LMFDB - Embedded newform 140.15.d.a.41.16

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			41.16
Character	$\chi$	$=$	140.41
Dual form			140.15.d.a.41.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-376.610i q^{3} -34938.6i q^{5} +(489531. - 662255. i) q^{7} +4.64113e6 q^{9} +O(q^{10})$	\(q-376.610i q^{3} -34938.6i q^{5} +(489531. - 662255. i) q^{7} +4.64113e6 q^{9} -1.32384e7 q^{11} -3.65880e7i q^{13} -1.31582e7 q^{15} +4.09147e8i q^{17} -8.08426e8i q^{19} +(-2.49412e8 - 1.84362e8i) q^{21} -2.19985e9 q^{23} -1.22070e9 q^{25} -3.54921e9i q^{27} -2.49747e10 q^{29} +1.84728e10i q^{31} +4.98573e9i q^{33} +(-2.31383e10 - 1.71035e10i) q^{35} -7.52525e10 q^{37} -1.37794e10 q^{39} -1.36052e11i q^{41} +5.14358e9 q^{43} -1.62155e11i q^{45} -1.55451e11i q^{47} +(-1.98941e11 - 6.48389e11i) q^{49} +1.54089e11 q^{51} +1.23014e12 q^{53} +4.62532e11i q^{55} -3.04461e11 q^{57} +3.86218e12i q^{59} -7.36099e10i q^{61} +(2.27198e12 - 3.07362e12i) q^{63} -1.27833e12 q^{65} -2.57292e11 q^{67} +8.28486e11i q^{69} -1.65089e13 q^{71} -7.41500e12i q^{73} +4.59729e11i q^{75} +(-6.48063e12 + 8.76723e12i) q^{77} +1.17517e13 q^{79} +2.08617e13 q^{81} -1.27759e13i q^{83} +1.42950e13 q^{85} +9.40572e12i q^{87} -9.27267e12i q^{89} +(-2.42306e13 - 1.79110e13i) q^{91} +6.95705e12 q^{93} -2.82452e13 q^{95} +5.77768e13i q^{97} -6.14414e13 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	376.610i		−	0.172204i	−0.996286	−	0.0861020i	$-0.972559\pi$
$3$		−	376.610i		−	0.172204i	0.996286	−	0.0861020i	$-0.0274411\pi$
$4$		0			0
$5$		−	34938.6i		−	0.447214i
$5$		−	34938.6i		−	0.447214i
$6$		0			0
$7$	489531.	−	662255.i	0.594421	−	0.804154i
$7$	489531.	−	662255.i	0.594421	−	0.804154i
$8$		0			0
$9$	4.64113e6			0.970346
$10$		0			0
$11$	−1.32384e7			−0.679341			−0.339671	−	0.940544i	$-0.610316\pi$
$11$	−1.32384e7			−0.679341			−0.339671	+	0.940544i	$0.610316\pi$
$12$		0			0
$13$		−	3.65880e7i		−	0.583090i	−0.956557	−	0.291545i	$-0.905831\pi$
$13$		−	3.65880e7i		−	0.583090i	0.956557	−	0.291545i	$-0.0941693\pi$
$14$		0			0
$15$	−1.31582e7			−0.0770120
$16$		0			0
$17$			4.09147e8i			0.997097i	0.866862	+	0.498548i	$0.166133\pi$
$17$			4.09147e8i			0.997097i	−0.866862	+	0.498548i	$0.833867\pi$
$18$		0			0
$19$		−	8.08426e8i		−	0.904409i	−0.891914	−	0.452205i	$-0.850638\pi$
$19$		−	8.08426e8i		−	0.904409i	0.891914	−	0.452205i	$-0.149362\pi$
$20$		0			0
$21$	−2.49412e8	−	1.84362e8i	−0.138479	−	0.102362i
$22$		0			0
$23$	−2.19985e9			−0.646098			−0.323049	−	0.946382i	$-0.604708\pi$
$23$	−2.19985e9			−0.646098			−0.323049	+	0.946382i	$0.604708\pi$
$24$		0			0
$25$	−1.22070e9			−0.200000
$26$		0			0
$27$		−	3.54921e9i		−	0.339301i
$28$		0			0
$29$	−2.49747e10			−1.44782			−0.723909	−	0.689896i	$-0.757656\pi$
$29$	−2.49747e10			−1.44782			−0.723909	+	0.689896i	$0.757656\pi$
$30$		0			0
$31$			1.84728e10i			0.671431i	0.941963	+	0.335715i	$0.108978\pi$
$31$			1.84728e10i			0.671431i	−0.941963	+	0.335715i	$0.891022\pi$
$32$		0			0
$33$			4.98573e9i			0.116985i
$34$		0			0
$35$	−2.31383e10	−	1.71035e10i	−0.359629	−	0.265833i
$36$		0			0
$37$	−7.52525e10			−0.792700			−0.396350	−	0.918100i	$-0.629723\pi$
$37$	−7.52525e10			−0.792700			−0.396350	+	0.918100i	$0.629723\pi$
$38$		0			0
$39$	−1.37794e10			−0.100410
$40$		0			0
$41$		−	1.36052e11i		−	0.698583i	−0.937014	−	0.349292i	$-0.886422\pi$
$41$		−	1.36052e11i		−	0.698583i	0.937014	−	0.349292i	$-0.113578\pi$
$42$		0			0
$43$	5.14358e9			0.0189229			0.00946143	−	0.999955i	$-0.496988\pi$
$43$	5.14358e9			0.0189229			0.00946143	+	0.999955i	$0.496988\pi$
$44$		0			0
$45$		−	1.62155e11i		−	0.433952i
$46$		0			0
$47$		−	1.55451e11i		−	0.306838i	−0.988161	−	0.153419i	$-0.950972\pi$
$47$		−	1.55451e11i		−	0.306838i	0.988161	−	0.153419i	$-0.0490284\pi$
$48$		0			0
$49$	−1.98941e11	−	6.48389e11i	−0.293328	−	0.956012i
$50$		0			0
$51$	1.54089e11			0.171704
$52$		0			0
$53$	1.23014e12			1.04718			0.523591	−	0.851970i	$-0.324592\pi$
$53$	1.23014e12			1.04718			0.523591	+	0.851970i	$0.324592\pi$
$54$		0			0
$55$			4.62532e11i			0.303811i
$56$		0			0
$57$	−3.04461e11			−0.155743
$58$		0			0
$59$			3.86218e12i			1.55192i	0.630784	+	0.775958i	$0.282733\pi$
$59$			3.86218e12i			1.55192i	−0.630784	+	0.775958i	$0.717267\pi$
$60$		0			0
$61$		−	7.36099e10i		−	0.0234222i	−0.999931	−	0.0117111i	$-0.996272\pi$
$61$		−	7.36099e10i		−	0.0234222i	0.999931	−	0.0117111i	$-0.00372784\pi$
$62$		0			0
$63$	2.27198e12	−	3.07362e12i	0.576794	−	0.780307i
$64$		0			0
$65$	−1.27833e12			−0.260766
$66$		0			0
$67$	−2.57292e11			−0.0424525			−0.0212262	−	0.999775i	$-0.506757\pi$
$67$	−2.57292e11			−0.0424525			−0.0212262	+	0.999775i	$0.506757\pi$
$68$		0			0
$69$			8.28486e11i			0.111261i
$70$		0			0
$71$	−1.65089e13			−1.81513			−0.907567	−	0.419908i	$-0.862062\pi$
$71$	−1.65089e13			−1.81513			−0.907567	+	0.419908i	$0.862062\pi$
$72$		0			0
$73$		−	7.41500e12i		−	0.671199i	−0.942005	−	0.335599i	$-0.891061\pi$
$73$		−	7.41500e12i		−	0.671199i	0.942005	−	0.335599i	$-0.108939\pi$
$74$		0			0
$75$			4.59729e11i			0.0344408i
$76$		0			0
$77$	−6.48063e12	+	8.76723e12i	−0.403815	+	0.546295i
$78$		0			0
$79$	1.17517e13			0.611944			0.305972	−	0.952040i	$-0.401019\pi$
$79$	1.17517e13			0.611944			0.305972	+	0.952040i	$0.401019\pi$
$80$		0			0
$81$	2.08617e13			0.911917
$82$		0			0
$83$		−	1.27759e13i		−	0.470808i	−0.971898	−	0.235404i	$-0.924359\pi$
$83$		−	1.27759e13i		−	0.470808i	0.971898	−	0.235404i	$-0.0756413\pi$
$84$		0			0
$85$	1.42950e13			0.445915
$86$		0			0
$87$			9.40572e12i			0.249320i
$88$		0			0
$89$		−	9.27267e12i		−	0.209640i	−0.994491	−	0.104820i	$-0.966573\pi$
$89$		−	9.27267e12i		−	0.209640i	0.994491	−	0.104820i	$-0.0334267\pi$
$90$		0			0
$91$	−2.42306e13	−	1.79110e13i	−0.468894	−	0.346601i
$92$		0			0
$93$	6.95705e12			0.115623
$94$		0			0
$95$	−2.82452e13			−0.404464
$96$		0			0
$97$			5.77768e13i			0.715075i	0.933899	+	0.357537i	$0.116383\pi$
$97$			5.77768e13i			0.715075i	−0.933899	+	0.357537i	$0.883617\pi$
$98$		0			0
$99$	−6.14414e13			−0.659196
$100$		0			0
$101$		−	1.28801e14i		−	1.20135i	−0.799492	−	0.600677i	$-0.794898\pi$
$101$		−	1.28801e14i		−	1.20135i	0.799492	−	0.600677i	$-0.205102\pi$
$102$		0			0
$103$			1.36747e14i			1.11188i	0.831224	+	0.555938i	$0.187641\pi$
$103$			1.36747e14i			1.11188i	−0.831224	+	0.555938i	$0.812359\pi$
$104$		0			0
$105$	−6.44136e12	+	8.71410e12i	−0.0457775	+	0.0619295i
$106$		0			0
$107$	−3.44571e13			−0.214581			−0.107291	−	0.994228i	$-0.534218\pi$
$107$	−3.44571e13			−0.214581			−0.107291	+	0.994228i	$0.534218\pi$
$108$		0			0
$109$	2.92179e14			1.59832			0.799159	−	0.601120i	$-0.205278\pi$
$109$	2.92179e14			1.59832			0.799159	+	0.601120i	$0.205278\pi$
$110$		0			0
$111$			2.83408e13i			0.136506i
$112$		0			0
$113$	−3.39355e14			−1.44246			−0.721232	−	0.692694i	$-0.756424\pi$
$113$	−3.39355e14			−1.44246			−0.721232	+	0.692694i	$0.756424\pi$
$114$		0			0
$115$			7.68597e13i			0.288944i
$116$		0			0
$117$		−	1.69810e14i		−	0.565799i
$118$		0			0
$119$	2.70960e14	+	2.00290e14i	0.801820	+	0.592695i
$120$		0			0
$121$	−2.04494e14			−0.538495
$122$		0			0
$123$	−5.12386e13			−0.120299
$124$		0			0
$125$			4.26496e13i			0.0894427i
$126$		0			0
$127$	4.38635e14			0.823147			0.411574	−	0.911377i	$-0.364979\pi$
$127$	4.38635e14			0.823147			0.411574	+	0.911377i	$0.364979\pi$
$128$		0			0
$129$		−	1.93713e12i		−	0.00325859i
$130$		0			0
$131$		−	1.13794e15i		−	1.71878i	−0.511323	−	0.859388i	$-0.670845\pi$
$131$		−	1.13794e15i		−	1.71878i	0.511323	−	0.859388i	$-0.329155\pi$
$132$		0			0
$133$	−5.35385e14	−	3.95750e14i	−0.727284	−	0.537600i
$134$		0			0
$135$	−1.24004e14			−0.151740
$136$		0			0
$137$	−4.06504e14			−0.448766			−0.224383	−	0.974501i	$-0.572037\pi$
$137$	−4.06504e14			−0.448766			−0.224383	+	0.974501i	$0.572037\pi$
$138$		0			0
$139$			1.93142e15i			1.92652i	0.268569	+	0.963261i	$0.413449\pi$
$139$			1.93142e15i			1.92652i	−0.268569	+	0.963261i	$0.586551\pi$
$140$		0			0
$141$	−5.85445e13			−0.0528387
$142$		0			0
$143$			4.84369e14i			0.396117i
$144$		0			0
$145$			8.72579e14i			0.647484i
$146$		0			0
$147$	−2.44190e14	+	7.49234e13i	−0.164629	+	0.0505122i
$148$		0			0
$149$	−1.78272e15			−1.09340			−0.546699	−	0.837329i	$-0.684116\pi$
$149$	−1.78272e15			−1.09340			−0.546699	+	0.837329i	$0.684116\pi$
$150$		0			0
$151$	4.00837e14			0.223939			0.111969	−	0.993712i	$-0.464284\pi$
$151$	4.00837e14			0.223939			0.111969	+	0.993712i	$0.464284\pi$
$152$		0			0
$153$			1.89891e15i			0.967529i
$154$		0			0
$155$	6.45414e14			0.300273
$156$		0			0
$157$		−	6.11308e14i		−	0.259993i	−0.991514	−	0.129997i	$-0.958503\pi$
$157$		−	6.11308e14i		−	0.259993i	0.991514	−	0.129997i	$-0.0414967\pi$
$158$		0			0
$159$		−	4.63282e14i		−	0.180329i
$160$		0			0
$161$	−1.07690e15	+	1.45686e15i	−0.384054	+	0.519562i
$162$		0			0
$163$	−7.70874e14			−0.252156			−0.126078	−	0.992020i	$-0.540239\pi$
$163$	−7.70874e14			−0.252156			−0.126078	+	0.992020i	$0.540239\pi$
$164$		0			0
$165$	1.74194e14			0.0523174
$166$		0			0
$167$			4.78990e14i			0.132224i	0.997812	+	0.0661122i	$0.0210595\pi$
$167$			4.78990e14i			0.132224i	−0.997812	+	0.0661122i	$0.978940\pi$
$168$		0			0
$169$	2.59869e15			0.660006
$170$		0			0
$171$		−	3.75201e15i		−	0.877590i
$172$		0			0
$173$			3.36394e15i			0.725313i	0.931923	+	0.362657i	$0.118130\pi$
$173$			3.36394e15i			0.725313i	−0.931923	+	0.362657i	$0.881870\pi$
$174$		0			0
$175$	−5.97572e14	+	8.08417e14i	−0.118884	+	0.160831i
$176$		0			0
$177$	1.45454e15			0.267246
$178$		0			0
$179$	−6.41429e15			−1.08937			−0.544687	−	0.838639i	$-0.683352\pi$
$179$	−6.41429e15			−1.08937			−0.544687	+	0.838639i	$0.683352\pi$
$180$		0			0
$181$			4.48200e15i			0.704242i	0.935955	+	0.352121i	$0.114539\pi$
$181$			4.48200e15i			0.704242i	−0.935955	+	0.352121i	$0.885461\pi$
$182$		0			0
$183$	−2.77223e13			−0.00403340
$184$		0			0
$185$			2.62921e15i			0.354506i
$186$		0			0
$187$		−	5.41647e15i		−	0.677369i
$188$		0			0
$189$	−2.35049e15	−	1.73745e15i	−0.272851	−	0.201688i
$190$		0			0
$191$	−1.30679e15			−0.140919			−0.0704597	−	0.997515i	$-0.522447\pi$
$191$	−1.30679e15			−0.140919			−0.0704597	+	0.997515i	$0.522447\pi$
$192$		0			0
$193$	−1.23015e16			−1.23327			−0.616633	−	0.787251i	$-0.711504\pi$
$193$	−1.23015e16			−1.23327			−0.616633	+	0.787251i	$0.711504\pi$
$194$		0			0
$195$			4.81433e14i			0.0449049i
$196$		0			0
$197$	−2.24263e16			−1.94758			−0.973788	−	0.227456i	$-0.926959\pi$
$197$	−2.24263e16			−1.94758			−0.973788	+	0.227456i	$0.926959\pi$
$198$		0			0
$199$		−	3.96014e15i		−	0.320434i	−0.987082	−	0.160217i	$-0.948781\pi$
$199$		−	3.96014e15i		−	0.320434i	0.987082	−	0.160217i	$-0.0512195\pi$
$200$		0			0
$201$			9.68989e13i			0.00731049i
$202$		0			0
$203$	−1.22259e16	+	1.65396e16i	−0.860613	+	1.16427i
$204$		0			0
$205$	−4.75346e15			−0.312416
$206$		0			0
$207$	−1.02098e16			−0.626939
$208$		0			0
$209$			1.07023e16i			0.614403i
$210$		0			0
$211$	−1.82383e16			−0.979509			−0.489754	−	0.871861i	$-0.662914\pi$
$211$	−1.82383e16			−0.979509			−0.489754	+	0.871861i	$0.662914\pi$
$212$		0			0
$213$			6.21740e15i			0.312573i
$214$		0			0
$215$		−	1.79709e14i		−	0.00846256i
$216$		0			0
$217$	1.22337e16	+	9.04302e15i	0.539934	+	0.399113i
$218$		0			0
$219$	−2.79256e15			−0.115583
$220$		0			0
$221$	1.49699e16			0.581397
$222$		0			0
$223$			4.72358e16i			1.72241i	0.508255	+	0.861207i	$0.330291\pi$
$223$			4.72358e16i			1.72241i	−0.508255	+	0.861207i	$0.669709\pi$
$224$		0			0
$225$	−5.66545e15			−0.194069
$226$		0			0
$227$			3.33486e16i			1.07373i	0.843667	+	0.536867i	$0.180392\pi$
$227$			3.33486e16i			1.07373i	−0.843667	+	0.536867i	$0.819608\pi$
$228$		0			0
$229$		−	2.19054e16i		−	0.663290i	−0.943404	−	0.331645i	$-0.892396\pi$
$229$		−	2.19054e16i		−	0.663290i	0.943404	−	0.331645i	$-0.107604\pi$
$230$		0			0
$231$	3.30183e15	+	2.44067e15i	0.0940742	+	0.0695385i
$232$		0			0
$233$	2.10095e16			0.563539			0.281770	−	0.959482i	$-0.409079\pi$
$233$	2.10095e16			0.563539			0.281770	+	0.959482i	$0.409079\pi$
$234$		0			0
$235$	−5.43124e15			−0.137222
$236$		0			0
$237$		−	4.42582e15i		−	0.105379i
$238$		0			0
$239$	−7.18935e16			−1.61400			−0.807002	−	0.590549i	$-0.798911\pi$
$239$	−7.18935e16			−1.61400			−0.807002	+	0.590549i	$0.798911\pi$
$240$		0			0
$241$			5.65645e16i			1.19791i	0.800783	+	0.598955i	$0.204417\pi$
$241$			5.65645e16i			1.19791i	−0.800783	+	0.598955i	$0.795583\pi$
$242$		0			0
$243$		−	2.48325e16i		−	0.496337i
$244$		0			0
$245$	−2.26538e16	+	6.95073e15i	−0.427542	+	0.131180i
$246$		0			0
$247$	−2.95787e16			−0.527352
$248$		0			0
$249$	−4.81152e15			−0.0810750
$250$		0			0
$251$		−	2.35358e16i		−	0.374984i	−0.982266	−	0.187492i	$-0.939964\pi$
$251$		−	2.35358e16i		−	0.374984i	0.982266	−	0.187492i	$-0.0600358\pi$
$252$		0			0
$253$	2.91226e16			0.438921
$254$		0			0
$255$		−	5.38365e15i		−	0.0767884i
$256$		0			0
$257$			1.18377e17i			1.59859i	0.600939	+	0.799295i	$0.294793\pi$
$257$			1.18377e17i			1.59859i	−0.600939	+	0.799295i	$0.705207\pi$
$258$		0			0
$259$	−3.68384e16	+	4.98364e16i	−0.471197	+	0.637453i
$260$		0			0
$261$	−1.15911e17			−1.40488
$262$		0			0
$263$	−1.08211e17			−1.24331			−0.621656	−	0.783290i	$-0.713540\pi$
$263$	−1.08211e17			−1.24331			−0.621656	+	0.783290i	$0.713540\pi$
$264$		0			0
$265$		−	4.29792e16i		−	0.468314i
$266$		0			0
$267$	−3.49218e15			−0.0361009
$268$		0			0
$269$			4.39221e16i			0.430940i	0.976510	+	0.215470i	$0.0691283\pi$
$269$			4.39221e16i			0.430940i	−0.976510	+	0.215470i	$0.930872\pi$
$270$		0			0
$271$			1.09543e17i			1.02047i	0.860034	+	0.510236i	$0.170442\pi$
$271$			1.09543e17i			1.02047i	−0.860034	+	0.510236i	$0.829558\pi$
$272$		0			0
$273$	−6.74546e15	+	9.12550e15i	−0.0596861	+	0.0807455i
$274$		0			0
$275$	1.61602e16			0.135868
$276$		0			0
$277$	−7.43695e16			−0.594342			−0.297171	−	0.954824i	$-0.596043\pi$
$277$	−7.43695e16			−0.594342			−0.297171	+	0.954824i	$0.596043\pi$
$278$		0			0
$279$			8.57348e16i			0.651520i
$280$		0			0
$281$	−2.07315e17			−1.49860			−0.749302	−	0.662228i	$-0.769611\pi$
$281$	−2.07315e17			−1.49860			−0.749302	+	0.662228i	$0.769611\pi$
$282$		0			0
$283$		−	9.39605e16i		−	0.646309i	−0.946346	−	0.323154i	$-0.895257\pi$
$283$		−	9.39605e16i		−	0.646309i	0.946346	−	0.323154i	$-0.104743\pi$
$284$		0			0
$285$			1.06374e16i			0.0696504i
$286$		0			0
$287$	−9.01012e16	−	6.66017e16i	−0.561769	−	0.415253i
$288$		0			0
$289$	9.76213e14			0.00579775
$290$		0			0
$291$	2.17593e16			0.123139
$292$		0			0
$293$		−	2.53081e17i		−	1.36517i	−0.730808	−	0.682583i	$-0.760856\pi$
$293$		−	2.53081e17i		−	1.36517i	0.730808	−	0.682583i	$-0.239144\pi$
$294$		0			0
$295$	1.34939e17			0.694038
$296$		0			0
$297$			4.69860e16i			0.230501i
$298$		0			0
$299$			8.04883e16i			0.376734i
$300$		0			0
$301$	2.51794e15	−	3.40637e15i	0.0112481	−	0.0152169i
$302$		0			0
$303$	−4.85079e16			−0.206878
$304$		0			0
$305$	−2.57183e15			−0.0104747
$306$		0			0
$307$			1.10960e17i			0.431716i	0.976425	+	0.215858i	$0.0692548\pi$
$307$			1.10960e17i			0.431716i	−0.976425	+	0.215858i	$0.930745\pi$
$308$		0			0
$309$	5.15002e16			0.191469
$310$		0			0
$311$			2.71059e17i			0.963255i	0.876376	+	0.481628i	$0.159954\pi$
$311$			2.71059e17i			0.963255i	−0.876376	+	0.481628i	$0.840046\pi$
$312$		0			0
$313$			1.94281e17i			0.660115i	0.943961	+	0.330058i	$0.107068\pi$
$313$			1.94281e17i			0.660115i	−0.943961	+	0.330058i	$0.892932\pi$
$314$		0			0
$315$	−1.07388e17	−	7.93797e16i	−0.348964	−	0.257950i
$316$		0			0
$317$	1.21598e17			0.378016			0.189008	−	0.981976i	$-0.439473\pi$
$317$	1.21598e17			0.378016			0.189008	+	0.981976i	$0.439473\pi$
$318$		0			0
$319$	3.30626e17			0.983562
$320$		0			0
$321$			1.29769e16i			0.0369518i
$322$		0			0
$323$	3.30765e17			0.901784
$324$		0			0
$325$			4.46631e16i			0.116618i
$326$		0			0
$327$		−	1.10038e17i		−	0.275237i
$328$		0			0
$329$	−1.02948e17	−	7.60982e16i	−0.246745	−	0.182391i
$330$		0			0
$331$	4.51454e17			1.03709			0.518546	−	0.855049i	$-0.326473\pi$
$331$	4.51454e17			1.03709			0.518546	+	0.855049i	$0.326473\pi$
$332$		0			0
$333$	−3.49257e17			−0.769193
$334$		0			0
$335$			8.98942e15i			0.0189853i
$336$		0			0
$337$	−5.98786e17			−1.21300			−0.606502	−	0.795082i	$-0.707428\pi$
$337$	−5.98786e17			−1.21300			−0.606502	+	0.795082i	$0.707428\pi$
$338$		0			0
$339$			1.27804e17i			0.248398i
$340$		0			0
$341$		−	2.44551e17i		−	0.456131i
$342$		0			0
$343$	−5.26787e17	−	1.85657e17i	−0.943141	−	0.332393i
$344$		0			0
$345$	2.89461e16			0.0497573
$346$		0			0
$347$	5.01105e17			0.827223			0.413611	−	0.910454i	$-0.364267\pi$
$347$	5.01105e17			0.827223			0.413611	+	0.910454i	$0.364267\pi$
$348$		0			0
$349$		−	3.04905e17i		−	0.483489i	−0.970340	−	0.241744i	$-0.922280\pi$
$349$		−	3.04905e17i		−	0.483489i	0.970340	−	0.241744i	$-0.0777196\pi$
$350$		0			0
$351$	−1.29859e17			−0.197843
$352$		0			0
$353$		−	6.13804e17i		−	0.898684i	−0.893360	−	0.449342i	$-0.851658\pi$
$353$		−	6.13804e17i		−	0.898684i	0.893360	−	0.449342i	$-0.148342\pi$
$354$		0			0
$355$			5.76796e17i			0.811752i
$356$		0			0
$357$	7.54314e16	−	1.02046e17i	0.102065	−	0.138077i
$358$		0			0
$359$	−1.21590e17			−0.158211			−0.0791057	−	0.996866i	$-0.525206\pi$
$359$	−1.21590e17			−0.158211			−0.0791057	+	0.996866i	$0.525206\pi$
$360$		0			0
$361$	1.45454e17			0.182044
$362$		0			0
$363$			7.70143e16i			0.0927311i
$364$		0			0
$365$	−2.59069e17			−0.300169
$366$		0			0
$367$		−	7.57552e17i		−	0.844792i	−0.906411	−	0.422396i	$-0.861189\pi$
$367$		−	7.57552e17i		−	0.844792i	0.906411	−	0.422396i	$-0.138811\pi$
$368$		0			0
$369$		−	6.31436e17i		−	0.677867i
$370$		0			0
$371$	6.02190e17	−	8.14665e17i	0.622467	−	0.842096i
$372$		0			0
$373$	1.95099e18			1.94220			0.971100	−	0.238675i	$-0.0767129\pi$
$373$	1.95099e18			1.94220			0.971100	+	0.238675i	$0.0767129\pi$
$374$		0			0
$375$	1.60623e16			0.0154024
$376$		0			0
$377$			9.13774e17i			0.844208i
$378$		0			0
$379$	1.03550e18			0.921881			0.460940	−	0.887431i	$-0.347512\pi$
$379$	1.03550e18			0.921881			0.460940	+	0.887431i	$0.347512\pi$
$380$		0			0
$381$		−	1.65195e17i		−	0.141749i
$382$		0			0
$383$			4.74023e17i			0.392110i	0.980593	+	0.196055i	$0.0628131\pi$
$383$			4.74023e17i			0.392110i	−0.980593	+	0.196055i	$0.937187\pi$
$384$		0			0
$385$	3.06314e17	+	2.26424e17i	0.244311	+	0.180591i
$386$		0			0
$387$	2.38721e16			0.0183617
$388$		0			0
$389$	8.05988e17			0.597973			0.298986	−	0.954257i	$-0.403352\pi$
$389$	8.05988e17			0.597973			0.298986	+	0.954257i	$0.403352\pi$
$390$		0			0
$391$		−	9.00064e17i		−	0.644223i
$392$		0			0
$393$	−4.28559e17			−0.295980
$394$		0			0
$395$		−	4.10588e17i		−	0.273670i
$396$		0			0
$397$		−	1.61822e18i		−	1.04113i	−0.853823	−	0.520563i	$-0.825722\pi$
$397$		−	1.61822e18i		−	1.04113i	0.853823	−	0.520563i	$-0.174278\pi$
$398$		0			0
$399$	−1.49043e17	+	2.01631e17i	−0.0925769	+	0.125241i
$400$		0			0
$401$	1.54640e18			0.927496			0.463748	−	0.885967i	$-0.346504\pi$
$401$	1.54640e18			0.927496			0.463748	+	0.885967i	$0.346504\pi$
$402$		0			0
$403$	6.75884e17			0.391505
$404$		0			0
$405$		−	7.28879e17i		−	0.407822i
$406$		0			0
$407$	9.96225e17			0.538514
$408$		0			0
$409$		−	1.81098e18i		−	0.945910i	−0.881086	−	0.472955i	$-0.843187\pi$
$409$		−	1.81098e18i		−	0.945910i	0.881086	−	0.472955i	$-0.156813\pi$
$410$		0			0
$411$			1.53093e17i			0.0772794i
$412$		0			0
$413$	2.55775e18	+	1.89066e18i	1.24798	+	0.922491i
$414$		0			0
$415$	−4.46370e17			−0.210552
$416$		0			0
$417$	7.27394e17			0.331755
$418$		0			0
$419$		−	1.04602e18i		−	0.461363i	−0.973029	−	0.230682i	$-0.925904\pi$
$419$		−	1.04602e18i		−	0.461363i	0.973029	−	0.230682i	$-0.0740956\pi$
$420$		0			0
$421$	1.30966e18			0.558707			0.279354	−	0.960188i	$-0.409880\pi$
$421$	1.30966e18			0.558707			0.279354	+	0.960188i	$0.409880\pi$
$422$		0			0
$423$		−	7.21470e17i		−	0.297739i
$424$		0			0
$425$		−	4.99448e17i		−	0.199419i
$426$		0			0
$427$	−4.87486e16	−	3.60344e16i	−0.0188351	−	0.0139226i
$428$		0			0
$429$	1.82418e17			0.0682130
$430$		0			0
$431$	−9.36240e17			−0.338880			−0.169440	−	0.985540i	$-0.554196\pi$
$431$	−9.36240e17			−0.338880			−0.169440	+	0.985540i	$0.554196\pi$
$432$		0			0
$433$			2.36294e18i			0.828013i	0.910274	+	0.414006i	$0.135871\pi$
$433$			2.36294e18i			0.828013i	−0.910274	+	0.414006i	$0.864129\pi$
$434$		0			0
$435$	3.28622e17			0.111499
$436$		0			0
$437$			1.77842e18i			0.584337i
$438$		0			0
$439$		−	3.01140e18i		−	0.958333i	−0.877724	−	0.479167i	$-0.840939\pi$
$439$		−	3.01140e18i		−	0.958333i	0.877724	−	0.479167i	$-0.159061\pi$
$440$		0			0
$441$	−9.23314e17	−	3.00926e18i	−0.284629	−	0.927662i
$442$		0			0
$443$	−1.89470e18			−0.565867			−0.282934	−	0.959139i	$-0.591308\pi$
$443$	−1.89470e18			−0.565867			−0.282934	+	0.959139i	$0.591308\pi$
$444$		0			0
$445$	−3.23974e17			−0.0937540
$446$		0			0
$447$			6.71389e17i			0.188288i
$448$		0			0
$449$	2.88811e17			0.0785036			0.0392518	−	0.999229i	$-0.487503\pi$
$449$	2.88811e17			0.0785036			0.0392518	+	0.999229i	$0.487503\pi$
$450$		0			0
$451$			1.80112e18i			0.474577i
$452$		0			0
$453$		−	1.50959e17i		−	0.0385631i
$454$		0			0
$455$	−6.25784e17	+	8.46583e17i	−0.155005	+	0.209696i
$456$		0			0
$457$	3.74159e18			0.898757			0.449379	−	0.893341i	$-0.351645\pi$
$457$	3.74159e18			0.898757			0.449379	+	0.893341i	$0.351645\pi$
$458$		0			0
$459$	1.45215e18			0.338316
$460$		0			0
$461$		−	4.22981e18i		−	0.955905i	−0.878386	−	0.477953i	$-0.841379\pi$
$461$		−	4.22981e18i		−	0.955905i	0.878386	−	0.477953i	$-0.158621\pi$
$462$		0			0
$463$	−8.19224e17			−0.179612			−0.0898061	−	0.995959i	$-0.528625\pi$
$463$	−8.19224e17			−0.179612			−0.0898061	+	0.995959i	$0.528625\pi$
$464$		0			0
$465$		−	2.43069e17i		−	0.0517082i
$466$		0			0
$467$		−	8.85627e18i		−	1.82824i	−0.405447	−	0.914119i	$-0.632884\pi$
$467$		−	8.85627e18i		−	1.82824i	0.405447	−	0.914119i	$-0.367116\pi$
$468$		0			0
$469$	−1.25953e17	+	1.70393e17i	−0.0252346	+	0.0341383i
$470$		0			0
$471$	−2.30225e17			−0.0447719
$472$		0			0
$473$	−6.80930e16			−0.0128551
$474$		0			0
$475$			9.86848e17i			0.180882i
$476$		0			0
$477$	5.70923e18			1.01613
$478$		0			0
$479$			5.22175e18i			0.902543i	0.892387	+	0.451271i	$0.149029\pi$
$479$			5.22175e18i			0.902543i	−0.892387	+	0.451271i	$0.850971\pi$
$480$		0			0
$481$			2.75334e18i			0.462215i
$482$		0			0
$483$	5.48670e17	+	4.05570e17i	0.0894707	+	0.0661357i
$484$		0			0
$485$	2.01864e18			0.319791
$486$		0			0
$487$	−6.44199e18			−0.991556			−0.495778	−	0.868449i	$-0.665117\pi$
$487$	−6.44199e18			−0.991556			−0.495778	+	0.868449i	$0.665117\pi$
$488$		0			0
$489$			2.90319e17i			0.0434223i
$490$		0			0
$491$	8.33912e18			1.21213			0.606066	−	0.795414i	$-0.292747\pi$
$491$	8.33912e18			1.21213			0.606066	+	0.795414i	$0.292747\pi$
$492$		0			0
$493$		−	1.02183e19i		−	1.44361i
$494$		0			0
$495$			2.14667e18i			0.294801i
$496$		0			0
$497$	−8.08160e18	+	1.09331e19i	−1.07895	+	1.45965i
$498$		0			0
$499$	−1.36981e18			−0.177810			−0.0889050	−	0.996040i	$-0.528337\pi$
$499$	−1.36981e18			−0.177810			−0.0889050	+	0.996040i	$0.528337\pi$
$500$		0			0
$501$	1.80393e17			0.0227696
$502$		0			0
$503$		−	4.82000e18i		−	0.591659i	−0.955241	−	0.295829i	$-0.904404\pi$
$503$		−	4.82000e18i		−	0.591659i	0.955241	−	0.295829i	$-0.0955960\pi$
$504$		0			0
$505$	−4.50013e18			−0.537261
$506$		0			0
$507$		−	9.78694e17i		−	0.113656i
$508$		0			0
$509$			2.71705e18i			0.306954i	0.988152	+	0.153477i	$0.0490471\pi$
$509$			2.71705e18i			0.306954i	−0.988152	+	0.153477i	$0.950953\pi$
$510$		0			0
$511$	−4.91063e18	−	3.62987e18i	−0.539747	−	0.398975i
$512$		0			0
$513$	−2.86928e18			−0.306867
$514$		0			0
$515$	4.77773e18			0.497246
$516$		0			0
$517$			2.05793e18i			0.208448i
$518$		0			0
$519$	1.26689e18			0.124902
$520$		0			0
$521$			8.98814e18i			0.862593i	0.902210	+	0.431296i	$0.141944\pi$
$521$			8.98814e18i			0.862593i	−0.902210	+	0.431296i	$0.858056\pi$
$522$		0			0
$523$			1.47434e18i			0.137748i	0.997625	+	0.0688741i	$0.0219407\pi$
$523$			1.47434e18i			0.137748i	−0.997625	+	0.0688741i	$0.978059\pi$
$524$		0			0
$525$	3.04458e17	+	2.25052e17i	0.0276957	+	0.0204723i
$526$		0			0
$527$	−7.55811e18			−0.669482
$528$		0			0
$529$	−6.75349e18			−0.582557
$530$		0			0
$531$			1.79249e19i			1.50590i
$532$		0			0
$533$	−4.97788e18			−0.407337
$534$		0			0
$535$			1.20388e18i			0.0959637i
$536$		0			0
$537$			2.41569e18i			0.187595i
$538$		0			0
$539$	2.63367e18	+	8.58366e18i	0.199269	+	0.649458i
$540$		0			0
$541$	−1.76737e19			−1.30301			−0.651503	−	0.758646i	$-0.725861\pi$
$541$	−1.76737e19			−1.30301			−0.651503	+	0.758646i	$0.725861\pi$
$542$		0			0
$543$	1.68797e18			0.121273
$544$		0			0
$545$		−	1.02083e19i		−	0.714790i
$546$		0			0
$547$	−2.38329e19			−1.62654			−0.813271	−	0.581885i	$-0.802315\pi$
$547$	−2.38329e19			−1.62654			−0.813271	+	0.581885i	$0.802315\pi$
$548$		0			0
$549$		−	3.41634e17i		−	0.0227276i
$550$		0			0
$551$			2.01902e19i			1.30942i
$552$		0			0
$553$	5.75283e18	−	7.78264e18i	0.363752	−	0.492097i
$554$		0			0
$555$	9.90188e17			0.0610474
$556$		0			0
$557$	4.59298e17			0.0276127			0.0138063	−	0.999905i	$-0.495605\pi$
$557$	4.59298e17			0.0276127			0.0138063	+	0.999905i	$0.495605\pi$
$558$		0			0
$559$		−	1.88194e17i		−	0.0110337i
$560$		0			0
$561$	−2.03990e18			−0.116646
$562$		0			0
$563$		−	1.55890e18i		−	0.0869478i	−0.999055	−	0.0434739i	$-0.986157\pi$
$563$		−	1.55890e18i		−	0.0869478i	0.999055	−	0.0434739i	$-0.0138425\pi$
$564$		0			0
$565$			1.18566e19i			0.645089i
$566$		0			0
$567$	1.02125e19	−	1.38158e19i	0.542062	−	0.733322i
$568$		0			0
$569$	−2.27884e18			−0.118012			−0.0590062	−	0.998258i	$-0.518793\pi$
$569$	−2.27884e18			−0.118012			−0.0590062	+	0.998258i	$0.518793\pi$
$570$		0			0
$571$	−4.24088e18			−0.214291			−0.107145	−	0.994243i	$-0.534171\pi$
$571$	−4.24088e18			−0.214291			−0.107145	+	0.994243i	$0.534171\pi$
$572$		0			0
$573$			4.92149e17i			0.0242669i
$574$		0			0
$575$	2.68537e18			0.129220
$576$		0			0
$577$			1.28174e19i			0.601962i	0.953630	+	0.300981i	$0.0973141\pi$
$577$			1.28174e19i			0.601962i	−0.953630	+	0.300981i	$0.902686\pi$
$578$		0			0
$579$			4.63287e18i			0.212373i
$580$		0			0
$581$	−8.46088e18	−	6.25418e18i	−0.378602	−	0.279858i
$582$		0			0
$583$	−1.62851e19			−0.711394
$584$		0			0
$585$	−5.93292e18			−0.253033
$586$		0			0
$587$			1.77989e19i			0.741184i	0.928796	+	0.370592i	$0.120845\pi$
$587$			1.77989e19i			0.741184i	−0.928796	+	0.370592i	$0.879155\pi$
$588$		0			0
$589$	1.49339e19			0.607248
$590$		0			0
$591$			8.44598e18i			0.335381i
$592$		0			0
$593$			2.39009e19i			0.926895i	0.886124	+	0.463448i	$0.153388\pi$
$593$			2.39009e19i			0.926895i	−0.886124	+	0.463448i	$0.846612\pi$
$594$		0			0
$595$	6.99786e18	−	9.46696e18i	0.265061	−	0.358585i
$596$		0			0
$597$	−1.49143e18			−0.0551801
$598$		0			0
$599$	−4.25217e19			−1.53682			−0.768410	−	0.639958i	$-0.778952\pi$
$599$	−4.25217e19			−1.53682			−0.768410	+	0.639958i	$0.778952\pi$
$600$		0			0
$601$		−	1.28268e19i		−	0.452896i	−0.974023	−	0.226448i	$-0.927289\pi$
$601$		−	1.28268e19i		−	0.452896i	0.974023	−	0.226448i	$-0.0727113\pi$
$602$		0			0
$603$	−1.19413e18			−0.0411936
$604$		0			0
$605$			7.14471e18i			0.240822i
$606$		0			0
$607$			8.52263e18i			0.280707i	0.990101	+	0.140353i	$0.0448239\pi$
$607$			8.52263e18i			0.280707i	−0.990101	+	0.140353i	$0.955176\pi$
$608$		0			0
$609$	6.22899e18	+	4.60439e18i	0.200492	+	0.148201i
$610$		0			0
$611$	−5.68765e18			−0.178914
$612$		0			0
$613$	−3.78825e19			−1.16470			−0.582352	−	0.812937i	$-0.697867\pi$
$613$	−3.78825e19			−1.16470			−0.582352	+	0.812937i	$0.697867\pi$
$614$		0			0
$615$			1.79020e18i			0.0537993i
$616$		0			0
$617$	1.35750e19			0.398791			0.199395	−	0.979919i	$-0.436102\pi$
$617$	1.35750e19			0.398791			0.199395	+	0.979919i	$0.436102\pi$
$618$		0			0
$619$			3.18750e19i			0.915410i	0.889104	+	0.457705i	$0.151328\pi$
$619$			3.18750e19i			0.915410i	−0.889104	+	0.457705i	$0.848672\pi$
$620$		0			0
$621$			7.80774e18i			0.219222i
$622$		0			0
$623$	−6.14087e18	−	4.53926e18i	−0.168583	−	0.124615i
$624$		0			0
$625$	1.49012e18			0.0400000
$626$		0			0
$627$	4.03059e18			0.105803
$628$		0			0
$629$		−	3.07893e19i		−	0.790398i
$630$		0			0
$631$	−1.88659e17			−0.00473666			−0.00236833	−	0.999997i	$-0.500754\pi$
$631$	−1.88659e17			−0.00473666			−0.00236833	+	0.999997i	$0.500754\pi$
$632$		0			0
$633$			6.86875e18i			0.168675i
$634$		0			0
$635$		−	1.53253e19i		−	0.368123i
$636$		0			0
$637$	−2.37233e19	+	7.27888e18i	−0.557441	+	0.171036i
$638$		0			0
$639$	−7.66198e19			−1.76131
$640$		0			0
$641$	5.07000e19			1.14025			0.570127	−	0.821556i	$-0.306894\pi$
$641$	5.07000e19			1.14025			0.570127	+	0.821556i	$0.306894\pi$
$642$		0			0
$643$			8.02501e19i			1.76591i	0.469456	+	0.882956i	$0.344450\pi$
$643$			8.02501e19i			1.76591i	−0.469456	+	0.882956i	$0.655550\pi$
$644$		0			0
$645$	−6.76804e16			−0.00145729
$646$		0			0
$647$		−	7.24097e19i		−	1.52569i	−0.646581	−	0.762846i	$-0.723802\pi$
$647$		−	7.24097e19i		−	1.52569i	0.646581	−	0.762846i	$-0.276198\pi$
$648$		0			0
$649$		−	5.11292e19i		−	1.05428i
$650$		0			0
$651$	3.40569e18	−	4.60735e18i	0.0687288	−	0.0929788i
$652$		0			0
$653$	3.54473e19			0.700150			0.350075	−	0.936722i	$-0.386156\pi$
$653$	3.54473e19			0.700150			0.350075	+	0.936722i	$0.386156\pi$
$654$		0			0
$655$	−3.97579e19			−0.768660
$656$		0			0
$657$		−	3.44140e19i		−	0.651295i
$658$		0			0
$659$	−3.20803e19			−0.594348			−0.297174	−	0.954823i	$-0.596044\pi$
$659$	−3.20803e19			−0.594348			−0.297174	+	0.954823i	$0.596044\pi$
$660$		0			0
$661$		−	5.75463e19i		−	1.04378i	−0.853014	−	0.521888i	$-0.825228\pi$
$661$		−	5.75463e19i		−	1.04378i	0.853014	−	0.521888i	$-0.174772\pi$
$662$		0			0
$663$		−	5.63782e18i		−	0.100119i
$664$		0			0
$665$	−1.38269e19	+	1.87056e19i	−0.240422	+	0.325252i
$666$		0			0
$667$	5.49406e19			0.935432
$668$		0			0
$669$	1.77895e19			0.296606
$670$		0			0
$671$			9.74481e17i			0.0159117i
$672$		0			0
$673$	8.76195e19			1.40118			0.700592	−	0.713562i	$-0.252919\pi$
$673$	8.76195e19			1.40118			0.700592	+	0.713562i	$0.252919\pi$
$674$		0			0
$675$			4.33254e18i			0.0678603i
$676$		0			0
$677$			2.86040e18i			0.0438840i	0.999759	+	0.0219420i	$0.00698491\pi$
$677$			2.86040e18i			0.0438840i	−0.999759	+	0.0219420i	$0.993015\pi$
$678$		0			0
$679$	3.82630e19	+	2.82835e19i	0.575030	+	0.425055i
$680$		0			0
$681$	1.25594e19			0.184901
$682$		0			0
$683$	−4.64674e19			−0.670198			−0.335099	−	0.942183i	$-0.608770\pi$
$683$	−4.64674e19			−0.670198			−0.335099	+	0.942183i	$0.608770\pi$
$684$		0			0
$685$			1.42026e19i			0.200694i
$686$		0			0
$687$	−8.24981e18			−0.114221
$688$		0			0
$689$		−	4.50083e19i		−	0.610602i
$690$		0			0
$691$			3.97924e19i			0.528997i	0.964386	+	0.264499i	$0.0852065\pi$
$691$			3.97924e19i			0.528997i	−0.964386	+	0.264499i	$0.914794\pi$
$692$		0			0
$693$	−3.00775e19	+	4.06899e19i	−0.391840	+	0.530095i
$694$		0			0
$695$	6.74811e19			0.861566
$696$		0			0
$697$	5.56654e19			0.696555
$698$		0			0
$699$		−	7.91239e18i		−	0.0970437i
$700$		0			0
$701$	2.31758e18			0.0278617			0.0139309	−	0.999903i	$-0.495566\pi$
$701$	2.31758e18			0.0278617			0.0139309	+	0.999903i	$0.495566\pi$
$702$		0			0
$703$			6.08360e19i			0.716925i
$704$		0			0
$705$			2.04546e18i			0.0236302i
$706$		0			0
$707$	−8.52994e19	−	6.30523e19i	−0.966073	−	0.714109i
$708$		0			0
$709$	7.86158e19			0.872943			0.436472	−	0.899718i	$-0.356228\pi$
$709$	7.86158e19			0.872943			0.436472	+	0.899718i	$0.356228\pi$
$710$		0			0
$711$	5.45413e19			0.593798
$712$		0			0
$713$		−	4.06375e19i		−	0.433810i
$714$		0			0
$715$	1.69231e19			0.177149
$716$		0			0
$717$			2.70758e19i			0.277938i
$718$		0			0
$719$		−	7.48871e18i		−	0.0753884i	−0.999289	−	0.0376942i	$-0.987999\pi$
$719$		−	7.48871e18i		−	0.0753884i	0.999289	−	0.0376942i	$-0.0120013\pi$
$720$		0			0
$721$	9.05612e19	+	6.69417e19i	0.894119	+	0.660922i
$722$		0			0
$723$	2.13028e19			0.206285
$724$		0			0
$725$	3.04867e19			0.289563
$726$		0			0
$727$		−	6.06413e19i		−	0.564973i	−0.959271	−	0.282486i	$-0.908841\pi$
$727$		−	6.06413e19i		−	0.564973i	0.959271	−	0.282486i	$-0.0911592\pi$
$728$		0			0
$729$	9.04288e19			0.826445
$730$		0			0
$731$			2.10448e18i			0.0188679i
$732$		0			0
$733$		−	1.92331e20i		−	1.69169i	−0.533427	−	0.845846i	$-0.679096\pi$
$733$		−	1.92331e20i		−	1.69169i	0.533427	−	0.845846i	$-0.320904\pi$
$734$		0			0
$735$	2.61772e18	+	8.53165e18i	0.0225897	+	0.0736244i
$736$		0			0
$737$	3.40615e18			0.0288397
$738$		0			0
$739$	7.65239e19			0.635750			0.317875	−	0.948133i	$-0.397031\pi$
$739$	7.65239e19			0.635750			0.317875	+	0.948133i	$0.397031\pi$
$740$		0			0
$741$			1.11396e19i			0.0908122i
$742$		0			0
$743$	−2.05061e19			−0.164045			−0.0820224	−	0.996630i	$-0.526138\pi$
$743$	−2.05061e19			−0.164045			−0.0820224	+	0.996630i	$0.526138\pi$
$744$		0			0
$745$			6.22855e19i			0.488982i
$746$		0			0
$747$		−	5.92945e19i		−	0.456846i
$748$		0			0
$749$	−1.68678e19	+	2.28194e19i	−0.127552	+	0.172556i
$750$		0			0
$751$	1.34951e19			0.100160			0.0500801	−	0.998745i	$-0.484052\pi$
$751$	1.34951e19			0.100160			0.0500801	+	0.998745i	$0.484052\pi$
$752$		0			0
$753$	−8.86381e18			−0.0645737
$754$		0			0
$755$		−	1.40047e19i		−	0.100148i
$756$		0			0
$757$	−1.18589e20			−0.832481			−0.416241	−	0.909254i	$-0.636653\pi$
$757$	−1.18589e20			−0.832481			−0.416241	+	0.909254i	$0.636653\pi$
$758$		0			0
$759$		−	1.09679e19i		−	0.0755840i
$760$		0			0
$761$		−	1.02573e20i		−	0.693972i	−0.937870	−	0.346986i	$-0.887205\pi$
$761$		−	1.02573e20i		−	0.693972i	0.937870	−	0.346986i	$-0.112795\pi$
$762$		0			0
$763$	1.43031e20	−	1.93497e20i	0.950074	−	1.28529i
$764$		0			0
$765$	6.63451e19			0.432692
$766$		0			0
$767$	1.41310e20			0.904907
$768$		0			0
$769$		−	1.72229e20i		−	1.08298i	−0.840706	−	0.541492i	$-0.817860\pi$
$769$		−	1.72229e20i		−	1.08298i	0.840706	−	0.541492i	$-0.182140\pi$
$770$		0			0
$771$	4.45821e19			0.275284
$772$		0			0
$773$		−	1.73182e20i		−	1.05014i	−0.851059	−	0.525070i	$-0.824039\pi$
$773$		−	1.73182e20i		−	1.05014i	0.851059	−	0.525070i	$-0.175961\pi$
$774$		0			0
$775$		−	2.25498e19i		−	0.134286i
$776$		0			0
$777$	1.87689e19	+	1.38737e19i	0.109772	+	0.0811421i
$778$		0			0
$779$	−1.09988e20			−0.631805
$780$		0			0
$781$	2.18551e20			1.23310
$782$		0			0
$783$			8.86404e19i			0.491247i
$784$		0			0
$785$	−2.13582e19			−0.116273
$786$		0			0
$787$		−	1.83461e20i		−	0.981114i	−0.871409	−	0.490557i	$-0.836793\pi$
$787$		−	1.83461e20i		−	0.981114i	0.871409	−	0.490557i	$-0.163207\pi$
$788$		0			0
$789$			4.07533e19i			0.214103i
$790$		0			0
$791$	−1.66125e20	+	2.24740e20i	−0.857430	+	1.15996i
$792$		0			0
$793$	−2.69324e18			−0.0136573
$794$		0			0
$795$	−1.61864e19			−0.0806456
$796$		0			0
$797$		−	1.47676e20i		−	0.722941i	−0.932384	−	0.361471i	$-0.882275\pi$
$797$		−	1.47676e20i		−	0.722941i	0.932384	−	0.361471i	$-0.117725\pi$
$798$		0			0
$799$	6.36024e19			0.305947
$800$		0			0
$801$		−	4.30357e19i		−	0.203424i
$802$		0			0
$803$			9.81631e19i			0.455973i
$804$		0			0
$805$	5.09007e19	+	3.76252e19i	0.232355	+	0.171754i
$806$		0			0
$807$	1.65415e19			0.0742096
$808$		0			0
$809$	3.65042e18			0.0160955			0.00804773	−	0.999968i	$-0.497438\pi$
$809$	3.65042e18			0.0160955			0.00804773	+	0.999968i	$0.497438\pi$
$810$		0			0
$811$			2.54150e20i			1.10140i	0.834704	+	0.550699i	$0.185639\pi$
$811$			2.54150e20i			1.10140i	−0.834704	+	0.550699i	$0.814361\pi$
$812$		0			0
$813$	4.12552e19			0.175729
$814$		0			0
$815$			2.69332e19i			0.112768i
$816$		0			0
$817$		−	4.15821e18i		−	0.0171140i
$818$		0			0
$819$	−1.12458e20	−	8.31273e19i	−0.454990	−	0.336323i
$820$		0			0
$821$	1.25235e19			0.0498109			0.0249054	−	0.999690i	$-0.492072\pi$
$821$	1.25235e19			0.0498109			0.0249054	+	0.999690i	$0.492072\pi$
$822$		0			0
$823$	−4.08184e20			−1.59609			−0.798044	−	0.602599i	$-0.794132\pi$
$823$	−4.08184e20			−1.59609			−0.798044	+	0.602599i	$0.794132\pi$
$824$		0			0
$825$		−	6.08610e18i		−	0.0233971i
$826$		0			0
$827$	−1.81108e20			−0.684542			−0.342271	−	0.939601i	$-0.611196\pi$
$827$	−1.81108e20			−0.684542			−0.342271	+	0.939601i	$0.611196\pi$
$828$		0			0
$829$			4.72986e19i			0.175779i	0.996130	+	0.0878895i	$0.0280122\pi$
$829$			4.72986e19i			0.175779i	−0.996130	+	0.0878895i	$0.971988\pi$
$830$		0			0
$831$			2.80083e19i			0.102348i
$832$		0			0
$833$	2.65287e20	−	8.13964e19i	0.953237	−	0.292476i
$834$		0			0
$835$	1.67352e19			0.0591325
$836$		0			0
$837$	6.55640e19			0.227817
$838$		0			0
$839$			1.45819e20i			0.498288i	0.968466	+	0.249144i	$0.0801493\pi$
$839$			1.45819e20i			0.498288i	−0.968466	+	0.249144i	$0.919851\pi$
$840$		0			0
$841$	3.26176e20			1.09618
$842$		0			0
$843$			7.80770e19i			0.258066i
$844$		0			0
$845$		−	9.07945e19i		−	0.295164i
$846$		0			0
$847$	−1.00106e20	+	1.35427e20i	−0.320093	+	0.433033i
$848$		0			0
$849$	−3.53865e19			−0.111297
$850$		0			0
$851$	1.65544e20			0.512162
$852$		0			0
$853$		−	4.93528e20i		−	1.50200i	−0.660304	−	0.750998i	$-0.729573\pi$
$853$		−	4.93528e20i		−	1.50200i	0.660304	−	0.750998i	$-0.270427\pi$
$854$		0			0
$855$	−1.31090e20			−0.392470
$856$		0			0
$857$		−	2.29628e20i		−	0.676332i	−0.941086	−	0.338166i	$-0.890193\pi$
$857$		−	2.29628e20i		−	0.676332i	0.941086	−	0.338166i	$-0.109807\pi$
$858$		0			0
$859$			3.49194e20i			1.01185i	0.862578	+	0.505924i	$0.168848\pi$
$859$			3.49194e20i			1.01185i	−0.862578	+	0.505924i	$0.831152\pi$
$860$		0			0
$861$	−2.50829e19	+	3.39330e19i	−0.0715082	+	0.0967388i
$862$		0			0
$863$	5.73647e20			1.60905			0.804524	−	0.593921i	$-0.202421\pi$
$863$	5.73647e20			1.60905			0.804524	+	0.593921i	$0.202421\pi$
$864$		0			0
$865$	1.17531e20			0.324370
$866$		0			0
$867$		−	3.67652e17i		−	0.000998396i
$868$		0			0
$869$	−1.55574e20			−0.415719
$870$		0			0
$871$			9.41382e18i			0.0247536i
$872$		0			0
$873$			2.68150e20i			0.693870i
$874$		0			0
$875$	2.82449e19	+	2.08783e19i	0.0719257	+	0.0531666i
$876$		0			0
$877$	1.39426e20			0.349418			0.174709	−	0.984620i	$-0.444102\pi$
$877$	1.39426e20			0.349418			0.174709	+	0.984620i	$0.444102\pi$
$878$		0			0
$879$	−9.53128e19			−0.235087
$880$		0			0
$881$			5.70218e20i			1.38423i	0.721785	+	0.692117i	$0.243322\pi$
$881$			5.70218e20i			1.38423i	−0.721785	+	0.692117i	$0.756678\pi$
$882$		0			0
$883$	−6.64830e20			−1.58850			−0.794248	−	0.607594i	$-0.792135\pi$
$883$	−6.64830e20			−1.58850			−0.794248	+	0.607594i	$0.792135\pi$
$884$		0			0
$885$		−	5.08194e19i		−	0.119516i
$886$		0			0
$887$			3.11408e20i			0.720883i	0.932782	+	0.360442i	$0.117374\pi$
$887$			3.11408e20i			0.720883i	−0.932782	+	0.360442i	$0.882626\pi$
$888$		0			0
$889$	2.14726e20	−	2.90489e20i	0.489296	−	0.661937i
$890$		0			0
$891$	−2.76177e20			−0.619503
$892$		0			0
$893$	−1.25671e20			−0.277507
$894$		0			0
$895$			2.24106e20i			0.487183i
$896$		0			0
$897$	3.03127e19			0.0648750
$898$		0			0
$899$		−	4.61353e20i		−	0.972109i
$900$		0			0
$901$			5.03307e20i			1.04414i
$902$		0			0
$903$	−1.28287e18	−	9.48283e17i	−0.00262041	−	0.00193697i
$904$		0			0
$905$	1.56595e20			0.314947
$906$		0			0
$907$	−9.14837e20			−1.81173			−0.905864	−	0.423569i	$-0.860777\pi$
$907$	−9.14837e20			−1.81173			−0.905864	+	0.423569i	$0.860777\pi$
$908$		0			0
$909$		−	5.97784e20i		−	1.16573i
$910$		0			0
$911$	3.76134e18			0.00722293			0.00361147	−	0.999993i	$-0.498850\pi$
$911$	3.76134e18			0.00722293			0.00361147	+	0.999993i	$0.498850\pi$
$912$		0			0
$913$			1.69132e20i			0.319839i
$914$		0			0
$915$			9.68576e17i			0.00180379i
$916$		0			0
$917$	−7.53606e20	−	5.57056e20i	−1.38216	−	1.02168i
$918$		0			0
$919$	−1.01888e21			−1.84040			−0.920202	−	0.391443i	$-0.871976\pi$
$919$	−1.01888e21			−1.84040			−0.920202	+	0.391443i	$0.871976\pi$
$920$		0			0
$921$	4.17887e19			0.0743432
$922$		0			0
$923$			6.04027e20i			1.05839i
$924$		0			0
$925$	9.18609e19			0.158540
$926$		0			0
$927$			6.34659e20i			1.07890i
$928$		0			0
$929$		−	3.13715e20i		−	0.525322i	−0.964888	−	0.262661i	$-0.915400\pi$
$929$		−	3.13715e20i		−	0.525322i	0.964888	−	0.262661i	$-0.0846001\pi$
$930$		0			0
$931$	−5.24175e20	+	1.60829e20i	−0.864626	+	0.265288i
$932$		0			0
$933$	1.02084e20			0.165876
$934$		0			0
$935$	−1.89244e20			−0.302929
$936$		0			0
$937$		−	4.40176e20i		−	0.694144i	−0.937838	−	0.347072i	$-0.887176\pi$
$937$		−	4.40176e20i		−	0.694144i	0.937838	−	0.347072i	$-0.112824\pi$
$938$		0			0
$939$	7.31682e19			0.113675
$940$		0			0
$941$			9.32718e20i			1.42765i	0.700322	+	0.713827i	$0.253040\pi$
$941$			9.32718e20i			1.42765i	−0.700322	+	0.713827i	$0.746960\pi$
$942$		0			0
$943$			2.99294e20i			0.451353i
$944$		0			0
$945$	−6.07040e19	+	8.21226e19i	−0.0901976	+	0.122023i
$946$		0			0
$947$	−3.18691e20			−0.466574			−0.233287	−	0.972408i	$-0.574948\pi$
$947$	−3.18691e20			−0.466574			−0.233287	+	0.972408i	$0.574948\pi$
$948$		0			0
$949$	−2.71300e20			−0.391369
$950$		0			0
$951$		−	4.57949e19i		−	0.0650958i
$952$		0			0
$953$	−6.25335e20			−0.875915			−0.437958	−	0.898996i	$-0.644298\pi$
$953$	−6.25335e20			−0.875915			−0.437958	+	0.898996i	$0.644298\pi$
$954$		0			0
$955$			4.56572e19i			0.0630211i
$956$		0			0
$957$		−	1.24517e20i		−	0.169373i
$958$		0			0
$959$	−1.98996e20	+	2.69209e20i	−0.266756	+	0.360877i
$960$		0			0
$961$	4.15699e20			0.549181
$962$		0			0
$963$	−1.59920e20			−0.208218
$964$		0			0
$965$			4.29797e20i			0.551534i
$966$		0			0
$967$	−9.25313e20			−1.17032			−0.585158	−	0.810919i	$-0.698968\pi$
$967$	−9.25313e20			−1.17032			−0.585158	+	0.810919i	$0.698968\pi$
$968$		0			0
$969$		−	1.24570e20i		−	0.155291i
$970$		0			0
$971$			7.31515e20i			0.898853i	0.893317	+	0.449426i	$0.148372\pi$
$971$			7.31515e20i			0.898853i	−0.893317	+	0.449426i	$0.851628\pi$
$972$		0			0
$973$	1.27910e21	+	9.45492e20i	1.54922	+	1.14516i
$974$		0			0
$975$	1.68206e19			0.0200821
$976$		0			0
$977$	2.88986e20			0.340106			0.170053	−	0.985435i	$-0.445606\pi$
$977$	2.88986e20			0.340106			0.170053	+	0.985435i	$0.445606\pi$
$978$		0			0
$979$			1.22756e20i			0.142417i
$980$		0			0
$981$	1.35604e21			1.55092
$982$		0			0
$983$		−	4.87190e20i		−	0.549318i	−0.961542	−	0.274659i	$-0.911435\pi$
$983$		−	4.87190e20i		−	0.549318i	0.961542	−	0.274659i	$-0.0885650\pi$
$984$		0			0
$985$			7.83544e20i			0.870983i
$986$		0			0
$987$	−2.86594e19	+	3.87714e19i	−0.0314084	+	0.0424905i
$988$		0			0
$989$	−1.13151e19			−0.0122260
$990$		0			0
$991$	1.63532e21			1.74216			0.871079	−	0.491142i	$-0.163420\pi$
$991$	1.63532e21			1.74216			0.871079	+	0.491142i	$0.163420\pi$
$992$		0			0
$993$		−	1.70022e20i		−	0.178592i
$994$		0			0
$995$	−1.38362e20			−0.143303
$996$		0			0
$997$		−	1.78541e21i		−	1.82335i	−0.410908	−	0.911677i	$-0.634788\pi$
$997$		−	1.78541e21i		−	1.82335i	0.410908	−	0.911677i	$-0.365212\pi$
$998$		0			0
$999$			2.67087e20i			0.268964i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.15.d.a.41.16	✓	36
7.6	odd	2	inner	140.15.d.a.41.21	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.15.d.a.41.16	✓	36	1.1	even	1	trivial
140.15.d.a.41.21	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)

\(f(q)\)	\(=\)	\(q-376.610i q^{3} -34938.6i q^{5} +(489531. - 662255. i) q^{7} +4.64113e6 q^{9} +O(q^{10})\)	\(q-376.610i q^{3} -34938.6i q^{5} +(489531. - 662255. i) q^{7} +4.64113e6 q^{9} -1.32384e7 q^{11} -3.65880e7i q^{13} -1.31582e7 q^{15} +4.09147e8i q^{17} -8.08426e8i q^{19} +(-2.49412e8 - 1.84362e8i) q^{21} -2.19985e9 q^{23} -1.22070e9 q^{25} -3.54921e9i q^{27} -2.49747e10 q^{29} +1.84728e10i q^{31} +4.98573e9i q^{33} +(-2.31383e10 - 1.71035e10i) q^{35} -7.52525e10 q^{37} -1.37794e10 q^{39} -1.36052e11i q^{41} +5.14358e9 q^{43} -1.62155e11i q^{45} -1.55451e11i q^{47} +(-1.98941e11 - 6.48389e11i) q^{49} +1.54089e11 q^{51} +1.23014e12 q^{53} +4.62532e11i q^{55} -3.04461e11 q^{57} +3.86218e12i q^{59} -7.36099e10i q^{61} +(2.27198e12 - 3.07362e12i) q^{63} -1.27833e12 q^{65} -2.57292e11 q^{67} +8.28486e11i q^{69} -1.65089e13 q^{71} -7.41500e12i q^{73} +4.59729e11i q^{75} +(-6.48063e12 + 8.76723e12i) q^{77} +1.17517e13 q^{79} +2.08617e13 q^{81} -1.27759e13i q^{83} +1.42950e13 q^{85} +9.40572e12i q^{87} -9.27267e12i q^{89} +(-2.42306e13 - 1.79110e13i) q^{91} +6.95705e12 q^{93} -2.82452e13 q^{95} +5.77768e13i q^{97} -6.14414e13 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 1364266 q^{7} - 54790830 q^{9}+O(q^{10}) \) 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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