# Properties

 Label 1008.2.cs.q Level $1008$ Weight $2$ Character orbit 1008.cs 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.cs (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.04892052375$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 7 x^{2} + 49$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 112) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{2} ) q^{5} + \beta_{3} q^{7} +O(q^{10})$$ $$q + ( 1 - \beta_{2} ) q^{5} + \beta_{3} q^{7} + ( -\beta_{1} + \beta_{3} ) q^{11} + ( 2 + 4 \beta_{2} ) q^{13} + ( -6 - 3 \beta_{2} ) q^{17} -\beta_{1} q^{19} + ( \beta_{1} + 2 \beta_{3} ) q^{23} + 2 \beta_{2} q^{25} + ( -\beta_{1} - \beta_{3} ) q^{31} + ( \beta_{1} + 2 \beta_{3} ) q^{35} + ( -7 - 7 \beta_{2} ) q^{37} + ( 2 + 4 \beta_{2} ) q^{41} + ( 4 \beta_{1} + 2 \beta_{3} ) q^{43} -3 \beta_{1} q^{47} + 7 q^{49} -3 \beta_{2} q^{53} + 3 \beta_{3} q^{55} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{59} + ( 1 - \beta_{2} ) q^{61} + ( 6 + 6 \beta_{2} ) q^{65} + ( -\beta_{1} + \beta_{3} ) q^{67} + ( -4 \beta_{1} - 2 \beta_{3} ) q^{71} + ( -6 - 3 \beta_{2} ) q^{73} + ( 14 + 7 \beta_{2} ) q^{77} + ( -\beta_{1} - 2 \beta_{3} ) q^{79} -9 q^{85} + ( -1 + \beta_{2} ) q^{89} + ( -4 \beta_{1} - 2 \beta_{3} ) q^{91} + ( -\beta_{1} + \beta_{3} ) q^{95} + ( -2 - 4 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 6q^{5} + O(q^{10})$$ $$4q + 6q^{5} - 18q^{17} - 4q^{25} - 14q^{37} + 28q^{49} + 6q^{53} + 6q^{61} + 12q^{65} - 18q^{73} + 42q^{77} - 36q^{85} - 6q^{89} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7 x^{2} + 49$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/7$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$7 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$7 \beta_{3}$$

## Character values

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

 $$n$$ $$127$$ $$577$$ $$757$$ $$785$$ $$\chi(n)$$ $$-1$$ $$1 + \beta_{2}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
271.1
 1.32288 − 2.29129i −1.32288 + 2.29129i 1.32288 + 2.29129i −1.32288 − 2.29129i
0 0 0 1.50000 + 0.866025i 0 −2.64575 0 0 0
271.2 0 0 0 1.50000 + 0.866025i 0 2.64575 0 0 0
703.1 0 0 0 1.50000 0.866025i 0 −2.64575 0 0 0
703.2 0 0 0 1.50000 0.866025i 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
4.b odd 2 1 inner
7.d odd 6 1 inner
28.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.cs.q 4
3.b odd 2 1 112.2.p.c 4
4.b odd 2 1 inner 1008.2.cs.q 4
7.c even 3 1 7056.2.b.s 4
7.d odd 6 1 inner 1008.2.cs.q 4
7.d odd 6 1 7056.2.b.s 4
12.b even 2 1 112.2.p.c 4
21.c even 2 1 784.2.p.g 4
21.g even 6 1 112.2.p.c 4
21.g even 6 1 784.2.f.d 4
21.h odd 6 1 784.2.f.d 4
21.h odd 6 1 784.2.p.g 4
24.f even 2 1 448.2.p.c 4
24.h odd 2 1 448.2.p.c 4
28.f even 6 1 inner 1008.2.cs.q 4
28.f even 6 1 7056.2.b.s 4
28.g odd 6 1 7056.2.b.s 4
84.h odd 2 1 784.2.p.g 4
84.j odd 6 1 112.2.p.c 4
84.j odd 6 1 784.2.f.d 4
84.n even 6 1 784.2.f.d 4
84.n even 6 1 784.2.p.g 4
168.s odd 6 1 3136.2.f.f 4
168.v even 6 1 3136.2.f.f 4
168.ba even 6 1 448.2.p.c 4
168.ba even 6 1 3136.2.f.f 4
168.be odd 6 1 448.2.p.c 4
168.be odd 6 1 3136.2.f.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.p.c 4 3.b odd 2 1
112.2.p.c 4 12.b even 2 1
112.2.p.c 4 21.g even 6 1
112.2.p.c 4 84.j odd 6 1
448.2.p.c 4 24.f even 2 1
448.2.p.c 4 24.h odd 2 1
448.2.p.c 4 168.ba even 6 1
448.2.p.c 4 168.be odd 6 1
784.2.f.d 4 21.g even 6 1
784.2.f.d 4 21.h odd 6 1
784.2.f.d 4 84.j odd 6 1
784.2.f.d 4 84.n even 6 1
784.2.p.g 4 21.c even 2 1
784.2.p.g 4 21.h odd 6 1
784.2.p.g 4 84.h odd 2 1
784.2.p.g 4 84.n even 6 1
1008.2.cs.q 4 1.a even 1 1 trivial
1008.2.cs.q 4 4.b odd 2 1 inner
1008.2.cs.q 4 7.d odd 6 1 inner
1008.2.cs.q 4 28.f even 6 1 inner
3136.2.f.f 4 168.s odd 6 1
3136.2.f.f 4 168.v even 6 1
3136.2.f.f 4 168.ba even 6 1
3136.2.f.f 4 168.be odd 6 1
7056.2.b.s 4 7.c even 3 1
7056.2.b.s 4 7.d odd 6 1
7056.2.b.s 4 28.f even 6 1
7056.2.b.s 4 28.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1008, [\chi])$$:

 $$T_{5}^{2} - 3 T_{5} + 3$$ $$T_{11}^{4} - 21 T_{11}^{2} + 441$$ $$T_{13}^{2} + 12$$ $$T_{17}^{2} + 9 T_{17} + 27$$ $$T_{19}^{4} + 7 T_{19}^{2} + 49$$