LMFDB - Embedded newform 140.15.d.a.41.11

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			41.11
Character	$\chi$	$=$	140.41
Dual form			140.15.d.a.41.26

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2006.39i q^{3} +34938.6i q^{5} +(770567. + 290602. i) q^{7} +757366. q^{9} +O(q^{10})$	\(q-2006.39i q^{3} +34938.6i q^{5} +(770567. + 290602. i) q^{7} +757366. q^{9} +8.85683e6 q^{11} -1.08874e8i q^{13} +7.01004e7 q^{15} +1.83621e8i q^{17} -2.90421e8i q^{19} +(5.83061e8 - 1.54606e9i) q^{21} +1.47703e9 q^{23} -1.22070e9 q^{25} -1.11161e10i q^{27} -3.38222e10 q^{29} +1.88913e10i q^{31} -1.77703e10i q^{33} +(-1.01532e10 + 2.69225e10i) q^{35} +1.25726e10 q^{37} -2.18445e11 q^{39} -2.75602e11i q^{41} +3.87665e11 q^{43} +2.64613e10i q^{45} +1.28928e11i q^{47} +(5.09324e11 + 4.47856e11i) q^{49} +3.68416e11 q^{51} -1.28617e12 q^{53} +3.09445e11i q^{55} -5.82697e11 q^{57} -2.40725e12i q^{59} +4.10235e12i q^{61} +(5.83601e11 + 2.20092e11i) q^{63} +3.80391e12 q^{65} +5.61828e12 q^{67} -2.96349e12i q^{69} +1.44515e13 q^{71} -1.79187e13i q^{73} +2.44921e12i q^{75} +(6.82479e12 + 2.57381e12i) q^{77} +1.10440e13 q^{79} -1.86807e13 q^{81} -1.83928e13i q^{83} -6.41547e12 q^{85} +6.78606e13i q^{87} -7.53526e13i q^{89} +(3.16391e13 - 8.38950e13i) q^{91} +3.79033e13 q^{93} +1.01469e13 q^{95} +7.51742e13i q^{97} +6.70786e12 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	2006.39i		−	0.917417i	−0.888587	−	0.458708i	$-0.848312\pi$
$3$		−	2006.39i		−	0.917417i	0.888587	−	0.458708i	$-0.151688\pi$
$4$		0			0
$5$			34938.6i			0.447214i
$5$			34938.6i			0.447214i
$6$		0			0
$7$	770567.	+	290602.i	0.935673	+	0.352868i
$7$	770567.	+	290602.i	0.935673	+	0.352868i
$8$		0			0
$9$	757366.			0.158346
$10$		0			0
$11$	8.85683e6			0.454496			0.227248	−	0.973837i	$-0.427027\pi$
$11$	8.85683e6			0.454496			0.227248	+	0.973837i	$0.427027\pi$
$12$		0			0
$13$		−	1.08874e8i		−	1.73509i	−0.497358	−	0.867545i	$-0.665696\pi$
$13$		−	1.08874e8i		−	1.73509i	0.497358	−	0.867545i	$-0.334304\pi$
$14$		0			0
$15$	7.01004e7			0.410281
$16$		0			0
$17$			1.83621e8i			0.447487i	0.974648	+	0.223744i	$0.0718278\pi$
$17$			1.83621e8i			0.447487i	−0.974648	+	0.223744i	$0.928172\pi$
$18$		0			0
$19$		−	2.90421e8i		−	0.324902i	−0.986717	−	0.162451i	$-0.948060\pi$
$19$		−	2.90421e8i		−	0.324902i	0.986717	−	0.162451i	$-0.0519399\pi$
$20$		0			0
$21$	5.83061e8	−	1.54606e9i	0.323727	−	0.858402i
$22$		0			0
$23$	1.47703e9			0.433804			0.216902	−	0.976193i	$-0.430405\pi$
$23$	1.47703e9			0.433804			0.216902	+	0.976193i	$0.430405\pi$
$24$		0			0
$25$	−1.22070e9			−0.200000
$26$		0			0
$27$		−	1.11161e10i		−	1.06269i
$28$		0			0
$29$	−3.38222e10			−1.96072			−0.980362	−	0.197207i	$-0.936813\pi$
$29$	−3.38222e10			−1.96072			−0.980362	+	0.197207i	$0.936813\pi$
$30$		0			0
$31$			1.88913e10i			0.686642i	0.939218	+	0.343321i	$0.111552\pi$
$31$			1.88913e10i			0.686642i	−0.939218	+	0.343321i	$0.888448\pi$
$32$		0			0
$33$		−	1.77703e10i		−	0.416962i
$34$		0			0
$35$	−1.01532e10	+	2.69225e10i	−0.157807	+	0.418446i
$36$		0			0
$37$	1.25726e10			0.132438			0.0662189	−	0.997805i	$-0.478906\pi$
$37$	1.25726e10			0.132438			0.0662189	+	0.997805i	$0.478906\pi$
$38$		0			0
$39$	−2.18445e11			−1.59180
$40$		0			0
$41$		−	2.75602e11i		−	1.41512i	−0.706651	−	0.707562i	$-0.749795\pi$
$41$		−	2.75602e11i		−	1.41512i	0.706651	−	0.707562i	$-0.250205\pi$
$42$		0			0
$43$	3.87665e11			1.42619			0.713095	−	0.701067i	$-0.247293\pi$
$43$	3.87665e11			1.42619			0.713095	+	0.701067i	$0.247293\pi$
$44$		0			0
$45$			2.64613e10i			0.0708147i
$46$		0			0
$47$			1.28928e11i			0.254485i	0.991872	+	0.127242i	$0.0406126\pi$
$47$			1.28928e11i			0.254485i	−0.991872	+	0.127242i	$0.959387\pi$
$48$		0			0
$49$	5.09324e11	+	4.47856e11i	0.750969	+	0.660338i
$50$		0			0
$51$	3.68416e11			0.410532
$52$		0			0
$53$	−1.28617e12			−1.09488			−0.547441	−	0.836844i	$-0.684398\pi$
$53$	−1.28617e12			−1.09488			−0.547441	+	0.836844i	$0.684398\pi$
$54$		0			0
$55$			3.09445e11i			0.203257i
$56$		0			0
$57$	−5.82697e11			−0.298071
$58$		0			0
$59$		−	2.40725e12i		−	0.967292i	−0.875264	−	0.483646i	$-0.839312\pi$
$59$		−	2.40725e12i		−	0.967292i	0.875264	−	0.483646i	$-0.160688\pi$
$60$		0			0
$61$			4.10235e12i			1.30534i	0.757642	+	0.652671i	$0.226352\pi$
$61$			4.10235e12i			1.30534i	−0.757642	+	0.652671i	$0.773648\pi$
$62$		0			0
$63$	5.83601e11	+	2.20092e11i	0.148160	+	0.0558753i
$64$		0			0
$65$	3.80391e12			0.775956
$66$		0			0
$67$	5.61828e12			0.927000			0.463500	−	0.886097i	$-0.346593\pi$
$67$	5.61828e12			0.927000			0.463500	+	0.886097i	$0.346593\pi$
$68$		0			0
$69$		−	2.96349e12i		−	0.397979i
$70$		0			0
$71$	1.44515e13			1.58893			0.794465	−	0.607310i	$-0.207752\pi$
$71$	1.44515e13			1.58893			0.794465	+	0.607310i	$0.207752\pi$
$72$		0			0
$73$		−	1.79187e13i		−	1.62198i	−0.585058	−	0.810991i	$-0.698928\pi$
$73$		−	1.79187e13i		−	1.62198i	0.585058	−	0.810991i	$-0.301072\pi$
$74$		0			0
$75$			2.44921e12i			0.183483i
$76$		0			0
$77$	6.82479e12	+	2.57381e12i	0.425259	+	0.160377i
$78$		0			0
$79$	1.10440e13			0.575090			0.287545	−	0.957767i	$-0.407161\pi$
$79$	1.10440e13			0.575090			0.287545	+	0.957767i	$0.407161\pi$
$80$		0			0
$81$	−1.86807e13			−0.816580
$82$		0			0
$83$		−	1.83928e13i		−	0.677800i	−0.940822	−	0.338900i	$-0.889945\pi$
$83$		−	1.83928e13i		−	0.677800i	0.940822	−	0.338900i	$-0.110055\pi$
$84$		0			0
$85$	−6.41547e12			−0.200122
$86$		0			0
$87$			6.78606e13i			1.79880i
$88$		0			0
$89$		−	7.53526e13i		−	1.70360i	−0.523865	−	0.851801i	$-0.675511\pi$
$89$		−	7.53526e13i		−	1.70360i	0.523865	−	0.851801i	$-0.324489\pi$
$90$		0			0
$91$	3.16391e13	−	8.38950e13i	0.612258	−	1.62348i
$92$		0			0
$93$	3.79033e13			0.629937
$94$		0			0
$95$	1.01469e13			0.145301
$96$		0			0
$97$			7.51742e13i			0.930393i	0.885207	+	0.465197i	$0.154016\pi$
$97$			7.51742e13i			0.930393i	−0.885207	+	0.465197i	$0.845984\pi$
$98$		0			0
$99$	6.70786e12			0.0719677
$100$		0			0
$101$			4.94271e13i			0.461016i	0.973070	+	0.230508i	$0.0740387\pi$
$101$			4.94271e13i			0.461016i	−0.973070	+	0.230508i	$0.925961\pi$
$102$		0			0
$103$			2.22986e13i			0.181308i	0.995882	+	0.0906542i	$0.0288958\pi$
$103$			2.22986e13i			0.181308i	−0.995882	+	0.0906542i	$0.971104\pi$
$104$		0			0
$105$	5.40171e13	+	2.03713e13i	0.383889	+	0.144775i
$106$		0			0
$107$	1.31092e14			0.816373			0.408186	−	0.912899i	$-0.366161\pi$
$107$	1.31092e14			0.816373			0.408186	+	0.912899i	$0.366161\pi$
$108$		0			0
$109$	−1.99299e14			−1.09023			−0.545117	−	0.838360i	$-0.683515\pi$
$109$	−1.99299e14			−1.09023			−0.545117	+	0.838360i	$0.683515\pi$
$110$		0			0
$111$		−	2.52255e13i		−	0.121501i
$112$		0			0
$113$	1.91985e14			0.816051			0.408025	−	0.912971i	$-0.366217\pi$
$113$	1.91985e14			0.816051			0.408025	+	0.912971i	$0.366217\pi$
$114$		0			0
$115$			5.16052e13i			0.194003i
$116$		0			0
$117$		−	8.24577e13i		−	0.274745i
$118$		0			0
$119$	−5.33607e13	+	1.41493e14i	−0.157904	+	0.418702i
$120$		0			0
$121$	−3.01306e14			−0.793434
$122$		0			0
$123$	−5.52964e14			−1.29826
$124$		0			0
$125$		−	4.26496e13i		−	0.0894427i
$126$		0			0
$127$	−9.42553e14			−1.76880			−0.884402	−	0.466725i	$-0.845434\pi$
$127$	−9.42553e14			−1.76880			−0.884402	+	0.466725i	$0.845434\pi$
$128$		0			0
$129$		−	7.77808e14i		−	1.30841i
$130$		0			0
$131$			5.64200e14i			0.852185i	0.904680	+	0.426092i	$0.140110\pi$
$131$			5.64200e14i			0.852185i	−0.904680	+	0.426092i	$0.859890\pi$
$132$		0			0
$133$	8.43968e13	−	2.23789e14i	0.114647	−	0.304002i
$134$		0			0
$135$	3.88380e14			0.475248
$136$		0			0
$137$	−8.91744e14			−0.984456			−0.492228	−	0.870466i	$-0.663817\pi$
$137$	−8.91744e14			−0.984456			−0.492228	+	0.870466i	$0.663817\pi$
$138$		0			0
$139$			8.45867e14i			0.843720i	0.906661	+	0.421860i	$0.138623\pi$
$139$			8.45867e14i			0.843720i	−0.906661	+	0.421860i	$0.861377\pi$
$140$		0			0
$141$	2.58680e14			0.233468
$142$		0			0
$143$		−	9.64282e14i		−	0.788591i
$144$		0			0
$145$		−	1.18170e15i		−	0.876862i
$146$		0			0
$147$	8.98575e14	−	1.02190e15i	0.605805	−	0.688951i
$148$		0			0
$149$	1.41023e15			0.864943			0.432471	−	0.901648i	$-0.357642\pi$
$149$	1.41023e15			0.864943			0.432471	+	0.901648i	$0.357642\pi$
$150$		0			0
$151$	−2.90780e15			−1.62452			−0.812262	−	0.583293i	$-0.801764\pi$
$151$	−2.90780e15			−1.62452			−0.812262	+	0.583293i	$0.801764\pi$
$152$		0			0
$153$			1.39069e14i			0.0708580i
$154$		0			0
$155$	−6.60035e14			−0.307076
$156$		0			0
$157$			6.67559e14i			0.283917i	0.989873	+	0.141959i	$0.0453400\pi$
$157$			6.67559e14i			0.283917i	−0.989873	+	0.141959i	$0.954660\pi$
$158$		0			0
$159$			2.58056e15i			1.00446i
$160$		0			0
$161$	1.13815e15	+	4.29227e14i	0.405899	+	0.153076i
$162$		0			0
$163$	3.78971e15			1.23963			0.619816	−	0.784747i	$-0.287207\pi$
$163$	3.78971e15			1.23963			0.619816	+	0.784747i	$0.287207\pi$
$164$		0			0
$165$	6.20868e14			0.186471
$166$		0			0
$167$		−	5.30773e15i		−	1.46519i	−0.680665	−	0.732595i	$-0.738309\pi$
$167$		−	5.30773e15i		−	1.46519i	0.680665	−	0.732595i	$-0.261691\pi$
$168$		0			0
$169$	−7.91625e15			−2.01054
$170$		0			0
$171$		−	2.19955e14i		−	0.0514471i
$172$		0			0
$173$		−	6.79281e15i		−	1.46463i	−0.680968	−	0.732313i	$-0.738441\pi$
$173$		−	6.79281e15i		−	1.46463i	0.680968	−	0.732313i	$-0.261559\pi$
$174$		0			0
$175$	−9.40634e14	−	3.54738e14i	−0.187135	−	0.0705735i
$176$		0			0
$177$	−4.82989e15			−0.887410
$178$		0			0
$179$	8.63556e15			1.46662			0.733312	−	0.679892i	$-0.237973\pi$
$179$	8.63556e15			1.46662			0.733312	+	0.679892i	$0.237973\pi$
$180$		0			0
$181$		−	1.78640e13i		−	0.00280692i	−0.999999	−	0.00140346i	$-0.999553\pi$
$181$		−	1.78640e13i		−	0.00280692i	0.999999	−	0.00140346i	$-0.000446735\pi$
$182$		0			0
$183$	8.23092e15			1.19754
$184$		0			0
$185$			4.39268e14i			0.0592280i
$186$		0			0
$187$			1.62630e15i			0.203381i
$188$		0			0
$189$	3.23035e15	−	8.56568e15i	0.374988	−	0.994327i
$190$		0			0
$191$	−1.47301e16			−1.58845			−0.794223	−	0.607626i	$-0.792122\pi$
$191$	−1.47301e16			−1.58845			−0.794223	+	0.607626i	$0.792122\pi$
$192$		0			0
$193$	3.83850e15			0.384823			0.192411	−	0.981314i	$-0.438369\pi$
$193$	3.83850e15			0.384823			0.192411	+	0.981314i	$0.438369\pi$
$194$		0			0
$195$		−	7.63214e15i		−	0.711875i
$196$		0			0
$197$	1.11741e16			0.970399			0.485200	−	0.874403i	$-0.338747\pi$
$197$	1.11741e16			0.970399			0.485200	+	0.874403i	$0.338747\pi$
$198$		0			0
$199$		−	1.92598e16i		−	1.55841i	−0.626771	−	0.779203i	$-0.715624\pi$
$199$		−	1.92598e16i		−	1.55841i	0.626771	−	0.779203i	$-0.284376\pi$
$200$		0			0
$201$		−	1.12725e16i		−	0.850446i
$202$		0			0
$203$	−2.60623e16	−	9.82880e15i	−1.83460	−	0.691876i
$204$		0			0
$205$	9.62912e15			0.632863
$206$		0			0
$207$	1.11865e15			0.0686913
$208$		0			0
$209$		−	2.57221e15i		−	0.147667i
$210$		0			0
$211$	−1.16224e16			−0.624192			−0.312096	−	0.950051i	$-0.601031\pi$
$211$	−1.16224e16			−0.624192			−0.312096	+	0.950051i	$0.601031\pi$
$212$		0			0
$213$		−	2.89954e16i		−	1.45771i
$214$		0			0
$215$			1.35445e16i			0.637812i
$216$		0			0
$217$	−5.48985e15	+	1.45570e16i	−0.242294	+	0.642472i
$218$		0			0
$219$	−3.59519e16			−1.48803
$220$		0			0
$221$	1.99917e16			0.776431
$222$		0			0
$223$		−	1.10252e16i		−	0.402023i	−0.979589	−	0.201012i	$-0.935577\pi$
$223$		−	1.10252e16i		−	0.402023i	0.979589	−	0.201012i	$-0.0644229\pi$
$224$		0			0
$225$	−9.24519e14			−0.0316693
$226$		0			0
$227$		−	4.90317e15i		−	0.157869i	−0.996880	−	0.0789344i	$-0.974848\pi$
$227$		−	4.90317e15i		−	0.157869i	0.996880	−	0.0789344i	$-0.0251518\pi$
$228$		0			0
$229$		−	5.14686e16i		−	1.55845i	−0.626743	−	0.779226i	$-0.715612\pi$
$229$		−	5.14686e16i		−	1.55845i	0.626743	−	0.779226i	$-0.284388\pi$
$230$		0			0
$231$	5.16407e15	−	1.36932e16i	0.147132	−	0.390140i
$232$		0			0
$233$	3.39167e16			0.909750			0.454875	−	0.890555i	$-0.349684\pi$
$233$	3.39167e16			0.909750			0.454875	+	0.890555i	$0.349684\pi$
$234$		0			0
$235$	−4.50455e15			−0.113809
$236$		0			0
$237$		−	2.21585e16i		−	0.527597i
$238$		0			0
$239$	−2.57650e16			−0.578423			−0.289212	−	0.957265i	$-0.593393\pi$
$239$	−2.57650e16			−0.578423			−0.289212	+	0.957265i	$0.593393\pi$
$240$		0			0
$241$			9.86825e15i			0.208988i	0.994526	+	0.104494i	$0.0333223\pi$
$241$			9.86825e15i			0.208988i	−0.994526	+	0.104494i	$0.966678\pi$
$242$		0			0
$243$		−	1.56870e16i		−	0.313542i
$244$		0			0
$245$	−1.56475e16	+	1.77951e16i	−0.295312	+	0.335843i
$246$		0			0
$247$	−3.16194e16			−0.563734
$248$		0			0
$249$	−3.69031e16			−0.621825
$250$		0			0
$251$		−	1.03110e17i		−	1.64280i	−0.570355	−	0.821399i	$-0.693194\pi$
$251$		−	1.03110e17i		−	1.64280i	0.570355	−	0.821399i	$-0.306806\pi$
$252$		0			0
$253$	1.30818e16			0.197162
$254$		0			0
$255$			1.28719e16i			0.183596i
$256$		0			0
$257$		−	4.15396e14i		−	0.00560958i	−0.999996	−	0.00280479i	$-0.999107\pi$
$257$		−	4.15396e14i		−	0.00560958i	0.999996	−	0.00280479i	$-0.000892794\pi$
$258$		0			0
$259$	9.68801e15	+	3.65361e15i	0.123919	+	0.0467330i
$260$		0			0
$261$	−2.56158e16			−0.310474
$262$		0			0
$263$	−1.49714e17			−1.72017			−0.860086	−	0.510150i	$-0.829590\pi$
$263$	−1.49714e17			−1.72017			−0.860086	+	0.510150i	$0.829590\pi$
$264$		0			0
$265$		−	4.49370e16i		−	0.489647i
$266$		0			0
$267$	−1.51187e17			−1.56291
$268$		0			0
$269$		−	8.96591e16i		−	0.879687i	−0.898074	−	0.439844i	$-0.855034\pi$
$269$		−	8.96591e16i		−	0.879687i	0.898074	−	0.439844i	$-0.144966\pi$
$270$		0			0
$271$		−	1.18860e17i		−	1.10727i	−0.832761	−	0.553633i	$-0.813241\pi$
$271$		−	1.18860e17i		−	1.10727i	0.832761	−	0.553633i	$-0.186759\pi$
$272$		0			0
$273$	−1.68326e17	−	6.34804e16i	−1.48941	−	0.561695i
$274$		0			0
$275$	−1.08116e16			−0.0908991
$276$		0			0
$277$	1.26661e17			1.01224			0.506120	−	0.862463i	$-0.331079\pi$
$277$	1.26661e17			1.01224			0.506120	+	0.862463i	$0.331079\pi$
$278$		0			0
$279$			1.43076e16i			0.108727i
$280$		0			0
$281$	−1.68947e17			−1.22125			−0.610627	−	0.791918i	$-0.709082\pi$
$281$	−1.68947e17			−1.22125			−0.610627	+	0.791918i	$0.709082\pi$
$282$		0			0
$283$			1.11560e17i			0.767367i	0.923465	+	0.383683i	$0.125345\pi$
$283$			1.11560e17i			0.767367i	−0.923465	+	0.383683i	$0.874655\pi$
$284$		0			0
$285$		−	2.03586e16i		−	0.133301i
$286$		0			0
$287$	8.00903e16	−	2.12370e17i	0.499352	−	1.32409i
$288$		0			0
$289$	1.34661e17			0.799755
$290$		0			0
$291$	1.50829e17			0.853558
$292$		0			0
$293$		−	3.04307e16i		−	0.164149i	−0.996626	−	0.0820745i	$-0.973845\pi$
$293$		−	3.04307e16i		−	0.164149i	0.996626	−	0.0820745i	$-0.0261545\pi$
$294$		0			0
$295$	8.41059e16			0.432586
$296$		0			0
$297$		−	9.84532e16i		−	0.482986i
$298$		0			0
$299$		−	1.60810e17i		−	0.752690i
$300$		0			0
$301$	2.98722e17	+	1.12656e17i	1.33445	+	0.503257i
$302$		0			0
$303$	9.91701e16			0.422943
$304$		0			0
$305$	−1.43330e17			−0.583767
$306$		0			0
$307$			2.21363e17i			0.861264i	0.902528	+	0.430632i	$0.141709\pi$
$307$			2.21363e17i			0.861264i	−0.902528	+	0.430632i	$0.858291\pi$
$308$		0			0
$309$	4.47398e16			0.166335
$310$		0			0
$311$		−	8.12341e16i		−	0.288679i	−0.989528	−	0.144340i	$-0.953894\pi$
$311$		−	8.12341e16i		−	0.288679i	0.989528	−	0.144340i	$-0.0461058\pi$
$312$		0			0
$313$		−	1.11291e17i		−	0.378138i	−0.981964	−	0.189069i	$-0.939453\pi$
$313$		−	1.11291e17i		−	0.378138i	0.981964	−	0.189069i	$-0.0605469\pi$
$314$		0			0
$315$	−7.68969e15	+	2.03902e16i	−0.0249882	+	0.0662594i
$316$		0			0
$317$	3.32396e17			1.03333			0.516667	−	0.856187i	$-0.327173\pi$
$317$	3.32396e17			1.03333			0.516667	+	0.856187i	$0.327173\pi$
$318$		0			0
$319$	−2.99558e17			−0.891140
$320$		0			0
$321$		−	2.63021e17i		−	0.748954i
$322$		0			0
$323$	5.33274e16			0.145390
$324$		0			0
$325$			1.32903e17i			0.347018i
$326$		0			0
$327$			3.99872e17i			1.00020i
$328$		0			0
$329$	−3.74666e16	+	9.93475e16i	−0.0897994	+	0.238114i
$330$		0			0
$331$	3.17813e17			0.730089			0.365044	−	0.930990i	$-0.381054\pi$
$331$	3.17813e17			0.730089			0.365044	+	0.930990i	$0.381054\pi$
$332$		0			0
$333$	9.52204e15			0.0209711
$334$		0			0
$335$			1.96295e17i			0.414567i
$336$		0			0
$337$	−7.55271e16			−0.153001			−0.0765004	−	0.997070i	$-0.524375\pi$
$337$	−7.55271e16			−0.153001			−0.0765004	+	0.997070i	$0.524375\pi$
$338$		0			0
$339$		−	3.85196e17i		−	0.748659i
$340$		0			0
$341$			1.67317e17i			0.312076i
$342$		0			0
$343$	2.62321e17	+	4.93114e17i	0.469649	+	0.882853i
$344$		0			0
$345$	1.03540e17			0.177982
$346$		0			0
$347$	2.51646e17			0.415416			0.207708	−	0.978191i	$-0.433400\pi$
$347$	2.51646e17			0.415416			0.207708	+	0.978191i	$0.433400\pi$
$348$		0			0
$349$		−	9.45946e17i		−	1.49999i	−0.661444	−	0.749995i	$-0.730056\pi$
$349$		−	9.45946e17i		−	1.49999i	0.661444	−	0.749995i	$-0.269944\pi$
$350$		0			0
$351$	−1.21026e18			−1.84386
$352$		0			0
$353$		−	1.15318e18i		−	1.68839i	−0.536033	−	0.844197i	$-0.680078\pi$
$353$		−	1.15318e18i		−	1.68839i	0.536033	−	0.844197i	$-0.319922\pi$
$354$		0			0
$355$			5.04915e17i			0.710591i
$356$		0			0
$357$	2.83889e17	+	1.07062e17i	0.384124	+	0.144864i
$358$		0			0
$359$	5.55541e17			0.722861			0.361431	−	0.932399i	$-0.382288\pi$
$359$	5.55541e17			0.722861			0.361431	+	0.932399i	$0.382288\pi$
$360$		0			0
$361$	7.14663e17			0.894439
$362$		0			0
$363$			6.04538e17i			0.727909i
$364$		0			0
$365$	6.26053e17			0.725373
$366$		0			0
$367$			1.09826e18i			1.22474i	0.790572	+	0.612369i	$0.209784\pi$
$367$			1.09826e18i			1.22474i	−0.790572	+	0.612369i	$0.790216\pi$
$368$		0			0
$369$		−	2.08731e17i		−	0.224080i
$370$		0			0
$371$	−9.91081e17	−	3.73764e17i	−1.02445	−	0.386349i
$372$		0			0
$373$	−1.90418e18			−1.89560			−0.947798	−	0.318871i	$-0.896696\pi$
$373$	−1.90418e18			−1.89560			−0.947798	+	0.318871i	$0.896696\pi$
$374$		0			0
$375$	−8.55718e16			−0.0820563
$376$		0			0
$377$			3.68238e18i			3.40203i
$378$		0			0
$379$	−1.70956e18			−1.52198			−0.760991	−	0.648763i	$-0.775287\pi$
$379$	−1.70956e18			−1.52198			−0.760991	+	0.648763i	$0.775287\pi$
$380$		0			0
$381$			1.89113e18i			1.62273i
$382$		0			0
$383$		−	1.10977e17i		−	0.0918001i	−0.998946	−	0.0459000i	$-0.985384\pi$
$383$		−	1.10977e17i		−	0.0918001i	0.998946	−	0.0459000i	$-0.0146156\pi$
$384$		0			0
$385$	−8.99253e16	+	2.38448e17i	−0.0717227	+	0.190182i
$386$		0			0
$387$	2.93604e17			0.225832
$388$		0			0
$389$	6.94096e17			0.514958			0.257479	−	0.966284i	$-0.417108\pi$
$389$	6.94096e17			0.514958			0.257479	+	0.966284i	$0.417108\pi$
$390$		0			0
$391$			2.71214e17i			0.194122i
$392$		0			0
$393$	1.13200e18			0.781808
$394$		0			0
$395$			3.85861e17i			0.257188i
$396$		0			0
$397$			1.34898e18i			0.867903i	0.900936	+	0.433952i	$0.142881\pi$
$397$			1.34898e18i			0.867903i	−0.900936	+	0.433952i	$0.857119\pi$
$398$		0			0
$399$	−4.49007e17	−	1.69333e17i	−0.278897	−	0.105179i
$400$		0			0
$401$	−4.04584e17			−0.242660			−0.121330	−	0.992612i	$-0.538716\pi$
$401$	−4.04584e17			−0.242660			−0.121330	+	0.992612i	$0.538716\pi$
$402$		0			0
$403$	2.05678e18			1.19139
$404$		0			0
$405$		−	6.52678e17i		−	0.365186i
$406$		0			0
$407$	1.11353e17			0.0601924
$408$		0			0
$409$			2.80954e18i			1.46748i	0.679431	+	0.733740i	$0.262227\pi$
$409$			2.80954e18i			1.46748i	−0.679431	+	0.733740i	$0.737773\pi$
$410$		0			0
$411$			1.78919e18i			0.903157i
$412$		0			0
$413$	6.99552e17	−	1.85495e18i	0.341326	−	0.905069i
$414$		0			0
$415$	6.42618e17			0.303121
$416$		0			0
$417$	1.69714e18			0.774043
$418$		0			0
$419$		−	1.61619e17i		−	0.0712844i	−0.999365	−	0.0356422i	$-0.988652\pi$
$419$		−	1.61619e17i		−	0.0712844i	0.999365	−	0.0356422i	$-0.0113477\pi$
$420$		0			0
$421$	1.41661e18			0.604331			0.302165	−	0.953256i	$-0.402291\pi$
$421$	1.41661e18			0.604331			0.302165	+	0.953256i	$0.402291\pi$
$422$		0			0
$423$			9.76455e16i			0.0402967i
$424$		0			0
$425$		−	2.24147e17i		−	0.0894975i
$426$		0			0
$427$	−1.19215e18	+	3.16114e18i	−0.460613	+	1.22137i
$428$		0			0
$429$	−1.93473e18			−0.723467
$430$		0			0
$431$	1.39545e18			0.505094			0.252547	−	0.967585i	$-0.418732\pi$
$431$	1.39545e18			0.505094			0.252547	+	0.967585i	$0.418732\pi$
$432$		0			0
$433$		−	6.52852e17i		−	0.228770i	−0.993436	−	0.114385i	$-0.963510\pi$
$433$		−	6.52852e17i		−	0.228770i	0.993436	−	0.114385i	$-0.0364898\pi$
$434$		0			0
$435$	−2.37095e18			−0.804448
$436$		0			0
$437$		−	4.28959e17i		−	0.140944i
$438$		0			0
$439$		−	2.15122e18i		−	0.684595i	−0.939592	−	0.342297i	$-0.888795\pi$
$439$		−	2.15122e18i		−	0.684595i	0.939592	−	0.342297i	$-0.111205\pi$
$440$		0			0
$441$	3.85745e17	+	3.39191e17i	0.118913	+	0.104562i
$442$		0			0
$443$	4.43332e18			1.32405			0.662023	−	0.749483i	$-0.269698\pi$
$443$	4.43332e18			1.32405			0.662023	+	0.749483i	$0.269698\pi$
$444$		0			0
$445$	2.63271e18			0.761874
$446$		0			0
$447$		−	2.82948e18i		−	0.793513i
$448$		0			0
$449$	−1.60459e17			−0.0436153			−0.0218077	−	0.999762i	$-0.506942\pi$
$449$	−1.60459e17			−0.0436153			−0.0218077	+	0.999762i	$0.506942\pi$
$450$		0			0
$451$		−	2.44096e18i		−	0.643168i
$452$		0			0
$453$			5.83418e18i			1.49036i
$454$		0			0
$455$	2.93117e18	+	1.10542e18i	0.726041	+	0.273810i
$456$		0			0
$457$	−9.22451e17			−0.221580			−0.110790	−	0.993844i	$-0.535338\pi$
$457$	−9.22451e17			−0.221580			−0.110790	+	0.993844i	$0.535338\pi$
$458$		0			0
$459$	2.04115e18			0.475539
$460$		0			0
$461$			4.37697e18i			0.989161i	0.869132	+	0.494580i	$0.164678\pi$
$461$			4.37697e18i			0.989161i	−0.869132	+	0.494580i	$0.835322\pi$
$462$		0			0
$463$	−8.00960e18			−1.75608			−0.878039	−	0.478589i	$-0.841148\pi$
$463$	−8.00960e18			−1.75608			−0.878039	+	0.478589i	$0.841148\pi$
$464$		0			0
$465$			1.32429e18i			0.281716i
$466$		0			0
$467$		−	5.33661e17i		−	0.110166i	−0.998482	−	0.0550829i	$-0.982458\pi$
$467$		−	5.33661e17i		−	0.110166i	0.998482	−	0.0550829i	$-0.0175423\pi$
$468$		0			0
$469$	4.32926e18	+	1.63268e18i	0.867369	+	0.327109i
$470$		0			0
$471$	1.33938e18			0.260471
$472$		0			0
$473$	3.43349e18			0.648198
$474$		0			0
$475$			3.54517e17i			0.0649804i
$476$		0			0
$477$	−9.74102e17			−0.173371
$478$		0			0
$479$		−	5.39512e17i		−	0.0932508i	−0.998912	−	0.0466254i	$-0.985153\pi$
$479$		−	5.39512e17i		−	0.0932508i	0.998912	−	0.0466254i	$-0.0148467\pi$
$480$		0			0
$481$		−	1.36883e18i		−	0.229792i
$482$		0			0
$483$	8.61197e17	−	2.28357e18i	0.140434	−	0.372379i
$484$		0			0
$485$	−2.62648e18			−0.416085
$486$		0			0
$487$	−1.05828e18			−0.162892			−0.0814458	−	0.996678i	$-0.525954\pi$
$487$	−1.05828e18			−0.162892			−0.0814458	+	0.996678i	$0.525954\pi$
$488$		0			0
$489$		−	7.60364e18i		−	1.13726i
$490$		0			0
$491$	3.95441e18			0.574793			0.287396	−	0.957812i	$-0.407210\pi$
$491$	3.95441e18			0.574793			0.287396	+	0.957812i	$0.407210\pi$
$492$		0			0
$493$		−	6.21049e18i		−	0.877399i
$494$		0			0
$495$			2.34363e17i			0.0321850i
$496$		0			0
$497$	1.11359e19	+	4.19963e18i	1.48672	+	0.560682i
$498$		0			0
$499$	2.48126e18			0.322084			0.161042	−	0.986948i	$-0.448515\pi$
$499$	2.48126e18			0.322084			0.161042	+	0.986948i	$0.448515\pi$
$500$		0			0
$501$	−1.06494e19			−1.34419
$502$		0			0
$503$			6.75401e18i			0.829059i	0.910036	+	0.414530i	$0.136054\pi$
$503$			6.75401e18i			0.829059i	−0.910036	+	0.414530i	$0.863946\pi$
$504$		0			0
$505$	−1.72691e18			−0.206172
$506$		0			0
$507$			1.58831e19i			1.84450i
$508$		0			0
$509$		−	3.70799e18i		−	0.418904i	−0.977819	−	0.209452i	$-0.932832\pi$
$509$		−	3.70799e18i		−	0.418904i	0.977819	−	0.209452i	$-0.0671680\pi$
$510$		0			0
$511$	5.20720e18	−	1.38075e19i	0.572345	−	1.51765i
$512$		0			0
$513$	−3.22834e18			−0.345269
$514$		0			0
$515$	−7.79082e17			−0.0810835
$516$		0			0
$517$			1.14189e18i			0.115662i
$518$		0			0
$519$	−1.36290e19			−1.34367
$520$		0			0
$521$		−	7.64999e18i		−	0.734170i	−0.930187	−	0.367085i	$-0.880356\pi$
$521$		−	7.64999e18i		−	0.734170i	0.930187	−	0.367085i	$-0.119644\pi$
$522$		0			0
$523$		−	9.41224e18i		−	0.879389i	−0.898147	−	0.439695i	$-0.855087\pi$
$523$		−	9.41224e18i		−	0.879389i	0.898147	−	0.439695i	$-0.144913\pi$
$524$		0			0
$525$	−7.11744e17	+	1.88728e18i	−0.0647454	+	0.171680i
$526$		0			0
$527$	−3.46885e18			−0.307263
$528$		0			0
$529$	−9.41123e18			−0.811814
$530$		0			0
$531$		−	1.82317e18i		−	0.153167i
$532$		0			0
$533$	−3.00060e19			−2.45537
$534$		0			0
$535$			4.58015e18i			0.365093i
$536$		0			0
$537$		−	1.73263e19i		−	1.34551i
$538$		0			0
$539$	4.51100e18	+	3.96659e18i	0.341312	+	0.300121i
$540$		0			0
$541$	−1.15399e19			−0.850786			−0.425393	−	0.905009i	$-0.639864\pi$
$541$	−1.15399e19			−0.850786			−0.425393	+	0.905009i	$0.639864\pi$
$542$		0			0
$543$	−3.58423e16			−0.00257511
$544$		0			0
$545$		−	6.96322e18i		−	0.487567i
$546$		0			0
$547$	6.99840e18			0.477625			0.238813	−	0.971066i	$-0.423242\pi$
$547$	6.99840e18			0.477625			0.238813	+	0.971066i	$0.423242\pi$
$548$		0			0
$549$			3.10698e18i			0.206696i
$550$		0			0
$551$			9.82268e18i			0.637043i
$552$		0			0
$553$	8.51013e18	+	3.20940e18i	0.538096	+	0.202931i
$554$		0			0
$555$	8.81342e17			0.0543368
$556$		0			0
$557$	1.55504e18			0.0934877			0.0467438	−	0.998907i	$-0.485116\pi$
$557$	1.55504e18			0.0934877			0.0467438	+	0.998907i	$0.485116\pi$
$558$		0			0
$559$		−	4.22068e19i		−	2.47457i
$560$		0			0
$561$	3.26300e18			0.186585
$562$		0			0
$563$		−	3.03814e19i		−	1.69453i	−0.531170	−	0.847265i	$-0.678247\pi$
$563$		−	3.03814e19i		−	1.69453i	0.531170	−	0.847265i	$-0.321753\pi$
$564$		0			0
$565$			6.70766e18i			0.364949i
$566$		0			0
$567$	−1.43948e19	−	5.42865e18i	−0.764052	−	0.288145i
$568$		0			0
$569$	−1.73910e19			−0.900612			−0.450306	−	0.892874i	$-0.648685\pi$
$569$	−1.73910e19			−0.900612			−0.450306	+	0.892874i	$0.648685\pi$
$570$		0			0
$571$	1.87196e19			0.945899			0.472949	−	0.881090i	$-0.343189\pi$
$571$	1.87196e19			0.945899			0.472949	+	0.881090i	$0.343189\pi$
$572$		0			0
$573$			2.95544e19i			1.45727i
$574$		0			0
$575$	−1.80301e18			−0.0867608
$576$		0			0
$577$		−	2.88523e19i		−	1.35503i	−0.735507	−	0.677517i	$-0.763056\pi$
$577$		−	2.88523e19i		−	1.35503i	0.735507	−	0.677517i	$-0.236944\pi$
$578$		0			0
$579$		−	7.70154e18i		−	0.353043i
$580$		0			0
$581$	5.34498e18	−	1.41729e19i	0.239174	−	0.634199i
$582$		0			0
$583$	−1.13914e19			−0.497619
$584$		0			0
$585$	2.88095e18			0.122870
$586$		0			0
$587$		−	8.82078e18i		−	0.367316i	−0.982990	−	0.183658i	$-0.941206\pi$
$587$		−	8.82078e18i		−	0.367316i	0.982990	−	0.183658i	$-0.0587939\pi$
$588$		0			0
$589$	5.48643e18			0.223091
$590$		0			0
$591$		−	2.24197e19i		−	0.890261i
$592$		0			0
$593$			1.37019e19i			0.531370i	0.964060	+	0.265685i	$0.0855980\pi$
$593$			1.37019e19i			0.531370i	−0.964060	+	0.265685i	$0.914402\pi$
$594$		0			0
$595$	−4.94355e18	−	1.86435e18i	−0.187249	−	0.0706167i
$596$		0			0
$597$	−3.86427e19			−1.42971
$598$		0			0
$599$	5.30275e19			1.91652			0.958260	−	0.285898i	$-0.0922917\pi$
$599$	5.30275e19			1.91652			0.958260	+	0.285898i	$0.0922917\pi$
$600$		0			0
$601$			1.04258e19i			0.368118i	0.982915	+	0.184059i	$0.0589237\pi$
$601$			1.04258e19i			0.368118i	−0.982915	+	0.184059i	$0.941076\pi$
$602$		0			0
$603$	4.25509e18			0.146787
$604$		0			0
$605$		−	1.05272e19i		−	0.354834i
$606$		0			0
$607$		−	9.40920e18i		−	0.309907i	−0.987922	−	0.154954i	$-0.950477\pi$
$607$		−	9.40920e18i		−	0.309907i	0.987922	−	0.154954i	$-0.0495228\pi$
$608$		0			0
$609$	−1.97204e19	+	5.22912e19i	−0.634739	+	1.68309i
$610$		0			0
$611$	1.40369e19			0.441554
$612$		0			0
$613$	1.43126e19			0.440042			0.220021	−	0.975495i	$-0.429387\pi$
$613$	1.43126e19			0.440042			0.220021	+	0.975495i	$0.429387\pi$
$614$		0			0
$615$		−	1.93198e19i		−	0.580599i
$616$		0			0
$617$	−3.96067e19			−1.16352			−0.581758	−	0.813362i	$-0.697635\pi$
$617$	−3.96067e19			−1.16352			−0.581758	+	0.813362i	$0.697635\pi$
$618$		0			0
$619$			5.57456e19i			1.60094i	0.599370	+	0.800472i	$0.295418\pi$
$619$			5.57456e19i			1.60094i	−0.599370	+	0.800472i	$0.704582\pi$
$620$		0			0
$621$		−	1.64188e19i		−	0.460998i
$622$		0			0
$623$	2.18976e19	−	5.80642e19i	0.601146	−	1.59401i
$624$		0			0
$625$	1.49012e18			0.0400000
$626$		0			0
$627$	−5.16085e18			−0.135472
$628$		0			0
$629$			2.30859e18i			0.0592643i
$630$		0			0
$631$	−7.22895e19			−1.81497			−0.907485	−	0.420084i	$-0.862001\pi$
$631$	−7.22895e19			−1.81497			−0.907485	+	0.420084i	$0.862001\pi$
$632$		0			0
$633$			2.33190e19i			0.572644i
$634$		0			0
$635$		−	3.29315e19i		−	0.791033i
$636$		0			0
$637$	4.87601e19	−	5.54524e19i	1.14575	−	1.30300i
$638$		0			0
$639$	1.09451e19			0.251601
$640$		0			0
$641$	6.67139e19			1.50041			0.750206	−	0.661204i	$-0.229954\pi$
$641$	6.67139e19			1.50041			0.750206	+	0.661204i	$0.229954\pi$
$642$		0			0
$643$			4.09889e19i			0.901965i	0.892533	+	0.450982i	$0.148926\pi$
$643$			4.09889e19i			0.901965i	−0.892533	+	0.450982i	$0.851074\pi$
$644$		0			0
$645$	2.71755e19			0.585139
$646$		0			0
$647$		−	6.63251e19i		−	1.39749i	−0.715372	−	0.698744i	$-0.753743\pi$
$647$		−	6.63251e19i		−	1.39749i	0.715372	−	0.698744i	$-0.246257\pi$
$648$		0			0
$649$		−	2.13206e19i		−	0.439630i
$650$		0			0
$651$	2.92071e19	+	1.10148e19i	0.589415	+	0.222284i
$652$		0			0
$653$	−9.77442e17			−0.0193063			−0.00965314	−	0.999953i	$-0.503073\pi$
$653$	−9.77442e17			−0.0193063			−0.00965314	+	0.999953i	$0.503073\pi$
$654$		0			0
$655$	−1.97123e19			−0.381109
$656$		0			0
$657$		−	1.35710e19i		−	0.256835i
$658$		0			0
$659$	−1.29733e19			−0.240355			−0.120178	−	0.992752i	$-0.538346\pi$
$659$	−1.29733e19			−0.240355			−0.120178	+	0.992752i	$0.538346\pi$
$660$		0			0
$661$			4.08852e18i			0.0741577i	0.999312	+	0.0370788i	$0.0118053\pi$
$661$			4.08852e18i			0.0741577i	−0.999312	+	0.0370788i	$0.988195\pi$
$662$		0			0
$663$		−	4.01111e19i		−	0.712311i
$664$		0			0
$665$	7.81885e18	+	2.94870e18i	0.135954	+	0.0512719i
$666$		0			0
$667$	−4.99564e19			−0.850570
$668$		0			0
$669$	−2.21208e19			−0.368823
$670$		0			0
$671$			3.63339e19i			0.593272i
$672$		0			0
$673$	1.03092e20			1.64861			0.824304	−	0.566147i	$-0.191567\pi$
$673$	1.03092e20			1.64861			0.824304	+	0.566147i	$0.191567\pi$
$674$		0			0
$675$			1.35694e19i			0.212537i
$676$		0			0
$677$			6.73357e19i			1.03306i	0.856270	+	0.516529i	$0.172776\pi$
$677$			6.73357e19i			1.03306i	−0.856270	+	0.516529i	$0.827224\pi$
$678$		0			0
$679$	−2.18457e19	+	5.79268e19i	−0.328306	+	0.870544i
$680$		0			0
$681$	−9.83768e18			−0.144831
$682$		0			0
$683$	−1.63429e19			−0.235713			−0.117857	−	0.993031i	$-0.537602\pi$
$683$	−1.63429e19			−0.235713			−0.117857	+	0.993031i	$0.537602\pi$
$684$		0			0
$685$		−	3.11563e19i		−	0.440262i
$686$		0			0
$687$	−1.03266e20			−1.42975
$688$		0			0
$689$			1.40031e20i			1.89972i
$690$		0			0
$691$			1.47080e19i			0.195527i	0.995210	+	0.0977636i	$0.0311689\pi$
$691$			1.47080e19i			0.195527i	−0.995210	+	0.0977636i	$0.968831\pi$
$692$		0			0
$693$	5.16886e18	+	1.94932e18i	0.0673383	+	0.0253951i
$694$		0			0
$695$	−2.95534e19			−0.377323
$696$		0			0
$697$	5.06063e19			0.633250
$698$		0			0
$699$		−	6.80501e19i		−	0.834620i
$700$		0			0
$701$	−2.38567e19			−0.286804			−0.143402	−	0.989665i	$-0.545804\pi$
$701$	−2.38567e19			−0.286804			−0.143402	+	0.989665i	$0.545804\pi$
$702$		0			0
$703$		−	3.65134e18i		−	0.0430293i
$704$		0			0
$705$			9.03789e18i			0.104410i
$706$		0			0
$707$	−1.43636e19	+	3.80869e19i	−0.162678	+	0.431360i
$708$		0			0
$709$	8.90723e19			0.989051			0.494526	−	0.869163i	$-0.335342\pi$
$709$	8.90723e19			0.989051			0.494526	+	0.869163i	$0.335342\pi$
$710$		0			0
$711$	8.36433e18			0.0910634
$712$		0			0
$713$			2.79030e19i			0.297868i
$714$		0			0
$715$	3.36906e19			0.352669
$716$		0			0
$717$			5.16947e19i			0.530655i
$718$		0			0
$719$			4.86979e18i			0.0490239i	0.999700	+	0.0245120i	$0.00780318\pi$
$719$			4.86979e18i			0.0490239i	−0.999700	+	0.0245120i	$0.992197\pi$
$720$		0			0
$721$	−6.48002e18	+	1.71826e19i	−0.0639779	+	0.169645i
$722$		0			0
$723$	1.97996e19			0.191729
$724$		0			0
$725$	4.12869e19			0.392145
$726$		0			0
$727$			1.16356e20i			1.08404i	0.840365	+	0.542021i	$0.182341\pi$
$727$			1.16356e20i			1.08404i	−0.840365	+	0.542021i	$0.817659\pi$
$728$		0			0
$729$	−1.20824e20			−1.10423
$730$		0			0
$731$			7.11836e19i			0.638202i
$732$		0			0
$733$			2.20668e20i			1.94094i	0.241219	+	0.970471i	$0.422453\pi$
$733$			2.20668e20i			1.94094i	−0.241219	+	0.970471i	$0.577547\pi$
$734$		0			0
$735$	3.57038e19	+	3.13949e19i	0.308108	+	0.270924i
$736$		0			0
$737$	4.97602e19			0.421318
$738$		0			0
$739$	4.56345e19			0.379125			0.189562	−	0.981869i	$-0.439293\pi$
$739$	4.56345e19			0.379125			0.189562	+	0.981869i	$0.439293\pi$
$740$		0			0
$741$			6.34408e19i			0.517179i
$742$		0			0
$743$	2.58330e19			0.206659			0.103329	−	0.994647i	$-0.467050\pi$
$743$	2.58330e19			0.206659			0.103329	+	0.994647i	$0.467050\pi$
$744$		0			0
$745$			4.92715e19i			0.386814i
$746$		0			0
$747$		−	1.39301e19i		−	0.107327i
$748$		0			0
$749$	1.01015e20	+	3.80955e19i	0.763858	+	0.288072i
$750$		0			0
$751$	1.12202e20			0.832764			0.416382	−	0.909190i	$-0.363298\pi$
$751$	1.12202e20			0.832764			0.416382	+	0.909190i	$0.363298\pi$
$752$		0			0
$753$	−2.06879e20			−1.50713
$754$		0			0
$755$		−	1.01594e20i		−	0.726509i
$756$		0			0
$757$	5.92944e19			0.416239			0.208120	−	0.978103i	$-0.433266\pi$
$757$	5.92944e19			0.416239			0.208120	+	0.978103i	$0.433266\pi$
$758$		0			0
$759$		−	2.62472e19i		−	0.180880i
$760$		0			0
$761$			6.84547e19i			0.463138i	0.972818	+	0.231569i	$0.0743860\pi$
$761$			6.84547e19i			0.463138i	−0.972818	+	0.231569i	$0.925614\pi$
$762$		0			0
$763$	−1.53573e20	−	5.79166e19i	−1.02010	−	0.384708i
$764$		0			0
$765$	−4.85886e18			−0.0316887
$766$		0			0
$767$	−2.62088e20			−1.67834
$768$		0			0
$769$		−	4.70776e17i		−	0.00296026i	−0.999999	−	0.00148013i	$-0.999529\pi$
$769$		−	4.70776e17i		−	0.00296026i	0.999999	−	0.00148013i	$-0.000471140\pi$
$770$		0			0
$771$	−8.33447e17			−0.00514632
$772$		0			0
$773$		−	1.66792e20i		−	1.01139i	−0.862711	−	0.505696i	$-0.831236\pi$
$773$		−	1.66792e20i		−	1.01139i	0.862711	−	0.505696i	$-0.168764\pi$
$774$		0			0
$775$		−	2.30607e19i		−	0.137328i
$776$		0			0
$777$	7.33057e18	−	1.94379e19i	0.0428737	−	0.113685i
$778$		0			0
$779$	−8.00404e19			−0.459777
$780$		0			0
$781$	1.27995e20			0.722161
$782$		0			0
$783$			3.75971e20i			2.08363i
$784$		0			0
$785$	−2.33235e19			−0.126972
$786$		0			0
$787$			2.06214e19i			0.110279i	0.998479	+	0.0551397i	$0.0175604\pi$
$787$			2.06214e19i			0.110279i	−0.998479	+	0.0551397i	$0.982440\pi$
$788$		0			0
$789$			3.00384e20i			1.57811i
$790$		0			0
$791$	1.47937e20	+	5.57910e19i	0.763557	+	0.287958i
$792$		0			0
$793$	4.46641e20			2.26489
$794$		0			0
$795$	−9.01611e19			−0.449210
$796$		0			0
$797$		−	3.22491e19i		−	0.157874i	−0.996880	−	0.0789368i	$-0.974847\pi$
$797$		−	3.22491e19i		−	0.157874i	0.996880	−	0.0789368i	$-0.0251525\pi$
$798$		0			0
$799$	−2.36739e19			−0.113879
$800$		0			0
$801$		−	5.70695e19i		−	0.269759i
$802$		0			0
$803$		−	1.58703e20i		−	0.737184i
$804$		0			0
$805$	−1.49966e19	+	3.97653e19i	−0.0684574	+	0.181524i
$806$		0			0
$807$	−1.79891e20			−0.807040
$808$		0			0
$809$	−3.27026e19			−0.144193			−0.0720963	−	0.997398i	$-0.522969\pi$
$809$	−3.27026e19			−0.144193			−0.0720963	+	0.997398i	$0.522969\pi$
$810$		0			0
$811$		−	3.23178e20i		−	1.40054i	−0.713877	−	0.700271i	$-0.753062\pi$
$811$		−	3.23178e20i		−	1.40054i	0.713877	−	0.700271i	$-0.246938\pi$
$812$		0			0
$813$	−2.38480e20			−1.01582
$814$		0			0
$815$			1.32407e20i			0.554380i
$816$		0			0
$817$		−	1.12586e20i		−	0.463372i
$818$		0			0
$819$	2.39624e19	−	6.35392e19i	0.0969488	−	0.257072i
$820$		0			0
$821$	2.55331e20			1.01555			0.507776	−	0.861489i	$-0.330468\pi$
$821$	2.55331e20			1.01555			0.507776	+	0.861489i	$0.330468\pi$
$822$		0			0
$823$	2.75790e20			1.07840			0.539200	−	0.842178i	$-0.318727\pi$
$823$	2.75790e20			1.07840			0.539200	+	0.842178i	$0.318727\pi$
$824$		0			0
$825$			2.16922e19i			0.0833924i
$826$		0			0
$827$	3.19644e20			1.20817			0.604085	−	0.796920i	$-0.293539\pi$
$827$	3.19644e20			1.20817			0.604085	+	0.796920i	$0.293539\pi$
$828$		0			0
$829$			3.35633e20i			1.24733i	0.781691	+	0.623666i	$0.214358\pi$
$829$			3.35633e20i			1.24733i	−0.781691	+	0.623666i	$0.785642\pi$
$830$		0			0
$831$		−	2.54131e20i		−	0.928646i
$832$		0			0
$833$	−8.22360e19	+	9.35228e19i	−0.295493	+	0.336049i
$834$		0			0
$835$	1.85445e20			0.655252
$836$		0			0
$837$	2.09997e20			0.729685
$838$		0			0
$839$		−	3.96825e19i		−	0.135602i	−0.997699	−	0.0678010i	$-0.978402\pi$
$839$		−	3.96825e19i		−	0.135602i	0.997699	−	0.0678010i	$-0.0215983\pi$
$840$		0			0
$841$	8.46386e20			2.84444
$842$		0			0
$843$			3.38973e20i			1.12040i
$844$		0			0
$845$		−	2.76582e20i		−	0.899141i
$846$		0			0
$847$	−2.32177e20	−	8.75601e19i	−0.742395	−	0.279977i
$848$		0			0
$849$	2.23833e20			0.703995
$850$		0			0
$851$	1.85700e19			0.0574521
$852$		0			0
$853$			1.67266e20i			0.509054i	0.967066	+	0.254527i	$0.0819198\pi$
$853$			1.67266e20i			0.509054i	−0.967066	+	0.254527i	$0.918080\pi$
$854$		0			0
$855$	7.68490e18			0.0230078
$856$		0			0
$857$			3.11187e20i			0.916549i	0.888811	+	0.458274i	$0.151532\pi$
$857$			3.11187e20i			0.916549i	−0.888811	+	0.458274i	$0.848468\pi$
$858$		0			0
$859$		−	2.46507e20i		−	0.714295i	−0.934048	−	0.357148i	$-0.883749\pi$
$859$		−	2.46507e20i		−	0.714295i	0.934048	−	0.357148i	$-0.116251\pi$
$860$		0			0
$861$	−4.26096e20	−	1.60692e20i	−1.21475	−	0.458114i
$862$		0			0
$863$	−5.69820e20			−1.59831			−0.799157	−	0.601122i	$-0.794720\pi$
$863$	−5.69820e20			−1.59831			−0.799157	+	0.601122i	$0.794720\pi$
$864$		0			0
$865$	2.37331e20			0.655001
$866$		0			0
$867$		−	2.70183e20i		−	0.733709i
$868$		0			0
$869$	9.78147e19			0.261376
$870$		0			0
$871$		−	6.11687e20i		−	1.60843i
$872$		0			0
$873$			5.69344e19i			0.147324i
$874$		0			0
$875$	1.23941e19	−	3.28644e19i	0.0315614	−	0.0836892i
$876$		0			0
$877$	−4.94267e20			−1.23869			−0.619347	−	0.785117i	$-0.712603\pi$
$877$	−4.94267e20			−1.23869			−0.619347	+	0.785117i	$0.712603\pi$
$878$		0			0
$879$	−6.10558e19			−0.150593
$880$		0			0
$881$			5.71359e20i			1.38700i	0.720454	+	0.693502i	$0.243933\pi$
$881$			5.71359e20i			1.38700i	−0.720454	+	0.693502i	$0.756067\pi$
$882$		0			0
$883$	−4.59593e20			−1.09812			−0.549058	−	0.835784i	$-0.685014\pi$
$883$	−4.59593e20			−1.09812			−0.549058	+	0.835784i	$0.685014\pi$
$884$		0			0
$885$		−	1.68749e20i		−	0.396862i
$886$		0			0
$887$			3.18121e20i			0.736422i	0.929742	+	0.368211i	$0.120030\pi$
$887$			3.18121e20i			0.736422i	−0.929742	+	0.368211i	$0.879970\pi$
$888$		0			0
$889$	−7.26301e20	−	2.73908e20i	−1.65502	−	0.624154i
$890$		0			0
$891$	−1.65452e20			−0.371132
$892$		0			0
$893$	3.74433e19			0.0826826
$894$		0			0
$895$			3.01714e20i			0.655895i
$896$		0			0
$897$	−3.22649e20			−0.690530
$898$		0			0
$899$		−	6.38946e20i		−	1.34631i
$900$		0			0
$901$		−	2.36168e20i		−	0.489946i
$902$		0			0
$903$	2.26032e20	−	5.99353e20i	0.461696	−	1.22425i
$904$		0			0
$905$	6.24144e17			0.00125529
$906$		0			0
$907$	−6.90164e20			−1.36679			−0.683394	−	0.730050i	$-0.739497\pi$
$907$	−6.90164e20			−1.36679			−0.683394	+	0.730050i	$0.739497\pi$
$908$		0			0
$909$			3.74344e19i			0.0730001i
$910$		0			0
$911$	5.82612e20			1.11879			0.559397	−	0.828900i	$-0.311033\pi$
$911$	5.82612e20			1.11879			0.559397	+	0.828900i	$0.311033\pi$
$912$		0			0
$913$		−	1.62902e20i		−	0.308057i
$914$		0			0
$915$			2.87577e20i			0.535557i
$916$		0			0
$917$	−1.63957e20	+	4.34754e20i	−0.300708	+	0.797366i
$918$		0			0
$919$	−3.03715e20			−0.548602			−0.274301	−	0.961644i	$-0.588447\pi$
$919$	−3.03715e20			−0.548602			−0.274301	+	0.961644i	$0.588447\pi$
$920$		0			0
$921$	4.44141e20			0.790138
$922$		0			0
$923$		−	1.57340e21i		−	2.75694i
$924$		0			0
$925$	−1.53474e19			−0.0264876
$926$		0			0
$927$			1.68882e19i			0.0287095i
$928$		0			0
$929$		−	7.95762e20i		−	1.33252i	−0.745720	−	0.666260i	$-0.767894\pi$
$929$		−	7.95762e20i		−	1.33252i	0.745720	−	0.666260i	$-0.232106\pi$
$930$		0			0
$931$	1.30067e20	−	1.47918e20i	0.214545	−	0.243991i
$932$		0			0
$933$	−1.62987e20			−0.264839
$934$		0			0
$935$	−5.68207e19			−0.0909548
$936$		0			0
$937$			9.60106e20i			1.51406i	0.653382	+	0.757028i	$0.273350\pi$
$937$			9.60106e20i			1.51406i	−0.653382	+	0.757028i	$0.726650\pi$
$938$		0			0
$939$	−2.23293e20			−0.346910
$940$		0			0
$941$			4.86216e20i			0.744221i	0.928188	+	0.372110i	$0.121366\pi$
$941$			4.86216e20i			0.744221i	−0.928188	+	0.372110i	$0.878634\pi$
$942$		0			0
$943$		−	4.07071e20i		−	0.613887i
$944$		0			0
$945$	2.99273e20	+	1.12864e20i	0.444677	+	0.167700i
$946$		0			0
$947$	−2.09381e20			−0.306540			−0.153270	−	0.988184i	$-0.548980\pi$
$947$	−2.09381e20			−0.306540			−0.153270	+	0.988184i	$0.548980\pi$
$948$		0			0
$949$	−1.95089e21			−2.81429
$950$		0			0
$951$		−	6.66915e20i		−	0.947997i
$952$		0			0
$953$	1.98446e20			0.277966			0.138983	−	0.990295i	$-0.455617\pi$
$953$	1.98446e20			0.277966			0.138983	+	0.990295i	$0.455617\pi$
$954$		0			0
$955$		−	5.14649e20i		−	0.710375i
$956$		0			0
$957$			6.01030e20i			0.817547i
$958$		0			0
$959$	−6.87149e20	−	2.59142e20i	−0.921129	−	0.347383i
$960$		0			0
$961$	4.00062e20			0.528523
$962$		0			0
$963$	9.92843e19			0.129270
$964$		0			0
$965$			1.34112e20i			0.172098i
$966$		0			0
$967$	5.40027e20			0.683015			0.341507	−	0.939879i	$-0.389063\pi$
$967$	5.40027e20			0.683015			0.341507	+	0.939879i	$0.389063\pi$
$968$		0			0
$969$		−	1.06996e20i		−	0.133383i
$970$		0			0
$971$		−	8.42153e20i		−	1.03480i	−0.855744	−	0.517400i	$-0.826900\pi$
$971$		−	8.42153e20i		−	1.03480i	0.855744	−	0.517400i	$-0.173100\pi$
$972$		0			0
$973$	−2.45810e20	+	6.51797e20i	−0.297722	+	0.789446i
$974$		0			0
$975$	2.66656e20			0.318360
$976$		0			0
$977$	−5.99897e20			−0.706016			−0.353008	−	0.935620i	$-0.614841\pi$
$977$	−5.99897e20			−0.706016			−0.353008	+	0.935620i	$0.614841\pi$
$978$		0			0
$979$		−	6.67385e20i		−	0.774280i
$980$		0			0
$981$	−1.50942e20			−0.172635
$982$		0			0
$983$		−	1.12901e20i		−	0.127299i	−0.997972	−	0.0636493i	$-0.979726\pi$
$983$		−	1.12901e20i		−	0.127299i	0.997972	−	0.0636493i	$-0.0202739\pi$
$984$		0			0
$985$			3.90408e20i			0.433976i
$986$		0			0
$987$	1.99330e20	+	7.51727e19i	0.218450	+	0.0823835i
$988$		0			0
$989$	5.72592e20			0.618688
$990$		0			0
$991$	−1.18626e21			−1.26376			−0.631881	−	0.775065i	$-0.717717\pi$
$991$	−1.18626e21			−1.26376			−0.631881	+	0.775065i	$0.717717\pi$
$992$		0			0
$993$		−	6.37657e20i		−	0.669796i
$994$		0			0
$995$	6.72911e20			0.696941
$996$		0			0
$997$			1.20656e21i			1.23221i	0.787665	+	0.616104i	$0.211290\pi$
$997$			1.20656e21i			1.23221i	−0.787665	+	0.616104i	$0.788710\pi$
$998$		0			0
$999$		−	1.39758e20i		−	0.140740i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.15.d.a.41.11	✓	36
7.6	odd	2	inner	140.15.d.a.41.26	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.15.d.a.41.11	✓	36	1.1	even	1	trivial
140.15.d.a.41.26	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)

\(f(q)\)	\(=\)	\(q-2006.39i q^{3} +34938.6i q^{5} +(770567. + 290602. i) q^{7} +757366. q^{9} +O(q^{10})\)	\(q-2006.39i q^{3} +34938.6i q^{5} +(770567. + 290602. i) q^{7} +757366. q^{9} +8.85683e6 q^{11} -1.08874e8i q^{13} +7.01004e7 q^{15} +1.83621e8i q^{17} -2.90421e8i q^{19} +(5.83061e8 - 1.54606e9i) q^{21} +1.47703e9 q^{23} -1.22070e9 q^{25} -1.11161e10i q^{27} -3.38222e10 q^{29} +1.88913e10i q^{31} -1.77703e10i q^{33} +(-1.01532e10 + 2.69225e10i) q^{35} +1.25726e10 q^{37} -2.18445e11 q^{39} -2.75602e11i q^{41} +3.87665e11 q^{43} +2.64613e10i q^{45} +1.28928e11i q^{47} +(5.09324e11 + 4.47856e11i) q^{49} +3.68416e11 q^{51} -1.28617e12 q^{53} +3.09445e11i q^{55} -5.82697e11 q^{57} -2.40725e12i q^{59} +4.10235e12i q^{61} +(5.83601e11 + 2.20092e11i) q^{63} +3.80391e12 q^{65} +5.61828e12 q^{67} -2.96349e12i q^{69} +1.44515e13 q^{71} -1.79187e13i q^{73} +2.44921e12i q^{75} +(6.82479e12 + 2.57381e12i) q^{77} +1.10440e13 q^{79} -1.86807e13 q^{81} -1.83928e13i q^{83} -6.41547e12 q^{85} +6.78606e13i q^{87} -7.53526e13i q^{89} +(3.16391e13 - 8.38950e13i) q^{91} +3.79033e13 q^{93} +1.01469e13 q^{95} +7.51742e13i q^{97} +6.70786e12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 1364266 q^{7} - 54790830 q^{9}+O(q^{10}) \) 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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