Properties

Label 4032.2.p.h
Level 4032
Weight 2
Character orbit 4032.p
Analytic conductor 32.196
Analytic rank 0
Dimension 8
CM discriminant -56
Inner twists 8

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Newspace parameters

Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.629407744.1
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 448)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{5} -\beta_{4} q^{7} +O(q^{10})\) \( q -\beta_{1} q^{5} -\beta_{4} q^{7} + ( -\beta_{1} + \beta_{6} ) q^{13} + \beta_{5} q^{19} -3 \beta_{2} q^{23} + ( 5 - \beta_{7} ) q^{25} + ( 2 \beta_{3} + \beta_{5} ) q^{35} -7 q^{49} + ( 4 \beta_{3} - \beta_{5} ) q^{59} + ( \beta_{1} + 2 \beta_{6} ) q^{61} + ( 6 - 3 \beta_{7} ) q^{65} -6 \beta_{4} q^{71} -2 \beta_{4} q^{79} + ( -\beta_{3} - 2 \beta_{5} ) q^{83} + ( -3 \beta_{3} + 2 \beta_{5} ) q^{91} + ( 9 \beta_{2} - 6 \beta_{4} ) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 40q^{25} - 56q^{49} + 48q^{65} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{6} + 2 x^{4} - 8 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{5} - 10 \nu^{3} + 28 \nu \)\()/12\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + 2 \nu^{4} - 2 \nu^{2} - 4 \)\()/6\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + 2 \nu^{3} - 4 \nu \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} - \nu^{4} + 4 \nu^{2} - 7 \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} - 4 \nu^{5} + 2 \nu^{3} + 4 \nu \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{7} + \nu^{5} + 2 \nu^{3} + 10 \nu \)\()/3\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{6} + 2 \nu^{4} + 2 \nu^{2} + 4 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6} + \beta_{5} + \beta_{3} + 2 \beta_{1}\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} + 2 \beta_{4} - \beta_{2} + 2\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{6} + \beta_{3} - \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{7} + 3 \beta_{2}\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(\beta_{6} - 3 \beta_{5} + 5 \beta_{3} + 2 \beta_{1}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(-\beta_{7} + 2 \beta_{4} + 5 \beta_{2} + 10\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-\beta_{6} + \beta_{5} + 7 \beta_{3} + 4 \beta_{1}\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1567.1
0.767178 + 1.18804i
0.767178 1.18804i
1.38255 + 0.297594i
1.38255 0.297594i
−1.38255 0.297594i
−1.38255 + 0.297594i
−0.767178 1.18804i
−0.767178 + 1.18804i
0 0 0 −3.91044 0 2.64575i 0 0 0
1567.2 0 0 0 −3.91044 0 2.64575i 0 0 0
1567.3 0 0 0 −2.16991 0 2.64575i 0 0 0
1567.4 0 0 0 −2.16991 0 2.64575i 0 0 0
1567.5 0 0 0 2.16991 0 2.64575i 0 0 0
1567.6 0 0 0 2.16991 0 2.64575i 0 0 0
1567.7 0 0 0 3.91044 0 2.64575i 0 0 0
1567.8 0 0 0 3.91044 0 2.64575i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1567.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)
4.b odd 2 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
28.d even 2 1 inner
56.e 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.p.h 8
3.b odd 2 1 448.2.e.a 8
4.b odd 2 1 inner 4032.2.p.h 8
7.b odd 2 1 inner 4032.2.p.h 8
8.b even 2 1 inner 4032.2.p.h 8
8.d odd 2 1 inner 4032.2.p.h 8
12.b even 2 1 448.2.e.a 8
21.c even 2 1 448.2.e.a 8
24.f even 2 1 448.2.e.a 8
24.h odd 2 1 448.2.e.a 8
28.d even 2 1 inner 4032.2.p.h 8
48.i odd 4 2 1792.2.f.k 8
48.k even 4 2 1792.2.f.k 8
56.e even 2 1 inner 4032.2.p.h 8
56.h odd 2 1 CM 4032.2.p.h 8
84.h odd 2 1 448.2.e.a 8
168.e odd 2 1 448.2.e.a 8
168.i even 2 1 448.2.e.a 8
336.v odd 4 2 1792.2.f.k 8
336.y even 4 2 1792.2.f.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
448.2.e.a 8 3.b odd 2 1
448.2.e.a 8 12.b even 2 1
448.2.e.a 8 21.c even 2 1
448.2.e.a 8 24.f even 2 1
448.2.e.a 8 24.h odd 2 1
448.2.e.a 8 84.h odd 2 1
448.2.e.a 8 168.e odd 2 1
448.2.e.a 8 168.i even 2 1
1792.2.f.k 8 48.i odd 4 2
1792.2.f.k 8 48.k even 4 2
1792.2.f.k 8 336.v odd 4 2
1792.2.f.k 8 336.y even 4 2
4032.2.p.h 8 1.a even 1 1 trivial
4032.2.p.h 8 4.b odd 2 1 inner
4032.2.p.h 8 7.b odd 2 1 inner
4032.2.p.h 8 8.b even 2 1 inner
4032.2.p.h 8 8.d odd 2 1 inner
4032.2.p.h 8 28.d even 2 1 inner
4032.2.p.h 8 56.e even 2 1 inner
4032.2.p.h 8 56.h odd 2 1 CM

Hecke kernels

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

\( T_{5}^{4} - 20 T_{5}^{2} + 72 \)
\( T_{11} \)
\( T_{13}^{4} - 52 T_{13}^{2} + 648 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 + 22 T^{4} + 625 T^{8} )^{2} \)
$7$ \( ( 1 + 7 T^{2} )^{4} \)
$11$ \( ( 1 + 11 T^{2} )^{8} \)
$13$ \( ( 1 + 310 T^{4} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 - 17 T^{2} )^{8} \)
$19$ \( ( 1 - 650 T^{4} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 - 10 T^{2} + 529 T^{4} )^{4} \)
$29$ \( ( 1 - 29 T^{2} )^{8} \)
$31$ \( ( 1 + 31 T^{2} )^{8} \)
$37$ \( ( 1 - 37 T^{2} )^{8} \)
$41$ \( ( 1 - 41 T^{2} )^{8} \)
$43$ \( ( 1 + 43 T^{2} )^{8} \)
$47$ \( ( 1 + 47 T^{2} )^{8} \)
$53$ \( ( 1 - 53 T^{2} )^{8} \)
$59$ \( ( 1 - 1130 T^{4} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 7370 T^{4} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 + 67 T^{2} )^{8} \)
$71$ \( ( 1 + 110 T^{2} + 5041 T^{4} )^{4} \)
$73$ \( ( 1 - 73 T^{2} )^{8} \)
$79$ \( ( 1 - 130 T^{2} + 6241 T^{4} )^{4} \)
$83$ \( ( 1 - 13130 T^{4} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 - 89 T^{2} )^{8} \)
$97$ \( ( 1 - 97 T^{2} )^{8} \)
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