Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7,26,Mod(2,7)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2]))
N = Newforms(chi, 26, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7.2");
S:= CuspForms(chi, 26);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 7 \) |
Weight: | \( k \) | \(=\) | \( 26 \) |
Character orbit: | \([\chi]\) | \(=\) | 7.c (of order \(3\), degree \(2\), 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: | \(27.7197745967\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{3})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | −5390.74 | + | 9337.04i | 821428. | + | 1.42275e6i | −4.13429e7 | − | 7.16081e7i | −2.66949e8 | + | 4.62368e8i | −1.77124e10 | −3.37771e10 | + | 1.41483e10i | 5.29710e11 | −9.25843e11 | + | 1.60361e12i | −2.87810e12 | − | 4.98502e12i | ||||
2.2 | −5310.35 | + | 9197.79i | −231399. | − | 400795.i | −3.96223e7 | − | 6.86279e7i | 3.63378e8 | − | 6.29389e8i | 4.91524e9 | 6.81365e9 | − | 3.59811e10i | 4.85262e11 | 3.16553e11 | − | 5.48286e11i | 3.85933e12 | + | 6.68455e12i | ||||
2.3 | −4255.77 | + | 7371.20i | −176711. | − | 306072.i | −1.94459e7 | − | 3.36812e7i | −3.85356e8 | + | 6.67455e8i | 3.00816e9 | 3.54675e10 | − | 9.11739e9i | 4.54286e10 | 3.61191e11 | − | 6.25601e11i | −3.27997e12 | − | 5.68107e12i | ||||
2.4 | −3543.50 | + | 6137.53i | −783073. | − | 1.35632e6i | −8.33560e6 | − | 1.44377e7i | −3.53207e7 | + | 6.11772e7i | 1.10993e10 | −3.36185e10 | + | 1.45213e10i | −1.19652e11 | −8.02763e11 | + | 1.39043e12i | −2.50318e11 | − | 4.33563e11i | ||||
2.5 | −3477.76 | + | 6023.65i | 109497. | + | 189654.i | −7.41236e6 | − | 1.28386e7i | 1.47246e8 | − | 2.55038e8i | −1.52321e9 | −7.40825e9 | + | 3.58634e10i | −1.30275e11 | 3.99665e11 | − | 6.92241e11i | 1.02417e12 | + | 1.77392e12i | ||||
2.6 | −2461.92 | + | 4264.17i | 704248. | + | 1.21979e6i | 4.65511e6 | + | 8.06289e6i | 3.67518e8 | − | 6.36561e8i | −6.93521e9 | 3.54576e10 | − | 9.15587e9i | −2.11059e11 | −5.68287e11 | + | 9.84302e11i | 1.80960e12 | + | 3.13432e12i | ||||
2.7 | −1395.52 | + | 2417.11i | 308966. | + | 535144.i | 1.28823e7 | + | 2.23127e7i | −2.17112e8 | + | 3.76049e8i | −1.72467e9 | −2.47591e10 | − | 2.69825e10i | −1.65562e11 | 2.32725e11 | − | 4.03091e11i | −6.05969e11 | − | 1.04957e12i | ||||
2.8 | −183.599 | + | 318.003i | −540247. | − | 935735.i | 1.67098e7 | + | 2.89422e7i | −1.54088e8 | + | 2.66888e8i | 3.96756e8 | 3.42104e10 | + | 1.30659e10i | −2.45928e10 | −1.60090e11 | + | 2.77283e11i | −5.65808e10 | − | 9.80009e10i | ||||
2.9 | −149.849 | + | 259.546i | −523783. | − | 907218.i | 1.67323e7 | + | 2.89812e7i | 4.32383e8 | − | 7.48910e8i | 3.13954e8 | −1.77836e9 | − | 3.65774e10i | −2.00855e10 | −1.25052e11 | + | 2.16597e11i | 1.29585e11 | + | 2.24447e11i | ||||
2.10 | 1480.81 | − | 2564.84i | 106081. | + | 183738.i | 1.23916e7 | + | 2.14629e7i | 1.83250e8 | − | 3.17398e8i | 6.28345e8 | −1.30897e10 | + | 3.42013e10i | 1.72774e11 | 4.01138e11 | − | 6.94791e11i | −5.42717e11 | − | 9.40013e11i | ||||
2.11 | 1652.55 | − | 2862.31i | 724994. | + | 1.25573e6i | 1.13153e7 | + | 1.95988e7i | −3.86443e8 | + | 6.69339e8i | 4.79237e9 | 3.22371e10 | + | 1.73734e10i | 1.85698e11 | −6.27589e11 | + | 1.08702e12i | 1.27724e12 | + | 2.21224e12i | ||||
2.12 | 3074.28 | − | 5324.82i | −523530. | − | 906781.i | −2.12523e6 | − | 3.68100e6i | −4.19055e8 | + | 7.25824e8i | −6.43792e9 | −3.45827e10 | − | 1.20460e10i | 1.80178e11 | −1.24523e11 | + | 2.15680e11i | 2.57659e12 | + | 4.46278e12i | ||||
2.13 | 3337.78 | − | 5781.20i | 693072. | + | 1.20044e6i | −5.50432e6 | − | 9.53376e6i | 3.37317e8 | − | 5.84251e8i | 9.25329e9 | −2.63901e10 | − | 2.53896e10i | 1.50506e11 | −5.37054e11 | + | 9.30204e11i | −2.25178e12 | − | 3.90020e12i | ||||
2.14 | 4138.61 | − | 7168.28i | −39686.9 | − | 68739.8i | −1.74790e7 | − | 3.02745e7i | 3.57400e7 | − | 6.19035e7i | −6.56995e8 | 2.92584e10 | − | 2.20231e10i | −1.16173e10 | 4.20494e11 | − | 7.28317e11i | −2.95828e11 | − | 5.12389e11i | ||||
2.15 | 4808.59 | − | 8328.72i | −811290. | − | 1.40520e6i | −2.94678e7 | − | 5.10397e7i | 3.71120e8 | − | 6.42799e8i | −1.56046e10 | −4.49765e9 | + | 3.63434e10i | −2.44095e11 | −8.92738e11 | + | 1.54627e12i | −3.56913e12 | − | 6.18191e12i | ||||
2.16 | 5651.37 | − | 9788.46i | 427154. | + | 739852.i | −4.70988e7 | − | 8.15775e7i | −2.29544e8 | + | 3.97582e8i | 9.65602e9 | −2.11525e10 | + | 2.98938e10i | −6.85434e11 | 5.87238e10 | − | 1.01713e11i | 2.59448e12 | + | 4.49377e12i | ||||
4.1 | −5390.74 | − | 9337.04i | 821428. | − | 1.42275e6i | −4.13429e7 | + | 7.16081e7i | −2.66949e8 | − | 4.62368e8i | −1.77124e10 | −3.37771e10 | − | 1.41483e10i | 5.29710e11 | −9.25843e11 | − | 1.60361e12i | −2.87810e12 | + | 4.98502e12i | ||||
4.2 | −5310.35 | − | 9197.79i | −231399. | + | 400795.i | −3.96223e7 | + | 6.86279e7i | 3.63378e8 | + | 6.29389e8i | 4.91524e9 | 6.81365e9 | + | 3.59811e10i | 4.85262e11 | 3.16553e11 | + | 5.48286e11i | 3.85933e12 | − | 6.68455e12i | ||||
4.3 | −4255.77 | − | 7371.20i | −176711. | + | 306072.i | −1.94459e7 | + | 3.36812e7i | −3.85356e8 | − | 6.67455e8i | 3.00816e9 | 3.54675e10 | + | 9.11739e9i | 4.54286e10 | 3.61191e11 | + | 6.25601e11i | −3.27997e12 | + | 5.68107e12i | ||||
4.4 | −3543.50 | − | 6137.53i | −783073. | + | 1.35632e6i | −8.33560e6 | + | 1.44377e7i | −3.53207e7 | − | 6.11772e7i | 1.10993e10 | −3.36185e10 | − | 1.45213e10i | −1.19652e11 | −8.02763e11 | − | 1.39043e12i | −2.50318e11 | + | 4.33563e11i | ||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 7.26.c.a | ✓ | 32 |
7.c | even | 3 | 1 | inner | 7.26.c.a | ✓ | 32 |
7.c | even | 3 | 1 | 49.26.a.f | 16 | ||
7.d | odd | 6 | 1 | 49.26.a.g | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
7.26.c.a | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
7.26.c.a | ✓ | 32 | 7.c | even | 3 | 1 | inner |
49.26.a.f | 16 | 7.c | even | 3 | 1 | ||
49.26.a.g | 16 | 7.d | odd | 6 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{26}^{\mathrm{new}}(7, [\chi])\).