Properties

Label 4032.2.b.m
Level $4032$
Weight $2$
Character orbit 4032.b
Analytic conductor $32.196$
Analytic rank $0$
Dimension $4$
CM discriminant -7
Inner twists $8$

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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: \(4\)
Coefficient field: \(\Q(i, \sqrt{7})\)
Defining polynomial: \( x^{4} - 3x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{7} - \beta_1 q^{11} + 2 \beta_1 q^{23} + 5 q^{25} - \beta_{2} q^{29} + 6 q^{37} + 2 \beta_{3} q^{43} - 7 q^{49} + \beta_{2} q^{53} + 6 \beta_{3} q^{67} - 4 \beta_1 q^{71} + \beta_{2} q^{77} + 6 \beta_{3} q^{79}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 20 q^{25} + 24 q^{37} - 28 q^{49}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 3x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu^{3} - 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -2\nu^{3} + 10\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{2} + 5\beta_1 ) / 8 \) Copy content Toggle raw display

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
−1.32288 + 0.500000i
1.32288 0.500000i
1.32288 + 0.500000i
−1.32288 0.500000i
0 0 0 0 0 2.64575i 0 0 0
3583.2 0 0 0 0 0 2.64575i 0 0 0
3583.3 0 0 0 0 0 2.64575i 0 0 0
3583.4 0 0 0 0 0 2.64575i 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner
84.h odd 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.m 4
3.b odd 2 1 inner 4032.2.b.m 4
4.b odd 2 1 inner 4032.2.b.m 4
7.b odd 2 1 CM 4032.2.b.m 4
8.b even 2 1 252.2.b.c 4
8.d odd 2 1 252.2.b.c 4
12.b even 2 1 inner 4032.2.b.m 4
21.c even 2 1 inner 4032.2.b.m 4
24.f even 2 1 252.2.b.c 4
24.h odd 2 1 252.2.b.c 4
28.d even 2 1 inner 4032.2.b.m 4
56.e even 2 1 252.2.b.c 4
56.h odd 2 1 252.2.b.c 4
84.h odd 2 1 inner 4032.2.b.m 4
168.e odd 2 1 252.2.b.c 4
168.i even 2 1 252.2.b.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.b.c 4 8.b even 2 1
252.2.b.c 4 8.d odd 2 1
252.2.b.c 4 24.f even 2 1
252.2.b.c 4 24.h odd 2 1
252.2.b.c 4 56.e even 2 1
252.2.b.c 4 56.h odd 2 1
252.2.b.c 4 168.e odd 2 1
252.2.b.c 4 168.i even 2 1
4032.2.b.m 4 1.a even 1 1 trivial
4032.2.b.m 4 3.b odd 2 1 inner
4032.2.b.m 4 4.b odd 2 1 inner
4032.2.b.m 4 7.b odd 2 1 CM
4032.2.b.m 4 12.b even 2 1 inner
4032.2.b.m 4 21.c even 2 1 inner
4032.2.b.m 4 28.d even 2 1 inner
4032.2.b.m 4 84.h odd 2 1 inner

Hecke kernels

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

\( T_{5} \) Copy content Toggle raw display
\( T_{11}^{2} + 16 \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 7)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 112)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T - 6)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 28)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 112)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 252)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 256)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 252)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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