Properties

Label 9075.2.a.bk
Level $9075$
Weight $2$
Character orbit 9075.a
Self dual yes
Analytic conductor $72.464$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9075.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.4642398343\)
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: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} - q^{3} + 3 q^{4} + \beta q^{6} -\beta q^{7} -\beta q^{8} + q^{9} +O(q^{10})\) \( q -\beta q^{2} - q^{3} + 3 q^{4} + \beta q^{6} -\beta q^{7} -\beta q^{8} + q^{9} -3 q^{12} -2 \beta q^{13} + 5 q^{14} - q^{16} -\beta q^{17} -\beta q^{18} + \beta q^{19} + \beta q^{21} - q^{23} + \beta q^{24} + 10 q^{26} - q^{27} -3 \beta q^{28} -2 \beta q^{29} + 10 q^{31} + 3 \beta q^{32} + 5 q^{34} + 3 q^{36} -7 q^{37} -5 q^{38} + 2 \beta q^{39} -3 \beta q^{41} -5 q^{42} + \beta q^{46} -3 q^{47} + q^{48} -2 q^{49} + \beta q^{51} -6 \beta q^{52} + 14 q^{53} + \beta q^{54} + 5 q^{56} -\beta q^{57} + 10 q^{58} -5 q^{59} -2 \beta q^{61} -10 \beta q^{62} -\beta q^{63} -13 q^{64} + 2 q^{67} -3 \beta q^{68} + q^{69} -13 q^{71} -\beta q^{72} -6 \beta q^{73} + 7 \beta q^{74} + 3 \beta q^{76} -10 q^{78} + 7 \beta q^{79} + q^{81} + 15 q^{82} -2 \beta q^{83} + 3 \beta q^{84} + 2 \beta q^{87} + 6 q^{89} + 10 q^{91} -3 q^{92} -10 q^{93} + 3 \beta q^{94} -3 \beta q^{96} -17 q^{97} + 2 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 6 q^{4} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{3} + 6 q^{4} + 2 q^{9} - 6 q^{12} + 10 q^{14} - 2 q^{16} - 2 q^{23} + 20 q^{26} - 2 q^{27} + 20 q^{31} + 10 q^{34} + 6 q^{36} - 14 q^{37} - 10 q^{38} - 10 q^{42} - 6 q^{47} + 2 q^{48} - 4 q^{49} + 28 q^{53} + 10 q^{56} + 20 q^{58} - 10 q^{59} - 26 q^{64} + 4 q^{67} + 2 q^{69} - 26 q^{71} - 20 q^{78} + 2 q^{81} + 30 q^{82} + 12 q^{89} + 20 q^{91} - 6 q^{92} - 20 q^{93} - 34 q^{97} + O(q^{100}) \)

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
1.61803
−0.618034
−2.23607 −1.00000 3.00000 0 2.23607 −2.23607 −2.23607 1.00000 0
1.2 2.23607 −1.00000 3.00000 0 −2.23607 2.23607 2.23607 1.00000 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 9075.2.a.bk 2
5.b even 2 1 9075.2.a.br yes 2
11.b odd 2 1 inner 9075.2.a.bk 2
55.d odd 2 1 9075.2.a.br yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9075.2.a.bk 2 1.a even 1 1 trivial
9075.2.a.bk 2 11.b odd 2 1 inner
9075.2.a.br yes 2 5.b even 2 1
9075.2.a.br yes 2 55.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9075))\):

\( T_{2}^{2} - 5 \)
\( T_{7}^{2} - 5 \)
\( T_{13}^{2} - 20 \)
\( T_{17}^{2} - 5 \)
\( T_{19}^{2} - 5 \)
\( T_{23} + 1 \)
\( T_{37} + 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -5 + T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( -5 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( -20 + T^{2} \)
$17$ \( -5 + T^{2} \)
$19$ \( -5 + T^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( -20 + T^{2} \)
$31$ \( ( -10 + T )^{2} \)
$37$ \( ( 7 + T )^{2} \)
$41$ \( -45 + T^{2} \)
$43$ \( T^{2} \)
$47$ \( ( 3 + T )^{2} \)
$53$ \( ( -14 + T )^{2} \)
$59$ \( ( 5 + T )^{2} \)
$61$ \( -20 + T^{2} \)
$67$ \( ( -2 + T )^{2} \)
$71$ \( ( 13 + T )^{2} \)
$73$ \( -180 + T^{2} \)
$79$ \( -245 + T^{2} \)
$83$ \( -20 + T^{2} \)
$89$ \( ( -6 + T )^{2} \)
$97$ \( ( 17 + T )^{2} \)
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