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Properties

Label 560.6.a.f.1.1
Level $560$
Weight $6$
Character 560.1
Self dual yes
Analytic conductor $89.815$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [560,6,Mod(1,560)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("560.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(560, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 560.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,11,0,-25,0,-49,0,-122] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(89.8149390953\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 560.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+11.0000 q^{3} -25.0000 q^{5} -49.0000 q^{7} -122.000 q^{9} +267.000 q^{11} -1087.00 q^{13} -275.000 q^{15} -513.000 q^{17} +802.000 q^{19} -539.000 q^{21} +1290.00 q^{23} +625.000 q^{25} -4015.00 q^{27} +1779.00 q^{29} +2584.00 q^{31} +2937.00 q^{33} +1225.00 q^{35} +13862.0 q^{37} -11957.0 q^{39} -11904.0 q^{41} +598.000 q^{43} +3050.00 q^{45} +17019.0 q^{47} +2401.00 q^{49} -5643.00 q^{51} +27852.0 q^{53} -6675.00 q^{55} +8822.00 q^{57} -30912.0 q^{59} -1780.00 q^{61} +5978.00 q^{63} +27175.0 q^{65} -25052.0 q^{67} +14190.0 q^{69} +51984.0 q^{71} +47690.0 q^{73} +6875.00 q^{75} -13083.0 q^{77} +102121. q^{79} -14519.0 q^{81} +83676.0 q^{83} +12825.0 q^{85} +19569.0 q^{87} -32400.0 q^{89} +53263.0 q^{91} +28424.0 q^{93} -20050.0 q^{95} -148645. q^{97} -32574.0 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 11.0000 0.705650 0.352825 − 0.935689i \(-0.385221\pi\)
0.352825 + 0.935689i \(0.385221\pi\)
\(4\) 0 0
\(5\) −25.0000 −0.447214
\(6\) 0 0
\(7\) −49.0000 −0.377964
\(8\) 0 0
\(9\) −122.000 −0.502058
\(10\) 0 0
\(11\) 267.000 0.665318 0.332659 − 0.943047i \(-0.392054\pi\)
0.332659 + 0.943047i \(0.392054\pi\)
\(12\) 0 0
\(13\) −1087.00 −1.78390 −0.891951 − 0.452131i \(-0.850664\pi\)
−0.891951 + 0.452131i \(0.850664\pi\)
\(14\) 0 0
\(15\) −275.000 −0.315576
\(16\) 0 0
\(17\) −513.000 −0.430522 −0.215261 − 0.976557i \(-0.569060\pi\)
−0.215261 + 0.976557i \(0.569060\pi\)
\(18\) 0 0
\(19\) 802.000 0.509672 0.254836 − 0.966984i \(-0.417979\pi\)
0.254836 + 0.966984i \(0.417979\pi\)
\(20\) 0 0
\(21\) −539.000 −0.266711
\(22\) 0 0
\(23\) 1290.00 0.508476 0.254238 − 0.967142i \(-0.418175\pi\)
0.254238 + 0.967142i \(0.418175\pi\)
\(24\) 0 0
\(25\) 625.000 0.200000
\(26\) 0 0
\(27\) −4015.00 −1.05993
\(28\) 0 0
\(29\) 1779.00 0.392809 0.196404 − 0.980523i \(-0.437074\pi\)
0.196404 + 0.980523i \(0.437074\pi\)
\(30\) 0 0
\(31\) 2584.00 0.482935 0.241467 − 0.970409i \(-0.422371\pi\)
0.241467 + 0.970409i \(0.422371\pi\)
\(32\) 0 0
\(33\) 2937.00 0.469482
\(34\) 0 0
\(35\) 1225.00 0.169031
\(36\) 0 0
\(37\) 13862.0 1.66464 0.832322 − 0.554292i \(-0.187011\pi\)
0.832322 + 0.554292i \(0.187011\pi\)
\(38\) 0 0
\(39\) −11957.0 −1.25881
\(40\) 0 0
\(41\) −11904.0 −1.10594 −0.552972 − 0.833200i \(-0.686506\pi\)
−0.552972 + 0.833200i \(0.686506\pi\)
\(42\) 0 0
\(43\) 598.000 0.0493208 0.0246604 − 0.999696i \(-0.492150\pi\)
0.0246604 + 0.999696i \(0.492150\pi\)
\(44\) 0 0
\(45\) 3050.00 0.224527
\(46\) 0 0
\(47\) 17019.0 1.12380 0.561900 − 0.827205i \(-0.310070\pi\)
0.561900 + 0.827205i \(0.310070\pi\)
\(48\) 0 0
\(49\) 2401.00 0.142857
\(50\) 0 0
\(51\) −5643.00 −0.303798
\(52\) 0 0
\(53\) 27852.0 1.36197 0.680984 − 0.732299i \(-0.261552\pi\)
0.680984 + 0.732299i \(0.261552\pi\)
\(54\) 0 0
\(55\) −6675.00 −0.297539
\(56\) 0 0
\(57\) 8822.00 0.359650
\(58\) 0 0
\(59\) −30912.0 −1.15610 −0.578052 − 0.816000i \(-0.696187\pi\)
−0.578052 + 0.816000i \(0.696187\pi\)
\(60\) 0 0
\(61\) −1780.00 −0.0612485 −0.0306242 − 0.999531i \(-0.509750\pi\)
−0.0306242 + 0.999531i \(0.509750\pi\)
\(62\) 0 0
\(63\) 5978.00 0.189760
\(64\) 0 0
\(65\) 27175.0 0.797786
\(66\) 0 0
\(67\) −25052.0 −0.681797 −0.340899 − 0.940100i \(-0.610731\pi\)
−0.340899 + 0.940100i \(0.610731\pi\)
\(68\) 0 0
\(69\) 14190.0 0.358806
\(70\) 0 0
\(71\) 51984.0 1.22384 0.611919 − 0.790921i \(-0.290398\pi\)
0.611919 + 0.790921i \(0.290398\pi\)
\(72\) 0 0
\(73\) 47690.0 1.04742 0.523709 − 0.851897i \(-0.324548\pi\)
0.523709 + 0.851897i \(0.324548\pi\)
\(74\) 0 0
\(75\) 6875.00 0.141130
\(76\) 0 0
\(77\) −13083.0 −0.251467
\(78\) 0 0
\(79\) 102121. 1.84097 0.920486 − 0.390775i \(-0.127793\pi\)
0.920486 + 0.390775i \(0.127793\pi\)
\(80\) 0 0
\(81\) −14519.0 −0.245881
\(82\) 0 0
\(83\) 83676.0 1.33323 0.666616 − 0.745401i \(-0.267742\pi\)
0.666616 + 0.745401i \(0.267742\pi\)
\(84\) 0 0
\(85\) 12825.0 0.192535
\(86\) 0 0
\(87\) 19569.0 0.277185
\(88\) 0 0
\(89\) −32400.0 −0.433581 −0.216790 − 0.976218i \(-0.569559\pi\)
−0.216790 + 0.976218i \(0.569559\pi\)
\(90\) 0 0
\(91\) 53263.0 0.674252
\(92\) 0 0
\(93\) 28424.0 0.340783
\(94\) 0 0
\(95\) −20050.0 −0.227932
\(96\) 0 0
\(97\) −148645. −1.60406 −0.802031 − 0.597283i \(-0.796247\pi\)
−0.802031 + 0.597283i \(0.796247\pi\)
\(98\) 0 0
\(99\) −32574.0 −0.334028
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 560.6.a.f.1.1 1
4.3 odd 2 70.6.a.f.1.1 ✓ 1
12.11 even 2 630.6.a.e.1.1 1
20.3 even 4 350.6.c.b.99.1 2
20.7 even 4 350.6.c.b.99.2 2
20.19 odd 2 350.6.a.d.1.1 1
28.27 even 2 490.6.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.6.a.f.1.1 ✓ 1 4.3 odd 2
350.6.a.d.1.1 1 20.19 odd 2
350.6.c.b.99.1 2 20.3 even 4
350.6.c.b.99.2 2 20.7 even 4
490.6.a.l.1.1 1 28.27 even 2
560.6.a.f.1.1 1 1.1 even 1 trivial
630.6.a.e.1.1 1 12.11 even 2