↑

Properties

Label 138.6.d.a
Level $138$
Weight $6$
Character orbit 138.d
Analytic conductor $22.133$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [138,6,Mod(137,138)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(138, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("138.137");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 138.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: \(22.1329671342\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{3} - 640 q^{4} + 80 q^{6} + 504 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{3} - 640 q^{4} + 80 q^{6} + 504 q^{9} + 64 q^{12} - 1048 q^{13} + 10240 q^{16} + 1280 q^{18} - 1280 q^{24} + 30480 q^{25} + 1700 q^{27} - 22576 q^{31} - 8064 q^{36} + 55608 q^{39} + 1088 q^{46} - 1024 q^{48} - 23224 q^{49} + 16768 q^{52} + 25456 q^{54} + 210400 q^{55} - 83168 q^{58} - 163840 q^{64} + 99076 q^{69} + 167520 q^{70} - 20480 q^{72} + 241160 q^{73} - 255604 q^{75} - 233440 q^{78} + 78512 q^{81} - 8832 q^{82} - 460296 q^{85} - 4136 q^{87} + 500704 q^{93} - 138272 q^{94} + 20480 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
137.1 − 4.00000i −11.7474 + 10.2469i −16.0000 −107.494 40.9875 + 46.9896i − 192.413i 64.0000i 33.0032 − 240.748i 429.977i
137.2 − 4.00000i −11.7474 + 10.2469i −16.0000 107.494 40.9875 + 46.9896i 192.413i 64.0000i 33.0032 − 240.748i − 429.977i
137.3 4.00000i −11.7474 − 10.2469i −16.0000 −107.494 40.9875 − 46.9896i 192.413i − 64.0000i 33.0032 + 240.748i − 429.977i
137.4 4.00000i −11.7474 − 10.2469i −16.0000 107.494 40.9875 − 46.9896i − 192.413i − 64.0000i 33.0032 + 240.748i 429.977i
137.5 − 4.00000i −1.79544 + 15.4847i −16.0000 −69.6024 61.9389 + 7.18174i 161.717i 64.0000i −236.553 − 55.6036i 278.410i
137.6 − 4.00000i −1.79544 + 15.4847i −16.0000 69.6024 61.9389 + 7.18174i − 161.717i 64.0000i −236.553 − 55.6036i − 278.410i
137.7 4.00000i −1.79544 − 15.4847i −16.0000 −69.6024 61.9389 − 7.18174i − 161.717i − 64.0000i −236.553 + 55.6036i − 278.410i
137.8 4.00000i −1.79544 − 15.4847i −16.0000 69.6024 61.9389 − 7.18174i 161.717i − 64.0000i −236.553 + 55.6036i 278.410i
137.9 − 4.00000i −15.1862 + 3.51856i −16.0000 −0.0328033 14.0742 + 60.7447i 156.501i 64.0000i 218.240 − 106.867i 0.131213i
137.10 − 4.00000i −15.1862 + 3.51856i −16.0000 0.0328033 14.0742 + 60.7447i − 156.501i 64.0000i 218.240 − 106.867i − 0.131213i
137.11 4.00000i −15.1862 − 3.51856i −16.0000 −0.0328033 14.0742 − 60.7447i − 156.501i − 64.0000i 218.240 + 106.867i − 0.131213i
137.12 4.00000i −15.1862 − 3.51856i −16.0000 0.0328033 14.0742 − 60.7447i 156.501i − 64.0000i 218.240 + 106.867i 0.131213i
137.13 − 4.00000i 15.5676 + 0.805444i −16.0000 −10.2990 3.22177 − 62.2705i 139.754i 64.0000i 241.703 + 25.0777i 41.1960i
137.14 − 4.00000i 15.5676 + 0.805444i −16.0000 10.2990 3.22177 − 62.2705i − 139.754i 64.0000i 241.703 + 25.0777i − 41.1960i
137.15 4.00000i 15.5676 − 0.805444i −16.0000 −10.2990 3.22177 + 62.2705i − 139.754i − 64.0000i 241.703 − 25.0777i − 41.1960i
137.16 4.00000i 15.5676 − 0.805444i −16.0000 10.2990 3.22177 + 62.2705i 139.754i − 64.0000i 241.703 − 25.0777i 41.1960i
137.17 − 4.00000i −14.2654 − 6.28478i −16.0000 −46.4212 −25.1391 + 57.0616i 104.982i 64.0000i 164.003 + 179.310i 185.685i
137.18 − 4.00000i −14.2654 − 6.28478i −16.0000 46.4212 −25.1391 + 57.0616i − 104.982i 64.0000i 164.003 + 179.310i − 185.685i
137.19 4.00000i −14.2654 + 6.28478i −16.0000 −46.4212 −25.1391 − 57.0616i − 104.982i − 64.0000i 164.003 − 179.310i − 185.685i
137.20 4.00000i −14.2654 + 6.28478i −16.0000 46.4212 −25.1391 − 57.0616i 104.982i − 64.0000i 164.003 − 179.310i 185.685i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 137.40
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 138.6.d.a ✓ 40
3.b odd 2 1 inner 138.6.d.a ✓ 40
23.b odd 2 1 inner 138.6.d.a ✓ 40
69.c even 2 1 inner 138.6.d.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.6.d.a ✓ 40 1.a even 1 1 trivial
138.6.d.a ✓ 40 3.b odd 2 1 inner
138.6.d.a ✓ 40 23.b odd 2 1 inner
138.6.d.a ✓ 40 69.c even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(138, [\chi])\).