^{41} +4.54451e11 q^{43} -4.68524e11i q^{45} +4.20385e9i q^{47} +(4.18218e9 - 6.78210e11i) q^{49} -1.70639e12 q^{51} +1.36796e12 q^{53} +7.42013e11i q^{55} +3.40065e12 q^{57} +2.36823e12i q^{59} +3.98635e12i q^{61} +(-7.83309e12 + 7.78494e12i) q^{63} -1.78702e12 q^{65} +7.42846e12 q^{67} +1.57794e12i q^{69} -9.43543e11 q^{71} +1.50515e13i q^{73} +5.20668e12i q^{75} +(1.24055e13 - 1.23292e13i) q^{77} +3.07309e13 q^{79} +9.28105e13 q^{81} +3.87814e13i q^{83} +1.39776e13 q^{85} +1.60549e13i q^{87} -2.82570e13i q^{89} +(2.96928e13 + 2.98765e13i) q^{91} +1.50797e14 q^{93} -2.78558e13 q^{95} +4.24375e13i q^{97} -2.84796e14 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 1364266 q^{7} - 54790830 q^{9}+O(q^{10}) \) 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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