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Properties

Label 560.6.a.a.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,-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: \(1\)
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)
 
\(f(q)\) \(=\) \(q-11.0000 q^{3} -25.0000 q^{5} +49.0000 q^{7} -122.000 q^{9} -83.0000 q^{11} -83.0000 q^{13} +275.000 q^{15} -177.000 q^{17} +2082.00 q^{19} -539.000 q^{21} +3170.00 q^{23} +625.000 q^{25} +4015.00 q^{27} -8681.00 q^{29} -1636.00 q^{31} +913.000 q^{33} -1225.00 q^{35} +4298.00 q^{37} +913.000 q^{39} +2356.00 q^{41} -8738.00 q^{43} +3050.00 q^{45} +3641.00 q^{47} +2401.00 q^{49} +1947.00 q^{51} +33268.0 q^{53} +2075.00 q^{55} -22902.0 q^{57} +30968.0 q^{59} +4560.00 q^{61} -5978.00 q^{63} +2075.00 q^{65} -37788.0 q^{67} -34870.0 q^{69} +59304.0 q^{71} -8910.00 q^{73} -6875.00 q^{75} -4067.00 q^{77} -27589.0 q^{79} -14519.0 q^{81} -67676.0 q^{83} +4425.00 q^{85} +95491.0 q^{87} +10700.0 q^{89} -4067.00 q^{91} +17996.0 q^{93} -52050.0 q^{95} +65075.0 q^{97} +10126.0 q^{99} +O(q^{100})\)

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 \(-0.614779\pi\)
−0.352825 + 0.935689i \(0.614779\pi\)
\(4\) 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
\(11\) −83.0000 −0.206822 −0.103411 − 0.994639i \(-0.532976\pi\)
−0.103411 + 0.994639i \(0.532976\pi\)
\(12\) 0 0
\(13\) −83.0000 −0.136213 −0.0681067 − 0.997678i \(-0.521696\pi\)
−0.0681067 + 0.997678i \(0.521696\pi\)
\(14\) 0 0
\(15\) 275.000 0.315576
\(16\) 0 0
\(17\) −177.000 −0.148543 −0.0742713 − 0.997238i \(-0.523663\pi\)
−0.0742713 + 0.997238i \(0.523663\pi\)
\(18\) 0 0
\(19\) 2082.00 1.32311 0.661556 − 0.749896i \(-0.269896\pi\)
0.661556 + 0.749896i \(0.269896\pi\)
\(20\) 0 0
\(21\) −539.000 −0.266711
\(22\) 0 0
\(23\) 3170.00 1.24951 0.624755 − 0.780821i \(-0.285199\pi\)
0.624755 + 0.780821i \(0.285199\pi\)
\(24\) 0 0
\(25\) 625.000 0.200000
\(26\) 0 0
\(27\) 4015.00 1.05993
\(28\) 0 0
\(29\) −8681.00 −1.91679 −0.958395 − 0.285444i \(-0.907859\pi\)
−0.958395 + 0.285444i \(0.907859\pi\)
\(30\) 0 0
\(31\) −1636.00 −0.305759 −0.152879 − 0.988245i \(-0.548855\pi\)
−0.152879 + 0.988245i \(0.548855\pi\)
\(32\) 0 0
\(33\) 913.000 0.145944
\(34\) 0 0
\(35\) −1225.00 −0.169031
\(36\) 0 0
\(37\) 4298.00 0.516134 0.258067 − 0.966127i \(-0.416915\pi\)
0.258067 + 0.966127i \(0.416915\pi\)
\(38\) 0 0
\(39\) 913.000 0.0961190
\(40\) 0 0
\(41\) 2356.00 0.218885 0.109442 − 0.993993i \(-0.465093\pi\)
0.109442 + 0.993993i \(0.465093\pi\)
\(42\) 0 0
\(43\) −8738.00 −0.720677 −0.360339 − 0.932822i \(-0.617339\pi\)
−0.360339 + 0.932822i \(0.617339\pi\)
\(44\) 0 0
\(45\) 3050.00 0.224527
\(46\) 0 0
\(47\) 3641.00 0.240423 0.120212 − 0.992748i \(-0.461643\pi\)
0.120212 + 0.992748i \(0.461643\pi\)
\(48\) 0 0
\(49\) 2401.00 0.142857
\(50\) 0 0
\(51\) 1947.00 0.104819
\(52\) 0 0
\(53\) 33268.0 1.62681 0.813405 − 0.581697i \(-0.197611\pi\)
0.813405 + 0.581697i \(0.197611\pi\)
\(54\) 0 0
\(55\) 2075.00 0.0924935
\(56\) 0 0
\(57\) −22902.0 −0.933655
\(58\) 0 0
\(59\) 30968.0 1.15820 0.579099 − 0.815257i \(-0.303404\pi\)
0.579099 + 0.815257i \(0.303404\pi\)
\(60\) 0 0
\(61\) 4560.00 0.156906 0.0784531 − 0.996918i \(-0.475002\pi\)
0.0784531 + 0.996918i \(0.475002\pi\)
\(62\) 0 0
\(63\) −5978.00 −0.189760
\(64\) 0 0
\(65\) 2075.00 0.0609165
\(66\) 0 0
\(67\) −37788.0 −1.02841 −0.514206 − 0.857667i \(-0.671913\pi\)
−0.514206 + 0.857667i \(0.671913\pi\)
\(68\) 0 0
\(69\) −34870.0 −0.881717
\(70\) 0 0
\(71\) 59304.0 1.39617 0.698085 − 0.716015i \(-0.254036\pi\)
0.698085 + 0.716015i \(0.254036\pi\)
\(72\) 0 0
\(73\) −8910.00 −0.195691 −0.0978454 − 0.995202i \(-0.531195\pi\)
−0.0978454 + 0.995202i \(0.531195\pi\)
\(74\) 0 0
\(75\) −6875.00 −0.141130
\(76\) 0 0
\(77\) −4067.00 −0.0781713
\(78\) 0 0
\(79\) −27589.0 −0.497357 −0.248678 − 0.968586i \(-0.579996\pi\)
−0.248678 + 0.968586i \(0.579996\pi\)
\(80\) 0 0
\(81\) −14519.0 −0.245881
\(82\) 0 0
\(83\) −67676.0 −1.07830 −0.539150 − 0.842210i \(-0.681254\pi\)
−0.539150 + 0.842210i \(0.681254\pi\)
\(84\) 0 0
\(85\) 4425.00 0.0664303
\(86\) 0 0
\(87\) 95491.0 1.35258
\(88\) 0 0
\(89\) 10700.0 0.143189 0.0715944 − 0.997434i \(-0.477191\pi\)
0.0715944 + 0.997434i \(0.477191\pi\)
\(90\) 0 0
\(91\) −4067.00 −0.0514838
\(92\) 0 0
\(93\) 17996.0 0.215759
\(94\) 0 0
\(95\) −52050.0 −0.591714
\(96\) 0 0
\(97\) 65075.0 0.702239 0.351119 − 0.936331i \(-0.385801\pi\)
0.351119 + 0.936331i \(0.385801\pi\)
\(98\) 0 0
\(99\) 10126.0 0.103836
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.a.1.1 1
4.3 odd 2 70.6.a.d.1.1 ✓ 1
12.11 even 2 630.6.a.l.1.1 1
20.3 even 4 350.6.c.c.99.2 2
20.7 even 4 350.6.c.c.99.1 2
20.19 odd 2 350.6.a.h.1.1 1
28.27 even 2 490.6.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.6.a.d.1.1 ✓ 1 4.3 odd 2
350.6.a.h.1.1 1 20.19 odd 2
350.6.c.c.99.1 2 20.7 even 4
350.6.c.c.99.2 2 20.3 even 4
490.6.a.c.1.1 1 28.27 even 2
560.6.a.a.1.1 1 1.1 even 1 trivial
630.6.a.l.1.1 1 12.11 even 2