# Properties

 Label 4018.2.a.w Level 4018 Weight 2 Character orbit 4018.a Self dual yes Analytic conductor 32.084 Analytic rank 0 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} + ( -1 + 2 \beta ) q^{5} - q^{6} - q^{8} -2 q^{9} +O(q^{10})$$ $$q - q^{2} + q^{3} + q^{4} + ( -1 + 2 \beta ) q^{5} - q^{6} - q^{8} -2 q^{9} + ( 1 - 2 \beta ) q^{10} + ( 2 - 4 \beta ) q^{11} + q^{12} + ( 4 - 4 \beta ) q^{13} + ( -1 + 2 \beta ) q^{15} + q^{16} + 3 q^{17} + 2 q^{18} -4 q^{19} + ( -1 + 2 \beta ) q^{20} + ( -2 + 4 \beta ) q^{22} + ( 2 + 4 \beta ) q^{23} - q^{24} + ( -4 + 4 \beta ) q^{26} -5 q^{27} + ( 5 - 2 \beta ) q^{29} + ( 1 - 2 \beta ) q^{30} + ( 3 + 2 \beta ) q^{31} - q^{32} + ( 2 - 4 \beta ) q^{33} -3 q^{34} -2 q^{36} + ( -6 + 4 \beta ) q^{37} + 4 q^{38} + ( 4 - 4 \beta ) q^{39} + ( 1 - 2 \beta ) q^{40} - q^{41} + 9 q^{43} + ( 2 - 4 \beta ) q^{44} + ( 2 - 4 \beta ) q^{45} + ( -2 - 4 \beta ) q^{46} -8 \beta q^{47} + q^{48} + 3 q^{51} + ( 4 - 4 \beta ) q^{52} + ( -3 + 6 \beta ) q^{53} + 5 q^{54} -10 q^{55} -4 q^{57} + ( -5 + 2 \beta ) q^{58} + ( 6 + 4 \beta ) q^{59} + ( -1 + 2 \beta ) q^{60} + ( 5 + 6 \beta ) q^{61} + ( -3 - 2 \beta ) q^{62} + q^{64} + ( -12 + 4 \beta ) q^{65} + ( -2 + 4 \beta ) q^{66} + ( -4 + 12 \beta ) q^{67} + 3 q^{68} + ( 2 + 4 \beta ) q^{69} + ( 1 - 2 \beta ) q^{71} + 2 q^{72} + 6 q^{73} + ( 6 - 4 \beta ) q^{74} -4 q^{76} + ( -4 + 4 \beta ) q^{78} + ( 1 + 6 \beta ) q^{79} + ( -1 + 2 \beta ) q^{80} + q^{81} + q^{82} + ( 2 + 8 \beta ) q^{83} + ( -3 + 6 \beta ) q^{85} -9 q^{86} + ( 5 - 2 \beta ) q^{87} + ( -2 + 4 \beta ) q^{88} -3 q^{89} + ( -2 + 4 \beta ) q^{90} + ( 2 + 4 \beta ) q^{92} + ( 3 + 2 \beta ) q^{93} + 8 \beta q^{94} + ( 4 - 8 \beta ) q^{95} - q^{96} + q^{97} + ( -4 + 8 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} - 2q^{8} - 4q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} - 2q^{8} - 4q^{9} + 2q^{12} + 4q^{13} + 2q^{16} + 6q^{17} + 4q^{18} - 8q^{19} + 8q^{23} - 2q^{24} - 4q^{26} - 10q^{27} + 8q^{29} + 8q^{31} - 2q^{32} - 6q^{34} - 4q^{36} - 8q^{37} + 8q^{38} + 4q^{39} - 2q^{41} + 18q^{43} - 8q^{46} - 8q^{47} + 2q^{48} + 6q^{51} + 4q^{52} + 10q^{54} - 20q^{55} - 8q^{57} - 8q^{58} + 16q^{59} + 16q^{61} - 8q^{62} + 2q^{64} - 20q^{65} + 4q^{67} + 6q^{68} + 8q^{69} + 4q^{72} + 12q^{73} + 8q^{74} - 8q^{76} - 4q^{78} + 8q^{79} + 2q^{81} + 2q^{82} + 12q^{83} - 18q^{86} + 8q^{87} - 6q^{89} + 8q^{92} + 8q^{93} + 8q^{94} - 2q^{96} + 2q^{97} + O(q^{100})$$

## 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
 −0.618034 1.61803
−1.00000 1.00000 1.00000 −2.23607 −1.00000 0 −1.00000 −2.00000 2.23607
1.2 −1.00000 1.00000 1.00000 2.23607 −1.00000 0 −1.00000 −2.00000 −2.23607
 $$n$$: 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.w yes 2
7.b odd 2 1 4018.2.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4018.2.a.t 2 7.b odd 2 1
4018.2.a.w yes 2 1.a even 1 1 trivial

## 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4018))$$:

 $$T_{3} - 1$$ $$T_{5}^{2} - 5$$ $$T_{11}^{2} - 20$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$( 1 - T + 3 T^{2} )^{2}$$
$5$ $$1 + 5 T^{2} + 25 T^{4}$$
$7$ 1
$11$ $$1 + 2 T^{2} + 121 T^{4}$$
$13$ $$1 - 4 T + 10 T^{2} - 52 T^{3} + 169 T^{4}$$
$17$ $$( 1 - 3 T + 17 T^{2} )^{2}$$
$19$ $$( 1 + 4 T + 19 T^{2} )^{2}$$
$23$ $$1 - 8 T + 42 T^{2} - 184 T^{3} + 529 T^{4}$$
$29$ $$1 - 8 T + 69 T^{2} - 232 T^{3} + 841 T^{4}$$
$31$ $$1 - 8 T + 73 T^{2} - 248 T^{3} + 961 T^{4}$$
$37$ $$1 + 8 T + 70 T^{2} + 296 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + T )^{2}$$
$43$ $$( 1 - 9 T + 43 T^{2} )^{2}$$
$47$ $$1 + 8 T + 30 T^{2} + 376 T^{3} + 2209 T^{4}$$
$53$ $$1 + 61 T^{2} + 2809 T^{4}$$
$59$ $$1 - 16 T + 162 T^{2} - 944 T^{3} + 3481 T^{4}$$
$61$ $$1 - 16 T + 141 T^{2} - 976 T^{3} + 3721 T^{4}$$
$67$ $$1 - 4 T - 42 T^{2} - 268 T^{3} + 4489 T^{4}$$
$71$ $$1 + 137 T^{2} + 5041 T^{4}$$
$73$ $$( 1 - 6 T + 73 T^{2} )^{2}$$
$79$ $$1 - 8 T + 129 T^{2} - 632 T^{3} + 6241 T^{4}$$
$83$ $$1 - 12 T + 122 T^{2} - 996 T^{3} + 6889 T^{4}$$
$89$ $$( 1 + 3 T + 89 T^{2} )^{2}$$
$97$ $$( 1 - T + 97 T^{2} )^{2}$$