Properties

Label 7942.2.a.y
Level $7942$
Weight $2$
Character orbit 7942.a
Self dual yes
Analytic conductor $63.417$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7942.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.4171892853\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta q^{3} + q^{4} + \beta q^{6} - \beta q^{7} + q^{8} + 3 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta q^{3} + q^{4} + \beta q^{6} - \beta q^{7} + q^{8} + 3 q^{9} - q^{11} + \beta q^{12} + ( - 2 \beta + 1) q^{13} - \beta q^{14} + q^{16} + ( - \beta - 2) q^{17} + 3 q^{18} - 6 q^{21} - q^{22} + (2 \beta - 4) q^{23} + \beta q^{24} - 5 q^{25} + ( - 2 \beta + 1) q^{26} - \beta q^{28} + (2 \beta - 3) q^{29} - 2 q^{31} + q^{32} - \beta q^{33} + ( - \beta - 2) q^{34} + 3 q^{36} + ( - \beta - 6) q^{37} + (\beta - 12) q^{39} + ( - \beta + 10) q^{41} - 6 q^{42} + (\beta + 7) q^{43} - q^{44} + (2 \beta - 4) q^{46} + (\beta - 9) q^{47} + \beta q^{48} - q^{49} - 5 q^{50} + ( - 2 \beta - 6) q^{51} + ( - 2 \beta + 1) q^{52} + (4 \beta - 2) q^{53} - \beta q^{56} + (2 \beta - 3) q^{58} + ( - 2 \beta + 2) q^{59} + (2 \beta + 3) q^{61} - 2 q^{62} - 3 \beta q^{63} + q^{64} - \beta q^{66} + (\beta - 12) q^{67} + ( - \beta - 2) q^{68} + ( - 4 \beta + 12) q^{69} + ( - \beta - 5) q^{71} + 3 q^{72} + ( - \beta - 12) q^{73} + ( - \beta - 6) q^{74} - 5 \beta q^{75} + \beta q^{77} + (\beta - 12) q^{78} + ( - \beta - 2) q^{79} - 9 q^{81} + ( - \beta + 10) q^{82} + ( - \beta - 3) q^{83} - 6 q^{84} + (\beta + 7) q^{86} + ( - 3 \beta + 12) q^{87} - q^{88} + 9 q^{89} + ( - \beta + 12) q^{91} + (2 \beta - 4) q^{92} - 2 \beta q^{93} + (\beta - 9) q^{94} + \beta q^{96} + ( - 2 \beta + 1) q^{97} - q^{98} - 3 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 6 q^{9} - 2 q^{11} + 2 q^{13} + 2 q^{16} - 4 q^{17} + 6 q^{18} - 12 q^{21} - 2 q^{22} - 8 q^{23} - 10 q^{25} + 2 q^{26} - 6 q^{29} - 4 q^{31} + 2 q^{32} - 4 q^{34} + 6 q^{36} - 12 q^{37} - 24 q^{39} + 20 q^{41} - 12 q^{42} + 14 q^{43} - 2 q^{44} - 8 q^{46} - 18 q^{47} - 2 q^{49} - 10 q^{50} - 12 q^{51} + 2 q^{52} - 4 q^{53} - 6 q^{58} + 4 q^{59} + 6 q^{61} - 4 q^{62} + 2 q^{64} - 24 q^{67} - 4 q^{68} + 24 q^{69} - 10 q^{71} + 6 q^{72} - 24 q^{73} - 12 q^{74} - 24 q^{78} - 4 q^{79} - 18 q^{81} + 20 q^{82} - 6 q^{83} - 12 q^{84} + 14 q^{86} + 24 q^{87} - 2 q^{88} + 18 q^{89} + 24 q^{91} - 8 q^{92} - 18 q^{94} + 2 q^{97} - 2 q^{98} - 6 q^{99}+O(q^{100}) \) 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
−2.44949
2.44949
1.00000 −2.44949 1.00000 0 −2.44949 2.44949 1.00000 3.00000 0
1.2 1.00000 2.44949 1.00000 0 2.44949 −2.44949 1.00000 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(11\) \(1\)
\(19\) \(-1\)

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 7942.2.a.y 2
19.b odd 2 1 7942.2.a.v 2
19.d odd 6 2 418.2.e.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.e.g 4 19.d odd 6 2
7942.2.a.v 2 19.b odd 2 1
7942.2.a.y 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7942))\):

\( T_{3}^{2} - 6 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} - 23 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 6 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 6 \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 2T - 23 \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 2 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 8T - 8 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T - 15 \) Copy content Toggle raw display
$31$ \( (T + 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 12T + 30 \) Copy content Toggle raw display
$41$ \( T^{2} - 20T + 94 \) Copy content Toggle raw display
$43$ \( T^{2} - 14T + 43 \) Copy content Toggle raw display
$47$ \( T^{2} + 18T + 75 \) Copy content Toggle raw display
$53$ \( T^{2} + 4T - 92 \) Copy content Toggle raw display
$59$ \( T^{2} - 4T - 20 \) Copy content Toggle raw display
$61$ \( T^{2} - 6T - 15 \) Copy content Toggle raw display
$67$ \( T^{2} + 24T + 138 \) Copy content Toggle raw display
$71$ \( T^{2} + 10T + 19 \) Copy content Toggle raw display
$73$ \( T^{2} + 24T + 138 \) Copy content Toggle raw display
$79$ \( T^{2} + 4T - 2 \) Copy content Toggle raw display
$83$ \( T^{2} + 6T + 3 \) Copy content Toggle raw display
$89$ \( (T - 9)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 2T - 23 \) Copy content Toggle raw display
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