Properties

Label 4032.2.c.q
Level 4032
Weight 2
Character orbit 4032.c
Analytic conductor 32.196
Analytic rank 0
Dimension 8
CM no
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.897122304.10
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{12} \)
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.

\(f(q)\) \(=\) \( q -\beta_{5} q^{5} - q^{7} +O(q^{10})\) \( q -\beta_{5} q^{5} - q^{7} + \beta_{5} q^{11} + ( -\beta_{4} - \beta_{6} ) q^{13} + \beta_{1} q^{17} + \beta_{4} q^{19} -\beta_{2} q^{23} + ( -5 - \beta_{7} ) q^{25} -\beta_{3} q^{29} + 2 q^{31} + \beta_{5} q^{35} + ( \beta_{4} + 4 \beta_{6} ) q^{37} -\beta_{1} q^{41} + ( 2 \beta_{4} + \beta_{6} ) q^{43} + ( -\beta_{1} - \beta_{2} ) q^{47} + q^{49} + ( 10 + \beta_{7} ) q^{55} + \beta_{3} q^{59} + ( \beta_{4} + \beta_{6} ) q^{61} + ( \beta_{1} - 3 \beta_{2} ) q^{65} -5 \beta_{6} q^{67} + ( 2 \beta_{1} - \beta_{2} ) q^{71} + ( 6 - \beta_{7} ) q^{73} -\beta_{5} q^{77} + ( 4 - \beta_{7} ) q^{79} + ( -\beta_{3} - 2 \beta_{5} ) q^{83} + ( -\beta_{4} + 4 \beta_{6} ) q^{85} + ( -\beta_{1} + 2 \beta_{2} ) q^{89} + ( \beta_{4} + \beta_{6} ) q^{91} + 2 \beta_{2} q^{95} + ( 2 - \beta_{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{7} + O(q^{10}) \) \( 8q - 8q^{7} - 40q^{25} + 16q^{31} + 8q^{49} + 80q^{55} + 48q^{73} + 32q^{79} + 16q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 8 x^{6} + 51 x^{4} - 104 x^{2} + 169\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 6 \nu^{7} + 17 \nu^{5} + 85 \nu^{3} - 403 \nu \)\()/663\)
\(\beta_{2}\)\(=\)\((\)\( -14 \nu^{7} + 255 \nu^{5} - 1377 \nu^{3} + 4771 \nu \)\()/663\)
\(\beta_{3}\)\(=\)\((\)\( 32 \nu^{7} - 204 \nu^{5} + 1632 \nu^{3} - 676 \nu \)\()/663\)
\(\beta_{4}\)\(=\)\((\)\( 32 \nu^{6} - 204 \nu^{4} + 1632 \nu^{2} - 2002 \)\()/663\)
\(\beta_{5}\)\(=\)\((\)\( 40 \nu^{7} - 255 \nu^{5} + 1377 \nu^{3} - 845 \nu \)\()/663\)
\(\beta_{6}\)\(=\)\((\)\( 2 \nu^{6} - 16 \nu^{4} + 76 \nu^{2} - 104 \)\()/39\)
\(\beta_{7}\)\(=\)\((\)\( 4 \nu^{6} + 400 \)\()/51\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - 3 \beta_{1}\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{7} - 6 \beta_{6} + 8 \beta_{4} + 16\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(-4 \beta_{5} + 5 \beta_{3}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(4 \beta_{7} - 24 \beta_{6} + 19 \beta_{4} - 38\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-32 \beta_{5} + 27 \beta_{3} + 5 \beta_{2} + 81 \beta_{1}\)\()/8\)
\(\nu^{6}\)\(=\)\((\)\(51 \beta_{7} - 400\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(204 \beta_{5} - 151 \beta_{3} + 53 \beta_{2} + 453 \beta_{1}\)\()/8\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2017.1
1.30421 + 0.752986i
−1.30421 + 0.752986i
−2.07341 1.19709i
2.07341 1.19709i
2.07341 + 1.19709i
−2.07341 + 1.19709i
−1.30421 0.752986i
1.30421 0.752986i
0 0 0 4.11439i 0 −1.00000 0 0 0
2017.2 0 0 0 4.11439i 0 −1.00000 0 0 0
2017.3 0 0 0 1.75265i 0 −1.00000 0 0 0
2017.4 0 0 0 1.75265i 0 −1.00000 0 0 0
2017.5 0 0 0 1.75265i 0 −1.00000 0 0 0
2017.6 0 0 0 1.75265i 0 −1.00000 0 0 0
2017.7 0 0 0 4.11439i 0 −1.00000 0 0 0
2017.8 0 0 0 4.11439i 0 −1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2017.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4032.2.c.q 8
3.b odd 2 1 inner 4032.2.c.q 8
4.b odd 2 1 4032.2.c.r yes 8
8.b even 2 1 inner 4032.2.c.q 8
8.d odd 2 1 4032.2.c.r yes 8
12.b even 2 1 4032.2.c.r yes 8
24.f even 2 1 4032.2.c.r yes 8
24.h odd 2 1 inner 4032.2.c.q 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4032.2.c.q 8 1.a even 1 1 trivial
4032.2.c.q 8 3.b odd 2 1 inner
4032.2.c.q 8 8.b even 2 1 inner
4032.2.c.q 8 24.h odd 2 1 inner
4032.2.c.r yes 8 4.b odd 2 1
4032.2.c.r yes 8 8.d odd 2 1
4032.2.c.r yes 8 12.b even 2 1
4032.2.c.r yes 8 24.f even 2 1

Hecke kernels

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

\( T_{5}^{4} + 20 T_{5}^{2} + 52 \)
\( T_{11}^{4} + 20 T_{11}^{2} + 52 \)
\( T_{13}^{4} + 32 T_{13}^{2} + 64 \)
\( T_{17}^{4} - 44 T_{17}^{2} + 52 \)
\( T_{23}^{4} - 60 T_{23}^{2} + 468 \)
\( T_{31} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 + 2 T^{4} + 625 T^{8} )^{2} \)
$7$ \( ( 1 + T )^{8} \)
$11$ \( ( 1 - 24 T^{2} + 338 T^{4} - 2904 T^{6} + 14641 T^{8} )^{2} \)
$13$ \( ( 1 - 20 T^{2} + 246 T^{4} - 3380 T^{6} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 + 24 T^{2} + 290 T^{4} + 6936 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )^{4}( 1 + 8 T + 19 T^{2} )^{4} \)
$23$ \( ( 1 + 32 T^{2} + 882 T^{4} + 16928 T^{6} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 - 4 T + 14 T^{2} - 116 T^{3} + 841 T^{4} )^{2}( 1 + 4 T + 14 T^{2} + 116 T^{3} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 2 T + 31 T^{2} )^{8} \)
$37$ \( ( 1 + 4 T^{2} - 330 T^{4} + 5476 T^{6} + 1874161 T^{8} )^{2} \)
$41$ \( ( 1 + 120 T^{2} + 6530 T^{4} + 201720 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 - 68 T^{2} + 4086 T^{4} - 125732 T^{6} + 3418801 T^{8} )^{2} \)
$47$ \( ( 1 + 60 T^{2} + 4550 T^{4} + 132540 T^{6} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 - 53 T^{2} )^{8} \)
$59$ \( ( 1 - 108 T^{2} + 9110 T^{4} - 375948 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 212 T^{2} + 18486 T^{4} - 788852 T^{6} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 - 34 T^{2} + 4489 T^{4} )^{4} \)
$71$ \( ( 1 + 96 T^{2} + 12338 T^{4} + 483936 T^{6} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 - 12 T + 134 T^{2} - 876 T^{3} + 5329 T^{4} )^{4} \)
$79$ \( ( 1 - 8 T + 126 T^{2} - 632 T^{3} + 6241 T^{4} )^{4} \)
$83$ \( ( 1 - 92 T^{2} + 8982 T^{4} - 633788 T^{6} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 + 120 T^{2} + 5570 T^{4} + 950520 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 - 4 T + 150 T^{2} - 388 T^{3} + 9409 T^{4} )^{4} \)
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