Properties

Label 5733.2.a.bq
Level $5733$
Weight $2$
Character orbit 5733.a
Self dual yes
Analytic conductor $45.778$
Analytic rank $0$
Dimension $5$
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] = [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: \(5\)
Coefficient field: 5.5.375116.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 6x^{3} + 7x^{2} + 2x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 273)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + ( - \beta_{4} + 1) q^{4} + ( - \beta_{2} + 1) q^{5} + (\beta_{4} - \beta_{3} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + ( - \beta_{4} + 1) q^{4} + ( - \beta_{2} + 1) q^{5} + (\beta_{4} - \beta_{3} + \beta_1) q^{8} + (\beta_{4} - 2 \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{10} + ( - \beta_{4} + \beta_{3}) q^{11} - q^{13} + (2 \beta_{3} - \beta_{2} - \beta_1) q^{16} + (\beta_1 + 3) q^{17} + ( - \beta_{3} - \beta_{2} - 1) q^{19} + ( - 2 \beta_{4} - \beta_{2} + \beta_1 + 3) q^{20} + (2 \beta_{4} - 2 \beta_{3} + \beta_1 - 3) q^{22} + ( - 2 \beta_{4} - \beta_1) q^{23} + ( - \beta_{4} - 2 \beta_{3} + 3) q^{25} + \beta_{3} q^{26} + ( - 2 \beta_{3} + \beta_{2} + 2) q^{29} + (\beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{31} + (\beta_{4} + \beta_{3} - 4) q^{32} + ( - 3 \beta_{3} - \beta_{2} - 1) q^{34} + (\beta_{4} - 2 \beta_{2} + 2 \beta_1 + 4) q^{37} + ( - \beta_{2} + 2 \beta_1 + 4) q^{38} + (\beta_{4} - 4 \beta_{3} - 2) q^{40} + ( - \beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{41} + ( - 2 \beta_{4} - 2 \beta_{3} + \beta_1 + 2) q^{43} + ( - 2 \beta_{4} + 5 \beta_{3} - \beta_{2} - 2 \beta_1 + 5) q^{44} + (2 \beta_{4} - 4 \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{46} + (3 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 1) q^{47} + ( - \beta_{4} - 5 \beta_{3} + \beta_1 + 6) q^{50} + (\beta_{4} - 1) q^{52} + ( - 3 \beta_{4} + 2 \beta_{3} + 3 \beta_{2}) q^{53} + ( - 3 \beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{55} + ( - 3 \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 + 5) q^{58} + (3 \beta_{4} - 3 \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 4) q^{59} + ( - \beta_1 - 5) q^{61} + (2 \beta_{4} + 2 \beta_{3} + \beta_1 - 4) q^{62} + (2 \beta_{3} + 2 \beta_{2} + \beta_1 - 3) q^{64} + (\beta_{2} - 1) q^{65} + ( - \beta_{4} - \beta_{3} - 2 \beta_1 + 4) q^{67} + ( - 2 \beta_{4} - \beta_{2} + 4) q^{68} + (\beta_{4} + \beta_{3} - 2 \beta_{2} - \beta_1 + 6) q^{71} + ( - \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1 - 3) q^{73} + (\beta_{4} - 4 \beta_{3} - 4 \beta_{2} + 3 \beta_1) q^{74} + (\beta_{4} - 3 \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{76} + (\beta_{4} - 2 \beta_{3} + 3 \beta_{2} + \beta_1 + 1) q^{79} + ( - \beta_{4} + 4 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 6) q^{80} + (2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - \beta_1 - 6) q^{82} + ( - \beta_{4} + 2 \beta_{3} - \beta_{2} - 4 \beta_1 + 1) q^{83} + ( - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{85} + ( - 6 \beta_{3} - \beta_{2} + 2 \beta_1 + 5) q^{86} + (4 \beta_{4} - 6 \beta_{3} + \beta_{2} + 2 \beta_1 - 6) q^{88} + (\beta_{2} - \beta_1 + 5) q^{89} + ( - 3 \beta_{4} + 4 \beta_{3} - \beta_{2} - 2 \beta_1 + 9) q^{92} + ( - 3 \beta_{4} + 8 \beta_{3} + 2 \beta_{2} - 5 \beta_1 - 3) q^{94} + ( - 4 \beta_{3} + \beta_{2} + 2 \beta_1 + 7) q^{95} + (3 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 6 q^{4} + 3 q^{5} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 6 q^{4} + 3 q^{5} - 3 q^{8} - 2 q^{10} + q^{11} - 5 q^{13} + 13 q^{17} - 7 q^{19} + 13 q^{20} - 19 q^{22} + 4 q^{23} + 16 q^{25} + 12 q^{29} - 6 q^{31} - 21 q^{32} - 7 q^{34} + 11 q^{37} + 14 q^{38} - 11 q^{40} + 10 q^{41} + 10 q^{43} + 29 q^{44} + q^{46} - 4 q^{47} + 29 q^{50} - 6 q^{52} + 9 q^{53} + 12 q^{55} + 34 q^{58} + 7 q^{59} - 23 q^{61} - 24 q^{62} - 13 q^{64} - 3 q^{65} + 25 q^{67} + 20 q^{68} + 27 q^{71} - 18 q^{73} - 15 q^{74} - 2 q^{76} + 8 q^{79} + 41 q^{80} - 26 q^{82} + 12 q^{83} + 10 q^{85} + 19 q^{86} - 36 q^{88} + 29 q^{89} + 50 q^{92} + 2 q^{94} + 33 q^{95} - 13 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 6x^{3} + 7x^{2} + 2x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{4} - 5\nu^{2} + 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{4} + 6\nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 6\nu^{2} + 2\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{4} + \nu^{3} + 6\nu^{2} - 6\nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + 3\beta_{3} + 2\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{3} - \beta_{2} + 6\beta _1 + 15 \) 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
0.507274
2.17362
−0.562376
1.32173
−2.44025
−2.53680 0 4.43536 2.52225 0 0 −6.17804 0 −6.39846
1.2 −1.32155 0 −0.253495 −2.02568 0 0 2.97812 0 2.67705
1.3 −0.0776754 0 −1.99397 2.20243 0 0 0.310233 0 −0.171074
1.4 1.78646 0 1.19144 −3.42992 0 0 −1.44447 0 −6.12741
1.5 2.14957 0 2.62066 3.73093 0 0 1.33415 0 8.01989
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

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

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 5733.2.a.bq 5
3.b odd 2 1 1911.2.a.t 5
7.b odd 2 1 5733.2.a.bp 5
7.c even 3 2 819.2.j.g 10
21.c even 2 1 1911.2.a.u 5
21.h odd 6 2 273.2.i.e 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.i.e 10 21.h odd 6 2
819.2.j.g 10 7.c even 3 2
1911.2.a.t 5 3.b odd 2 1
1911.2.a.u 5 21.c even 2 1
5733.2.a.bp 5 7.b odd 2 1
5733.2.a.bq 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5733))\):

\( T_{2}^{5} - 8T_{2}^{3} + T_{2}^{2} + 13T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{5} - 3T_{5}^{4} - 16T_{5}^{3} + 47T_{5}^{2} + 48T_{5} - 144 \) Copy content Toggle raw display
\( T_{11}^{5} - T_{11}^{4} - 23T_{11}^{3} - 38T_{11}^{2} - 12T_{11} + 1 \) Copy content Toggle raw display
\( T_{17}^{5} - 13T_{17}^{4} + 55T_{17}^{3} - 82T_{17}^{2} + 22T_{17} + 1 \) Copy content Toggle raw display
\( T_{19}^{5} + 7T_{19}^{4} - 6T_{19}^{3} - 106T_{19}^{2} - 139T_{19} + 19 \) Copy content Toggle raw display
\( T_{31}^{5} + 6T_{31}^{4} - 39T_{31}^{3} - 251T_{31}^{2} + 120T_{31} + 1376 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} - 8 T^{3} + T^{2} + 13 T + 1 \) Copy content Toggle raw display
$3$ \( T^{5} \) Copy content Toggle raw display
$5$ \( T^{5} - 3 T^{4} - 16 T^{3} + 47 T^{2} + \cdots - 144 \) Copy content Toggle raw display
$7$ \( T^{5} \) Copy content Toggle raw display
$11$ \( T^{5} - T^{4} - 23 T^{3} - 38 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( (T + 1)^{5} \) Copy content Toggle raw display
$17$ \( T^{5} - 13 T^{4} + 55 T^{3} - 82 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{5} + 7 T^{4} - 6 T^{3} - 106 T^{2} + \cdots + 19 \) Copy content Toggle raw display
$23$ \( T^{5} - 4 T^{4} - 47 T^{3} + \cdots - 1108 \) Copy content Toggle raw display
$29$ \( T^{5} - 12 T^{4} + 2 T^{3} + 231 T^{2} + \cdots + 251 \) Copy content Toggle raw display
$31$ \( T^{5} + 6 T^{4} - 39 T^{3} + \cdots + 1376 \) Copy content Toggle raw display
$37$ \( T^{5} - 11 T^{4} - 74 T^{3} + \cdots + 7456 \) Copy content Toggle raw display
$41$ \( T^{5} - 10 T^{4} - 13 T^{3} + \cdots - 224 \) Copy content Toggle raw display
$43$ \( T^{5} - 10 T^{4} - 37 T^{3} + 413 T^{2} + \cdots - 76 \) Copy content Toggle raw display
$47$ \( T^{5} + 4 T^{4} - 171 T^{3} + \cdots - 18392 \) Copy content Toggle raw display
$53$ \( T^{5} - 9 T^{4} - 209 T^{3} + \cdots - 84029 \) Copy content Toggle raw display
$59$ \( T^{5} - 7 T^{4} - 225 T^{3} + \cdots - 71807 \) Copy content Toggle raw display
$61$ \( T^{5} + 23 T^{4} + 199 T^{3} + \cdots + 1051 \) Copy content Toggle raw display
$67$ \( T^{5} - 25 T^{4} + 177 T^{3} + \cdots + 11303 \) Copy content Toggle raw display
$71$ \( T^{5} - 27 T^{4} + 205 T^{3} + \cdots - 761 \) Copy content Toggle raw display
$73$ \( T^{5} + 18 T^{4} + 61 T^{3} + \cdots - 7468 \) Copy content Toggle raw display
$79$ \( T^{5} - 8 T^{4} - 229 T^{3} + \cdots + 18572 \) Copy content Toggle raw display
$83$ \( T^{5} - 12 T^{4} - 145 T^{3} + \cdots - 17248 \) Copy content Toggle raw display
$89$ \( T^{5} - 29 T^{4} + 302 T^{3} + \cdots - 2144 \) Copy content Toggle raw display
$97$ \( T^{5} + 13 T^{4} - 186 T^{3} + \cdots + 14308 \) Copy content Toggle raw display
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