LMFDB - Embedded newform 140.15.d.a.41.3

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			41.3
Character	$\chi$	$=$	140.41
Dual form			140.15.d.a.41.34

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3891.45i q^{3} -34938.6i q^{5} +(-806096. + 168618. i) q^{7} -1.03604e7 q^{9} +O(q^{10})$	\(q-3891.45i q^{3} -34938.6i q^{5} +(-806096. + 168618. i) q^{7} -1.03604e7 q^{9} -1.36469e7 q^{11} +3.11609e7i q^{13} -1.35962e8 q^{15} +1.27150e8i q^{17} +1.25158e9i q^{19} +(6.56168e8 + 3.13689e9i) q^{21} -1.28222e9 q^{23} -1.22070e9 q^{25} +2.17044e10i q^{27} -1.34497e10 q^{29} -4.30838e9i q^{31} +5.31061e10i q^{33} +(5.89126e9 + 2.81638e10i) q^{35} +3.62580e10 q^{37} +1.21261e11 q^{39} -1.83852e11i q^{41} +2.10703e10 q^{43} +3.61979e11i q^{45} -9.67091e11i q^{47} +(6.21359e11 - 2.71844e11i) q^{49} +4.94800e11 q^{51} -7.91927e11 q^{53} +4.76801e11i q^{55} +4.87048e12 q^{57} +2.02256e12i q^{59} +3.80641e12i q^{61} +(8.35151e12 - 1.74695e12i) q^{63} +1.08872e12 q^{65} -6.56159e12 q^{67} +4.98970e12i q^{69} +2.60416e12 q^{71} -7.69276e12i q^{73} +4.75031e12i q^{75} +(1.10007e13 - 2.30110e12i) q^{77} +3.17570e13 q^{79} +3.49081e13 q^{81} -3.51557e12i q^{83} +4.44245e12 q^{85} +5.23390e13i q^{87} -2.09454e13i q^{89} +(-5.25428e12 - 2.51187e13i) q^{91} -1.67659e13 q^{93} +4.37285e13 q^{95} +2.50980e13i q^{97} +1.41387e14 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	3891.45i		−	1.77936i	−0.456588	−	0.889678i	$-0.650929\pi$
$3$		−	3891.45i		−	1.77936i	0.456588	−	0.889678i	$-0.349071\pi$
$4$		0			0
$5$		−	34938.6i		−	0.447214i
$5$		−	34938.6i		−	0.447214i
$6$		0			0
$7$	−806096.	+	168618.i	−0.978815	+	0.204747i
$7$	−806096.	+	168618.i	−0.978815	+	0.204747i
$8$		0			0
$9$	−1.03604e7			−2.16611
$10$		0			0
$11$	−1.36469e7			−0.700299			−0.350150	−	0.936694i	$-0.613869\pi$
$11$	−1.36469e7			−0.700299			−0.350150	+	0.936694i	$0.613869\pi$
$12$		0			0
$13$			3.11609e7i			0.496600i	0.968683	+	0.248300i	$0.0798719\pi$
$13$			3.11609e7i			0.496600i	−0.968683	+	0.248300i	$0.920128\pi$
$14$		0			0
$15$	−1.35962e8			−0.795752
$16$		0			0
$17$			1.27150e8i			0.309867i	0.987925	+	0.154934i	$0.0495163\pi$
$17$			1.27150e8i			0.309867i	−0.987925	+	0.154934i	$0.950484\pi$
$18$		0			0
$19$			1.25158e9i			1.40018i	0.714054	+	0.700091i	$0.246857\pi$
$19$			1.25158e9i			1.40018i	−0.714054	+	0.700091i	$0.753143\pi$
$20$		0			0
$21$	6.56168e8	+	3.13689e9i	0.364317	+	1.74166i
$22$		0			0
$23$	−1.28222e9			−0.376589			−0.188295	−	0.982113i	$-0.560296\pi$
$23$	−1.28222e9			−0.376589			−0.188295	+	0.982113i	$0.560296\pi$
$24$		0			0
$25$	−1.22070e9			−0.200000
$26$		0			0
$27$			2.17044e10i			2.07492i
$28$		0			0
$29$	−1.34497e10			−0.779701			−0.389850	−	0.920878i	$-0.627473\pi$
$29$	−1.34497e10			−0.779701			−0.389850	+	0.920878i	$0.627473\pi$
$30$		0			0
$31$		−	4.30838e9i		−	0.156597i	−0.996930	−	0.0782983i	$-0.975051\pi$
$31$		−	4.30838e9i		−	0.156597i	0.996930	−	0.0782983i	$-0.0249486\pi$
$32$		0			0
$33$			5.31061e10i			1.24608i
$34$		0			0
$35$	5.89126e9	+	2.81638e10i	0.0915655	+	0.437739i
$36$		0			0
$37$	3.62580e10			0.381937			0.190969	−	0.981596i	$-0.438837\pi$
$37$	3.62580e10			0.381937			0.190969	+	0.981596i	$0.438837\pi$
$38$		0			0
$39$	1.21261e11			0.883629
$40$		0			0
$41$		−	1.83852e11i		−	0.944020i	−0.881593	−	0.472010i	$-0.843529\pi$
$41$		−	1.83852e11i		−	0.944020i	0.881593	−	0.472010i	$-0.156471\pi$
$42$		0			0
$43$	2.10703e10			0.0775160			0.0387580	−	0.999249i	$-0.487660\pi$
$43$	2.10703e10			0.0775160			0.0387580	+	0.999249i	$0.487660\pi$
$44$		0			0
$45$			3.61979e11i			0.968713i
$46$		0			0
$47$		−	9.67091e11i		−	1.90890i	−0.298378	−	0.954448i	$-0.596446\pi$
$47$		−	9.67091e11i		−	1.90890i	0.298378	−	0.954448i	$-0.403554\pi$
$48$		0			0
$49$	6.21359e11	−	2.71844e11i	0.916158	−	0.400818i
$50$		0			0
$51$	4.94800e11			0.551364
$52$		0			0
$53$	−7.91927e11			−0.674146			−0.337073	−	0.941478i	$-0.609437\pi$
$53$	−7.91927e11			−0.674146			−0.337073	+	0.941478i	$0.609437\pi$
$54$		0			0
$55$			4.76801e11i			0.313183i
$56$		0			0
$57$	4.87048e12			2.49142
$58$		0			0
$59$			2.02256e12i			0.812712i	0.913715	+	0.406356i	$0.133201\pi$
$59$			2.02256e12i			0.812712i	−0.913715	+	0.406356i	$0.866799\pi$
$60$		0			0
$61$			3.80641e12i			1.21118i	0.795779	+	0.605588i	$0.207062\pi$
$61$			3.80641e12i			1.21118i	−0.795779	+	0.605588i	$0.792938\pi$
$62$		0			0
$63$	8.35151e12	−	1.74695e12i	2.12022	−	0.443504i
$64$		0			0
$65$	1.08872e12			0.222086
$66$		0			0
$67$	−6.56159e12			−1.08264			−0.541322	−	0.840815i	$-0.682076\pi$
$67$	−6.56159e12			−1.08264			−0.541322	+	0.840815i	$0.682076\pi$
$68$		0			0
$69$			4.98970e12i			0.670086i
$70$		0			0
$71$	2.60416e12			0.286325			0.143162	−	0.989699i	$-0.454273\pi$
$71$	2.60416e12			0.286325			0.143162	+	0.989699i	$0.454273\pi$
$72$		0			0
$73$		−	7.69276e12i		−	0.696341i	−0.937431	−	0.348170i	$-0.886803\pi$
$73$		−	7.69276e12i		−	0.696341i	0.937431	−	0.348170i	$-0.113197\pi$
$74$		0			0
$75$			4.75031e12i			0.355871i
$76$		0			0
$77$	1.10007e13	−	2.30110e12i	0.685464	−	0.143384i
$78$		0			0
$79$	3.17570e13			1.65368			0.826838	−	0.562440i	$-0.190137\pi$
$79$	3.17570e13			1.65368			0.826838	+	0.562440i	$0.190137\pi$
$80$		0			0
$81$	3.49081e13			1.52592
$82$		0			0
$83$		−	3.51557e12i		−	0.129553i	−0.997900	−	0.0647767i	$-0.979366\pi$
$83$		−	3.51557e12i		−	0.129553i	0.997900	−	0.0647767i	$-0.0206335\pi$
$84$		0			0
$85$	4.44245e12			0.138577
$86$		0			0
$87$			5.23390e13i			1.38737i
$88$		0			0
$89$		−	2.09454e13i		−	0.473541i	−0.971566	−	0.236771i	$-0.923911\pi$
$89$		−	2.09454e13i		−	0.473541i	0.971566	−	0.236771i	$-0.0760890\pi$
$90$		0			0
$91$	−5.25428e12	−	2.51187e13i	−0.101677	−	0.486080i
$92$		0			0
$93$	−1.67659e13			−0.278641
$94$		0			0
$95$	4.37285e13			0.626181
$96$		0			0
$97$			2.50980e13i			0.310626i	0.987865	+	0.155313i	$0.0496386\pi$
$97$			2.50980e13i			0.310626i	−0.987865	+	0.155313i	$0.950361\pi$
$98$		0			0
$99$	1.41387e14			1.51692
$100$		0			0
$101$		−	2.64871e13i		−	0.247050i	−0.992341	−	0.123525i	$-0.960580\pi$
$101$		−	2.64871e13i		−	0.247050i	0.992341	−	0.123525i	$-0.0394199\pi$
$102$		0			0
$103$			1.43961e14i			1.17053i	0.810842	+	0.585265i	$0.199010\pi$
$103$			1.43961e14i			1.17053i	−0.810842	+	0.585265i	$0.800990\pi$
$104$		0			0
$105$	1.09598e14	−	2.29256e13i	0.778894	−	0.162928i
$106$		0			0
$107$	−2.45944e14			−1.53162			−0.765809	−	0.643068i	$-0.777661\pi$
$107$	−2.45944e14			−1.53162			−0.765809	+	0.643068i	$0.777661\pi$
$108$		0			0
$109$	1.96969e14			1.07749			0.538744	−	0.842470i	$-0.318899\pi$
$109$	1.96969e14			1.07749			0.538744	+	0.842470i	$0.318899\pi$
$110$		0			0
$111$		−	1.41096e14i		−	0.679602i
$112$		0			0
$113$	−1.32198e14			−0.561922			−0.280961	−	0.959719i	$-0.590653\pi$
$113$	−1.32198e14			−0.561922			−0.280961	+	0.959719i	$0.590653\pi$
$114$		0			0
$115$			4.47989e13i			0.168416i
$116$		0			0
$117$		−	3.22841e14i		−	1.07569i
$118$		0			0
$119$	−2.14398e13	−	1.02495e14i	−0.0634442	−	0.303302i
$120$		0			0
$121$	−1.93513e14			−0.509581
$122$		0			0
$123$	−7.15451e14			−1.67975
$124$		0			0
$125$			4.26496e13i			0.0894427i
$126$		0			0
$127$	3.87118e14			0.726470			0.363235	−	0.931698i	$-0.381672\pi$
$127$	3.87118e14			0.726470			0.363235	+	0.931698i	$0.381672\pi$
$128$		0			0
$129$		−	8.19941e13i		−	0.137929i
$130$		0			0
$131$			4.47346e14i			0.675685i	0.941203	+	0.337843i	$0.109697\pi$
$131$			4.47346e14i			0.675685i	−0.941203	+	0.337843i	$0.890303\pi$
$132$		0			0
$133$	−2.11039e14	−	1.00890e15i	−0.286683	−	1.37052i
$134$		0			0
$135$	7.58322e14			0.927934
$136$		0			0
$137$	−1.55429e15			−1.71588			−0.857941	−	0.513747i	$-0.828257\pi$
$137$	−1.55429e15			−1.71588			−0.857941	+	0.513747i	$0.828257\pi$
$138$		0			0
$139$			1.50133e14i			0.149752i	0.997193	+	0.0748758i	$0.0238560\pi$
$139$			1.50133e14i			0.149752i	−0.997193	+	0.0748758i	$0.976144\pi$
$140$		0			0
$141$	−3.76339e15			−3.39661
$142$		0			0
$143$		−	4.25249e14i		−	0.347769i
$144$		0			0
$145$			4.69915e14i			0.348693i
$146$		0			0
$147$	−1.05787e15	−	2.41799e15i	−0.713198	−	1.63017i
$148$		0			0
$149$	2.26975e15			1.39211			0.696055	−	0.717989i	$-0.254937\pi$
$149$	2.26975e15			1.39211			0.696055	+	0.717989i	$0.254937\pi$
$150$		0			0
$151$	−2.87527e14			−0.160635			−0.0803174	−	0.996769i	$-0.525593\pi$
$151$	−2.87527e14			−0.160635			−0.0803174	+	0.996769i	$0.525593\pi$
$152$		0			0
$153$		−	1.31733e15i		−	0.671206i
$154$		0			0
$155$	−1.50529e14			−0.0700321
$156$		0			0
$157$			3.24387e15i			1.37964i	0.723981	+	0.689820i	$0.242310\pi$
$157$			3.24387e15i			1.37964i	−0.723981	+	0.689820i	$0.757690\pi$
$158$		0			0
$159$			3.08175e15i			1.19955i
$160$		0			0
$161$	1.03359e15	−	2.16205e14i	0.368611	−	0.0771054i
$162$		0			0
$163$	−2.38920e15			−0.781518			−0.390759	−	0.920493i	$-0.627787\pi$
$163$	−2.38920e15			−0.781518			−0.390759	+	0.920493i	$0.627787\pi$
$164$		0			0
$165$	1.85545e15			0.557265
$166$		0			0
$167$		−	6.94499e15i		−	1.91715i	−0.284837	−	0.958576i	$-0.591939\pi$
$167$		−	6.94499e15i		−	1.91715i	0.284837	−	0.958576i	$-0.408061\pi$
$168$		0			0
$169$	2.96637e15			0.753388
$170$		0			0
$171$		−	1.29669e16i		−	3.03295i
$172$		0			0
$173$			8.18968e15i			1.76581i	0.469551	+	0.882905i	$0.344416\pi$
$173$			8.18968e15i			1.76581i	−0.469551	+	0.882905i	$0.655584\pi$
$174$		0			0
$175$	9.84004e14	−	2.05832e14i	0.195763	−	0.0409493i
$176$		0			0
$177$	7.87068e15			1.44610
$178$		0			0
$179$	2.34807e15			0.398785			0.199393	−	0.979920i	$-0.436103\pi$
$179$	2.34807e15			0.398785			0.199393	+	0.979920i	$0.436103\pi$
$180$		0			0
$181$		−	5.55918e15i		−	0.873496i	−0.899584	−	0.436748i	$-0.856130\pi$
$181$		−	5.55918e15i		−	0.873496i	0.899584	−	0.436748i	$-0.143870\pi$
$182$		0			0
$183$	1.48125e16			2.15511
$184$		0			0
$185$		−	1.26680e15i		−	0.170808i
$186$		0			0
$187$		−	1.73520e15i		−	0.217000i
$188$		0			0
$189$	−3.65975e15	−	1.74959e16i	−0.424834	−	2.03097i
$190$		0			0
$191$	−3.55935e13			−0.00383829			−0.00191914	−	0.999998i	$-0.500611\pi$
$191$	−3.55935e13			−0.00383829			−0.00191914	+	0.999998i	$0.500611\pi$
$192$		0			0
$193$	1.40942e16			1.41299			0.706496	−	0.707717i	$-0.250275\pi$
$193$	1.40942e16			1.41299			0.706496	+	0.707717i	$0.250275\pi$
$194$		0			0
$195$		−	4.23670e15i		−	0.395171i
$196$		0			0
$197$	−1.04357e15			−0.0906267			−0.0453133	−	0.998973i	$-0.514429\pi$
$197$	−1.04357e15			−0.0906267			−0.0453133	+	0.998973i	$0.514429\pi$
$198$		0			0
$199$			1.00542e16i			0.813539i	0.913531	+	0.406769i	$0.133345\pi$
$199$			1.00542e16i			0.813539i	−0.913531	+	0.406769i	$0.866655\pi$
$200$		0			0
$201$			2.55341e16i			1.92641i
$202$		0			0
$203$	1.08418e16	−	2.26786e15i	0.763183	−	0.159641i
$204$		0			0
$205$	−6.42353e15			−0.422179
$206$		0			0
$207$	1.32844e16			0.815733
$208$		0			0
$209$		−	1.70802e16i		−	0.980547i
$210$		0			0
$211$	1.03011e16			0.553231			0.276615	−	0.960981i	$-0.410787\pi$
$211$	1.03011e16			0.553231			0.276615	+	0.960981i	$0.410787\pi$
$212$		0			0
$213$		−	1.01340e16i		−	0.509474i
$214$		0			0
$215$		−	7.36166e14i		−	0.0346662i
$216$		0			0
$217$	7.26469e14	+	3.47297e15i	0.0320626	+	0.153279i
$218$		0			0
$219$	−2.99360e16			−1.23904
$220$		0			0
$221$	−3.96213e15			−0.153880
$222$		0			0
$223$		−	2.51576e16i		−	0.917350i	−0.888604	−	0.458675i	$-0.848324\pi$
$223$		−	2.51576e16i		−	0.917350i	0.888604	−	0.458675i	$-0.151676\pi$
$224$		0			0
$225$	1.26470e16			0.433222
$226$		0			0
$227$		−	1.12088e16i		−	0.360893i	−0.983585	−	0.180447i	$-0.942246\pi$
$227$		−	1.12088e16i		−	0.360893i	0.983585	−	0.180447i	$-0.0577543\pi$
$228$		0			0
$229$		−	3.58900e16i		−	1.08674i	−0.839494	−	0.543369i	$-0.817149\pi$
$229$		−	3.58900e16i		−	1.08674i	0.839494	−	0.543369i	$-0.182851\pi$
$230$		0			0
$231$	−8.95462e15	−	4.28086e16i	−0.255131	−	1.21968i
$232$		0			0
$233$	3.83435e16			1.02849			0.514245	−	0.857643i	$-0.328072\pi$
$233$	3.83435e16			1.02849			0.514245	+	0.857643i	$0.328072\pi$
$234$		0			0
$235$	−3.37888e16			−0.853684
$236$		0			0
$237$		−	1.23581e17i		−	2.94248i
$238$		0			0
$239$	−7.99687e16			−1.79529			−0.897645	−	0.440719i	$-0.854723\pi$
$239$	−7.99687e16			−1.79529			−0.897645	+	0.440719i	$0.854723\pi$
$240$		0			0
$241$		−	5.48430e16i		−	1.16145i	−0.814099	−	0.580727i	$-0.802768\pi$
$241$		−	5.48430e16i		−	1.16145i	0.814099	−	0.580727i	$-0.197232\pi$
$242$		0			0
$243$		−	3.20318e16i		−	0.640231i
$244$		0			0
$245$	−9.49784e15	−	2.17094e16i	−0.179251	−	0.409718i
$246$		0			0
$247$	−3.90005e16			−0.695331
$248$		0			0
$249$	−1.36807e16			−0.230522
$250$		0			0
$251$			1.65433e16i			0.263576i	0.991278	+	0.131788i	$0.0420717\pi$
$251$			1.65433e16i			0.263576i	−0.991278	+	0.131788i	$0.957928\pi$
$252$		0			0
$253$	1.74983e16			0.263725
$254$		0			0
$255$		−	1.72876e16i		−	0.246577i
$256$		0			0
$257$		−	1.09187e17i		−	1.47449i	−0.675627	−	0.737244i	$-0.736127\pi$
$257$		−	1.09187e17i		−	1.47449i	0.675627	−	0.737244i	$-0.263873\pi$
$258$		0			0
$259$	−2.92275e16	+	6.11374e15i	−0.373846	+	0.0782004i
$260$		0			0
$261$	1.39345e17			1.68892
$262$		0			0
$263$	−1.39006e17			−1.59714			−0.798571	−	0.601900i	$-0.794410\pi$
$263$	−1.39006e17			−1.59714			−0.798571	+	0.601900i	$0.794410\pi$
$264$		0			0
$265$			2.76688e16i			0.301487i
$266$		0			0
$267$	−8.15079e16			−0.842599
$268$		0			0
$269$			1.30478e17i			1.28018i	0.768300	+	0.640089i	$0.221103\pi$
$269$			1.30478e17i			1.28018i	−0.768300	+	0.640089i	$0.778897\pi$
$270$		0			0
$271$		−	5.71386e16i		−	0.532285i	−0.963934	−	0.266143i	$-0.914251\pi$
$271$		−	5.71386e16i		−	0.532285i	0.963934	−	0.266143i	$-0.0857492\pi$
$272$		0			0
$273$	−9.77483e16	+	2.04468e16i	−0.864909	+	0.180920i
$274$		0			0
$275$	1.66588e16			0.140060
$276$		0			0
$277$	1.65628e17			1.32366			0.661830	−	0.749654i	$-0.269780\pi$
$277$	1.65628e17			1.32366			0.661830	+	0.749654i	$0.269780\pi$
$278$		0			0
$279$			4.46367e16i			0.339205i
$280$		0			0
$281$	−9.57798e15			−0.0692357			−0.0346178	−	0.999401i	$-0.511021\pi$
$281$	−9.57798e15			−0.0692357			−0.0346178	+	0.999401i	$0.511021\pi$
$282$		0			0
$283$		−	2.90512e16i		−	0.199829i	−0.994996	−	0.0999147i	$-0.968143\pi$
$283$		−	2.90512e16i		−	0.199829i	0.994996	−	0.0999147i	$-0.0318570\pi$
$284$		0			0
$285$		−	1.70167e17i		−	1.11420i
$286$		0			0
$287$	3.10007e16	+	1.48202e17i	0.193285	+	0.924021i
$288$		0			0
$289$	1.52211e17			0.903982
$290$		0			0
$291$	9.76677e16			0.552714
$292$		0			0
$293$			3.00430e16i			0.162058i	0.996712	+	0.0810290i	$0.0258206\pi$
$293$			3.00430e16i			0.162058i	−0.996712	+	0.0810290i	$0.974179\pi$
$294$		0			0
$295$	7.06652e16			0.363456
$296$		0			0
$297$		−	2.96197e17i		−	1.45307i
$298$		0			0
$299$		−	3.99552e16i		−	0.187014i
$300$		0			0
$301$	−1.69847e16	+	3.55283e15i	−0.0758739	+	0.0158712i
$302$		0			0
$303$	−1.03073e17			−0.439590
$304$		0			0
$305$	1.32991e17			0.541654
$306$		0			0
$307$		−	4.53117e17i		−	1.76296i	−0.472224	−	0.881478i	$-0.656549\pi$
$307$		−	4.53117e17i		−	1.76296i	0.472224	−	0.881478i	$-0.343451\pi$
$308$		0			0
$309$	5.60215e17			2.08279
$310$		0			0
$311$		−	2.90908e17i		−	1.03379i	−0.856049	−	0.516895i	$-0.827088\pi$
$311$		−	2.90908e17i		−	1.03379i	0.856049	−	0.516895i	$-0.172912\pi$
$312$		0			0
$313$		−	1.37513e17i		−	0.467233i	−0.972329	−	0.233616i	$-0.924944\pi$
$313$		−	1.37513e17i		−	0.467233i	0.972329	−	0.233616i	$-0.0750560\pi$
$314$		0			0
$315$	−6.10360e16	−	2.91790e17i	−0.198341	−	0.948191i
$316$		0			0
$317$	5.47401e17			1.70173			0.850866	−	0.525383i	$-0.176078\pi$
$317$	5.47401e17			1.70173			0.850866	+	0.525383i	$0.176078\pi$
$318$		0			0
$319$	1.83547e17			0.546024
$320$		0			0
$321$			9.57081e17i			2.72529i
$322$		0			0
$323$	−1.59139e17			−0.433870
$324$		0			0
$325$		−	3.80382e16i		−	0.0993201i
$326$		0			0
$327$		−	7.66495e17i		−	1.91723i
$328$		0			0
$329$	1.63069e17	+	7.79568e17i	0.390840	+	1.86846i
$330$		0			0
$331$	−1.37550e17			−0.315984			−0.157992	−	0.987440i	$-0.550502\pi$
$331$	−1.37550e17			−0.315984			−0.157992	+	0.987440i	$0.550502\pi$
$332$		0			0
$333$	−3.75649e17			−0.827318
$334$		0			0
$335$			2.29253e17i			0.484173i
$336$		0			0
$337$	8.86449e17			1.79574			0.897872	−	0.440257i	$-0.145113\pi$
$337$	8.86449e17			1.79574			0.897872	+	0.440257i	$0.145113\pi$
$338$		0			0
$339$			5.14443e17i			0.999860i
$340$		0			0
$341$			5.87958e16i			0.109664i
$342$		0			0
$343$	−4.55038e17	+	3.23905e17i	−0.814683	+	0.579907i
$344$		0			0
$345$	1.74333e17			0.299672
$346$		0			0
$347$	−8.86058e17			−1.46270			−0.731351	−	0.682001i	$-0.761110\pi$
$347$	−8.86058e17			−1.46270			−0.731351	+	0.682001i	$0.761110\pi$
$348$		0			0
$349$			6.76408e17i			1.07258i	0.844033	+	0.536291i	$0.180175\pi$
$349$			6.76408e17i			1.07258i	−0.844033	+	0.536291i	$0.819825\pi$
$350$		0			0
$351$	−6.76330e17			−1.03041
$352$		0			0
$353$			9.77340e17i			1.43094i	0.698641	+	0.715472i	$0.253788\pi$
$353$			9.77340e17i			1.43094i	−0.698641	+	0.715472i	$0.746212\pi$
$354$		0			0
$355$		−	9.09855e16i		−	0.128048i
$356$		0			0
$357$	−3.98856e17	+	8.34320e16i	−0.539683	+	0.112890i
$358$		0			0
$359$	−2.22501e17			−0.289514			−0.144757	−	0.989467i	$-0.546240\pi$
$359$	−2.22501e17			−0.289514			−0.144757	+	0.989467i	$0.546240\pi$
$360$		0			0
$361$	−7.67454e17			−0.960510
$362$		0			0
$363$			7.53048e17i			0.906726i
$364$		0			0
$365$	−2.68774e17			−0.311413
$366$		0			0
$367$		−	1.14046e18i		−	1.27180i	−0.771771	−	0.635900i	$-0.780629\pi$
$367$		−	1.14046e18i		−	1.27180i	0.771771	−	0.635900i	$-0.219371\pi$
$368$		0			0
$369$			1.90479e18i			2.04485i
$370$		0			0
$371$	6.38369e17	−	1.33533e17i	0.659864	−	0.138029i
$372$		0			0
$373$	8.41096e17			0.837305			0.418652	−	0.908147i	$-0.362503\pi$
$373$	8.41096e17			0.837305			0.418652	+	0.908147i	$0.362503\pi$
$374$		0			0
$375$	1.65969e17			0.159150
$376$		0			0
$377$		−	4.19106e17i		−	0.387200i
$378$		0			0
$379$	−1.12653e18			−1.00293			−0.501464	−	0.865179i	$-0.667205\pi$
$379$	−1.12653e18			−1.00293			−0.501464	+	0.865179i	$0.667205\pi$
$380$		0			0
$381$		−	1.50645e18i		−	1.29265i
$382$		0			0
$383$			1.79571e17i			0.148540i	0.997238	+	0.0742701i	$0.0236627\pi$
$383$			1.79571e17i			0.148540i	−0.997238	+	0.0742701i	$0.976337\pi$
$384$		0			0
$385$	−8.03971e16	−	3.84348e17i	−0.0641233	−	0.306549i
$386$		0			0
$387$	−2.18297e17			−0.167908
$388$		0			0
$389$	−1.64987e18			−1.22406			−0.612029	−	0.790835i	$-0.709647\pi$
$389$	−1.64987e18			−1.22406			−0.612029	+	0.790835i	$0.709647\pi$
$390$		0			0
$391$		−	1.63035e17i		−	0.116693i
$392$		0			0
$393$	1.74083e18			1.20228
$394$		0			0
$395$		−	1.10955e18i		−	0.739546i
$396$		0			0
$397$		−	1.18744e18i		−	0.763970i	−0.924169	−	0.381985i	$-0.875241\pi$
$397$		−	1.18744e18i		−	0.763970i	0.924169	−	0.381985i	$-0.124759\pi$
$398$		0			0
$399$	−3.92607e18	+	8.21248e17i	−2.43864	+	0.510111i
$400$		0			0
$401$	−1.77612e18			−1.06528			−0.532638	−	0.846343i	$-0.678799\pi$
$401$	−1.77612e18			−1.06528			−0.532638	+	0.846343i	$0.678799\pi$
$402$		0			0
$403$	1.34253e17			0.0777659
$404$		0			0
$405$		−	1.21964e18i		−	0.682412i
$406$		0			0
$407$	−4.94808e17			−0.267470
$408$		0			0
$409$			2.08860e18i			1.09092i	0.838138	+	0.545458i	$0.183644\pi$
$409$			2.08860e18i			1.09092i	−0.838138	+	0.545458i	$0.816356\pi$
$410$		0			0
$411$			6.04844e18i			3.05317i
$412$		0			0
$413$	−3.41039e17	−	1.63038e18i	−0.166400	−	0.795494i
$414$		0			0
$415$	−1.22829e17			−0.0579380
$416$		0			0
$417$	5.84234e17			0.266461
$418$		0			0
$419$			2.60929e18i			1.15087i	0.817849	+	0.575433i	$0.195166\pi$
$419$			2.60929e18i			1.15087i	−0.817849	+	0.575433i	$0.804834\pi$
$420$		0			0
$421$	4.26644e18			1.82008			0.910042	−	0.414516i	$-0.136049\pi$
$421$	4.26644e18			1.82008			0.910042	+	0.414516i	$0.136049\pi$
$422$		0			0
$423$			1.00195e19i			4.13488i
$424$		0			0
$425$		−	1.55213e17i		−	0.0619734i
$426$		0			0
$427$	−6.41828e17	−	3.06833e18i	−0.247984	−	1.18552i
$428$		0			0
$429$	−1.65484e18			−0.618805
$430$		0			0
$431$	−5.96156e16			−0.0215784			−0.0107892	−	0.999942i	$-0.503434\pi$
$431$	−5.96156e16			−0.0215784			−0.0107892	+	0.999942i	$0.503434\pi$
$432$		0			0
$433$			4.12945e18i			1.44703i	0.690310	+	0.723514i	$0.257474\pi$
$433$			4.12945e18i			1.44703i	−0.690310	+	0.723514i	$0.742526\pi$
$434$		0			0
$435$	1.82865e18			0.620449
$436$		0			0
$437$		−	1.60481e18i		−	0.527294i
$438$		0			0
$439$			4.50958e18i			1.43511i	0.696503	+	0.717554i	$0.254738\pi$
$439$			4.50958e18i			1.43511i	−0.696503	+	0.717554i	$0.745262\pi$
$440$		0			0
$441$	−6.43755e18	+	2.81642e18i	−1.98450	+	0.868216i
$442$		0			0
$443$	3.62728e18			1.08332			0.541658	−	0.840599i	$-0.317797\pi$
$443$	3.62728e18			1.08332			0.541658	+	0.840599i	$0.317797\pi$
$444$		0			0
$445$	−7.31801e17			−0.211774
$446$		0			0
$447$		−	8.83261e18i		−	2.47706i
$448$		0			0
$449$	1.73694e18			0.472129			0.236065	−	0.971737i	$-0.424142\pi$
$449$	1.73694e18			0.472129			0.236065	+	0.971737i	$0.424142\pi$
$450$		0			0
$451$			2.50900e18i			0.661097i
$452$		0			0
$453$			1.11890e18i			0.285827i
$454$		0			0
$455$	−8.77612e17	+	1.83577e17i	−0.217382	+	0.0454715i
$456$		0			0
$457$	5.90386e18			1.41815			0.709076	−	0.705132i	$-0.249112\pi$
$457$	5.90386e18			1.41815			0.709076	+	0.705132i	$0.249112\pi$
$458$		0			0
$459$	−2.75973e18			−0.642950
$460$		0			0
$461$		−	6.46605e18i		−	1.46128i	−0.682764	−	0.730639i	$-0.739222\pi$
$461$		−	6.46605e18i		−	1.46128i	0.682764	−	0.730639i	$-0.260778\pi$
$462$		0			0
$463$	3.21581e18			0.705055			0.352528	−	0.935801i	$-0.385322\pi$
$463$	3.21581e18			0.705055			0.352528	+	0.935801i	$0.385322\pi$
$464$		0			0
$465$			5.85775e17i			0.124612i
$466$		0			0
$467$			9.66059e17i			0.199428i	0.995016	+	0.0997139i	$0.0317927\pi$
$467$			9.66059e17i			0.199428i	−0.995016	+	0.0997139i	$0.968207\pi$
$468$		0			0
$469$	5.28928e18	−	1.10640e18i	1.05971	−	0.221668i
$470$		0			0
$471$	1.26234e19			2.45487
$472$		0			0
$473$	−2.87543e17			−0.0542844
$474$		0			0
$475$		−	1.52781e18i		−	0.280036i
$476$		0			0
$477$	8.20471e18			1.46027
$478$		0			0
$479$			1.99789e18i			0.345322i	0.984981	+	0.172661i	$0.0552365\pi$
$479$			1.99789e18i			0.345322i	−0.984981	+	0.172661i	$0.944764\pi$
$480$		0			0
$481$			1.12983e18i			0.189670i
$482$		0			0
$483$	−8.41352e17	−	4.02218e18i	−0.137198	−	0.655891i
$484$		0			0
$485$	8.76889e17			0.138916
$486$		0			0
$487$	−5.67757e18			−0.873896			−0.436948	−	0.899487i	$-0.643941\pi$
$487$	−5.67757e18			−0.873896			−0.436948	+	0.899487i	$0.643941\pi$
$488$		0			0
$489$			9.29745e18i			1.39060i
$490$		0			0
$491$	1.06510e19			1.54817			0.774084	−	0.633083i	$-0.218211\pi$
$491$	1.06510e19			1.54817			0.774084	+	0.633083i	$0.218211\pi$
$492$		0			0
$493$		−	1.71014e18i		−	0.241604i
$494$		0			0
$495$		−	4.93987e18i		−	0.678389i
$496$		0			0
$497$	−2.09920e18	+	4.39107e17i	−0.280259	+	0.0586240i
$498$		0			0
$499$	−2.85229e17			−0.0370246			−0.0185123	−	0.999829i	$-0.505893\pi$
$499$	−2.85229e17			−0.0370246			−0.0185123	+	0.999829i	$0.505893\pi$
$500$		0			0
$501$	−2.70261e19			−3.41130
$502$		0			0
$503$		−	6.83996e18i		−	0.839610i	−0.907614	−	0.419805i	$-0.862098\pi$
$503$		−	6.83996e18i		−	0.839610i	0.907614	−	0.419805i	$-0.137902\pi$
$504$		0			0
$505$	−9.25421e17			−0.110484
$506$		0			0
$507$		−	1.15435e19i		−	1.34055i
$508$		0			0
$509$			1.05113e19i			1.18749i	0.804652	+	0.593747i	$0.202352\pi$
$509$			1.05113e19i			1.18749i	−0.804652	+	0.593747i	$0.797648\pi$
$510$		0			0
$511$	1.29713e18	+	6.20110e18i	0.142573	+	0.681589i
$512$		0			0
$513$	−2.71649e19			−2.90527
$514$		0			0
$515$	5.02977e18			0.523477
$516$		0			0
$517$			1.31977e19i			1.33680i
$518$		0			0
$519$	3.18697e19			3.14201
$520$		0			0
$521$			1.06112e18i			0.101836i	0.998703	+	0.0509181i	$0.0162148\pi$
$521$			1.06112e18i			0.101836i	−0.998703	+	0.0509181i	$0.983785\pi$
$522$		0			0
$523$		−	1.35785e19i		−	1.26864i	−0.773070	−	0.634320i	$-0.781280\pi$
$523$		−	1.35785e19i		−	1.26864i	0.773070	−	0.634320i	$-0.218720\pi$
$524$		0			0
$525$	−8.00986e17	−	3.82921e18i	−0.0728635	−	0.348332i
$526$		0			0
$527$	5.47812e17			0.0485241
$528$		0			0
$529$	−9.94875e18			−0.858181
$530$		0			0
$531$		−	2.09546e19i		−	1.76042i
$532$		0			0
$533$	5.72900e18			0.468801
$534$		0			0
$535$			8.59294e18i			0.684960i
$536$		0			0
$537$		−	9.13739e18i		−	0.709581i
$538$		0			0
$539$	−8.47960e18	+	3.70982e18i	−0.641585	+	0.280693i
$540$		0			0
$541$	−1.05914e19			−0.780860			−0.390430	−	0.920633i	$-0.627674\pi$
$541$	−1.05914e19			−0.780860			−0.390430	+	0.920633i	$0.627674\pi$
$542$		0			0
$543$	−2.16333e19			−1.55426
$544$		0			0
$545$		−	6.88181e18i		−	0.481867i
$546$		0			0
$547$	1.57912e19			1.07772			0.538859	−	0.842396i	$-0.318856\pi$
$547$	1.57912e19			1.07772			0.538859	+	0.842396i	$0.318856\pi$
$548$		0			0
$549$		−	3.94361e19i		−	2.62354i
$550$		0			0
$551$		−	1.68335e19i		−	1.09172i
$552$		0			0
$553$	−2.55992e19	+	5.35480e18i	−1.61864	+	0.338585i
$554$		0			0
$555$	−4.92970e18			−0.303927
$556$		0			0
$557$	−5.95294e18			−0.357886			−0.178943	−	0.983859i	$-0.557268\pi$
$557$	−5.95294e18			−0.357886			−0.178943	+	0.983859i	$0.557268\pi$
$558$		0			0
$559$			6.56570e17i			0.0384945i
$560$		0			0
$561$	−6.75246e18			−0.386120
$562$		0			0
$563$		−	1.16392e19i		−	0.649177i	−0.945855	−	0.324589i	$-0.894774\pi$
$563$		−	1.16392e19i		−	0.649177i	0.945855	−	0.324589i	$-0.105226\pi$
$564$		0			0
$565$			4.61881e18i			0.251299i
$566$		0			0
$567$	−2.81393e19	+	5.88613e18i	−1.49359	+	0.312427i
$568$		0			0
$569$	2.03087e19			1.05171			0.525856	−	0.850574i	$-0.323745\pi$
$569$	2.03087e19			1.05171			0.525856	+	0.850574i	$0.323745\pi$
$570$		0			0
$571$	1.00277e19			0.506699			0.253350	−	0.967375i	$-0.418468\pi$
$571$	1.00277e19			0.506699			0.253350	+	0.967375i	$0.418468\pi$
$572$		0			0
$573$			1.38510e17i			0.00682968i
$574$		0			0
$575$	1.56521e18			0.0753178
$576$		0			0
$577$		−	2.31488e19i		−	1.08717i	−0.839354	−	0.543586i	$-0.817066\pi$
$577$		−	2.31488e19i		−	1.08717i	0.839354	−	0.543586i	$-0.182934\pi$
$578$		0			0
$579$		−	5.48470e19i		−	2.51422i
$580$		0			0
$581$	5.92787e17	+	2.83388e18i	0.0265256	+	0.126809i
$582$		0			0
$583$	1.08073e19			0.472104
$584$		0			0
$585$	−1.12796e19			−0.481063
$586$		0			0
$587$			2.49307e19i			1.03817i	0.854723	+	0.519085i	$0.173727\pi$
$587$			2.49307e19i			1.03817i	−0.854723	+	0.519085i	$0.826273\pi$
$588$		0			0
$589$	5.39230e18			0.219264
$590$		0			0
$591$			4.06098e18i			0.161257i
$592$		0			0
$593$		−	1.34802e19i		−	0.522772i	−0.965234	−	0.261386i	$-0.915820\pi$
$593$		−	1.34802e19i		−	0.522772i	0.965234	−	0.261386i	$-0.0841795\pi$
$594$		0			0
$595$	−3.58104e18	+	7.49076e17i	−0.135641	+	0.0283731i
$596$		0			0
$597$	3.91256e19			1.44757
$598$		0			0
$599$	−1.50814e19			−0.545072			−0.272536	−	0.962146i	$-0.587862\pi$
$599$	−1.50814e19			−0.545072			−0.272536	+	0.962146i	$0.587862\pi$
$600$		0			0
$601$			2.03853e19i			0.719774i	0.932996	+	0.359887i	$0.117185\pi$
$601$			2.03853e19i			0.719774i	−0.932996	+	0.359887i	$0.882815\pi$
$602$		0			0
$603$	6.79809e19			2.34512
$604$		0			0
$605$			6.76107e18i			0.227891i
$606$		0			0
$607$			6.95337e18i			0.229021i	0.993422	+	0.114510i	$0.0365299\pi$
$607$			6.95337e18i			0.229021i	−0.993422	+	0.114510i	$0.963470\pi$
$608$		0			0
$609$	−8.82529e18	−	4.21903e19i	−0.284058	−	1.35797i
$610$		0			0
$611$	3.01354e19			0.947958
$612$		0			0
$613$	5.36435e19			1.64928			0.824639	−	0.565659i	$-0.191378\pi$
$613$	5.36435e19			1.64928			0.824639	+	0.565659i	$0.191378\pi$
$614$		0			0
$615$			2.49968e19i			0.751207i
$616$		0			0
$617$	5.27806e19			1.55052			0.775262	−	0.631640i	$-0.217618\pi$
$617$	5.27806e19			1.55052			0.775262	+	0.631640i	$0.217618\pi$
$618$		0			0
$619$			2.13317e19i			0.612620i	0.951932	+	0.306310i	$0.0990943\pi$
$619$			2.13317e19i			0.612620i	−0.951932	+	0.306310i	$0.900906\pi$
$620$		0			0
$621$		−	2.78299e19i		−	0.781394i
$622$		0			0
$623$	3.53176e18	+	1.68840e19i	0.0969560	+	0.463510i
$624$		0			0
$625$	1.49012e18			0.0400000
$626$		0			0
$627$	−6.64667e19			−1.74474
$628$		0			0
$629$			4.61022e18i			0.118350i
$630$		0			0
$631$	6.59464e19			1.65571			0.827857	−	0.560939i	$-0.189560\pi$
$631$	6.59464e19			1.65571			0.827857	+	0.560939i	$0.189560\pi$
$632$		0			0
$633$		−	4.00862e19i		−	0.984394i
$634$		0			0
$635$		−	1.35254e19i		−	0.324887i
$636$		0			0
$637$	8.47092e18	+	1.93621e19i	0.199046	+	0.454964i
$638$		0			0
$639$	−2.69802e19			−0.620211
$640$		0			0
$641$	−4.14245e19			−0.931647			−0.465823	−	0.884878i	$-0.654242\pi$
$641$	−4.14245e19			−0.931647			−0.465823	+	0.884878i	$0.654242\pi$
$642$		0			0
$643$		−	6.23055e19i		−	1.37104i	−0.728055	−	0.685519i	$-0.759575\pi$
$643$		−	6.23055e19i		−	1.37104i	0.728055	−	0.685519i	$-0.240425\pi$
$644$		0			0
$645$	−2.86476e18			−0.0616836
$646$		0			0
$647$			6.03594e19i			1.27179i	0.771776	+	0.635894i	$0.219369\pi$
$647$			6.03594e19i			1.27179i	−0.771776	+	0.635894i	$0.780631\pi$
$648$		0			0
$649$		−	2.76015e19i		−	0.569142i
$650$		0			0
$651$	1.35149e19	−	2.82702e18i	0.272738	−	0.0570508i
$652$		0			0
$653$	2.56327e19			0.506292			0.253146	−	0.967428i	$-0.418535\pi$
$653$	2.56327e19			0.506292			0.253146	+	0.967428i	$0.418535\pi$
$654$		0			0
$655$	1.56296e19			0.302176
$656$		0			0
$657$			7.97003e19i			1.50835i
$658$		0			0
$659$	6.52826e19			1.20948			0.604741	−	0.796422i	$-0.293277\pi$
$659$	6.52826e19			1.20948			0.604741	+	0.796422i	$0.293277\pi$
$660$		0			0
$661$		−	3.47244e19i		−	0.629831i	−0.949120	−	0.314916i	$-0.898024\pi$
$661$		−	3.47244e19i		−	0.629831i	0.949120	−	0.314916i	$-0.101976\pi$
$662$		0			0
$663$			1.54184e19i			0.273807i
$664$		0			0
$665$	−3.52494e19	+	7.37340e18i	−0.612915	+	0.128208i
$666$		0			0
$667$	1.72455e19			0.293627
$668$		0			0
$669$	−9.78996e19			−1.63229
$670$		0			0
$671$		−	5.19455e19i		−	0.848185i
$672$		0			0
$673$	1.18451e20			1.89423			0.947115	−	0.320895i	$-0.103984\pi$
$673$	1.18451e20			1.89423			0.947115	+	0.320895i	$0.103984\pi$
$674$		0			0
$675$		−	2.64947e19i		−	0.414985i
$676$		0			0
$677$			1.16154e20i			1.78203i	0.453975	+	0.891014i	$0.350006\pi$
$677$			1.16154e20i			1.78203i	−0.453975	+	0.891014i	$0.649994\pi$
$678$		0			0
$679$	−4.23197e18	−	2.02314e19i	−0.0635996	−	0.304045i
$680$		0			0
$681$	−4.36185e19			−0.642157
$682$		0			0
$683$	7.88900e19			1.13783			0.568914	−	0.822397i	$-0.307364\pi$
$683$	7.88900e19			1.13783			0.568914	+	0.822397i	$0.307364\pi$
$684$		0			0
$685$			5.43046e19i			0.767366i
$686$		0			0
$687$	−1.39664e20			−1.93369
$688$		0			0
$689$		−	2.46772e19i		−	0.334781i
$690$		0			0
$691$			6.00439e19i			0.798220i	0.916903	+	0.399110i	$0.130681\pi$
$691$			6.00439e19i			0.798220i	−0.916903	+	0.399110i	$0.869319\pi$
$692$		0			0
$693$	−1.13972e20	+	2.38404e19i	−1.48479	+	0.310585i
$694$		0			0
$695$	5.24541e18			0.0669709
$696$		0			0
$697$	2.33769e19			0.292521
$698$		0			0
$699$		−	1.49212e20i		−	1.83005i
$700$		0			0
$701$	−3.19197e18			−0.0383736			−0.0191868	−	0.999816i	$-0.506108\pi$
$701$	−3.19197e18			−0.0383736			−0.0191868	+	0.999816i	$0.506108\pi$
$702$		0			0
$703$			4.53799e19i			0.534782i
$704$		0			0
$705$			1.31487e20i			1.51901i
$706$		0			0
$707$	4.46619e18	+	2.13511e19i	0.0505826	+	0.241816i
$708$		0			0
$709$	−5.47470e19			−0.607906			−0.303953	−	0.952687i	$-0.598307\pi$
$709$	−5.47470e19			−0.607906			−0.303953	+	0.952687i	$0.598307\pi$
$710$		0			0
$711$	−3.29017e20			−3.58204
$712$		0			0
$713$			5.52429e18i			0.0589726i
$714$		0			0
$715$	−1.48576e19			−0.155527
$716$		0			0
$717$			3.11194e20i			3.19446i
$718$		0			0
$719$		−	5.61470e19i		−	0.565229i	−0.959234	−	0.282615i	$-0.908798\pi$
$719$		−	5.61470e19i		−	0.565229i	0.959234	−	0.282615i	$-0.0912018\pi$
$720$		0			0
$721$	−2.42743e19	−	1.16046e20i	−0.239662	−	1.14573i
$722$		0			0
$723$	−2.13419e20			−2.06664
$724$		0			0
$725$	1.64181e19			0.155940
$726$		0			0
$727$		−	1.52592e20i		−	1.42164i	−0.703374	−	0.710820i	$-0.748324\pi$
$727$		−	1.52592e20i		−	1.42164i	0.703374	−	0.710820i	$-0.251676\pi$
$728$		0			0
$729$	4.23145e19			0.386720
$730$		0			0
$731$			2.67910e18i			0.0240197i
$732$		0			0
$733$			1.17184e20i			1.03072i	0.856974	+	0.515360i	$0.172342\pi$
$733$			1.17184e20i			1.03072i	−0.856974	+	0.515360i	$0.827658\pi$
$734$		0			0
$735$	−8.44811e19	+	3.69604e19i	−0.729035	+	0.318952i
$736$		0			0
$737$	8.95451e19			0.758175
$738$		0			0
$739$	1.60334e20			1.33203			0.666015	−	0.745938i	$-0.267998\pi$
$739$	1.60334e20			1.33203			0.666015	+	0.745938i	$0.267998\pi$
$740$		0			0
$741$			1.51769e20i			1.23724i
$742$		0			0
$743$	−3.96327e19			−0.317053			−0.158526	−	0.987355i	$-0.550674\pi$
$743$	−3.96327e19			−0.317053			−0.158526	+	0.987355i	$0.550674\pi$
$744$		0			0
$745$		−	7.93016e19i		−	0.622570i
$746$		0			0
$747$			3.64228e19i			0.280627i
$748$		0			0
$749$	1.98255e20	−	4.14706e19i	1.49917	−	0.313594i
$750$		0			0
$751$	8.98086e19			0.666558			0.333279	−	0.942828i	$-0.391845\pi$
$751$	8.98086e19			0.666558			0.333279	+	0.942828i	$0.391845\pi$
$752$		0			0
$753$	6.43773e19			0.468995
$754$		0			0
$755$			1.00458e19i			0.0718381i
$756$		0			0
$757$	−2.64087e19			−0.185386			−0.0926928	−	0.995695i	$-0.529547\pi$
$757$	−2.64087e19			−0.185386			−0.0926928	+	0.995695i	$0.529547\pi$
$758$		0			0
$759$		−	6.80937e19i		−	0.469261i
$760$		0			0
$761$			1.44652e20i			0.978660i	0.872099	+	0.489330i	$0.162759\pi$
$761$			1.44652e20i			0.978660i	−0.872099	+	0.489330i	$0.837241\pi$
$762$		0			0
$763$	−1.58776e20	+	3.32124e19i	−1.05466	+	0.220612i
$764$		0			0
$765$	−4.60257e19			−0.300172
$766$		0			0
$767$	−6.30247e19			−0.403593
$768$		0			0
$769$			3.86371e19i			0.242952i	0.992594	+	0.121476i	$0.0387627\pi$
$769$			3.86371e19i			0.242952i	−0.992594	+	0.121476i	$0.961237\pi$
$770$		0			0
$771$	−4.24898e20			−2.62364
$772$		0			0
$773$		−	1.56977e20i		−	0.951875i	−0.879479	−	0.475938i	$-0.842109\pi$
$773$		−	1.56977e20i		−	0.951875i	0.879479	−	0.475938i	$-0.157891\pi$
$774$		0			0
$775$			5.25925e18i			0.0313193i
$776$		0			0
$777$	2.37913e19	+	1.13737e20i	0.139146	+	0.665205i
$778$		0			0
$779$	2.30106e20			1.32180
$780$		0			0
$781$	−3.55386e19			−0.200513
$782$		0			0
$783$		−	2.91919e20i		−	1.61782i
$784$		0			0
$785$	1.13336e20			0.616994
$786$		0			0
$787$			2.16871e20i			1.15979i	0.814692	+	0.579893i	$0.196906\pi$
$787$			2.16871e20i			1.15979i	−0.814692	+	0.579893i	$0.803094\pi$
$788$		0			0
$789$			5.40936e20i			2.84189i
$790$		0			0
$791$	1.06564e20	−	2.22909e19i	0.550018	−	0.115052i
$792$		0			0
$793$	−1.18611e20			−0.601470
$794$		0			0
$795$	1.07672e20			0.536453
$796$		0			0
$797$			1.55330e20i			0.760410i	0.924902	+	0.380205i	$0.124146\pi$
$797$			1.55330e20i			0.760410i	−0.924902	+	0.380205i	$0.875854\pi$
$798$		0			0
$799$	1.22966e20			0.591504
$800$		0			0
$801$			2.17003e20i			1.02574i
$802$		0			0
$803$			1.04982e20i			0.487647i
$804$		0			0
$805$	−7.55389e18	−	3.61123e19i	−0.0344826	−	0.164848i
$806$		0			0
$807$	5.07748e20			2.27789
$808$		0			0
$809$	−1.74502e19			−0.0769414			−0.0384707	−	0.999260i	$-0.512249\pi$
$809$	−1.74502e19			−0.0769414			−0.0384707	+	0.999260i	$0.512249\pi$
$810$		0			0
$811$			8.34257e19i			0.361538i	0.983526	+	0.180769i	$0.0578587\pi$
$811$			8.34257e19i			0.361538i	−0.983526	+	0.180769i	$0.942141\pi$
$812$		0			0
$813$	−2.22352e20			−0.947125
$814$		0			0
$815$			8.34751e19i			0.349505i
$816$		0			0
$817$			2.63712e19i			0.108537i
$818$		0			0
$819$	5.44367e19	+	2.60241e20i	0.220244	+	1.05290i
$820$		0			0
$821$	3.87887e20			1.54278			0.771388	−	0.636365i	$-0.219563\pi$
$821$	3.87887e20			1.54278			0.771388	+	0.636365i	$0.219563\pi$
$822$		0			0
$823$	4.06964e20			1.59132			0.795660	−	0.605744i	$-0.207124\pi$
$823$	4.06964e20			1.59132			0.795660	+	0.605744i	$0.207124\pi$
$824$		0			0
$825$		−	6.48268e19i		−	0.249216i
$826$		0			0
$827$	2.90084e19			0.109644			0.0548221	−	0.998496i	$-0.482541\pi$
$827$	2.90084e19			0.109644			0.0548221	+	0.998496i	$0.482541\pi$
$828$		0			0
$829$		−	1.37634e20i		−	0.511497i	−0.966743	−	0.255748i	$-0.917678\pi$
$829$		−	1.37634e20i		−	0.511497i	0.966743	−	0.255748i	$-0.0823218\pi$
$830$		0			0
$831$		−	6.44535e20i		−	2.35526i
$832$		0			0
$833$	3.45651e19	+	7.90061e19i	0.124200	+	0.283887i
$834$		0			0
$835$	−2.42648e20			−0.857376
$836$		0			0
$837$	9.35109e19			0.324926
$838$		0			0
$839$			3.26934e20i			1.11719i	0.829442	+	0.558593i	$0.188659\pi$
$839$			3.26934e20i			1.11719i	−0.829442	+	0.558593i	$0.811341\pi$
$840$		0			0
$841$	−1.16663e20			−0.392067
$842$		0			0
$843$			3.72723e19i			0.123195i
$844$		0			0
$845$		−	1.03641e20i		−	0.336925i
$846$		0			0
$847$	1.55990e20	−	3.26298e19i	0.498785	−	0.104335i
$848$		0			0
$849$	−1.13051e20			−0.355568
$850$		0			0
$851$	−4.64908e19			−0.143833
$852$		0			0
$853$			5.01046e20i			1.52487i	0.647062	+	0.762437i	$0.275997\pi$
$853$			5.01046e20i			1.52487i	−0.647062	+	0.762437i	$0.724003\pi$
$854$		0			0
$855$	−4.53046e20			−1.35638
$856$		0			0
$857$		−	6.24275e20i		−	1.83870i	−0.393445	−	0.919348i	$-0.628717\pi$
$857$		−	6.24275e20i		−	1.83870i	0.393445	−	0.919348i	$-0.371283\pi$
$858$		0			0
$859$		−	4.11296e20i		−	1.19180i	−0.803060	−	0.595899i	$-0.796796\pi$
$859$		−	4.11296e20i		−	1.19180i	0.803060	−	0.595899i	$-0.203204\pi$
$860$		0			0
$861$	5.76723e20	−	1.20638e20i	1.64416	−	0.343923i
$862$		0			0
$863$	−6.08059e20			−1.70557			−0.852786	−	0.522260i	$-0.825089\pi$
$863$	−6.08059e20			−1.70557			−0.852786	+	0.522260i	$0.825089\pi$
$864$		0			0
$865$	2.86136e20			0.789695
$866$		0			0
$867$		−	5.92320e20i		−	1.60851i
$868$		0			0
$869$	−4.33384e20			−1.15807
$870$		0			0
$871$		−	2.04465e20i		−	0.537641i
$872$		0			0
$873$		−	2.60026e20i		−	0.672849i
$874$		0			0
$875$	−7.19148e18	−	3.43797e19i	−0.0183131	−	0.0875479i
$876$		0			0
$877$	3.31839e20			0.831630			0.415815	−	0.909449i	$-0.363496\pi$
$877$	3.31839e20			0.831630			0.415815	+	0.909449i	$0.363496\pi$
$878$		0			0
$879$	1.16911e20			0.288359
$880$		0			0
$881$			6.00515e20i			1.45778i	0.684630	+	0.728891i	$0.259964\pi$
$881$			6.00515e20i			1.45778i	−0.684630	+	0.728891i	$0.740036\pi$
$882$		0			0
$883$	3.67973e20			0.879208			0.439604	−	0.898192i	$-0.355119\pi$
$883$	3.67973e20			0.879208			0.439604	+	0.898192i	$0.355119\pi$
$884$		0			0
$885$		−	2.74990e20i		−	0.646717i
$886$		0			0
$887$		−	3.38909e20i		−	0.784546i	−0.919849	−	0.392273i	$-0.871689\pi$
$887$		−	3.38909e20i		−	0.784546i	0.919849	−	0.392273i	$-0.128311\pi$
$888$		0			0
$889$	−3.12055e20	+	6.52750e19i	−0.711080	+	0.148742i
$890$		0			0
$891$	−4.76386e20			−1.06860
$892$		0			0
$893$	1.21039e21			2.67280
$894$		0			0
$895$		−	8.20381e19i		−	0.178342i
$896$		0			0
$897$	−1.55484e20			−0.332765
$898$		0			0
$899$			5.79466e19i			0.122098i
$900$		0			0
$901$		−	1.00694e20i		−	0.208896i
$902$		0			0
$903$	1.38257e19	+	6.60951e19i	0.0282404	+	0.135007i
$904$		0			0
$905$	−1.94230e20			−0.390639
$906$		0			0
$907$	7.56692e20			1.49854			0.749270	−	0.662265i	$-0.230405\pi$
$907$	7.56692e20			1.49854			0.749270	+	0.662265i	$0.230405\pi$
$908$		0			0
$909$			2.74418e20i			0.535137i
$910$		0			0
$911$	−7.50640e20			−1.44146			−0.720731	−	0.693215i	$-0.756194\pi$
$911$	−7.50640e20			−1.44146			−0.720731	+	0.693215i	$0.756194\pi$
$912$		0			0
$913$			4.79764e19i			0.0907261i
$914$		0			0
$915$		−	5.17526e20i		−	0.963795i
$916$		0			0
$917$	−7.54304e19	−	3.60604e20i	−0.138344	−	0.661371i
$918$		0			0
$919$	−8.46822e20			−1.52962			−0.764810	−	0.644255i	$-0.777167\pi$
$919$	−8.46822e20			−1.52962			−0.764810	+	0.644255i	$0.777167\pi$
$920$		0			0
$921$	−1.76328e21			−3.13693
$922$		0			0
$923$			8.11480e19i			0.142189i
$924$		0			0
$925$	−4.42603e19			−0.0763874
$926$		0			0
$927$		−	1.49149e21i		−	2.53550i
$928$		0			0
$929$		−	1.17622e20i		−	0.196960i	−0.995139	−	0.0984798i	$-0.968602\pi$
$929$		−	1.17622e20i		−	0.196960i	0.995139	−	0.0984798i	$-0.0313980\pi$
$930$		0			0
$931$	3.40236e20	+	7.77683e20i	0.561219	+	1.28279i
$932$		0			0
$933$	−1.13205e21			−1.83948
$934$		0			0
$935$	−6.06255e19			−0.0970452
$936$		0			0
$937$			6.36289e20i			1.00341i	0.865040	+	0.501703i	$0.167293\pi$
$937$			6.36289e20i			1.00341i	−0.865040	+	0.501703i	$0.832707\pi$
$938$		0			0
$939$	−5.35125e20			−0.831374
$940$		0			0
$941$			5.20910e20i			0.797326i	0.917098	+	0.398663i	$0.130526\pi$
$941$			5.20910e20i			0.797326i	−0.917098	+	0.398663i	$0.869474\pi$
$942$		0			0
$943$			2.35739e20i			0.355508i
$944$		0			0
$945$	−6.11280e20	+	1.27866e20i	−0.908276	+	0.189991i
$946$		0			0
$947$	7.17223e20			1.05004			0.525018	−	0.851091i	$-0.324059\pi$
$947$	7.17223e20			1.05004			0.525018	+	0.851091i	$0.324059\pi$
$948$		0			0
$949$	2.39713e20			0.345803
$950$		0			0
$951$		−	2.13019e21i		−	3.02799i
$952$		0			0
$953$	−5.59563e20			−0.783789			−0.391894	−	0.920010i	$-0.628180\pi$
$953$	−5.59563e20			−0.783789			−0.391894	+	0.920010i	$0.628180\pi$
$954$		0			0
$955$			1.24359e18i			0.00171653i
$956$		0			0
$957$		−	7.14263e20i		−	0.971571i
$958$		0			0
$959$	1.25291e21	−	2.62081e20i	1.67953	−	0.351321i
$960$		0			0
$961$	7.38382e20			0.975478
$962$		0			0
$963$	2.54809e21			3.31765
$964$		0			0
$965$		−	4.92432e20i		−	0.631910i
$966$		0			0
$967$	−5.03698e20			−0.637067			−0.318533	−	0.947912i	$-0.603190\pi$
$967$	−5.03698e20			−0.637067			−0.318533	+	0.947912i	$0.603190\pi$
$968$		0			0
$969$			6.19283e20i			0.772010i
$970$		0			0
$971$		−	2.60549e20i		−	0.320151i	−0.987105	−	0.160076i	$-0.948826\pi$
$971$		−	2.60549e20i		−	0.320151i	0.987105	−	0.160076i	$-0.0511738\pi$
$972$		0			0
$973$	−2.53150e19	−	1.21021e20i	−0.0306611	−	0.146579i
$974$		0			0
$975$	−1.48024e20			−0.176726
$976$		0			0
$977$	−6.53864e20			−0.769530			−0.384765	−	0.923015i	$-0.625718\pi$
$977$	−6.53864e20			−0.769530			−0.384765	+	0.923015i	$0.625718\pi$
$978$		0			0
$979$			2.85838e20i			0.331621i
$980$		0			0
$981$	−2.04068e21			−2.33396
$982$		0			0
$983$		−	3.95350e20i		−	0.445767i	−0.974845	−	0.222883i	$-0.928453\pi$
$983$		−	3.95350e20i		−	0.445767i	0.974845	−	0.222883i	$-0.0715469\pi$
$984$		0			0
$985$			3.64607e19i			0.0405295i
$986$		0			0
$987$	3.03365e21	−	6.34574e20i	3.32465	−	0.695444i
$988$		0			0
$989$	−2.70168e19			−0.0291917
$990$		0			0
$991$	−1.79783e21			−1.91528			−0.957641	−	0.287966i	$-0.907021\pi$
$991$	−1.79783e21			−1.91528			−0.957641	+	0.287966i	$0.907021\pi$
$992$		0			0
$993$			5.35270e20i			0.562249i
$994$		0			0
$995$	3.51281e20			0.363825
$996$		0			0
$997$		−	9.23671e20i		−	0.943303i	−0.881785	−	0.471652i	$-0.843658\pi$
$997$		−	9.23671e20i		−	0.943303i	0.881785	−	0.471652i	$-0.156342\pi$
$998$		0			0
$999$			7.86960e20i			0.792491i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.15.d.a.41.3	✓	36
7.6	odd	2	inner	140.15.d.a.41.34	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.15.d.a.41.3	✓	36	1.1	even	1	trivial
140.15.d.a.41.34	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)

\(f(q)\)	\(=\)	\(q-3891.45i q^{3} -34938.6i q^{5} +(-806096. + 168618. i) q^{7} -1.03604e7 q^{9} +O(q^{10})\)	\(q-3891.45i q^{3} -34938.6i q^{5} +(-806096. + 168618. i) q^{7} -1.03604e7 q^{9} -1.36469e7 q^{11} +3.11609e7i q^{13} -1.35962e8 q^{15} +1.27150e8i q^{17} +1.25158e9i q^{19} +(6.56168e8 + 3.13689e9i) q^{21} -1.28222e9 q^{23} -1.22070e9 q^{25} +2.17044e10i q^{27} -1.34497e10 q^{29} -4.30838e9i q^{31} +5.31061e10i q^{33} +(5.89126e9 + 2.81638e10i) q^{35} +3.62580e10 q^{37} +1.21261e11 q^{39} -1.83852e11i q^{41} +2.10703e10 q^{43} +3.61979e11i q^{45} -9.67091e11i q^{47} +(6.21359e11 - 2.71844e11i) q^{49} +4.94800e11 q^{51} -7.91927e11 q^{53} +4.76801e11i q^{55} +4.87048e12 q^{57} +2.02256e12i q^{59} +3.80641e12i q^{61} +(8.35151e12 - 1.74695e12i) q^{63} +1.08872e12 q^{65} -6.56159e12 q^{67} +4.98970e12i q^{69} +2.60416e12 q^{71} -7.69276e12i q^{73} +4.75031e12i q^{75} +(1.10007e13 - 2.30110e12i) q^{77} +3.17570e13 q^{79} +3.49081e13 q^{81} -3.51557e12i q^{83} +4.44245e12 q^{85} +5.23390e13i q^{87} -2.09454e13i q^{89} +(-5.25428e12 - 2.51187e13i) q^{91} -1.67659e13 q^{93} +4.37285e13 q^{95} +2.50980e13i q^{97} +1.41387e14 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 1364266 q^{7} - 54790830 q^{9}+O(q^{10}) \) 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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