# Properties

 Label 1008.3.dc.a Level $1008$ Weight $3$ Character orbit 1008.dc Analytic conductor $27.466$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1008 = 2^{4} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1008.dc (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$27.4660106475$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2x^{2} + 4$$ 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: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{5} + ( - 3 \beta_{2} - 5) q^{7}+O(q^{10})$$ q + b1 * q^5 + (-3*b2 - 5) * q^7 $$q + \beta_1 q^{5} + ( - 3 \beta_{2} - 5) q^{7} + ( - 5 \beta_{3} + 5 \beta_1) q^{11} + 15 q^{13} + (8 \beta_{3} - 8 \beta_1) q^{17} + (13 \beta_{2} - 13) q^{19} + 16 \beta_1 q^{23} - 23 \beta_{2} q^{25} - 16 \beta_{3} q^{29} + 3 \beta_{2} q^{31} + ( - 3 \beta_{3} - 5 \beta_1) q^{35} + (17 \beta_{2} - 17) q^{37} - 57 \beta_{3} q^{41} + 85 q^{43} - 51 \beta_1 q^{47} + (39 \beta_{2} + 16) q^{49} + ( - 24 \beta_{3} + 24 \beta_1) q^{53} + 10 q^{55} + ( - 64 \beta_{3} + 64 \beta_1) q^{59} + ( - 72 \beta_{2} + 72) q^{61} + 15 \beta_1 q^{65} + 43 \beta_{2} q^{67} - 37 \beta_{3} q^{71} + 95 \beta_{2} q^{73} + (25 \beta_{3} - 40 \beta_1) q^{77} + ( - 69 \beta_{2} + 69) q^{79} - 43 \beta_{3} q^{83} - 16 q^{85} + 96 \beta_1 q^{89} + ( - 45 \beta_{2} - 75) q^{91} + (13 \beta_{3} - 13 \beta_1) q^{95} + 16 q^{97}+O(q^{100})$$ q + b1 * q^5 + (-3*b2 - 5) * q^7 + (-5*b3 + 5*b1) * q^11 + 15 * q^13 + (8*b3 - 8*b1) * q^17 + (13*b2 - 13) * q^19 + 16*b1 * q^23 - 23*b2 * q^25 - 16*b3 * q^29 + 3*b2 * q^31 + (-3*b3 - 5*b1) * q^35 + (17*b2 - 17) * q^37 - 57*b3 * q^41 + 85 * q^43 - 51*b1 * q^47 + (39*b2 + 16) * q^49 + (-24*b3 + 24*b1) * q^53 + 10 * q^55 + (-64*b3 + 64*b1) * q^59 + (-72*b2 + 72) * q^61 + 15*b1 * q^65 + 43*b2 * q^67 - 37*b3 * q^71 + 95*b2 * q^73 + (25*b3 - 40*b1) * q^77 + (-69*b2 + 69) * q^79 - 43*b3 * q^83 - 16 * q^85 + 96*b1 * q^89 + (-45*b2 - 75) * q^91 + (13*b3 - 13*b1) * q^95 + 16 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 26 q^{7}+O(q^{10})$$ 4 * q - 26 * q^7 $$4 q - 26 q^{7} + 60 q^{13} - 26 q^{19} - 46 q^{25} + 6 q^{31} - 34 q^{37} + 340 q^{43} + 142 q^{49} + 40 q^{55} + 144 q^{61} + 86 q^{67} + 190 q^{73} + 138 q^{79} - 64 q^{85} - 390 q^{91} + 64 q^{97}+O(q^{100})$$ 4 * q - 26 * q^7 + 60 * q^13 - 26 * q^19 - 46 * q^25 + 6 * q^31 - 34 * q^37 + 340 * q^43 + 142 * q^49 + 40 * q^55 + 144 * q^61 + 86 * q^67 + 190 * q^73 + 138 * q^79 - 64 * q^85 - 390 * q^91 + 64 * q^97

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
305.1
 −1.22474 + 0.707107i 1.22474 − 0.707107i −1.22474 − 0.707107i 1.22474 + 0.707107i
0 0 0 −1.22474 + 0.707107i 0 −6.50000 + 2.59808i 0 0 0
305.2 0 0 0 1.22474 0.707107i 0 −6.50000 + 2.59808i 0 0 0
737.1 0 0 0 −1.22474 0.707107i 0 −6.50000 2.59808i 0 0 0
737.2 0 0 0 1.22474 + 0.707107i 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
3.b odd 2 1 inner
7.c even 3 1 inner
21.h 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.dc.a 4
3.b odd 2 1 inner 1008.3.dc.a 4
4.b odd 2 1 126.3.s.b 4
7.c even 3 1 inner 1008.3.dc.a 4
12.b even 2 1 126.3.s.b 4
21.h odd 6 1 inner 1008.3.dc.a 4
28.d even 2 1 882.3.s.c 4
28.f even 6 1 882.3.b.d 2
28.f even 6 1 882.3.s.c 4
28.g odd 6 1 126.3.s.b 4
28.g odd 6 1 882.3.b.c 2
84.h odd 2 1 882.3.s.c 4
84.j odd 6 1 882.3.b.d 2
84.j odd 6 1 882.3.s.c 4
84.n even 6 1 126.3.s.b 4
84.n even 6 1 882.3.b.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.s.b 4 4.b odd 2 1
126.3.s.b 4 12.b even 2 1
126.3.s.b 4 28.g odd 6 1
126.3.s.b 4 84.n even 6 1
882.3.b.c 2 28.g odd 6 1
882.3.b.c 2 84.n even 6 1
882.3.b.d 2 28.f even 6 1
882.3.b.d 2 84.j odd 6 1
882.3.s.c 4 28.d even 2 1
882.3.s.c 4 28.f even 6 1
882.3.s.c 4 84.h odd 2 1
882.3.s.c 4 84.j odd 6 1
1008.3.dc.a 4 1.a even 1 1 trivial
1008.3.dc.a 4 3.b odd 2 1 inner
1008.3.dc.a 4 7.c even 3 1 inner
1008.3.dc.a 4 21.h odd 6 1 inner

## Hecke kernels

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

 $$T_{5}^{4} - 2T_{5}^{2} + 4$$ T5^4 - 2*T5^2 + 4 $$T_{11}^{4} - 50T_{11}^{2} + 2500$$ T11^4 - 50*T11^2 + 2500

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 2T^{2} + 4$$
$7$ $$(T^{2} + 13 T + 49)^{2}$$
$11$ $$T^{4} - 50T^{2} + 2500$$
$13$ $$(T - 15)^{4}$$
$17$ $$T^{4} - 128 T^{2} + 16384$$
$19$ $$(T^{2} + 13 T + 169)^{2}$$
$23$ $$T^{4} - 512 T^{2} + 262144$$
$29$ $$(T^{2} + 512)^{2}$$
$31$ $$(T^{2} - 3 T + 9)^{2}$$
$37$ $$(T^{2} + 17 T + 289)^{2}$$
$41$ $$(T^{2} + 6498)^{2}$$
$43$ $$(T - 85)^{4}$$
$47$ $$T^{4} - 5202 T^{2} + \cdots + 27060804$$
$53$ $$T^{4} - 1152 T^{2} + \cdots + 1327104$$
$59$ $$T^{4} - 8192 T^{2} + \cdots + 67108864$$
$61$ $$(T^{2} - 72 T + 5184)^{2}$$
$67$ $$(T^{2} - 43 T + 1849)^{2}$$
$71$ $$(T^{2} + 2738)^{2}$$
$73$ $$(T^{2} - 95 T + 9025)^{2}$$
$79$ $$(T^{2} - 69 T + 4761)^{2}$$
$83$ $$(T^{2} + 3698)^{2}$$
$89$ $$T^{4} - 18432 T^{2} + \cdots + 339738624$$
$97$ $$(T - 16)^{4}$$