LMFDB - Embedded newform 60.8.h.b.59.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [60,8,Mod(59,60)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(60, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("60.59");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 60 = 2^{2} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	60.h (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:	$18.7431015290$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 2x^{2} + x + 1 $ x^4 - x^3 + 2*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			59.1
Root			$-0.309017 - 0.535233i$ of defining polynomial
Character	$\chi$	$=$	60.59
Dual form			60.8.h.b.59.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.11803 - 11.2583i) q^{2} +46.7654i q^{3} +(-125.500 + 25.1744i) q^{4} -279.508 q^{5} +(526.500 - 52.2853i) q^{6} +(423.735 + 1384.77i) q^{8} -2187.00 q^{9} +O(q^{10})$	\(q+(-1.11803 - 11.2583i) q^{2} +46.7654i q^{3} +(-125.500 + 25.1744i) q^{4} -279.508 q^{5} +(526.500 - 52.2853i) q^{6} +(423.735 + 1384.77i) q^{8} -2187.00 q^{9} +(312.500 + 3146.80i) q^{10} +(-1177.29 - 5869.05i) q^{12} -13071.3i q^{15} +(15116.5 - 6318.77i) q^{16} +30987.4 q^{17} +(2445.14 + 24622.0i) q^{18} -58660.2i q^{19} +(35078.3 - 7036.46i) q^{20} +61498.2i q^{23} +(-64759.5 + 19816.1i) q^{24} +78125.0 q^{25} -102276. i q^{27} +(-147161. + 14614.2i) q^{30} -259513. i q^{31} +(-88039.6 - 163122. i) q^{32} +(-34645.0 - 348867. i) q^{34} +(274468. - 55056.4i) q^{36} +(-660416. + 65584.1i) q^{38} +(-118438. - 387056. i) q^{40} +611285. q^{45} +(692367. - 68757.1i) q^{46} +368958. i q^{47} +(295500. + 706929. i) q^{48} -823543. q^{49} +(-87346.4 - 879557. i) q^{50} +1.44914e6i q^{51} +1.77105e6 q^{53} +(-1.15146e6 + 114348. i) q^{54} +2.74327e6 q^{57} +(329062. + 1.64045e6i) q^{60} +2.77452e6 q^{61} +(-2.92168e6 + 290144. i) q^{62} +(-1.73805e6 + 1.17355e6i) q^{64} +(-3.88892e6 + 780090. i) q^{68} -2.87599e6 q^{69} +(-926708. - 3.02850e6i) q^{72} +3.65354e6i q^{75} +(1.47674e6 + 7.36186e6i) q^{76} -126778. i q^{79} +(-4.22519e6 + 1.76615e6i) q^{80} +4.78297e6 q^{81} -1.01397e7i q^{83} -8.66125e6 q^{85} +(-683438. - 6.88205e6i) q^{90} +(-1.54818e6 - 7.71802e6i) q^{92} +1.21362e7 q^{93} +(4.15385e6 - 412508. i) q^{94} +1.63960e7i q^{95} +(7.62846e6 - 4.11720e6i) q^{96} +(920749. + 9.27172e6i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 502 q^{4} + 2106 q^{6} - 8748 q^{9}+O(q^{10}) $ 4 * q - 502 * q^4 + 2106 * q^6 - 8748 * q^9	$ 4 q - 502 q^{4} + 2106 q^{6} - 8748 q^{9} + 1250 q^{10} + 60466 q^{16} - 259038 q^{24} + 312500 q^{25} - 138580 q^{34} + 1097874 q^{36} - 473750 q^{40} + 2769468 q^{46} - 3294172 q^{49} - 4605822 q^{54} + 1316250 q^{60} + 11098072 q^{61} - 6952198 q^{64} - 11503944 q^{69} + 5906940 q^{76} + 19131876 q^{81} - 34645000 q^{85} - 2733750 q^{90} + 16615404 q^{94} + 30513834 q^{96}+O(q^{100}) $ 4 * q - 502 * q^4 + 2106 * q^6 - 8748 * q^9 + 1250 * q^10 + 60466 * q^16 - 259038 * q^24 + 312500 * q^25 - 138580 * q^34 + 1097874 * q^36 - 473750 * q^40 + 2769468 * q^46 - 3294172 * q^49 - 4605822 * q^54 + 1316250 * q^60 + 11098072 * q^61 - 6952198 * q^64 - 11503944 * q^69 + 5906940 * q^76 + 19131876 * q^81 - 34645000 * q^85 - 2733750 * q^90 + 16615404 * q^94 + 30513834 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/60\mathbb{Z}\right)^\times$.

$n$	$31$	$37$	$41$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	−1.11803	−	11.2583i	−0.0988212	−	0.995105i
$2$	−1.11803	−	11.2583i	−0.0988212	−	0.995105i
$3$			46.7654i			1.00000i
$3$			46.7654i			1.00000i
$4$	−125.500	+	25.1744i	−0.980469	+	0.196675i
$5$	−279.508			−1.00000
$5$	−279.508			−1.00000
$6$	526.500	−	52.2853i	0.995105	−	0.0988212i
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$	423.735	+	1384.77i	0.292603	+	0.956234i
$9$	−2187.00			−1.00000
$10$	312.500	+	3146.80i	0.0988212	+	0.995105i
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	−1177.29	−	5869.05i	−0.196675	−	0.980469i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		−	13071.3i		−	1.00000i
$16$	15116.5	−	6318.77i	0.922638	−	0.385667i
$17$	30987.4			1.52973			0.764864	−	0.644192i	$-0.222806\pi$
$17$	30987.4			1.52973			0.764864	+	0.644192i	$0.222806\pi$
$18$	2445.14	+	24622.0i	0.0988212	+	0.995105i
$19$		−	58660.2i		−	1.96203i	−0.193928	−	0.981016i	$-0.562123\pi$
$19$		−	58660.2i		−	1.96203i	0.193928	−	0.981016i	$-0.437877\pi$
$20$	35078.3	−	7036.46i	0.980469	−	0.196675i
$21$		0			0
$22$		0			0
$23$			61498.2i			1.05394i	0.849885	+	0.526969i	$0.176672\pi$
$23$			61498.2i			1.05394i	−0.849885	+	0.526969i	$0.823328\pi$
$24$	−64759.5	+	19816.1i	−0.956234	+	0.292603i
$25$	78125.0			1.00000
$26$		0			0
$27$		−	102276.i		−	1.00000i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$	−147161.	+	14614.2i	−0.995105	+	0.0988212i
$31$		−	259513.i		−	1.56456i	−0.622924	−	0.782282i	$-0.714056\pi$
$31$		−	259513.i		−	1.56456i	0.622924	−	0.782282i	$-0.285944\pi$
$32$	−88039.6	−	163122.i	−0.474956	−	0.880010i
$33$		0			0
$34$	−34645.0	−	348867.i	−0.151170	−	1.52224i
$35$		0			0
$36$	274468.	−	55056.4i	0.980469	−	0.196675i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	−660416.	+	65584.1i	−1.95243	+	0.193890i
$39$		0			0
$40$	−118438.	−	387056.i	−0.292603	−	0.956234i
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$	611285.			1.00000
$46$	692367.	−	68757.1i	1.04878	−	0.104151i
$47$			368958.i			0.518364i	0.965829	+	0.259182i	$0.0834529\pi$
$47$			368958.i			0.518364i	−0.965829	+	0.259182i	$0.916547\pi$
$48$	295500.	+	706929.i	0.385667	+	0.922638i
$49$	−823543.			−1.00000
$50$	−87346.4	−	879557.i	−0.0988212	−	0.995105i
$51$			1.44914e6i			1.52973i
$52$		0			0
$53$	1.77105e6			1.63405			0.817025	−	0.576602i	$-0.195622\pi$
$53$	1.77105e6			1.63405			0.817025	+	0.576602i	$0.195622\pi$
$54$	−1.15146e6	+	114348.i	−0.995105	+	0.0988212i
$55$		0			0
$56$		0			0
$57$	2.74327e6			1.96203
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$	329062.	+	1.64045e6i	0.196675	+	0.980469i
$61$	2.77452e6			1.56507			0.782534	−	0.622608i	$-0.213927\pi$
$61$	2.77452e6			1.56507			0.782534	+	0.622608i	$0.213927\pi$
$62$	−2.92168e6	+	290144.i	−1.55691	+	0.154612i
$63$		0			0
$64$	−1.73805e6	+	1.17355e6i	−0.828767	+	0.559594i
$65$		0			0
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$	−3.88892e6	+	780090.i	−1.49985	+	0.300859i
$69$	−2.87599e6			−1.05394
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	−926708.	−	3.02850e6i	−0.292603	−	0.956234i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$			3.65354e6i			1.00000i
$76$	1.47674e6	+	7.36186e6i	0.385882	+	1.92371i
$77$		0			0
$78$		0			0
$79$		−	126778.i		−	0.0289301i	−0.999895	−	0.0144650i	$-0.995395\pi$
$79$		−	126778.i		−	0.0289301i	0.999895	−	0.0144650i	$-0.00460453\pi$
$80$	−4.22519e6	+	1.76615e6i	−0.922638	+	0.385667i
$81$	4.78297e6			1.00000
$82$		0			0
$83$		−	1.01397e7i		−	1.94650i	−0.229754	−	0.973249i	$-0.573792\pi$
$83$		−	1.01397e7i		−	1.94650i	0.229754	−	0.973249i	$-0.426208\pi$
$84$		0			0
$85$	−8.66125e6			−1.52973
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	−683438.	−	6.88205e6i	−0.0988212	−	0.995105i
$91$		0			0
$92$	−1.54818e6	−	7.71802e6i	−0.207283	−	1.03335i
$93$	1.21362e7			1.56456
$94$	4.15385e6	−	412508.i	0.515826	−	0.0512253i
$95$			1.63960e7i			1.96203i
$96$	7.62846e6	−	4.11720e6i	0.880010	−	0.474956i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	920749.	+	9.27172e6i	0.0988212	+	0.995105i
$99$		0			0
$100$	−9.80469e6	+	1.96675e6i	−0.980469	+	0.196675i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$	1.63149e7	−	1.62019e6i	1.52224	−	0.151170i
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$	−1.98010e6	−	1.99391e7i	−0.161479	−	1.62605i
$107$		−	1.30835e7i		−	1.03248i	−0.856444	−	0.516240i	$-0.827331\pi$
$107$		−	1.30835e7i		−	1.03248i	0.856444	−	0.516240i	$-0.172669\pi$
$108$	2.57473e6	+	1.28356e7i	0.196675	+	0.980469i
$109$	2.45614e7			1.81661			0.908304	−	0.418311i	$-0.137378\pi$
$109$	2.45614e7			1.81661			0.908304	+	0.418311i	$0.137378\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−3.05597e7			−1.99239			−0.996196	−	0.0871436i	$-0.972226\pi$
$113$	−3.05597e7			−1.99239			−0.996196	+	0.0871436i	$0.972226\pi$
$114$	−3.06706e6	−	3.08846e7i	−0.193890	−	1.95243i
$115$		−	1.71893e7i		−	1.05394i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$	1.81008e7	−	5.53877e6i	0.956234	−	0.292603i
$121$	−1.94872e7			−1.00000
$122$	−3.10201e6	−	3.12364e7i	−0.154662	−	1.55741i
$123$		0			0
$124$	6.53309e6	+	3.25689e7i	0.307711	+	1.53401i
$125$	−2.18366e7			−1.00000
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	1.51555e7	+	1.82555e7i	0.638755	+	0.769410i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$			2.85870e7i			1.00000i
$136$	1.31305e7	+	4.29106e7i	0.447604	+	1.46278i
$137$	−3.05552e7			−1.01523			−0.507614	−	0.861585i	$-0.669472\pi$
$137$	−3.05552e7			−1.01523			−0.507614	+	0.861585i	$0.669472\pi$
$138$	3.21545e6	+	3.23788e7i	0.104151	+	1.04878i
$139$		−	2.33454e7i		−	0.737310i	−0.929566	−	0.368655i	$-0.879818\pi$
$139$		−	2.33454e7i		−	0.737310i	0.929566	−	0.368655i	$-0.120182\pi$
$140$		0			0
$141$	−1.72545e7			−0.518364
$142$		0			0
$143$		0			0
$144$	−3.30598e7	+	1.38192e7i	−0.922638	+	0.385667i
$145$		0			0
$146$		0			0
$147$		−	3.85133e7i		−	1.00000i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$	4.11328e7	−	4.08479e6i	0.995105	−	0.0988212i
$151$		−	5.76834e7i		−	1.36343i	−0.731620	−	0.681713i	$-0.761235\pi$
$151$		−	5.76834e7i		−	1.36343i	0.731620	−	0.681713i	$-0.238765\pi$
$152$	8.12312e7	−	2.48564e7i	1.87616	−	0.574097i
$153$	−6.77695e7			−1.52973
$154$		0			0
$155$			7.25361e7i			1.56456i
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$	−1.42731e6	+	141742.i	−0.0287885	+	0.00285891i
$159$			8.28238e7i			1.63405i
$160$	2.46078e7	+	4.55940e7i	0.474956	+	0.880010i
$161$		0			0
$162$	−5.34752e6	−	5.38482e7i	−0.0988212	−	0.995105i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		0			0
$165$		0			0
$166$	−1.14157e8	+	1.13366e7i	−1.93697	+	0.192355i
$167$		−	9.45035e7i		−	1.57015i	−0.619403	−	0.785073i	$-0.712625\pi$
$167$		−	9.45035e7i		−	1.57015i	0.619403	−	0.785073i	$-0.287375\pi$
$168$		0			0
$169$	6.27485e7			1.00000
$170$	9.68357e6	+	9.75112e7i	0.151170	+	1.52224i
$171$			1.28290e8i			1.96203i
$172$		0			0
$173$	1.00422e8			1.47458			0.737290	−	0.675576i	$-0.236105\pi$
$173$	1.00422e8			1.47458			0.737290	+	0.675576i	$0.236105\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	−7.67163e7	+	1.53887e7i	−0.980469	+	0.196675i
$181$	−6.47262e7			−0.811344			−0.405672	−	0.914019i	$-0.632962\pi$
$181$	−6.47262e7			−0.811344			−0.405672	+	0.914019i	$0.632962\pi$
$182$		0			0
$183$			1.29751e8i			1.56507i
$184$	−8.51611e7	+	2.60589e7i	−1.00781	+	0.308386i
$185$		0			0
$186$	−1.35687e7	−	1.36634e8i	−0.154612	−	1.55691i
$187$		0			0
$188$	−9.28829e6	−	4.63042e7i	−0.101949	−	0.508239i
$189$		0			0
$190$	1.84592e8	−	1.83313e7i	1.95243	−	0.193890i
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	−5.48817e7	−	8.12805e7i	−0.559594	−	0.828767i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$	1.03355e8	−	2.07322e7i	0.980469	−	0.196675i
$197$	1.93173e8			1.80017			0.900085	−	0.435714i	$-0.143504\pi$
$197$	1.93173e8			1.80017			0.900085	+	0.435714i	$0.143504\pi$
$198$		0			0
$199$		−	1.04929e8i		−	0.943866i	−0.881634	−	0.471933i	$-0.843556\pi$
$199$		−	1.04929e8i		−	0.943866i	0.881634	−	0.471933i	$-0.156444\pi$
$200$	3.31043e7	+	1.08186e8i	0.292603	+	0.956234i
$201$		0			0
$202$		0			0
$203$		0			0
$204$	−3.64812e7	−	1.81867e8i	−0.300859	−	1.49985i
$205$		0			0
$206$		0			0
$207$		−	1.34497e8i		−	1.05394i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		−	2.59188e8i		−	1.89945i	−0.313091	−	0.949723i	$-0.601365\pi$
$211$		−	2.59188e8i		−	1.89945i	0.313091	−	0.949723i	$-0.398635\pi$
$212$	−2.22267e8	+	4.45851e7i	−1.60214	+	0.321377i
$213$		0			0
$214$	−1.47299e8	+	1.46278e7i	−1.02743	+	0.102031i
$215$		0			0
$216$	1.41629e8	−	4.33379e7i	0.956234	−	0.292603i
$217$		0			0
$218$	−2.74605e7	−	2.76521e8i	−0.179519	−	1.80772i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	−1.70859e8			−1.00000
$226$	3.41668e7	+	3.44051e8i	0.196890	+	1.98264i
$227$			9.17231e7i			0.520461i	0.965547	+	0.260231i	$0.0837986\pi$
$227$			9.17231e7i			0.520461i	−0.965547	+	0.260231i	$0.916201\pi$
$228$	−3.44280e8	+	6.90601e7i	−1.92371	+	0.385882i
$229$	2.95829e8			1.62786			0.813930	−	0.580963i	$-0.197324\pi$
$229$	2.95829e8			1.62786			0.813930	+	0.580963i	$0.197324\pi$
$230$	−1.93522e8	+	1.92182e7i	−1.04878	+	0.104151i
$231$		0			0
$232$		0			0
$233$	1.95156e8			1.01073			0.505366	−	0.862905i	$-0.331357\pi$
$233$	1.95156e8			1.01073			0.505366	+	0.862905i	$0.331357\pi$
$234$		0			0
$235$		−	1.03127e8i		−	0.518364i
$236$		0			0
$237$	5.92883e6			0.0289301
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$	−8.25947e7	−	1.97593e8i	−0.385667	−	0.922638i
$241$	−1.89514e8			−0.872131			−0.436066	−	0.899915i	$-0.643628\pi$
$241$	−1.89514e8			−0.872131			−0.436066	+	0.899915i	$0.643628\pi$
$242$	2.17873e7	+	2.19393e8i	0.0988212	+	0.995105i
$243$			2.23677e8i			1.00000i
$244$	−3.48202e8	+	6.98468e7i	−1.53450	+	0.307810i
$245$	2.30187e8			1.00000
$246$		0			0
$247$		0			0
$248$	3.59367e8	−	1.09965e8i	1.49609	−	0.457797i
$249$	4.74189e8			1.94650
$250$	2.44141e7	+	2.45844e8i	0.0988212	+	0.995105i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		−	4.05047e8i		−	1.52973i
$256$	1.88582e8	−	1.91035e8i	0.702522	−	0.711662i
$257$	−2.51231e7			−0.0923226			−0.0461613	−	0.998934i	$-0.514699\pi$
$257$	−2.51231e7			−0.0923226			−0.0461613	+	0.998934i	$0.514699\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$			2.21533e8i			0.750919i	0.926839	+	0.375460i	$0.122515\pi$
$263$			2.21533e8i			0.750919i	−0.926839	+	0.375460i	$0.877485\pi$
$264$		0			0
$265$	−4.95024e8			−1.63405
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$	3.21842e8	−	3.19612e7i	0.995105	−	0.0988212i
$271$			6.52523e8i			1.99160i	0.0915282	+	0.995802i	$0.470825\pi$
$271$			6.52523e8i			1.99160i	−0.0915282	+	0.995802i	$0.529175\pi$
$272$	4.68421e8	−	1.95803e8i	1.41139	−	0.589966i
$273$		0			0
$274$	3.41618e7	+	3.44001e8i	0.100326	+	1.01026i
$275$		0			0
$276$	3.60936e8	−	7.24012e7i	1.03335	−	0.207283i
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$	−2.62830e8	+	2.61010e7i	−0.733701	+	0.0728618i
$279$			5.67555e8i			1.56456i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$	1.92911e7	+	1.94256e8i	0.0512253	+	0.515826i
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$		0			0
$285$	−7.66766e8			−1.96203
$286$		0			0
$287$		0			0
$288$	1.92543e8	+	3.56748e8i	0.474956	+	0.880010i
$289$	5.49882e8			1.34007
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−8.39487e8			−1.94974			−0.974871	−	0.222770i	$-0.928490\pi$
$293$	−8.39487e8			−1.94974			−0.974871	+	0.222770i	$0.928490\pi$
$294$	−4.33595e8	+	4.30592e7i	−0.995105	+	0.0988212i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−9.19758e7	−	4.58520e8i	−0.196675	−	0.980469i
$301$		0			0
$302$	−6.49418e8	+	6.44920e7i	−1.35675	+	0.134735i
$303$		0			0
$304$	−3.70660e8	−	8.86737e8i	−0.756691	−	1.81024i
$305$	−7.75501e8			−1.56507
$306$	7.57686e7	+	7.62972e8i	0.151170	+	1.52224i
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$		0			0
$310$	8.16636e8	−	8.10979e7i	1.55691	−	0.154612i
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$	3.19156e6	+	1.59107e7i	0.00568982	+	0.0283651i
$317$	−1.13342e9			−1.99840			−0.999199	−	0.0400185i	$-0.987258\pi$
$317$	−1.13342e9			−1.99840			−0.999199	+	0.0400185i	$0.987258\pi$
$318$	9.32458e8	−	9.25999e7i	1.62605	−	0.161479i
$319$		0			0
$320$	4.85800e8	−	3.28018e8i	0.828767	−	0.559594i
$321$	6.11856e8			1.03248
$322$		0			0
$323$		−	1.81773e9i		−	3.00138i
$324$	−6.00263e8	+	1.20408e8i	−0.980469	+	0.196675i
$325$		0			0
$326$		0			0
$327$			1.14863e9i			1.81661i
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	1.30725e9i		−	1.98135i	−0.136250	−	0.990675i	$-0.543505\pi$
$331$		−	1.30725e9i		−	1.98135i	0.136250	−	0.990675i	$-0.456495\pi$
$332$	2.55262e8	+	1.27254e9i	0.382827	+	1.90848i
$333$		0			0
$334$	−1.06395e9	+	1.05658e8i	−1.56246	+	0.155164i
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	−7.01550e7	−	7.06444e8i	−0.0988212	−	0.995105i
$339$		−	1.42914e9i		−	1.99239i
$340$	1.08699e9	−	2.18042e8i	1.49985	−	0.300859i
$341$		0			0
$342$	1.44433e9	−	1.43432e8i	1.95243	−	0.193890i
$343$		0			0
$344$		0			0
$345$	8.03863e8			1.05394
$346$	−1.12275e8	−	1.13059e9i	−0.145720	−	1.46736i
$347$			1.42881e9i			1.83578i	0.396833	+	0.917891i	$0.370109\pi$
$347$			1.42881e9i			1.83578i	−0.396833	+	0.917891i	$0.629891\pi$
$348$		0			0
$349$	1.57069e9			1.97789			0.988946	−	0.148275i	$-0.0473720\pi$
$349$	1.57069e9			1.97789			0.988946	+	0.148275i	$0.0473720\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	9.46341e8			1.14508			0.572541	−	0.819876i	$-0.305958\pi$
$353$	9.46341e8			1.14508			0.572541	+	0.819876i	$0.305958\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	2.59023e8	+	8.46492e8i	0.292603	+	0.956234i
$361$	−2.54715e9			−2.84957
$362$	7.23661e7	+	7.28709e8i	0.0801780	+	0.807372i
$363$		−	9.11325e8i		−	1.00000i
$364$		0			0
$365$		0			0
$366$	1.46078e9	−	1.45066e8i	1.55741	−	0.154662i
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$	3.88593e8	+	9.29637e8i	0.406469	+	0.972403i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−1.52310e9	+	3.05522e8i	−1.53401	+	0.307711i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		−	1.02120e9i		−	1.00000i
$376$	−5.10924e8	+	1.56340e8i	−0.495677	+	0.151675i
$377$		0			0
$378$		0			0
$379$			1.59301e9i			1.50308i	0.659687	+	0.751540i	$0.270689\pi$
$379$			1.59301e9i			1.50308i	−0.659687	+	0.751540i	$0.729311\pi$
$380$	−4.12760e8	−	2.05770e9i	−0.385882	−	1.92371i
$381$		0			0
$382$		0			0
$383$		−	9.46849e7i		−	0.0861162i	−0.999073	−	0.0430581i	$-0.986290\pi$
$383$		−	9.46849e7i		−	0.0861162i	0.999073	−	0.0430581i	$-0.0137101\pi$
$384$	−8.53723e8	+	7.08751e8i	−0.769410	+	0.638755i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$			1.90567e9i			1.61224i
$392$	−3.48964e8	−	1.14042e9i	−0.292603	−	0.956234i
$393$		0			0
$394$	−2.15973e8	−	2.17480e9i	−0.177895	−	1.79136i
$395$			3.54356e7i			0.0289301i
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	−1.18133e9	+	1.17314e8i	−0.939246	+	0.0932740i
$399$		0			0
$400$	1.18098e9	−	4.93654e8i	0.922638	−	0.385667i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−1.33688e9			−1.00000
$406$		0			0
$407$		0			0
$408$	−2.00673e9	+	6.14051e8i	−1.46278	+	0.447604i
$409$	2.72520e9			1.96955			0.984774	−	0.173837i	$-0.0556167\pi$
$409$	2.72520e9			1.96955			0.984774	+	0.173837i	$0.0556167\pi$
$410$		0			0
$411$		−	1.42893e9i		−	1.01523i
$412$		0			0
$413$		0			0
$414$	−1.51421e9	+	1.50372e8i	−1.04878	+	0.104151i
$415$			2.83415e9i			1.94650i
$416$		0			0
$417$	1.09176e9			0.737310
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−2.78101e9			−1.81642			−0.908209	−	0.418517i	$-0.862550\pi$
$421$	−2.78101e9			−1.81642			−0.908209	+	0.418517i	$0.862550\pi$
$422$	−2.91803e9	+	2.89781e8i	−1.89015	+	0.187705i
$423$		−	8.06911e8i		−	0.518364i
$424$	7.50456e8	+	2.45251e9i	0.478129	+	1.56253i
$425$	2.42089e9			1.52973
$426$		0			0
$427$		0			0
$428$	3.29370e8	+	1.64198e9i	0.203063	+	1.01231i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	−6.46258e8	−	1.54605e9i	−0.385667	−	0.922638i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$	−3.08246e9	+	6.18319e8i	−1.78113	+	0.357281i
$437$	3.60750e9			2.06786
$438$		0			0
$439$			3.26650e9i			1.84271i	0.388724	+	0.921354i	$0.372916\pi$
$439$			3.26650e9i			1.84271i	−0.388724	+	0.921354i	$0.627084\pi$
$440$		0			0
$441$	1.80109e9			1.00000
$442$		0			0
$443$			3.65092e9i			1.99521i	0.0691508	+	0.997606i	$0.477971\pi$
$443$			3.65092e9i			1.99521i	−0.0691508	+	0.997606i	$0.522029\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	1.91027e8	+	1.92359e9i	0.0988212	+	0.995105i
$451$		0			0
$452$	3.83524e9	−	7.69322e8i	1.95348	−	0.391853i
$453$	2.69758e9			1.36343
$454$	1.03265e9	−	1.02550e8i	0.517914	−	0.0514326i
$455$		0			0
$456$	1.16242e9	+	3.79881e9i	0.574097	+	1.87616i
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$	−3.30747e8	−	3.33054e9i	−0.160867	−	1.61989i
$459$		−	3.16927e9i		−	1.52973i
$460$	4.32729e8	+	2.15725e9i	0.207283	+	1.03335i
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$	−3.39218e9			−1.56456
$466$	−2.18191e8	−	2.19713e9i	−0.0998818	−	1.00578i
$467$		−	3.76912e9i		−	1.71250i	−0.516559	−	0.856251i	$-0.672787\pi$
$467$		−	3.76912e9i		−	1.71250i	0.516559	−	0.856251i	$-0.327213\pi$
$468$		0			0
$469$		0			0
$470$	−1.16104e9	+	1.15299e8i	−0.515826	+	0.0512253i
$471$		0			0
$472$		0			0
$473$		0			0
$474$	−6.62863e6	−	6.67487e7i	−0.00285891	−	0.0287885i
$475$		−	4.58283e9i		−	1.96203i
$476$		0			0
$477$	−3.87329e9			−1.63405
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$	−2.13222e9	+	1.15079e9i	−0.880010	+	0.474956i
$481$		0			0
$482$	2.11883e8	+	2.13361e9i	0.0861850	+	0.867862i
$483$		0			0
$484$	2.44564e9	−	4.90578e8i	0.980469	−	0.196675i
$485$		0			0
$486$	2.51823e9	−	2.50079e8i	0.995105	−	0.0988212i
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$	1.17566e9	+	3.84208e9i	0.457944	+	1.49657i
$489$		0			0
$490$	−2.57357e8	−	2.59152e9i	−0.0988212	−	0.995105i
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−1.63980e9	−	3.92293e9i	−0.603401	−	1.44353i
$497$		0			0
$498$	−5.30159e8	−	5.33858e9i	−0.192355	−	1.93697i
$499$		−	5.21271e9i		−	1.87807i	−0.343820	−	0.939035i	$-0.611721\pi$
$499$		−	5.21271e9i		−	1.87807i	0.343820	−	0.939035i	$-0.388279\pi$
$500$	2.74049e9	−	5.49723e8i	0.980469	−	0.196675i
$501$	4.41949e9			1.57015
$502$		0			0
$503$			2.94048e9i			1.03022i	0.857123	+	0.515111i	$0.172249\pi$
$503$			2.94048e9i			1.03022i	−0.857123	+	0.515111i	$0.827751\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			2.93446e9i			1.00000i
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$	−4.56015e9	+	4.52856e8i	−1.52224	+	0.151170i
$511$		0			0
$512$	−2.36158e9	−	1.90953e9i	−0.777603	−	0.628756i
$513$	−5.99952e9			−1.96203
$514$	2.80885e7	+	2.82845e8i	0.00912342	+	0.0918707i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$			4.69628e9i			1.47458i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$		0			0
$526$	2.49409e9	−	2.47681e8i	0.747244	−	0.0742067i
$527$		−	8.04164e9i		−	2.39336i
$528$		0			0
$529$	−3.77203e8			−0.110785
$530$	5.53453e8	+	5.57314e9i	0.161479	+	1.62605i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$			3.65696e9i			1.03248i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$	−7.19660e8	−	3.58767e9i	−0.196675	−	0.980469i
$541$	2.18411e9			0.593039			0.296520	−	0.955027i	$-0.404174\pi$
$541$	2.18411e9			0.593039			0.296520	+	0.955027i	$0.404174\pi$
$542$	7.34632e9	−	7.29543e8i	1.98186	−	0.196813i
$543$		−	3.02694e9i		−	0.811344i
$544$	−2.72812e9	−	5.05473e9i	−0.726553	−	1.34618i
$545$	−6.86513e9			−1.81661
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$	3.83468e9	−	7.69209e8i	0.995399	−	0.199670i
$549$	−6.06787e9			−1.56507
$550$		0			0
$551$		0			0
$552$	−1.21866e9	−	3.98259e9i	−0.308386	−	1.00781i
$553$		0			0
$554$		0			0
$555$		0			0
$556$	5.87707e8	+	2.92985e9i	0.145010	+	0.722909i
$557$	−2.51182e9			−0.615880			−0.307940	−	0.951406i	$-0.599640\pi$
$557$	−2.51182e9			−0.615880			−0.307940	+	0.951406i	$0.599640\pi$
$558$	6.38972e9	−	6.34546e8i	1.55691	−	0.154612i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$			8.07718e9i			1.90757i	0.300491	+	0.953785i	$0.402849\pi$
$563$			8.07718e9i			1.90757i	−0.300491	+	0.953785i	$0.597151\pi$
$564$	2.16543e9	−	4.34370e8i	0.508239	−	0.101949i
$565$	8.54170e9			1.99239
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$	8.57271e8	+	8.63251e9i	0.193890	+	1.95243i
$571$		−	2.88827e9i		−	0.649250i	−0.945843	−	0.324625i	$-0.894762\pi$
$571$		−	2.88827e9i		−	0.649250i	0.945843	−	0.324625i	$-0.105238\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			4.80455e9i			1.05394i
$576$	3.80111e9	−	2.56656e9i	0.828767	−	0.559594i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−6.14787e8	−	6.19075e9i	−0.132427	−	1.33351i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	9.38575e8	+	9.45122e9i	0.192676	+	1.94020i
$587$		−	8.62543e9i		−	1.76014i	−0.474843	−	0.880071i	$-0.657495\pi$
$587$		−	8.62543e9i		−	1.76014i	0.474843	−	0.880071i	$-0.342505\pi$
$588$	9.69549e8	+	4.83342e9i	0.196675	+	0.980469i
$589$	−1.52231e10			−3.06972
$590$		0			0
$591$			9.03379e9i			1.80017i
$592$		0			0
$593$	−6.09501e9			−1.20028			−0.600141	−	0.799895i	$-0.704889\pi$
$593$	−6.09501e9			−1.20028			−0.600141	+	0.799895i	$0.704889\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	4.90705e9			0.943866
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	−5.05934e9	+	1.54813e9i	−0.956234	+	0.292603i
$601$	−8.06970e8			−0.151634			−0.0758170	−	0.997122i	$-0.524156\pi$
$601$	−8.06970e8			−0.151634			−0.0758170	+	0.997122i	$0.524156\pi$
$602$		0			0
$603$		0			0
$604$	1.45214e9	+	7.23926e9i	0.268152	+	1.33680i
$605$	5.44683e9			1.00000
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$	−9.56877e9	+	5.16442e9i	−1.72661	+	0.931878i
$609$		0			0
$610$	8.67037e8	+	8.73085e9i	0.154662	+	1.55741i
$611$		0			0
$612$	8.50507e9	−	1.70606e9i	1.49985	−	0.300859i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−6.50554e9			−1.11503			−0.557513	−	0.830168i	$-0.688245\pi$
$617$	−6.50554e9			−1.11503			−0.557513	+	0.830168i	$0.688245\pi$
$618$		0			0
$619$		−	3.03073e9i		−	0.513605i	−0.966464	−	0.256803i	$-0.917331\pi$
$619$		−	3.03073e9i		−	0.513605i	0.966464	−	0.256803i	$-0.0826691\pi$
$620$	−1.82605e9	−	9.10328e9i	−0.307711	−	1.53401i
$621$	6.28978e9			1.05394
$622$		0			0
$623$		0			0
$624$		0			0
$625$	6.10352e9			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		−	1.52804e9i		−	0.242121i	−0.992645	−	0.121060i	$-0.961370\pi$
$631$		−	1.52804e9i		−	0.242121i	0.992645	−	0.121060i	$-0.0386295\pi$
$632$	1.75559e8	−	5.37204e7i	0.0276639	−	0.00846504i
$633$	1.21210e10			1.89945
$634$	1.26720e9	+	1.27604e10i	0.197484	+	1.98862i
$635$		0			0
$636$	−2.08504e9	−	1.03944e10i	−0.321377	−	1.60214i
$637$		0			0
$638$		0			0
$639$		0			0
$640$	−4.23608e9	−	5.10256e9i	−0.638755	−	0.769410i
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$	−6.84076e8	−	6.88848e9i	−0.102031	−	1.02743i
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$	−2.04646e10	+	2.03228e9i	−2.98668	+	0.296599i
$647$			4.54741e9i			0.660084i	0.943966	+	0.330042i	$0.107063\pi$
$647$			4.54741e9i			0.660084i	−0.943966	+	0.330042i	$0.892937\pi$
$648$	2.02671e9	+	6.62333e9i	0.292603	+	0.956234i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−4.89451e9			−0.687880			−0.343940	−	0.938992i	$-0.611762\pi$
$653$	−4.89451e9			−0.687880			−0.343940	+	0.938992i	$0.611762\pi$
$654$	1.29316e10	−	1.28420e9i	1.80772	−	0.179519i
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−6.94746e8			−0.0935667			−0.0467833	−	0.998905i	$-0.514897\pi$
$661$	−6.94746e8			−0.0935667			−0.0467833	+	0.998905i	$0.514897\pi$
$662$	−1.47175e10	+	1.46155e9i	−1.97165	+	0.195799i
$663$		0			0
$664$	1.40413e10	−	4.29656e9i	1.86131	−	0.569552i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	2.37907e9	+	1.18602e10i	0.308808	+	1.53948i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$		−	7.99030e9i		−	1.00000i
$676$	−7.87494e9	+	1.57966e9i	−0.980469	+	0.196675i
$677$	1.58437e10			1.96244			0.981221	−	0.192886i	$-0.0617848\pi$
$677$	1.58437e10			1.96244			0.981221	+	0.192886i	$0.0617848\pi$
$678$	−1.60897e10	+	1.59782e9i	−1.98264	+	0.196890i
$679$		0			0
$680$	−3.67007e9	−	1.19939e10i	−0.447604	−	1.46278i
$681$	−4.28946e9			−0.520461
$682$		0			0
$683$		−	8.91509e9i		−	1.07066i	−0.844642	−	0.535332i	$-0.820186\pi$
$683$		−	8.91509e9i		−	1.07066i	0.844642	−	0.535332i	$-0.179814\pi$
$684$	−3.22962e9	−	1.61004e10i	−0.385882	−	1.92371i
$685$	8.54044e9			1.01523
$686$		0			0
$687$			1.38346e10i			1.62786i
$688$		0			0
$689$		0			0
$690$	−8.98746e8	−	9.05015e9i	−0.104151	−	1.04878i
$691$		−	1.49163e10i		−	1.71984i	−0.510427	−	0.859921i	$-0.670513\pi$
$691$		−	1.49163e10i		−	1.71984i	0.510427	−	0.859921i	$-0.329487\pi$
$692$	−1.26030e10	+	2.52807e9i	−1.44578	+	0.290013i
$693$		0			0
$694$	1.60860e10	−	1.59746e9i	1.82680	−	0.181414i
$695$			6.52524e9i			0.737310i
$696$		0			0
$697$		0			0
$698$	−1.75609e9	−	1.76834e10i	−0.195458	−	1.96821i
$699$			9.12654e9i			1.01073i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	4.82277e9			0.518364
$706$	−1.05804e9	−	1.06542e10i	−0.113158	−	1.13948i
$707$		0			0
$708$		0			0
$709$	−7.89129e9			−0.831546			−0.415773	−	0.909469i	$-0.636489\pi$
$709$	−7.89129e9			−0.831546			−0.415773	+	0.909469i	$0.636489\pi$
$710$		0			0
$711$			2.77264e8i			0.0289301i
$712$		0			0
$713$	1.59596e10			1.64895
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	9.24049e9	−	3.86257e9i	0.922638	−	0.385667i
$721$		0			0
$722$	2.84780e9	+	2.86766e10i	0.281598	+	2.83562i
$723$		−	8.86270e9i		−	0.872131i
$724$	8.12314e9	−	1.62944e9i	0.795497	−	0.159571i
$725$		0			0
$726$	−1.02600e10	+	1.01889e9i	−0.995105	+	0.0988212i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−1.04604e10			−1.00000
$730$		0			0
$731$		0			0
$732$	−3.26641e9	−	1.62838e10i	−0.307810	−	1.53450i
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$			1.07648e10i			1.00000i
$736$	1.00317e10	−	5.41428e9i	0.927475	−	0.500574i
$737$		0			0
$738$		0			0
$739$		−	1.90914e10i		−	1.74013i	−0.492939	−	0.870064i	$-0.664077\pi$
$739$		−	1.90914e10i		−	1.74013i	0.492939	−	0.870064i	$-0.335923\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	2.01273e10i		−	1.80022i	−0.435664	−	0.900109i	$-0.643486\pi$
$743$		−	2.01273e10i		−	1.80022i	0.435664	−	0.900109i	$-0.356514\pi$
$744$	5.14254e9	+	1.68059e10i	0.457797	+	1.49609i
$745$		0			0
$746$		0			0
$747$			2.21756e10i			1.94650i
$748$		0			0
$749$		0			0
$750$	−1.14970e10	+	1.14173e9i	−0.995105	+	0.0988212i
$751$			1.20430e10i			1.03752i	0.854921	+	0.518759i	$0.173606\pi$
$751$			1.20430e10i			1.03752i	−0.854921	+	0.518759i	$0.826394\pi$
$752$	2.33136e9	+	5.57735e9i	0.199916	+	0.478262i
$753$		0			0
$754$		0			0
$755$			1.61230e10i			1.36343i
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	1.79347e10	−	1.78104e9i	1.49572	−	0.148536i
$759$		0			0
$760$	−2.27048e10	+	6.94757e9i	−1.87616	+	0.574097i
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	1.89422e10			1.52973
$766$	−1.06599e9	+	1.05861e8i	−0.0856947	+	0.00851010i
$767$		0			0
$768$	8.93384e9	+	8.81909e9i	0.711662	+	0.702522i
$769$	1.34627e10			1.06755			0.533776	−	0.845626i	$-0.320773\pi$
$769$	1.34627e10			1.06755			0.533776	+	0.845626i	$0.320773\pi$
$770$		0			0
$771$		−	1.17489e9i		−	0.0923226i
$772$		0			0
$773$	1.37209e10			1.06845			0.534224	−	0.845343i	$-0.320604\pi$
$773$	1.37209e10			1.06845			0.534224	+	0.845343i	$0.320604\pi$
$774$		0			0
$775$		−	2.02745e10i		−	1.56456i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	2.14547e10	−	2.13060e9i	1.60435	−	0.159323i
$783$		0			0
$784$	−1.24491e10	+	5.20378e9i	−0.922638	+	0.385667i
$785$		0			0
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$	−2.42432e10	+	4.86300e9i	−1.76501	+	0.354048i
$789$	−1.03601e10			−0.750919
$790$	3.98946e8	−	3.96182e7i	0.0287885	−	0.00285891i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		−	2.31500e10i		−	1.63405i
$796$	2.64153e9	+	1.31686e10i	0.185635	+	0.925431i
$797$	−2.57438e10			−1.80123			−0.900615	−	0.434619i	$-0.856883\pi$
$797$	−2.57438e10			−1.80123			−0.900615	+	0.434619i	$0.856883\pi$
$798$		0			0
$799$			1.14331e10i			0.792955i
$800$	−6.87809e9	−	1.27439e10i	−0.474956	−	0.880010i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	1.49468e9	+	1.50510e10i	0.0988212	+	0.995105i
$811$			6.02932e9i			0.396913i	0.980110	+	0.198456i	$0.0635928\pi$
$811$			6.02932e9i			0.396913i	−0.980110	+	0.198456i	$0.936407\pi$
$812$		0			0
$813$	−3.05155e10			−1.99160
$814$		0			0
$815$		0			0
$816$	9.15678e9	+	2.19059e10i	0.589966	+	1.41139i
$817$		0			0
$818$	−3.04687e9	−	3.06812e10i	−0.194633	−	1.95991i
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$	−1.60873e10	+	1.59759e9i	−1.01026	+	0.100326i
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	4.20316e8i		−	0.0258408i	−0.999917	−	0.0129204i	$-0.995887\pi$
$827$		−	4.20316e8i		−	0.0258408i	0.999917	−	0.0129204i	$-0.00411281\pi$
$828$	3.38587e9	+	1.68793e10i	0.207283	+	1.03335i
$829$	−3.14260e10			−1.91579			−0.957897	−	0.287112i	$-0.907305\pi$
$829$	−3.14260e10			−1.91579			−0.957897	+	0.287112i	$0.907305\pi$
$830$	3.19077e10	−	3.16867e9i	1.93697	−	0.192355i
$831$		0			0
$832$		0			0
$833$	−2.55195e10			−1.52973
$834$	−1.22062e9	−	1.22914e10i	−0.0728618	−	0.733701i
$835$			2.64145e10i			1.57015i
$836$		0			0
$837$	−2.65419e10			−1.56456
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	1.72499e10			1.00000
$842$	3.10927e9	+	3.13096e10i	0.179501	+	1.80753i
$843$		0			0
$844$	6.52491e9	+	3.25281e10i	0.373573	+	1.86235i
$845$	−1.75387e10			−1.00000
$846$	−9.08447e9	+	9.02154e8i	−0.515826	+	0.0512253i
$847$		0			0
$848$	2.67721e10	−	1.11909e10i	1.50764	−	0.630200i
$849$		0			0
$850$	−2.70664e9	−	2.72552e10i	−0.151170	−	1.52224i
$851$		0			0
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		−	3.58581e10i		−	1.96203i
$856$	1.81177e10	−	5.54395e9i	0.987292	−	0.302107i
$857$	2.48634e10			1.34936			0.674680	−	0.738111i	$-0.264282\pi$
$857$	2.48634e10			1.34936			0.674680	+	0.738111i	$0.264282\pi$
$858$		0			0
$859$			3.50262e10i			1.88546i	0.333559	+	0.942729i	$0.391750\pi$
$859$			3.50262e10i			1.88546i	−0.333559	+	0.942729i	$0.608250\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			3.67908e10i			1.94850i	0.225462	+	0.974252i	$0.427611\pi$
$863$			3.67908e10i			1.94850i	−0.225462	+	0.974252i	$0.572389\pi$
$864$	−1.66834e10	+	9.00433e9i	−0.880010	+	0.474956i
$865$	−2.80689e10			−1.47458
$866$		0			0
$867$			2.57154e10i			1.34007i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	1.04075e10	+	3.40121e10i	0.531546	+	1.73710i
$873$		0			0
$874$	−4.03330e9	−	4.06144e10i	−0.204348	−	2.05774i
$875$		0			0
$876$		0			0
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$	3.67753e10	−	3.65206e9i	1.83369	−	0.182099i
$879$		−	3.92589e10i		−	1.94974i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−2.01368e9	−	2.02772e10i	−0.0988212	−	0.995105i
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$		0			0
$886$	4.11033e10	−	4.08185e9i	1.98545	−	0.197169i
$887$		−	2.04987e10i		−	0.986267i	−0.869954	−	0.493134i	$-0.835851\pi$
$887$		−	2.04987e10i		−	0.986267i	0.869954	−	0.493134i	$-0.164149\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2.16432e10			1.01705
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	2.14429e10	−	4.30128e9i	0.980469	−	0.196675i
$901$	5.48803e10			2.49965
$902$		0			0
$903$		0			0
$904$	−1.29492e10	−	4.23183e10i	−0.582980	−	1.90519i
$905$	1.80915e10			0.811344
$906$	−3.01599e9	−	3.03703e10i	−0.134735	−	1.35675i
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$	−2.30907e9	−	1.15112e10i	−0.102362	−	0.510296i
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$	4.14686e10	−	1.73341e10i	1.81024	−	0.756691i
$913$		0			0
$914$		0			0
$915$		−	3.62666e10i		−	1.56507i
$916$	−3.71266e10	+	7.44732e9i	−1.59607	+	0.320159i
$917$		0			0
$918$	−3.56806e10	+	3.54335e9i	−1.52224	+	0.151170i
$919$		−	1.78137e10i		−	0.757093i	−0.925582	−	0.378547i	$-0.876424\pi$
$919$		−	1.78137e10i		−	0.757093i	0.925582	−	0.378547i	$-0.123576\pi$
$920$	2.38033e10	−	7.28369e9i	1.00781	−	0.308386i
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$	3.79257e9	+	3.81903e10i	0.154612	+	1.55691i
$931$			4.83092e10i			1.96203i
$932$	−2.44921e10	+	4.91293e9i	−0.990991	+	0.198786i
$933$		0			0
$934$	−4.24340e10	+	4.21401e9i	−1.70412	+	0.169232i
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$	2.59616e9	+	1.29424e10i	0.101949	+	0.508239i
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	2.47631e10i		−	0.947501i	−0.880659	−	0.473751i	$-0.842900\pi$
$947$		−	2.47631e10i		−	0.947501i	0.880659	−	0.473751i	$-0.157100\pi$
$948$	−7.44068e8	+	1.49255e8i	−0.0283651	+	0.00568982i
$949$		0			0
$950$	−5.15950e10	+	5.12376e9i	−1.95243	+	0.193890i
$951$		−	5.30046e10i		−	1.99840i
$952$		0			0
$953$	−3.95286e10			−1.47940			−0.739701	−	0.672935i	$-0.765033\pi$
$953$	−3.95286e10			−1.47940			−0.739701	+	0.672935i	$0.765033\pi$
$954$	4.33047e9	+	4.36068e10i	0.161479	+	1.62605i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	1.53399e10	+	2.27186e10i	0.559594	+	0.828767i
$961$	−3.98344e10			−1.44786
$962$		0			0
$963$			2.86137e10i			1.03248i
$964$	2.37840e10	−	4.77090e9i	0.855097	−	0.171526i
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	−8.25739e9	−	2.69853e10i	−0.292603	−	0.956234i
$969$	8.50068e10			3.00138
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	−5.63094e9	−	2.80715e10i	−0.196675	−	0.980469i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	4.19410e10	−	1.75315e10i	1.44399	−	0.603595i
$977$	−2.80111e10			−0.960948			−0.480474	−	0.877009i	$-0.659535\pi$
$977$	−2.80111e10			−0.960948			−0.480474	+	0.877009i	$0.659535\pi$
$978$		0			0
$979$		0			0
$980$	−2.88885e10	+	5.79482e9i	−0.980469	+	0.196675i
$981$	−5.37159e10			−1.81661
$982$		0			0
$983$			2.24740e10i			0.754646i	0.926082	+	0.377323i	$0.123155\pi$
$983$			2.24740e10i			0.754646i	−0.926082	+	0.377323i	$0.876845\pi$
$984$		0			0
$985$	−5.39934e10			−1.80017
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		−	5.08986e10i		−	1.66130i	−0.556796	−	0.830650i	$-0.687969\pi$
$991$		−	5.08986e10i		−	1.66130i	0.556796	−	0.830650i	$-0.312031\pi$
$992$	−4.23323e10	+	2.28474e10i	−1.37683	+	0.743099i
$993$	6.11341e10			1.98135
$994$		0			0
$995$			2.93286e10i			0.943866i
$996$	−5.95107e10	+	1.19374e10i	−1.90848	+	0.382827i
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$	−5.86865e10	+	5.82799e9i	−1.86888	+	0.185593i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.8.h.b.59.1	✓	4
3.2	odd	2	inner	60.8.h.b.59.4	yes	4
4.3	odd	2	inner	60.8.h.b.59.2	yes	4
5.4	even	2	inner	60.8.h.b.59.4	yes	4
12.11	even	2	inner	60.8.h.b.59.3	yes	4
15.14	odd	2	CM	60.8.h.b.59.1	✓	4
20.19	odd	2	inner	60.8.h.b.59.3	yes	4
60.59	even	2	inner	60.8.h.b.59.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.8.h.b.59.1	✓	4	1.1	even	1	trivial
60.8.h.b.59.1	✓	4	15.14	odd	2	CM
60.8.h.b.59.2	yes	4	4.3	odd	2	inner
60.8.h.b.59.2	yes	4	60.59	even	2	inner
60.8.h.b.59.3	yes	4	12.11	even	2	inner
60.8.h.b.59.3	yes	4	20.19	odd	2	inner
60.8.h.b.59.4	yes	4	3.2	odd	2	inner
60.8.h.b.59.4	yes	4	5.4	even	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:	\( N \)	\(=\)	\( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	60.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.11803 - 11.2583i) q^{2} +46.7654i q^{3} +(-125.500 + 25.1744i) q^{4} -279.508 q^{5} +(526.500 - 52.2853i) q^{6} +(423.735 + 1384.77i) q^{8} -2187.00 q^{9} +O(q^{10})\)	\(q+(-1.11803 - 11.2583i) q^{2} +46.7654i q^{3} +(-125.500 + 25.1744i) q^{4} -279.508 q^{5} +(526.500 - 52.2853i) q^{6} +(423.735 + 1384.77i) q^{8} -2187.00 q^{9} +(312.500 + 3146.80i) q^{10} +(-1177.29 - 5869.05i) q^{12} -13071.3i q^{15} +(15116.5 - 6318.77i) q^{16} +30987.4 q^{17} +(2445.14 + 24622.0i) q^{18} -58660.2i q^{19} +(35078.3 - 7036.46i) q^{20} +61498.2i q^{23} +(-64759.5 + 19816.1i) q^{24} +78125.0 q^{25} -102276. i q^{27} +(-147161. + 14614.2i) q^{30} -259513. i q^{31} +(-88039.6 - 163122. i) q^{32} +(-34645.0 - 348867. i) q^{34} +(274468. - 55056.4i) q^{36} +(-660416. + 65584.1i) q^{38} +(-118438. - 387056. i) q^{40} +611285. q^{45} +(692367. - 68757.1i) q^{46} +368958. i q^{47} +(295500. + 706929. i) q^{48} -823543. q^{49} +(-87346.4 - 879557. i) q^{50} +1.44914e6i q^{51} +1.77105e6 q^{53} +(-1.15146e6 + 114348. i) q^{54} +2.74327e6 q^{57} +(329062. + 1.64045e6i) q^{60} +2.77452e6 q^{61} +(-2.92168e6 + 290144. i) q^{62} +(-1.73805e6 + 1.17355e6i) q^{64} +(-3.88892e6 + 780090. i) q^{68} -2.87599e6 q^{69} +(-926708. - 3.02850e6i) q^{72} +3.65354e6i q^{75} +(1.47674e6 + 7.36186e6i) q^{76} -126778. i q^{79} +(-4.22519e6 + 1.76615e6i) q^{80} +4.78297e6 q^{81} -1.01397e7i q^{83} -8.66125e6 q^{85} +(-683438. - 6.88205e6i) q^{90} +(-1.54818e6 - 7.71802e6i) q^{92} +1.21362e7 q^{93} +(4.15385e6 - 412508. i) q^{94} +1.63960e7i q^{95} +(7.62846e6 - 4.11720e6i) q^{96} +(920749. + 9.27172e6i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 502 q^{4} + 2106 q^{6} - 8748 q^{9}+O(q^{10}) \) 4 * q - 502 * q^4 + 2106 * q^6 - 8748 * q^9	\( 4 q - 502 q^{4} + 2106 q^{6} - 8748 q^{9} + 1250 q^{10} + 60466 q^{16} - 259038 q^{24} + 312500 q^{25} - 138580 q^{34} + 1097874 q^{36} - 473750 q^{40} + 2769468 q^{46} - 3294172 q^{49} - 4605822 q^{54} + 1316250 q^{60} + 11098072 q^{61} - 6952198 q^{64} - 11503944 q^{69} + 5906940 q^{76} + 19131876 q^{81} - 34645000 q^{85} - 2733750 q^{90} + 16615404 q^{94} + 30513834 q^{96}+O(q^{100}) \) 4 * q - 502 * q^4 + 2106 * q^6 - 8748 * q^9 + 1250 * q^10 + 60466 * q^16 - 259038 * q^24 + 312500 * q^25 - 138580 * q^34 + 1097874 * q^36 - 473750 * q^40 + 2769468 * q^46 - 3294172 * q^49 - 4605822 * q^54 + 1316250 * q^60 + 11098072 * q^61 - 6952198 * q^64 - 11503944 * q^69 + 5906940 * q^76 + 19131876 * q^81 - 34645000 * q^85 - 2733750 * q^90 + 16615404 * q^94 + 30513834 * q^96

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