Properties

Label 9075.2.a.z
Level $9075$
Weight $2$
Character orbit 9075.a
Self dual yes
Analytic conductor $72.464$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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: \( 1 \)
Twist minimal: no (minimal twist has level 825)
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.

\(f(q)\) \(=\) \( q -\beta q^{2} - q^{3} + ( -1 + \beta ) q^{4} + \beta q^{6} + ( -2 - 2 \beta ) q^{7} + ( -1 + 2 \beta ) q^{8} + q^{9} +O(q^{10})\) \( q -\beta q^{2} - q^{3} + ( -1 + \beta ) q^{4} + \beta q^{6} + ( -2 - 2 \beta ) q^{7} + ( -1 + 2 \beta ) q^{8} + q^{9} + ( 1 - \beta ) q^{12} + 2 \beta q^{13} + ( 2 + 4 \beta ) q^{14} -3 \beta q^{16} + 2 q^{17} -\beta q^{18} + 5 q^{19} + ( 2 + 2 \beta ) q^{21} + ( 3 + \beta ) q^{23} + ( 1 - 2 \beta ) q^{24} + ( -2 - 2 \beta ) q^{26} - q^{27} -2 \beta q^{28} + ( -4 + 3 \beta ) q^{29} + 7 q^{31} + ( 5 - \beta ) q^{32} -2 \beta q^{34} + ( -1 + \beta ) q^{36} + ( -5 + 4 \beta ) q^{37} -5 \beta q^{38} -2 \beta q^{39} + ( 5 + 4 \beta ) q^{41} + ( -2 - 4 \beta ) q^{42} + ( 1 + 5 \beta ) q^{43} + ( -1 - 4 \beta ) q^{46} + ( 2 - 5 \beta ) q^{47} + 3 \beta q^{48} + ( 1 + 12 \beta ) q^{49} -2 q^{51} + 2 q^{52} + ( -7 + \beta ) q^{53} + \beta q^{54} + ( -2 - 6 \beta ) q^{56} -5 q^{57} + ( -3 + \beta ) q^{58} + ( -3 + 6 \beta ) q^{59} + 7 q^{61} -7 \beta q^{62} + ( -2 - 2 \beta ) q^{63} + ( 1 + 2 \beta ) q^{64} + ( -10 - \beta ) q^{67} + ( -2 + 2 \beta ) q^{68} + ( -3 - \beta ) q^{69} -8 q^{71} + ( -1 + 2 \beta ) q^{72} + ( -11 + 4 \beta ) q^{73} + ( -4 + \beta ) q^{74} + ( -5 + 5 \beta ) q^{76} + ( 2 + 2 \beta ) q^{78} + ( -5 + 5 \beta ) q^{79} + q^{81} + ( -4 - 9 \beta ) q^{82} + ( -2 + 6 \beta ) q^{83} + 2 \beta q^{84} + ( -5 - 6 \beta ) q^{86} + ( 4 - 3 \beta ) q^{87} + ( 9 - 3 \beta ) q^{89} + ( -4 - 8 \beta ) q^{91} + ( -2 + 3 \beta ) q^{92} -7 q^{93} + ( 5 + 3 \beta ) q^{94} + ( -5 + \beta ) q^{96} + ( 9 - 4 \beta ) q^{97} + ( -12 - 13 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 2 q^{3} - q^{4} + q^{6} - 6 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q - q^{2} - 2 q^{3} - q^{4} + q^{6} - 6 q^{7} + 2 q^{9} + q^{12} + 2 q^{13} + 8 q^{14} - 3 q^{16} + 4 q^{17} - q^{18} + 10 q^{19} + 6 q^{21} + 7 q^{23} - 6 q^{26} - 2 q^{27} - 2 q^{28} - 5 q^{29} + 14 q^{31} + 9 q^{32} - 2 q^{34} - q^{36} - 6 q^{37} - 5 q^{38} - 2 q^{39} + 14 q^{41} - 8 q^{42} + 7 q^{43} - 6 q^{46} - q^{47} + 3 q^{48} + 14 q^{49} - 4 q^{51} + 4 q^{52} - 13 q^{53} + q^{54} - 10 q^{56} - 10 q^{57} - 5 q^{58} + 14 q^{61} - 7 q^{62} - 6 q^{63} + 4 q^{64} - 21 q^{67} - 2 q^{68} - 7 q^{69} - 16 q^{71} - 18 q^{73} - 7 q^{74} - 5 q^{76} + 6 q^{78} - 5 q^{79} + 2 q^{81} - 17 q^{82} + 2 q^{83} + 2 q^{84} - 16 q^{86} + 5 q^{87} + 15 q^{89} - 16 q^{91} - q^{92} - 14 q^{93} + 13 q^{94} - 9 q^{96} + 14 q^{97} - 37 q^{98} + O(q^{100}) \)

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
1.61803
−0.618034
−1.61803 −1.00000 0.618034 0 1.61803 −5.23607 2.23607 1.00000 0
1.2 0.618034 −1.00000 −1.61803 0 −0.618034 −0.763932 −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

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 9075.2.a.z 2
5.b even 2 1 9075.2.a.by 2
11.b odd 2 1 9075.2.a.bt 2
11.c even 5 2 825.2.n.b 4
55.d odd 2 1 9075.2.a.bc 2
55.j even 10 2 825.2.n.d yes 4
55.k odd 20 4 825.2.bx.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.n.b 4 11.c even 5 2
825.2.n.d yes 4 55.j even 10 2
825.2.bx.c 8 55.k odd 20 4
9075.2.a.z 2 1.a even 1 1 trivial
9075.2.a.bc 2 55.d odd 2 1
9075.2.a.bt 2 11.b odd 2 1
9075.2.a.by 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9075))\):

\( T_{2}^{2} + T_{2} - 1 \)
\( T_{7}^{2} + 6 T_{7} + 4 \)
\( T_{13}^{2} - 2 T_{13} - 4 \)
\( T_{17} - 2 \)
\( T_{19} - 5 \)
\( T_{23}^{2} - 7 T_{23} + 11 \)
\( T_{37}^{2} + 6 T_{37} - 11 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T + T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( 4 + 6 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( -4 - 2 T + T^{2} \)
$17$ \( ( -2 + T )^{2} \)
$19$ \( ( -5 + T )^{2} \)
$23$ \( 11 - 7 T + T^{2} \)
$29$ \( -5 + 5 T + T^{2} \)
$31$ \( ( -7 + T )^{2} \)
$37$ \( -11 + 6 T + T^{2} \)
$41$ \( 29 - 14 T + T^{2} \)
$43$ \( -19 - 7 T + T^{2} \)
$47$ \( -31 + T + T^{2} \)
$53$ \( 41 + 13 T + T^{2} \)
$59$ \( -45 + T^{2} \)
$61$ \( ( -7 + T )^{2} \)
$67$ \( 109 + 21 T + T^{2} \)
$71$ \( ( 8 + T )^{2} \)
$73$ \( 61 + 18 T + T^{2} \)
$79$ \( -25 + 5 T + T^{2} \)
$83$ \( -44 - 2 T + T^{2} \)
$89$ \( 45 - 15 T + T^{2} \)
$97$ \( 29 - 14 T + T^{2} \)
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