# Properties

 Label 1008.2.cs.o 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: $$\Q(\sqrt{-3}, \sqrt{5})$$ Defining polynomial: $$x^{4} - x^{3} + 2 x^{2} + x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{5} + ( -3 - \beta_{1} ) q^{7} +O(q^{10})$$ $$q -\beta_{2} q^{5} + ( -3 - \beta_{1} ) q^{7} + ( \beta_{2} - \beta_{3} ) q^{11} + ( 2 + 4 \beta_{1} ) q^{13} + 4 \beta_{1} q^{19} -2 \beta_{2} q^{23} + ( 10 + 10 \beta_{1} ) q^{25} + ( 2 \beta_{2} - \beta_{3} ) q^{29} + ( 1 + \beta_{1} ) q^{31} + ( 2 \beta_{2} + \beta_{3} ) q^{35} + 4 \beta_{1} q^{37} + 2 \beta_{3} q^{41} + ( 4 + 8 \beta_{1} ) q^{43} + ( -2 \beta_{2} + 4 \beta_{3} ) q^{47} + ( 8 + 5 \beta_{1} ) q^{49} + ( \beta_{2} + \beta_{3} ) q^{53} -15 q^{55} + ( -\beta_{2} - \beta_{3} ) q^{59} + ( 12 + 6 \beta_{1} ) q^{61} + ( 2 \beta_{2} - 4 \beta_{3} ) q^{65} + ( 4 - 4 \beta_{1} ) q^{67} -2 \beta_{3} q^{71} + ( -4 + 4 \beta_{1} ) q^{73} + ( -3 \beta_{2} + 2 \beta_{3} ) q^{77} + ( -14 - 7 \beta_{1} ) q^{79} + ( 2 \beta_{2} - \beta_{3} ) q^{83} -2 \beta_{2} q^{89} + ( -2 - 10 \beta_{1} ) q^{91} + ( 4 \beta_{2} - 4 \beta_{3} ) q^{95} + ( -3 - 6 \beta_{1} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 10q^{7} + O(q^{10})$$ $$4q - 10q^{7} - 8q^{19} + 20q^{25} + 2q^{31} - 8q^{37} + 22q^{49} - 60q^{55} + 36q^{61} + 24q^{67} - 24q^{73} - 42q^{79} + 12q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 2 x^{2} + x + 1$$:

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
271.1
 0.809017 + 1.40126i −0.309017 − 0.535233i 0.809017 − 1.40126i −0.309017 + 0.535233i
0 0 0 −3.35410 1.93649i 0 −2.50000 0.866025i 0 0 0
271.2 0 0 0 3.35410 + 1.93649i 0 −2.50000 0.866025i 0 0 0
703.1 0 0 0 −3.35410 + 1.93649i 0 −2.50000 + 0.866025i 0 0 0
703.2 0 0 0 3.35410 1.93649i 0 −2.50000 + 0.866025i 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
28.f even 6 1 inner
84.j odd 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.o 4
3.b odd 2 1 inner 1008.2.cs.o 4
4.b odd 2 1 1008.2.cs.p yes 4
7.c even 3 1 7056.2.b.v 4
7.d odd 6 1 1008.2.cs.p yes 4
7.d odd 6 1 7056.2.b.o 4
12.b even 2 1 1008.2.cs.p yes 4
21.g even 6 1 1008.2.cs.p yes 4
21.g even 6 1 7056.2.b.o 4
21.h odd 6 1 7056.2.b.v 4
28.f even 6 1 inner 1008.2.cs.o 4
28.f even 6 1 7056.2.b.v 4
28.g odd 6 1 7056.2.b.o 4
84.j odd 6 1 inner 1008.2.cs.o 4
84.j odd 6 1 7056.2.b.v 4
84.n even 6 1 7056.2.b.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.cs.o 4 1.a even 1 1 trivial
1008.2.cs.o 4 3.b odd 2 1 inner
1008.2.cs.o 4 28.f even 6 1 inner
1008.2.cs.o 4 84.j odd 6 1 inner
1008.2.cs.p yes 4 4.b odd 2 1
1008.2.cs.p yes 4 7.d odd 6 1
1008.2.cs.p yes 4 12.b even 2 1
1008.2.cs.p yes 4 21.g even 6 1
7056.2.b.o 4 7.d odd 6 1
7056.2.b.o 4 21.g even 6 1
7056.2.b.o 4 28.g odd 6 1
7056.2.b.o 4 84.n even 6 1
7056.2.b.v 4 7.c even 3 1
7056.2.b.v 4 21.h odd 6 1
7056.2.b.v 4 28.f even 6 1
7056.2.b.v 4 84.j odd 6 1

## Hecke kernels

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

 $$T_{5}^{4} - 15 T_{5}^{2} + 225$$ $$T_{11}^{4} - 15 T_{11}^{2} + 225$$ $$T_{13}^{2} + 12$$ $$T_{17}$$ $$T_{19}^{2} + 4 T_{19} + 16$$