Properties

Label 2842.2.a.s
Level $2842$
Weight $2$
Character orbit 2842.a
Self dual yes
Analytic conductor $22.693$
Analytic rank $1$
Dimension $5$
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: \(5\)
Coefficient field: 5.5.369849.1
Defining polynomial: \( x^{5} - x^{4} - 6x^{3} + 3x^{2} + 3x - 1 \) 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + (\beta_{3} - \beta_1) q^{3} + q^{4} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{5} + ( - \beta_{3} + \beta_1) q^{6} - q^{8} + (\beta_{2} + 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + (\beta_{3} - \beta_1) q^{3} + q^{4} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{5} + ( - \beta_{3} + \beta_1) q^{6} - q^{8} + (\beta_{2} + 2 \beta_1) q^{9} + (\beta_{4} - \beta_{3} - \beta_{2} + 1) q^{10} + ( - \beta_{4} - \beta_{3} + 2 \beta_1) q^{11} + (\beta_{3} - \beta_1) q^{12} + (2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{13} + (\beta_{4} - 2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{15} + q^{16} + (\beta_{4} - \beta_{3} + \beta_1 - 1) q^{17} + ( - \beta_{2} - 2 \beta_1) q^{18} + (\beta_{3} - \beta_1 - 1) q^{19} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{20} + (\beta_{4} + \beta_{3} - 2 \beta_1) q^{22} + (\beta_{4} - 3 \beta_{3} - 3) q^{23} + ( - \beta_{3} + \beta_1) q^{24} + (2 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{25} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{26} + (2 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 2) q^{27} + q^{29} + ( - \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{30} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{31} - q^{32} + (\beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 5) q^{33} + ( - \beta_{4} + \beta_{3} - \beta_1 + 1) q^{34} + (\beta_{2} + 2 \beta_1) q^{36} + ( - \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 4) q^{37} + ( - \beta_{3} + \beta_1 + 1) q^{38} + (\beta_{4} + 2 \beta_{2} + 3 \beta_1 + 5) q^{39} + (\beta_{4} - \beta_{3} - \beta_{2} + 1) q^{40} + (2 \beta_{3} + 2 \beta_{2} - \beta_1) q^{41} + (2 \beta_{4} - 3 \beta_{2} - 2 \beta_1 + 3) q^{43} + ( - \beta_{4} - \beta_{3} + 2 \beta_1) q^{44} + ( - \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{45} + ( - \beta_{4} + 3 \beta_{3} + 3) q^{46} + ( - 2 \beta_{4} - \beta_{3} + 3 \beta_1 - 5) q^{47} + (\beta_{3} - \beta_1) q^{48} + ( - 2 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{50} + (\beta_{3} - \beta_{2} - \beta_1 - 2) q^{51} + (2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{52} + (\beta_{3} + 3 \beta_1 + 2) q^{53} + ( - 2 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 2) q^{54} + (\beta_{4} + 2 \beta_{2}) q^{55} + ( - \beta_{3} + \beta_{2} + 3 \beta_1 + 3) q^{57} - q^{58} + ( - 2 \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{59} + (\beta_{4} - 2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{60} + (2 \beta_{4} + 2 \beta_{3} + 4 \beta_1 + 3) q^{61} + (2 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{62} + q^{64} + (4 \beta_{4} - 5 \beta_{3} - 2 \beta_1 - 3) q^{65} + ( - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 5) q^{66} + (2 \beta_{4} - \beta_{3} - 3 \beta_{2} - \beta_1 + 5) q^{67} + (\beta_{4} - \beta_{3} + \beta_1 - 1) q^{68} + ( - 3 \beta_{4} - \beta_{3} + 3 \beta_1 - 5) q^{69} + ( - 3 \beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{71} + ( - \beta_{2} - 2 \beta_1) q^{72} + ( - \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - 2) q^{73} + (\beta_{3} + 3 \beta_{2} - 2 \beta_1 + 4) q^{74} + ( - 5 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + 5 \beta_1 - 5) q^{75} + (\beta_{3} - \beta_1 - 1) q^{76} + ( - \beta_{4} - 2 \beta_{2} - 3 \beta_1 - 5) q^{78} + ( - 5 \beta_{3} - \beta_{2} - 2 \beta_1 - 1) q^{79} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{80} + ( - 5 \beta_{4} - \beta_{3} + 3 \beta_{2} + 8 \beta_1 + 1) q^{81} + ( - 2 \beta_{3} - 2 \beta_{2} + \beta_1) q^{82} + (3 \beta_{4} - 2 \beta_{3} - \beta_{2} - 10) q^{83} + ( - \beta_{4} + 2 \beta_{3} - \beta_{2} + 3 \beta_1 - 2) q^{85} + ( - 2 \beta_{4} + 3 \beta_{2} + 2 \beta_1 - 3) q^{86} + (\beta_{3} - \beta_1) q^{87} + (\beta_{4} + \beta_{3} - 2 \beta_1) q^{88} + (3 \beta_{4} - 4 \beta_{3} - 3 \beta_{2} + 1) q^{89} + (\beta_{3} + \beta_{2} - 3 \beta_1 - 1) q^{90} + (\beta_{4} - 3 \beta_{3} - 3) q^{92} + ( - 3 \beta_{4} + 3 \beta_{3} - 2 \beta_1 - 3) q^{93} + (2 \beta_{4} + \beta_{3} - 3 \beta_1 + 5) q^{94} + (2 \beta_{4} - 3 \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{95} + ( - \beta_{3} + \beta_1) q^{96} + (2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 2) q^{97} + ( - 3 \beta_{4} - 2 \beta_{3} + 6 \beta_{2} + 11 \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{2} - 3 q^{3} + 5 q^{4} - 5 q^{5} + 3 q^{6} - 5 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 5 q^{2} - 3 q^{3} + 5 q^{4} - 5 q^{5} + 3 q^{6} - 5 q^{8} + 4 q^{9} + 5 q^{10} + 4 q^{11} - 3 q^{12} - 2 q^{13} + 6 q^{15} + 5 q^{16} - 2 q^{17} - 4 q^{18} - 8 q^{19} - 5 q^{20} - 4 q^{22} - 9 q^{23} + 3 q^{24} + 8 q^{25} + 2 q^{26} - 15 q^{27} + 5 q^{29} - 6 q^{30} + 15 q^{31} - 5 q^{32} - 29 q^{33} + 2 q^{34} + 4 q^{36} - 22 q^{37} + 8 q^{38} + 32 q^{39} + 5 q^{40} - q^{41} + 7 q^{43} + 4 q^{44} + 8 q^{45} + 9 q^{46} - 20 q^{47} - 3 q^{48} - 8 q^{50} - 15 q^{51} - 2 q^{52} + 11 q^{53} + 15 q^{54} + 4 q^{55} + 22 q^{57} - 5 q^{58} - 13 q^{59} + 6 q^{60} + 15 q^{61} - 15 q^{62} + 5 q^{64} - 7 q^{65} + 29 q^{66} + 20 q^{67} - 2 q^{68} - 20 q^{69} - 4 q^{71} - 4 q^{72} + 22 q^{74} - 20 q^{75} - 8 q^{76} - 32 q^{78} + q^{79} - 5 q^{80} + 21 q^{81} + q^{82} - 48 q^{83} - 13 q^{85} - 7 q^{86} - 3 q^{87} - 4 q^{88} + 7 q^{89} - 8 q^{90} - 9 q^{92} - 23 q^{93} + 20 q^{94} + 11 q^{95} + 3 q^{96} - 6 q^{97} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 6x^{3} + 3x^{2} + 3x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{4} - 6\nu^{2} - 3\nu + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - \nu^{3} - 6\nu^{2} + 3\nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 7\nu^{2} - 2\nu + 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{4} + \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{3} + \beta_{2} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{4} + 7\beta_{2} + 9\beta _1 + 10 \) 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
2.68337
−0.697329
0.840202
−2.12550
0.299252
−1.00000 −3.31071 1.00000 0.889787 3.31071 0 −1.00000 7.96079 −0.889787
1.2 −1.00000 −1.73672 1.00000 −4.25045 1.73672 0 −1.00000 0.0161818 4.25045
1.3 −1.00000 −0.650011 1.00000 −2.94407 0.650011 0 −1.00000 −2.57749 2.94407
1.4 −1.00000 0.655021 1.00000 2.17277 −0.655021 0 −1.00000 −2.57095 −2.17277
1.5 −1.00000 2.04241 1.00000 −0.868034 −2.04241 0 −1.00000 1.17146 0.868034
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.s 5
7.b odd 2 1 2842.2.a.v 5
7.d odd 6 2 406.2.e.c 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
406.2.e.c 10 7.d odd 6 2
2842.2.a.s 5 1.a even 1 1 trivial
2842.2.a.v 5 7.b odd 2 1

Hecke kernels

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

\( T_{3}^{5} + 3T_{3}^{4} - 5T_{3}^{3} - 13T_{3}^{2} + 2T_{3} + 5 \) Copy content Toggle raw display
\( T_{5}^{5} + 5T_{5}^{4} - 4T_{5}^{3} - 31T_{5}^{2} + 3T_{5} + 21 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{5} + 3 T^{4} - 5 T^{3} - 13 T^{2} + \cdots + 5 \) Copy content Toggle raw display
$5$ \( T^{5} + 5 T^{4} - 4 T^{3} - 31 T^{2} + \cdots + 21 \) Copy content Toggle raw display
$7$ \( T^{5} \) Copy content Toggle raw display
$11$ \( T^{5} - 4 T^{4} - 27 T^{3} + 76 T^{2} + \cdots - 21 \) Copy content Toggle raw display
$13$ \( T^{5} + 2 T^{4} - 48 T^{3} - 83 T^{2} + \cdots + 875 \) Copy content Toggle raw display
$17$ \( T^{5} + 2 T^{4} - 19 T^{3} - 16 T^{2} + \cdots + 21 \) Copy content Toggle raw display
$19$ \( T^{5} + 8 T^{4} + 17 T^{3} - 22 T - 7 \) Copy content Toggle raw display
$23$ \( T^{5} + 9 T^{4} - 25 T^{3} - 371 T^{2} + \cdots - 435 \) Copy content Toggle raw display
$29$ \( (T - 1)^{5} \) Copy content Toggle raw display
$31$ \( T^{5} - 15 T^{4} + 20 T^{3} + \cdots - 2083 \) Copy content Toggle raw display
$37$ \( T^{5} + 22 T^{4} + 49 T^{3} + \cdots - 37693 \) Copy content Toggle raw display
$41$ \( T^{5} + T^{4} - 60 T^{3} + 119 T^{2} + \cdots - 213 \) Copy content Toggle raw display
$43$ \( T^{5} - 7 T^{4} - 111 T^{3} + \cdots - 10547 \) Copy content Toggle raw display
$47$ \( T^{5} + 20 T^{4} + 71 T^{3} + \cdots + 2421 \) Copy content Toggle raw display
$53$ \( T^{5} - 11 T^{4} - 29 T^{3} + \cdots - 3435 \) Copy content Toggle raw display
$59$ \( T^{5} + 13 T^{4} - 31 T^{3} + \cdots + 225 \) Copy content Toggle raw display
$61$ \( T^{5} - 15 T^{4} - 178 T^{3} + \cdots - 1327 \) Copy content Toggle raw display
$67$ \( T^{5} - 20 T^{4} + 54 T^{3} + \cdots - 7193 \) Copy content Toggle raw display
$71$ \( T^{5} + 4 T^{4} - 91 T^{3} + \cdots - 3675 \) Copy content Toggle raw display
$73$ \( T^{5} - 165 T^{3} + 312 T^{2} + \cdots - 24575 \) Copy content Toggle raw display
$79$ \( T^{5} - T^{4} - 215 T^{3} + 473 T^{2} + \cdots - 5041 \) Copy content Toggle raw display
$83$ \( T^{5} + 48 T^{4} + 833 T^{3} + \cdots - 2349 \) Copy content Toggle raw display
$89$ \( T^{5} - 7 T^{4} - 119 T^{3} + \cdots + 4335 \) Copy content Toggle raw display
$97$ \( T^{5} + 6 T^{4} - 141 T^{3} + \cdots + 10525 \) Copy content Toggle raw display
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