Properties

Label 1932.2.a.j
Level $1932$
Weight $2$
Character orbit 1932.a
Self dual yes
Analytic conductor $15.427$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1932.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.4270976705\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1101.1
Defining polynomial: \( x^{3} - x^{2} - 9x + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + q^{3} + \beta_1 q^{5} - q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + \beta_1 q^{5} - q^{7} + q^{9} + q^{11} + ( - \beta_{2} + \beta_1 - 1) q^{13} + \beta_1 q^{15} + ( - \beta_{2} + 1) q^{17} + 3 q^{19} - q^{21} + q^{23} + (\beta_{2} - \beta_1 + 2) q^{25} + q^{27} + (3 \beta_{2} - 2 \beta_1 + 3) q^{29} + (\beta_{2} + 2 \beta_1 + 1) q^{31} + q^{33} - \beta_1 q^{35} + (3 \beta_{2} + 3) q^{37} + ( - \beta_{2} + \beta_1 - 1) q^{39} + ( - 2 \beta_{2} - 2 \beta_1 + 1) q^{41} + ( - \beta_{2} - \beta_1 + 5) q^{43} + \beta_1 q^{45} + ( - \beta_{2} + 4) q^{47} + q^{49} + ( - \beta_{2} + 1) q^{51} + (\beta_{2} - \beta_1) q^{53} + \beta_1 q^{55} + 3 q^{57} + (\beta_{2} + 3 \beta_1 + 4) q^{59} + ( - 2 \beta_{2} - \beta_1 - 1) q^{61} - q^{63} + ( - \beta_{2} - 2 \beta_1 + 5) q^{65} + ( - 2 \beta_{2} + \beta_1 + 2) q^{67} + q^{69} + ( - 3 \beta_{2} - \beta_1 + 3) q^{71} + ( - \beta_{2} - 3) q^{73} + (\beta_{2} - \beta_1 + 2) q^{75} - q^{77} + ( - \beta_{2} + 2 \beta_1 - 5) q^{79} + q^{81} + (3 \beta_{2} - 2 \beta_1 + 7) q^{83} + ( - 2 \beta_{2} + \beta_1 - 2) q^{85} + (3 \beta_{2} - 2 \beta_1 + 3) q^{87} + (\beta_{2} - 3 \beta_1 + 5) q^{89} + (\beta_{2} - \beta_1 + 1) q^{91} + (\beta_{2} + 2 \beta_1 + 1) q^{93} + 3 \beta_1 q^{95} + (\beta_{2} - 3) q^{97} + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9} + 3 q^{11} - q^{13} + q^{15} + 4 q^{17} + 9 q^{19} - 3 q^{21} + 3 q^{23} + 4 q^{25} + 3 q^{27} + 4 q^{29} + 4 q^{31} + 3 q^{33} - q^{35} + 6 q^{37} - q^{39} + 3 q^{41} + 15 q^{43} + q^{45} + 13 q^{47} + 3 q^{49} + 4 q^{51} - 2 q^{53} + q^{55} + 9 q^{57} + 14 q^{59} - 2 q^{61} - 3 q^{63} + 14 q^{65} + 9 q^{67} + 3 q^{69} + 11 q^{71} - 8 q^{73} + 4 q^{75} - 3 q^{77} - 12 q^{79} + 3 q^{81} + 16 q^{83} - 3 q^{85} + 4 q^{87} + 11 q^{89} + q^{91} + 4 q^{93} + 3 q^{95} - 10 q^{97} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 9x + 12 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + \nu - 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - \beta _1 + 7 \) 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
−3.11903
1.43163
2.68740
0 1.00000 0 −3.11903 0 −1.00000 0 1.00000 0
1.2 0 1.00000 0 1.43163 0 −1.00000 0 1.00000 0
1.3 0 1.00000 0 2.68740 0 −1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(1\)
\(23\) \(-1\)

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 1932.2.a.j 3
3.b odd 2 1 5796.2.a.o 3
4.b odd 2 1 7728.2.a.br 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1932.2.a.j 3 1.a even 1 1 trivial
5796.2.a.o 3 3.b odd 2 1
7728.2.a.br 3 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1932))\):

\( T_{5}^{3} - T_{5}^{2} - 9T_{5} + 12 \) Copy content Toggle raw display
\( T_{11} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - T^{2} - 9T + 12 \) Copy content Toggle raw display
$7$ \( (T + 1)^{3} \) Copy content Toggle raw display
$11$ \( (T - 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + T^{2} - 15 T - 18 \) Copy content Toggle raw display
$17$ \( T^{3} - 4 T^{2} - 5 T + 12 \) Copy content Toggle raw display
$19$ \( (T - 3)^{3} \) Copy content Toggle raw display
$23$ \( (T - 1)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} - 4 T^{2} - 99 T + 534 \) Copy content Toggle raw display
$31$ \( T^{3} - 4 T^{2} - 51 T + 18 \) Copy content Toggle raw display
$37$ \( T^{3} - 6 T^{2} - 81 T + 162 \) Copy content Toggle raw display
$41$ \( T^{3} - 3 T^{2} - 93 T + 423 \) Copy content Toggle raw display
$43$ \( T^{3} - 15 T^{2} + 51 T + 36 \) Copy content Toggle raw display
$47$ \( T^{3} - 13 T^{2} + 46 T - 36 \) Copy content Toggle raw display
$53$ \( T^{3} + 2 T^{2} - 14 T + 3 \) Copy content Toggle raw display
$59$ \( T^{3} - 14 T^{2} - 42 T + 411 \) Copy content Toggle raw display
$61$ \( T^{3} + 2 T^{2} - 58 T + 127 \) Copy content Toggle raw display
$67$ \( T^{3} - 9 T^{2} - 15 T - 4 \) Copy content Toggle raw display
$71$ \( T^{3} - 11 T^{2} - 75 T + 744 \) Copy content Toggle raw display
$73$ \( T^{3} + 8 T^{2} + 11 T - 8 \) Copy content Toggle raw display
$79$ \( T^{3} + 12 T^{2} + 9 T - 38 \) Copy content Toggle raw display
$83$ \( T^{3} - 16 T^{2} - 19 T + 802 \) Copy content Toggle raw display
$89$ \( T^{3} - 11 T^{2} - 41 T - 6 \) Copy content Toggle raw display
$97$ \( T^{3} + 10 T^{2} + 23 T + 2 \) Copy content Toggle raw display
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