# Properties

 Label 1008.2.t.e Level $1008$ Weight $2$ Character orbit 1008.t Analytic conductor $8.049$ Analytic rank $0$ Dimension $2$ 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.t (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.04892052375$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 504) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$127$$ $$577$$ $$757$$ $$785$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$ $$-1 + \zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.50000 0.866025i 0 1.00000 0 2.50000 0.866025i 0 1.50000 2.59808i 0
961.1 0 1.50000 + 0.866025i 0 1.00000 0 2.50000 + 0.866025i 0 1.50000 + 2.59808i 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
63.g 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.t.e 2
3.b odd 2 1 3024.2.t.c 2
4.b odd 2 1 504.2.t.a yes 2
7.c even 3 1 1008.2.q.b 2
9.c even 3 1 1008.2.q.b 2
9.d odd 6 1 3024.2.q.d 2
12.b even 2 1 1512.2.t.a 2
21.h odd 6 1 3024.2.q.d 2
28.g odd 6 1 504.2.q.a 2
36.f odd 6 1 504.2.q.a 2
36.h even 6 1 1512.2.q.b 2
63.g even 3 1 inner 1008.2.t.e 2
63.n odd 6 1 3024.2.t.c 2
84.n even 6 1 1512.2.q.b 2
252.o even 6 1 1512.2.t.a 2
252.bl odd 6 1 504.2.t.a yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.q.a 2 28.g odd 6 1
504.2.q.a 2 36.f odd 6 1
504.2.t.a yes 2 4.b odd 2 1
504.2.t.a yes 2 252.bl odd 6 1
1008.2.q.b 2 7.c even 3 1
1008.2.q.b 2 9.c even 3 1
1008.2.t.e 2 1.a even 1 1 trivial
1008.2.t.e 2 63.g even 3 1 inner
1512.2.q.b 2 36.h even 6 1
1512.2.q.b 2 84.n even 6 1
1512.2.t.a 2 12.b even 2 1
1512.2.t.a 2 252.o even 6 1
3024.2.q.d 2 9.d odd 6 1
3024.2.q.d 2 21.h odd 6 1
3024.2.t.c 2 3.b odd 2 1
3024.2.t.c 2 63.n odd 6 1

## Hecke kernels

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

 $$T_{5} - 1$$ $$T_{11} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 3 T + 3 T^{2}$$
$5$ $$( 1 - T + 5 T^{2} )^{2}$$
$7$ $$1 - 5 T + 7 T^{2}$$
$11$ $$( 1 - 3 T + 11 T^{2} )^{2}$$
$13$ $$1 + T - 12 T^{2} + 13 T^{3} + 169 T^{4}$$
$17$ $$1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4}$$
$19$ $$1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4}$$
$23$ $$( 1 + T + 23 T^{2} )^{2}$$
$29$ $$1 + 9 T + 52 T^{2} + 261 T^{3} + 841 T^{4}$$
$31$ $$( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} )$$
$37$ $$1 + 5 T - 12 T^{2} + 185 T^{3} + 1369 T^{4}$$
$41$ $$1 + 7 T + 8 T^{2} + 287 T^{3} + 1681 T^{4}$$
$43$ $$1 - 3 T - 34 T^{2} - 129 T^{3} + 1849 T^{4}$$
$47$ $$1 - 8 T + 17 T^{2} - 376 T^{3} + 2209 T^{4}$$
$53$ $$1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4}$$
$59$ $$1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4}$$
$61$ $$1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4}$$
$67$ $$1 - 12 T + 77 T^{2} - 804 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 8 T + 71 T^{2} )^{2}$$
$73$ $$1 - 13 T + 96 T^{2} - 949 T^{3} + 5329 T^{4}$$
$79$ $$1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4}$$
$83$ $$1 + 13 T + 86 T^{2} + 1079 T^{3} + 6889 T^{4}$$
$89$ $$1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4}$$
$97$ $$1 - 17 T + 192 T^{2} - 1649 T^{3} + 9409 T^{4}$$