# Properties

 Label 4018.2.a.bd Level 4018 Weight 2 Character orbit 4018.a Self dual yes Analytic conductor 32.084 Analytic rank 1 Dimension 3 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: $$3$$ Coefficient field: 3.3.404.1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 5 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 3$$

## 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
 −1.65544 2.86620 −0.210756
−1.00000 −3.05137 1.00000 −2.65544 3.05137 0 −1.00000 6.31088 2.65544
1.2 −1.00000 0.517304 1.00000 1.86620 −0.517304 0 −1.00000 −2.73240 −1.86620
1.3 −1.00000 2.53407 1.00000 −1.21076 −2.53407 0 −1.00000 3.42151 1.21076
 $$n$$: 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.bd 3
7.b odd 2 1 4018.2.a.be yes 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4018.2.a.bd 3 1.a even 1 1 trivial
4018.2.a.be yes 3 7.b odd 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4018))$$:

 $$T_{3}^{3} - 8 T_{3} + 4$$ $$T_{5}^{3} + 2 T_{5}^{2} - 4 T_{5} - 6$$ $$T_{11}^{3} - 2 T_{11}^{2} - 6 T_{11} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{3}$$
$3$ $$1 + T^{2} + 4 T^{3} + 3 T^{4} + 27 T^{6}$$
$5$ $$1 + 2 T + 11 T^{2} + 14 T^{3} + 55 T^{4} + 50 T^{5} + 125 T^{6}$$
$7$ 1
$11$ $$1 - 2 T + 27 T^{2} - 46 T^{3} + 297 T^{4} - 242 T^{5} + 1331 T^{6}$$
$13$ $$1 - 2 T + 23 T^{2} - 56 T^{3} + 299 T^{4} - 338 T^{5} + 2197 T^{6}$$
$17$ $$1 + 12 T + 91 T^{2} + 436 T^{3} + 1547 T^{4} + 3468 T^{5} + 4913 T^{6}$$
$19$ $$1 + 4 T + 45 T^{2} + 116 T^{3} + 855 T^{4} + 1444 T^{5} + 6859 T^{6}$$
$23$ $$1 + 8 T + 33 T^{2} + 76 T^{3} + 759 T^{4} + 4232 T^{5} + 12167 T^{6}$$
$29$ $$1 + 13 T^{2} - 66 T^{3} + 377 T^{4} + 24389 T^{6}$$
$31$ $$1 + 10 T + 97 T^{2} + 612 T^{3} + 3007 T^{4} + 9610 T^{5} + 29791 T^{6}$$
$37$ $$1 - 18 T + 187 T^{2} - 1324 T^{3} + 6919 T^{4} - 24642 T^{5} + 50653 T^{6}$$
$41$ $$( 1 - T )^{3}$$
$43$ $$1 + 12 T + 145 T^{2} + 936 T^{3} + 6235 T^{4} + 22188 T^{5} + 79507 T^{6}$$
$47$ $$1 + 22 T + 285 T^{2} + 2320 T^{3} + 13395 T^{4} + 48598 T^{5} + 103823 T^{6}$$
$53$ $$1 + 10 T + 173 T^{2} + 1002 T^{3} + 9169 T^{4} + 28090 T^{5} + 148877 T^{6}$$
$59$ $$1 + 12 T + 177 T^{2} + 1362 T^{3} + 10443 T^{4} + 41772 T^{5} + 205379 T^{6}$$
$61$ $$1 - 24 T + 271 T^{2} - 2202 T^{3} + 16531 T^{4} - 89304 T^{5} + 226981 T^{6}$$
$67$ $$1 - 4 T + 91 T^{2} - 294 T^{3} + 6097 T^{4} - 17956 T^{5} + 300763 T^{6}$$
$71$ $$1 + 4 T + 117 T^{2} + 712 T^{3} + 8307 T^{4} + 20164 T^{5} + 357911 T^{6}$$
$73$ $$1 + 12 T + 235 T^{2} + 1720 T^{3} + 17155 T^{4} + 63948 T^{5} + 389017 T^{6}$$
$79$ $$1 - 8 T + 173 T^{2} - 880 T^{3} + 13667 T^{4} - 49928 T^{5} + 493039 T^{6}$$
$83$ $$1 - 24 T + 397 T^{2} - 4138 T^{3} + 32951 T^{4} - 165336 T^{5} + 571787 T^{6}$$
$89$ $$1 + 6 T + 103 T^{2} + 676 T^{3} + 9167 T^{4} + 47526 T^{5} + 704969 T^{6}$$
$97$ $$1 - 18 T + 271 T^{2} - 3196 T^{3} + 26287 T^{4} - 169362 T^{5} + 912673 T^{6}$$