Properties

Label 9075.2.a.ci
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: 3.3.469.1
Defining polynomial: \(x^{3} - x^{2} - 5 x + 4\)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 5 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)

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
−2.16425
0.772866
2.39138
−2.16425 −1.00000 2.68397 0 2.16425 −0.480279 −1.48028 1.00000 0
1.2 0.772866 −1.00000 −1.40268 0 −0.772866 −1.62981 −2.62981 1.00000 0
1.3 2.39138 −1.00000 3.71871 0 −2.39138 5.11009 4.11009 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.ci 3
5.b even 2 1 1815.2.a.l 3
11.b odd 2 1 9075.2.a.ce 3
15.d odd 2 1 5445.2.a.bc 3
55.d odd 2 1 1815.2.a.n yes 3
165.d even 2 1 5445.2.a.ba 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.a.l 3 5.b even 2 1
1815.2.a.n yes 3 55.d odd 2 1
5445.2.a.ba 3 165.d even 2 1
5445.2.a.bc 3 15.d odd 2 1
9075.2.a.ce 3 11.b odd 2 1
9075.2.a.ci 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9075))\):

\( T_{2}^{3} - T_{2}^{2} - 5 T_{2} + 4 \)
\( T_{7}^{3} - 3 T_{7}^{2} - 10 T_{7} - 4 \)
\( T_{13}^{3} + 4 T_{13}^{2} - 24 T_{13} - 88 \)
\( T_{17}^{3} + 4 T_{17}^{2} - 7 T_{17} - 14 \)
\( T_{19}^{3} - 5 T_{19}^{2} - 4 T_{19} + 16 \)
\( T_{23}^{3} + 6 T_{23}^{2} - T_{23} - 34 \)
\( T_{37}^{3} + 3 T_{37}^{2} - 114 T_{37} - 548 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - 5 T - T^{2} + T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( T^{3} \)
$7$ \( -4 - 10 T - 3 T^{2} + T^{3} \)
$11$ \( T^{3} \)
$13$ \( -88 - 24 T + 4 T^{2} + T^{3} \)
$17$ \( -14 - 7 T + 4 T^{2} + T^{3} \)
$19$ \( 16 - 4 T - 5 T^{2} + T^{3} \)
$23$ \( -34 - T + 6 T^{2} + T^{3} \)
$29$ \( -16 + 12 T + 10 T^{2} + T^{3} \)
$31$ \( -61 - 29 T + T^{2} + T^{3} \)
$37$ \( -548 - 114 T + 3 T^{2} + T^{3} \)
$41$ \( 392 - 52 T - 10 T^{2} + T^{3} \)
$43$ \( 32 - 20 T - 2 T^{2} + T^{3} \)
$47$ \( -292 + 31 T + 16 T^{2} + T^{3} \)
$53$ \( 586 - 139 T + T^{3} \)
$59$ \( 1088 - 28 T - 18 T^{2} + T^{3} \)
$61$ \( 133 + 191 T + 27 T^{2} + T^{3} \)
$67$ \( 172 - 54 T - T^{2} + T^{3} \)
$71$ \( 2144 - 152 T - 14 T^{2} + T^{3} \)
$73$ \( -16 + 4 T + 7 T^{2} + T^{3} \)
$79$ \( 83 - 85 T + 9 T^{2} + T^{3} \)
$83$ \( ( -4 + T )^{3} \)
$89$ \( -968 + 320 T - 32 T^{2} + T^{3} \)
$97$ \( -172 - 54 T + T^{2} + T^{3} \)
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