Properties

Label 9075.2.a.cf
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: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 165)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - 2 \)
\(\beta_{2}\)\(=\)\( -\nu^{2} + 2 \nu + 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{1} + 2\)

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
2.17009
−1.48119
0.311108
−2.70928 −1.00000 5.34017 0 2.70928 1.07838 −9.04945 1.00000 0
1.2 −0.193937 −1.00000 −1.96239 0 0.193937 3.35026 0.768452 1.00000 0
1.3 1.90321 −1.00000 1.62222 0 −1.90321 −4.42864 −0.719004 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.cf 3
5.b even 2 1 1815.2.a.m 3
11.b odd 2 1 825.2.a.k 3
15.d odd 2 1 5445.2.a.z 3
33.d even 2 1 2475.2.a.bb 3
55.d odd 2 1 165.2.a.c 3
55.e even 4 2 825.2.c.g 6
165.d even 2 1 495.2.a.e 3
165.l odd 4 2 2475.2.c.r 6
220.g even 2 1 2640.2.a.be 3
385.h even 2 1 8085.2.a.bk 3
660.g odd 2 1 7920.2.a.cj 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.c 3 55.d odd 2 1
495.2.a.e 3 165.d even 2 1
825.2.a.k 3 11.b odd 2 1
825.2.c.g 6 55.e even 4 2
1815.2.a.m 3 5.b even 2 1
2475.2.a.bb 3 33.d even 2 1
2475.2.c.r 6 165.l odd 4 2
2640.2.a.be 3 220.g even 2 1
5445.2.a.z 3 15.d odd 2 1
7920.2.a.cj 3 660.g odd 2 1
8085.2.a.bk 3 385.h even 2 1
9075.2.a.cf 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9075))\):

\( T_{2}^{3} + T_{2}^{2} - 5 T_{2} - 1 \)
\( T_{7}^{3} - 16 T_{7} + 16 \)
\( T_{13}^{3} + 2 T_{13}^{2} - 12 T_{13} - 8 \)
\( T_{17}^{3} + 2 T_{17}^{2} - 52 T_{17} - 184 \)
\( T_{19}^{3} + 8 T_{19}^{2} - 16 T_{19} - 160 \)
\( T_{23}^{3} - 64 T_{23} + 128 \)
\( T_{37} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 - 5 T + T^{2} + T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( T^{3} \)
$7$ \( 16 - 16 T + T^{3} \)
$11$ \( T^{3} \)
$13$ \( -8 - 12 T + 2 T^{2} + T^{3} \)
$17$ \( -184 - 52 T + 2 T^{2} + T^{3} \)
$19$ \( -160 - 16 T + 8 T^{2} + T^{3} \)
$23$ \( 128 - 64 T + T^{3} \)
$29$ \( 40 + 12 T - 10 T^{2} + T^{3} \)
$31$ \( 128 - 32 T - 8 T^{2} + T^{3} \)
$37$ \( ( -2 + T )^{3} \)
$41$ \( -8 + 44 T - 14 T^{2} + T^{3} \)
$43$ \( 400 - 80 T - 4 T^{2} + T^{3} \)
$47$ \( 128 - 32 T - 8 T^{2} + T^{3} \)
$53$ \( -8 - 52 T - 6 T^{2} + T^{3} \)
$59$ \( 320 - 16 T - 12 T^{2} + T^{3} \)
$61$ \( 248 - 52 T - 6 T^{2} + T^{3} \)
$67$ \( 64 - 48 T - 4 T^{2} + T^{3} \)
$71$ \( 128 - 32 T - 8 T^{2} + T^{3} \)
$73$ \( -344 + 4 T + 14 T^{2} + T^{3} \)
$79$ \( -800 - 64 T + 12 T^{2} + T^{3} \)
$83$ \( 16 - 120 T + T^{3} \)
$89$ \( -200 - 52 T + 10 T^{2} + T^{3} \)
$97$ \( 8 + 108 T + 22 T^{2} + T^{3} \)
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