Properties

Label 4032.2.k.f
Level 4032
Weight 2
Character orbit 4032.k
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.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 504)
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 \(\zeta_{16}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 \zeta_{16} + 2 \zeta_{16}^{7} ) q^{5} + ( 1 + \zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{3} - \zeta_{16}^{5} - \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{7} +O(q^{10})\) \( q + ( -2 \zeta_{16} + 2 \zeta_{16}^{7} ) q^{5} + ( 1 + \zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{3} - \zeta_{16}^{5} - \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{7} + ( \zeta_{16}^{2} + 2 \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{11} + ( -2 \zeta_{16} - 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{13} + ( -2 \zeta_{16} - 4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} + 2 \zeta_{16}^{7} ) q^{17} + ( -2 \zeta_{16} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{19} + ( -3 \zeta_{16}^{2} - 2 \zeta_{16}^{4} - 3 \zeta_{16}^{6} ) q^{23} + ( 3 + 4 \zeta_{16}^{2} - 4 \zeta_{16}^{6} ) q^{25} + ( \zeta_{16}^{2} - 4 \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{29} + ( 4 \zeta_{16} + 4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} + 4 \zeta_{16}^{7} ) q^{31} + ( -4 \zeta_{16} - 2 \zeta_{16}^{3} + 4 \zeta_{16}^{4} + 2 \zeta_{16}^{5} + 4 \zeta_{16}^{7} ) q^{35} -4 q^{37} + ( 2 \zeta_{16} - 4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{41} + ( -2 + 6 \zeta_{16}^{2} - 6 \zeta_{16}^{6} ) q^{43} + ( 4 \zeta_{16}^{3} - 4 \zeta_{16}^{5} ) q^{47} + ( -1 - 2 \zeta_{16} + 4 \zeta_{16}^{2} - 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} - 4 \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{49} + ( -\zeta_{16}^{2} + 4 \zeta_{16}^{4} - \zeta_{16}^{6} ) q^{53} + ( -2 \zeta_{16} - 6 \zeta_{16}^{3} - 6 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{55} + ( 8 \zeta_{16}^{2} + 8 \zeta_{16}^{4} + 8 \zeta_{16}^{6} ) q^{65} + ( 4 - 4 \zeta_{16}^{2} + 4 \zeta_{16}^{6} ) q^{67} + ( \zeta_{16}^{2} + 6 \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{71} + ( -6 \zeta_{16} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - 6 \zeta_{16}^{7} ) q^{73} + ( 2 \zeta_{16} + 3 \zeta_{16}^{2} + 4 \zeta_{16}^{4} + 3 \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{77} + ( -8 - 4 \zeta_{16}^{2} + 4 \zeta_{16}^{6} ) q^{79} + ( 8 \zeta_{16} - 4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} - 8 \zeta_{16}^{7} ) q^{83} + ( 8 + 12 \zeta_{16}^{2} - 12 \zeta_{16}^{6} ) q^{85} + ( -2 \zeta_{16} - 4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} + 2 \zeta_{16}^{7} ) q^{89} + ( -2 \zeta_{16} - 4 \zeta_{16}^{2} - 6 \zeta_{16}^{3} - 6 \zeta_{16}^{5} + 4 \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{91} -8 \zeta_{16}^{4} q^{95} + ( -2 \zeta_{16} + 6 \zeta_{16}^{3} + 6 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{7} + O(q^{10}) \) \( 8q + 8q^{7} + 24q^{25} - 32q^{37} - 16q^{43} - 8q^{49} + 32q^{67} - 64q^{79} + 64q^{85} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3905.1
0.923880 + 0.382683i
0.923880 0.382683i
0.382683 0.923880i
0.382683 + 0.923880i
−0.382683 0.923880i
−0.382683 + 0.923880i
−0.923880 + 0.382683i
−0.923880 0.382683i
0 0 0 −3.69552 0 2.41421 1.08239i 0 0 0
3905.2 0 0 0 −3.69552 0 2.41421 + 1.08239i 0 0 0
3905.3 0 0 0 −1.53073 0 −0.414214 2.61313i 0 0 0
3905.4 0 0 0 −1.53073 0 −0.414214 + 2.61313i 0 0 0
3905.5 0 0 0 1.53073 0 −0.414214 2.61313i 0 0 0
3905.6 0 0 0 1.53073 0 −0.414214 + 2.61313i 0 0 0
3905.7 0 0 0 3.69552 0 2.41421 1.08239i 0 0 0
3905.8 0 0 0 3.69552 0 2.41421 + 1.08239i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3905.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c 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.k.f 8
3.b odd 2 1 inner 4032.2.k.f 8
4.b odd 2 1 4032.2.k.e 8
7.b odd 2 1 inner 4032.2.k.f 8
8.b even 2 1 504.2.k.a 8
8.d odd 2 1 1008.2.k.c 8
12.b even 2 1 4032.2.k.e 8
21.c even 2 1 inner 4032.2.k.f 8
24.f even 2 1 1008.2.k.c 8
24.h odd 2 1 504.2.k.a 8
28.d even 2 1 4032.2.k.e 8
56.e even 2 1 1008.2.k.c 8
56.h odd 2 1 504.2.k.a 8
56.j odd 6 2 3528.2.bl.b 16
56.p even 6 2 3528.2.bl.b 16
84.h odd 2 1 4032.2.k.e 8
168.e odd 2 1 1008.2.k.c 8
168.i even 2 1 504.2.k.a 8
168.s odd 6 2 3528.2.bl.b 16
168.ba even 6 2 3528.2.bl.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.k.a 8 8.b even 2 1
504.2.k.a 8 24.h odd 2 1
504.2.k.a 8 56.h odd 2 1
504.2.k.a 8 168.i even 2 1
1008.2.k.c 8 8.d odd 2 1
1008.2.k.c 8 24.f even 2 1
1008.2.k.c 8 56.e even 2 1
1008.2.k.c 8 168.e odd 2 1
3528.2.bl.b 16 56.j odd 6 2
3528.2.bl.b 16 56.p even 6 2
3528.2.bl.b 16 168.s odd 6 2
3528.2.bl.b 16 168.ba even 6 2
4032.2.k.e 8 4.b odd 2 1
4032.2.k.e 8 12.b even 2 1
4032.2.k.e 8 28.d even 2 1
4032.2.k.e 8 84.h odd 2 1
4032.2.k.f 8 1.a even 1 1 trivial
4032.2.k.f 8 3.b odd 2 1 inner
4032.2.k.f 8 7.b odd 2 1 inner
4032.2.k.f 8 21.c 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 \(S_{2}^{\mathrm{new}}(4032, [\chi])\):

\( T_{5}^{4} - 16 T_{5}^{2} + 32 \)
\( T_{43}^{2} + 4 T_{43} - 68 \)
\( T_{67}^{2} - 8 T_{67} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 + 4 T^{2} + 22 T^{4} + 100 T^{6} + 625 T^{8} )^{2} \)
$7$ \( ( 1 - 4 T + 10 T^{2} - 28 T^{3} + 49 T^{4} )^{2} \)
$11$ \( ( 1 - 32 T^{2} + 466 T^{4} - 3872 T^{6} + 14641 T^{8} )^{2} \)
$13$ \( ( 1 - 20 T^{2} + 310 T^{4} - 3380 T^{6} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 - 12 T^{2} + 582 T^{4} - 3468 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 - 44 T^{2} + 1078 T^{4} - 15884 T^{6} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 - 48 T^{2} + 1346 T^{4} - 25392 T^{6} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 - 80 T^{2} + 3154 T^{4} - 67280 T^{6} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 + 4 T^{2} - 122 T^{4} + 3844 T^{6} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 + 4 T + 37 T^{2} )^{8} \)
$41$ \( ( 1 + 84 T^{2} + 3558 T^{4} + 141204 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 + 4 T + 18 T^{2} + 172 T^{3} + 1849 T^{4} )^{4} \)
$47$ \( ( 1 + 124 T^{2} + 7750 T^{4} + 273916 T^{6} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 - 176 T^{2} + 13234 T^{4} - 494384 T^{6} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 + 59 T^{2} )^{8} \)
$61$ \( ( 1 - 61 T^{2} )^{8} \)
$67$ \( ( 1 - 8 T + 118 T^{2} - 536 T^{3} + 4489 T^{4} )^{4} \)
$71$ \( ( 1 - 208 T^{2} + 20610 T^{4} - 1048528 T^{6} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 - 132 T^{2} + 8742 T^{4} - 703428 T^{6} + 28398241 T^{8} )^{2} \)
$79$ \( ( 1 + 16 T + 190 T^{2} + 1264 T^{3} + 6241 T^{4} )^{4} \)
$83$ \( ( 1 + 12 T^{2} + 13302 T^{4} + 82668 T^{6} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 + 276 T^{2} + 34854 T^{4} + 2186196 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 - 228 T^{2} + 31686 T^{4} - 2145252 T^{6} + 88529281 T^{8} )^{2} \)
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