Properties

Label 2842.2.a.r
Level $2842$
Weight $2$
Character orbit 2842.a
Self dual yes
Analytic conductor $22.693$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(4\)
Coefficient field: 4.4.11348.1
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 5x^{2} + x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - \nu^{2} - 5\nu + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu^{2} - 3\nu + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{3} - 3\beta_{2} + 7\beta _1 + 9 ) / 2 \) 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
−0.589216
−1.77571
2.64119
0.723742
1.00000 −3.39434 1.00000 2.65282 −3.39434 0 1.00000 8.52156 2.65282
1.2 1.00000 −1.12631 1.00000 −0.153156 −1.12631 0 1.00000 −1.73143 −0.153156
1.3 1.00000 0.757235 1.00000 −3.97587 0.757235 0 1.00000 −2.42659 −3.97587
1.4 1.00000 2.76342 1.00000 2.47620 2.76342 0 1.00000 4.63646 2.47620
\(n\): 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.r 4
7.b odd 2 1 406.2.a.g 4
21.c even 2 1 3654.2.a.bg 4
28.d even 2 1 3248.2.a.x 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
406.2.a.g 4 7.b odd 2 1
2842.2.a.r 4 1.a even 1 1 trivial
3248.2.a.x 4 28.d even 2 1
3654.2.a.bg 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2842))\):

\( T_{3}^{4} + T_{3}^{3} - 10T_{3}^{2} - 4T_{3} + 8 \) Copy content Toggle raw display
\( T_{5}^{4} - T_{5}^{3} - 14T_{5}^{2} + 24T_{5} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + T^{3} - 10 T^{2} - 4 T + 8 \) Copy content Toggle raw display
$5$ \( T^{4} - T^{3} - 14 T^{2} + 24 T + 4 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 7 T^{3} + 8 T^{2} + 16 T - 16 \) Copy content Toggle raw display
$13$ \( T^{4} - 7 T^{3} - 26 T^{2} + 176 T + 28 \) Copy content Toggle raw display
$17$ \( T^{4} - 20 T^{2} + 36 T - 16 \) Copy content Toggle raw display
$19$ \( T^{4} + 2 T^{3} - 20 T^{2} - 8 T + 32 \) Copy content Toggle raw display
$23$ \( T^{4} - 6 T^{3} - 8 T^{2} + 64 T - 32 \) Copy content Toggle raw display
$29$ \( (T - 1)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} - 7 T^{3} - 62 T^{2} + 388 T + 356 \) Copy content Toggle raw display
$37$ \( T^{4} + 12 T^{3} + 24 T^{2} - 64 T - 16 \) Copy content Toggle raw display
$41$ \( T^{4} + 4 T^{3} - 148 T^{2} + \cdots - 448 \) Copy content Toggle raw display
$43$ \( T^{4} + 5 T^{3} - 32 T^{2} - 64 T + 256 \) Copy content Toggle raw display
$47$ \( T^{4} - 11 T^{3} - 10 T^{2} + \cdots - 188 \) Copy content Toggle raw display
$53$ \( T^{4} - 5 T^{3} - 46 T^{2} + 260 T - 248 \) Copy content Toggle raw display
$59$ \( T^{4} + 16 T^{3} - 16 T^{2} + \cdots - 2752 \) Copy content Toggle raw display
$61$ \( T^{4} - 34 T^{3} + 368 T^{2} + \cdots - 1376 \) Copy content Toggle raw display
$67$ \( T^{4} - 2 T^{3} - 320 T^{2} + \cdots + 24832 \) Copy content Toggle raw display
$71$ \( T^{4} - 24 T^{3} - 56 T^{2} + \cdots - 28544 \) Copy content Toggle raw display
$73$ \( T^{4} - 24 T^{3} + 132 T^{2} + \cdots - 64 \) Copy content Toggle raw display
$79$ \( T^{4} + 9 T^{3} - 188 T^{2} + \cdots - 544 \) Copy content Toggle raw display
$83$ \( T^{4} - 8 T^{3} - 296 T^{2} + \cdots + 17024 \) Copy content Toggle raw display
$89$ \( T^{4} + 2 T^{3} - 124 T^{2} + \cdots + 824 \) Copy content Toggle raw display
$97$ \( T^{4} - 4 T^{3} - 148 T^{2} + \cdots - 448 \) Copy content Toggle raw display
show more
show less