Properties

Label 3762.2.a.bg
Level $3762$
Weight $2$
Character orbit 3762.a
Self dual yes
Analytic conductor $30.040$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3762 = 2 \cdot 3^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3762.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.0397212404\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
Defining polynomial: \( x^{3} - 6x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + (\beta_{2} + \beta_1 + 1) q^{5} + (\beta_{2} - \beta_1 - 2) q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + (\beta_{2} + \beta_1 + 1) q^{5} + (\beta_{2} - \beta_1 - 2) q^{7} + q^{8} + (\beta_{2} + \beta_1 + 1) q^{10} + q^{11} + (\beta_{2} - \beta_1) q^{13} + (\beta_{2} - \beta_1 - 2) q^{14} + q^{16} + (\beta_{2} + 3) q^{17} - q^{19} + (\beta_{2} + \beta_1 + 1) q^{20} + q^{22} + (\beta_{2} + 5) q^{23} + (3 \beta_1 + 4) q^{25} + (\beta_{2} - \beta_1) q^{26} + (\beta_{2} - \beta_1 - 2) q^{28} + ( - 2 \beta_{2} - \beta_1 - 2) q^{29} + (3 \beta_1 - 1) q^{31} + q^{32} + (\beta_{2} + 3) q^{34} + ( - 3 \beta_{2} - 6 \beta_1) q^{35} + ( - 2 \beta_{2} - 2) q^{37} - q^{38} + (\beta_{2} + \beta_1 + 1) q^{40} + (2 \beta_{2} + 3 \beta_1 + 3) q^{41} + (\beta_{2} + 3 \beta_1 - 7) q^{43} + q^{44} + (\beta_{2} + 5) q^{46} + ( - 2 \beta_{2} - 4 \beta_1 + 4) q^{47} + ( - 2 \beta_{2} + \beta_1 + 9) q^{49} + (3 \beta_1 + 4) q^{50} + (\beta_{2} - \beta_1) q^{52} + ( - \beta_{2} + 4 \beta_1 + 3) q^{53} + (\beta_{2} + \beta_1 + 1) q^{55} + (\beta_{2} - \beta_1 - 2) q^{56} + ( - 2 \beta_{2} - \beta_1 - 2) q^{58} + ( - \beta_{2} - 2 \beta_1 + 1) q^{59} + (2 \beta_{2} + 8) q^{61} + (3 \beta_1 - 1) q^{62} + q^{64} + ( - \beta_{2} - 4 \beta_1 + 2) q^{65} + ( - 3 \beta_{2} + \beta_1) q^{67} + (\beta_{2} + 3) q^{68} + ( - 3 \beta_{2} - 6 \beta_1) q^{70} + (\beta_{2} + 5 \beta_1 - 7) q^{71} + ( - \beta_{2} - 1) q^{73} + ( - 2 \beta_{2} - 2) q^{74} - q^{76} + (\beta_{2} - \beta_1 - 2) q^{77} + ( - 4 \beta_{2} - 2 \beta_1 - 2) q^{79} + (\beta_{2} + \beta_1 + 1) q^{80} + (2 \beta_{2} + 3 \beta_1 + 3) q^{82} + ( - \beta_{2} + \beta_1 + 11) q^{83} + (2 \beta_{2} + 2 \beta_1 + 8) q^{85} + (\beta_{2} + 3 \beta_1 - 7) q^{86} + q^{88} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{89} + ( - \beta_1 + 12) q^{91} + (\beta_{2} + 5) q^{92} + ( - 2 \beta_{2} - 4 \beta_1 + 4) q^{94} + ( - \beta_{2} - \beta_1 - 1) q^{95} + ( - 4 \beta_{2} + 4) q^{97} + ( - 2 \beta_{2} + \beta_1 + 9) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 6 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 6 q^{7} + 3 q^{8} + 3 q^{10} + 3 q^{11} - 6 q^{14} + 3 q^{16} + 9 q^{17} - 3 q^{19} + 3 q^{20} + 3 q^{22} + 15 q^{23} + 12 q^{25} - 6 q^{28} - 6 q^{29} - 3 q^{31} + 3 q^{32} + 9 q^{34} - 6 q^{37} - 3 q^{38} + 3 q^{40} + 9 q^{41} - 21 q^{43} + 3 q^{44} + 15 q^{46} + 12 q^{47} + 27 q^{49} + 12 q^{50} + 9 q^{53} + 3 q^{55} - 6 q^{56} - 6 q^{58} + 3 q^{59} + 24 q^{61} - 3 q^{62} + 3 q^{64} + 6 q^{65} + 9 q^{68} - 21 q^{71} - 3 q^{73} - 6 q^{74} - 3 q^{76} - 6 q^{77} - 6 q^{79} + 3 q^{80} + 9 q^{82} + 33 q^{83} + 24 q^{85} - 21 q^{86} + 3 q^{88} - 6 q^{89} + 36 q^{91} + 15 q^{92} + 12 q^{94} - 3 q^{95} + 12 q^{97} + 27 q^{98}+O(q^{100}) \) 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
−0.523976
−2.14510
2.66908
1.00000 0 1.00000 −2.72545 0 −4.67750 1.00000 0 −2.72545
1.2 1.00000 0 1.00000 1.60147 0 2.89167 1.00000 0 1.60147
1.3 1.00000 0 1.00000 4.12398 0 −4.21417 1.00000 0 4.12398
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(11\) \(-1\)
\(19\) \(1\)

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 3762.2.a.bg 3
3.b odd 2 1 418.2.a.g 3
12.b even 2 1 3344.2.a.q 3
33.d even 2 1 4598.2.a.bo 3
57.d even 2 1 7942.2.a.bi 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.a.g 3 3.b odd 2 1
3344.2.a.q 3 12.b even 2 1
3762.2.a.bg 3 1.a even 1 1 trivial
4598.2.a.bo 3 33.d even 2 1
7942.2.a.bi 3 57.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3762))\):

\( T_{5}^{3} - 3T_{5}^{2} - 9T_{5} + 18 \) Copy content Toggle raw display
\( T_{7}^{3} + 6T_{7}^{2} - 6T_{7} - 57 \) Copy content Toggle raw display
\( T_{13}^{3} - 18T_{13} - 29 \) Copy content Toggle raw display
\( T_{17}^{3} - 9T_{17}^{2} + 18T_{17} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) 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
$19$ \( (T + 1)^{3} \) 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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