# Properties

 Label 4018.2.a.x Level 4018 Weight 2 Character orbit 4018.a Self dual yes Analytic conductor 32.084 Analytic rank 1 Dimension 2 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 574) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} -2 q^{3} + q^{4} + ( -1 - \beta ) q^{5} -2 q^{6} + q^{8} + q^{9} +O(q^{10})$$ $$q + q^{2} -2 q^{3} + q^{4} + ( -1 - \beta ) q^{5} -2 q^{6} + q^{8} + q^{9} + ( -1 - \beta ) q^{10} + ( -1 - \beta ) q^{11} -2 q^{12} + ( 2 + 2 \beta ) q^{13} + ( 2 + 2 \beta ) q^{15} + q^{16} + q^{18} + 2 \beta q^{19} + ( -1 - \beta ) q^{20} + ( -1 - \beta ) q^{22} -2 q^{23} -2 q^{24} + ( -1 + 2 \beta ) q^{25} + ( 2 + 2 \beta ) q^{26} + 4 q^{27} + ( 1 - 5 \beta ) q^{29} + ( 2 + 2 \beta ) q^{30} + ( 2 - 2 \beta ) q^{31} + q^{32} + ( 2 + 2 \beta ) q^{33} + q^{36} + ( -2 + 4 \beta ) q^{37} + 2 \beta q^{38} + ( -4 - 4 \beta ) q^{39} + ( -1 - \beta ) q^{40} - q^{41} + ( 2 + 6 \beta ) q^{43} + ( -1 - \beta ) q^{44} + ( -1 - \beta ) q^{45} -2 q^{46} + ( -4 + 2 \beta ) q^{47} -2 q^{48} + ( -1 + 2 \beta ) q^{50} + ( 2 + 2 \beta ) q^{52} + ( -1 + 5 \beta ) q^{53} + 4 q^{54} + ( 4 + 2 \beta ) q^{55} -4 \beta q^{57} + ( 1 - 5 \beta ) q^{58} + ( -1 - 7 \beta ) q^{59} + ( 2 + 2 \beta ) q^{60} + ( -5 - \beta ) q^{61} + ( 2 - 2 \beta ) q^{62} + q^{64} + ( -8 - 4 \beta ) q^{65} + ( 2 + 2 \beta ) q^{66} + ( -5 + 3 \beta ) q^{67} + 4 q^{69} + ( 6 - 2 \beta ) q^{71} + q^{72} + ( 2 + 4 \beta ) q^{73} + ( -2 + 4 \beta ) q^{74} + ( 2 - 4 \beta ) q^{75} + 2 \beta q^{76} + ( -4 - 4 \beta ) q^{78} + ( -1 - \beta ) q^{80} -11 q^{81} - q^{82} + ( -3 - \beta ) q^{83} + ( 2 + 6 \beta ) q^{86} + ( -2 + 10 \beta ) q^{87} + ( -1 - \beta ) q^{88} + ( -6 - 4 \beta ) q^{89} + ( -1 - \beta ) q^{90} -2 q^{92} + ( -4 + 4 \beta ) q^{93} + ( -4 + 2 \beta ) q^{94} + ( -6 - 2 \beta ) q^{95} -2 q^{96} -2 q^{97} + ( -1 - \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 4q^{3} + 2q^{4} - 2q^{5} - 4q^{6} + 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{2} - 4q^{3} + 2q^{4} - 2q^{5} - 4q^{6} + 2q^{8} + 2q^{9} - 2q^{10} - 2q^{11} - 4q^{12} + 4q^{13} + 4q^{15} + 2q^{16} + 2q^{18} - 2q^{20} - 2q^{22} - 4q^{23} - 4q^{24} - 2q^{25} + 4q^{26} + 8q^{27} + 2q^{29} + 4q^{30} + 4q^{31} + 2q^{32} + 4q^{33} + 2q^{36} - 4q^{37} - 8q^{39} - 2q^{40} - 2q^{41} + 4q^{43} - 2q^{44} - 2q^{45} - 4q^{46} - 8q^{47} - 4q^{48} - 2q^{50} + 4q^{52} - 2q^{53} + 8q^{54} + 8q^{55} + 2q^{58} - 2q^{59} + 4q^{60} - 10q^{61} + 4q^{62} + 2q^{64} - 16q^{65} + 4q^{66} - 10q^{67} + 8q^{69} + 12q^{71} + 2q^{72} + 4q^{73} - 4q^{74} + 4q^{75} - 8q^{78} - 2q^{80} - 22q^{81} - 2q^{82} - 6q^{83} + 4q^{86} - 4q^{87} - 2q^{88} - 12q^{89} - 2q^{90} - 4q^{92} - 8q^{93} - 8q^{94} - 12q^{95} - 4q^{96} - 4q^{97} - 2q^{99} + O(q^{100})$$

## 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
 1.73205 −1.73205
1.00000 −2.00000 1.00000 −2.73205 −2.00000 0 1.00000 1.00000 −2.73205
1.2 1.00000 −2.00000 1.00000 0.732051 −2.00000 0 1.00000 1.00000 0.732051
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.x 2
7.b odd 2 1 574.2.a.k 2
21.c even 2 1 5166.2.a.bo 2
28.d even 2 1 4592.2.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.a.k 2 7.b odd 2 1
4018.2.a.x 2 1.a even 1 1 trivial
4592.2.a.m 2 28.d even 2 1
5166.2.a.bo 2 21.c even 2 1

## Atkin-Lehner signs

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

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T )^{2}$$
$3$ $$( 1 + 2 T + 3 T^{2} )^{2}$$
$5$ $$1 + 2 T + 8 T^{2} + 10 T^{3} + 25 T^{4}$$
$7$ 1
$11$ $$1 + 2 T + 20 T^{2} + 22 T^{3} + 121 T^{4}$$
$13$ $$1 - 4 T + 18 T^{2} - 52 T^{3} + 169 T^{4}$$
$17$ $$( 1 + 17 T^{2} )^{2}$$
$19$ $$1 + 26 T^{2} + 361 T^{4}$$
$23$ $$( 1 + 2 T + 23 T^{2} )^{2}$$
$29$ $$1 - 2 T - 16 T^{2} - 58 T^{3} + 841 T^{4}$$
$31$ $$1 - 4 T + 54 T^{2} - 124 T^{3} + 961 T^{4}$$
$37$ $$1 + 4 T + 30 T^{2} + 148 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + T )^{2}$$
$43$ $$1 - 4 T - 18 T^{2} - 172 T^{3} + 1849 T^{4}$$
$47$ $$1 + 8 T + 98 T^{2} + 376 T^{3} + 2209 T^{4}$$
$53$ $$1 + 2 T + 32 T^{2} + 106 T^{3} + 2809 T^{4}$$
$59$ $$1 + 2 T - 28 T^{2} + 118 T^{3} + 3481 T^{4}$$
$61$ $$1 + 10 T + 144 T^{2} + 610 T^{3} + 3721 T^{4}$$
$67$ $$1 + 10 T + 132 T^{2} + 670 T^{3} + 4489 T^{4}$$
$71$ $$1 - 12 T + 166 T^{2} - 852 T^{3} + 5041 T^{4}$$
$73$ $$1 - 4 T + 102 T^{2} - 292 T^{3} + 5329 T^{4}$$
$79$ $$( 1 + 79 T^{2} )^{2}$$
$83$ $$1 + 6 T + 172 T^{2} + 498 T^{3} + 6889 T^{4}$$
$89$ $$1 + 12 T + 166 T^{2} + 1068 T^{3} + 7921 T^{4}$$
$97$ $$( 1 + 2 T + 97 T^{2} )^{2}$$