# Properties

 Label 1008.2.k.a Level $1008$ Weight $2$ Character orbit 1008.k Analytic conductor $8.049$ 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: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1008.k (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.04892052375$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 8 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2\cdot 3$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{7} +O(q^{10})$$ $$q -\beta_{2} q^{7} + ( \beta_{1} - \beta_{3} ) q^{11} + ( \beta_{1} + \beta_{3} ) q^{23} -5 q^{25} + ( 2 \beta_{1} - \beta_{3} ) q^{29} -4 \beta_{2} q^{37} -2 \beta_{2} q^{43} + 7 q^{49} + ( 2 \beta_{1} + \beta_{3} ) q^{53} + 4 q^{67} + ( \beta_{1} - 3 \beta_{3} ) q^{71} + ( 2 \beta_{1} - 3 \beta_{3} ) q^{77} -8 q^{79} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 20q^{25} + 28q^{49} + 16q^{67} - 32q^{79} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2 \nu^{3} - 16 \nu$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 4$$ $$\beta_{3}$$ $$=$$ $$-\nu^{3} - 5 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{3} - 3 \beta_{1}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 4$$ $$\nu^{3}$$ $$=$$ $$($$$$-16 \beta_{3} + 15 \beta_{1}$$$$)/6$$

## 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 −2.64575 0 0 0
881.2 0 0 0 0 0 −2.64575 0 0 0
881.3 0 0 0 0 0 2.64575 0 0 0
881.4 0 0 0 0 0 2.64575 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 1008.2.k.a 4
3.b odd 2 1 inner 1008.2.k.a 4
4.b odd 2 1 63.2.c.a 4
7.b odd 2 1 CM 1008.2.k.a 4
8.b even 2 1 4032.2.k.b 4
8.d odd 2 1 4032.2.k.c 4
12.b even 2 1 63.2.c.a 4
20.d odd 2 1 1575.2.b.a 4
20.e even 4 2 1575.2.g.d 8
21.c even 2 1 inner 1008.2.k.a 4
24.f even 2 1 4032.2.k.c 4
24.h odd 2 1 4032.2.k.b 4
28.d even 2 1 63.2.c.a 4
28.f even 6 2 441.2.p.b 8
28.g odd 6 2 441.2.p.b 8
36.f odd 6 2 567.2.o.f 8
36.h even 6 2 567.2.o.f 8
56.e even 2 1 4032.2.k.c 4
56.h odd 2 1 4032.2.k.b 4
60.h even 2 1 1575.2.b.a 4
60.l odd 4 2 1575.2.g.d 8
84.h odd 2 1 63.2.c.a 4
84.j odd 6 2 441.2.p.b 8
84.n even 6 2 441.2.p.b 8
140.c even 2 1 1575.2.b.a 4
140.j odd 4 2 1575.2.g.d 8
168.e odd 2 1 4032.2.k.c 4
168.i even 2 1 4032.2.k.b 4
252.s odd 6 2 567.2.o.f 8
252.bi even 6 2 567.2.o.f 8
420.o odd 2 1 1575.2.b.a 4
420.w even 4 2 1575.2.g.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.c.a 4 4.b odd 2 1
63.2.c.a 4 12.b even 2 1
63.2.c.a 4 28.d even 2 1
63.2.c.a 4 84.h odd 2 1
441.2.p.b 8 28.f even 6 2
441.2.p.b 8 28.g odd 6 2
441.2.p.b 8 84.j odd 6 2
441.2.p.b 8 84.n even 6 2
567.2.o.f 8 36.f odd 6 2
567.2.o.f 8 36.h even 6 2
567.2.o.f 8 252.s odd 6 2
567.2.o.f 8 252.bi even 6 2
1008.2.k.a 4 1.a even 1 1 trivial
1008.2.k.a 4 3.b odd 2 1 inner
1008.2.k.a 4 7.b odd 2 1 CM
1008.2.k.a 4 21.c even 2 1 inner
1575.2.b.a 4 20.d odd 2 1
1575.2.b.a 4 60.h even 2 1
1575.2.b.a 4 140.c even 2 1
1575.2.b.a 4 420.o odd 2 1
1575.2.g.d 8 20.e even 4 2
1575.2.g.d 8 60.l odd 4 2
1575.2.g.d 8 140.j odd 4 2
1575.2.g.d 8 420.w even 4 2
4032.2.k.b 4 8.b even 2 1
4032.2.k.b 4 24.h odd 2 1
4032.2.k.b 4 56.h odd 2 1
4032.2.k.b 4 168.i even 2 1
4032.2.k.c 4 8.d odd 2 1
4032.2.k.c 4 24.f even 2 1
4032.2.k.c 4 56.e even 2 1
4032.2.k.c 4 168.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}$$ acting on $$S_{2}^{\mathrm{new}}(1008, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$( -7 + T^{2} )^{2}$$
$11$ $$36 + 44 T^{2} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$324 + 92 T^{2} + T^{4}$$
$29$ $$2916 + 116 T^{2} + T^{4}$$
$31$ $$T^{4}$$
$37$ $$( -112 + T^{2} )^{2}$$
$41$ $$T^{4}$$
$43$ $$( -28 + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$36 + 212 T^{2} + T^{4}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$( -4 + T )^{4}$$
$71$ $$12996 + 284 T^{2} + T^{4}$$
$73$ $$T^{4}$$
$79$ $$( 8 + T )^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$