Properties

Label 2842.2.a.p
Level $2842$
Weight $2$
Character orbit 2842.a
Self dual yes
Analytic conductor $22.693$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2842 = 2 \cdot 7^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2842.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.6934842544\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.9248.1
Defining polynomial: \( x^{4} - 5x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - \beta_{3} q^{3} + q^{4} - \beta_1 q^{5} + \beta_{3} q^{6} - q^{8} - \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - \beta_{3} q^{3} + q^{4} - \beta_1 q^{5} + \beta_{3} q^{6} - q^{8} - \beta_{2} q^{9} + \beta_1 q^{10} + ( - \beta_{2} - 3) q^{11} - \beta_{3} q^{12} + (2 \beta_{3} + \beta_1) q^{13} + (\beta_{2} + 1) q^{15} + q^{16} + (\beta_{3} - 3 \beta_1) q^{17} + \beta_{2} q^{18} + ( - 2 \beta_{3} + 4 \beta_1) q^{19} - \beta_1 q^{20} + (\beta_{2} + 3) q^{22} + (4 \beta_{2} + 2) q^{23} + \beta_{3} q^{24} + (\beta_{2} - 2) q^{25} + ( - 2 \beta_{3} - \beta_1) q^{26} + (\beta_{3} + 2 \beta_1) q^{27} + q^{29} + ( - \beta_{2} - 1) q^{30} + (2 \beta_{3} + \beta_1) q^{31} - q^{32} + (\beta_{3} + 2 \beta_1) q^{33} + ( - \beta_{3} + 3 \beta_1) q^{34} - \beta_{2} q^{36} + ( - 2 \beta_{2} + 2) q^{37} + (2 \beta_{3} - 4 \beta_1) q^{38} + (\beta_{2} - 7) q^{39} + \beta_1 q^{40} + (\beta_{3} - 3 \beta_1) q^{41} + (3 \beta_{2} + 3) q^{43} + ( - \beta_{2} - 3) q^{44} + (\beta_{3} + \beta_1) q^{45} + ( - 4 \beta_{2} - 2) q^{46} - \beta_1 q^{47} - \beta_{3} q^{48} + ( - \beta_{2} + 2) q^{50} + 4 \beta_{2} q^{51} + (2 \beta_{3} + \beta_1) q^{52} + (\beta_{2} - 9) q^{53} + ( - \beta_{3} - 2 \beta_1) q^{54} + (\beta_{3} + 4 \beta_1) q^{55} + ( - 6 \beta_{2} + 2) q^{57} - q^{58} + ( - \beta_{3} + 3 \beta_1) q^{59} + (\beta_{2} + 1) q^{60} + 2 \beta_1 q^{61} + ( - 2 \beta_{3} - \beta_1) q^{62} + q^{64} + ( - 3 \beta_{2} - 5) q^{65} + ( - \beta_{3} - 2 \beta_1) q^{66} + ( - 2 \beta_{2} - 10) q^{67} + (\beta_{3} - 3 \beta_1) q^{68} + (6 \beta_{3} - 8 \beta_1) q^{69} + ( - 6 \beta_{2} - 4) q^{71} + \beta_{2} q^{72} + (5 \beta_{3} + 5 \beta_1) q^{73} + (2 \beta_{2} - 2) q^{74} + (4 \beta_{3} - 2 \beta_1) q^{75} + ( - 2 \beta_{3} + 4 \beta_1) q^{76} + ( - \beta_{2} + 7) q^{78} + ( - 5 \beta_{2} - 5) q^{79} - \beta_1 q^{80} + (2 \beta_{2} - 5) q^{81} + ( - \beta_{3} + 3 \beta_1) q^{82} + (3 \beta_{3} + 3 \beta_1) q^{83} + (2 \beta_{2} + 8) q^{85} + ( - 3 \beta_{2} - 3) q^{86} - \beta_{3} q^{87} + (\beta_{2} + 3) q^{88} + (3 \beta_{3} - 3 \beta_1) q^{89} + ( - \beta_{3} - \beta_1) q^{90} + (4 \beta_{2} + 2) q^{92} + (\beta_{2} - 7) q^{93} + \beta_1 q^{94} + ( - 2 \beta_{2} - 10) q^{95} + \beta_{3} q^{96} + (\beta_{3} - 7 \beta_1) q^{97} + (2 \beta_{2} + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 2 q^{9} - 10 q^{11} + 2 q^{15} + 4 q^{16} - 2 q^{18} + 10 q^{22} - 10 q^{25} + 4 q^{29} - 2 q^{30} - 4 q^{32} + 2 q^{36} + 12 q^{37} - 30 q^{39} + 6 q^{43} - 10 q^{44} + 10 q^{50} - 8 q^{51} - 38 q^{53} + 20 q^{57} - 4 q^{58} + 2 q^{60} + 4 q^{64} - 14 q^{65} - 36 q^{67} - 4 q^{71} - 2 q^{72} - 12 q^{74} + 30 q^{78} - 10 q^{79} - 24 q^{81} + 28 q^{85} - 6 q^{86} + 10 q^{88} - 30 q^{93} - 36 q^{95} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 4\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 4\beta_1 \) 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
−0.662153
2.13578
−2.13578
0.662153
−1.00000 −2.35829 1.00000 0.662153 2.35829 0 −1.00000 2.56155 −0.662153
1.2 −1.00000 −1.19935 1.00000 −2.13578 1.19935 0 −1.00000 −1.56155 2.13578
1.3 −1.00000 1.19935 1.00000 2.13578 −1.19935 0 −1.00000 −1.56155 −2.13578
1.4 −1.00000 2.35829 1.00000 −0.662153 −2.35829 0 −1.00000 2.56155 0.662153
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)
\(29\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2842.2.a.p 4
7.b odd 2 1 inner 2842.2.a.p 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2842.2.a.p 4 1.a even 1 1 trivial
2842.2.a.p 4 7.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2842))\):

\( T_{3}^{4} - 7T_{3}^{2} + 8 \) Copy content Toggle raw display
\( T_{5}^{4} - 5T_{5}^{2} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 7T^{2} + 8 \) Copy content Toggle raw display
$5$ \( T^{4} - 5T^{2} + 2 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 5 T + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 37T^{2} + 338 \) Copy content Toggle raw display
$17$ \( T^{4} - 46T^{2} + 512 \) Copy content Toggle raw display
$19$ \( T^{4} - 92T^{2} + 2048 \) Copy content Toggle raw display
$23$ \( (T^{2} - 68)^{2} \) Copy content Toggle raw display
$29$ \( (T - 1)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} - 37T^{2} + 338 \) Copy content Toggle raw display
$37$ \( (T^{2} - 6 T - 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 46T^{2} + 512 \) Copy content Toggle raw display
$43$ \( (T^{2} - 3 T - 36)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 5T^{2} + 2 \) Copy content Toggle raw display
$53$ \( (T^{2} + 19 T + 86)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 46T^{2} + 512 \) Copy content Toggle raw display
$61$ \( T^{4} - 20T^{2} + 32 \) Copy content Toggle raw display
$67$ \( (T^{2} + 18 T + 64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 2 T - 152)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 350 T^{2} + 20000 \) Copy content Toggle raw display
$79$ \( (T^{2} + 5 T - 100)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 126T^{2} + 2592 \) Copy content Toggle raw display
$89$ \( T^{4} - 90T^{2} + 648 \) Copy content Toggle raw display
$97$ \( T^{4} - 238T^{2} + 9248 \) Copy content Toggle raw display
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