Properties

Label 237.2.n.a
Level $237$
Weight $2$
Character orbit 237.n
Analytic conductor $1.892$
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] = [237,2,Mod(29,237)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(237, base_ring=CyclotomicField(78))
 
chi = DirichletCharacter(H, H._module([39, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("237.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 237 = 3 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 237.n (of order \(78\), degree \(24\), 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: \(1.89245452790\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{78}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 3 q^{3} + 2 q^{4} + 6 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 3 q^{3} + 2 q^{4} + 6 q^{7} - 3 q^{9} + 5 q^{13} + 4 q^{16} + 8 q^{19} - 12 q^{21} + 5 q^{25} - 12 q^{28} + 7 q^{31} - 6 q^{36} + 21 q^{37} - 15 q^{39} - 21 q^{43} - 12 q^{48} - 5 q^{49} - 20 q^{52} - 177 q^{63} - 16 q^{64} - 121 q^{67} - 10 q^{73} - 15 q^{75} - 10 q^{76} + 17 q^{79} + 9 q^{81} + 90 q^{84} + 195 q^{93} + 209 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 0 −1.56482 0.742517i −1.83996 0.783933i 0 0 0.193617 + 2.39838i 0 1.89734 + 2.32381i 0
35.1 0 −0.139372 1.72643i 1.59889 1.20148i 0 0 3.90354 + 1.85225i 0 −2.96115 + 0.481234i 0
47.1 0 1.24916 1.19983i 1.97410 + 0.320823i 0 0 0.0572213 + 0.0165743i 0 0.120798 2.99757i 0
53.1 0 −1.73065 + 0.0697427i 1.89707 0.633336i 0 0 −2.34544 3.70902i 0 2.99027 0.241400i 0
59.1 0 0.548485 + 1.64291i −1.69038 1.06893i 0 0 5.15347 + 1.05209i 0 −2.39833 + 1.80223i 0
68.1 0 0.678906 + 1.59345i −1.99351 + 0.160933i 0 0 −1.50887 + 2.00794i 0 −2.07817 + 2.16361i 0
74.1 0 −1.04052 1.38468i 0.857385 + 1.80690i 0 0 −1.94747 + 4.57087i 0 −0.834652 + 2.88155i 0
77.1 0 −0.277840 + 1.70962i 0.556435 + 1.92104i 0 0 0.936046 + 0.764258i 0 −2.84561 0.950004i 0
86.1 0 −0.925722 + 1.46391i −0.400051 1.95958i 0 0 3.55040 + 0.143076i 0 −1.28608 2.71035i 0
107.1 0 1.34166 1.09543i 1.38545 + 1.44240i 0 0 −0.790218 4.86241i 0 0.600077 2.93937i 0
113.1 0 −0.925722 1.46391i −0.400051 + 1.95958i 0 0 3.55040 0.143076i 0 −1.28608 + 2.71035i 0
116.1 0 1.24916 + 1.19983i 1.97410 0.320823i 0 0 0.0572213 0.0165743i 0 0.120798 + 2.99757i 0
122.1 0 0.678906 1.59345i −1.99351 0.160933i 0 0 −1.50887 2.00794i 0 −2.07817 2.16361i 0
149.1 0 −0.139372 + 1.72643i 1.59889 + 1.20148i 0 0 3.90354 1.85225i 0 −2.96115 0.481234i 0
161.1 0 −1.73065 0.0697427i 1.89707 + 0.633336i 0 0 −2.34544 + 3.70902i 0 2.99027 + 0.241400i 0
164.1 0 1.69705 + 0.346455i −0.0805319 + 1.99838i 0 0 −0.419301 1.25596i 0 2.75994 + 1.17590i 0
188.1 0 −1.56482 + 0.742517i −1.83996 + 0.783933i 0 0 0.193617 2.39838i 0 1.89734 2.32381i 0
197.1 0 −0.277840 1.70962i 0.556435 1.92104i 0 0 0.936046 0.764258i 0 −2.84561 + 0.950004i 0
206.1 0 1.34166 + 1.09543i 1.38545 1.44240i 0 0 −0.790218 + 4.86241i 0 0.600077 + 2.93937i 0
212.1 0 1.66367 + 0.481887i −1.26489 + 1.54921i 0 0 −3.78300 + 3.63362i 0 2.53557 + 1.60340i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.1
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.h odd 78 1 inner
237.n even 78 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 237.2.n.a 24
3.b odd 2 1 CM 237.2.n.a 24
79.h odd 78 1 inner 237.2.n.a 24
237.n even 78 1 inner 237.2.n.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
237.2.n.a 24 1.a even 1 1 trivial
237.2.n.a 24 3.b odd 2 1 CM
237.2.n.a 24 79.h odd 78 1 inner
237.2.n.a 24 237.n even 78 1 inner

Hecke kernels

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