Properties

Label 9075.2.a.bc
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: 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.

\(f(q)\) \(=\) \( q - \beta q^{2} + q^{3} + (\beta - 1) q^{4} - \beta q^{6} + ( - 2 \beta - 2) q^{7} + (2 \beta - 1) q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + q^{3} + (\beta - 1) q^{4} - \beta q^{6} + ( - 2 \beta - 2) q^{7} + (2 \beta - 1) q^{8} + q^{9} + (\beta - 1) q^{12} + 2 \beta q^{13} + (4 \beta + 2) q^{14} - 3 \beta q^{16} + 2 q^{17} - \beta q^{18} - 5 q^{19} + ( - 2 \beta - 2) q^{21} + ( - \beta - 3) q^{23} + (2 \beta - 1) q^{24} + ( - 2 \beta - 2) q^{26} + q^{27} - 2 \beta q^{28} + ( - 3 \beta + 4) q^{29} + 7 q^{31} + ( - \beta + 5) q^{32} - 2 \beta q^{34} + (\beta - 1) q^{36} + ( - 4 \beta + 5) q^{37} + 5 \beta q^{38} + 2 \beta q^{39} + ( - 4 \beta - 5) q^{41} + (4 \beta + 2) q^{42} + (5 \beta + 1) q^{43} + (4 \beta + 1) q^{46} + (5 \beta - 2) q^{47} - 3 \beta q^{48} + (12 \beta + 1) q^{49} + 2 q^{51} + 2 q^{52} + ( - \beta + 7) q^{53} - \beta q^{54} + ( - 6 \beta - 2) q^{56} - 5 q^{57} + ( - \beta + 3) q^{58} + (6 \beta - 3) q^{59} - 7 q^{61} - 7 \beta q^{62} + ( - 2 \beta - 2) q^{63} + (2 \beta + 1) q^{64} + (\beta + 10) q^{67} + (2 \beta - 2) q^{68} + ( - \beta - 3) q^{69} - 8 q^{71} + (2 \beta - 1) q^{72} + (4 \beta - 11) q^{73} + ( - \beta + 4) q^{74} + ( - 5 \beta + 5) q^{76} + ( - 2 \beta - 2) q^{78} + ( - 5 \beta + 5) q^{79} + q^{81} + (9 \beta + 4) q^{82} + (6 \beta - 2) q^{83} - 2 \beta q^{84} + ( - 6 \beta - 5) q^{86} + ( - 3 \beta + 4) q^{87} + ( - 3 \beta + 9) q^{89} + ( - 8 \beta - 4) q^{91} + ( - 3 \beta + 2) q^{92} + 7 q^{93} + ( - 3 \beta - 5) q^{94} + ( - \beta + 5) q^{96} + (4 \beta - 9) q^{97} + ( - 13 \beta - 12) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 2 q^{3} - q^{4} - q^{6} - 6 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 2 q^{3} - q^{4} - q^{6} - 6 q^{7} + 2 q^{9} - q^{12} + 2 q^{13} + 8 q^{14} - 3 q^{16} + 4 q^{17} - q^{18} - 10 q^{19} - 6 q^{21} - 7 q^{23} - 6 q^{26} + 2 q^{27} - 2 q^{28} + 5 q^{29} + 14 q^{31} + 9 q^{32} - 2 q^{34} - q^{36} + 6 q^{37} + 5 q^{38} + 2 q^{39} - 14 q^{41} + 8 q^{42} + 7 q^{43} + 6 q^{46} + q^{47} - 3 q^{48} + 14 q^{49} + 4 q^{51} + 4 q^{52} + 13 q^{53} - q^{54} - 10 q^{56} - 10 q^{57} + 5 q^{58} - 14 q^{61} - 7 q^{62} - 6 q^{63} + 4 q^{64} + 21 q^{67} - 2 q^{68} - 7 q^{69} - 16 q^{71} - 18 q^{73} + 7 q^{74} + 5 q^{76} - 6 q^{78} + 5 q^{79} + 2 q^{81} + 17 q^{82} + 2 q^{83} - 2 q^{84} - 16 q^{86} + 5 q^{87} + 15 q^{89} - 16 q^{91} + q^{92} + 14 q^{93} - 13 q^{94} + 9 q^{96} - 14 q^{97} - 37 q^{98}+O(q^{100}) \) 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 −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.bc 2
5.b even 2 1 9075.2.a.bt 2
11.b odd 2 1 9075.2.a.by 2
11.d odd 10 2 825.2.n.d yes 4
55.d odd 2 1 9075.2.a.z 2
55.h odd 10 2 825.2.n.b 4
55.l even 20 4 825.2.bx.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.n.b 4 55.h odd 10 2
825.2.n.d yes 4 11.d odd 10 2
825.2.bx.c 8 55.l even 20 4
9075.2.a.z 2 55.d odd 2 1
9075.2.a.bc 2 1.a even 1 1 trivial
9075.2.a.bt 2 5.b even 2 1
9075.2.a.by 2 11.b odd 2 1

Hecke kernels

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

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