# Properties

 Label 1008.2.r.e 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: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{3} ) q^{3} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{5} + ( -1 + \beta_{2} ) q^{7} + ( -1 + 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{3} ) q^{3} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{5} + ( -1 + \beta_{2} ) q^{7} + ( -1 + 2 \beta_{3} ) q^{9} + ( 2 - 2 \beta_{2} ) q^{11} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{13} + ( 4 - 2 \beta_{1} - \beta_{2} ) q^{15} + 2 q^{17} + ( -5 + 2 \beta_{1} - \beta_{3} ) q^{19} + ( 1 + \beta_{1} - \beta_{2} ) q^{21} -\beta_{2} q^{23} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{25} + ( 5 - \beta_{3} ) q^{27} + ( 2 + 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{29} + 6 \beta_{2} q^{31} + ( -2 - 2 \beta_{1} + 2 \beta_{2} ) q^{33} + ( 1 - 2 \beta_{1} + \beta_{3} ) q^{35} + ( 2 + 8 \beta_{1} - 4 \beta_{3} ) q^{37} + ( -8 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{39} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{41} + ( 2 + 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{43} + ( -8 + \beta_{1} + 5 \beta_{2} - 3 \beta_{3} ) q^{45} + ( 4 \beta_{1} - 8 \beta_{3} ) q^{47} -\beta_{2} q^{49} + ( -2 - 2 \beta_{3} ) q^{51} + ( -6 - 4 \beta_{1} + 2 \beta_{3} ) q^{53} + ( -2 + 4 \beta_{1} - 2 \beta_{3} ) q^{55} + ( 7 - 2 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} ) q^{57} -2 \beta_{2} q^{59} + ( -9 + \beta_{1} + 9 \beta_{2} - 2 \beta_{3} ) q^{61} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{63} + ( 12 - 2 \beta_{1} - 12 \beta_{2} + 4 \beta_{3} ) q^{65} + ( -2 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} ) q^{67} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{69} + ( -5 + 4 \beta_{1} - 2 \beta_{3} ) q^{71} + ( -2 - 4 \beta_{1} + 2 \beta_{3} ) q^{73} + ( -2 - 6 \beta_{2} + 4 \beta_{3} ) q^{75} + 2 \beta_{2} q^{77} + ( 3 - 2 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} ) q^{79} + ( -7 - 4 \beta_{3} ) q^{81} + ( 2 - 2 \beta_{2} ) q^{83} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{85} + ( -6 - 4 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{87} + ( -12 - 4 \beta_{1} + 2 \beta_{3} ) q^{89} + ( 4 \beta_{1} - 2 \beta_{3} ) q^{91} + ( 6 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} ) q^{93} + ( -6 \beta_{1} + 11 \beta_{2} - 6 \beta_{3} ) q^{95} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{97} + ( -2 + 4 \beta_{1} + 2 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{3} - 2q^{5} - 2q^{7} - 4q^{9} + O(q^{10})$$ $$4q - 4q^{3} - 2q^{5} - 2q^{7} - 4q^{9} + 4q^{11} + 14q^{15} + 8q^{17} - 20q^{19} + 2q^{21} - 2q^{23} - 4q^{25} + 20q^{27} + 4q^{29} + 12q^{31} - 4q^{33} + 4q^{35} + 8q^{37} - 24q^{39} + 4q^{43} - 22q^{45} - 2q^{49} - 8q^{51} - 24q^{53} - 8q^{55} + 20q^{57} - 4q^{59} - 18q^{61} + 2q^{63} + 24q^{65} - 16q^{67} + 2q^{69} - 20q^{71} - 8q^{73} - 20q^{75} + 4q^{77} + 6q^{79} - 28q^{81} + 4q^{83} - 4q^{85} - 28q^{87} - 48q^{89} - 12q^{93} + 22q^{95} + 4q^{97} - 4q^{99} + O(q^{100})$$

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

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

## Character values

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

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

## 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
 −1.22474 + 0.707107i 1.22474 − 0.707107i 1.22474 + 0.707107i −1.22474 − 0.707107i
0 −1.00000 1.41421i 0 −1.72474 + 2.98735i 0 −0.500000 0.866025i 0 −1.00000 + 2.82843i 0
337.2 0 −1.00000 + 1.41421i 0 0.724745 1.25529i 0 −0.500000 0.866025i 0 −1.00000 2.82843i 0
673.1 0 −1.00000 1.41421i 0 0.724745 + 1.25529i 0 −0.500000 + 0.866025i 0 −1.00000 + 2.82843i 0
673.2 0 −1.00000 + 1.41421i 0 −1.72474 2.98735i 0 −0.500000 + 0.866025i 0 −1.00000 2.82843i 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.e 4
3.b odd 2 1 3024.2.r.e 4
4.b odd 2 1 126.2.f.c 4
9.c even 3 1 inner 1008.2.r.e 4
9.c even 3 1 9072.2.a.bk 2
9.d odd 6 1 3024.2.r.e 4
9.d odd 6 1 9072.2.a.bd 2
12.b even 2 1 378.2.f.d 4
28.d even 2 1 882.2.f.j 4
28.f even 6 1 882.2.e.n 4
28.f even 6 1 882.2.h.l 4
28.g odd 6 1 882.2.e.m 4
28.g odd 6 1 882.2.h.k 4
36.f odd 6 1 126.2.f.c 4
36.f odd 6 1 1134.2.a.p 2
36.h even 6 1 378.2.f.d 4
36.h even 6 1 1134.2.a.i 2
84.h odd 2 1 2646.2.f.k 4
84.j odd 6 1 2646.2.e.k 4
84.j odd 6 1 2646.2.h.n 4
84.n even 6 1 2646.2.e.l 4
84.n even 6 1 2646.2.h.m 4
252.n even 6 1 882.2.e.n 4
252.o even 6 1 2646.2.e.l 4
252.r odd 6 1 2646.2.h.n 4
252.s odd 6 1 2646.2.f.k 4
252.s odd 6 1 7938.2.a.bm 2
252.u odd 6 1 882.2.h.k 4
252.bb even 6 1 2646.2.h.m 4
252.bi even 6 1 882.2.f.j 4
252.bi even 6 1 7938.2.a.bn 2
252.bj even 6 1 882.2.h.l 4
252.bl odd 6 1 882.2.e.m 4
252.bn odd 6 1 2646.2.e.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.c 4 4.b odd 2 1
126.2.f.c 4 36.f odd 6 1
378.2.f.d 4 12.b even 2 1
378.2.f.d 4 36.h even 6 1
882.2.e.m 4 28.g odd 6 1
882.2.e.m 4 252.bl odd 6 1
882.2.e.n 4 28.f even 6 1
882.2.e.n 4 252.n even 6 1
882.2.f.j 4 28.d even 2 1
882.2.f.j 4 252.bi even 6 1
882.2.h.k 4 28.g odd 6 1
882.2.h.k 4 252.u odd 6 1
882.2.h.l 4 28.f even 6 1
882.2.h.l 4 252.bj even 6 1
1008.2.r.e 4 1.a even 1 1 trivial
1008.2.r.e 4 9.c even 3 1 inner
1134.2.a.i 2 36.h even 6 1
1134.2.a.p 2 36.f odd 6 1
2646.2.e.k 4 84.j odd 6 1
2646.2.e.k 4 252.bn odd 6 1
2646.2.e.l 4 84.n even 6 1
2646.2.e.l 4 252.o even 6 1
2646.2.f.k 4 84.h odd 2 1
2646.2.f.k 4 252.s odd 6 1
2646.2.h.m 4 84.n even 6 1
2646.2.h.m 4 252.bb even 6 1
2646.2.h.n 4 84.j odd 6 1
2646.2.h.n 4 252.r odd 6 1
3024.2.r.e 4 3.b odd 2 1
3024.2.r.e 4 9.d odd 6 1
7938.2.a.bm 2 252.s odd 6 1
7938.2.a.bn 2 252.bi even 6 1
9072.2.a.bd 2 9.d odd 6 1
9072.2.a.bk 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 $$S_{2}^{\mathrm{new}}(1008, [\chi])$$:

 $$T_{5}^{4} + 2 T_{5}^{3} + 9 T_{5}^{2} - 10 T_{5} + 25$$ $$T_{11}^{2} - 2 T_{11} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + 2 T + 3 T^{2} )^{2}$$
$5$ $$1 + 2 T - T^{2} - 10 T^{3} - 20 T^{4} - 50 T^{5} - 25 T^{6} + 250 T^{7} + 625 T^{8}$$
$7$ $$( 1 + T + T^{2} )^{2}$$
$11$ $$( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} )^{2}$$
$13$ $$1 - 2 T^{2} - 165 T^{4} - 338 T^{6} + 28561 T^{8}$$
$17$ $$( 1 - 2 T + 17 T^{2} )^{4}$$
$19$ $$( 1 + 10 T + 57 T^{2} + 190 T^{3} + 361 T^{4} )^{2}$$
$23$ $$( 1 + T - 22 T^{2} + 23 T^{3} + 529 T^{4} )^{2}$$
$29$ $$1 - 4 T - 22 T^{2} + 80 T^{3} + 139 T^{4} + 2320 T^{5} - 18502 T^{6} - 97556 T^{7} + 707281 T^{8}$$
$31$ $$( 1 - 6 T + 5 T^{2} - 186 T^{3} + 961 T^{4} )^{2}$$
$37$ $$( 1 - 4 T - 18 T^{2} - 148 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$1 + 14 T^{2} - 1485 T^{4} + 23534 T^{6} + 2825761 T^{8}$$
$43$ $$1 - 4 T - 50 T^{2} + 80 T^{3} + 1819 T^{4} + 3440 T^{5} - 92450 T^{6} - 318028 T^{7} + 3418801 T^{8}$$
$47$ $$1 + 2 T^{2} - 2205 T^{4} + 4418 T^{6} + 4879681 T^{8}$$
$53$ $$( 1 + 12 T + 118 T^{2} + 636 T^{3} + 2809 T^{4} )^{2}$$
$59$ $$( 1 + 2 T - 55 T^{2} + 118 T^{3} + 3481 T^{4} )^{2}$$
$61$ $$1 + 18 T + 127 T^{2} + 1350 T^{3} + 15324 T^{4} + 82350 T^{5} + 472567 T^{6} + 4085658 T^{7} + 13845841 T^{8}$$
$67$ $$1 + 16 T + 82 T^{2} + 640 T^{3} + 8635 T^{4} + 42880 T^{5} + 368098 T^{6} + 4812208 T^{7} + 20151121 T^{8}$$
$71$ $$( 1 + 10 T + 143 T^{2} + 710 T^{3} + 5041 T^{4} )^{2}$$
$73$ $$( 1 + 4 T + 126 T^{2} + 292 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$1 - 6 T - 107 T^{2} + 90 T^{3} + 11364 T^{4} + 7110 T^{5} - 667787 T^{6} - 2958234 T^{7} + 38950081 T^{8}$$
$83$ $$( 1 - 2 T - 79 T^{2} - 166 T^{3} + 6889 T^{4} )^{2}$$
$89$ $$( 1 + 24 T + 298 T^{2} + 2136 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$1 - 4 T - 158 T^{2} + 80 T^{3} + 19315 T^{4} + 7760 T^{5} - 1486622 T^{6} - 3650692 T^{7} + 88529281 T^{8}$$