# Properties

 Label 1008.2.r.f Level $1008$ Weight $2$ Character orbit 1008.r 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.r (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.04892052375$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 126) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{2} + \beta_{3} ) q^{3} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{5} + ( 1 - \beta_{2} ) q^{7} + ( 2 + \beta_{1} - 2 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{2} + \beta_{3} ) q^{3} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{5} + ( 1 - \beta_{2} ) q^{7} + ( 2 + \beta_{1} - 2 \beta_{2} ) q^{9} + ( 1 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{11} -2 \beta_{2} q^{13} + ( -1 - 2 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{15} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{17} -5 q^{19} + ( 1 - \beta_{1} + \beta_{3} ) q^{21} + ( -1 + 2 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{23} + ( -7 + 3 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} ) q^{25} + ( 5 - 2 \beta_{1} + 2 \beta_{3} ) q^{27} + ( 4 - 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{29} + 2 \beta_{2} q^{31} + ( -1 - 2 \beta_{1} + 5 \beta_{2} + \beta_{3} ) q^{33} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{35} + 2 q^{37} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{39} + ( -1 + 2 \beta_{1} - 7 \beta_{2} - \beta_{3} ) q^{41} + ( -1 + 3 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} ) q^{43} + ( 1 - 4 \beta_{1} - 8 \beta_{2} - \beta_{3} ) q^{45} -\beta_{2} q^{49} + ( 5 + \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{51} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{53} + 6 q^{55} + ( -5 \beta_{2} - 5 \beta_{3} ) q^{57} + ( 3 - 6 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{59} + ( 4 + 3 \beta_{1} - 7 \beta_{2} - 6 \beta_{3} ) q^{61} + ( -2 \beta_{2} + \beta_{3} ) q^{63} + ( -4 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{65} + ( 3 - 6 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} ) q^{67} + ( 7 - 4 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{69} + ( -3 + 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{71} + ( 2 + 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{73} + ( -13 + 4 \beta_{1} + 15 \beta_{2} - 7 \beta_{3} ) q^{75} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{77} + ( 2 + 3 \beta_{1} - 5 \beta_{2} - 6 \beta_{3} ) q^{79} + ( -\beta_{2} + 5 \beta_{3} ) q^{81} + ( 8 - 4 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} ) q^{83} -6 \beta_{2} q^{85} + ( 8 - 2 \beta_{1} - 10 \beta_{2} + 4 \beta_{3} ) q^{87} + ( 8 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{89} -2 q^{91} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{93} + ( -5 + 10 \beta_{1} + 10 \beta_{2} - 5 \beta_{3} ) q^{95} + ( 1 - 3 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} ) q^{97} + ( -8 + 5 \beta_{1} + \beta_{2} - \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{3} - 3 q^{5} + 2 q^{7} + 5 q^{9} + O(q^{10})$$ $$4 q + q^{3} - 3 q^{5} + 2 q^{7} + 5 q^{9} + 3 q^{11} - 4 q^{13} - 15 q^{15} - 6 q^{17} - 20 q^{19} + 2 q^{21} - 9 q^{23} - 11 q^{25} + 16 q^{27} + 6 q^{29} + 4 q^{31} + 3 q^{33} - 6 q^{35} + 8 q^{37} + 2 q^{39} - 15 q^{41} + q^{43} - 15 q^{45} - 2 q^{49} + 15 q^{51} - 12 q^{53} + 24 q^{55} - 5 q^{57} - 3 q^{59} + 11 q^{61} - 5 q^{63} - 6 q^{65} + 13 q^{67} + 21 q^{69} - 6 q^{71} + 14 q^{73} - 11 q^{75} - 3 q^{77} + 7 q^{79} - 7 q^{81} + 12 q^{83} - 12 q^{85} + 6 q^{87} + 36 q^{89} - 8 q^{91} - 2 q^{93} + 15 q^{95} - q^{97} - 24 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$:

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 −1.18614 + 1.26217i 1.68614 − 0.396143i −1.18614 − 1.26217i 1.68614 + 0.396143i
0 −1.18614 1.26217i 0 0.686141 1.18843i 0 0.500000 + 0.866025i 0 −0.186141 + 2.99422i 0
337.2 0 1.68614 + 0.396143i 0 −2.18614 + 3.78651i 0 0.500000 + 0.866025i 0 2.68614 + 1.33591i 0
673.1 0 −1.18614 + 1.26217i 0 0.686141 + 1.18843i 0 0.500000 0.866025i 0 −0.186141 2.99422i 0
673.2 0 1.68614 0.396143i 0 −2.18614 3.78651i 0 0.500000 0.866025i 0 2.68614 1.33591i 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.r.f 4
3.b odd 2 1 3024.2.r.f 4
4.b odd 2 1 126.2.f.d 4
9.c even 3 1 inner 1008.2.r.f 4
9.c even 3 1 9072.2.a.bm 2
9.d odd 6 1 3024.2.r.f 4
9.d odd 6 1 9072.2.a.bb 2
12.b even 2 1 378.2.f.c 4
28.d even 2 1 882.2.f.k 4
28.f even 6 1 882.2.e.k 4
28.f even 6 1 882.2.h.n 4
28.g odd 6 1 882.2.e.l 4
28.g odd 6 1 882.2.h.m 4
36.f odd 6 1 126.2.f.d 4
36.f odd 6 1 1134.2.a.k 2
36.h even 6 1 378.2.f.c 4
36.h even 6 1 1134.2.a.n 2
84.h odd 2 1 2646.2.f.j 4
84.j odd 6 1 2646.2.e.m 4
84.j odd 6 1 2646.2.h.l 4
84.n even 6 1 2646.2.e.n 4
84.n even 6 1 2646.2.h.k 4
252.n even 6 1 882.2.e.k 4
252.o even 6 1 2646.2.e.n 4
252.r odd 6 1 2646.2.h.l 4
252.s odd 6 1 2646.2.f.j 4
252.s odd 6 1 7938.2.a.bs 2
252.u odd 6 1 882.2.h.m 4
252.bb even 6 1 2646.2.h.k 4
252.bi even 6 1 882.2.f.k 4
252.bi even 6 1 7938.2.a.bh 2
252.bj even 6 1 882.2.h.n 4
252.bl odd 6 1 882.2.e.l 4
252.bn odd 6 1 2646.2.e.m 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.d 4 4.b odd 2 1
126.2.f.d 4 36.f odd 6 1
378.2.f.c 4 12.b even 2 1
378.2.f.c 4 36.h even 6 1
882.2.e.k 4 28.f even 6 1
882.2.e.k 4 252.n even 6 1
882.2.e.l 4 28.g odd 6 1
882.2.e.l 4 252.bl odd 6 1
882.2.f.k 4 28.d even 2 1
882.2.f.k 4 252.bi even 6 1
882.2.h.m 4 28.g odd 6 1
882.2.h.m 4 252.u odd 6 1
882.2.h.n 4 28.f even 6 1
882.2.h.n 4 252.bj even 6 1
1008.2.r.f 4 1.a even 1 1 trivial
1008.2.r.f 4 9.c even 3 1 inner
1134.2.a.k 2 36.f odd 6 1
1134.2.a.n 2 36.h even 6 1
2646.2.e.m 4 84.j odd 6 1
2646.2.e.m 4 252.bn odd 6 1
2646.2.e.n 4 84.n even 6 1
2646.2.e.n 4 252.o even 6 1
2646.2.f.j 4 84.h odd 2 1
2646.2.f.j 4 252.s odd 6 1
2646.2.h.k 4 84.n even 6 1
2646.2.h.k 4 252.bb even 6 1
2646.2.h.l 4 84.j odd 6 1
2646.2.h.l 4 252.r odd 6 1
3024.2.r.f 4 3.b odd 2 1
3024.2.r.f 4 9.d odd 6 1
7938.2.a.bh 2 252.bi even 6 1
7938.2.a.bs 2 252.s odd 6 1
9072.2.a.bb 2 9.d odd 6 1
9072.2.a.bm 2 9.c even 3 1

## Hecke kernels

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

 $$T_{5}^{4} + 3 T_{5}^{3} + 15 T_{5}^{2} - 18 T_{5} + 36$$ $$T_{11}^{4} - 3 T_{11}^{3} + 15 T_{11}^{2} + 18 T_{11} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 - 3 T - 2 T^{2} - T^{3} + T^{4}$$
$5$ $$36 - 18 T + 15 T^{2} + 3 T^{3} + T^{4}$$
$7$ $$( 1 - T + T^{2} )^{2}$$
$11$ $$36 + 18 T + 15 T^{2} - 3 T^{3} + T^{4}$$
$13$ $$( 4 + 2 T + T^{2} )^{2}$$
$17$ $$( -6 + 3 T + T^{2} )^{2}$$
$19$ $$( 5 + T )^{4}$$
$23$ $$144 + 108 T + 69 T^{2} + 9 T^{3} + T^{4}$$
$29$ $$576 + 144 T + 60 T^{2} - 6 T^{3} + T^{4}$$
$31$ $$( 4 - 2 T + T^{2} )^{2}$$
$37$ $$( -2 + T )^{4}$$
$41$ $$2304 + 720 T + 177 T^{2} + 15 T^{3} + T^{4}$$
$43$ $$5476 + 74 T + 75 T^{2} - T^{3} + T^{4}$$
$47$ $$T^{4}$$
$53$ $$( -24 + 6 T + T^{2} )^{2}$$
$59$ $$5184 - 216 T + 81 T^{2} + 3 T^{3} + T^{4}$$
$61$ $$1936 + 484 T + 165 T^{2} - 11 T^{3} + T^{4}$$
$67$ $$1024 + 416 T + 201 T^{2} - 13 T^{3} + T^{4}$$
$71$ $$( -72 + 3 T + T^{2} )^{2}$$
$73$ $$( -62 - 7 T + T^{2} )^{2}$$
$79$ $$3844 + 434 T + 111 T^{2} - 7 T^{3} + T^{4}$$
$83$ $$9216 + 1152 T + 240 T^{2} - 12 T^{3} + T^{4}$$
$89$ $$( 48 - 18 T + T^{2} )^{2}$$
$97$ $$5476 - 74 T + 75 T^{2} + T^{3} + T^{4}$$