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: \(x^{3} - x^{2} - 5 x + 4\)
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + ( -2 + \beta_{1} ) q^{5} -\beta_{2} q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + ( -2 + \beta_{1} ) q^{5} -\beta_{2} q^{7} - q^{8} + ( 2 - \beta_{1} ) q^{10} + q^{11} + ( 2 + \beta_{2} ) q^{13} + \beta_{2} q^{14} + q^{16} + ( -2 - 2 \beta_{2} ) q^{17} - q^{19} + ( -2 + \beta_{1} ) q^{20} - q^{22} + 2 \beta_{2} q^{23} + ( 3 - 4 \beta_{1} + \beta_{2} ) q^{25} + ( -2 - \beta_{2} ) q^{26} -\beta_{2} q^{28} + ( -2 + \beta_{1} + 2 \beta_{2} ) q^{29} + ( -2 \beta_{1} + \beta_{2} ) q^{31} - q^{32} + ( 2 + 2 \beta_{2} ) q^{34} + ( -\beta_{1} + \beta_{2} ) q^{35} + ( -6 + 2 \beta_{1} - 2 \beta_{2} ) q^{37} + q^{38} + ( 2 - \beta_{1} ) q^{40} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{41} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{43} + q^{44} -2 \beta_{2} q^{46} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{47} + ( -3 + \beta_{1} - 2 \beta_{2} ) q^{49} + ( -3 + 4 \beta_{1} - \beta_{2} ) q^{50} + ( 2 + \beta_{2} ) q^{52} + ( 6 - 2 \beta_{1} ) q^{53} + ( -2 + \beta_{1} ) q^{55} + \beta_{2} q^{56} + ( 2 - \beta_{1} - 2 \beta_{2} ) q^{58} + ( 4 - 4 \beta_{1} - 4 \beta_{2} ) q^{59} + 2 q^{61} + ( 2 \beta_{1} - \beta_{2} ) q^{62} + q^{64} + ( -4 + 3 \beta_{1} - \beta_{2} ) q^{65} -\beta_{2} q^{67} + ( -2 - 2 \beta_{2} ) q^{68} + ( \beta_{1} - \beta_{2} ) q^{70} + ( \beta_{1} - 2 \beta_{2} ) q^{71} + ( 2 + 2 \beta_{1} + 4 \beta_{2} ) q^{73} + ( 6 - 2 \beta_{1} + 2 \beta_{2} ) q^{74} - q^{76} -\beta_{2} q^{77} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{79} + ( -2 + \beta_{1} ) q^{80} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{82} + ( -5 \beta_{1} + 2 \beta_{2} ) q^{83} + ( 4 - 4 \beta_{1} + 2 \beta_{2} ) q^{85} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{86} - q^{88} + ( -2 - 8 \beta_{1} + 2 \beta_{2} ) q^{89} + ( -4 - \beta_{1} ) q^{91} + 2 \beta_{2} q^{92} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{94} + ( 2 - \beta_{1} ) q^{95} + ( -6 + 2 \beta_{1} - 2 \beta_{2} ) q^{97} + ( 3 - \beta_{1} + 2 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} - 5 q^{5} + q^{7} - 3 q^{8} + O(q^{10}) \) \( 3 q - 3 q^{2} + 3 q^{4} - 5 q^{5} + q^{7} - 3 q^{8} + 5 q^{10} + 3 q^{11} + 5 q^{13} - q^{14} + 3 q^{16} - 4 q^{17} - 3 q^{19} - 5 q^{20} - 3 q^{22} - 2 q^{23} + 4 q^{25} - 5 q^{26} + q^{28} - 7 q^{29} - 3 q^{31} - 3 q^{32} + 4 q^{34} - 2 q^{35} - 14 q^{37} + 3 q^{38} + 5 q^{40} + 3 q^{41} - 5 q^{43} + 3 q^{44} + 2 q^{46} - 6 q^{49} - 4 q^{50} + 5 q^{52} + 16 q^{53} - 5 q^{55} - q^{56} + 7 q^{58} + 12 q^{59} + 6 q^{61} + 3 q^{62} + 3 q^{64} - 8 q^{65} + q^{67} - 4 q^{68} + 2 q^{70} + 3 q^{71} + 4 q^{73} + 14 q^{74} - 3 q^{76} + q^{77} + 2 q^{79} - 5 q^{80} - 3 q^{82} - 7 q^{83} + 6 q^{85} + 5 q^{86} - 3 q^{88} - 16 q^{89} - 13 q^{91} - 2 q^{92} + 5 q^{95} - 14 q^{97} + 6 q^{98} + O(q^{100}) \)

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

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

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
−2.16425
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3762))\):

\( T_{5}^{3} + 5 T_{5}^{2} + 3 T_{5} - 2 \)
\( T_{7}^{3} - T_{7}^{2} - 7 T_{7} - 4 \)
\( T_{13}^{3} - 5 T_{13}^{2} + T_{13} + 14 \)
\( T_{17}^{3} + 4 T_{17}^{2} - 24 T_{17} - 88 \)

Hecke characteristic polynomials

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