Properties

Label 7942.2.a.bj
Level $7942$
Weight $2$
Character orbit 7942.a
Self dual yes
Analytic conductor $63.417$
Analytic rank $0$
Dimension $3$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1940.1
Defining polynomial: \( x^{3} - 8x - 2 \) 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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta_1 q^{3} + q^{4} + 2 q^{5} + \beta_1 q^{6} + (\beta_1 + 2) q^{7} + q^{8} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta_1 q^{3} + q^{4} + 2 q^{5} + \beta_1 q^{6} + (\beta_1 + 2) q^{7} + q^{8} + (\beta_{2} + 2) q^{9} + 2 q^{10} + q^{11} + \beta_1 q^{12} + ( - \beta_{2} - 2) q^{13} + (\beta_1 + 2) q^{14} + 2 \beta_1 q^{15} + q^{16} + (\beta_{2} - \beta_1 + 3) q^{17} + (\beta_{2} + 2) q^{18} + 2 q^{20} + (\beta_{2} + 2 \beta_1 + 5) q^{21} + q^{22} + ( - 2 \beta_1 + 4) q^{23} + \beta_1 q^{24} - q^{25} + ( - \beta_{2} - 2) q^{26} + (2 \beta_1 + 2) q^{27} + (\beta_1 + 2) q^{28} + q^{29} + 2 \beta_1 q^{30} + (\beta_{2} - 2 \beta_1 - 1) q^{31} + q^{32} + \beta_1 q^{33} + (\beta_{2} - \beta_1 + 3) q^{34} + (2 \beta_1 + 4) q^{35} + (\beta_{2} + 2) q^{36} + ( - \beta_{2} + 3 \beta_1 - 1) q^{37} + ( - 5 \beta_1 - 2) q^{39} + 2 q^{40} + ( - \beta_{2} - \beta_1 - 3) q^{41} + (\beta_{2} + 2 \beta_1 + 5) q^{42} + ( - \beta_{2} - \beta_1 + 2) q^{43} + q^{44} + (2 \beta_{2} + 4) q^{45} + ( - 2 \beta_1 + 4) q^{46} + ( - 2 \beta_{2} + \beta_1 - 3) q^{47} + \beta_1 q^{48} + (\beta_{2} + 4 \beta_1 + 2) q^{49} - q^{50} + ( - \beta_{2} + 6 \beta_1 - 3) q^{51} + ( - \beta_{2} - 2) q^{52} - 4 \beta_1 q^{53} + (2 \beta_1 + 2) q^{54} + 2 q^{55} + (\beta_1 + 2) q^{56} + q^{58} + 2 \beta_1 q^{60} + ( - \beta_{2} - 2 \beta_1 + 4) q^{61} + (\beta_{2} - 2 \beta_1 - 1) q^{62} + (2 \beta_{2} + 5 \beta_1 + 6) q^{63} + q^{64} + ( - 2 \beta_{2} - 4) q^{65} + \beta_1 q^{66} - 3 \beta_1 q^{67} + (\beta_{2} - \beta_1 + 3) q^{68} + ( - 2 \beta_{2} + 4 \beta_1 - 10) q^{69} + (2 \beta_1 + 4) q^{70} + ( - \beta_{2} - \beta_1 + 2) q^{71} + (\beta_{2} + 2) q^{72} + ( - \beta_{2} - \beta_1 + 3) q^{73} + ( - \beta_{2} + 3 \beta_1 - 1) q^{74} - \beta_1 q^{75} + (\beta_1 + 2) q^{77} + ( - 5 \beta_1 - 2) q^{78} + (3 \beta_1 - 2) q^{79} + 2 q^{80} + ( - \beta_{2} + 2 \beta_1 + 4) q^{81} + ( - \beta_{2} - \beta_1 - 3) q^{82} + (\beta_{2} + 3 \beta_1 + 2) q^{83} + (\beta_{2} + 2 \beta_1 + 5) q^{84} + (2 \beta_{2} - 2 \beta_1 + 6) q^{85} + ( - \beta_{2} - \beta_1 + 2) q^{86} + \beta_1 q^{87} + q^{88} + (2 \beta_{2} - 2 \beta_1 - 1) q^{89} + (2 \beta_{2} + 4) q^{90} + ( - 2 \beta_{2} - 5 \beta_1 - 6) q^{91} + ( - 2 \beta_1 + 4) q^{92} + ( - 2 \beta_{2} + 2 \beta_1 - 8) q^{93} + ( - 2 \beta_{2} + \beta_1 - 3) q^{94} + \beta_1 q^{96} + ( - 2 \beta_{2} + 2 \beta_1 + 3) q^{97} + (\beta_{2} + 4 \beta_1 + 2) q^{98} + (\beta_{2} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 6 q^{7} + 3 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 6 q^{7} + 3 q^{8} + 7 q^{9} + 6 q^{10} + 3 q^{11} - 7 q^{13} + 6 q^{14} + 3 q^{16} + 10 q^{17} + 7 q^{18} + 6 q^{20} + 16 q^{21} + 3 q^{22} + 12 q^{23} - 3 q^{25} - 7 q^{26} + 6 q^{27} + 6 q^{28} + 3 q^{29} - 2 q^{31} + 3 q^{32} + 10 q^{34} + 12 q^{35} + 7 q^{36} - 4 q^{37} - 6 q^{39} + 6 q^{40} - 10 q^{41} + 16 q^{42} + 5 q^{43} + 3 q^{44} + 14 q^{45} + 12 q^{46} - 11 q^{47} + 7 q^{49} - 3 q^{50} - 10 q^{51} - 7 q^{52} + 6 q^{54} + 6 q^{55} + 6 q^{56} + 3 q^{58} + 11 q^{61} - 2 q^{62} + 20 q^{63} + 3 q^{64} - 14 q^{65} + 10 q^{68} - 32 q^{69} + 12 q^{70} + 5 q^{71} + 7 q^{72} + 8 q^{73} - 4 q^{74} + 6 q^{77} - 6 q^{78} - 6 q^{79} + 6 q^{80} + 11 q^{81} - 10 q^{82} + 7 q^{83} + 16 q^{84} + 20 q^{85} + 5 q^{86} + 3 q^{88} - q^{89} + 14 q^{90} - 20 q^{91} + 12 q^{92} - 26 q^{93} - 11 q^{94} + 7 q^{97} + 7 q^{98} + 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 8x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) 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
−2.69399
−0.252000
2.94600
1.00000 −2.69399 1.00000 2.00000 −2.69399 −0.693995 1.00000 4.25761 2.00000
1.2 1.00000 −0.252000 1.00000 2.00000 −0.252000 1.74800 1.00000 −2.93650 2.00000
1.3 1.00000 2.94600 1.00000 2.00000 2.94600 4.94600 1.00000 5.67889 2.00000
\(n\): 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.bj 3
19.b odd 2 1 7942.2.a.bd 3
19.d odd 6 2 418.2.e.i 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.e.i 6 19.d odd 6 2
7942.2.a.bd 3 19.b odd 2 1
7942.2.a.bj 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7942))\):

\( T_{3}^{3} - 8T_{3} - 2 \) Copy content Toggle raw display
\( T_{5} - 2 \) Copy content Toggle raw display
\( T_{13}^{3} + 7T_{13}^{2} - 5T_{13} - 71 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 8T - 2 \) Copy content Toggle raw display
$5$ \( (T - 2)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 6 T^{2} + 4 T + 6 \) Copy content Toggle raw display
$11$ \( (T - 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 7 T^{2} - 5 T - 71 \) Copy content Toggle raw display
$17$ \( T^{3} - 10 T^{2} + 10 T + 50 \) Copy content Toggle raw display
$19$ \( T^{3} \) Copy content Toggle raw display
$23$ \( T^{3} - 12 T^{2} + 16 T + 80 \) Copy content Toggle raw display
$29$ \( (T - 1)^{3} \) Copy content Toggle raw display
$31$ \( T^{3} + 2 T^{2} - 40 T - 116 \) Copy content Toggle raw display
$37$ \( T^{3} + 4 T^{2} - 70 T + 150 \) Copy content Toggle raw display
$41$ \( T^{3} + 10 T^{2} - 2 T - 54 \) Copy content Toggle raw display
$43$ \( T^{3} - 5 T^{2} - 27 T + 81 \) Copy content Toggle raw display
$47$ \( T^{3} + 11 T^{2} - 41 T - 501 \) Copy content Toggle raw display
$53$ \( T^{3} - 128T + 128 \) Copy content Toggle raw display
$59$ \( T^{3} \) Copy content Toggle raw display
$61$ \( T^{3} - 11 T^{2} - 25 T + 375 \) Copy content Toggle raw display
$67$ \( T^{3} - 72T + 54 \) Copy content Toggle raw display
$71$ \( T^{3} - 5 T^{2} - 27 T + 81 \) Copy content Toggle raw display
$73$ \( T^{3} - 8 T^{2} - 14 T + 102 \) Copy content Toggle raw display
$79$ \( T^{3} + 6 T^{2} - 60 T - 190 \) Copy content Toggle raw display
$83$ \( T^{3} - 7 T^{2} - 95 T - 205 \) Copy content Toggle raw display
$89$ \( T^{3} + T^{2} - 93 T + 43 \) Copy content Toggle raw display
$97$ \( T^{3} - 7 T^{2} - 77 T + 131 \) Copy content Toggle raw display
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