Properties

 Label 1008.4.k.a Level $1008$ Weight $4$ Character orbit 1008.k Analytic conductor $59.474$ Analytic rank $0$ Dimension $4$ CM discriminant -7 Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1008 = 2^{4} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1008.k (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$59.4739252858$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 8 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$2\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 63) 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.

 $$f(q)$$ $$=$$ $$q -7 \beta_{2} q^{7} +O(q^{10})$$ $$q -7 \beta_{2} q^{7} + ( -6 \beta_{1} - 7 \beta_{3} ) q^{11} + ( -16 \beta_{1} + 9 \beta_{3} ) q^{23} -125 q^{25} + ( 13 \beta_{1} - 24 \beta_{3} ) q^{29} -4 \beta_{2} q^{37} + 202 \beta_{2} q^{43} + 343 q^{49} + ( -9 \beta_{1} + 76 \beta_{3} ) q^{53} -740 q^{67} + ( -44 \beta_{1} + 97 \beta_{3} ) q^{71} + ( -133 \beta_{1} - 28 \beta_{3} ) q^{77} + 1384 q^{79} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 500q^{25} + 1372q^{49} - 2960q^{67} + 5536q^{79} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 8 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 17 \nu$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 4$$ $$\beta_{3}$$ $$=$$ $$($$$$-5 \nu^{3} - 31 \nu$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 5 \beta_{1}$$$$)/18$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 4$$ $$\nu^{3}$$ $$=$$ $$($$$$-17 \beta_{3} - 31 \beta_{1}$$$$)/18$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$577$$ $$757$$ $$785$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
881.1
 − 1.16372i 1.16372i 2.57794i − 2.57794i
0 0 0 0 0 −18.5203 0 0 0
881.2 0 0 0 0 0 −18.5203 0 0 0
881.3 0 0 0 0 0 18.5203 0 0 0
881.4 0 0 0 0 0 18.5203 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
21.c even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.4.k.a 4
3.b odd 2 1 inner 1008.4.k.a 4
4.b odd 2 1 63.4.c.a 4
7.b odd 2 1 CM 1008.4.k.a 4
12.b even 2 1 63.4.c.a 4
21.c even 2 1 inner 1008.4.k.a 4
28.d even 2 1 63.4.c.a 4
28.f even 6 2 441.4.p.b 8
28.g odd 6 2 441.4.p.b 8
84.h odd 2 1 63.4.c.a 4
84.j odd 6 2 441.4.p.b 8
84.n even 6 2 441.4.p.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.c.a 4 4.b odd 2 1
63.4.c.a 4 12.b even 2 1
63.4.c.a 4 28.d even 2 1
63.4.c.a 4 84.h odd 2 1
441.4.p.b 8 28.f even 6 2
441.4.p.b 8 28.g odd 6 2
441.4.p.b 8 84.j odd 6 2
441.4.p.b 8 84.n even 6 2
1008.4.k.a 4 1.a even 1 1 trivial
1008.4.k.a 4 3.b odd 2 1 inner
1008.4.k.a 4 7.b odd 2 1 CM
1008.4.k.a 4 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}$$ acting on $$S_{4}^{\mathrm{new}}(1008, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$( -343 + T^{2} )^{2}$$
$11$ $$3849444 + 5324 T^{2} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$516834756 + 48668 T^{2} + T^{4}$$
$29$ $$450373284 + 97556 T^{2} + T^{4}$$
$31$ $$T^{4}$$
$37$ $$( -112 + T^{2} )^{2}$$
$41$ $$T^{4}$$
$43$ $$( -285628 + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$2534719716 + 595508 T^{2} + T^{4}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$( 740 + T )^{4}$$
$71$ $$58795580484 + 1431644 T^{2} + T^{4}$$
$73$ $$T^{4}$$
$79$ $$( -1384 + T )^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$