# Properties

 Label 1008.2.k.b Level $1008$ Weight $2$ Character orbit 1008.k Analytic conductor $8.049$ Analytic rank $0$ Dimension $4$ CM no 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: $$\Q(\sqrt{-2}, \sqrt{3})$$ Defining polynomial: $$x^{4} + 4 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: no (minimal twist has level 252) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{5} + ( 1 + \beta_{1} ) q^{7} +O(q^{10})$$ $$q -\beta_{3} q^{5} + ( 1 + \beta_{1} ) q^{7} -\beta_{2} q^{11} -2 \beta_{1} q^{13} -\beta_{3} q^{17} -2 \beta_{1} q^{19} + \beta_{2} q^{23} + 7 q^{25} + \beta_{2} q^{29} + ( 2 \beta_{2} - \beta_{3} ) q^{35} -8 q^{37} + \beta_{3} q^{41} + 2 q^{43} -2 \beta_{3} q^{47} + ( -5 + 2 \beta_{1} ) q^{49} + 3 \beta_{2} q^{53} -6 \beta_{1} q^{55} -4 \beta_{3} q^{59} -4 \beta_{1} q^{61} -4 \beta_{2} q^{65} -8 q^{67} + \beta_{2} q^{71} + 2 \beta_{1} q^{73} + ( -\beta_{2} - 3 \beta_{3} ) q^{77} + 4 q^{79} + 2 \beta_{3} q^{83} + 12 q^{85} + 3 \beta_{3} q^{89} + ( 12 - 2 \beta_{1} ) q^{91} -4 \beta_{2} q^{95} -2 \beta_{1} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{7} + O(q^{10})$$ $$4q + 4q^{7} + 28q^{25} - 32q^{37} + 8q^{43} - 20q^{49} - 32q^{67} + 16q^{79} + 48q^{85} + 48q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 4 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} + 5 \nu$$ $$\beta_{2}$$ $$=$$ $$-3 \nu^{3} - 9 \nu$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} + 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 3 \beta_{1}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 4$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{2} - 9 \beta_{1}$$$$)/6$$

## 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
 − 0.517638i 0.517638i − 1.93185i 1.93185i
0 0 0 −3.46410 0 1.00000 2.44949i 0 0 0
881.2 0 0 0 −3.46410 0 1.00000 + 2.44949i 0 0 0
881.3 0 0 0 3.46410 0 1.00000 2.44949i 0 0 0
881.4 0 0 0 3.46410 0 1.00000 + 2.44949i 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
3.b odd 2 1 inner
7.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.b 4
3.b odd 2 1 inner 1008.2.k.b 4
4.b odd 2 1 252.2.f.a 4
7.b odd 2 1 inner 1008.2.k.b 4
8.b even 2 1 4032.2.k.d 4
8.d odd 2 1 4032.2.k.a 4
12.b even 2 1 252.2.f.a 4
20.d odd 2 1 6300.2.d.c 4
20.e even 4 2 6300.2.f.b 8
21.c even 2 1 inner 1008.2.k.b 4
24.f even 2 1 4032.2.k.a 4
24.h odd 2 1 4032.2.k.d 4
28.d even 2 1 252.2.f.a 4
28.f even 6 2 1764.2.t.b 8
28.g odd 6 2 1764.2.t.b 8
36.f odd 6 2 2268.2.x.i 8
36.h even 6 2 2268.2.x.i 8
56.e even 2 1 4032.2.k.a 4
56.h odd 2 1 4032.2.k.d 4
60.h even 2 1 6300.2.d.c 4
60.l odd 4 2 6300.2.f.b 8
84.h odd 2 1 252.2.f.a 4
84.j odd 6 2 1764.2.t.b 8
84.n even 6 2 1764.2.t.b 8
140.c even 2 1 6300.2.d.c 4
140.j odd 4 2 6300.2.f.b 8
168.e odd 2 1 4032.2.k.a 4
168.i even 2 1 4032.2.k.d 4
252.s odd 6 2 2268.2.x.i 8
252.bi even 6 2 2268.2.x.i 8
420.o odd 2 1 6300.2.d.c 4
420.w even 4 2 6300.2.f.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.f.a 4 4.b odd 2 1
252.2.f.a 4 12.b even 2 1
252.2.f.a 4 28.d even 2 1
252.2.f.a 4 84.h odd 2 1
1008.2.k.b 4 1.a even 1 1 trivial
1008.2.k.b 4 3.b odd 2 1 inner
1008.2.k.b 4 7.b odd 2 1 inner
1008.2.k.b 4 21.c even 2 1 inner
1764.2.t.b 8 28.f even 6 2
1764.2.t.b 8 28.g odd 6 2
1764.2.t.b 8 84.j odd 6 2
1764.2.t.b 8 84.n even 6 2
2268.2.x.i 8 36.f odd 6 2
2268.2.x.i 8 36.h even 6 2
2268.2.x.i 8 252.s odd 6 2
2268.2.x.i 8 252.bi even 6 2
4032.2.k.a 4 8.d odd 2 1
4032.2.k.a 4 24.f even 2 1
4032.2.k.a 4 56.e even 2 1
4032.2.k.a 4 168.e odd 2 1
4032.2.k.d 4 8.b even 2 1
4032.2.k.d 4 24.h odd 2 1
4032.2.k.d 4 56.h odd 2 1
4032.2.k.d 4 168.i even 2 1
6300.2.d.c 4 20.d odd 2 1
6300.2.d.c 4 60.h even 2 1
6300.2.d.c 4 140.c even 2 1
6300.2.d.c 4 420.o odd 2 1
6300.2.f.b 8 20.e even 4 2
6300.2.f.b 8 60.l odd 4 2
6300.2.f.b 8 140.j odd 4 2
6300.2.f.b 8 420.w even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 12$$ acting on $$S_{2}^{\mathrm{new}}(1008, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -12 + T^{2} )^{2}$$
$7$ $$( 7 - 2 T + T^{2} )^{2}$$
$11$ $$( 18 + T^{2} )^{2}$$
$13$ $$( 24 + T^{2} )^{2}$$
$17$ $$( -12 + T^{2} )^{2}$$
$19$ $$( 24 + T^{2} )^{2}$$
$23$ $$( 18 + T^{2} )^{2}$$
$29$ $$( 18 + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$( 8 + T )^{4}$$
$41$ $$( -12 + T^{2} )^{2}$$
$43$ $$( -2 + T )^{4}$$
$47$ $$( -48 + T^{2} )^{2}$$
$53$ $$( 162 + T^{2} )^{2}$$
$59$ $$( -192 + T^{2} )^{2}$$
$61$ $$( 96 + T^{2} )^{2}$$
$67$ $$( 8 + T )^{4}$$
$71$ $$( 18 + T^{2} )^{2}$$
$73$ $$( 24 + T^{2} )^{2}$$
$79$ $$( -4 + T )^{4}$$
$83$ $$( -48 + T^{2} )^{2}$$
$89$ $$( -108 + T^{2} )^{2}$$
$97$ $$( 24 + T^{2} )^{2}$$