Properties

Label 56.10.a.b.1.3
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 = [3,0,84] 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: \( x^{3} - x^{2} - 1266x - 4032 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-33.3426\) 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)
 
\(f(q)\) \(=\) \(q+230.056 q^{3} -2532.41 q^{5} +2401.00 q^{7} +33242.7 q^{9} -5309.19 q^{11} -138145. q^{13} -582597. q^{15} +129547. q^{17} -886436. q^{19} +552364. q^{21} -1.67081e6 q^{23} +4.46000e6 q^{25} +3.11948e6 q^{27} -4.30491e6 q^{29} +2.81186e6 q^{31} -1.22141e6 q^{33} -6.08033e6 q^{35} -1.40879e7 q^{37} -3.17811e7 q^{39} +1.56714e7 q^{41} -4.41164e7 q^{43} -8.41842e7 q^{45} +4.79580e7 q^{47} +5.76480e6 q^{49} +2.98030e7 q^{51} +3.77845e7 q^{53} +1.34451e7 q^{55} -2.03930e8 q^{57} -2.30062e7 q^{59} -7.39313e7 q^{61} +7.98156e7 q^{63} +3.49841e8 q^{65} +1.59722e8 q^{67} -3.84378e8 q^{69} -4.39691e7 q^{71} -1.04436e8 q^{73} +1.02605e9 q^{75} -1.27474e7 q^{77} -3.20994e8 q^{79} +6.33390e7 q^{81} +2.06084e8 q^{83} -3.28066e8 q^{85} -9.90370e8 q^{87} +2.25453e8 q^{89} -3.31686e8 q^{91} +6.46884e8 q^{93} +2.24482e9 q^{95} +9.23936e8 q^{97} -1.76492e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 84 q^{3} - 2958 q^{5} + 7203 q^{7} + 34479 q^{9} - 8340 q^{11} + 50202 q^{13} - 499344 q^{15} - 145674 q^{17} - 1214460 q^{19} + 201684 q^{21} - 334080 q^{23} + 735237 q^{25} + 1531656 q^{27} - 7164198 q^{29}+ \cdots + 1478623356 q^{99}+O(q^{100}) \) 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
\(3\) 230.056 1.63979 0.819894 − 0.572516i \(-0.194032\pi\)
0.819894 + 0.572516i \(0.194032\pi\)
\(4\) 0 0
\(5\) −2532.41 −1.81205 −0.906024 − 0.423226i \(-0.860898\pi\)
−0.906024 + 0.423226i \(0.860898\pi\)
\(6\) 0 0
\(7\) 2401.00 0.377964
\(8\) 0 0
\(9\) 33242.7 1.68890
\(10\) 0 0
\(11\) −5309.19 −0.109335 −0.0546677 − 0.998505i \(-0.517410\pi\)
−0.0546677 + 0.998505i \(0.517410\pi\)
\(12\) 0 0
\(13\) −138145. −1.34150 −0.670749 − 0.741684i \(-0.734027\pi\)
−0.670749 + 0.741684i \(0.734027\pi\)
\(14\) 0 0
\(15\) −582597. −2.97137
\(16\) 0 0
\(17\) 129547. 0.376190 0.188095 − 0.982151i \(-0.439769\pi\)
0.188095 + 0.982151i \(0.439769\pi\)
\(18\) 0 0
\(19\) −886436. −1.56047 −0.780237 − 0.625485i \(-0.784901\pi\)
−0.780237 + 0.625485i \(0.784901\pi\)
\(20\) 0 0
\(21\) 552364. 0.619781
\(22\) 0 0
\(23\) −1.67081e6 −1.24495 −0.622473 − 0.782641i \(-0.713872\pi\)
−0.622473 + 0.782641i \(0.713872\pi\)
\(24\) 0 0
\(25\) 4.46000e6 2.28352
\(26\) 0 0
\(27\) 3.11948e6 1.12965
\(28\) 0 0
\(29\) −4.30491e6 −1.13025 −0.565124 − 0.825006i \(-0.691172\pi\)
−0.565124 + 0.825006i \(0.691172\pi\)
\(30\) 0 0
\(31\) 2.81186e6 0.546847 0.273423 − 0.961894i \(-0.411844\pi\)
0.273423 + 0.961894i \(0.411844\pi\)
\(32\) 0 0
\(33\) −1.22141e6 −0.179287
\(34\) 0 0
\(35\) −6.08033e6 −0.684890
\(36\) 0 0
\(37\) −1.40879e7 −1.23577 −0.617887 − 0.786267i \(-0.712011\pi\)
−0.617887 + 0.786267i \(0.712011\pi\)
\(38\) 0 0
\(39\) −3.17811e7 −2.19977
\(40\) 0 0
\(41\) 1.56714e7 0.866124 0.433062 − 0.901364i \(-0.357433\pi\)
0.433062 + 0.901364i \(0.357433\pi\)
\(42\) 0 0
\(43\) −4.41164e7 −1.96785 −0.983924 − 0.178585i \(-0.942848\pi\)
−0.983924 + 0.178585i \(0.942848\pi\)
\(44\) 0 0
\(45\) −8.41842e7 −3.06037
\(46\) 0 0
\(47\) 4.79580e7 1.43358 0.716789 − 0.697290i \(-0.245611\pi\)
0.716789 + 0.697290i \(0.245611\pi\)
\(48\) 0 0
\(49\) 5.76480e6 0.142857
\(50\) 0 0
\(51\) 2.98030e7 0.616871
\(52\) 0 0
\(53\) 3.77845e7 0.657767 0.328884 − 0.944370i \(-0.393328\pi\)
0.328884 + 0.944370i \(0.393328\pi\)
\(54\) 0 0
\(55\) 1.34451e7 0.198121
\(56\) 0 0
\(57\) −2.03930e8 −2.55884
\(58\) 0 0
\(59\) −2.30062e7 −0.247179 −0.123589 − 0.992333i \(-0.539441\pi\)
−0.123589 + 0.992333i \(0.539441\pi\)
\(60\) 0 0
\(61\) −7.39313e7 −0.683666 −0.341833 − 0.939761i \(-0.611048\pi\)
−0.341833 + 0.939761i \(0.611048\pi\)
\(62\) 0 0
\(63\) 7.98156e7 0.638345
\(64\) 0 0
\(65\) 3.49841e8 2.43086
\(66\) 0 0
\(67\) 1.59722e8 0.968339 0.484170 − 0.874974i \(-0.339122\pi\)
0.484170 + 0.874974i \(0.339122\pi\)
\(68\) 0 0
\(69\) −3.84378e8 −2.04145
\(70\) 0 0
\(71\) −4.39691e7 −0.205345 −0.102673 − 0.994715i \(-0.532739\pi\)
−0.102673 + 0.994715i \(0.532739\pi\)
\(72\) 0 0
\(73\) −1.04436e8 −0.430427 −0.215213 − 0.976567i \(-0.569045\pi\)
−0.215213 + 0.976567i \(0.569045\pi\)
\(74\) 0 0
\(75\) 1.02605e9 3.74449
\(76\) 0 0
\(77\) −1.27474e7 −0.0413249
\(78\) 0 0
\(79\) −3.20994e8 −0.927204 −0.463602 − 0.886044i \(-0.653443\pi\)
−0.463602 + 0.886044i \(0.653443\pi\)
\(80\) 0 0
\(81\) 6.33390e7 0.163489
\(82\) 0 0
\(83\) 2.06084e8 0.476643 0.238322 − 0.971186i \(-0.423403\pi\)
0.238322 + 0.971186i \(0.423403\pi\)
\(84\) 0 0
\(85\) −3.28066e8 −0.681674
\(86\) 0 0
\(87\) −9.90370e8 −1.85337
\(88\) 0 0
\(89\) 2.25453e8 0.380891 0.190445 − 0.981698i \(-0.439007\pi\)
0.190445 + 0.981698i \(0.439007\pi\)
\(90\) 0 0
\(91\) −3.31686e8 −0.507039
\(92\) 0 0
\(93\) 6.46884e8 0.896712
\(94\) 0 0
\(95\) 2.24482e9 2.82765
\(96\) 0 0
\(97\) 9.23936e8 1.05967 0.529833 − 0.848102i \(-0.322254\pi\)
0.529833 + 0.848102i \(0.322254\pi\)
\(98\) 0 0
\(99\) −1.76492e8 −0.184657
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.b.1.3 ✓ 3
4.3 odd 2 112.10.a.g.1.1 3
7.6 odd 2 392.10.a.c.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.10.a.b.1.3 ✓ 3 1.1 even 1 trivial
112.10.a.g.1.1 3 4.3 odd 2
392.10.a.c.1.1 3 7.6 odd 2