Properties

Label 1080.6.a.e
Level $1080$
Weight $6$
Character orbit 1080.a
Self dual yes
Analytic conductor $173.215$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1080.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(173.214525398\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.15881.1
Defining polynomial: \( x^{3} - 29x - 55 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2}\cdot 11 \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 25 q^{5} + (\beta_{2} - 10) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 25 q^{5} + (\beta_{2} - 10) q^{7} + ( - 3 \beta_{2} + \beta_1 + 220) q^{11} + ( - 3 \beta_{2} + \beta_1 - 426) q^{13} + (7 \beta_{2} + 2 \beta_1 - 577) q^{17} + (3 \beta_{2} + 9 \beta_1 + 337) q^{19} + (8 \beta_{2} + 21 \beta_1 - 811) q^{23} + 625 q^{25} + ( - 3 \beta_{2} - 32 \beta_1 - 1262) q^{29} + ( - 11 \beta_{2} + 16 \beta_1 + 1377) q^{31} + ( - 25 \beta_{2} + 250) q^{35} + ( - 68 \beta_{2} - 32 \beta_1 - 2374) q^{37} + (77 \beta_{2} - 12 \beta_1 - 5784) q^{41} + ( - \beta_{2} - 44 \beta_1 + 3852) q^{43} + (6 \beta_{2} - 37 \beta_1 + 6516) q^{47} + ( - 98 \beta_{2} + 15 \beta_1 - 2619) q^{49} + (151 \beta_{2} + 103 \beta_1 + 1655) q^{53} + (75 \beta_{2} - 25 \beta_1 - 5500) q^{55} + (153 \beta_{2} - 45 \beta_1 + 10702) q^{59} + (126 \beta_{2} + 142 \beta_1 - 5439) q^{61} + (75 \beta_{2} - 25 \beta_1 + 10650) q^{65} + (250 \beta_{2} + 84 \beta_1 - 17086) q^{67} + ( - 273 \beta_{2} - 127 \beta_1 + 28024) q^{71} + ( - 175 \beta_{2} - 75 \beta_1 - 53964) q^{73} + (548 \beta_{2} + 23 \beta_1 - 47656) q^{77} + (553 \beta_{2} - 135 \beta_1 + 12147) q^{79} + ( - 112 \beta_{2} + 497 \beta_1 + 21527) q^{83} + ( - 175 \beta_{2} - 50 \beta_1 + 14425) q^{85} + ( - 637 \beta_{2} - 194 \beta_1 + 20528) q^{89} + ( - 98 \beta_{2} + 23 \beta_1 - 41196) q^{91} + ( - 75 \beta_{2} - 225 \beta_1 - 8425) q^{95} + (284 \beta_{2} - 92 \beta_1 + 4920) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 75 q^{5} - 30 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 75 q^{5} - 30 q^{7} + 660 q^{11} - 1278 q^{13} - 1731 q^{17} + 1011 q^{19} - 2433 q^{23} + 1875 q^{25} - 3786 q^{29} + 4131 q^{31} + 750 q^{35} - 7122 q^{37} - 17352 q^{41} + 11556 q^{43} + 19548 q^{47} - 7857 q^{49} + 4965 q^{53} - 16500 q^{55} + 32106 q^{59} - 16317 q^{61} + 31950 q^{65} - 51258 q^{67} + 84072 q^{71} - 161892 q^{73} - 142968 q^{77} + 36441 q^{79} + 64581 q^{83} + 43275 q^{85} + 61584 q^{89} - 123588 q^{91} - 25275 q^{95} + 14760 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 29x - 55 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -12\nu^{2} + 232 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -18\nu^{2} + 66\nu + 348 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{2} - 3\beta_1 ) / 132 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta _1 + 232 ) / 12 \) 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
−3.82250
6.15876
−2.33626
0 0 0 −25.0000 0 −177.292 0 0 0
1.2 0 0 0 −25.0000 0 61.7318 0 0 0
1.3 0 0 0 −25.0000 0 85.5605 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)

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 1080.6.a.e 3
3.b odd 2 1 1080.6.a.g yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.6.a.e 3 1.a even 1 1 trivial
1080.6.a.g yes 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1080))\):

\( T_{7}^{3} + 30T_{7}^{2} - 20832T_{7} + 936424 \) Copy content Toggle raw display
\( T_{11}^{3} - 660T_{11}^{2} - 114084T_{11} + 16969720 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T + 25)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 30 T^{2} - 20832 T + 936424 \) Copy content Toggle raw display
$11$ \( T^{3} - 660 T^{2} + \cdots + 16969720 \) Copy content Toggle raw display
$13$ \( T^{3} + 1278 T^{2} + \cdots - 62570968 \) Copy content Toggle raw display
$17$ \( T^{3} + 1731 T^{2} + \cdots - 362051059 \) Copy content Toggle raw display
$19$ \( T^{3} - 1011 T^{2} + \cdots + 1066478795 \) Copy content Toggle raw display
$23$ \( T^{3} + 2433 T^{2} + \cdots - 16297632125 \) Copy content Toggle raw display
$29$ \( T^{3} + 3786 T^{2} + \cdots - 100231188440 \) Copy content Toggle raw display
$31$ \( T^{3} - 4131 T^{2} + \cdots + 36775432739 \) Copy content Toggle raw display
$37$ \( T^{3} + 7122 T^{2} + \cdots - 11280290472 \) Copy content Toggle raw display
$41$ \( T^{3} + 17352 T^{2} + \cdots - 19821902472 \) Copy content Toggle raw display
$43$ \( T^{3} - 11556 T^{2} + \cdots + 74089212824 \) Copy content Toggle raw display
$47$ \( T^{3} - 19548 T^{2} + \cdots - 48232251200 \) Copy content Toggle raw display
$53$ \( T^{3} - 4965 T^{2} + \cdots - 6200694992835 \) Copy content Toggle raw display
$59$ \( T^{3} - 32106 T^{2} + \cdots + 9865368652616 \) Copy content Toggle raw display
$61$ \( T^{3} + 16317 T^{2} + \cdots - 15693428262193 \) Copy content Toggle raw display
$67$ \( T^{3} + 51258 T^{2} + \cdots - 20150295569000 \) Copy content Toggle raw display
$71$ \( T^{3} - 84072 T^{2} + \cdots + 46987871536152 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots + 119813275420344 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots + 306616858223481 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots + 623768894560265 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots + 152901460706184 \) Copy content Toggle raw display
$97$ \( T^{3} - 14760 T^{2} + \cdots + 36671348002304 \) Copy content Toggle raw display
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