Properties

Label 8379.2.a.bw
Level $8379$
Weight $2$
Character orbit 8379.a
Self dual yes
Analytic conductor $66.907$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 8379 = 3^{2} \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8379.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(66.9066518536\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.13068.1
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 171)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{3} + 2) q^{4} + ( - \beta_{2} - \beta_1) q^{5} + ( - 2 \beta_{2} - 3 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{3} + 2) q^{4} + ( - \beta_{2} - \beta_1) q^{5} + ( - 2 \beta_{2} - 3 \beta_1) q^{8} + (2 \beta_{3} + 2) q^{10} + ( - \beta_{2} + \beta_1) q^{11} - 2 q^{13} + (3 \beta_{3} + 4) q^{16} + ( - \beta_{2} - \beta_1) q^{17} - q^{19} + ( - 2 \beta_{2} - 6 \beta_1) q^{20} - 6 q^{22} - 2 \beta_{2} q^{23} + (\beta_{3} + 2) q^{25} + 2 \beta_1 q^{26} - 2 \beta_{2} q^{29} + ( - 2 \beta_{3} + 2) q^{31} + ( - 2 \beta_{2} - 7 \beta_1) q^{32} + (2 \beta_{3} + 2) q^{34} + (2 \beta_{3} + 4) q^{37} + \beta_1 q^{38} + (4 \beta_{3} + 16) q^{40} - 2 \beta_1 q^{41} + ( - 3 \beta_{3} - 1) q^{43} + (2 \beta_{2} + 4 \beta_1) q^{44} + (2 \beta_{3} - 4) q^{46} + ( - 3 \beta_{2} - \beta_1) q^{47} + ( - 2 \beta_{2} - 5 \beta_1) q^{50} + ( - 2 \beta_{3} - 4) q^{52} + (2 \beta_{2} - 4 \beta_1) q^{53} + ( - 3 \beta_{3} + 3) q^{55} + (2 \beta_{3} - 4) q^{58} + 4 \beta_1 q^{59} + (3 \beta_{3} - 5) q^{61} + (4 \beta_{2} + 4 \beta_1) q^{62} + (3 \beta_{3} + 16) q^{64} + (2 \beta_{2} + 2 \beta_1) q^{65} - 4 q^{67} + ( - 2 \beta_{2} - 6 \beta_1) q^{68} - 4 \beta_{2} q^{71} + (3 \beta_{3} - 5) q^{73} + ( - 4 \beta_{2} - 10 \beta_1) q^{74} + ( - \beta_{3} - 2) q^{76} - 4 q^{79} + ( - 4 \beta_{2} - 16 \beta_1) q^{80} + (2 \beta_{3} + 8) q^{82} + ( - 2 \beta_{2} - 4 \beta_1) q^{83} + (\beta_{3} + 7) q^{85} + (6 \beta_{2} + 10 \beta_1) q^{86} - 6 \beta_{3} q^{88} + ( - 2 \beta_{2} + 4 \beta_1) q^{89} - 2 \beta_1 q^{92} + (4 \beta_{3} - 2) q^{94} + (\beta_{2} + \beta_1) q^{95} + ( - 4 \beta_{3} + 6) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 10 q^{4} + 12 q^{10} - 8 q^{13} + 22 q^{16} - 4 q^{19} - 24 q^{22} + 10 q^{25} + 4 q^{31} + 12 q^{34} + 20 q^{37} + 72 q^{40} - 10 q^{43} - 12 q^{46} - 20 q^{52} + 6 q^{55} - 12 q^{58} - 14 q^{61} + 70 q^{64} - 16 q^{67} - 14 q^{73} - 10 q^{76} - 16 q^{79} + 36 q^{82} + 30 q^{85} - 12 q^{88} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 6x^{2} - x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} - \nu^{2} - 5\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{3} + 2\nu^{2} + 4\nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + \nu^{2} + 7\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 2\beta_{2} + 3\beta _1 + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} + \beta_{2} + 5\beta _1 + 4 \) 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
3.04374
−0.548230
−1.82405
0.328543
−2.71519 0 5.37228 −3.22060 0 0 −9.15640 0 8.74456
1.2 −1.27582 0 −0.372281 2.15121 0 0 3.02661 0 −2.74456
1.3 1.27582 0 −0.372281 −2.15121 0 0 −3.02661 0 −2.74456
1.4 2.71519 0 5.37228 3.22060 0 0 9.15640 0 8.74456
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(19\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8379.2.a.bw 4
3.b odd 2 1 inner 8379.2.a.bw 4
7.b odd 2 1 171.2.a.e 4
21.c even 2 1 171.2.a.e 4
28.d even 2 1 2736.2.a.bf 4
35.c odd 2 1 4275.2.a.bp 4
84.h odd 2 1 2736.2.a.bf 4
105.g even 2 1 4275.2.a.bp 4
133.c even 2 1 3249.2.a.bf 4
399.h odd 2 1 3249.2.a.bf 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.a.e 4 7.b odd 2 1
171.2.a.e 4 21.c even 2 1
2736.2.a.bf 4 28.d even 2 1
2736.2.a.bf 4 84.h odd 2 1
3249.2.a.bf 4 133.c even 2 1
3249.2.a.bf 4 399.h odd 2 1
4275.2.a.bp 4 35.c odd 2 1
4275.2.a.bp 4 105.g even 2 1
8379.2.a.bw 4 1.a even 1 1 trivial
8379.2.a.bw 4 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8379))\):

\( T_{2}^{4} - 9T_{2}^{2} + 12 \) Copy content Toggle raw display
\( T_{5}^{4} - 15T_{5}^{2} + 48 \) Copy content Toggle raw display
\( T_{11}^{4} - 27T_{11}^{2} + 108 \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display
\( T_{17}^{4} - 15T_{17}^{2} + 48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 9T^{2} + 12 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 15T^{2} + 48 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 27T^{2} + 108 \) Copy content Toggle raw display
$13$ \( (T + 2)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 15T^{2} + 48 \) Copy content Toggle raw display
$19$ \( (T + 1)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 48T^{2} + 48 \) Copy content Toggle raw display
$29$ \( T^{4} - 48T^{2} + 48 \) Copy content Toggle raw display
$31$ \( (T^{2} - 2 T - 32)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 10 T - 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 36T^{2} + 192 \) Copy content Toggle raw display
$43$ \( (T^{2} + 5 T - 68)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 99T^{2} + 1452 \) Copy content Toggle raw display
$53$ \( T^{4} - 240 T^{2} + 13872 \) Copy content Toggle raw display
$59$ \( T^{4} - 144T^{2} + 3072 \) Copy content Toggle raw display
$61$ \( (T^{2} + 7 T - 62)^{2} \) Copy content Toggle raw display
$67$ \( (T + 4)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} - 192T^{2} + 768 \) Copy content Toggle raw display
$73$ \( (T^{2} + 7 T - 62)^{2} \) Copy content Toggle raw display
$79$ \( (T + 4)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} - 144T^{2} + 432 \) Copy content Toggle raw display
$89$ \( T^{4} - 240 T^{2} + 13872 \) Copy content Toggle raw display
$97$ \( (T^{2} - 8 T - 116)^{2} \) Copy content Toggle raw display
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