Properties

Label 4032.2.k.a
Level 4032
Weight 2
Character orbit 4032.k
Analytic conductor 32.196
Analytic rank 0
Dimension 4
CM no
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{3} q^{5} + ( -1 + \beta_{1} ) q^{7} +O(q^{10})\) \( q -\beta_{3} q^{5} + ( -1 + \beta_{1} ) q^{7} -\beta_{2} q^{11} -2 \beta_{1} q^{13} + \beta_{3} q^{17} + 2 \beta_{1} q^{19} -\beta_{2} q^{23} + 7 q^{25} -\beta_{2} q^{29} + ( 2 \beta_{2} + \beta_{3} ) q^{35} + 8 q^{37} -\beta_{3} q^{41} + 2 q^{43} -2 \beta_{3} q^{47} + ( -5 - 2 \beta_{1} ) q^{49} -3 \beta_{2} q^{53} -6 \beta_{1} q^{55} + 4 \beta_{3} q^{59} -4 \beta_{1} q^{61} -4 \beta_{2} q^{65} -8 q^{67} -\beta_{2} q^{71} -2 \beta_{1} q^{73} + ( \beta_{2} - 3 \beta_{3} ) q^{77} -4 q^{79} -2 \beta_{3} q^{83} -12 q^{85} -3 \beta_{3} q^{89} + ( 12 + 2 \beta_{1} ) q^{91} + 4 \beta_{2} q^{95} + 2 \beta_{1} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{7} + O(q^{10}) \) \( 4q - 4q^{7} + 28q^{25} + 32q^{37} + 8q^{43} - 20q^{49} - 32q^{67} - 16q^{79} - 48q^{85} + 48q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 4 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} + 5 \nu \)
\(\beta_{2}\)\(=\)\( -3 \nu^{3} - 9 \nu \)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} + 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 3 \beta_{1}\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 4\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-5 \beta_{2} - 9 \beta_{1}\)\()/6\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3905.1
0.517638i
0.517638i
1.93185i
1.93185i
0 0 0 −3.46410 0 −1.00000 2.44949i 0 0 0
3905.2 0 0 0 −3.46410 0 −1.00000 + 2.44949i 0 0 0
3905.3 0 0 0 3.46410 0 −1.00000 2.44949i 0 0 0
3905.4 0 0 0 3.46410 0 −1.00000 + 2.44949i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4032.2.k.a 4
3.b odd 2 1 inner 4032.2.k.a 4
4.b odd 2 1 4032.2.k.d 4
7.b odd 2 1 inner 4032.2.k.a 4
8.b even 2 1 252.2.f.a 4
8.d odd 2 1 1008.2.k.b 4
12.b even 2 1 4032.2.k.d 4
21.c even 2 1 inner 4032.2.k.a 4
24.f even 2 1 1008.2.k.b 4
24.h odd 2 1 252.2.f.a 4
28.d even 2 1 4032.2.k.d 4
40.f even 2 1 6300.2.d.c 4
40.i odd 4 2 6300.2.f.b 8
56.e even 2 1 1008.2.k.b 4
56.h odd 2 1 252.2.f.a 4
56.j odd 6 2 1764.2.t.b 8
56.p even 6 2 1764.2.t.b 8
72.j odd 6 2 2268.2.x.i 8
72.n even 6 2 2268.2.x.i 8
84.h odd 2 1 4032.2.k.d 4
120.i odd 2 1 6300.2.d.c 4
120.w even 4 2 6300.2.f.b 8
168.e odd 2 1 1008.2.k.b 4
168.i even 2 1 252.2.f.a 4
168.s odd 6 2 1764.2.t.b 8
168.ba even 6 2 1764.2.t.b 8
280.c odd 2 1 6300.2.d.c 4
280.s even 4 2 6300.2.f.b 8
504.bn odd 6 2 2268.2.x.i 8
504.cc even 6 2 2268.2.x.i 8
840.u even 2 1 6300.2.d.c 4
840.bp odd 4 2 6300.2.f.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.f.a 4 8.b even 2 1
252.2.f.a 4 24.h odd 2 1
252.2.f.a 4 56.h odd 2 1
252.2.f.a 4 168.i even 2 1
1008.2.k.b 4 8.d odd 2 1
1008.2.k.b 4 24.f even 2 1
1008.2.k.b 4 56.e even 2 1
1008.2.k.b 4 168.e odd 2 1
1764.2.t.b 8 56.j odd 6 2
1764.2.t.b 8 56.p even 6 2
1764.2.t.b 8 168.s odd 6 2
1764.2.t.b 8 168.ba even 6 2
2268.2.x.i 8 72.j odd 6 2
2268.2.x.i 8 72.n even 6 2
2268.2.x.i 8 504.bn odd 6 2
2268.2.x.i 8 504.cc even 6 2
4032.2.k.a 4 1.a even 1 1 trivial
4032.2.k.a 4 3.b odd 2 1 inner
4032.2.k.a 4 7.b odd 2 1 inner
4032.2.k.a 4 21.c even 2 1 inner
4032.2.k.d 4 4.b odd 2 1
4032.2.k.d 4 12.b even 2 1
4032.2.k.d 4 28.d even 2 1
4032.2.k.d 4 84.h odd 2 1
6300.2.d.c 4 40.f even 2 1
6300.2.d.c 4 120.i odd 2 1
6300.2.d.c 4 280.c odd 2 1
6300.2.d.c 4 840.u even 2 1
6300.2.f.b 8 40.i odd 4 2
6300.2.f.b 8 120.w even 4 2
6300.2.f.b 8 280.s even 4 2
6300.2.f.b 8 840.bp odd 4 2

Hecke kernels

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

\( T_{5}^{2} - 12 \)
\( T_{43} - 2 \)
\( T_{67} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 - 2 T^{2} + 25 T^{4} )^{2} \)
$7$ \( ( 1 + 2 T + 7 T^{2} )^{2} \)
$11$ \( ( 1 - 4 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 2 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 + 22 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 14 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 - 28 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 40 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 31 T^{2} )^{4} \)
$37$ \( ( 1 - 8 T + 37 T^{2} )^{4} \)
$41$ \( ( 1 + 70 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 2 T + 43 T^{2} )^{4} \)
$47$ \( ( 1 + 46 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 + 56 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 - 74 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 26 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 + 8 T + 67 T^{2} )^{4} \)
$71$ \( ( 1 - 124 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 122 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 + 4 T + 79 T^{2} )^{4} \)
$83$ \( ( 1 + 118 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 + 70 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 170 T^{2} + 9409 T^{4} )^{2} \)
show more
show less