# Properties

 Label 4018.2.a.s Level $4018$ Weight $2$ Character orbit 4018.a Self dual yes Analytic conductor $32.084$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4018 = 2 \cdot 7^{2} \cdot 41$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4018.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.0838915322$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 574) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + 3q^{3} + q^{4} + q^{5} + 3q^{6} + q^{8} + 6q^{9} + O(q^{10})$$ $$q + q^{2} + 3q^{3} + q^{4} + q^{5} + 3q^{6} + q^{8} + 6q^{9} + q^{10} - 2q^{11} + 3q^{12} + 3q^{15} + q^{16} + 3q^{17} + 6q^{18} + 8q^{19} + q^{20} - 2q^{22} - 4q^{23} + 3q^{24} - 4q^{25} + 9q^{27} - 5q^{29} + 3q^{30} + 3q^{31} + q^{32} - 6q^{33} + 3q^{34} + 6q^{36} + 10q^{37} + 8q^{38} + q^{40} + q^{41} - 5q^{43} - 2q^{44} + 6q^{45} - 4q^{46} - 6q^{47} + 3q^{48} - 4q^{50} + 9q^{51} - 9q^{53} + 9q^{54} - 2q^{55} + 24q^{57} - 5q^{58} + 10q^{59} + 3q^{60} - 13q^{61} + 3q^{62} + q^{64} - 6q^{66} - 2q^{67} + 3q^{68} - 12q^{69} + 9q^{71} + 6q^{72} - 4q^{73} + 10q^{74} - 12q^{75} + 8q^{76} - 11q^{79} + q^{80} + 9q^{81} + q^{82} + 14q^{83} + 3q^{85} - 5q^{86} - 15q^{87} - 2q^{88} + q^{89} + 6q^{90} - 4q^{92} + 9q^{93} - 6q^{94} + 8q^{95} + 3q^{96} - 7q^{97} - 12q^{99} + O(q^{100})$$

## 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
1.00000 3.00000 1.00000 1.00000 3.00000 0 1.00000 6.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$-1$$
$$41$$ $$-1$$

## 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 4018.2.a.s 1
7.b odd 2 1 574.2.a.g 1
21.c even 2 1 5166.2.a.o 1
28.d even 2 1 4592.2.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.a.g 1 7.b odd 2 1
4018.2.a.s 1 1.a even 1 1 trivial
4592.2.a.l 1 28.d even 2 1
5166.2.a.o 1 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4018))$$:

 $$T_{3} - 3$$ $$T_{5} - 1$$ $$T_{11} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$-3 + T$$
$5$ $$-1 + T$$
$7$ $$T$$
$11$ $$2 + T$$
$13$ $$T$$
$17$ $$-3 + T$$
$19$ $$-8 + T$$
$23$ $$4 + T$$
$29$ $$5 + T$$
$31$ $$-3 + T$$
$37$ $$-10 + T$$
$41$ $$-1 + T$$
$43$ $$5 + T$$
$47$ $$6 + T$$
$53$ $$9 + T$$
$59$ $$-10 + T$$
$61$ $$13 + T$$
$67$ $$2 + T$$
$71$ $$-9 + T$$
$73$ $$4 + T$$
$79$ $$11 + T$$
$83$ $$-14 + T$$
$89$ $$-1 + T$$
$97$ $$7 + T$$