# Properties

 Label 1008.4.a.be Level $1008$ Weight $4$ Character orbit 1008.a Self dual yes Analytic conductor $59.474$ 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: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1008.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$59.4739252858$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{19})$$ Defining polynomial: $$x^{2} - 19$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 63) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{19}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{5} + 7 q^{7} +O(q^{10})$$ $$q + \beta q^{5} + 7 q^{7} + 5 \beta q^{11} + 82 q^{13} -9 \beta q^{17} + 20 q^{19} + 15 \beta q^{23} -49 q^{25} + 28 \beta q^{29} -156 q^{31} + 7 \beta q^{35} + 186 q^{37} -19 \beta q^{41} -164 q^{43} -54 \beta q^{47} + 49 q^{49} -18 \beta q^{53} + 380 q^{55} -18 \beta q^{59} + 790 q^{61} + 82 \beta q^{65} + 44 q^{67} -51 \beta q^{71} + 126 q^{73} + 35 \beta q^{77} + 712 q^{79} + 168 \beta q^{83} -684 q^{85} -167 \beta q^{89} + 574 q^{91} + 20 \beta q^{95} + 798 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 14q^{7} + O(q^{10})$$ $$2q + 14q^{7} + 164q^{13} + 40q^{19} - 98q^{25} - 312q^{31} + 372q^{37} - 328q^{43} + 98q^{49} + 760q^{55} + 1580q^{61} + 88q^{67} + 252q^{73} + 1424q^{79} - 1368q^{85} + 1148q^{91} + 1596q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −4.35890 4.35890
0 0 0 −8.71780 0 7.00000 0 0 0
1.2 0 0 0 8.71780 0 7.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$-1$$

## 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 1008.4.a.be 2
3.b odd 2 1 inner 1008.4.a.be 2
4.b odd 2 1 63.4.a.d 2
12.b even 2 1 63.4.a.d 2
20.d odd 2 1 1575.4.a.t 2
28.d even 2 1 441.4.a.q 2
28.f even 6 2 441.4.e.s 4
28.g odd 6 2 441.4.e.r 4
60.h even 2 1 1575.4.a.t 2
84.h odd 2 1 441.4.a.q 2
84.j odd 6 2 441.4.e.s 4
84.n even 6 2 441.4.e.r 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.a.d 2 4.b odd 2 1
63.4.a.d 2 12.b even 2 1
441.4.a.q 2 28.d even 2 1
441.4.a.q 2 84.h odd 2 1
441.4.e.r 4 28.g odd 6 2
441.4.e.r 4 84.n even 6 2
441.4.e.s 4 28.f even 6 2
441.4.e.s 4 84.j odd 6 2
1008.4.a.be 2 1.a even 1 1 trivial
1008.4.a.be 2 3.b odd 2 1 inner
1575.4.a.t 2 20.d odd 2 1
1575.4.a.t 2 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1008))$$:

 $$T_{5}^{2} - 76$$ $$T_{11}^{2} - 1900$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-76 + T^{2}$$
$7$ $$( -7 + T )^{2}$$
$11$ $$-1900 + T^{2}$$
$13$ $$( -82 + T )^{2}$$
$17$ $$-6156 + T^{2}$$
$19$ $$( -20 + T )^{2}$$
$23$ $$-17100 + T^{2}$$
$29$ $$-59584 + T^{2}$$
$31$ $$( 156 + T )^{2}$$
$37$ $$( -186 + T )^{2}$$
$41$ $$-27436 + T^{2}$$
$43$ $$( 164 + T )^{2}$$
$47$ $$-221616 + T^{2}$$
$53$ $$-24624 + T^{2}$$
$59$ $$-24624 + T^{2}$$
$61$ $$( -790 + T )^{2}$$
$67$ $$( -44 + T )^{2}$$
$71$ $$-197676 + T^{2}$$
$73$ $$( -126 + T )^{2}$$
$79$ $$( -712 + T )^{2}$$
$83$ $$-2145024 + T^{2}$$
$89$ $$-2119564 + T^{2}$$
$97$ $$( -798 + T )^{2}$$