# Properties

 Label 1008.3.cg.b 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 2) q^{5} + ( - 3 \zeta_{6} - 5) q^{7}+O(q^{10})$$ q + (z - 2) * q^5 + (-3*z - 5) * q^7 $$q + (\zeta_{6} - 2) q^{5} + ( - 3 \zeta_{6} - 5) q^{7} + ( - 3 \zeta_{6} + 3) q^{11} + (8 \zeta_{6} - 4) q^{13} + (10 \zeta_{6} + 10) q^{17} + (6 \zeta_{6} - 12) q^{19} - 36 \zeta_{6} q^{23} + (22 \zeta_{6} - 22) q^{25} + 51 q^{29} + (7 \zeta_{6} + 7) q^{31} + ( - 2 \zeta_{6} + 13) q^{35} - 22 \zeta_{6} q^{37} + ( - 28 \zeta_{6} + 14) q^{41} - 10 q^{43} + ( - 52 \zeta_{6} + 104) q^{47} + (39 \zeta_{6} + 16) q^{49} + ( - 51 \zeta_{6} + 51) q^{53} + (6 \zeta_{6} - 3) q^{55} + (43 \zeta_{6} + 43) q^{59} + ( - 40 \zeta_{6} + 80) q^{61} - 12 \zeta_{6} q^{65} + ( - 68 \zeta_{6} + 68) q^{67} + ( - 12 \zeta_{6} - 12) q^{73} + (15 \zeta_{6} - 24) q^{77} + 125 \zeta_{6} q^{79} + (178 \zeta_{6} - 89) q^{83} - 30 q^{85} + (42 \zeta_{6} - 84) q^{89} + ( - 52 \zeta_{6} + 44) q^{91} + ( - 18 \zeta_{6} + 18) q^{95} + ( - 170 \zeta_{6} + 85) q^{97} +O(q^{100})$$ q + (z - 2) * q^5 + (-3*z - 5) * q^7 + (-3*z + 3) * q^11 + (8*z - 4) * q^13 + (10*z + 10) * q^17 + (6*z - 12) * q^19 - 36*z * q^23 + (22*z - 22) * q^25 + 51 * q^29 + (7*z + 7) * q^31 + (-2*z + 13) * q^35 - 22*z * q^37 + (-28*z + 14) * q^41 - 10 * q^43 + (-52*z + 104) * q^47 + (39*z + 16) * q^49 + (-51*z + 51) * q^53 + (6*z - 3) * q^55 + (43*z + 43) * q^59 + (-40*z + 80) * q^61 - 12*z * q^65 + (-68*z + 68) * q^67 + (-12*z - 12) * q^73 + (15*z - 24) * q^77 + 125*z * q^79 + (178*z - 89) * q^83 - 30 * q^85 + (42*z - 84) * q^89 + (-52*z + 44) * q^91 + (-18*z + 18) * q^95 + (-170*z + 85) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{5} - 13 q^{7}+O(q^{10})$$ 2 * q - 3 * q^5 - 13 * q^7 $$2 q - 3 q^{5} - 13 q^{7} + 3 q^{11} + 30 q^{17} - 18 q^{19} - 36 q^{23} - 22 q^{25} + 102 q^{29} + 21 q^{31} + 24 q^{35} - 22 q^{37} - 20 q^{43} + 156 q^{47} + 71 q^{49} + 51 q^{53} + 129 q^{59} + 120 q^{61} - 12 q^{65} + 68 q^{67} - 36 q^{73} - 33 q^{77} + 125 q^{79} - 60 q^{85} - 126 q^{89} + 36 q^{91} + 18 q^{95}+O(q^{100})$$ 2 * q - 3 * q^5 - 13 * q^7 + 3 * q^11 + 30 * q^17 - 18 * q^19 - 36 * q^23 - 22 * q^25 + 102 * q^29 + 21 * q^31 + 24 * q^35 - 22 * q^37 - 20 * q^43 + 156 * q^47 + 71 * q^49 + 51 * q^53 + 129 * q^59 + 120 * q^61 - 12 * q^65 + 68 * q^67 - 36 * q^73 - 33 * q^77 + 125 * q^79 - 60 * q^85 - 126 * q^89 + 36 * q^91 + 18 * 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 −1.50000 0.866025i 0 −6.50000 + 2.59808i 0 0 0
577.1 0 0 0 −1.50000 + 0.866025i 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.b 2
3.b odd 2 1 336.3.bh.b 2
4.b odd 2 1 252.3.z.b 2
7.d odd 6 1 inner 1008.3.cg.b 2
12.b even 2 1 84.3.m.a 2
21.g even 6 1 336.3.bh.b 2
21.g even 6 1 2352.3.f.c 2
21.h odd 6 1 2352.3.f.c 2
28.d even 2 1 1764.3.z.e 2
28.f even 6 1 252.3.z.b 2
28.f even 6 1 1764.3.d.c 2
28.g odd 6 1 1764.3.d.c 2
28.g odd 6 1 1764.3.z.e 2
60.h even 2 1 2100.3.bd.b 2
60.l odd 4 2 2100.3.be.c 4
84.h odd 2 1 588.3.m.a 2
84.j odd 6 1 84.3.m.a 2
84.j odd 6 1 588.3.d.a 2
84.n even 6 1 588.3.d.a 2
84.n even 6 1 588.3.m.a 2
420.be odd 6 1 2100.3.bd.b 2
420.br even 12 2 2100.3.be.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.3.m.a 2 12.b even 2 1
84.3.m.a 2 84.j odd 6 1
252.3.z.b 2 4.b odd 2 1
252.3.z.b 2 28.f even 6 1
336.3.bh.b 2 3.b odd 2 1
336.3.bh.b 2 21.g even 6 1
588.3.d.a 2 84.j odd 6 1
588.3.d.a 2 84.n even 6 1
588.3.m.a 2 84.h odd 2 1
588.3.m.a 2 84.n even 6 1
1008.3.cg.b 2 1.a even 1 1 trivial
1008.3.cg.b 2 7.d odd 6 1 inner
1764.3.d.c 2 28.f even 6 1
1764.3.d.c 2 28.g odd 6 1
1764.3.z.e 2 28.d even 2 1
1764.3.z.e 2 28.g odd 6 1
2100.3.bd.b 2 60.h even 2 1
2100.3.bd.b 2 420.be odd 6 1
2100.3.be.c 4 60.l odd 4 2
2100.3.be.c 4 420.br even 12 2
2352.3.f.c 2 21.g even 6 1
2352.3.f.c 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 $$S_{3}^{\mathrm{new}}(1008, [\chi])$$:

 $$T_{5}^{2} + 3T_{5} + 3$$ T5^2 + 3*T5 + 3 $$T_{11}^{2} - 3T_{11} + 9$$ T11^2 - 3*T11 + 9 $$T_{13}^{2} + 48$$ T13^2 + 48

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 3T + 3$$
$7$ $$T^{2} + 13T + 49$$
$11$ $$T^{2} - 3T + 9$$
$13$ $$T^{2} + 48$$
$17$ $$T^{2} - 30T + 300$$
$19$ $$T^{2} + 18T + 108$$
$23$ $$T^{2} + 36T + 1296$$
$29$ $$(T - 51)^{2}$$
$31$ $$T^{2} - 21T + 147$$
$37$ $$T^{2} + 22T + 484$$
$41$ $$T^{2} + 588$$
$43$ $$(T + 10)^{2}$$
$47$ $$T^{2} - 156T + 8112$$
$53$ $$T^{2} - 51T + 2601$$
$59$ $$T^{2} - 129T + 5547$$
$61$ $$T^{2} - 120T + 4800$$
$67$ $$T^{2} - 68T + 4624$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 36T + 432$$
$79$ $$T^{2} - 125T + 15625$$
$83$ $$T^{2} + 23763$$
$89$ $$T^{2} + 126T + 5292$$
$97$ $$T^{2} + 21675$$