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Properties

Label 62.8.g.b
Level $62$
Weight $8$
Character orbit 62.g
Analytic conductor $19.368$
Analytic rank $0$
Dimension $72$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [62,8,Mod(7,62)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(62, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([28])) N = Newforms(chi, 8, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("62.7"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Level: \( N \) \(=\) \( 62 = 2 \cdot 31 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 62.g (of order \(15\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72,144] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(19.3678715800\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(9\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 144 q^{2} + 53 q^{3} - 1152 q^{4} + 97 q^{5} + 656 q^{6} - 3516 q^{7} + 9216 q^{8} + 4100 q^{9} - 3616 q^{10} - 6385 q^{11} + 3392 q^{12} - 7108 q^{13} - 25872 q^{14} - 37456 q^{15} - 73728 q^{16}+ \cdots + 20676441 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −2.47214 − 7.60845i −85.5242 + 18.1787i −51.7771 + 37.6183i 85.4938 − 148.080i 349.740 + 605.767i 403.909 + 179.832i 414.217 + 300.946i 4986.00 − 2219.91i −1338.01 − 284.403i
7.2 −2.47214 − 7.60845i −49.8104 + 10.5875i −51.7771 + 37.6183i −198.975 + 344.634i 203.693 + 352.806i 769.064 + 342.409i 414.217 + 300.946i 371.054 − 165.204i 3114.03 + 661.907i
7.3 −2.47214 − 7.60845i −34.2192 + 7.27353i −51.7771 + 37.6183i 141.827 − 245.651i 139.935 + 242.374i 360.576 + 160.539i 414.217 + 300.946i −879.871 + 391.744i −2219.64 − 471.799i
7.4 −2.47214 − 7.60845i −27.9180 + 5.93415i −51.7771 + 37.6183i −188.074 + 325.754i 114.167 + 197.743i −1642.75 − 731.401i 414.217 + 300.946i −1253.72 + 558.194i 2943.43 + 625.645i
7.5 −2.47214 − 7.60845i −20.1699 + 4.28724i −51.7771 + 37.6183i 104.818 − 181.551i 82.4820 + 142.863i −243.185 − 108.273i 414.217 + 300.946i −1609.48 + 716.587i −1640.45 − 348.687i
7.6 −2.47214 − 7.60845i 34.2775 − 7.28590i −51.7771 + 37.6183i −132.279 + 229.114i −140.173 − 242.787i 455.779 + 202.926i 414.217 + 300.946i −876.063 + 390.048i 2070.21 + 440.037i
7.7 −2.47214 − 7.60845i 49.2561 − 10.4697i −51.7771 + 37.6183i 158.162 − 273.945i −201.426 − 348.880i −1044.21 − 464.913i 414.217 + 300.946i 318.621 − 141.859i −2475.30 − 526.141i
7.8 −2.47214 − 7.60845i 61.5772 − 13.0886i −51.7771 + 37.6183i −130.560 + 226.137i −251.812 − 436.150i −320.081 − 142.509i 414.217 + 300.946i 1622.52 − 722.391i 2043.32 + 434.320i
7.9 −2.47214 − 7.60845i 66.6958 − 14.1766i −51.7771 + 37.6183i 217.612 − 376.916i −272.743 − 472.405i 1419.54 + 632.018i 414.217 + 300.946i 2249.43 − 1001.51i −3405.71 − 723.907i
9.1 −2.47214 + 7.60845i −85.5242 − 18.1787i −51.7771 − 37.6183i 85.4938 + 148.080i 349.740 − 605.767i 403.909 − 179.832i 414.217 − 300.946i 4986.00 + 2219.91i −1338.01 + 284.403i
9.2 −2.47214 + 7.60845i −49.8104 − 10.5875i −51.7771 − 37.6183i −198.975 − 344.634i 203.693 − 352.806i 769.064 − 342.409i 414.217 − 300.946i 371.054 + 165.204i 3114.03 − 661.907i
9.3 −2.47214 + 7.60845i −34.2192 − 7.27353i −51.7771 − 37.6183i 141.827 + 245.651i 139.935 − 242.374i 360.576 − 160.539i 414.217 − 300.946i −879.871 − 391.744i −2219.64 + 471.799i
9.4 −2.47214 + 7.60845i −27.9180 − 5.93415i −51.7771 − 37.6183i −188.074 − 325.754i 114.167 − 197.743i −1642.75 + 731.401i 414.217 − 300.946i −1253.72 − 558.194i 2943.43 − 625.645i
9.5 −2.47214 + 7.60845i −20.1699 − 4.28724i −51.7771 − 37.6183i 104.818 + 181.551i 82.4820 − 142.863i −243.185 + 108.273i 414.217 − 300.946i −1609.48 − 716.587i −1640.45 + 348.687i
9.6 −2.47214 + 7.60845i 34.2775 + 7.28590i −51.7771 − 37.6183i −132.279 − 229.114i −140.173 + 242.787i 455.779 − 202.926i 414.217 − 300.946i −876.063 − 390.048i 2070.21 − 440.037i
9.7 −2.47214 + 7.60845i 49.2561 + 10.4697i −51.7771 − 37.6183i 158.162 + 273.945i −201.426 + 348.880i −1044.21 + 464.913i 414.217 − 300.946i 318.621 + 141.859i −2475.30 + 526.141i
9.8 −2.47214 + 7.60845i 61.5772 + 13.0886i −51.7771 − 37.6183i −130.560 − 226.137i −251.812 + 436.150i −320.081 + 142.509i 414.217 − 300.946i 1622.52 + 722.391i 2043.32 − 434.320i
9.9 −2.47214 + 7.60845i 66.6958 + 14.1766i −51.7771 − 37.6183i 217.612 + 376.916i −272.743 + 472.405i 1419.54 − 632.018i 414.217 − 300.946i 2249.43 + 1001.51i −3405.71 + 723.907i
19.1 6.47214 + 4.70228i −64.0587 − 28.5208i 19.7771 + 60.8676i −246.786 + 427.446i −280.484 − 485.813i 122.648 − 136.214i −158.217 + 486.941i 1826.70 + 2028.75i −3607.20 + 1606.03i
19.2 6.47214 + 4.70228i −59.9156 − 26.6762i 19.7771 + 60.8676i 85.4015 − 147.920i −262.343 − 454.392i −1205.22 + 1338.54i −158.217 + 486.941i 1414.88 + 1571.38i 1248.29 − 555.775i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.9
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.g even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 62.8.g.b ✓ 72
31.g even 15 1 inner 62.8.g.b ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
62.8.g.b ✓ 72 1.a even 1 1 trivial
62.8.g.b ✓ 72 31.g even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{72} - 53 T_{3}^{71} - 10487 T_{3}^{70} + 594124 T_{3}^{69} + 14614744 T_{3}^{68} + \cdots + 51\!\cdots\!25 \) acting on \(S_{8}^{\mathrm{new}}(62, [\chi])\). Copy content Toggle raw display