Properties

Label 1080.6.a.h
Level $1080$
Weight $6$
Character orbit 1080.a
Self dual yes
Analytic conductor $173.215$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \( x^{3} - 460x - 1125 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
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.

\(f(q)\) \(=\) \( q + 25 q^{5} + (\beta_1 + 47) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 25 q^{5} + (\beta_1 + 47) q^{7} + (2 \beta_{2} + \beta_1 - 54) q^{11} + (3 \beta_{2} + 3 \beta_1 + 18) q^{13} + ( - 5 \beta_{2} + 8 \beta_1 - 199) q^{17} + ( - 7 \beta_{2} - 10 \beta_1 - 544) q^{19} + (2 \beta_{2} - 7 \beta_1 + 546) q^{23} + 625 q^{25} + ( - 15 \beta_{2} - 112 \beta_1 - 223) q^{29} + ( - 28 \beta_{2} - 79 \beta_1 - 962) q^{31} + (25 \beta_1 + 1175) q^{35} + (13 \beta_{2} - 88 \beta_1 - 2076) q^{37} + ( - 57 \beta_{2} + 119 \beta_1 - 4547) q^{41} + ( - 10 \beta_{2} + 137 \beta_1 - 744) q^{43} + (53 \beta_{2} + 74 \beta_1 - 4945) q^{47} + ( - 8 \beta_{2} + 55 \beta_1 - 11320) q^{49} + (3 \beta_{2} + 119 \beta_1 + 361) q^{53} + (50 \beta_{2} + 25 \beta_1 - 1350) q^{55} + ( - 83 \beta_{2} + 39 \beta_1 - 16067) q^{59} + (140 \beta_{2} - 509 \beta_1 + 1929) q^{61} + (75 \beta_{2} + 75 \beta_1 + 450) q^{65} + ( - 168 \beta_{2} - 335 \beta_1 + 1497) q^{67} + (13 \beta_{2} - 743 \beta_1 - 9577) q^{71} + (219 \beta_{2} + 532 \beta_1 - 11932) q^{73} + (162 \beta_{2} - 90 \beta_1 - 2340) q^{77} + (67 \beta_{2} + 399 \beta_1 + 8564) q^{79} + (541 \beta_{2} + 171 \beta_1 - 16025) q^{83} + ( - 125 \beta_{2} + 200 \beta_1 - 4975) q^{85} + ( - 217 \beta_{2} - 159 \beta_1 + 13683) q^{89} + (231 \beta_{2} - 24 \beta_1 + 6060) q^{91} + ( - 175 \beta_{2} - 250 \beta_1 - 13600) q^{95} + ( - 603 \beta_{2} + 1542 \beta_1 - 15874) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 75 q^{5} + 140 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 75 q^{5} + 140 q^{7} - 163 q^{11} + 51 q^{13} - 605 q^{17} - 1622 q^{19} + 1645 q^{23} + 1875 q^{25} - 557 q^{29} - 2807 q^{31} + 3500 q^{35} - 6140 q^{37} - 13760 q^{41} - 2369 q^{43} - 14909 q^{47} - 34015 q^{49} + 964 q^{53} - 4075 q^{55} - 48240 q^{59} + 6296 q^{61} + 1275 q^{65} + 4826 q^{67} - 27988 q^{71} - 36328 q^{73} - 6930 q^{77} + 25293 q^{79} - 48246 q^{83} - 15125 q^{85} + 41208 q^{89} + 18204 q^{91} - 40550 q^{95} - 49164 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 460x - 1125 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{2} + 15\nu + 305 ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{2} + 15\nu - 920 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 3\beta _1 + 1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{2} - 5\beta _1 + 1225 ) / 4 \) 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
−20.1005
22.5793
−2.47876
0 0 0 25.0000 0 −33.1079 0 0 0
1.2 0 0 0 25.0000 0 73.7730 0 0 0
1.3 0 0 0 25.0000 0 99.3349 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.h yes 3
3.b odd 2 1 1080.6.a.f 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.6.a.f 3 3.b odd 2 1
1080.6.a.h yes 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(1080))\):

\( T_{7}^{3} - 140T_{7}^{2} + 1597T_{7} + 242622 \) Copy content Toggle raw display
\( T_{11}^{3} + 163T_{11}^{2} - 129312T_{11} - 18306324 \) 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
$7$ \( T^{3} - 140 T^{2} + 1597 T + 242622 \) Copy content Toggle raw display
$11$ \( T^{3} + 163 T^{2} + \cdots - 18306324 \) Copy content Toggle raw display
$13$ \( T^{3} - 51 T^{2} - 322641 T - 59054589 \) Copy content Toggle raw display
$17$ \( T^{3} + 605 T^{2} + \cdots - 897539376 \) Copy content Toggle raw display
$19$ \( T^{3} + 1622 T^{2} + \cdots + 142738684 \) Copy content Toggle raw display
$23$ \( T^{3} - 1645 T^{2} + \cdots + 159386316 \) Copy content Toggle raw display
$29$ \( T^{3} + 557 T^{2} + \cdots - 174014485380 \) Copy content Toggle raw display
$31$ \( T^{3} + 2807 T^{2} + \cdots + 7296447744 \) Copy content Toggle raw display
$37$ \( T^{3} + 6140 T^{2} + \cdots - 88793131806 \) Copy content Toggle raw display
$41$ \( T^{3} + 13760 T^{2} + \cdots - 2105614536000 \) Copy content Toggle raw display
$43$ \( T^{3} + 2369 T^{2} + \cdots + 99827900592 \) Copy content Toggle raw display
$47$ \( T^{3} + 14909 T^{2} + \cdots - 857156082128 \) Copy content Toggle raw display
$53$ \( T^{3} - 964 T^{2} + \cdots + 227607713808 \) Copy content Toggle raw display
$59$ \( T^{3} + 48240 T^{2} + \cdots - 911884153536 \) Copy content Toggle raw display
$61$ \( T^{3} - 6296 T^{2} + \cdots + 32138860458870 \) Copy content Toggle raw display
$67$ \( T^{3} - 4826 T^{2} + \cdots + 17472859220516 \) Copy content Toggle raw display
$71$ \( T^{3} + 27988 T^{2} + \cdots - 68617099591632 \) Copy content Toggle raw display
$73$ \( T^{3} + 36328 T^{2} + \cdots - 60737955374814 \) Copy content Toggle raw display
$79$ \( T^{3} - 25293 T^{2} + \cdots + 12677984881325 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots - 304428707967672 \) Copy content Toggle raw display
$89$ \( T^{3} - 41208 T^{2} + \cdots + 39239500298432 \) Copy content Toggle raw display
$97$ \( T^{3} + 49164 T^{2} + \cdots - 21\!\cdots\!10 \) Copy content Toggle raw display
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