Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [12,16,Mod(11,12)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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")
//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);
Level: | \( N \) | \(=\) | \( 12 = 2^{2} \cdot 3 \) |
Weight: | \( k \) | \(=\) | \( 16 \) |
Character orbit: | \([\chi]\) | \(=\) | 12.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
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.
comment: embeddings in the coefficient field
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 |
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 \(S_{16}^{\mathrm{new}}(12, [\chi])\).