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Properties

Label 23.37.b.c
Level $23$
Weight $37$
Character orbit 23.b
Analytic conductor $188.810$
Analytic rank $0$
Dimension $68$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [23,37,Mod(22,23)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(23, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 37, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("23.22");
 
S:= CuspForms(chi, 37);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 37 \)
Character orbit: \([\chi]\) \(=\) 23.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: \(188.809894917\)
Analytic rank: \(0\)
Dimension: \(68\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q - 169832 q^{2} + 420071358 q^{3} + 2233707075048 q^{4} - 262008021589200 q^{6} + 47\!\cdots\!12 q^{8}+ \cdots + 29\!\cdots\!90 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q - 169832 q^{2} + 420071358 q^{3} + 2233707075048 q^{4} - 262008021589200 q^{6} + 47\!\cdots\!12 q^{8}+ \cdots + 55\!\cdots\!28 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1 −489228. 1.09821e7 1.70625e11 2.06652e12i −5.37275e12 − 2.78671e15i −4.98548e16 −1.49974e17 − 1.01100e18i
22.2 −489228. 1.09821e7 1.70625e11 − 2.06652e12i −5.37275e12 2.78671e15i −4.98548e16 −1.49974e17 1.01100e18i
22.3 −484017. −4.64426e8 1.65553e11 3.72485e12i 2.24790e14 4.69428e14i −4.68693e16 6.55965e16 − 1.80289e18i
22.4 −484017. −4.64426e8 1.65553e11 − 3.72485e12i 2.24790e14 − 4.69428e14i −4.68693e16 6.55965e16 1.80289e18i
22.5 −479695. 3.36313e8 1.61388e11 5.78517e12i −1.61328e14 3.89623e14i −4.44525e16 −3.69883e16 − 2.77511e18i
22.6 −479695. 3.36313e8 1.61388e11 − 5.78517e12i −1.61328e14 − 3.89623e14i −4.44525e16 −3.69883e16 2.77511e18i
22.7 −444581. 6.99679e8 1.28933e11 − 3.58455e11i −3.11064e14 − 2.16806e15i −2.67699e16 3.39457e17 1.59363e17i
22.8 −444581. 6.99679e8 1.28933e11 3.58455e11i −3.11064e14 2.16806e15i −2.67699e16 3.39457e17 − 1.59363e17i
22.9 −369775. −8.91776e7 6.80142e10 1.41364e12i 3.29757e13 2.56323e14i 2.60790e14 −1.42142e17 − 5.22730e17i
22.10 −369775. −8.91776e7 6.80142e10 − 1.41364e12i 3.29757e13 − 2.56323e14i 2.60790e14 −1.42142e17 5.22730e17i
22.11 −359295. −3.03855e8 6.03737e10 − 5.22108e12i 1.09174e14 2.99567e15i 2.99861e15 −5.77666e16 1.87591e18i
22.12 −359295. −3.03855e8 6.03737e10 5.22108e12i 1.09174e14 − 2.99567e15i 2.99861e15 −5.77666e16 − 1.87591e18i
22.13 −359253. −4.43320e8 6.03436e10 7.17873e12i 1.59264e14 2.68871e15i 3.00908e15 4.64378e16 − 2.57898e18i
22.14 −359253. −4.43320e8 6.03436e10 − 7.17873e12i 1.59264e14 − 2.68871e15i 3.00908e15 4.64378e16 2.57898e18i
22.15 −355814. 4.11127e8 5.78844e10 − 2.31241e12i −1.46285e14 − 1.37033e15i 3.85529e15 1.89306e16 8.22790e17i
22.16 −355814. 4.11127e8 5.78844e10 2.31241e12i −1.46285e14 1.37033e15i 3.85529e15 1.89306e16 − 8.22790e17i
22.17 −277285. 1.78977e8 8.16767e9 5.05491e12i −4.96277e13 8.18328e13i 1.67901e16 −1.18062e17 − 1.40165e18i
22.18 −277285. 1.78977e8 8.16767e9 − 5.05491e12i −4.96277e13 − 8.18328e13i 1.67901e16 −1.18062e17 1.40165e18i
22.19 −269386. 5.83675e8 3.84927e9 − 6.49898e12i −1.57234e14 2.49713e15i 1.74751e16 1.90582e17 1.75073e18i
22.20 −269386. 5.83675e8 3.84927e9 6.49898e12i −1.57234e14 − 2.49713e15i 1.74751e16 1.90582e17 − 1.75073e18i
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 22.68
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 23.37.b.c ✓ 68
23.b odd 2 1 inner 23.37.b.c ✓ 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.37.b.c ✓ 68 1.a even 1 1 trivial
23.37.b.c ✓ 68 23.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{34} + 84916 T_{2}^{33} - 1723052509746 T_{2}^{32} + \cdots + 20\!\cdots\!00 \) acting on \(S_{37}^{\mathrm{new}}(23, [\chi])\). Copy content Toggle raw display