Properties

Label 7.26.c.a
Level $7$
Weight $26$
Character orbit 7.c
Analytic conductor $27.720$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 4050 q^{2} + 531440 q^{3} - 286295596 q^{4} + 288173088 q^{5} - 13063290884 q^{6} - 55218840880 q^{7} + 274366746720 q^{8} - 5146896583216 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 4050 q^{2} + 531440 q^{3} - 286295596 q^{4} + 288173088 q^{5} - 13063290884 q^{6} - 55218840880 q^{7} + 274366746720 q^{8} - 5146896583216 q^{9} - 918803280822 q^{10} + 253661467680 q^{11} + 59498382182260 q^{12} + 136259290951840 q^{13} + 669930284417310 q^{14} - 28\!\cdots\!16 q^{15}+ \cdots + 42\!\cdots\!56 q^{99}+O(q^{100}) \) 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.

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
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.16
Significant digits:
Format:

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 \(S_{26}^{\mathrm{new}}(7, [\chi])\).