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Properties

Label 62.7.f.a
Level $62$
Weight $7$
Character orbit 62.f
Analytic conductor $14.263$
Analytic rank $0$
Dimension $64$
Inner twists $2$

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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] = [62,7,Mod(15,62)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(62, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([7])) N = Newforms(chi, 7, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("62.15"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Level: \( N \) \(=\) \( 62 = 2 \cdot 31 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 62.f (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.2633531844\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(16\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 512 q^{4} - 288 q^{5} - 692 q^{7} + 3248 q^{9} + 1344 q^{10} + 3200 q^{11} - 3600 q^{13} + 4560 q^{14} + 9520 q^{15} - 16384 q^{16} - 1400 q^{17} - 11200 q^{18} + 2912 q^{19} - 9216 q^{20} + 4860 q^{21}+ \cdots + 13577088 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1 −4.57649 + 3.32502i −25.5799 + 35.2077i 9.88854 − 30.4338i −242.487 − 246.181i 114.010 − 350.886i 55.9381 + 172.160i −359.977 − 1107.90i 1109.74 − 806.272i
15.2 −4.57649 + 3.32502i −16.9719 + 23.3599i 9.88854 − 30.4338i 12.8309 − 163.338i −163.532 + 503.299i 55.9381 + 172.160i −32.3638 − 99.6054i −58.7205 + 42.6630i
15.3 −4.57649 + 3.32502i −12.3480 + 16.9955i 9.88854 − 30.4338i 59.5260 − 118.837i −30.1125 + 92.6767i 55.9381 + 172.160i 88.8978 + 273.599i −272.420 + 197.925i
15.4 −4.57649 + 3.32502i 1.22115 − 1.68076i 9.88854 − 30.4338i −33.3114 11.7523i 91.0827 − 280.324i 55.9381 + 172.160i 223.940 + 689.215i 152.449 − 110.761i
15.5 −4.57649 + 3.32502i 8.28172 − 11.3988i 9.88854 − 30.4338i −177.402 79.7034i −37.1950 + 114.474i 55.9381 + 172.160i 163.927 + 504.517i 811.879 − 589.865i
15.6 −4.57649 + 3.32502i 12.2169 − 16.8151i 9.88854 − 30.4338i 201.726 117.576i 109.915 − 338.284i 55.9381 + 172.160i 91.7775 + 282.462i −923.196 + 670.741i
15.7 −4.57649 + 3.32502i 20.4297 − 28.1190i 9.88854 − 30.4338i 128.397 196.615i −202.944 + 624.597i 55.9381 + 172.160i −148.035 − 455.605i −587.607 + 426.921i
15.8 −4.57649 + 3.32502i 30.6389 − 42.1708i 9.88854 − 30.4338i −102.487 294.869i 114.954 − 353.792i 55.9381 + 172.160i −614.363 − 1890.82i 469.030 − 340.771i
15.9 4.57649 − 3.32502i −24.4446 + 33.6452i 9.88854 − 30.4338i 110.694 235.256i −126.529 + 389.418i −55.9381 − 172.160i −309.183 − 951.568i 506.591 − 368.060i
15.10 4.57649 − 3.32502i −22.4820 + 30.9438i 9.88854 − 30.4338i −120.991 216.367i 61.2460 − 188.496i −55.9381 − 172.160i −226.806 − 698.038i −553.716 + 402.298i
15.11 4.57649 − 3.32502i −3.06561 + 4.21946i 9.88854 − 30.4338i −100.632 29.5035i 61.2265 − 188.436i −55.9381 − 172.160i 216.868 + 667.450i −460.541 + 334.602i
15.12 4.57649 − 3.32502i −3.00199 + 4.13188i 9.88854 − 30.4338i 176.963 28.8912i 7.81304 − 24.0461i −55.9381 − 172.160i 217.213 + 668.512i 809.870 − 588.405i
15.13 4.57649 − 3.32502i 0.519266 − 0.714708i 9.88854 − 30.4338i −131.510 − 4.99742i −80.4012 + 247.449i −55.9381 − 172.160i 225.032 + 692.578i −601.854 + 437.272i
15.14 4.57649 − 3.32502i 15.4410 − 21.2527i 9.88854 − 30.4338i 6.26454 − 148.604i 109.443 − 336.831i −55.9381 − 172.160i 12.0203 + 36.9947i 28.6696 − 20.8297i
15.15 4.57649 − 3.32502i 24.8780 − 34.2416i 9.88854 − 30.4338i 150.694 − 239.426i 33.3766 − 102.723i −55.9381 − 172.160i −328.299 − 1010.40i 689.650 − 501.060i
15.16 4.57649 − 3.32502i 30.0446 − 41.3528i 9.88854 − 30.4338i −171.272 − 289.150i −121.313 + 373.364i −55.9381 − 172.160i −582.105 − 1791.54i −783.825 + 569.482i
23.1 −1.74806 + 5.37999i −43.6742 + 14.1906i −25.8885 − 18.8091i 144.799 − 259.773i 285.748 + 207.608i 146.448 − 106.400i 1116.29 − 811.032i −253.118 + 779.019i
23.2 −1.74806 + 5.37999i −35.2660 + 11.4586i −25.8885 − 18.8091i −54.3091 − 209.761i −333.682 − 242.434i 146.448 − 106.400i 522.615 − 379.702i 94.9357 − 292.182i
23.3 −1.74806 + 5.37999i −12.9095 + 4.19455i −25.8885 − 18.8091i −246.570 − 76.7853i 34.8772 + 25.3397i 146.448 − 106.400i −440.712 + 320.196i 431.020 − 1326.54i
23.4 −1.74806 + 5.37999i −5.83008 + 1.89431i −25.8885 − 18.8091i 226.620 − 34.6772i −495.718 − 360.160i 146.448 − 106.400i −559.372 + 406.407i −396.147 + 1219.21i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.16
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.f odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 62.7.f.a ✓ 64
31.f odd 10 1 inner 62.7.f.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
62.7.f.a ✓ 64 1.a even 1 1 trivial
62.7.f.a ✓ 64 31.f odd 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(62, [\chi])\).