Properties

Label 2738.2.a.c
Level $2738$
Weight $2$
Character orbit 2738.a
Self dual yes
Analytic conductor $21.863$
Analytic rank $2$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2738,2,Mod(1,2738)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2738, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2738.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2738 = 2 \cdot 37^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2738.a (trivial)

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: yes
Analytic conductor: \(21.8630400734\)
Analytic rank: \(2\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + q^{2} - 2 q^{3} + q^{4} - 3 q^{5} - 2 q^{6} - 4 q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - 2 q^{3} + q^{4} - 3 q^{5} - 2 q^{6} - 4 q^{7} + q^{8} + q^{9} - 3 q^{10} - 6 q^{11} - 2 q^{12} - 2 q^{13} - 4 q^{14} + 6 q^{15} + q^{16} - 3 q^{17} + q^{18} - 2 q^{19} - 3 q^{20} + 8 q^{21} - 6 q^{22} - 6 q^{23} - 2 q^{24} + 4 q^{25} - 2 q^{26} + 4 q^{27} - 4 q^{28} - 3 q^{29} + 6 q^{30} - 2 q^{31} + q^{32} + 12 q^{33} - 3 q^{34} + 12 q^{35} + q^{36} - 2 q^{38} + 4 q^{39} - 3 q^{40} + 3 q^{41} + 8 q^{42} + 4 q^{43} - 6 q^{44} - 3 q^{45} - 6 q^{46} - 6 q^{47} - 2 q^{48} + 9 q^{49} + 4 q^{50} + 6 q^{51} - 2 q^{52} - 6 q^{53} + 4 q^{54} + 18 q^{55} - 4 q^{56} + 4 q^{57} - 3 q^{58} + 6 q^{60} + q^{61} - 2 q^{62} - 4 q^{63} + q^{64} + 6 q^{65} + 12 q^{66} + 2 q^{67} - 3 q^{68} + 12 q^{69} + 12 q^{70} - 12 q^{71} + q^{72} - 10 q^{73} - 8 q^{75} - 2 q^{76} + 24 q^{77} + 4 q^{78} - 14 q^{79} - 3 q^{80} - 11 q^{81} + 3 q^{82} + 6 q^{83} + 8 q^{84} + 9 q^{85} + 4 q^{86} + 6 q^{87} - 6 q^{88} - 3 q^{89} - 3 q^{90} + 8 q^{91} - 6 q^{92} + 4 q^{93} - 6 q^{94} + 6 q^{95} - 2 q^{96} + 13 q^{97} + 9 q^{98} - 6 q^{99}+O(q^{100}) \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
1.00000 −2.00000 1.00000 −3.00000 −2.00000 −4.00000 1.00000 1.00000 −3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(37\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2738.2.a.c 1
37.b even 2 1 2738.2.a.a 1
37.e even 6 2 74.2.c.b 2
111.h odd 6 2 666.2.f.d 2
148.j odd 6 2 592.2.i.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.c.b 2 37.e even 6 2
592.2.i.a 2 148.j odd 6 2
666.2.f.d 2 111.h odd 6 2
2738.2.a.a 1 37.b even 2 1
2738.2.a.c 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2738))\):

\( T_{3} + 2 \) Copy content Toggle raw display
\( T_{5} + 3 \) Copy content Toggle raw display
\( T_{7} + 4 \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display
\( T_{17} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T + 2 \) Copy content Toggle raw display
$5$ \( T + 3 \) Copy content Toggle raw display
$7$ \( T + 4 \) Copy content Toggle raw display
$11$ \( T + 6 \) Copy content Toggle raw display
$13$ \( T + 2 \) Copy content Toggle raw display
$17$ \( T + 3 \) Copy content Toggle raw display
$19$ \( T + 2 \) Copy content Toggle raw display
$23$ \( T + 6 \) Copy content Toggle raw display
$29$ \( T + 3 \) Copy content Toggle raw display
$31$ \( T + 2 \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T - 3 \) Copy content Toggle raw display
$43$ \( T - 4 \) Copy content Toggle raw display
$47$ \( T + 6 \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 1 \) Copy content Toggle raw display
$67$ \( T - 2 \) Copy content Toggle raw display
$71$ \( T + 12 \) Copy content Toggle raw display
$73$ \( T + 10 \) Copy content Toggle raw display
$79$ \( T + 14 \) Copy content Toggle raw display
$83$ \( T - 6 \) Copy content Toggle raw display
$89$ \( T + 3 \) Copy content Toggle raw display
$97$ \( T - 13 \) Copy content Toggle raw display
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