Properties

Label 37.2.a.b
Level $37$
Weight $2$
Character orbit 37.a
Self dual yes
Analytic conductor $0.295$
Analytic rank $0$
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] = [37,2,Mod(1,37)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(37, 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("37.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 37.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: \(0.295446487479\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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)
 
\(f(q)\) \(=\) \( q + q^{3} - 2 q^{4} - q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - 2 q^{4} - q^{7} - 2 q^{9} + 3 q^{11} - 2 q^{12} - 4 q^{13} + 4 q^{16} + 6 q^{17} + 2 q^{19} - q^{21} + 6 q^{23} - 5 q^{25} - 5 q^{27} + 2 q^{28} - 6 q^{29} - 4 q^{31} + 3 q^{33} + 4 q^{36} + q^{37} - 4 q^{39} - 9 q^{41} + 8 q^{43} - 6 q^{44} + 3 q^{47} + 4 q^{48} - 6 q^{49} + 6 q^{51} + 8 q^{52} - 3 q^{53} + 2 q^{57} + 12 q^{59} + 8 q^{61} + 2 q^{63} - 8 q^{64} - 4 q^{67} - 12 q^{68} + 6 q^{69} - 15 q^{71} + 11 q^{73} - 5 q^{75} - 4 q^{76} - 3 q^{77} - 10 q^{79} + q^{81} + 9 q^{83} + 2 q^{84} - 6 q^{87} + 6 q^{89} + 4 q^{91} - 12 q^{92} - 4 q^{93} + 8 q^{97} - 6 q^{99}+O(q^{100}) \) 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
0 1.00000 −2.00000 0 0 −1.00000 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 37.2.a.b 1
3.b odd 2 1 333.2.a.b 1
4.b odd 2 1 592.2.a.a 1
5.b even 2 1 925.2.a.b 1
5.c odd 4 2 925.2.b.e 2
7.b odd 2 1 1813.2.a.b 1
8.b even 2 1 2368.2.a.d 1
8.d odd 2 1 2368.2.a.m 1
11.b odd 2 1 4477.2.a.a 1
12.b even 2 1 5328.2.a.k 1
13.b even 2 1 6253.2.a.b 1
15.d odd 2 1 8325.2.a.p 1
37.b even 2 1 1369.2.a.c 1
37.d odd 4 2 1369.2.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.a.b 1 1.a even 1 1 trivial
333.2.a.b 1 3.b odd 2 1
592.2.a.a 1 4.b odd 2 1
925.2.a.b 1 5.b even 2 1
925.2.b.e 2 5.c odd 4 2
1369.2.a.c 1 37.b even 2 1
1369.2.b.a 2 37.d odd 4 2
1813.2.a.b 1 7.b odd 2 1
2368.2.a.d 1 8.b even 2 1
2368.2.a.m 1 8.d odd 2 1
4477.2.a.a 1 11.b odd 2 1
5328.2.a.k 1 12.b even 2 1
6253.2.a.b 1 13.b even 2 1
8325.2.a.p 1 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(37))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 1 \) Copy content Toggle raw display
$11$ \( T - 3 \) Copy content Toggle raw display
$13$ \( T + 4 \) Copy content Toggle raw display
$17$ \( T - 6 \) Copy content Toggle raw display
$19$ \( T - 2 \) Copy content Toggle raw display
$23$ \( T - 6 \) Copy content Toggle raw display
$29$ \( T + 6 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T - 1 \) Copy content Toggle raw display
$41$ \( T + 9 \) Copy content Toggle raw display
$43$ \( T - 8 \) Copy content Toggle raw display
$47$ \( T - 3 \) Copy content Toggle raw display
$53$ \( T + 3 \) Copy content Toggle raw display
$59$ \( T - 12 \) Copy content Toggle raw display
$61$ \( T - 8 \) Copy content Toggle raw display
$67$ \( T + 4 \) Copy content Toggle raw display
$71$ \( T + 15 \) Copy content Toggle raw display
$73$ \( T - 11 \) Copy content Toggle raw display
$79$ \( T + 10 \) Copy content Toggle raw display
$83$ \( T - 9 \) Copy content Toggle raw display
$89$ \( T - 6 \) Copy content Toggle raw display
$97$ \( T - 8 \) Copy content Toggle raw display
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