Properties

 Label 1008.4.a.w 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 126) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + 22q^{5} + 7q^{7} + O(q^{10})$$ $$q + 22q^{5} + 7q^{7} - 26q^{11} - 54q^{13} + 74q^{17} - 116q^{19} + 58q^{23} + 359q^{25} + 208q^{29} + 252q^{31} + 154q^{35} + 50q^{37} + 126q^{41} - 164q^{43} + 444q^{47} + 49q^{49} + 12q^{53} - 572q^{55} - 124q^{59} - 162q^{61} - 1188q^{65} + 860q^{67} + 238q^{71} - 146q^{73} - 182q^{77} + 984q^{79} - 656q^{83} + 1628q^{85} - 954q^{89} - 378q^{91} - 2552q^{95} + 526q^{97} + O(q^{100})$$

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 22.0000 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.w 1
3.b odd 2 1 1008.4.a.a 1
4.b odd 2 1 126.4.a.e 1
12.b even 2 1 126.4.a.f yes 1
28.d even 2 1 882.4.a.a 1
28.f even 6 2 882.4.g.x 2
28.g odd 6 2 882.4.g.n 2
84.h odd 2 1 882.4.a.s 1
84.j odd 6 2 882.4.g.a 2
84.n even 6 2 882.4.g.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.a.e 1 4.b odd 2 1
126.4.a.f yes 1 12.b even 2 1
882.4.a.a 1 28.d even 2 1
882.4.a.s 1 84.h odd 2 1
882.4.g.a 2 84.j odd 6 2
882.4.g.m 2 84.n even 6 2
882.4.g.n 2 28.g odd 6 2
882.4.g.x 2 28.f even 6 2
1008.4.a.a 1 3.b odd 2 1
1008.4.a.w 1 1.a even 1 1 trivial

Hecke kernels

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

 $$T_{5} - 22$$ $$T_{11} + 26$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-22 + T$$
$7$ $$-7 + T$$
$11$ $$26 + T$$
$13$ $$54 + T$$
$17$ $$-74 + T$$
$19$ $$116 + T$$
$23$ $$-58 + T$$
$29$ $$-208 + T$$
$31$ $$-252 + T$$
$37$ $$-50 + T$$
$41$ $$-126 + T$$
$43$ $$164 + T$$
$47$ $$-444 + T$$
$53$ $$-12 + T$$
$59$ $$124 + T$$
$61$ $$162 + T$$
$67$ $$-860 + T$$
$71$ $$-238 + T$$
$73$ $$146 + T$$
$79$ $$-984 + T$$
$83$ $$656 + T$$
$89$ $$954 + T$$
$97$ $$-526 + T$$