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Properties

Label 560.6.a.h.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 = [1,0,17,0,25,0,49,0,46] 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)
 
\(f(q)\) \(=\) \(q+17.0000 q^{3} +25.0000 q^{5} +49.0000 q^{7} +46.0000 q^{9} +715.000 q^{11} +331.000 q^{13} +425.000 q^{15} -1699.00 q^{17} +1718.00 q^{19} +833.000 q^{21} +3950.00 q^{23} +625.000 q^{25} -3349.00 q^{27} +4579.00 q^{29} -6756.00 q^{31} +12155.0 q^{33} +1225.00 q^{35} -16518.0 q^{37} +5627.00 q^{39} +18876.0 q^{41} -2258.00 q^{43} +1150.00 q^{45} +537.000 q^{47} +2401.00 q^{49} -28883.0 q^{51} -10984.0 q^{53} +17875.0 q^{55} +29206.0 q^{57} +25956.0 q^{59} +39188.0 q^{61} +2254.00 q^{63} +8275.00 q^{65} -4416.00 q^{67} +67150.0 q^{69} +31880.0 q^{71} -5018.00 q^{73} +10625.0 q^{75} +35035.0 q^{77} +27977.0 q^{79} -68111.0 q^{81} -37644.0 q^{83} -42475.0 q^{85} +77843.0 q^{87} -17216.0 q^{89} +16219.0 q^{91} -114852. q^{93} +42950.0 q^{95} -63175.0 q^{97} +32890.0 q^{99} +O(q^{100})\)

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
\(3\) 17.0000 1.09055 0.545275 − 0.838257i \(-0.316425\pi\)
0.545275 + 0.838257i \(0.316425\pi\)
\(4\) 0 0
\(5\) 25.0000 0.447214
\(6\) 0 0
\(7\) 49.0000 0.377964
\(8\) 0 0
\(9\) 46.0000 0.189300
\(10\) 0 0
\(11\) 715.000 1.78166 0.890829 − 0.454339i \(-0.150124\pi\)
0.890829 + 0.454339i \(0.150124\pi\)
\(12\) 0 0
\(13\) 331.000 0.543212 0.271606 − 0.962408i \(-0.412445\pi\)
0.271606 + 0.962408i \(0.412445\pi\)
\(14\) 0 0
\(15\) 425.000 0.487709
\(16\) 0 0
\(17\) −1699.00 −1.42584 −0.712920 − 0.701245i \(-0.752628\pi\)
−0.712920 + 0.701245i \(0.752628\pi\)
\(18\) 0 0
\(19\) 1718.00 1.09179 0.545895 − 0.837854i \(-0.316190\pi\)
0.545895 + 0.837854i \(0.316190\pi\)
\(20\) 0 0
\(21\) 833.000 0.412189
\(22\) 0 0
\(23\) 3950.00 1.55696 0.778480 − 0.627669i \(-0.215991\pi\)
0.778480 + 0.627669i \(0.215991\pi\)
\(24\) 0 0
\(25\) 625.000 0.200000
\(26\) 0 0
\(27\) −3349.00 −0.884109
\(28\) 0 0
\(29\) 4579.00 1.01106 0.505529 − 0.862810i \(-0.331298\pi\)
0.505529 + 0.862810i \(0.331298\pi\)
\(30\) 0 0
\(31\) −6756.00 −1.26266 −0.631329 − 0.775516i \(-0.717490\pi\)
−0.631329 + 0.775516i \(0.717490\pi\)
\(32\) 0 0
\(33\) 12155.0 1.94299
\(34\) 0 0
\(35\) 1225.00 0.169031
\(36\) 0 0
\(37\) −16518.0 −1.98360 −0.991798 − 0.127816i \(-0.959203\pi\)
−0.991798 + 0.127816i \(0.959203\pi\)
\(38\) 0 0
\(39\) 5627.00 0.592400
\(40\) 0 0
\(41\) 18876.0 1.75368 0.876840 − 0.480782i \(-0.159647\pi\)
0.876840 + 0.480782i \(0.159647\pi\)
\(42\) 0 0
\(43\) −2258.00 −0.186231 −0.0931157 − 0.995655i \(-0.529683\pi\)
−0.0931157 + 0.995655i \(0.529683\pi\)
\(44\) 0 0
\(45\) 1150.00 0.0846577
\(46\) 0 0
\(47\) 537.000 0.0354593 0.0177296 − 0.999843i \(-0.494356\pi\)
0.0177296 + 0.999843i \(0.494356\pi\)
\(48\) 0 0
\(49\) 2401.00 0.142857
\(50\) 0 0
\(51\) −28883.0 −1.55495
\(52\) 0 0
\(53\) −10984.0 −0.537119 −0.268560 − 0.963263i \(-0.586548\pi\)
−0.268560 + 0.963263i \(0.586548\pi\)
\(54\) 0 0
\(55\) 17875.0 0.796782
\(56\) 0 0
\(57\) 29206.0 1.19065
\(58\) 0 0
\(59\) 25956.0 0.970751 0.485375 − 0.874306i \(-0.338683\pi\)
0.485375 + 0.874306i \(0.338683\pi\)
\(60\) 0 0
\(61\) 39188.0 1.34843 0.674215 − 0.738535i \(-0.264482\pi\)
0.674215 + 0.738535i \(0.264482\pi\)
\(62\) 0 0
\(63\) 2254.00 0.0715488
\(64\) 0 0
\(65\) 8275.00 0.242932
\(66\) 0 0
\(67\) −4416.00 −0.120183 −0.0600914 − 0.998193i \(-0.519139\pi\)
−0.0600914 + 0.998193i \(0.519139\pi\)
\(68\) 0 0
\(69\) 67150.0 1.69794
\(70\) 0 0
\(71\) 31880.0 0.750538 0.375269 − 0.926916i \(-0.377550\pi\)
0.375269 + 0.926916i \(0.377550\pi\)
\(72\) 0 0
\(73\) −5018.00 −0.110211 −0.0551053 − 0.998481i \(-0.517549\pi\)
−0.0551053 + 0.998481i \(0.517549\pi\)
\(74\) 0 0
\(75\) 10625.0 0.218110
\(76\) 0 0
\(77\) 35035.0 0.673403
\(78\) 0 0
\(79\) 27977.0 0.504352 0.252176 − 0.967681i \(-0.418854\pi\)
0.252176 + 0.967681i \(0.418854\pi\)
\(80\) 0 0
\(81\) −68111.0 −1.15347
\(82\) 0 0
\(83\) −37644.0 −0.599792 −0.299896 − 0.953972i \(-0.596952\pi\)
−0.299896 + 0.953972i \(0.596952\pi\)
\(84\) 0 0
\(85\) −42475.0 −0.637655
\(86\) 0 0
\(87\) 77843.0 1.10261
\(88\) 0 0
\(89\) −17216.0 −0.230387 −0.115193 − 0.993343i \(-0.536749\pi\)
−0.115193 + 0.993343i \(0.536749\pi\)
\(90\) 0 0
\(91\) 16219.0 0.205315
\(92\) 0 0
\(93\) −114852. −1.37699
\(94\) 0 0
\(95\) 42950.0 0.488263
\(96\) 0 0
\(97\) −63175.0 −0.681736 −0.340868 − 0.940111i \(-0.610721\pi\)
−0.340868 + 0.940111i \(0.610721\pi\)
\(98\) 0 0
\(99\) 32890.0 0.337269
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.h.1.1 1
4.3 odd 2 70.6.a.e.1.1 ✓ 1
12.11 even 2 630.6.a.b.1.1 1
20.3 even 4 350.6.c.a.99.1 2
20.7 even 4 350.6.c.a.99.2 2
20.19 odd 2 350.6.a.e.1.1 1
28.27 even 2 490.6.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.6.a.e.1.1 ✓ 1 4.3 odd 2
350.6.a.e.1.1 1 20.19 odd 2
350.6.c.a.99.1 2 20.3 even 4
350.6.c.a.99.2 2 20.7 even 4
490.6.a.m.1.1 1 28.27 even 2
560.6.a.h.1.1 1 1.1 even 1 trivial
630.6.a.b.1.1 1 12.11 even 2