LMFDB - Embedded newform 140.15.d.a.41.5

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			41.5
Character	$\chi$	$=$	140.41
Dual form			140.15.d.a.41.32

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3089.94i q^{3} -34938.6i q^{5} +(775860. - 276159. i) q^{7} -4.76477e6 q^{9} +O(q^{10})$	\(q-3089.94i q^{3} -34938.6i q^{5} +(775860. - 276159. i) q^{7} -4.76477e6 q^{9} -1.57426e7 q^{11} -7.52264e7i q^{13} -1.07958e8 q^{15} -5.31929e8i q^{17} +6.79635e8i q^{19} +(-8.53315e8 - 2.39736e9i) q^{21} +2.16074e9 q^{23} -1.22070e9 q^{25} -5.62383e7i q^{27} -3.70703e9 q^{29} -3.57385e9i q^{31} +4.86438e10i q^{33} +(-9.64860e9 - 2.71074e10i) q^{35} +5.03725e9 q^{37} -2.32445e11 q^{39} +2.33128e11i q^{41} -2.46665e11 q^{43} +1.66474e11i q^{45} -4.68156e11i q^{47} +(5.25695e11 - 4.28522e11i) q^{49} -1.64363e12 q^{51} -1.75321e11 q^{53} +5.50025e11i q^{55} +2.10003e12 q^{57} -3.59189e12i q^{59} -3.47300e12i q^{61} +(-3.69679e12 + 1.31583e12i) q^{63} -2.62830e12 q^{65} -5.66050e12 q^{67} -6.67655e12i q^{69} +7.81227e10 q^{71} +7.35820e12i q^{73} +3.77190e12i q^{75} +(-1.22141e13 + 4.34747e12i) q^{77} -2.41833e13 q^{79} -2.29635e13 q^{81} +7.63289e12i q^{83} -1.85848e13 q^{85} +1.14545e13i q^{87} -1.57376e13i q^{89} +(-2.07744e13 - 5.83652e13i) q^{91} -1.10430e13 q^{93} +2.37455e13 q^{95} +6.35298e13i q^{97} +7.50100e13 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	3089.94i		−	1.41287i	−0.707779	−	0.706434i	$-0.750303\pi$
$3$		−	3089.94i		−	1.41287i	0.707779	−	0.706434i	$-0.249697\pi$
$4$		0			0
$5$		−	34938.6i		−	0.447214i
$5$		−	34938.6i		−	0.447214i
$6$		0			0
$7$	775860.	−	276159.i	0.942101	−	0.335330i
$7$	775860.	−	276159.i	0.942101	−	0.335330i
$8$		0			0
$9$	−4.76477e6			−0.996195
$10$		0			0
$11$	−1.57426e7			−0.807846			−0.403923	−	0.914793i	$-0.632354\pi$
$11$	−1.57426e7			−0.807846			−0.403923	+	0.914793i	$0.632354\pi$
$12$		0			0
$13$		−	7.52264e7i		−	1.19886i	−0.800429	−	0.599428i	$-0.795395\pi$
$13$		−	7.52264e7i		−	1.19886i	0.800429	−	0.599428i	$-0.204605\pi$
$14$		0			0
$15$	−1.07958e8			−0.631854
$16$		0			0
$17$		−	5.31929e8i		−	1.29632i	−0.761505	−	0.648159i	$-0.775539\pi$
$17$		−	5.31929e8i		−	1.29632i	0.761505	−	0.648159i	$-0.224461\pi$
$18$		0			0
$19$			6.79635e8i			0.760327i	0.924919	+	0.380163i	$0.124132\pi$
$19$			6.79635e8i			0.760327i	−0.924919	+	0.380163i	$0.875868\pi$
$20$		0			0
$21$	−8.53315e8	−	2.39736e9i	−0.473778	−	1.33106i
$22$		0			0
$23$	2.16074e9			0.634610			0.317305	−	0.948324i	$-0.397222\pi$
$23$	2.16074e9			0.634610			0.317305	+	0.948324i	$0.397222\pi$
$24$		0			0
$25$	−1.22070e9			−0.200000
$26$		0			0
$27$		−	5.62383e7i		−	0.00537633i
$28$		0			0
$29$	−3.70703e9			−0.214902			−0.107451	−	0.994210i	$-0.534269\pi$
$29$	−3.70703e9			−0.214902			−0.107451	+	0.994210i	$0.534269\pi$
$30$		0			0
$31$		−	3.57385e9i		−	0.129899i	−0.997889	−	0.0649493i	$-0.979311\pi$
$31$		−	3.57385e9i		−	0.129899i	0.997889	−	0.0649493i	$-0.0206886\pi$
$32$		0			0
$33$			4.86438e10i			1.14138i
$34$		0			0
$35$	−9.64860e9	−	2.71074e10i	−0.149964	−	0.421320i
$36$		0			0
$37$	5.03725e9			0.0530617			0.0265309	−	0.999648i	$-0.491554\pi$
$37$	5.03725e9			0.0530617			0.0265309	+	0.999648i	$0.491554\pi$
$38$		0			0
$39$	−2.32445e11			−1.69382
$40$		0			0
$41$			2.33128e11i			1.19704i	0.801108	+	0.598519i	$0.204244\pi$
$41$			2.33128e11i			1.19704i	−0.801108	+	0.598519i	$0.795756\pi$
$42$		0			0
$43$	−2.46665e11			−0.907464			−0.453732	−	0.891138i	$-0.649908\pi$
$43$	−2.46665e11			−0.907464			−0.453732	+	0.891138i	$0.649908\pi$
$44$		0			0
$45$			1.66474e11i			0.445512i
$46$		0			0
$47$		−	4.68156e11i		−	0.924072i	−0.886861	−	0.462036i	$-0.847119\pi$
$47$		−	4.68156e11i		−	0.924072i	0.886861	−	0.462036i	$-0.152881\pi$
$48$		0			0
$49$	5.25695e11	−	4.28522e11i	0.775107	−	0.631830i
$50$		0			0
$51$	−1.64363e12			−1.83152
$52$		0			0
$53$	−1.75321e11			−0.149246			−0.0746230	−	0.997212i	$-0.523775\pi$
$53$	−1.75321e11			−0.149246			−0.0746230	+	0.997212i	$0.523775\pi$
$54$		0			0
$55$			5.50025e11i			0.361280i
$56$		0			0
$57$	2.10003e12			1.07424
$58$		0			0
$59$		−	3.59189e12i		−	1.44331i	−0.692254	−	0.721654i	$-0.743382\pi$
$59$		−	3.59189e12i		−	1.44331i	0.692254	−	0.721654i	$-0.256618\pi$
$60$		0			0
$61$		−	3.47300e12i		−	1.10509i	−0.833484	−	0.552543i	$-0.813657\pi$
$61$		−	3.47300e12i		−	1.10509i	0.833484	−	0.552543i	$-0.186343\pi$
$62$		0			0
$63$	−3.69679e12	+	1.31583e12i	−0.938516	+	0.334054i
$64$		0			0
$65$	−2.62830e12			−0.536144
$66$		0			0
$67$	−5.66050e12			−0.933966			−0.466983	−	0.884266i	$-0.654659\pi$
$67$	−5.66050e12			−0.933966			−0.466983	+	0.884266i	$0.654659\pi$
$68$		0			0
$69$		−	6.67655e12i		−	0.896620i
$70$		0			0
$71$	7.81227e10			0.00858952			0.00429476	−	0.999991i	$-0.498633\pi$
$71$	7.81227e10			0.00858952			0.00429476	+	0.999991i	$0.498633\pi$
$72$		0			0
$73$			7.35820e12i			0.666057i	0.942917	+	0.333029i	$0.108071\pi$
$73$			7.35820e12i			0.666057i	−0.942917	+	0.333029i	$0.891929\pi$
$74$		0			0
$75$			3.77190e12i			0.282574i
$76$		0			0
$77$	−1.22141e13	+	4.34747e12i	−0.761072	+	0.270895i
$78$		0			0
$79$	−2.41833e13			−1.25929			−0.629644	−	0.776884i	$-0.716799\pi$
$79$	−2.41833e13			−1.25929			−0.629644	+	0.776884i	$0.716799\pi$
$80$		0			0
$81$	−2.29635e13			−1.00379
$82$		0			0
$83$			7.63289e12i			0.281282i	0.990061	+	0.140641i	$0.0449164\pi$
$83$			7.63289e12i			0.281282i	−0.990061	+	0.140641i	$0.955084\pi$
$84$		0			0
$85$	−1.85848e13			−0.579731
$86$		0			0
$87$			1.14545e13i			0.303628i
$88$		0			0
$89$		−	1.57376e13i		−	0.355801i	−0.984048	−	0.177900i	$-0.943069\pi$
$89$		−	1.57376e13i		−	0.355801i	0.984048	−	0.177900i	$-0.0569305\pi$
$90$		0			0
$91$	−2.07744e13	−	5.83652e13i	−0.402013	−	1.12944i
$92$		0			0
$93$	−1.10430e13			−0.183530
$94$		0			0
$95$	2.37455e13			0.340028
$96$		0			0
$97$			6.35298e13i			0.786277i	0.919479	+	0.393138i	$0.128611\pi$
$97$			6.35298e13i			0.786277i	−0.919479	+	0.393138i	$0.871389\pi$
$98$		0			0
$99$	7.50100e13			0.804772
$100$		0			0
$101$		−	4.34554e13i		−	0.405316i	−0.979250	−	0.202658i	$-0.935042\pi$
$101$		−	4.34554e13i		−	0.405316i	0.979250	−	0.202658i	$-0.0649580\pi$
$102$		0			0
$103$		−	1.16435e14i		−	0.946723i	−0.880868	−	0.473361i	$-0.843041\pi$
$103$		−	1.16435e14i		−	0.946723i	0.880868	−	0.473361i	$-0.156959\pi$
$104$		0			0
$105$	−8.37604e13	+	2.98136e13i	−0.595270	+	0.211880i
$106$		0			0
$107$	−1.08317e14			−0.674544			−0.337272	−	0.941407i	$-0.609504\pi$
$107$	−1.08317e14			−0.674544			−0.337272	+	0.941407i	$0.609504\pi$
$108$		0			0
$109$	−2.17181e14			−1.18806			−0.594028	−	0.804444i	$-0.702463\pi$
$109$	−2.17181e14			−1.18806			−0.594028	+	0.804444i	$0.702463\pi$
$110$		0			0
$111$		−	1.55648e13i		−	0.0749692i
$112$		0			0
$113$	3.04644e14			1.29492			0.647460	−	0.762099i	$-0.275831\pi$
$113$	3.04644e14			1.29492			0.647460	+	0.762099i	$0.275831\pi$
$114$		0			0
$115$		−	7.54930e13i		−	0.283806i
$116$		0			0
$117$			3.58436e14i			1.19429i
$118$		0			0
$119$	−1.46897e14	−	4.12703e14i	−0.434695	−	1.22126i
$120$		0			0
$121$	−1.31920e14			−0.347385
$122$		0			0
$123$	7.20353e14			1.69126
$124$		0			0
$125$			4.26496e13i			0.0894427i
$126$		0			0
$127$	6.77662e14			1.27171			0.635853	−	0.771810i	$-0.280648\pi$
$127$	6.77662e14			1.27171			0.635853	+	0.771810i	$0.280648\pi$
$128$		0			0
$129$			7.62182e14i			1.28213i
$130$		0			0
$131$			9.82022e14i			1.48328i	0.670800	+	0.741638i	$0.265951\pi$
$131$			9.82022e14i			1.48328i	−0.670800	+	0.741638i	$0.734049\pi$
$132$		0			0
$133$	1.87687e14	+	5.27302e14i	0.254961	+	0.716304i
$134$		0			0
$135$	−1.96489e12			−0.00240437
$136$		0			0
$137$	1.23068e15			1.35863			0.679315	−	0.733847i	$-0.262277\pi$
$137$	1.23068e15			1.35863			0.679315	+	0.733847i	$0.262277\pi$
$138$		0			0
$139$			1.92042e13i			0.0191555i	0.999954	+	0.00957773i	$0.00304873\pi$
$139$			1.92042e13i			0.0191555i	−0.999954	+	0.00957773i	$0.996951\pi$
$140$		0			0
$141$	−1.44658e15			−1.30559
$142$		0			0
$143$			1.18426e15i			0.968490i
$144$		0			0
$145$			1.29518e14i			0.0961071i
$146$		0			0
$147$	−1.32411e15	−	1.62437e15i	−0.892692	−	1.09512i
$148$		0			0
$149$	1.76137e14			0.108031			0.0540153	−	0.998540i	$-0.482798\pi$
$149$	1.76137e14			0.108031			0.0540153	+	0.998540i	$0.482798\pi$
$150$		0			0
$151$	1.37739e15			0.769520			0.384760	−	0.923017i	$-0.374284\pi$
$151$	1.37739e15			0.769520			0.384760	+	0.923017i	$0.374284\pi$
$152$		0			0
$153$			2.53452e15i			1.29138i
$154$		0			0
$155$	−1.24865e14			−0.0580924
$156$		0			0
$157$		−	1.82894e15i		−	0.777859i	−0.921267	−	0.388930i	$-0.872845\pi$
$157$		−	1.82894e15i		−	0.777859i	0.921267	−	0.388930i	$-0.127155\pi$
$158$		0			0
$159$			5.41731e14i			0.210865i
$160$		0			0
$161$	1.67643e15	−	5.96707e14i	0.597866	−	0.212804i
$162$		0			0
$163$	6.09632e15			1.99414			0.997068	−	0.0765178i	$-0.0243802\pi$
$163$	6.09632e15			1.99414			0.997068	+	0.0765178i	$0.0243802\pi$
$164$		0			0
$165$	1.69954e15			0.510440
$166$		0			0
$167$			2.22881e15i			0.615259i	0.951506	+	0.307629i	$0.0995357\pi$
$167$			2.22881e15i			0.615259i	−0.951506	+	0.307629i	$0.900464\pi$
$168$		0			0
$169$	−1.72163e15			−0.437254
$170$		0			0
$171$		−	3.23830e15i		−	0.757434i
$172$		0			0
$173$			6.97553e15i			1.50402i	0.659150	+	0.752012i	$0.270916\pi$
$173$			6.97553e15i			1.50402i	−0.659150	+	0.752012i	$0.729084\pi$
$174$		0			0
$175$	−9.47095e14	+	3.37108e14i	−0.188420	+	0.0670661i
$176$		0			0
$177$	−1.10987e16			−2.03920
$178$		0			0
$179$	−1.00613e16			−1.70877			−0.854384	−	0.519641i	$-0.826066\pi$
$179$	−1.00613e16			−1.70877			−0.854384	+	0.519641i	$0.826066\pi$
$180$		0			0
$181$		−	3.88310e15i		−	0.610138i	−0.952330	−	0.305069i	$-0.901320\pi$
$181$		−	3.88310e15i		−	0.610138i	0.952330	−	0.305069i	$-0.0986796\pi$
$182$		0			0
$183$	−1.07314e16			−1.56134
$184$		0			0
$185$		−	1.75994e14i		−	0.0237299i
$186$		0			0
$187$			8.37396e15i			1.04722i
$188$		0			0
$189$	−1.55307e13	−	4.36331e13i	−0.00180285	−	0.00506504i
$190$		0			0
$191$	−1.72302e16			−1.85804			−0.929021	−	0.370026i	$-0.879349\pi$
$191$	−1.72302e16			−1.85804			−0.929021	+	0.370026i	$0.879349\pi$
$192$		0			0
$193$	−1.15600e15			−0.115893			−0.0579463	−	0.998320i	$-0.518455\pi$
$193$	−1.15600e15			−0.115893			−0.0579463	+	0.998320i	$0.518455\pi$
$194$		0			0
$195$			8.12130e15i			0.757501i
$196$		0			0
$197$	−5.93225e15			−0.515176			−0.257588	−	0.966255i	$-0.582928\pi$
$197$	−5.93225e15			−0.515176			−0.257588	+	0.966255i	$0.582928\pi$
$198$		0			0
$199$			1.74142e16i			1.40907i	0.709671	+	0.704534i	$0.248844\pi$
$199$			1.74142e16i			1.40907i	−0.709671	+	0.704534i	$0.751156\pi$
$200$		0			0
$201$			1.74906e16i			1.31957i
$202$		0			0
$203$	−2.87614e15	+	1.02373e15i	−0.202459	+	0.0720632i
$204$		0			0
$205$	8.14517e15			0.535332
$206$		0			0
$207$	−1.02954e16			−0.632195
$208$		0			0
$209$		−	1.06992e16i		−	0.614227i
$210$		0			0
$211$	9.61369e15			0.516313			0.258156	−	0.966103i	$-0.416885\pi$
$211$	9.61369e15			0.516313			0.258156	+	0.966103i	$0.416885\pi$
$212$		0			0
$213$		−	2.41395e14i		−	0.0121359i
$214$		0			0
$215$			8.61814e15i			0.405830i
$216$		0			0
$217$	−9.86951e14	−	2.77281e15i	−0.0435590	−	0.122378i
$218$		0			0
$219$	2.27364e16			0.941051
$220$		0			0
$221$	−4.00151e16			−1.55410
$222$		0			0
$223$		−	2.38018e16i		−	0.867913i	−0.900934	−	0.433957i	$-0.857117\pi$
$223$		−	2.38018e16i		−	0.867913i	0.900934	−	0.433957i	$-0.142883\pi$
$224$		0			0
$225$	5.81637e15			0.199239
$226$		0			0
$227$			5.03251e16i			1.62033i	0.586201	+	0.810166i	$0.300623\pi$
$227$			5.03251e16i			1.62033i	−0.586201	+	0.810166i	$0.699377\pi$
$228$		0			0
$229$			2.56530e15i			0.0776767i	0.999246	+	0.0388383i	$0.0123657\pi$
$229$			2.56530e15i			0.0776767i	−0.999246	+	0.0388383i	$0.987634\pi$
$230$		0			0
$231$	1.34334e16	+	3.77408e16i	0.382739	+	1.07529i
$232$		0			0
$233$	1.78026e16			0.477520			0.238760	−	0.971079i	$-0.423259\pi$
$233$	1.78026e16			0.477520			0.238760	+	0.971079i	$0.423259\pi$
$234$		0			0
$235$	−1.63567e16			−0.413258
$236$		0			0
$237$			7.47248e16i			1.77921i
$238$		0			0
$239$	5.01154e16			1.12509			0.562543	−	0.826768i	$-0.309823\pi$
$239$	5.01154e16			1.12509			0.562543	+	0.826768i	$0.309823\pi$
$240$		0			0
$241$			1.86064e16i			0.394042i	0.980399	+	0.197021i	$0.0631267\pi$
$241$			1.86064e16i			0.394042i	−0.980399	+	0.197021i	$0.936873\pi$
$242$		0			0
$243$			7.06869e16i			1.41285i
$244$		0			0
$245$	−1.49719e16	−	1.83670e16i	−0.282563	−	0.346638i
$246$		0			0
$247$	5.11265e16			0.911522
$248$		0			0
$249$	2.35852e16			0.397414
$250$		0			0
$251$			4.38184e16i			0.698136i	0.937097	+	0.349068i	$0.113502\pi$
$251$			4.38184e16i			0.698136i	−0.937097	+	0.349068i	$0.886498\pi$
$252$		0			0
$253$	−3.40157e16			−0.512667
$254$		0			0
$255$			5.74261e16i			0.819083i
$256$		0			0
$257$			7.12803e16i			0.962582i	0.876561	+	0.481291i	$0.159832\pi$
$257$			7.12803e16i			0.962582i	−0.876561	+	0.481291i	$0.840168\pi$
$258$		0			0
$259$	3.90820e15	−	1.39108e15i	0.0499895	−	0.0177932i
$260$		0			0
$261$	1.76632e16			0.214084
$262$		0			0
$263$	3.64303e16			0.418575			0.209287	−	0.977854i	$-0.432886\pi$
$263$	3.64303e16			0.418575			0.209287	+	0.977854i	$0.432886\pi$
$264$		0			0
$265$			6.12546e15i			0.0667448i
$266$		0			0
$267$	−4.86281e16			−0.502700
$268$		0			0
$269$			1.19390e17i			1.17139i	0.810532	+	0.585694i	$0.199178\pi$
$269$			1.19390e17i			1.17139i	−0.810532	+	0.585694i	$0.800822\pi$
$270$		0			0
$271$		−	5.40417e16i		−	0.503436i	−0.967801	−	0.251718i	$-0.919005\pi$
$271$		−	5.40417e16i		−	0.503436i	0.967801	−	0.251718i	$-0.0809955\pi$
$272$		0			0
$273$	−1.80345e17	+	6.41918e16i	−1.59575	+	0.567991i
$274$		0			0
$275$	1.92171e16			0.161569
$276$		0			0
$277$	−5.68548e16			−0.454369			−0.227185	−	0.973852i	$-0.572952\pi$
$277$	−5.68548e16			−0.454369			−0.227185	+	0.973852i	$0.572952\pi$
$278$		0			0
$279$			1.70286e16i			0.129404i
$280$		0			0
$281$	9.03456e15			0.0653075			0.0326537	−	0.999467i	$-0.489604\pi$
$281$	9.03456e15			0.0653075			0.0326537	+	0.999467i	$0.489604\pi$
$282$		0			0
$283$			4.80913e15i			0.0330797i	0.999863	+	0.0165399i	$0.00526504\pi$
$283$			4.80913e15i			0.0330797i	−0.999863	+	0.0165399i	$0.994735\pi$
$284$		0			0
$285$		−	7.33721e16i		−	0.480415i
$286$		0			0
$287$	6.43805e16	+	1.80875e17i	0.401404	+	1.12773i
$288$		0			0
$289$	−1.14571e17			−0.680438
$290$		0			0
$291$	1.96303e17			1.11091
$292$		0			0
$293$			2.59916e16i			0.140204i	0.997540	+	0.0701020i	$0.0223325\pi$
$293$			2.59916e16i			0.140204i	−0.997540	+	0.0701020i	$0.977668\pi$
$294$		0			0
$295$	−1.25495e17			−0.645467
$296$		0			0
$297$			8.85339e14i			0.00434325i
$298$		0			0
$299$		−	1.62544e17i		−	0.760805i
$300$		0			0
$301$	−1.91378e17	+	6.81189e16i	−0.854922	+	0.304300i
$302$		0			0
$303$	−1.34275e17			−0.572658
$304$		0			0
$305$	−1.21342e17			−0.494210
$306$		0			0
$307$			2.99027e16i			0.116344i	0.998307	+	0.0581718i	$0.0185271\pi$
$307$			2.99027e16i			0.116344i	−0.998307	+	0.0581718i	$0.981473\pi$
$308$		0			0
$309$	−3.59777e17			−1.33759
$310$		0			0
$311$		−	1.49970e16i		−	0.0532943i	−0.999645	−	0.0266472i	$-0.991517\pi$
$311$		−	1.49970e16i		−	0.0532943i	0.999645	−	0.0266472i	$-0.00848306\pi$
$312$		0			0
$313$		−	5.20903e17i		−	1.76989i	−0.465694	−	0.884946i	$-0.654195\pi$
$313$		−	5.20903e17i		−	1.76989i	0.465694	−	0.884946i	$-0.345805\pi$
$314$		0			0
$315$	4.59733e16	+	1.29161e17i	0.149394	+	0.419717i
$316$		0			0
$317$	−2.02502e17			−0.629527			−0.314764	−	0.949170i	$-0.601925\pi$
$317$	−2.02502e17			−0.629527			−0.314764	+	0.949170i	$0.601925\pi$
$318$		0			0
$319$	5.83584e16			0.173608
$320$		0			0
$321$			3.34693e17i			0.953042i
$322$		0			0
$323$	3.61517e17			0.985625
$324$		0			0
$325$			9.18291e16i			0.239771i
$326$		0			0
$327$			6.71077e17i			1.67857i
$328$		0			0
$329$	−1.29286e17	−	3.63224e17i	−0.309870	−	0.870569i
$330$		0			0
$331$	3.01652e17			0.692963			0.346481	−	0.938057i	$-0.387376\pi$
$331$	3.01652e17			0.692963			0.346481	+	0.938057i	$0.387376\pi$
$332$		0			0
$333$	−2.40013e16			−0.0528598
$334$		0			0
$335$			1.97770e17i			0.417682i
$336$		0			0
$337$	3.67931e14			0.000745345			0.000372672	−	1.00000i	$-0.499881\pi$
$337$	3.67931e14			0.000745345			0.000372672		1.00000i	$0.499881\pi$
$338$		0			0
$339$		−	9.41331e17i		−	1.82955i
$340$		0			0
$341$			5.62618e16i			0.104938i
$342$		0			0
$343$	2.89526e17	−	4.77649e17i	0.518357	−	0.855164i
$344$		0			0
$345$	−2.33269e17			−0.400980
$346$		0			0
$347$	−3.05684e17			−0.504622			−0.252311	−	0.967646i	$-0.581191\pi$
$347$	−3.05684e17			−0.504622			−0.252311	+	0.967646i	$0.581191\pi$
$348$		0			0
$349$		−	7.11901e17i		−	1.12886i	−0.825480	−	0.564432i	$-0.809095\pi$
$349$		−	7.11901e17i		−	1.12886i	0.825480	−	0.564432i	$-0.190905\pi$
$350$		0			0
$351$	−4.23061e15			−0.00644544
$352$		0			0
$353$		−	7.56828e17i		−	1.10809i	−0.832487	−	0.554044i	$-0.813084\pi$
$353$		−	7.56828e17i		−	1.10809i	0.832487	−	0.554044i	$-0.186916\pi$
$354$		0			0
$355$		−	2.72950e15i		−	0.00384135i
$356$		0			0
$357$	−1.27523e18	+	4.53903e17i	−1.72548	+	0.614166i
$358$		0			0
$359$	1.38964e18			1.80818			0.904091	−	0.427340i	$-0.140549\pi$
$359$	1.38964e18			1.80818			0.904091	+	0.427340i	$0.140549\pi$
$360$		0			0
$361$	3.37103e17			0.421903
$362$		0			0
$363$			4.07624e17i			0.490810i
$364$		0			0
$365$	2.57085e17			0.297870
$366$		0			0
$367$			1.23299e18i			1.37498i	0.726192	+	0.687492i	$0.241288\pi$
$367$			1.23299e18i			1.37498i	−0.726192	+	0.687492i	$0.758712\pi$
$368$		0			0
$369$		−	1.11080e18i		−	1.19248i
$370$		0			0
$371$	−1.36024e17	+	4.84164e16i	−0.140605	+	0.0500467i
$372$		0			0
$373$	1.44322e17			0.143672			0.0718358	−	0.997416i	$-0.477114\pi$
$373$	1.44322e17			0.143672			0.0718358	+	0.997416i	$0.477114\pi$
$374$		0			0
$375$	1.31785e17			0.126371
$376$		0			0
$377$			2.78867e17i			0.257636i
$378$		0			0
$379$	1.86321e18			1.65877			0.829387	−	0.558675i	$-0.188690\pi$
$379$	1.86321e18			1.65877			0.829387	+	0.558675i	$0.188690\pi$
$380$		0			0
$381$		−	2.09394e18i		−	1.79675i
$382$		0			0
$383$		−	2.57237e17i		−	0.212785i	−0.994324	−	0.106393i	$-0.966070\pi$
$383$		−	2.57237e17i		−	0.212785i	0.994324	−	0.106393i	$-0.0339300\pi$
$384$		0			0
$385$	1.51894e17	+	4.26742e17i	0.121148	+	0.340362i
$386$		0			0
$387$	1.17530e18			0.904010
$388$		0			0
$389$	−1.74839e18			−1.29715			−0.648575	−	0.761150i	$-0.724635\pi$
$389$	−1.74839e18			−1.29715			−0.648575	+	0.761150i	$0.724635\pi$
$390$		0			0
$391$		−	1.14936e18i		−	0.822656i
$392$		0			0
$393$	3.03439e18			2.09567
$394$		0			0
$395$			8.44928e17i			0.563171i
$396$		0			0
$397$			1.86550e18i			1.20022i	0.799917	+	0.600111i	$0.204877\pi$
$397$			1.86550e18i			1.20022i	−0.799917	+	0.600111i	$0.795123\pi$
$398$		0			0
$399$	1.62933e18	−	5.79943e17i	1.01204	−	0.360226i
$400$		0			0
$401$	1.66297e18			0.997409			0.498704	−	0.866772i	$-0.333809\pi$
$401$	1.66297e18			0.997409			0.498704	+	0.866772i	$0.333809\pi$
$402$		0			0
$403$	−2.68848e17			−0.155730
$404$		0			0
$405$			8.02312e17i			0.448909i
$406$		0			0
$407$	−7.92995e16			−0.0428657
$408$		0			0
$409$		−	2.58012e18i		−	1.34765i	−0.738891	−	0.673824i	$-0.764650\pi$
$409$		−	2.58012e18i		−	1.34765i	0.738891	−	0.673824i	$-0.235350\pi$
$410$		0			0
$411$		−	3.80273e18i		−	1.91956i
$412$		0			0
$413$	−9.91933e17	−	2.78680e18i	−0.483985	−	1.35974i
$414$		0			0
$415$	2.66682e17			0.125793
$416$		0			0
$417$	5.93398e16			0.0270641
$418$		0			0
$419$		−	4.14996e18i		−	1.83040i	−0.403002	−	0.915199i	$-0.632033\pi$
$419$		−	4.14996e18i		−	1.83040i	0.403002	−	0.915199i	$-0.367967\pi$
$420$		0			0
$421$	6.40811e17			0.273373			0.136686	−	0.990614i	$-0.456355\pi$
$421$	6.40811e17			0.273373			0.136686	+	0.990614i	$0.456355\pi$
$422$		0			0
$423$			2.23066e18i			0.920556i
$424$		0			0
$425$			6.49327e17i			0.259263i
$426$		0			0
$427$	−9.59101e17	−	2.69457e18i	−0.370569	−	1.04110i
$428$		0			0
$429$	3.65930e18			1.36835
$430$		0			0
$431$	−5.23310e18			−1.89417			−0.947084	−	0.320986i	$-0.895986\pi$
$431$	−5.23310e18			−1.89417			−0.947084	+	0.320986i	$0.895986\pi$
$432$		0			0
$433$		−	2.00754e18i		−	0.703477i	−0.936098	−	0.351738i	$-0.885591\pi$
$433$		−	2.00754e18i		−	0.703477i	0.936098	−	0.351738i	$-0.114409\pi$
$434$		0			0
$435$	4.00204e17			0.135787
$436$		0			0
$437$			1.46851e18i			0.482511i
$438$		0			0
$439$			1.78484e18i			0.567998i	0.958825	+	0.283999i	$0.0916613\pi$
$439$			1.78484e18i			0.567998i	−0.958825	+	0.283999i	$0.908339\pi$
$440$		0			0
$441$	−2.50482e18	+	2.04181e18i	−0.772157	+	0.629426i
$442$		0			0
$443$	8.10509e17			0.242065			0.121032	−	0.992649i	$-0.461379\pi$
$443$	8.10509e17			0.242065			0.121032	+	0.992649i	$0.461379\pi$
$444$		0			0
$445$	−5.49847e17			−0.159119
$446$		0			0
$447$		−	5.44253e17i		−	0.152633i
$448$		0			0
$449$	2.07469e18			0.563934			0.281967	−	0.959424i	$-0.409013\pi$
$449$	2.07469e18			0.563934			0.281967	+	0.959424i	$0.409013\pi$
$450$		0			0
$451$		−	3.67005e18i		−	0.967022i
$452$		0			0
$453$		−	4.25607e18i		−	1.08723i
$454$		0			0
$455$	−2.03920e18	+	7.25829e17i	−0.505102	+	0.179786i
$456$		0			0
$457$	−1.44680e18			−0.347533			−0.173766	−	0.984787i	$-0.555594\pi$
$457$	−1.44680e18			−0.347533			−0.173766	+	0.984787i	$0.555594\pi$
$458$		0			0
$459$	−2.99148e16			−0.00696943
$460$		0			0
$461$			3.70643e18i			0.837624i	0.908073	+	0.418812i	$0.137553\pi$
$461$			3.70643e18i			0.837624i	−0.908073	+	0.418812i	$0.862447\pi$
$462$		0			0
$463$	−3.85093e18			−0.844304			−0.422152	−	0.906525i	$-0.638725\pi$
$463$	−3.85093e18			−0.844304			−0.422152	+	0.906525i	$0.638725\pi$
$464$		0			0
$465$			3.85826e17i			0.0820769i
$466$		0			0
$467$			5.56822e18i			1.14947i	0.818339	+	0.574736i	$0.194895\pi$
$467$			5.56822e18i			1.14947i	−0.818339	+	0.574736i	$0.805105\pi$
$468$		0			0
$469$	−4.39176e18	+	1.56320e18i	−0.879890	+	0.313187i
$470$		0			0
$471$	−5.65131e18			−1.09901
$472$		0			0
$473$	3.88316e18			0.733091
$474$		0			0
$475$		−	8.29632e17i		−	0.152065i
$476$		0			0
$477$	8.35363e17			0.148678
$478$		0			0
$479$			2.39765e18i			0.414416i	0.978297	+	0.207208i	$0.0664377\pi$
$479$			2.39765e18i			0.414416i	−0.978297	+	0.207208i	$0.933562\pi$
$480$		0			0
$481$		−	3.78934e17i		−	0.0636133i
$482$		0			0
$483$	−1.84379e18	−	5.18007e18i	−0.300664	−	0.844706i
$484$		0			0
$485$	2.21964e18			0.351634
$486$		0			0
$487$	−1.01530e19			−1.56276			−0.781379	−	0.624057i	$-0.785483\pi$
$487$	−1.01530e19			−1.56276			−0.781379	+	0.624057i	$0.785483\pi$
$488$		0			0
$489$		−	1.88373e19i		−	2.81745i
$490$		0			0
$491$	1.27577e19			1.85440			0.927199	−	0.374570i	$-0.122210\pi$
$491$	1.27577e19			1.85440			0.927199	+	0.374570i	$0.122210\pi$
$492$		0			0
$493$			1.97188e18i			0.278581i
$494$		0			0
$495$		−	2.62074e18i		−	0.359905i
$496$		0			0
$497$	6.06123e16	−	2.15743e16i	0.00809219	−	0.00288033i
$498$		0			0
$499$	−1.27837e19			−1.65941			−0.829705	−	0.558202i	$-0.811491\pi$
$499$	−1.27837e19			−1.65941			−0.829705	+	0.558202i	$0.811491\pi$
$500$		0			0
$501$	6.88689e18			0.869279
$502$		0			0
$503$		−	1.72898e18i		−	0.212233i	−0.994354	−	0.106116i	$-0.966158\pi$
$503$		−	1.72898e18i		−	0.212233i	0.994354	−	0.106116i	$-0.0338416\pi$
$504$		0			0
$505$	−1.51827e18			−0.181263
$506$		0			0
$507$			5.31974e18i			0.617782i
$508$		0			0
$509$		−	4.65773e18i		−	0.526198i	−0.964769	−	0.263099i	$-0.915255\pi$
$509$		−	4.65773e18i		−	0.526198i	0.964769	−	0.263099i	$-0.0847446\pi$
$510$		0			0
$511$	2.03203e18	+	5.70894e18i	0.223349	+	0.627493i
$512$		0			0
$513$	3.82215e16			0.00408777
$514$		0			0
$515$	−4.06807e18			−0.423387
$516$		0			0
$517$			7.37001e18i			0.746508i
$518$		0			0
$519$	2.15540e19			2.12499
$520$		0			0
$521$		−	1.23746e18i		−	0.118759i	−0.998235	−	0.0593795i	$-0.981088\pi$
$521$		−	1.23746e18i		−	0.118759i	0.998235	−	0.0593795i	$-0.0189122\pi$
$522$		0			0
$523$			6.16234e18i			0.575750i	0.957668	+	0.287875i	$0.0929488\pi$
$523$			6.16234e18i			0.575750i	−0.957668	+	0.287875i	$0.907051\pi$
$524$		0			0
$525$	1.04164e18	+	2.92647e18i	0.0947555	+	0.266213i
$526$		0			0
$527$	−1.90103e18			−0.168390
$528$		0			0
$529$	−6.92406e18			−0.597270
$530$		0			0
$531$			1.71145e19i			1.43782i
$532$		0			0
$533$	1.75374e19			1.43508
$534$		0			0
$535$			3.78444e18i			0.301665i
$536$		0			0
$537$			3.10889e19i			2.41426i
$538$		0			0
$539$	−8.27583e18	+	6.74606e18i	−0.626167	+	0.510421i
$540$		0			0
$541$	1.86972e19			1.37847			0.689234	−	0.724539i	$-0.257947\pi$
$541$	1.86972e19			1.37847			0.689234	+	0.724539i	$0.257947\pi$
$542$		0			0
$543$	−1.19985e19			−0.862044
$544$		0			0
$545$			7.58800e18i			0.531315i
$546$		0			0
$547$	2.37829e19			1.62313			0.811564	−	0.584263i	$-0.198616\pi$
$547$	2.37829e19			1.62313			0.811564	+	0.584263i	$0.198616\pi$
$548$		0			0
$549$			1.65481e19i			1.10088i
$550$		0			0
$551$		−	2.51943e18i		−	0.163396i
$552$		0			0
$553$	−1.87628e19	+	6.67842e18i	−1.18638	+	0.422278i
$554$		0			0
$555$	−5.43812e17			−0.0335272
$556$		0			0
$557$	−1.06326e19			−0.639225			−0.319612	−	0.947548i	$-0.603553\pi$
$557$	−1.06326e19			−0.639225			−0.319612	+	0.947548i	$0.603553\pi$
$558$		0			0
$559$			1.85558e19i			1.08792i
$560$		0			0
$561$	2.58750e19			1.47959
$562$		0			0
$563$		−	1.69660e19i		−	0.946282i	−0.880987	−	0.473141i	$-0.843120\pi$
$563$		−	1.69660e19i		−	0.946282i	0.880987	−	0.473141i	$-0.156880\pi$
$564$		0			0
$565$		−	1.06438e19i		−	0.579106i
$566$		0			0
$567$	−1.78165e19	+	6.34158e18i	−0.945672	+	0.336602i
$568$		0			0
$569$	−9.02876e18			−0.467565			−0.233783	−	0.972289i	$-0.575110\pi$
$569$	−9.02876e18			−0.467565			−0.233783	+	0.972289i	$0.575110\pi$
$570$		0			0
$571$	−2.99318e19			−1.51245			−0.756223	−	0.654314i	$-0.772957\pi$
$571$	−2.99318e19			−1.51245			−0.756223	+	0.654314i	$0.772957\pi$
$572$		0			0
$573$			5.32402e19i			2.62517i
$574$		0			0
$575$	−2.63762e18			−0.126922
$576$		0			0
$577$		−	3.34309e19i		−	1.57006i	−0.619455	−	0.785032i	$-0.712646\pi$
$577$		−	3.34309e19i		−	1.57006i	0.619455	−	0.785032i	$-0.287354\pi$
$578$		0			0
$579$			3.57196e18i			0.163741i
$580$		0			0
$581$	2.10789e18	+	5.92205e18i	0.0943225	+	0.264996i
$582$		0			0
$583$	2.76001e18			0.120568
$584$		0			0
$585$	1.25232e19			0.534104
$586$		0			0
$587$		−	2.53884e19i		−	1.05723i	−0.848862	−	0.528614i	$-0.822712\pi$
$587$		−	2.53884e19i		−	1.05723i	0.848862	−	0.528614i	$-0.177288\pi$
$588$		0			0
$589$	2.42891e18			0.0987654
$590$		0			0
$591$			1.83303e19i			0.727875i
$592$		0			0
$593$		−	1.98760e18i		−	0.0770809i	−0.999257	−	0.0385405i	$-0.987729\pi$
$593$		−	1.98760e18i		−	0.0770809i	0.999257	−	0.0385405i	$-0.0122708\pi$
$594$		0			0
$595$	−1.44192e19	+	5.13237e18i	−0.546165	+	0.194401i
$596$		0			0
$597$	5.38088e19			1.99083
$598$		0			0
$599$	−1.82849e19			−0.660854			−0.330427	−	0.943832i	$-0.607193\pi$
$599$	−1.82849e19			−0.660854			−0.330427	+	0.943832i	$0.607193\pi$
$600$		0			0
$601$			2.16830e19i			0.765593i	0.923833	+	0.382796i	$0.125039\pi$
$601$			2.16830e19i			0.765593i	−0.923833	+	0.382796i	$0.874961\pi$
$602$		0			0
$603$	2.69710e19			0.930412
$604$		0			0
$605$			4.60908e18i			0.155355i
$606$		0			0
$607$		−	4.00215e19i		−	1.31817i	−0.752067	−	0.659087i	$-0.770943\pi$
$607$		−	4.00215e19i		−	1.31817i	0.752067	−	0.659087i	$-0.229057\pi$
$608$		0			0
$609$	3.16327e18	+	8.88710e18i	0.101816	+	0.286048i
$610$		0			0
$611$	−3.52177e19			−1.10783
$612$		0			0
$613$	−5.45335e19			−1.67664			−0.838321	−	0.545177i	$-0.816463\pi$
$613$	−5.45335e19			−1.67664			−0.838321	+	0.545177i	$0.816463\pi$
$614$		0			0
$615$		−	2.51681e19i		−	0.756353i
$616$		0			0
$617$	−1.42541e19			−0.418739			−0.209369	−	0.977837i	$-0.567141\pi$
$617$	−1.42541e19			−0.418739			−0.209369	+	0.977837i	$0.567141\pi$
$618$		0			0
$619$			6.83851e19i			1.96394i	0.189048	+	0.981968i	$0.439460\pi$
$619$			6.83851e19i			1.96394i	−0.189048	+	0.981968i	$0.560540\pi$
$620$		0			0
$621$		−	1.21516e17i		−	0.00341187i
$622$		0			0
$623$	−4.34607e18	−	1.22101e19i	−0.119311	−	0.335200i
$624$		0			0
$625$	1.49012e18			0.0400000
$626$		0			0
$627$	−3.30600e19			−0.867821
$628$		0			0
$629$		−	2.67946e18i		−	0.0687848i
$630$		0			0
$631$	−1.89419e19			−0.475575			−0.237787	−	0.971317i	$-0.576422\pi$
$631$	−1.89419e19			−0.475575			−0.237787	+	0.971317i	$0.576422\pi$
$632$		0			0
$633$		−	2.97057e19i		−	0.729482i
$634$		0			0
$635$		−	2.36765e19i		−	0.568725i
$636$		0			0
$637$	−3.22361e19	−	3.95462e19i	−0.757473	−	0.929241i
$638$		0			0
$639$	−3.72237e17			−0.00855684
$640$		0			0
$641$	−8.56420e19			−1.92611			−0.963054	−	0.269309i	$-0.913205\pi$
$641$	−8.56420e19			−1.92611			−0.963054	+	0.269309i	$0.913205\pi$
$642$		0			0
$643$			2.41399e19i			0.531201i	0.964083	+	0.265600i	$0.0855702\pi$
$643$			2.41399e19i			0.531201i	−0.964083	+	0.265600i	$0.914430\pi$
$644$		0			0
$645$	2.66295e19			0.573384
$646$		0			0
$647$			4.28913e19i			0.903731i	0.892086	+	0.451865i	$0.149241\pi$
$647$			4.28913e19i			0.903731i	−0.892086	+	0.451865i	$0.850759\pi$
$648$		0			0
$649$			5.65458e19i			1.16597i
$650$		0			0
$651$	−8.56782e18	+	3.04962e18i	−0.172903	+	0.0615430i
$652$		0			0
$653$	−9.10456e19			−1.79832			−0.899160	−	0.437621i	$-0.855821\pi$
$653$	−9.10456e19			−1.79832			−0.899160	+	0.437621i	$0.855821\pi$
$654$		0			0
$655$	3.43104e19			0.663341
$656$		0			0
$657$		−	3.50601e19i		−	0.663523i
$658$		0			0
$659$	1.64365e19			0.304518			0.152259	−	0.988341i	$-0.451345\pi$
$659$	1.64365e19			0.304518			0.152259	+	0.988341i	$0.451345\pi$
$660$		0			0
$661$		−	9.87267e19i		−	1.79071i	−0.445357	−	0.895353i	$-0.646923\pi$
$661$		−	9.87267e19i		−	1.79071i	0.445357	−	0.895353i	$-0.353077\pi$
$662$		0			0
$663$			1.23644e20i			2.19573i
$664$		0			0
$665$	1.84232e19	−	6.55752e18i	0.320341	−	0.114022i
$666$		0			0
$667$	−8.00992e18			−0.136379
$668$		0			0
$669$	−7.35462e19			−1.22625
$670$		0			0
$671$			5.46742e19i			0.892740i
$672$		0			0
$673$	6.48391e19			1.03689			0.518443	−	0.855112i	$-0.326512\pi$
$673$	6.48391e19			1.03689			0.518443	+	0.855112i	$0.326512\pi$
$674$		0			0
$675$			6.86503e16i			0.00107527i
$676$		0			0
$677$		−	1.17893e20i		−	1.80871i	−0.426783	−	0.904354i	$-0.640353\pi$
$677$		−	1.17893e20i		−	1.80871i	0.426783	−	0.904354i	$-0.359647\pi$
$678$		0			0
$679$	1.75443e19	+	4.92903e19i	0.263663	+	0.740752i
$680$		0			0
$681$	1.55502e20			2.28931
$682$		0			0
$683$	3.58734e19			0.517401			0.258701	−	0.965958i	$-0.416706\pi$
$683$	3.58734e19			0.517401			0.258701	+	0.965958i	$0.416706\pi$
$684$		0			0
$685$		−	4.29982e19i		−	0.607598i
$686$		0			0
$687$	7.92664e18			0.109747
$688$		0			0
$689$			1.31888e19i			0.178924i
$690$		0			0
$691$		−	2.55368e19i		−	0.339485i	−0.985488	−	0.169742i	$-0.945706\pi$
$691$		−	2.55368e19i		−	0.339485i	0.985488	−	0.169742i	$-0.0542935\pi$
$692$		0			0
$693$	5.81973e19	−	2.07147e19i	0.758176	−	0.269864i
$694$		0			0
$695$	6.70967e17			0.00856658
$696$		0			0
$697$	1.24008e20			1.55174
$698$		0			0
$699$		−	5.50090e19i		−	0.674673i
$700$		0			0
$701$	1.31674e19			0.158298			0.0791489	−	0.996863i	$-0.474780\pi$
$701$	1.31674e19			0.158298			0.0791489	+	0.996863i	$0.474780\pi$
$702$		0			0
$703$			3.42349e18i			0.0403443i
$704$		0			0
$705$			5.05413e19i			0.583878i
$706$		0			0
$707$	−1.20006e19	−	3.37153e19i	−0.135915	−	0.381849i
$708$		0			0
$709$	−8.10238e19			−0.899681			−0.449841	−	0.893109i	$-0.648519\pi$
$709$	−8.10238e19			−0.899681			−0.449841	+	0.893109i	$0.648519\pi$
$710$		0			0
$711$	1.15228e20			1.25450
$712$		0			0
$713$		−	7.72214e18i		−	0.0824349i
$714$		0			0
$715$	4.13764e19			0.433122
$716$		0			0
$717$		−	1.54854e20i		−	1.58960i
$718$		0			0
$719$		−	5.94415e19i		−	0.598395i	−0.954191	−	0.299197i	$-0.903281\pi$
$719$		−	5.94415e19i		−	0.598395i	0.954191	−	0.299197i	$-0.0967189\pi$
$720$		0			0
$721$	−3.21546e19	−	9.03372e19i	−0.317465	−	0.891908i
$722$		0			0
$723$	5.74926e19			0.556729
$724$		0			0
$725$	4.52519e18			0.0429804
$726$		0			0
$727$		−	1.16087e20i		−	1.08154i	−0.841170	−	0.540770i	$-0.818133\pi$
$727$		−	1.16087e20i		−	1.08154i	0.841170	−	0.540770i	$-0.181867\pi$
$728$		0			0
$729$	1.08585e20			0.992375
$730$		0			0
$731$			1.31209e20i			1.17636i
$732$		0			0
$733$		−	2.25120e19i		−	0.198009i	−0.995087	−	0.0990047i	$-0.968434\pi$
$733$		−	2.25120e19i		−	0.198009i	0.995087	−	0.0990047i	$-0.0315659\pi$
$734$		0			0
$735$	−5.67531e19	+	4.62624e19i	−0.489754	+	0.399224i
$736$		0			0
$737$	8.91111e19			0.754500
$738$		0			0
$739$	1.00102e20			0.831635			0.415818	−	0.909448i	$-0.363495\pi$
$739$	1.00102e20			0.831635			0.415818	+	0.909448i	$0.363495\pi$
$740$		0			0
$741$		−	1.57978e20i		−	1.28786i
$742$		0			0
$743$	7.04894e19			0.563900			0.281950	−	0.959429i	$-0.409019\pi$
$743$	7.04894e19			0.563900			0.281950	+	0.959429i	$0.409019\pi$
$744$		0			0
$745$		−	6.15398e18i		−	0.0483128i
$746$		0			0
$747$		−	3.63689e19i		−	0.280212i
$748$		0			0
$749$	−8.40389e19	+	2.99127e19i	−0.635488	+	0.226195i
$750$		0			0
$751$	7.69311e19			0.570982			0.285491	−	0.958381i	$-0.407843\pi$
$751$	7.69311e19			0.570982			0.285491	+	0.958381i	$0.407843\pi$
$752$		0			0
$753$	1.35396e20			0.986374
$754$		0			0
$755$		−	4.81242e19i		−	0.344140i
$756$		0			0
$757$	−1.30417e20			−0.915511			−0.457756	−	0.889078i	$-0.651347\pi$
$757$	−1.30417e20			−0.915511			−0.457756	+	0.889078i	$0.651347\pi$
$758$		0			0
$759$			1.05106e20i			0.724330i
$760$		0			0
$761$			1.88228e20i			1.27348i	0.771078	+	0.636740i	$0.219718\pi$
$761$			1.88228e20i			1.27348i	−0.771078	+	0.636740i	$0.780282\pi$
$762$		0			0
$763$	−1.68502e20	+	5.99766e19i	−1.11927	+	0.398391i
$764$		0			0
$765$	8.85524e19			0.577525
$766$		0			0
$767$	−2.70205e20			−1.73032
$768$		0			0
$769$		−	2.18523e20i		−	1.37408i	−0.726619	−	0.687040i	$-0.758910\pi$
$769$		−	2.18523e20i		−	1.37408i	0.726619	−	0.687040i	$-0.241090\pi$
$770$		0			0
$771$	2.20252e20			1.36000
$772$		0			0
$773$		−	1.85050e20i		−	1.12210i	−0.827781	−	0.561052i	$-0.810397\pi$
$773$		−	1.85050e20i		−	1.12210i	0.827781	−	0.561052i	$-0.189603\pi$
$774$		0			0
$775$			4.36261e18i			0.0259797i
$776$		0			0
$777$	−4.29836e18	−	1.20761e19i	−0.0251395	−	0.0706285i
$778$		0			0
$779$	−1.58442e20			−0.910141
$780$		0			0
$781$	−1.22986e18			−0.00693901
$782$		0			0
$783$			2.08477e17i			0.00115538i
$784$		0			0
$785$	−6.39004e19			−0.347869
$786$		0			0
$787$		−	1.40601e20i		−	0.751908i	−0.926638	−	0.375954i	$-0.877315\pi$
$787$		−	1.40601e20i		−	0.751908i	0.926638	−	0.375954i	$-0.122685\pi$
$788$		0			0
$789$		−	1.12568e20i		−	0.591390i
$790$		0			0
$791$	2.36361e20	−	8.41301e19i	1.21995	−	0.434226i
$792$		0			0
$793$	−2.61262e20			−1.32484
$794$		0			0
$795$	1.89273e19			0.0943016
$796$		0			0
$797$			1.63563e20i			0.800715i	0.916359	+	0.400357i	$0.131114\pi$
$797$			1.63563e20i			0.800715i	−0.916359	+	0.400357i	$0.868886\pi$
$798$		0			0
$799$	−2.49026e20			−1.19789
$800$		0			0
$801$			7.49858e19i			0.354447i
$802$		0			0
$803$		−	1.15837e20i		−	0.538071i
$804$		0			0
$805$	−2.08481e19	−	5.85720e19i	−0.0951688	−	0.267374i
$806$		0			0
$807$	3.68907e20			1.65502
$808$		0			0
$809$	−3.00503e20			−1.32498			−0.662489	−	0.749071i	$-0.730500\pi$
$809$	−3.00503e20			−1.32498			−0.662489	+	0.749071i	$0.730500\pi$
$810$		0			0
$811$		−	1.34985e19i		−	0.0584976i	−0.999572	−	0.0292488i	$-0.990688\pi$
$811$		−	1.34985e19i		−	0.0584976i	0.999572	−	0.0292488i	$-0.00931151\pi$
$812$		0			0
$813$	−1.66986e20			−0.711288
$814$		0			0
$815$		−	2.12997e20i		−	0.891805i
$816$		0			0
$817$		−	1.67642e20i		−	0.689969i
$818$		0			0
$819$	9.89854e19	+	2.78097e20i	0.400483	+	1.12514i
$820$		0			0
$821$	−1.73764e19			−0.0691128			−0.0345564	−	0.999403i	$-0.511002\pi$
$821$	−1.73764e19			−0.0691128			−0.0345564	+	0.999403i	$0.511002\pi$
$822$		0			0
$823$	4.31978e20			1.68913			0.844564	−	0.535455i	$-0.179860\pi$
$823$	4.31978e20			1.68913			0.844564	+	0.535455i	$0.179860\pi$
$824$		0			0
$825$		−	5.93796e19i		−	0.228276i
$826$		0			0
$827$	−3.80493e20			−1.43816			−0.719082	−	0.694925i	$-0.755437\pi$
$827$	−3.80493e20			−1.43816			−0.719082	+	0.694925i	$0.755437\pi$
$828$		0			0
$829$		−	3.80126e20i		−	1.41269i	−0.707869	−	0.706344i	$-0.750343\pi$
$829$		−	3.80126e20i		−	1.41269i	0.707869	−	0.706344i	$-0.249657\pi$
$830$		0			0
$831$			1.75678e20i			0.641964i
$832$		0			0
$833$	−2.27943e20	−	2.79633e20i	−0.819052	−	1.00478i
$834$		0			0
$835$	7.78714e19			0.275152
$836$		0			0
$837$	−2.00987e17			−0.000698378
$838$		0			0
$839$			5.86569e19i			0.200441i	0.994965	+	0.100220i	$0.0319548\pi$
$839$			5.86569e19i			0.200441i	−0.994965	+	0.100220i	$0.968045\pi$
$840$		0			0
$841$	−2.83816e20			−0.953817
$842$		0			0
$843$		−	2.79163e19i		−	0.0922708i
$844$		0			0
$845$			6.01514e19i			0.195546i
$846$		0			0
$847$	−1.02351e20	+	3.64308e19i	−0.327272	+	0.116489i
$848$		0			0
$849$	1.48599e19			0.0467372
$850$		0			0
$851$	1.08842e19			0.0336735
$852$		0			0
$853$			1.27523e20i			0.388101i	0.980992	+	0.194050i	$0.0621626\pi$
$853$			1.27523e20i			0.388101i	−0.980992	+	0.194050i	$0.937837\pi$
$854$		0			0
$855$	−1.13142e20			−0.338735
$856$		0			0
$857$		−	2.76964e19i		−	0.0815751i	−0.999168	−	0.0407876i	$-0.987013\pi$
$857$		−	2.76964e19i		−	0.0815751i	0.999168	−	0.0407876i	$-0.0129867\pi$
$858$		0			0
$859$			1.10929e20i			0.321436i	0.987000	+	0.160718i	$0.0513810\pi$
$859$			1.10929e20i			0.321436i	−0.987000	+	0.160718i	$0.948619\pi$
$860$		0			0
$861$	5.58893e20	−	1.98932e20i	1.59333	−	0.567130i
$862$		0			0
$863$	−2.50541e20			−0.702754			−0.351377	−	0.936234i	$-0.614286\pi$
$863$	−2.50541e20			−0.702754			−0.351377	+	0.936234i	$0.614286\pi$
$864$		0			0
$865$	2.43715e20			0.672620
$866$		0			0
$867$			3.54017e20i			0.961369i
$868$		0			0
$869$	3.80708e20			1.01731
$870$		0			0
$871$			4.25819e20i			1.11969i
$872$		0			0
$873$		−	3.02705e20i		−	0.783285i
$874$		0			0
$875$	1.17781e19	+	3.30901e19i	0.0299929	+	0.0842640i
$876$		0			0
$877$	−2.49364e20			−0.624938			−0.312469	−	0.949928i	$-0.601156\pi$
$877$	−2.49364e20			−0.624938			−0.312469	+	0.949928i	$0.601156\pi$
$878$		0			0
$879$	8.03127e19			0.198090
$880$		0			0
$881$			9.11795e19i			0.221343i	0.993857	+	0.110672i	$0.0353002\pi$
$881$			9.11795e19i			0.221343i	−0.993857	+	0.110672i	$0.964700\pi$
$882$		0			0
$883$	−8.33346e20			−1.99113			−0.995567	−	0.0940506i	$-0.970018\pi$
$883$	−8.33346e20			−1.99113			−0.995567	+	0.0940506i	$0.970018\pi$
$884$		0			0
$885$			3.87774e20i			0.911959i
$886$		0			0
$887$			3.17911e20i			0.735936i	0.929838	+	0.367968i	$0.119946\pi$
$887$			3.17911e20i			0.735936i	−0.929838	+	0.367968i	$0.880054\pi$
$888$		0			0
$889$	5.25771e20	−	1.87142e20i	1.19808	−	0.426442i
$890$		0			0
$891$	3.61506e20			0.810908
$892$		0			0
$893$	3.18175e20			0.702597
$894$		0			0
$895$			3.51528e20i			0.764185i
$896$		0			0
$897$	−5.02252e20			−1.07492
$898$		0			0
$899$			1.32484e19i			0.0279155i
$900$		0			0
$901$			9.32582e19i			0.193470i
$902$		0			0
$903$	2.10483e20	+	5.91347e20i	0.429936	+	1.20789i
$904$		0			0
$905$	−1.35670e20			−0.272862
$906$		0			0
$907$	−3.12258e20			−0.618391			−0.309196	−	0.950998i	$-0.600060\pi$
$907$	−3.12258e20			−0.618391			−0.309196	+	0.950998i	$0.600060\pi$
$908$		0			0
$909$			2.07055e20i			0.403774i
$910$		0			0
$911$	−1.05084e20			−0.201794			−0.100897	−	0.994897i	$-0.532171\pi$
$911$	−1.05084e20			−0.201794			−0.100897	+	0.994897i	$0.532171\pi$
$912$		0			0
$913$		−	1.20162e20i		−	0.227233i
$914$		0			0
$915$			3.74939e20i			0.698253i
$916$		0			0
$917$	2.71194e20	+	7.61912e20i	0.497388	+	1.39740i
$918$		0			0
$919$	1.83499e20			0.331456			0.165728	−	0.986172i	$-0.447003\pi$
$919$	1.83499e20			0.331456			0.165728	+	0.986172i	$0.447003\pi$
$920$		0			0
$921$	9.23977e19			0.164378
$922$		0			0
$923$		−	5.87689e18i		−	0.0102976i
$924$		0			0
$925$	−6.14899e18			−0.0106123
$926$		0			0
$927$			5.54785e20i			0.943120i
$928$		0			0
$929$		−	7.01194e20i		−	1.17416i	−0.809528	−	0.587082i	$-0.800277\pi$
$929$		−	7.01194e20i		−	1.17416i	0.809528	−	0.587082i	$-0.199723\pi$
$930$		0			0
$931$	2.91238e20	+	3.57281e20i	0.480397	+	0.589335i
$932$		0			0
$933$	−4.63398e19			−0.0752979
$934$		0			0
$935$	2.92574e20			0.468333
$936$		0			0
$937$			6.57961e20i			1.03758i	0.854901	+	0.518792i	$0.173618\pi$
$937$			6.57961e20i			1.03758i	−0.854901	+	0.518792i	$0.826382\pi$
$938$		0			0
$939$	−1.60956e21			−2.50062
$940$		0			0
$941$		−	6.87802e20i		−	1.05278i	−0.850245	−	0.526388i	$-0.823546\pi$
$941$		−	6.87802e20i		−	1.05278i	0.850245	−	0.526388i	$-0.176454\pi$
$942$		0			0
$943$			5.03729e20i			0.759652i
$944$		0			0
$945$	−1.52448e18	+	5.42621e17i	−0.00226516	+	0.000806258i
$946$		0			0
$947$	−1.14751e21			−1.67999			−0.839996	−	0.542592i	$-0.817443\pi$
$947$	−1.14751e21			−1.67999			−0.839996	+	0.542592i	$0.817443\pi$
$948$		0			0
$949$	5.53531e20			0.798506
$950$		0			0
$951$			6.25719e20i			0.889439i
$952$		0			0
$953$	1.07144e21			1.50078			0.750391	−	0.660994i	$-0.229865\pi$
$953$	1.07144e21			1.50078			0.750391	+	0.660994i	$0.229865\pi$
$954$		0			0
$955$			6.01997e20i			0.830942i
$956$		0			0
$957$		−	1.80324e20i		−	0.245285i
$958$		0			0
$959$	9.54836e20	−	3.39864e20i	1.27997	−	0.455590i
$960$		0			0
$961$	7.44172e20			0.983126
$962$		0			0
$963$	5.16106e20			0.671977
$964$		0			0
$965$			4.03889e19i			0.0518287i
$966$		0			0
$967$	−1.52166e21			−1.92456			−0.962279	−	0.272065i	$-0.912293\pi$
$967$	−1.52166e21			−1.92456			−0.962279	+	0.272065i	$0.912293\pi$
$968$		0			0
$969$		−	1.11707e21i		−	1.39256i
$970$		0			0
$971$			7.61706e20i			0.935950i	0.883742	+	0.467975i	$0.155016\pi$
$971$			7.61706e20i			0.935950i	−0.883742	+	0.467975i	$0.844984\pi$
$972$		0			0
$973$	5.30341e18	+	1.48998e19i	0.00642341	+	0.0180464i
$974$		0			0
$975$	2.83746e20			0.338765
$976$		0			0
$977$	2.62058e20			0.308414			0.154207	−	0.988039i	$-0.450718\pi$
$977$	2.62058e20			0.308414			0.154207	+	0.988039i	$0.450718\pi$
$978$		0			0
$979$			2.47750e20i			0.287432i
$980$		0			0
$981$	1.03482e21			1.18353
$982$		0			0
$983$		−	1.06164e20i		−	0.119702i	−0.998207	−	0.0598512i	$-0.980937\pi$
$983$		−	1.06164e20i		−	0.119702i	0.998207	−	0.0598512i	$-0.0190626\pi$
$984$		0			0
$985$			2.07264e20i			0.230394i
$986$		0			0
$987$	−1.12234e21	+	3.99485e20i	−1.23000	+	0.437805i
$988$		0			0
$989$	−5.32979e20			−0.575885
$990$		0			0
$991$	6.22420e20			0.663083			0.331541	−	0.943441i	$-0.392431\pi$
$991$	6.22420e20			0.663083			0.331541	+	0.943441i	$0.392431\pi$
$992$		0			0
$993$		−	9.32086e20i		−	0.979064i
$994$		0			0
$995$	6.08427e20			0.630154
$996$		0			0
$997$			6.34533e20i			0.648019i	0.946054	+	0.324010i	$0.105031\pi$
$997$			6.34533e20i			0.648019i	−0.946054	+	0.324010i	$0.894969\pi$
$998$		0			0
$999$		−	2.83286e17i		−	0.000285277i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.15.d.a.41.5	✓	36
7.6	odd	2	inner	140.15.d.a.41.32	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.15.d.a.41.5	✓	36	1.1	even	1	trivial
140.15.d.a.41.32	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)

\(f(q)\)	\(=\)	\(q-3089.94i q^{3} -34938.6i q^{5} +(775860. - 276159. i) q^{7} -4.76477e6 q^{9} +O(q^{10})\)	\(q-3089.94i q^{3} -34938.6i q^{5} +(775860. - 276159. i) q^{7} -4.76477e6 q^{9} -1.57426e7 q^{11} -7.52264e7i q^{13} -1.07958e8 q^{15} -5.31929e8i q^{17} +6.79635e8i q^{19} +(-8.53315e8 - 2.39736e9i) q^{21} +2.16074e9 q^{23} -1.22070e9 q^{25} -5.62383e7i q^{27} -3.70703e9 q^{29} -3.57385e9i q^{31} +4.86438e10i q^{33} +(-9.64860e9 - 2.71074e10i) q^{35} +5.03725e9 q^{37} -2.32445e11 q^{39} +2.33128e11i q^{41} -2.46665e11 q^{43} +1.66474e11i q^{45} -4.68156e11i q^{47} +(5.25695e11 - 4.28522e11i) q^{49} -1.64363e12 q^{51} -1.75321e11 q^{53} +5.50025e11i q^{55} +2.10003e12 q^{57} -3.59189e12i q^{59} -3.47300e12i q^{61} +(-3.69679e12 + 1.31583e12i) q^{63} -2.62830e12 q^{65} -5.66050e12 q^{67} -6.67655e12i q^{69} +7.81227e10 q^{71} +7.35820e12i q^{73} +3.77190e12i q^{75} +(-1.22141e13 + 4.34747e12i) q^{77} -2.41833e13 q^{79} -2.29635e13 q^{81} +7.63289e12i q^{83} -1.85848e13 q^{85} +1.14545e13i q^{87} -1.57376e13i q^{89} +(-2.07744e13 - 5.83652e13i) q^{91} -1.10430e13 q^{93} +2.37455e13 q^{95} +6.35298e13i q^{97} +7.50100e13 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 1364266 q^{7} - 54790830 q^{9}+O(q^{10}) \) 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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