# Properties

 Label 1008.3.d.a Level $1008$ Weight $3$ Character orbit 1008.d Analytic conductor $27.466$ Analytic rank $0$ Dimension $4$ 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.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$27.4660106475$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 8 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: no (minimal twist has level 126) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 2 \beta_{1} - \beta_{3} ) q^{5} + \beta_{2} q^{7} +O(q^{10})$$ $$q + ( 2 \beta_{1} - \beta_{3} ) q^{5} + \beta_{2} q^{7} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{11} + ( -8 - 4 \beta_{2} ) q^{13} + ( -2 \beta_{1} + 5 \beta_{3} ) q^{17} -20 q^{19} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{23} + ( -33 + 8 \beta_{2} ) q^{25} + ( -4 \beta_{1} - 5 \beta_{3} ) q^{29} + ( -4 - 8 \beta_{2} ) q^{31} + ( -\beta_{1} + 5 \beta_{3} ) q^{35} + 38 q^{37} + ( 6 \beta_{1} + 7 \beta_{3} ) q^{41} + ( -20 - 24 \beta_{2} ) q^{43} -4 \beta_{3} q^{47} + 7 q^{49} + ( -24 \beta_{1} + 7 \beta_{3} ) q^{53} + ( 116 - 16 \beta_{2} ) q^{55} + ( -8 \beta_{1} - 4 \beta_{3} ) q^{59} + ( 58 - 16 \beta_{2} ) q^{61} + ( -12 \beta_{1} - 12 \beta_{3} ) q^{65} + ( 48 + 32 \beta_{2} ) q^{67} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{71} + ( -24 - 20 \beta_{2} ) q^{73} + ( 2 \beta_{1} - 10 \beta_{3} ) q^{77} + ( -76 + 16 \beta_{2} ) q^{79} + ( 8 \beta_{1} - 24 \beta_{3} ) q^{83} + ( 82 - 56 \beta_{2} ) q^{85} + ( -18 \beta_{1} + 23 \beta_{3} ) q^{89} + ( -28 - 8 \beta_{2} ) q^{91} + ( -40 \beta_{1} + 20 \beta_{3} ) q^{95} + ( -72 - 44 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 32q^{13} - 80q^{19} - 132q^{25} - 16q^{31} + 152q^{37} - 80q^{43} + 28q^{49} + 464q^{55} + 232q^{61} + 192q^{67} - 96q^{73} - 304q^{79} + 328q^{85} - 112q^{91} - 288q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 8 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2 \nu^{3} - 16 \nu$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 4$$ $$\beta_{3}$$ $$=$$ $$-\nu^{3} - 5 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{3} - 3 \beta_{1}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 4$$ $$\nu^{3}$$ $$=$$ $$($$$$-16 \beta_{3} + 15 \beta_{1}$$$$)/6$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 2.57794i 1.16372i − 1.16372i − 2.57794i
0 0 0 8.89753i 0 −2.64575 0 0 0
449.2 0 0 0 6.06910i 0 2.64575 0 0 0
449.3 0 0 0 6.06910i 0 2.64575 0 0 0
449.4 0 0 0 8.89753i 0 −2.64575 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.3.d.a 4
3.b odd 2 1 inner 1008.3.d.a 4
4.b odd 2 1 126.3.b.a 4
8.b even 2 1 4032.3.d.j 4
8.d odd 2 1 4032.3.d.i 4
12.b even 2 1 126.3.b.a 4
20.d odd 2 1 3150.3.e.e 4
20.e even 4 2 3150.3.c.b 8
24.f even 2 1 4032.3.d.i 4
24.h odd 2 1 4032.3.d.j 4
28.d even 2 1 882.3.b.f 4
28.f even 6 2 882.3.s.i 8
28.g odd 6 2 882.3.s.e 8
36.f odd 6 2 1134.3.q.c 8
36.h even 6 2 1134.3.q.c 8
60.h even 2 1 3150.3.e.e 4
60.l odd 4 2 3150.3.c.b 8
84.h odd 2 1 882.3.b.f 4
84.j odd 6 2 882.3.s.i 8
84.n even 6 2 882.3.s.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.b.a 4 4.b odd 2 1
126.3.b.a 4 12.b even 2 1
882.3.b.f 4 28.d even 2 1
882.3.b.f 4 84.h odd 2 1
882.3.s.e 8 28.g odd 6 2
882.3.s.e 8 84.n even 6 2
882.3.s.i 8 28.f even 6 2
882.3.s.i 8 84.j odd 6 2
1008.3.d.a 4 1.a even 1 1 trivial
1008.3.d.a 4 3.b odd 2 1 inner
1134.3.q.c 8 36.f odd 6 2
1134.3.q.c 8 36.h even 6 2
3150.3.c.b 8 20.e even 4 2
3150.3.c.b 8 60.l odd 4 2
3150.3.e.e 4 20.d odd 2 1
3150.3.e.e 4 60.h even 2 1
4032.3.d.i 4 8.d odd 2 1
4032.3.d.i 4 24.f even 2 1
4032.3.d.j 4 8.b even 2 1
4032.3.d.j 4 24.h odd 2 1

## Hecke kernels

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

 $$T_{5}^{4} + 116 T_{5}^{2} + 2916$$ $$T_{11}^{4} + 464 T_{11}^{2} + 46656$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$2916 + 116 T^{2} + T^{4}$$
$7$ $$( -7 + T^{2} )^{2}$$
$11$ $$46656 + 464 T^{2} + T^{4}$$
$13$ $$( -48 + 16 T + T^{2} )^{2}$$
$17$ $$79524 + 788 T^{2} + T^{4}$$
$19$ $$( 20 + T )^{4}$$
$23$ $$46656 + 464 T^{2} + T^{4}$$
$29$ $$248004 + 1892 T^{2} + T^{4}$$
$31$ $$( -432 + 8 T + T^{2} )^{2}$$
$37$ $$( -38 + T )^{4}$$
$41$ $$910116 + 3924 T^{2} + T^{4}$$
$43$ $$( -3632 + 40 T + T^{2} )^{2}$$
$47$ $$( 288 + T^{2} )^{2}$$
$53$ $$64738116 + 16164 T^{2} + T^{4}$$
$59$ $$9216 + 3392 T^{2} + T^{4}$$
$61$ $$( 1572 - 116 T + T^{2} )^{2}$$
$67$ $$( -4864 - 96 T + T^{2} )^{2}$$
$71$ $$46656 + 464 T^{2} + T^{4}$$
$73$ $$( -2224 + 48 T + T^{2} )^{2}$$
$79$ $$( 3984 + 152 T + T^{2} )^{2}$$
$83$ $$53231616 + 18176 T^{2} + T^{4}$$
$89$ $$443556 + 19476 T^{2} + T^{4}$$
$97$ $$( -8368 + 144 T + T^{2} )^{2}$$