Properties

Label 2842.2.a.n
Level $2842$
Weight $2$
Character orbit 2842.a
Self dual yes
Analytic conductor $22.693$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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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: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 406)
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.

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

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 4 \) 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
−0.363328
3.12489
−1.76156
−1.00000 −3.14134 1.00000 −2.36333 3.14134 0 −1.00000 6.86799 2.36333
1.2 −1.00000 −0.484862 1.00000 1.12489 0.484862 0 −1.00000 −2.76491 −1.12489
1.3 −1.00000 2.62620 1.00000 −3.76156 −2.62620 0 −1.00000 3.89692 3.76156
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 2842.2.a.n 3
7.b odd 2 1 406.2.a.f 3
21.c even 2 1 3654.2.a.bc 3
28.d even 2 1 3248.2.a.u 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
406.2.a.f 3 7.b odd 2 1
2842.2.a.n 3 1.a even 1 1 trivial
3248.2.a.u 3 28.d even 2 1
3654.2.a.bc 3 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2842))\):

\( T_{3}^{3} + T_{3}^{2} - 8T_{3} - 4 \) Copy content Toggle raw display
\( T_{5}^{3} + 5T_{5}^{2} + 2T_{5} - 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + T^{2} - 8T - 4 \) Copy content Toggle raw display
$5$ \( T^{3} + 5 T^{2} + 2 T - 10 \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - T^{2} - 24 T - 20 \) Copy content Toggle raw display
$13$ \( T^{3} + 7 T^{2} + 10 T + 2 \) Copy content Toggle raw display
$17$ \( T^{3} + 8 T^{2} + 2 T - 64 \) Copy content Toggle raw display
$19$ \( T^{3} - 8 T^{2} - 12 T + 80 \) Copy content Toggle raw display
$23$ \( T^{3} - 6 T^{2} - 28 T + 136 \) Copy content Toggle raw display
$29$ \( (T - 1)^{3} \) Copy content Toggle raw display
$31$ \( T^{3} + 5 T^{2} - 22 T - 106 \) Copy content Toggle raw display
$37$ \( T^{3} - 10 T^{2} + 8 T + 80 \) Copy content Toggle raw display
$41$ \( T^{3} + 16 T^{2} + 18 T - 352 \) Copy content Toggle raw display
$43$ \( T^{3} + 15 T^{2} + 32 T - 32 \) Copy content Toggle raw display
$47$ \( T^{3} + 5 T^{2} + 2 T - 10 \) Copy content Toggle raw display
$53$ \( T^{3} - 11 T^{2} + 16 T + 68 \) Copy content Toggle raw display
$59$ \( T^{3} + 28 T^{2} + 242 T + 664 \) Copy content Toggle raw display
$61$ \( T^{3} - 10 T^{2} + 8 T + 80 \) Copy content Toggle raw display
$67$ \( T^{3} + 14 T^{2} - 32 T - 128 \) Copy content Toggle raw display
$71$ \( T^{3} - 8 T^{2} - 100 T - 176 \) Copy content Toggle raw display
$73$ \( T^{3} - 16 T^{2} + 66 T - 80 \) Copy content Toggle raw display
$79$ \( T^{3} + 3 T^{2} - 72 T - 40 \) Copy content Toggle raw display
$83$ \( T^{3} - 12 T^{2} - 142 T + 568 \) Copy content Toggle raw display
$89$ \( T^{3} - 10 T^{2} - 158 T + 220 \) Copy content Toggle raw display
$97$ \( T^{3} - 16 T^{2} - 78 T + 976 \) Copy content Toggle raw display
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