Properties

Label 4032.2.h.e
Level 4032
Weight 2
Character orbit 4032.h
Analytic conductor 32.196
Analytic rank 0
Dimension 4
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.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1008)
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 \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{5} + \zeta_{8}^{2} q^{7} +O(q^{10})\) \( q + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{5} + \zeta_{8}^{2} q^{7} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{11} + 4 q^{13} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{17} + 8 \zeta_{8}^{2} q^{19} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{23} + 3 q^{25} + ( 5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{29} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{35} -2 q^{37} + ( 5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{41} + 8 \zeta_{8}^{2} q^{43} - q^{49} + ( -7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{53} -8 \zeta_{8}^{2} q^{55} + ( 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{59} -6 q^{61} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{65} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{71} -8 q^{73} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{77} + ( 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{83} -2 q^{85} + ( -7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{89} + 4 \zeta_{8}^{2} q^{91} + ( 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{95} -16 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 16q^{13} + 12q^{25} - 8q^{37} - 4q^{49} - 24q^{61} - 32q^{73} - 8q^{85} - 64q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
575.1
−0.707107 + 0.707107i
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0 0 0 1.41421i 0 1.00000i 0 0 0
575.2 0 0 0 1.41421i 0 1.00000i 0 0 0
575.3 0 0 0 1.41421i 0 1.00000i 0 0 0
575.4 0 0 0 1.41421i 0 1.00000i 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
3.b odd 2 1 inner
4.b odd 2 1 inner
12.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.h.e 4
3.b odd 2 1 inner 4032.2.h.e 4
4.b odd 2 1 inner 4032.2.h.e 4
8.b even 2 1 1008.2.h.a 4
8.d odd 2 1 1008.2.h.a 4
12.b even 2 1 inner 4032.2.h.e 4
24.f even 2 1 1008.2.h.a 4
24.h odd 2 1 1008.2.h.a 4
56.e even 2 1 7056.2.h.j 4
56.h odd 2 1 7056.2.h.j 4
168.e odd 2 1 7056.2.h.j 4
168.i even 2 1 7056.2.h.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.h.a 4 8.b even 2 1
1008.2.h.a 4 8.d odd 2 1
1008.2.h.a 4 24.f even 2 1
1008.2.h.a 4 24.h odd 2 1
4032.2.h.e 4 1.a even 1 1 trivial
4032.2.h.e 4 3.b odd 2 1 inner
4032.2.h.e 4 4.b odd 2 1 inner
4032.2.h.e 4 12.b even 2 1 inner
7056.2.h.j 4 56.e even 2 1
7056.2.h.j 4 56.h odd 2 1
7056.2.h.j 4 168.e odd 2 1
7056.2.h.j 4 168.i even 2 1

Hecke kernels

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

\( T_{5}^{2} + 2 \)
\( T_{11}^{2} - 32 \)
\( T_{13} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 - 8 T^{2} + 25 T^{4} )^{2} \)
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( ( 1 - 10 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 4 T + 13 T^{2} )^{4} \)
$17$ \( ( 1 - 32 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 + 26 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 + 14 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 8 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 31 T^{2} )^{4} \)
$37$ \( ( 1 + 2 T + 37 T^{2} )^{4} \)
$41$ \( ( 1 - 32 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 22 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 + 47 T^{2} )^{4} \)
$53$ \( ( 1 - 8 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 - 10 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 6 T + 61 T^{2} )^{4} \)
$67$ \( ( 1 - 67 T^{2} )^{4} \)
$71$ \( ( 1 + 110 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 + 8 T + 73 T^{2} )^{4} \)
$79$ \( ( 1 - 79 T^{2} )^{4} \)
$83$ \( ( 1 + 38 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 80 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 + 16 T + 97 T^{2} )^{4} \)
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