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Properties

Label 560.6.a.i.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 = [1,0,23,0,25,0,-49,0,286] 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)
 
\(f(q)\) \(=\) \(q+23.0000 q^{3} +25.0000 q^{5} -49.0000 q^{7} +286.000 q^{9} -555.000 q^{11} -241.000 q^{13} +575.000 q^{15} -1491.00 q^{17} +2038.00 q^{19} -1127.00 q^{21} +1230.00 q^{23} +625.000 q^{25} +989.000 q^{27} -5001.00 q^{29} -5696.00 q^{31} -12765.0 q^{33} -1225.00 q^{35} -5602.00 q^{37} -5543.00 q^{39} -2424.00 q^{41} -602.000 q^{43} +7150.00 q^{45} +23163.0 q^{47} +2401.00 q^{49} -34293.0 q^{51} -25296.0 q^{53} -13875.0 q^{55} +46874.0 q^{57} -5724.00 q^{59} -36112.0 q^{61} -14014.0 q^{63} -6025.00 q^{65} -66104.0 q^{67} +28290.0 q^{69} -16080.0 q^{71} -80482.0 q^{73} +14375.0 q^{75} +27195.0 q^{77} +64147.0 q^{79} -46751.0 q^{81} +106284. q^{83} -37275.0 q^{85} -115023. q^{87} -71676.0 q^{89} +11809.0 q^{91} -131008. q^{93} +50950.0 q^{95} +151025. q^{97} -158730. q^{99} +O(q^{100})\)

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
\(3\) 23.0000 1.47545 0.737725 − 0.675101i \(-0.235900\pi\)
0.737725 + 0.675101i \(0.235900\pi\)
\(4\) 0 0
\(5\) 25.0000 0.447214
\(6\) 0 0
\(7\) −49.0000 −0.377964
\(8\) 0 0
\(9\) 286.000 1.17695
\(10\) 0 0
\(11\) −555.000 −1.38297 −0.691483 − 0.722393i \(-0.743042\pi\)
−0.691483 + 0.722393i \(0.743042\pi\)
\(12\) 0 0
\(13\) −241.000 −0.395511 −0.197756 − 0.980251i \(-0.563365\pi\)
−0.197756 + 0.980251i \(0.563365\pi\)
\(14\) 0 0
\(15\) 575.000 0.659842
\(16\) 0 0
\(17\) −1491.00 −1.25128 −0.625641 − 0.780111i \(-0.715163\pi\)
−0.625641 + 0.780111i \(0.715163\pi\)
\(18\) 0 0
\(19\) 2038.00 1.29515 0.647575 − 0.762002i \(-0.275783\pi\)
0.647575 + 0.762002i \(0.275783\pi\)
\(20\) 0 0
\(21\) −1127.00 −0.557668
\(22\) 0 0
\(23\) 1230.00 0.484826 0.242413 − 0.970173i \(-0.422061\pi\)
0.242413 + 0.970173i \(0.422061\pi\)
\(24\) 0 0
\(25\) 625.000 0.200000
\(26\) 0 0
\(27\) 989.000 0.261088
\(28\) 0 0
\(29\) −5001.00 −1.10424 −0.552118 − 0.833766i \(-0.686180\pi\)
−0.552118 + 0.833766i \(0.686180\pi\)
\(30\) 0 0
\(31\) −5696.00 −1.06455 −0.532275 − 0.846572i \(-0.678663\pi\)
−0.532275 + 0.846572i \(0.678663\pi\)
\(32\) 0 0
\(33\) −12765.0 −2.04050
\(34\) 0 0
\(35\) −1225.00 −0.169031
\(36\) 0 0
\(37\) −5602.00 −0.672727 −0.336363 − 0.941732i \(-0.609197\pi\)
−0.336363 + 0.941732i \(0.609197\pi\)
\(38\) 0 0
\(39\) −5543.00 −0.583557
\(40\) 0 0
\(41\) −2424.00 −0.225202 −0.112601 − 0.993640i \(-0.535918\pi\)
−0.112601 + 0.993640i \(0.535918\pi\)
\(42\) 0 0
\(43\) −602.000 −0.0496507 −0.0248253 − 0.999692i \(-0.507903\pi\)
−0.0248253 + 0.999692i \(0.507903\pi\)
\(44\) 0 0
\(45\) 7150.00 0.526350
\(46\) 0 0
\(47\) 23163.0 1.52950 0.764751 − 0.644326i \(-0.222862\pi\)
0.764751 + 0.644326i \(0.222862\pi\)
\(48\) 0 0
\(49\) 2401.00 0.142857
\(50\) 0 0
\(51\) −34293.0 −1.84621
\(52\) 0 0
\(53\) −25296.0 −1.23698 −0.618489 − 0.785793i \(-0.712255\pi\)
−0.618489 + 0.785793i \(0.712255\pi\)
\(54\) 0 0
\(55\) −13875.0 −0.618481
\(56\) 0 0
\(57\) 46874.0 1.91093
\(58\) 0 0
\(59\) −5724.00 −0.214077 −0.107038 − 0.994255i \(-0.534137\pi\)
−0.107038 + 0.994255i \(0.534137\pi\)
\(60\) 0 0
\(61\) −36112.0 −1.24259 −0.621294 − 0.783578i \(-0.713393\pi\)
−0.621294 + 0.783578i \(0.713393\pi\)
\(62\) 0 0
\(63\) −14014.0 −0.444847
\(64\) 0 0
\(65\) −6025.00 −0.176878
\(66\) 0 0
\(67\) −66104.0 −1.79904 −0.899520 − 0.436880i \(-0.856083\pi\)
−0.899520 + 0.436880i \(0.856083\pi\)
\(68\) 0 0
\(69\) 28290.0 0.715336
\(70\) 0 0
\(71\) −16080.0 −0.378565 −0.189282 − 0.981923i \(-0.560616\pi\)
−0.189282 + 0.981923i \(0.560616\pi\)
\(72\) 0 0
\(73\) −80482.0 −1.76763 −0.883816 − 0.467836i \(-0.845034\pi\)
−0.883816 + 0.467836i \(0.845034\pi\)
\(74\) 0 0
\(75\) 14375.0 0.295090
\(76\) 0 0
\(77\) 27195.0 0.522712
\(78\) 0 0
\(79\) 64147.0 1.15640 0.578201 − 0.815895i \(-0.303755\pi\)
0.578201 + 0.815895i \(0.303755\pi\)
\(80\) 0 0
\(81\) −46751.0 −0.791732
\(82\) 0 0
\(83\) 106284. 1.69345 0.846726 − 0.532030i \(-0.178571\pi\)
0.846726 + 0.532030i \(0.178571\pi\)
\(84\) 0 0
\(85\) −37275.0 −0.559591
\(86\) 0 0
\(87\) −115023. −1.62925
\(88\) 0 0
\(89\) −71676.0 −0.959177 −0.479588 − 0.877494i \(-0.659214\pi\)
−0.479588 + 0.877494i \(0.659214\pi\)
\(90\) 0 0
\(91\) 11809.0 0.149489
\(92\) 0 0
\(93\) −131008. −1.57069
\(94\) 0 0
\(95\) 50950.0 0.579209
\(96\) 0 0
\(97\) 151025. 1.62974 0.814872 − 0.579641i \(-0.196807\pi\)
0.814872 + 0.579641i \(0.196807\pi\)
\(98\) 0 0
\(99\) −158730. −1.62769
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.i.1.1 1
4.3 odd 2 70.6.a.a.1.1 ✓ 1
12.11 even 2 630.6.a.j.1.1 1
20.3 even 4 350.6.c.h.99.2 2
20.7 even 4 350.6.c.h.99.1 2
20.19 odd 2 350.6.a.n.1.1 1
28.27 even 2 490.6.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.6.a.a.1.1 ✓ 1 4.3 odd 2
350.6.a.n.1.1 1 20.19 odd 2
350.6.c.h.99.1 2 20.7 even 4
350.6.c.h.99.2 2 20.3 even 4
490.6.a.i.1.1 1 28.27 even 2
560.6.a.i.1.1 1 1.1 even 1 trivial
630.6.a.j.1.1 1 12.11 even 2