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Properties

Label 36.27.d.d
Level $36$
Weight $27$
Character orbit 36.d
Analytic conductor $154.185$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [36,27,Mod(19,36)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(36, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 27, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("36.19");
 
S:= CuspForms(chi, 27);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 27 \)
Character orbit: \([\chi]\) \(=\) 36.d (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: \(154.185451463\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 120559416 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 120559416 q^{4} - 4695415493280 q^{10} - 15\!\cdots\!00 q^{13}+ \cdots + 14\!\cdots\!80 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −7676.54 − 2860.01i 0 5.07495e7 + 4.39100e7i −2.77474e8 0 8.09400e10i −2.63998e11 − 4.82221e11i 0 2.13004e12 + 7.93580e11i
19.2 −7676.54 + 2860.01i 0 5.07495e7 − 4.39100e7i −2.77474e8 0 − 8.09400e10i −2.63998e11 + 4.82221e11i 0 2.13004e12 − 7.93580e11i
19.3 −7486.27 − 3326.34i 0 4.49797e7 + 4.98038e7i 1.72143e9 0 − 8.72680e10i −1.71066e11 − 5.22463e11i 0 −1.28871e13 − 5.72609e12i
19.4 −7486.27 + 3326.34i 0 4.49797e7 − 4.98038e7i 1.72143e9 0 8.72680e10i −1.71066e11 + 5.22463e11i 0 −1.28871e13 + 5.72609e12i
19.5 −5471.08 − 6097.22i 0 −7.24337e6 + 6.67168e7i −2.11499e9 0 − 1.50066e11i 4.46416e11 − 3.20849e11i 0 1.15713e13 + 1.28956e13i
19.6 −5471.08 + 6097.22i 0 −7.24337e6 − 6.67168e7i −2.11499e9 0 1.50066e11i 4.46416e11 + 3.20849e11i 0 1.15713e13 − 1.28956e13i
19.7 −5121.78 − 6393.45i 0 −1.46435e7 + 6.54917e7i −6.77003e7 0 1.81935e11i 4.93719e11 − 2.41812e11i 0 3.46746e11 + 4.32839e11i
19.8 −5121.78 + 6393.45i 0 −1.46435e7 − 6.54917e7i −6.77003e7 0 − 1.81935e11i 4.93719e11 + 2.41812e11i 0 3.46746e11 − 4.32839e11i
19.9 −3842.94 − 7234.68i 0 −3.75725e7 + 5.56049e7i 3.80000e8 0 − 4.47086e10i 5.46673e11 + 5.81385e10i 0 −1.46032e12 − 2.74918e12i
19.10 −3842.94 + 7234.68i 0 −3.75725e7 − 5.56049e7i 3.80000e8 0 4.47086e10i 5.46673e11 − 5.81385e10i 0 −1.46032e12 + 2.74918e12i
19.11 −591.241 − 8170.64i 0 −6.64097e7 + 9.66163e6i 1.47908e9 0 − 3.03775e10i 1.18206e11 + 5.36897e11i 0 −8.74490e11 − 1.20850e13i
19.12 −591.241 + 8170.64i 0 −6.64097e7 − 9.66163e6i 1.47908e9 0 3.03775e10i 1.18206e11 − 5.36897e11i 0 −8.74490e11 + 1.20850e13i
19.13 591.241 − 8170.64i 0 −6.64097e7 − 9.66163e6i −1.47908e9 0 3.03775e10i −1.18206e11 + 5.36897e11i 0 −8.74490e11 + 1.20850e13i
19.14 591.241 + 8170.64i 0 −6.64097e7 + 9.66163e6i −1.47908e9 0 − 3.03775e10i −1.18206e11 − 5.36897e11i 0 −8.74490e11 − 1.20850e13i
19.15 3842.94 − 7234.68i 0 −3.75725e7 − 5.56049e7i −3.80000e8 0 4.47086e10i −5.46673e11 + 5.81385e10i 0 −1.46032e12 + 2.74918e12i
19.16 3842.94 + 7234.68i 0 −3.75725e7 + 5.56049e7i −3.80000e8 0 − 4.47086e10i −5.46673e11 − 5.81385e10i 0 −1.46032e12 − 2.74918e12i
19.17 5121.78 − 6393.45i 0 −1.46435e7 − 6.54917e7i 6.77003e7 0 − 1.81935e11i −4.93719e11 − 2.41812e11i 0 3.46746e11 − 4.32839e11i
19.18 5121.78 + 6393.45i 0 −1.46435e7 + 6.54917e7i 6.77003e7 0 1.81935e11i −4.93719e11 + 2.41812e11i 0 3.46746e11 + 4.32839e11i
19.19 5471.08 − 6097.22i 0 −7.24337e6 − 6.67168e7i 2.11499e9 0 1.50066e11i −4.46416e11 − 3.20849e11i 0 1.15713e13 − 1.28956e13i
19.20 5471.08 + 6097.22i 0 −7.24337e6 + 6.67168e7i 2.11499e9 0 − 1.50066e11i −4.46416e11 + 3.20849e11i 0 1.15713e13 + 1.28956e13i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.24
Significant digits:
Format:

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 36.27.d.d ✓ 24
3.b odd 2 1 inner 36.27.d.d ✓ 24
4.b odd 2 1 inner 36.27.d.d ✓ 24
12.b even 2 1 inner 36.27.d.d ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.27.d.d ✓ 24 1.a even 1 1 trivial
36.27.d.d ✓ 24 3.b odd 2 1 inner
36.27.d.d ✓ 24 4.b odd 2 1 inner
36.27.d.d ✓ 24 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + \cdots + 14\!\cdots\!00 \) acting on \(S_{27}^{\mathrm{new}}(36, [\chi])\). Copy content Toggle raw display