Properties

Label 7942.2.a.bh
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: 3.3.148.1
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) 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.

\(f(q)\) \(=\) \( q + q^{2} + (\beta_1 - 1) q^{3} + q^{4} + 2 \beta_1 q^{5} + (\beta_1 - 1) q^{6} + (2 \beta_{2} + \beta_1 + 1) q^{7} + q^{8} + (\beta_{2} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + (\beta_1 - 1) q^{3} + q^{4} + 2 \beta_1 q^{5} + (\beta_1 - 1) q^{6} + (2 \beta_{2} + \beta_1 + 1) q^{7} + q^{8} + (\beta_{2} - \beta_1) q^{9} + 2 \beta_1 q^{10} - q^{11} + (\beta_1 - 1) q^{12} + (\beta_{2} - \beta_1 - 2) q^{13} + (2 \beta_{2} + \beta_1 + 1) q^{14} + (2 \beta_{2} + 4) q^{15} + q^{16} + ( - \beta_{2} - 2 \beta_1 + 6) q^{17} + (\beta_{2} - \beta_1) q^{18} + 2 \beta_1 q^{20} + ( - \beta_{2} + 3 \beta_1 - 1) q^{21} - q^{22} + (4 \beta_{2} + 2 \beta_1 + 2) q^{23} + (\beta_1 - 1) q^{24} + (4 \beta_{2} + 4 \beta_1 + 3) q^{25} + (\beta_{2} - \beta_1 - 2) q^{26} + ( - 2 \beta_{2} - 2 \beta_1) q^{27} + (2 \beta_{2} + \beta_1 + 1) q^{28} + ( - 2 \beta_{2} - 2 \beta_1 - 1) q^{29} + (2 \beta_{2} + 4) q^{30} + (\beta_{2} + 3 \beta_1 + 1) q^{31} + q^{32} + ( - \beta_1 + 1) q^{33} + ( - \beta_{2} - 2 \beta_1 + 6) q^{34} + (2 \beta_{2} + 8 \beta_1) q^{35} + (\beta_{2} - \beta_1) q^{36} + (\beta_{2} - 2 \beta_1) q^{37} + ( - 2 \beta_{2} - \beta_1 - 1) q^{39} + 2 \beta_1 q^{40} - \beta_{2} q^{41} + ( - \beta_{2} + 3 \beta_1 - 1) q^{42} + ( - \beta_{2} - 2 \beta_1 + 1) q^{43} - q^{44} + ( - 2 \beta_{2} - 6) q^{45} + (4 \beta_{2} + 2 \beta_1 + 2) q^{46} + ( - 2 \beta_{2} - 3 \beta_1 - 2) q^{47} + (\beta_1 - 1) q^{48} + (\beta_{2} + 3 \beta_1 + 4) q^{49} + (4 \beta_{2} + 4 \beta_1 + 3) q^{50} + ( - \beta_{2} + 5 \beta_1 - 9) q^{51} + (\beta_{2} - \beta_1 - 2) q^{52} + (2 \beta_1 - 2) q^{53} + ( - 2 \beta_{2} - 2 \beta_1) q^{54} - 2 \beta_1 q^{55} + (2 \beta_{2} + \beta_1 + 1) q^{56} + ( - 2 \beta_{2} - 2 \beta_1 - 1) q^{58} - 2 \beta_{2} q^{59} + (2 \beta_{2} + 4) q^{60} + ( - 3 \beta_{2} - 3 \beta_1 + 8) q^{61} + (\beta_{2} + 3 \beta_1 + 1) q^{62} + ( - 2 \beta_{2} - 5 \beta_1 + 5) q^{63} + q^{64} + ( - 2 \beta_{2} - 4 \beta_1 - 6) q^{65} + ( - \beta_1 + 1) q^{66} + ( - 4 \beta_{2} + \beta_1 - 5) q^{67} + ( - \beta_{2} - 2 \beta_1 + 6) q^{68} + ( - 2 \beta_{2} + 6 \beta_1 - 2) q^{69} + (2 \beta_{2} + 8 \beta_1) q^{70} + (5 \beta_{2} - 1) q^{71} + (\beta_{2} - \beta_1) q^{72} + (\beta_{2} + 4 \beta_1 - 10) q^{73} + (\beta_{2} - 2 \beta_1) q^{74} + (7 \beta_1 + 1) q^{75} + ( - 2 \beta_{2} - \beta_1 - 1) q^{77} + ( - 2 \beta_{2} - \beta_1 - 1) q^{78} + ( - 4 \beta_{2} - 7 \beta_1 + 9) q^{79} + 2 \beta_1 q^{80} + ( - 3 \beta_{2} + \beta_1 - 2) q^{81} - \beta_{2} q^{82} + ( - 5 \beta_{2} - 2 \beta_1 + 3) q^{83} + ( - \beta_{2} + 3 \beta_1 - 1) q^{84} + ( - 4 \beta_{2} + 6 \beta_1 - 6) q^{85} + ( - \beta_{2} - 2 \beta_1 + 1) q^{86} + ( - 3 \beta_1 - 1) q^{87} - q^{88} + (4 \beta_{2} - 2 \beta_1 - 7) q^{89} + ( - 2 \beta_{2} - 6) q^{90} + ( - 6 \beta_{2} - 7 \beta_1 + 3) q^{91} + (4 \beta_{2} + 2 \beta_1 + 2) q^{92} + (2 \beta_{2} + 2 \beta_1 + 4) q^{93} + ( - 2 \beta_{2} - 3 \beta_1 - 2) q^{94} + (\beta_1 - 1) q^{96} + (10 \beta_{2} + 4 \beta_1 + 3) q^{97} + (\beta_{2} + 3 \beta_1 + 4) q^{98} + ( - \beta_{2} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{5} - 2 q^{6} + 4 q^{7} + 3 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{5} - 2 q^{6} + 4 q^{7} + 3 q^{8} - q^{9} + 2 q^{10} - 3 q^{11} - 2 q^{12} - 7 q^{13} + 4 q^{14} + 12 q^{15} + 3 q^{16} + 16 q^{17} - q^{18} + 2 q^{20} - 3 q^{22} + 8 q^{23} - 2 q^{24} + 13 q^{25} - 7 q^{26} - 2 q^{27} + 4 q^{28} - 5 q^{29} + 12 q^{30} + 6 q^{31} + 3 q^{32} + 2 q^{33} + 16 q^{34} + 8 q^{35} - q^{36} - 2 q^{37} - 4 q^{39} + 2 q^{40} + q^{43} - 3 q^{44} - 18 q^{45} + 8 q^{46} - 9 q^{47} - 2 q^{48} + 15 q^{49} + 13 q^{50} - 22 q^{51} - 7 q^{52} - 4 q^{53} - 2 q^{54} - 2 q^{55} + 4 q^{56} - 5 q^{58} + 12 q^{60} + 21 q^{61} + 6 q^{62} + 10 q^{63} + 3 q^{64} - 22 q^{65} + 2 q^{66} - 14 q^{67} + 16 q^{68} + 8 q^{70} - 3 q^{71} - q^{72} - 26 q^{73} - 2 q^{74} + 10 q^{75} - 4 q^{77} - 4 q^{78} + 20 q^{79} + 2 q^{80} - 5 q^{81} + 7 q^{83} - 12 q^{85} + q^{86} - 6 q^{87} - 3 q^{88} - 23 q^{89} - 18 q^{90} + 2 q^{91} + 8 q^{92} + 14 q^{93} - 9 q^{94} - 2 q^{96} + 13 q^{97} + 15 q^{98} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) 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
−1.48119
0.311108
2.17009
1.00000 −2.48119 1.00000 −2.96239 −2.48119 2.86907 1.00000 3.15633 −2.96239
1.2 1.00000 −0.688892 1.00000 0.622216 −0.688892 −3.11753 1.00000 −2.52543 0.622216
1.3 1.00000 1.17009 1.00000 4.34017 1.17009 4.24846 1.00000 −1.63090 4.34017
\(n\): 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.bh 3
19.b odd 2 1 7942.2.a.be 3
19.c even 3 2 418.2.e.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.e.h 6 19.c even 3 2
7942.2.a.be 3 19.b odd 2 1
7942.2.a.bh 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7942))\):

\( T_{3}^{3} + 2T_{3}^{2} - 2T_{3} - 2 \) Copy content Toggle raw display
\( T_{5}^{3} - 2T_{5}^{2} - 12T_{5} + 8 \) Copy content Toggle raw display
\( T_{13}^{3} + 7T_{13}^{2} + 7T_{13} - 19 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + 2 T^{2} - 2 T - 2 \) Copy content Toggle raw display
$5$ \( T^{3} - 2 T^{2} - 12 T + 8 \) Copy content Toggle raw display
$7$ \( T^{3} - 4 T^{2} - 10 T + 38 \) Copy content Toggle raw display
$11$ \( (T + 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 7 T^{2} + 7 T - 19 \) Copy content Toggle raw display
$17$ \( T^{3} - 16 T^{2} + 72 T - 62 \) Copy content Toggle raw display
$19$ \( T^{3} \) Copy content Toggle raw display
$23$ \( T^{3} - 8 T^{2} - 40 T + 304 \) Copy content Toggle raw display
$29$ \( T^{3} + 5 T^{2} - 13 T - 25 \) Copy content Toggle raw display
$31$ \( T^{3} - 6 T^{2} - 16 T - 4 \) Copy content Toggle raw display
$37$ \( T^{3} + 2 T^{2} - 20 T - 50 \) Copy content Toggle raw display
$41$ \( T^{3} - 4T - 2 \) Copy content Toggle raw display
$43$ \( T^{3} - T^{2} - 13 T + 23 \) Copy content Toggle raw display
$47$ \( T^{3} + 9 T^{2} - 7 T - 13 \) Copy content Toggle raw display
$53$ \( T^{3} + 4 T^{2} - 8 T - 16 \) Copy content Toggle raw display
$59$ \( T^{3} - 16T - 16 \) Copy content Toggle raw display
$61$ \( T^{3} - 21 T^{2} + 99 T + 13 \) Copy content Toggle raw display
$67$ \( T^{3} + 14 T^{2} - 10 T - 274 \) Copy content Toggle raw display
$71$ \( T^{3} + 3 T^{2} - 97 T + 151 \) Copy content Toggle raw display
$73$ \( T^{3} + 26 T^{2} + 176 T + 122 \) Copy content Toggle raw display
$79$ \( T^{3} - 20 T^{2} - 38 T + 1658 \) Copy content Toggle raw display
$83$ \( T^{3} - 7 T^{2} - 77 T - 131 \) Copy content Toggle raw display
$89$ \( T^{3} + 23 T^{2} + 83 T - 403 \) Copy content Toggle raw display
$97$ \( T^{3} - 13 T^{2} - 317 T + 4225 \) Copy content Toggle raw display
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