Properties

Label 9075.2.a.cw
Level $9075$
Weight $2$
Character orbit 9075.a
Self dual yes
Analytic conductor $72.464$
Analytic rank $0$
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: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{11})\)
Defining polynomial: \( x^{4} - 7x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} + (\beta_{3} + 2) q^{4} - \beta_1 q^{6} + ( - \beta_{2} + \beta_1) q^{7} + (2 \beta_{2} + \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{3} + (\beta_{3} + 2) q^{4} - \beta_1 q^{6} + ( - \beta_{2} + \beta_1) q^{7} + (2 \beta_{2} + \beta_1) q^{8} + q^{9} + ( - \beta_{3} - 2) q^{12} + ( - 3 \beta_{2} + 2 \beta_1) q^{13} + 2 q^{14} + (\beta_{3} + 4) q^{16} + ( - 2 \beta_{2} + 4 \beta_1) q^{17} + \beta_1 q^{18} + (4 \beta_{2} - \beta_1) q^{19} + (\beta_{2} - \beta_1) q^{21} + 8 q^{23} + ( - 2 \beta_{2} - \beta_1) q^{24} + ( - \beta_{3} + 2) q^{26} - q^{27} + 2 \beta_{2} q^{28} - 4 \beta_1 q^{29} + \beta_{3} q^{31} + ( - 2 \beta_{2} + 3 \beta_1) q^{32} + (2 \beta_{3} + 12) q^{34} + (\beta_{3} + 2) q^{36} + 5 q^{37} + (3 \beta_{3} + 4) q^{38} + (3 \beta_{2} - 2 \beta_1) q^{39} - 4 \beta_{2} q^{41} - 2 q^{42} + ( - 3 \beta_{2} + 6 \beta_1) q^{43} + 8 \beta_1 q^{46} + ( - 2 \beta_{3} - 4) q^{47} + ( - \beta_{3} - 4) q^{48} + ( - \beta_{3} - 4) q^{49} + (2 \beta_{2} - 4 \beta_1) q^{51} + (4 \beta_{2} - 3 \beta_1) q^{52} + (2 \beta_{3} - 6) q^{53} - \beta_1 q^{54} + 2 \beta_{3} q^{56} + ( - 4 \beta_{2} + \beta_1) q^{57} + ( - 4 \beta_{3} - 16) q^{58} + 4 q^{59} + ( - 2 \beta_{2} - \beta_1) q^{61} + (2 \beta_{2} + \beta_1) q^{62} + ( - \beta_{2} + \beta_1) q^{63} - \beta_{3} q^{64} + ( - 3 \beta_{3} - 4) q^{67} + (8 \beta_{2} + 6 \beta_1) q^{68} - 8 q^{69} + (2 \beta_{3} + 6) q^{71} + (2 \beta_{2} + \beta_1) q^{72} - 3 \beta_{2} q^{73} + 5 \beta_1 q^{74} + ( - 2 \beta_{2} + 9 \beta_1) q^{76} + (\beta_{3} - 2) q^{78} + (\beta_{2} + 2 \beta_1) q^{79} + q^{81} + ( - 4 \beta_{3} - 8) q^{82} + (2 \beta_{2} + 2 \beta_1) q^{83} - 2 \beta_{2} q^{84} + (3 \beta_{3} + 18) q^{86} + 4 \beta_1 q^{87} + (4 \beta_{3} - 4) q^{89} + ( - 3 \beta_{3} + 7) q^{91} + (8 \beta_{3} + 16) q^{92} - \beta_{3} q^{93} + ( - 4 \beta_{2} - 6 \beta_1) q^{94} + (2 \beta_{2} - 3 \beta_1) q^{96} + ( - \beta_{3} - 7) q^{97} + ( - 2 \beta_{2} - 5 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 6 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} + 6 q^{4} + 4 q^{9} - 6 q^{12} + 8 q^{14} + 14 q^{16} + 32 q^{23} + 10 q^{26} - 4 q^{27} - 2 q^{31} + 44 q^{34} + 6 q^{36} + 20 q^{37} + 10 q^{38} - 8 q^{42} - 12 q^{47} - 14 q^{48} - 14 q^{49} - 28 q^{53} - 4 q^{56} - 56 q^{58} + 16 q^{59} + 2 q^{64} - 10 q^{67} - 32 q^{69} + 20 q^{71} - 10 q^{78} + 4 q^{81} - 24 q^{82} + 66 q^{86} - 24 q^{89} + 34 q^{91} + 48 q^{92} + 2 q^{93} - 26 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 7x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 5\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} + 5\beta_1 \) 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
−2.52434
−0.792287
0.792287
2.52434
−2.52434 −1.00000 4.37228 0 2.52434 −0.792287 −5.98844 1.00000 0
1.2 −0.792287 −1.00000 −1.37228 0 0.792287 −2.52434 2.67181 1.00000 0
1.3 0.792287 −1.00000 −1.37228 0 −0.792287 2.52434 −2.67181 1.00000 0
1.4 2.52434 −1.00000 4.37228 0 −2.52434 0.792287 5.98844 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.cw 4
5.b even 2 1 9075.2.a.db yes 4
11.b odd 2 1 inner 9075.2.a.cw 4
55.d odd 2 1 9075.2.a.db yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9075.2.a.cw 4 1.a even 1 1 trivial
9075.2.a.cw 4 11.b odd 2 1 inner
9075.2.a.db yes 4 5.b even 2 1
9075.2.a.db yes 4 55.d odd 2 1

Hecke kernels

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

\( T_{2}^{4} - 7T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{4} - 7T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{13}^{4} - 46T_{13}^{2} + 1 \) Copy content Toggle raw display
\( T_{17}^{2} - 44 \) Copy content Toggle raw display
\( T_{19}^{4} - 79T_{19}^{2} + 1156 \) Copy content Toggle raw display
\( T_{23} - 8 \) Copy content Toggle raw display
\( T_{37} - 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 7T^{2} + 4 \) Copy content Toggle raw display
$3$ \( (T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 7T^{2} + 4 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 46T^{2} + 1 \) Copy content Toggle raw display
$17$ \( (T^{2} - 44)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 79T^{2} + 1156 \) Copy content Toggle raw display
$23$ \( (T - 8)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} - 112T^{2} + 1024 \) Copy content Toggle raw display
$31$ \( (T^{2} + T - 8)^{2} \) Copy content Toggle raw display
$37$ \( (T - 5)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 99)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 6 T - 24)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 14 T + 16)^{2} \) Copy content Toggle raw display
$59$ \( (T - 4)^{4} \) Copy content Toggle raw display
$61$ \( T^{4} - 43T^{2} + 256 \) Copy content Toggle raw display
$67$ \( (T^{2} + 5 T - 68)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 10 T - 8)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 46T^{2} + 1 \) Copy content Toggle raw display
$83$ \( T^{4} - 76T^{2} + 256 \) Copy content Toggle raw display
$89$ \( (T^{2} + 12 T - 96)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 13 T + 34)^{2} \) Copy content Toggle raw display
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