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Properties

Label 12.16.b.a
Level $12$
Weight $16$
Character orbit 12.b
Analytic conductor $17.123$
Analytic rank $0$
Dimension $28$
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] = [12,16,Mod(11,12)] 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("12.11"); S:= CuspForms(chi, 16); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(12, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 16, names="a")
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 12.b (of order \(2\), degree \(1\), 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: \(17.1232206120\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 26968 q^{4} + 823656 q^{6} - 3812052 q^{9} - 43831600 q^{10} - 226414248 q^{12} + 124527272 q^{13} - 1459325408 q^{16} - 4234777584 q^{18} - 7261350648 q^{21} - 2533670160 q^{22} - 18487781856 q^{24}+ \cdots - 824166397720648 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −172.746 − 54.1000i 3105.31 + 2169.32i 26914.4 + 18691.1i − 192307.i −419070. − 542739.i − 366926.i −3.63816e6 − 4.68489e6i 4.93700e6 + 1.34728e7i −1.04038e7 + 3.32203e7i
11.2 −172.746 + 54.1000i 3105.31 − 2169.32i 26914.4 − 18691.1i 192307.i −419070. + 542739.i 366926.i −3.63816e6 + 4.68489e6i 4.93700e6 − 1.34728e7i −1.04038e7 − 3.32203e7i
11.3 −171.504 − 57.9158i −3751.22 − 526.541i 26059.5 + 19865.6i 172529.i 612856. + 307559.i − 3.89480e6i −3.31879e6 − 4.91630e6i 1.37944e7 + 3.95035e6i 9.99218e6 − 2.95896e7i
11.4 −171.504 + 57.9158i −3751.22 + 526.541i 26059.5 − 19865.6i − 172529.i 612856. − 307559.i 3.89480e6i −3.31879e6 + 4.91630e6i 1.37944e7 − 3.95035e6i 9.99218e6 + 2.95896e7i
11.5 −166.441 − 71.1726i −8.61090 − 3787.99i 22636.9 + 23692.0i − 99785.5i −268168. + 631087.i 1.78364e6i −2.08148e6 − 5.55444e6i −1.43488e7 + 65235.9i −7.10200e6 + 1.66083e7i
11.6 −166.441 + 71.1726i −8.61090 + 3787.99i 22636.9 − 23692.0i 99785.5i −268168. − 631087.i − 1.78364e6i −2.08148e6 + 5.55444e6i −1.43488e7 − 65235.9i −7.10200e6 − 1.66083e7i
11.7 −127.986 − 128.014i −692.213 + 3724.21i −7.18349 + 32768.0i 230590.i 565345. − 388034.i 3.60781e6i 4.19568e6 − 4.19292e6i −1.33906e7 − 5.15589e6i 2.95188e7 − 2.95123e7i
11.8 −127.986 + 128.014i −692.213 − 3724.21i −7.18349 − 32768.0i − 230590.i 565345. + 388034.i − 3.60781e6i 4.19568e6 + 4.19292e6i −1.33906e7 + 5.15589e6i 2.95188e7 + 2.95123e7i
11.9 −83.3106 − 160.709i 3555.47 − 1306.74i −18886.7 + 26777.5i 93360.9i −506212. − 462530.i − 1.14542e6i 5.87684e6 + 804409.i 1.09338e7 − 9.29212e6i 1.50039e7 − 7.77795e6i
11.10 −83.3106 + 160.709i 3555.47 + 1306.74i −18886.7 − 26777.5i − 93360.9i −506212. + 462530.i 1.14542e6i 5.87684e6 − 804409.i 1.09338e7 + 9.29212e6i 1.50039e7 + 7.77795e6i
11.11 −80.5973 − 162.087i −2767.36 + 2586.62i −19776.2 + 26127.5i − 332581.i 642299. + 240078.i − 2.13481e6i 5.82882e6 + 1.09965e6i 967701. − 1.43162e7i −5.39069e7 + 2.68051e7i
11.12 −80.5973 + 162.087i −2767.36 − 2586.62i −19776.2 − 26127.5i 332581.i 642299. − 240078.i 2.13481e6i 5.82882e6 − 1.09965e6i 967701. + 1.43162e7i −5.39069e7 − 2.68051e7i
11.13 −35.8416 − 177.436i −2291.54 − 3016.25i −30198.8 + 12719.1i 33476.6i −453057. + 514708.i 694274.i 3.33920e6 + 4.90246e6i −3.84657e6 + 1.38237e7i 5.93994e6 − 1.19985e6i
11.14 −35.8416 + 177.436i −2291.54 + 3016.25i −30198.8 − 12719.1i − 33476.6i −453057. − 514708.i − 694274.i 3.33920e6 − 4.90246e6i −3.84657e6 − 1.38237e7i 5.93994e6 + 1.19985e6i
11.15 35.8416 − 177.436i 2291.54 + 3016.25i −30198.8 − 12719.1i 33476.6i 617322. − 298494.i − 694274.i −3.33920e6 + 4.90246e6i −3.84657e6 + 1.38237e7i 5.93994e6 + 1.19985e6i
11.16 35.8416 + 177.436i 2291.54 − 3016.25i −30198.8 + 12719.1i − 33476.6i 617322. + 298494.i 694274.i −3.33920e6 − 4.90246e6i −3.84657e6 − 1.38237e7i 5.93994e6 − 1.19985e6i
11.17 80.5973 − 162.087i 2767.36 − 2586.62i −19776.2 − 26127.5i − 332581.i −196215. − 657027.i 2.13481e6i −5.82882e6 + 1.09965e6i 967701. − 1.43162e7i −5.39069e7 − 2.68051e7i
11.18 80.5973 + 162.087i 2767.36 + 2586.62i −19776.2 + 26127.5i 332581.i −196215. + 657027.i − 2.13481e6i −5.82882e6 − 1.09965e6i 967701. + 1.43162e7i −5.39069e7 + 2.68051e7i
11.19 83.3106 − 160.709i −3555.47 + 1306.74i −18886.7 − 26777.5i 93360.9i −86203.8 + 680260.i 1.14542e6i −5.87684e6 + 804409.i 1.09338e7 − 9.29212e6i 1.50039e7 + 7.77795e6i
11.20 83.3106 + 160.709i −3555.47 − 1306.74i −18886.7 + 26777.5i − 93360.9i −86203.8 − 680260.i − 1.14542e6i −5.87684e6 − 804409.i 1.09338e7 + 9.29212e6i 1.50039e7 − 7.77795e6i
See all 28 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 11.28
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
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 12.16.b.a ✓ 28
3.b odd 2 1 inner 12.16.b.a ✓ 28
4.b odd 2 1 inner 12.16.b.a ✓ 28
12.b even 2 1 inner 12.16.b.a ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.16.b.a ✓ 28 1.a even 1 1 trivial
12.16.b.a ✓ 28 3.b odd 2 1 inner
12.16.b.a ✓ 28 4.b odd 2 1 inner
12.16.b.a ✓ 28 12.b even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{16}^{\mathrm{new}}(12, [\chi])\).