Properties

Label 4032.2.a.bx
Level 4032
Weight 2
Character orbit 4032.a
Self dual yes
Analytic conductor 32.196
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2016)
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.

\(f(q)\) \(=\) \( q + 2 \beta q^{5} + q^{7} +O(q^{10})\) \( q + 2 \beta q^{5} + q^{7} + 2 \beta q^{11} + ( 2 - 4 \beta ) q^{13} + ( -4 + 2 \beta ) q^{17} + ( 4 - 4 \beta ) q^{19} + ( -4 + 6 \beta ) q^{23} + ( -1 + 4 \beta ) q^{25} + ( 8 - 4 \beta ) q^{29} + ( 4 - 4 \beta ) q^{31} + 2 \beta q^{35} + ( -2 + 4 \beta ) q^{37} + ( -4 - 2 \beta ) q^{41} + 8 \beta q^{43} + ( 8 - 4 \beta ) q^{47} + q^{49} + 8 q^{53} + ( 4 + 4 \beta ) q^{55} + 4 \beta q^{59} + ( -2 + 4 \beta ) q^{61} + ( -8 - 4 \beta ) q^{65} + 8 q^{67} + ( 4 + 2 \beta ) q^{71} + ( -2 - 8 \beta ) q^{73} + 2 \beta q^{77} + 8 \beta q^{79} -8 \beta q^{83} + ( 4 - 4 \beta ) q^{85} + ( -12 + 6 \beta ) q^{89} + ( 2 - 4 \beta ) q^{91} -8 q^{95} + ( 6 - 8 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} + 2q^{7} + O(q^{10}) \) \( 2q + 2q^{5} + 2q^{7} + 2q^{11} - 6q^{17} + 4q^{19} - 2q^{23} + 2q^{25} + 12q^{29} + 4q^{31} + 2q^{35} - 10q^{41} + 8q^{43} + 12q^{47} + 2q^{49} + 16q^{53} + 12q^{55} + 4q^{59} - 20q^{65} + 16q^{67} + 10q^{71} - 12q^{73} + 2q^{77} + 8q^{79} - 8q^{83} + 4q^{85} - 18q^{89} - 16q^{95} + 4q^{97} + O(q^{100}) \)

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 0 0 −1.23607 0 1.00000 0 0 0
1.2 0 0 0 3.23607 0 1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 4032.2.a.bx 2
3.b odd 2 1 4032.2.a.bp 2
4.b odd 2 1 4032.2.a.bu 2
8.b even 2 1 2016.2.a.q yes 2
8.d odd 2 1 2016.2.a.p 2
12.b even 2 1 4032.2.a.bo 2
24.f even 2 1 2016.2.a.u yes 2
24.h odd 2 1 2016.2.a.v yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2016.2.a.p 2 8.d odd 2 1
2016.2.a.q yes 2 8.b even 2 1
2016.2.a.u yes 2 24.f even 2 1
2016.2.a.v yes 2 24.h odd 2 1
4032.2.a.bo 2 12.b even 2 1
4032.2.a.bp 2 3.b odd 2 1
4032.2.a.bu 2 4.b odd 2 1
4032.2.a.bx 2 1.a even 1 1 trivial

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4032))\):

\( T_{5}^{2} - 2 T_{5} - 4 \)
\( T_{11}^{2} - 2 T_{11} - 4 \)
\( T_{13}^{2} - 20 \)
\( T_{17}^{2} + 6 T_{17} + 4 \)
\( T_{19}^{2} - 4 T_{19} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 2 T + 6 T^{2} - 10 T^{3} + 25 T^{4} \)
$7$ \( ( 1 - T )^{2} \)
$11$ \( 1 - 2 T + 18 T^{2} - 22 T^{3} + 121 T^{4} \)
$13$ \( 1 + 6 T^{2} + 169 T^{4} \)
$17$ \( 1 + 6 T + 38 T^{2} + 102 T^{3} + 289 T^{4} \)
$19$ \( 1 - 4 T + 22 T^{2} - 76 T^{3} + 361 T^{4} \)
$23$ \( 1 + 2 T + 2 T^{2} + 46 T^{3} + 529 T^{4} \)
$29$ \( 1 - 12 T + 74 T^{2} - 348 T^{3} + 841 T^{4} \)
$31$ \( 1 - 4 T + 46 T^{2} - 124 T^{3} + 961 T^{4} \)
$37$ \( 1 + 54 T^{2} + 1369 T^{4} \)
$41$ \( 1 + 10 T + 102 T^{2} + 410 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 8 T + 22 T^{2} - 344 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 12 T + 110 T^{2} - 564 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 - 8 T + 53 T^{2} )^{2} \)
$59$ \( 1 - 4 T + 102 T^{2} - 236 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 102 T^{2} + 3721 T^{4} \)
$67$ \( ( 1 - 8 T + 67 T^{2} )^{2} \)
$71$ \( 1 - 10 T + 162 T^{2} - 710 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 12 T + 102 T^{2} + 876 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 8 T + 94 T^{2} - 632 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 8 T + 102 T^{2} + 664 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 18 T + 214 T^{2} + 1602 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 4 T + 118 T^{2} - 388 T^{3} + 9409 T^{4} \)
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