Properties

Label 2842.2.a.h
Level $2842$
Weight $2$
Character orbit 2842.a
Self dual yes
Analytic conductor $22.693$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + 2 \beta q^{3} + q^{4} - 2 \beta q^{6} - q^{8} + 5 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + 2 \beta q^{3} + q^{4} - 2 \beta q^{6} - q^{8} + 5 q^{9} + 2 \beta q^{12} + q^{16} - 5 q^{18} + 8 q^{23} - 2 \beta q^{24} - 5 q^{25} + 4 \beta q^{27} + q^{29} + \beta q^{31} - q^{32} + 5 q^{36} + 8 q^{37} + 6 \beta q^{41} + 4 q^{43} - 8 q^{46} - \beta q^{47} + 2 \beta q^{48} + 5 q^{50} - 6 q^{53} - 4 \beta q^{54} - q^{58} - \beta q^{59} + 5 \beta q^{61} - \beta q^{62} + q^{64} - 2 q^{67} + 16 \beta q^{69} - 5 q^{72} + 6 \beta q^{73} - 8 q^{74} - 10 \beta q^{75} + 2 q^{79} + q^{81} - 6 \beta q^{82} - \beta q^{83} - 4 q^{86} + 2 \beta q^{87} + 2 \beta q^{89} + 8 q^{92} + 4 q^{93} + \beta q^{94} - 2 \beta q^{96} + 4 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 10 q^{9} + 2 q^{16} - 10 q^{18} + 16 q^{23} - 10 q^{25} + 2 q^{29} - 2 q^{32} + 10 q^{36} + 16 q^{37} + 8 q^{43} - 16 q^{46} + 10 q^{50} - 12 q^{53} - 2 q^{58} + 2 q^{64} - 4 q^{67} - 10 q^{72} - 16 q^{74} + 4 q^{79} + 2 q^{81} - 8 q^{86} + 16 q^{92} + 8 q^{93}+O(q^{100}) \) 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
−1.41421
1.41421
−1.00000 −2.82843 1.00000 0 2.82843 0 −1.00000 5.00000 0
1.2 −1.00000 2.82843 1.00000 0 −2.82843 0 −1.00000 5.00000 0
\(n\): 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.h 2
7.b odd 2 1 inner 2842.2.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2842.2.a.h 2 1.a even 1 1 trivial
2842.2.a.h 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2842))\):

\( T_{3}^{2} - 8 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 8 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( (T - 8)^{2} \) Copy content Toggle raw display
$29$ \( (T - 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 2 \) Copy content Toggle raw display
$37$ \( (T - 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 72 \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 2 \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 2 \) Copy content Toggle raw display
$61$ \( T^{2} - 50 \) Copy content Toggle raw display
$67$ \( (T + 2)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 72 \) Copy content Toggle raw display
$79$ \( (T - 2)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 2 \) Copy content Toggle raw display
$89$ \( T^{2} - 8 \) Copy content Toggle raw display
$97$ \( T^{2} - 32 \) Copy content Toggle raw display
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