Properties

Label 9075.2.a.ba
Level $9075$
Weight $2$
Character orbit 9075.a
Self dual yes
Analytic conductor $72.464$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} - q^{3} + (\beta - 1) q^{4} + \beta q^{6} + ( - \beta + 1) q^{7} + (2 \beta - 1) q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} - q^{3} + (\beta - 1) q^{4} + \beta q^{6} + ( - \beta + 1) q^{7} + (2 \beta - 1) q^{8} + q^{9} + ( - \beta + 1) q^{12} + ( - \beta + 2) q^{13} + q^{14} - 3 \beta q^{16} + (2 \beta - 3) q^{17} - \beta q^{18} + ( - 2 \beta - 1) q^{19} + (\beta - 1) q^{21} + ( - 3 \beta + 2) q^{23} + ( - 2 \beta + 1) q^{24} + ( - \beta + 1) q^{26} - q^{27} + (\beta - 2) q^{28} + ( - 2 \beta + 3) q^{29} + (3 \beta - 6) q^{31} + ( - \beta + 5) q^{32} + (\beta - 2) q^{34} + (\beta - 1) q^{36} + (8 \beta - 4) q^{37} + (3 \beta + 2) q^{38} + (\beta - 2) q^{39} + (5 \beta - 3) q^{41} - q^{42} + (4 \beta - 2) q^{43} + (\beta + 3) q^{46} + (6 \beta + 2) q^{47} + 3 \beta q^{48} + ( - \beta - 5) q^{49} + ( - 2 \beta + 3) q^{51} + (2 \beta - 3) q^{52} + ( - 5 \beta + 8) q^{53} + \beta q^{54} + (\beta - 3) q^{56} + (2 \beta + 1) q^{57} + ( - \beta + 2) q^{58} + ( - 6 \beta + 5) q^{59} + (2 \beta + 2) q^{61} + (3 \beta - 3) q^{62} + ( - \beta + 1) q^{63} + (2 \beta + 1) q^{64} + ( - 2 \beta - 2) q^{67} + ( - 3 \beta + 5) q^{68} + (3 \beta - 2) q^{69} + ( - 2 \beta - 3) q^{71} + (2 \beta - 1) q^{72} + ( - \beta - 4) q^{73} + ( - 4 \beta - 8) q^{74} + ( - \beta - 1) q^{76} + (\beta - 1) q^{78} + ( - 6 \beta + 6) q^{79} + q^{81} + ( - 2 \beta - 5) q^{82} + ( - 3 \beta + 4) q^{83} + ( - \beta + 2) q^{84} + ( - 2 \beta - 4) q^{86} + (2 \beta - 3) q^{87} + ( - 7 \beta - 3) q^{89} + ( - 2 \beta + 3) q^{91} + (2 \beta - 5) q^{92} + ( - 3 \beta + 6) q^{93} + ( - 8 \beta - 6) q^{94} + (\beta - 5) q^{96} + ( - 10 \beta + 2) q^{97} + (6 \beta + 1) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 2 q^{3} - q^{4} + q^{6} + q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - 2 q^{3} - q^{4} + q^{6} + q^{7} + 2 q^{9} + q^{12} + 3 q^{13} + 2 q^{14} - 3 q^{16} - 4 q^{17} - q^{18} - 4 q^{19} - q^{21} + q^{23} + q^{26} - 2 q^{27} - 3 q^{28} + 4 q^{29} - 9 q^{31} + 9 q^{32} - 3 q^{34} - q^{36} + 7 q^{38} - 3 q^{39} - q^{41} - 2 q^{42} + 7 q^{46} + 10 q^{47} + 3 q^{48} - 11 q^{49} + 4 q^{51} - 4 q^{52} + 11 q^{53} + q^{54} - 5 q^{56} + 4 q^{57} + 3 q^{58} + 4 q^{59} + 6 q^{61} - 3 q^{62} + q^{63} + 4 q^{64} - 6 q^{67} + 7 q^{68} - q^{69} - 8 q^{71} - 9 q^{73} - 20 q^{74} - 3 q^{76} - q^{78} + 6 q^{79} + 2 q^{81} - 12 q^{82} + 5 q^{83} + 3 q^{84} - 10 q^{86} - 4 q^{87} - 13 q^{89} + 4 q^{91} - 8 q^{92} + 9 q^{93} - 20 q^{94} - 9 q^{96} - 6 q^{97} + 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

\( T_{2}^{2} + T_{2} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} - T_{7} - 1 \) Copy content Toggle raw display
\( T_{13}^{2} - 3T_{13} + 1 \) Copy content Toggle raw display
\( T_{17}^{2} + 4T_{17} - 1 \) Copy content Toggle raw display
\( T_{19}^{2} + 4T_{19} - 1 \) Copy content Toggle raw display
\( T_{23}^{2} - T_{23} - 11 \) Copy content Toggle raw display
\( T_{37}^{2} - 80 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 3T + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$23$ \( T^{2} - T - 11 \) Copy content Toggle raw display
$29$ \( T^{2} - 4T - 1 \) Copy content Toggle raw display
$31$ \( T^{2} + 9T + 9 \) Copy content Toggle raw display
$37$ \( T^{2} - 80 \) Copy content Toggle raw display
$41$ \( T^{2} + T - 31 \) Copy content Toggle raw display
$43$ \( T^{2} - 20 \) Copy content Toggle raw display
$47$ \( T^{2} - 10T - 20 \) Copy content Toggle raw display
$53$ \( T^{2} - 11T - 1 \) Copy content Toggle raw display
$59$ \( T^{2} - 4T - 41 \) Copy content Toggle raw display
$61$ \( T^{2} - 6T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$71$ \( T^{2} + 8T + 11 \) Copy content Toggle raw display
$73$ \( T^{2} + 9T + 19 \) Copy content Toggle raw display
$79$ \( T^{2} - 6T - 36 \) Copy content Toggle raw display
$83$ \( T^{2} - 5T - 5 \) Copy content Toggle raw display
$89$ \( T^{2} + 13T - 19 \) Copy content Toggle raw display
$97$ \( T^{2} + 6T - 116 \) Copy content Toggle raw display
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