# Properties

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

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## 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: $$4$$ 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: $$2^{2}\cdot 3$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{5} + 7 \beta_1 q^{7}+O(q^{10})$$ q + b3 * q^5 + 7*b1 * q^7 $$q + \beta_{3} q^{5} + 7 \beta_1 q^{7} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{11} + (2 \beta_1 + 1) q^{13} + ( - \beta_{3} + \beta_{2}) q^{17} + ( - 17 \beta_1 - 34) q^{19} + ( - \beta_{3} + 2 \beta_{2}) q^{23} + ( - \beta_1 - 1) q^{25} + ( - 8 \beta_{3} + 4 \beta_{2}) q^{29} + ( - 7 \beta_1 + 7) q^{31} + ( - 7 \beta_{3} + 7 \beta_{2}) q^{35} - 47 \beta_1 q^{37} - 14 \beta_{2} q^{41} - 31 q^{43} + 17 \beta_{3} q^{47} + ( - 49 \beta_1 - 49) q^{49} + (9 \beta_{3} + 9 \beta_{2}) q^{53} + ( - 96 \beta_1 - 48) q^{55} + (17 \beta_{3} - 17 \beta_{2}) q^{59} + ( - 48 \beta_1 - 96) q^{61} + ( - \beta_{3} + 2 \beta_{2}) q^{65} + ( - 31 \beta_1 - 31) q^{67} + ( - 14 \beta_{3} + 7 \beta_{2}) q^{71} + (47 \beta_1 - 47) q^{73} + (28 \beta_{3} - 14 \beta_{2}) q^{77} - 41 \beta_1 q^{79} - \beta_{2} q^{83} - 24 q^{85} + 12 \beta_{3} q^{89} + ( - 7 \beta_1 - 14) q^{91} + ( - 17 \beta_{3} - 17 \beta_{2}) q^{95} + ( - 48 \beta_1 - 24) q^{97}+O(q^{100})$$ q + b3 * q^5 + 7*b1 * q^7 + (-2*b3 - 2*b2) * q^11 + (2*b1 + 1) * q^13 + (-b3 + b2) * q^17 + (-17*b1 - 34) * q^19 + (-b3 + 2*b2) * q^23 + (-b1 - 1) * q^25 + (-8*b3 + 4*b2) * q^29 + (-7*b1 + 7) * q^31 + (-7*b3 + 7*b2) * q^35 - 47*b1 * q^37 - 14*b2 * q^41 - 31 * q^43 + 17*b3 * q^47 + (-49*b1 - 49) * q^49 + (9*b3 + 9*b2) * q^53 + (-96*b1 - 48) * q^55 + (17*b3 - 17*b2) * q^59 + (-48*b1 - 96) * q^61 + (-b3 + 2*b2) * q^65 + (-31*b1 - 31) * q^67 + (-14*b3 + 7*b2) * q^71 + (47*b1 - 47) * q^73 + (28*b3 - 14*b2) * q^77 - 41*b1 * q^79 - b2 * q^83 - 24 * q^85 + 12*b3 * q^89 + (-7*b1 - 14) * q^91 + (-17*b3 - 17*b2) * q^95 + (-48*b1 - 24) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 14 q^{7}+O(q^{10})$$ 4 * q - 14 * q^7 $$4 q - 14 q^{7} - 102 q^{19} - 2 q^{25} + 42 q^{31} + 94 q^{37} - 124 q^{43} - 98 q^{49} - 288 q^{61} - 62 q^{67} - 282 q^{73} + 82 q^{79} - 96 q^{85} - 42 q^{91}+O(q^{100})$$ 4 * q - 14 * q^7 - 102 * q^19 - 2 * q^25 + 42 * q^31 + 94 * q^37 - 124 * q^43 - 98 * q^49 - 288 * q^61 - 62 * q^67 - 282 * q^73 + 82 * q^79 - 96 * q^85 - 42 * q^91

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

 $$\beta_{1}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{2}$$ $$=$$ $$\nu^{3} + 4\nu$$ v^3 + 4*v $$\beta_{3}$$ $$=$$ $$-\nu^{3} + 2\nu$$ -v^3 + 2*v
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 6$$ (b3 + b2) / 6 $$\nu^{2}$$ $$=$$ $$2\beta_1$$ 2*b1 $$\nu^{3}$$ $$=$$ $$( -2\beta_{3} + \beta_{2} ) / 3$$ (-2*b3 + b2) / 3

## Character values

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

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

## 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
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
0 0 0 −4.24264 2.44949i 0 −3.50000 + 6.06218i 0 0 0
145.2 0 0 0 4.24264 + 2.44949i 0 −3.50000 + 6.06218i 0 0 0
577.1 0 0 0 −4.24264 + 2.44949i 0 −3.50000 6.06218i 0 0 0
577.2 0 0 0 4.24264 2.44949i 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
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 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.i 4
3.b odd 2 1 inner 1008.3.cg.i 4
4.b odd 2 1 126.3.n.b 4
7.d odd 6 1 inner 1008.3.cg.i 4
12.b even 2 1 126.3.n.b 4
21.g even 6 1 inner 1008.3.cg.i 4
28.d even 2 1 882.3.n.c 4
28.f even 6 1 126.3.n.b 4
28.f even 6 1 882.3.c.c 4
28.g odd 6 1 882.3.c.c 4
28.g odd 6 1 882.3.n.c 4
84.h odd 2 1 882.3.n.c 4
84.j odd 6 1 126.3.n.b 4
84.j odd 6 1 882.3.c.c 4
84.n even 6 1 882.3.c.c 4
84.n even 6 1 882.3.n.c 4

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

## Hecke kernels

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

 $$T_{5}^{4} - 24T_{5}^{2} + 576$$ T5^4 - 24*T5^2 + 576 $$T_{11}^{4} + 288T_{11}^{2} + 82944$$ T11^4 + 288*T11^2 + 82944 $$T_{13}^{2} + 3$$ T13^2 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 24T^{2} + 576$$
$7$ $$(T^{2} + 7 T + 49)^{2}$$
$11$ $$T^{4} + 288 T^{2} + 82944$$
$13$ $$(T^{2} + 3)^{2}$$
$17$ $$T^{4} - 24T^{2} + 576$$
$19$ $$(T^{2} + 51 T + 867)^{2}$$
$23$ $$T^{4} + 72T^{2} + 5184$$
$29$ $$(T^{2} - 1152)^{2}$$
$31$ $$(T^{2} - 21 T + 147)^{2}$$
$37$ $$(T^{2} - 47 T + 2209)^{2}$$
$41$ $$(T^{2} + 4704)^{2}$$
$43$ $$(T + 31)^{4}$$
$47$ $$T^{4} - 6936 T^{2} + \cdots + 48108096$$
$53$ $$T^{4} + 5832 T^{2} + \cdots + 34012224$$
$59$ $$T^{4} - 6936 T^{2} + \cdots + 48108096$$
$61$ $$(T^{2} + 144 T + 6912)^{2}$$
$67$ $$(T^{2} + 31 T + 961)^{2}$$
$71$ $$(T^{2} - 3528)^{2}$$
$73$ $$(T^{2} + 141 T + 6627)^{2}$$
$79$ $$(T^{2} - 41 T + 1681)^{2}$$
$83$ $$(T^{2} + 24)^{2}$$
$89$ $$T^{4} - 3456 T^{2} + \cdots + 11943936$$
$97$ $$(T^{2} + 1728)^{2}$$
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