Properties

Label 5445.2.a.bz
Level $5445$
Weight $2$
Character orbit 5445.a
Self dual yes
Analytic conductor $43.479$
Analytic rank $0$
Dimension $6$
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: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.437199552.1
Defining polynomial: \(x^{6} - 13 x^{4} + 49 x^{2} - 48\)
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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 2 + \beta_{2} ) q^{4} - q^{5} + ( -\beta_{1} + \beta_{3} + \beta_{4} ) q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 2 + \beta_{2} ) q^{4} - q^{5} + ( -\beta_{1} + \beta_{3} + \beta_{4} ) q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} -\beta_{1} q^{10} -2 \beta_{1} q^{13} + ( -2 + 2 \beta_{2} + 2 \beta_{5} ) q^{14} + ( 1 + 2 \beta_{2} + \beta_{5} ) q^{16} + ( \beta_{1} + \beta_{3} - 2 \beta_{4} ) q^{17} + ( \beta_{1} + \beta_{3} - 3 \beta_{4} ) q^{19} + ( -2 - \beta_{2} ) q^{20} + ( 1 - \beta_{2} - \beta_{5} ) q^{23} + q^{25} + ( -8 - 2 \beta_{2} ) q^{26} + ( 2 \beta_{3} + 6 \beta_{4} ) q^{28} + 2 \beta_{1} q^{29} + ( 3 + 2 \beta_{5} ) q^{31} + ( \beta_{3} + 4 \beta_{4} ) q^{32} + ( 3 + 4 \beta_{2} - \beta_{5} ) q^{34} + ( \beta_{1} - \beta_{3} - \beta_{4} ) q^{35} + ( 2 - \beta_{2} - \beta_{5} ) q^{37} + ( 2 + 4 \beta_{2} - 2 \beta_{5} ) q^{38} + ( -\beta_{1} - \beta_{3} ) q^{40} -2 \beta_{4} q^{41} + ( 2 \beta_{3} - 2 \beta_{4} ) q^{43} + ( \beta_{1} - 2 \beta_{3} - 4 \beta_{4} ) q^{46} + ( 1 - 3 \beta_{2} + \beta_{5} ) q^{47} + ( 11 + \beta_{2} - \beta_{5} ) q^{49} + \beta_{1} q^{50} + ( -6 \beta_{1} - 2 \beta_{3} ) q^{52} + ( 3 - \beta_{2} + \beta_{5} ) q^{53} + ( 12 + 2 \beta_{2} + 4 \beta_{5} ) q^{56} + ( 8 + 2 \beta_{2} ) q^{58} + ( 2 + 4 \beta_{2} - 2 \beta_{5} ) q^{59} + ( 4 \beta_{1} + \beta_{4} ) q^{61} + ( \beta_{1} + 2 \beta_{3} + 8 \beta_{4} ) q^{62} + ( 3 - \beta_{2} + 3 \beta_{5} ) q^{64} + 2 \beta_{1} q^{65} + ( 2 - \beta_{2} - \beta_{5} ) q^{67} + ( 6 \beta_{1} + \beta_{3} ) q^{68} + ( 2 - 2 \beta_{2} - 2 \beta_{5} ) q^{70} + ( 2 \beta_{2} - 2 \beta_{5} ) q^{71} + ( -3 \beta_{1} + \beta_{3} - \beta_{4} ) q^{73} + ( 2 \beta_{1} - 2 \beta_{3} - 4 \beta_{4} ) q^{74} + ( 6 \beta_{1} - 2 \beta_{4} ) q^{76} + ( -2 \beta_{1} + 7 \beta_{4} ) q^{79} + ( -1 - 2 \beta_{2} - \beta_{5} ) q^{80} + ( -2 - 2 \beta_{5} ) q^{82} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{83} + ( -\beta_{1} - \beta_{3} + 2 \beta_{4} ) q^{85} + 6 \beta_{2} q^{86} + ( -8 - 2 \beta_{5} ) q^{89} + ( 4 - 4 \beta_{2} - 4 \beta_{5} ) q^{91} + ( -4 - 3 \beta_{2} - 4 \beta_{5} ) q^{92} + ( -3 \beta_{1} - 2 \beta_{3} + 4 \beta_{4} ) q^{94} + ( -\beta_{1} - \beta_{3} + 3 \beta_{4} ) q^{95} + ( 2 - 5 \beta_{2} - \beta_{5} ) q^{97} + ( 13 \beta_{1} - 4 \beta_{4} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 14q^{4} - 6q^{5} + O(q^{10}) \) \( 6q + 14q^{4} - 6q^{5} - 8q^{14} + 10q^{16} - 14q^{20} + 4q^{23} + 6q^{25} - 52q^{26} + 18q^{31} + 26q^{34} + 10q^{37} + 20q^{38} + 68q^{49} + 16q^{53} + 76q^{56} + 52q^{58} + 20q^{59} + 16q^{64} + 10q^{67} + 8q^{70} + 4q^{71} - 10q^{80} - 12q^{82} + 12q^{86} - 48q^{89} + 16q^{91} - 30q^{92} + 2q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 13 x^{4} + 49 x^{2} - 48\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} - 9 \nu^{3} + 17 \nu \)\()/4\)
\(\beta_{5}\)\(=\)\( \nu^{4} - 8 \nu^{2} + 11 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{5} + 8 \beta_{2} + 21\)
\(\nu^{5}\)\(=\)\(4 \beta_{4} + 9 \beta_{3} + 28 \beta_{1}\)

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
−2.63162
−2.13353
−1.23396
1.23396
2.13353
2.63162
−2.63162 0 4.92542 −1.00000 0 −4.16741 −7.69860 0 2.63162
1.2 −2.13353 0 2.55193 −1.00000 0 4.82155 −1.17756 0 2.13353
1.3 −1.23396 0 −0.477352 −1.00000 0 3.79281 3.05694 0 1.23396
1.4 1.23396 0 −0.477352 −1.00000 0 −3.79281 −3.05694 0 −1.23396
1.5 2.13353 0 2.55193 −1.00000 0 −4.82155 1.17756 0 −2.13353
1.6 2.63162 0 4.92542 −1.00000 0 4.16741 7.69860 0 −2.63162
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.bz 6
3.b odd 2 1 1815.2.a.y 6
11.b odd 2 1 inner 5445.2.a.bz 6
15.d odd 2 1 9075.2.a.dq 6
33.d even 2 1 1815.2.a.y 6
165.d even 2 1 9075.2.a.dq 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.a.y 6 3.b odd 2 1
1815.2.a.y 6 33.d even 2 1
5445.2.a.bz 6 1.a even 1 1 trivial
5445.2.a.bz 6 11.b odd 2 1 inner
9075.2.a.dq 6 15.d odd 2 1
9075.2.a.dq 6 165.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5445))\):

\( T_{2}^{6} - 13 T_{2}^{4} + 49 T_{2}^{2} - 48 \)
\( T_{7}^{6} - 55 T_{7}^{4} + 988 T_{7}^{2} - 5808 \)
\( T_{23}^{3} - 2 T_{23}^{2} - 29 T_{23} + 66 \)
\( T_{53}^{3} - 8 T_{53}^{2} + T_{53} + 54 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -48 + 49 T^{2} - 13 T^{4} + T^{6} \)
$3$ \( T^{6} \)
$5$ \( ( 1 + T )^{6} \)
$7$ \( -5808 + 988 T^{2} - 55 T^{4} + T^{6} \)
$11$ \( T^{6} \)
$13$ \( -3072 + 784 T^{2} - 52 T^{4} + T^{6} \)
$17$ \( -16428 + 2065 T^{2} - 82 T^{4} + T^{6} \)
$19$ \( -17328 + 3448 T^{2} - 115 T^{4} + T^{6} \)
$23$ \( ( 66 - 29 T - 2 T^{2} + T^{3} )^{2} \)
$29$ \( -3072 + 784 T^{2} - 52 T^{4} + T^{6} \)
$31$ \( ( 341 - 45 T - 9 T^{2} + T^{3} )^{2} \)
$37$ \( ( 92 - 22 T - 5 T^{2} + T^{3} )^{2} \)
$41$ \( ( -12 + T^{2} )^{3} \)
$43$ \( -15552 + 6912 T^{2} - 192 T^{4} + T^{6} \)
$47$ \( ( -216 - 69 T + T^{3} )^{2} \)
$53$ \( ( 54 + T - 8 T^{2} + T^{3} )^{2} \)
$59$ \( ( 912 - 116 T - 10 T^{2} + T^{3} )^{2} \)
$61$ \( -309123 + 15691 T^{2} - 241 T^{4} + T^{6} \)
$67$ \( ( 92 - 22 T - 5 T^{2} + T^{3} )^{2} \)
$71$ \( ( -96 - 80 T - 2 T^{2} + T^{3} )^{2} \)
$73$ \( -62208 + 5616 T^{2} - 147 T^{4} + T^{6} \)
$79$ \( -1179387 + 42091 T^{2} - 409 T^{4} + T^{6} \)
$83$ \( -110592 + 8464 T^{2} - 184 T^{4} + T^{6} \)
$89$ \( ( -216 + 120 T + 24 T^{2} + T^{3} )^{2} \)
$97$ \( ( 844 - 226 T - T^{2} + T^{3} )^{2} \)
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