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Properties

Label 237.4.l.a
Level $237$
Weight $4$
Character orbit 237.l
Analytic conductor $13.983$
Analytic rank $0$
Dimension $24$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [237,4,Mod(14,237)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(237, base_ring=CyclotomicField(26))
 
chi = DirichletCharacter(H, H._module([13, 19]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("237.14");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 237 = 3 \cdot 79 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 237.l (of order \(26\), degree \(12\), 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: \(13.9834526714\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(2\) over \(\Q(\zeta_{26})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{26}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 16 q^{4} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 16 q^{4} + 54 q^{9} + 140 q^{13} - 128 q^{16} - 112 q^{19} + 324 q^{21} - 250 q^{25} + 616 q^{31} + 432 q^{36} + 1258 q^{49} + 1120 q^{52} - 7020 q^{63} - 1024 q^{64} + 9680 q^{67} - 2380 q^{73} + 4928 q^{76} + 884 q^{79} - 1458 q^{81} - 14256 q^{84} - 10530 q^{93} + 14630 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1 0 −3.44569 + 3.88938i 7.08365 − 3.71779i 0 0 −0.829453 + 0.936259i 0 −3.25449 − 26.8031i 0
14.2 0 3.44569 − 3.88938i 7.08365 − 3.71779i 0 0 22.6590 − 25.5767i 0 −3.25449 − 26.8031i 0
17.1 0 −3.44569 − 3.88938i 7.08365 + 3.71779i 0 0 −0.829453 − 0.936259i 0 −3.25449 + 26.8031i 0
17.2 0 3.44569 + 3.88938i 7.08365 + 3.71779i 0 0 22.6590 + 25.5767i 0 −3.25449 + 26.8031i 0
41.1 0 −5.15827 + 0.626327i 4.54452 − 6.58387i 0 0 −7.58910 + 0.921483i 0 26.2154 − 6.46152i 0
41.2 0 5.15827 − 0.626327i 4.54452 − 6.58387i 0 0 −29.5389 + 3.58667i 0 26.2154 − 6.46152i 0
71.1 0 −2.41477 + 4.60096i −5.98809 − 5.30498i 0 0 7.48028 − 14.2525i 0 −15.3377 − 22.2206i 0
71.2 0 2.41477 − 4.60096i −5.98809 − 5.30498i 0 0 10.9731 − 20.9075i 0 −15.3377 − 22.2206i 0
137.1 0 −4.27635 − 2.95175i 0.964293 − 7.94167i 0 0 28.8516 + 19.9148i 0 9.57433 + 25.2454i 0
137.2 0 4.27635 + 2.95175i 0.964293 − 7.94167i 0 0 −20.9735 − 14.4769i 0 9.57433 + 25.2454i 0
140.1 0 −1.24352 + 5.04516i −2.83684 + 7.48013i 0 0 8.75864 − 35.5352i 0 −23.9073 − 12.5475i 0
140.2 0 1.24352 − 5.04516i −2.83684 + 7.48013i 0 0 −2.41082 + 9.78108i 0 −23.9073 − 12.5475i 0
170.1 0 −4.85849 − 1.84258i −7.76753 + 1.91453i 0 0 −34.5023 − 13.0850i 0 20.2098 + 17.9043i 0
170.2 0 4.85849 + 1.84258i −7.76753 + 1.91453i 0 0 17.1214 + 6.49328i 0 20.2098 + 17.9043i 0
173.1 0 −4.27635 + 2.95175i 0.964293 + 7.94167i 0 0 28.8516 − 19.9148i 0 9.57433 − 25.2454i 0
173.2 0 4.27635 − 2.95175i 0.964293 + 7.94167i 0 0 −20.9735 + 14.4769i 0 9.57433 − 25.2454i 0
185.1 0 −5.15827 − 0.626327i 4.54452 + 6.58387i 0 0 −7.58910 − 0.921483i 0 26.2154 + 6.46152i 0
185.2 0 5.15827 + 0.626327i 4.54452 + 6.58387i 0 0 −29.5389 − 3.58667i 0 26.2154 + 6.46152i 0
191.1 0 −4.85849 + 1.84258i −7.76753 − 1.91453i 0 0 −34.5023 + 13.0850i 0 20.2098 − 17.9043i 0
191.2 0 4.85849 − 1.84258i −7.76753 − 1.91453i 0 0 17.1214 − 6.49328i 0 20.2098 − 17.9043i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 14.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
79.f odd 26 1 inner
237.l even 26 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 237.4.l.a ✓ 24
3.b odd 2 1 CM 237.4.l.a ✓ 24
79.f odd 26 1 inner 237.4.l.a ✓ 24
237.l even 26 1 inner 237.4.l.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
237.4.l.a ✓ 24 1.a even 1 1 trivial
237.4.l.a ✓ 24 3.b odd 2 1 CM
237.4.l.a ✓ 24 79.f odd 26 1 inner
237.4.l.a ✓ 24 237.l even 26 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{4}^{\mathrm{new}}(237, [\chi])\). Copy content Toggle raw display