Properties

Label 9075.2.a.bx
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: \(x^{2} - x - 1\)
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.

\(f(q)\) \(=\) \( q + \beta q^{2} + q^{3} + ( -1 + \beta ) q^{4} + \beta q^{6} + ( 2 - 2 \beta ) q^{7} + ( 1 - 2 \beta ) q^{8} + q^{9} +O(q^{10})\) \( q + \beta q^{2} + q^{3} + ( -1 + \beta ) q^{4} + \beta q^{6} + ( 2 - 2 \beta ) q^{7} + ( 1 - 2 \beta ) q^{8} + q^{9} + ( -1 + \beta ) q^{12} + ( 4 - 2 \beta ) q^{13} -2 q^{14} -3 \beta q^{16} + ( -3 + 4 \beta ) q^{17} + \beta q^{18} + ( -4 + 4 \beta ) q^{19} + ( 2 - 2 \beta ) q^{21} + ( 1 - 6 \beta ) q^{23} + ( 1 - 2 \beta ) q^{24} + ( -2 + 2 \beta ) q^{26} + q^{27} + ( -4 + 2 \beta ) q^{28} -2 \beta q^{29} + ( -3 + 6 \beta ) q^{31} + ( -5 + \beta ) q^{32} + ( 4 + \beta ) q^{34} + ( -1 + \beta ) q^{36} + ( -2 - 2 \beta ) q^{37} + 4 q^{38} + ( 4 - 2 \beta ) q^{39} + ( -6 - 4 \beta ) q^{41} -2 q^{42} + ( -4 + 2 \beta ) q^{43} + ( -6 - 5 \beta ) q^{46} + ( -5 + 6 \beta ) q^{47} -3 \beta q^{48} + ( 1 - 4 \beta ) q^{49} + ( -3 + 4 \beta ) q^{51} + ( -6 + 4 \beta ) q^{52} + ( -5 + 8 \beta ) q^{53} + \beta q^{54} + ( 6 - 2 \beta ) q^{56} + ( -4 + 4 \beta ) q^{57} + ( -2 - 2 \beta ) q^{58} + ( 8 - 6 \beta ) q^{59} + ( -1 - 4 \beta ) q^{61} + ( 6 + 3 \beta ) q^{62} + ( 2 - 2 \beta ) q^{63} + ( 1 + 2 \beta ) q^{64} + ( -10 + 2 \beta ) q^{67} + ( 7 - 3 \beta ) q^{68} + ( 1 - 6 \beta ) q^{69} + ( -12 + 4 \beta ) q^{71} + ( 1 - 2 \beta ) q^{72} + ( -8 + 4 \beta ) q^{73} + ( -2 - 4 \beta ) q^{74} + ( 8 - 4 \beta ) q^{76} + ( -2 + 2 \beta ) q^{78} + ( -9 + 6 \beta ) q^{79} + q^{81} + ( -4 - 10 \beta ) q^{82} -4 q^{83} + ( -4 + 2 \beta ) q^{84} + ( 2 - 2 \beta ) q^{86} -2 \beta q^{87} + ( 6 + 2 \beta ) q^{89} + ( 12 - 8 \beta ) q^{91} + ( -7 + \beta ) q^{92} + ( -3 + 6 \beta ) q^{93} + ( 6 + \beta ) q^{94} + ( -5 + \beta ) q^{96} + ( -14 - 2 \beta ) q^{97} + ( -4 - 3 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + 2q^{3} - q^{4} + q^{6} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + q^{2} + 2q^{3} - q^{4} + q^{6} + 2q^{7} + 2q^{9} - q^{12} + 6q^{13} - 4q^{14} - 3q^{16} - 2q^{17} + q^{18} - 4q^{19} + 2q^{21} - 4q^{23} - 2q^{26} + 2q^{27} - 6q^{28} - 2q^{29} - 9q^{32} + 9q^{34} - q^{36} - 6q^{37} + 8q^{38} + 6q^{39} - 16q^{41} - 4q^{42} - 6q^{43} - 17q^{46} - 4q^{47} - 3q^{48} - 2q^{49} - 2q^{51} - 8q^{52} - 2q^{53} + q^{54} + 10q^{56} - 4q^{57} - 6q^{58} + 10q^{59} - 6q^{61} + 15q^{62} + 2q^{63} + 4q^{64} - 18q^{67} + 11q^{68} - 4q^{69} - 20q^{71} - 12q^{73} - 8q^{74} + 12q^{76} - 2q^{78} - 12q^{79} + 2q^{81} - 18q^{82} - 8q^{83} - 6q^{84} + 2q^{86} - 2q^{87} + 14q^{89} + 16q^{91} - 13q^{92} + 13q^{94} - 9q^{96} - 30q^{97} - 11q^{98} + O(q^{100}) \)

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 1.00000 −1.61803 0 −0.618034 3.23607 2.23607 1.00000 0
1.2 1.61803 1.00000 0.618034 0 1.61803 −1.23607 −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

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.bx 2
5.b even 2 1 1815.2.a.f 2
11.b odd 2 1 9075.2.a.bd 2
15.d odd 2 1 5445.2.a.x 2
55.d odd 2 1 1815.2.a.j yes 2
165.d even 2 1 5445.2.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.a.f 2 5.b even 2 1
1815.2.a.j yes 2 55.d odd 2 1
5445.2.a.o 2 165.d even 2 1
5445.2.a.x 2 15.d odd 2 1
9075.2.a.bd 2 11.b odd 2 1
9075.2.a.bx 2 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2}^{2} - T_{2} - 1 \)
\( T_{7}^{2} - 2 T_{7} - 4 \)
\( T_{13}^{2} - 6 T_{13} + 4 \)
\( T_{17}^{2} + 2 T_{17} - 19 \)
\( T_{19}^{2} + 4 T_{19} - 16 \)
\( T_{23}^{2} + 4 T_{23} - 41 \)
\( T_{37}^{2} + 6 T_{37} + 4 \)

Hecke characteristic polynomials

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