Properties

Label 5733.2.a.s
Level $5733$
Weight $2$
Character orbit 5733.a
Self dual yes
Analytic conductor $45.778$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( - \beta + 3) q^{5} - 2 \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + ( - \beta + 3) q^{5} - 2 \beta q^{8} + (3 \beta - 2) q^{10} - 3 \beta q^{11} + q^{13} - 4 q^{16} + \beta q^{17} + (3 \beta + 3) q^{19} - 6 q^{22} + (2 \beta + 3) q^{23} + ( - 6 \beta + 6) q^{25} + \beta q^{26} + (2 \beta - 3) q^{29} + ( - 3 \beta + 1) q^{31} + 2 q^{34} + (3 \beta - 2) q^{37} + (3 \beta + 6) q^{38} + ( - 6 \beta + 4) q^{40} + (2 \beta + 6) q^{41} - 5 q^{43} + (3 \beta + 4) q^{46} + ( - \beta + 3) q^{47} + (6 \beta - 12) q^{50} + ( - 2 \beta + 3) q^{53} + ( - 9 \beta + 6) q^{55} + ( - 3 \beta + 4) q^{58} + ( - 4 \beta + 6) q^{59} - 6 q^{61} + (\beta - 6) q^{62} + 8 q^{64} + ( - \beta + 3) q^{65} + ( - 6 \beta - 6) q^{67} + (5 \beta + 6) q^{71} + (3 \beta + 5) q^{73} + ( - 2 \beta + 6) q^{74} + (6 \beta + 7) q^{79} + (4 \beta - 12) q^{80} + (6 \beta + 4) q^{82} + (3 \beta + 9) q^{83} + (3 \beta - 2) q^{85} - 5 \beta q^{86} + 12 q^{88} + ( - \beta + 3) q^{89} + (3 \beta - 2) q^{94} + (6 \beta + 3) q^{95} + ( - 9 \beta + 1) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{5} - 4 q^{10} + 2 q^{13} - 8 q^{16} + 6 q^{19} - 12 q^{22} + 6 q^{23} + 12 q^{25} - 6 q^{29} + 2 q^{31} + 4 q^{34} - 4 q^{37} + 12 q^{38} + 8 q^{40} + 12 q^{41} - 10 q^{43} + 8 q^{46} + 6 q^{47} - 24 q^{50} + 6 q^{53} + 12 q^{55} + 8 q^{58} + 12 q^{59} - 12 q^{61} - 12 q^{62} + 16 q^{64} + 6 q^{65} - 12 q^{67} + 12 q^{71} + 10 q^{73} + 12 q^{74} + 14 q^{79} - 24 q^{80} + 8 q^{82} + 18 q^{83} - 4 q^{85} + 24 q^{88} + 6 q^{89} - 4 q^{94} + 6 q^{95} + 2 q^{97}+O(q^{100}) \) 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
−1.41421
1.41421
−1.41421 0 0 4.41421 0 0 2.82843 0 −6.24264
1.2 1.41421 0 0 1.58579 0 0 −2.82843 0 2.24264
\(n\): e.g. 2-40 or 990-1000
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.s 2
3.b odd 2 1 637.2.a.g 2
7.b odd 2 1 819.2.a.h 2
21.c even 2 1 91.2.a.c 2
21.g even 6 2 637.2.e.f 4
21.h odd 6 2 637.2.e.g 4
39.d odd 2 1 8281.2.a.v 2
84.h odd 2 1 1456.2.a.q 2
105.g even 2 1 2275.2.a.j 2
168.e odd 2 1 5824.2.a.bk 2
168.i even 2 1 5824.2.a.bl 2
273.g even 2 1 1183.2.a.d 2
273.o odd 4 2 1183.2.c.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.c 2 21.c even 2 1
637.2.a.g 2 3.b odd 2 1
637.2.e.f 4 21.g even 6 2
637.2.e.g 4 21.h odd 6 2
819.2.a.h 2 7.b odd 2 1
1183.2.a.d 2 273.g even 2 1
1183.2.c.d 4 273.o odd 4 2
1456.2.a.q 2 84.h odd 2 1
2275.2.a.j 2 105.g even 2 1
5733.2.a.s 2 1.a even 1 1 trivial
5824.2.a.bk 2 168.e odd 2 1
5824.2.a.bl 2 168.i even 2 1
8281.2.a.v 2 39.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5733))\):

\( T_{2}^{2} - 2 \) Copy content Toggle raw display
\( T_{5}^{2} - 6T_{5} + 7 \) Copy content Toggle raw display
\( T_{11}^{2} - 18 \) Copy content Toggle raw display
\( T_{17}^{2} - 2 \) Copy content Toggle raw display
\( T_{19}^{2} - 6T_{19} - 9 \) Copy content Toggle raw display
\( T_{31}^{2} - 2T_{31} - 17 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 6T + 7 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 18 \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 2 \) Copy content Toggle raw display
$19$ \( T^{2} - 6T - 9 \) Copy content Toggle raw display
$23$ \( T^{2} - 6T + 1 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 1 \) Copy content Toggle raw display
$31$ \( T^{2} - 2T - 17 \) Copy content Toggle raw display
$37$ \( T^{2} + 4T - 14 \) Copy content Toggle raw display
$41$ \( T^{2} - 12T + 28 \) Copy content Toggle raw display
$43$ \( (T + 5)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 6T + 7 \) Copy content Toggle raw display
$53$ \( T^{2} - 6T + 1 \) Copy content Toggle raw display
$59$ \( T^{2} - 12T + 4 \) Copy content Toggle raw display
$61$ \( (T + 6)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 12T - 36 \) Copy content Toggle raw display
$71$ \( T^{2} - 12T - 14 \) Copy content Toggle raw display
$73$ \( T^{2} - 10T + 7 \) Copy content Toggle raw display
$79$ \( T^{2} - 14T - 23 \) Copy content Toggle raw display
$83$ \( T^{2} - 18T + 63 \) Copy content Toggle raw display
$89$ \( T^{2} - 6T + 7 \) Copy content Toggle raw display
$97$ \( T^{2} - 2T - 161 \) Copy content Toggle raw display
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