LMFDB - Embedded newform 140.15.d.a.41.15

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			41.15
Character	$\chi$	$=$	140.41
Dual form			140.15.d.a.41.22

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-770.711i q^{3} +34938.6i q^{5} +(503431. - 651752. i) q^{7} +4.18897e6 q^{9} +O(q^{10})$	\(q-770.711i q^{3} +34938.6i q^{5} +(503431. - 651752. i) q^{7} +4.18897e6 q^{9} -7.83359e6 q^{11} +3.96837e6i q^{13} +2.69275e7 q^{15} -1.72182e8i q^{17} +5.17601e8i q^{19} +(-5.02312e8 - 3.88000e8i) q^{21} +5.03953e9 q^{23} -1.22070e9 q^{25} -6.91477e9i q^{27} +1.44785e10 q^{29} +2.06389e8i q^{31} +6.03743e9i q^{33} +(2.27713e10 + 1.75892e10i) q^{35} +1.08919e11 q^{37} +3.05846e9 q^{39} -1.67850e11i q^{41} -2.73948e11 q^{43} +1.46357e11i q^{45} +5.56659e11i q^{47} +(-1.71337e11 - 6.56224e11i) q^{49} -1.32703e11 q^{51} -1.32580e12 q^{53} -2.73694e11i q^{55} +3.98921e11 q^{57} +7.33926e11i q^{59} +6.19462e10i q^{61} +(2.10886e12 - 2.73017e12i) q^{63} -1.38649e11 q^{65} +4.34717e12 q^{67} -3.88402e12i q^{69} -4.75246e12 q^{71} +1.69903e13i q^{73} +9.40809e11i q^{75} +(-3.94367e12 + 5.10555e12i) q^{77} +2.29814e13 q^{79} +1.47064e13 q^{81} +2.52379e12i q^{83} +6.01580e12 q^{85} -1.11587e13i q^{87} +8.19078e13i q^{89} +(2.58639e12 + 1.99780e12i) q^{91} +1.59067e11 q^{93} -1.80842e13 q^{95} -1.84404e13i q^{97} -3.28147e13 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	770.711i		−	0.352405i	−0.984354	−	0.176203i	$-0.943619\pi$
$3$		−	770.711i		−	0.352405i	0.984354	−	0.176203i	$-0.0563814\pi$
$4$		0			0
$5$			34938.6i			0.447214i
$5$			34938.6i			0.447214i
$6$		0			0
$7$	503431.	−	651752.i	0.611299	−	0.791400i
$7$	503431.	−	651752.i	0.611299	−	0.791400i
$8$		0			0
$9$	4.18897e6			0.875810
$10$		0			0
$11$	−7.83359e6			−0.401987			−0.200993	−	0.979593i	$-0.564417\pi$
$11$	−7.83359e6			−0.401987			−0.200993	+	0.979593i	$0.564417\pi$
$12$		0			0
$13$			3.96837e6i			0.0632424i	0.999500	+	0.0316212i	$0.0100670\pi$
$13$			3.96837e6i			0.0632424i	−0.999500	+	0.0316212i	$0.989933\pi$
$14$		0			0
$15$	2.69275e7			0.157600
$16$		0			0
$17$		−	1.72182e8i		−	0.419610i	−0.977743	−	0.209805i	$-0.932717\pi$
$17$		−	1.72182e8i		−	0.419610i	0.977743	−	0.209805i	$-0.0672829\pi$
$18$		0			0
$19$			5.17601e8i			0.579055i	0.957170	+	0.289528i	$0.0934982\pi$
$19$			5.17601e8i			0.579055i	−0.957170	+	0.289528i	$0.906502\pi$
$20$		0			0
$21$	−5.02312e8	−	3.88000e8i	−0.278893	−	0.215425i
$22$		0			0
$23$	5.03953e9			1.48011			0.740057	−	0.672545i	$-0.234799\pi$
$23$	5.03953e9			1.48011			0.740057	+	0.672545i	$0.234799\pi$
$24$		0			0
$25$	−1.22070e9			−0.200000
$26$		0			0
$27$		−	6.91477e9i		−	0.661046i
$28$		0			0
$29$	1.44785e10			0.839338			0.419669	−	0.907677i	$-0.362146\pi$
$29$	1.44785e10			0.839338			0.419669	+	0.907677i	$0.362146\pi$
$30$		0			0
$31$			2.06389e8i			0.00750163i	0.999993	+	0.00375082i	$0.00119392\pi$
$31$			2.06389e8i			0.00750163i	−0.999993	+	0.00375082i	$0.998806\pi$
$32$		0			0
$33$			6.03743e9i			0.141662i
$34$		0			0
$35$	2.27713e10	+	1.75892e10i	0.353925	+	0.273381i
$36$		0			0
$37$	1.08919e11			1.14734			0.573672	−	0.819085i	$-0.305518\pi$
$37$	1.08919e11			1.14734			0.573672	+	0.819085i	$0.305518\pi$
$38$		0			0
$39$	3.05846e9			0.0222870
$40$		0			0
$41$		−	1.67850e11i		−	0.861857i	−0.902386	−	0.430929i	$-0.858186\pi$
$41$		−	1.67850e11i		−	0.861857i	0.902386	−	0.430929i	$-0.141814\pi$
$42$		0			0
$43$	−2.73948e11			−1.00783			−0.503916	−	0.863753i	$-0.668108\pi$
$43$	−2.73948e11			−1.00783			−0.503916	+	0.863753i	$0.668108\pi$
$44$		0			0
$45$			1.46357e11i			0.391674i
$46$		0			0
$47$			5.56659e11i			1.09876i	0.835572	+	0.549381i	$0.185137\pi$
$47$			5.56659e11i			1.09876i	−0.835572	+	0.549381i	$0.814863\pi$
$48$		0			0
$49$	−1.71337e11	−	6.56224e11i	−0.252626	−	0.967564i
$50$		0			0
$51$	−1.32703e11			−0.147873
$52$		0			0
$53$	−1.32580e12			−1.12862			−0.564309	−	0.825564i	$-0.690857\pi$
$53$	−1.32580e12			−1.12862			−0.564309	+	0.825564i	$0.690857\pi$
$54$		0			0
$55$		−	2.73694e11i		−	0.179774i
$56$		0			0
$57$	3.98921e11			0.204062
$58$		0			0
$59$			7.33926e11i			0.294909i	0.989069	+	0.147454i	$0.0471080\pi$
$59$			7.33926e11i			0.294909i	−0.989069	+	0.147454i	$0.952892\pi$
$60$		0			0
$61$			6.19462e10i			0.0197109i	0.999951	+	0.00985543i	$0.00313713\pi$
$61$			6.19462e10i			0.0197109i	−0.999951	+	0.00985543i	$0.996863\pi$
$62$		0			0
$63$	2.10886e12	−	2.73017e12i	0.535382	−	0.693116i
$64$		0			0
$65$	−1.38649e11			−0.0282829
$66$		0			0
$67$	4.34717e12			0.717270			0.358635	−	0.933478i	$-0.383242\pi$
$67$	4.34717e12			0.717270			0.358635	+	0.933478i	$0.383242\pi$
$68$		0			0
$69$		−	3.88402e12i		−	0.521600i
$70$		0			0
$71$	−4.75246e12			−0.522528			−0.261264	−	0.965267i	$-0.584139\pi$
$71$	−4.75246e12			−0.522528			−0.261264	+	0.965267i	$0.584139\pi$
$72$		0			0
$73$			1.69903e13i			1.53795i	0.639280	+	0.768974i	$0.279232\pi$
$73$			1.69903e13i			1.53795i	−0.639280	+	0.768974i	$0.720768\pi$
$74$		0			0
$75$			9.40809e11i			0.0704811i
$76$		0			0
$77$	−3.94367e12	+	5.10555e12i	−0.245734	+	0.318132i
$78$		0			0
$79$	2.29814e13			1.19671			0.598353	−	0.801232i	$-0.295822\pi$
$79$	2.29814e13			1.19671			0.598353	+	0.801232i	$0.295822\pi$
$80$		0			0
$81$	1.47064e13			0.642854
$82$		0			0
$83$			2.52379e12i			0.0930050i	0.998918	+	0.0465025i	$0.0148075\pi$
$83$			2.52379e12i			0.0930050i	−0.998918	+	0.0465025i	$0.985192\pi$
$84$		0			0
$85$	6.01580e12			0.187655
$86$		0			0
$87$		−	1.11587e13i		−	0.295787i
$88$		0			0
$89$			8.19078e13i			1.85181i	0.377762	+	0.925903i	$0.376694\pi$
$89$			8.19078e13i			1.85181i	−0.377762	+	0.925903i	$0.623306\pi$
$90$		0			0
$91$	2.58639e12	+	1.99780e12i	0.0500500	+	0.0386600i
$92$		0			0
$93$	1.59067e11			0.00264362
$94$		0			0
$95$	−1.80842e13			−0.258961
$96$		0			0
$97$		−	1.84404e13i		−	0.228228i	−0.993468	−	0.114114i	$-0.963597\pi$
$97$		−	1.84404e13i		−	0.228228i	0.993468	−	0.114114i	$-0.0364028\pi$
$98$		0			0
$99$	−3.28147e13			−0.352064
$100$		0			0
$101$		−	8.94673e13i		−	0.834477i	−0.908797	−	0.417239i	$-0.862998\pi$
$101$		−	8.94673e13i		−	0.834477i	0.908797	−	0.417239i	$-0.137002\pi$
$102$		0			0
$103$		−	1.05320e14i		−	0.856348i	−0.903696	−	0.428174i	$-0.859157\pi$
$103$		−	1.05320e14i		−	0.856348i	0.903696	−	0.428174i	$-0.140843\pi$
$104$		0			0
$105$	1.35562e13	−	1.75501e13i	0.0963411	−	0.124725i
$106$		0			0
$107$	5.27584e12			0.0328553			0.0164276	−	0.999865i	$-0.494771\pi$
$107$	5.27584e12			0.0328553			0.0164276	+	0.999865i	$0.494771\pi$
$108$		0			0
$109$	1.57337e14			0.860685			0.430343	−	0.902666i	$-0.358393\pi$
$109$	1.57337e14			0.860685			0.430343	+	0.902666i	$0.358393\pi$
$110$		0			0
$111$		−	8.39454e13i		−	0.404330i
$112$		0			0
$113$	−1.47340e14			−0.626283			−0.313141	−	0.949706i	$-0.601381\pi$
$113$	−1.47340e14			−0.626283			−0.313141	+	0.949706i	$0.601381\pi$
$114$		0			0
$115$			1.76074e14i			0.661927i
$116$		0			0
$117$			1.66234e13i			0.0553884i
$118$		0			0
$119$	−1.12220e14	−	8.66819e13i	−0.332079	−	0.256507i
$120$		0			0
$121$	−3.18385e14			−0.838407
$122$		0			0
$123$	−1.29364e14			−0.303723
$124$		0			0
$125$		−	4.26496e13i		−	0.0894427i
$126$		0			0
$127$	2.79736e14			0.524956			0.262478	−	0.964938i	$-0.415460\pi$
$127$	2.79736e14			0.524956			0.262478	+	0.964938i	$0.415460\pi$
$128$		0			0
$129$			2.11134e14i			0.355166i
$130$		0			0
$131$		−	9.19351e14i		−	1.38862i	−0.719678	−	0.694308i	$-0.755711\pi$
$131$		−	9.19351e14i		−	1.38862i	0.719678	−	0.694308i	$-0.244289\pi$
$132$		0			0
$133$	3.37347e14	+	2.60577e14i	0.458264	+	0.353976i
$134$		0			0
$135$	2.41592e14			0.295629
$136$		0			0
$137$	3.94459e14			0.435469			0.217735	−	0.976008i	$-0.430133\pi$
$137$	3.94459e14			0.435469			0.217735	+	0.976008i	$0.430133\pi$
$138$		0			0
$139$		−	1.60634e14i		−	0.160227i	−0.996786	−	0.0801134i	$-0.974472\pi$
$139$		−	1.60634e14i		−	0.160227i	0.996786	−	0.0801134i	$-0.0255282\pi$
$140$		0			0
$141$	4.29023e14			0.387210
$142$		0			0
$143$		−	3.10866e13i		−	0.0254226i
$144$		0			0
$145$			5.05857e14i			0.375364i
$146$		0			0
$147$	−5.05759e14	+	1.32051e14i	−0.340975	+	0.0890269i
$148$		0			0
$149$	2.30721e15			1.41509			0.707544	−	0.706670i	$-0.249803\pi$
$149$	2.30721e15			1.41509			0.707544	+	0.706670i	$0.249803\pi$
$150$		0			0
$151$	−4.53699e14			−0.253472			−0.126736	−	0.991936i	$-0.540450\pi$
$151$	−4.53699e14			−0.253472			−0.126736	+	0.991936i	$0.540450\pi$
$152$		0			0
$153$		−	7.21267e14i		−	0.367499i
$154$		0			0
$155$	−7.21095e12			−0.00335483
$156$		0			0
$157$		−	1.19728e15i		−	0.509213i	−0.967045	−	0.254606i	$-0.918054\pi$
$157$		−	1.19728e15i		−	0.509213i	0.967045	−	0.254606i	$-0.0819459\pi$
$158$		0			0
$159$			1.02181e15i			0.397731i
$160$		0			0
$161$	2.53706e15	−	3.28452e15i	0.904792	−	1.17136i
$162$		0			0
$163$	−3.50825e15			−1.14757			−0.573783	−	0.819007i	$-0.694525\pi$
$163$	−3.50825e15			−1.14757			−0.573783	+	0.819007i	$0.694525\pi$
$164$		0			0
$165$	−2.10939e14			−0.0633533
$166$		0			0
$167$		−	6.38487e15i		−	1.76253i	−0.472620	−	0.881266i	$-0.656692\pi$
$167$		−	6.38487e15i		−	1.76253i	0.472620	−	0.881266i	$-0.343308\pi$
$168$		0			0
$169$	3.92163e15			0.996000
$170$		0			0
$171$			2.16822e15i			0.507143i
$172$		0			0
$173$			7.79514e15i			1.68074i	0.542012	+	0.840371i	$0.317663\pi$
$173$			7.79514e15i			1.68074i	−0.542012	+	0.840371i	$0.682337\pi$
$174$		0			0
$175$	−6.14540e14	+	7.95595e14i	−0.122260	+	0.158280i
$176$		0			0
$177$	5.65644e14			0.103928
$178$		0			0
$179$	−1.14757e15			−0.194898			−0.0974488	−	0.995241i	$-0.531068\pi$
$179$	−1.14757e15			−0.194898			−0.0974488	+	0.995241i	$0.531068\pi$
$180$		0			0
$181$		−	8.73696e15i		−	1.37281i	−0.727220	−	0.686405i	$-0.759188\pi$
$181$		−	8.73696e15i		−	1.37281i	0.727220	−	0.686405i	$-0.240812\pi$
$182$		0			0
$183$	4.77426e13			0.00694621
$184$		0			0
$185$			3.80549e15i			0.513108i
$186$		0			0
$187$			1.34880e15i			0.168678i
$188$		0			0
$189$	−4.50671e15	−	3.48111e15i	−0.523151	−	0.404097i
$190$		0			0
$191$	1.50850e16			1.62672			0.813361	−	0.581760i	$-0.197636\pi$
$191$	1.50850e16			1.62672			0.813361	+	0.581760i	$0.197636\pi$
$192$		0			0
$193$	1.48517e16			1.48894			0.744468	−	0.667658i	$-0.232703\pi$
$193$	1.48517e16			1.48894			0.744468	+	0.667658i	$0.232703\pi$
$194$		0			0
$195$			1.06858e14i			0.00996703i
$196$		0			0
$197$	5.11863e15			0.444519			0.222259	−	0.974988i	$-0.428657\pi$
$197$	5.11863e15			0.444519			0.222259	+	0.974988i	$0.428657\pi$
$198$		0			0
$199$		−	1.70818e16i		−	1.38217i	−0.722773	−	0.691086i	$-0.757133\pi$
$199$		−	1.70818e16i		−	1.38217i	0.722773	−	0.691086i	$-0.242867\pi$
$200$		0			0
$201$		−	3.35041e15i		−	0.252770i
$202$		0			0
$203$	7.28892e15	−	9.43637e15i	0.513087	−	0.664252i
$204$		0			0
$205$	5.86445e15			0.385434
$206$		0			0
$207$	2.11104e16			1.29630
$208$		0			0
$209$		−	4.05467e15i		−	0.232773i
$210$		0			0
$211$	5.38227e15			0.289060			0.144530	−	0.989500i	$-0.453833\pi$
$211$	5.38227e15			0.289060			0.144530	+	0.989500i	$0.453833\pi$
$212$		0			0
$213$			3.66277e15i			0.184142i
$214$		0			0
$215$		−	9.57133e15i		−	0.450716i
$216$		0			0
$217$	1.34515e14	+	1.03903e14i	0.00593679	+	0.00458574i
$218$		0			0
$219$	1.30946e16			0.541981
$220$		0			0
$221$	6.83282e14			0.0265371
$222$		0			0
$223$			8.54826e15i			0.311705i	0.987780	+	0.155853i	$0.0498125\pi$
$223$			8.54826e15i			0.311705i	−0.987780	+	0.155853i	$0.950188\pi$
$224$		0			0
$225$	−5.11349e15			−0.175162
$226$		0			0
$227$		−	1.68796e16i		−	0.543478i	−0.962371	−	0.271739i	$-0.912401\pi$
$227$		−	1.68796e16i		−	0.543478i	0.962371	−	0.271739i	$-0.0875988\pi$
$228$		0			0
$229$		−	1.90690e16i		−	0.577405i	−0.957419	−	0.288703i	$-0.906776\pi$
$229$		−	1.90690e16i		−	0.577405i	0.957419	−	0.288703i	$-0.0932239\pi$
$230$		0			0
$231$	3.93490e15	+	3.03943e15i	0.112112	+	0.0865981i
$232$		0			0
$233$	3.61787e16			0.970423			0.485212	−	0.874397i	$-0.338743\pi$
$233$	3.61787e16			0.970423			0.485212	+	0.874397i	$0.338743\pi$
$234$		0			0
$235$	−1.94488e16			−0.491382
$236$		0			0
$237$		−	1.77120e16i		−	0.421726i
$238$		0			0
$239$	2.89873e16			0.650763			0.325381	−	0.945583i	$-0.394507\pi$
$239$	2.89873e16			0.650763			0.325381	+	0.945583i	$0.394507\pi$
$240$		0			0
$241$		−	2.43253e16i		−	0.515156i	−0.966257	−	0.257578i	$-0.917076\pi$
$241$		−	2.43253e16i		−	0.515156i	0.966257	−	0.257578i	$-0.0829245\pi$
$242$		0			0
$243$		−	4.44076e16i		−	0.887591i
$244$		0			0
$245$	2.29275e16	−	5.98627e15i	0.432708	−	0.112978i
$246$		0			0
$247$	−2.05403e15			−0.0366209
$248$		0			0
$249$	1.94511e15			0.0327755
$250$		0			0
$251$		−	2.15520e16i		−	0.343377i	−0.985151	−	0.171688i	$-0.945078\pi$
$251$		−	2.15520e16i		−	0.343377i	0.985151	−	0.171688i	$-0.0549222\pi$
$252$		0			0
$253$	−3.94776e16			−0.594986
$254$		0			0
$255$		−	4.63644e15i		−	0.0661308i
$256$		0			0
$257$			3.68776e15i			0.0498002i	0.999690	+	0.0249001i	$0.00792677\pi$
$257$			3.68776e15i			0.0498002i	−0.999690	+	0.0249001i	$0.992073\pi$
$258$		0			0
$259$	5.48335e16	−	7.09885e16i	0.701370	−	0.908007i
$260$		0			0
$261$	6.06500e16			0.735101
$262$		0			0
$263$	1.47755e17			1.69766			0.848831	−	0.528664i	$-0.177307\pi$
$263$	1.47755e17			1.69766			0.848831	+	0.528664i	$0.177307\pi$
$264$		0			0
$265$		−	4.63215e16i		−	0.504733i
$266$		0			0
$267$	6.31272e16			0.652586
$268$		0			0
$269$		−	8.38492e15i		−	0.0822684i	−0.999154	−	0.0411342i	$-0.986903\pi$
$269$		−	8.38492e15i		−	0.0822684i	0.999154	−	0.0411342i	$-0.0130971\pi$
$270$		0			0
$271$		−	1.34235e16i		−	0.125049i	−0.998043	−	0.0625246i	$-0.980085\pi$
$271$		−	1.34235e16i		−	0.125049i	0.998043	−	0.0625246i	$-0.0199152\pi$
$272$		0			0
$273$	1.53973e15	−	1.99336e15i	0.0136240	−	0.0176379i
$274$		0			0
$275$	9.56249e15			0.0803974
$276$		0			0
$277$	5.61996e16			0.449133			0.224566	−	0.974459i	$-0.427903\pi$
$277$	5.61996e16			0.449133			0.224566	+	0.974459i	$0.427903\pi$
$278$		0			0
$279$			8.64560e14i			0.00657001i
$280$		0			0
$281$	1.02490e17			0.740860			0.370430	−	0.928860i	$-0.379210\pi$
$281$	1.02490e17			0.740860			0.370430	+	0.928860i	$0.379210\pi$
$282$		0			0
$283$		−	1.87106e17i		−	1.28701i	−0.765440	−	0.643507i	$-0.777479\pi$
$283$		−	1.87106e17i		−	1.28701i	0.765440	−	0.643507i	$-0.222521\pi$
$284$		0			0
$285$			1.39377e16i			0.0912594i
$286$		0			0
$287$	−1.09397e17	−	8.45011e16i	−0.682073	−	0.526853i
$288$		0			0
$289$	1.38731e17			0.823927
$290$		0			0
$291$	−1.42122e16			−0.0804286
$292$		0			0
$293$		−	1.44228e17i		−	0.777995i	−0.921239	−	0.388997i	$-0.872822\pi$
$293$		−	1.44228e17i		−	0.777995i	0.921239	−	0.388997i	$-0.127178\pi$
$294$		0			0
$295$	−2.56423e16			−0.131887
$296$		0			0
$297$			5.41675e16i			0.265732i
$298$		0			0
$299$			1.99987e16i			0.0936059i
$300$		0			0
$301$	−1.37914e17	+	1.78546e17i	−0.616087	+	0.797598i
$302$		0			0
$303$	−6.89534e16			−0.294074
$304$		0			0
$305$	−2.16431e15			−0.00881496
$306$		0			0
$307$			3.77798e17i			1.46991i	0.678115	+	0.734956i	$0.262797\pi$
$307$			3.77798e17i			1.46991i	−0.678115	+	0.734956i	$0.737203\pi$
$308$		0			0
$309$	−8.11713e16			−0.301782
$310$		0			0
$311$		−	6.70064e15i		−	0.0238119i	−0.999929	−	0.0119059i	$-0.996210\pi$
$311$		−	6.70064e15i		−	0.0238119i	0.999929	−	0.0119059i	$-0.00378987\pi$
$312$		0			0
$313$		−	3.24500e17i		−	1.10257i	−0.834318	−	0.551283i	$-0.814139\pi$
$313$		−	3.24500e17i		−	1.10257i	0.834318	−	0.551283i	$-0.185861\pi$
$314$		0			0
$315$	9.53882e16	+	7.36805e16i	0.309971	+	0.239430i
$316$		0			0
$317$	2.27827e17			0.708257			0.354128	−	0.935197i	$-0.384778\pi$
$317$	2.27827e17			0.708257			0.354128	+	0.935197i	$0.384778\pi$
$318$		0			0
$319$	−1.13418e17			−0.337403
$320$		0			0
$321$		−	4.06615e15i		−	0.0115784i
$322$		0			0
$323$	8.91217e16			0.242977
$324$		0			0
$325$		−	4.84420e15i		−	0.0126485i
$326$		0			0
$327$		−	1.21261e17i		−	0.303310i
$328$		0			0
$329$	3.62803e17	+	2.80239e17i	0.869560	+	0.671673i
$330$		0			0
$331$	−1.18934e17			−0.273218			−0.136609	−	0.990625i	$-0.543620\pi$
$331$	−1.18934e17			−0.273218			−0.136609	+	0.990625i	$0.543620\pi$
$332$		0			0
$333$	4.56261e17			1.00486
$334$		0			0
$335$			1.51884e17i			0.320773i
$336$		0			0
$337$	−4.70737e17			−0.953606			−0.476803	−	0.879010i	$-0.658204\pi$
$337$	−4.70737e17			−0.953606			−0.476803	+	0.879010i	$0.658204\pi$
$338$		0			0
$339$			1.13556e17i			0.220706i
$340$		0			0
$341$		−	1.61677e15i		−	0.00301556i
$342$		0			0
$343$	−5.13952e17	−	2.18694e17i	−0.920160	−	0.391543i
$344$		0			0
$345$	1.35702e17			0.233267
$346$		0			0
$347$	−5.79348e17			−0.956385			−0.478193	−	0.878255i	$-0.658708\pi$
$347$	−5.79348e17			−0.956385			−0.478193	+	0.878255i	$0.658708\pi$
$348$		0			0
$349$			1.60163e17i			0.253971i	0.991905	+	0.126986i	$0.0405302\pi$
$349$			1.60163e17i			0.253971i	−0.991905	+	0.126986i	$0.959470\pi$
$350$		0			0
$351$	2.74404e16			0.0418061
$352$		0			0
$353$			8.57101e17i			1.25490i	0.778657	+	0.627450i	$0.215901\pi$
$353$			8.57101e17i			1.25490i	−0.778657	+	0.627450i	$0.784099\pi$
$354$		0			0
$355$		−	1.66044e17i		−	0.233682i
$356$		0			0
$357$	−6.68067e16	+	8.64892e16i	−0.0903946	+	0.117027i
$358$		0			0
$359$	1.19216e18			1.55121			0.775607	−	0.631216i	$-0.217444\pi$
$359$	1.19216e18			1.55121			0.775607	+	0.631216i	$0.217444\pi$
$360$		0			0
$361$	5.31096e17			0.664695
$362$		0			0
$363$			2.45383e17i			0.295459i
$364$		0			0
$365$	−5.93617e17			−0.687791
$366$		0			0
$367$		−	6.74476e17i		−	0.752150i	−0.926589	−	0.376075i	$-0.877274\pi$
$367$		−	6.74476e17i		−	0.752150i	0.926589	−	0.376075i	$-0.122726\pi$
$368$		0			0
$369$		−	7.03121e17i		−	0.754824i
$370$		0			0
$371$	−6.67449e17	+	8.64092e17i	−0.689923	+	0.893187i
$372$		0			0
$373$	−9.35853e17			−0.931636			−0.465818	−	0.884881i	$-0.654240\pi$
$373$	−9.35853e17			−0.931636			−0.465818	+	0.884881i	$0.654240\pi$
$374$		0			0
$375$	−3.28705e16			−0.0315201
$376$		0			0
$377$			5.74559e16i			0.0530818i
$378$		0			0
$379$	−1.61834e18			−1.44077			−0.720386	−	0.693573i	$-0.756036\pi$
$379$	−1.61834e18			−1.44077			−0.720386	+	0.693573i	$0.756036\pi$
$380$		0			0
$381$		−	2.15596e17i		−	0.184997i
$382$		0			0
$383$			9.40531e17i			0.778004i	0.921237	+	0.389002i	$0.127180\pi$
$383$			9.40531e17i			0.778004i	−0.921237	+	0.389002i	$0.872820\pi$
$384$		0			0
$385$	−1.78381e17	−	1.37786e17i	−0.142273	−	0.109896i
$386$		0			0
$387$	−1.14756e18			−0.882670
$388$		0			0
$389$	−2.11285e18			−1.56755			−0.783777	−	0.621043i	$-0.786709\pi$
$389$	−2.11285e18			−1.56755			−0.783777	+	0.621043i	$0.786709\pi$
$390$		0			0
$391$		−	8.67717e17i		−	0.621070i
$392$		0			0
$393$	−7.08553e17			−0.489356
$394$		0			0
$395$			8.02939e17i			0.535183i
$396$		0			0
$397$			2.69799e18i			1.73583i	0.496715	+	0.867914i	$0.334540\pi$
$397$			2.69799e18i			1.73583i	−0.496715	+	0.867914i	$0.665460\pi$
$398$		0			0
$399$	2.00829e17	−	2.59997e17i	0.124743	−	0.161495i
$400$		0			0
$401$	−1.50143e18			−0.900525			−0.450262	−	0.892896i	$-0.648670\pi$
$401$	−1.50143e18			−0.900525			−0.450262	+	0.892896i	$0.648670\pi$
$402$		0			0
$403$	−8.19029e14			−0.000474421
$404$		0			0
$405$			5.13822e17i			0.287493i
$406$		0			0
$407$	−8.53230e17			−0.461217
$408$		0			0
$409$		−	7.06058e17i		−	0.368788i	−0.982852	−	0.184394i	$-0.940968\pi$
$409$		−	7.06058e17i		−	0.368788i	0.982852	−	0.184394i	$-0.0590323\pi$
$410$		0			0
$411$		−	3.04014e17i		−	0.153462i
$412$		0			0
$413$	4.78337e17	+	3.69481e17i	0.233391	+	0.180278i
$414$		0			0
$415$	−8.81775e16			−0.0415931
$416$		0			0
$417$	−1.23803e17			−0.0564648
$418$		0			0
$419$		−	1.79927e18i		−	0.793595i	−0.917906	−	0.396798i	$-0.870122\pi$
$419$		−	1.79927e18i		−	0.793595i	0.917906	−	0.396798i	$-0.129878\pi$
$420$		0			0
$421$	−1.23577e18			−0.527183			−0.263591	−	0.964634i	$-0.584907\pi$
$421$	−1.23577e18			−0.527183			−0.263591	+	0.964634i	$0.584907\pi$
$422$		0			0
$423$			2.33183e18i			0.962308i
$424$		0			0
$425$			2.10183e17i			0.0839220i
$426$		0			0
$427$	4.03735e16	+	3.11856e16i	0.0155992	+	0.0120492i
$428$		0			0
$429$	−2.39587e16			−0.00895907
$430$		0			0
$431$	3.33735e18			1.20798			0.603991	−	0.796991i	$-0.293576\pi$
$431$	3.33735e18			1.20798			0.603991	+	0.796991i	$0.293576\pi$
$432$		0			0
$433$			2.57057e18i			0.900771i	0.892834	+	0.450386i	$0.148714\pi$
$433$			2.57057e18i			0.900771i	−0.892834	+	0.450386i	$0.851286\pi$
$434$		0			0
$435$	3.89870e17			0.132280
$436$		0			0
$437$			2.60847e18i			0.857068i
$438$		0			0
$439$			5.11066e18i			1.62639i	0.581990	+	0.813196i	$0.302274\pi$
$439$			5.11066e18i			1.62639i	−0.581990	+	0.813196i	$0.697726\pi$
$440$		0			0
$441$	−7.17727e17	−	2.74891e18i	−0.221253	−	0.847403i
$442$		0			0
$443$	−5.15379e17			−0.153922			−0.0769610	−	0.997034i	$-0.524522\pi$
$443$	−5.15379e17			−0.153922			−0.0769610	+	0.997034i	$0.524522\pi$
$444$		0			0
$445$	−2.86174e18			−0.828153
$446$		0			0
$447$		−	1.77819e18i		−	0.498684i
$448$		0			0
$449$	1.68838e18			0.458929			0.229464	−	0.973317i	$-0.426303\pi$
$449$	1.68838e18			0.458929			0.229464	+	0.973317i	$0.426303\pi$
$450$		0			0
$451$			1.31487e18i			0.346455i
$452$		0			0
$453$			3.49671e17i			0.0893248i
$454$		0			0
$455$	−6.98003e16	+	9.03647e16i	−0.0172893	+	0.0223830i
$456$		0			0
$457$	1.28703e17			0.0309155			0.0154577	−	0.999881i	$-0.495079\pi$
$457$	1.28703e17			0.0309155			0.0154577	+	0.999881i	$0.495079\pi$
$458$		0			0
$459$	−1.19060e18			−0.277381
$460$		0			0
$461$			4.82729e18i			1.09093i	0.838134	+	0.545465i	$0.183647\pi$
$461$			4.82729e18i			1.09093i	−0.838134	+	0.545465i	$0.816353\pi$
$462$		0			0
$463$	3.59084e17			0.0787279			0.0393639	−	0.999225i	$-0.487467\pi$
$463$	3.59084e17			0.0787279			0.0393639	+	0.999225i	$0.487467\pi$
$464$		0			0
$465$			5.55756e15i			0.00118226i
$466$		0			0
$467$			7.28496e16i			0.0150387i	0.999972	+	0.00751933i	$0.00239350\pi$
$467$			7.28496e16i			0.0150387i	−0.999972	+	0.00751933i	$0.997606\pi$
$468$		0			0
$469$	2.18850e18	−	2.83327e18i	0.438467	−	0.567647i
$470$		0			0
$471$	−9.22759e17			−0.179449
$472$		0			0
$473$	2.14599e18			0.405135
$474$		0			0
$475$		−	6.31837e17i		−	0.115811i
$476$		0			0
$477$	−5.55374e18			−0.988455
$478$		0			0
$479$		−	7.59843e17i		−	0.131333i	−0.997842	−	0.0656667i	$-0.979083\pi$
$479$		−	7.59843e17i		−	0.131333i	0.997842	−	0.0656667i	$-0.0209174\pi$
$480$		0			0
$481$			4.32233e17i			0.0725608i
$482$		0			0
$483$	−2.53141e18	−	1.95534e18i	−0.412794	−	0.318854i
$484$		0			0
$485$	6.44281e17			0.102066
$486$		0			0
$487$	8.47512e17			0.130450			0.0652249	−	0.997871i	$-0.479224\pi$
$487$	8.47512e17			0.130450			0.0652249	+	0.997871i	$0.479224\pi$
$488$		0			0
$489$			2.70385e18i			0.404409i
$490$		0			0
$491$	−3.90619e18			−0.567783			−0.283892	−	0.958856i	$-0.591626\pi$
$491$	−3.90619e18			−0.567783			−0.283892	+	0.958856i	$0.591626\pi$
$492$		0			0
$493$		−	2.49294e18i		−	0.352195i
$494$		0			0
$495$		−	1.14650e18i		−	0.157448i
$496$		0			0
$497$	−2.39253e18	+	3.09742e18i	−0.319421	+	0.413529i
$498$		0			0
$499$	−3.42827e18			−0.445011			−0.222505	−	0.974931i	$-0.571424\pi$
$499$	−3.42827e18			−0.445011			−0.222505	+	0.974931i	$0.571424\pi$
$500$		0			0
$501$	−4.92089e18			−0.621126
$502$		0			0
$503$		−	6.57144e18i		−	0.806649i	−0.915057	−	0.403324i	$-0.867855\pi$
$503$		−	6.57144e18i		−	0.806649i	0.915057	−	0.403324i	$-0.132145\pi$
$504$		0			0
$505$	3.12586e18			0.373190
$506$		0			0
$507$		−	3.02244e18i		−	0.350996i
$508$		0			0
$509$		−	5.67295e18i		−	0.640892i	−0.947267	−	0.320446i	$-0.896167\pi$
$509$		−	5.67295e18i		−	0.640892i	0.947267	−	0.320446i	$-0.103833\pi$
$510$		0			0
$511$	1.10735e19	+	8.55346e18i	1.21713	+	0.940146i
$512$		0			0
$513$	3.57909e18			0.382782
$514$		0			0
$515$	3.67973e18			0.382971
$516$		0			0
$517$		−	4.36063e18i		−	0.441688i
$518$		0			0
$519$	6.00779e18			0.592303
$520$		0			0
$521$		−	1.40399e19i		−	1.34741i	−0.739001	−	0.673704i	$-0.764702\pi$
$521$		−	1.40399e19i		−	1.34741i	0.739001	−	0.673704i	$-0.235298\pi$
$522$		0			0
$523$		−	1.44081e19i		−	1.34615i	−0.739574	−	0.673075i	$-0.764973\pi$
$523$		−	1.44081e19i		−	1.34615i	0.739574	−	0.673075i	$-0.235027\pi$
$524$		0			0
$525$	6.13174e17	+	4.73633e17i	0.0557787	+	0.0430850i
$526$		0			0
$527$	3.55366e16			0.00314776
$528$		0			0
$529$	1.38040e19			1.19074
$530$		0			0
$531$			3.07440e18i			0.258284i
$532$		0			0
$533$	6.66092e17			0.0545059
$534$		0			0
$535$			1.84330e17i			0.0146933i
$536$		0			0
$537$			8.84442e17i			0.0686830i
$538$		0			0
$539$	1.34218e18	+	5.14059e18i	0.101553	+	0.388948i
$540$		0			0
$541$	7.21925e18			0.532244			0.266122	−	0.963939i	$-0.414258\pi$
$541$	7.21925e18			0.532244			0.266122	+	0.963939i	$0.414258\pi$
$542$		0			0
$543$	−6.73367e18			−0.483785
$544$		0			0
$545$			5.49712e18i			0.384910i
$546$		0			0
$547$	3.83853e18			0.261971			0.130985	−	0.991384i	$-0.458186\pi$
$547$	3.83853e18			0.261971			0.130985	+	0.991384i	$0.458186\pi$
$548$		0			0
$549$			2.59491e17i			0.0172630i
$550$		0			0
$551$			7.49408e18i			0.486023i
$552$		0			0
$553$	1.15696e19	−	1.49782e19i	0.731546	−	0.947073i
$554$		0			0
$555$	2.93293e18			0.180822
$556$		0			0
$557$	2.06352e19			1.24058			0.620288	−	0.784374i	$-0.287016\pi$
$557$	2.06352e19			1.24058			0.620288	+	0.784374i	$0.287016\pi$
$558$		0			0
$559$		−	1.08712e18i		−	0.0637377i
$560$		0			0
$561$	1.03954e18			0.0594430
$562$		0			0
$563$		−	1.28174e18i		−	0.0714891i	−0.999361	−	0.0357445i	$-0.988620\pi$
$563$		−	1.28174e18i		−	0.0714891i	0.999361	−	0.0357445i	$-0.0113803\pi$
$564$		0			0
$565$		−	5.14784e18i		−	0.280082i
$566$		0			0
$567$	7.40368e18	−	9.58495e18i	0.392976	−	0.508755i
$568$		0			0
$569$	−1.78558e19			−0.924683			−0.462341	−	0.886702i	$-0.652990\pi$
$569$	−1.78558e19			−0.924683			−0.462341	+	0.886702i	$0.652990\pi$
$570$		0			0
$571$	1.37130e19			0.692915			0.346458	−	0.938066i	$-0.387384\pi$
$571$	1.37130e19			0.692915			0.346458	+	0.938066i	$0.387384\pi$
$572$		0			0
$573$		−	1.16262e19i		−	0.573265i
$574$		0			0
$575$	−6.15177e18			−0.296023
$576$		0			0
$577$			1.50914e19i			0.708760i	0.935102	+	0.354380i	$0.115308\pi$
$577$			1.50914e19i			0.708760i	−0.935102	+	0.354380i	$0.884692\pi$
$578$		0			0
$579$		−	1.14464e19i		−	0.524709i
$580$		0			0
$581$	1.64488e18	+	1.27055e18i	0.0736041	+	0.0568539i
$582$		0			0
$583$	1.03858e19			0.453689
$584$		0			0
$585$	−5.80797e17			−0.0247704
$586$		0			0
$587$		−	2.32291e19i		−	0.967309i	−0.875259	−	0.483654i	$-0.839309\pi$
$587$		−	2.32291e19i		−	0.967309i	0.875259	−	0.483654i	$-0.160691\pi$
$588$		0			0
$589$	−1.06827e17			−0.00434386
$590$		0			0
$591$		−	3.94498e18i		−	0.156651i
$592$		0			0
$593$			9.84602e18i			0.381837i	0.981606	+	0.190918i	$0.0611466\pi$
$593$			9.84602e18i			0.381837i	−0.981606	+	0.190918i	$0.938853\pi$
$594$		0			0
$595$	3.02854e18	−	3.92081e18i	0.114714	−	0.148510i
$596$		0			0
$597$	−1.31651e19			−0.487085
$598$		0			0
$599$	3.42308e19			1.23717			0.618585	−	0.785718i	$-0.287706\pi$
$599$	3.42308e19			1.23717			0.618585	+	0.785718i	$0.287706\pi$
$600$		0			0
$601$			2.95416e18i			0.104307i	0.998639	+	0.0521534i	$0.0166085\pi$
$601$			2.95416e18i			0.104307i	−0.998639	+	0.0521534i	$0.983392\pi$
$602$		0			0
$603$	1.82102e19			0.628193
$604$		0			0
$605$		−	1.11239e19i		−	0.374947i
$606$		0			0
$607$		−	2.94882e19i		−	0.971243i	−0.874169	−	0.485621i	$-0.838593\pi$
$607$		−	2.94882e19i		−	0.971243i	0.874169	−	0.485621i	$-0.161407\pi$
$608$		0			0
$609$	−7.27271e18	−	5.61765e18i	−0.234086	−	0.180815i
$610$		0			0
$611$	−2.20903e18			−0.0694884
$612$		0			0
$613$	−3.74093e19			−1.15015			−0.575077	−	0.818099i	$-0.695028\pi$
$613$	−3.74093e19			−1.15015			−0.575077	+	0.818099i	$0.695028\pi$
$614$		0			0
$615$		−	4.51980e18i		−	0.135829i
$616$		0			0
$617$	7.81743e18			0.229651			0.114825	−	0.993386i	$-0.463369\pi$
$617$	7.81743e18			0.229651			0.114825	+	0.993386i	$0.463369\pi$
$618$		0			0
$619$			4.66814e18i			0.134063i	0.997751	+	0.0670315i	$0.0213528\pi$
$619$			4.66814e18i			0.134063i	−0.997751	+	0.0670315i	$0.978647\pi$
$620$		0			0
$621$		−	3.48472e19i		−	0.978423i
$622$		0			0
$623$	5.33836e19	+	4.12350e19i	1.46552	+	1.13201i
$624$		0			0
$625$	1.49012e18			0.0400000
$626$		0			0
$627$	−3.12498e18			−0.0820304
$628$		0			0
$629$		−	1.87540e19i		−	0.481437i
$630$		0			0
$631$	3.08861e19			0.775455			0.387728	−	0.921774i	$-0.373260\pi$
$631$	3.08861e19			0.775455			0.387728	+	0.921774i	$0.373260\pi$
$632$		0			0
$633$		−	4.14817e18i		−	0.101866i
$634$		0			0
$635$			9.77359e18i			0.234768i
$636$		0			0
$637$	2.60414e18	−	6.79929e17i	0.0611911	−	0.0159767i
$638$		0			0
$639$	−1.99079e19			−0.457636
$640$		0			0
$641$	−8.19200e18			−0.184240			−0.0921200	−	0.995748i	$-0.529364\pi$
$641$	−8.19200e18			−0.184240			−0.0921200	+	0.995748i	$0.529364\pi$
$642$		0			0
$643$			3.52480e19i			0.775635i	0.921736	+	0.387818i	$0.126771\pi$
$643$			3.52480e19i			0.775635i	−0.921736	+	0.387818i	$0.873229\pi$
$644$		0			0
$645$	−7.37673e18			−0.158835
$646$		0			0
$647$		−	5.89543e19i		−	1.24218i	−0.783738	−	0.621091i	$-0.786690\pi$
$647$		−	5.89543e19i		−	1.24218i	0.783738	−	0.621091i	$-0.213310\pi$
$648$		0			0
$649$		−	5.74927e18i		−	0.118550i
$650$		0			0
$651$	8.00791e16	−	1.03672e17i	0.00161604	−	0.00209216i
$652$		0			0
$653$	1.55419e19			0.306981			0.153490	−	0.988150i	$-0.450949\pi$
$653$	1.55419e19			0.306981			0.153490	+	0.988150i	$0.450949\pi$
$654$		0			0
$655$	3.21208e19			0.621008
$656$		0			0
$657$			7.11720e19i			1.34695i
$658$		0			0
$659$	−3.92727e19			−0.727600			−0.363800	−	0.931477i	$-0.618521\pi$
$659$	−3.92727e19			−0.727600			−0.363800	+	0.931477i	$0.618521\pi$
$660$		0			0
$661$			6.73688e19i			1.22194i	0.791655	+	0.610968i	$0.209219\pi$
$661$			6.73688e19i			1.22194i	−0.791655	+	0.610968i	$0.790781\pi$
$662$		0			0
$663$		−	5.26613e17i		−	0.00935183i
$664$		0			0
$665$	−9.10417e18	+	1.17864e19i	−0.158303	+	0.204942i
$666$		0			0
$667$	7.29647e19			1.24232
$668$		0			0
$669$	6.58824e18			0.109847
$670$		0			0
$671$		−	4.85261e17i		−	0.00792351i
$672$		0			0
$673$	−6.62284e19			−1.05910			−0.529552	−	0.848277i	$-0.677640\pi$
$673$	−6.62284e19			−1.05910			−0.529552	+	0.848277i	$0.677640\pi$
$674$		0			0
$675$			8.44088e18i			0.132209i
$676$		0			0
$677$			1.49946e19i			0.230046i	0.993363	+	0.115023i	$0.0366942\pi$
$677$			1.49946e19i			0.230046i	−0.993363	+	0.115023i	$0.963306\pi$
$678$		0			0
$679$	−1.20186e19	−	9.28347e18i	−0.180619	−	0.139515i
$680$		0			0
$681$	−1.30093e19			−0.191525
$682$		0			0
$683$	−4.70196e19			−0.678163			−0.339082	−	0.940757i	$-0.610116\pi$
$683$	−4.70196e19			−0.678163			−0.339082	+	0.940757i	$0.610116\pi$
$684$		0			0
$685$			1.37818e19i			0.194748i
$686$		0			0
$687$	−1.46967e19			−0.203481
$688$		0			0
$689$		−	5.26126e18i		−	0.0713765i
$690$		0			0
$691$		−	7.97635e18i		−	0.106037i	−0.998594	−	0.0530185i	$-0.983116\pi$
$691$		−	7.97635e18i		−	0.106037i	0.998594	−	0.0530185i	$-0.0168842\pi$
$692$		0			0
$693$	−1.65199e19	+	2.13870e19i	−0.215217	+	0.278624i
$694$		0			0
$695$	5.61234e18			0.0716556
$696$		0			0
$697$	−2.89009e19			−0.361644
$698$		0			0
$699$		−	2.78833e19i		−	0.341982i
$700$		0			0
$701$	2.20815e19			0.265462			0.132731	−	0.991152i	$-0.457625\pi$
$701$	2.20815e19			0.265462			0.132731	+	0.991152i	$0.457625\pi$
$702$		0			0
$703$			5.63769e19i			0.664376i
$704$		0			0
$705$			1.49894e19i			0.173166i
$706$		0			0
$707$	−5.83104e19	−	4.50406e19i	−0.660405	−	0.510115i
$708$		0			0
$709$	−1.11064e20			−1.23325			−0.616623	−	0.787258i	$-0.711500\pi$
$709$	−1.11064e20			−1.23325			−0.616623	+	0.787258i	$0.711500\pi$
$710$		0			0
$711$	9.62687e19			1.04809
$712$		0			0
$713$			1.04011e18i			0.0111033i
$714$		0			0
$715$	1.08612e18			0.0113693
$716$		0			0
$717$		−	2.23408e19i		−	0.229332i
$718$		0			0
$719$			1.02019e20i			1.02702i	0.858084	+	0.513509i	$0.171655\pi$
$719$			1.02019e20i			1.02702i	−0.858084	+	0.513509i	$0.828345\pi$
$720$		0			0
$721$	−6.86425e19	−	5.30214e19i	−0.677714	−	0.523485i
$722$		0			0
$723$	−1.87478e19			−0.181544
$724$		0			0
$725$	−1.76739e19			−0.167868
$726$		0			0
$727$		−	1.49560e20i		−	1.39340i	−0.717363	−	0.696700i	$-0.754651\pi$
$727$		−	1.49560e20i		−	1.39340i	0.717363	−	0.696700i	$-0.245349\pi$
$728$		0			0
$729$	3.61151e19			0.330062
$730$		0			0
$731$			4.71689e19i			0.422897i
$732$		0			0
$733$			1.32451e20i			1.16500i	0.812830	+	0.582502i	$0.197926\pi$
$733$			1.32451e20i			1.16500i	−0.812830	+	0.582502i	$0.802074\pi$
$734$		0			0
$735$	−4.61368e18	−	1.76705e19i	−0.0398141	−	0.152489i
$736$		0			0
$737$	−3.40539e19			−0.288333
$738$		0			0
$739$	−1.44541e20			−1.20083			−0.600415	−	0.799689i	$-0.704998\pi$
$739$	−1.44541e20			−1.20083			−0.600415	+	0.799689i	$0.704998\pi$
$740$		0			0
$741$			1.58306e18i			0.0129054i
$742$		0			0
$743$	−1.69119e20			−1.35291			−0.676457	−	0.736483i	$-0.736485\pi$
$743$	−1.69119e20			−1.35291			−0.676457	+	0.736483i	$0.736485\pi$
$744$		0			0
$745$			8.06106e19i			0.632846i
$746$		0			0
$747$			1.05721e19i			0.0814547i
$748$		0			0
$749$	2.65602e18	−	3.43854e18i	0.0200844	−	0.0260017i
$750$		0			0
$751$	−1.10839e20			−0.822645			−0.411323	−	0.911490i	$-0.634933\pi$
$751$	−1.10839e20			−0.822645			−0.411323	+	0.911490i	$0.634933\pi$
$752$		0			0
$753$	−1.66103e19			−0.121008
$754$		0			0
$755$		−	1.58516e19i		−	0.113356i
$756$		0			0
$757$	−2.15332e20			−1.51160			−0.755800	−	0.654803i	$-0.772752\pi$
$757$	−2.15332e20			−1.51160			−0.755800	+	0.654803i	$0.772752\pi$
$758$		0			0
$759$			3.04258e19i			0.209676i
$760$		0			0
$761$			2.33971e20i			1.58296i	0.611195	+	0.791480i	$0.290689\pi$
$761$			2.33971e20i			1.58296i	−0.611195	+	0.791480i	$0.709311\pi$
$762$		0			0
$763$	7.92082e19	−	1.02544e20i	0.526136	−	0.681146i
$764$		0			0
$765$	2.52000e19			0.164350
$766$		0			0
$767$	−2.91249e18			−0.0186507
$768$		0			0
$769$			2.83984e20i			1.78571i	0.450348	+	0.892853i	$0.351300\pi$
$769$			2.83984e20i			1.78571i	−0.450348	+	0.892853i	$0.648700\pi$
$770$		0			0
$771$	2.84220e18			0.0175499
$772$		0			0
$773$		−	1.26794e20i		−	0.768854i	−0.923155	−	0.384427i	$-0.874399\pi$
$773$		−	1.26794e20i		−	0.768854i	0.923155	−	0.384427i	$-0.125601\pi$
$774$		0			0
$775$		−	2.51940e17i		−	0.00150033i
$776$		0			0
$777$	−5.47116e19	−	4.22607e19i	−0.319987	−	0.247167i
$778$		0			0
$779$	8.68796e19			0.499063
$780$		0			0
$781$	3.72288e19			0.210049
$782$		0			0
$783$		−	1.00115e20i		−	0.554841i
$784$		0			0
$785$	4.18314e19			0.227727
$786$		0			0
$787$			4.92876e19i			0.263581i	0.991278	+	0.131791i	$0.0420727\pi$
$787$			4.92876e19i			0.263581i	−0.991278	+	0.131791i	$0.957927\pi$
$788$		0			0
$789$		−	1.13876e20i		−	0.598265i
$790$		0			0
$791$	−7.41754e19	+	9.60289e19i	−0.382846	+	0.495640i
$792$		0			0
$793$	−2.45825e17			−0.00124656
$794$		0			0
$795$	−3.57005e19			−0.177871
$796$		0			0
$797$			1.62628e20i			0.796135i	0.917356	+	0.398067i	$0.130319\pi$
$797$			1.62628e20i			0.796135i	−0.917356	+	0.398067i	$0.869681\pi$
$798$		0			0
$799$	9.58467e19			0.461052
$800$		0			0
$801$			3.43110e20i			1.62183i
$802$		0			0
$803$		−	1.33095e20i		−	0.618235i
$804$		0			0
$805$	1.14756e20	+	8.86411e19i	0.523849	+	0.404635i
$806$		0			0
$807$	−6.46235e18			−0.0289918
$808$		0			0
$809$	−1.23057e20			−0.542584			−0.271292	−	0.962497i	$-0.587451\pi$
$809$	−1.23057e20			−0.542584			−0.271292	+	0.962497i	$0.587451\pi$
$810$		0			0
$811$			8.81628e19i			0.382067i	0.981584	+	0.191033i	$0.0611839\pi$
$811$			8.81628e19i			0.382067i	−0.981584	+	0.191033i	$0.938816\pi$
$812$		0			0
$813$	−1.03456e19			−0.0440680
$814$		0			0
$815$		−	1.22573e20i		−	0.513207i
$816$		0			0
$817$		−	1.41796e20i		−	0.583591i
$818$		0			0
$819$	1.08343e19	+	8.36873e18i	0.0438343	+	0.0338589i
$820$		0			0
$821$	1.16965e20			0.465214			0.232607	−	0.972571i	$-0.425274\pi$
$821$	1.16965e20			0.465214			0.232607	+	0.972571i	$0.425274\pi$
$822$		0			0
$823$	−1.25853e20			−0.492113			−0.246056	−	0.969256i	$-0.579135\pi$
$823$	−1.25853e20			−0.492113			−0.246056	+	0.969256i	$0.579135\pi$
$824$		0			0
$825$		−	7.36991e18i		−	0.0283325i
$826$		0			0
$827$	6.34716e19			0.239906			0.119953	−	0.992780i	$-0.461726\pi$
$827$	6.34716e19			0.239906			0.119953	+	0.992780i	$0.461726\pi$
$828$		0			0
$829$		−	1.01675e20i		−	0.377862i	−0.981990	−	0.188931i	$-0.939498\pi$
$829$		−	1.01675e20i		−	0.377862i	0.981990	−	0.188931i	$-0.0605022\pi$
$830$		0			0
$831$		−	4.33136e19i		−	0.158277i
$832$		0			0
$833$	−1.12990e20	+	2.95012e19i	−0.406000	+	0.106005i
$834$		0			0
$835$	2.23078e20			0.788228
$836$		0			0
$837$	1.42714e18			0.00495892
$838$		0			0
$839$		−	2.03208e20i		−	0.694396i	−0.937792	−	0.347198i	$-0.887133\pi$
$839$		−	2.03208e20i		−	0.694396i	0.937792	−	0.347198i	$-0.112867\pi$
$840$		0			0
$841$	−8.79318e19			−0.295511
$842$		0			0
$843$		−	7.89899e19i		−	0.261083i
$844$		0			0
$845$			1.37016e20i			0.445425i
$846$		0			0
$847$	−1.60285e20	+	2.07508e20i	−0.512517	+	0.663515i
$848$		0			0
$849$	−1.44205e20			−0.453551
$850$		0			0
$851$	5.48903e20			1.69820
$852$		0			0
$853$			7.46861e19i			0.227298i	0.993521	+	0.113649i	$0.0362540\pi$
$853$			7.46861e19i			0.227298i	−0.993521	+	0.113649i	$0.963746\pi$
$854$		0			0
$855$	−7.57544e19			−0.226801
$856$		0			0
$857$			2.08583e20i			0.614345i	0.951654	+	0.307173i	$0.0993829\pi$
$857$			2.08583e20i			0.614345i	−0.951654	+	0.307173i	$0.900617\pi$
$858$		0			0
$859$		−	1.65183e20i		−	0.478644i	−0.970940	−	0.239322i	$-0.923075\pi$
$859$		−	1.65183e20i		−	0.478644i	0.970940	−	0.239322i	$-0.0769252\pi$
$860$		0			0
$861$	−6.51259e19	+	8.43132e19i	−0.185666	+	0.240366i
$862$		0			0
$863$	2.97803e20			0.835320			0.417660	−	0.908603i	$-0.362850\pi$
$863$	2.97803e20			0.835320			0.417660	+	0.908603i	$0.362850\pi$
$864$		0			0
$865$	−2.72351e20			−0.751651
$866$		0			0
$867$		−	1.06922e20i		−	0.290356i
$868$		0			0
$869$	−1.80027e20			−0.481060
$870$		0			0
$871$			1.72512e19i			0.0453619i
$872$		0			0
$873$		−	7.72463e19i		−	0.199884i
$874$		0			0
$875$	−2.77970e19	−	2.14711e19i	−0.0707849	−	0.0546763i
$876$		0			0
$877$	−4.10284e20			−1.02822			−0.514112	−	0.857723i	$-0.671878\pi$
$877$	−4.10284e20			−1.02822			−0.514112	+	0.857723i	$0.671878\pi$
$878$		0			0
$879$	−1.11158e20			−0.274170
$880$		0			0
$881$			3.66861e20i			0.890574i	0.895388	+	0.445287i	$0.146898\pi$
$881$			3.66861e20i			0.890574i	−0.895388	+	0.445287i	$0.853102\pi$
$882$		0			0
$883$	7.85178e20			1.87604			0.938022	−	0.346575i	$-0.112655\pi$
$883$	7.85178e20			1.87604			0.938022	+	0.346575i	$0.112655\pi$
$884$		0			0
$885$			1.97628e19i			0.0464778i
$886$		0			0
$887$		−	8.18383e20i		−	1.89449i	−0.320518	−	0.947243i	$-0.603857\pi$
$887$		−	8.18383e20i		−	1.89449i	0.320518	−	0.947243i	$-0.396143\pi$
$888$		0			0
$889$	1.40828e20	−	1.82319e20i	0.320905	−	0.415450i
$890$		0			0
$891$	−1.15204e20			−0.258419
$892$		0			0
$893$	−2.88127e20			−0.636244
$894$		0			0
$895$		−	4.00943e19i		−	0.0871609i
$896$		0			0
$897$	1.54132e19			0.0329872
$898$		0			0
$899$			2.98821e18i			0.00629641i
$900$		0			0
$901$			2.28279e20i			0.473579i
$902$		0			0
$903$	1.37607e20	+	1.06292e20i	0.281078	+	0.217112i
$904$		0			0
$905$	3.05257e20			0.613939
$906$		0			0
$907$	6.02113e20			1.19241			0.596207	−	0.802831i	$-0.296674\pi$
$907$	6.02113e20			1.19241			0.596207	+	0.802831i	$0.296674\pi$
$908$		0			0
$909$		−	3.74776e20i		−	0.730844i
$910$		0			0
$911$	8.43054e20			1.61892			0.809462	−	0.587172i	$-0.199759\pi$
$911$	8.43054e20			1.61892			0.809462	+	0.587172i	$0.199759\pi$
$912$		0			0
$913$		−	1.97703e19i		−	0.0373868i
$914$		0			0
$915$			1.66806e18i			0.00310644i
$916$		0			0
$917$	−5.99188e20	−	4.62830e20i	−1.09895	−	0.848860i
$918$		0			0
$919$	8.57474e20			1.54886			0.774431	−	0.632659i	$-0.218036\pi$
$919$	8.57474e20			1.54886			0.774431	+	0.632659i	$0.218036\pi$
$920$		0			0
$921$	2.91173e20			0.518005
$922$		0			0
$923$		−	1.88595e19i		−	0.0330459i
$924$		0			0
$925$	−1.32958e20			−0.229469
$926$		0			0
$927$		−	4.41183e20i		−	0.749999i
$928$		0			0
$929$		−	5.63120e20i		−	0.942955i	−0.881878	−	0.471478i	$-0.843721\pi$
$929$		−	5.63120e20i		−	0.942955i	0.881878	−	0.471478i	$-0.156279\pi$
$930$		0			0
$931$	3.39662e20	−	8.86843e19i	0.560273	−	0.146285i
$932$		0			0
$933$	−5.16426e18			−0.00839144
$934$		0			0
$935$	−4.71253e19			−0.0754350
$936$		0			0
$937$		−	4.10251e20i		−	0.646952i	−0.946236	−	0.323476i	$-0.895148\pi$
$937$		−	4.10251e20i		−	0.646952i	0.946236	−	0.323476i	$-0.104852\pi$
$938$		0			0
$939$	−2.50096e20			−0.388550
$940$		0			0
$941$		−	1.20591e20i		−	0.184582i	−0.995732	−	0.0922908i	$-0.970581\pi$
$941$		−	1.20591e20i		−	0.184582i	0.995732	−	0.0922908i	$-0.0294189\pi$
$942$		0			0
$943$		−	8.45887e20i		−	1.27565i
$944$		0			0
$945$	1.21625e20	−	1.57458e20i	0.180718	−	0.233960i
$946$		0			0
$947$	1.88960e20			0.276643			0.138321	−	0.990387i	$-0.455829\pi$
$947$	1.88960e20			0.276643			0.138321	+	0.990387i	$0.455829\pi$
$948$		0			0
$949$	−6.74238e19			−0.0972635
$950$		0			0
$951$		−	1.75589e20i		−	0.249594i
$952$		0			0
$953$	1.23585e20			0.173107			0.0865534	−	0.996247i	$-0.472415\pi$
$953$	1.23585e20			0.173107			0.0865534	+	0.996247i	$0.472415\pi$
$954$		0			0
$955$			5.27050e20i			0.727492i
$956$		0			0
$957$			8.74128e19i			0.118903i
$958$		0			0
$959$	1.98583e20	−	2.57089e20i	0.266202	−	0.344630i
$960$		0			0
$961$	7.56901e20			0.999944
$962$		0			0
$963$	2.21004e19			0.0287750
$964$		0			0
$965$			5.18898e20i			0.665872i
$966$		0			0
$967$	−4.14274e20			−0.523965			−0.261983	−	0.965073i	$-0.584376\pi$
$967$	−4.14274e20			−0.523965			−0.261983	+	0.965073i	$0.584376\pi$
$968$		0			0
$969$		−	6.86871e19i		−	0.0856266i
$970$		0			0
$971$			3.90832e20i			0.480237i	0.970744	+	0.240118i	$0.0771863\pi$
$971$			3.90832e20i			0.480237i	−0.970744	+	0.240118i	$0.922814\pi$
$972$		0			0
$973$	−1.04694e20	−	8.08684e19i	−0.126803	−	0.0979465i
$974$		0			0
$975$	−3.73347e18			−0.00445739
$976$		0			0
$977$	−1.00915e21			−1.18766			−0.593830	−	0.804591i	$-0.702385\pi$
$977$	−1.00915e21			−1.18766			−0.593830	+	0.804591i	$0.702385\pi$
$978$		0			0
$979$		−	6.41632e20i		−	0.744402i
$980$		0			0
$981$	6.59079e20			0.753797
$982$		0			0
$983$			1.10350e21i			1.24422i	0.782929	+	0.622111i	$0.213725\pi$
$983$			1.10350e21i			1.24422i	−0.782929	+	0.622111i	$0.786275\pi$
$984$		0			0
$985$			1.78838e20i			0.198795i
$986$		0			0
$987$	2.15983e20	−	2.79616e20i	0.236701	−	0.306438i
$988$		0			0
$989$	−1.38057e21			−1.49171
$990$		0			0
$991$	8.09884e20			0.862794			0.431397	−	0.902162i	$-0.358021\pi$
$991$	8.09884e20			0.862794			0.431397	+	0.902162i	$0.358021\pi$
$992$		0			0
$993$			9.16635e19i			0.0962834i
$994$		0			0
$995$	5.96813e20			0.618126
$996$		0			0
$997$		−	1.20008e21i		−	1.22558i	−0.790244	−	0.612792i	$-0.790046\pi$
$997$		−	1.20008e21i		−	1.22558i	0.790244	−	0.612792i	$-0.209954\pi$
$998$		0			0
$999$		−	7.53153e20i		−	0.758447i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.15.d.a.41.15	✓	36
7.6	odd	2	inner	140.15.d.a.41.22	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.15.d.a.41.15	✓	36	1.1	even	1	trivial
140.15.d.a.41.22	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)

\(f(q)\)	\(=\)	\(q-770.711i q^{3} +34938.6i q^{5} +(503431. - 651752. i) q^{7} +4.18897e6 q^{9} +O(q^{10})\)	\(q-770.711i q^{3} +34938.6i q^{5} +(503431. - 651752. i) q^{7} +4.18897e6 q^{9} -7.83359e6 q^{11} +3.96837e6i q^{13} +2.69275e7 q^{15} -1.72182e8i q^{17} +5.17601e8i q^{19} +(-5.02312e8 - 3.88000e8i) q^{21} +5.03953e9 q^{23} -1.22070e9 q^{25} -6.91477e9i q^{27} +1.44785e10 q^{29} +2.06389e8i q^{31} +6.03743e9i q^{33} +(2.27713e10 + 1.75892e10i) q^{35} +1.08919e11 q^{37} +3.05846e9 q^{39} -1.67850e11i q^{41} -2.73948e11 q^{43} +1.46357e11i q^{45} +5.56659e11i q^{47} +(-1.71337e11 - 6.56224e11i) q^{49} -1.32703e11 q^{51} -1.32580e12 q^{53} -2.73694e11i q^{55} +3.98921e11 q^{57} +7.33926e11i q^{59} +6.19462e10i q^{61} +(2.10886e12 - 2.73017e12i) q^{63} -1.38649e11 q^{65} +4.34717e12 q^{67} -3.88402e12i q^{69} -4.75246e12 q^{71} +1.69903e13i q^{73} +9.40809e11i q^{75} +(-3.94367e12 + 5.10555e12i) q^{77} +2.29814e13 q^{79} +1.47064e13 q^{81} +2.52379e12i q^{83} +6.01580e12 q^{85} -1.11587e13i q^{87} +8.19078e13i q^{89} +(2.58639e12 + 1.99780e12i) q^{91} +1.59067e11 q^{93} -1.80842e13 q^{95} -1.84404e13i q^{97} -3.28147e13 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 1364266 q^{7} - 54790830 q^{9}+O(q^{10}) \) 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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