Properties

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

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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: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1344)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} - q^{7} +O(q^{10})\) \( q + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} - q^{7} + ( 1 - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{11} -4 \zeta_{12}^{3} q^{13} + ( 1 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{17} + ( 2 - 4 \zeta_{12}^{2} ) q^{19} + ( 3 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{23} + ( 1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{25} + ( -2 + 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{29} + 2 q^{31} + ( -1 + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{35} -2 \zeta_{12}^{3} q^{37} + ( -1 - 10 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{41} + ( 4 - 8 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{43} -4 q^{47} + q^{49} + ( -2 + 4 \zeta_{12}^{2} ) q^{53} -2 q^{55} + 8 \zeta_{12}^{3} q^{59} + ( -6 + 12 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{61} + ( 4 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{65} + ( 6 - 12 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{67} + ( 5 - 10 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{71} + ( -12 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{73} + ( -1 + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{77} + ( -2 - 12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{79} + ( -2 + 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{83} + ( 4 - 8 \zeta_{12}^{2} + 10 \zeta_{12}^{3} ) q^{85} + ( -7 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{89} + 4 \zeta_{12}^{3} q^{91} + ( -6 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{95} + ( 8 - 12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{7} + O(q^{10}) \) \( 4q - 4q^{7} + 4q^{17} + 12q^{23} + 4q^{25} + 8q^{31} - 4q^{41} - 16q^{47} + 4q^{49} - 8q^{55} + 16q^{65} + 20q^{71} - 48q^{73} - 8q^{79} - 28q^{89} - 24q^{95} + 32q^{97} + O(q^{100}) \)

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
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 0 0 2.73205i 0 −1.00000 0 0 0
2017.2 0 0 0 0.732051i 0 −1.00000 0 0 0
2017.3 0 0 0 0.732051i 0 −1.00000 0 0 0
2017.4 0 0 0 2.73205i 0 −1.00000 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
8.b 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.c.l 4
3.b odd 2 1 1344.2.c.e 4
4.b odd 2 1 4032.2.c.o 4
8.b even 2 1 inner 4032.2.c.l 4
8.d odd 2 1 4032.2.c.o 4
12.b even 2 1 1344.2.c.h yes 4
24.f even 2 1 1344.2.c.h yes 4
24.h odd 2 1 1344.2.c.e 4
48.i odd 4 1 5376.2.a.t 2
48.i odd 4 1 5376.2.a.z 2
48.k even 4 1 5376.2.a.n 2
48.k even 4 1 5376.2.a.bd 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1344.2.c.e 4 3.b odd 2 1
1344.2.c.e 4 24.h odd 2 1
1344.2.c.h yes 4 12.b even 2 1
1344.2.c.h yes 4 24.f even 2 1
4032.2.c.l 4 1.a even 1 1 trivial
4032.2.c.l 4 8.b even 2 1 inner
4032.2.c.o 4 4.b odd 2 1
4032.2.c.o 4 8.d odd 2 1
5376.2.a.n 2 48.k even 4 1
5376.2.a.t 2 48.i odd 4 1
5376.2.a.z 2 48.i odd 4 1
5376.2.a.bd 2 48.k even 4 1

Hecke kernels

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

\( T_{5}^{4} + 8 T_{5}^{2} + 4 \)
\( T_{11}^{4} + 8 T_{11}^{2} + 4 \)
\( T_{13}^{2} + 16 \)
\( T_{17}^{2} - 2 T_{17} - 26 \)
\( T_{23}^{2} - 6 T_{23} + 6 \)
\( T_{31} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 12 T^{2} + 74 T^{4} - 300 T^{6} + 625 T^{8} \)
$7$ \( ( 1 + T )^{4} \)
$11$ \( 1 - 36 T^{2} + 554 T^{4} - 4356 T^{6} + 14641 T^{8} \)
$13$ \( ( 1 - 6 T + 13 T^{2} )^{2}( 1 + 6 T + 13 T^{2} )^{2} \)
$17$ \( ( 1 - 2 T + 8 T^{2} - 34 T^{3} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2} \)
$23$ \( ( 1 - 6 T + 52 T^{2} - 138 T^{3} + 529 T^{4} )^{2} \)
$29$ \( 1 - 60 T^{2} + 1814 T^{4} - 50460 T^{6} + 707281 T^{8} \)
$31$ \( ( 1 - 2 T + 31 T^{2} )^{4} \)
$37$ \( ( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2} \)
$41$ \( ( 1 + 2 T + 8 T^{2} + 82 T^{3} + 1681 T^{4} )^{2} \)
$43$ \( 1 - 68 T^{2} + 4086 T^{4} - 125732 T^{6} + 3418801 T^{8} \)
$47$ \( ( 1 + 4 T + 47 T^{2} )^{4} \)
$53$ \( ( 1 - 94 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 - 54 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( 1 - 20 T^{2} + 5814 T^{4} - 74420 T^{6} + 13845841 T^{8} \)
$67$ \( 1 - 20 T^{2} + 2166 T^{4} - 89780 T^{6} + 20151121 T^{8} \)
$71$ \( ( 1 - 10 T + 92 T^{2} - 710 T^{3} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 + 24 T + 278 T^{2} + 1752 T^{3} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 + 4 T + 54 T^{2} + 316 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( 1 - 236 T^{2} + 25974 T^{4} - 1625804 T^{6} + 47458321 T^{8} \)
$89$ \( ( 1 + 14 T + 200 T^{2} + 1246 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 16 T + 150 T^{2} - 1552 T^{3} + 9409 T^{4} )^{2} \)
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