Properties

Label 4032.2.c.n
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 448)
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.

\(f(q)\) \(=\) \( q + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + q^{7} +O(q^{10})\) \( q + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + q^{7} + ( -2 + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{11} + ( 1 - 2 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{13} -2 q^{17} + ( 1 - 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{19} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{23} + ( 1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{25} + ( 2 - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{29} + 4 q^{31} + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{35} + ( 2 - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{37} + 2 q^{41} + ( 2 - 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{43} + ( 4 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{47} + q^{49} + 12 \zeta_{12}^{3} q^{53} + ( 8 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{55} + ( 5 - 10 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{59} + ( -7 + 14 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{61} + ( -8 + 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{65} + 8 \zeta_{12}^{3} q^{67} + ( 4 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{71} + ( 6 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{73} + ( -2 + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{77} + ( -4 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{79} + ( 5 - 10 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{83} + ( -2 + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{85} + ( 2 + 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{89} + ( 1 - 2 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{91} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{95} + ( 2 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{7} + O(q^{10}) \) \( 4q + 4q^{7} - 8q^{17} + 4q^{25} + 16q^{31} + 8q^{41} + 16q^{47} + 4q^{49} + 32q^{55} - 32q^{65} + 16q^{71} + 24q^{73} - 16q^{79} + 8q^{89} + 8q^{97} + O(q^{100}) \)

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.n 4
3.b odd 2 1 448.2.b.d yes 4
4.b odd 2 1 4032.2.c.k 4
8.b even 2 1 inner 4032.2.c.n 4
8.d odd 2 1 4032.2.c.k 4
12.b even 2 1 448.2.b.c 4
21.c even 2 1 3136.2.b.h 4
24.f even 2 1 448.2.b.c 4
24.h odd 2 1 448.2.b.d yes 4
48.i odd 4 1 1792.2.a.i 2
48.i odd 4 1 1792.2.a.s 2
48.k even 4 1 1792.2.a.k 2
48.k even 4 1 1792.2.a.q 2
84.h odd 2 1 3136.2.b.g 4
168.e odd 2 1 3136.2.b.g 4
168.i even 2 1 3136.2.b.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
448.2.b.c 4 12.b even 2 1
448.2.b.c 4 24.f even 2 1
448.2.b.d yes 4 3.b odd 2 1
448.2.b.d yes 4 24.h odd 2 1
1792.2.a.i 2 48.i odd 4 1
1792.2.a.k 2 48.k even 4 1
1792.2.a.q 2 48.k even 4 1
1792.2.a.s 2 48.i odd 4 1
3136.2.b.g 4 84.h odd 2 1
3136.2.b.g 4 168.e odd 2 1
3136.2.b.h 4 21.c even 2 1
3136.2.b.h 4 168.i even 2 1
4032.2.c.k 4 4.b odd 2 1
4032.2.c.k 4 8.d odd 2 1
4032.2.c.n 4 1.a even 1 1 trivial
4032.2.c.n 4 8.b even 2 1 inner

Hecke kernels

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

\( T_{5}^{4} + 8 T_{5}^{2} + 4 \)
\( T_{11}^{4} + 32 T_{11}^{2} + 64 \)
\( T_{13}^{4} + 56 T_{13}^{2} + 484 \)
\( T_{17} + 2 \)
\( T_{23}^{2} - 12 \)
\( T_{31} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 12 T^{2} + 74 T^{4} - 300 T^{6} + 625 T^{8} \)
$7$ \( ( 1 - T )^{4} \)
$11$ \( 1 - 12 T^{2} + 86 T^{4} - 1452 T^{6} + 14641 T^{8} \)
$13$ \( 1 + 4 T^{2} + 42 T^{4} + 676 T^{6} + 28561 T^{8} \)
$17$ \( ( 1 + 2 T + 17 T^{2} )^{4} \)
$19$ \( 1 - 52 T^{2} + 1290 T^{4} - 18772 T^{6} + 130321 T^{8} \)
$23$ \( ( 1 + 34 T^{2} + 529 T^{4} )^{2} \)
$29$ \( 1 - 84 T^{2} + 3254 T^{4} - 70644 T^{6} + 707281 T^{8} \)
$31$ \( ( 1 - 4 T + 31 T^{2} )^{4} \)
$37$ \( 1 - 116 T^{2} + 5910 T^{4} - 158804 T^{6} + 1874161 T^{8} \)
$41$ \( ( 1 - 2 T + 41 T^{2} )^{4} \)
$43$ \( 1 - 140 T^{2} + 8406 T^{4} - 258860 T^{6} + 3418801 T^{8} \)
$47$ \( ( 1 - 8 T + 62 T^{2} - 376 T^{3} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 + 38 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( 1 - 84 T^{2} + 8426 T^{4} - 292404 T^{6} + 12117361 T^{8} \)
$61$ \( 1 + 52 T^{2} + 7530 T^{4} + 193492 T^{6} + 13845841 T^{8} \)
$67$ \( ( 1 - 70 T^{2} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 - 8 T + 110 T^{2} - 568 T^{3} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 12 T + 134 T^{2} - 876 T^{3} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 + 8 T + 126 T^{2} + 632 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( 1 - 164 T^{2} + 17802 T^{4} - 1129796 T^{6} + 47458321 T^{8} \)
$89$ \( ( 1 - 4 T - 10 T^{2} - 356 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 4 T + 150 T^{2} - 388 T^{3} + 9409 T^{4} )^{2} \)
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