# Properties

 Label 1008.2.cz.f Level $1008$ Weight $2$ Character orbit 1008.cz Analytic conductor $8.049$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1008 = 2^{4} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1008.cz (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: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 - 2 \beta_{1} ) q^{3} + ( 2 + \beta_{1} + \beta_{3} ) q^{5} + ( 1 + \beta_{2} ) q^{7} -3 q^{9} +O(q^{10})$$ $$q + ( -1 - 2 \beta_{1} ) q^{3} + ( 2 + \beta_{1} + \beta_{3} ) q^{5} + ( 1 + \beta_{2} ) q^{7} -3 q^{9} + ( -1 + \beta_{1} ) q^{11} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{13} + ( -3 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{15} + ( 2 + \beta_{1} + 2 \beta_{3} ) q^{17} + ( 1 + \beta_{1} ) q^{19} + ( -1 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{21} + ( -6 - 3 \beta_{1} - \beta_{3} ) q^{23} + ( 4 + 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{25} + ( 3 + 6 \beta_{1} ) q^{27} + ( 3 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{29} + 4 q^{31} + ( 3 + 3 \beta_{1} ) q^{33} + ( 2 + 7 \beta_{1} + 2 \beta_{2} ) q^{35} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{37} + ( 3 + 3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{39} + ( -3 + 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{41} + ( 2 + \beta_{1} - 2 \beta_{3} ) q^{43} + ( -6 - 3 \beta_{1} - 3 \beta_{3} ) q^{45} + ( -2 \beta_{2} + 4 \beta_{3} ) q^{47} + ( -5 + 2 \beta_{2} ) q^{49} + ( -3 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{51} + ( -3 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{53} + ( -3 + \beta_{2} - 2 \beta_{3} ) q^{55} + ( 1 - \beta_{1} ) q^{57} + 6 q^{59} + ( -4 - 8 \beta_{1} ) q^{61} + ( -3 - 3 \beta_{2} ) q^{63} + 3 q^{65} + ( 9 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{69} + ( -2 - 4 \beta_{1} + 4 \beta_{2} ) q^{71} + ( -10 - 5 \beta_{1} ) q^{73} + ( 4 - 4 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} ) q^{75} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{77} + ( -4 - 8 \beta_{1} - 2 \beta_{2} ) q^{79} + 9 q^{81} + ( 3 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{83} + ( 15 + 15 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{85} + ( 6 + 3 \beta_{1} + 3 \beta_{3} ) q^{87} + ( -1 + \beta_{1} - 6 \beta_{2} + 6 \beta_{3} ) q^{89} + ( 5 + 7 \beta_{1} - 2 \beta_{2} ) q^{91} + ( -4 - 8 \beta_{1} ) q^{93} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{95} + ( -2 - \beta_{1} + 2 \beta_{3} ) q^{97} + ( 3 - 3 \beta_{1} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 6q^{5} + 4q^{7} - 12q^{9} + O(q^{10})$$ $$4q + 6q^{5} + 4q^{7} - 12q^{9} - 6q^{11} - 6q^{13} + 6q^{15} + 6q^{17} + 2q^{19} - 18q^{23} + 8q^{25} - 6q^{29} + 16q^{31} + 6q^{33} - 6q^{35} - 2q^{37} + 6q^{39} - 18q^{41} + 6q^{43} - 18q^{45} - 20q^{49} + 6q^{51} + 6q^{53} - 12q^{55} + 6q^{57} + 24q^{59} - 12q^{63} + 12q^{65} - 18q^{69} - 30q^{73} + 24q^{75} - 6q^{77} + 36q^{81} - 6q^{83} + 30q^{85} + 18q^{87} - 6q^{89} + 6q^{91} - 6q^{97} + 18q^{99} + O(q^{100})$$

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

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
367.1
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
0 1.73205i 0 −0.621320 0.358719i 0 1.00000 2.44949i 0 −3.00000 0
367.2 0 1.73205i 0 3.62132 + 2.09077i 0 1.00000 + 2.44949i 0 −3.00000 0
607.1 0 1.73205i 0 −0.621320 + 0.358719i 0 1.00000 + 2.44949i 0 −3.00000 0
607.2 0 1.73205i 0 3.62132 2.09077i 0 1.00000 2.44949i 0 −3.00000 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
252.bj 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.cz.f yes 4
3.b odd 2 1 3024.2.cz.f 4
4.b odd 2 1 1008.2.cz.e yes 4
7.d odd 6 1 1008.2.bf.e 4
9.c even 3 1 1008.2.bf.f yes 4
9.d odd 6 1 3024.2.bf.e 4
12.b even 2 1 3024.2.cz.e 4
21.g even 6 1 3024.2.bf.f 4
28.f even 6 1 1008.2.bf.f yes 4
36.f odd 6 1 1008.2.bf.e 4
36.h even 6 1 3024.2.bf.f 4
63.i even 6 1 3024.2.cz.e 4
63.t odd 6 1 1008.2.cz.e yes 4
84.j odd 6 1 3024.2.bf.e 4
252.r odd 6 1 3024.2.cz.f 4
252.bj even 6 1 inner 1008.2.cz.f yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.bf.e 4 7.d odd 6 1
1008.2.bf.e 4 36.f odd 6 1
1008.2.bf.f yes 4 9.c even 3 1
1008.2.bf.f yes 4 28.f even 6 1
1008.2.cz.e yes 4 4.b odd 2 1
1008.2.cz.e yes 4 63.t odd 6 1
1008.2.cz.f yes 4 1.a even 1 1 trivial
1008.2.cz.f yes 4 252.bj even 6 1 inner
3024.2.bf.e 4 9.d odd 6 1
3024.2.bf.e 4 84.j odd 6 1
3024.2.bf.f 4 21.g even 6 1
3024.2.bf.f 4 36.h even 6 1
3024.2.cz.e 4 12.b even 2 1
3024.2.cz.e 4 63.i even 6 1
3024.2.cz.f 4 3.b odd 2 1
3024.2.cz.f 4 252.r odd 6 1

## Hecke kernels

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

 $$T_{5}^{4} - 6 T_{5}^{3} + 9 T_{5}^{2} + 18 T_{5} + 9$$ $$T_{11}^{2} + 3 T_{11} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 3 + T^{2} )^{2}$$
$5$ $$9 + 18 T + 9 T^{2} - 6 T^{3} + T^{4}$$
$7$ $$( 7 - 2 T + T^{2} )^{2}$$
$11$ $$( 3 + 3 T + T^{2} )^{2}$$
$13$ $$9 - 18 T + 9 T^{2} + 6 T^{3} + T^{4}$$
$17$ $$441 + 126 T - 9 T^{2} - 6 T^{3} + T^{4}$$
$19$ $$( 1 - T + T^{2} )^{2}$$
$23$ $$441 + 378 T + 129 T^{2} + 18 T^{3} + T^{4}$$
$29$ $$81 - 54 T + 45 T^{2} + 6 T^{3} + T^{4}$$
$31$ $$( -4 + T )^{4}$$
$37$ $$289 - 34 T + 21 T^{2} + 2 T^{3} + T^{4}$$
$41$ $$9 + 54 T + 111 T^{2} + 18 T^{3} + T^{4}$$
$43$ $$441 + 126 T - 9 T^{2} - 6 T^{3} + T^{4}$$
$47$ $$( -72 + T^{2} )^{2}$$
$53$ $$81 + 54 T + 45 T^{2} - 6 T^{3} + T^{4}$$
$59$ $$( -6 + T )^{4}$$
$61$ $$( 48 + T^{2} )^{2}$$
$67$ $$T^{4}$$
$71$ $$7056 + 216 T^{2} + T^{4}$$
$73$ $$( 75 + 15 T + T^{2} )^{2}$$
$79$ $$576 + 144 T^{2} + T^{4}$$
$83$ $$3969 - 378 T + 99 T^{2} + 6 T^{3} + T^{4}$$
$89$ $$45369 - 1278 T - 201 T^{2} + 6 T^{3} + T^{4}$$
$97$ $$441 - 126 T - 9 T^{2} + 6 T^{3} + T^{4}$$