Properties

Label 2842.2.a.u
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: 5.5.974241.1
Defining polynomial: \( x^{5} - x^{4} - 9x^{3} - 2x^{2} + 11x + 3 \) 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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 9x^{3} - 2x^{2} + 11x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 2\nu^{3} - 6\nu^{2} + 2\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 3\nu^{3} - 4\nu^{2} + 7\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} + \beta_{3} + 2\beta_{2} + 9\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{4} + 3\beta_{3} + 10\beta_{2} + 28\beta _1 + 25 \) 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
−1.76647
1.13397
−0.275357
−1.56579
3.47364
−1.00000 −2.42629 1.00000 0.260736 2.42629 0 −1.00000 2.88687 −0.260736
1.2 −1.00000 −1.07327 1.00000 2.57622 1.07327 0 −1.00000 −1.84808 −2.57622
1.3 −1.00000 1.16240 1.00000 −1.76651 −1.16240 0 −1.00000 −1.64882 1.76651
1.4 −1.00000 2.23997 1.00000 2.71905 −2.23997 0 −1.00000 2.01749 −2.71905
1.5 −1.00000 3.09718 1.00000 −2.78950 −3.09718 0 −1.00000 6.59255 2.78950
\(n\): 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.u 5
7.b odd 2 1 2842.2.a.t 5
7.d odd 6 2 406.2.e.d 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
406.2.e.d 10 7.d odd 6 2
2842.2.a.t 5 7.b odd 2 1
2842.2.a.u 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2842))\):

\( T_{3}^{5} - 3T_{3}^{4} - 7T_{3}^{3} + 21T_{3}^{2} + 6T_{3} - 21 \) Copy content Toggle raw display
\( T_{5}^{5} - T_{5}^{4} - 12T_{5}^{3} + 9T_{5}^{2} + 33T_{5} - 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{5} - 3 T^{4} - 7 T^{3} + 21 T^{2} + \cdots - 21 \) Copy content Toggle raw display
$5$ \( T^{5} - T^{4} - 12 T^{3} + 9 T^{2} + \cdots - 9 \) Copy content Toggle raw display
$7$ \( T^{5} \) Copy content Toggle raw display
$11$ \( T^{5} - 4 T^{4} - 15 T^{3} + 82 T^{2} + \cdots - 27 \) Copy content Toggle raw display
$13$ \( T^{5} - 46 T^{3} - 27 T^{2} + \cdots + 129 \) Copy content Toggle raw display
$17$ \( T^{5} - 10 T^{4} + 3 T^{3} + 176 T^{2} + \cdots - 69 \) Copy content Toggle raw display
$19$ \( T^{5} - 8 T^{4} - 29 T^{3} + 148 T^{2} + \cdots + 379 \) Copy content Toggle raw display
$23$ \( T^{5} - 9 T^{4} - 27 T^{3} + 387 T^{2} + \cdots + 567 \) Copy content Toggle raw display
$29$ \( (T + 1)^{5} \) Copy content Toggle raw display
$31$ \( T^{5} - 3 T^{4} - 52 T^{3} - 3 T^{2} + \cdots + 477 \) Copy content Toggle raw display
$37$ \( T^{5} - 10 T^{4} - 17 T^{3} + \cdots + 1979 \) Copy content Toggle raw display
$41$ \( T^{5} - 19 T^{4} + 1897 T^{2} + \cdots + 18957 \) Copy content Toggle raw display
$43$ \( T^{5} - 3 T^{4} - 121 T^{3} + \cdots - 557 \) Copy content Toggle raw display
$47$ \( T^{5} - 36 T^{4} + 399 T^{3} + \cdots + 74601 \) Copy content Toggle raw display
$53$ \( T^{5} + 3 T^{4} - 171 T^{3} + \cdots - 117 \) Copy content Toggle raw display
$59$ \( T^{5} + 5 T^{4} - 297 T^{3} + \cdots - 65799 \) Copy content Toggle raw display
$61$ \( T^{5} - 3 T^{4} - 154 T^{3} + \cdots - 15547 \) Copy content Toggle raw display
$67$ \( T^{5} + 2 T^{4} - 224 T^{3} + \cdots - 26561 \) Copy content Toggle raw display
$71$ \( T^{5} - 2 T^{4} - 105 T^{3} + \cdots + 423 \) Copy content Toggle raw display
$73$ \( T^{5} + 2 T^{4} - 105 T^{3} + \cdots + 387 \) Copy content Toggle raw display
$79$ \( T^{5} - 17 T^{4} - 147 T^{3} + \cdots - 84693 \) Copy content Toggle raw display
$83$ \( T^{5} - 30 T^{4} + 327 T^{3} + \cdots - 2133 \) Copy content Toggle raw display
$89$ \( T^{5} - 29 T^{4} + 135 T^{3} + \cdots + 48681 \) Copy content Toggle raw display
$97$ \( T^{5} - 187 T^{3} + 186 T^{2} + \cdots - 9 \) Copy content Toggle raw display
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