Properties

Label 7942.2.a.bi
Level $7942$
Weight $2$
Character orbit 7942.a
Self dual yes
Analytic conductor $63.417$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
Defining polynomial: \( x^{3} - 6x - 3 \) 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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 6x - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) 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
2.66908
−0.523976
−2.14510
1.00000 −2.66908 1.00000 −4.12398 −2.66908 −4.21417 1.00000 4.12398 −4.12398
1.2 1.00000 0.523976 1.00000 2.72545 0.523976 −4.67750 1.00000 −2.72545 2.72545
1.3 1.00000 2.14510 1.00000 −1.60147 2.14510 2.89167 1.00000 1.60147 −1.60147
\(n\): 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.bi 3
19.b odd 2 1 418.2.a.g 3
57.d even 2 1 3762.2.a.bg 3
76.d even 2 1 3344.2.a.q 3
209.d even 2 1 4598.2.a.bo 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.a.g 3 19.b odd 2 1
3344.2.a.q 3 76.d even 2 1
3762.2.a.bg 3 57.d even 2 1
4598.2.a.bo 3 209.d even 2 1
7942.2.a.bi 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7942))\):

\( T_{3}^{3} - 6T_{3} + 3 \) Copy content Toggle raw display
\( T_{5}^{3} + 3T_{5}^{2} - 9T_{5} - 18 \) Copy content Toggle raw display
\( T_{13}^{3} - 18T_{13} + 29 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 6T + 3 \) Copy content Toggle raw display
$5$ \( T^{3} + 3 T^{2} - 9 T - 18 \) Copy content Toggle raw display
$7$ \( T^{3} + 6 T^{2} - 6 T - 57 \) Copy content Toggle raw display
$11$ \( (T + 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 18T + 29 \) Copy content Toggle raw display
$17$ \( T^{3} + 9 T^{2} + 18 T - 4 \) Copy content Toggle raw display
$19$ \( T^{3} \) Copy content Toggle raw display
$23$ \( T^{3} + 15 T^{2} + 66 T + 76 \) Copy content Toggle raw display
$29$ \( T^{3} + 6 T^{2} - 24 T - 147 \) Copy content Toggle raw display
$31$ \( T^{3} - 3 T^{2} - 51 T + 134 \) Copy content Toggle raw display
$37$ \( T^{3} - 6 T^{2} - 24 T + 96 \) Copy content Toggle raw display
$41$ \( T^{3} - 9 T^{2} - 45 T + 122 \) Copy content Toggle raw display
$43$ \( T^{3} + 21 T^{2} + 93 T - 184 \) Copy content Toggle raw display
$47$ \( T^{3} + 12 T^{2} - 60 T - 672 \) Copy content Toggle raw display
$53$ \( T^{3} - 9 T^{2} - 90 T + 452 \) Copy content Toggle raw display
$59$ \( T^{3} - 3 T^{2} - 24 T + 64 \) Copy content Toggle raw display
$61$ \( T^{3} - 24 T^{2} + 156 T - 192 \) Copy content Toggle raw display
$67$ \( T^{3} - 96T - 123 \) Copy content Toggle raw display
$71$ \( T^{3} + 21 T^{2} + 3 T - 1306 \) Copy content Toggle raw display
$73$ \( T^{3} + 3 T^{2} - 6 T - 12 \) Copy content Toggle raw display
$79$ \( T^{3} - 6 T^{2} - 132 T + 944 \) Copy content Toggle raw display
$83$ \( T^{3} + 33 T^{2} + 345 T + 1104 \) Copy content Toggle raw display
$89$ \( T^{3} + 6 T^{2} - 36 T - 144 \) Copy content Toggle raw display
$97$ \( T^{3} + 12 T^{2} - 96 T - 256 \) Copy content Toggle raw display
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