LMFDB - Embedded newform 140.15.d.a.41.13

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			41.13
Character	$\chi$	$=$	140.41
Dual form			140.15.d.a.41.24

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1557.79i q^{3} -34938.6i q^{5} +(-188159. + 801760. i) q^{7} +2.35627e6 q^{9} +O(q^{10})$	\(q-1557.79i q^{3} -34938.6i q^{5} +(-188159. + 801760. i) q^{7} +2.35627e6 q^{9} -6.25796e6 q^{11} -7.13992e7i q^{13} -5.44269e7 q^{15} +1.43033e8i q^{17} +1.07178e9i q^{19} +(1.24897e9 + 2.93113e8i) q^{21} -1.64574e9 q^{23} -1.22070e9 q^{25} -1.11214e10i q^{27} +6.22362e9 q^{29} -4.38762e10i q^{31} +9.74857e9i q^{33} +(2.80123e10 + 6.57402e9i) q^{35} -9.52892e10 q^{37} -1.11225e11 q^{39} +1.20434e11i q^{41} +3.83871e11 q^{43} -8.23245e10i q^{45} +9.30979e11i q^{47} +(-6.07415e11 - 3.01718e11i) q^{49} +2.22816e11 q^{51} -5.02191e10 q^{53} +2.18644e11i q^{55} +1.66961e12 q^{57} +2.90690e12i q^{59} +5.76056e11i q^{61} +(-4.43354e11 + 1.88916e12i) q^{63} -2.49459e12 q^{65} +8.65860e12 q^{67} +2.56371e12i q^{69} +2.78666e12 q^{71} +1.78765e12i q^{73} +1.90160e12i q^{75} +(1.17749e12 - 5.01738e12i) q^{77} -2.59978e13 q^{79} -6.05486e12 q^{81} +7.49187e12i q^{83} +4.99738e12 q^{85} -9.69509e12i q^{87} -1.45143e13i q^{89} +(5.72450e13 + 1.34344e13i) q^{91} -6.83499e13 q^{93} +3.74464e13 q^{95} +2.16872e12i q^{97} -1.47454e13 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	1557.79i		−	0.712295i	−0.934430	−	0.356147i	$-0.884090\pi$
$3$		−	1557.79i		−	0.712295i	0.934430	−	0.356147i	$-0.115910\pi$
$4$		0			0
$5$		−	34938.6i		−	0.447214i
$5$		−	34938.6i		−	0.447214i
$6$		0			0
$7$	−188159.	+	801760.i	−0.228476	+	0.973550i
$7$	−188159.	+	801760.i	−0.228476	+	0.973550i
$8$		0			0
$9$	2.35627e6			0.492637
$10$		0			0
$11$	−6.25796e6			−0.321132			−0.160566	−	0.987025i	$-0.551332\pi$
$11$	−6.25796e6			−0.321132			−0.160566	+	0.987025i	$0.551332\pi$
$12$		0			0
$13$		−	7.13992e7i		−	1.13786i	−0.822385	−	0.568932i	$-0.807357\pi$
$13$		−	7.13992e7i		−	1.13786i	0.822385	−	0.568932i	$-0.192643\pi$
$14$		0			0
$15$	−5.44269e7			−0.318548
$16$		0			0
$17$			1.43033e8i			0.348574i	0.984695	+	0.174287i	$0.0557620\pi$
$17$			1.43033e8i			0.348574i	−0.984695	+	0.174287i	$0.944238\pi$
$18$		0			0
$19$			1.07178e9i			1.19903i	0.800363	+	0.599515i	$0.204640\pi$
$19$			1.07178e9i			1.19903i	−0.800363	+	0.599515i	$0.795360\pi$
$20$		0			0
$21$	1.24897e9	+	2.93113e8i	0.693454	+	0.162742i
$22$		0			0
$23$	−1.64574e9			−0.483354			−0.241677	−	0.970357i	$-0.577697\pi$
$23$	−1.64574e9			−0.483354			−0.241677	+	0.970357i	$0.577697\pi$
$24$		0			0
$25$	−1.22070e9			−0.200000
$26$		0			0
$27$		−	1.11214e10i		−	1.06320i
$28$		0			0
$29$	6.22362e9			0.360792			0.180396	−	0.983594i	$-0.442262\pi$
$29$	6.22362e9			0.360792			0.180396	+	0.983594i	$0.442262\pi$
$30$		0			0
$31$		−	4.38762e10i		−	1.59477i	−0.603472	−	0.797384i	$-0.706217\pi$
$31$		−	4.38762e10i		−	1.59477i	0.603472	−	0.797384i	$-0.293783\pi$
$32$		0			0
$33$			9.74857e9i			0.228741i
$34$		0			0
$35$	2.80123e10	+	6.57402e9i	0.435385	+	0.102177i
$36$		0			0
$37$	−9.52892e10			−1.00376			−0.501882	−	0.864936i	$-0.667359\pi$
$37$	−9.52892e10			−1.00376			−0.501882	+	0.864936i	$0.667359\pi$
$38$		0			0
$39$	−1.11225e11			−0.810494
$40$		0			0
$41$			1.20434e11i			0.618389i	0.950999	+	0.309195i	$0.100059\pi$
$41$			1.20434e11i			0.618389i	−0.950999	+	0.309195i	$0.899941\pi$
$42$		0			0
$43$	3.83871e11			1.41223			0.706116	−	0.708096i	$-0.250446\pi$
$43$	3.83871e11			1.41223			0.706116	+	0.708096i	$0.250446\pi$
$44$		0			0
$45$		−	8.23245e10i		−	0.220314i
$46$		0			0
$47$			9.30979e11i			1.83762i	0.394704	+	0.918809i	$0.370847\pi$
$47$			9.30979e11i			1.83762i	−0.394704	+	0.918809i	$0.629153\pi$
$48$		0			0
$49$	−6.07415e11	−	3.01718e11i	−0.895598	−	0.444865i
$50$		0			0
$51$	2.22816e11			0.248287
$52$		0			0
$53$	−5.02191e10			−0.0427501			−0.0213751	−	0.999772i	$-0.506804\pi$
$53$	−5.02191e10			−0.0427501			−0.0213751	+	0.999772i	$0.506804\pi$
$54$		0			0
$55$			2.18644e11i			0.143615i
$56$		0			0
$57$	1.66961e12			0.854063
$58$		0			0
$59$			2.90690e12i			1.16806i	0.811731	+	0.584032i	$0.198526\pi$
$59$			2.90690e12i			1.16806i	−0.811731	+	0.584032i	$0.801474\pi$
$60$		0			0
$61$			5.76056e11i			0.183297i	0.995791	+	0.0916486i	$0.0292137\pi$
$61$			5.76056e11i			0.183297i	−0.995791	+	0.0916486i	$0.970786\pi$
$62$		0			0
$63$	−4.43354e11	+	1.88916e12i	−0.112555	+	0.479606i
$64$		0			0
$65$	−2.49459e12			−0.508868
$66$		0			0
$67$	8.65860e12			1.42864			0.714322	−	0.699817i	$-0.246735\pi$
$67$	8.65860e12			1.42864			0.714322	+	0.699817i	$0.246735\pi$
$68$		0			0
$69$			2.56371e12i			0.344290i
$70$		0			0
$71$	2.78666e12			0.306391			0.153196	−	0.988196i	$-0.451044\pi$
$71$	2.78666e12			0.306391			0.153196	+	0.988196i	$0.451044\pi$
$72$		0			0
$73$			1.78765e12i			0.161817i	0.996722	+	0.0809083i	$0.0257821\pi$
$73$			1.78765e12i			0.161817i	−0.996722	+	0.0809083i	$0.974218\pi$
$74$		0			0
$75$			1.90160e12i			0.142459i
$76$		0			0
$77$	1.17749e12	−	5.01738e12i	0.0733709	−	0.312638i
$78$		0			0
$79$	−2.59978e13			−1.35378			−0.676889	−	0.736085i	$-0.736672\pi$
$79$	−2.59978e13			−1.35378			−0.676889	+	0.736085i	$0.736672\pi$
$80$		0			0
$81$	−6.05486e12			−0.264673
$82$		0			0
$83$			7.49187e12i			0.276085i	0.990426	+	0.138043i	$0.0440811\pi$
$83$			7.49187e12i			0.276085i	−0.990426	+	0.138043i	$0.955919\pi$
$84$		0			0
$85$	4.99738e12			0.155887
$86$		0			0
$87$		−	9.69509e12i		−	0.256990i
$88$		0			0
$89$		−	1.45143e13i		−	0.328146i	−0.986448	−	0.164073i	$-0.947537\pi$
$89$		−	1.45143e13i		−	0.328146i	0.986448	−	0.164073i	$-0.0524632\pi$
$90$		0			0
$91$	5.72450e13	+	1.34344e13i	1.10777	+	0.259974i
$92$		0			0
$93$	−6.83499e13			−1.13594
$94$		0			0
$95$	3.74464e13			0.536223
$96$		0			0
$97$			2.16872e12i			0.0268411i	0.999910	+	0.0134206i	$0.00427203\pi$
$97$			2.16872e12i			0.0268411i	−0.999910	+	0.0134206i	$0.995728\pi$
$98$		0			0
$99$	−1.47454e13			−0.158201
$100$		0			0
$101$			8.59906e13i			0.802050i	0.916067	+	0.401025i	$0.131346\pi$
$101$			8.59906e13i			0.802050i	−0.916067	+	0.401025i	$0.868654\pi$
$102$		0			0
$103$			1.23597e14i			1.00495i	0.864591	+	0.502477i	$0.167578\pi$
$103$			1.23597e14i			1.00495i	−0.864591	+	0.502477i	$0.832422\pi$
$104$		0			0
$105$	1.02409e13	−	4.36373e13i	0.0727804	−	0.310122i
$106$		0			0
$107$	1.70196e14			1.05989			0.529947	−	0.848031i	$-0.322212\pi$
$107$	1.70196e14			1.05989			0.529947	+	0.848031i	$0.322212\pi$
$108$		0			0
$109$	1.31714e14			0.720518			0.360259	−	0.932852i	$-0.382688\pi$
$109$	1.31714e14			0.720518			0.360259	+	0.932852i	$0.382688\pi$
$110$		0			0
$111$			1.48440e14i			0.714976i
$112$		0			0
$113$	3.17106e13			0.134789			0.0673946	−	0.997726i	$-0.478531\pi$
$113$	3.17106e13			0.134789			0.0673946	+	0.997726i	$0.478531\pi$
$114$		0			0
$115$			5.74996e13i			0.216162i
$116$		0			0
$117$		−	1.68236e14i		−	0.560553i
$118$		0			0
$119$	−1.14678e14	−	2.69131e13i	−0.339354	−	0.0796406i
$120$		0			0
$121$	−3.40588e14			−0.896874
$122$		0			0
$123$	1.87611e14			0.440475
$124$		0			0
$125$			4.26496e13i			0.0894427i
$126$		0			0
$127$	1.13869e14			0.213689			0.106844	−	0.994276i	$-0.465925\pi$
$127$	1.13869e14			0.213689			0.106844	+	0.994276i	$0.465925\pi$
$128$		0			0
$129$		−	5.97990e14i		−	1.00593i
$130$		0			0
$131$		−	1.10173e15i		−	1.66409i	−0.554707	−	0.832046i	$-0.687170\pi$
$131$		−	1.10173e15i		−	1.66409i	0.554707	−	0.832046i	$-0.312830\pi$
$132$		0			0
$133$	−8.59310e14	−	2.01666e14i	−1.16732	−	0.273949i
$134$		0			0
$135$	−3.88566e14			−0.475476
$136$		0			0
$137$	7.23999e13			0.0799271			0.0399636	−	0.999201i	$-0.487276\pi$
$137$	7.23999e13			0.0799271			0.0399636	+	0.999201i	$0.487276\pi$
$138$		0			0
$139$		−	6.08995e14i		−	0.607449i	−0.952760	−	0.303725i	$-0.901770\pi$
$139$		−	6.08995e14i		−	0.607449i	0.952760	−	0.303725i	$-0.0982303\pi$
$140$		0			0
$141$	1.45027e15			1.30892
$142$		0			0
$143$			4.46813e14i			0.365404i
$144$		0			0
$145$		−	2.17444e14i		−	0.161351i
$146$		0			0
$147$	−4.70012e14	+	9.46224e14i	−0.316875	+	0.637929i
$148$		0			0
$149$	9.45035e14			0.579621			0.289810	−	0.957084i	$-0.406408\pi$
$149$	9.45035e14			0.579621			0.289810	+	0.957084i	$0.406408\pi$
$150$		0			0
$151$	3.22574e15			1.80215			0.901074	−	0.433665i	$-0.142780\pi$
$151$	3.22574e15			1.80215			0.901074	+	0.433665i	$0.142780\pi$
$152$		0			0
$153$			3.37025e14i			0.171720i
$154$		0			0
$155$	−1.53297e15			−0.713202
$156$		0			0
$157$		−	4.28222e15i		−	1.82126i	−0.413227	−	0.910628i	$-0.635598\pi$
$157$		−	4.28222e15i		−	1.82126i	0.413227	−	0.910628i	$-0.364402\pi$
$158$		0			0
$159$			7.82307e13i			0.0304507i
$160$		0			0
$161$	3.09661e14	−	1.31948e15i	0.110435	−	0.470569i
$162$		0			0
$163$	1.91155e15			0.625277			0.312638	−	0.949872i	$-0.398787\pi$
$163$	1.91155e15			0.625277			0.312638	+	0.949872i	$0.398787\pi$
$164$		0			0
$165$	3.40601e14			0.102296
$166$		0			0
$167$			3.87265e15i			1.06904i	0.845157	+	0.534519i	$0.179507\pi$
$167$			3.87265e15i			1.06904i	−0.845157	+	0.534519i	$0.820493\pi$
$168$		0			0
$169$	−1.16047e15			−0.294733
$170$		0			0
$171$			2.52540e15i			0.590686i
$172$		0			0
$173$			3.61925e15i			0.780361i	0.920738	+	0.390181i	$0.127587\pi$
$173$			3.61925e15i			0.780361i	−0.920738	+	0.390181i	$0.872413\pi$
$174$		0			0
$175$	2.29687e14	−	9.78711e14i	0.0456951	−	0.194710i
$176$		0			0
$177$	4.52834e15			0.832006
$178$		0			0
$179$	6.59112e15			1.11941			0.559703	−	0.828693i	$-0.310915\pi$
$179$	6.59112e15			1.11941			0.559703	+	0.828693i	$0.310915\pi$
$180$		0			0
$181$		−	6.97180e14i		−	0.109546i	−0.998499	−	0.0547728i	$-0.982557\pi$
$181$		−	6.97180e14i		−	0.109546i	0.998499	−	0.0547728i	$-0.0174435\pi$
$182$		0			0
$183$	8.97374e14			0.130562
$184$		0			0
$185$			3.32927e15i			0.448897i
$186$		0			0
$187$		−	8.95097e14i		−	0.111938i
$188$		0			0
$189$	8.91670e15	+	2.09260e15i	1.03507	+	0.242915i
$190$		0			0
$191$	6.30750e15			0.680179			0.340090	−	0.940393i	$-0.389543\pi$
$191$	6.30750e15			0.680179			0.340090	+	0.940393i	$0.389543\pi$
$192$		0			0
$193$	−1.16713e16			−1.17008			−0.585042	−	0.811003i	$-0.698922\pi$
$193$	−1.16713e16			−1.17008			−0.585042	+	0.811003i	$0.698922\pi$
$194$		0			0
$195$			3.88604e15i			0.362464i
$196$		0			0
$197$	1.83007e16			1.58930			0.794648	−	0.607071i	$-0.207656\pi$
$197$	1.83007e16			1.58930			0.794648	+	0.607071i	$0.207656\pi$
$198$		0			0
$199$			1.44241e15i			0.116712i	0.998296	+	0.0583560i	$0.0185859\pi$
$199$			1.44241e15i			0.116712i	−0.998296	+	0.0583560i	$0.981414\pi$
$200$		0			0
$201$		−	1.34883e16i		−	1.01762i
$202$		0			0
$203$	−1.17103e15	+	4.98985e15i	−0.0824323	+	0.351249i
$204$		0			0
$205$	4.20779e15			0.276552
$206$		0			0
$207$	−3.87779e15			−0.238118
$208$		0			0
$209$		−	6.70715e15i		−	0.385047i
$210$		0			0
$211$	2.71712e16			1.45925			0.729627	−	0.683845i	$-0.239694\pi$
$211$	2.71712e16			1.45925			0.729627	+	0.683845i	$0.239694\pi$
$212$		0			0
$213$		−	4.34103e15i		−	0.218241i
$214$		0			0
$215$		−	1.34119e16i		−	0.631569i
$216$		0			0
$217$	3.51782e16	+	8.25573e15i	1.55259	+	0.364366i
$218$		0			0
$219$	2.78478e15			0.115261
$220$		0			0
$221$	1.02125e16			0.396629
$222$		0			0
$223$		−	1.09049e16i		−	0.397639i	−0.980036	−	0.198819i	$-0.936289\pi$
$223$		−	1.09049e16i		−	0.397639i	0.980036	−	0.198819i	$-0.0637107\pi$
$224$		0			0
$225$	−2.87630e15			−0.0985273
$226$		0			0
$227$		−	4.75743e16i		−	1.53176i	−0.642982	−	0.765881i	$-0.722303\pi$
$227$		−	4.75743e16i		−	1.53176i	0.642982	−	0.765881i	$-0.277697\pi$
$228$		0			0
$229$			2.79956e16i			0.847698i	0.905733	+	0.423849i	$0.139321\pi$
$229$			2.79956e16i			0.847698i	−0.905733	+	0.423849i	$0.860679\pi$
$230$		0			0
$231$	−7.81601e15	−	1.83429e15i	−0.222690	−	0.0522617i
$232$		0			0
$233$	5.51922e16			1.48043			0.740213	−	0.672373i	$-0.234725\pi$
$233$	5.51922e16			1.48043			0.740213	+	0.672373i	$0.234725\pi$
$234$		0			0
$235$	3.25271e16			0.821807
$236$		0			0
$237$			4.04991e16i			0.964289i
$238$		0			0
$239$	−1.20163e16			−0.269764			−0.134882	−	0.990862i	$-0.543066\pi$
$239$	−1.20163e16			−0.269764			−0.134882	+	0.990862i	$0.543066\pi$
$240$		0			0
$241$			1.70819e16i			0.361758i	0.983505	+	0.180879i	$0.0578942\pi$
$241$			1.70819e16i			0.361758i	−0.983505	+	0.180879i	$0.942106\pi$
$242$		0			0
$243$		−	4.37612e16i		−	0.874672i
$244$		0			0
$245$	−1.05416e16	+	2.12222e16i	−0.198950	+	0.400524i
$246$		0			0
$247$	7.65242e16			1.36433
$248$		0			0
$249$	1.16707e16			0.196654
$250$		0			0
$251$		−	9.05455e16i		−	1.44262i	−0.692615	−	0.721308i	$-0.743541\pi$
$251$		−	9.05455e16i		−	1.44262i	0.692615	−	0.721308i	$-0.256459\pi$
$252$		0			0
$253$	1.02989e16			0.155220
$254$		0			0
$255$		−	7.78486e15i		−	0.111037i
$256$		0			0
$257$		−	1.10055e17i		−	1.48620i	−0.669181	−	0.743099i	$-0.733355\pi$
$257$		−	1.10055e17i		−	1.48620i	0.669181	−	0.743099i	$-0.266645\pi$
$258$		0			0
$259$	1.79296e16	−	7.63991e16i	0.229336	−	0.977214i
$260$		0			0
$261$	1.46645e16			0.177740
$262$		0			0
$263$	1.31781e17			1.51413			0.757065	−	0.653340i	$-0.226633\pi$
$263$	1.31781e17			1.51413			0.757065	+	0.653340i	$0.226633\pi$
$264$		0			0
$265$			1.75458e15i			0.0191184i
$266$		0			0
$267$	−2.26102e16			−0.233736
$268$		0			0
$269$			5.44616e16i			0.534348i	0.963648	+	0.267174i	$0.0860899\pi$
$269$			5.44616e16i			0.534348i	−0.963648	+	0.267174i	$0.913910\pi$
$270$		0			0
$271$			1.39494e17i			1.29948i	0.760155	+	0.649741i	$0.225123\pi$
$271$			1.39494e17i			1.29948i	−0.760155	+	0.649741i	$0.774877\pi$
$272$		0			0
$273$	2.09280e16	−	8.91756e16i	0.185178	−	0.789056i
$274$		0			0
$275$	7.63911e15			0.0642264
$276$		0			0
$277$	8.22591e16			0.657394			0.328697	−	0.944435i	$-0.393391\pi$
$277$	8.22591e16			0.657394			0.328697	+	0.944435i	$0.393391\pi$
$278$		0			0
$279$		−	1.03384e17i		−	0.785641i
$280$		0			0
$281$	6.69583e16			0.484016			0.242008	−	0.970274i	$-0.422194\pi$
$281$	6.69583e16			0.484016			0.242008	+	0.970274i	$0.422194\pi$
$282$		0			0
$283$			4.42814e16i			0.304590i	0.988335	+	0.152295i	$0.0486665\pi$
$283$			4.42814e16i			0.304590i	−0.988335	+	0.152295i	$0.951334\pi$
$284$		0			0
$285$		−	5.83336e16i		−	0.381949i
$286$		0			0
$287$	−9.65591e16	−	2.26608e16i	−0.602033	−	0.141287i
$288$		0			0
$289$	1.47919e17			0.878496
$290$		0			0
$291$	3.37840e15			0.0191188
$292$		0			0
$293$		−	2.62084e16i		−	0.141373i	−0.997499	−	0.0706867i	$-0.977481\pi$
$293$		−	2.62084e16i		−	0.141373i	0.997499	−	0.0706867i	$-0.0225191\pi$
$294$		0			0
$295$	1.01563e17			0.522374
$296$		0			0
$297$			6.95973e16i			0.341427i
$298$		0			0
$299$			1.17504e17i			0.549991i
$300$		0			0
$301$	−7.22290e16	+	3.07772e17i	−0.322661	+	1.37488i
$302$		0			0
$303$	1.33955e17			0.571296
$304$		0			0
$305$	2.01266e16			0.0819730
$306$		0			0
$307$			3.40875e17i			1.32625i	0.748508	+	0.663126i	$0.230771\pi$
$307$			3.40875e17i			1.32625i	−0.748508	+	0.663126i	$0.769229\pi$
$308$		0			0
$309$	1.92537e17			0.715823
$310$		0			0
$311$		−	2.14637e17i		−	0.762751i	−0.924420	−	0.381375i	$-0.875451\pi$
$311$		−	2.14637e17i		−	0.762751i	0.924420	−	0.381375i	$-0.124549\pi$
$312$		0			0
$313$			2.63012e17i			0.893646i	0.894622	+	0.446823i	$0.147445\pi$
$313$			2.63012e17i			0.893646i	−0.894622	+	0.446823i	$0.852555\pi$
$314$		0			0
$315$	6.60045e16	+	1.54901e16i	0.214486	+	0.0503363i
$316$		0			0
$317$	1.62845e17			0.506243			0.253122	−	0.967434i	$-0.418543\pi$
$317$	1.62845e17			0.506243			0.253122	+	0.967434i	$0.418543\pi$
$318$		0			0
$319$	−3.89472e16			−0.115862
$320$		0			0
$321$		−	2.65129e17i		−	0.754956i
$322$		0			0
$323$	−1.53300e17			−0.417951
$324$		0			0
$325$			8.71573e16i			0.227573i
$326$		0			0
$327$		−	2.05182e17i		−	0.513221i
$328$		0			0
$329$	−7.46422e17	−	1.75173e17i	−1.78901	−	0.419851i
$330$		0			0
$331$	−1.02072e17			−0.234483			−0.117242	−	0.993103i	$-0.537405\pi$
$331$	−1.02072e17			−0.234483			−0.117242	+	0.993103i	$0.537405\pi$
$332$		0			0
$333$	−2.24527e17			−0.494491
$334$		0			0
$335$		−	3.02519e17i		−	0.638909i
$336$		0			0
$337$	4.72551e17			0.957280			0.478640	−	0.878011i	$-0.341130\pi$
$337$	4.72551e17			0.957280			0.478640	+	0.878011i	$0.341130\pi$
$338$		0			0
$339$		−	4.93984e16i		−	0.0960096i
$340$		0			0
$341$			2.74576e17i			0.512131i
$342$		0			0
$343$	3.56196e17	−	4.30230e17i	0.637720	−	0.770268i
$344$		0			0
$345$	8.95722e16			0.153971
$346$		0			0
$347$	4.03527e17			0.666140			0.333070	−	0.942902i	$-0.391915\pi$
$347$	4.03527e17			0.666140			0.333070	+	0.942902i	$0.391915\pi$
$348$		0			0
$349$			4.42338e17i			0.701417i	0.936485	+	0.350709i	$0.114059\pi$
$349$			4.42338e17i			0.701417i	−0.936485	+	0.350709i	$0.885941\pi$
$350$		0			0
$351$	−7.94060e17			−1.20977
$352$		0			0
$353$			4.87624e17i			0.713941i	0.934116	+	0.356970i	$0.116190\pi$
$353$			4.87624e17i			0.713941i	−0.934116	+	0.356970i	$0.883810\pi$
$354$		0			0
$355$		−	9.73621e16i		−	0.137022i
$356$		0			0
$357$	−4.19249e16	+	1.78645e17i	−0.0567276	+	0.241720i
$358$		0			0
$359$	−1.38063e18			−1.79646			−0.898230	−	0.439525i	$-0.855147\pi$
$359$	−1.38063e18			−1.79646			−0.898230	+	0.439525i	$0.855147\pi$
$360$		0			0
$361$	−3.49705e17			−0.437675
$362$		0			0
$363$			5.30564e17i			0.638839i
$364$		0			0
$365$	6.24580e16			0.0723666
$366$		0			0
$367$		−	1.01340e18i		−	1.13011i	−0.825055	−	0.565053i	$-0.808856\pi$
$367$		−	1.01340e18i		−	1.13011i	0.825055	−	0.565053i	$-0.191144\pi$
$368$		0			0
$369$			2.83774e17i			0.304641i
$370$		0			0
$371$	9.44919e15	−	4.02636e16i	0.00976736	−	0.0416194i
$372$		0			0
$373$	−1.10631e18			−1.10132			−0.550662	−	0.834728i	$-0.685625\pi$
$373$	−1.10631e18			−1.10132			−0.550662	+	0.834728i	$0.685625\pi$
$374$		0			0
$375$	6.64391e16			0.0637096
$376$		0			0
$377$		−	4.44362e17i		−	0.410532i
$378$		0			0
$379$	−8.10335e17			−0.721424			−0.360712	−	0.932677i	$-0.617466\pi$
$379$	−8.10335e17			−0.721424			−0.360712	+	0.932677i	$0.617466\pi$
$380$		0			0
$381$		−	1.77384e17i		−	0.152209i
$382$		0			0
$383$		−	8.51408e17i		−	0.704281i	−0.935947	−	0.352141i	$-0.885454\pi$
$383$		−	8.51408e17i		−	0.704281i	0.935947	−	0.352141i	$-0.114546\pi$
$384$		0			0
$385$	−1.75300e17	−	4.11400e16i	−0.139816	−	0.0328125i
$386$		0			0
$387$	9.04502e17			0.695717
$388$		0			0
$389$	−1.49100e18			−1.10619			−0.553097	−	0.833117i	$-0.686554\pi$
$389$	−1.49100e18			−1.10619			−0.553097	+	0.833117i	$0.686554\pi$
$390$		0			0
$391$		−	2.35395e17i		−	0.168485i
$392$		0			0
$393$	−1.71627e18			−1.18532
$394$		0			0
$395$			9.08327e17i			0.605428i
$396$		0			0
$397$			1.06149e18i			0.682939i	0.939893	+	0.341470i	$0.110925\pi$
$397$			1.06149e18i			0.682939i	−0.939893	+	0.341470i	$0.889075\pi$
$398$		0			0
$399$	−3.14152e17	+	1.33862e18i	−0.195133	+	0.831473i
$400$		0			0
$401$	2.94003e18			1.76336			0.881679	−	0.471850i	$-0.156414\pi$
$401$	2.94003e18			1.76336			0.881679	+	0.471850i	$0.156414\pi$
$402$		0			0
$403$	−3.13273e18			−1.81463
$404$		0			0
$405$			2.11548e17i			0.118365i
$406$		0			0
$407$	5.96316e17			0.322341
$408$		0			0
$409$		−	6.18611e17i		−	0.323113i	−0.986863	−	0.161557i	$-0.948349\pi$
$409$		−	6.18611e17i		−	0.323113i	0.986863	−	0.161557i	$-0.0516514\pi$
$410$		0			0
$411$		−	1.12784e17i		−	0.0569317i
$412$		0			0
$413$	−2.33064e18	−	5.46962e17i	−1.13717	−	0.266874i
$414$		0			0
$415$	2.61755e17			0.123469
$416$		0			0
$417$	−9.48685e17			−0.432683
$418$		0			0
$419$		−	1.32218e17i		−	0.0583167i	−0.999575	−	0.0291583i	$-0.990717\pi$
$419$		−	1.32218e17i		−	0.0583167i	0.999575	−	0.0291583i	$-0.00928270\pi$
$420$		0			0
$421$	−3.27327e17			−0.139639			−0.0698195	−	0.997560i	$-0.522242\pi$
$421$	−3.27327e17			−0.139639			−0.0698195	+	0.997560i	$0.522242\pi$
$422$		0			0
$423$			2.19363e18i			0.905277i
$424$		0			0
$425$		−	1.74601e17i		−	0.0697148i
$426$		0			0
$427$	−4.61859e17	−	1.08390e17i	−0.178449	−	0.0418790i
$428$		0			0
$429$	6.96040e17			0.260276
$430$		0			0
$431$	−4.28779e18			−1.55200			−0.776002	−	0.630731i	$-0.782755\pi$
$431$	−4.28779e18			−1.55200			−0.776002	+	0.630731i	$0.782755\pi$
$432$		0			0
$433$			3.60067e18i			1.26174i	0.775891	+	0.630868i	$0.217301\pi$
$433$			3.60067e18i			1.26174i	−0.775891	+	0.630868i	$0.782699\pi$
$434$		0			0
$435$	−3.38732e17			−0.114930
$436$		0			0
$437$		−	1.76387e18i		−	0.579556i
$438$		0			0
$439$		−	1.47758e18i		−	0.470220i	−0.971969	−	0.235110i	$-0.924455\pi$
$439$		−	1.47758e18i		−	0.470220i	0.971969	−	0.235110i	$-0.0755450\pi$
$440$		0			0
$441$	−1.43123e18	−	7.10926e17i	−0.441204	−	0.219157i
$442$		0			0
$443$	−7.30698e17			−0.218229			−0.109114	−	0.994029i	$-0.534801\pi$
$443$	−7.30698e17			−0.218229			−0.109114	+	0.994029i	$0.534801\pi$
$444$		0			0
$445$	−5.07109e17			−0.146751
$446$		0			0
$447$		−	1.47216e18i		−	0.412861i
$448$		0			0
$449$	−2.10562e18			−0.572342			−0.286171	−	0.958178i	$-0.592383\pi$
$449$	−2.10562e18			−0.572342			−0.286171	+	0.958178i	$0.592383\pi$
$450$		0			0
$451$		−	7.53671e17i		−	0.198585i
$452$		0			0
$453$		−	5.02502e18i		−	1.28366i
$454$		0			0
$455$	4.69380e17	−	2.00006e18i	0.116264	−	0.495408i
$456$		0			0
$457$	3.04463e17			0.0731343			0.0365672	−	0.999331i	$-0.488358\pi$
$457$	3.04463e17			0.0731343			0.0365672	+	0.999331i	$0.488358\pi$
$458$		0			0
$459$	1.59073e18			0.370603
$460$		0			0
$461$			9.17806e17i			0.207417i	0.994608	+	0.103709i	$0.0330709\pi$
$461$			9.17806e17i			0.207417i	−0.994608	+	0.103709i	$0.966929\pi$
$462$		0			0
$463$	6.00669e18			1.31695			0.658473	−	0.752604i	$-0.271203\pi$
$463$	6.00669e18			1.31695			0.658473	+	0.752604i	$0.271203\pi$
$464$		0			0
$465$			2.38805e18i			0.508010i
$466$		0			0
$467$			5.76624e18i			1.19035i	0.803596	+	0.595175i	$0.202917\pi$
$467$			5.76624e18i			1.19035i	−0.803596	+	0.595175i	$0.797083\pi$
$468$		0			0
$469$	−1.62920e18	+	6.94212e18i	−0.326410	+	1.39086i
$470$		0			0
$471$	−6.67079e18			−1.29727
$472$		0			0
$473$	−2.40225e18			−0.453513
$474$		0			0
$475$		−	1.30832e18i		−	0.239806i
$476$		0			0
$477$	−1.18329e17			−0.0210603
$478$		0			0
$479$			5.70447e18i			0.985977i	0.870036	+	0.492988i	$0.164096\pi$
$479$			5.70447e18i			0.985977i	−0.870036	+	0.492988i	$0.835904\pi$
$480$		0			0
$481$			6.80358e18i			1.14215i
$482$		0			0
$483$	−2.05548e18	−	4.82386e17i	−0.335184	−	0.0786619i
$484$		0			0
$485$	7.57718e16			0.0120037
$486$		0			0
$487$	5.23748e18			0.806157			0.403079	−	0.915165i	$-0.367940\pi$
$487$	5.23748e18			0.806157			0.403079	+	0.915165i	$0.367940\pi$
$488$		0			0
$489$		−	2.97779e18i		−	0.445381i
$490$		0			0
$491$	1.18560e19			1.72332			0.861662	−	0.507483i	$-0.169424\pi$
$491$	1.18560e19			1.72332			0.861662	+	0.507483i	$0.169424\pi$
$492$		0			0
$493$			8.90186e17i			0.125763i
$494$		0			0
$495$			5.15183e17i			0.0707498i
$496$		0			0
$497$	−5.24337e17	+	2.23424e18i	−0.0700029	+	0.298287i
$498$		0			0
$499$	−1.05261e19			−1.36636			−0.683179	−	0.730251i	$-0.739403\pi$
$499$	−1.05261e19			−1.36636			−0.683179	+	0.730251i	$0.739403\pi$
$500$		0			0
$501$	6.03277e18			0.761469
$502$		0			0
$503$			6.59349e18i			0.809355i	0.914459	+	0.404678i	$0.132616\pi$
$503$			6.59349e18i			0.809355i	−0.914459	+	0.404678i	$0.867384\pi$
$504$		0			0
$505$	3.00439e18			0.358688
$506$		0			0
$507$			1.80777e18i			0.209936i
$508$		0			0
$509$		−	1.13838e19i		−	1.28607i	−0.765838	−	0.643034i	$-0.777676\pi$
$509$		−	1.13838e19i		−	1.28607i	0.765838	−	0.643034i	$-0.222324\pi$
$510$		0			0
$511$	−1.43327e18	−	3.36364e17i	−0.157537	−	0.0369712i
$512$		0			0
$513$	1.19197e19			1.27481
$514$		0			0
$515$	4.31829e18			0.449429
$516$		0			0
$517$		−	5.82603e18i		−	0.590118i
$518$		0			0
$519$	5.63802e18			0.555847
$520$		0			0
$521$		−	1.85144e19i		−	1.77683i	−0.459038	−	0.888417i	$-0.651806\pi$
$521$		−	1.85144e19i		−	1.77683i	0.459038	−	0.888417i	$-0.348194\pi$
$522$		0			0
$523$		−	2.39178e18i		−	0.223465i	−0.993738	−	0.111732i	$-0.964360\pi$
$523$		−	2.39178e18i		−	0.223465i	0.993738	−	0.111732i	$-0.0356399\pi$
$524$		0			0
$525$	−1.52462e18	−	3.57803e17i	−0.138691	−	0.0325484i
$526$		0			0
$527$	6.27576e18			0.555894
$528$		0			0
$529$	−8.88439e18			−0.766369
$530$		0			0
$531$			6.84944e18i			0.575431i
$532$		0			0
$533$	8.59889e18			0.703642
$534$		0			0
$535$		−	5.94639e18i		−	0.473999i
$536$		0			0
$537$		−	1.02676e19i		−	0.797347i
$538$		0			0
$539$	3.80118e18	+	1.88814e18i	0.287605	+	0.142860i
$540$		0			0
$541$	−1.80203e19			−1.32856			−0.664280	−	0.747484i	$-0.731262\pi$
$541$	−1.80203e19			−1.32856			−0.664280	+	0.747484i	$0.731262\pi$
$542$		0			0
$543$	−1.08606e18			−0.0780288
$544$		0			0
$545$		−	4.60188e18i		−	0.322225i
$546$		0			0
$547$	2.79104e19			1.90482			0.952411	−	0.304817i	$-0.0985952\pi$
$547$	2.79104e19			1.90482			0.952411	+	0.304817i	$0.0985952\pi$
$548$		0			0
$549$			1.35734e18i			0.0902989i
$550$		0			0
$551$			6.67035e18i			0.432601i
$552$		0			0
$553$	4.89174e18	−	2.08440e19i	0.309305	−	1.31797i
$554$		0			0
$555$	5.18630e18			0.319747
$556$		0			0
$557$	−1.56217e19			−0.939166			−0.469583	−	0.882888i	$-0.655596\pi$
$557$	−1.56217e19			−0.939166			−0.469583	+	0.882888i	$0.655596\pi$
$558$		0			0
$559$		−	2.74081e19i		−	1.60693i
$560$		0			0
$561$	−1.39437e18			−0.0797330
$562$		0			0
$563$		−	3.25617e18i		−	0.181613i	−0.995869	−	0.0908067i	$-0.971055\pi$
$563$		−	3.25617e18i		−	0.181613i	0.995869	−	0.0908067i	$-0.0289445\pi$
$564$		0			0
$565$		−	1.10792e18i		−	0.0602796i
$566$		0			0
$567$	1.13928e18	−	4.85455e18i	0.0604713	−	0.257672i
$568$		0			0
$569$	3.12755e19			1.61964			0.809819	−	0.586680i	$-0.199565\pi$
$569$	3.12755e19			1.61964			0.809819	+	0.586680i	$0.199565\pi$
$570$		0			0
$571$	6.82786e18			0.345010			0.172505	−	0.985009i	$-0.444814\pi$
$571$	6.82786e18			0.345010			0.172505	+	0.985009i	$0.444814\pi$
$572$		0			0
$573$		−	9.82574e18i		−	0.484488i
$574$		0			0
$575$	2.00895e18			0.0966708
$576$		0			0
$577$			1.74078e19i			0.817549i	0.912636	+	0.408774i	$0.134044\pi$
$577$			1.74078e19i			0.817549i	−0.912636	+	0.408774i	$0.865956\pi$
$578$		0			0
$579$			1.81814e19i			0.833444i
$580$		0			0
$581$	−6.00668e18	−	1.40967e18i	−0.268783	−	0.0630788i
$582$		0			0
$583$	3.14269e17			0.0137284
$584$		0			0
$585$	−5.87791e18			−0.250687
$586$		0			0
$587$			2.57991e19i			1.07433i	0.843477	+	0.537165i	$0.180505\pi$
$587$			2.57991e19i			1.07433i	−0.843477	+	0.537165i	$0.819495\pi$
$588$		0			0
$589$	4.70257e19			1.91218
$590$		0			0
$591$		−	2.85086e19i		−	1.13205i
$592$		0			0
$593$			3.09734e19i			1.20118i	0.799559	+	0.600588i	$0.205067\pi$
$593$			3.09734e19i			1.20118i	−0.799559	+	0.600588i	$0.794933\pi$
$594$		0			0
$595$	−9.40304e17	+	4.00670e18i	−0.0356164	+	0.151764i
$596$		0			0
$597$	2.24696e18			0.0831334
$598$		0			0
$599$	−9.39636e18			−0.339603			−0.169802	−	0.985478i	$-0.554313\pi$
$599$	−9.39636e18			−0.339603			−0.169802	+	0.985478i	$0.554313\pi$
$600$		0			0
$601$		−	7.02680e16i		−	0.00248106i	−0.999999	−	0.00124053i	$-0.999605\pi$
$601$		−	7.02680e16i		−	0.00248106i	0.999999	−	0.00124053i	$-0.000394872\pi$
$602$		0			0
$603$	2.04020e19			0.703802
$604$		0			0
$605$			1.18996e19i			0.401094i
$606$		0			0
$607$		−	2.07626e19i		−	0.683851i	−0.939727	−	0.341926i	$-0.888921\pi$
$607$		−	2.07626e19i		−	0.683851i	0.939727	−	0.341926i	$-0.111079\pi$
$608$		0			0
$609$	7.77313e18	+	1.82422e18i	0.250193	+	0.0587160i
$610$		0			0
$611$	6.64712e19			2.09096
$612$		0			0
$613$	3.57196e19			1.09820			0.549102	−	0.835756i	$-0.314970\pi$
$613$	3.57196e19			1.09820			0.549102	+	0.835756i	$0.314970\pi$
$614$		0			0
$615$		−	6.55484e18i		−	0.196987i
$616$		0			0
$617$	−3.75243e19			−1.10234			−0.551171	−	0.834392i	$-0.685819\pi$
$617$	−3.75243e19			−1.10234			−0.551171	+	0.834392i	$0.685819\pi$
$618$		0			0
$619$			4.93461e19i			1.41716i	0.705631	+	0.708579i	$0.250664\pi$
$619$			4.93461e19i			1.41716i	−0.705631	+	0.708579i	$0.749336\pi$
$620$		0			0
$621$			1.83029e19i			0.513900i
$622$		0			0
$623$	1.16370e19	+	2.73101e18i	0.319466	+	0.0749733i
$624$		0			0
$625$	1.49012e18			0.0400000
$626$		0			0
$627$	−1.04483e19			−0.274267
$628$		0			0
$629$		−	1.36295e19i		−	0.349886i
$630$		0			0
$631$	9.40786e18			0.236203			0.118102	−	0.993002i	$-0.462319\pi$
$631$	9.40786e18			0.236203			0.118102	+	0.993002i	$0.462319\pi$
$632$		0			0
$633$		−	4.23269e19i		−	1.03942i
$634$		0			0
$635$		−	3.97844e18i		−	0.0955644i
$636$		0			0
$637$	−2.15424e19	+	4.33690e19i	−0.506195	+	1.01907i
$638$		0			0
$639$	6.56612e18			0.150940
$640$		0			0
$641$	1.63387e18			0.0367461			0.0183730	−	0.999831i	$-0.494151\pi$
$641$	1.63387e18			0.0367461			0.0183730	+	0.999831i	$0.494151\pi$
$642$		0			0
$643$			5.66177e19i			1.24588i	0.782270	+	0.622939i	$0.214062\pi$
$643$			5.66177e19i			1.24588i	−0.782270	+	0.622939i	$0.785938\pi$
$644$		0			0
$645$	−2.08929e19			−0.449863
$646$		0			0
$647$		−	3.35028e19i		−	0.705913i	−0.935640	−	0.352957i	$-0.885176\pi$
$647$		−	3.35028e19i		−	0.705913i	0.935640	−	0.352957i	$-0.114824\pi$
$648$		0			0
$649$		−	1.81913e19i		−	0.375103i
$650$		0			0
$651$	1.28607e19	−	5.48002e19i	0.259536	−	1.10590i
$652$		0			0
$653$	−4.17847e19			−0.825326			−0.412663	−	0.910884i	$-0.635401\pi$
$653$	−4.17847e19			−0.825326			−0.412663	+	0.910884i	$0.635401\pi$
$654$		0			0
$655$	−3.84930e19			−0.744204
$656$		0			0
$657$			4.21219e18i			0.0797168i
$658$		0			0
$659$	5.32597e19			0.986736			0.493368	−	0.869821i	$-0.335766\pi$
$659$	5.32597e19			0.986736			0.493368	+	0.869821i	$0.335766\pi$
$660$		0			0
$661$			5.83573e19i			1.05849i	0.848470	+	0.529243i	$0.177524\pi$
$661$			5.83573e19i			1.05849i	−0.848470	+	0.529243i	$0.822476\pi$
$662$		0			0
$663$		−	1.59089e19i		−	0.282517i
$664$		0			0
$665$	−7.04590e18	+	3.00231e19i	−0.122514	+	0.522040i
$666$		0			0
$667$	−1.02424e19			−0.174390
$668$		0			0
$669$	−1.69876e19			−0.283236
$670$		0			0
$671$		−	3.60494e18i		−	0.0588627i
$672$		0			0
$673$	1.91204e18			0.0305768			0.0152884	−	0.999883i	$-0.495133\pi$
$673$	1.91204e18			0.0305768			0.0152884	+	0.999883i	$0.495133\pi$
$674$		0			0
$675$			1.35759e19i			0.212639i
$676$		0			0
$677$			3.32144e19i			0.509573i	0.966997	+	0.254786i	$0.0820051\pi$
$677$			3.32144e19i			0.509573i	−0.966997	+	0.254786i	$0.917995\pi$
$678$		0			0
$679$	−1.73879e18	−	4.08065e17i	−0.0261312	−	0.00613254i
$680$		0			0
$681$	−7.41107e19			−1.09107
$682$		0			0
$683$	−9.75354e19			−1.40675			−0.703376	−	0.710818i	$-0.748325\pi$
$683$	−9.75354e19			−1.40675			−0.703376	+	0.710818i	$0.748325\pi$
$684$		0			0
$685$		−	2.52955e18i		−	0.0357445i
$686$		0			0
$687$	4.36112e19			0.603811
$688$		0			0
$689$			3.58560e18i			0.0486438i
$690$		0			0
$691$		−	1.29570e20i		−	1.72250i	−0.508184	−	0.861249i	$-0.669683\pi$
$691$		−	1.29570e20i		−	1.72250i	0.508184	−	0.861249i	$-0.330317\pi$
$692$		0			0
$693$	2.77449e18	−	1.18223e19i	0.0361452	−	0.154017i
$694$		0			0
$695$	−2.12774e19			−0.271660
$696$		0			0
$697$	−1.72261e19			−0.215554
$698$		0			0
$699$		−	8.59778e19i		−	1.05450i
$700$		0			0
$701$	9.68567e19			1.16440			0.582202	−	0.813044i	$-0.302191\pi$
$701$	9.68567e19			1.16440			0.582202	+	0.813044i	$0.302191\pi$
$702$		0			0
$703$		−	1.02129e20i		−	1.20354i
$704$		0			0
$705$		−	5.06703e19i		−	0.585369i
$706$		0			0
$707$	−6.89438e19	−	1.61800e19i	−0.780835	−	0.183249i
$708$		0			0
$709$	−2.73814e19			−0.304041			−0.152021	−	0.988377i	$-0.548578\pi$
$709$	−2.73814e19			−0.304041			−0.152021	+	0.988377i	$0.548578\pi$
$710$		0			0
$711$	−6.12578e19			−0.666921
$712$		0			0
$713$			7.22087e19i			0.770837i
$714$		0			0
$715$	1.56110e19			0.163414
$716$		0			0
$717$			1.87188e19i			0.192151i
$718$		0			0
$719$			1.04566e20i			1.05266i	0.850280	+	0.526331i	$0.176433\pi$
$719$			1.04566e20i			1.05266i	−0.850280	+	0.526331i	$0.823567\pi$
$720$		0			0
$721$	−9.90948e19	−	2.32559e19i	−0.978372	−	0.229607i
$722$		0			0
$723$	2.66100e19			0.257678
$724$		0			0
$725$	−7.59720e18			−0.0721585
$726$		0			0
$727$			1.18622e20i			1.10516i	0.833461	+	0.552579i	$0.186356\pi$
$727$			1.18622e20i			1.10516i	−0.833461	+	0.552579i	$0.813644\pi$
$728$		0			0
$729$	−9.71309e19			−0.887697
$730$		0			0
$731$			5.49064e19i			0.492267i
$732$		0			0
$733$			1.67217e20i			1.47079i	0.677637	+	0.735396i	$0.263004\pi$
$733$			1.67217e20i			1.47079i	−0.677637	+	0.735396i	$0.736996\pi$
$734$		0			0
$735$	3.30597e19	+	1.64215e19i	0.285291	+	0.141711i
$736$		0			0
$737$	−5.41851e19			−0.458784
$738$		0			0
$739$	1.28344e20			1.06627			0.533134	−	0.846031i	$-0.321014\pi$
$739$	1.28344e20			1.06627			0.533134	+	0.846031i	$0.321014\pi$
$740$		0			0
$741$		−	1.19209e20i		−	0.971807i
$742$		0			0
$743$	−9.14458e19			−0.731547			−0.365773	−	0.930704i	$-0.619195\pi$
$743$	−9.14458e19			−0.731547			−0.365773	+	0.930704i	$0.619195\pi$
$744$		0			0
$745$		−	3.30182e19i		−	0.259214i
$746$		0			0
$747$			1.76528e19i			0.136010i
$748$		0			0
$749$	−3.20239e19	+	1.36456e20i	−0.242160	+	1.03186i
$750$		0			0
$751$	−1.34188e19			−0.0995943			−0.0497971	−	0.998759i	$-0.515857\pi$
$751$	−1.34188e19			−0.0995943			−0.0497971	+	0.998759i	$0.515857\pi$
$752$		0			0
$753$	−1.41051e20			−1.02757
$754$		0			0
$755$		−	1.12703e20i		−	0.805945i
$756$		0			0
$757$	−2.42539e20			−1.70259			−0.851295	−	0.524687i	$-0.824182\pi$
$757$	−2.42539e20			−1.70259			−0.851295	+	0.524687i	$0.824182\pi$
$758$		0			0
$759$		−	1.60436e19i		−	0.110563i
$760$		0			0
$761$		−	1.09090e20i		−	0.738063i	−0.929417	−	0.369032i	$-0.879689\pi$
$761$		−	1.09090e20i		−	0.738063i	0.929417	−	0.369032i	$-0.120311\pi$
$762$		0			0
$763$	−2.47831e19	+	1.05603e20i	−0.164621	+	0.701460i
$764$		0			0
$765$	1.17752e19			0.0767956
$766$		0			0
$767$	2.07551e20			1.32910
$768$		0			0
$769$		−	3.36970e19i		−	0.211888i	−0.994372	−	0.105944i	$-0.966214\pi$
$769$		−	3.36970e19i		−	0.211888i	0.994372	−	0.105944i	$-0.0337864\pi$
$770$		0			0
$771$	−1.71442e20			−1.05861
$772$		0			0
$773$		−	6.83999e19i		−	0.414762i	−0.978260	−	0.207381i	$-0.933506\pi$
$773$		−	6.83999e19i		−	0.414762i	0.978260	−	0.207381i	$-0.0664940\pi$
$774$		0			0
$775$			5.35598e19i			0.318954i
$776$		0			0
$777$	−1.19014e20	−	2.79305e19i	−0.696065	−	0.163355i
$778$		0			0
$779$	−1.29079e20			−0.741468
$780$		0			0
$781$	−1.74388e19			−0.0983921
$782$		0			0
$783$		−	6.92155e19i		−	0.383593i
$784$		0			0
$785$	−1.49614e20			−0.814491
$786$		0			0
$787$			6.45669e19i			0.345292i	0.984984	+	0.172646i	$0.0552317\pi$
$787$			6.45669e19i			0.345292i	−0.984984	+	0.172646i	$0.944768\pi$
$788$		0			0
$789$		−	2.05287e20i		−	1.07851i
$790$		0			0
$791$	−5.96665e18	+	2.54243e19i	−0.0307960	+	0.131224i
$792$		0			0
$793$	4.11300e19			0.208567
$794$		0			0
$795$	2.73327e18			0.0136180
$796$		0			0
$797$			1.65536e20i			0.810370i	0.914235	+	0.405185i	$0.132793\pi$
$797$			1.65536e20i			0.810370i	−0.914235	+	0.405185i	$0.867207\pi$
$798$		0			0
$799$	−1.33161e20			−0.640545
$800$		0			0
$801$		−	3.41996e19i		−	0.161656i
$802$		0			0
$803$		−	1.11871e19i		−	0.0519645i
$804$		0			0
$805$	−4.61009e19	−	1.08191e19i	−0.210445	−	0.0493878i
$806$		0			0
$807$	8.48396e19			0.380613
$808$		0			0
$809$	−3.36920e19			−0.148555			−0.0742774	−	0.997238i	$-0.523665\pi$
$809$	−3.36920e19			−0.148555			−0.0742774	+	0.997238i	$0.523665\pi$
$810$		0			0
$811$			2.58082e20i			1.11844i	0.829020	+	0.559219i	$0.188899\pi$
$811$			2.58082e20i			1.11844i	−0.829020	+	0.559219i	$0.811101\pi$
$812$		0			0
$813$	2.17302e20			0.925614
$814$		0			0
$815$		−	6.67868e19i		−	0.279632i
$816$		0			0
$817$			4.11425e20i			1.69331i
$818$		0			0
$819$	1.34885e20	+	3.16551e19i	0.545726	+	0.128073i
$820$		0			0
$821$	−1.67762e20			−0.667256			−0.333628	−	0.942705i	$-0.608273\pi$
$821$	−1.67762e20			−0.667256			−0.333628	+	0.942705i	$0.608273\pi$
$822$		0			0
$823$	4.07513e20			1.59346			0.796732	−	0.604333i	$-0.206560\pi$
$823$	4.07513e20			1.59346			0.796732	+	0.604333i	$0.206560\pi$
$824$		0			0
$825$		−	1.19001e19i		−	0.0457481i
$826$		0			0
$827$	1.12084e20			0.423646			0.211823	−	0.977308i	$-0.432060\pi$
$827$	1.12084e20			0.423646			0.211823	+	0.977308i	$0.432060\pi$
$828$		0			0
$829$			2.86742e20i			1.06564i	0.846229	+	0.532819i	$0.178867\pi$
$829$			2.86742e20i			1.06564i	−0.846229	+	0.532819i	$0.821133\pi$
$830$		0			0
$831$		−	1.28142e20i		−	0.468258i
$832$		0			0
$833$	4.31557e19	−	8.68806e19i	0.155068	−	0.312182i
$834$		0			0
$835$	1.35305e20			0.478088
$836$		0			0
$837$	−4.87966e20			−1.69555
$838$		0			0
$839$			2.39430e20i			0.818172i	0.912496	+	0.409086i	$0.134152\pi$
$839$			2.39430e20i			0.818172i	−0.912496	+	0.409086i	$0.865848\pi$
$840$		0			0
$841$	−2.58825e20			−0.869829
$842$		0			0
$843$		−	1.04307e20i		−	0.344762i
$844$		0			0
$845$			4.05453e19i			0.131808i
$846$		0			0
$847$	6.40848e19	−	2.73070e20i	0.204914	−	0.873151i
$848$		0			0
$849$	6.89810e19			0.216958
$850$		0			0
$851$	1.56821e20			0.485173
$852$		0			0
$853$			9.21328e19i			0.280396i	0.990124	+	0.140198i	$0.0447739\pi$
$853$			9.21328e19i			0.280396i	−0.990124	+	0.140198i	$0.955226\pi$
$854$		0			0
$855$	8.82338e19			0.264163
$856$		0			0
$857$			3.85524e20i			1.13550i	0.823203	+	0.567748i	$0.192185\pi$
$857$			3.85524e20i			1.13550i	−0.823203	+	0.567748i	$0.807815\pi$
$858$		0			0
$859$			7.97159e18i			0.0230990i	0.999933	+	0.0115495i	$0.00367640\pi$
$859$			7.97159e18i			0.0230990i	−0.999933	+	0.0115495i	$0.996324\pi$
$860$		0			0
$861$	−3.53007e19	+	1.50419e20i	−0.100638	+	0.428825i
$862$		0			0
$863$	5.92487e20			1.66189			0.830947	−	0.556352i	$-0.187799\pi$
$863$	5.92487e20			1.66189			0.830947	+	0.556352i	$0.187799\pi$
$864$		0			0
$865$	1.26451e20			0.348988
$866$		0			0
$867$		−	2.30427e20i		−	0.625748i
$868$		0			0
$869$	1.62693e20			0.434742
$870$		0			0
$871$		−	6.18217e20i		−	1.62560i
$872$		0			0
$873$			5.11007e18i			0.0132229i
$874$		0			0
$875$	−3.41948e19	−	8.02493e18i	−0.0870769	−	0.0204355i
$876$		0			0
$877$	6.97422e19			0.174783			0.0873914	−	0.996174i	$-0.472147\pi$
$877$	6.97422e19			0.174783			0.0873914	+	0.996174i	$0.472147\pi$
$878$		0			0
$879$	−4.08272e19			−0.100700
$880$		0			0
$881$		−	5.86686e20i		−	1.42421i	−0.702072	−	0.712106i	$-0.747742\pi$
$881$		−	5.86686e20i		−	1.42421i	0.702072	−	0.712106i	$-0.252258\pi$
$882$		0			0
$883$	−4.99823e20			−1.19424			−0.597119	−	0.802153i	$-0.703688\pi$
$883$	−4.99823e20			−1.19424			−0.597119	+	0.802153i	$0.703688\pi$
$884$		0			0
$885$		−	1.58214e20i		−	0.372084i
$886$		0			0
$887$			1.96472e20i			0.454816i	0.973800	+	0.227408i	$0.0730250\pi$
$887$			1.96472e20i			0.454816i	−0.973800	+	0.227408i	$0.926975\pi$
$888$		0			0
$889$	−2.14256e19	+	9.12960e19i	−0.0488226	+	0.208036i
$890$		0			0
$891$	3.78911e19			0.0849949
$892$		0			0
$893$	−9.97805e20			−2.20336
$894$		0			0
$895$		−	2.30284e20i		−	0.500614i
$896$		0			0
$897$	1.83047e20			0.391755
$898$		0			0
$899$		−	2.73069e20i		−	0.575380i
$900$		0			0
$901$		−	7.18300e18i		−	0.0149016i
$902$		0			0
$903$	4.79444e20	+	1.12517e20i	0.979318	+	0.229829i
$904$		0			0
$905$	−2.43585e19			−0.0489903
$906$		0			0
$907$	9.10389e20			1.80292			0.901459	−	0.432864i	$-0.142497\pi$
$907$	9.10389e20			1.80292			0.901459	+	0.432864i	$0.142497\pi$
$908$		0			0
$909$			2.02617e20i			0.395119i
$910$		0			0
$911$	2.40567e19			0.0461963			0.0230982	−	0.999733i	$-0.492647\pi$
$911$	2.40567e19			0.0461963			0.0230982	+	0.999733i	$0.492647\pi$
$912$		0			0
$913$		−	4.68838e19i		−	0.0886599i
$914$		0			0
$915$		−	3.13529e19i		−	0.0583889i
$916$		0			0
$917$	8.83325e20	+	2.07301e20i	1.62008	+	0.380204i
$918$		0			0
$919$	−8.30894e19			−0.150085			−0.0750425	−	0.997180i	$-0.523909\pi$
$919$	−8.30894e19			−0.150085			−0.0750425	+	0.997180i	$0.523909\pi$
$920$		0			0
$921$	5.31011e20			0.944682
$922$		0			0
$923$		−	1.98966e20i		−	0.348631i
$924$		0			0
$925$	1.16320e20			0.200753
$926$		0			0
$927$			2.91226e20i			0.495077i
$928$		0			0
$929$			4.61272e20i			0.772408i	0.922413	+	0.386204i	$0.126214\pi$
$929$			4.61272e20i			0.772408i	−0.922413	+	0.386204i	$0.873786\pi$
$930$		0			0
$931$	3.23375e20	−	6.51015e20i	0.533407	−	1.07385i
$932$		0			0
$933$	−3.34360e20			−0.543303
$934$		0			0
$935$	−3.12734e19			−0.0500603
$936$		0			0
$937$		−	1.64011e20i		−	0.258639i	−0.991603	−	0.129320i	$-0.958721\pi$
$937$		−	1.64011e20i		−	0.258639i	0.991603	−	0.129320i	$-0.0412793\pi$
$938$		0			0
$939$	4.09717e20			0.636539
$940$		0			0
$941$		−	9.82548e20i		−	1.50393i	−0.659205	−	0.751963i	$-0.729107\pi$
$941$		−	9.82548e20i		−	1.50393i	0.659205	−	0.751963i	$-0.270893\pi$
$942$		0			0
$943$		−	1.98202e20i		−	0.298901i
$944$		0			0
$945$	7.31124e19	−	3.11537e20i	0.108635	−	0.462900i
$946$		0			0
$947$	1.14230e21			1.67236			0.836180	−	0.548455i	$-0.184784\pi$
$947$	1.14230e21			1.67236			0.836180	+	0.548455i	$0.184784\pi$
$948$		0			0
$949$	1.27637e20			0.184125
$950$		0			0
$951$		−	2.53678e20i		−	0.360594i
$952$		0			0
$953$	−4.77204e20			−0.668427			−0.334213	−	0.942497i	$-0.608471\pi$
$953$	−4.77204e20			−0.668427			−0.334213	+	0.942497i	$0.608471\pi$
$954$		0			0
$955$		−	2.20375e20i		−	0.304185i
$956$		0			0
$957$			6.06714e19i			0.0825279i
$958$		0			0
$959$	−1.36227e19	+	5.80474e19i	−0.0182614	+	0.0778130i
$960$		0			0
$961$	−1.16818e21			−1.54328
$962$		0			0
$963$	4.01026e20			0.522142
$964$		0			0
$965$			4.07777e20i			0.523277i
$966$		0			0
$967$	9.42159e19			0.119162			0.0595812	−	0.998223i	$-0.481023\pi$
$967$	9.42159e19			0.119162			0.0595812	+	0.998223i	$0.481023\pi$
$968$		0			0
$969$			2.38809e20i			0.297704i
$970$		0			0
$971$		−	5.91369e20i		−	0.726647i	−0.931663	−	0.363324i	$-0.881642\pi$
$971$		−	5.91369e20i		−	0.726647i	0.931663	−	0.363324i	$-0.118358\pi$
$972$		0			0
$973$	4.88268e20	+	1.14588e20i	0.591382	+	0.138787i
$974$		0			0
$975$	1.35773e20			0.162099
$976$		0			0
$977$	1.48673e21			1.74973			0.874866	−	0.484366i	$-0.160950\pi$
$977$	1.48673e21			1.74973			0.874866	+	0.484366i	$0.160950\pi$
$978$		0			0
$979$			9.08300e19i			0.105378i
$980$		0			0
$981$	3.10352e20			0.354954
$982$		0			0
$983$			1.03572e21i			1.16780i	0.811826	+	0.583899i	$0.198474\pi$
$983$			1.03572e21i			1.16780i	−0.811826	+	0.583899i	$0.801526\pi$
$984$		0			0
$985$		−	6.39401e20i		−	0.710754i
$986$		0			0
$987$	−2.72882e20	+	1.16277e21i	−0.299057	+	1.27430i
$988$		0			0
$989$	−6.31750e20			−0.682608
$990$		0			0
$991$	1.39386e21			1.48492			0.742460	−	0.669891i	$-0.233659\pi$
$991$	1.39386e21			1.48492			0.742460	+	0.669891i	$0.233659\pi$
$992$		0			0
$993$			1.59007e20i			0.167021i
$994$		0			0
$995$	5.03956e19			0.0521952
$996$		0			0
$997$		−	4.59357e20i		−	0.469121i	−0.972102	−	0.234560i	$-0.924635\pi$
$997$		−	4.59357e20i		−	0.469121i	0.972102	−	0.234560i	$-0.0753650\pi$
$998$		0			0
$999$			1.05975e21i			1.06720i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.15.d.a.41.13	✓	36
7.6	odd	2	inner	140.15.d.a.41.24	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.15.d.a.41.13	✓	36	1.1	even	1	trivial
140.15.d.a.41.24	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)

\(f(q)\)	\(=\)	\(q-1557.79i q^{3} -34938.6i q^{5} +(-188159. + 801760. i) q^{7} +2.35627e6 q^{9} +O(q^{10})\)	\(q-1557.79i q^{3} -34938.6i q^{5} +(-188159. + 801760. i) q^{7} +2.35627e6 q^{9} -6.25796e6 q^{11} -7.13992e7i q^{13} -5.44269e7 q^{15} +1.43033e8i q^{17} +1.07178e9i q^{19} +(1.24897e9 + 2.93113e8i) q^{21} -1.64574e9 q^{23} -1.22070e9 q^{25} -1.11214e10i q^{27} +6.22362e9 q^{29} -4.38762e10i q^{31} +9.74857e9i q^{33} +(2.80123e10 + 6.57402e9i) q^{35} -9.52892e10 q^{37} -1.11225e11 q^{39} +1.20434e11i q^{41} +3.83871e11 q^{43} -8.23245e10i q^{45} +9.30979e11i q^{47} +(-6.07415e11 - 3.01718e11i) q^{49} +2.22816e11 q^{51} -5.02191e10 q^{53} +2.18644e11i q^{55} +1.66961e12 q^{57} +2.90690e12i q^{59} +5.76056e11i q^{61} +(-4.43354e11 + 1.88916e12i) q^{63} -2.49459e12 q^{65} +8.65860e12 q^{67} +2.56371e12i q^{69} +2.78666e12 q^{71} +1.78765e12i q^{73} +1.90160e12i q^{75} +(1.17749e12 - 5.01738e12i) q^{77} -2.59978e13 q^{79} -6.05486e12 q^{81} +7.49187e12i q^{83} +4.99738e12 q^{85} -9.69509e12i q^{87} -1.45143e13i q^{89} +(5.72450e13 + 1.34344e13i) q^{91} -6.83499e13 q^{93} +3.74464e13 q^{95} +2.16872e12i q^{97} -1.47454e13 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 1364266 q^{7} - 54790830 q^{9}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more