Properties

 Label 1008.2.s.r Level $1008$ Weight $2$ Character orbit 1008.s 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.s (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: $$\Q(\sqrt{-3}, \sqrt{-19})$$ Defining polynomial: $$x^{4} - x^{3} - 4 x^{2} - 5 x + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 168) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{5} + ( 1 + \beta_{1} - \beta_{3} ) q^{7} +O(q^{10})$$ $$q + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{5} + ( 1 + \beta_{1} - \beta_{3} ) q^{7} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{11} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{13} + ( -4 + 4 \beta_{2} ) q^{17} + ( 1 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{19} -4 \beta_{2} q^{23} + ( -9 - \beta_{1} + 10 \beta_{2} + 2 \beta_{3} ) q^{25} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{29} + ( -1 + \beta_{2} ) q^{31} + ( -11 + 3 \beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{35} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{37} + ( -4 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{41} + ( 4 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{43} + 6 \beta_{2} q^{47} + ( -4 + 3 \beta_{1} - 3 \beta_{3} ) q^{49} + ( 6 - \beta_{1} - 5 \beta_{2} + 2 \beta_{3} ) q^{53} + ( -15 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{55} + ( -2 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{59} -10 \beta_{2} q^{61} + ( -2 + 4 \beta_{1} + 14 \beta_{2} - 2 \beta_{3} ) q^{65} + ( 3 + \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{67} + 2 q^{71} + ( 1 - \beta_{1} + 2 \beta_{3} ) q^{73} + ( -5 + 2 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} ) q^{77} + ( -2 + 4 \beta_{1} + 5 \beta_{2} - 2 \beta_{3} ) q^{79} + ( 3 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{83} + ( 4 - 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{85} + ( 2 - 4 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{89} + ( -3 + 4 \beta_{1} + 11 \beta_{2} - \beta_{3} ) q^{91} + ( 14 - 2 \beta_{1} - 12 \beta_{2} + 4 \beta_{3} ) q^{95} + ( 13 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - q^{5} + 6q^{7} + O(q^{10})$$ $$4q - q^{5} + 6q^{7} + q^{11} + 10q^{13} - 8q^{17} - 5q^{19} - 8q^{23} - 19q^{25} - 6q^{29} - 2q^{31} - 30q^{35} - 3q^{37} - 12q^{41} + 14q^{43} + 12q^{47} - 10q^{49} + 11q^{53} - 58q^{55} - 5q^{59} - 20q^{61} + 26q^{65} + 7q^{67} + 8q^{71} + q^{73} - 27q^{77} + 8q^{79} + 14q^{83} + 8q^{85} - 6q^{89} + 15q^{91} + 26q^{95} + 50q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 4 x^{2} - 5 x + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu^{2} - 4 \nu - 5$$$$)/20$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 4 \nu + 5$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 5 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-4 \beta_{3} + 4 \beta_{1} + 5$$

Character values

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

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

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 −1.63746 + 1.52274i 2.13746 − 0.656712i −1.63746 − 1.52274i 2.13746 + 0.656712i
0 0 0 −2.13746 + 3.70219i 0 1.50000 + 2.17945i 0 0 0
289.2 0 0 0 1.63746 2.83616i 0 1.50000 2.17945i 0 0 0
865.1 0 0 0 −2.13746 3.70219i 0 1.50000 2.17945i 0 0 0
865.2 0 0 0 1.63746 + 2.83616i 0 1.50000 + 2.17945i 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
7.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.s.r 4
3.b odd 2 1 336.2.q.g 4
4.b odd 2 1 504.2.s.i 4
7.c even 3 1 inner 1008.2.s.r 4
7.c even 3 1 7056.2.a.cu 2
7.d odd 6 1 7056.2.a.ch 2
12.b even 2 1 168.2.q.c 4
21.c even 2 1 2352.2.q.bf 4
21.g even 6 1 2352.2.a.ba 2
21.g even 6 1 2352.2.q.bf 4
21.h odd 6 1 336.2.q.g 4
21.h odd 6 1 2352.2.a.bf 2
24.f even 2 1 1344.2.q.w 4
24.h odd 2 1 1344.2.q.x 4
28.d even 2 1 3528.2.s.bk 4
28.f even 6 1 3528.2.a.bd 2
28.f even 6 1 3528.2.s.bk 4
28.g odd 6 1 504.2.s.i 4
28.g odd 6 1 3528.2.a.bk 2
84.h odd 2 1 1176.2.q.l 4
84.j odd 6 1 1176.2.a.n 2
84.j odd 6 1 1176.2.q.l 4
84.n even 6 1 168.2.q.c 4
84.n even 6 1 1176.2.a.k 2
168.s odd 6 1 1344.2.q.x 4
168.s odd 6 1 9408.2.a.dp 2
168.v even 6 1 1344.2.q.w 4
168.v even 6 1 9408.2.a.ec 2
168.ba even 6 1 9408.2.a.dw 2
168.be odd 6 1 9408.2.a.dj 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.q.c 4 12.b even 2 1
168.2.q.c 4 84.n even 6 1
336.2.q.g 4 3.b odd 2 1
336.2.q.g 4 21.h odd 6 1
504.2.s.i 4 4.b odd 2 1
504.2.s.i 4 28.g odd 6 1
1008.2.s.r 4 1.a even 1 1 trivial
1008.2.s.r 4 7.c even 3 1 inner
1176.2.a.k 2 84.n even 6 1
1176.2.a.n 2 84.j odd 6 1
1176.2.q.l 4 84.h odd 2 1
1176.2.q.l 4 84.j odd 6 1
1344.2.q.w 4 24.f even 2 1
1344.2.q.w 4 168.v even 6 1
1344.2.q.x 4 24.h odd 2 1
1344.2.q.x 4 168.s odd 6 1
2352.2.a.ba 2 21.g even 6 1
2352.2.a.bf 2 21.h odd 6 1
2352.2.q.bf 4 21.c even 2 1
2352.2.q.bf 4 21.g even 6 1
3528.2.a.bd 2 28.f even 6 1
3528.2.a.bk 2 28.g odd 6 1
3528.2.s.bk 4 28.d even 2 1
3528.2.s.bk 4 28.f even 6 1
7056.2.a.ch 2 7.d odd 6 1
7056.2.a.cu 2 7.c even 3 1
9408.2.a.dj 2 168.be odd 6 1
9408.2.a.dp 2 168.s odd 6 1
9408.2.a.dw 2 168.ba even 6 1
9408.2.a.ec 2 168.v even 6 1

Hecke kernels

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

 $$T_{5}^{4} + T_{5}^{3} + 15 T_{5}^{2} - 14 T_{5} + 196$$ $$T_{11}^{4} - T_{11}^{3} + 15 T_{11}^{2} + 14 T_{11} + 196$$ $$T_{13}^{2} - 5 T_{13} - 8$$ $$T_{17}^{2} + 4 T_{17} + 16$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$196 - 14 T + 15 T^{2} + T^{3} + T^{4}$$
$7$ $$( 7 - 3 T + T^{2} )^{2}$$
$11$ $$196 + 14 T + 15 T^{2} - T^{3} + T^{4}$$
$13$ $$( -8 - 5 T + T^{2} )^{2}$$
$17$ $$( 16 + 4 T + T^{2} )^{2}$$
$19$ $$64 - 40 T + 33 T^{2} + 5 T^{3} + T^{4}$$
$23$ $$( 16 + 4 T + T^{2} )^{2}$$
$29$ $$( -12 + 3 T + T^{2} )^{2}$$
$31$ $$( 1 + T + T^{2} )^{2}$$
$37$ $$144 - 36 T + 21 T^{2} + 3 T^{3} + T^{4}$$
$41$ $$( -48 + 6 T + T^{2} )^{2}$$
$43$ $$( -2 - 7 T + T^{2} )^{2}$$
$47$ $$( 36 - 6 T + T^{2} )^{2}$$
$53$ $$256 - 176 T + 105 T^{2} - 11 T^{3} + T^{4}$$
$59$ $$64 - 40 T + 33 T^{2} + 5 T^{3} + T^{4}$$
$61$ $$( 100 + 10 T + T^{2} )^{2}$$
$67$ $$4 + 14 T + 51 T^{2} - 7 T^{3} + T^{4}$$
$71$ $$( -2 + T )^{4}$$
$73$ $$196 + 14 T + 15 T^{2} - T^{3} + T^{4}$$
$79$ $$1681 + 328 T + 105 T^{2} - 8 T^{3} + T^{4}$$
$83$ $$( -2 - 7 T + T^{2} )^{2}$$
$89$ $$2304 - 288 T + 84 T^{2} + 6 T^{3} + T^{4}$$
$97$ $$( 142 - 25 T + T^{2} )^{2}$$