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Properties

Label 644.2.c.a
Level $644$
Weight $2$
Character orbit 644.c
Analytic conductor $5.142$
Analytic rank $0$
Dimension $36$
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] = [644,2,Mod(183,644)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(644, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("644.183");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 644 = 2^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 644.c (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: \(5.14236589017\)
Analytic rank: \(0\)
Dimension: \(36\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + q^{2} - 3 q^{4} + q^{6} - 36 q^{7} - 2 q^{8} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + q^{2} - 3 q^{4} + q^{6} - 36 q^{7} - 2 q^{8} - 36 q^{9} + 8 q^{11} - 3 q^{12} - q^{14} + 8 q^{15} + 9 q^{16} - 8 q^{18} + 8 q^{19} - 14 q^{20} + 24 q^{22} - 8 q^{23} - 4 q^{24} - 36 q^{25} + 7 q^{26} + 3 q^{28} - 8 q^{29} - 9 q^{32} + 14 q^{34} + 14 q^{36} - q^{42} - 40 q^{43} + 16 q^{44} - 9 q^{46} - 39 q^{48} + 36 q^{49} - 43 q^{50} + 25 q^{52} - 33 q^{54} + 2 q^{56} + 19 q^{58} + 76 q^{60} - 5 q^{62} + 36 q^{63} + 12 q^{64} + 50 q^{66} + 8 q^{67} - 60 q^{68} + 28 q^{69} - 36 q^{72} + 36 q^{74} - 8 q^{77} + 17 q^{78} + 40 q^{79} - 50 q^{80} + 28 q^{81} - 17 q^{82} + 3 q^{84} - 24 q^{85} - 16 q^{86} - 28 q^{88} - 98 q^{90} + 19 q^{92} + 8 q^{93} + 35 q^{94} + 91 q^{96} + q^{98} - 24 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
183.1 −1.41198 − 0.0793747i − 1.23564i 1.98740 + 0.224152i 3.38329i −0.0980783 + 1.74470i −1.00000 −2.78838 − 0.474248i 1.47320 0.268548 − 4.77716i
183.2 −1.41198 + 0.0793747i 1.23564i 1.98740 − 0.224152i − 3.38329i −0.0980783 − 1.74470i −1.00000 −2.78838 + 0.474248i 1.47320 0.268548 + 4.77716i
183.3 −1.38455 − 0.288145i 2.18249i 1.83394 + 0.797901i − 1.38768i 0.628873 − 3.02176i −1.00000 −2.30927 − 1.63317i −1.76325 −0.399855 + 1.92132i
183.4 −1.38455 + 0.288145i − 2.18249i 1.83394 − 0.797901i 1.38768i 0.628873 + 3.02176i −1.00000 −2.30927 + 1.63317i −1.76325 −0.399855 − 1.92132i
183.5 −1.28172 − 0.597664i − 0.481494i 1.28560 + 1.53207i − 0.116571i −0.287772 + 0.617139i −1.00000 −0.732108 − 2.73204i 2.76816 −0.0696705 + 0.149412i
183.6 −1.28172 + 0.597664i 0.481494i 1.28560 − 1.53207i 0.116571i −0.287772 − 0.617139i −1.00000 −0.732108 + 2.73204i 2.76816 −0.0696705 − 0.149412i
183.7 −1.13221 − 0.847404i 2.81043i 0.563811 + 1.91888i 1.87060i 2.38157 − 3.18201i −1.00000 0.987717 − 2.65036i −4.89854 1.58515 − 2.11792i
183.8 −1.13221 + 0.847404i − 2.81043i 0.563811 − 1.91888i − 1.87060i 2.38157 + 3.18201i −1.00000 0.987717 + 2.65036i −4.89854 1.58515 + 2.11792i
183.9 −0.805628 − 1.16231i − 2.66076i −0.701926 + 1.87278i 2.47812i −3.09262 + 2.14358i −1.00000 2.74224 − 0.692909i −4.07963 2.88034 − 1.99644i
183.10 −0.805628 + 1.16231i 2.66076i −0.701926 − 1.87278i − 2.47812i −3.09262 − 2.14358i −1.00000 2.74224 + 0.692909i −4.07963 2.88034 + 1.99644i
183.11 −0.766428 − 1.18852i − 1.47961i −0.825175 + 1.82184i − 0.833230i −1.75855 + 1.13401i −1.00000 2.79773 − 0.415566i 0.810767 −0.990314 + 0.638611i
183.12 −0.766428 + 1.18852i 1.47961i −0.825175 − 1.82184i 0.833230i −1.75855 − 1.13401i −1.00000 2.79773 + 0.415566i 0.810767 −0.990314 − 0.638611i
183.13 −0.413125 − 1.35253i 2.02108i −1.65866 + 1.11752i − 1.88306i 2.73356 − 0.834958i −1.00000 2.19671 + 1.78170i −1.08476 −2.54689 + 0.777941i
183.14 −0.413125 + 1.35253i − 2.02108i −1.65866 − 1.11752i 1.88306i 2.73356 + 0.834958i −1.00000 2.19671 − 1.78170i −1.08476 −2.54689 − 0.777941i
183.15 −0.340486 − 1.37261i 0.155030i −1.76814 + 0.934711i 1.40532i 0.212796 − 0.0527855i −1.00000 1.88502 + 2.10872i 2.97597 1.92896 − 0.478490i
183.16 −0.340486 + 1.37261i − 0.155030i −1.76814 − 0.934711i − 1.40532i 0.212796 + 0.0527855i −1.00000 1.88502 − 2.10872i 2.97597 1.92896 + 0.478490i
183.17 −0.104849 − 1.41032i − 2.22315i −1.97801 + 0.295741i − 3.61192i −3.13536 + 0.233094i −1.00000 0.624482 + 2.75863i −1.94239 −5.09396 + 0.378705i
183.18 −0.104849 + 1.41032i 2.22315i −1.97801 − 0.295741i 3.61192i −3.13536 − 0.233094i −1.00000 0.624482 − 2.75863i −1.94239 −5.09396 − 0.378705i
183.19 0.236103 − 1.39437i 3.31913i −1.88851 − 0.658429i 3.46030i 4.62808 + 0.783658i −1.00000 −1.36397 + 2.47782i −8.01664 4.82492 + 0.816988i
183.20 0.236103 + 1.39437i − 3.31913i −1.88851 + 0.658429i − 3.46030i 4.62808 − 0.783658i −1.00000 −1.36397 − 2.47782i −8.01664 4.82492 − 0.816988i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 183.36
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 644.2.c.a ✓ 36
4.b odd 2 1 644.2.c.b yes 36
23.b odd 2 1 644.2.c.b yes 36
92.b even 2 1 inner 644.2.c.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
644.2.c.a ✓ 36 1.a even 1 1 trivial
644.2.c.a ✓ 36 92.b even 2 1 inner
644.2.c.b yes 36 4.b odd 2 1
644.2.c.b yes 36 23.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{18} - 4 T_{11}^{17} - 108 T_{11}^{16} + 344 T_{11}^{15} + 4870 T_{11}^{14} - 11200 T_{11}^{13} + \cdots - 61341696 \) acting on \(S_{2}^{\mathrm{new}}(644, [\chi])\). Copy content Toggle raw display