Properties

Label 4032.2.b.b
Level 4032
Weight 2
Character orbit 4032.b
Analytic conductor 32.196
Analytic rank 0
Dimension 2
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.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 112)
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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3583.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 3.46410i 0 −2.00000 + 1.73205i 0 0 0
3583.2 0 0 0 3.46410i 0 −2.00000 1.73205i 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
28.d 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.b.b 2
3.b odd 2 1 448.2.f.a 2
4.b odd 2 1 4032.2.b.h 2
7.b odd 2 1 4032.2.b.h 2
8.b even 2 1 1008.2.b.b 2
8.d odd 2 1 1008.2.b.g 2
12.b even 2 1 448.2.f.c 2
21.c even 2 1 448.2.f.c 2
24.f even 2 1 112.2.f.a 2
24.h odd 2 1 112.2.f.b yes 2
28.d even 2 1 inner 4032.2.b.b 2
48.i odd 4 2 1792.2.e.c 4
48.k even 4 2 1792.2.e.a 4
56.e even 2 1 1008.2.b.b 2
56.h odd 2 1 1008.2.b.g 2
84.h odd 2 1 448.2.f.a 2
120.i odd 2 1 2800.2.k.b 2
120.m even 2 1 2800.2.k.e 2
120.q odd 4 2 2800.2.e.c 4
120.w even 4 2 2800.2.e.b 4
168.e odd 2 1 112.2.f.b yes 2
168.i even 2 1 112.2.f.a 2
168.s odd 6 1 784.2.p.a 2
168.s odd 6 1 784.2.p.b 2
168.v even 6 1 784.2.p.e 2
168.v even 6 1 784.2.p.f 2
168.ba even 6 1 784.2.p.e 2
168.ba even 6 1 784.2.p.f 2
168.be odd 6 1 784.2.p.a 2
168.be odd 6 1 784.2.p.b 2
336.v odd 4 2 1792.2.e.c 4
336.y even 4 2 1792.2.e.a 4
840.b odd 2 1 2800.2.k.b 2
840.u even 2 1 2800.2.k.e 2
840.bm even 4 2 2800.2.e.b 4
840.bp odd 4 2 2800.2.e.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.f.a 2 24.f even 2 1
112.2.f.a 2 168.i even 2 1
112.2.f.b yes 2 24.h odd 2 1
112.2.f.b yes 2 168.e odd 2 1
448.2.f.a 2 3.b odd 2 1
448.2.f.a 2 84.h odd 2 1
448.2.f.c 2 12.b even 2 1
448.2.f.c 2 21.c even 2 1
784.2.p.a 2 168.s odd 6 1
784.2.p.a 2 168.be odd 6 1
784.2.p.b 2 168.s odd 6 1
784.2.p.b 2 168.be odd 6 1
784.2.p.e 2 168.v even 6 1
784.2.p.e 2 168.ba even 6 1
784.2.p.f 2 168.v even 6 1
784.2.p.f 2 168.ba even 6 1
1008.2.b.b 2 8.b even 2 1
1008.2.b.b 2 56.e even 2 1
1008.2.b.g 2 8.d odd 2 1
1008.2.b.g 2 56.h odd 2 1
1792.2.e.a 4 48.k even 4 2
1792.2.e.a 4 336.y even 4 2
1792.2.e.c 4 48.i odd 4 2
1792.2.e.c 4 336.v odd 4 2
2800.2.e.b 4 120.w even 4 2
2800.2.e.b 4 840.bm even 4 2
2800.2.e.c 4 120.q odd 4 2
2800.2.e.c 4 840.bp odd 4 2
2800.2.k.b 2 120.i odd 2 1
2800.2.k.b 2 840.b odd 2 1
2800.2.k.e 2 120.m even 2 1
2800.2.k.e 2 840.u even 2 1
4032.2.b.b 2 1.a even 1 1 trivial
4032.2.b.b 2 28.d even 2 1 inner
4032.2.b.h 2 4.b odd 2 1
4032.2.b.h 2 7.b odd 2 1

Hecke kernels

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

\( T_{5}^{2} + 12 \)
\( T_{11}^{2} + 12 \)
\( T_{19} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 2 T^{2} + 25 T^{4} \)
$7$ \( 1 + 4 T + 7 T^{2} \)
$11$ \( 1 - 10 T^{2} + 121 T^{4} \)
$13$ \( 1 - 14 T^{2} + 169 T^{4} \)
$17$ \( ( 1 - 17 T^{2} )^{2} \)
$19$ \( ( 1 + 2 T + 19 T^{2} )^{2} \)
$23$ \( 1 - 34 T^{2} + 529 T^{4} \)
$29$ \( ( 1 - 6 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 8 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 2 T + 37 T^{2} )^{2} \)
$41$ \( 1 - 34 T^{2} + 1681 T^{4} \)
$43$ \( ( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} ) \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( ( 1 - 6 T + 53 T^{2} )^{2} \)
$59$ \( ( 1 - 6 T + 59 T^{2} )^{2} \)
$61$ \( 1 - 110 T^{2} + 3721 T^{4} \)
$67$ \( ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \)
$71$ \( 1 - 130 T^{2} + 5041 T^{4} \)
$73$ \( 1 - 98 T^{2} + 5329 T^{4} \)
$79$ \( 1 - 146 T^{2} + 6241 T^{4} \)
$83$ \( ( 1 + 6 T + 83 T^{2} )^{2} \)
$89$ \( 1 - 130 T^{2} + 7921 T^{4} \)
$97$ \( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \)
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