Properties

Label 9075.2.a.cz
Level $9075$
Weight $2$
Character orbit 9075.a
Self dual yes
Analytic conductor $72.464$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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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: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{7})\)
Defining polynomial: \(x^{4} - 5 x^{2} + 1\)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + q^{3} -\beta_{2} q^{4} + \beta_{3} q^{6} -2 \beta_{3} q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} + q^{9} +O(q^{10})\) \( q + \beta_{3} q^{2} + q^{3} -\beta_{2} q^{4} + \beta_{3} q^{6} -2 \beta_{3} q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} + q^{9} -\beta_{2} q^{12} + 2 \beta_{3} q^{13} + ( -4 + 2 \beta_{2} ) q^{14} + ( 1 + \beta_{2} ) q^{16} + ( 3 \beta_{1} - \beta_{3} ) q^{17} + \beta_{3} q^{18} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{19} -2 \beta_{3} q^{21} + ( -3 + 2 \beta_{2} ) q^{23} + ( \beta_{1} + \beta_{3} ) q^{24} + ( 4 - 2 \beta_{2} ) q^{26} + q^{27} + ( -2 \beta_{1} - 6 \beta_{3} ) q^{28} -2 \beta_{1} q^{29} + ( -1 + 2 \beta_{2} ) q^{31} + ( -3 \beta_{1} - 4 \beta_{3} ) q^{32} + ( -5 + \beta_{2} ) q^{34} -\beta_{2} q^{36} + ( -6 - 2 \beta_{2} ) q^{37} + ( 6 - 2 \beta_{2} ) q^{38} + 2 \beta_{3} q^{39} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{41} + ( -4 + 2 \beta_{2} ) q^{42} + 2 \beta_{1} q^{43} + ( -2 \beta_{1} - 9 \beta_{3} ) q^{46} + ( -3 - 2 \beta_{2} ) q^{47} + ( 1 + \beta_{2} ) q^{48} + ( 1 - 4 \beta_{2} ) q^{49} + ( 3 \beta_{1} - \beta_{3} ) q^{51} + ( 2 \beta_{1} + 6 \beta_{3} ) q^{52} -5 q^{53} + \beta_{3} q^{54} + ( -2 + 2 \beta_{2} ) q^{56} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{57} + 2 q^{58} -2 \beta_{2} q^{59} + ( -5 \beta_{1} - 5 \beta_{3} ) q^{61} + ( -2 \beta_{1} - 7 \beta_{3} ) q^{62} -2 \beta_{3} q^{63} + ( -7 + 2 \beta_{2} ) q^{64} + ( 4 + 6 \beta_{2} ) q^{67} + ( -7 \beta_{1} - 6 \beta_{3} ) q^{68} + ( -3 + 2 \beta_{2} ) q^{69} + ( -2 - 4 \beta_{2} ) q^{71} + ( \beta_{1} + \beta_{3} ) q^{72} + ( 8 \beta_{1} + 4 \beta_{3} ) q^{73} + 2 \beta_{1} q^{74} + ( 6 \beta_{1} + 8 \beta_{3} ) q^{76} + ( 4 - 2 \beta_{2} ) q^{78} + ( \beta_{1} - \beta_{3} ) q^{79} + q^{81} + ( -4 + 4 \beta_{2} ) q^{82} + ( -6 \beta_{1} - 2 \beta_{3} ) q^{83} + ( -2 \beta_{1} - 6 \beta_{3} ) q^{84} -2 q^{86} -2 \beta_{1} q^{87} + ( 6 + 6 \beta_{2} ) q^{89} + ( -8 + 4 \beta_{2} ) q^{91} + ( -10 + 5 \beta_{2} ) q^{92} + ( -1 + 2 \beta_{2} ) q^{93} + ( 2 \beta_{1} + 3 \beta_{3} ) q^{94} + ( -3 \beta_{1} - 4 \beta_{3} ) q^{96} + 2 \beta_{2} q^{97} + ( 4 \beta_{1} + 13 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 2 q^{4} + 4 q^{9} + O(q^{10}) \) \( 4 q + 4 q^{3} + 2 q^{4} + 4 q^{9} + 2 q^{12} - 20 q^{14} + 2 q^{16} - 16 q^{23} + 20 q^{26} + 4 q^{27} - 8 q^{31} - 22 q^{34} + 2 q^{36} - 20 q^{37} + 28 q^{38} - 20 q^{42} - 8 q^{47} + 2 q^{48} + 12 q^{49} - 20 q^{53} - 12 q^{56} + 8 q^{58} + 4 q^{59} - 32 q^{64} + 4 q^{67} - 16 q^{69} + 20 q^{78} + 4 q^{81} - 24 q^{82} - 8 q^{86} + 12 q^{89} - 40 q^{91} - 50 q^{92} - 8 q^{93} - 4 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 5 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 5 \beta_{1}\)

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
0.456850
2.18890
−2.18890
−0.456850
−2.18890 1.00000 2.79129 0 −2.18890 4.37780 −1.73205 1.00000 0
1.2 −0.456850 1.00000 −1.79129 0 −0.456850 0.913701 1.73205 1.00000 0
1.3 0.456850 1.00000 −1.79129 0 0.456850 −0.913701 −1.73205 1.00000 0
1.4 2.18890 1.00000 2.79129 0 2.18890 −4.37780 1.73205 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9075.2.a.cz 4
5.b even 2 1 9075.2.a.cs 4
5.c odd 4 2 1815.2.c.f 8
11.b odd 2 1 inner 9075.2.a.cz 4
55.d odd 2 1 9075.2.a.cs 4
55.e even 4 2 1815.2.c.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.c.f 8 5.c odd 4 2
1815.2.c.f 8 55.e even 4 2
9075.2.a.cs 4 5.b even 2 1
9075.2.a.cs 4 55.d odd 2 1
9075.2.a.cz 4 1.a even 1 1 trivial
9075.2.a.cz 4 11.b odd 2 1 inner

Hecke kernels

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

\( T_{2}^{4} - 5 T_{2}^{2} + 1 \)
\( T_{7}^{4} - 20 T_{7}^{2} + 16 \)
\( T_{13}^{4} - 20 T_{13}^{2} + 16 \)
\( T_{17}^{4} - 62 T_{17}^{2} + 625 \)
\( T_{19}^{2} - 28 \)
\( T_{23}^{2} + 8 T_{23} - 5 \)
\( T_{37}^{2} + 10 T_{37} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 5 T^{2} + T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( T^{4} \)
$7$ \( 16 - 20 T^{2} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( 16 - 20 T^{2} + T^{4} \)
$17$ \( 625 - 62 T^{2} + T^{4} \)
$19$ \( ( -28 + T^{2} )^{2} \)
$23$ \( ( -5 + 8 T + T^{2} )^{2} \)
$29$ \( 16 - 20 T^{2} + T^{4} \)
$31$ \( ( -17 + 4 T + T^{2} )^{2} \)
$37$ \( ( 4 + 10 T + T^{2} )^{2} \)
$41$ \( ( -48 + T^{2} )^{2} \)
$43$ \( 16 - 20 T^{2} + T^{4} \)
$47$ \( ( -17 + 4 T + T^{2} )^{2} \)
$53$ \( ( 5 + T )^{4} \)
$59$ \( ( -20 - 2 T + T^{2} )^{2} \)
$61$ \( ( -75 + T^{2} )^{2} \)
$67$ \( ( -188 - 2 T + T^{2} )^{2} \)
$71$ \( ( -84 + T^{2} )^{2} \)
$73$ \( 6400 - 272 T^{2} + T^{4} \)
$79$ \( ( -7 + T^{2} )^{2} \)
$83$ \( 400 - 152 T^{2} + T^{4} \)
$89$ \( ( -180 - 6 T + T^{2} )^{2} \)
$97$ \( ( -20 + 2 T + T^{2} )^{2} \)
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