LMFDB - Embedded newform 140.15.d.a.41.9

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			41.9
Character	$\chi$	$=$	140.41
Dual form			140.15.d.a.41.28

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2419.29i q^{3} -34938.6i q^{5} +(57888.7 - 821506. i) q^{7} -1.06999e6 q^{9} +O(q^{10})$	\(q-2419.29i q^{3} -34938.6i q^{5} +(57888.7 - 821506. i) q^{7} -1.06999e6 q^{9} +2.36010e7 q^{11} +7.29941e7i q^{13} -8.45265e7 q^{15} +2.60695e7i q^{17} +2.55942e8i q^{19} +(-1.98746e9 - 1.40049e8i) q^{21} -5.14900e7 q^{23} -1.22070e9 q^{25} -8.98277e9i q^{27} +5.92847e9 q^{29} -1.93367e10i q^{31} -5.70977e10i q^{33} +(-2.87022e10 - 2.02255e9i) q^{35} +4.03397e9 q^{37} +1.76594e11 q^{39} -1.96990e11i q^{41} -9.05715e10 q^{43} +3.73838e10i q^{45} +2.50899e10i q^{47} +(-6.71521e11 - 9.51118e10i) q^{49} +6.30697e10 q^{51} -6.93770e11 q^{53} -8.24586e11i q^{55} +6.19197e11 q^{57} -1.32725e12i q^{59} -2.34311e12i q^{61} +(-6.19402e10 + 8.79001e11i) q^{63} +2.55031e12 q^{65} +4.54824e12 q^{67} +1.24569e11i q^{69} +1.24938e11 q^{71} -1.06067e13i q^{73} +2.95323e12i q^{75} +(1.36623e12 - 1.93884e13i) q^{77} -1.56285e13 q^{79} -2.68496e13 q^{81} -1.02931e13i q^{83} +9.10832e11 q^{85} -1.43427e13i q^{87} -7.64916e13i q^{89} +(5.99651e13 + 4.22554e12i) q^{91} -4.67812e13 q^{93} +8.94223e12 q^{95} -1.09501e14i q^{97} -2.52528e13 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	2419.29i		−	1.10621i	−0.833110	−	0.553107i	$-0.813442\pi$
$3$		−	2419.29i		−	1.10621i	0.833110	−	0.553107i	$-0.186558\pi$
$4$		0			0
$5$		−	34938.6i		−	0.447214i
$5$		−	34938.6i		−	0.447214i
$6$		0			0
$7$	57888.7	−	821506.i	0.0702923	−	0.997526i
$7$	57888.7	−	821506.i	0.0702923	−	0.997526i
$8$		0			0
$9$	−1.06999e6			−0.223708
$10$		0			0
$11$	2.36010e7			1.21111			0.605553	−	0.795805i	$-0.292952\pi$
$11$	2.36010e7			1.21111			0.605553	+	0.795805i	$0.292952\pi$
$12$		0			0
$13$			7.29941e7i			1.16328i	0.813446	+	0.581640i	$0.197589\pi$
$13$			7.29941e7i			1.16328i	−0.813446	+	0.581640i	$0.802411\pi$
$14$		0			0
$15$	−8.45265e7			−0.494714
$16$		0			0
$17$			2.60695e7i			0.0635317i	0.999495	+	0.0317659i	$0.0101131\pi$
$17$			2.60695e7i			0.0635317i	−0.999495	+	0.0317659i	$0.989887\pi$
$18$		0			0
$19$			2.55942e8i			0.286329i	0.989699	+	0.143165i	$0.0457278\pi$
$19$			2.55942e8i			0.286329i	−0.989699	+	0.143165i	$0.954272\pi$
$20$		0			0
$21$	−1.98746e9	−	1.40049e8i	−1.10348	−	0.0777582i
$22$		0			0
$23$	−5.14900e7			−0.0151227			−0.00756133	−	0.999971i	$-0.502407\pi$
$23$	−5.14900e7			−0.0151227			−0.00756133	+	0.999971i	$0.502407\pi$
$24$		0			0
$25$	−1.22070e9			−0.200000
$26$		0			0
$27$		−	8.98277e9i		−	0.858745i
$28$		0			0
$29$	5.92847e9			0.343682			0.171841	−	0.985125i	$-0.445028\pi$
$29$	5.92847e9			0.343682			0.171841	+	0.985125i	$0.445028\pi$
$30$		0			0
$31$		−	1.93367e10i		−	0.702832i	−0.936219	−	0.351416i	$-0.885700\pi$
$31$		−	1.93367e10i		−	0.702832i	0.936219	−	0.351416i	$-0.114300\pi$
$32$		0			0
$33$		−	5.70977e10i		−	1.33974i
$34$		0			0
$35$	−2.87022e10	−	2.02255e9i	−0.446107	−	0.0314357i
$36$		0			0
$37$	4.03397e9			0.0424933			0.0212467	−	0.999774i	$-0.493236\pi$
$37$	4.03397e9			0.0424933			0.0212467	+	0.999774i	$0.493236\pi$
$38$		0			0
$39$	1.76594e11			1.28684
$40$		0			0
$41$		−	1.96990e11i		−	1.01148i	−0.862687	−	0.505739i	$-0.831220\pi$
$41$		−	1.96990e11i		−	1.01148i	0.862687	−	0.505739i	$-0.168780\pi$
$42$		0			0
$43$	−9.05715e10			−0.333206			−0.166603	−	0.986024i	$-0.553280\pi$
$43$	−9.05715e10			−0.333206			−0.166603	+	0.986024i	$0.553280\pi$
$44$		0			0
$45$			3.73838e10i			0.100045i
$46$		0			0
$47$			2.50899e10i			0.0495238i	0.999693	+	0.0247619i	$0.00788276\pi$
$47$			2.50899e10i			0.0495238i	−0.999693	+	0.0247619i	$0.992117\pi$
$48$		0			0
$49$	−6.71521e11	−	9.51118e10i	−0.990118	−	0.140237i
$50$		0			0
$51$	6.30697e10			0.0702797
$52$		0			0
$53$	−6.93770e11			−0.590588			−0.295294	−	0.955406i	$-0.595418\pi$
$53$	−6.93770e11			−0.590588			−0.295294	+	0.955406i	$0.595418\pi$
$54$		0			0
$55$		−	8.24586e11i		−	0.541623i
$56$		0			0
$57$	6.19197e11			0.316741
$58$		0			0
$59$		−	1.32725e12i		−	0.533322i	−0.963790	−	0.266661i	$-0.914080\pi$
$59$		−	1.32725e12i		−	0.533322i	0.963790	−	0.266661i	$-0.0859204\pi$
$60$		0			0
$61$		−	2.34311e12i		−	0.745562i	−0.927919	−	0.372781i	$-0.878404\pi$
$61$		−	2.34311e12i		−	0.745562i	0.927919	−	0.372781i	$-0.121596\pi$
$62$		0			0
$63$	−6.19402e10	+	8.79001e11i	−0.0157249	+	0.223154i
$64$		0			0
$65$	2.55031e12			0.520235
$66$		0			0
$67$	4.54824e12			0.750446			0.375223	−	0.926935i	$-0.377566\pi$
$67$	4.54824e12			0.750446			0.375223	+	0.926935i	$0.377566\pi$
$68$		0			0
$69$			1.24569e11i			0.0167289i
$70$		0			0
$71$	1.24938e11			0.0137368			0.00686841	−	0.999976i	$-0.497814\pi$
$71$	1.24938e11			0.0137368			0.00686841	+	0.999976i	$0.497814\pi$
$72$		0			0
$73$		−	1.06067e13i		−	0.960107i	−0.877239	−	0.480053i	$-0.840617\pi$
$73$		−	1.06067e13i		−	0.960107i	0.877239	−	0.480053i	$-0.159383\pi$
$74$		0			0
$75$			2.95323e12i			0.221243i
$76$		0			0
$77$	1.36623e12	−	1.93884e13i	0.0851313	−	1.20811i
$78$		0			0
$79$	−1.56285e13			−0.813821			−0.406911	−	0.913468i	$-0.633394\pi$
$79$	−1.56285e13			−0.813821			−0.406911	+	0.913468i	$0.633394\pi$
$80$		0			0
$81$	−2.68496e13			−1.17366
$82$		0			0
$83$		−	1.02931e13i		−	0.379315i	−0.981850	−	0.189657i	$-0.939262\pi$
$83$		−	1.02931e13i		−	0.379315i	0.981850	−	0.189657i	$-0.0607377\pi$
$84$		0			0
$85$	9.10832e11			0.0284123
$86$		0			0
$87$		−	1.43427e13i		−	0.380186i
$88$		0			0
$89$		−	7.64916e13i		−	1.72935i	−0.502329	−	0.864677i	$-0.667523\pi$
$89$		−	7.64916e13i		−	1.72935i	0.502329	−	0.864677i	$-0.332477\pi$
$90$		0			0
$91$	5.99651e13	+	4.22554e12i	1.16040	+	0.0817696i
$92$		0			0
$93$	−4.67812e13			−0.777482
$94$		0			0
$95$	8.94223e12			0.128050
$96$		0			0
$97$		−	1.09501e14i		−	1.35524i	−0.735411	−	0.677621i	$-0.763011\pi$
$97$		−	1.09501e14i		−	1.35524i	0.735411	−	0.677621i	$-0.236989\pi$
$98$		0			0
$99$	−2.52528e13			−0.270934
$100$		0			0
$101$			2.01991e14i			1.88401i	0.335599	+	0.942005i	$0.391061\pi$
$101$			2.01991e14i			1.88401i	−0.335599	+	0.942005i	$0.608939\pi$
$102$		0			0
$103$			2.65230e13i			0.215656i	0.994170	+	0.107828i	$0.0343896\pi$
$103$			2.65230e13i			0.215656i	−0.994170	+	0.107828i	$0.965610\pi$
$104$		0			0
$105$	−4.89313e12	+	6.94390e13i	−0.0347745	+	0.493490i
$106$		0			0
$107$	2.69697e14			1.67954			0.839768	−	0.542946i	$-0.182691\pi$
$107$	2.69697e14			1.67954			0.839768	+	0.542946i	$0.182691\pi$
$108$		0			0
$109$	−1.57407e14			−0.861072			−0.430536	−	0.902573i	$-0.641675\pi$
$109$	−1.57407e14			−0.861072			−0.430536	+	0.902573i	$0.641675\pi$
$110$		0			0
$111$		−	9.75934e12i		−	0.0470067i
$112$		0			0
$113$	−7.96067e13			−0.338377			−0.169188	−	0.985584i	$-0.554115\pi$
$113$	−7.96067e13			−0.338377			−0.169188	+	0.985584i	$0.554115\pi$
$114$		0			0
$115$			1.79899e12i			0.00676306i
$116$		0			0
$117$		−	7.81028e13i		−	0.260235i
$118$		0			0
$119$	2.14163e13	+	1.50913e12i	0.0633746	+	0.00446579i
$120$		0			0
$121$	1.77258e14			0.466776
$122$		0			0
$123$	−4.76575e14			−1.11891
$124$		0			0
$125$			4.26496e13i			0.0894427i
$126$		0			0
$127$	−2.75865e14			−0.517690			−0.258845	−	0.965919i	$-0.583342\pi$
$127$	−2.75865e14			−0.517690			−0.258845	+	0.965919i	$0.583342\pi$
$128$		0			0
$129$			2.19119e14i			0.368597i
$130$		0			0
$131$			4.30219e14i			0.649816i	0.945746	+	0.324908i	$0.105333\pi$
$131$			4.30219e14i			0.649816i	−0.945746	+	0.324908i	$0.894667\pi$
$132$		0			0
$133$	2.10258e14	+	1.48161e13i	0.285621	+	0.0201267i
$134$		0			0
$135$	−3.13845e14			−0.384042
$136$		0			0
$137$	−7.32788e14			−0.808974			−0.404487	−	0.914544i	$-0.632550\pi$
$137$	−7.32788e14			−0.808974			−0.404487	+	0.914544i	$0.632550\pi$
$138$		0			0
$139$		−	5.42497e14i		−	0.541120i	−0.962703	−	0.270560i	$-0.912791\pi$
$139$		−	5.42497e14i		−	0.541120i	0.962703	−	0.270560i	$-0.0872089\pi$
$140$		0			0
$141$	6.06997e13			0.0547839
$142$		0			0
$143$			1.72274e15i			1.40886i
$144$		0			0
$145$		−	2.07132e14i		−	0.153699i
$146$		0			0
$147$	−2.30103e14	+	1.62460e15i	−0.155132	+	1.09528i
$148$		0			0
$149$	1.90629e15			1.16919			0.584597	−	0.811324i	$-0.301253\pi$
$149$	1.90629e15			1.16919			0.584597	+	0.811324i	$0.301253\pi$
$150$		0			0
$151$	6.18243e14			0.345399			0.172699	−	0.984975i	$-0.444751\pi$
$151$	6.18243e14			0.345399			0.172699	+	0.984975i	$0.444751\pi$
$152$		0			0
$153$		−	2.78941e13i		−	0.0142125i
$154$		0			0
$155$	−6.75598e14			−0.314316
$156$		0			0
$157$			1.29669e15i			0.551493i	0.961230	+	0.275747i	$0.0889250\pi$
$157$			1.29669e15i			0.551493i	−0.961230	+	0.275747i	$0.911075\pi$
$158$		0			0
$159$			1.67843e15i			0.653316i
$160$		0			0
$161$	−2.98069e12	+	4.22993e13i	−0.00106301	+	0.0150853i
$162$		0			0
$163$	−5.37747e15			−1.75900			−0.879498	−	0.475902i	$-0.842122\pi$
$163$	−5.37747e15			−1.75900			−0.879498	+	0.475902i	$0.842122\pi$
$164$		0			0
$165$	−1.99491e15			−0.599150
$166$		0			0
$167$		−	3.24476e15i		−	0.895710i	−0.894106	−	0.447855i	$-0.852188\pi$
$167$		−	3.24476e15i		−	0.895710i	0.894106	−	0.447855i	$-0.147812\pi$
$168$		0			0
$169$	−1.39077e15			−0.353222
$170$		0			0
$171$		−	2.73854e14i		−	0.0640541i
$172$		0			0
$173$		−	2.50334e15i		−	0.539755i	−0.962895	−	0.269877i	$-0.913017\pi$
$173$		−	2.50334e15i		−	0.539755i	0.962895	−	0.269877i	$-0.0869832\pi$
$174$		0			0
$175$	−7.06649e13	+	1.00281e15i	−0.0140585	+	0.199505i
$176$		0			0
$177$	−3.21101e15			−0.589968
$178$		0			0
$179$	−7.60864e14			−0.129222			−0.0646109	−	0.997911i	$-0.520581\pi$
$179$	−7.60864e14			−0.129222			−0.0646109	+	0.997911i	$0.520581\pi$
$180$		0			0
$181$		−	8.05042e15i		−	1.26494i	−0.774587	−	0.632468i	$-0.782042\pi$
$181$		−	8.05042e15i		−	1.26494i	0.774587	−	0.632468i	$-0.217958\pi$
$182$		0			0
$183$	−5.66866e15			−0.824750
$184$		0			0
$185$		−	1.40941e14i		−	0.0190036i
$186$		0			0
$187$			6.15267e14i			0.0769436i
$188$		0			0
$189$	−7.37940e15	−	5.20001e14i	−0.856621	−	0.0603631i
$190$		0			0
$191$	6.24411e15			0.673344			0.336672	−	0.941622i	$-0.390699\pi$
$191$	6.24411e15			0.673344			0.336672	+	0.941622i	$0.390699\pi$
$192$		0			0
$193$	−1.10943e16			−1.11224			−0.556118	−	0.831103i	$-0.687710\pi$
$193$	−1.10943e16			−1.11224			−0.556118	+	0.831103i	$0.687710\pi$
$194$		0			0
$195$		−	6.16994e15i		−	0.575491i
$196$		0			0
$197$	−3.00607e15			−0.261057			−0.130528	−	0.991445i	$-0.541667\pi$
$197$	−3.00607e15			−0.261057			−0.130528	+	0.991445i	$0.541667\pi$
$198$		0			0
$199$		−	1.53878e16i		−	1.24510i	−0.782579	−	0.622551i	$-0.786096\pi$
$199$		−	1.53878e16i		−	1.24510i	0.782579	−	0.622551i	$-0.213904\pi$
$200$		0			0
$201$		−	1.10035e16i		−	0.830153i
$202$		0			0
$203$	3.43192e14	−	4.87027e15i	0.0241582	−	0.342832i
$204$		0			0
$205$	−6.88253e15			−0.452347
$206$		0			0
$207$	5.50937e13			0.00338306
$208$		0			0
$209$			6.04048e15i			0.346775i
$210$		0			0
$211$	−1.90124e16			−1.02108			−0.510541	−	0.859853i	$-0.670555\pi$
$211$	−1.90124e16			−1.02108			−0.510541	+	0.859853i	$0.670555\pi$
$212$		0			0
$213$		−	3.02261e14i		−	0.0151959i
$214$		0			0
$215$			3.16444e15i			0.149014i
$216$		0			0
$217$	−1.58853e16	−	1.11938e15i	−0.701094	−	0.0494036i
$218$		0			0
$219$	−2.56606e16			−1.06208
$220$		0			0
$221$	−1.90292e15			−0.0739053
$222$		0			0
$223$			1.20790e16i			0.440452i	0.975449	+	0.220226i	$0.0706795\pi$
$223$			1.20790e16i			0.440452i	−0.975449	+	0.220226i	$0.929320\pi$
$224$		0			0
$225$	1.30614e15			0.0447416
$226$		0			0
$227$			1.00605e16i			0.323921i	0.986797	+	0.161960i	$0.0517817\pi$
$227$			1.00605e16i			0.323921i	−0.986797	+	0.161960i	$0.948218\pi$
$228$		0			0
$229$			5.93067e16i			1.79579i	0.440211	+	0.897895i	$0.354904\pi$
$229$			5.93067e16i			1.79579i	−0.440211	+	0.897895i	$0.645096\pi$
$230$		0			0
$231$	−4.69061e16	−	3.30531e15i	−1.33643	−	0.0941734i
$232$		0			0
$233$	−6.15427e16			−1.65077			−0.825383	−	0.564574i	$-0.809041\pi$
$233$	−6.15427e16			−1.65077			−0.825383	+	0.564574i	$0.809041\pi$
$234$		0			0
$235$	8.76605e14			0.0221477
$236$		0			0
$237$			3.78100e16i			0.900260i
$238$		0			0
$239$	−8.66474e15			−0.194523			−0.0972614	−	0.995259i	$-0.531008\pi$
$239$	−8.66474e15			−0.194523			−0.0972614	+	0.995259i	$0.531008\pi$
$240$		0			0
$241$			4.58690e16i			0.971405i	0.874124	+	0.485703i	$0.161436\pi$
$241$			4.58690e16i			0.971405i	−0.874124	+	0.485703i	$0.838564\pi$
$242$		0			0
$243$			2.19927e16i			0.439576i
$244$		0			0
$245$	−3.32307e15	+	2.34620e16i	−0.0627158	+	0.442794i
$246$		0			0
$247$	−1.86822e16			−0.333081
$248$		0			0
$249$	−2.49020e16			−0.419603
$250$		0			0
$251$			5.87266e16i			0.935662i	0.883818	+	0.467831i	$0.154964\pi$
$251$			5.87266e16i			0.935662i	−0.883818	+	0.467831i	$0.845036\pi$
$252$		0			0
$253$	−1.21522e15			−0.0183151
$254$		0			0
$255$		−	2.20357e15i		−	0.0314300i
$256$		0			0
$257$		−	8.85464e16i		−	1.19575i	−0.801591	−	0.597873i	$-0.796013\pi$
$257$		−	8.85464e16i		−	1.19575i	0.801591	−	0.597873i	$-0.203987\pi$
$258$		0			0
$259$	2.33521e14	−	3.31393e15i	0.00298695	−	0.0423882i
$260$		0			0
$261$	−6.34339e15			−0.0768844
$262$		0			0
$263$	5.06234e16			0.581649			0.290824	−	0.956776i	$-0.406070\pi$
$263$	5.06234e16			0.581649			0.290824	+	0.956776i	$0.406070\pi$
$264$		0			0
$265$			2.42393e16i			0.264119i
$266$		0			0
$267$	−1.85055e17			−1.91303
$268$		0			0
$269$		−	1.42121e17i		−	1.39442i	−0.716868	−	0.697209i	$-0.754425\pi$
$269$		−	1.42121e17i		−	1.39442i	0.716868	−	0.697209i	$-0.245575\pi$
$270$		0			0
$271$			5.96752e16i			0.555915i	0.960593	+	0.277958i	$0.0896575\pi$
$271$			5.96752e16i			0.555915i	−0.960593	+	0.277958i	$0.910342\pi$
$272$		0			0
$273$	1.02228e16	−	1.45073e17i	0.0904546	−	1.28365i
$274$		0			0
$275$	−2.88098e16			−0.242221
$276$		0			0
$277$	−2.14495e17			−1.71419			−0.857097	−	0.515156i	$-0.827734\pi$
$277$	−2.14495e17			−1.71419			−0.857097	+	0.515156i	$0.827734\pi$
$278$		0			0
$279$			2.06901e16i			0.157229i
$280$		0			0
$281$	−6.67468e16			−0.482487			−0.241244	−	0.970465i	$-0.577555\pi$
$281$	−6.67468e16			−0.482487			−0.241244	+	0.970465i	$0.577555\pi$
$282$		0			0
$283$			1.62262e17i			1.11612i	0.829801	+	0.558060i	$0.188454\pi$
$283$			1.62262e17i			1.11612i	−0.829801	+	0.558060i	$0.811546\pi$
$284$		0			0
$285$		−	2.16338e16i		−	0.141651i
$286$		0			0
$287$	−1.61828e17	−	1.14035e16i	−1.00898	−	0.0710991i
$288$		0			0
$289$	1.67698e17			0.995964
$290$		0			0
$291$	−2.64915e17			−1.49919
$292$		0			0
$293$			1.24106e16i			0.0669453i	0.999440	+	0.0334726i	$0.0106567\pi$
$293$			1.24106e16i			0.0669453i	−0.999440	+	0.0334726i	$0.989343\pi$
$294$		0			0
$295$	−4.63723e16			−0.238509
$296$		0			0
$297$		−	2.12003e17i		−	1.04003i
$298$		0			0
$299$		−	3.75847e15i		−	0.0175919i
$300$		0			0
$301$	−5.24307e15	+	7.44050e16i	−0.0234218	+	0.332382i
$302$		0			0
$303$	4.88675e17			2.08412
$304$		0			0
$305$	−8.18648e16			−0.333425
$306$		0			0
$307$			2.79918e17i			1.08909i	0.838733	+	0.544543i	$0.183297\pi$
$307$			2.79918e17i			1.08909i	−0.838733	+	0.544543i	$0.816703\pi$
$308$		0			0
$309$	6.41667e16			0.238561
$310$		0			0
$311$			3.71475e16i			0.132010i	0.997819	+	0.0660049i	$0.0210253\pi$
$311$			3.71475e16i			0.132010i	−0.997819	+	0.0660049i	$0.978975\pi$
$312$		0			0
$313$		−	3.32477e17i		−	1.12967i	−0.825204	−	0.564835i	$-0.808940\pi$
$313$		−	3.32477e17i		−	1.12967i	0.825204	−	0.564835i	$-0.191060\pi$
$314$		0			0
$315$	3.07110e16	+	2.16410e15i	0.0997977	+	0.00703240i
$316$		0			0
$317$	−3.42965e17			−1.06619			−0.533095	−	0.846056i	$-0.678971\pi$
$317$	−3.42965e17			−1.06619			−0.533095	+	0.846056i	$0.678971\pi$
$318$		0			0
$319$	1.39918e17			0.416235
$320$		0			0
$321$		−	6.52474e17i		−	1.85792i
$322$		0			0
$323$	−6.67228e15			−0.0181910
$324$		0			0
$325$		−	8.91042e16i		−	0.232656i
$326$		0			0
$327$			3.80814e17i			0.952529i
$328$		0			0
$329$	2.06115e16	+	1.45242e15i	0.0494013	+	0.00348114i
$330$		0			0
$331$	−4.32418e16			−0.0993363			−0.0496681	−	0.998766i	$-0.515816\pi$
$331$	−4.32418e16			−0.0993363			−0.0496681	+	0.998766i	$0.515816\pi$
$332$		0			0
$333$	−4.31630e15			−0.00950609
$334$		0			0
$335$		−	1.58909e17i		−	0.335610i
$336$		0			0
$337$	1.72746e17			0.349944			0.174972	−	0.984573i	$-0.444016\pi$
$337$	1.72746e17			0.349944			0.174972	+	0.984573i	$0.444016\pi$
$338$		0			0
$339$			1.92592e17i			0.374317i
$340$		0			0
$341$		−	4.56367e17i		−	0.851204i
$342$		0			0
$343$	−1.17008e17	+	5.46152e17i	−0.209488	+	0.977811i
$344$		0			0
$345$	4.35227e15			0.00748138
$346$		0			0
$347$	−3.99541e17			−0.659560			−0.329780	−	0.944058i	$-0.606975\pi$
$347$	−3.99541e17			−0.659560			−0.329780	+	0.944058i	$0.606975\pi$
$348$		0			0
$349$		−	2.71055e17i		−	0.429812i	−0.976635	−	0.214906i	$-0.931055\pi$
$349$		−	2.71055e17i		−	0.429812i	0.976635	−	0.214906i	$-0.0689446\pi$
$350$		0			0
$351$	6.55690e17			0.998961
$352$		0			0
$353$		−	4.60483e15i		−	0.00674203i	−0.999994	−	0.00337101i	$-0.998927\pi$
$353$		−	4.60483e15i		−	0.00674203i	0.999994	−	0.00337101i	$-0.00107303\pi$
$354$		0			0
$355$		−	4.36516e15i		−	0.00614330i
$356$		0			0
$357$	3.65102e15	−	5.18121e16i	0.00494012	−	0.0701058i
$358$		0			0
$359$	9.38255e17			1.22084			0.610421	−	0.792077i	$-0.291000\pi$
$359$	9.38255e17			1.22084			0.610421	+	0.792077i	$0.291000\pi$
$360$		0			0
$361$	7.33501e17			0.918016
$362$		0			0
$363$		−	4.28838e17i		−	0.516354i
$364$		0			0
$365$	−3.70582e17			−0.429373
$366$		0			0
$367$		−	9.54260e15i		−	0.0106415i	−0.999986	−	0.00532077i	$-0.998306\pi$
$367$		−	9.54260e15i		−	0.0106415i	0.999986	−	0.00532077i	$-0.00169366\pi$
$368$		0			0
$369$			2.10776e17i			0.226276i
$370$		0			0
$371$	−4.01614e16	+	5.69936e17i	−0.0415137	+	0.589127i
$372$		0			0
$373$	5.49907e17			0.547429			0.273714	−	0.961811i	$-0.411748\pi$
$373$	5.49907e17			0.547429			0.273714	+	0.961811i	$0.411748\pi$
$374$		0			0
$375$	1.03182e17			0.0989427
$376$		0			0
$377$			4.32744e17i			0.399799i
$378$		0			0
$379$	1.41176e18			1.25686			0.628428	−	0.777868i	$-0.283699\pi$
$379$	1.41176e18			1.25686			0.628428	+	0.777868i	$0.283699\pi$
$380$		0			0
$381$			6.67396e17i			0.572676i
$382$		0			0
$383$			7.08650e17i			0.586192i	0.956083	+	0.293096i	$0.0946856\pi$
$383$			7.08650e17i			0.586192i	−0.956083	+	0.293096i	$0.905314\pi$
$384$		0			0
$385$	−6.77402e17	−	4.77342e16i	−0.540283	−	0.0380719i
$386$		0			0
$387$	9.69104e16			0.0745407
$388$		0			0
$389$	1.91394e18			1.41997			0.709987	−	0.704214i	$-0.248701\pi$
$389$	1.91394e18			1.41997			0.709987	+	0.704214i	$0.248701\pi$
$390$		0			0
$391$		−	1.34232e15i		−	0.000960769i
$392$		0			0
$393$	1.04082e18			0.718835
$394$		0			0
$395$			5.46039e17i			0.363952i
$396$		0			0
$397$		−	7.20475e17i		−	0.463538i	−0.972771	−	0.231769i	$-0.925549\pi$
$397$		−	7.20475e17i		−	0.463538i	0.972771	−	0.231769i	$-0.0744513\pi$
$398$		0			0
$399$	3.58445e16	−	5.08674e17i	0.0222645	−	0.315958i
$400$		0			0
$401$	1.67507e18			1.00467			0.502335	−	0.864673i	$-0.332474\pi$
$401$	1.67507e18			1.00467			0.502335	+	0.864673i	$0.332474\pi$
$402$		0			0
$403$	1.41147e18			0.817591
$404$		0			0
$405$			9.38088e17i			0.524878i
$406$		0			0
$407$	9.52058e16			0.0514639
$408$		0			0
$409$			7.98494e17i			0.417069i	0.978015	+	0.208535i	$0.0668694\pi$
$409$			7.98494e17i			0.417069i	−0.978015	+	0.208535i	$0.933131\pi$
$410$		0			0
$411$			1.77283e18i			0.894898i
$412$		0			0
$413$	−1.09035e18	−	7.68329e16i	−0.532003	−	0.0374884i
$414$		0			0
$415$	−3.59626e17			−0.169635
$416$		0			0
$417$	−1.31246e18			−0.598595
$418$		0			0
$419$			1.29089e18i			0.569364i	0.958622	+	0.284682i	$0.0918879\pi$
$419$			1.29089e18i			0.569364i	−0.958622	+	0.284682i	$0.908112\pi$
$420$		0			0
$421$	3.39935e18			1.45018			0.725089	−	0.688655i	$-0.241798\pi$
$421$	3.39935e18			1.45018			0.725089	+	0.688655i	$0.241798\pi$
$422$		0			0
$423$		−	2.68459e16i		−	0.0110789i
$424$		0			0
$425$		−	3.18232e16i		−	0.0127063i
$426$		0			0
$427$	−1.92488e18	−	1.35640e17i	−0.743717	−	0.0524072i
$428$		0			0
$429$	4.16779e18			1.55849
$430$		0			0
$431$	8.00962e17			0.289915			0.144958	−	0.989438i	$-0.453695\pi$
$431$	8.00962e17			0.289915			0.144958	+	0.989438i	$0.453695\pi$
$432$		0			0
$433$		−	1.47752e18i		−	0.517746i	−0.965911	−	0.258873i	$-0.916649\pi$
$433$		−	1.47752e18i		−	0.517746i	0.965911	−	0.258873i	$-0.0833511\pi$
$434$		0			0
$435$	−5.01113e17			−0.170024
$436$		0			0
$437$		−	1.31784e16i		−	0.00433006i
$438$		0			0
$439$		−	3.63345e18i		−	1.15629i	−0.815934	−	0.578145i	$-0.803777\pi$
$439$		−	3.63345e18i		−	1.15629i	0.815934	−	0.578145i	$-0.196223\pi$
$440$		0			0
$441$	7.18519e17	+	1.01768e17i	0.221497	+	0.0313721i
$442$		0			0
$443$	2.14654e18			0.641082			0.320541	−	0.947235i	$-0.396135\pi$
$443$	2.14654e18			0.641082			0.320541	+	0.947235i	$0.396135\pi$
$444$		0			0
$445$	−2.67251e18			−0.773390
$446$		0			0
$447$		−	4.61188e18i		−	1.29338i
$448$		0			0
$449$	6.76471e18			1.83876			0.919379	−	0.393372i	$-0.128692\pi$
$449$	6.76471e18			1.83876			0.919379	+	0.393372i	$0.128692\pi$
$450$		0			0
$451$		−	4.64916e18i		−	1.22501i
$452$		0			0
$453$		−	1.49571e18i		−	0.382085i
$454$		0			0
$455$	1.47634e17	−	2.09510e18i	0.0365685	−	0.518948i
$456$		0			0
$457$	−5.23603e18			−1.25773			−0.628867	−	0.777513i	$-0.716481\pi$
$457$	−5.23603e18			−1.25773			−0.628867	+	0.777513i	$0.716481\pi$
$458$		0			0
$459$	2.34177e17			0.0545575
$460$		0			0
$461$			7.51535e17i			0.169841i	0.996388	+	0.0849206i	$0.0270637\pi$
$461$			7.51535e17i			0.169841i	−0.996388	+	0.0849206i	$0.972936\pi$
$462$		0			0
$463$	4.59306e18			1.00701			0.503506	−	0.863992i	$-0.332043\pi$
$463$	4.59306e18			1.00701			0.503506	+	0.863992i	$0.332043\pi$
$464$		0			0
$465$			1.63447e18i			0.347701i
$466$		0			0
$467$			2.84675e18i			0.587668i	0.955857	+	0.293834i	$0.0949312\pi$
$467$			2.84675e18i			0.587668i	−0.955857	+	0.293834i	$0.905069\pi$
$468$		0			0
$469$	2.63291e17	−	3.73640e18i	0.0527505	−	0.748589i
$470$		0			0
$471$	3.13708e18			0.610069
$472$		0			0
$473$	−2.13758e18			−0.403547
$474$		0			0
$475$		−	3.12429e17i		−	0.0572658i
$476$		0			0
$477$	7.42325e17			0.132119
$478$		0			0
$479$		−	7.76846e18i		−	1.34272i	−0.741131	−	0.671361i	$-0.765710\pi$
$479$		−	7.76846e18i		−	1.34272i	0.741131	−	0.671361i	$-0.234290\pi$
$480$		0			0
$481$			2.94456e17i			0.0494317i
$482$		0			0
$483$	1.02334e17	+	7.21115e15i	0.0166875	+	0.00117591i
$484$		0			0
$485$	−3.82582e18			−0.606083
$486$		0			0
$487$	−7.08599e18			−1.09068			−0.545341	−	0.838215i	$-0.683600\pi$
$487$	−7.08599e18			−1.09068			−0.545341	+	0.838215i	$0.683600\pi$
$488$		0			0
$489$			1.30097e19i			1.94583i
$490$		0			0
$491$	−2.37273e18			−0.344888			−0.172444	−	0.985019i	$-0.555166\pi$
$491$	−2.37273e18			−0.344888			−0.172444	+	0.985019i	$0.555166\pi$
$492$		0			0
$493$			1.54552e17i			0.0218347i
$494$		0			0
$495$			8.82296e17i			0.121165i
$496$		0			0
$497$	7.23250e15	−	1.02637e17i	0.000965593	−	0.0137028i
$498$		0			0
$499$	3.52671e18			0.457789			0.228895	−	0.973451i	$-0.426489\pi$
$499$	3.52671e18			0.457789			0.228895	+	0.973451i	$0.426489\pi$
$500$		0			0
$501$	−7.85002e18			−0.990847
$502$		0			0
$503$		−	1.35049e16i		−	0.00165774i	−1.00000		0.000828869i	$-0.999736\pi$
$503$		−	1.35049e16i		−	0.00165774i	1.00000		0.000828869i	$-0.000263837\pi$
$504$		0			0
$505$	7.05729e18			0.842555
$506$		0			0
$507$			3.36467e18i			0.390739i
$508$		0			0
$509$			2.41337e18i			0.272646i	0.990664	+	0.136323i	$0.0435285\pi$
$509$			2.41337e18i			0.272646i	−0.990664	+	0.136323i	$0.956472\pi$
$510$		0			0
$511$	−8.71345e18	−	6.14007e17i	−0.957732	−	0.0674881i
$512$		0			0
$513$	2.29907e18			0.245884
$514$		0			0
$515$	9.26674e17			0.0964443
$516$		0			0
$517$			5.92147e17i			0.0599785i
$518$		0			0
$519$	−6.05629e18			−0.597084
$520$		0			0
$521$			4.44690e18i			0.426770i	0.976968	+	0.213385i	$0.0684488\pi$
$521$			4.44690e18i			0.426770i	−0.976968	+	0.213385i	$0.931551\pi$
$522$		0			0
$523$			9.98277e17i			0.0932694i	0.998912	+	0.0466347i	$0.0148497\pi$
$523$			9.98277e17i			0.0932694i	−0.998912	+	0.0466347i	$0.985150\pi$
$524$		0			0
$525$	2.42610e18	+	1.70959e17i	0.220695	+	0.0155516i
$526$		0			0
$527$	5.04100e17			0.0446521
$528$		0			0
$529$	−1.15902e19			−0.999771
$530$		0			0
$531$			1.42014e18i			0.119308i
$532$		0			0
$533$	1.43791e19			1.17663
$534$		0			0
$535$		−	9.42282e18i		−	0.751111i
$536$		0			0
$537$			1.84075e18i			0.142947i
$538$		0			0
$539$	−1.58486e19	−	2.24474e18i	−1.19914	−	0.169841i
$540$		0			0
$541$	−7.68533e18			−0.566607			−0.283303	−	0.959030i	$-0.591430\pi$
$541$	−7.68533e18			−0.566607			−0.283303	+	0.959030i	$0.591430\pi$
$542$		0			0
$543$	−1.94763e19			−1.39929
$544$		0			0
$545$			5.49958e18i			0.385083i
$546$		0			0
$547$	1.38466e19			0.945000			0.472500	−	0.881331i	$-0.343352\pi$
$547$	1.38466e19			0.945000			0.472500	+	0.881331i	$0.343352\pi$
$548$		0			0
$549$			2.50710e18i			0.166788i
$550$		0			0
$551$			1.51734e18i			0.0984062i
$552$		0			0
$553$	−9.04716e17	+	1.28389e19i	−0.0572053	+	0.811808i
$554$		0			0
$555$	−3.40977e17			−0.0210220
$556$		0			0
$557$	−1.99069e19			−1.19679			−0.598394	−	0.801202i	$-0.704194\pi$
$557$	−1.99069e19			−1.19679			−0.598394	+	0.801202i	$0.704194\pi$
$558$		0			0
$559$		−	6.61119e18i		−	0.387612i
$560$		0			0
$561$	1.48851e18			0.0851161
$562$		0			0
$563$			1.98876e19i			1.10924i	0.832105	+	0.554618i	$0.187135\pi$
$563$			1.98876e19i			1.10924i	−0.832105	+	0.554618i	$0.812865\pi$
$564$		0			0
$565$			2.78134e18i			0.151327i
$566$		0			0
$567$	−1.55429e18	+	2.20571e19i	−0.0824994	+	1.17076i
$568$		0			0
$569$	−5.64882e18			−0.292531			−0.146265	−	0.989245i	$-0.546725\pi$
$569$	−5.64882e18			−0.292531			−0.146265	+	0.989245i	$0.546725\pi$
$570$		0			0
$571$	−1.96810e19			−0.994478			−0.497239	−	0.867614i	$-0.665653\pi$
$571$	−1.96810e19			−0.994478			−0.497239	+	0.867614i	$0.665653\pi$
$572$		0			0
$573$		−	1.51063e19i		−	0.744862i
$574$		0			0
$575$	6.28540e16			0.00302453
$576$		0			0
$577$			7.92437e18i			0.372164i	0.982534	+	0.186082i	$0.0595789\pi$
$577$			7.92437e18i			0.372164i	−0.982534	+	0.186082i	$0.940421\pi$
$578$		0			0
$579$			2.68402e19i			1.23037i
$580$		0			0
$581$	−8.45585e18	−	5.95855e17i	−0.378377	−	0.0266629i
$582$		0			0
$583$	−1.63737e19			−0.715264
$584$		0			0
$585$	−2.72880e18			−0.116381
$586$		0			0
$587$		−	1.69358e19i		−	0.705242i	−0.935766	−	0.352621i	$-0.885291\pi$
$587$		−	1.69358e19i		−	0.705242i	0.935766	−	0.352621i	$-0.114709\pi$
$588$		0			0
$589$	4.94908e18			0.201241
$590$		0			0
$591$			7.27255e18i			0.288785i
$592$		0			0
$593$		−	3.03132e19i		−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$593$		−	3.03132e19i		−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$594$		0			0
$595$	5.27269e16	−	7.48254e17i	0.00199716	−	0.0283420i
$596$		0			0
$597$	−3.72275e19			−1.37735
$598$		0			0
$599$	−2.88946e19			−1.04431			−0.522155	−	0.852850i	$-0.674872\pi$
$599$	−2.88946e19			−1.04431			−0.522155	+	0.852850i	$0.674872\pi$
$600$		0			0
$601$			2.78888e18i			0.0984709i	0.998787	+	0.0492355i	$0.0156785\pi$
$601$			2.78888e18i			0.0984709i	−0.998787	+	0.0492355i	$0.984322\pi$
$602$		0			0
$603$	−4.86655e18			−0.167881
$604$		0			0
$605$		−	6.19314e18i		−	0.208749i
$606$		0			0
$607$		−	3.72198e19i		−	1.22589i	−0.790124	−	0.612947i	$-0.789984\pi$
$607$		−	3.72198e19i		−	1.22589i	0.790124	−	0.612947i	$-0.210016\pi$
$608$		0			0
$609$	−1.17826e19	−	8.30279e17i	−0.379245	−	0.0267241i
$610$		0			0
$611$	−1.83141e18			−0.0576100
$612$		0			0
$613$	1.95835e19			0.602097			0.301049	−	0.953609i	$-0.402663\pi$
$613$	1.95835e19			0.602097			0.301049	+	0.953609i	$0.402663\pi$
$614$		0			0
$615$			1.66508e19i			0.500392i
$616$		0			0
$617$	5.79679e19			1.70291			0.851454	−	0.524429i	$-0.175721\pi$
$617$	5.79679e19			1.70291			0.851454	+	0.524429i	$0.175721\pi$
$618$		0			0
$619$		−	5.14273e19i		−	1.47693i	−0.674293	−	0.738464i	$-0.735552\pi$
$619$		−	5.14273e19i		−	1.47693i	0.674293	−	0.738464i	$-0.264448\pi$
$620$		0			0
$621$			4.62523e17i			0.0129865i
$622$		0			0
$623$	−6.28383e19	−	4.42800e18i	−1.72508	−	0.121560i
$624$		0			0
$625$	1.49012e18			0.0400000
$626$		0			0
$627$	1.46137e19			0.383607
$628$		0			0
$629$			1.05164e17i			0.00269968i
$630$		0			0
$631$	2.47831e19			0.622228			0.311114	−	0.950373i	$-0.399298\pi$
$631$	2.47831e19			0.622228			0.311114	+	0.950373i	$0.399298\pi$
$632$		0			0
$633$			4.59966e19i			1.12953i
$634$		0			0
$635$			9.63832e18i			0.231518i
$636$		0			0
$637$	6.94261e18	−	4.90171e19i	0.163135	−	1.15179i
$638$		0			0
$639$	−1.33682e17			−0.00307304
$640$		0			0
$641$	−1.27115e19			−0.285885			−0.142942	−	0.989731i	$-0.545656\pi$
$641$	−1.27115e19			−0.285885			−0.142942	+	0.989731i	$0.545656\pi$
$642$		0			0
$643$		−	6.82171e19i		−	1.50112i	−0.660800	−	0.750562i	$-0.729783\pi$
$643$		−	6.82171e19i		−	1.50112i	0.660800	−	0.750562i	$-0.270217\pi$
$644$		0			0
$645$	7.65569e18			0.164841
$646$		0			0
$647$		−	5.33042e19i		−	1.12313i	−0.827431	−	0.561567i	$-0.810199\pi$
$647$		−	5.33042e19i		−	1.12313i	0.827431	−	0.561567i	$-0.189801\pi$
$648$		0			0
$649$		−	3.13245e19i		−	0.645909i
$650$		0			0
$651$	−2.70810e18	+	3.84310e19i	−0.0546510	+	0.775559i
$652$		0			0
$653$	5.43150e18			0.107282			0.0536410	−	0.998560i	$-0.482917\pi$
$653$	5.43150e18			0.107282			0.0536410	+	0.998560i	$0.482917\pi$
$654$		0			0
$655$	1.50312e19			0.290606
$656$		0			0
$657$			1.13490e19i			0.214783i
$658$		0			0
$659$	−4.43291e19			−0.821279			−0.410640	−	0.911798i	$-0.634695\pi$
$659$	−4.43291e19			−0.821279			−0.410640	+	0.911798i	$0.634695\pi$
$660$		0			0
$661$			7.33935e19i			1.33121i	0.746303	+	0.665606i	$0.231827\pi$
$661$			7.33935e19i			1.33121i	−0.746303	+	0.665606i	$0.768173\pi$
$662$		0			0
$663$			4.60372e18i			0.0817550i
$664$		0			0
$665$	5.17654e17	−	7.34610e18i	0.00900095	−	0.127734i
$666$		0			0
$667$	−3.05257e17			−0.00519739
$668$		0			0
$669$	2.92227e19			0.487234
$670$		0			0
$671$		−	5.52997e19i		−	0.902954i
$672$		0			0
$673$	−3.83519e19			−0.613313			−0.306656	−	0.951820i	$-0.599210\pi$
$673$	−3.83519e19			−0.613313			−0.306656	+	0.951820i	$0.599210\pi$
$674$		0			0
$675$			1.09653e19i			0.171749i
$676$		0			0
$677$		−	6.63585e18i		−	0.101807i	−0.998704	−	0.0509033i	$-0.983790\pi$
$677$		−	6.63585e18i		−	0.101807i	0.998704	−	0.0509033i	$-0.0162100\pi$
$678$		0			0
$679$	−8.99559e19	−	6.33888e18i	−1.35189	−	0.0952630i
$680$		0			0
$681$	2.43393e19			0.358325
$682$		0			0
$683$	−8.82825e19			−1.27330			−0.636649	−	0.771154i	$-0.719680\pi$
$683$	−8.82825e19			−1.27330			−0.636649	+	0.771154i	$0.719680\pi$
$684$		0			0
$685$			2.56026e19i			0.361784i
$686$		0			0
$687$	1.43480e20			1.98653
$688$		0			0
$689$		−	5.06411e19i		−	0.687019i
$690$		0			0
$691$			2.43186e19i			0.323290i	0.986849	+	0.161645i	$0.0516799\pi$
$691$			2.43186e19i			0.323290i	−0.986849	+	0.161645i	$0.948320\pi$
$692$		0			0
$693$	−1.46185e18	+	2.07453e19i	−0.0190445	+	0.270264i
$694$		0			0
$695$	−1.89541e19			−0.241996
$696$		0			0
$697$	5.13543e18			0.0642610
$698$		0			0
$699$			1.48890e20i			1.82610i
$700$		0			0
$701$	1.40697e20			1.69144			0.845721	−	0.533625i	$-0.179171\pi$
$701$	1.40697e20			1.69144			0.845721	+	0.533625i	$0.179171\pi$
$702$		0			0
$703$			1.03246e18i			0.0121671i
$704$		0			0
$705$		−	2.12076e18i		−	0.0245001i
$706$		0			0
$707$	1.65937e20	+	1.16930e19i	1.87935	+	0.132431i
$708$		0			0
$709$	1.26449e20			1.40408			0.702042	−	0.712136i	$-0.252272\pi$
$709$	1.26449e20			1.40408			0.702042	+	0.712136i	$0.252272\pi$
$710$		0			0
$711$	1.67224e19			0.182058
$712$		0			0
$713$			9.95649e17i			0.0106287i
$714$		0			0
$715$	6.01899e19			0.630059
$716$		0			0
$717$			2.09625e19i			0.215184i
$718$		0			0
$719$			1.21775e20i			1.22590i	0.790122	+	0.612949i	$0.210017\pi$
$719$			1.21775e20i			1.22590i	−0.790122	+	0.612949i	$0.789983\pi$
$720$		0			0
$721$	2.17888e19	+	1.53538e18i	0.215122	+	0.0151589i
$722$		0			0
$723$	1.10970e20			1.07458
$724$		0			0
$725$	−7.23690e18			−0.0687364
$726$		0			0
$727$			1.13343e20i			1.05598i	0.849251	+	0.527989i	$0.177054\pi$
$727$			1.13343e20i			1.05598i	−0.849251	+	0.527989i	$0.822946\pi$
$728$		0			0
$729$	−7.52143e19			−0.687397
$730$		0			0
$731$		−	2.36116e18i		−	0.0211691i
$732$		0			0
$733$			1.56583e20i			1.37726i	0.725112	+	0.688631i	$0.241788\pi$
$733$			1.56583e20i			1.37726i	−0.725112	+	0.688631i	$0.758212\pi$
$734$		0			0
$735$	5.67613e19	+	8.03946e18i	0.489825	+	0.0693770i
$736$		0			0
$737$	1.07343e20			0.908869
$738$		0			0
$739$	−1.53428e20			−1.27465			−0.637327	−	0.770594i	$-0.719960\pi$
$739$	−1.53428e20			−1.27465			−0.637327	+	0.770594i	$0.719960\pi$
$740$		0			0
$741$			4.51977e19i			0.368459i
$742$		0			0
$743$	−1.01652e20			−0.813198			−0.406599	−	0.913607i	$-0.633285\pi$
$743$	−1.01652e20			−0.813198			−0.406599	+	0.913607i	$0.633285\pi$
$744$		0			0
$745$		−	6.66032e19i		−	0.522879i
$746$		0			0
$747$			1.10135e19i			0.0848557i
$748$		0			0
$749$	1.56124e19	−	2.21557e20i	0.118058	−	1.67538i
$750$		0			0
$751$	2.57594e19			0.191186			0.0955930	−	0.995420i	$-0.469525\pi$
$751$	2.57594e19			0.191186			0.0955930	+	0.995420i	$0.469525\pi$
$752$		0			0
$753$	1.42077e20			1.03504
$754$		0			0
$755$		−	2.16005e19i		−	0.154467i
$756$		0			0
$757$	1.78342e20			1.25194			0.625968	−	0.779849i	$-0.284704\pi$
$757$	1.78342e20			1.25194			0.625968	+	0.779849i	$0.284704\pi$
$758$		0			0
$759$			2.93996e18i			0.0202604i
$760$		0			0
$761$		−	8.37612e19i		−	0.566696i	−0.959017	−	0.283348i	$-0.908555\pi$
$761$		−	8.37612e19i		−	0.566696i	0.959017	−	0.283348i	$-0.0914451\pi$
$762$		0			0
$763$	−9.11210e18	+	1.29311e20i	−0.0605267	+	0.858942i
$764$		0			0
$765$	−9.74579e17			−0.00635604
$766$		0			0
$767$	9.68817e19			0.620403
$768$		0			0
$769$		−	1.37237e18i		−	0.00862953i	−0.999991	−	0.00431477i	$-0.998627\pi$
$769$		−	1.37237e18i		−	0.00862953i	0.999991	−	0.00431477i	$-0.00137344\pi$
$770$		0			0
$771$	−2.14219e20			−1.32275
$772$		0			0
$773$		−	2.45143e20i		−	1.48649i	−0.669019	−	0.743245i	$-0.733285\pi$
$773$		−	2.45143e20i		−	1.48649i	0.669019	−	0.743245i	$-0.266715\pi$
$774$		0			0
$775$			2.36044e19i			0.140566i
$776$		0			0
$777$	−8.01736e18	−	5.64956e17i	−0.0468904	−	0.00330421i
$778$		0			0
$779$	5.04178e19			0.289616
$780$		0			0
$781$	2.94867e18			0.0166367
$782$		0			0
$783$		−	5.32541e19i		−	0.295135i
$784$		0			0
$785$	4.53046e19			0.246635
$786$		0			0
$787$		−	1.73196e20i		−	0.926220i	−0.886301	−	0.463110i	$-0.846734\pi$
$787$		−	1.73196e20i		−	0.926220i	0.886301	−	0.463110i	$-0.153266\pi$
$788$		0			0
$789$		−	1.22473e20i		−	0.643428i
$790$		0			0
$791$	−4.60833e18	+	6.53974e19i	−0.0237853	+	0.337540i
$792$		0			0
$793$	1.71033e20			0.867298
$794$		0			0
$795$	5.86419e19			0.292172
$796$		0			0
$797$		−	3.91880e20i		−	1.91842i	−0.282688	−	0.959212i	$-0.591226\pi$
$797$		−	3.91880e20i		−	1.91842i	0.282688	−	0.959212i	$-0.408774\pi$
$798$		0			0
$799$	−6.54082e17			−0.00314633
$800$		0			0
$801$			8.18451e19i			0.386870i
$802$		0			0
$803$		−	2.50328e20i		−	1.16279i
$804$		0			0
$805$	1.47788e18	+	1.04141e17i	0.00674633	+	0.000475391i
$806$		0			0
$807$	−3.43832e20			−1.54252
$808$		0			0
$809$	3.84501e20			1.69534			0.847672	−	0.530520i	$-0.178003\pi$
$809$	3.84501e20			1.69534			0.847672	+	0.530520i	$0.178003\pi$
$810$		0			0
$811$			3.64414e19i			0.157924i	0.996878	+	0.0789621i	$0.0251606\pi$
$811$			3.64414e19i			0.157924i	−0.996878	+	0.0789621i	$0.974839\pi$
$812$		0			0
$813$	1.44371e20			0.614961
$814$		0			0
$815$			1.87881e20i			0.786647i
$816$		0			0
$817$		−	2.31810e19i		−	0.0954066i
$818$		0			0
$819$	−6.41619e19	−	4.52127e18i	−0.259591	−	0.0182925i
$820$		0			0
$821$	4.22129e20			1.67897			0.839486	−	0.543382i	$-0.182856\pi$
$821$	4.22129e20			1.67897			0.839486	+	0.543382i	$0.182856\pi$
$822$		0			0
$823$	3.36637e20			1.31632			0.658162	−	0.752876i	$-0.271334\pi$
$823$	3.36637e20			1.31632			0.658162	+	0.752876i	$0.271334\pi$
$824$		0			0
$825$			6.96993e19i			0.267948i
$826$		0			0
$827$	−3.96032e20			−1.49690			−0.748449	−	0.663192i	$-0.769201\pi$
$827$	−3.96032e20			−1.49690			−0.748449	+	0.663192i	$0.769201\pi$
$828$		0			0
$829$		−	1.17027e20i		−	0.434914i	−0.976070	−	0.217457i	$-0.930224\pi$
$829$		−	1.17027e20i		−	0.434914i	0.976070	−	0.217457i	$-0.0697762\pi$
$830$		0			0
$831$			5.18926e20i			1.89626i
$832$		0			0
$833$	2.47952e18	−	1.75062e19i	0.00890949	−	0.0629039i
$834$		0			0
$835$	−1.13367e20			−0.400574
$836$		0			0
$837$	−1.73698e20			−0.603553
$838$		0			0
$839$			1.08663e20i			0.371321i	0.982614	+	0.185661i	$0.0594425\pi$
$839$			1.08663e20i			0.371321i	−0.982614	+	0.185661i	$0.940557\pi$
$840$		0			0
$841$	−2.62411e20			−0.881883
$842$		0			0
$843$			1.61480e20i			0.533734i
$844$		0			0
$845$			4.85914e19i			0.157966i
$846$		0			0
$847$	1.02612e19	−	1.45619e20i	0.0328107	−	0.465621i
$848$		0			0
$849$	3.92558e20			1.23467
$850$		0			0
$851$	−2.07709e17			−0.000642612
$852$		0			0
$853$		−	1.05948e20i		−	0.322440i	−0.986919	−	0.161220i	$-0.948457\pi$
$853$		−	1.05948e20i		−	0.322440i	0.986919	−	0.161220i	$-0.0515428\pi$
$854$		0			0
$855$	−9.56808e18			−0.0286459
$856$		0			0
$857$		−	7.04947e19i		−	0.207630i	−0.994597	−	0.103815i	$-0.966895\pi$
$857$		−	7.04947e19i		−	0.207630i	0.994597	−	0.103815i	$-0.0331050\pi$
$858$		0			0
$859$		−	5.98830e20i		−	1.73521i	−0.497257	−	0.867604i	$-0.665659\pi$
$859$		−	5.98830e20i		−	1.73521i	0.497257	−	0.867604i	$-0.334341\pi$
$860$		0			0
$861$	−2.75883e19	+	3.91509e20i	−0.0786507	+	1.11614i
$862$		0			0
$863$	−4.36360e20			−1.22397			−0.611983	−	0.790871i	$-0.709628\pi$
$863$	−4.36360e20			−1.22397			−0.611983	+	0.790871i	$0.709628\pi$
$864$		0			0
$865$	−8.74630e19			−0.241386
$866$		0			0
$867$		−	4.05710e20i		−	1.10175i
$868$		0			0
$869$	−3.68850e20			−0.985623
$870$		0			0
$871$			3.31995e20i			0.872979i
$872$		0			0
$873$			1.17165e20i			0.303178i
$874$		0			0
$875$	3.50369e19	+	2.46893e18i	0.0892215	+	0.00628713i
$876$		0			0
$877$	−4.25346e20			−1.06597			−0.532985	−	0.846124i	$-0.678930\pi$
$877$	−4.25346e20			−1.06597			−0.532985	+	0.846124i	$0.678930\pi$
$878$		0			0
$879$	3.00249e19			0.0740558
$880$		0			0
$881$			2.21139e20i			0.536826i	0.963304	+	0.268413i	$0.0864992\pi$
$881$			2.21139e20i			0.536826i	−0.963304	+	0.268413i	$0.913501\pi$
$882$		0			0
$883$	1.07875e20			0.257747			0.128874	−	0.991661i	$-0.458864\pi$
$883$	1.07875e20			0.257747			0.128874	+	0.991661i	$0.458864\pi$
$884$		0			0
$885$			1.12188e20i			0.263842i
$886$		0			0
$887$		−	1.63863e20i		−	0.379328i	−0.981849	−	0.189664i	$-0.939260\pi$
$887$		−	1.63863e20i		−	0.379328i	0.981849	−	0.189664i	$-0.0607399\pi$
$888$		0			0
$889$	−1.59694e19	+	2.26624e20i	−0.0363896	+	0.516410i
$890$		0			0
$891$	−6.33679e20			−1.42143
$892$		0			0
$893$	−6.42155e18			−0.0141801
$894$		0			0
$895$			2.65835e19i			0.0577897i
$896$		0			0
$897$	−9.09282e18			−0.0194604
$898$		0			0
$899$		−	1.14637e20i		−	0.241551i
$900$		0			0
$901$		−	1.80863e19i		−	0.0375211i
$902$		0			0
$903$	1.80007e20	+	1.26845e19i	0.367685	+	0.0259095i
$904$		0			0
$905$	−2.81270e20			−0.565696
$906$		0			0
$907$	−4.12066e20			−0.816049			−0.408024	−	0.912971i	$-0.633782\pi$
$907$	−4.12066e20			−0.816049			−0.408024	+	0.912971i	$0.633782\pi$
$908$		0			0
$909$		−	2.16128e20i		−	0.421468i
$910$		0			0
$911$	−4.66571e20			−0.895960			−0.447980	−	0.894043i	$-0.647857\pi$
$911$	−4.66571e20			−0.895960			−0.447980	+	0.894043i	$0.647857\pi$
$912$		0			0
$913$		−	2.42928e20i		−	0.459390i
$914$		0			0
$915$			1.98055e20i			0.368840i
$916$		0			0
$917$	3.53427e20	+	2.49048e19i	0.648208	+	0.0456770i
$918$		0			0
$919$	6.63727e19			0.119889			0.0599447	−	0.998202i	$-0.480908\pi$
$919$	6.63727e19			0.119889			0.0599447	+	0.998202i	$0.480908\pi$
$920$		0			0
$921$	6.77202e20			1.20476
$922$		0			0
$923$			9.11975e18i			0.0159798i
$924$		0			0
$925$	−4.92428e18			−0.00849867
$926$		0			0
$927$		−	2.83792e19i		−	0.0482439i
$928$		0			0
$929$			7.10983e20i			1.19055i	0.803520	+	0.595277i	$0.202958\pi$
$929$			7.10983e20i			1.19055i	−0.803520	+	0.595277i	$0.797042\pi$
$930$		0			0
$931$	2.43431e19	−	1.71870e20i	0.0401539	−	0.283500i
$932$		0			0
$933$	8.98704e19			0.146031
$934$		0			0
$935$	2.14966e19			0.0344102
$936$		0			0
$937$			1.58607e20i			0.250118i	0.992149	+	0.125059i	$0.0399120\pi$
$937$			1.58607e20i			0.250118i	−0.992149	+	0.125059i	$0.960088\pi$
$938$		0			0
$939$	−8.04358e20			−1.24966
$940$		0			0
$941$			1.64286e20i			0.251462i	0.992064	+	0.125731i	$0.0401276\pi$
$941$			1.64286e20i			0.251462i	−0.992064	+	0.125731i	$0.959872\pi$
$942$		0			0
$943$			1.01430e19i			0.0152962i
$944$		0			0
$945$	−1.81681e19	+	2.57826e20i	−0.0269952	+	0.383092i
$946$		0			0
$947$	−1.14993e21			−1.68353			−0.841767	−	0.539842i	$-0.818484\pi$
$947$	−1.14993e21			−1.68353			−0.841767	+	0.539842i	$0.818484\pi$
$948$		0			0
$949$	7.74226e20			1.11687
$950$		0			0
$951$			8.29730e20i			1.17943i
$952$		0			0
$953$	−1.08539e21			−1.52032			−0.760160	−	0.649736i	$-0.774879\pi$
$953$	−1.08539e21			−1.52032			−0.760160	+	0.649736i	$0.774879\pi$
$954$		0			0
$955$		−	2.18160e20i		−	0.301129i
$956$		0			0
$957$		−	3.38502e20i		−	0.460445i
$958$		0			0
$959$	−4.24201e19	+	6.01990e20i	−0.0568646	+	0.806973i
$960$		0			0
$961$	3.83034e20			0.506027
$962$		0			0
$963$	−2.88572e20			−0.375725
$964$		0			0
$965$			3.87617e20i			0.497407i
$966$		0			0
$967$	−1.22257e21			−1.54628			−0.773139	−	0.634236i	$-0.781315\pi$
$967$	−1.22257e21			−1.54628			−0.773139	+	0.634236i	$0.781315\pi$
$968$		0			0
$969$			1.61422e19i			0.0201231i
$970$		0			0
$971$			7.91110e20i			0.972081i	0.873936	+	0.486040i	$0.161559\pi$
$971$			7.91110e20i			0.972081i	−0.873936	+	0.486040i	$0.838441\pi$
$972$		0			0
$973$	−4.45665e20	−	3.14045e19i	−0.539782	−	0.0380366i
$974$		0			0
$975$	−2.15569e20			−0.257367
$976$		0			0
$977$	2.12376e19			0.0249945			0.0124972	−	0.999922i	$-0.496022\pi$
$977$	2.12376e19			0.0249945			0.0124972	+	0.999922i	$0.496022\pi$
$978$		0			0
$979$		−	1.80528e21i		−	2.09443i
$980$		0			0
$981$	1.68424e20			0.192628
$982$		0			0
$983$			3.25456e20i			0.366960i	0.983023	+	0.183480i	$0.0587362\pi$
$983$			3.25456e20i			0.366960i	−0.983023	+	0.183480i	$0.941264\pi$
$984$		0			0
$985$			1.05028e20i			0.116748i
$986$		0			0
$987$	3.51382e18	−	4.98651e19i	0.00385088	−	0.0546483i
$988$		0			0
$989$	4.66353e18			0.00503896
$990$		0			0
$991$	−1.05652e21			−1.12555			−0.562773	−	0.826611i	$-0.690266\pi$
$991$	−1.05652e21			−1.12555			−0.562773	+	0.826611i	$0.690266\pi$
$992$		0			0
$993$			1.04614e20i			0.109887i
$994$		0			0
$995$	−5.37628e20			−0.556827
$996$		0			0
$997$		−	1.09238e21i		−	1.11560i	−0.829976	−	0.557799i	$-0.811646\pi$
$997$		−	1.09238e21i		−	1.11560i	0.829976	−	0.557799i	$-0.188354\pi$
$998$		0			0
$999$		−	3.62363e19i		−	0.0364909i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.15.d.a.41.9	✓	36
7.6	odd	2	inner	140.15.d.a.41.28	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.15.d.a.41.9	✓	36	1.1	even	1	trivial
140.15.d.a.41.28	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)

\(f(q)\)	\(=\)	\(q-2419.29i q^{3} -34938.6i q^{5} +(57888.7 - 821506. i) q^{7} -1.06999e6 q^{9} +O(q^{10})\)	\(q-2419.29i q^{3} -34938.6i q^{5} +(57888.7 - 821506. i) q^{7} -1.06999e6 q^{9} +2.36010e7 q^{11} +7.29941e7i q^{13} -8.45265e7 q^{15} +2.60695e7i q^{17} +2.55942e8i q^{19} +(-1.98746e9 - 1.40049e8i) q^{21} -5.14900e7 q^{23} -1.22070e9 q^{25} -8.98277e9i q^{27} +5.92847e9 q^{29} -1.93367e10i q^{31} -5.70977e10i q^{33} +(-2.87022e10 - 2.02255e9i) q^{35} +4.03397e9 q^{37} +1.76594e11 q^{39} -1.96990e11i q^{41} -9.05715e10 q^{43} +3.73838e10i q^{45} +2.50899e10i q^{47} +(-6.71521e11 - 9.51118e10i) q^{49} +6.30697e10 q^{51} -6.93770e11 q^{53} -8.24586e11i q^{55} +6.19197e11 q^{57} -1.32725e12i q^{59} -2.34311e12i q^{61} +(-6.19402e10 + 8.79001e11i) q^{63} +2.55031e12 q^{65} +4.54824e12 q^{67} +1.24569e11i q^{69} +1.24938e11 q^{71} -1.06067e13i q^{73} +2.95323e12i q^{75} +(1.36623e12 - 1.93884e13i) q^{77} -1.56285e13 q^{79} -2.68496e13 q^{81} -1.02931e13i q^{83} +9.10832e11 q^{85} -1.43427e13i q^{87} -7.64916e13i q^{89} +(5.99651e13 + 4.22554e12i) q^{91} -4.67812e13 q^{93} +8.94223e12 q^{95} -1.09501e14i q^{97} -2.52528e13 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 1364266 q^{7} - 54790830 q^{9}+O(q^{10}) \) 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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