Properties

Label 2842.2.a.q
Level $2842$
Weight $2$
Character orbit 2842.a
Self dual yes
Analytic conductor $22.693$
Analytic rank $1$
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: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q - q^{2} + (\beta_{3} + 2 \beta_1) q^{3} + q^{4} - \beta_1 q^{5} + ( - \beta_{3} - 2 \beta_1) q^{6} - q^{8} + (3 \beta_{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + (\beta_{3} + 2 \beta_1) q^{3} + q^{4} - \beta_1 q^{5} + ( - \beta_{3} - 2 \beta_1) q^{6} - q^{8} + (3 \beta_{2} + 3) q^{9} + \beta_1 q^{10} + ( - 2 \beta_{2} - 3) q^{11} + (\beta_{3} + 2 \beta_1) q^{12} + 3 \beta_{3} q^{13} + ( - 2 \beta_{2} - 3) q^{15} + q^{16} + ( - 4 \beta_{3} - 4 \beta_1) q^{17} + ( - 3 \beta_{2} - 3) q^{18} + ( - 3 \beta_{3} - \beta_1) q^{19} - \beta_1 q^{20} + (2 \beta_{2} + 3) q^{22} + 2 q^{23} + ( - \beta_{3} - 2 \beta_1) q^{24} + (\beta_{2} - 3) q^{25} - 3 \beta_{3} q^{26} + 9 \beta_1 q^{27} - q^{29} + (2 \beta_{2} + 3) q^{30} + ( - 4 \beta_{3} + \beta_1) q^{31} - q^{32} + ( - 3 \beta_{3} - 12 \beta_1) q^{33} + (4 \beta_{3} + 4 \beta_1) q^{34} + (3 \beta_{2} + 3) q^{36} + ( - 4 \beta_{2} - 2) q^{37} + (3 \beta_{3} + \beta_1) q^{38} - 3 \beta_{2} q^{39} + \beta_1 q^{40} + 4 \beta_{3} q^{41} + (\beta_{2} - 2) q^{43} + ( - 2 \beta_{2} - 3) q^{44} + ( - 3 \beta_{3} - 9 \beta_1) q^{45} - 2 q^{46} + ( - 4 \beta_{3} + \beta_1) q^{47} + (\beta_{3} + 2 \beta_1) q^{48} + ( - \beta_{2} + 3) q^{50} + ( - 4 \beta_{2} - 12) q^{51} + 3 \beta_{3} q^{52} + (2 \beta_{2} - 11) q^{53} - 9 \beta_1 q^{54} + (2 \beta_{3} + 7 \beta_1) q^{55} + (\beta_{2} - 3) q^{57} + q^{58} + ( - 2 \beta_{3} - 6 \beta_1) q^{59} + ( - 2 \beta_{2} - 3) q^{60} - 6 \beta_1 q^{61} + (4 \beta_{3} - \beta_1) q^{62} + q^{64} + 3 q^{65} + (3 \beta_{3} + 12 \beta_1) q^{66} + ( - 2 \beta_{2} + 8) q^{67} + ( - 4 \beta_{3} - 4 \beta_1) q^{68} + (2 \beta_{3} + 4 \beta_1) q^{69} - 4 q^{71} + ( - 3 \beta_{2} - 3) q^{72} + (6 \beta_{3} - 2 \beta_1) q^{73} + (4 \beta_{2} + 2) q^{74} + ( - 3 \beta_{3} - 3 \beta_1) q^{75} + ( - 3 \beta_{3} - \beta_1) q^{76} + 3 \beta_{2} q^{78} + 9 \beta_{2} q^{79} - \beta_1 q^{80} + (9 \beta_{2} + 18) q^{81} - 4 \beta_{3} q^{82} + 8 \beta_{3} q^{83} + (4 \beta_{2} + 4) q^{85} + ( - \beta_{2} + 2) q^{86} + ( - \beta_{3} - 2 \beta_1) q^{87} + (2 \beta_{2} + 3) q^{88} + ( - 6 \beta_{3} + 4 \beta_1) q^{89} + (3 \beta_{3} + 9 \beta_1) q^{90} + 2 q^{92} + (6 \beta_{2} + 3) q^{93} + (4 \beta_{3} - \beta_1) q^{94} + (\beta_{2} - 1) q^{95} + ( - \beta_{3} - 2 \beta_1) q^{96} + ( - 6 \beta_{3} + 2 \beta_1) q^{97} + ( - 15 \beta_{2} - 27) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 12 q^{9} - 12 q^{11} - 12 q^{15} + 4 q^{16} - 12 q^{18} + 12 q^{22} + 8 q^{23} - 12 q^{25} - 4 q^{29} + 12 q^{30} - 4 q^{32} + 12 q^{36} - 8 q^{37} - 8 q^{43} - 12 q^{44} - 8 q^{46} + 12 q^{50} - 48 q^{51} - 44 q^{53} - 12 q^{57} + 4 q^{58} - 12 q^{60} + 4 q^{64} + 12 q^{65} + 32 q^{67} - 16 q^{71} - 12 q^{72} + 8 q^{74} + 72 q^{81} + 16 q^{85} + 8 q^{86} + 12 q^{88} + 8 q^{92} + 12 q^{93} - 4 q^{95} - 108 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{24} + \zeta_{24}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 4\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 4\beta_1 \) 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
−1.93185
0.517638
−0.517638
1.93185
−1.00000 −3.34607 1.00000 1.93185 3.34607 0 −1.00000 8.19615 −1.93185
1.2 −1.00000 −0.896575 1.00000 −0.517638 0.896575 0 −1.00000 −2.19615 0.517638
1.3 −1.00000 0.896575 1.00000 0.517638 −0.896575 0 −1.00000 −2.19615 −0.517638
1.4 −1.00000 3.34607 1.00000 −1.93185 −3.34607 0 −1.00000 8.19615 1.93185
\(n\): 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.q 4
7.b odd 2 1 inner 2842.2.a.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2842.2.a.q 4 1.a even 1 1 trivial
2842.2.a.q 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2842))\):

\( T_{3}^{4} - 12T_{3}^{2} + 9 \) Copy content Toggle raw display
\( T_{5}^{4} - 4T_{5}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 12T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} - 4T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 6 T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 36T^{2} + 81 \) Copy content Toggle raw display
$17$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 28T^{2} + 4 \) Copy content Toggle raw display
$23$ \( (T - 2)^{4} \) Copy content Toggle raw display
$29$ \( (T + 1)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} - 84T^{2} + 1089 \) Copy content Toggle raw display
$37$ \( (T^{2} + 4 T - 44)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 64T^{2} + 256 \) Copy content Toggle raw display
$43$ \( (T^{2} + 4 T + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 84T^{2} + 1089 \) Copy content Toggle raw display
$53$ \( (T^{2} + 22 T + 109)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 112T^{2} + 64 \) Copy content Toggle raw display
$61$ \( T^{4} - 144T^{2} + 1296 \) Copy content Toggle raw display
$67$ \( (T^{2} - 16 T + 52)^{2} \) Copy content Toggle raw display
$71$ \( (T + 4)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 208T^{2} + 7744 \) Copy content Toggle raw display
$79$ \( (T^{2} - 243)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 256T^{2} + 4096 \) Copy content Toggle raw display
$89$ \( T^{4} - 304 T^{2} + 21904 \) Copy content Toggle raw display
$97$ \( T^{4} - 208T^{2} + 7744 \) Copy content Toggle raw display
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