Properties

Label 5445.2.a.q
Level $5445$
Weight $2$
Character orbit 5445.a
Self dual yes
Analytic conductor $43.479$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5445.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.4785439006\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1815)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 q^{4} + q^{5} -\beta q^{7} +O(q^{10})\) \( q -2 q^{4} + q^{5} -\beta q^{7} + 4 q^{16} -2 \beta q^{17} + 3 \beta q^{19} -2 q^{20} -6 q^{23} + q^{25} + 2 \beta q^{28} + 4 \beta q^{29} + q^{31} -\beta q^{35} -5 q^{37} + 2 \beta q^{41} -6 \beta q^{43} + 12 q^{47} -4 q^{49} -6 q^{53} + 7 \beta q^{61} -8 q^{64} -5 q^{67} + 4 \beta q^{68} + 6 q^{71} + \beta q^{73} -6 \beta q^{76} + 9 \beta q^{79} + 4 q^{80} -4 \beta q^{83} -2 \beta q^{85} + 6 q^{89} + 12 q^{92} + 3 \beta q^{95} -13 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{4} + 2q^{5} + O(q^{10}) \) \( 2q - 4q^{4} + 2q^{5} + 8q^{16} - 4q^{20} - 12q^{23} + 2q^{25} + 2q^{31} - 10q^{37} + 24q^{47} - 8q^{49} - 12q^{53} - 16q^{64} - 10q^{67} + 12q^{71} + 8q^{80} + 12q^{89} + 24q^{92} - 26q^{97} + O(q^{100}) \)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
1.73205
−1.73205
0 0 −2.00000 1.00000 0 −1.73205 0 0 0
1.2 0 0 −2.00000 1.00000 0 1.73205 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(11\) \(1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5445.2.a.q 2
3.b odd 2 1 1815.2.a.g 2
11.b odd 2 1 inner 5445.2.a.q 2
15.d odd 2 1 9075.2.a.bl 2
33.d even 2 1 1815.2.a.g 2
165.d even 2 1 9075.2.a.bl 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.a.g 2 3.b odd 2 1
1815.2.a.g 2 33.d even 2 1
5445.2.a.q 2 1.a even 1 1 trivial
5445.2.a.q 2 11.b odd 2 1 inner
9075.2.a.bl 2 15.d odd 2 1
9075.2.a.bl 2 165.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5445))\):

\( T_{2} \)
\( T_{7}^{2} - 3 \)
\( T_{23} + 6 \)
\( T_{53} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( -3 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( T^{2} \)
$17$ \( -12 + T^{2} \)
$19$ \( -27 + T^{2} \)
$23$ \( ( 6 + T )^{2} \)
$29$ \( -48 + T^{2} \)
$31$ \( ( -1 + T )^{2} \)
$37$ \( ( 5 + T )^{2} \)
$41$ \( -12 + T^{2} \)
$43$ \( -108 + T^{2} \)
$47$ \( ( -12 + T )^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( T^{2} \)
$61$ \( -147 + T^{2} \)
$67$ \( ( 5 + T )^{2} \)
$71$ \( ( -6 + T )^{2} \)
$73$ \( -3 + T^{2} \)
$79$ \( -243 + T^{2} \)
$83$ \( -48 + T^{2} \)
$89$ \( ( -6 + T )^{2} \)
$97$ \( ( 13 + T )^{2} \)
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