Properties

Label 5445.2.a.z
Level $5445$
Weight $2$
Character orbit 5445.a
Self dual yes
Analytic conductor $43.479$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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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: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 1 + \beta_{1} + \beta_{2} ) q^{4} - q^{5} + ( -\beta_{1} + \beta_{2} ) q^{7} + ( -2 - 2 \beta_{1} - \beta_{2} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 1 + \beta_{1} + \beta_{2} ) q^{4} - q^{5} + ( -\beta_{1} + \beta_{2} ) q^{7} + ( -2 - 2 \beta_{1} - \beta_{2} ) q^{8} + \beta_{1} q^{10} + ( \beta_{1} + \beta_{2} ) q^{13} + ( 4 - \beta_{1} + \beta_{2} ) q^{14} + ( 3 + 4 \beta_{1} ) q^{16} + ( -2 + 3 \beta_{1} + \beta_{2} ) q^{17} + ( -2 - 2 \beta_{2} ) q^{19} + ( -1 - \beta_{1} - \beta_{2} ) q^{20} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{23} + q^{25} + ( -2 - 3 \beta_{1} - \beta_{2} ) q^{26} + ( 4 - 3 \beta_{1} - \beta_{2} ) q^{28} + ( -4 + 2 \beta_{1} ) q^{29} + ( 4 - 2 \beta_{1} - 2 \beta_{2} ) q^{31} + ( -8 - 3 \beta_{1} - 2 \beta_{2} ) q^{32} + ( -8 - 3 \beta_{1} - 3 \beta_{2} ) q^{34} + ( \beta_{1} - \beta_{2} ) q^{35} -2 q^{37} + ( -2 + 6 \beta_{1} ) q^{38} + ( 2 + 2 \beta_{1} + \beta_{2} ) q^{40} + ( -4 - 2 \beta_{1} ) q^{41} + ( -\beta_{1} - 3 \beta_{2} ) q^{43} + ( -8 + 2 \beta_{1} - 2 \beta_{2} ) q^{46} + ( 4 - 2 \beta_{1} - 2 \beta_{2} ) q^{47} + ( 5 - 4 \beta_{1} ) q^{49} -\beta_{1} q^{50} + ( 8 + 5 \beta_{1} + \beta_{2} ) q^{52} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{53} + ( 3 \beta_{1} + \beta_{2} ) q^{56} + ( -6 + 2 \beta_{1} - 2 \beta_{2} ) q^{58} + ( -4 - 2 \beta_{1} + 2 \beta_{2} ) q^{59} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{61} + ( 4 + 2 \beta_{1} + 2 \beta_{2} ) q^{62} + ( 1 + 7 \beta_{1} + 3 \beta_{2} ) q^{64} + ( -\beta_{1} - \beta_{2} ) q^{65} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{67} + ( 10 + 11 \beta_{1} + \beta_{2} ) q^{68} + ( -4 + \beta_{1} - \beta_{2} ) q^{70} + ( -4 + 2 \beta_{1} + 2 \beta_{2} ) q^{71} + ( 4 + 3 \beta_{1} - \beta_{2} ) q^{73} + 2 \beta_{1} q^{74} + ( -14 - 4 \beta_{1} - 2 \beta_{2} ) q^{76} + ( -6 + 4 \beta_{1} + 2 \beta_{2} ) q^{79} + ( -3 - 4 \beta_{1} ) q^{80} + ( 6 + 6 \beta_{1} + 2 \beta_{2} ) q^{82} + ( 2 - 3 \beta_{1} - 3 \beta_{2} ) q^{83} + ( 2 - 3 \beta_{1} - \beta_{2} ) q^{85} + ( 7 \beta_{1} + \beta_{2} ) q^{86} + ( 2 + 4 \beta_{1} ) q^{89} + ( 4 - 2 \beta_{1} - 2 \beta_{2} ) q^{91} + ( -8 + 6 \beta_{1} + 2 \beta_{2} ) q^{92} + ( 4 + 2 \beta_{1} + 2 \beta_{2} ) q^{94} + ( 2 + 2 \beta_{2} ) q^{95} + ( 6 + 2 \beta_{1} + 2 \beta_{2} ) q^{97} + ( 12 - \beta_{1} + 4 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{2} + 5q^{4} - 3q^{5} - 9q^{8} + O(q^{10}) \) \( 3q - q^{2} + 5q^{4} - 3q^{5} - 9q^{8} + q^{10} + 2q^{13} + 12q^{14} + 13q^{16} - 2q^{17} - 8q^{19} - 5q^{20} + 3q^{25} - 10q^{26} + 8q^{28} - 10q^{29} + 8q^{31} - 29q^{32} - 30q^{34} - 6q^{37} + 9q^{40} - 14q^{41} - 4q^{43} - 24q^{46} + 8q^{47} + 11q^{49} - q^{50} + 30q^{52} + 6q^{53} + 4q^{56} - 18q^{58} - 12q^{59} + 6q^{61} + 16q^{62} + 13q^{64} - 2q^{65} - 4q^{67} + 42q^{68} - 12q^{70} - 8q^{71} + 14q^{73} + 2q^{74} - 48q^{76} - 12q^{79} - 13q^{80} + 26q^{82} + 2q^{85} + 8q^{86} + 10q^{89} + 8q^{91} - 16q^{92} + 16q^{94} + 8q^{95} + 22q^{97} + 39q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - 2 \)
\(\beta_{2}\)\(=\)\( -\nu^{2} + 2 \nu + 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{1} + 2\)

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
2.17009
−1.48119
0.311108
−2.70928 0 5.34017 −1.00000 0 −1.07838 −9.04945 0 2.70928
1.2 −0.193937 0 −1.96239 −1.00000 0 −3.35026 0.768452 0 0.193937
1.3 1.90321 0 1.62222 −1.00000 0 4.42864 −0.719004 0 −1.90321
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 5445.2.a.z 3
3.b odd 2 1 1815.2.a.m 3
11.b odd 2 1 495.2.a.e 3
15.d odd 2 1 9075.2.a.cf 3
33.d even 2 1 165.2.a.c 3
44.c even 2 1 7920.2.a.cj 3
55.d odd 2 1 2475.2.a.bb 3
55.e even 4 2 2475.2.c.r 6
132.d odd 2 1 2640.2.a.be 3
165.d even 2 1 825.2.a.k 3
165.l odd 4 2 825.2.c.g 6
231.h odd 2 1 8085.2.a.bk 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.c 3 33.d even 2 1
495.2.a.e 3 11.b odd 2 1
825.2.a.k 3 165.d even 2 1
825.2.c.g 6 165.l odd 4 2
1815.2.a.m 3 3.b odd 2 1
2475.2.a.bb 3 55.d odd 2 1
2475.2.c.r 6 55.e even 4 2
2640.2.a.be 3 132.d odd 2 1
5445.2.a.z 3 1.a even 1 1 trivial
7920.2.a.cj 3 44.c even 2 1
8085.2.a.bk 3 231.h odd 2 1
9075.2.a.cf 3 15.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5445))\):

\( T_{2}^{3} + T_{2}^{2} - 5 T_{2} - 1 \)
\( T_{7}^{3} - 16 T_{7} - 16 \)
\( T_{23}^{3} - 64 T_{23} + 128 \)
\( T_{53}^{3} - 6 T_{53}^{2} - 52 T_{53} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 - 5 T + T^{2} + T^{3} \)
$3$ \( T^{3} \)
$5$ \( ( 1 + T )^{3} \)
$7$ \( -16 - 16 T + T^{3} \)
$11$ \( T^{3} \)
$13$ \( 8 - 12 T - 2 T^{2} + T^{3} \)
$17$ \( -184 - 52 T + 2 T^{2} + T^{3} \)
$19$ \( -160 - 16 T + 8 T^{2} + T^{3} \)
$23$ \( 128 - 64 T + T^{3} \)
$29$ \( -40 + 12 T + 10 T^{2} + T^{3} \)
$31$ \( 128 - 32 T - 8 T^{2} + T^{3} \)
$37$ \( ( 2 + T )^{3} \)
$41$ \( 8 + 44 T + 14 T^{2} + T^{3} \)
$43$ \( -400 - 80 T + 4 T^{2} + T^{3} \)
$47$ \( 128 - 32 T - 8 T^{2} + T^{3} \)
$53$ \( -8 - 52 T - 6 T^{2} + T^{3} \)
$59$ \( -320 - 16 T + 12 T^{2} + T^{3} \)
$61$ \( 248 - 52 T - 6 T^{2} + T^{3} \)
$67$ \( -64 - 48 T + 4 T^{2} + T^{3} \)
$71$ \( -128 - 32 T + 8 T^{2} + T^{3} \)
$73$ \( 344 + 4 T - 14 T^{2} + T^{3} \)
$79$ \( -800 - 64 T + 12 T^{2} + T^{3} \)
$83$ \( 16 - 120 T + T^{3} \)
$89$ \( 200 - 52 T - 10 T^{2} + T^{3} \)
$97$ \( -8 + 108 T - 22 T^{2} + T^{3} \)
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