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Properties

Label 560.6.a.b.1.1
Level $560$
Weight $6$
Character 560.1
Self dual yes
Analytic conductor $89.815$
Analytic rank $1$
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,-4,0,25,0,49,0,-227] 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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
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)
 
\(f(q)\) \(=\) \(q-4.00000 q^{3} +25.0000 q^{5} +49.0000 q^{7} -227.000 q^{9} -124.000 q^{11} +766.000 q^{13} -100.000 q^{15} -1102.00 q^{17} +764.000 q^{19} -196.000 q^{21} -168.000 q^{23} +625.000 q^{25} +1880.00 q^{27} -6866.00 q^{29} +4096.00 q^{31} +496.000 q^{33} +1225.00 q^{35} -4682.00 q^{37} -3064.00 q^{39} +13130.0 q^{41} -18220.0 q^{43} -5675.00 q^{45} +7104.00 q^{47} +2401.00 q^{49} +4408.00 q^{51} -20026.0 q^{53} -3100.00 q^{55} -3056.00 q^{57} +38964.0 q^{59} -56274.0 q^{61} -11123.0 q^{63} +19150.0 q^{65} +24060.0 q^{67} +672.000 q^{69} +31896.0 q^{71} -23670.0 q^{73} -2500.00 q^{75} -6076.00 q^{77} -37744.0 q^{79} +47641.0 q^{81} +68204.0 q^{83} -27550.0 q^{85} +27464.0 q^{87} -19078.0 q^{89} +37534.0 q^{91} -16384.0 q^{93} +19100.0 q^{95} -115646. q^{97} +28148.0 q^{99} +O(q^{100})\)

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
\(3\) −4.00000 −0.256600 −0.128300 − 0.991735i \(-0.540952\pi\)
−0.128300 + 0.991735i \(0.540952\pi\)
\(4\) 0 0
\(5\) 25.0000 0.447214
\(6\) 0 0
\(7\) 49.0000 0.377964
\(8\) 0 0
\(9\) −227.000 −0.934156
\(10\) 0 0
\(11\) −124.000 −0.308987 −0.154493 − 0.987994i \(-0.549375\pi\)
−0.154493 + 0.987994i \(0.549375\pi\)
\(12\) 0 0
\(13\) 766.000 1.25710 0.628551 − 0.777769i \(-0.283648\pi\)
0.628551 + 0.777769i \(0.283648\pi\)
\(14\) 0 0
\(15\) −100.000 −0.114755
\(16\) 0 0
\(17\) −1102.00 −0.924824 −0.462412 − 0.886665i \(-0.653016\pi\)
−0.462412 + 0.886665i \(0.653016\pi\)
\(18\) 0 0
\(19\) 764.000 0.485522 0.242761 − 0.970086i \(-0.421947\pi\)
0.242761 + 0.970086i \(0.421947\pi\)
\(20\) 0 0
\(21\) −196.000 −0.0969857
\(22\) 0 0
\(23\) −168.000 −0.0662201 −0.0331100 − 0.999452i \(-0.510541\pi\)
−0.0331100 + 0.999452i \(0.510541\pi\)
\(24\) 0 0
\(25\) 625.000 0.200000
\(26\) 0 0
\(27\) 1880.00 0.496305
\(28\) 0 0
\(29\) −6866.00 −1.51603 −0.758017 − 0.652235i \(-0.773832\pi\)
−0.758017 + 0.652235i \(0.773832\pi\)
\(30\) 0 0
\(31\) 4096.00 0.765519 0.382759 − 0.923848i \(-0.374974\pi\)
0.382759 + 0.923848i \(0.374974\pi\)
\(32\) 0 0
\(33\) 496.000 0.0792861
\(34\) 0 0
\(35\) 1225.00 0.169031
\(36\) 0 0
\(37\) −4682.00 −0.562247 −0.281123 − 0.959672i \(-0.590707\pi\)
−0.281123 + 0.959672i \(0.590707\pi\)
\(38\) 0 0
\(39\) −3064.00 −0.322572
\(40\) 0 0
\(41\) 13130.0 1.21985 0.609923 − 0.792461i \(-0.291200\pi\)
0.609923 + 0.792461i \(0.291200\pi\)
\(42\) 0 0
\(43\) −18220.0 −1.50272 −0.751359 − 0.659894i \(-0.770601\pi\)
−0.751359 + 0.659894i \(0.770601\pi\)
\(44\) 0 0
\(45\) −5675.00 −0.417767
\(46\) 0 0
\(47\) 7104.00 0.469092 0.234546 − 0.972105i \(-0.424640\pi\)
0.234546 + 0.972105i \(0.424640\pi\)
\(48\) 0 0
\(49\) 2401.00 0.142857
\(50\) 0 0
\(51\) 4408.00 0.237310
\(52\) 0 0
\(53\) −20026.0 −0.979275 −0.489637 − 0.871926i \(-0.662871\pi\)
−0.489637 + 0.871926i \(0.662871\pi\)
\(54\) 0 0
\(55\) −3100.00 −0.138183
\(56\) 0 0
\(57\) −3056.00 −0.124585
\(58\) 0 0
\(59\) 38964.0 1.45725 0.728624 − 0.684914i \(-0.240160\pi\)
0.728624 + 0.684914i \(0.240160\pi\)
\(60\) 0 0
\(61\) −56274.0 −1.93635 −0.968174 − 0.250280i \(-0.919477\pi\)
−0.968174 + 0.250280i \(0.919477\pi\)
\(62\) 0 0
\(63\) −11123.0 −0.353078
\(64\) 0 0
\(65\) 19150.0 0.562193
\(66\) 0 0
\(67\) 24060.0 0.654800 0.327400 − 0.944886i \(-0.393828\pi\)
0.327400 + 0.944886i \(0.393828\pi\)
\(68\) 0 0
\(69\) 672.000 0.0169921
\(70\) 0 0
\(71\) 31896.0 0.750914 0.375457 − 0.926840i \(-0.377486\pi\)
0.375457 + 0.926840i \(0.377486\pi\)
\(72\) 0 0
\(73\) −23670.0 −0.519866 −0.259933 − 0.965627i \(-0.583700\pi\)
−0.259933 + 0.965627i \(0.583700\pi\)
\(74\) 0 0
\(75\) −2500.00 −0.0513200
\(76\) 0 0
\(77\) −6076.00 −0.116786
\(78\) 0 0
\(79\) −37744.0 −0.680425 −0.340212 − 0.940349i \(-0.610499\pi\)
−0.340212 + 0.940349i \(0.610499\pi\)
\(80\) 0 0
\(81\) 47641.0 0.806805
\(82\) 0 0
\(83\) 68204.0 1.08671 0.543356 − 0.839502i \(-0.317153\pi\)
0.543356 + 0.839502i \(0.317153\pi\)
\(84\) 0 0
\(85\) −27550.0 −0.413594
\(86\) 0 0
\(87\) 27464.0 0.389014
\(88\) 0 0
\(89\) −19078.0 −0.255304 −0.127652 − 0.991819i \(-0.540744\pi\)
−0.127652 + 0.991819i \(0.540744\pi\)
\(90\) 0 0
\(91\) 37534.0 0.475140
\(92\) 0 0
\(93\) −16384.0 −0.196432
\(94\) 0 0
\(95\) 19100.0 0.217132
\(96\) 0 0
\(97\) −115646. −1.24796 −0.623981 − 0.781440i \(-0.714486\pi\)
−0.623981 + 0.781440i \(0.714486\pi\)
\(98\) 0 0
\(99\) 28148.0 0.288642
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.b.1.1 1
4.3 odd 2 280.6.a.b.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.6.a.b.1.1 ✓ 1 4.3 odd 2
560.6.a.b.1.1 1 1.1 even 1 trivial