Properties

Label 2842.2.a.i
Level $2842$
Weight $2$
Character orbit 2842.a
Self dual yes
Analytic conductor $22.693$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \( x^{2} - 3 \) 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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - 2 q^{3} + q^{4} + ( - \beta - 1) q^{5} - 2 q^{6} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - 2 q^{3} + q^{4} + ( - \beta - 1) q^{5} - 2 q^{6} + q^{8} + q^{9} + ( - \beta - 1) q^{10} - 2 \beta q^{11} - 2 q^{12} + (\beta + 1) q^{13} + (2 \beta + 2) q^{15} + q^{16} + (3 \beta + 1) q^{17} + q^{18} + 2 \beta q^{19} + ( - \beta - 1) q^{20} - 2 \beta q^{22} + (4 \beta - 2) q^{23} - 2 q^{24} + (2 \beta - 1) q^{25} + (\beta + 1) q^{26} + 4 q^{27} - q^{29} + (2 \beta + 2) q^{30} + ( - 3 \beta - 3) q^{31} + q^{32} + 4 \beta q^{33} + (3 \beta + 1) q^{34} + q^{36} - 4 q^{37} + 2 \beta q^{38} + ( - 2 \beta - 2) q^{39} + ( - \beta - 1) q^{40} + ( - 5 \beta + 1) q^{41} + (4 \beta + 4) q^{43} - 2 \beta q^{44} + ( - \beta - 1) q^{45} + (4 \beta - 2) q^{46} + ( - 5 \beta - 1) q^{47} - 2 q^{48} + (2 \beta - 1) q^{50} + ( - 6 \beta - 2) q^{51} + (\beta + 1) q^{52} + ( - 4 \beta - 2) q^{53} + 4 q^{54} + (2 \beta + 6) q^{55} - 4 \beta q^{57} - q^{58} + (\beta - 9) q^{59} + (2 \beta + 2) q^{60} - 4 q^{61} + ( - 3 \beta - 3) q^{62} + q^{64} + ( - 2 \beta - 4) q^{65} + 4 \beta q^{66} + 4 \beta q^{67} + (3 \beta + 1) q^{68} + ( - 8 \beta + 4) q^{69} + ( - 4 \beta + 2) q^{71} + q^{72} + ( - \beta - 7) q^{73} - 4 q^{74} + ( - 4 \beta + 2) q^{75} + 2 \beta q^{76} + ( - 2 \beta - 2) q^{78} + ( - 2 \beta - 6) q^{79} + ( - \beta - 1) q^{80} - 11 q^{81} + ( - 5 \beta + 1) q^{82} + (9 \beta - 1) q^{83} + ( - 4 \beta - 10) q^{85} + (4 \beta + 4) q^{86} + 2 q^{87} - 2 \beta q^{88} + (3 \beta - 11) q^{89} + ( - \beta - 1) q^{90} + (4 \beta - 2) q^{92} + (6 \beta + 6) q^{93} + ( - 5 \beta - 1) q^{94} + ( - 2 \beta - 6) q^{95} - 2 q^{96} + ( - \beta + 5) q^{97} - 2 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 4 q^{3} + 2 q^{4} - 2 q^{5} - 4 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 4 q^{3} + 2 q^{4} - 2 q^{5} - 4 q^{6} + 2 q^{8} + 2 q^{9} - 2 q^{10} - 4 q^{12} + 2 q^{13} + 4 q^{15} + 2 q^{16} + 2 q^{17} + 2 q^{18} - 2 q^{20} - 4 q^{23} - 4 q^{24} - 2 q^{25} + 2 q^{26} + 8 q^{27} - 2 q^{29} + 4 q^{30} - 6 q^{31} + 2 q^{32} + 2 q^{34} + 2 q^{36} - 8 q^{37} - 4 q^{39} - 2 q^{40} + 2 q^{41} + 8 q^{43} - 2 q^{45} - 4 q^{46} - 2 q^{47} - 4 q^{48} - 2 q^{50} - 4 q^{51} + 2 q^{52} - 4 q^{53} + 8 q^{54} + 12 q^{55} - 2 q^{58} - 18 q^{59} + 4 q^{60} - 8 q^{61} - 6 q^{62} + 2 q^{64} - 8 q^{65} + 2 q^{68} + 8 q^{69} + 4 q^{71} + 2 q^{72} - 14 q^{73} - 8 q^{74} + 4 q^{75} - 4 q^{78} - 12 q^{79} - 2 q^{80} - 22 q^{81} + 2 q^{82} - 2 q^{83} - 20 q^{85} + 8 q^{86} + 4 q^{87} - 22 q^{89} - 2 q^{90} - 4 q^{92} + 12 q^{93} - 2 q^{94} - 12 q^{95} - 4 q^{96} + 10 q^{97}+O(q^{100}) \) 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
1.73205
−1.73205
1.00000 −2.00000 1.00000 −2.73205 −2.00000 0 1.00000 1.00000 −2.73205
1.2 1.00000 −2.00000 1.00000 0.732051 −2.00000 0 1.00000 1.00000 0.732051
\(n\): e.g. 2-40 or 990-1000
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.i 2
7.b odd 2 1 406.2.a.e 2
21.c even 2 1 3654.2.a.y 2
28.d even 2 1 3248.2.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
406.2.a.e 2 7.b odd 2 1
2842.2.a.i 2 1.a even 1 1 trivial
3248.2.a.o 2 28.d even 2 1
3654.2.a.y 2 21.c even 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( (T + 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2T - 2 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 12 \) Copy content Toggle raw display
$13$ \( T^{2} - 2T - 2 \) Copy content Toggle raw display
$17$ \( T^{2} - 2T - 26 \) Copy content Toggle raw display
$19$ \( T^{2} - 12 \) Copy content Toggle raw display
$23$ \( T^{2} + 4T - 44 \) Copy content Toggle raw display
$29$ \( (T + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 6T - 18 \) Copy content Toggle raw display
$37$ \( (T + 4)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 2T - 74 \) Copy content Toggle raw display
$43$ \( T^{2} - 8T - 32 \) Copy content Toggle raw display
$47$ \( T^{2} + 2T - 74 \) Copy content Toggle raw display
$53$ \( T^{2} + 4T - 44 \) Copy content Toggle raw display
$59$ \( T^{2} + 18T + 78 \) Copy content Toggle raw display
$61$ \( (T + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 48 \) Copy content Toggle raw display
$71$ \( T^{2} - 4T - 44 \) Copy content Toggle raw display
$73$ \( T^{2} + 14T + 46 \) Copy content Toggle raw display
$79$ \( T^{2} + 12T + 24 \) Copy content Toggle raw display
$83$ \( T^{2} + 2T - 242 \) Copy content Toggle raw display
$89$ \( T^{2} + 22T + 94 \) Copy content Toggle raw display
$97$ \( T^{2} - 10T + 22 \) Copy content Toggle raw display
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