Properties

Label 8280.2.a.bv
Level $8280$
Weight $2$
Character orbit 8280.a
Self dual yes
Analytic conductor $66.116$
Analytic rank $0$
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: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 3 x^{6} - 18 x^{5} + 46 x^{4} + 60 x^{3} - 76 x^{2} - 51 x - 7\)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -11 \nu^{6} + 24 \nu^{5} + 238 \nu^{4} - 352 \nu^{3} - 1172 \nu^{2} + 264 \nu + 553 \)\()/224\)
\(\beta_{2}\)\(=\)\((\)\( 5 \nu^{6} - 16 \nu^{5} - 84 \nu^{4} + 244 \nu^{3} + 212 \nu^{2} - 386 \nu - 133 \)\()/14\)
\(\beta_{3}\)\(=\)\((\)\( -13 \nu^{6} + 40 \nu^{5} + 226 \nu^{4} - 608 \nu^{3} - 652 \nu^{2} + 984 \nu + 383 \)\()/32\)
\(\beta_{4}\)\(=\)\((\)\( 25 \nu^{6} - 80 \nu^{5} - 434 \nu^{4} + 1248 \nu^{3} + 1228 \nu^{2} - 2280 \nu - 651 \)\()/56\)
\(\beta_{5}\)\(=\)\((\)\( 111 \nu^{6} - 344 \nu^{5} - 1974 \nu^{4} + 5344 \nu^{3} + 6308 \nu^{2} - 9608 \nu - 4501 \)\()/224\)
\(\beta_{6}\)\(=\)\((\)\( -99 \nu^{6} + 328 \nu^{5} + 1694 \nu^{4} - 5072 \nu^{3} - 4612 \nu^{2} + 8872 \nu + 2737 \)\()/112\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_{1} + 12\)\()/2\)
\(\nu^{3}\)\(=\)\(\beta_{6} + \beta_{5} + 2 \beta_{4} + 7 \beta_{3} + 6 \beta_{2} - 4 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\((\)\(4 \beta_{6} + 28 \beta_{5} - 24 \beta_{4} + 15 \beta_{3} + 21 \beta_{2} + 21 \beta_{1} + 154\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(52 \beta_{6} + 40 \beta_{5} + 78 \beta_{4} + 187 \beta_{3} + 177 \beta_{2} - 83 \beta_{1} + 88\)\()/2\)
\(\nu^{6}\)\(=\)\(68 \beta_{6} + 208 \beta_{5} - 132 \beta_{4} + 101 \beta_{3} + 187 \beta_{2} + 179 \beta_{1} + 1109\)

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \)
$3$ \( T^{7} \)
$5$ \( ( 1 + T )^{7} \)
$7$ \( 256 + 732 T + 516 T^{2} - 77 T^{3} - 134 T^{4} - 16 T^{5} + 6 T^{6} + T^{7} \)
$11$ \( -384 + 1728 T - 1928 T^{2} + 556 T^{3} + 128 T^{4} - 50 T^{5} - 2 T^{6} + T^{7} \)
$13$ \( 14336 - 14368 T + 2072 T^{2} + 1412 T^{3} - 248 T^{4} - 58 T^{5} + 6 T^{6} + T^{7} \)
$17$ \( 2688 + 6564 T - 7840 T^{2} + 1599 T^{3} + 364 T^{4} - 88 T^{5} - 4 T^{6} + T^{7} \)
$19$ \( -3072 - 6720 T + 6656 T^{2} + 1540 T^{3} - 516 T^{4} - 74 T^{5} + 8 T^{6} + T^{7} \)
$23$ \( ( 1 + T )^{7} \)
$29$ \( -336 - 4188 T + 296 T^{2} + 2509 T^{3} - 102 T^{4} - 98 T^{5} + 2 T^{6} + T^{7} \)
$31$ \( -294144 - 86352 T + 45280 T^{2} + 7009 T^{3} - 1224 T^{4} - 162 T^{5} + 8 T^{6} + T^{7} \)
$37$ \( 65536 + 60680 T + 3326 T^{2} - 7033 T^{3} - 1634 T^{4} - 40 T^{5} + 16 T^{6} + T^{7} \)
$41$ \( 390144 - 180768 T - 14948 T^{2} + 10989 T^{3} - 2 T^{4} - 190 T^{5} + 2 T^{6} + T^{7} \)
$43$ \( 248832 - 217728 T + 26784 T^{2} + 9360 T^{3} - 1176 T^{4} - 164 T^{5} + 10 T^{6} + T^{7} \)
$47$ \( -4096 - 12544 T - 10496 T^{2} - 244 T^{3} + 1492 T^{4} - 162 T^{5} - 8 T^{6} + T^{7} \)
$53$ \( 310472 + 41764 T - 50978 T^{2} + 1431 T^{3} + 1740 T^{4} - 144 T^{5} - 10 T^{6} + T^{7} \)
$59$ \( -19528 + 17836 T + 17586 T^{2} - 18743 T^{3} + 2850 T^{4} + 26 T^{5} - 24 T^{6} + T^{7} \)
$61$ \( 1757504 - 440368 T - 130992 T^{2} + 21820 T^{3} + 1988 T^{4} - 278 T^{5} - 8 T^{6} + T^{7} \)
$67$ \( -383968 + 209664 T + 224154 T^{2} + 1271 T^{3} - 4578 T^{4} - 176 T^{5} + 20 T^{6} + T^{7} \)
$71$ \( 1146496 - 317512 T - 117750 T^{2} + 16401 T^{3} + 2426 T^{4} - 290 T^{5} - 8 T^{6} + T^{7} \)
$73$ \( 2552736 - 948624 T - 90152 T^{2} + 32396 T^{3} + 808 T^{4} - 326 T^{5} - 2 T^{6} + T^{7} \)
$79$ \( 73728 - 233472 T + 14336 T^{2} + 18944 T^{3} - 256 T^{4} - 272 T^{5} + 2 T^{6} + T^{7} \)
$83$ \( 300512 + 199088 T - 93734 T^{2} - 5929 T^{3} + 2780 T^{4} - 28 T^{5} - 22 T^{6} + T^{7} \)
$89$ \( 3662848 + 306432 T - 344704 T^{2} + 15232 T^{3} + 4704 T^{4} - 276 T^{5} - 16 T^{6} + T^{7} \)
$97$ \( -284928 - 311232 T - 12800 T^{2} + 16416 T^{3} + 480 T^{4} - 240 T^{5} - 4 T^{6} + T^{7} \)
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