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Properties

Label 5733.2.a.by
Level $5733$
Weight $2$
Character orbit 5733.a
Self dual yes
Analytic conductor $45.778$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5733,2,Mod(1,5733)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5733, 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("5733.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5733 = 3^{2} \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5733.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: \(45.7782354788\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 18x^{8} + 110x^{6} - 265x^{4} + 243x^{2} - 63 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 819)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + \beta_{5} q^{5} + (\beta_{7} - \beta_{5} + 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + \beta_{5} q^{5} + (\beta_{7} - \beta_{5} + 2 \beta_1) q^{8} + ( - \beta_{6} - \beta_{4} + \beta_{3} + \cdots - 1) q^{10}+ \cdots + (4 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 16 q^{4} - 4 q^{10} + 10 q^{13} + 32 q^{16} + 6 q^{19} + 10 q^{22} + 24 q^{25} + 12 q^{31} + 34 q^{34} + 26 q^{37} - 70 q^{40} + 40 q^{43} - 6 q^{46} + 16 q^{52} - 24 q^{55} + 36 q^{58} - 22 q^{61} + 78 q^{64} + 26 q^{67} + 24 q^{73} + 4 q^{76} + 28 q^{79} - 28 q^{82} - 20 q^{85} + 12 q^{88} + 4 q^{94} + 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 18x^{8} + 110x^{6} - 265x^{4} + 243x^{2} - 63 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{8} + 14\nu^{6} - 59\nu^{4} + 74\nu^{2} - 17 ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{8} + 19\nu^{6} - 109\nu^{4} + 184\nu^{2} - 57 ) / 10 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{9} + 33\nu^{7} - 178\nu^{5} + 338\nu^{3} - 159\nu ) / 15 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{8} - 19\nu^{6} + 119\nu^{4} - 264\nu^{2} + 137 ) / 10 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{9} + 33\nu^{7} - 178\nu^{5} + 353\nu^{3} - 249\nu ) / 15 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{9} + 19\nu^{7} - 119\nu^{5} + 274\nu^{3} - 197\nu ) / 10 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -4\nu^{9} + 66\nu^{7} - 341\nu^{5} + 556\nu^{3} - 198\nu ) / 15 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{5} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + \beta_{4} + 8\beta_{2} + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{9} + 8\beta_{7} - 10\beta_{5} + 40\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10\beta_{6} + 12\beta_{4} - \beta_{3} + 58\beta_{2} + 160 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12\beta_{9} + 4\beta_{8} + 54\beta_{7} - 81\beta_{5} + 275\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 81\beta_{6} + 109\beta_{4} - 19\beta_{3} + 414\beta_{2} + 1103 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 109\beta_{9} + 66\beta_{8} + 348\beta_{7} - 623\beta_{5} + 1912\beta_1 \) 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
−2.70790
−2.55095
−1.57133
−1.12683
−0.648951
0.648951
1.12683
1.57133
2.55095
2.70790
−2.70790 0 5.33272 4.07829 0 0 −9.02468 0 −11.0436
1.2 −2.55095 0 4.50732 −1.71239 0 0 −6.39604 0 4.36821
1.3 −1.57133 0 0.469085 −1.34037 0 0 2.40558 0 2.10616
1.4 −1.12683 0 −0.730265 −3.42178 0 0 3.07653 0 3.85575
1.5 −0.648951 0 −1.57886 1.98246 0 0 2.32251 0 −1.28652
1.6 0.648951 0 −1.57886 −1.98246 0 0 −2.32251 0 −1.28652
1.7 1.12683 0 −0.730265 3.42178 0 0 −3.07653 0 3.85575
1.8 1.57133 0 0.469085 1.34037 0 0 −2.40558 0 2.10616
1.9 2.55095 0 4.50732 1.71239 0 0 6.39604 0 4.36821
1.10 2.70790 0 5.33272 −4.07829 0 0 9.02468 0 −11.0436
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(1\)
\(13\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5733.2.a.by 10
3.b odd 2 1 inner 5733.2.a.by 10
7.b odd 2 1 5733.2.a.bz 10
7.c even 3 2 819.2.j.j ✓ 20
21.c even 2 1 5733.2.a.bz 10
21.h odd 6 2 819.2.j.j ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.j.j ✓ 20 7.c even 3 2
819.2.j.j ✓ 20 21.h odd 6 2
5733.2.a.by 10 1.a even 1 1 trivial
5733.2.a.by 10 3.b odd 2 1 inner
5733.2.a.bz 10 7.b odd 2 1
5733.2.a.bz 10 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5733))\):

\( T_{2}^{10} - 18T_{2}^{8} + 110T_{2}^{6} - 265T_{2}^{4} + 243T_{2}^{2} - 63 \) Copy content Toggle raw display
\( T_{5}^{10} - 37T_{5}^{8} + 464T_{5}^{6} - 2383T_{5}^{4} + 5232T_{5}^{2} - 4032 \) Copy content Toggle raw display
\( T_{11}^{10} - 79T_{11}^{8} + 1625T_{11}^{6} - 2560T_{11}^{4} + 1116T_{11}^{2} - 63 \) Copy content Toggle raw display
\( T_{17}^{10} - 81T_{17}^{8} + 1799T_{17}^{6} - 6658T_{17}^{4} + 2358T_{17}^{2} - 63 \) Copy content Toggle raw display
\( T_{19}^{5} - 3T_{19}^{4} - 32T_{19}^{3} + 116T_{19}^{2} - 49T_{19} - 49 \) Copy content Toggle raw display
\( T_{31}^{5} - 6T_{31}^{4} - 73T_{31}^{3} + 219T_{31}^{2} + 232T_{31} + 28 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 18 T^{8} + \cdots - 63 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} - 37 T^{8} + \cdots - 4032 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( T^{10} - 79 T^{8} + \cdots - 63 \) Copy content Toggle raw display
$13$ \( (T - 1)^{10} \) Copy content Toggle raw display
$17$ \( T^{10} - 81 T^{8} + \cdots - 63 \) Copy content Toggle raw display
$19$ \( (T^{5} - 3 T^{4} - 32 T^{3} + \cdots - 49)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} - 134 T^{8} + \cdots - 8347248 \) Copy content Toggle raw display
$29$ \( T^{10} - 146 T^{8} + \cdots - 642663 \) Copy content Toggle raw display
$31$ \( (T^{5} - 6 T^{4} - 73 T^{3} + \cdots + 28)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} - 13 T^{4} + 48 T^{3} + \cdots + 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} - 228 T^{8} + \cdots - 197568 \) Copy content Toggle raw display
$43$ \( (T^{5} - 20 T^{4} + \cdots + 3748)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} - 186 T^{8} + \cdots - 100800 \) Copy content Toggle raw display
$53$ \( T^{10} - 167 T^{8} + \cdots - 3087 \) Copy content Toggle raw display
$59$ \( T^{10} - 211 T^{8} + \cdots - 46486503 \) Copy content Toggle raw display
$61$ \( (T^{5} + 11 T^{4} + \cdots + 287)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} - 13 T^{4} + \cdots - 14797)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} - 215 T^{8} + \cdots - 22743 \) Copy content Toggle raw display
$73$ \( (T^{5} - 12 T^{4} + \cdots - 364)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} - 14 T^{4} + \cdots + 3620)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} - 368 T^{8} + \cdots - 326592 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 674476992 \) Copy content Toggle raw display
$97$ \( (T^{5} - 9 T^{4} + \cdots - 178556)^{2} \) Copy content Toggle raw display
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