Properties

Label 4032.2.c.f
Level $4032$
Weight $2$
Character orbit 4032.c
Analytic conductor $32.196$
Analytic rank $1$
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.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.1956820950\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 i q^{5} + q^{7} +O(q^{10})\) \( q + 4 i q^{5} + q^{7} -2 i q^{11} -4 i q^{13} -2 q^{17} + 6 i q^{19} -11 q^{25} -8 i q^{29} -8 q^{31} + 4 i q^{35} -8 i q^{37} -10 q^{41} + 2 i q^{43} -8 q^{47} + q^{49} + 8 q^{55} + 10 i q^{59} + 4 i q^{61} + 16 q^{65} + 2 i q^{67} -8 q^{71} -6 q^{73} -2 i q^{77} + 8 q^{79} -6 i q^{83} -8 i q^{85} -10 q^{89} -4 i q^{91} -24 q^{95} + 2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{7} + O(q^{10}) \) \( 2 q + 2 q^{7} - 4 q^{17} - 22 q^{25} - 16 q^{31} - 20 q^{41} - 16 q^{47} + 2 q^{49} + 16 q^{55} + 32 q^{65} - 16 q^{71} - 12 q^{73} + 16 q^{79} - 20 q^{89} - 48 q^{95} + 4 q^{97} + O(q^{100}) \)

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
1.00000i
1.00000i
0 0 0 4.00000i 0 1.00000 0 0 0
2017.2 0 0 0 4.00000i 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.f 2
3.b odd 2 1 448.2.b.b yes 2
4.b odd 2 1 4032.2.c.b 2
8.b even 2 1 inner 4032.2.c.f 2
8.d odd 2 1 4032.2.c.b 2
12.b even 2 1 448.2.b.a 2
21.c even 2 1 3136.2.b.a 2
24.f even 2 1 448.2.b.a 2
24.h odd 2 1 448.2.b.b yes 2
48.i odd 4 1 1792.2.a.d 1
48.i odd 4 1 1792.2.a.e 1
48.k even 4 1 1792.2.a.a 1
48.k even 4 1 1792.2.a.h 1
84.h odd 2 1 3136.2.b.c 2
168.e odd 2 1 3136.2.b.c 2
168.i even 2 1 3136.2.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
448.2.b.a 2 12.b even 2 1
448.2.b.a 2 24.f even 2 1
448.2.b.b yes 2 3.b odd 2 1
448.2.b.b yes 2 24.h odd 2 1
1792.2.a.a 1 48.k even 4 1
1792.2.a.d 1 48.i odd 4 1
1792.2.a.e 1 48.i odd 4 1
1792.2.a.h 1 48.k even 4 1
3136.2.b.a 2 21.c even 2 1
3136.2.b.a 2 168.i even 2 1
3136.2.b.c 2 84.h odd 2 1
3136.2.b.c 2 168.e odd 2 1
4032.2.c.b 2 4.b odd 2 1
4032.2.c.b 2 8.d odd 2 1
4032.2.c.f 2 1.a even 1 1 trivial
4032.2.c.f 2 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 \(S_{2}^{\mathrm{new}}(4032, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 16 + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( 4 + T^{2} \)
$13$ \( 16 + T^{2} \)
$17$ \( ( 2 + T )^{2} \)
$19$ \( 36 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( 64 + T^{2} \)
$31$ \( ( 8 + T )^{2} \)
$37$ \( 64 + T^{2} \)
$41$ \( ( 10 + T )^{2} \)
$43$ \( 4 + T^{2} \)
$47$ \( ( 8 + T )^{2} \)
$53$ \( T^{2} \)
$59$ \( 100 + T^{2} \)
$61$ \( 16 + T^{2} \)
$67$ \( 4 + T^{2} \)
$71$ \( ( 8 + T )^{2} \)
$73$ \( ( 6 + T )^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( 36 + T^{2} \)
$89$ \( ( 10 + T )^{2} \)
$97$ \( ( -2 + T )^{2} \)
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