Properties

Label 112.6.a.g.1.1
Level $112$
Weight $6$
Character 112.1
Self dual yes
Analytic conductor $17.963$
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] = [112,6,Mod(1,112)] 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("112.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 112.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,14,0,-56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(17.9629878191\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 112.1

$q$-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)
 
\(f(q)\) \(=\) \(q+14.0000 q^{3} -56.0000 q^{5} +49.0000 q^{7} -47.0000 q^{9} -232.000 q^{11} -140.000 q^{13} -784.000 q^{15} -1722.00 q^{17} +98.0000 q^{19} +686.000 q^{21} -1824.00 q^{23} +11.0000 q^{25} -4060.00 q^{27} +3418.00 q^{29} +7644.00 q^{31} -3248.00 q^{33} -2744.00 q^{35} -10398.0 q^{37} -1960.00 q^{39} -17962.0 q^{41} -10880.0 q^{43} +2632.00 q^{45} -9324.00 q^{47} +2401.00 q^{49} -24108.0 q^{51} +2262.00 q^{53} +12992.0 q^{55} +1372.00 q^{57} +2730.00 q^{59} +25648.0 q^{61} -2303.00 q^{63} +7840.00 q^{65} +48404.0 q^{67} -25536.0 q^{69} +58560.0 q^{71} +68082.0 q^{73} +154.000 q^{75} -11368.0 q^{77} -31784.0 q^{79} -45419.0 q^{81} +20538.0 q^{83} +96432.0 q^{85} +47852.0 q^{87} -50582.0 q^{89} -6860.00 q^{91} +107016. q^{93} -5488.00 q^{95} -58506.0 q^{97} +10904.0 q^{99} +O(q^{100})\)

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
\(3\) 14.0000 0.898100 0.449050 − 0.893507i \(-0.351762\pi\)
0.449050 + 0.893507i \(0.351762\pi\)
\(4\) 0 0
\(5\) −56.0000 −1.00176 −0.500879 − 0.865517i \(-0.666990\pi\)
−0.500879 + 0.865517i \(0.666990\pi\)
\(6\) 0 0
\(7\) 49.0000 0.377964
\(8\) 0 0
\(9\) −47.0000 −0.193416
\(10\) 0 0
\(11\) −232.000 −0.578104 −0.289052 − 0.957313i \(-0.593340\pi\)
−0.289052 + 0.957313i \(0.593340\pi\)
\(12\) 0 0
\(13\) −140.000 −0.229757 −0.114879 − 0.993380i \(-0.536648\pi\)
−0.114879 + 0.993380i \(0.536648\pi\)
\(14\) 0 0
\(15\) −784.000 −0.899680
\(16\) 0 0
\(17\) −1722.00 −1.44514 −0.722572 − 0.691296i \(-0.757040\pi\)
−0.722572 + 0.691296i \(0.757040\pi\)
\(18\) 0 0
\(19\) 98.0000 0.0622791 0.0311395 − 0.999515i \(-0.490086\pi\)
0.0311395 + 0.999515i \(0.490086\pi\)
\(20\) 0 0
\(21\) 686.000 0.339450
\(22\) 0 0
\(23\) −1824.00 −0.718961 −0.359480 − 0.933153i \(-0.617046\pi\)
−0.359480 + 0.933153i \(0.617046\pi\)
\(24\) 0 0
\(25\) 11.0000 0.00352000
\(26\) 0 0
\(27\) −4060.00 −1.07181
\(28\) 0 0
\(29\) 3418.00 0.754705 0.377352 − 0.926070i \(-0.376835\pi\)
0.377352 + 0.926070i \(0.376835\pi\)
\(30\) 0 0
\(31\) 7644.00 1.42862 0.714310 − 0.699830i \(-0.246741\pi\)
0.714310 + 0.699830i \(0.246741\pi\)
\(32\) 0 0
\(33\) −3248.00 −0.519196
\(34\) 0 0
\(35\) −2744.00 −0.378629
\(36\) 0 0
\(37\) −10398.0 −1.24866 −0.624332 − 0.781159i \(-0.714629\pi\)
−0.624332 + 0.781159i \(0.714629\pi\)
\(38\) 0 0
\(39\) −1960.00 −0.206345
\(40\) 0 0
\(41\) −17962.0 −1.66876 −0.834382 − 0.551186i \(-0.814175\pi\)
−0.834382 + 0.551186i \(0.814175\pi\)
\(42\) 0 0
\(43\) −10880.0 −0.897342 −0.448671 − 0.893697i \(-0.648102\pi\)
−0.448671 + 0.893697i \(0.648102\pi\)
\(44\) 0 0
\(45\) 2632.00 0.193756
\(46\) 0 0
\(47\) −9324.00 −0.615684 −0.307842 − 0.951438i \(-0.599607\pi\)
−0.307842 + 0.951438i \(0.599607\pi\)
\(48\) 0 0
\(49\) 2401.00 0.142857
\(50\) 0 0
\(51\) −24108.0 −1.29788
\(52\) 0 0
\(53\) 2262.00 0.110612 0.0553061 − 0.998469i \(-0.482387\pi\)
0.0553061 + 0.998469i \(0.482387\pi\)
\(54\) 0 0
\(55\) 12992.0 0.579121
\(56\) 0 0
\(57\) 1372.00 0.0559329
\(58\) 0 0
\(59\) 2730.00 0.102102 0.0510508 − 0.998696i \(-0.483743\pi\)
0.0510508 + 0.998696i \(0.483743\pi\)
\(60\) 0 0
\(61\) 25648.0 0.882529 0.441264 − 0.897377i \(-0.354530\pi\)
0.441264 + 0.897377i \(0.354530\pi\)
\(62\) 0 0
\(63\) −2303.00 −0.0731042
\(64\) 0 0
\(65\) 7840.00 0.230161
\(66\) 0 0
\(67\) 48404.0 1.31733 0.658664 − 0.752437i \(-0.271122\pi\)
0.658664 + 0.752437i \(0.271122\pi\)
\(68\) 0 0
\(69\) −25536.0 −0.645699
\(70\) 0 0
\(71\) 58560.0 1.37865 0.689327 − 0.724450i \(-0.257906\pi\)
0.689327 + 0.724450i \(0.257906\pi\)
\(72\) 0 0
\(73\) 68082.0 1.49529 0.747645 − 0.664099i \(-0.231185\pi\)
0.747645 + 0.664099i \(0.231185\pi\)
\(74\) 0 0
\(75\) 154.000 0.00316131
\(76\) 0 0
\(77\) −11368.0 −0.218503
\(78\) 0 0
\(79\) −31784.0 −0.572982 −0.286491 − 0.958083i \(-0.592489\pi\)
−0.286491 + 0.958083i \(0.592489\pi\)
\(80\) 0 0
\(81\) −45419.0 −0.769175
\(82\) 0 0
\(83\) 20538.0 0.327237 0.163619 − 0.986524i \(-0.447683\pi\)
0.163619 + 0.986524i \(0.447683\pi\)
\(84\) 0 0
\(85\) 96432.0 1.44768
\(86\) 0 0
\(87\) 47852.0 0.677801
\(88\) 0 0
\(89\) −50582.0 −0.676894 −0.338447 − 0.940985i \(-0.609902\pi\)
−0.338447 + 0.940985i \(0.609902\pi\)
\(90\) 0 0
\(91\) −6860.00 −0.0868402
\(92\) 0 0
\(93\) 107016. 1.28304
\(94\) 0 0
\(95\) −5488.00 −0.0623886
\(96\) 0 0
\(97\) −58506.0 −0.631351 −0.315676 − 0.948867i \(-0.602231\pi\)
−0.315676 + 0.948867i \(0.602231\pi\)
\(98\) 0 0
\(99\) 10904.0 0.111814
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 112.6.a.g.1.1 1
3.2 odd 2 1008.6.a.y.1.1 1
4.3 odd 2 7.6.a.a.1.1 ✓ 1
7.6 odd 2 784.6.a.c.1.1 1
8.3 odd 2 448.6.a.m.1.1 1
8.5 even 2 448.6.a.c.1.1 1
12.11 even 2 63.6.a.e.1.1 1
20.3 even 4 175.6.b.a.99.2 2
20.7 even 4 175.6.b.a.99.1 2
20.19 odd 2 175.6.a.b.1.1 1
28.3 even 6 49.6.c.b.30.1 2
28.11 odd 6 49.6.c.c.30.1 2
28.19 even 6 49.6.c.b.18.1 2
28.23 odd 6 49.6.c.c.18.1 2
28.27 even 2 49.6.a.a.1.1 1
44.43 even 2 847.6.a.b.1.1 1
84.83 odd 2 441.6.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.6.a.a.1.1 ✓ 1 4.3 odd 2
49.6.a.a.1.1 1 28.27 even 2
49.6.c.b.18.1 2 28.19 even 6
49.6.c.b.30.1 2 28.3 even 6
49.6.c.c.18.1 2 28.23 odd 6
49.6.c.c.30.1 2 28.11 odd 6
63.6.a.e.1.1 1 12.11 even 2
112.6.a.g.1.1 1 1.1 even 1 trivial
175.6.a.b.1.1 1 20.19 odd 2
175.6.b.a.99.1 2 20.7 even 4
175.6.b.a.99.2 2 20.3 even 4
441.6.a.k.1.1 1 84.83 odd 2
448.6.a.c.1.1 1 8.5 even 2
448.6.a.m.1.1 1 8.3 odd 2
784.6.a.c.1.1 1 7.6 odd 2
847.6.a.b.1.1 1 44.43 even 2
1008.6.a.y.1.1 1 3.2 odd 2