Properties

Label 2842.2.a.z
Level $2842$
Weight $2$
Character orbit 2842.a
Self dual yes
Analytic conductor $22.693$
Analytic rank $0$
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: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.1019601.1
Defining polynomial: \( x^{5} - 10x^{3} - x^{2} + 24x + 7 \) 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.

\(f(q)\) \(=\) \( q + q^{2} + (\beta_{3} + 1) q^{3} + q^{4} + ( - \beta_{3} + \beta_1 + 1) q^{5} + (\beta_{3} + 1) q^{6} + q^{8} + (\beta_{4} - \beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + (\beta_{3} + 1) q^{3} + q^{4} + ( - \beta_{3} + \beta_1 + 1) q^{5} + (\beta_{3} + 1) q^{6} + q^{8} + (\beta_{4} - \beta_{2} + 2) q^{9} + ( - \beta_{3} + \beta_1 + 1) q^{10} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{11} + (\beta_{3} + 1) q^{12} + (\beta_{3} + \beta_{2} + 2) q^{13} + (2 \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{15} + q^{16} + ( - \beta_{2} - \beta_1 + 2) q^{17} + (\beta_{4} - \beta_{2} + 2) q^{18} + ( - \beta_{3} - 2 \beta_1) q^{19} + ( - \beta_{3} + \beta_1 + 1) q^{20} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{22} + ( - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{23} + (\beta_{3} + 1) q^{24} + ( - \beta_{4} - 3 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{25} + (\beta_{3} + \beta_{2} + 2) q^{26} + (2 \beta_{4} + \beta_{3} + \beta_{2} + 3) q^{27} - q^{29} + (2 \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{30} + ( - 2 \beta_{4} + \beta_{3} - \beta_{2} + 3) q^{31} + q^{32} + ( - 3 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{33} + ( - \beta_{2} - \beta_1 + 2) q^{34} + (\beta_{4} - \beta_{2} + 2) q^{36} + (2 \beta_{4} + \beta_{2} - 2) q^{37} + ( - \beta_{3} - 2 \beta_1) q^{38} + (\beta_{4} - \beta_{2} + \beta_1 + 4) q^{39} + ( - \beta_{3} + \beta_1 + 1) q^{40} + (\beta_{4} - \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 5) q^{41} + (3 \beta_{4} - 2 \beta_{3} - \beta_{2} - 1) q^{43} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{44} + (3 \beta_{4} - 3 \beta_{3} - \beta_{2}) q^{45} + ( - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{46} + ( - 3 \beta_{3} + 2 \beta_1 + 2) q^{47} + (\beta_{3} + 1) q^{48} + ( - \beta_{4} - 3 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{50} + ( - \beta_{4} + 3 \beta_{3} - \beta_{2} - 2 \beta_1 + 3) q^{51} + (\beta_{3} + \beta_{2} + 2) q^{52} + (\beta_{3} + 2 \beta_{2} - 2 \beta_1 + 1) q^{53} + (2 \beta_{4} + \beta_{3} + \beta_{2} + 3) q^{54} + ( - 4 \beta_{4} + 5 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 2) q^{55} + ( - 3 \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 - 6) q^{57} - q^{58} + ( - 2 \beta_{3} + \beta_{2} - 3) q^{59} + (2 \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{60} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 3) q^{61} + ( - 2 \beta_{4} + \beta_{3} - \beta_{2} + 3) q^{62} + q^{64} + ( - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{65} + ( - 3 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{66} + ( - \beta_{4} + \beta_{3} - \beta_{2} - 4 \beta_1) q^{67} + ( - \beta_{2} - \beta_1 + 2) q^{68} + ( - 2 \beta_{4} - 3 \beta_{2} - 3 \beta_1 + 4) q^{69} + (\beta_{3} - 4 \beta_1 - 4) q^{71} + (\beta_{4} - \beta_{2} + 2) q^{72} + ( - 3 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + \beta_1) q^{73} + (2 \beta_{4} + \beta_{2} - 2) q^{74} + ( - 3 \beta_{4} + 6 \beta_{3} + 4 \beta_{2} - \beta_1 - 6) q^{75} + ( - \beta_{3} - 2 \beta_1) q^{76} + (\beta_{4} - \beta_{2} + \beta_1 + 4) q^{78} + ( - 2 \beta_{4} + \beta_{2} - 3) q^{79} + ( - \beta_{3} + \beta_1 + 1) q^{80} + (2 \beta_{4} + 3 \beta_{3} + 4 \beta_{2} + 3 \beta_1 + 3) q^{81} + (\beta_{4} - \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 5) q^{82} + ( - \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + \beta_1 + 6) q^{83} + (2 \beta_{4} - 4 \beta_{3} + \beta_1 - 3) q^{85} + (3 \beta_{4} - 2 \beta_{3} - \beta_{2} - 1) q^{86} + ( - \beta_{3} - 1) q^{87} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{88} + (\beta_{4} - 3 \beta_{3} - 2 \beta_{2} + 5 \beta_1 + 3) q^{89} + (3 \beta_{4} - 3 \beta_{3} - \beta_{2}) q^{90} + ( - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{92} + ( - 3 \beta_{4} + \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 5) q^{93} + ( - 3 \beta_{3} + 2 \beta_1 + 2) q^{94} + (2 \beta_{4} - 5 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 3) q^{95} + (\beta_{3} + 1) q^{96} + (3 \beta_{4} - 2 \beta_{3} - \beta_{2} + 8) q^{97} + (\beta_{4} + 2 \beta_{3} + \beta_{2} - 3 \beta_1 - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{2} + 3 q^{3} + 5 q^{4} + 7 q^{5} + 3 q^{6} + 5 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 5 q^{2} + 3 q^{3} + 5 q^{4} + 7 q^{5} + 3 q^{6} + 5 q^{8} + 8 q^{9} + 7 q^{10} + 3 q^{12} + 10 q^{13} - 10 q^{15} + 5 q^{16} + 8 q^{17} + 8 q^{18} + 2 q^{19} + 7 q^{20} + q^{23} + 3 q^{24} + 12 q^{25} + 10 q^{26} + 15 q^{27} - 5 q^{29} - 10 q^{30} + 11 q^{31} + 5 q^{32} + 9 q^{33} + 8 q^{34} + 8 q^{36} - 8 q^{37} + 2 q^{38} + 18 q^{39} + 7 q^{40} + 23 q^{41} - 3 q^{43} + 4 q^{45} + q^{46} + 16 q^{47} + 3 q^{48} + 12 q^{50} + 7 q^{51} + 10 q^{52} + 7 q^{53} + 15 q^{54} + 6 q^{55} - 34 q^{57} - 5 q^{58} - 9 q^{59} - 10 q^{60} + 15 q^{61} + 11 q^{62} + 5 q^{64} + 5 q^{65} + 9 q^{66} - 4 q^{67} + 8 q^{68} + 14 q^{69} - 22 q^{71} + 8 q^{72} - 8 q^{74} - 34 q^{75} + 2 q^{76} + 18 q^{78} - 13 q^{79} + 7 q^{80} + 17 q^{81} + 23 q^{82} + 28 q^{83} - 7 q^{85} - 3 q^{86} - 3 q^{87} + 17 q^{89} + 4 q^{90} + q^{92} + 17 q^{93} + 16 q^{94} - 9 q^{95} + 3 q^{96} + 42 q^{97} - 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 10x^{3} - x^{2} + 24x + 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{3} + 5\nu + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + \nu^{2} - 5\nu - 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} + 2\nu^{3} - 5\nu^{2} - 10\nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 5\beta_{3} + 7\beta_{2} + 20 \) 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
−0.298978
−2.48141
2.20194
−1.80092
2.37936
1.00000 −2.44245 1.00000 4.14347 −2.44245 0 1.00000 2.96555 4.14347
1.2 1.00000 −0.714579 1.00000 0.233169 −0.714579 0 1.00000 −2.48938 0.233169
1.3 1.00000 0.515089 1.00000 3.68685 0.515089 0 1.00000 −2.73468 3.68685
1.4 1.00000 2.40697 1.00000 −2.20789 2.40697 0 1.00000 2.79350 −2.20789
1.5 1.00000 3.23497 1.00000 1.14439 3.23497 0 1.00000 7.46501 1.14439
\(n\): 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.z 5
7.b odd 2 1 2842.2.a.x 5
7.c even 3 2 406.2.e.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
406.2.e.a 10 7.c even 3 2
2842.2.a.x 5 7.b odd 2 1
2842.2.a.z 5 1.a even 1 1 trivial

Hecke kernels

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

\( T_{3}^{5} - 3T_{3}^{4} - 7T_{3}^{3} + 19T_{3}^{2} + 6T_{3} - 7 \) Copy content Toggle raw display
\( T_{5}^{5} - 7T_{5}^{4} + 6T_{5}^{3} + 35T_{5}^{2} - 47T_{5} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{5} - 3 T^{4} - 7 T^{3} + 19 T^{2} + \cdots - 7 \) Copy content Toggle raw display
$5$ \( T^{5} - 7 T^{4} + 6 T^{3} + 35 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{5} \) Copy content Toggle raw display
$11$ \( T^{5} - 33 T^{3} - 54 T^{2} - 12 T + 9 \) Copy content Toggle raw display
$13$ \( T^{5} - 10 T^{4} + 28 T^{3} + \cdots + 103 \) Copy content Toggle raw display
$17$ \( T^{5} - 8 T^{4} + 9 T^{3} + 22 T^{2} + \cdots + 3 \) Copy content Toggle raw display
$19$ \( T^{5} - 2 T^{4} - 63 T^{3} + \cdots - 1623 \) Copy content Toggle raw display
$23$ \( T^{5} - T^{4} - 39 T^{3} - 97 T^{2} + \cdots + 33 \) Copy content Toggle raw display
$29$ \( (T + 1)^{5} \) Copy content Toggle raw display
$31$ \( T^{5} - 11 T^{4} - 24 T^{3} + \cdots + 387 \) Copy content Toggle raw display
$37$ \( T^{5} + 8 T^{4} - 47 T^{3} - 422 T^{2} + \cdots + 349 \) Copy content Toggle raw display
$41$ \( T^{5} - 23 T^{4} + 132 T^{3} + \cdots + 7113 \) Copy content Toggle raw display
$43$ \( T^{5} + 3 T^{4} - 163 T^{3} + \cdots + 16843 \) Copy content Toggle raw display
$47$ \( T^{5} - 16 T^{4} + 9 T^{3} + 584 T^{2} + \cdots + 63 \) Copy content Toggle raw display
$53$ \( T^{5} - 7 T^{4} - 57 T^{3} - 49 T^{2} + \cdots + 183 \) Copy content Toggle raw display
$59$ \( T^{5} + 9 T^{4} - 39 T^{3} - 291 T^{2} + \cdots + 657 \) Copy content Toggle raw display
$61$ \( T^{5} - 15 T^{4} - 10 T^{3} + \cdots - 5327 \) Copy content Toggle raw display
$67$ \( T^{5} + 4 T^{4} - 174 T^{3} + \cdots + 11481 \) Copy content Toggle raw display
$71$ \( T^{5} + 22 T^{4} + 51 T^{3} + \cdots + 18189 \) Copy content Toggle raw display
$73$ \( T^{5} - 247 T^{3} - 226 T^{2} + \cdots - 341 \) Copy content Toggle raw display
$79$ \( T^{5} + 13 T^{4} - 33 T^{3} + \cdots + 1563 \) Copy content Toggle raw display
$83$ \( T^{5} - 28 T^{4} + 213 T^{3} + \cdots + 25851 \) Copy content Toggle raw display
$89$ \( T^{5} - 17 T^{4} - 195 T^{3} + \cdots - 162771 \) Copy content Toggle raw display
$97$ \( T^{5} - 42 T^{4} + 539 T^{3} + \cdots + 15007 \) Copy content Toggle raw display
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