# Properties

 Label 1008.3.cg.a Level $1008$ Weight $3$ Character orbit 1008.cg Analytic conductor $27.466$ 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: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1008.cg (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$27.4660106475$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (3 \zeta_{6} - 6) q^{5} + ( - 3 \zeta_{6} - 5) q^{7}+O(q^{10})$$ q + (3*z - 6) * q^5 + (-3*z - 5) * q^7 $$q + (3 \zeta_{6} - 6) q^{5} + ( - 3 \zeta_{6} - 5) q^{7} + (15 \zeta_{6} - 15) q^{11} + ( - 16 \zeta_{6} + 8) q^{13} + ( - 6 \zeta_{6} - 6) q^{17} + ( - 6 \zeta_{6} + 12) q^{19} + ( - 2 \zeta_{6} + 2) q^{25} + 9 q^{29} + (7 \zeta_{6} + 7) q^{31} + ( - 6 \zeta_{6} + 39) q^{35} - 10 \zeta_{6} q^{37} + (12 \zeta_{6} - 6) q^{41} + 74 q^{43} + (39 \zeta_{6} + 16) q^{49} + ( - 33 \zeta_{6} + 33) q^{53} + ( - 90 \zeta_{6} + 45) q^{55} + (9 \zeta_{6} + 9) q^{59} + ( - 52 \zeta_{6} + 104) q^{61} + 72 \zeta_{6} q^{65} + (76 \zeta_{6} - 76) q^{67} + 84 q^{71} + ( - 36 \zeta_{6} - 36) q^{73} + ( - 75 \zeta_{6} + 120) q^{77} - 43 \zeta_{6} q^{79} + ( - 138 \zeta_{6} + 69) q^{83} + 54 q^{85} + ( - 42 \zeta_{6} + 84) q^{89} + (104 \zeta_{6} - 88) q^{91} + (54 \zeta_{6} - 54) q^{95} + (214 \zeta_{6} - 107) q^{97} +O(q^{100})$$ q + (3*z - 6) * q^5 + (-3*z - 5) * q^7 + (15*z - 15) * q^11 + (-16*z + 8) * q^13 + (-6*z - 6) * q^17 + (-6*z + 12) * q^19 + (-2*z + 2) * q^25 + 9 * q^29 + (7*z + 7) * q^31 + (-6*z + 39) * q^35 - 10*z * q^37 + (12*z - 6) * q^41 + 74 * q^43 + (39*z + 16) * q^49 + (-33*z + 33) * q^53 + (-90*z + 45) * q^55 + (9*z + 9) * q^59 + (-52*z + 104) * q^61 + 72*z * q^65 + (76*z - 76) * q^67 + 84 * q^71 + (-36*z - 36) * q^73 + (-75*z + 120) * q^77 - 43*z * q^79 + (-138*z + 69) * q^83 + 54 * q^85 + (-42*z + 84) * q^89 + (104*z - 88) * q^91 + (54*z - 54) * q^95 + (214*z - 107) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 9 q^{5} - 13 q^{7}+O(q^{10})$$ 2 * q - 9 * q^5 - 13 * q^7 $$2 q - 9 q^{5} - 13 q^{7} - 15 q^{11} - 18 q^{17} + 18 q^{19} + 2 q^{25} + 18 q^{29} + 21 q^{31} + 72 q^{35} - 10 q^{37} + 148 q^{43} + 71 q^{49} + 33 q^{53} + 27 q^{59} + 156 q^{61} + 72 q^{65} - 76 q^{67} + 168 q^{71} - 108 q^{73} + 165 q^{77} - 43 q^{79} + 108 q^{85} + 126 q^{89} - 72 q^{91} - 54 q^{95}+O(q^{100})$$ 2 * q - 9 * q^5 - 13 * q^7 - 15 * q^11 - 18 * q^17 + 18 * q^19 + 2 * q^25 + 18 * q^29 + 21 * q^31 + 72 * q^35 - 10 * q^37 + 148 * q^43 + 71 * q^49 + 33 * q^53 + 27 * q^59 + 156 * q^61 + 72 * q^65 - 76 * q^67 + 168 * q^71 - 108 * q^73 + 165 * q^77 - 43 * q^79 + 108 * q^85 + 126 * q^89 - 72 * q^91 - 54 * q^95

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
145.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 −4.50000 2.59808i 0 −6.50000 + 2.59808i 0 0 0
577.1 0 0 0 −4.50000 + 2.59808i 0 −6.50000 2.59808i 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.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.3.cg.a 2
3.b odd 2 1 336.3.bh.d 2
4.b odd 2 1 63.3.m.d 2
7.d odd 6 1 inner 1008.3.cg.a 2
12.b even 2 1 21.3.f.a 2
21.g even 6 1 336.3.bh.d 2
21.g even 6 1 2352.3.f.a 2
21.h odd 6 1 2352.3.f.a 2
28.d even 2 1 441.3.m.g 2
28.f even 6 1 63.3.m.d 2
28.f even 6 1 441.3.d.a 2
28.g odd 6 1 441.3.d.a 2
28.g odd 6 1 441.3.m.g 2
60.h even 2 1 525.3.o.h 2
60.l odd 4 2 525.3.s.e 4
84.h odd 2 1 147.3.f.a 2
84.j odd 6 1 21.3.f.a 2
84.j odd 6 1 147.3.d.c 2
84.n even 6 1 147.3.d.c 2
84.n even 6 1 147.3.f.a 2
420.be odd 6 1 525.3.o.h 2
420.br even 12 2 525.3.s.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.f.a 2 12.b even 2 1
21.3.f.a 2 84.j odd 6 1
63.3.m.d 2 4.b odd 2 1
63.3.m.d 2 28.f even 6 1
147.3.d.c 2 84.j odd 6 1
147.3.d.c 2 84.n even 6 1
147.3.f.a 2 84.h odd 2 1
147.3.f.a 2 84.n even 6 1
336.3.bh.d 2 3.b odd 2 1
336.3.bh.d 2 21.g even 6 1
441.3.d.a 2 28.f even 6 1
441.3.d.a 2 28.g odd 6 1
441.3.m.g 2 28.d even 2 1
441.3.m.g 2 28.g odd 6 1
525.3.o.h 2 60.h even 2 1
525.3.o.h 2 420.be odd 6 1
525.3.s.e 4 60.l odd 4 2
525.3.s.e 4 420.br even 12 2
1008.3.cg.a 2 1.a even 1 1 trivial
1008.3.cg.a 2 7.d odd 6 1 inner
2352.3.f.a 2 21.g even 6 1
2352.3.f.a 2 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1008, [\chi])$$:

 $$T_{5}^{2} + 9T_{5} + 27$$ T5^2 + 9*T5 + 27 $$T_{11}^{2} + 15T_{11} + 225$$ T11^2 + 15*T11 + 225 $$T_{13}^{2} + 192$$ T13^2 + 192

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 9T + 27$$
$7$ $$T^{2} + 13T + 49$$
$11$ $$T^{2} + 15T + 225$$
$13$ $$T^{2} + 192$$
$17$ $$T^{2} + 18T + 108$$
$19$ $$T^{2} - 18T + 108$$
$23$ $$T^{2}$$
$29$ $$(T - 9)^{2}$$
$31$ $$T^{2} - 21T + 147$$
$37$ $$T^{2} + 10T + 100$$
$41$ $$T^{2} + 108$$
$43$ $$(T - 74)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 33T + 1089$$
$59$ $$T^{2} - 27T + 243$$
$61$ $$T^{2} - 156T + 8112$$
$67$ $$T^{2} + 76T + 5776$$
$71$ $$(T - 84)^{2}$$
$73$ $$T^{2} + 108T + 3888$$
$79$ $$T^{2} + 43T + 1849$$
$83$ $$T^{2} + 14283$$
$89$ $$T^{2} - 126T + 5292$$
$97$ $$T^{2} + 34347$$