Properties

Label 57.2.a.b
Level $57$
Weight $2$
Character orbit 57.a
Self dual yes
Analytic conductor $0.455$
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] = [57,2,Mod(1,57)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(57, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("57.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 57 = 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 57.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: \(0.455147291521\)
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)
 
\(f(q)\) \(=\) \( q - 2 q^{2} + q^{3} + 2 q^{4} + q^{5} - 2 q^{6} + 3 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + q^{3} + 2 q^{4} + q^{5} - 2 q^{6} + 3 q^{7} + q^{9} - 2 q^{10} - 3 q^{11} + 2 q^{12} - 6 q^{13} - 6 q^{14} + q^{15} - 4 q^{16} + 3 q^{17} - 2 q^{18} - q^{19} + 2 q^{20} + 3 q^{21} + 6 q^{22} + 4 q^{23} - 4 q^{25} + 12 q^{26} + q^{27} + 6 q^{28} - 10 q^{29} - 2 q^{30} + 2 q^{31} + 8 q^{32} - 3 q^{33} - 6 q^{34} + 3 q^{35} + 2 q^{36} + 8 q^{37} + 2 q^{38} - 6 q^{39} - 8 q^{41} - 6 q^{42} - q^{43} - 6 q^{44} + q^{45} - 8 q^{46} + 3 q^{47} - 4 q^{48} + 2 q^{49} + 8 q^{50} + 3 q^{51} - 12 q^{52} - 6 q^{53} - 2 q^{54} - 3 q^{55} - q^{57} + 20 q^{58} + 2 q^{60} + 7 q^{61} - 4 q^{62} + 3 q^{63} - 8 q^{64} - 6 q^{65} + 6 q^{66} + 8 q^{67} + 6 q^{68} + 4 q^{69} - 6 q^{70} + 12 q^{71} - 11 q^{73} - 16 q^{74} - 4 q^{75} - 2 q^{76} - 9 q^{77} + 12 q^{78} - 4 q^{80} + q^{81} + 16 q^{82} + 4 q^{83} + 6 q^{84} + 3 q^{85} + 2 q^{86} - 10 q^{87} + 10 q^{89} - 2 q^{90} - 18 q^{91} + 8 q^{92} + 2 q^{93} - 6 q^{94} - q^{95} + 8 q^{96} - 2 q^{97} - 4 q^{98} - 3 q^{99}+O(q^{100}) \) 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
−2.00000 1.00000 2.00000 1.00000 −2.00000 3.00000 0 1.00000 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(19\) \( +1 \)

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 57.2.a.b 1
3.b odd 2 1 171.2.a.c 1
4.b odd 2 1 912.2.a.d 1
5.b even 2 1 1425.2.a.i 1
5.c odd 4 2 1425.2.c.a 2
7.b odd 2 1 2793.2.a.a 1
8.b even 2 1 3648.2.a.h 1
8.d odd 2 1 3648.2.a.y 1
11.b odd 2 1 6897.2.a.g 1
12.b even 2 1 2736.2.a.h 1
13.b even 2 1 9633.2.a.p 1
15.d odd 2 1 4275.2.a.a 1
19.b odd 2 1 1083.2.a.d 1
21.c even 2 1 8379.2.a.q 1
57.d even 2 1 3249.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.a.b 1 1.a even 1 1 trivial
171.2.a.c 1 3.b odd 2 1
912.2.a.d 1 4.b odd 2 1
1083.2.a.d 1 19.b odd 2 1
1425.2.a.i 1 5.b even 2 1
1425.2.c.a 2 5.c odd 4 2
2736.2.a.h 1 12.b even 2 1
2793.2.a.a 1 7.b odd 2 1
3249.2.a.a 1 57.d even 2 1
3648.2.a.h 1 8.b even 2 1
3648.2.a.y 1 8.d odd 2 1
4275.2.a.a 1 15.d odd 2 1
6897.2.a.g 1 11.b odd 2 1
8379.2.a.q 1 21.c even 2 1
9633.2.a.p 1 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(57))\):

\( T_{2} + 2 \) Copy content Toggle raw display
\( T_{5} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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