LMFDB - Embedded newform 140.15.d.a.41.6

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			41.6
Character	$\chi$	$=$	140.41
Dual form			140.15.d.a.41.31

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3008.95i q^{3} +34938.6i q^{5} +(-209395. - 796478. i) q^{7} -4.27082e6 q^{9} +O(q^{10})$	\(q-3008.95i q^{3} +34938.6i q^{5} +(-209395. - 796478. i) q^{7} -4.27082e6 q^{9} -3.27038e7 q^{11} +2.83664e6i q^{13} +1.05128e8 q^{15} +1.16828e8i q^{17} -7.25611e8i q^{19} +(-2.39656e9 + 6.30059e8i) q^{21} -5.44236e9 q^{23} -1.22070e9 q^{25} -1.54104e9i q^{27} -2.49314e10 q^{29} -1.90777e10i q^{31} +9.84043e10i q^{33} +(2.78278e10 - 7.31596e9i) q^{35} +1.63093e11 q^{37} +8.53530e9 q^{39} -6.25329e9i q^{41} -1.86792e11 q^{43} -1.49216e11i q^{45} +1.72163e11i q^{47} +(-5.90531e11 + 3.33557e11i) q^{49} +3.51530e11 q^{51} +1.06932e12 q^{53} -1.14263e12i q^{55} -2.18333e12 q^{57} +8.95174e10i q^{59} -1.59224e12i q^{61} +(8.94287e11 + 3.40161e12i) q^{63} -9.91080e10 q^{65} -4.05312e12 q^{67} +1.63758e13i q^{69} +1.01026e13 q^{71} +2.29257e12i q^{73} +3.67304e12i q^{75} +(6.84802e12 + 2.60479e13i) q^{77} -2.61502e13 q^{79} -2.50641e13 q^{81} +3.43401e13i q^{83} -4.08181e12 q^{85} +7.50173e13i q^{87} -3.86635e13i q^{89} +(2.25932e12 - 5.93977e11i) q^{91} -5.74039e13 q^{93} +2.53518e13 q^{95} +1.82890e13i q^{97} +1.39672e14 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	3008.95i		−	1.37583i	−0.725789	−	0.687917i	$-0.758525\pi$
$3$		−	3008.95i		−	1.37583i	0.725789	−	0.687917i	$-0.241475\pi$
$4$		0			0
$5$			34938.6i			0.447214i
$5$			34938.6i			0.447214i
$6$		0			0
$7$	−209395.	−	796478.i	−0.254261	−	0.967136i
$7$	−209395.	−	796478.i	−0.254261	−	0.967136i
$8$		0			0
$9$	−4.27082e6			−0.892921
$10$		0			0
$11$	−3.27038e7			−1.67822			−0.839112	−	0.543958i	$-0.816925\pi$
$11$	−3.27038e7			−1.67822			−0.839112	+	0.543958i	$0.816925\pi$
$12$		0			0
$13$			2.83664e6i			0.0452064i	0.999745	+	0.0226032i	$0.00719544\pi$
$13$			2.83664e6i			0.0452064i	−0.999745	+	0.0226032i	$0.992805\pi$
$14$		0			0
$15$	1.05128e8			0.615292
$16$		0			0
$17$			1.16828e8i			0.284712i	0.989816	+	0.142356i	$0.0454677\pi$
$17$			1.16828e8i			0.284712i	−0.989816	+	0.142356i	$0.954532\pi$
$18$		0			0
$19$		−	7.25611e8i		−	0.811762i	−0.913926	−	0.405881i	$-0.866965\pi$
$19$		−	7.25611e8i		−	0.811762i	0.913926	−	0.405881i	$-0.133035\pi$
$20$		0			0
$21$	−2.39656e9	+	6.30059e8i	−1.33062	+	0.349821i
$22$		0			0
$23$	−5.44236e9			−1.59843			−0.799213	−	0.601047i	$-0.794750\pi$
$23$	−5.44236e9			−1.59843			−0.799213	+	0.601047i	$0.794750\pi$
$24$		0			0
$25$	−1.22070e9			−0.200000
$26$		0			0
$27$		−	1.54104e9i		−	0.147322i
$28$		0			0
$29$	−2.49314e10			−1.44531			−0.722654	−	0.691210i	$-0.757078\pi$
$29$	−2.49314e10			−1.44531			−0.722654	+	0.691210i	$0.757078\pi$
$30$		0			0
$31$		−	1.90777e10i		−	0.693417i	−0.937973	−	0.346708i	$-0.887299\pi$
$31$		−	1.90777e10i		−	0.693417i	0.937973	−	0.346708i	$-0.112701\pi$
$32$		0			0
$33$			9.84043e10i			2.30896i
$34$		0			0
$35$	2.78278e10	−	7.31596e9i	0.432516	−	0.113709i
$36$		0			0
$37$	1.63093e11			1.71800			0.859002	−	0.511972i	$-0.171085\pi$
$37$	1.63093e11			1.71800			0.859002	+	0.511972i	$0.171085\pi$
$38$		0			0
$39$	8.53530e9			0.0621966
$40$		0			0
$41$		−	6.25329e9i		−	0.0321086i	−0.999871	−	0.0160543i	$-0.994890\pi$
$41$		−	6.25329e9i		−	0.0321086i	0.999871	−	0.0160543i	$-0.00511046\pi$
$42$		0			0
$43$	−1.86792e11			−0.687193			−0.343597	−	0.939117i	$-0.611645\pi$
$43$	−1.86792e11			−0.687193			−0.343597	+	0.939117i	$0.611645\pi$
$44$		0			0
$45$		−	1.49216e11i		−	0.399327i
$46$		0			0
$47$			1.72163e11i			0.339824i	0.985459	+	0.169912i	$0.0543483\pi$
$47$			1.72163e11i			0.339824i	−0.985459	+	0.169912i	$0.945652\pi$
$48$		0			0
$49$	−5.90531e11	+	3.33557e11i	−0.870703	+	0.491810i
$50$		0			0
$51$	3.51530e11			0.391716
$52$		0			0
$53$	1.06932e12			0.910282			0.455141	−	0.890419i	$-0.349589\pi$
$53$	1.06932e12			0.910282			0.455141	+	0.890419i	$0.349589\pi$
$54$		0			0
$55$		−	1.14263e12i		−	0.750525i
$56$		0			0
$57$	−2.18333e12			−1.11685
$58$		0			0
$59$			8.95174e10i			0.0359702i	0.999838	+	0.0179851i	$0.00572515\pi$
$59$			8.95174e10i			0.0359702i	−0.999838	+	0.0179851i	$0.994275\pi$
$60$		0			0
$61$		−	1.59224e12i		−	0.506640i	−0.967383	−	0.253320i	$-0.918478\pi$
$61$		−	1.59224e12i		−	0.506640i	0.967383	−	0.253320i	$-0.0815225\pi$
$62$		0			0
$63$	8.94287e11	+	3.40161e12i	0.227035	+	0.863576i
$64$		0			0
$65$	−9.91080e10			−0.0202169
$66$		0			0
$67$	−4.05312e12			−0.668753			−0.334377	−	0.942440i	$-0.608526\pi$
$67$	−4.05312e12			−0.668753			−0.334377	+	0.942440i	$0.608526\pi$
$68$		0			0
$69$			1.63758e13i			2.19917i
$70$		0			0
$71$	1.01026e13			1.11078			0.555388	−	0.831591i	$-0.312570\pi$
$71$	1.01026e13			1.11078			0.555388	+	0.831591i	$0.312570\pi$
$72$		0			0
$73$			2.29257e12i			0.207521i	0.994602	+	0.103761i	$0.0330876\pi$
$73$			2.29257e12i			0.207521i	−0.994602	+	0.103761i	$0.966912\pi$
$74$		0			0
$75$			3.67304e12i			0.275167i
$76$		0			0
$77$	6.84802e12	+	2.60479e13i	0.426707	+	1.62307i
$78$		0			0
$79$	−2.61502e13			−1.36171			−0.680857	−	0.732416i	$-0.738392\pi$
$79$	−2.61502e13			−1.36171			−0.680857	+	0.732416i	$0.738392\pi$
$80$		0			0
$81$	−2.50641e13			−1.09561
$82$		0			0
$83$			3.43401e13i			1.26548i	0.774365	+	0.632739i	$0.218069\pi$
$83$			3.43401e13i			1.26548i	−0.774365	+	0.632739i	$0.781931\pi$
$84$		0			0
$85$	−4.08181e12			−0.127327
$86$		0			0
$87$			7.50173e13i			1.98851i
$88$		0			0
$89$		−	3.86635e13i		−	0.874120i	−0.899432	−	0.437060i	$-0.856020\pi$
$89$		−	3.86635e13i		−	0.874120i	0.899432	−	0.437060i	$-0.143980\pi$
$90$		0			0
$91$	2.25932e12	−	5.93977e11i	0.0437207	−	0.0114942i
$92$		0			0
$93$	−5.74039e13			−0.954027
$94$		0			0
$95$	2.53518e13			0.363031
$96$		0			0
$97$			1.82890e13i			0.226354i	0.993575	+	0.113177i	$0.0361026\pi$
$97$			1.82890e13i			0.226354i	−0.993575	+	0.113177i	$0.963897\pi$
$98$		0			0
$99$	1.39672e14			1.49852
$100$		0			0
$101$			1.53506e14i			1.43178i	0.698213	+	0.715890i	$0.253979\pi$
$101$			1.53506e14i			1.43178i	−0.698213	+	0.715890i	$0.746021\pi$
$102$		0			0
$103$			1.19177e13i			0.0969018i	0.998826	+	0.0484509i	$0.0154284\pi$
$103$			1.19177e13i			0.0969018i	−0.998826	+	0.0484509i	$0.984572\pi$
$104$		0			0
$105$	−2.20134e13	−	8.37324e13i	−0.156445	−	0.595071i
$106$		0			0
$107$	−9.12571e12			−0.0568303			−0.0284152	−	0.999596i	$-0.509046\pi$
$107$	−9.12571e12			−0.0568303			−0.0284152	+	0.999596i	$0.509046\pi$
$108$		0			0
$109$	−1.83304e13			−0.100273			−0.0501367	−	0.998742i	$-0.515966\pi$
$109$	−1.83304e13			−0.100273			−0.0501367	+	0.998742i	$0.515966\pi$
$110$		0			0
$111$		−	4.90740e14i		−	2.36369i
$112$		0			0
$113$	−6.19933e13			−0.263509			−0.131755	−	0.991282i	$-0.542061\pi$
$113$	−6.19933e13			−0.263509			−0.131755	+	0.991282i	$0.542061\pi$
$114$		0			0
$115$		−	1.90148e14i		−	0.714838i
$116$		0			0
$117$		−	1.21148e13i		−	0.0403658i
$118$		0			0
$119$	9.30511e13	−	2.44632e13i	0.275355	−	0.0723911i
$120$		0			0
$121$	6.89792e14			1.81644
$122$		0			0
$123$	−1.88158e13			−0.0441761
$124$		0			0
$125$		−	4.26496e13i		−	0.0894427i
$126$		0			0
$127$	−4.83147e14			−0.906677			−0.453339	−	0.891338i	$-0.649767\pi$
$127$	−4.83147e14			−0.906677			−0.453339	+	0.891338i	$0.649767\pi$
$128$		0			0
$129$			5.62048e14i			0.945464i
$130$		0			0
$131$		−	5.94901e14i		−	0.898556i	−0.893392	−	0.449278i	$-0.851681\pi$
$131$		−	5.94901e14i		−	0.898556i	0.893392	−	0.449278i	$-0.148319\pi$
$132$		0			0
$133$	−5.77933e14	+	1.51939e14i	−0.785084	+	0.206400i
$134$		0			0
$135$	5.38419e13			0.0658846
$136$		0			0
$137$	1.20749e15			1.33302			0.666512	−	0.745494i	$-0.267786\pi$
$137$	1.20749e15			1.33302			0.666512	+	0.745494i	$0.267786\pi$
$138$		0			0
$139$		−	1.43802e15i		−	1.43437i	−0.696885	−	0.717183i	$-0.745431\pi$
$139$		−	1.43802e15i		−	1.43437i	0.696885	−	0.717183i	$-0.254569\pi$
$140$		0			0
$141$	5.18029e14			0.467541
$142$		0			0
$143$		−	9.27689e13i		−	0.0758665i
$144$		0			0
$145$		−	8.71067e14i		−	0.646361i
$146$		0			0
$147$	1.00366e15	+	1.77688e15i	0.676649	+	1.19794i
$148$		0			0
$149$	−2.40320e14			−0.147396			−0.0736979	−	0.997281i	$-0.523480\pi$
$149$	−2.40320e14			−0.147396			−0.0736979	+	0.997281i	$0.523480\pi$
$150$		0			0
$151$	5.71870e14			0.319491			0.159745	−	0.987158i	$-0.448933\pi$
$151$	5.71870e14			0.319491			0.159745	+	0.987158i	$0.448933\pi$
$152$		0			0
$153$		−	4.98952e14i		−	0.254225i
$154$		0			0
$155$	6.66547e14			0.310105
$156$		0			0
$157$		−	1.79623e14i		−	0.0763951i	−0.999270	−	0.0381975i	$-0.987838\pi$
$157$		−	1.79623e14i		−	0.0763951i	0.999270	−	0.0381975i	$-0.0121616\pi$
$158$		0			0
$159$		−	3.21753e15i		−	1.25240i
$160$		0			0
$161$	1.13960e15	+	4.33472e15i	0.406418	+	1.54590i
$162$		0			0
$163$	1.21703e15			0.398098			0.199049	−	0.979990i	$-0.436215\pi$
$163$	1.21703e15			0.398098			0.199049	+	0.979990i	$0.436215\pi$
$164$		0			0
$165$	−3.43810e15			−1.03260
$166$		0			0
$167$		−	2.46806e15i		−	0.681304i	−0.940189	−	0.340652i	$-0.889352\pi$
$167$		−	2.46806e15i		−	0.681304i	0.940189	−	0.340652i	$-0.110648\pi$
$168$		0			0
$169$	3.92933e15			0.997956
$170$		0			0
$171$			3.09895e15i			0.724840i
$172$		0			0
$173$		−	7.50723e14i		−	0.161866i	−0.996720	−	0.0809332i	$-0.974210\pi$
$173$		−	7.50723e14i		−	0.161866i	0.996720	−	0.0809332i	$-0.0257900\pi$
$174$		0			0
$175$	2.55609e14	+	9.72263e14i	0.0508522	+	0.193427i
$176$		0			0
$177$	2.69353e14			0.0494891
$178$		0			0
$179$	8.71031e15			1.47932			0.739660	−	0.672980i	$-0.234986\pi$
$179$	8.71031e15			1.47932			0.739660	+	0.672980i	$0.234986\pi$
$180$		0			0
$181$			9.74851e15i			1.53175i	0.642989	+	0.765875i	$0.277694\pi$
$181$			9.74851e15i			1.53175i	−0.642989	+	0.765875i	$0.722306\pi$
$182$		0			0
$183$	−4.79097e15			−0.697052
$184$		0			0
$185$			5.69825e15i			0.768315i
$186$		0			0
$187$		−	3.82073e15i		−	0.477810i
$188$		0			0
$189$	−1.22741e15	+	3.22687e14i	−0.142481	+	0.0374583i
$190$		0			0
$191$	1.31238e16			1.41522			0.707611	−	0.706602i	$-0.249773\pi$
$191$	1.31238e16			1.41522			0.707611	+	0.706602i	$0.249773\pi$
$192$		0			0
$193$	−6.70006e14			−0.0671703			−0.0335852	−	0.999436i	$-0.510693\pi$
$193$	−6.70006e14			−0.0671703			−0.0335852	+	0.999436i	$0.510693\pi$
$194$		0			0
$195$			2.98211e14i			0.0278152i
$196$		0			0
$197$	1.12531e16			0.977261			0.488630	−	0.872491i	$-0.337497\pi$
$197$	1.12531e16			0.977261			0.488630	+	0.872491i	$0.337497\pi$
$198$		0			0
$199$			1.80197e16i			1.45807i	0.684479	+	0.729033i	$0.260030\pi$
$199$			1.80197e16i			1.45807i	−0.684479	+	0.729033i	$0.739970\pi$
$200$		0			0
$201$			1.21956e16i			0.920094i
$202$		0			0
$203$	5.22051e15	+	1.98573e16i	0.367486	+	1.39781i
$204$		0			0
$205$	2.18481e14			0.0143594
$206$		0			0
$207$	2.32433e16			1.42727
$208$		0			0
$209$			2.37303e16i			1.36232i
$210$		0			0
$211$	−5.14470e15			−0.276301			−0.138151	−	0.990411i	$-0.544116\pi$
$211$	−5.14470e15			−0.276301			−0.138151	+	0.990411i	$0.544116\pi$
$212$		0			0
$213$		−	3.03984e16i		−	1.52824i
$214$		0			0
$215$		−	6.52624e15i		−	0.307322i
$216$		0			0
$217$	−1.51950e16	+	3.99477e15i	−0.670628	+	0.176309i
$218$		0			0
$219$	6.89823e15			0.285515
$220$		0			0
$221$	−3.31399e14			−0.0128708
$222$		0			0
$223$		−	3.03072e16i		−	1.10513i	−0.833471	−	0.552563i	$-0.813650\pi$
$223$		−	3.03072e16i		−	1.10513i	0.833471	−	0.552563i	$-0.186350\pi$
$224$		0			0
$225$	5.21340e15			0.178584
$226$		0			0
$227$		−	2.03584e16i		−	0.655486i	−0.944767	−	0.327743i	$-0.893712\pi$
$227$		−	2.03584e16i		−	0.655486i	0.944767	−	0.327743i	$-0.106288\pi$
$228$		0			0
$229$		−	4.73241e15i		−	0.143296i	−0.997430	−	0.0716480i	$-0.977174\pi$
$229$		−	4.73241e15i		−	0.143296i	0.997430	−	0.0716480i	$-0.0228258\pi$
$230$		0			0
$231$	7.83768e16	−	2.06054e16i	2.23308	−	0.587078i
$232$		0			0
$233$	3.88544e16			1.04219			0.521097	−	0.853497i	$-0.325523\pi$
$233$	3.88544e16			1.04219			0.521097	+	0.853497i	$0.325523\pi$
$234$		0			0
$235$	−6.01511e15			−0.151974
$236$		0			0
$237$			7.86848e16i			1.87349i
$238$		0			0
$239$	4.96741e16			1.11518			0.557590	−	0.830117i	$-0.311726\pi$
$239$	4.96741e16			1.11518			0.557590	+	0.830117i	$0.311726\pi$
$240$		0			0
$241$		−	5.73356e16i		−	1.21424i	−0.794610	−	0.607120i	$-0.792325\pi$
$241$		−	5.73356e16i		−	1.21424i	0.794610	−	0.607120i	$-0.207675\pi$
$242$		0			0
$243$			6.80459e16i			1.36006i
$244$		0			0
$245$	−1.16540e16	−	2.06323e16i	−0.219944	−	0.389390i
$246$		0			0
$247$	2.05830e15			0.0366969
$248$		0			0
$249$	1.03328e17			1.74109
$250$		0			0
$251$		−	4.99179e16i		−	0.795318i	−0.917533	−	0.397659i	$-0.869823\pi$
$251$		−	4.99179e16i		−	0.795318i	0.917533	−	0.397659i	$-0.130177\pi$
$252$		0			0
$253$	1.77986e17			2.68252
$254$		0			0
$255$			1.22820e16i			0.175181i
$256$		0			0
$257$		−	4.51434e16i		−	0.609624i	−0.952413	−	0.304812i	$-0.901406\pi$
$257$		−	4.51434e16i		−	0.609624i	0.952413	−	0.304812i	$-0.0985937\pi$
$258$		0			0
$259$	−3.41509e16	−	1.29900e17i	−0.436822	−	1.66154i
$260$		0			0
$261$	1.06477e17			1.29055
$262$		0			0
$263$	1.14484e16			0.131539			0.0657697	−	0.997835i	$-0.479050\pi$
$263$	1.14484e16			0.131539			0.0657697	+	0.997835i	$0.479050\pi$
$264$		0			0
$265$			3.73604e16i			0.407090i
$266$		0			0
$267$	−1.16337e17			−1.20264
$268$		0			0
$269$			1.81028e17i			1.77615i	0.459696	+	0.888076i	$0.347958\pi$
$269$			1.81028e17i			1.77615i	−0.459696	+	0.888076i	$0.652042\pi$
$270$		0			0
$271$		−	2.91200e16i		−	0.271273i	−0.990759	−	0.135636i	$-0.956692\pi$
$271$		−	2.91200e16i		−	0.271273i	0.990759	−	0.135636i	$-0.0433079\pi$
$272$		0			0
$273$	−1.78725e15	−	6.79818e15i	−0.0158142	−	0.0601525i
$274$		0			0
$275$	3.99217e16			0.335645
$276$		0			0
$277$	5.82733e16			0.465705			0.232853	−	0.972512i	$-0.425194\pi$
$277$	5.82733e16			0.465705			0.232853	+	0.972512i	$0.425194\pi$
$278$		0			0
$279$			8.14774e16i			0.619167i
$280$		0			0
$281$	−2.48433e17			−1.79583			−0.897916	−	0.440166i	$-0.854919\pi$
$281$	−2.48433e17			−1.79583			−0.897916	+	0.440166i	$0.854919\pi$
$282$		0			0
$283$		−	1.82139e17i		−	1.25285i	−0.779483	−	0.626423i	$-0.784518\pi$
$283$		−	1.82139e17i		−	1.25285i	0.779483	−	0.626423i	$-0.215482\pi$
$284$		0			0
$285$		−	7.62824e16i		−	0.499471i
$286$		0			0
$287$	−4.98061e15	+	1.30941e15i	−0.0310534	+	0.00816397i
$288$		0			0
$289$	1.54729e17			0.918939
$290$		0			0
$291$	5.50306e16			0.311425
$292$		0			0
$293$			3.32869e17i			1.79556i	0.440443	+	0.897781i	$0.354821\pi$
$293$			3.32869e17i			1.79556i	−0.440443	+	0.897781i	$0.645179\pi$
$294$		0			0
$295$	−3.12761e15			−0.0160864
$296$		0			0
$297$			5.03981e16i			0.247240i
$298$		0			0
$299$		−	1.54380e16i		−	0.0722592i
$300$		0			0
$301$	3.91133e16	+	1.48776e17i	0.174726	+	0.664609i
$302$		0			0
$303$	4.61893e17			1.96989
$304$		0			0
$305$	5.56305e16			0.226576
$306$		0			0
$307$		−	2.56812e17i		−	0.999186i	−0.866260	−	0.499593i	$-0.833483\pi$
$307$		−	2.56812e17i		−	0.999186i	0.866260	−	0.499593i	$-0.166517\pi$
$308$		0			0
$309$	3.58598e16			0.133321
$310$		0			0
$311$			1.57377e17i			0.559265i	0.960107	+	0.279632i	$0.0902125\pi$
$311$			1.57377e17i			0.559265i	−0.960107	+	0.279632i	$0.909787\pi$
$312$		0			0
$313$			5.75416e17i			1.95511i	0.210674	+	0.977556i	$0.432434\pi$
$313$			5.75416e17i			1.95511i	−0.210674	+	0.977556i	$0.567566\pi$
$314$		0			0
$315$	−1.18847e17	+	3.12451e16i	−0.386203	+	0.101533i
$316$		0			0
$317$	−4.51448e17			−1.40344			−0.701718	−	0.712455i	$-0.747583\pi$
$317$	−4.51448e17			−1.40344			−0.701718	+	0.712455i	$0.747583\pi$
$318$		0			0
$319$	8.15352e17			2.42555
$320$		0			0
$321$			2.74588e16i			0.0781891i
$322$		0			0
$323$	8.47719e16			0.231118
$324$		0			0
$325$		−	3.46269e15i		−	0.00904129i
$326$		0			0
$327$			5.51552e16i			0.137960i
$328$		0			0
$329$	1.37124e17	−	3.60500e16i	0.328656	−	0.0864040i
$330$		0			0
$331$	−1.86131e17			−0.427585			−0.213792	−	0.976879i	$-0.568582\pi$
$331$	−1.86131e17			−0.427585			−0.213792	+	0.976879i	$0.568582\pi$
$332$		0			0
$333$	−6.96542e17			−1.53404
$334$		0			0
$335$		−	1.41610e17i		−	0.299076i
$336$		0			0
$337$	−9.28836e17			−1.88161			−0.940805	−	0.338947i	$-0.889929\pi$
$337$	−9.28836e17			−1.88161			−0.940805	+	0.338947i	$0.889929\pi$
$338$		0			0
$339$			1.86535e17i			0.362545i
$340$		0			0
$341$			6.23914e17i			1.16371i
$342$		0			0
$343$	3.89325e17	+	4.00499e17i	0.697033	+	0.717039i
$344$		0			0
$345$	−5.72147e17			−0.983499
$346$		0			0
$347$	−8.86675e17			−1.46372			−0.731861	−	0.681454i	$-0.761348\pi$
$347$	−8.86675e17			−1.46372			−0.731861	+	0.681454i	$0.761348\pi$
$348$		0			0
$349$		−	8.55909e17i		−	1.35722i	−0.734500	−	0.678609i	$-0.762583\pi$
$349$		−	8.55909e17i		−	1.35722i	0.734500	−	0.678609i	$-0.237417\pi$
$350$		0			0
$351$	4.37138e15			0.00665992
$352$		0			0
$353$			3.99098e17i			0.584328i	0.956368	+	0.292164i	$0.0943754\pi$
$353$			3.99098e17i			0.584328i	−0.956368	+	0.292164i	$0.905625\pi$
$354$		0			0
$355$			3.52972e17i			0.496754i
$356$		0			0
$357$	−7.36087e16	−	2.79986e17i	−0.0995982	−	0.378843i
$358$		0			0
$359$	−7.06315e17			−0.919046			−0.459523	−	0.888166i	$-0.651980\pi$
$359$	−7.06315e17			−0.919046			−0.459523	+	0.888166i	$0.651980\pi$
$360$		0			0
$361$	2.72495e17			0.341042
$362$		0			0
$363$		−	2.07555e18i		−	2.49912i
$364$		0			0
$365$	−8.00990e16			−0.0928063
$366$		0			0
$367$		−	8.21419e17i		−	0.916015i	−0.888948	−	0.458007i	$-0.848563\pi$
$367$		−	8.21419e17i		−	0.916015i	0.888948	−	0.458007i	$-0.151437\pi$
$368$		0			0
$369$			2.67066e16i			0.0286705i
$370$		0			0
$371$	−2.23910e17	−	8.51688e17i	−0.231449	−	0.880366i
$372$		0			0
$373$	8.06411e17			0.802776			0.401388	−	0.915908i	$-0.368528\pi$
$373$	8.06411e17			0.802776			0.401388	+	0.915908i	$0.368528\pi$
$374$		0			0
$375$	−1.28331e17			−0.123058
$376$		0			0
$377$		−	7.07213e16i		−	0.0653372i
$378$		0			0
$379$	5.82738e16			0.0518799			0.0259399	−	0.999664i	$-0.491742\pi$
$379$	5.82738e16			0.0518799			0.0259399	+	0.999664i	$0.491742\pi$
$380$		0			0
$381$			1.45376e18i			1.24744i
$382$		0			0
$383$		−	1.54521e18i		−	1.27819i	−0.769126	−	0.639097i	$-0.779308\pi$
$383$		−	1.54521e18i		−	1.27819i	0.769126	−	0.639097i	$-0.220692\pi$
$384$		0			0
$385$	−9.10076e17	+	2.39260e17i	−0.725859	+	0.190829i
$386$		0			0
$387$	7.97754e17			0.613609
$388$		0			0
$389$	2.52182e18			1.87097			0.935487	−	0.353362i	$-0.114961\pi$
$389$	2.52182e18			1.87097			0.935487	+	0.353362i	$0.114961\pi$
$390$		0			0
$391$		−	6.35822e17i		−	0.455091i
$392$		0			0
$393$	−1.79003e18			−1.23627
$394$		0			0
$395$		−	9.13651e17i		−	0.608977i
$396$		0			0
$397$		−	2.89839e18i		−	1.86476i	−0.361474	−	0.932382i	$-0.617726\pi$
$397$		−	2.89839e18i		−	1.86476i	0.361474	−	0.932382i	$-0.382274\pi$
$398$		0			0
$399$	4.57178e17	+	1.73897e18i	0.283972	+	1.08015i
$400$		0			0
$401$	1.29220e17			0.0775030			0.0387515	−	0.999249i	$-0.487662\pi$
$401$	1.29220e17			0.0775030			0.0387515	+	0.999249i	$0.487662\pi$
$402$		0			0
$403$	5.41165e16			0.0313469
$404$		0			0
$405$		−	8.75704e17i		−	0.489973i
$406$		0			0
$407$	−5.33378e18			−2.88320
$408$		0			0
$409$		−	2.33201e18i		−	1.21806i	−0.793149	−	0.609028i	$-0.791560\pi$
$409$		−	2.33201e18i		−	1.21806i	0.793149	−	0.609028i	$-0.208440\pi$
$410$		0			0
$411$		−	3.63327e18i		−	1.83402i
$412$		0			0
$413$	7.12986e16	−	1.87445e16i	0.0347881	−	0.00914583i
$414$		0			0
$415$	−1.19979e18			−0.565939
$416$		0			0
$417$	−4.32692e18			−1.97345
$418$		0			0
$419$			4.06585e18i			1.79330i	0.442738	+	0.896651i	$0.354007\pi$
$419$			4.06585e18i			1.79330i	−0.442738	+	0.896651i	$0.645993\pi$
$420$		0			0
$421$	−2.37980e18			−1.01523			−0.507617	−	0.861583i	$-0.669473\pi$
$421$	−2.37980e18			−1.01523			−0.507617	+	0.861583i	$0.669473\pi$
$422$		0			0
$423$		−	7.35275e17i		−	0.303436i
$424$		0			0
$425$		−	1.42613e17i		−	0.0569423i
$426$		0			0
$427$	−1.26818e18	+	3.33407e17i	−0.489989	+	0.128819i
$428$		0			0
$429$	−2.79137e17			−0.104380
$430$		0			0
$431$	2.01459e18			0.729197			0.364598	−	0.931165i	$-0.381206\pi$
$431$	2.01459e18			0.729197			0.364598	+	0.931165i	$0.381206\pi$
$432$		0			0
$433$			4.13712e18i			1.44972i	0.688898	+	0.724859i	$0.258095\pi$
$433$			4.13712e18i			1.44972i	−0.688898	+	0.724859i	$0.741905\pi$
$434$		0			0
$435$	−2.62100e18			−0.889287
$436$		0			0
$437$			3.94904e18i			1.29754i
$438$		0			0
$439$			4.27028e18i			1.35895i	0.733697	+	0.679477i	$0.237793\pi$
$439$			4.27028e18i			1.35895i	−0.733697	+	0.679477i	$0.762207\pi$
$440$		0			0
$441$	2.52205e18	−	1.42456e18i	0.777469	−	0.439148i
$442$		0			0
$443$	−3.23194e18			−0.965245			−0.482622	−	0.875829i	$-0.660316\pi$
$443$	−3.23194e18			−0.965245			−0.482622	+	0.875829i	$0.660316\pi$
$444$		0			0
$445$	1.35085e18			0.390918
$446$		0			0
$447$			7.23110e17i			0.202792i
$448$		0			0
$449$	2.77124e18			0.753269			0.376634	−	0.926362i	$-0.377081\pi$
$449$	2.77124e18			0.753269			0.376634	+	0.926362i	$0.377081\pi$
$450$		0			0
$451$			2.04507e17i			0.0538855i
$452$		0			0
$453$		−	1.72073e18i		−	0.439567i
$454$		0			0
$455$	2.07527e16	+	7.89373e16i	0.00514038	+	0.0195525i
$456$		0			0
$457$	−5.75134e18			−1.38151			−0.690757	−	0.723087i	$-0.742723\pi$
$457$	−5.75134e18			−1.38151			−0.690757	+	0.723087i	$0.742723\pi$
$458$		0			0
$459$	1.80037e17			0.0419444
$460$		0			0
$461$			4.22605e18i			0.955055i	0.878617	+	0.477527i	$0.158467\pi$
$461$			4.22605e18i			0.955055i	−0.878617	+	0.477527i	$0.841533\pi$
$462$		0			0
$463$	−6.54283e18			−1.43449			−0.717247	−	0.696819i	$-0.754598\pi$
$463$	−6.54283e18			−1.43449			−0.717247	+	0.696819i	$0.754598\pi$
$464$		0			0
$465$		−	2.00561e18i		−	0.426654i
$466$		0			0
$467$			6.41064e18i			1.32338i	0.749779	+	0.661688i	$0.230160\pi$
$467$			6.41064e18i			1.32338i	−0.749779	+	0.661688i	$0.769840\pi$
$468$		0			0
$469$	8.48703e17	+	3.22822e18i	0.170038	+	0.646775i
$470$		0			0
$471$	−5.40478e17			−0.105107
$472$		0			0
$473$	6.10881e18			1.15326
$474$		0			0
$475$			8.85756e17i			0.162352i
$476$		0			0
$477$	−4.56686e18			−0.812810
$478$		0			0
$479$		−	2.77195e18i		−	0.479111i	−0.970883	−	0.239555i	$-0.922998\pi$
$479$		−	2.77195e18i		−	0.479111i	0.970883	−	0.239555i	$-0.0770017\pi$
$480$		0			0
$481$			4.62637e17i			0.0776649i
$482$		0			0
$483$	1.30430e19	−	3.42901e18i	2.12690	−	0.559164i
$484$		0			0
$485$	−6.38991e17			−0.101228
$486$		0			0
$487$	6.79197e18			1.04542			0.522712	−	0.852509i	$-0.324920\pi$
$487$	6.79197e18			1.04542			0.522712	+	0.852509i	$0.324920\pi$
$488$		0			0
$489$		−	3.66200e18i		−	0.547717i
$490$		0			0
$491$	−5.07744e17			−0.0738031			−0.0369015	−	0.999319i	$-0.511749\pi$
$491$	−5.07744e17			−0.0738031			−0.0369015	+	0.999319i	$0.511749\pi$
$492$		0			0
$493$		−	2.91269e18i		−	0.411496i
$494$		0			0
$495$			4.87994e18i			0.670160i
$496$		0			0
$497$	−2.11544e18	−	8.04653e18i	−0.282427	−	1.07427i
$498$		0			0
$499$	1.11810e18			0.145137			0.0725684	−	0.997363i	$-0.476880\pi$
$499$	1.11810e18			0.145137			0.0725684	+	0.997363i	$0.476880\pi$
$500$		0			0
$501$	−7.42628e18			−0.937361
$502$		0			0
$503$			8.06326e18i			0.989771i	0.868958	+	0.494886i	$0.164790\pi$
$503$			8.06326e18i			0.989771i	−0.868958	+	0.494886i	$0.835210\pi$
$504$		0			0
$505$	−5.36329e18			−0.640312
$506$		0			0
$507$		−	1.18232e19i		−	1.37302i
$508$		0			0
$509$		−	1.30879e19i		−	1.47858i	−0.673387	−	0.739290i	$-0.735161\pi$
$509$		−	1.30879e19i		−	1.47858i	0.673387	−	0.739290i	$-0.264839\pi$
$510$		0			0
$511$	1.82598e18	−	4.80052e17i	0.200701	−	0.0527645i
$512$		0			0
$513$	−1.11820e18			−0.119591
$514$		0			0
$515$	−4.16387e17			−0.0433358
$516$		0			0
$517$		−	5.63038e18i		−	0.570301i
$518$		0			0
$519$	−2.25889e18			−0.222701
$520$		0			0
$521$			1.35176e19i			1.29728i	0.761095	+	0.648641i	$0.224662\pi$
$521$			1.35176e19i			1.29728i	−0.761095	+	0.648641i	$0.775338\pi$
$522$		0			0
$523$			1.50426e19i			1.40544i	0.711467	+	0.702720i	$0.248031\pi$
$523$			1.50426e19i			1.40544i	−0.711467	+	0.702720i	$0.751969\pi$
$524$		0			0
$525$	2.92549e18	−	7.69115e17i	0.266124	−	0.0699642i
$526$		0			0
$527$	2.22881e18			0.197424
$528$		0			0
$529$	1.80265e19			1.55497
$530$		0			0
$531$		−	3.82312e17i		−	0.0321186i
$532$		0			0
$533$	1.77383e16			0.00145152
$534$		0			0
$535$		−	3.18839e17i		−	0.0254153i
$536$		0			0
$537$		−	2.62089e19i		−	2.03530i
$538$		0			0
$539$	1.93126e19	−	1.09086e19i	1.46123	−	0.825367i
$540$		0			0
$541$	6.37530e17			0.0470024			0.0235012	−	0.999724i	$-0.492519\pi$
$541$	6.37530e17			0.0470024			0.0235012	+	0.999724i	$0.492519\pi$
$542$		0			0
$543$	2.93328e19			2.10744
$544$		0			0
$545$		−	6.40437e17i		−	0.0448437i
$546$		0			0
$547$	−5.42226e18			−0.370057			−0.185029	−	0.982733i	$-0.559238\pi$
$547$	−5.42226e18			−0.370057			−0.185029	+	0.982733i	$0.559238\pi$
$548$		0			0
$549$			6.80016e18i			0.452389i
$550$		0			0
$551$			1.80905e19i			1.17325i
$552$		0			0
$553$	5.47573e18	+	2.08281e19i	0.346231	+	1.31696i
$554$		0			0
$555$	1.71458e19			1.05707
$556$		0			0
$557$	2.60582e19			1.56660			0.783299	−	0.621645i	$-0.213535\pi$
$557$	2.60582e19			1.56660			0.783299	+	0.621645i	$0.213535\pi$
$558$		0			0
$559$		−	5.29861e17i		−	0.0310655i
$560$		0			0
$561$	−1.14964e19			−0.657388
$562$		0			0
$563$		−	5.08642e18i		−	0.283696i	−0.989888	−	0.141848i	$-0.954696\pi$
$563$		−	5.08642e18i		−	0.283696i	0.989888	−	0.141848i	$-0.0453045\pi$
$564$		0			0
$565$		−	2.16596e18i		−	0.117845i
$566$		0			0
$567$	5.24830e18	+	1.99630e19i	0.278572	+	1.05961i
$568$		0			0
$569$	−2.35380e19			−1.21894			−0.609471	−	0.792809i	$-0.708618\pi$
$569$	−2.35380e19			−1.21894			−0.609471	+	0.792809i	$0.708618\pi$
$570$		0			0
$571$	5.21931e18			0.263731			0.131865	−	0.991268i	$-0.457903\pi$
$571$	5.21931e18			0.263731			0.131865	+	0.991268i	$0.457903\pi$
$572$		0			0
$573$		−	3.94888e19i		−	1.94711i
$574$		0			0
$575$	6.64351e18			0.319685
$576$		0			0
$577$			1.72033e19i			0.807944i	0.914771	+	0.403972i	$0.132371\pi$
$577$			1.72033e19i			0.807944i	−0.914771	+	0.403972i	$0.867629\pi$
$578$		0			0
$579$			2.01601e18i			0.0924153i
$580$		0			0
$581$	2.73511e19	−	7.19063e18i	1.22389	−	0.321762i
$582$		0			0
$583$	−3.49708e19			−1.52766
$584$		0			0
$585$	4.23272e17			0.0180521
$586$		0			0
$587$			1.07610e19i			0.448112i	0.974576	+	0.224056i	$0.0719298\pi$
$587$			1.07610e19i			0.448112i	−0.974576	+	0.224056i	$0.928070\pi$
$588$		0			0
$589$	−1.38430e19			−0.562889
$590$		0			0
$591$		−	3.38602e19i		−	1.34455i
$592$		0			0
$593$			1.74345e18i			0.0676125i	0.999428	+	0.0338063i	$0.0107629\pi$
$593$			1.74345e18i			0.0676125i	−0.999428	+	0.0338063i	$0.989237\pi$
$594$		0			0
$595$	8.54710e17	+	3.25107e18i	0.0323743	+	0.123142i
$596$		0			0
$597$	5.42205e19			2.00606
$598$		0			0
$599$	−3.66164e19			−1.32339			−0.661694	−	0.749774i	$-0.730162\pi$
$599$	−3.66164e19			−1.32339			−0.661694	+	0.749774i	$0.730162\pi$
$600$		0			0
$601$		−	2.51753e19i		−	0.888900i	−0.895804	−	0.444450i	$-0.853399\pi$
$601$		−	2.51753e19i		−	0.888900i	0.895804	−	0.444450i	$-0.146601\pi$
$602$		0			0
$603$	1.73101e19			0.597144
$604$		0			0
$605$			2.41003e19i			0.812335i
$606$		0			0
$607$			4.60472e19i			1.51664i	0.651883	+	0.758320i	$0.273979\pi$
$607$			4.60472e19i			1.51664i	−0.651883	+	0.758320i	$0.726021\pi$
$608$		0			0
$609$	5.97496e19	−	1.57082e19i	1.92315	−	0.505599i
$610$		0			0
$611$	−4.88363e17			−0.0153622
$612$		0			0
$613$	3.79378e19			1.16640			0.583202	−	0.812327i	$-0.301800\pi$
$613$	3.79378e19			1.16640			0.583202	+	0.812327i	$0.301800\pi$
$614$		0			0
$615$		−	6.57398e17i		−	0.0197562i
$616$		0			0
$617$	−4.79518e19			−1.40867			−0.704335	−	0.709868i	$-0.748755\pi$
$617$	−4.79518e19			−1.40867			−0.704335	+	0.709868i	$0.748755\pi$
$618$		0			0
$619$			5.52445e19i			1.58655i	0.608862	+	0.793276i	$0.291626\pi$
$619$			5.52445e19i			1.58655i	−0.608862	+	0.793276i	$0.708374\pi$
$620$		0			0
$621$			8.38692e18i			0.235484i
$622$		0			0
$623$	−3.07946e19	+	8.09594e18i	−0.845392	+	0.222255i
$624$		0			0
$625$	1.49012e18			0.0400000
$626$		0			0
$627$	7.14032e19			1.87433
$628$		0			0
$629$			1.90539e19i			0.489136i
$630$		0			0
$631$	3.22117e18			0.0808738			0.0404369	−	0.999182i	$-0.487125\pi$
$631$	3.22117e18			0.0808738			0.0404369	+	0.999182i	$0.487125\pi$
$632$		0			0
$633$			1.54802e19i			0.380145i
$634$		0			0
$635$		−	1.68804e19i		−	0.405478i
$636$		0			0
$637$	−9.46179e17	−	1.67512e18i	−0.0222330	−	0.0393614i
$638$		0			0
$639$	−4.31465e19			−0.991836
$640$		0			0
$641$	3.64212e19			0.819121			0.409561	−	0.912283i	$-0.365682\pi$
$641$	3.64212e19			0.819121			0.409561	+	0.912283i	$0.365682\pi$
$642$		0			0
$643$		−	7.74596e18i		−	0.170451i	−0.996362	−	0.0852253i	$-0.972839\pi$
$643$		−	7.74596e18i		−	0.170451i	0.996362	−	0.0852253i	$-0.0271610\pi$
$644$		0			0
$645$	−1.96371e19			−0.422824
$646$		0			0
$647$			2.49352e19i			0.525392i	0.964879	+	0.262696i	$0.0846117\pi$
$647$			2.49352e19i			0.525392i	−0.964879	+	0.262696i	$0.915388\pi$
$648$		0			0
$649$		−	2.92756e18i		−	0.0603661i
$650$		0			0
$651$	1.20201e19	+	4.57209e19i	0.242572	+	0.922673i
$652$		0			0
$653$	−6.34696e19			−1.25364			−0.626821	−	0.779163i	$-0.715644\pi$
$653$	−6.34696e19			−1.25364			−0.626821	+	0.779163i	$0.715644\pi$
$654$		0			0
$655$	2.07850e19			0.401847
$656$		0			0
$657$		−	9.79114e18i		−	0.185300i
$658$		0			0
$659$	−5.98059e19			−1.10802			−0.554008	−	0.832511i	$-0.686902\pi$
$659$	−5.98059e19			−1.10802			−0.554008	+	0.832511i	$0.686902\pi$
$660$		0			0
$661$			4.65569e19i			0.844451i	0.906491	+	0.422225i	$0.138751\pi$
$661$			4.65569e19i			0.844451i	−0.906491	+	0.422225i	$0.861249\pi$
$662$		0			0
$663$			9.97164e17i			0.0177081i
$664$		0			0
$665$	−5.30854e18	−	2.01922e19i	−0.0923047	−	0.351100i
$666$		0			0
$667$	1.35686e20			2.31022
$668$		0			0
$669$	−9.11928e19			−1.52047
$670$		0			0
$671$			5.20723e19i			0.850255i
$672$		0			0
$673$	−4.79299e19			−0.766481			−0.383241	−	0.923649i	$-0.625192\pi$
$673$	−4.79299e19			−0.766481			−0.383241	+	0.923649i	$0.625192\pi$
$674$		0			0
$675$			1.88116e18i			0.0294645i
$676$		0			0
$677$		−	3.17378e19i		−	0.486919i	−0.969911	−	0.243459i	$-0.921718\pi$
$677$		−	3.17378e19i		−	0.486919i	0.969911	−	0.243459i	$-0.0782822\pi$
$678$		0			0
$679$	1.45668e19	−	3.82962e18i	0.218915	−	0.0575529i
$680$		0			0
$681$	−6.12575e19			−0.901840
$682$		0			0
$683$	−3.54666e19			−0.511534			−0.255767	−	0.966739i	$-0.582328\pi$
$683$	−3.54666e19			−0.511534			−0.255767	+	0.966739i	$0.582328\pi$
$684$		0			0
$685$			4.21878e19i			0.596147i
$686$		0			0
$687$	−1.42396e19			−0.197152
$688$		0			0
$689$			3.03327e18i			0.0411506i
$690$		0			0
$691$		−	2.41806e19i		−	0.321456i	−0.986999	−	0.160728i	$-0.948616\pi$
$691$		−	2.41806e19i		−	0.321456i	0.986999	−	0.160728i	$-0.0513841\pi$
$692$		0			0
$693$	−2.92466e19	−	1.11246e20i	−0.381016	−	1.44927i
$694$		0			0
$695$	5.02422e19			0.641468
$696$		0			0
$697$	7.30561e17			0.00914170
$698$		0			0
$699$		−	1.16911e20i		−	1.43389i
$700$		0			0
$701$	−1.40516e20			−1.68927			−0.844637	−	0.535340i	$-0.820184\pi$
$701$	−1.40516e20			−1.68927			−0.844637	+	0.535340i	$0.820184\pi$
$702$		0			0
$703$		−	1.18342e20i		−	1.39461i
$704$		0			0
$705$			1.80992e19i			0.209091i
$706$		0			0
$707$	1.22264e20	−	3.21434e19i	1.38473	−	0.364046i
$708$		0			0
$709$	−1.73124e20			−1.92236			−0.961178	−	0.275929i	$-0.911015\pi$
$709$	−1.73124e20			−1.92236			−0.961178	+	0.275929i	$0.911015\pi$
$710$		0			0
$711$	1.11683e20			1.21590
$712$		0			0
$713$			1.03828e20i			1.10838i
$714$		0			0
$715$	3.24121e18			0.0339285
$716$		0			0
$717$		−	1.49467e20i		−	1.53430i
$718$		0			0
$719$		−	4.48404e19i		−	0.451405i	−0.974196	−	0.225703i	$-0.927532\pi$
$719$		−	4.48404e19i		−	0.451405i	0.974196	−	0.225703i	$-0.0724678\pi$
$720$		0			0
$721$	9.49218e18	−	2.49551e18i	0.0937172	−	0.0246384i
$722$		0			0
$723$	−1.72520e20			−1.67059
$724$		0			0
$725$	3.04338e19			0.289062
$726$		0			0
$727$			5.19007e19i			0.483540i	0.970334	+	0.241770i	$0.0777280\pi$
$727$			5.19007e19i			0.483540i	−0.970334	+	0.241770i	$0.922272\pi$
$728$		0			0
$729$	8.48659e19			0.775605
$730$		0			0
$731$		−	2.18226e19i		−	0.195652i
$732$		0			0
$733$		−	1.57422e20i		−	1.38465i	−0.721587	−	0.692323i	$-0.756587\pi$
$733$		−	1.57422e20i		−	1.38465i	0.721587	−	0.692323i	$-0.243413\pi$
$734$		0			0
$735$	−6.20815e19	+	3.50663e19i	−0.535736	+	0.302607i
$736$		0			0
$737$	1.32553e20			1.12232
$738$		0			0
$739$	−4.29564e19			−0.356876			−0.178438	−	0.983951i	$-0.557104\pi$
$739$	−4.29564e19			−0.356876			−0.178438	+	0.983951i	$0.557104\pi$
$740$		0			0
$741$		−	6.19331e18i		−	0.0504888i
$742$		0			0
$743$	1.31473e20			1.05176			0.525879	−	0.850560i	$-0.323737\pi$
$743$	1.31473e20			1.05176			0.525879	+	0.850560i	$0.323737\pi$
$744$		0			0
$745$		−	8.39642e18i		−	0.0659174i
$746$		0			0
$747$		−	1.46660e20i		−	1.12997i
$748$		0			0
$749$	1.91088e18	+	7.26842e18i	0.0144497	+	0.0549626i
$750$		0			0
$751$	−1.14035e20			−0.846364			−0.423182	−	0.906045i	$-0.639087\pi$
$751$	−1.14035e20			−0.846364			−0.423182	+	0.906045i	$0.639087\pi$
$752$		0			0
$753$	−1.50201e20			−1.09423
$754$		0			0
$755$			1.99803e19i			0.142881i
$756$		0			0
$757$	−1.51523e20			−1.06367			−0.531837	−	0.846847i	$-0.678498\pi$
$757$	−1.51523e20			−1.06367			−0.531837	+	0.846847i	$0.678498\pi$
$758$		0			0
$759$		−	5.35552e20i		−	3.69070i
$760$		0			0
$761$			1.74811e20i			1.18271i	0.806413	+	0.591353i	$0.201406\pi$
$761$			1.74811e20i			1.18271i	−0.806413	+	0.591353i	$0.798594\pi$
$762$		0			0
$763$	3.83829e18	+	1.45997e19i	0.0254956	+	0.0969781i
$764$		0			0
$765$	1.74327e19			0.113693
$766$		0			0
$767$	−2.53928e17			−0.00162609
$768$		0			0
$769$			1.52426e20i			0.958463i	0.877689	+	0.479232i	$0.159085\pi$
$769$			1.52426e20i			0.958463i	−0.877689	+	0.479232i	$0.840915\pi$
$770$		0			0
$771$	−1.35834e20			−0.838742
$772$		0			0
$773$			2.98327e20i			1.80899i	0.426487	+	0.904494i	$0.359751\pi$
$773$			2.98327e20i			1.80899i	−0.426487	+	0.904494i	$0.640249\pi$
$774$		0			0
$775$			2.32882e19i			0.138683i
$776$		0			0
$777$	−3.90864e20	+	1.02758e20i	−2.28601	+	0.600995i
$778$		0			0
$779$	−4.53746e18			−0.0260646
$780$		0			0
$781$	−3.30395e20			−1.86413
$782$		0			0
$783$			3.84204e19i			0.212926i
$784$		0			0
$785$	6.27579e18			0.0341649
$786$		0			0
$787$		−	4.24211e19i		−	0.226861i	−0.993546	−	0.113430i	$-0.963816\pi$
$787$		−	4.24211e19i		−	0.226861i	0.993546	−	0.113430i	$-0.0361839\pi$
$788$		0			0
$789$		−	3.44478e19i		−	0.180977i
$790$		0			0
$791$	1.29811e19	+	4.93763e19i	0.0670001	+	0.254849i
$792$		0			0
$793$	4.51660e18			0.0229034
$794$		0			0
$795$	1.12416e20			0.560089
$796$		0			0
$797$		−	1.41668e20i		−	0.693526i	−0.937953	−	0.346763i	$-0.887281\pi$
$797$		−	1.41668e20i		−	0.693526i	0.937953	−	0.346763i	$-0.112719\pi$
$798$		0			0
$799$	−2.01134e19			−0.0967518
$800$		0			0
$801$			1.65125e20i			0.780520i
$802$		0			0
$803$		−	7.49758e19i		−	0.348267i
$804$		0			0
$805$	−1.51449e20	+	3.98161e19i	−0.691345	+	0.181756i
$806$		0			0
$807$	5.44705e20			2.44369
$808$		0			0
$809$	−2.26380e20			−0.998157			−0.499078	−	0.866557i	$-0.666328\pi$
$809$	−2.26380e20			−0.998157			−0.499078	+	0.866557i	$0.666328\pi$
$810$		0			0
$811$			4.25708e20i			1.84487i	0.386152	+	0.922435i	$0.373804\pi$
$811$			4.25708e20i			1.84487i	−0.386152	+	0.922435i	$0.626196\pi$
$812$		0			0
$813$	−8.76207e19			−0.373227
$814$		0			0
$815$			4.25214e19i			0.178035i
$816$		0			0
$817$			1.35538e20i			0.557837i
$818$		0			0
$819$	−9.64913e18	+	2.53677e18i	−0.0390392	+	0.0102634i
$820$		0			0
$821$	2.29976e20			0.914703			0.457352	−	0.889286i	$-0.348798\pi$
$821$	2.29976e20			0.914703			0.457352	+	0.889286i	$0.348798\pi$
$822$		0			0
$823$	4.21872e20			1.64961			0.824806	−	0.565415i	$-0.191284\pi$
$823$	4.21872e20			1.64961			0.824806	+	0.565415i	$0.191284\pi$
$824$		0			0
$825$		−	1.20122e20i		−	0.461792i
$826$		0			0
$827$	−2.42405e20			−0.916226			−0.458113	−	0.888894i	$-0.651475\pi$
$827$	−2.42405e20			−0.916226			−0.458113	+	0.888894i	$0.651475\pi$
$828$		0			0
$829$			3.30352e20i			1.22771i	0.789420	+	0.613853i	$0.210381\pi$
$829$			3.30352e20i			1.22771i	−0.789420	+	0.613853i	$0.789619\pi$
$830$		0			0
$831$		−	1.75341e20i		−	0.640733i
$832$		0			0
$833$	−3.89688e19	−	6.89906e19i	−0.140024	−	0.247899i
$834$		0			0
$835$	8.62305e19			0.304688
$836$		0			0
$837$	−2.93996e19			−0.102156
$838$		0			0
$839$		−	1.16858e20i		−	0.399324i	−0.979865	−	0.199662i	$-0.936016\pi$
$839$		−	1.16858e20i		−	0.399324i	0.979865	−	0.199662i	$-0.0639845\pi$
$840$		0			0
$841$	3.24016e20			1.08892
$842$		0			0
$843$			7.47524e20i			2.47077i
$844$		0			0
$845$			1.37285e20i			0.446300i
$846$		0			0
$847$	−1.44439e20	−	5.49404e20i	−0.461849	−	1.75674i
$848$		0			0
$849$	−5.48047e20			−1.72371
$850$		0			0
$851$	−8.87614e20			−2.74610
$852$		0			0
$853$		−	2.84193e20i		−	0.864909i	−0.901656	−	0.432455i	$-0.857648\pi$
$853$		−	2.84193e20i		−	0.864909i	0.901656	−	0.432455i	$-0.142352\pi$
$854$		0			0
$855$	−1.08273e20			−0.324158
$856$		0			0
$857$			1.63856e20i			0.482611i	0.970449	+	0.241305i	$0.0775755\pi$
$857$			1.63856e20i			0.482611i	−0.970449	+	0.241305i	$0.922424\pi$
$858$		0			0
$859$			3.35765e20i			0.972934i	0.873699	+	0.486467i	$0.161715\pi$
$859$			3.35765e20i			0.972934i	−0.873699	+	0.486467i	$0.838285\pi$
$860$		0			0
$861$	3.93994e18	+	1.49864e19i	0.0112323	+	0.0427243i
$862$		0			0
$863$	−1.39296e20			−0.390717			−0.195358	−	0.980732i	$-0.562587\pi$
$863$	−1.39296e20			−0.390717			−0.195358	+	0.980732i	$0.562587\pi$
$864$		0			0
$865$	2.62292e19			0.0723889
$866$		0			0
$867$		−	4.65572e20i		−	1.26431i
$868$		0			0
$869$	8.55213e20			2.28526
$870$		0			0
$871$		−	1.14972e19i		−	0.0302319i
$872$		0			0
$873$		−	7.81089e19i		−	0.202116i
$874$		0			0
$875$	−3.39695e19	+	8.93061e18i	−0.0865032	+	0.0227418i
$876$		0			0
$877$	−1.43354e20			−0.359264			−0.179632	−	0.983734i	$-0.557491\pi$
$877$	−1.43354e20			−0.359264			−0.179632	+	0.983734i	$0.557491\pi$
$878$		0			0
$879$	1.00159e21			2.47040
$880$		0			0
$881$		−	6.19120e20i		−	1.50295i	−0.659763	−	0.751473i	$-0.729343\pi$
$881$		−	6.19120e20i		−	1.50295i	0.659763	−	0.751473i	$-0.270657\pi$
$882$		0			0
$883$	6.34083e20			1.51503			0.757515	−	0.652817i	$-0.226413\pi$
$883$	6.34083e20			1.51503			0.757515	+	0.652817i	$0.226413\pi$
$884$		0			0
$885$			9.41082e18i			0.0221322i
$886$		0			0
$887$			4.07879e20i			0.944203i	0.881544	+	0.472102i	$0.156504\pi$
$887$			4.07879e20i			0.944203i	−0.881544	+	0.472102i	$0.843496\pi$
$888$		0			0
$889$	1.01168e20	+	3.84815e20i	0.230533	+	0.876880i
$890$		0			0
$891$	8.19693e20			1.83868
$892$		0			0
$893$	1.24923e20			0.275856
$894$		0			0
$895$			3.04326e20i			0.661572i
$896$		0			0
$897$	−4.64522e19			−0.0994167
$898$		0			0
$899$			4.75634e20i			1.00220i
$900$		0			0
$901$			1.24927e20i			0.259168i
$902$		0			0
$903$	4.47658e20	−	1.17690e20i	0.914392	−	0.240395i
$904$		0			0
$905$	−3.40599e20			−0.685020
$906$		0			0
$907$	−6.14426e20			−1.21680			−0.608400	−	0.793631i	$-0.708188\pi$
$907$	−6.14426e20			−1.21680			−0.608400	+	0.793631i	$0.708188\pi$
$908$		0			0
$909$		−	6.55597e20i		−	1.27847i
$910$		0			0
$911$	7.70704e20			1.47999			0.739995	−	0.672612i	$-0.234828\pi$
$911$	7.70704e20			1.47999			0.739995	+	0.672612i	$0.234828\pi$
$912$		0			0
$913$		−	1.12305e21i		−	2.12375i
$914$		0			0
$915$		−	1.67389e20i		−	0.311731i
$916$		0			0
$917$	−4.73825e20	+	1.24569e20i	−0.869026	+	0.228468i
$918$		0			0
$919$	1.01140e20			0.182689			0.0913446	−	0.995819i	$-0.470884\pi$
$919$	1.01140e20			0.182689			0.0913446	+	0.995819i	$0.470884\pi$
$920$		0			0
$921$	−7.72734e20			−1.37472
$922$		0			0
$923$			2.86575e19i			0.0502142i
$924$		0			0
$925$	−1.99089e20			−0.343601
$926$		0			0
$927$		−	5.08983e19i		−	0.0865257i
$928$		0			0
$929$		−	7.60774e20i		−	1.27393i	−0.770892	−	0.636965i	$-0.780189\pi$
$929$		−	7.60774e20i		−	1.27393i	0.770892	−	0.636965i	$-0.219811\pi$
$930$		0			0
$931$	2.42033e20	+	4.28496e20i	0.399233	+	0.706804i
$932$		0			0
$933$	4.73538e20			0.769456
$934$		0			0
$935$	1.33491e20			0.213683
$936$		0			0
$937$		−	4.43334e20i		−	0.699124i	−0.936913	−	0.349562i	$-0.886330\pi$
$937$		−	4.43334e20i		−	0.699124i	0.936913	−	0.349562i	$-0.113670\pi$
$938$		0			0
$939$	1.73140e21			2.68991
$940$		0			0
$941$			7.69272e20i			1.17748i	0.808324	+	0.588738i	$0.200375\pi$
$941$			7.69272e20i			1.17748i	−0.808324	+	0.588738i	$0.799625\pi$
$942$		0			0
$943$			3.40327e19i			0.0513233i
$944$		0			0
$945$	−1.12742e19	−	4.28838e19i	−0.0167519	−	0.0637193i
$946$		0			0
$947$	5.84910e20			0.856326			0.428163	−	0.903702i	$-0.359161\pi$
$947$	5.84910e20			0.856326			0.428163	+	0.903702i	$0.359161\pi$
$948$		0			0
$949$	−6.50318e18			−0.00938129
$950$		0			0
$951$			1.35838e21i			1.93090i
$952$		0			0
$953$	3.17896e20			0.445281			0.222641	−	0.974901i	$-0.428532\pi$
$953$	3.17896e20			0.445281			0.222641	+	0.974901i	$0.428532\pi$
$954$		0			0
$955$			4.58525e20i			0.632907i
$956$		0			0
$957$		−	2.45335e21i		−	3.33716i
$958$		0			0
$959$	−2.52841e20	−	9.61736e20i	−0.338936	−	1.28922i
$960$		0			0
$961$	3.92985e20			0.519173
$962$		0			0
$963$	3.89742e19			0.0507450
$964$		0			0
$965$		−	2.34090e19i		−	0.0300395i
$966$		0			0
$967$	8.91495e20			1.12754			0.563772	−	0.825930i	$-0.309350\pi$
$967$	8.91495e20			1.12754			0.563772	+	0.825930i	$0.309350\pi$
$968$		0			0
$969$		−	2.55074e20i		−	0.317980i
$970$		0			0
$971$			1.07079e21i			1.31573i	0.753134	+	0.657867i	$0.228541\pi$
$971$			1.07079e21i			1.31573i	−0.753134	+	0.657867i	$0.771459\pi$
$972$		0			0
$973$	−1.14535e21	+	3.01113e20i	−1.38723	+	0.364703i
$974$		0			0
$975$	−1.04191e19			−0.0124393
$976$		0			0
$977$	1.75026e20			0.205988			0.102994	−	0.994682i	$-0.467158\pi$
$977$	1.75026e20			0.205988			0.102994	+	0.994682i	$0.467158\pi$
$978$		0			0
$979$			1.26444e21i			1.46697i
$980$		0			0
$981$	7.82857e19			0.0895363
$982$		0			0
$983$		−	9.85936e20i		−	1.11167i	−0.831294	−	0.555833i	$-0.812400\pi$
$983$		−	9.85936e20i		−	1.11167i	0.831294	−	0.555833i	$-0.187600\pi$
$984$		0			0
$985$			3.93169e20i			0.437044i
$986$		0			0
$987$	−1.08473e20	−	4.12598e20i	−0.118878	−	0.452176i
$988$		0			0
$989$	1.01659e21			1.09843
$990$		0			0
$991$	4.15492e20			0.442637			0.221318	−	0.975202i	$-0.428964\pi$
$991$	4.15492e20			0.442637			0.221318	+	0.975202i	$0.428964\pi$
$992$		0			0
$993$			5.60058e20i			0.588286i
$994$		0			0
$995$	−6.29584e20			−0.652067
$996$		0			0
$997$		−	1.33037e21i		−	1.35865i	−0.733838	−	0.679324i	$-0.762273\pi$
$997$		−	1.33037e21i		−	1.35865i	0.733838	−	0.679324i	$-0.237727\pi$
$998$		0			0
$999$		−	2.51334e20i		−	0.253101i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.15.d.a.41.6	✓	36
7.6	odd	2	inner	140.15.d.a.41.31	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.15.d.a.41.6	✓	36	1.1	even	1	trivial
140.15.d.a.41.31	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)

\(f(q)\)	\(=\)	\(q-3008.95i q^{3} +34938.6i q^{5} +(-209395. - 796478. i) q^{7} -4.27082e6 q^{9} +O(q^{10})\)	\(q-3008.95i q^{3} +34938.6i q^{5} +(-209395. - 796478. i) q^{7} -4.27082e6 q^{9} -3.27038e7 q^{11} +2.83664e6i q^{13} +1.05128e8 q^{15} +1.16828e8i q^{17} -7.25611e8i q^{19} +(-2.39656e9 + 6.30059e8i) q^{21} -5.44236e9 q^{23} -1.22070e9 q^{25} -1.54104e9i q^{27} -2.49314e10 q^{29} -1.90777e10i q^{31} +9.84043e10i q^{33} +(2.78278e10 - 7.31596e9i) q^{35} +1.63093e11 q^{37} +8.53530e9 q^{39} -6.25329e9i q^{41} -1.86792e11 q^{43} -1.49216e11i q^{45} +1.72163e11i q^{47} +(-5.90531e11 + 3.33557e11i) q^{49} +3.51530e11 q^{51} +1.06932e12 q^{53} -1.14263e12i q^{55} -2.18333e12 q^{57} +8.95174e10i q^{59} -1.59224e12i q^{61} +(8.94287e11 + 3.40161e12i) q^{63} -9.91080e10 q^{65} -4.05312e12 q^{67} +1.63758e13i q^{69} +1.01026e13 q^{71} +2.29257e12i q^{73} +3.67304e12i q^{75} +(6.84802e12 + 2.60479e13i) q^{77} -2.61502e13 q^{79} -2.50641e13 q^{81} +3.43401e13i q^{83} -4.08181e12 q^{85} +7.50173e13i q^{87} -3.86635e13i q^{89} +(2.25932e12 - 5.93977e11i) q^{91} -5.74039e13 q^{93} +2.53518e13 q^{95} +1.82890e13i q^{97} +1.39672e14 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 1364266 q^{7} - 54790830 q^{9}+O(q^{10}) \) 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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