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

Related objects

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: $$\Q(i, \sqrt{7})$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{7} +O(q^{10})$$ $$q + \beta_{3} q^{7} -\beta_{1} q^{11} + 2 \beta_{1} q^{23} + 5 q^{25} -\beta_{2} q^{29} + 6 q^{37} + 2 \beta_{3} q^{43} -7 q^{49} + \beta_{2} q^{53} + 6 \beta_{3} q^{67} -4 \beta_{1} q^{71} + \beta_{2} q^{77} + 6 \beta_{3} q^{79} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 20q^{25} + 24q^{37} - 28q^{49} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 3 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{3} - 2 \nu$$ $$\beta_{2}$$ $$=$$ $$-2 \nu^{3} + 10 \nu$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{2} + 5 \beta_{1}$$$$)/8$$

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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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 $$S_{2}^{\mathrm{new}}(4032, [\chi])$$:

 $$T_{5}$$ $$T_{11}^{2} + 16$$ $$T_{19}$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 - 5 T^{2} )^{4}$$
$7$ $$( 1 + 7 T^{2} )^{2}$$
$11$ $$( 1 - 6 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 13 T^{2} )^{4}$$
$17$ $$( 1 - 17 T^{2} )^{4}$$
$19$ $$( 1 + 19 T^{2} )^{4}$$
$23$ $$( 1 + 18 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 54 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 + 31 T^{2} )^{4}$$
$37$ $$( 1 - 6 T + 37 T^{2} )^{4}$$
$41$ $$( 1 - 41 T^{2} )^{4}$$
$43$ $$( 1 - 12 T + 43 T^{2} )^{2}( 1 + 12 T + 43 T^{2} )^{2}$$
$47$ $$( 1 + 47 T^{2} )^{4}$$
$53$ $$( 1 - 6 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$( 1 + 59 T^{2} )^{4}$$
$61$ $$( 1 - 61 T^{2} )^{4}$$
$67$ $$( 1 - 4 T + 67 T^{2} )^{2}( 1 + 4 T + 67 T^{2} )^{2}$$
$71$ $$( 1 + 114 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 73 T^{2} )^{4}$$
$79$ $$( 1 - 8 T + 79 T^{2} )^{2}( 1 + 8 T + 79 T^{2} )^{2}$$
$83$ $$( 1 + 83 T^{2} )^{4}$$
$89$ $$( 1 - 89 T^{2} )^{4}$$
$97$ $$( 1 - 97 T^{2} )^{4}$$