LMFDB - Embedded newform 60.6.h.b.59.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [60,6,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, 6, names="a")

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

chi := DirichletCharacter("60.59");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 60 = 2^{2} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
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:	$9.62302918878$
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}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			59.1
Root			$0.809017 - 1.40126i$ of defining polynomial
Character	$\chi$	$=$	60.59
Dual form			60.6.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)

$f(q)$	$=$	$q+(-5.59017 - 0.866025i) q^{2} -15.5885i q^{3} +(30.5000 + 9.68246i) q^{4} -55.9017 q^{5} +(-13.5000 + 87.1421i) q^{6} +(-162.115 - 80.5404i) q^{8} -243.000 q^{9} +O(q^{10})$	\(q+(-5.59017 - 0.866025i) q^{2} -15.5885i q^{3} +(30.5000 + 9.68246i) q^{4} -55.9017 q^{5} +(-13.5000 + 87.1421i) q^{6} +(-162.115 - 80.5404i) q^{8} -243.000 q^{9} +(312.500 + 48.4123i) q^{10} +(150.935 - 475.448i) q^{12} +871.421i q^{15} +(836.500 + 590.630i) q^{16} +648.460 q^{17} +(1358.41 + 210.444i) q^{18} +2285.06i q^{19} +(-1705.00 - 541.266i) q^{20} +4880.92i q^{23} +(-1255.50 + 2527.12i) q^{24} +3125.00 q^{25} +3788.00i q^{27} +(754.673 - 4871.39i) q^{30} -6932.64i q^{31} +(-4164.68 - 4026.15i) q^{32} +(-3625.00 - 561.583i) q^{34} +(-7411.50 - 2352.84i) q^{36} +(1978.92 - 12773.9i) q^{38} +(9062.50 + 4502.34i) q^{40} +13584.1 q^{45} +(4227.00 - 27285.2i) q^{46} +27757.8i q^{47} +(9207.01 - 13039.7i) q^{48} -16807.0 q^{49} +(-17469.3 - 2706.33i) q^{50} -10108.5i q^{51} -40897.7 q^{53} +(3280.50 - 21175.5i) q^{54} +35620.6 q^{57} +(-8437.50 + 26578.3i) q^{60} -34802.0 q^{61} +(-6003.84 + 38754.6i) q^{62} +(19794.5 + 26113.6i) q^{64} +(19778.0 + 6278.68i) q^{68} +76086.0 q^{69} +(39393.9 + 19571.3i) q^{72} -48713.9i q^{75} +(-22125.0 + 69694.3i) q^{76} +86019.0i q^{79} +(-46761.8 - 33017.2i) q^{80} +59049.0 q^{81} -102569. i q^{83} -36250.0 q^{85} +(-75937.5 - 11764.2i) q^{90} +(-47259.3 + 148868. i) q^{92} -108069. q^{93} +(24039.0 - 155171. i) q^{94} -127739. i q^{95} +(-62761.5 + 64920.9i) q^{96} +(93954.0 + 14555.3i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 122 q^{4} - 54 q^{6} - 972 q^{9}+O(q^{10}) $ 4 * q + 122 * q^4 - 54 * q^6 - 972 * q^9	$ 4 q + 122 q^{4} - 54 q^{6} - 972 q^{9} + 1250 q^{10} + 3346 q^{16} - 5022 q^{24} + 12500 q^{25} - 14500 q^{34} - 29646 q^{36} + 36250 q^{40} + 16908 q^{46} - 67228 q^{49} + 13122 q^{54} - 33750 q^{60} - 139208 q^{61} + 79178 q^{64} + 304344 q^{69} - 88500 q^{76} + 236196 q^{81} - 145000 q^{85} - 303750 q^{90} + 96156 q^{94} - 251046 q^{96}+O(q^{100}) $ 4 * q + 122 * q^4 - 54 * q^6 - 972 * q^9 + 1250 * q^10 + 3346 * q^16 - 5022 * q^24 + 12500 * q^25 - 14500 * q^34 - 29646 * q^36 + 36250 * q^40 + 16908 * q^46 - 67228 * q^49 + 13122 * q^54 - 33750 * q^60 - 139208 * q^61 + 79178 * q^64 + 304344 * q^69 - 88500 * q^76 + 236196 * q^81 - 145000 * q^85 - 303750 * q^90 + 96156 * q^94 - 251046 * 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 $

$2$	−5.59017	−	0.866025i	−0.988212	−	0.153093i
$2$	−5.59017	−	0.866025i	−0.988212	−	0.153093i
$3$		−	15.5885i		−	1.00000i
$3$		−	15.5885i		−	1.00000i
$4$	30.5000	+	9.68246i	0.953125	+	0.302577i
$5$	−55.9017			−1.00000
$5$	−55.9017			−1.00000
$6$	−13.5000	+	87.1421i	−0.153093	+	0.988212i
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$	−162.115	−	80.5404i	−0.895567	−	0.444927i
$9$	−243.000			−1.00000
$10$	312.500	+	48.4123i	0.988212	+	0.153093i
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	150.935	−	475.448i	0.302577	−	0.953125i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$			871.421i			1.00000i
$16$	836.500	+	590.630i	0.816895	+	0.576787i
$17$	648.460			0.544203			0.272101	−	0.962269i	$-0.412281\pi$
$17$	648.460			0.544203			0.272101	+	0.962269i	$0.412281\pi$
$18$	1358.41	+	210.444i	0.988212	+	0.153093i
$19$			2285.06i			1.45216i	0.687612	+	0.726079i	$0.258659\pi$
$19$			2285.06i			1.45216i	−0.687612	+	0.726079i	$0.741341\pi$
$20$	−1705.00	−	541.266i	−0.953125	−	0.302577i
$21$		0			0
$22$		0			0
$23$			4880.92i			1.92390i	0.273229	+	0.961949i	$0.411908\pi$
$23$			4880.92i			1.92390i	−0.273229	+	0.961949i	$0.588092\pi$
$24$	−1255.50	+	2527.12i	−0.444927	+	0.895567i
$25$	3125.00			1.00000
$26$		0			0
$27$			3788.00i			1.00000i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$	754.673	−	4871.39i	0.153093	−	0.988212i
$31$		−	6932.64i		−	1.29567i	−0.761781	−	0.647835i	$-0.775675\pi$
$31$		−	6932.64i		−	1.29567i	0.761781	−	0.647835i	$-0.224325\pi$
$32$	−4164.68	−	4026.15i	−0.718963	−	0.695049i
$33$		0			0
$34$	−3625.00	−	561.583i	−0.537788	−	0.0833137i
$35$		0			0
$36$	−7411.50	−	2352.84i	−0.953125	−	0.302577i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	1978.92	−	12773.9i	0.222315	−	1.43504i
$39$		0			0
$40$	9062.50	+	4502.34i	0.895567	+	0.444927i
$41$		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
$45$	13584.1			1.00000
$46$	4227.00	−	27285.2i	0.294536	−	1.90122i
$47$			27757.8i			1.83291i	0.400138	+	0.916455i	$0.368962\pi$
$47$			27757.8i			1.83291i	−0.400138	+	0.916455i	$0.631038\pi$
$48$	9207.01	−	13039.7i	0.576787	−	0.816895i
$49$	−16807.0			−1.00000
$50$	−17469.3	−	2706.33i	−0.988212	−	0.153093i
$51$		−	10108.5i		−	0.544203i
$52$		0			0
$53$	−40897.7			−1.99990			−0.999952	−	0.00982492i	$-0.996873\pi$
$53$	−40897.7			−1.99990			−0.999952	+	0.00982492i	$0.996873\pi$
$54$	3280.50	−	21175.5i	0.153093	−	0.988212i
$55$		0			0
$56$		0			0
$57$	35620.6			1.45216
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$	−8437.50	+	26578.3i	−0.302577	+	0.953125i
$61$	−34802.0			−1.19751			−0.598756	−	0.800932i	$-0.704338\pi$
$61$	−34802.0			−1.19751			−0.598756	+	0.800932i	$0.704338\pi$
$62$	−6003.84	+	38754.6i	−0.198358	+	1.28040i
$63$		0			0
$64$	19794.5	+	26113.6i	0.604080	+	0.796924i
$65$		0			0
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$	19778.0	+	6278.68i	0.518693	+	0.164663i
$69$	76086.0			1.92390
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	39393.9	+	19571.3i	0.895567	+	0.444927i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$		−	48713.9i		−	1.00000i
$76$	−22125.0	+	69694.3i	−0.439389	+	1.38409i
$77$		0			0
$78$		0			0
$79$			86019.0i			1.55070i	0.631535	+	0.775348i	$0.282425\pi$
$79$			86019.0i			1.55070i	−0.631535	+	0.775348i	$0.717575\pi$
$80$	−46761.8	−	33017.2i	−0.816895	−	0.576787i
$81$	59049.0			1.00000
$82$		0			0
$83$		−	102569.i		−	1.63425i	−0.576459	−	0.817126i	$-0.695566\pi$
$83$		−	102569.i		−	1.63425i	0.576459	−	0.817126i	$-0.304434\pi$
$84$		0			0
$85$	−36250.0			−0.544203
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	−75937.5	−	11764.2i	−0.988212	−	0.153093i
$91$		0			0
$92$	−47259.3	+	148868.i	−0.582127	+	1.83372i
$93$	−108069.			−1.29567
$94$	24039.0	−	155171.i	0.280606	−	1.81130i
$95$		−	127739.i		−	1.45216i
$96$	−62761.5	+	64920.9i	−0.695049	+	0.718963i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	93954.0	+	14555.3i	0.988212	+	0.153093i
$99$		0			0
$100$	95312.5	+	30257.7i	0.953125	+	0.302577i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$	−8754.21	+	56508.2i	−0.0833137	+	0.537788i
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$	228625.	+	35418.4i	1.97633	+	0.306171i
$107$			115656.i			0.976582i	0.872681	+	0.488291i	$0.162379\pi$
$107$			115656.i			0.976582i	−0.872681	+	0.488291i	$0.837621\pi$
$108$	−36677.1	+	115534.i	−0.302577	+	0.953125i
$109$	−125986.			−1.01568			−0.507839	−	0.861452i	$-0.669556\pi$
$109$	−125986.			−1.01568			−0.507839	+	0.861452i	$0.669556\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	236777.			1.74439			0.872196	−	0.489157i	$-0.162696\pi$
$113$	236777.			1.74439			0.872196	+	0.489157i	$0.162696\pi$
$114$	−199125.	−	30848.3i	−1.43504	−	0.222315i
$115$		−	272852.i		−	1.92390i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$	70184.6	−	141270.i	0.444927	−	0.895567i
$121$	−161051.			−1.00000
$122$	194549.	+	30139.4i	1.18339	+	0.183331i
$123$		0			0
$124$	67125.0	−	211446.i	0.392040	−	1.23494i
$125$	−174693.			−1.00000
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	−88039.6	−	163122.i	−0.474956	−	0.880010i
$129$		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
$135$		−	211755.i		−	1.00000i
$136$	−105125.	−	52227.2i	−0.487370	−	0.242130i
$137$	30075.1			0.136901			0.0684504	−	0.997655i	$-0.478195\pi$
$137$	30075.1			0.136901			0.0684504	+	0.997655i	$0.478195\pi$
$138$	−425334.	−	65892.4i	−1.90122	−	0.294536i
$139$			299885.i			1.31649i	0.752803	+	0.658245i	$0.228701\pi$
$139$			299885.i			1.31649i	−0.752803	+	0.658245i	$0.771299\pi$
$140$		0			0
$141$	432702.			1.83291
$142$		0			0
$143$		0			0
$144$	−203270.	−	143523.i	−0.816895	−	0.576787i
$145$		0			0
$146$		0			0
$147$			261995.i			1.00000i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$	−42187.5	+	272319.i	−0.153093	+	0.988212i
$151$			48683.4i			0.173755i	0.996219	+	0.0868777i	$0.0276890\pi$
$151$			48683.4i			0.173755i	−0.996219	+	0.0868777i	$0.972311\pi$
$152$	184040.	−	370442.i	0.646104	−	1.30050i
$153$	−157576.			−0.544203
$154$		0			0
$155$			387546.i			1.29567i
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$	74494.6	−	480861.i	0.237401	−	1.53242i
$159$			637532.i			1.99990i
$160$	232813.	+	225069.i	0.718963	+	0.695049i
$161$		0			0
$162$	−330094.	−	51137.9i	−0.988212	−	0.153093i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		0			0
$165$		0			0
$166$	−88827.0	+	573376.i	−0.250193	+	1.61499i
$167$			140424.i			0.389629i	0.980840	+	0.194814i	$0.0624105\pi$
$167$			140424.i			0.389629i	−0.980840	+	0.194814i	$0.937590\pi$
$168$		0			0
$169$	371293.			1.00000
$170$	202644.	+	31393.4i	0.537788	+	0.0833137i
$171$		−	555270.i		−	1.45216i
$172$		0			0
$173$	734660.			1.86626			0.933128	−	0.359544i	$-0.117068\pi$
$173$	734660.			1.86626			0.933128	+	0.359544i	$0.117068\pi$
$174$		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	$0.5\pi$
$180$	414315.	+	131528.i	0.953125	+	0.302577i
$181$	−879098.			−1.99453			−0.997266	−	0.0739001i	$-0.976455\pi$
$181$	−879098.			−1.99453			−0.997266	+	0.0739001i	$0.976455\pi$
$182$		0			0
$183$			542509.i			1.19751i
$184$	393111.	−	791270.i	0.855994	−	1.72298i
$185$		0			0
$186$	604125.	+	93590.6i	1.28040	+	0.198358i
$187$		0			0
$188$	−268764.	+	846614.i	−0.554596	+	1.74699i
$189$		0			0
$190$	−110625.	+	714081.i	−0.222315	+	1.43504i
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	407071.	−	308566.i	0.796924	−	0.604080i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$	−512614.	−	162733.i	−0.953125	−	0.302577i
$197$	−781483.			−1.43468			−0.717339	−	0.696725i	$-0.754640\pi$
$197$	−781483.			−1.43468			−0.717339	+	0.696725i	$0.754640\pi$
$198$		0			0
$199$			1.10833e6i			1.98398i	0.126321	+	0.991989i	$0.459683\pi$
$199$			1.10833e6i			1.98398i	−0.126321	+	0.991989i	$0.540317\pi$
$200$	−506609.	−	251689.i	−0.895567	−	0.444927i
$201$		0			0
$202$		0			0
$203$		0			0
$204$	97875.0	−	308309.i	0.164663	−	0.518693i
$205$		0			0
$206$		0			0
$207$		−	1.18606e6i		−	1.92390i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		−	1.26186e6i		−	1.95121i	−0.219534	−	0.975605i	$-0.570454\pi$
$211$		−	1.26186e6i		−	1.95121i	0.219534	−	0.975605i	$-0.429546\pi$
$212$	−1.24738e6	−	395990.i	−1.90616	−	0.605124i
$213$		0			0
$214$	100161.	−	646536.i	0.149508	−	0.965070i
$215$		0			0
$216$	305086.	−	614091.i	0.444927	−	0.895567i
$217$		0			0
$218$	704283.	+	109107.i	1.00370	+	0.155493i
$219$		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
$225$	−759375.			−1.00000
$226$	−1.32362e6	−	205055.i	−1.72383	−	0.267054i
$227$		−	1.55170e6i		−	1.99868i	−0.0363479	−	0.999339i	$-0.511572\pi$
$227$		−	1.55170e6i		−	1.99868i	0.0363479	−	0.999339i	$-0.488428\pi$
$228$	1.08643e6	+	344895.i	1.38409	+	0.439389i
$229$	1.42581e6			1.79669			0.898347	−	0.439286i	$-0.144769\pi$
$229$	1.42581e6			1.79669			0.898347	+	0.439286i	$0.144769\pi$
$230$	−236296.	+	1.52529e6i	−0.294536	+	1.90122i
$231$		0			0
$232$		0			0
$233$	−1.36523e6			−1.64747			−0.823733	−	0.566978i	$-0.808112\pi$
$233$	−1.36523e6			−1.64747			−0.823733	+	0.566978i	$0.808112\pi$
$234$		0			0
$235$		−	1.55171e6i		−	1.83291i
$236$		0			0
$237$	1.34090e6			1.55070
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$	−514688.	+	728944.i	−0.576787	+	0.816895i
$241$	571202.			0.633501			0.316750	−	0.948509i	$-0.397408\pi$
$241$	571202.			0.633501			0.316750	+	0.948509i	$0.397408\pi$
$242$	900302.	+	139474.i	0.988212	+	0.153093i
$243$		−	920483.i		−	1.00000i
$244$	−1.06146e6	−	336969.i	−1.14138	−	0.362339i
$245$	939540.			1.00000
$246$		0			0
$247$		0			0
$248$	−558357.	+	1.12388e6i	−0.576478	+	1.16036i
$249$	−1.59889e6			−1.63425
$250$	976562.	+	151288.i	0.988212	+	0.153093i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$			565082.i			0.544203i
$256$	350888.	+	988124.i	0.334633	+	0.942348i
$257$	−981254.			−0.926720			−0.463360	−	0.886170i	$-0.653356\pi$
$257$	−981254.			−0.926720			−0.463360	+	0.886170i	$0.653356\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$			388121.i			0.346002i	0.984922	+	0.173001i	$0.0553463\pi$
$263$			388121.i			0.346002i	−0.984922	+	0.173001i	$0.944654\pi$
$264$		0			0
$265$	2.28625e6			1.99990
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$	−183386.	+	1.18375e6i	−0.153093	+	0.988212i
$271$			2.24250e6i			1.85485i	0.374011	+	0.927424i	$0.377982\pi$
$271$			2.24250e6i			1.85485i	−0.374011	+	0.927424i	$0.622018\pi$
$272$	542437.	+	383000.i	0.444556	+	0.313889i
$273$		0			0
$274$	−168125.	−	26045.8i	−0.135287	−	0.0209586i
$275$		0			0
$276$	2.32062e6	+	736700.i	1.83372	+	0.582127i
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$	259708.	−	1.67641e6i	0.201546	−	1.30097i
$279$			1.68463e6i			1.29567i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$	−2.41888e6	−	374731.i	−1.81130	−	0.280606i
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$		0			0
$285$	−1.99125e6			−1.45216
$286$		0			0
$287$		0			0
$288$	1.01202e6	+	978355.i	0.718963	+	0.695049i
$289$	−999357.			−0.703843
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.44175e6			−0.981117			−0.490558	−	0.871408i	$-0.663207\pi$
$293$	−1.44175e6			−0.981117			−0.490558	+	0.871408i	$0.663207\pi$
$294$	226894.	−	1.46460e6i	0.153093	−	0.988212i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	471671.	−	1.48577e6i	0.302577	−	0.953125i
$301$		0			0
$302$	42161.1	−	272148.i	0.0266008	−	0.171707i
$303$		0			0
$304$	−1.34962e6	+	1.91145e6i	−0.837586	+	1.18626i
$305$	1.94549e6			1.19751
$306$	880875.	+	136465.i	0.537788	+	0.0833137i
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$		0			0
$310$	335625.	−	2.16645e6i	0.198358	−	1.28040i
$311$		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
$316$	−832875.	+	2.62358e6i	−0.469204	+	1.47801i
$317$	895657.			0.500603			0.250302	−	0.968168i	$-0.419470\pi$
$317$	895657.			0.500603			0.250302	+	0.968168i	$0.419470\pi$
$318$	552119.	−	3.56391e6i	0.306171	−	1.97633i
$319$		0			0
$320$	−1.10655e6	−	1.45979e6i	−0.604080	−	0.796924i
$321$	1.80290e6			0.976582
$322$		0			0
$323$			1.48177e6i			0.790268i
$324$	1.80099e6	+	571739.i	0.953125	+	0.302577i
$325$		0			0
$326$		0			0
$327$			1.96393e6i			1.01568i
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	1.26170e6i		−	0.632975i	−0.948597	−	0.316488i	$-0.897496\pi$
$331$		−	1.26170e6i		−	0.632975i	0.948597	−	0.316488i	$-0.102504\pi$
$332$	993116.	−	3.12834e6i	0.494487	−	1.55765i
$333$		0			0
$334$	121611.	−	784996.i	0.0596495	−	0.385036i
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	−2.07559e6	−	321549.i	−0.988212	−	0.153093i
$339$		−	3.69099e6i		−	1.74439i
$340$	−1.10562e6	−	350989.i	−0.518693	−	0.164663i
$341$		0			0
$342$	−480878.	+	3.10405e6i	−0.222315	+	1.43504i
$343$		0			0
$344$		0			0
$345$	−4.25334e6			−1.92390
$346$	−4.10688e6	−	636234.i	−1.84426	−	0.285711i
$347$			4.43054e6i			1.97530i	0.156676	+	0.987650i	$0.449922\pi$
$347$			4.43054e6i			1.97530i	−0.156676	+	0.987650i	$0.550078\pi$
$348$		0			0
$349$	2.44389e6			1.07403			0.537016	−	0.843572i	$-0.319551\pi$
$349$	2.44389e6			1.07403			0.537016	+	0.843572i	$0.319551\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	4.57851e6			1.95563			0.977816	−	0.209467i	$-0.0671729\pi$
$353$	4.57851e6			1.95563			0.977816	+	0.209467i	$0.0671729\pi$
$354$		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	$0.5\pi$
$360$	−2.20219e6	−	1.09407e6i	−0.895567	−	0.444927i
$361$	−2.74540e6			−1.10876
$362$	4.91431e6	+	761321.i	1.97102	+	0.305349i
$363$			2.51054e6i			1.00000i
$364$		0			0
$365$		0			0
$366$	469827.	−	3.03272e6i	0.183331	−	1.18339i
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$	−2.88282e6	+	4.08289e6i	−1.10968	+	1.57162i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−3.29611e6	−	1.04638e6i	−1.23494	−	0.392040i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$			2.72319e6i			1.00000i
$376$	2.23563e6	−	4.49996e6i	0.815511	−	1.64149i
$377$		0			0
$378$		0			0
$379$		−	4.86853e6i		−	1.74101i	−0.492164	−	0.870503i	$-0.663794\pi$
$379$		−	4.86853e6i		−	1.74101i	0.492164	−	0.870503i	$-0.336206\pi$
$380$	1.23683e6	−	3.89603e6i	0.439389	−	1.38409i
$381$		0			0
$382$		0			0
$383$		−	2.33085e6i		−	0.811928i	−0.913889	−	0.405964i	$-0.866936\pi$
$383$		−	2.33085e6i		−	0.811928i	0.913889	−	0.405964i	$-0.133064\pi$
$384$	−2.54282e6	+	1.37240e6i	−0.880010	+	0.474956i
$385$		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
$391$			3.16508e6i			1.04699i
$392$	2.72467e6	+	1.35364e6i	0.895567	+	0.444927i
$393$		0			0
$394$	4.36862e6	+	676784.i	1.41776	+	0.219639i
$395$		−	4.80861e6i		−	1.55070i
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	959843.	−	6.19576e6i	0.303733	−	1.96059i
$399$		0			0
$400$	2.61406e6	+	1.84572e6i	0.816895	+	0.576787i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−3.30094e6			−1.00000
$406$		0			0
$407$		0			0
$408$	−814141.	+	1.63874e6i	−0.242130	+	0.487370i
$409$	−2.31501e6			−0.684298			−0.342149	−	0.939646i	$-0.611155\pi$
$409$	−2.31501e6			−0.684298			−0.342149	+	0.939646i	$0.611155\pi$
$410$		0			0
$411$		−	468825.i		−	0.136901i
$412$		0			0
$413$		0			0
$414$	−1.02716e6	+	6.63030e6i	−0.294536	+	1.90122i
$415$			5.73376e6i			1.63425i
$416$		0			0
$417$	4.67475e6			1.31649
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−2.59460e6			−0.713453			−0.356727	−	0.934209i	$-0.616107\pi$
$421$	−2.59460e6			−0.713453			−0.356727	+	0.934209i	$0.616107\pi$
$422$	−1.09280e6	+	7.05399e6i	−0.298717	+	1.92821i
$423$		−	6.74516e6i		−	1.83291i
$424$	6.63012e6	+	3.29391e6i	1.79105	+	0.889811i
$425$	2.02644e6			0.544203
$426$		0			0
$427$		0			0
$428$	−1.11983e6	+	3.52751e6i	−0.295491	+	0.930804i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	−2.23730e6	+	3.16866e6i	−0.576787	+	0.816895i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$	−3.84257e6	−	1.21985e6i	−0.968068	−	0.307321i
$437$	−1.11532e7			−2.79380
$438$		0			0
$439$		−	3.05559e6i		−	0.756718i	−0.925659	−	0.378359i	$-0.876489\pi$
$439$		−	3.05559e6i		−	0.756718i	0.925659	−	0.378359i	$-0.123511\pi$
$440$		0			0
$441$	4.08410e6			1.00000
$442$		0			0
$443$		−	4.82529e6i		−	1.16819i	−0.811685	−	0.584096i	$-0.801449\pi$
$443$		−	4.82529e6i		−	1.16819i	0.811685	−	0.584096i	$-0.198551\pi$
$444$		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			$\pi$
$450$	4.24504e6	+	657638.i	0.988212	+	0.153093i
$451$		0			0
$452$	7.22171e6	+	2.29259e6i	1.66262	+	0.527812i
$453$	758899.			0.173755
$454$	−1.34381e6	+	8.67426e6i	−0.305984	+	1.97512i
$455$		0			0
$456$	−5.77462e6	−	2.86889e6i	−1.30050	−	0.646104i
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$	−7.97054e6	−	1.23479e6i	−1.77551	−	0.275062i
$459$			2.45636e6i			0.544203i
$460$	2.64188e6	−	8.32198e6i	0.582127	−	1.83372i
$461$		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
$465$	6.04125e6			1.29567
$466$	7.63188e6	+	1.18232e6i	1.62805	+	0.252216i
$467$			1.53274e6i			0.325219i	0.986691	+	0.162609i	$0.0519910\pi$
$467$			1.53274e6i			0.325219i	−0.986691	+	0.162609i	$0.948009\pi$
$468$		0			0
$469$		0			0
$470$	−1.34382e6	+	8.67433e6i	−0.280606	+	1.81130i
$471$		0			0
$472$		0			0
$473$		0			0
$474$	−7.49588e6	−	1.16126e6i	−1.53242	−	0.237401i
$475$			7.14081e6i			1.45216i
$476$		0			0
$477$	9.93814e6			1.99990
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$	3.50847e6	−	3.62919e6i	0.695049	−	0.718963i
$481$		0			0
$482$	−3.19312e6	−	494675.i	−0.626033	−	0.0969846i
$483$		0			0
$484$	−4.91206e6	−	1.55937e6i	−0.953125	−	0.302577i
$485$		0			0
$486$	−797162.	+	5.14566e6i	−0.153093	+	0.988212i
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$	5.64192e6	+	2.80297e6i	1.07245	+	0.532805i
$489$		0			0
$490$	−5.25219e6	−	813665.i	−0.988212	−	0.153093i
$491$		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
$496$	4.09463e6	−	5.79915e6i	0.747326	−	1.05843i
$497$		0			0
$498$	8.93804e6	+	1.38468e6i	1.61499	+	0.250193i
$499$		−	8.51208e6i		−	1.53033i	−0.643836	−	0.765164i	$-0.722658\pi$
$499$		−	8.51208e6i		−	1.53033i	0.643836	−	0.765164i	$-0.277342\pi$
$500$	−5.32813e6	−	1.69146e6i	−0.953125	−	0.302577i
$501$	2.18900e6			0.389629
$502$		0			0
$503$			4.27820e6i			0.753947i	0.926224	+	0.376973i	$0.123035\pi$
$503$			4.27820e6i			0.753947i	−0.926224	+	0.376973i	$0.876965\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	5.78789e6i		−	1.00000i
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$	489375.	−	3.15890e6i	0.0833137	−	0.537788i
$511$		0			0
$512$	−1.10579e6	−	5.82766e6i	−0.186422	−	0.982470i
$513$	−8.65580e6			−1.45216
$514$	5.48538e6	+	849791.i	0.915796	+	0.141874i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		−	1.14522e7i		−	1.86626i
$520$		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
$526$	336123.	−	2.16966e6i	0.0529705	−	0.341923i
$527$		−	4.49554e6i		−	0.705107i
$528$		0			0
$529$	−1.73870e7			−2.70138
$530$	−1.27805e7	−	1.97995e6i	−1.97633	−	0.306171i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		−	6.46536e6i		−	0.976582i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$	2.05031e6	−	6.45854e6i	0.302577	−	0.953125i
$541$	−2.90520e6			−0.426759			−0.213380	−	0.976969i	$-0.568447\pi$
$541$	−2.90520e6			−0.426759			−0.213380	+	0.976969i	$0.568447\pi$
$542$	1.94206e6	−	1.25359e7i	0.283965	−	1.83298i
$543$			1.37038e7i			1.99453i
$544$	−2.70063e6	−	2.61080e6i	−0.391261	−	0.378247i
$545$	7.04283e6			1.01568
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$	917291.	+	291201.i	0.130484	+	0.0414230i
$549$	8.45689e6			1.19751
$550$		0			0
$551$		0			0
$552$	−1.23347e7	−	6.12799e6i	−1.72298	−	0.855994i
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−2.90362e6	+	9.14650e6i	−0.398340	+	1.25478i
$557$	−9.13982e6			−1.24824			−0.624122	−	0.781327i	$-0.714543\pi$
$557$	−9.13982e6			−1.24824			−0.624122	+	0.781327i	$0.714543\pi$
$558$	1.45893e6	−	9.41738e6i	0.198358	−	1.28040i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$			6.43962e6i			0.856228i	0.903725	+	0.428114i	$0.140822\pi$
$563$			6.43962e6i			0.856228i	−0.903725	+	0.428114i	$0.859178\pi$
$564$	1.31974e7	+	4.18962e6i	1.74699	+	0.554596i
$565$	−1.32362e7			−1.74439
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$	1.11314e7	+	1.72447e6i	1.43504	+	0.222315i
$571$			1.55808e7i			1.99986i	0.0117539	+	0.999931i	$0.496259\pi$
$571$			1.55808e7i			1.99986i	−0.0117539	+	0.999931i	$0.503741\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			1.52529e7i			1.92390i
$576$	−4.81006e6	−	6.34560e6i	−0.604080	−	0.796924i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	5.58658e6	+	865469.i	0.695546	+	0.107754i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	8.05962e6	+	1.24859e6i	0.969551	+	0.150202i
$587$			5.24868e6i			0.628716i	0.949304	+	0.314358i	$0.101789\pi$
$587$			5.24868e6i			0.628716i	−0.949304	+	0.314358i	$0.898211\pi$
$588$	−2.53676e6	+	7.99085e6i	−0.302577	+	0.953125i
$589$	1.58415e7			1.88152
$590$		0			0
$591$			1.21821e7i			1.43468i
$592$		0			0
$593$	7.59970e6			0.887483			0.443741	−	0.896155i	$-0.353651\pi$
$593$	7.59970e6			0.887483			0.443741	+	0.896155i	$0.353651\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	1.72772e7			1.98398
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	−3.92344e6	+	7.89726e6i	−0.444927	+	0.895567i
$601$	1.70270e7			1.92288			0.961440	−	0.275016i	$-0.0886832\pi$
$601$	1.70270e7			1.92288			0.961440	+	0.275016i	$0.0886832\pi$
$602$		0			0
$603$		0			0
$604$	−471375.	+	1.48484e6i	−0.0525744	+	0.165611i
$605$	9.00302e6			1.00000
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$	9.20000e6	−	9.51654e6i	1.00932	−	1.04405i
$609$		0			0
$610$	−1.08756e7	−	1.68484e6i	−1.18339	−	0.183331i
$611$		0			0
$612$	−4.80606e6	−	1.52572e6i	−0.518693	−	0.164663i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.44700e7			−1.53023			−0.765114	−	0.643895i	$-0.777318\pi$
$617$	−1.44700e7			−1.53023			−0.765114	+	0.643895i	$0.777318\pi$
$618$		0			0
$619$			1.24580e7i			1.30683i	0.756999	+	0.653417i	$0.226665\pi$
$619$			1.24580e7i			1.30683i	−0.756999	+	0.653417i	$0.773335\pi$
$620$	−3.75240e6	+	1.18202e7i	−0.392040	+	1.23494i
$621$	−1.84889e7			−1.92390
$622$		0			0
$623$		0			0
$624$		0			0
$625$	9.76562e6			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		−	1.90433e7i		−	1.90400i	−0.306091	−	0.952002i	$-0.599021\pi$
$631$		−	1.90433e7i		−	1.90400i	0.306091	−	0.952002i	$-0.400979\pi$
$632$	6.92800e6	−	1.39450e7i	0.689946	−	1.38875i
$633$	−1.96704e7			−1.95121
$634$	−5.00688e6	−	775662.i	−0.494702	−	0.0766389i
$635$		0			0
$636$	−6.17288e6	+	1.94447e7i	−0.605124	+	1.90616i
$637$		0			0
$638$		0			0
$639$		0			0
$640$	4.92156e6	+	9.11879e6i	0.474956	+	0.880010i
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$	−1.00785e7	−	1.56136e6i	−0.965070	−	0.149508i
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$	1.28325e6	−	8.28334e6i	0.120985	−	0.780952i
$647$		−	1.30121e7i		−	1.22204i	−0.791615	−	0.611021i	$-0.790759\pi$
$647$		−	1.30121e7i		−	1.22204i	0.791615	−	0.611021i	$-0.209241\pi$
$648$	−9.57272e6	−	4.75583e6i	−0.895567	−	0.444927i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	4.28697e6			0.393430			0.196715	−	0.980461i	$-0.436973\pi$
$653$	4.28697e6			0.393430			0.196715	+	0.980461i	$0.436973\pi$
$654$	1.70081e6	−	1.09787e7i	0.155493	−	1.00370i
$655$		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
$661$	1.80233e7			1.60447			0.802233	−	0.597011i	$-0.203645\pi$
$661$	1.80233e7			1.60447			0.802233	+	0.597011i	$0.203645\pi$
$662$	−1.09267e6	+	7.05313e6i	−0.0969041	+	0.625514i
$663$		0			0
$664$	−8.26091e6	+	1.66279e7i	−0.727123	+	1.46358i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−1.35965e6	+	4.28294e6i	−0.117893	+	0.371365i
$669$		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
$675$			1.18375e7i			1.00000i
$676$	1.13244e7	+	3.59503e6i	0.953125	+	0.302577i
$677$	−2.04280e6			−0.171299			−0.0856496	−	0.996325i	$-0.527297\pi$
$677$	−2.04280e6			−0.171299			−0.0856496	+	0.996325i	$0.527297\pi$
$678$	−3.19649e6	+	2.06333e7i	−0.267054	+	1.72383i
$679$		0			0
$680$	5.87667e6	+	2.91959e6i	0.487370	+	0.242130i
$681$	−2.41886e7			−1.99868
$682$		0			0
$683$		−	1.97320e7i		−	1.61853i	−0.587447	−	0.809263i	$-0.699867\pi$
$683$		−	1.97320e7i		−	1.61853i	0.587447	−	0.809263i	$-0.300133\pi$
$684$	5.37638e6	−	1.69357e7i	0.439389	−	1.38409i
$685$	−1.68125e6			−0.136901
$686$		0			0
$687$		−	2.22262e7i		−	1.79669i
$688$		0			0
$689$		0			0
$690$	2.37769e7	+	3.68350e6i	1.90122	+	0.294536i
$691$		−	1.69149e7i		−	1.34764i	−0.738895	−	0.673821i	$-0.764652\pi$
$691$		−	1.69149e7i		−	1.34764i	0.738895	−	0.673821i	$-0.235348\pi$
$692$	2.24071e7	+	7.11332e6i	1.77878	+	0.564686i
$693$		0			0
$694$	3.83696e6	−	2.47675e7i	0.302405	−	1.95201i
$695$		−	1.67641e7i		−	1.31649i
$696$		0			0
$697$		0			0
$698$	−1.36617e7	−	2.11647e6i	−1.06137	−	0.164427i
$699$			2.12818e7i			1.64747i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−2.41888e7			−1.83291
$706$	−2.55946e7	−	3.96510e6i	−1.93258	−	0.299394i
$707$		0			0
$708$		0			0
$709$	−1.49230e7			−1.11491			−0.557455	−	0.830207i	$-0.688222\pi$
$709$	−1.49230e7			−1.11491			−0.557455	+	0.830207i	$0.688222\pi$
$710$		0			0
$711$		−	2.09026e7i		−	1.55070i
$712$		0			0
$713$	3.38377e7			2.49274
$714$		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	$0.5\pi$
$720$	1.13631e7	+	8.02318e6i	0.816895	+	0.576787i
$721$		0			0
$722$	1.53473e7	+	2.37759e6i	1.09569	+	0.169744i
$723$		−	8.90416e6i		−	0.633501i
$724$	−2.68125e7	−	8.51183e6i	−1.90104	−	0.603499i
$725$		0			0
$726$	2.17419e6	−	1.40343e7i	0.153093	−	0.988212i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−1.43489e7			−1.00000
$730$		0			0
$731$		0			0
$732$	−5.25283e6	+	1.65465e7i	−0.362339	+	1.14138i
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		−	1.46460e7i		−	1.00000i
$736$	1.96513e7	−	2.03274e7i	1.33720	−	1.38321i
$737$		0			0
$738$		0			0
$739$			2.77022e7i			1.86596i	0.359926	+	0.932981i	$0.382802\pi$
$739$			2.77022e7i			1.86596i	−0.359926	+	0.932981i	$0.617198\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			2.93216e6i			0.194857i	0.995243	+	0.0974283i	$0.0310617\pi$
$743$			2.93216e6i			0.194857i	−0.995243	+	0.0974283i	$0.968938\pi$
$744$	1.75196e7	+	8.70393e6i	1.16036	+	0.576478i
$745$		0			0
$746$		0			0
$747$			2.49242e7i			1.63425i
$748$		0			0
$749$		0			0
$750$	2.35835e6	−	1.52231e7i	0.153093	−	0.988212i
$751$		−	2.29849e7i		−	1.48711i	−0.668676	−	0.743554i	$-0.733139\pi$
$751$		−	2.29849e7i		−	1.48711i	0.668676	−	0.743554i	$-0.266861\pi$
$752$	−1.63946e7	+	2.32194e7i	−1.05720	+	1.49729i
$753$		0			0
$754$		0			0
$755$		−	2.72148e6i		−	0.173755i
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	−4.21627e6	+	2.72159e7i	−0.266536	+	1.72048i
$759$		0			0
$760$	−1.02881e7	+	2.07084e7i	−0.646104	+	1.30050i
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	8.80875e6			0.544203
$766$	−2.01858e6	+	1.30299e7i	−0.124301	+	0.802357i
$767$		0			0
$768$	1.54033e7	−	5.46981e6i	0.942348	−	0.334633i
$769$	3.22809e7			1.96848			0.984238	−	0.176851i	$-0.0565910\pi$
$769$	3.22809e7			1.96848			0.984238	+	0.176851i	$0.0565910\pi$
$770$		0			0
$771$			1.52962e7i			0.926720i
$772$		0			0
$773$	−1.30172e7			−0.783557			−0.391778	−	0.920060i	$-0.628140\pi$
$773$	−1.30172e7			−0.783557			−0.391778	+	0.920060i	$0.628140\pi$
$774$		0			0
$775$		−	2.16645e7i		−	1.29567i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	2.74104e6	−	1.76933e7i	0.160287	−	1.03465i
$783$		0			0
$784$	−1.40591e7	−	9.92672e6i	−0.816895	−	0.576787i
$785$		0			0
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$	−2.38352e7	−	7.56668e6i	−1.36743	−	0.434100i
$789$	6.05021e6			0.346002
$790$	−4.16438e6	+	2.68809e7i	−0.237401	+	1.53242i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		−	3.56391e7i		−	1.99990i
$796$	−1.07314e7	+	3.38041e7i	−0.600306	+	1.89098i
$797$	−3.00698e7			−1.67681			−0.838406	−	0.545046i	$-0.816512\pi$
$797$	−3.00698e7			−1.67681			−0.838406	+	0.545046i	$0.816512\pi$
$798$		0			0
$799$			1.79998e7i			0.997475i
$800$	−1.30146e7	−	1.25817e7i	−0.718963	−	0.695049i
$801$		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			$\pi$
$810$	1.84528e7	+	2.85870e6i	0.988212	+	0.153093i
$811$			3.23124e7i			1.72511i	0.505962	+	0.862556i	$0.331138\pi$
$811$			3.23124e7i			1.72511i	−0.505962	+	0.862556i	$0.668862\pi$
$812$		0			0
$813$	3.49571e7			1.85485
$814$		0			0
$815$		0			0
$816$	5.97037e6	−	8.45575e6i	0.313889	−	0.444556i
$817$		0			0
$818$	1.29413e7	+	2.00486e6i	0.676231	+	0.104761i
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$	−406014.	+	2.62081e6i	−0.0209586	+	0.135287i
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			3.84292e7i			1.95388i	0.213514	+	0.976940i	$0.431509\pi$
$827$			3.84292e7i			1.95388i	−0.213514	+	0.976940i	$0.568491\pi$
$828$	1.14840e7	−	3.61749e7i	0.582127	−	1.83372i
$829$	648686.			0.0327830			0.0163915	−	0.999866i	$-0.494782\pi$
$829$	648686.			0.0327830			0.0163915	+	0.999866i	$0.494782\pi$
$830$	4.96558e6	−	3.20527e7i	0.250193	−	1.61499i
$831$		0			0
$832$		0			0
$833$	−1.08987e7			−0.544203
$834$	−2.61326e7	−	4.04845e6i	−1.30097	−	0.201546i
$835$		−	7.84996e6i		−	0.389629i
$836$		0			0
$837$	2.62608e7			1.29567
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	2.05111e7			1.00000
$842$	1.45043e7	+	2.24699e6i	0.705043	+	0.109225i
$843$		0			0
$844$	1.22179e7	−	3.84866e7i	0.590391	−	1.85975i
$845$	−2.07559e7			−1.00000
$846$	−5.84148e6	+	3.77066e7i	−0.280606	+	1.81130i
$847$		0			0
$848$	−3.42109e7	−	2.41554e7i	−1.63371	−	1.15352i
$849$		0			0
$850$	−1.13281e7	−	1.75495e6i	−0.537788	−	0.0833137i
$851$		0			0
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$			3.10405e7i			1.45216i
$856$	9.31497e6	−	1.87496e7i	0.434507	−	0.874594i
$857$	3.56585e7			1.65848			0.829242	−	0.558890i	$-0.188773\pi$
$857$	3.56585e7			1.65848			0.829242	+	0.558890i	$0.188773\pi$
$858$		0			0
$859$			1.94848e7i			0.900975i	0.892783	+	0.450487i	$0.148750\pi$
$859$			1.94848e7i			0.900975i	−0.892783	+	0.450487i	$0.851250\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			1.65083e7i			0.754529i	0.926106	+	0.377264i	$0.123135\pi$
$863$			1.65083e7i			0.754529i	−0.926106	+	0.377264i	$0.876865\pi$
$864$	1.52510e7	−	1.57758e7i	0.695049	−	0.718963i
$865$	−4.10688e7			−1.86626
$866$		0			0
$867$			1.55784e7i			0.703843i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	2.04242e7	+	1.01470e7i	0.909608	+	0.451902i
$873$		0			0
$874$	6.23482e7	+	9.65895e6i	2.76087	+	0.427712i
$875$		0			0
$876$		0			0
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$	−2.64622e6	+	1.70813e7i	−0.115848	+	0.747797i
$879$			2.24747e7i			0.981117i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−2.28308e7	−	3.53694e6i	−0.988212	−	0.153093i
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$		0			0
$886$	−4.17882e6	+	2.69742e7i	−0.178842	+	1.15442i
$887$			3.41731e7i			1.45839i	0.684304	+	0.729197i	$0.260106\pi$
$887$			3.41731e7i			1.45839i	−0.684304	+	0.729197i	$0.739894\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−6.34283e7			−2.66167
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−2.31609e7	−	7.35262e6i	−0.953125	−	0.302577i
$901$	−2.65205e7			−1.08835
$902$		0			0
$903$		0			0
$904$	−3.83851e7	−	1.90701e7i	−1.56222	−	0.776126i
$905$	4.91431e7			1.99453
$906$	−4.24238e6	−	657226.i	−0.171707	−	0.0266008i
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$	1.50243e7	−	4.73268e7i	0.604754	−	1.90499i
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$	2.97966e7	+	2.10386e7i	1.18626	+	0.837586i
$913$		0			0
$914$		0			0
$915$		−	3.03272e7i		−	1.19751i
$916$	4.34873e7	+	1.38054e7i	1.71247	+	0.543638i
$917$		0			0
$918$	2.12727e6	−	1.37315e7i	0.0833137	−	0.537788i
$919$		−	4.72451e7i		−	1.84530i	−0.385634	−	0.922652i	$-0.626017\pi$
$919$		−	4.72451e7i		−	1.84530i	0.385634	−	0.922652i	$-0.373983\pi$
$920$	−2.19756e7	+	4.42333e7i	−0.855994	+	1.72298i
$921$		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			$\pi$
$930$	−3.37716e7	−	5.23188e6i	−1.28040	−	0.198358i
$931$		−	3.84050e7i		−	1.45216i
$932$	−4.16396e7	−	1.32188e7i	−1.57024	−	0.498485i
$933$		0			0
$934$	1.32739e6	−	8.56826e6i	0.0497887	−	0.321385i
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$	1.50244e7	−	4.73272e7i	0.554596	−	1.74699i
$941$		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
$947$		−	5.30532e6i		−	0.192237i	−0.995370	−	0.0961184i	$-0.969357\pi$
$947$		−	5.30532e6i		−	0.192237i	0.995370	−	0.0961184i	$-0.0306427\pi$
$948$	4.08975e7	+	1.29832e7i	1.47801	+	0.469204i
$949$		0			0
$950$	6.18413e6	−	3.99184e7i	0.222315	−	1.43504i
$951$		−	1.39619e7i		−	0.500603i
$952$		0			0
$953$	−8.15411e6			−0.290834			−0.145417	−	0.989370i	$-0.546452\pi$
$953$	−8.15411e6			−0.290834			−0.145417	+	0.989370i	$0.546452\pi$
$954$	−5.55559e7	−	8.60668e6i	−1.97633	−	0.306171i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−2.27559e7	+	1.72493e7i	−0.796924	+	0.604080i
$961$	−1.94323e7			−0.678761
$962$		0			0
$963$		−	2.81044e7i		−	0.976582i
$964$	1.74217e7	+	5.53064e6i	0.603805	+	0.191683i
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	2.61088e7	+	1.29711e7i	0.895567	+	0.444927i
$969$	2.30985e7			0.790268
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	8.91254e6	−	2.80747e7i	0.302577	−	0.953125i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−2.91119e7	−	2.05551e7i	−0.978240	−	0.690709i
$977$	2.09088e7			0.700797			0.350399	−	0.936601i	$-0.386046\pi$
$977$	2.09088e7			0.700797			0.350399	+	0.936601i	$0.386046\pi$
$978$		0			0
$979$		0			0
$980$	2.86560e7	+	9.09706e6i	0.953125	+	0.302577i
$981$	3.06146e7			1.01568
$982$		0			0
$983$			1.65328e7i			0.545711i	0.962055	+	0.272855i	$0.0879681\pi$
$983$			1.65328e7i			0.545711i	−0.962055	+	0.272855i	$0.912032\pi$
$984$		0			0
$985$	4.36862e7			1.43468
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$			5.49643e7i			1.77786i	0.458046	+	0.888929i	$0.348550\pi$
$991$			5.49643e7i			1.77786i	−0.458046	+	0.888929i	$0.651450\pi$
$992$	−2.79119e7	+	2.88722e7i	−0.900554	+	0.931538i
$993$	−1.96680e7			−0.632975
$994$		0			0
$995$		−	6.19576e7i		−	1.98398i
$996$	−4.87660e7	−	1.54811e7i	−1.55765	−	0.494487i
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$	−7.37168e6	+	4.75840e7i	−0.234283	+	1.51229i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.6.h.b.59.1	✓	4
3.2	odd	2	inner	60.6.h.b.59.4	yes	4
4.3	odd	2	inner	60.6.h.b.59.2	yes	4
5.4	even	2	inner	60.6.h.b.59.4	yes	4
12.11	even	2	inner	60.6.h.b.59.3	yes	4
15.14	odd	2	CM	60.6.h.b.59.1	✓	4
20.19	odd	2	inner	60.6.h.b.59.3	yes	4
60.59	even	2	inner	60.6.h.b.59.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.6.h.b.59.1	✓	4	1.1	even	1	trivial
60.6.h.b.59.1	✓	4	15.14	odd	2	CM
60.6.h.b.59.2	yes	4	4.3	odd	2	inner
60.6.h.b.59.2	yes	4	60.59	even	2	inner
60.6.h.b.59.3	yes	4	12.11	even	2	inner
60.6.h.b.59.3	yes	4	20.19	odd	2	inner
60.6.h.b.59.4	yes	4	3.2	odd	2	inner
60.6.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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	60.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-5.59017 - 0.866025i) q^{2} -15.5885i q^{3} +(30.5000 + 9.68246i) q^{4} -55.9017 q^{5} +(-13.5000 + 87.1421i) q^{6} +(-162.115 - 80.5404i) q^{8} -243.000 q^{9} +O(q^{10})\)	\(q+(-5.59017 - 0.866025i) q^{2} -15.5885i q^{3} +(30.5000 + 9.68246i) q^{4} -55.9017 q^{5} +(-13.5000 + 87.1421i) q^{6} +(-162.115 - 80.5404i) q^{8} -243.000 q^{9} +(312.500 + 48.4123i) q^{10} +(150.935 - 475.448i) q^{12} +871.421i q^{15} +(836.500 + 590.630i) q^{16} +648.460 q^{17} +(1358.41 + 210.444i) q^{18} +2285.06i q^{19} +(-1705.00 - 541.266i) q^{20} +4880.92i q^{23} +(-1255.50 + 2527.12i) q^{24} +3125.00 q^{25} +3788.00i q^{27} +(754.673 - 4871.39i) q^{30} -6932.64i q^{31} +(-4164.68 - 4026.15i) q^{32} +(-3625.00 - 561.583i) q^{34} +(-7411.50 - 2352.84i) q^{36} +(1978.92 - 12773.9i) q^{38} +(9062.50 + 4502.34i) q^{40} +13584.1 q^{45} +(4227.00 - 27285.2i) q^{46} +27757.8i q^{47} +(9207.01 - 13039.7i) q^{48} -16807.0 q^{49} +(-17469.3 - 2706.33i) q^{50} -10108.5i q^{51} -40897.7 q^{53} +(3280.50 - 21175.5i) q^{54} +35620.6 q^{57} +(-8437.50 + 26578.3i) q^{60} -34802.0 q^{61} +(-6003.84 + 38754.6i) q^{62} +(19794.5 + 26113.6i) q^{64} +(19778.0 + 6278.68i) q^{68} +76086.0 q^{69} +(39393.9 + 19571.3i) q^{72} -48713.9i q^{75} +(-22125.0 + 69694.3i) q^{76} +86019.0i q^{79} +(-46761.8 - 33017.2i) q^{80} +59049.0 q^{81} -102569. i q^{83} -36250.0 q^{85} +(-75937.5 - 11764.2i) q^{90} +(-47259.3 + 148868. i) q^{92} -108069. q^{93} +(24039.0 - 155171. i) q^{94} -127739. i q^{95} +(-62761.5 + 64920.9i) q^{96} +(93954.0 + 14555.3i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 122 q^{4} - 54 q^{6} - 972 q^{9}+O(q^{10}) \) 4 * q + 122 * q^4 - 54 * q^6 - 972 * q^9	\( 4 q + 122 q^{4} - 54 q^{6} - 972 q^{9} + 1250 q^{10} + 3346 q^{16} - 5022 q^{24} + 12500 q^{25} - 14500 q^{34} - 29646 q^{36} + 36250 q^{40} + 16908 q^{46} - 67228 q^{49} + 13122 q^{54} - 33750 q^{60} - 139208 q^{61} + 79178 q^{64} + 304344 q^{69} - 88500 q^{76} + 236196 q^{81} - 145000 q^{85} - 303750 q^{90} + 96156 q^{94} - 251046 q^{96}+O(q^{100}) \) 4 * q + 122 * q^4 - 54 * q^6 - 972 * q^9 + 1250 * q^10 + 3346 * q^16 - 5022 * q^24 + 12500 * q^25 - 14500 * q^34 - 29646 * q^36 + 36250 * q^40 + 16908 * q^46 - 67228 * q^49 + 13122 * q^54 - 33750 * q^60 - 139208 * q^61 + 79178 * q^64 + 304344 * q^69 - 88500 * q^76 + 236196 * q^81 - 145000 * q^85 - 303750 * q^90 + 96156 * q^94 - 251046 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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