Properties

Label 4032.2.i.a
Level 4032
Weight 2
Character orbit 4032.i
Analytic conductor 32.196
Analytic rank 0
Dimension 8
CM no
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.i (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.40960000.1
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q -\beta_{1} q^{5} + ( -\beta_{2} + \beta_{4} ) q^{7} +O(q^{10})\) \( q -\beta_{1} q^{5} + ( -\beta_{2} + \beta_{4} ) q^{7} + \beta_{3} q^{11} + 2 \beta_{2} q^{17} + 2 \beta_{7} q^{19} -\beta_{6} q^{23} + q^{25} + \beta_{7} q^{29} -4 \beta_{4} q^{31} + ( 2 \beta_{3} + \beta_{5} ) q^{35} -\beta_{5} q^{37} + 2 \beta_{2} q^{41} -2 \beta_{1} q^{43} -8 q^{47} + ( 3 + 2 \beta_{6} ) q^{49} + 3 \beta_{7} q^{53} -2 \beta_{4} q^{55} -2 \beta_{5} q^{59} + 6 \beta_{3} q^{61} + \beta_{1} q^{67} -3 \beta_{6} q^{71} -2 \beta_{6} q^{73} + ( \beta_{1} + \beta_{7} ) q^{77} -4 \beta_{2} q^{79} -2 \beta_{5} q^{85} -6 \beta_{2} q^{89} -4 \beta_{6} q^{95} + 2 \beta_{6} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 8q^{25} - 64q^{47} + 24q^{49} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 7 x^{4} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{6} + 16 \nu^{2} \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{4} + 7 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{7} - \nu^{5} + 13 \nu^{3} - 5 \nu \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{7} - \nu^{5} - 13 \nu^{3} - 5 \nu \)\()/3\)
\(\beta_{5}\)\(=\)\( 2 \nu^{6} + 12 \nu^{2} \)
\(\beta_{6}\)\(=\)\((\)\( 4 \nu^{7} + \nu^{5} + 29 \nu^{3} + 11 \nu \)\()/3\)
\(\beta_{7}\)\(=\)\((\)\( -4 \nu^{7} + \nu^{5} - 29 \nu^{3} + 11 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + \beta_{6} + \beta_{4} + \beta_{3}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{5} + 3 \beta_{1}\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{7} + \beta_{6} + 2 \beta_{4} - 2 \beta_{3}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{2} - 7\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-5 \beta_{7} - 5 \beta_{6} - 11 \beta_{4} - 11 \beta_{3}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(4 \beta_{5} - 9 \beta_{1}\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(13 \beta_{7} - 13 \beta_{6} - 29 \beta_{4} + 29 \beta_{3}\)\()/4\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1889.1
0.437016 + 0.437016i
−0.437016 0.437016i
−1.14412 1.14412i
1.14412 + 1.14412i
−0.437016 + 0.437016i
0.437016 0.437016i
1.14412 1.14412i
−1.14412 + 1.14412i
0 0 0 2.00000i 0 −2.23607 1.41421i 0 0 0
1889.2 0 0 0 2.00000i 0 −2.23607 + 1.41421i 0 0 0
1889.3 0 0 0 2.00000i 0 2.23607 1.41421i 0 0 0
1889.4 0 0 0 2.00000i 0 2.23607 + 1.41421i 0 0 0
1889.5 0 0 0 2.00000i 0 −2.23607 1.41421i 0 0 0
1889.6 0 0 0 2.00000i 0 −2.23607 + 1.41421i 0 0 0
1889.7 0 0 0 2.00000i 0 2.23607 1.41421i 0 0 0
1889.8 0 0 0 2.00000i 0 2.23607 + 1.41421i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1889.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
24.f even 2 1 inner
28.d even 2 1 inner
56.e even 2 1 inner
168.i 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.i.a 8
3.b odd 2 1 4032.2.i.b yes 8
4.b odd 2 1 4032.2.i.b yes 8
7.b odd 2 1 4032.2.i.b yes 8
8.b even 2 1 inner 4032.2.i.a 8
8.d odd 2 1 4032.2.i.b yes 8
12.b even 2 1 inner 4032.2.i.a 8
21.c even 2 1 inner 4032.2.i.a 8
24.f even 2 1 inner 4032.2.i.a 8
24.h odd 2 1 4032.2.i.b yes 8
28.d even 2 1 inner 4032.2.i.a 8
56.e even 2 1 inner 4032.2.i.a 8
56.h odd 2 1 4032.2.i.b yes 8
84.h odd 2 1 4032.2.i.b yes 8
168.e odd 2 1 4032.2.i.b yes 8
168.i even 2 1 inner 4032.2.i.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4032.2.i.a 8 1.a even 1 1 trivial
4032.2.i.a 8 8.b even 2 1 inner
4032.2.i.a 8 12.b even 2 1 inner
4032.2.i.a 8 21.c even 2 1 inner
4032.2.i.a 8 24.f even 2 1 inner
4032.2.i.a 8 28.d even 2 1 inner
4032.2.i.a 8 56.e even 2 1 inner
4032.2.i.a 8 168.i even 2 1 inner
4032.2.i.b yes 8 3.b odd 2 1
4032.2.i.b yes 8 4.b odd 2 1
4032.2.i.b yes 8 7.b odd 2 1
4032.2.i.b yes 8 8.d odd 2 1
4032.2.i.b yes 8 24.h odd 2 1
4032.2.i.b yes 8 56.h odd 2 1
4032.2.i.b yes 8 84.h odd 2 1
4032.2.i.b yes 8 168.e odd 2 1

Hecke kernels

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

\( T_{5}^{2} + 4 \)
\( T_{47} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 - 4 T + 5 T^{2} )^{4}( 1 + 4 T + 5 T^{2} )^{4} \)
$7$ \( ( 1 - 6 T^{2} + 49 T^{4} )^{2} \)
$11$ \( ( 1 + 20 T^{2} + 121 T^{4} )^{4} \)
$13$ \( ( 1 + 13 T^{2} )^{8} \)
$17$ \( ( 1 + 14 T^{2} + 289 T^{4} )^{4} \)
$19$ \( ( 1 - 2 T^{2} + 361 T^{4} )^{4} \)
$23$ \( ( 1 - 36 T^{2} + 529 T^{4} )^{4} \)
$29$ \( ( 1 + 48 T^{2} + 841 T^{4} )^{4} \)
$31$ \( ( 1 - 30 T^{2} + 961 T^{4} )^{4} \)
$37$ \( ( 1 - 54 T^{2} + 1369 T^{4} )^{4} \)
$41$ \( ( 1 + 62 T^{2} + 1681 T^{4} )^{4} \)
$43$ \( ( 1 - 70 T^{2} + 1849 T^{4} )^{4} \)
$47$ \( ( 1 + 8 T + 47 T^{2} )^{8} \)
$53$ \( ( 1 + 16 T^{2} + 2809 T^{4} )^{4} \)
$59$ \( ( 1 - 38 T^{2} + 3481 T^{4} )^{4} \)
$61$ \( ( 1 + 50 T^{2} + 3721 T^{4} )^{4} \)
$67$ \( ( 1 - 130 T^{2} + 4489 T^{4} )^{4} \)
$71$ \( ( 1 - 52 T^{2} + 5041 T^{4} )^{4} \)
$73$ \( ( 1 - 106 T^{2} + 5329 T^{4} )^{4} \)
$79$ \( ( 1 + 78 T^{2} + 6241 T^{4} )^{4} \)
$83$ \( ( 1 - 83 T^{2} )^{8} \)
$89$ \( ( 1 - 2 T^{2} + 7921 T^{4} )^{4} \)
$97$ \( ( 1 - 154 T^{2} + 9409 T^{4} )^{4} \)
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