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: \(x^{3} - 6 x - 3\)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 4\)

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7942))\):

\( T_{3}^{3} - 6 T_{3} + 3 \)
\( T_{5}^{3} + 3 T_{5}^{2} - 9 T_{5} - 18 \)
\( T_{13}^{3} - 18 T_{13} + 29 \)

Hecke characteristic polynomials

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