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Properties

Label 42.12.f.a
Level $42$
Weight $12$
Character orbit 42.f
Analytic conductor $32.270$
Analytic rank $0$
Dimension $60$
Inner twists $4$

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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] = [42,12,Mod(5,42)] 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("42.5"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(42, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 5])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 42.f (of order \(6\), degree \(2\), 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: \(32.2704135835\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(30\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 30720 q^{4} - 95298 q^{7} - 91434 q^{9} + 245184 q^{10} - 8480820 q^{15} - 31457280 q^{16} + 12170496 q^{18} + 1649724 q^{19} + 43254216 q^{21} + 20705664 q^{22} + 4521984 q^{24} - 336132444 q^{25}+ \cdots + 288758042496 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −27.7128 − 16.0000i −420.023 − 26.9724i 512.000 + 886.810i 956.472 − 1656.66i 11208.5 + 7467.85i −43035.3 − 11193.3i − 32768.0i 175692. + 22658.1i −53013.1 + 30607.1i
5.2 −27.7128 − 16.0000i −357.754 − 221.718i 512.000 + 886.810i 3644.72 − 6312.84i 6366.89 + 11868.5i 18530.2 − 40422.3i − 32768.0i 78829.4 + 158641.i −202011. + 116631.i
5.3 −27.7128 − 16.0000i −317.166 + 276.681i 512.000 + 886.810i 1584.15 − 2743.83i 13216.5 − 2592.95i 44235.4 − 4533.49i − 32768.0i 24042.2 − 175508.i −87802.6 + 50692.9i
5.4 −27.7128 − 16.0000i −296.285 − 298.935i 512.000 + 886.810i −3221.21 + 5579.29i 3427.95 + 13024.9i 25344.2 + 36537.6i − 32768.0i −1576.82 + 177140.i 178537. − 103079.i
5.5 −27.7128 − 16.0000i −289.106 + 305.883i 512.000 + 886.810i −6887.81 + 11930.0i 12906.1 − 3851.19i −11166.0 − 43042.4i − 32768.0i −9982.23 − 176866.i 381761. − 220410.i
5.6 −27.7128 − 16.0000i −131.210 + 399.914i 512.000 + 886.810i 6258.52 − 10840.1i 10034.8 − 8983.37i −41292.3 + 16500.6i − 32768.0i −142715. − 104945.i −346882. + 200273.i
5.7 −27.7128 − 16.0000i −76.1744 − 413.938i 512.000 + 886.810i −4602.68 + 7972.08i −4512.00 + 12690.2i −26134.7 − 35976.5i − 32768.0i −165542. + 63062.9i 255107. − 147286.i
5.8 −27.7128 − 16.0000i −13.7199 − 420.665i 512.000 + 886.810i 5294.36 − 9170.11i −6350.42 + 11877.3i 18339.4 + 40509.2i − 32768.0i −176771. + 11543.0i −293443. + 169420.i
5.9 −27.7128 − 16.0000i 76.3670 + 413.902i 512.000 + 886.810i −4176.66 + 7234.19i 4506.09 − 12692.3i −20634.7 + 39389.6i − 32768.0i −165483. + 63217.0i 231494. − 133653.i
5.10 −27.7128 − 16.0000i 142.282 + 396.110i 512.000 + 886.810i −497.386 + 861.497i 2394.71 − 13253.8i 43875.4 + 7230.30i − 32768.0i −136659. + 112719.i 27567.9 − 15916.3i
5.11 −27.7128 − 16.0000i 213.694 − 362.604i 512.000 + 886.810i −1022.28 + 1770.64i −11723.7 + 6629.68i −41388.4 + 16258.2i − 32768.0i −85816.7 − 154973.i 56660.5 − 32712.9i
5.12 −27.7128 − 16.0000i 300.225 + 294.978i 512.000 + 886.810i 407.491 − 705.796i −3600.42 − 12978.3i −18834.9 − 40281.2i − 32768.0i 3122.76 + 177119.i −22585.5 + 13039.7i
5.13 −27.7128 − 16.0000i 357.546 − 222.054i 512.000 + 886.810i 4010.83 − 6946.96i −13461.5 + 433.002i 8015.68 − 43738.7i − 32768.0i 78531.2 − 158789.i −222303. + 128347.i
5.14 −27.7128 − 16.0000i 410.418 − 93.2952i 512.000 + 886.810i −4829.27 + 8364.54i −12866.6 − 3981.22i 44317.7 + 3642.71i − 32768.0i 159739. − 76580.1i 267665. − 154537.i
5.15 −27.7128 − 16.0000i 420.827 + 7.21258i 512.000 + 886.810i 1974.84 − 3420.52i −11546.9 − 6933.11i −23996.1 + 37436.8i − 32768.0i 177043. + 6070.49i −109457. + 63194.8i
5.16 27.7128 + 16.0000i −411.940 − 86.3257i 512.000 + 886.810i −6258.52 + 10840.1i −10034.8 − 8983.37i −41292.3 + 16500.6i 32768.0i 162243. + 71122.1i −346882. + 200273.i
5.17 27.7128 + 16.0000i −409.456 + 97.4316i 512.000 + 886.810i 6887.81 − 11930.0i −12906.1 − 3851.19i −11166.0 − 43042.4i 32768.0i 158161. − 79787.9i 381761. − 220410.i
5.18 27.7128 + 16.0000i −398.196 + 136.334i 512.000 + 886.810i −1584.15 + 2743.83i −13216.5 − 2592.95i 44235.4 − 4533.49i 32768.0i 139973. − 108575.i −87802.6 + 50692.9i
5.19 27.7128 + 16.0000i −320.266 − 273.087i 512.000 + 886.810i 4176.66 − 7234.19i −4506.09 − 12692.3i −20634.7 + 39389.6i 32768.0i 27994.1 + 174921.i 231494. − 133653.i
5.20 27.7128 + 16.0000i −271.900 − 321.275i 512.000 + 886.810i 497.386 − 861.497i −2394.71 − 13253.8i 43875.4 + 7230.30i 32768.0i −29287.9 + 174709.i 27567.9 − 15916.3i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.30
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 42.12.f.a ✓ 60
3.b odd 2 1 inner 42.12.f.a ✓ 60
7.d odd 6 1 inner 42.12.f.a ✓ 60
21.g even 6 1 inner 42.12.f.a ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.12.f.a ✓ 60 1.a even 1 1 trivial
42.12.f.a ✓ 60 3.b odd 2 1 inner
42.12.f.a ✓ 60 7.d odd 6 1 inner
42.12.f.a ✓ 60 21.g even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(42, [\chi])\).