Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [348,2,Mod(59,348)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(348, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("348.59");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 348 = 2^{2} \cdot 3 \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 348.c (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: | \(2.77879399034\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
59.1 | −1.41245 | − | 0.0705195i | −0.494212 | − | 1.66005i | 1.99005 | + | 0.199211i | − | 3.75368i | 0.580986 | + | 2.37959i | − | 4.16013i | −2.79681 | − | 0.421714i | −2.51151 | + | 1.64083i | −0.264707 | + | 5.30190i | ||
59.2 | −1.41245 | + | 0.0705195i | −0.494212 | + | 1.66005i | 1.99005 | − | 0.199211i | 3.75368i | 0.580986 | − | 2.37959i | 4.16013i | −2.79681 | + | 0.421714i | −2.51151 | − | 1.64083i | −0.264707 | − | 5.30190i | ||||
59.3 | −1.38479 | − | 0.286955i | 1.43171 | + | 0.974781i | 1.83531 | + | 0.794747i | − | 1.32510i | −1.70291 | − | 1.76071i | 2.06888i | −2.31348 | − | 1.62721i | 1.09961 | + | 2.79121i | −0.380244 | + | 1.83499i | |||
59.4 | −1.38479 | + | 0.286955i | 1.43171 | − | 0.974781i | 1.83531 | − | 0.794747i | 1.32510i | −1.70291 | + | 1.76071i | − | 2.06888i | −2.31348 | + | 1.62721i | 1.09961 | − | 2.79121i | −0.380244 | − | 1.83499i | |||
59.5 | −1.34502 | − | 0.436932i | −0.312286 | − | 1.70367i | 1.61818 | + | 1.17537i | 2.54452i | −0.324355 | + | 2.42792i | − | 0.190467i | −1.66294 | − | 2.28794i | −2.80496 | + | 1.06406i | 1.11178 | − | 3.42245i | |||
59.6 | −1.34502 | + | 0.436932i | −0.312286 | + | 1.70367i | 1.61818 | − | 1.17537i | − | 2.54452i | −0.324355 | − | 2.42792i | 0.190467i | −1.66294 | + | 2.28794i | −2.80496 | − | 1.06406i | 1.11178 | + | 3.42245i | |||
59.7 | −1.30790 | − | 0.537964i | −1.67797 | + | 0.429454i | 1.42119 | + | 1.40720i | 1.34318i | 2.42564 | + | 0.341005i | − | 1.97166i | −1.10174 | − | 2.60503i | 2.63114 | − | 1.44122i | 0.722585 | − | 1.75675i | |||
59.8 | −1.30790 | + | 0.537964i | −1.67797 | − | 0.429454i | 1.42119 | − | 1.40720i | − | 1.34318i | 2.42564 | − | 0.341005i | 1.97166i | −1.10174 | + | 2.60503i | 2.63114 | + | 1.44122i | 0.722585 | + | 1.75675i | |||
59.9 | −1.10940 | − | 0.877053i | −1.15966 | − | 1.28654i | 0.461555 | + | 1.94601i | − | 1.35780i | 0.158175 | + | 2.44438i | 3.91076i | 1.19471 | − | 2.56372i | −0.310357 | + | 2.98390i | −1.19086 | + | 1.50634i | |||
59.10 | −1.10940 | + | 0.877053i | −1.15966 | + | 1.28654i | 0.461555 | − | 1.94601i | 1.35780i | 0.158175 | − | 2.44438i | − | 3.91076i | 1.19471 | + | 2.56372i | −0.310357 | − | 2.98390i | −1.19086 | − | 1.50634i | |||
59.11 | −0.946094 | − | 1.05115i | 1.21284 | − | 1.23653i | −0.209812 | + | 1.98896i | − | 2.17467i | −2.44724 | − | 0.104998i | 0.0947864i | 2.28919 | − | 1.66120i | −0.0580256 | − | 2.99944i | −2.28589 | + | 2.05744i | |||
59.12 | −0.946094 | + | 1.05115i | 1.21284 | + | 1.23653i | −0.209812 | − | 1.98896i | 2.17467i | −2.44724 | + | 0.104998i | − | 0.0947864i | 2.28919 | + | 1.66120i | −0.0580256 | + | 2.99944i | −2.28589 | − | 2.05744i | |||
59.13 | −0.479912 | − | 1.33029i | −1.72606 | + | 0.143940i | −1.53937 | + | 1.27685i | − | 4.13273i | 1.01984 | + | 2.22709i | − | 2.24214i | 2.43735 | + | 1.43504i | 2.95856 | − | 0.496898i | −5.49775 | + | 1.98335i | ||
59.14 | −0.479912 | + | 1.33029i | −1.72606 | − | 0.143940i | −1.53937 | − | 1.27685i | 4.13273i | 1.01984 | − | 2.22709i | 2.24214i | 2.43735 | − | 1.43504i | 2.95856 | + | 0.496898i | −5.49775 | − | 1.98335i | ||||
59.15 | −0.377291 | − | 1.36296i | 1.65882 | + | 0.498301i | −1.71530 | + | 1.02846i | 0.328095i | 0.0533041 | − | 2.44891i | − | 4.37145i | 2.04892 | + | 1.94986i | 2.50339 | + | 1.65319i | 0.447179 | − | 0.123787i | |||
59.16 | −0.377291 | + | 1.36296i | 1.65882 | − | 0.498301i | −1.71530 | − | 1.02846i | − | 0.328095i | 0.0533041 | + | 2.44891i | 4.37145i | 2.04892 | − | 1.94986i | 2.50339 | − | 1.65319i | 0.447179 | + | 0.123787i | |||
59.17 | −0.254328 | − | 1.39116i | 0.774651 | − | 1.54917i | −1.87063 | + | 0.707619i | 3.96336i | −2.35215 | − | 0.683665i | 3.59138i | 1.46016 | + | 2.42238i | −1.79983 | − | 2.40013i | 5.51365 | − | 1.00799i | ||||
59.18 | −0.254328 | + | 1.39116i | 0.774651 | + | 1.54917i | −1.87063 | − | 0.707619i | − | 3.96336i | −2.35215 | + | 0.683665i | − | 3.59138i | 1.46016 | − | 2.42238i | −1.79983 | + | 2.40013i | 5.51365 | + | 1.00799i | ||
59.19 | −0.0664333 | − | 1.41265i | −0.382087 | + | 1.68938i | −1.99117 | + | 0.187694i | − | 1.18625i | 2.41189 | + | 0.427525i | 1.07303i | 0.397427 | + | 2.80037i | −2.70802 | − | 1.29098i | −1.67575 | + | 0.0788063i | |||
59.20 | −0.0664333 | + | 1.41265i | −0.382087 | − | 1.68938i | −1.99117 | − | 0.187694i | 1.18625i | 2.41189 | − | 0.427525i | − | 1.07303i | 0.397427 | − | 2.80037i | −2.70802 | + | 1.29098i | −1.67575 | − | 0.0788063i | |||
See all 40 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 | 348.2.c.b | ✓ | 40 |
3.b | odd | 2 | 1 | inner | 348.2.c.b | ✓ | 40 |
4.b | odd | 2 | 1 | inner | 348.2.c.b | ✓ | 40 |
12.b | even | 2 | 1 | inner | 348.2.c.b | ✓ | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
348.2.c.b | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
348.2.c.b | ✓ | 40 | 3.b | odd | 2 | 1 | inner |
348.2.c.b | ✓ | 40 | 4.b | odd | 2 | 1 | inner |
348.2.c.b | ✓ | 40 | 12.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} + 65 T_{5}^{18} + 1706 T_{5}^{16} + 23366 T_{5}^{14} + 181893 T_{5}^{12} + 833653 T_{5}^{10} + \cdots + 102400 \) acting on \(S_{2}^{\mathrm{new}}(348, [\chi])\).