LMFDB - Embedded newform 60.12.h.a.59.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [60,12,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, 12, names="a")

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

chi := DirichletCharacter("60.59");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 60 = 2^{2} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 12 $
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:	$46.1005908336$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 9 $ x^4 - 4*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			59.3
Root			$-1.58114 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	60.59
Dual form			60.12.h.a.59.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+45.2548i q^{2} +(-36.3662 + 419.314i) q^{3} -2048.00 q^{4} -6987.71i q^{5} +(-18976.0 - 1645.75i) q^{6} +61433.6 q^{7} -92681.9i q^{8} +(-174502. - 30497.7i) q^{9} +O(q^{10})$	\(q+45.2548i q^{2} +(-36.3662 + 419.314i) q^{3} -2048.00 q^{4} -6987.71i q^{5} +(-18976.0 - 1645.75i) q^{6} +61433.6 q^{7} -92681.9i q^{8} +(-174502. - 30497.7i) q^{9} +316228. q^{10} +(74478.0 - 858756. i) q^{12} +2.78017e6i q^{14} +(2.93005e6 + 254116. i) q^{15} +4.19430e6 q^{16} +(1.38017e6 - 7.89706e6i) q^{18} +1.43108e7i q^{20} +(-2.23411e6 + 2.57600e7i) q^{21} +6.16350e7i q^{23} +(3.88628e7 + 3.37049e6i) q^{24} -4.88281e7 q^{25} +(1.91341e7 - 7.20621e7i) q^{27} -1.25816e8 q^{28} +2.15759e8i q^{29} +(-1.15000e7 + 1.32599e8i) q^{30} +1.89813e8i q^{32} -4.29280e8i q^{35} +(3.57380e8 + 6.24594e7i) q^{36} -6.47634e8 q^{40} -1.04605e9i q^{41} +(-1.16576e9 - 1.01104e8i) q^{42} +8.60868e8 q^{43} +(-2.13109e8 + 1.21937e9i) q^{45} -2.78928e9 q^{46} +1.64710e9i q^{47} +(-1.52531e8 + 1.75873e9i) q^{48} +1.79676e9 q^{49} -2.20971e9i q^{50} +(3.26116e9 + 8.65911e8i) q^{54} -5.69378e9i q^{56} -9.76413e9 q^{58} +(-6.00074e9 - 5.20431e8i) q^{60} +4.75443e9 q^{61} +(-1.07203e10 - 1.87358e9i) q^{63} -8.58993e9 q^{64} -2.13715e10 q^{67} +(-2.58444e10 - 2.24143e9i) q^{69} +1.94270e10 q^{70} +(-2.82659e9 + 1.61732e10i) q^{72} +(1.77569e9 - 2.04743e10i) q^{75} -2.93086e10i q^{80} +(2.95208e10 + 1.06438e10i) q^{81} +4.73389e10 q^{82} +6.93724e10i q^{83} +(4.57545e9 - 5.27564e10i) q^{84} +3.89584e10i q^{86} +(-9.04707e10 - 7.84632e9i) q^{87} +9.62796e10i q^{89} +(-5.51824e10 - 9.64423e9i) q^{90} -1.26228e11i q^{92} -7.45391e10 q^{94} +(-7.95911e10 - 6.90276e9i) q^{96} +8.13119e10i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8192 q^{4} - 75904 q^{6} - 698008 q^{9}+O(q^{10}) $ 4 * q - 8192 * q^4 - 75904 * q^6 - 698008 * q^9	$ 4 q - 8192 q^{4} - 75904 q^{6} - 698008 q^{9} + 16777216 q^{16} - 8936420 q^{21} + 155451392 q^{24} - 195312500 q^{25} - 46000000 q^{30} + 1429520384 q^{36} - 852437500 q^{45} - 11157121792 q^{46} + 7187026188 q^{49} + 13044633728 q^{54} + 19017715232 q^{61} - 34359738368 q^{64} - 103377706604 q^{69} + 77708000000 q^{70} + 118083345596 q^{81} + 18301788160 q^{84} - 298156262656 q^{94} - 318364450816 q^{96}+O(q^{100}) $ 4 * q - 8192 * q^4 - 75904 * q^6 - 698008 * q^9 + 16777216 * q^16 - 8936420 * q^21 + 155451392 * q^24 - 195312500 * q^25 - 46000000 * q^30 + 1429520384 * q^36 - 852437500 * q^45 - 11157121792 * q^46 + 7187026188 * q^49 + 13044633728 * q^54 + 19017715232 * q^61 - 34359738368 * q^64 - 103377706604 * q^69 + 77708000000 * q^70 + 118083345596 * q^81 + 18301788160 * q^84 - 298156262656 * q^94 - 318364450816 * 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 $

$2$			45.2548i			1.00000i
$2$			45.2548i			1.00000i
$3$	−36.3662	+	419.314i	−0.0864034	+	0.996260i
$3$	−36.3662	+	419.314i	−0.0864034	+	0.996260i
$4$	−2048.00			−1.00000
$5$		−	6987.71i		−	1.00000i
$5$		−	6987.71i		−	1.00000i
$6$	−18976.0	−	1645.75i	−0.996260	−	0.0864034i
$7$	61433.6			1.38155			0.690775	−	0.723070i	$-0.257270\pi$
$7$	61433.6			1.38155			0.690775	+	0.723070i	$0.257270\pi$
$8$		−	92681.9i		−	1.00000i
$9$	−174502.	−	30497.7i	−0.985069	−	0.172161i
$10$	316228.			1.00000
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	74478.0	−	858756.i	0.0864034	−	0.996260i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$			2.78017e6i			1.38155i
$15$	2.93005e6	+	254116.i	0.996260	+	0.0864034i
$16$	4.19430e6			1.00000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	1.38017e6	−	7.89706e6i	0.172161	−	0.985069i
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$			1.43108e7i			1.00000i
$21$	−2.23411e6	+	2.57600e7i	−0.119371	+	1.37638i
$22$		0			0
$23$			6.16350e7i			1.99675i	0.0569689	+	0.998376i	$0.481856\pi$
$23$			6.16350e7i			1.99675i	−0.0569689	+	0.998376i	$0.518144\pi$
$24$	3.88628e7	+	3.37049e6i	0.996260	+	0.0864034i
$25$	−4.88281e7			−1.00000
$26$		0			0
$27$	1.91341e7	−	7.20621e7i	0.256630	−	0.966510i
$28$	−1.25816e8			−1.38155
$29$			2.15759e8i			1.95335i	0.214730	+	0.976674i	$0.431113\pi$
$29$			2.15759e8i			1.95335i	−0.214730	+	0.976674i	$0.568887\pi$
$30$	−1.15000e7	+	1.32599e8i	−0.0864034	+	0.996260i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$			1.89813e8i			1.00000i
$33$		0			0
$34$		0			0
$35$		−	4.29280e8i		−	1.38155i
$36$	3.57380e8	+	6.24594e7i	0.985069	+	0.172161i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$	−6.47634e8			−1.00000
$41$		−	1.04605e9i		−	1.41007i	−0.709171	−	0.705037i	$-0.750931\pi$
$41$		−	1.04605e9i		−	1.41007i	0.709171	−	0.705037i	$-0.249069\pi$
$42$	−1.16576e9	−	1.01104e8i	−1.37638	−	0.119371i
$43$	8.60868e8			0.893017			0.446509	−	0.894779i	$-0.352667\pi$
$43$	8.60868e8			0.893017			0.446509	+	0.894779i	$0.352667\pi$
$44$		0			0
$45$	−2.13109e8	+	1.21937e9i	−0.172161	+	0.985069i
$46$	−2.78928e9			−1.99675
$47$			1.64710e9i			1.04756i	0.851852	+	0.523782i	$0.175479\pi$
$47$			1.64710e9i			1.04756i	−0.851852	+	0.523782i	$0.824521\pi$
$48$	−1.52531e8	+	1.75873e9i	−0.0864034	+	0.996260i
$49$	1.79676e9			0.908680
$50$		−	2.20971e9i		−	1.00000i
$51$		0			0
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	3.26116e9	+	8.65911e8i	0.966510	+	0.256630i
$55$		0			0
$56$		−	5.69378e9i		−	1.38155i
$57$		0			0
$58$	−9.76413e9			−1.95335
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$	−6.00074e9	−	5.20431e8i	−0.996260	−	0.0864034i
$61$	4.75443e9			0.720749			0.360375	−	0.932808i	$-0.382649\pi$
$61$	4.75443e9			0.720749			0.360375	+	0.932808i	$0.382649\pi$
$62$		0			0
$63$	−1.07203e10	−	1.87358e9i	−1.36092	−	0.237848i
$64$	−8.58993e9			−1.00000
$65$		0			0
$66$		0			0
$67$	−2.13715e10			−1.93385			−0.966925	−	0.255061i	$-0.917905\pi$
$67$	−2.13715e10			−1.93385			−0.966925	+	0.255061i	$0.917905\pi$
$68$		0			0
$69$	−2.58444e10	−	2.24143e9i	−1.98928	−	0.172526i
$70$	1.94270e10			1.38155
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	−2.82659e9	+	1.61732e10i	−0.172161	+	0.985069i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	1.77569e9	−	2.04743e10i	0.0864034	−	0.996260i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		−	2.93086e10i		−	1.00000i
$81$	2.95208e10	+	1.06438e10i	0.940721	+	0.339180i
$82$	4.73389e10			1.41007
$83$			6.93724e10i			1.93311i	0.256456	+	0.966556i	$0.417445\pi$
$83$			6.93724e10i			1.93311i	−0.256456	+	0.966556i	$0.582555\pi$
$84$	4.57545e9	−	5.27564e10i	0.119371	−	1.37638i
$85$		0			0
$86$			3.89584e10i			0.893017i
$87$	−9.04707e10	−	7.84632e9i	−1.94604	−	0.168776i
$88$		0			0
$89$			9.62796e10i			1.82763i	0.406127	+	0.913817i	$0.366879\pi$
$89$			9.62796e10i			1.82763i	−0.406127	+	0.913817i	$0.633121\pi$
$90$	−5.51824e10	−	9.64423e9i	−0.985069	−	0.172161i
$91$		0			0
$92$		−	1.26228e11i		−	1.99675i
$93$		0			0
$94$	−7.45391e10			−1.04756
$95$		0			0
$96$	−7.95911e10	−	6.90276e9i	−0.996260	−	0.0864034i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$			8.13119e10i			0.908680i
$99$		0			0
$100$	1.00000e11			1.00000
$101$			1.97675e11i			1.87147i	0.352701	+	0.935736i	$0.385263\pi$
$101$			1.97675e11i			1.87147i	−0.352701	+	0.935736i	$0.614737\pi$
$102$		0			0
$103$	9.13033e10			0.776036			0.388018	−	0.921652i	$-0.373160\pi$
$103$	9.13033e10			0.776036			0.388018	+	0.921652i	$0.373160\pi$
$104$		0			0
$105$	1.80003e11	+	1.56113e10i	1.37638	+	0.119371i
$106$		0			0
$107$		−	1.46128e11i		−	1.00721i	−0.863933	−	0.503606i	$-0.832006\pi$
$107$		−	1.46128e11i		−	1.00721i	0.863933	−	0.503606i	$-0.167994\pi$
$108$	−3.91867e10	+	1.47583e11i	−0.256630	+	0.966510i
$109$	−5.69612e10			−0.354596			−0.177298	−	0.984157i	$-0.556736\pi$
$109$	−5.69612e10			−0.354596			−0.177298	+	0.984157i	$0.556736\pi$
$110$		0			0
$111$		0			0
$112$	2.57671e11			1.38155
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$	4.30687e11			1.99675
$116$		−	4.41874e11i		−	1.95335i
$117$		0			0
$118$		0			0
$119$		0			0
$120$	2.35520e10	−	2.71562e11i	0.0864034	−	0.996260i
$121$	−2.85312e11			−1.00000
$122$			2.15161e11i			0.720749i
$123$	4.38624e11	+	3.80409e10i	1.40480	+	0.121835i
$124$		0			0
$125$			3.41197e11i			1.00000i
$126$	8.47888e10	−	4.85145e11i	0.237848	−	1.36092i
$127$	3.64971e11			0.980253			0.490127	−	0.871651i	$-0.336951\pi$
$127$	3.64971e11			0.980253			0.490127	+	0.871651i	$0.336951\pi$
$128$		−	3.88736e11i		−	1.00000i
$129$	−3.13065e10	+	3.60974e11i	−0.0771597	+	0.889678i
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$		−	9.67162e11i		−	1.93385i
$135$	−5.03549e11	−	1.33704e11i	−0.966510	−	0.256630i
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$	1.01436e11	−	1.16959e12i	0.172526	−	1.98928i
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$			8.79166e11i			1.38155i
$141$	−6.90651e11	−	5.98986e10i	−1.04365	−	0.0905132i
$142$		0			0
$143$		0			0
$144$	−7.31914e11	−	1.27917e11i	−0.985069	−	0.172161i
$145$	1.50766e12			1.95335
$146$		0			0
$147$	−6.53412e10	+	7.53406e11i	−0.0785130	+	0.905281i
$148$		0			0
$149$			1.60994e12i			1.79591i	0.440083	+	0.897957i	$0.354949\pi$
$149$			1.60994e12i			1.79591i	−0.440083	+	0.897957i	$0.645051\pi$
$150$	9.26562e11	+	8.03587e10i	0.996260	+	0.0864034i
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$	1.32636e12			1.00000
$161$			3.78646e12i			2.75861i
$162$	−4.81685e11	+	1.33596e12i	−0.339180	+	0.940721i
$163$	−2.51531e12			−1.71222			−0.856111	−	0.516792i	$-0.827126\pi$
$163$	−2.51531e12			−1.71222			−0.856111	+	0.516792i	$0.827126\pi$
$164$			2.14231e12i			1.41007i
$165$		0			0
$166$	−3.13943e12			−1.93311
$167$		−	1.83451e12i		−	1.09290i	−0.837492	−	0.546449i	$-0.815979\pi$
$167$		−	1.83451e12i		−	1.09290i	0.837492	−	0.546449i	$-0.184021\pi$
$168$	2.38748e12	+	2.07061e11i	1.37638	+	0.119371i
$169$	1.79216e12			1.00000
$170$		0			0
$171$		0			0
$172$	−1.76306e12			−0.893017
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$	3.55084e11	−	4.09424e12i	0.168776	−	1.94604i
$175$	−2.99969e12			−1.38155
$176$		0			0
$177$		0			0
$178$	−4.35712e12			−1.82763
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	4.36448e11	−	2.49727e12i	0.172161	−	0.985069i
$181$	−3.81600e12			−1.46008			−0.730039	−	0.683406i	$-0.760498\pi$
$181$	−3.81600e12			−1.46008			−0.730039	+	0.683406i	$0.760498\pi$
$182$		0			0
$183$	−1.72900e11	+	1.99360e12i	−0.0622752	+	0.718054i
$184$	5.71245e12			1.99675
$185$		0			0
$186$		0			0
$187$		0			0
$188$		−	3.37325e12i		−	1.04756i
$189$	1.17548e12	−	4.42703e12i	0.354547	−	1.33528i
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	3.12383e11	−	3.60188e12i	0.0864034	−	0.996260i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$	−3.67976e12			−0.908680
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$			4.52548e12i			1.00000i
$201$	7.77199e11	−	8.96136e12i	0.167091	−	1.92662i
$202$	−8.94573e12			−1.87147
$203$			1.32548e13i			2.69865i
$204$		0			0
$205$	−7.30950e12			−1.41007
$206$			4.13192e12i			0.776036i
$207$	1.87973e12	−	1.07554e13i	0.343762	−	1.96694i
$208$		0			0
$209$		0			0
$210$	−7.06486e11	+	8.14602e12i	−0.119371	+	1.37638i
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$	6.61298e12			1.00721
$215$		−	6.01549e12i		−	0.893017i
$216$	−6.67885e12	−	1.77339e12i	−0.966510	−	0.256630i
$217$		0			0
$218$		−	2.57777e12i		−	0.354596i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	1.51451e13			1.83906			0.919528	−	0.393025i	$-0.128572\pi$
$223$	1.51451e13			1.83906			0.919528	+	0.393025i	$0.128572\pi$
$224$			1.16609e13i			1.38155i
$225$	8.52061e12	+	1.48915e12i	0.985069	+	0.172161i
$226$		0			0
$227$		−	9.34693e12i		−	1.02926i	−0.857411	−	0.514632i	$-0.827929\pi$
$227$		−	9.34693e12i		−	1.02926i	0.857411	−	0.514632i	$-0.172071\pi$
$228$		0			0
$229$	1.59696e13			1.67571			0.837853	−	0.545896i	$-0.183811\pi$
$229$	1.59696e13			1.67571			0.837853	+	0.545896i	$0.183811\pi$
$230$			1.94907e13i			1.99675i
$231$		0			0
$232$	1.99969e13			1.95335
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$	1.15094e13			1.04756
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$	1.22895e13	+	1.06584e12i	0.996260	+	0.0864034i
$241$	−5.08395e12			−0.402817			−0.201408	−	0.979507i	$-0.564552\pi$
$241$	−5.08395e12			−0.402817			−0.201408	+	0.979507i	$0.564552\pi$
$242$		−	1.29117e13i		−	1.00000i
$243$	−5.53667e12	+	1.19914e13i	−0.419193	+	0.907897i
$244$	−9.73707e12			−0.720749
$245$		−	1.25552e13i		−	0.908680i
$246$	−1.72153e12	+	1.98499e13i	−0.121835	+	1.40480i
$247$		0			0
$248$		0			0
$249$	−2.90888e13	−	2.52281e12i	−1.92588	−	0.167027i
$250$	−1.54408e13			−1.00000
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	2.19551e13	+	3.83710e12i	1.36092	+	0.237848i
$253$		0			0
$254$			1.65167e13i			0.980253i
$255$		0			0
$256$	1.75922e13			1.00000
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$	−1.63358e13	−	1.41677e12i	−0.889678	−	0.0771597i
$259$		0			0
$260$		0			0
$261$	6.58015e12	−	3.76503e13i	0.336289	−	1.92418i
$262$		0			0
$263$			2.87228e13i			1.40757i	0.710412	+	0.703786i	$0.248509\pi$
$263$			2.87228e13i			1.40757i	−0.710412	+	0.703786i	$0.751491\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−4.03714e13	−	3.50132e12i	−1.82080	−	0.157914i
$268$	4.37688e13			1.93385
$269$		−	4.26165e13i		−	1.84476i	−0.386283	−	0.922380i	$-0.626241\pi$
$269$		−	4.26165e13i		−	1.84476i	0.386283	−	0.922380i	$-0.373759\pi$
$270$	6.05074e12	−	2.27880e13i	0.256630	−	0.966510i
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	5.29294e13	+	4.59045e12i	1.98928	+	0.172526i
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$	−3.97865e13			−1.38155
$281$			3.71997e13i			1.26664i	0.773888	+	0.633322i	$0.218309\pi$
$281$			3.71997e13i			1.26664i	−0.773888	+	0.633322i	$0.781691\pi$
$282$	2.71070e12	−	3.12553e13i	0.0905132	−	1.04365i
$283$	4.79043e13			1.56873			0.784367	−	0.620297i	$-0.212988\pi$
$283$	4.79043e13			1.56873			0.784367	+	0.620297i	$0.212988\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	6.42626e13i		−	1.94809i
$288$	5.78885e12	−	3.31227e13i	0.172161	−	0.985069i
$289$	−3.42719e13			−1.00000
$290$			6.82289e13i			1.95335i
$291$		0			0
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	−3.40953e13	−	2.95700e12i	−0.905281	−	0.0785130i
$295$		0			0
$296$		0			0
$297$		0			0
$298$	−7.28576e13			−1.79591
$299$		0			0
$300$	−3.63662e12	+	4.19314e13i	−0.0864034	+	0.996260i
$301$	5.28862e13			1.23375
$302$		0			0
$303$	−8.28878e13	−	7.18867e12i	−1.86447	−	0.161702i
$304$		0			0
$305$		−	3.32226e13i		−	0.720749i
$306$		0			0
$307$	2.23209e13			0.467144			0.233572	−	0.972340i	$-0.424959\pi$
$307$	2.23209e13			0.467144			0.233572	+	0.972340i	$0.424959\pi$
$308$		0			0
$309$	−3.32035e12	+	3.82848e13i	−0.0670521	+	0.773134i
$310$		0			0
$311$		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
$315$	−1.30921e13	+	7.49102e13i	−0.237848	+	1.36092i
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$			6.00240e13i			1.00000i
$321$	6.12734e13	+	5.31410e12i	1.00345	+	0.0870266i
$322$	−1.71355e14			−2.75861
$323$		0			0
$324$	−6.04587e13	−	2.17986e13i	−0.940721	−	0.339180i
$325$		0			0
$326$		−	1.13830e14i		−	1.71222i
$327$	2.07146e12	−	2.38846e13i	0.0306383	−	0.353270i
$328$	−9.69500e13			−1.41007
$329$			1.01187e14i			1.44726i
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		−	1.42075e14i		−	1.93311i
$333$		0			0
$334$	8.30205e13			1.09290
$335$			1.49338e14i			1.93385i
$336$	−9.37052e12	+	1.08045e14i	−0.119371	+	1.37638i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$			8.11039e13i			1.00000i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−1.10931e13			−0.126164
$344$		−	7.97868e13i		−	0.893017i
$345$	−1.56625e13	+	1.80593e14i	−0.172526	+	1.98928i
$346$		0			0
$347$		−	1.85643e14i		−	1.98092i	−0.137806	−	0.990459i	$-0.544005\pi$
$347$		−	1.85643e14i		−	1.98092i	0.137806	−	0.990459i	$-0.455995\pi$
$348$	1.85284e14	+	1.60693e13i	1.94604	+	0.168776i
$349$	−1.58337e14			−1.63698			−0.818491	−	0.574520i	$-0.805189\pi$
$349$	−1.58337e14			−1.63698			−0.818491	+	0.574520i	$0.805189\pi$
$350$		−	1.35750e14i		−	1.38155i
$351$		0			0
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$		−	1.97181e14i		−	1.82763i
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	1.13014e14	+	1.97514e13i	0.985069	+	0.172161i
$361$	1.16490e14			1.00000
$362$		−	1.72692e14i		−	1.46008i
$363$	1.03757e13	−	1.19635e14i	0.0864034	−	0.996260i
$364$		0			0
$365$		0			0
$366$	−9.02200e13	−	7.82458e12i	−0.718054	−	0.0622752i
$367$	2.01225e14			1.57768			0.788838	−	0.614601i	$-0.210683\pi$
$367$	2.01225e14			1.57768			0.788838	+	0.614601i	$0.210683\pi$
$368$			2.58516e14i			1.99675i
$369$	−3.19022e13	+	1.82538e14i	−0.242759	+	1.38902i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$	−1.43069e14	−	1.24080e13i	−0.996260	−	0.0864034i
$376$	1.52656e14			1.04756
$377$		0			0
$378$	2.00345e14	+	5.31960e13i	1.33528	+	0.354547i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	−1.32726e13	+	1.53038e14i	−0.0846972	+	0.976587i
$382$		0			0
$383$			2.87695e14i			1.78377i	0.452260	+	0.891886i	$0.350618\pi$
$383$			2.87695e14i			1.78377i	−0.452260	+	0.891886i	$0.649382\pi$
$384$	1.63003e14	+	1.41369e13i	0.996260	+	0.0864034i
$385$		0			0
$386$		0			0
$387$	−1.50223e14	−	2.62545e13i	−0.879683	−	0.153742i
$388$		0			0
$389$		−	2.15196e14i		−	1.22493i	−0.790498	−	0.612464i	$-0.790178\pi$
$389$		−	2.15196e14i		−	1.22493i	0.790498	−	0.612464i	$-0.209822\pi$
$390$		0			0
$391$		0			0
$392$		−	1.66527e14i		−	0.908680i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	−2.04800e14			−1.00000
$401$			1.67474e14i			0.806591i	0.915070	+	0.403295i	$0.132135\pi$
$401$			1.67474e14i			0.806591i	−0.915070	+	0.403295i	$0.867865\pi$
$402$	4.05545e14	+	3.51720e13i	1.92662	+	0.167091i
$403$		0			0
$404$		−	4.04838e14i		−	1.87147i
$405$	7.43760e13	−	2.06283e14i	0.339180	−	0.940721i
$406$	−5.99845e14			−2.69865
$407$		0			0
$408$		0			0
$409$	4.10356e14			1.77289			0.886447	−	0.462830i	$-0.153166\pi$
$409$	4.10356e14			1.77289			0.886447	+	0.462830i	$0.153166\pi$
$410$		−	3.30790e14i		−	1.41007i
$411$		0			0
$412$	−1.86989e14			−0.776036
$413$		0			0
$414$	4.86735e14	+	8.50667e13i	1.96694	+	0.343762i
$415$	4.84754e14			1.93311
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$	−3.68647e14	−	3.19719e13i	−1.37638	−	0.119371i
$421$	−4.51742e14			−1.66471			−0.832355	−	0.554243i	$-0.813008\pi$
$421$	−4.51742e14			−1.66471			−0.832355	+	0.554243i	$0.813008\pi$
$422$		0			0
$423$	5.02327e13	−	2.87422e14i	0.180349	−	1.03192i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	2.92082e14			0.995751
$428$			2.99269e14i			1.00721i
$429$		0			0
$430$	2.72230e14			0.893017
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	8.02543e13	−	3.02250e14i	0.256630	−	0.966510i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$	−5.48279e13	+	6.32183e14i	−0.168776	+	1.94604i
$436$	1.16657e14			0.354596
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	−3.13538e14	−	5.47970e13i	−0.895112	−	0.156439i
$442$		0			0
$443$		−	5.75608e14i		−	1.60290i	−0.598062	−	0.801450i	$-0.704063\pi$
$443$		−	5.75608e14i		−	1.60290i	0.598062	−	0.801450i	$-0.295937\pi$
$444$		0			0
$445$	6.72774e14			1.82763
$446$			6.85388e14i			1.83906i
$447$	−6.75071e14	−	5.85474e13i	−1.78920	−	0.155173i
$448$	−5.27710e14			−1.38155
$449$			1.30274e14i			0.336903i	0.985710	+	0.168451i	$0.0538766\pi$
$449$			1.30274e14i			0.336903i	−0.985710	+	0.168451i	$0.946123\pi$
$450$	−6.73911e13	+	3.85599e14i	−0.172161	+	0.985069i
$451$		0			0
$452$		0			0
$453$		0			0
$454$	4.22994e14			1.02926
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$			7.22700e14i			1.67571i
$459$		0			0
$460$	−8.82048e14			−1.99675
$461$		−	6.61905e14i		−	1.48061i	−0.672271	−	0.740305i	$-0.734681\pi$
$461$		−	6.61905e14i		−	1.48061i	0.672271	−	0.740305i	$-0.265319\pi$
$462$		0			0
$463$	9.03778e14			1.97409			0.987043	−	0.160453i	$-0.0512955\pi$
$463$	9.03778e14			1.97409			0.987043	+	0.160453i	$0.0512955\pi$
$464$			9.04958e14i			1.95335i
$465$		0			0
$466$		0			0
$467$		−	5.08978e13i		−	0.106037i	−0.998594	−	0.0530183i	$-0.983116\pi$
$467$		−	5.08978e13i		−	0.106037i	0.998594	−	0.0530183i	$-0.0168842\pi$
$468$		0			0
$469$	−1.31293e15			−2.67171
$470$			5.20858e14i			1.04756i
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$	−4.82345e13	+	5.56160e14i	−0.0864034	+	0.996260i
$481$		0			0
$482$		−	2.30073e14i		−	0.402817i
$483$	−1.58772e15	−	1.37699e14i	−2.74830	−	0.238354i
$484$	5.84318e14			1.00000
$485$		0			0
$486$	−5.42670e14	−	2.50561e14i	−0.907897	−	0.419193i
$487$	−5.83943e14			−0.965965			−0.482982	−	0.875630i	$-0.660447\pi$
$487$	−5.83943e14			−0.965965			−0.482982	+	0.875630i	$0.660447\pi$
$488$		−	4.40649e14i		−	0.720749i
$489$	9.14723e13	−	1.05471e15i	0.147942	−	1.70582i
$490$	5.68184e14			0.908680
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$	−8.98302e14	−	7.79077e13i	−1.40480	−	0.121835i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	1.14169e14	−	1.31641e15i	0.167027	−	1.92588i
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$		−	6.98771e14i		−	1.00000i
$501$	7.69237e14	+	6.67142e13i	1.08881	+	0.0944302i
$502$		0			0
$503$			1.00893e15i			1.39713i	0.715546	+	0.698565i	$0.246178\pi$
$503$			1.00893e15i			1.39713i	−0.715546	+	0.698565i	$0.753822\pi$
$504$	−1.73647e14	+	9.93576e14i	−0.237848	+	1.36092i
$505$	1.38129e15			1.87147
$506$		0			0
$507$	−6.51741e13	+	7.51479e14i	−0.0864034	+	0.996260i
$508$	−7.47461e14			−0.980253
$509$		−	1.59659e14i		−	0.207132i	−0.994623	−	0.103566i	$-0.966975\pi$
$509$		−	1.59659e14i		−	0.207132i	0.994623	−	0.103566i	$-0.0330252\pi$
$510$		0			0
$511$		0			0
$512$			7.96131e14i			1.00000i
$513$		0			0
$514$		0			0
$515$		−	6.38001e14i		−	0.776036i
$516$	6.41157e13	−	7.39275e14i	0.0771597	−	0.889678i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	1.68258e15i		−	1.92030i	−0.279491	−	0.960148i	$-0.590166\pi$
$521$		−	1.68258e15i		−	1.92030i	0.279491	−	0.960148i	$-0.409834\pi$
$522$	1.70386e15	+	2.97784e14i	1.92418	+	0.336289i
$523$	1.43931e14			0.160841			0.0804203	−	0.996761i	$-0.474374\pi$
$523$	1.43931e14			0.160841			0.0804203	+	0.996761i	$0.474374\pi$
$524$		0			0
$525$	1.09087e14	−	1.25781e15i	0.119371	−	1.37638i
$526$	−1.29985e15			−1.40757
$527$		0			0
$528$		0			0
$529$	−2.84606e15			−2.98702
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$	1.58452e14	−	1.82700e15i	0.157914	−	1.82080i
$535$	−1.02110e15			−1.00721
$536$			1.98075e15i			1.93385i
$537$		0			0
$538$	1.92860e15			1.84476
$539$		0			0
$540$	1.03127e15	+	2.73825e14i	0.966510	+	0.256630i
$541$	1.91231e15			1.77408			0.887041	−	0.461691i	$-0.152757\pi$
$541$	1.91231e15			1.77408			0.887041	+	0.461691i	$0.152757\pi$
$542$		0			0
$543$	1.38773e14	−	1.60010e15i	0.126156	−	1.45462i
$544$		0			0
$545$			3.98028e14i			0.354596i
$546$		0			0
$547$	1.55169e15			1.35479			0.677397	−	0.735618i	$-0.263108\pi$
$547$	1.55169e15			1.35479			0.677397	+	0.735618i	$0.263108\pi$
$548$		0			0
$549$	−8.29657e14	−	1.44999e14i	−0.709988	−	0.124085i
$550$		0			0
$551$		0			0
$552$	−2.07740e14	+	2.39531e15i	−0.172526	+	1.98928i
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		−	1.80053e15i		−	1.38155i
$561$		0			0
$562$	−1.68347e15			−1.26664
$563$			8.64381e14i			0.644035i	0.946734	+	0.322017i	$0.104361\pi$
$563$			8.64381e14i			0.644035i	−0.946734	+	0.322017i	$0.895639\pi$
$564$	1.41445e15	+	1.22672e14i	1.04365	+	0.0905132i
$565$		0			0
$566$			2.16790e15i			1.56873i
$567$	1.81357e15	+	6.53888e14i	1.29965	+	0.468594i
$568$		0			0
$569$			2.84490e15i			1.99963i	0.0193017	+	0.999814i	$0.493856\pi$
$569$			2.84490e15i			1.99963i	−0.0193017	+	0.999814i	$0.506144\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$	2.90820e15			1.94809
$575$		−	3.00952e15i		−	1.99675i
$576$	1.49896e15	+	2.61974e14i	0.985069	+	0.172161i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$		−	1.55097e15i		−	1.00000i
$579$		0			0
$580$	−3.08769e15			−1.95335
$581$			4.26179e15i			2.67069i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$			2.59935e15i			1.53941i	0.638398	+	0.769707i	$0.279598\pi$
$587$			2.59935e15i			1.53941i	−0.638398	+	0.769707i	$0.720402\pi$
$588$	1.33819e14	−	1.54297e15i	0.0785130	−	0.905281i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		−	3.29716e15i		−	1.79591i
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	−1.89760e15	−	1.64575e14i	−0.996260	−	0.0864034i
$601$	1.50680e15			0.783874			0.391937	−	0.919992i	$-0.371805\pi$
$601$	1.50680e15			0.783874			0.391937	+	0.919992i	$0.371805\pi$
$602$			2.39335e15i			1.23375i
$603$	3.72936e15	+	6.51781e14i	1.90498	+	0.332933i
$604$		0			0
$605$			1.99368e15i			1.00000i
$606$	3.25322e14	−	3.75107e15i	0.161702	−	1.86447i
$607$	1.77226e15			0.872949			0.436474	−	0.899717i	$-0.356227\pi$
$607$	1.77226e15			0.872949			0.436474	+	0.899717i	$0.356227\pi$
$608$		0			0
$609$	−5.55794e15	−	4.82028e14i	−2.68855	−	0.233172i
$610$	1.50348e15			0.720749
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$			1.01013e15i			0.467144i
$615$	2.65819e14	−	3.06498e15i	0.121835	−	1.40480i
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$	−1.73257e15	−	1.50262e14i	−0.773134	−	0.0670521i
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$	4.44155e15	+	1.17933e15i	1.92988	+	0.512427i
$622$		0			0
$623$			5.91480e15i			2.52497i
$624$		0			0
$625$	2.38419e15			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$	−3.39005e15	−	5.92479e14i	−1.36092	−	0.237848i
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	2.55031e15i		−	0.980253i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	−2.71638e15			−1.00000
$641$		−	4.78307e15i		−	1.74577i	−0.487924	−	0.872886i	$-0.662246\pi$
$641$		−	4.78307e15i		−	1.74577i	0.487924	−	0.872886i	$-0.337754\pi$
$642$	−2.40489e14	+	2.77292e15i	−0.0870266	+	1.00345i
$643$	−4.60509e15			−1.65226			−0.826129	−	0.563481i	$-0.809462\pi$
$643$	−4.60509e15			−1.65226			−0.826129	+	0.563481i	$0.809462\pi$
$644$		−	7.75466e15i		−	2.75861i
$645$	2.52238e15	+	2.18761e14i	0.889678	+	0.0771597i
$646$		0			0
$647$		−	4.60881e15i		−	1.59814i	−0.601237	−	0.799071i	$-0.705325\pi$
$647$		−	4.60881e15i		−	1.59814i	0.601237	−	0.799071i	$-0.294675\pi$
$648$	9.86490e14	−	2.73605e15i	0.339180	−	0.940721i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	5.15136e15			1.71222
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$	1.08090e15	+	9.37437e13i	0.353270	+	0.0306383i
$655$		0			0
$656$		−	4.38746e15i		−	1.41007i
$657$		0			0
$658$	−4.57920e15			−1.44726
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−5.79908e15			−1.78752			−0.893760	−	0.448545i	$-0.851942\pi$
$661$	−5.79908e15			−1.78752			−0.893760	+	0.448545i	$0.851942\pi$
$662$		0			0
$663$		0			0
$664$	6.42956e15			1.93311
$665$		0			0
$666$		0			0
$667$	−1.32983e16			−3.90035
$668$			3.75708e15i			1.09290i
$669$	−5.50769e14	+	6.35055e15i	−0.158901	+	1.83218i
$670$	−6.75825e15			−1.93385
$671$		0			0
$672$	−4.88957e15	−	4.24061e14i	−1.37638	−	0.119371i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	−9.34283e14	+	3.51866e15i	−0.256630	+	0.966510i
$676$	−3.67034e15			−1.00000
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	3.91930e15	+	3.39912e14i	1.02541	+	0.0889319i
$682$		0			0
$683$		−	7.75867e15i		−	1.99744i	−0.0505851	−	0.998720i	$-0.516109\pi$
$683$		−	7.75867e15i		−	1.99744i	0.0505851	−	0.998720i	$-0.483891\pi$
$684$		0			0
$685$		0			0
$686$		−	5.02015e14i		−	0.126164i
$687$	−5.80752e14	+	6.69627e15i	−0.144787	+	1.66944i
$688$	3.61074e15			0.893017
$689$		0			0
$690$	−8.17273e15	−	7.08802e14i	−1.98928	−	0.172526i
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$	8.40125e15			1.98092
$695$		0			0
$696$	−7.27212e14	+	8.38500e15i	−0.168776	+	1.94604i
$697$		0			0
$698$		−	7.16554e15i		−	1.63698i
$699$		0			0
$700$	6.14336e15			1.38155
$701$			8.93259e15i			1.99310i	0.0830124	+	0.996549i	$0.473546\pi$
$701$			8.93259e15i			1.99310i	−0.0830124	+	0.996549i	$0.526454\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−4.18554e14	+	4.82607e15i	−0.0905132	+	1.04365i
$706$		0			0
$707$			1.21439e16i			2.58553i
$708$		0			0
$709$	−8.48691e15			−1.77908			−0.889540	−	0.456857i	$-0.848975\pi$
$709$	−8.48691e15			−1.77908			−0.889540	+	0.456857i	$0.848975\pi$
$710$		0			0
$711$		0			0
$712$	8.92337e15			1.82763
$713$		0			0
$714$		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	$0.5\pi$
$720$	−8.93846e14	+	5.11441e15i	−0.172161	+	0.985069i
$721$	5.60909e15			1.07213
$722$			5.27175e15i			1.00000i
$723$	1.84884e14	−	2.13177e15i	0.0348047	−	0.401310i
$724$	7.81516e15			1.46008
$725$		−	1.05351e16i		−	1.95335i
$726$	5.41407e15	+	4.69551e14i	0.996260	+	0.0864034i
$727$	7.09246e15			1.29526			0.647631	−	0.761954i	$-0.275760\pi$
$727$	7.09246e15			1.29526			0.647631	+	0.761954i	$0.275760\pi$
$728$		0			0
$729$	−4.82683e15	−	2.75769e15i	−0.868282	−	0.496071i
$730$		0			0
$731$		0			0
$732$	3.54100e14	−	4.08289e15i	0.0622752	−	0.718054i
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$			9.10639e15i			1.57768i
$735$	5.26458e15	+	4.56585e14i	0.905281	+	0.0785130i
$736$	−1.16991e16			−1.99675
$737$		0			0
$738$	−8.26073e15	−	1.44373e15i	−1.38902	−	0.242759i
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	7.10987e15i		−	1.15192i	−0.817477	−	0.575961i	$-0.804628\pi$
$743$		−	7.10987e15i		−	1.15192i	0.817477	−	0.575961i	$-0.195372\pi$
$744$		0			0
$745$	1.12498e16			1.79591
$746$		0			0
$747$	2.11570e15	−	1.21056e16i	0.332806	−	1.90425i
$748$		0			0
$749$		−	8.97713e15i		−	1.39151i
$750$	5.61523e14	−	6.47455e15i	0.0864034	−	0.996260i
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$			6.90842e15i			1.04756i
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−2.40738e15	+	9.06656e15i	−0.354547	+	1.33528i
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	1.40208e16i		−	1.99139i	−0.0927031	−	0.995694i	$-0.529551\pi$
$761$		−	1.40208e16i		−	1.99139i	0.0927031	−	0.995694i	$-0.470449\pi$
$762$	−6.92570e15	−	6.00650e14i	−0.976587	−	0.0846972i
$763$	−3.49933e15			−0.489891
$764$		0			0
$765$		0			0
$766$	−1.30196e16			−1.78377
$767$		0			0
$768$	−6.39761e14	+	7.37666e15i	−0.0864034	+	0.996260i
$769$	−4.89705e15			−0.656659			−0.328330	−	0.944563i	$-0.606486\pi$
$769$	−4.89705e15			−0.656659			−0.328330	+	0.944563i	$0.606486\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$	1.18814e15	−	6.79832e15i	0.153742	−	0.879683i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	9.73864e15			1.22493
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	1.55480e16	+	4.12835e15i	1.88793	+	0.501288i
$784$	7.53614e15			0.908680
$785$		0			0
$786$		0			0
$787$	6.27825e15			0.741272			0.370636	−	0.928778i	$-0.379140\pi$
$787$	6.27825e15			0.741272			0.370636	+	0.928778i	$0.379140\pi$
$788$		0			0
$789$	−1.20439e16	−	1.04454e15i	−1.40231	−	0.121619i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		−	9.26819e15i		−	1.00000i
$801$	2.93631e15	−	1.68010e16i	0.314646	−	1.80034i
$802$	−7.57900e15			−0.806591
$803$		0			0
$804$	−1.59170e15	+	1.83529e16i	−0.167091	+	1.92662i
$805$	2.64587e16			2.75861
$806$		0			0
$807$	1.78697e16	+	1.54980e15i	1.83786	+	0.159394i
$808$	1.83209e16			1.87147
$809$		−	7.26088e15i		−	0.736669i	−0.929693	−	0.368335i	$-0.879928\pi$
$809$		−	7.26088e15i		−	0.736669i	0.929693	−	0.368335i	$-0.120072\pi$
$810$	9.33531e15	+	3.36587e15i	0.940721	+	0.339180i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		−	2.71459e16i		−	2.69865i
$813$		0			0
$814$		0			0
$815$			1.75763e16i			1.71222i
$816$		0			0
$817$		0			0
$818$			1.85706e16i			1.77289i
$819$		0			0
$820$	1.49699e16			1.41007
$821$			1.57679e15i			0.147532i	0.997276	+	0.0737660i	$0.0235018\pi$
$821$			1.57679e15i			0.147532i	−0.997276	+	0.0737660i	$0.976498\pi$
$822$		0			0
$823$	−1.52655e15			−0.140933			−0.0704663	−	0.997514i	$-0.522449\pi$
$823$	−1.52655e15			−0.140933			−0.0704663	+	0.997514i	$0.522449\pi$
$824$		−	8.46216e15i		−	0.776036i
$825$		0			0
$826$		0			0
$827$		−	7.18279e15i		−	0.645673i	−0.946455	−	0.322837i	$-0.895364\pi$
$827$		−	7.18279e15i		−	0.645673i	0.946455	−	0.322837i	$-0.104636\pi$
$828$	−3.84968e15	+	2.20271e16i	−0.343762	+	1.96694i
$829$	−1.91626e16			−1.69982			−0.849911	−	0.526926i	$-0.823344\pi$
$829$	−1.91626e16			−1.69982			−0.849911	+	0.526926i	$0.823344\pi$
$830$			2.19375e16i			1.93311i
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−1.28190e16			−1.09290
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$	1.44688e15	−	1.66830e16i	0.119371	−	1.37638i
$841$	−3.43513e16			−2.81556
$842$		−	2.04435e16i		−	1.66471i
$843$	−1.55984e16	−	1.35281e15i	−1.26191	−	0.109442i
$844$		0			0
$845$		−	1.25231e16i		−	1.00000i
$846$	1.30072e16	+	2.27327e15i	1.03192	+	0.180349i
$847$	−1.75277e16			−1.38155
$848$		0			0
$849$	−1.74210e15	+	2.00870e16i	−0.135544	+	1.56287i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$			1.32181e16i			0.995751i
$855$		0			0
$856$	−1.35434e16			−1.00721
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$			1.23197e16i			0.893017i
$861$	2.69462e16	+	2.33699e15i	1.94080	+	0.168321i
$862$		0			0
$863$		−	5.94976e15i		−	0.423097i	−0.977367	−	0.211549i	$-0.932149\pi$
$863$		−	5.94976e15i		−	0.423097i	0.977367	−	0.211549i	$-0.0678506\pi$
$864$	1.36783e16	+	3.63189e15i	0.966510	+	0.256630i
$865$		0			0
$866$		0			0
$867$	1.24634e15	−	1.43707e16i	0.0864034	−	0.996260i
$868$		0			0
$869$		0			0
$870$	−2.86094e16	−	2.48123e15i	−1.94604	−	0.168776i
$871$		0			0
$872$			5.27927e15i			0.354596i
$873$		0			0
$874$		0			0
$875$			2.09609e16i			1.38155i
$876$		0			0
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			1.95212e16i			1.23919i	0.784920	+	0.619597i	$0.212704\pi$
$881$			1.95212e16i			1.23919i	−0.784920	+	0.619597i	$0.787296\pi$
$882$	2.47983e15	−	1.41891e16i	0.156439	−	0.895112i
$883$	−2.77785e16			−1.74151			−0.870753	−	0.491721i	$-0.836368\pi$
$883$	−2.77785e16			−1.74151			−0.870753	+	0.491721i	$0.836368\pi$
$884$		0			0
$885$		0			0
$886$	2.60490e16			1.60290
$887$		−	2.27036e16i		−	1.38840i	−0.719781	−	0.694201i	$-0.755758\pi$
$887$		−	2.27036e16i		−	1.38840i	0.719781	−	0.694201i	$-0.244242\pi$
$888$		0			0
$889$	2.24215e16			1.35427
$890$			3.04463e16i			1.82763i
$891$		0			0
$892$	−3.10171e16			−1.83906
$893$		0			0
$894$	2.64955e15	−	3.05502e16i	0.155173	−	1.78920i
$895$		0			0
$896$		−	2.38814e16i		−	1.38155i
$897$		0			0
$898$	−5.89555e15			−0.336903
$899$		0			0
$900$	−1.74502e16	−	3.04977e15i	−0.985069	−	0.172161i
$901$		0			0
$902$		0			0
$903$	−1.92327e15	+	2.21759e16i	−0.106600	+	1.22913i
$904$		0			0
$905$			2.66651e16i			1.46008i
$906$		0			0
$907$	−2.64017e16			−1.42821			−0.714104	−	0.700039i	$-0.753166\pi$
$907$	−2.64017e16			−1.42821			−0.714104	+	0.700039i	$0.753166\pi$
$908$			1.91425e16i			1.02926i
$909$	6.02863e15	−	3.44946e16i	0.322194	−	1.84353i
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	1.39307e16	+	1.20818e15i	0.718054	+	0.0622752i
$916$	−3.27057e16			−1.67571
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		−	3.99169e16i		−	1.99675i
$921$	−8.11726e14	+	9.35947e15i	−0.0403628	+	0.465397i
$922$	2.99544e16			1.48061
$923$		0			0
$924$		0			0
$925$		0			0
$926$			4.09003e16i			1.97409i
$927$	−1.59326e16	−	2.78454e15i	−0.764449	−	0.133603i
$928$	−4.09537e16			−1.95335
$929$			3.31641e16i			1.57247i	0.617928	+	0.786235i	$0.287972\pi$
$929$			3.31641e16i			1.57247i	−0.617928	+	0.786235i	$0.712028\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$	2.30337e15			0.106037
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		−	5.94162e16i		−	2.67171i
$939$		0			0
$940$	−2.35713e16			−1.04756
$941$			1.92940e16i			0.852469i	0.904613	+	0.426234i	$0.140160\pi$
$941$			1.92940e16i			0.852469i	−0.904613	+	0.426234i	$0.859840\pi$
$942$		0			0
$943$	6.44733e16			2.81557
$944$		0			0
$945$	−3.09348e16	−	8.21389e15i	−1.33528	−	0.354547i
$946$		0			0
$947$			6.14809e15i			0.262310i	0.991362	+	0.131155i	$0.0418686\pi$
$947$			6.14809e15i			0.262310i	−0.991362	+	0.131155i	$0.958131\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−2.51689e16	−	2.18284e15i	−0.996260	−	0.0864034i
$961$	2.54085e16			1.00000
$962$		0			0
$963$	−4.45656e15	+	2.54995e16i	−0.173402	+	0.992174i
$964$	1.04119e16			0.402817
$965$		0			0
$966$	6.23155e15	−	7.18518e16i	0.238354	−	2.74830i
$967$	5.23755e16			1.99197			0.995985	−	0.0895254i	$-0.0285350\pi$
$967$	5.23755e16			1.99197			0.995985	+	0.0895254i	$0.0285350\pi$
$968$			2.64432e16i			1.00000i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	1.13391e16	−	2.45585e16i	0.419193	−	0.907897i
$973$		0			0
$974$		−	2.64263e16i		−	0.965965i
$975$		0			0
$976$	1.99415e16			0.720749
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$	4.77306e16	+	4.13956e15i	1.70582	+	0.147942i
$979$		0			0
$980$			2.57131e16i			0.908680i
$981$	9.93984e15	+	1.73719e15i	0.349301	+	0.0610474i
$982$		0			0
$983$		−	5.75498e16i		−	1.99986i	−0.0118977	−	0.999929i	$-0.503787\pi$
$983$		−	5.75498e16i		−	1.99986i	0.0118977	−	0.999929i	$-0.496213\pi$
$984$	3.52570e15	−	4.06525e16i	0.121835	−	1.40480i
$985$		0			0
$986$		0			0
$987$	−4.24292e16	−	3.67979e15i	−1.44185	−	0.125048i
$988$		0			0
$989$			5.30595e16i			1.78313i
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$	5.95739e16	+	5.16671e15i	1.92588	+	0.167027i
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.12.h.a.59.3	yes	4
3.2	odd	2	inner	60.12.h.a.59.1	✓	4
4.3	odd	2	inner	60.12.h.a.59.2	yes	4
5.4	even	2	inner	60.12.h.a.59.2	yes	4
12.11	even	2	inner	60.12.h.a.59.4	yes	4
15.14	odd	2	inner	60.12.h.a.59.4	yes	4
20.19	odd	2	CM	60.12.h.a.59.3	yes	4
60.59	even	2	inner	60.12.h.a.59.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.12.h.a.59.1	✓	4	3.2	odd	2	inner
60.12.h.a.59.1	✓	4	60.59	even	2	inner
60.12.h.a.59.2	yes	4	4.3	odd	2	inner
60.12.h.a.59.2	yes	4	5.4	even	2	inner
60.12.h.a.59.3	yes	4	1.1	even	1	trivial
60.12.h.a.59.3	yes	4	20.19	odd	2	CM
60.12.h.a.59.4	yes	4	12.11	even	2	inner
60.12.h.a.59.4	yes	4	15.14	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:	\( N \)	\(=\)	\( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	60.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+45.2548i q^{2} +(-36.3662 + 419.314i) q^{3} -2048.00 q^{4} -6987.71i q^{5} +(-18976.0 - 1645.75i) q^{6} +61433.6 q^{7} -92681.9i q^{8} +(-174502. - 30497.7i) q^{9} +O(q^{10})\)	\(q+45.2548i q^{2} +(-36.3662 + 419.314i) q^{3} -2048.00 q^{4} -6987.71i q^{5} +(-18976.0 - 1645.75i) q^{6} +61433.6 q^{7} -92681.9i q^{8} +(-174502. - 30497.7i) q^{9} +316228. q^{10} +(74478.0 - 858756. i) q^{12} +2.78017e6i q^{14} +(2.93005e6 + 254116. i) q^{15} +4.19430e6 q^{16} +(1.38017e6 - 7.89706e6i) q^{18} +1.43108e7i q^{20} +(-2.23411e6 + 2.57600e7i) q^{21} +6.16350e7i q^{23} +(3.88628e7 + 3.37049e6i) q^{24} -4.88281e7 q^{25} +(1.91341e7 - 7.20621e7i) q^{27} -1.25816e8 q^{28} +2.15759e8i q^{29} +(-1.15000e7 + 1.32599e8i) q^{30} +1.89813e8i q^{32} -4.29280e8i q^{35} +(3.57380e8 + 6.24594e7i) q^{36} -6.47634e8 q^{40} -1.04605e9i q^{41} +(-1.16576e9 - 1.01104e8i) q^{42} +8.60868e8 q^{43} +(-2.13109e8 + 1.21937e9i) q^{45} -2.78928e9 q^{46} +1.64710e9i q^{47} +(-1.52531e8 + 1.75873e9i) q^{48} +1.79676e9 q^{49} -2.20971e9i q^{50} +(3.26116e9 + 8.65911e8i) q^{54} -5.69378e9i q^{56} -9.76413e9 q^{58} +(-6.00074e9 - 5.20431e8i) q^{60} +4.75443e9 q^{61} +(-1.07203e10 - 1.87358e9i) q^{63} -8.58993e9 q^{64} -2.13715e10 q^{67} +(-2.58444e10 - 2.24143e9i) q^{69} +1.94270e10 q^{70} +(-2.82659e9 + 1.61732e10i) q^{72} +(1.77569e9 - 2.04743e10i) q^{75} -2.93086e10i q^{80} +(2.95208e10 + 1.06438e10i) q^{81} +4.73389e10 q^{82} +6.93724e10i q^{83} +(4.57545e9 - 5.27564e10i) q^{84} +3.89584e10i q^{86} +(-9.04707e10 - 7.84632e9i) q^{87} +9.62796e10i q^{89} +(-5.51824e10 - 9.64423e9i) q^{90} -1.26228e11i q^{92} -7.45391e10 q^{94} +(-7.95911e10 - 6.90276e9i) q^{96} +8.13119e10i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8192 q^{4} - 75904 q^{6} - 698008 q^{9}+O(q^{10}) \) 4 * q - 8192 * q^4 - 75904 * q^6 - 698008 * q^9	\( 4 q - 8192 q^{4} - 75904 q^{6} - 698008 q^{9} + 16777216 q^{16} - 8936420 q^{21} + 155451392 q^{24} - 195312500 q^{25} - 46000000 q^{30} + 1429520384 q^{36} - 852437500 q^{45} - 11157121792 q^{46} + 7187026188 q^{49} + 13044633728 q^{54} + 19017715232 q^{61} - 34359738368 q^{64} - 103377706604 q^{69} + 77708000000 q^{70} + 118083345596 q^{81} + 18301788160 q^{84} - 298156262656 q^{94} - 318364450816 q^{96}+O(q^{100}) \) 4 * q - 8192 * q^4 - 75904 * q^6 - 698008 * q^9 + 16777216 * q^16 - 8936420 * q^21 + 155451392 * q^24 - 195312500 * q^25 - 46000000 * q^30 + 1429520384 * q^36 - 852437500 * q^45 - 11157121792 * q^46 + 7187026188 * q^49 + 13044633728 * q^54 + 19017715232 * q^61 - 34359738368 * q^64 - 103377706604 * q^69 + 77708000000 * q^70 + 118083345596 * q^81 + 18301788160 * q^84 - 298156262656 * q^94 - 318364450816 * q^96

Introduction

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