Properties

Label 56.10.a.b.1.1
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 1.1
Root \(37.5620\) 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)
 
\(f(q)\) \(=\) \(q-195.372 q^{3} -426.015 q^{5} +2401.00 q^{7} +18487.2 q^{9} +44849.3 q^{11} +30461.0 q^{13} +83231.4 q^{15} +147701. q^{17} +235665. q^{19} -469088. q^{21} -361212. q^{23} -1.77164e6 q^{25} +233618. q^{27} -7.37105e6 q^{29} +655353. q^{31} -8.76230e6 q^{33} -1.02286e6 q^{35} -9.92020e6 q^{37} -5.95123e6 q^{39} +7.31053e6 q^{41} -2.82702e6 q^{43} -7.87584e6 q^{45} +1.25979e6 q^{47} +5.76480e6 q^{49} -2.88567e7 q^{51} -3.49253e7 q^{53} -1.91065e7 q^{55} -4.60423e7 q^{57} -7.71202e7 q^{59} -1.08670e8 q^{61} +4.43879e7 q^{63} -1.29769e7 q^{65} -2.28868e8 q^{67} +7.05708e7 q^{69} +2.96897e8 q^{71} +1.26106e8 q^{73} +3.46128e8 q^{75} +1.07683e8 q^{77} +3.36576e8 q^{79} -4.09527e8 q^{81} +6.60104e8 q^{83} -6.29229e7 q^{85} +1.44010e9 q^{87} -6.09127e8 q^{89} +7.31369e7 q^{91} -1.28038e8 q^{93} -1.00397e8 q^{95} +5.20519e8 q^{97} +8.29140e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 84 q^{3} - 2958 q^{5} + 7203 q^{7} + 34479 q^{9} - 8340 q^{11} + 50202 q^{13} - 499344 q^{15} - 145674 q^{17} - 1214460 q^{19} + 201684 q^{21} - 334080 q^{23} + 735237 q^{25} + 1531656 q^{27} - 7164198 q^{29}+ \cdots + 1478623356 q^{99}+O(q^{100}) \) 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
\(3\) −195.372 −1.39257 −0.696285 − 0.717766i \(-0.745165\pi\)
−0.696285 + 0.717766i \(0.745165\pi\)
\(4\) 0 0
\(5\) −426.015 −0.304832 −0.152416 − 0.988316i \(-0.548705\pi\)
−0.152416 + 0.988316i \(0.548705\pi\)
\(6\) 0 0
\(7\) 2401.00 0.377964
\(8\) 0 0
\(9\) 18487.2 0.939249
\(10\) 0 0
\(11\) 44849.3 0.923610 0.461805 − 0.886982i \(-0.347202\pi\)
0.461805 + 0.886982i \(0.347202\pi\)
\(12\) 0 0
\(13\) 30461.0 0.295801 0.147900 − 0.989002i \(-0.452748\pi\)
0.147900 + 0.989002i \(0.452748\pi\)
\(14\) 0 0
\(15\) 83231.4 0.424499
\(16\) 0 0
\(17\) 147701. 0.428907 0.214454 − 0.976734i \(-0.431203\pi\)
0.214454 + 0.976734i \(0.431203\pi\)
\(18\) 0 0
\(19\) 235665. 0.414862 0.207431 − 0.978250i \(-0.433490\pi\)
0.207431 + 0.978250i \(0.433490\pi\)
\(20\) 0 0
\(21\) −469088. −0.526342
\(22\) 0 0
\(23\) −361212. −0.269146 −0.134573 − 0.990904i \(-0.542966\pi\)
−0.134573 + 0.990904i \(0.542966\pi\)
\(24\) 0 0
\(25\) −1.77164e6 −0.907078
\(26\) 0 0
\(27\) 233618. 0.0845999
\(28\) 0 0
\(29\) −7.37105e6 −1.93526 −0.967628 − 0.252382i \(-0.918786\pi\)
−0.967628 + 0.252382i \(0.918786\pi\)
\(30\) 0 0
\(31\) 655353. 0.127452 0.0637262 − 0.997967i \(-0.479702\pi\)
0.0637262 + 0.997967i \(0.479702\pi\)
\(32\) 0 0
\(33\) −8.76230e6 −1.28619
\(34\) 0 0
\(35\) −1.02286e6 −0.115215
\(36\) 0 0
\(37\) −9.92020e6 −0.870187 −0.435093 − 0.900385i \(-0.643285\pi\)
−0.435093 + 0.900385i \(0.643285\pi\)
\(38\) 0 0
\(39\) −5.95123e6 −0.411923
\(40\) 0 0
\(41\) 7.31053e6 0.404038 0.202019 − 0.979382i \(-0.435250\pi\)
0.202019 + 0.979382i \(0.435250\pi\)
\(42\) 0 0
\(43\) −2.82702e6 −0.126102 −0.0630509 − 0.998010i \(-0.520083\pi\)
−0.0630509 + 0.998010i \(0.520083\pi\)
\(44\) 0 0
\(45\) −7.87584e6 −0.286313
\(46\) 0 0
\(47\) 1.25979e6 0.0376581 0.0188290 − 0.999823i \(-0.494006\pi\)
0.0188290 + 0.999823i \(0.494006\pi\)
\(48\) 0 0
\(49\) 5.76480e6 0.142857
\(50\) 0 0
\(51\) −2.88567e7 −0.597283
\(52\) 0 0
\(53\) −3.49253e7 −0.607993 −0.303996 − 0.952673i \(-0.598321\pi\)
−0.303996 + 0.952673i \(0.598321\pi\)
\(54\) 0 0
\(55\) −1.91065e7 −0.281545
\(56\) 0 0
\(57\) −4.60423e7 −0.577724
\(58\) 0 0
\(59\) −7.71202e7 −0.828580 −0.414290 − 0.910145i \(-0.635970\pi\)
−0.414290 + 0.910145i \(0.635970\pi\)
\(60\) 0 0
\(61\) −1.08670e8 −1.00491 −0.502455 − 0.864604i \(-0.667570\pi\)
−0.502455 + 0.864604i \(0.667570\pi\)
\(62\) 0 0
\(63\) 4.43879e7 0.355003
\(64\) 0 0
\(65\) −1.29769e7 −0.0901694
\(66\) 0 0
\(67\) −2.28868e8 −1.38755 −0.693774 − 0.720193i \(-0.744053\pi\)
−0.693774 + 0.720193i \(0.744053\pi\)
\(68\) 0 0
\(69\) 7.05708e7 0.374804
\(70\) 0 0
\(71\) 2.96897e8 1.38657 0.693287 − 0.720662i \(-0.256162\pi\)
0.693287 + 0.720662i \(0.256162\pi\)
\(72\) 0 0
\(73\) 1.26106e8 0.519738 0.259869 − 0.965644i \(-0.416321\pi\)
0.259869 + 0.965644i \(0.416321\pi\)
\(74\) 0 0
\(75\) 3.46128e8 1.26317
\(76\) 0 0
\(77\) 1.07683e8 0.349092
\(78\) 0 0
\(79\) 3.36576e8 0.972213 0.486107 − 0.873899i \(-0.338417\pi\)
0.486107 + 0.873899i \(0.338417\pi\)
\(80\) 0 0
\(81\) −4.09527e8 −1.05706
\(82\) 0 0
\(83\) 6.60104e8 1.52673 0.763363 − 0.645969i \(-0.223547\pi\)
0.763363 + 0.645969i \(0.223547\pi\)
\(84\) 0 0
\(85\) −6.29229e7 −0.130744
\(86\) 0 0
\(87\) 1.44010e9 2.69498
\(88\) 0 0
\(89\) −6.09127e8 −1.02909 −0.514545 − 0.857464i \(-0.672039\pi\)
−0.514545 + 0.857464i \(0.672039\pi\)
\(90\) 0 0
\(91\) 7.31369e7 0.111802
\(92\) 0 0
\(93\) −1.28038e8 −0.177486
\(94\) 0 0
\(95\) −1.00397e8 −0.126463
\(96\) 0 0
\(97\) 5.20519e8 0.596986 0.298493 − 0.954412i \(-0.403516\pi\)
0.298493 + 0.954412i \(0.403516\pi\)
\(98\) 0 0
\(99\) 8.29140e8 0.867500
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.1.1 ✓ 3
4.3 odd 2 112.10.a.g.1.3 3
7.6 odd 2 392.10.a.c.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.10.a.b.1.1 ✓ 3 1.1 even 1 trivial
112.10.a.g.1.3 3 4.3 odd 2
392.10.a.c.1.3 3 7.6 odd 2