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Properties

Label 5776.2.a.br
Level $5776$
Weight $2$
Character orbit 5776.a
Self dual yes
Analytic conductor $46.122$
Analytic rank $0$
Dimension $3$
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] = [5776,2,Mod(1,5776)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5776, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("5776.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5776 = 2^{4} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5776.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: \(46.1215922075\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{3} + (\beta_1 - 1) q^{5} + (\beta_{2} - \beta_1) q^{7} + (\beta_{2} + 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{3} + (\beta_1 - 1) q^{5} + (\beta_{2} - \beta_1) q^{7} + (\beta_{2} + 2 \beta_1) q^{9} + ( - \beta_{2} + 2 \beta_1) q^{11} + ( - 2 \beta_{2} + 3 \beta_1) q^{13} + (\beta_{2} + 1) q^{15} + ( - \beta_{2} + 2) q^{17} - q^{21} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{23} + (\beta_{2} - 2 \beta_1 - 2) q^{25} + (3 \beta_{2} + 2) q^{27} + ( - \beta_1 - 5) q^{29} + ( - \beta_{2} + 3 \beta_1 + 3) q^{31} + (\beta_{2} + \beta_1 + 3) q^{33} + ( - 2 \beta_{2} + 2 \beta_1 - 1) q^{35} + (2 \beta_{2} + \beta_1) q^{37} + (\beta_{2} + \beta_1 + 4) q^{39} + (3 \beta_{2} + \beta_1 - 4) q^{41} + ( - 5 \beta_{2} + 2 \beta_1) q^{43} + (\beta_{2} - \beta_1 + 5) q^{45} + (\beta_{2} + 2 \beta_1 + 2) q^{47} + ( - \beta_1 - 5) q^{49} + ( - \beta_{2} + \beta_1 + 1) q^{51} + ( - 3 \beta_{2} + 2 \beta_1 - 2) q^{53} + (3 \beta_{2} - 3 \beta_1 + 3) q^{55} + (2 \beta_{2} - 2 \beta_1 + 7) q^{59} + (4 \beta_{2} + 3) q^{61} + ( - 3 \beta_{2} + 2 \beta_1 - 1) q^{63} + (5 \beta_{2} - 5 \beta_1 + 4) q^{65} + ( - 6 \beta_{2} + 4 \beta_1 - 6) q^{67} + (2 \beta_1 + 4) q^{69} + (2 \beta_{2} - 2 \beta_1 + 10) q^{71} + ( - 4 \beta_{2} + 4 \beta_1) q^{73} + ( - \beta_{2} - 3 \beta_1 - 5) q^{75} + ( - \beta_{2} + 2 \beta_1 - 3) q^{77} + (7 \beta_{2} - \beta_1 + 3) q^{79} + ( - \beta_1 + 5) q^{81} + (6 \beta_{2} + 3 \beta_1) q^{83} + (\beta_{2} + \beta_1 - 3) q^{85} + ( - \beta_{2} - 6 \beta_1 - 7) q^{87} + ( - \beta_{2} - 2 \beta_1 - 5) q^{89} + ( - \beta_{2} + 3 \beta_1 - 5) q^{91} + (2 \beta_{2} + 5 \beta_1 + 8) q^{93} + ( - 2 \beta_{2} + 4 \beta_1 + 5) q^{97} + (5 \beta_{2} - \beta_1 + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - 3 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} - 3 q^{5} + 3 q^{15} + 6 q^{17} - 3 q^{21} + 6 q^{23} - 6 q^{25} + 6 q^{27} - 15 q^{29} + 9 q^{31} + 9 q^{33} - 3 q^{35} + 12 q^{39} - 12 q^{41} + 15 q^{45} + 6 q^{47} - 15 q^{49} + 3 q^{51} - 6 q^{53} + 9 q^{55} + 21 q^{59} + 9 q^{61} - 3 q^{63} + 12 q^{65} - 18 q^{67} + 12 q^{69} + 30 q^{71} - 15 q^{75} - 9 q^{77} + 9 q^{79} + 15 q^{81} - 9 q^{85} - 21 q^{87} - 15 q^{89} - 15 q^{91} + 24 q^{93} + 15 q^{97} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−1.53209
−0.347296
1.87939
0 −0.532089 0 −2.53209 0 1.87939 0 −2.71688 0
1.2 0 0.652704 0 −1.34730 0 −1.53209 0 −2.57398 0
1.3 0 2.87939 0 0.879385 0 −0.347296 0 5.29086 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(1\)

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 5776.2.a.br 3
4.b odd 2 1 361.2.a.g 3
12.b even 2 1 3249.2.a.z 3
19.b odd 2 1 5776.2.a.bi 3
19.e even 9 2 304.2.u.b 6
20.d odd 2 1 9025.2.a.bd 3
76.d even 2 1 361.2.a.h 3
76.f even 6 2 361.2.c.h 6
76.g odd 6 2 361.2.c.i 6
76.k even 18 2 361.2.e.a 6
76.k even 18 2 361.2.e.b 6
76.k even 18 2 361.2.e.h 6
76.l odd 18 2 19.2.e.a ✓ 6
76.l odd 18 2 361.2.e.f 6
76.l odd 18 2 361.2.e.g 6
228.b odd 2 1 3249.2.a.s 3
228.v even 18 2 171.2.u.c 6
380.d even 2 1 9025.2.a.x 3
380.ba odd 18 2 475.2.l.a 6
380.bj even 36 4 475.2.u.a 12
532.br even 18 2 931.2.x.b 6
532.bt odd 18 2 931.2.x.a 6
532.cc even 18 2 931.2.w.a 6
532.cd even 18 2 931.2.v.a 6
532.cf odd 18 2 931.2.v.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.2.e.a ✓ 6 76.l odd 18 2
171.2.u.c 6 228.v even 18 2
304.2.u.b 6 19.e even 9 2
361.2.a.g 3 4.b odd 2 1
361.2.a.h 3 76.d even 2 1
361.2.c.h 6 76.f even 6 2
361.2.c.i 6 76.g odd 6 2
361.2.e.a 6 76.k even 18 2
361.2.e.b 6 76.k even 18 2
361.2.e.f 6 76.l odd 18 2
361.2.e.g 6 76.l odd 18 2
361.2.e.h 6 76.k even 18 2
475.2.l.a 6 380.ba odd 18 2
475.2.u.a 12 380.bj even 36 4
931.2.v.a 6 532.cd even 18 2
931.2.v.b 6 532.cf odd 18 2
931.2.w.a 6 532.cc even 18 2
931.2.x.a 6 532.bt odd 18 2
931.2.x.b 6 532.br even 18 2
3249.2.a.s 3 228.b odd 2 1
3249.2.a.z 3 12.b even 2 1
5776.2.a.bi 3 19.b odd 2 1
5776.2.a.br 3 1.a even 1 1 trivial
9025.2.a.x 3 380.d even 2 1
9025.2.a.bd 3 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5776))\):

\( T_{3}^{3} - 3T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{3} + 3T_{5}^{2} - 3 \) Copy content Toggle raw display
\( T_{7}^{3} - 3T_{7} - 1 \) Copy content Toggle raw display
\( T_{11}^{3} - 9T_{11} + 9 \) Copy content Toggle raw display
\( T_{13}^{3} - 21T_{13} + 37 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 3T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{3} + 3T^{2} - 3 \) Copy content Toggle raw display
$7$ \( T^{3} - 3T - 1 \) Copy content Toggle raw display
$11$ \( T^{3} - 9T + 9 \) Copy content Toggle raw display
$13$ \( T^{3} - 21T + 37 \) Copy content Toggle raw display
$17$ \( T^{3} - 6 T^{2} + \cdots - 3 \) Copy content Toggle raw display
$19$ \( T^{3} \) Copy content Toggle raw display
$23$ \( T^{3} - 6T^{2} + 24 \) Copy content Toggle raw display
$29$ \( T^{3} + 15 T^{2} + \cdots + 111 \) Copy content Toggle raw display
$31$ \( T^{3} - 9 T^{2} + \cdots + 53 \) Copy content Toggle raw display
$37$ \( T^{3} - 21T - 17 \) Copy content Toggle raw display
$41$ \( T^{3} + 12 T^{2} + \cdots - 111 \) Copy content Toggle raw display
$43$ \( T^{3} - 57T - 163 \) Copy content Toggle raw display
$47$ \( T^{3} - 6 T^{2} + \cdots - 3 \) Copy content Toggle raw display
$53$ \( T^{3} + 6 T^{2} + \cdots - 51 \) Copy content Toggle raw display
$59$ \( T^{3} - 21 T^{2} + \cdots - 267 \) Copy content Toggle raw display
$61$ \( T^{3} - 9 T^{2} + \cdots + 181 \) Copy content Toggle raw display
$67$ \( T^{3} + 18 T^{2} + \cdots - 424 \) Copy content Toggle raw display
$71$ \( T^{3} - 30 T^{2} + \cdots - 888 \) Copy content Toggle raw display
$73$ \( T^{3} - 48T + 64 \) Copy content Toggle raw display
$79$ \( T^{3} - 9 T^{2} + \cdots + 809 \) Copy content Toggle raw display
$83$ \( T^{3} - 189T - 459 \) Copy content Toggle raw display
$89$ \( T^{3} + 15 T^{2} + \cdots + 57 \) Copy content Toggle raw display
$97$ \( T^{3} - 15 T^{2} + \cdots + 127 \) Copy content Toggle raw display
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