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Properties

Label 644.4.d.a
Level $644$
Weight $4$
Character orbit 644.d
Analytic conductor $37.997$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [644,4,Mod(321,644)]
 
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([0, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("644.321");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 644 = 2^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 644.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: \(37.9972300437\)
Analytic rank: \(0\)
Dimension: \(48\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 376 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 376 q^{9} + 104 q^{23} + 984 q^{25} - 184 q^{29} + 472 q^{35} - 1328 q^{39} - 776 q^{49} + 528 q^{71} - 1432 q^{77} + 3496 q^{81} + 1848 q^{85} + 6792 q^{93} + 1696 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
321.1 0 − 6.92386i 0 −17.9808 0 9.85072 + 15.6832i 0 −20.9399 0
321.2 0 6.92386i 0 −17.9808 0 9.85072 − 15.6832i 0 −20.9399 0
321.3 0 − 2.35909i 0 19.1754 0 10.6625 + 15.1430i 0 21.4347 0
321.4 0 2.35909i 0 19.1754 0 10.6625 − 15.1430i 0 21.4347 0
321.5 0 − 2.44013i 0 16.1511 0 −16.2145 + 8.94928i 0 21.0458 0
321.6 0 2.44013i 0 16.1511 0 −16.2145 − 8.94928i 0 21.0458 0
321.7 0 − 9.32715i 0 14.1188 0 3.84923 + 18.1158i 0 −59.9958 0
321.8 0 9.32715i 0 14.1188 0 3.84923 − 18.1158i 0 −59.9958 0
321.9 0 − 3.35562i 0 15.6084 0 18.3148 − 2.75124i 0 15.7398 0
321.10 0 3.35562i 0 15.6084 0 18.3148 + 2.75124i 0 15.7398 0
321.11 0 − 5.98837i 0 12.4329 0 2.12843 − 18.3975i 0 −8.86056 0
321.12 0 5.98837i 0 12.4329 0 2.12843 + 18.3975i 0 −8.86056 0
321.13 0 − 7.81315i 0 −9.06517 0 −18.3426 + 2.55938i 0 −34.0453 0
321.14 0 7.81315i 0 −9.06517 0 −18.3426 − 2.55938i 0 −34.0453 0
321.15 0 − 9.67686i 0 −6.94569 0 18.5132 − 0.509730i 0 −66.6416 0
321.16 0 9.67686i 0 −6.94569 0 18.5132 + 0.509730i 0 −66.6416 0
321.17 0 − 5.42252i 0 4.39383 0 −12.9663 + 13.2240i 0 −2.40370 0
321.18 0 5.42252i 0 4.39383 0 −12.9663 − 13.2240i 0 −2.40370 0
321.19 0 − 5.96550i 0 3.04184 0 3.95931 + 18.0921i 0 −8.58715 0
321.20 0 5.96550i 0 3.04184 0 3.95931 − 18.0921i 0 −8.58715 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 321.48
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 644.4.d.a ✓ 48
7.b odd 2 1 inner 644.4.d.a ✓ 48
23.b odd 2 1 inner 644.4.d.a ✓ 48
161.c even 2 1 inner 644.4.d.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
644.4.d.a ✓ 48 1.a even 1 1 trivial
644.4.d.a ✓ 48 7.b odd 2 1 inner
644.4.d.a ✓ 48 23.b odd 2 1 inner
644.4.d.a ✓ 48 161.c even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(644, [\chi])\).