# Properties

 Label 1008.4.a.r Level $1008$ Weight $4$ Character orbit 1008.a Self dual yes Analytic conductor $59.474$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 12q^{5} - 7q^{7} + O(q^{10})$$ $$q + 12q^{5} - 7q^{7} + 48q^{11} + 56q^{13} + 114q^{17} - 2q^{19} - 120q^{23} + 19q^{25} + 54q^{29} - 236q^{31} - 84q^{35} + 146q^{37} - 126q^{41} + 376q^{43} - 12q^{47} + 49q^{49} - 174q^{53} + 576q^{55} + 138q^{59} + 380q^{61} + 672q^{65} + 484q^{67} + 576q^{71} - 1150q^{73} - 336q^{77} - 776q^{79} + 378q^{83} + 1368q^{85} + 390q^{89} - 392q^{91} - 24q^{95} - 1330q^{97} + O(q^{100})$$

## 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
 0
0 0 0 12.0000 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.4.a.r 1
3.b odd 2 1 112.4.a.e 1
4.b odd 2 1 126.4.a.d 1
12.b even 2 1 14.4.a.b 1
21.c even 2 1 784.4.a.h 1
24.f even 2 1 448.4.a.k 1
24.h odd 2 1 448.4.a.g 1
28.d even 2 1 882.4.a.b 1
28.f even 6 2 882.4.g.v 2
28.g odd 6 2 882.4.g.p 2
60.h even 2 1 350.4.a.f 1
60.l odd 4 2 350.4.c.g 2
84.h odd 2 1 98.4.a.e 1
84.j odd 6 2 98.4.c.b 2
84.n even 6 2 98.4.c.c 2
132.d odd 2 1 1694.4.a.b 1
156.h even 2 1 2366.4.a.c 1
420.o odd 2 1 2450.4.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.b 1 12.b even 2 1
98.4.a.e 1 84.h odd 2 1
98.4.c.b 2 84.j odd 6 2
98.4.c.c 2 84.n even 6 2
112.4.a.e 1 3.b odd 2 1
126.4.a.d 1 4.b odd 2 1
350.4.a.f 1 60.h even 2 1
350.4.c.g 2 60.l odd 4 2
448.4.a.g 1 24.h odd 2 1
448.4.a.k 1 24.f even 2 1
784.4.a.h 1 21.c even 2 1
882.4.a.b 1 28.d even 2 1
882.4.g.p 2 28.g odd 6 2
882.4.g.v 2 28.f even 6 2
1008.4.a.r 1 1.a even 1 1 trivial
1694.4.a.b 1 132.d odd 2 1
2366.4.a.c 1 156.h even 2 1
2450.4.a.i 1 420.o odd 2 1

## Hecke kernels

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

 $$T_{5} - 12$$ $$T_{11} - 48$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-12 + T$$
$7$ $$7 + T$$
$11$ $$-48 + T$$
$13$ $$-56 + T$$
$17$ $$-114 + T$$
$19$ $$2 + T$$
$23$ $$120 + T$$
$29$ $$-54 + T$$
$31$ $$236 + T$$
$37$ $$-146 + T$$
$41$ $$126 + T$$
$43$ $$-376 + T$$
$47$ $$12 + T$$
$53$ $$174 + T$$
$59$ $$-138 + T$$
$61$ $$-380 + T$$
$67$ $$-484 + T$$
$71$ $$-576 + T$$
$73$ $$1150 + T$$
$79$ $$776 + T$$
$83$ $$-378 + T$$
$89$ $$-390 + T$$
$97$ $$1330 + T$$