Properties

Label 9075.2.a.cd
Level $9075$
Weight $2$
Character orbit 9075.a
Self dual yes
Analytic conductor $72.464$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) 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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + q^{3} + (\beta_{2} + 3) q^{4} + (\beta_1 - 1) q^{6} + (\beta_{2} + \beta_1 + 1) q^{7} + ( - 2 \beta_{2} + \beta_1 - 3) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + q^{3} + (\beta_{2} + 3) q^{4} + (\beta_1 - 1) q^{6} + (\beta_{2} + \beta_1 + 1) q^{7} + ( - 2 \beta_{2} + \beta_1 - 3) q^{8} + q^{9} + (\beta_{2} + 3) q^{12} + (\beta_{2} + 2) q^{13} + ( - \beta_{2} + 2 \beta_1 + 1) q^{14} + (3 \beta_{2} - 2 \beta_1 + 5) q^{16} + (2 \beta_{2} + \beta_1 - 1) q^{17} + (\beta_1 - 1) q^{18} + (\beta_{2} - \beta_1 + 1) q^{19} + (\beta_{2} + \beta_1 + 1) q^{21} + ( - \beta_{2} + 2 \beta_1 - 1) q^{23} + ( - 2 \beta_{2} + \beta_1 - 3) q^{24} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{26} + q^{27} + (2 \beta_{2} + \beta_1 + 7) q^{28} - 2 \beta_1 q^{29} + (\beta_{2} + 6) q^{31} + ( - 4 \beta_{2} + \beta_1 - 13) q^{32} + ( - 3 \beta_{2} + 1) q^{34} + (\beta_{2} + 3) q^{36} + (3 \beta_{2} + 1) q^{37} + ( - 3 \beta_{2} - 7) q^{38} + (\beta_{2} + 2) q^{39} + ( - \beta_1 - 1) q^{41} + ( - \beta_{2} + 2 \beta_1 + 1) q^{42} + ( - 3 \beta_{2} + 2 \beta_1 + 4) q^{43} + (4 \beta_{2} + \beta_1 + 11) q^{46} + ( - \beta_{2} + 3) q^{47} + (3 \beta_{2} - 2 \beta_1 + 5) q^{48} + 2 \beta_1 q^{49} + (2 \beta_{2} + \beta_1 - 1) q^{51} + (4 \beta_{2} - 2 \beta_1 + 12) q^{52} + (4 \beta_{2} + 4 \beta_1 + 2) q^{53} + (\beta_1 - 1) q^{54} + ( - \beta_{2} + 4 \beta_1 - 9) q^{56} + (\beta_{2} - \beta_1 + 1) q^{57} + ( - 2 \beta_{2} - 2 \beta_1 - 8) q^{58} + (\beta_{2} - 2 \beta_1 - 1) q^{59} + (\beta_{2} + 4 \beta_1) q^{61} + ( - 2 \beta_{2} + 6 \beta_1 - 8) q^{62} + (\beta_{2} + \beta_1 + 1) q^{63} + (3 \beta_{2} - 8 \beta_1 + 15) q^{64} + (\beta_{2} + 6 \beta_1 - 4) q^{67} + (2 \beta_{2} - \beta_1 + 7) q^{68} + ( - \beta_{2} + 2 \beta_1 - 1) q^{69} + (\beta_{2} + 9) q^{71} + ( - 2 \beta_{2} + \beta_1 - 3) q^{72} + (2 \beta_{2} - 2 \beta_1 - 2) q^{73} + ( - 6 \beta_{2} + \beta_1 - 7) q^{74} + (4 \beta_{2} - 5 \beta_1 + 11) q^{76} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{78} + (2 \beta_{2} - \beta_1 - 1) q^{79} + q^{81} + ( - \beta_{2} - 2 \beta_1 - 3) q^{82} + ( - 6 \beta_1 + 4) q^{83} + (2 \beta_{2} + \beta_1 + 7) q^{84} + (8 \beta_{2} + 6 \beta_1 + 10) q^{86} - 2 \beta_1 q^{87} + ( - 4 \beta_1 + 2) q^{89} + (\beta_{2} + 6) q^{91} + ( - 5 \beta_{2} + 8 \beta_1 - 13) q^{92} + (\beta_{2} + 6) q^{93} + (2 \beta_{2} + 3 \beta_1 - 1) q^{94} + ( - 4 \beta_{2} + \beta_1 - 13) q^{96} + (2 \beta_{2} + 6 \beta_1 - 11) q^{97} + (2 \beta_{2} + 2 \beta_1 + 8) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} + 3 q^{3} + 8 q^{4} - 2 q^{6} + 3 q^{7} - 6 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} + 3 q^{3} + 8 q^{4} - 2 q^{6} + 3 q^{7} - 6 q^{8} + 3 q^{9} + 8 q^{12} + 5 q^{13} + 6 q^{14} + 10 q^{16} - 4 q^{17} - 2 q^{18} + q^{19} + 3 q^{21} - 6 q^{24} - 8 q^{26} + 3 q^{27} + 20 q^{28} - 2 q^{29} + 17 q^{31} - 34 q^{32} + 6 q^{34} + 8 q^{36} - 18 q^{38} + 5 q^{39} - 4 q^{41} + 6 q^{42} + 17 q^{43} + 30 q^{46} + 10 q^{47} + 10 q^{48} + 2 q^{49} - 4 q^{51} + 30 q^{52} + 6 q^{53} - 2 q^{54} - 22 q^{56} + q^{57} - 24 q^{58} - 6 q^{59} + 3 q^{61} - 16 q^{62} + 3 q^{63} + 34 q^{64} - 7 q^{67} + 18 q^{68} + 26 q^{71} - 6 q^{72} - 10 q^{73} - 14 q^{74} + 24 q^{76} - 8 q^{78} - 6 q^{79} + 3 q^{81} - 10 q^{82} + 6 q^{83} + 20 q^{84} + 28 q^{86} - 2 q^{87} + 2 q^{89} + 17 q^{91} - 26 q^{92} + 17 q^{93} - 2 q^{94} - 34 q^{96} - 29 q^{97} + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 6x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 4 \) 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
−1.76156
−0.363328
3.12489
−2.76156 1.00000 5.62620 0 −2.76156 1.86464 −10.0140 1.00000 0
1.2 −1.36333 1.00000 −0.141336 0 −1.36333 −2.50466 2.91934 1.00000 0
1.3 2.12489 1.00000 2.51514 0 2.12489 3.64002 1.09461 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.cd 3
5.b even 2 1 9075.2.a.cj 3
11.b odd 2 1 825.2.a.m yes 3
33.d even 2 1 2475.2.a.z 3
55.d odd 2 1 825.2.a.i 3
55.e even 4 2 825.2.c.f 6
165.d even 2 1 2475.2.a.bd 3
165.l odd 4 2 2475.2.c.q 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.a.i 3 55.d odd 2 1
825.2.a.m yes 3 11.b odd 2 1
825.2.c.f 6 55.e even 4 2
2475.2.a.z 3 33.d even 2 1
2475.2.a.bd 3 165.d even 2 1
2475.2.c.q 6 165.l odd 4 2
9075.2.a.cd 3 1.a even 1 1 trivial
9075.2.a.cj 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9075))\):

\( T_{2}^{3} + 2T_{2}^{2} - 5T_{2} - 8 \) Copy content Toggle raw display
\( T_{7}^{3} - 3T_{7}^{2} - 7T_{7} + 17 \) Copy content Toggle raw display
\( T_{13}^{3} - 5T_{13}^{2} + 8 \) Copy content Toggle raw display
\( T_{17}^{3} + 4T_{17}^{2} - 25T_{17} + 22 \) Copy content Toggle raw display
\( T_{19}^{3} - T_{19}^{2} - 19T_{19} - 25 \) Copy content Toggle raw display
\( T_{23}^{3} - 43T_{23} + 58 \) Copy content Toggle raw display
\( T_{37}^{3} - 75T_{37} - 34 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 2 T^{2} - 5 T - 8 \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 3 T^{2} - 7 T + 17 \) Copy content Toggle raw display
$11$ \( T^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 5T^{2} + 8 \) Copy content Toggle raw display
$17$ \( T^{3} + 4 T^{2} - 25 T + 22 \) Copy content Toggle raw display
$19$ \( T^{3} - T^{2} - 19 T - 25 \) Copy content Toggle raw display
$23$ \( T^{3} - 43T + 58 \) Copy content Toggle raw display
$29$ \( T^{3} + 2 T^{2} - 24 T + 16 \) Copy content Toggle raw display
$31$ \( T^{3} - 17 T^{2} + 88 T - 136 \) Copy content Toggle raw display
$37$ \( T^{3} - 75T - 34 \) Copy content Toggle raw display
$41$ \( T^{3} + 4T^{2} - T - 2 \) Copy content Toggle raw display
$43$ \( T^{3} - 17 T^{2} - 32 T + 1100 \) Copy content Toggle raw display
$47$ \( T^{3} - 10 T^{2} + 25 T - 8 \) Copy content Toggle raw display
$53$ \( T^{3} - 6 T^{2} - 148 T + 824 \) Copy content Toggle raw display
$59$ \( T^{3} + 6 T^{2} - 31 T - 136 \) Copy content Toggle raw display
$61$ \( T^{3} - 3 T^{2} - 88 T - 244 \) Copy content Toggle raw display
$67$ \( T^{3} + 7 T^{2} - 192 T - 1588 \) Copy content Toggle raw display
$71$ \( T^{3} - 26 T^{2} + 217 T - 580 \) Copy content Toggle raw display
$73$ \( T^{3} + 10 T^{2} - 44 T - 472 \) Copy content Toggle raw display
$79$ \( T^{3} + 6 T^{2} - 37 T - 212 \) Copy content Toggle raw display
$83$ \( T^{3} - 6 T^{2} - 216 T + 1328 \) Copy content Toggle raw display
$89$ \( T^{3} - 2 T^{2} - 100 T + 328 \) Copy content Toggle raw display
$97$ \( T^{3} + 29 T^{2} + 75 T - 2153 \) Copy content Toggle raw display
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