Properties

Label 2842.2.a.o
Level $2842$
Weight $2$
Character orbit 2842.a
Self dual yes
Analytic conductor $22.693$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{13})\)
Defining polynomial: \( x^{4} - 2x^{3} - 9x^{2} + 10x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 9x^{2} + 10x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + \nu^{2} - 11\nu - 8 ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{3} + 3\nu^{2} + 17\nu - 9 ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{3} + 6\nu^{2} + 44\nu - 23 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - 2\beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 4\beta _1 + 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} - 11\beta_{2} + 3\beta _1 + 8 \) 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
−2.71699
0.888562
3.71699
0.111438
−1.00000 −1.41421 1.00000 −1.84240 1.41421 0 −1.00000 −1.00000 1.84240
1.2 −1.00000 −1.41421 1.00000 3.25662 1.41421 0 −1.00000 −1.00000 −3.25662
1.3 −1.00000 1.41421 1.00000 −3.25662 −1.41421 0 −1.00000 −1.00000 3.25662
1.4 −1.00000 1.41421 1.00000 1.84240 −1.41421 0 −1.00000 −1.00000 −1.84240
\(n\): e.g. 2-40 or 990-1000
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.o 4
7.b odd 2 1 inner 2842.2.a.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2842.2.a.o 4 1.a even 1 1 trivial
2842.2.a.o 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2842))\):

\( T_{3}^{2} - 2 \) Copy content Toggle raw display
\( T_{5}^{4} - 14T_{5}^{2} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - 14T^{2} + 36 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 2 T - 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 14T^{2} + 36 \) Copy content Toggle raw display
$17$ \( T^{4} - 38T^{2} + 36 \) Copy content Toggle raw display
$19$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$23$ \( (T - 2)^{4} \) Copy content Toggle raw display
$29$ \( (T - 1)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} - 118T^{2} + 3364 \) Copy content Toggle raw display
$37$ \( (T + 4)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 38T^{2} + 36 \) Copy content Toggle raw display
$43$ \( (T^{2} - 52)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 62T^{2} + 324 \) Copy content Toggle raw display
$53$ \( (T^{2} - 8 T - 36)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 22T^{2} + 4 \) Copy content Toggle raw display
$61$ \( T^{4} - 224T^{2} + 9216 \) Copy content Toggle raw display
$67$ \( (T^{2} - 12 T - 16)^{2} \) Copy content Toggle raw display
$71$ \( (T - 6)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 62T^{2} + 324 \) Copy content Toggle raw display
$79$ \( (T^{2} + 10 T + 12)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 118T^{2} + 3364 \) Copy content Toggle raw display
$89$ \( T^{4} - 238 T^{2} + 11236 \) Copy content Toggle raw display
$97$ \( T^{4} - 118T^{2} + 3364 \) Copy content Toggle raw display
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