Properties

Label 3762.2.a.bd
Level $3762$
Weight $2$
Character orbit 3762.a
Self dual yes
Analytic conductor $30.040$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.469.1
Defining polynomial: \( x^{3} - x^{2} - 5x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + (\beta_1 - 2) q^{5} - \beta_{2} q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + (\beta_1 - 2) q^{5} - \beta_{2} q^{7} - q^{8} + ( - \beta_1 + 2) q^{10} + q^{11} + (\beta_{2} + 2) q^{13} + \beta_{2} q^{14} + q^{16} + ( - 2 \beta_{2} - 2) q^{17} - q^{19} + (\beta_1 - 2) q^{20} - q^{22} + 2 \beta_{2} q^{23} + (\beta_{2} - 4 \beta_1 + 3) q^{25} + ( - \beta_{2} - 2) q^{26} - \beta_{2} q^{28} + (2 \beta_{2} + \beta_1 - 2) q^{29} + (\beta_{2} - 2 \beta_1) q^{31} - q^{32} + (2 \beta_{2} + 2) q^{34} + (\beta_{2} - \beta_1) q^{35} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{37} + q^{38} + ( - \beta_1 + 2) q^{40} + (\beta_{2} - 2 \beta_1 + 2) q^{41} + (2 \beta_{2} - 3 \beta_1) q^{43} + q^{44} - 2 \beta_{2} q^{46} + (2 \beta_{2} + 2 \beta_1) q^{47} + ( - 2 \beta_{2} + \beta_1 - 3) q^{49} + ( - \beta_{2} + 4 \beta_1 - 3) q^{50} + (\beta_{2} + 2) q^{52} + ( - 2 \beta_1 + 6) q^{53} + (\beta_1 - 2) q^{55} + \beta_{2} q^{56} + ( - 2 \beta_{2} - \beta_1 + 2) q^{58} + ( - 4 \beta_{2} - 4 \beta_1 + 4) q^{59} + 2 q^{61} + ( - \beta_{2} + 2 \beta_1) q^{62} + q^{64} + ( - \beta_{2} + 3 \beta_1 - 4) q^{65} - \beta_{2} q^{67} + ( - 2 \beta_{2} - 2) q^{68} + ( - \beta_{2} + \beta_1) q^{70} + ( - 2 \beta_{2} + \beta_1) q^{71} + (4 \beta_{2} + 2 \beta_1 + 2) q^{73} + (2 \beta_{2} - 2 \beta_1 + 6) q^{74} - q^{76} - \beta_{2} q^{77} + (2 \beta_{2} + 4 \beta_1) q^{79} + (\beta_1 - 2) q^{80} + ( - \beta_{2} + 2 \beta_1 - 2) q^{82} + (2 \beta_{2} - 5 \beta_1) q^{83} + (2 \beta_{2} - 4 \beta_1 + 4) q^{85} + ( - 2 \beta_{2} + 3 \beta_1) q^{86} - q^{88} + (2 \beta_{2} - 8 \beta_1 - 2) q^{89} + ( - \beta_1 - 4) q^{91} + 2 \beta_{2} q^{92} + ( - 2 \beta_{2} - 2 \beta_1) q^{94} + ( - \beta_1 + 2) q^{95} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{97} + (2 \beta_{2} - \beta_1 + 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} - 5 q^{5} + q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 3 q^{4} - 5 q^{5} + q^{7} - 3 q^{8} + 5 q^{10} + 3 q^{11} + 5 q^{13} - q^{14} + 3 q^{16} - 4 q^{17} - 3 q^{19} - 5 q^{20} - 3 q^{22} - 2 q^{23} + 4 q^{25} - 5 q^{26} + q^{28} - 7 q^{29} - 3 q^{31} - 3 q^{32} + 4 q^{34} - 2 q^{35} - 14 q^{37} + 3 q^{38} + 5 q^{40} + 3 q^{41} - 5 q^{43} + 3 q^{44} + 2 q^{46} - 6 q^{49} - 4 q^{50} + 5 q^{52} + 16 q^{53} - 5 q^{55} - q^{56} + 7 q^{58} + 12 q^{59} + 6 q^{61} + 3 q^{62} + 3 q^{64} - 8 q^{65} + q^{67} - 4 q^{68} + 2 q^{70} + 3 q^{71} + 4 q^{73} + 14 q^{74} - 3 q^{76} + q^{77} + 2 q^{79} - 5 q^{80} - 3 q^{82} - 7 q^{83} + 6 q^{85} + 5 q^{86} - 3 q^{88} - 16 q^{89} - 13 q^{91} - 2 q^{92} + 5 q^{95} - 14 q^{97} + 6 q^{98}+O(q^{100}) \) 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
−2.16425
0.772866
2.39138
−1.00000 0 1.00000 −4.16425 0 −0.683969 −1.00000 0 4.16425
1.2 −1.00000 0 1.00000 −1.22713 0 3.40268 −1.00000 0 1.22713
1.3 −1.00000 0 1.00000 0.391382 0 −1.71871 −1.00000 0 −0.391382
\(n\): 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.bd 3
3.b odd 2 1 418.2.a.h 3
12.b even 2 1 3344.2.a.p 3
33.d even 2 1 4598.2.a.bm 3
57.d even 2 1 7942.2.a.bc 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.a.h 3 3.b odd 2 1
3344.2.a.p 3 12.b even 2 1
3762.2.a.bd 3 1.a even 1 1 trivial
4598.2.a.bm 3 33.d even 2 1
7942.2.a.bc 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3762))\):

\( T_{5}^{3} + 5T_{5}^{2} + 3T_{5} - 2 \) Copy content Toggle raw display
\( T_{7}^{3} - T_{7}^{2} - 7T_{7} - 4 \) Copy content Toggle raw display
\( T_{13}^{3} - 5T_{13}^{2} + T_{13} + 14 \) Copy content Toggle raw display
\( T_{17}^{3} + 4T_{17}^{2} - 24T_{17} - 88 \) 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
$5$ \( T^{3} + 5 T^{2} + 3 T - 2 \) 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
$19$ \( (T + 1)^{3} \) 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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