Properties

Label 5445.2.a.x
Level $5445$
Weight $2$
Character orbit 5445.a
Self dual yes
Analytic conductor $43.479$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( -1 + \beta ) q^{4} + q^{5} + ( -2 + 2 \beta ) q^{7} + ( 1 - 2 \beta ) q^{8} +O(q^{10})\) \( q + \beta q^{2} + ( -1 + \beta ) q^{4} + q^{5} + ( -2 + 2 \beta ) q^{7} + ( 1 - 2 \beta ) q^{8} + \beta q^{10} + ( -4 + 2 \beta ) q^{13} + 2 q^{14} -3 \beta q^{16} + ( -3 + 4 \beta ) q^{17} + ( -4 + 4 \beta ) q^{19} + ( -1 + \beta ) q^{20} + ( 1 - 6 \beta ) q^{23} + q^{25} + ( 2 - 2 \beta ) q^{26} + ( 4 - 2 \beta ) q^{28} + 2 \beta q^{29} + ( -3 + 6 \beta ) q^{31} + ( -5 + \beta ) q^{32} + ( 4 + \beta ) q^{34} + ( -2 + 2 \beta ) q^{35} + ( 2 + 2 \beta ) q^{37} + 4 q^{38} + ( 1 - 2 \beta ) q^{40} + ( 6 + 4 \beta ) q^{41} + ( 4 - 2 \beta ) q^{43} + ( -6 - 5 \beta ) q^{46} + ( -5 + 6 \beta ) q^{47} + ( 1 - 4 \beta ) q^{49} + \beta q^{50} + ( 6 - 4 \beta ) q^{52} + ( -5 + 8 \beta ) q^{53} + ( -6 + 2 \beta ) q^{56} + ( 2 + 2 \beta ) q^{58} + ( -8 + 6 \beta ) q^{59} + ( -1 - 4 \beta ) q^{61} + ( 6 + 3 \beta ) q^{62} + ( 1 + 2 \beta ) q^{64} + ( -4 + 2 \beta ) q^{65} + ( 10 - 2 \beta ) q^{67} + ( 7 - 3 \beta ) q^{68} + 2 q^{70} + ( 12 - 4 \beta ) q^{71} + ( 8 - 4 \beta ) q^{73} + ( 2 + 4 \beta ) q^{74} + ( 8 - 4 \beta ) q^{76} + ( -9 + 6 \beta ) q^{79} -3 \beta q^{80} + ( 4 + 10 \beta ) q^{82} -4 q^{83} + ( -3 + 4 \beta ) q^{85} + ( -2 + 2 \beta ) q^{86} + ( -6 - 2 \beta ) q^{89} + ( 12 - 8 \beta ) q^{91} + ( -7 + \beta ) q^{92} + ( 6 + \beta ) q^{94} + ( -4 + 4 \beta ) q^{95} + ( 14 + 2 \beta ) q^{97} + ( -4 - 3 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{4} + 2q^{5} - 2q^{7} + O(q^{10}) \) \( 2q + q^{2} - q^{4} + 2q^{5} - 2q^{7} + q^{10} - 6q^{13} + 4q^{14} - 3q^{16} - 2q^{17} - 4q^{19} - q^{20} - 4q^{23} + 2q^{25} + 2q^{26} + 6q^{28} + 2q^{29} - 9q^{32} + 9q^{34} - 2q^{35} + 6q^{37} + 8q^{38} + 16q^{41} + 6q^{43} - 17q^{46} - 4q^{47} - 2q^{49} + q^{50} + 8q^{52} - 2q^{53} - 10q^{56} + 6q^{58} - 10q^{59} - 6q^{61} + 15q^{62} + 4q^{64} - 6q^{65} + 18q^{67} + 11q^{68} + 4q^{70} + 20q^{71} + 12q^{73} + 8q^{74} + 12q^{76} - 12q^{79} - 3q^{80} + 18q^{82} - 8q^{83} - 2q^{85} - 2q^{86} - 14q^{89} + 16q^{91} - 13q^{92} + 13q^{94} - 4q^{95} + 30q^{97} - 11q^{98} + O(q^{100}) \)

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
−0.618034
1.61803
−0.618034 0 −1.61803 1.00000 0 −3.23607 2.23607 0 −0.618034
1.2 1.61803 0 0.618034 1.00000 0 1.23607 −2.23607 0 1.61803
\(n\): 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.x 2
3.b odd 2 1 1815.2.a.f 2
11.b odd 2 1 5445.2.a.o 2
15.d odd 2 1 9075.2.a.bx 2
33.d even 2 1 1815.2.a.j yes 2
165.d even 2 1 9075.2.a.bd 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.a.f 2 3.b odd 2 1
1815.2.a.j yes 2 33.d even 2 1
5445.2.a.o 2 11.b odd 2 1
5445.2.a.x 2 1.a even 1 1 trivial
9075.2.a.bd 2 165.d even 2 1
9075.2.a.bx 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5445))\):

\( T_{2}^{2} - T_{2} - 1 \)
\( T_{7}^{2} + 2 T_{7} - 4 \)
\( T_{23}^{2} + 4 T_{23} - 41 \)
\( T_{53}^{2} + 2 T_{53} - 79 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 - T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( -4 + 2 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 4 + 6 T + T^{2} \)
$17$ \( -19 + 2 T + T^{2} \)
$19$ \( -16 + 4 T + T^{2} \)
$23$ \( -41 + 4 T + T^{2} \)
$29$ \( -4 - 2 T + T^{2} \)
$31$ \( -45 + T^{2} \)
$37$ \( 4 - 6 T + T^{2} \)
$41$ \( 44 - 16 T + T^{2} \)
$43$ \( 4 - 6 T + T^{2} \)
$47$ \( -41 + 4 T + T^{2} \)
$53$ \( -79 + 2 T + T^{2} \)
$59$ \( -20 + 10 T + T^{2} \)
$61$ \( -11 + 6 T + T^{2} \)
$67$ \( 76 - 18 T + T^{2} \)
$71$ \( 80 - 20 T + T^{2} \)
$73$ \( 16 - 12 T + T^{2} \)
$79$ \( -9 + 12 T + T^{2} \)
$83$ \( ( 4 + T )^{2} \)
$89$ \( 44 + 14 T + T^{2} \)
$97$ \( 220 - 30 T + T^{2} \)
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