Properties

Label 1932.2.a.h
Level $1932$
Weight $2$
Character orbit 1932.a
Self dual yes
Analytic conductor $15.427$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + (3 \beta - 1) q^{5} + q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + (3 \beta - 1) q^{5} + q^{7} + q^{9} + 3 q^{11} + (\beta - 2) q^{13} + (3 \beta - 1) q^{15} + ( - 4 \beta + 5) q^{17} + (2 \beta - 3) q^{19} + q^{21} - q^{23} + (3 \beta + 5) q^{25} + q^{27} - 5 q^{29} + ( - 2 \beta + 9) q^{31} + 3 q^{33} + (3 \beta - 1) q^{35} + ( - 2 \beta - 1) q^{37} + (\beta - 2) q^{39} + 3 q^{41} - \beta q^{43} + (3 \beta - 1) q^{45} - 2 \beta q^{47} + q^{49} + ( - 4 \beta + 5) q^{51} + ( - 7 \beta + 9) q^{53} + (9 \beta - 3) q^{55} + (2 \beta - 3) q^{57} + ( - 9 \beta + 1) q^{59} - 3 \beta q^{61} + q^{63} + ( - 4 \beta + 5) q^{65} + (7 \beta - 9) q^{67} - q^{69} + (\beta + 6) q^{71} + ( - 8 \beta + 7) q^{73} + (3 \beta + 5) q^{75} + 3 q^{77} + ( - 4 \beta - 3) q^{79} + q^{81} + (2 \beta + 9) q^{83} + (7 \beta - 17) q^{85} - 5 q^{87} + ( - 9 \beta + 8) q^{89} + (\beta - 2) q^{91} + ( - 2 \beta + 9) q^{93} + ( - 5 \beta + 9) q^{95} + ( - 8 \beta + 11) q^{97} + 3 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9} + 6 q^{11} - 3 q^{13} + q^{15} + 6 q^{17} - 4 q^{19} + 2 q^{21} - 2 q^{23} + 13 q^{25} + 2 q^{27} - 10 q^{29} + 16 q^{31} + 6 q^{33} + q^{35} - 4 q^{37} - 3 q^{39} + 6 q^{41} - q^{43} + q^{45} - 2 q^{47} + 2 q^{49} + 6 q^{51} + 11 q^{53} + 3 q^{55} - 4 q^{57} - 7 q^{59} - 3 q^{61} + 2 q^{63} + 6 q^{65} - 11 q^{67} - 2 q^{69} + 13 q^{71} + 6 q^{73} + 13 q^{75} + 6 q^{77} - 10 q^{79} + 2 q^{81} + 20 q^{83} - 27 q^{85} - 10 q^{87} + 7 q^{89} - 3 q^{91} + 16 q^{93} + 13 q^{95} + 14 q^{97} + 6 q^{99}+O(q^{100}) \) 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
−0.618034
1.61803
0 1.00000 0 −2.85410 0 1.00000 0 1.00000 0
1.2 0 1.00000 0 3.85410 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.h 2
3.b odd 2 1 5796.2.a.j 2
4.b odd 2 1 7728.2.a.bb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1932.2.a.h 2 1.a even 1 1 trivial
5796.2.a.j 2 3.b odd 2 1
7728.2.a.bb 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1932))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - T - 11 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( (T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 3T + 1 \) Copy content Toggle raw display
$17$ \( T^{2} - 6T - 11 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$23$ \( (T + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T + 5)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 16T + 59 \) Copy content Toggle raw display
$37$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$41$ \( (T - 3)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$47$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$53$ \( T^{2} - 11T - 31 \) Copy content Toggle raw display
$59$ \( T^{2} + 7T - 89 \) Copy content Toggle raw display
$61$ \( T^{2} + 3T - 9 \) Copy content Toggle raw display
$67$ \( T^{2} + 11T - 31 \) Copy content Toggle raw display
$71$ \( T^{2} - 13T + 41 \) Copy content Toggle raw display
$73$ \( T^{2} - 6T - 71 \) Copy content Toggle raw display
$79$ \( T^{2} + 10T + 5 \) Copy content Toggle raw display
$83$ \( T^{2} - 20T + 95 \) Copy content Toggle raw display
$89$ \( T^{2} - 7T - 89 \) Copy content Toggle raw display
$97$ \( T^{2} - 14T - 31 \) Copy content Toggle raw display
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