Properties

Label 4032.2.v.b
Level 4032
Weight 2
Character orbit 4032.v
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.v (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1008)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + 2 \zeta_{8} q^{5} - q^{7} +O(q^{10})\) \( q + 2 \zeta_{8} q^{5} - q^{7} + 4 \zeta_{8}^{3} q^{11} + ( 4 + 4 \zeta_{8}^{2} ) q^{13} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{17} + ( -4 + 4 \zeta_{8}^{2} ) q^{19} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{23} -\zeta_{8}^{2} q^{25} + 8 \zeta_{8}^{3} q^{29} -2 \zeta_{8} q^{35} + ( -5 + 5 \zeta_{8}^{2} ) q^{37} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{41} + ( -7 - 7 \zeta_{8}^{2} ) q^{43} + q^{49} -10 \zeta_{8} q^{53} -8 q^{55} -8 \zeta_{8}^{3} q^{59} + ( 4 + 4 \zeta_{8}^{2} ) q^{61} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{65} + ( -7 + 7 \zeta_{8}^{2} ) q^{67} + ( -5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{71} + 6 \zeta_{8}^{2} q^{73} -4 \zeta_{8}^{3} q^{77} + 4 \zeta_{8} q^{83} + ( 8 - 8 \zeta_{8}^{2} ) q^{85} + ( -12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{89} + ( -4 - 4 \zeta_{8}^{2} ) q^{91} + ( -8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{95} -10 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{7} + O(q^{10}) \) \( 4q - 4q^{7} + 16q^{13} - 16q^{19} - 20q^{37} - 28q^{43} + 4q^{49} - 32q^{55} + 16q^{61} - 28q^{67} + 32q^{85} - 16q^{91} - 40q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-\zeta_{8}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1583.1
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
0 0 0 −1.41421 1.41421i 0 −1.00000 0 0 0
1583.2 0 0 0 1.41421 + 1.41421i 0 −1.00000 0 0 0
3599.1 0 0 0 −1.41421 + 1.41421i 0 −1.00000 0 0 0
3599.2 0 0 0 1.41421 1.41421i 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
3.b odd 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4032.2.v.b 4
3.b odd 2 1 inner 4032.2.v.b 4
4.b odd 2 1 1008.2.v.a 4
12.b even 2 1 1008.2.v.a 4
16.e even 4 1 1008.2.v.a 4
16.f odd 4 1 inner 4032.2.v.b 4
48.i odd 4 1 1008.2.v.a 4
48.k even 4 1 inner 4032.2.v.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.v.a 4 4.b odd 2 1
1008.2.v.a 4 12.b even 2 1
1008.2.v.a 4 16.e even 4 1
1008.2.v.a 4 48.i odd 4 1
4032.2.v.b 4 1.a even 1 1 trivial
4032.2.v.b 4 3.b odd 2 1 inner
4032.2.v.b 4 16.f odd 4 1 inner
4032.2.v.b 4 48.k even 4 1 inner

Hecke kernels

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

\( T_{5}^{4} + 16 \)
\( T_{11}^{4} + 256 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 - 8 T^{2} + 25 T^{4} )( 1 + 8 T^{2} + 25 T^{4} ) \)
$7$ \( ( 1 + T )^{4} \)
$11$ \( 1 - 206 T^{4} + 14641 T^{8} \)
$13$ \( ( 1 - 8 T + 32 T^{2} - 104 T^{3} + 169 T^{4} )^{2} \)
$17$ \( ( 1 - 6 T + 17 T^{2} )^{2}( 1 + 6 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 + 8 T + 32 T^{2} + 152 T^{3} + 361 T^{4} )^{2} \)
$23$ \( ( 1 - 44 T^{2} + 529 T^{4} )^{2} \)
$29$ \( 1 - 1646 T^{4} + 707281 T^{8} \)
$31$ \( ( 1 - 31 T^{2} )^{4} \)
$37$ \( ( 1 - 2 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2} \)
$41$ \( ( 1 + 10 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 + 14 T + 98 T^{2} + 602 T^{3} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 + 47 T^{2} )^{4} \)
$53$ \( 1 - 5582 T^{4} + 7890481 T^{8} \)
$59$ \( 1 - 4046 T^{4} + 12117361 T^{8} \)
$61$ \( ( 1 - 8 T + 32 T^{2} - 488 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 + 14 T + 98 T^{2} + 938 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 - 92 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 16 T + 73 T^{2} )^{2}( 1 + 16 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 - 79 T^{2} )^{4} \)
$83$ \( 1 + 8722 T^{4} + 47458321 T^{8} \)
$89$ \( ( 1 - 110 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 + 10 T + 97 T^{2} )^{4} \)
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