Properties

Label 56.10.a.a.1.2
Level $56$
Weight $10$
Character 56.1
Self dual yes
Analytic conductor $28.842$
Analytic rank $1$
Dimension $3$
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] = [56,10,Mod(1,56)] 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("56.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(56, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 56.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-92] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(28.8420068252\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 823x - 4578 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-25.3451\) of defining polynomial
Character \(\chi\) \(=\) 56.1

$q$-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)
 
\(f(q)\) \(=\) \(q-7.32110 q^{3} +1191.17 q^{5} -2401.00 q^{7} -19629.4 q^{9} -18252.2 q^{11} +50333.5 q^{13} -8720.67 q^{15} +283542. q^{17} -273335. q^{19} +17578.0 q^{21} -1.06061e6 q^{23} -534240. q^{25} +287810. q^{27} -5.40121e6 q^{29} -1.75987e6 q^{31} +133626. q^{33} -2.86000e6 q^{35} -7.76468e6 q^{37} -368496. q^{39} +8.92276e6 q^{41} -3.46464e7 q^{43} -2.33819e7 q^{45} -3.66292e7 q^{47} +5.76480e6 q^{49} -2.07584e6 q^{51} -6.10693e7 q^{53} -2.17415e7 q^{55} +2.00111e6 q^{57} -9.02214e7 q^{59} +1.52451e8 q^{61} +4.71302e7 q^{63} +5.99557e7 q^{65} -2.18974e7 q^{67} +7.76485e6 q^{69} -2.33501e8 q^{71} +1.06705e8 q^{73} +3.91123e6 q^{75} +4.38236e7 q^{77} +2.89730e8 q^{79} +3.84258e8 q^{81} +4.73876e7 q^{83} +3.37747e8 q^{85} +3.95428e7 q^{87} +6.97049e8 q^{89} -1.20851e8 q^{91} +1.28842e7 q^{93} -3.25588e8 q^{95} +7.82411e8 q^{97} +3.58280e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 92 q^{3} + 274 q^{5} - 7203 q^{7} + 51871 q^{9} - 45364 q^{11} - 11158 q^{13} - 95584 q^{15} - 55866 q^{17} + 488772 q^{19} + 220892 q^{21} - 253888 q^{23} - 3872619 q^{25} - 10157192 q^{27} - 765318 q^{29}+ \cdots + 168732636 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −7.32110 −0.0521832 −0.0260916 − 0.999660i \(-0.508306\pi\)
−0.0260916 + 0.999660i \(0.508306\pi\)
\(4\) 0 0
\(5\) 1191.17 0.852332 0.426166 − 0.904645i \(-0.359864\pi\)
0.426166 + 0.904645i \(0.359864\pi\)
\(6\) 0 0
\(7\) −2401.00 −0.377964
\(8\) 0 0
\(9\) −19629.4 −0.997277
\(10\) 0 0
\(11\) −18252.2 −0.375880 −0.187940 − 0.982181i \(-0.560181\pi\)
−0.187940 + 0.982181i \(0.560181\pi\)
\(12\) 0 0
\(13\) 50333.5 0.488778 0.244389 − 0.969677i \(-0.421413\pi\)
0.244389 + 0.969677i \(0.421413\pi\)
\(14\) 0 0
\(15\) −8720.67 −0.0444774
\(16\) 0 0
\(17\) 283542. 0.823375 0.411688 − 0.911325i \(-0.364939\pi\)
0.411688 + 0.911325i \(0.364939\pi\)
\(18\) 0 0
\(19\) −273335. −0.481176 −0.240588 − 0.970627i \(-0.577340\pi\)
−0.240588 + 0.970627i \(0.577340\pi\)
\(20\) 0 0
\(21\) 17578.0 0.0197234
\(22\) 0 0
\(23\) −1.06061e6 −0.790281 −0.395140 − 0.918621i \(-0.629304\pi\)
−0.395140 + 0.918621i \(0.629304\pi\)
\(24\) 0 0
\(25\) −534240. −0.273531
\(26\) 0 0
\(27\) 287810. 0.104224
\(28\) 0 0
\(29\) −5.40121e6 −1.41808 −0.709039 − 0.705170i \(-0.750871\pi\)
−0.709039 + 0.705170i \(0.750871\pi\)
\(30\) 0 0
\(31\) −1.75987e6 −0.342257 −0.171129 − 0.985249i \(-0.554741\pi\)
−0.171129 + 0.985249i \(0.554741\pi\)
\(32\) 0 0
\(33\) 133626. 0.0196146
\(34\) 0 0
\(35\) −2.86000e6 −0.322151
\(36\) 0 0
\(37\) −7.76468e6 −0.681108 −0.340554 − 0.940225i \(-0.610615\pi\)
−0.340554 + 0.940225i \(0.610615\pi\)
\(38\) 0 0
\(39\) −368496. −0.0255060
\(40\) 0 0
\(41\) 8.92276e6 0.493142 0.246571 − 0.969125i \(-0.420696\pi\)
0.246571 + 0.969125i \(0.420696\pi\)
\(42\) 0 0
\(43\) −3.46464e7 −1.54543 −0.772716 − 0.634752i \(-0.781102\pi\)
−0.772716 + 0.634752i \(0.781102\pi\)
\(44\) 0 0
\(45\) −2.33819e7 −0.850011
\(46\) 0 0
\(47\) −3.66292e7 −1.09493 −0.547466 − 0.836828i \(-0.684408\pi\)
−0.547466 + 0.836828i \(0.684408\pi\)
\(48\) 0 0
\(49\) 5.76480e6 0.142857
\(50\) 0 0
\(51\) −2.07584e6 −0.0429664
\(52\) 0 0
\(53\) −6.10693e7 −1.06312 −0.531559 − 0.847021i \(-0.678394\pi\)
−0.531559 + 0.847021i \(0.678394\pi\)
\(54\) 0 0
\(55\) −2.17415e7 −0.320374
\(56\) 0 0
\(57\) 2.00111e6 0.0251093
\(58\) 0 0
\(59\) −9.02214e7 −0.969339 −0.484669 − 0.874697i \(-0.661060\pi\)
−0.484669 + 0.874697i \(0.661060\pi\)
\(60\) 0 0
\(61\) 1.52451e8 1.40976 0.704879 − 0.709327i \(-0.251001\pi\)
0.704879 + 0.709327i \(0.251001\pi\)
\(62\) 0 0
\(63\) 4.71302e7 0.376935
\(64\) 0 0
\(65\) 5.99557e7 0.416601
\(66\) 0 0
\(67\) −2.18974e7 −0.132757 −0.0663783 − 0.997795i \(-0.521144\pi\)
−0.0663783 + 0.997795i \(0.521144\pi\)
\(68\) 0 0
\(69\) 7.76485e6 0.0412394
\(70\) 0 0
\(71\) −2.33501e8 −1.09050 −0.545251 − 0.838273i \(-0.683566\pi\)
−0.545251 + 0.838273i \(0.683566\pi\)
\(72\) 0 0
\(73\) 1.06705e8 0.439775 0.219887 − 0.975525i \(-0.429431\pi\)
0.219887 + 0.975525i \(0.429431\pi\)
\(74\) 0 0
\(75\) 3.91123e6 0.0142737
\(76\) 0 0
\(77\) 4.38236e7 0.142069
\(78\) 0 0
\(79\) 2.89730e8 0.836896 0.418448 − 0.908241i \(-0.362574\pi\)
0.418448 + 0.908241i \(0.362574\pi\)
\(80\) 0 0
\(81\) 3.84258e8 0.991838
\(82\) 0 0
\(83\) 4.73876e7 0.109601 0.0548004 − 0.998497i \(-0.482548\pi\)
0.0548004 + 0.998497i \(0.482548\pi\)
\(84\) 0 0
\(85\) 3.37747e8 0.701789
\(86\) 0 0
\(87\) 3.95428e7 0.0739998
\(88\) 0 0
\(89\) 6.97049e8 1.17763 0.588815 − 0.808268i \(-0.299595\pi\)
0.588815 + 0.808268i \(0.299595\pi\)
\(90\) 0 0
\(91\) −1.20851e8 −0.184741
\(92\) 0 0
\(93\) 1.28842e7 0.0178601
\(94\) 0 0
\(95\) −3.25588e8 −0.410121
\(96\) 0 0
\(97\) 7.82411e8 0.897351 0.448676 − 0.893695i \(-0.351896\pi\)
0.448676 + 0.893695i \(0.351896\pi\)
\(98\) 0 0
\(99\) 3.58280e8 0.374856
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 56.10.a.a.1.2 ✓ 3
4.3 odd 2 112.10.a.i.1.2 3
7.6 odd 2 392.10.a.d.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.10.a.a.1.2 ✓ 3 1.1 even 1 trivial
112.10.a.i.1.2 3 4.3 odd 2
392.10.a.d.1.2 3 7.6 odd 2