# Properties

 Label 1008.3.cg.f 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: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 4) q^{5} + 7 \zeta_{6} q^{7}+O(q^{10})$$ q + (-2*z + 4) * q^5 + 7*z * q^7 $$q + ( - 2 \zeta_{6} + 4) q^{5} + 7 \zeta_{6} q^{7} + (10 \zeta_{6} - 10) q^{11} + (14 \zeta_{6} - 7) q^{13} + (4 \zeta_{6} + 4) q^{17} + (19 \zeta_{6} - 38) q^{19} - 40 \zeta_{6} q^{23} + (13 \zeta_{6} - 13) q^{25} - 16 q^{29} + ( - 3 \zeta_{6} - 3) q^{31} + (14 \zeta_{6} + 14) q^{35} - 5 \zeta_{6} q^{37} + ( - 28 \zeta_{6} + 14) q^{41} + 19 q^{43} + (30 \zeta_{6} - 60) q^{47} + (49 \zeta_{6} - 49) q^{49} + (32 \zeta_{6} - 32) q^{53} + (40 \zeta_{6} - 20) q^{55} + (24 \zeta_{6} + 24) q^{59} + ( - 12 \zeta_{6} + 24) q^{61} + 42 \zeta_{6} q^{65} + ( - 59 \zeta_{6} + 59) q^{67} - 26 q^{71} + ( - 11 \zeta_{6} - 11) q^{73} - 70 q^{77} + 47 \zeta_{6} q^{79} + ( - 28 \zeta_{6} + 14) q^{83} + 24 q^{85} + (68 \zeta_{6} - 136) q^{89} + (49 \zeta_{6} - 98) q^{91} + (114 \zeta_{6} - 114) q^{95} + ( - 56 \zeta_{6} + 28) q^{97} +O(q^{100})$$ q + (-2*z + 4) * q^5 + 7*z * q^7 + (10*z - 10) * q^11 + (14*z - 7) * q^13 + (4*z + 4) * q^17 + (19*z - 38) * q^19 - 40*z * q^23 + (13*z - 13) * q^25 - 16 * q^29 + (-3*z - 3) * q^31 + (14*z + 14) * q^35 - 5*z * q^37 + (-28*z + 14) * q^41 + 19 * q^43 + (30*z - 60) * q^47 + (49*z - 49) * q^49 + (32*z - 32) * q^53 + (40*z - 20) * q^55 + (24*z + 24) * q^59 + (-12*z + 24) * q^61 + 42*z * q^65 + (-59*z + 59) * q^67 - 26 * q^71 + (-11*z - 11) * q^73 - 70 * q^77 + 47*z * q^79 + (-28*z + 14) * q^83 + 24 * q^85 + (68*z - 136) * q^89 + (49*z - 98) * q^91 + (114*z - 114) * q^95 + (-56*z + 28) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{5} + 7 q^{7}+O(q^{10})$$ 2 * q + 6 * q^5 + 7 * q^7 $$2 q + 6 q^{5} + 7 q^{7} - 10 q^{11} + 12 q^{17} - 57 q^{19} - 40 q^{23} - 13 q^{25} - 32 q^{29} - 9 q^{31} + 42 q^{35} - 5 q^{37} + 38 q^{43} - 90 q^{47} - 49 q^{49} - 32 q^{53} + 72 q^{59} + 36 q^{61} + 42 q^{65} + 59 q^{67} - 52 q^{71} - 33 q^{73} - 140 q^{77} + 47 q^{79} + 48 q^{85} - 204 q^{89} - 147 q^{91} - 114 q^{95}+O(q^{100})$$ 2 * q + 6 * q^5 + 7 * q^7 - 10 * q^11 + 12 * q^17 - 57 * q^19 - 40 * q^23 - 13 * q^25 - 32 * q^29 - 9 * q^31 + 42 * q^35 - 5 * q^37 + 38 * q^43 - 90 * q^47 - 49 * q^49 - 32 * q^53 + 72 * q^59 + 36 * q^61 + 42 * q^65 + 59 * q^67 - 52 * q^71 - 33 * q^73 - 140 * q^77 + 47 * q^79 + 48 * q^85 - 204 * q^89 - 147 * q^91 - 114 * 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 3.00000 + 1.73205i 0 3.50000 6.06218i 0 0 0
577.1 0 0 0 3.00000 1.73205i 0 3.50000 + 6.06218i 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.f 2
3.b odd 2 1 336.3.bh.c 2
4.b odd 2 1 63.3.m.a 2
7.d odd 6 1 inner 1008.3.cg.f 2
12.b even 2 1 21.3.f.c 2
21.g even 6 1 336.3.bh.c 2
21.g even 6 1 2352.3.f.b 2
21.h odd 6 1 2352.3.f.b 2
28.d even 2 1 441.3.m.b 2
28.f even 6 1 63.3.m.a 2
28.f even 6 1 441.3.d.d 2
28.g odd 6 1 441.3.d.d 2
28.g odd 6 1 441.3.m.b 2
60.h even 2 1 525.3.o.b 2
60.l odd 4 2 525.3.s.d 4
84.h odd 2 1 147.3.f.e 2
84.j odd 6 1 21.3.f.c 2
84.j odd 6 1 147.3.d.a 2
84.n even 6 1 147.3.d.a 2
84.n even 6 1 147.3.f.e 2
420.be odd 6 1 525.3.o.b 2
420.br even 12 2 525.3.s.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.f.c 2 12.b even 2 1
21.3.f.c 2 84.j odd 6 1
63.3.m.a 2 4.b odd 2 1
63.3.m.a 2 28.f even 6 1
147.3.d.a 2 84.j odd 6 1
147.3.d.a 2 84.n even 6 1
147.3.f.e 2 84.h odd 2 1
147.3.f.e 2 84.n even 6 1
336.3.bh.c 2 3.b odd 2 1
336.3.bh.c 2 21.g even 6 1
441.3.d.d 2 28.f even 6 1
441.3.d.d 2 28.g odd 6 1
441.3.m.b 2 28.d even 2 1
441.3.m.b 2 28.g odd 6 1
525.3.o.b 2 60.h even 2 1
525.3.o.b 2 420.be odd 6 1
525.3.s.d 4 60.l odd 4 2
525.3.s.d 4 420.br even 12 2
1008.3.cg.f 2 1.a even 1 1 trivial
1008.3.cg.f 2 7.d odd 6 1 inner
2352.3.f.b 2 21.g even 6 1
2352.3.f.b 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 $$S_{3}^{\mathrm{new}}(1008, [\chi])$$:

 $$T_{5}^{2} - 6T_{5} + 12$$ T5^2 - 6*T5 + 12 $$T_{11}^{2} + 10T_{11} + 100$$ T11^2 + 10*T11 + 100 $$T_{13}^{2} + 147$$ T13^2 + 147

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 6T + 12$$
$7$ $$T^{2} - 7T + 49$$
$11$ $$T^{2} + 10T + 100$$
$13$ $$T^{2} + 147$$
$17$ $$T^{2} - 12T + 48$$
$19$ $$T^{2} + 57T + 1083$$
$23$ $$T^{2} + 40T + 1600$$
$29$ $$(T + 16)^{2}$$
$31$ $$T^{2} + 9T + 27$$
$37$ $$T^{2} + 5T + 25$$
$41$ $$T^{2} + 588$$
$43$ $$(T - 19)^{2}$$
$47$ $$T^{2} + 90T + 2700$$
$53$ $$T^{2} + 32T + 1024$$
$59$ $$T^{2} - 72T + 1728$$
$61$ $$T^{2} - 36T + 432$$
$67$ $$T^{2} - 59T + 3481$$
$71$ $$(T + 26)^{2}$$
$73$ $$T^{2} + 33T + 363$$
$79$ $$T^{2} - 47T + 2209$$
$83$ $$T^{2} + 588$$
$89$ $$T^{2} + 204T + 13872$$
$97$ $$T^{2} + 2352$$