# Properties

 Label 4018.2.a.bb 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: $$\Q(\sqrt{2})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + q^{2} + ( 1 + \beta ) q^{3} + q^{4} - q^{5} + ( 1 + \beta ) q^{6} + q^{8} + 2 \beta q^{9} +O(q^{10})$$ $$q + q^{2} + ( 1 + \beta ) q^{3} + q^{4} - q^{5} + ( 1 + \beta ) q^{6} + q^{8} + 2 \beta q^{9} - q^{10} + ( 2 + \beta ) q^{11} + ( 1 + \beta ) q^{12} + ( 2 - 3 \beta ) q^{13} + ( -1 - \beta ) q^{15} + q^{16} + ( 5 + 2 \beta ) q^{17} + 2 \beta q^{18} + 2 \beta q^{19} - q^{20} + ( 2 + \beta ) q^{22} -4 q^{23} + ( 1 + \beta ) q^{24} -4 q^{25} + ( 2 - 3 \beta ) q^{26} + ( 1 - \beta ) q^{27} + ( 1 - 2 \beta ) q^{29} + ( -1 - \beta ) q^{30} + ( 7 - \beta ) q^{31} + q^{32} + ( 4 + 3 \beta ) q^{33} + ( 5 + 2 \beta ) q^{34} + 2 \beta q^{36} + \beta q^{37} + 2 \beta q^{38} + ( -4 - \beta ) q^{39} - q^{40} - q^{41} + ( 1 + 5 \beta ) q^{43} + ( 2 + \beta ) q^{44} -2 \beta q^{45} -4 q^{46} + ( 10 + \beta ) q^{47} + ( 1 + \beta ) q^{48} -4 q^{50} + ( 9 + 7 \beta ) q^{51} + ( 2 - 3 \beta ) q^{52} + ( 3 - 4 \beta ) q^{53} + ( 1 - \beta ) q^{54} + ( -2 - \beta ) q^{55} + ( 4 + 2 \beta ) q^{57} + ( 1 - 2 \beta ) q^{58} + ( -2 + 2 \beta ) q^{59} + ( -1 - \beta ) q^{60} + ( -3 - 4 \beta ) q^{61} + ( 7 - \beta ) q^{62} + q^{64} + ( -2 + 3 \beta ) q^{65} + ( 4 + 3 \beta ) q^{66} + ( 6 + 2 \beta ) q^{67} + ( 5 + 2 \beta ) q^{68} + ( -4 - 4 \beta ) q^{69} + ( -3 - 3 \beta ) q^{71} + 2 \beta q^{72} + 4 q^{73} + \beta q^{74} + ( -4 - 4 \beta ) q^{75} + 2 \beta q^{76} + ( -4 - \beta ) q^{78} + ( 1 + 3 \beta ) q^{79} - q^{80} + ( -1 - 6 \beta ) q^{81} - q^{82} + ( -6 - 5 \beta ) q^{83} + ( -5 - 2 \beta ) q^{85} + ( 1 + 5 \beta ) q^{86} + ( -3 - \beta ) q^{87} + ( 2 + \beta ) q^{88} + ( 5 - 2 \beta ) q^{89} -2 \beta q^{90} -4 q^{92} + ( 5 + 6 \beta ) q^{93} + ( 10 + \beta ) q^{94} -2 \beta q^{95} + ( 1 + \beta ) q^{96} + ( 15 + 2 \beta ) q^{97} + ( 4 + 4 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} + 2q^{8} + O(q^{10})$$ $$2q + 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} + 2q^{8} - 2q^{10} + 4q^{11} + 2q^{12} + 4q^{13} - 2q^{15} + 2q^{16} + 10q^{17} - 2q^{20} + 4q^{22} - 8q^{23} + 2q^{24} - 8q^{25} + 4q^{26} + 2q^{27} + 2q^{29} - 2q^{30} + 14q^{31} + 2q^{32} + 8q^{33} + 10q^{34} - 8q^{39} - 2q^{40} - 2q^{41} + 2q^{43} + 4q^{44} - 8q^{46} + 20q^{47} + 2q^{48} - 8q^{50} + 18q^{51} + 4q^{52} + 6q^{53} + 2q^{54} - 4q^{55} + 8q^{57} + 2q^{58} - 4q^{59} - 2q^{60} - 6q^{61} + 14q^{62} + 2q^{64} - 4q^{65} + 8q^{66} + 12q^{67} + 10q^{68} - 8q^{69} - 6q^{71} + 8q^{73} - 8q^{75} - 8q^{78} + 2q^{79} - 2q^{80} - 2q^{81} - 2q^{82} - 12q^{83} - 10q^{85} + 2q^{86} - 6q^{87} + 4q^{88} + 10q^{89} - 8q^{92} + 10q^{93} + 20q^{94} + 2q^{96} + 30q^{97} + 8q^{99} + O(q^{100})$$

## 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
 −1.41421 1.41421
1.00000 −0.414214 1.00000 −1.00000 −0.414214 0 1.00000 −2.82843 −1.00000
1.2 1.00000 2.41421 1.00000 −1.00000 2.41421 0 1.00000 2.82843 −1.00000
 $$n$$: 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.bb yes 2
7.b odd 2 1 4018.2.a.z 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4018.2.a.z 2 7.b odd 2 1
4018.2.a.bb 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4018))$$:

 $$T_{3}^{2} - 2 T_{3} - 1$$ $$T_{5} + 1$$ $$T_{11}^{2} - 4 T_{11} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T )^{2}$$
$3$ $$1 - 2 T + 5 T^{2} - 6 T^{3} + 9 T^{4}$$
$5$ $$( 1 + T + 5 T^{2} )^{2}$$
$7$ 1
$11$ $$1 - 4 T + 24 T^{2} - 44 T^{3} + 121 T^{4}$$
$13$ $$1 - 4 T + 12 T^{2} - 52 T^{3} + 169 T^{4}$$
$17$ $$1 - 10 T + 51 T^{2} - 170 T^{3} + 289 T^{4}$$
$19$ $$1 + 30 T^{2} + 361 T^{4}$$
$23$ $$( 1 + 4 T + 23 T^{2} )^{2}$$
$29$ $$1 - 2 T + 51 T^{2} - 58 T^{3} + 841 T^{4}$$
$31$ $$1 - 14 T + 109 T^{2} - 434 T^{3} + 961 T^{4}$$
$37$ $$1 + 72 T^{2} + 1369 T^{4}$$
$41$ $$( 1 + T )^{2}$$
$43$ $$1 - 2 T + 37 T^{2} - 86 T^{3} + 1849 T^{4}$$
$47$ $$1 - 20 T + 192 T^{2} - 940 T^{3} + 2209 T^{4}$$
$53$ $$1 - 6 T + 83 T^{2} - 318 T^{3} + 2809 T^{4}$$
$59$ $$1 + 4 T + 114 T^{2} + 236 T^{3} + 3481 T^{4}$$
$61$ $$1 + 6 T + 99 T^{2} + 366 T^{3} + 3721 T^{4}$$
$67$ $$1 - 12 T + 162 T^{2} - 804 T^{3} + 4489 T^{4}$$
$71$ $$1 + 6 T + 133 T^{2} + 426 T^{3} + 5041 T^{4}$$
$73$ $$( 1 - 4 T + 73 T^{2} )^{2}$$
$79$ $$1 - 2 T + 141 T^{2} - 158 T^{3} + 6241 T^{4}$$
$83$ $$1 + 12 T + 152 T^{2} + 996 T^{3} + 6889 T^{4}$$
$89$ $$1 - 10 T + 195 T^{2} - 890 T^{3} + 7921 T^{4}$$
$97$ $$1 - 30 T + 411 T^{2} - 2910 T^{3} + 9409 T^{4}$$