Properties

Label 2842.2.a.bc
Level $2842$
Weight $2$
Character orbit 2842.a
Self dual yes
Analytic conductor $22.693$
Analytic rank $0$
Dimension $6$
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: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.373409792.1
Defining polynomial: \( x^{6} - 13x^{4} + 48x^{2} - 50 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 13x^{4} + 48x^{2} - 50 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 8\nu^{3} + 8\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} - 9 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{5} + 21\nu^{3} - 46\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{4} - 10\nu^{2} + 19 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + 2\beta_{2} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + 5\beta_{3} + 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{4} + 21\beta_{2} + 20\beta_1 \) 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
−1.93372
2.73329
−1.33784
1.33784
−2.73329
1.93372
1.00000 −3.06767 1.00000 −1.76375 −3.06767 0 1.00000 6.41061 −1.76375
1.2 1.00000 −2.21239 1.00000 −0.404393 −2.21239 0 1.00000 1.89468 −0.404393
1.3 1.00000 −0.833497 1.00000 3.96556 −0.833497 0 1.00000 −2.30528 3.96556
1.4 1.00000 0.833497 1.00000 −3.96556 0.833497 0 1.00000 −2.30528 −3.96556
1.5 1.00000 2.21239 1.00000 0.404393 2.21239 0 1.00000 1.89468 0.404393
1.6 1.00000 3.06767 1.00000 1.76375 3.06767 0 1.00000 6.41061 1.76375
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.bc 6
7.b odd 2 1 inner 2842.2.a.bc 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2842.2.a.bc 6 1.a even 1 1 trivial
2842.2.a.bc 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2842))\):

\( T_{3}^{6} - 15T_{3}^{4} + 56T_{3}^{2} - 32 \) Copy content Toggle raw display
\( T_{5}^{6} - 19T_{5}^{4} + 52T_{5}^{2} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 15 T^{4} + 56 T^{2} - 32 \) Copy content Toggle raw display
$5$ \( T^{6} - 19 T^{4} + 52 T^{2} - 8 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( (T^{3} + T^{2} - 36 T - 40)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} - 19 T^{4} + 52 T^{2} - 8 \) Copy content Toggle raw display
$17$ \( T^{6} - 28 T^{4} + 120 T^{2} + \cdots - 128 \) Copy content Toggle raw display
$19$ \( T^{6} - 52 T^{4} + 768 T^{2} + \cdots - 3200 \) Copy content Toggle raw display
$23$ \( (T^{3} - 10 T^{2} + 112)^{2} \) Copy content Toggle raw display
$29$ \( (T - 1)^{6} \) Copy content Toggle raw display
$31$ \( T^{6} - 99 T^{4} + 812 T^{2} + \cdots - 200 \) Copy content Toggle raw display
$37$ \( (T^{3} - 14 T^{2} - 12 T + 488)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 268 T^{4} + 23800 T^{2} + \cdots - 700928 \) Copy content Toggle raw display
$43$ \( (T^{3} + 3 T^{2} - 16 T - 16)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} - 147 T^{4} + 4748 T^{2} + \cdots - 42632 \) Copy content Toggle raw display
$53$ \( (T^{3} - 3 T^{2} - 16 T + 20)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 212 T^{4} + 10808 T^{2} + \cdots - 6272 \) Copy content Toggle raw display
$61$ \( T^{6} - 132 T^{4} + 4160 T^{2} + \cdots - 6272 \) Copy content Toggle raw display
$67$ \( (T^{3} - 6 T^{2} - 64 T + 128)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} - 4 T^{2} - 136 T - 448)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 28 T^{4} + 120 T^{2} + \cdots - 128 \) Copy content Toggle raw display
$79$ \( (T^{3} - 9 T^{2} + 8 T + 32)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} - 484 T^{4} + 62328 T^{2} + \cdots - 1229312 \) Copy content Toggle raw display
$89$ \( T^{6} - 296 T^{4} + 392 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$97$ \( T^{6} - 700 T^{4} + 144376 T^{2} + \cdots - 7311488 \) Copy content Toggle raw display
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