Properties

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

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.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.2312.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 84)
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.

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

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3583.1
1.28078 0.599676i
−0.780776 1.17915i
−0.780776 + 1.17915i
1.28078 + 0.599676i
0 0 0 3.33513i 0 1.56155 + 2.13578i 0 0 0
3583.2 0 0 0 1.69614i 0 −2.56155 0.662153i 0 0 0
3583.3 0 0 0 1.69614i 0 −2.56155 + 0.662153i 0 0 0
3583.4 0 0 0 3.33513i 0 1.56155 2.13578i 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
28.d 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.b.j 4
3.b odd 2 1 1344.2.b.e 4
4.b odd 2 1 4032.2.b.n 4
7.b odd 2 1 4032.2.b.n 4
8.b even 2 1 252.2.b.d 4
8.d odd 2 1 252.2.b.e 4
12.b even 2 1 1344.2.b.f 4
21.c even 2 1 1344.2.b.f 4
24.f even 2 1 84.2.b.a 4
24.h odd 2 1 84.2.b.b yes 4
28.d even 2 1 inner 4032.2.b.j 4
56.e even 2 1 252.2.b.d 4
56.h odd 2 1 252.2.b.e 4
84.h odd 2 1 1344.2.b.e 4
168.e odd 2 1 84.2.b.b yes 4
168.i even 2 1 84.2.b.a 4
168.s odd 6 2 588.2.o.a 8
168.v even 6 2 588.2.o.c 8
168.ba even 6 2 588.2.o.c 8
168.be odd 6 2 588.2.o.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.b.a 4 24.f even 2 1
84.2.b.a 4 168.i even 2 1
84.2.b.b yes 4 24.h odd 2 1
84.2.b.b yes 4 168.e odd 2 1
252.2.b.d 4 8.b even 2 1
252.2.b.d 4 56.e even 2 1
252.2.b.e 4 8.d odd 2 1
252.2.b.e 4 56.h odd 2 1
588.2.o.a 8 168.s odd 6 2
588.2.o.a 8 168.be odd 6 2
588.2.o.c 8 168.v even 6 2
588.2.o.c 8 168.ba even 6 2
1344.2.b.e 4 3.b odd 2 1
1344.2.b.e 4 84.h odd 2 1
1344.2.b.f 4 12.b even 2 1
1344.2.b.f 4 21.c even 2 1
4032.2.b.j 4 1.a even 1 1 trivial
4032.2.b.j 4 28.d even 2 1 inner
4032.2.b.n 4 4.b odd 2 1
4032.2.b.n 4 7.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 \(S_{2}^{\mathrm{new}}(4032, [\chi])\):

\( T_{5}^{4} + 14 T_{5}^{2} + 32 \)
\( T_{11}^{4} + 10 T_{11}^{2} + 8 \)
\( T_{19}^{2} - 6 T_{19} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 6 T^{2} + 42 T^{4} - 150 T^{6} + 625 T^{8} \)
$7$ \( 1 + 2 T - 2 T^{2} + 14 T^{3} + 49 T^{4} \)
$11$ \( 1 - 34 T^{2} + 514 T^{4} - 4114 T^{6} + 14641 T^{8} \)
$13$ \( 1 - 12 T^{2} + 102 T^{4} - 2028 T^{6} + 28561 T^{8} \)
$17$ \( 1 - 22 T^{2} + 682 T^{4} - 6358 T^{6} + 83521 T^{8} \)
$19$ \( ( 1 - 6 T + 30 T^{2} - 114 T^{3} + 361 T^{4} )^{2} \)
$23$ \( 1 - 82 T^{2} + 2722 T^{4} - 43378 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 + 2 T + 29 T^{2} )^{4} \)
$31$ \( ( 1 + 31 T^{2} )^{4} \)
$37$ \( ( 1 - 6 T + 66 T^{2} - 222 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( 1 - 102 T^{2} + 5130 T^{4} - 171462 T^{6} + 2825761 T^{8} \)
$43$ \( 1 - 24 T^{2} + 3774 T^{4} - 44376 T^{6} + 3418801 T^{8} \)
$47$ \( ( 1 + 4 T + 30 T^{2} + 188 T^{3} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 - 8 T + 54 T^{2} - 424 T^{3} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 + 4 T + 59 T^{2} )^{4} \)
$61$ \( 1 - 132 T^{2} + 10710 T^{4} - 491172 T^{6} + 13845841 T^{8} \)
$67$ \( 1 - 144 T^{2} + 10830 T^{4} - 646416 T^{6} + 20151121 T^{8} \)
$71$ \( 1 - 114 T^{2} + 8418 T^{4} - 574674 T^{6} + 25411681 T^{8} \)
$73$ \( 1 - 236 T^{2} + 24310 T^{4} - 1257644 T^{6} + 28398241 T^{8} \)
$79$ \( 1 - 288 T^{2} + 33150 T^{4} - 1797408 T^{6} + 38950081 T^{8} \)
$83$ \( ( 1 - 4 T + 102 T^{2} - 332 T^{3} + 6889 T^{4} )^{2} \)
$89$ \( 1 - 294 T^{2} + 36618 T^{4} - 2328774 T^{6} + 62742241 T^{8} \)
$97$ \( 1 - 204 T^{2} + 28950 T^{4} - 1919436 T^{6} + 88529281 T^{8} \)
show more
show less