Properties

Label 8280.2.a.bw
Level $8280$
Weight $2$
Character orbit 8280.a
Self dual yes
Analytic conductor $66.116$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8280.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(66.1161328736\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \( x^{7} - 3x^{6} - 18x^{5} + 46x^{4} + 60x^{3} - 76x^{2} - 51x - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5} + (\beta_{4} - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} + (\beta_{4} - 1) q^{7} - \beta_{3} q^{11} + ( - \beta_{6} + \beta_{3} + \beta_{2} - 2) q^{13} + (\beta_{6} - \beta_{4}) q^{17} + ( - \beta_{5} - \beta_1 - 1) q^{19} + q^{23} + q^{25} + ( - \beta_{4} - \beta_{2} + \beta_1 + 1) q^{29} + (\beta_{5} - \beta_{4} - \beta_{2} - \beta_1 - 1) q^{31} + (\beta_{4} - 1) q^{35} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 3) q^{37} + (\beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{41} + (\beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{43} + (\beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} - 1) q^{47} + (\beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 4) q^{49} + ( - \beta_{4} + \beta_{2} - 2 \beta_1 - 2) q^{53} - \beta_{3} q^{55} + (\beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{59} + ( - \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{3} + 1) q^{61} + ( - \beta_{6} + \beta_{3} + \beta_{2} - 2) q^{65} + ( - \beta_{5} + \beta_{4} - \beta_{2} + 2 \beta_1 - 2) q^{67} + ( - 2 \beta_{6} - 2 \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 3) q^{71} + (\beta_{5} + \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{73} + ( - \beta_{5} + 3 \beta_{3} + \beta_{2} - 2 \beta_1 - 3) q^{77} + (\beta_{6} + 2 \beta_{4} - \beta_{3} - \beta_1) q^{79} + (\beta_{6} - \beta_{4} - 2 \beta_{3} - 2) q^{83} + (\beta_{6} - \beta_{4}) q^{85} + (2 \beta_{6} - \beta_{5} - \beta_{3} - \beta_1 - 1) q^{89} + ( - 3 \beta_{5} + 2 \beta_{2} - \beta_1 - 3) q^{91} + ( - \beta_{5} - \beta_1 - 1) q^{95} + (\beta_{6} - \beta_{5} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 7 q^{5} - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 7 q^{5} - 6 q^{7} - 2 q^{11} - 6 q^{13} - 4 q^{17} - 8 q^{19} + 7 q^{23} + 7 q^{25} + 2 q^{29} - 8 q^{31} - 6 q^{35} - 16 q^{37} + 2 q^{41} - 10 q^{43} - 8 q^{47} + 19 q^{49} - 10 q^{53} - 2 q^{55} - 24 q^{59} + 8 q^{61} - 6 q^{65} - 20 q^{67} - 8 q^{71} + 2 q^{73} - 12 q^{77} - 2 q^{79} - 22 q^{83} - 4 q^{85} - 16 q^{89} - 20 q^{91} - 8 q^{95} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 3x^{6} - 18x^{5} + 46x^{4} + 60x^{3} - 76x^{2} - 51x - 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -11\nu^{6} + 24\nu^{5} + 238\nu^{4} - 352\nu^{3} - 1172\nu^{2} + 264\nu + 553 ) / 224 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5\nu^{6} - 16\nu^{5} - 84\nu^{4} + 244\nu^{3} + 212\nu^{2} - 386\nu - 133 ) / 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -13\nu^{6} + 40\nu^{5} + 226\nu^{4} - 608\nu^{3} - 652\nu^{2} + 984\nu + 383 ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 25\nu^{6} - 80\nu^{5} - 434\nu^{4} + 1248\nu^{3} + 1228\nu^{2} - 2280\nu - 651 ) / 56 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 111\nu^{6} - 344\nu^{5} - 1974\nu^{4} + 5344\nu^{3} + 6308\nu^{2} - 9608\nu - 4501 ) / 224 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -99\nu^{6} + 328\nu^{5} + 1694\nu^{4} - 5072\nu^{3} - 4612\nu^{2} + 8872\nu + 2737 ) / 112 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} - 2\beta_{4} + \beta_{3} + \beta_{2} + \beta _1 + 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} + 2\beta_{4} + 7\beta_{3} + 6\beta_{2} - 4\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{6} + 28\beta_{5} - 24\beta_{4} + 15\beta_{3} + 21\beta_{2} + 21\beta _1 + 154 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 52\beta_{6} + 40\beta_{5} + 78\beta_{4} + 187\beta_{3} + 177\beta_{2} - 83\beta _1 + 88 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 68\beta_{6} + 208\beta_{5} - 132\beta_{4} + 101\beta_{3} + 187\beta_{2} + 179\beta _1 + 1109 \) 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
−3.70204
−0.218266
3.22540
1.37429
4.09437
−0.330805
−1.44295
0 0 0 1.00000 0 −5.18676 0 0 0
1.2 0 0 0 1.00000 0 −2.94234 0 0 0
1.3 0 0 0 1.00000 0 −2.82985 0 0 0
1.4 0 0 0 1.00000 0 −0.958551 0 0 0
1.5 0 0 0 1.00000 0 −0.627340 0 0 0
1.6 0 0 0 1.00000 0 2.34984 0 0 0
1.7 0 0 0 1.00000 0 4.19500 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(5\) \(-1\)
\(23\) \(-1\)

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 8280.2.a.bw yes 7
3.b odd 2 1 8280.2.a.bv 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8280.2.a.bv 7 3.b odd 2 1
8280.2.a.bw yes 7 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8280))\):

\( T_{7}^{7} + 6T_{7}^{6} - 16T_{7}^{5} - 134T_{7}^{4} - 77T_{7}^{3} + 516T_{7}^{2} + 732T_{7} + 256 \) Copy content Toggle raw display
\( T_{11}^{7} + 2T_{11}^{6} - 50T_{11}^{5} - 128T_{11}^{4} + 556T_{11}^{3} + 1928T_{11}^{2} + 1728T_{11} + 384 \) Copy content Toggle raw display
\( T_{13}^{7} + 6T_{13}^{6} - 58T_{13}^{5} - 248T_{13}^{4} + 1412T_{13}^{3} + 2072T_{13}^{2} - 14368T_{13} + 14336 \) Copy content Toggle raw display
\( T_{17}^{7} + 4T_{17}^{6} - 88T_{17}^{5} - 364T_{17}^{4} + 1599T_{17}^{3} + 7840T_{17}^{2} + 6564T_{17} - 2688 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \) Copy content Toggle raw display
$3$ \( T^{7} \) Copy content Toggle raw display
$5$ \( (T - 1)^{7} \) Copy content Toggle raw display
$7$ \( T^{7} + 6 T^{6} - 16 T^{5} - 134 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( T^{7} + 2 T^{6} - 50 T^{5} - 128 T^{4} + \cdots + 384 \) Copy content Toggle raw display
$13$ \( T^{7} + 6 T^{6} - 58 T^{5} + \cdots + 14336 \) Copy content Toggle raw display
$17$ \( T^{7} + 4 T^{6} - 88 T^{5} + \cdots - 2688 \) Copy content Toggle raw display
$19$ \( T^{7} + 8 T^{6} - 74 T^{5} + \cdots - 3072 \) Copy content Toggle raw display
$23$ \( (T - 1)^{7} \) Copy content Toggle raw display
$29$ \( T^{7} - 2 T^{6} - 98 T^{5} + 102 T^{4} + \cdots + 336 \) Copy content Toggle raw display
$31$ \( T^{7} + 8 T^{6} - 162 T^{5} + \cdots - 294144 \) Copy content Toggle raw display
$37$ \( T^{7} + 16 T^{6} - 40 T^{5} + \cdots + 65536 \) Copy content Toggle raw display
$41$ \( T^{7} - 2 T^{6} - 190 T^{5} + \cdots - 390144 \) Copy content Toggle raw display
$43$ \( T^{7} + 10 T^{6} - 164 T^{5} + \cdots + 248832 \) Copy content Toggle raw display
$47$ \( T^{7} + 8 T^{6} - 162 T^{5} + \cdots + 4096 \) Copy content Toggle raw display
$53$ \( T^{7} + 10 T^{6} - 144 T^{5} + \cdots - 310472 \) Copy content Toggle raw display
$59$ \( T^{7} + 24 T^{6} + 26 T^{5} + \cdots + 19528 \) Copy content Toggle raw display
$61$ \( T^{7} - 8 T^{6} - 278 T^{5} + \cdots + 1757504 \) Copy content Toggle raw display
$67$ \( T^{7} + 20 T^{6} - 176 T^{5} + \cdots - 383968 \) Copy content Toggle raw display
$71$ \( T^{7} + 8 T^{6} - 290 T^{5} + \cdots - 1146496 \) Copy content Toggle raw display
$73$ \( T^{7} - 2 T^{6} - 326 T^{5} + \cdots + 2552736 \) Copy content Toggle raw display
$79$ \( T^{7} + 2 T^{6} - 272 T^{5} + \cdots + 73728 \) Copy content Toggle raw display
$83$ \( T^{7} + 22 T^{6} - 28 T^{5} + \cdots - 300512 \) Copy content Toggle raw display
$89$ \( T^{7} + 16 T^{6} - 276 T^{5} + \cdots - 3662848 \) Copy content Toggle raw display
$97$ \( T^{7} - 4 T^{6} - 240 T^{5} + \cdots - 284928 \) Copy content Toggle raw display
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