Properties

Label 7942.2.a.bc
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: 3.3.469.1
Defining polynomial: \( x^{3} - x^{2} - 5x + 4 \) 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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 5x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) 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
2.39138
0.772866
−2.16425
−1.00000 −2.39138 1.00000 −0.391382 2.39138 −1.71871 −1.00000 2.71871 0.391382
1.2 −1.00000 −0.772866 1.00000 1.22713 0.772866 3.40268 −1.00000 −2.40268 −1.22713
1.3 −1.00000 2.16425 1.00000 4.16425 −2.16425 −0.683969 −1.00000 1.68397 −4.16425
\(n\): 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.bc 3
19.b odd 2 1 418.2.a.h 3
57.d even 2 1 3762.2.a.bd 3
76.d even 2 1 3344.2.a.p 3
209.d even 2 1 4598.2.a.bm 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.a.h 3 19.b odd 2 1
3344.2.a.p 3 76.d even 2 1
3762.2.a.bd 3 57.d even 2 1
4598.2.a.bm 3 209.d even 2 1
7942.2.a.bc 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7942))\):

\( T_{3}^{3} + T_{3}^{2} - 5T_{3} - 4 \) Copy content Toggle raw display
\( T_{5}^{3} - 5T_{5}^{2} + 3T_{5} + 2 \) Copy content Toggle raw display
\( T_{13}^{3} + 5T_{13}^{2} + T_{13} - 14 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + T^{2} - 5T - 4 \) Copy content Toggle raw display
$5$ \( T^{3} - 5 T^{2} + 3 T + 2 \) Copy content Toggle raw display
$7$ \( T^{3} - T^{2} - 7T - 4 \) Copy content Toggle raw display
$11$ \( (T + 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 5T^{2} + T - 14 \) Copy content Toggle raw display
$17$ \( T^{3} - 4 T^{2} - 24 T + 88 \) Copy content Toggle raw display
$19$ \( T^{3} \) Copy content Toggle raw display
$23$ \( T^{3} - 2 T^{2} - 28 T - 32 \) Copy content Toggle raw display
$29$ \( T^{3} + 7 T^{2} - 19 T - 86 \) Copy content Toggle raw display
$31$ \( T^{3} - 3 T^{2} - 25 T + 76 \) Copy content Toggle raw display
$37$ \( T^{3} - 14 T^{2} + 16 T + 128 \) Copy content Toggle raw display
$41$ \( T^{3} - 3 T^{2} - 25 T - 22 \) Copy content Toggle raw display
$43$ \( T^{3} + 5 T^{2} - 67 T - 268 \) Copy content Toggle raw display
$47$ \( T^{3} - 52T + 128 \) Copy content Toggle raw display
$53$ \( T^{3} - 16 T^{2} + 64 T - 56 \) Copy content Toggle raw display
$59$ \( T^{3} - 12 T^{2} - 160 T + 1792 \) Copy content Toggle raw display
$61$ \( (T - 2)^{3} \) Copy content Toggle raw display
$67$ \( T^{3} + T^{2} - 7T + 4 \) Copy content Toggle raw display
$71$ \( T^{3} - 3 T^{2} - 31 T - 28 \) Copy content Toggle raw display
$73$ \( T^{3} - 4 T^{2} - 136 T + 56 \) Copy content Toggle raw display
$79$ \( T^{3} + 2 T^{2} - 116 T + 352 \) Copy content Toggle raw display
$83$ \( T^{3} - 7 T^{2} - 143 T + 1108 \) Copy content Toggle raw display
$89$ \( T^{3} + 16 T^{2} - 280 T - 4424 \) Copy content Toggle raw display
$97$ \( T^{3} - 14 T^{2} + 16 T + 128 \) Copy content Toggle raw display
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