# Properties

 Label 4018.2.a.bg Level 4018 Weight 2 Character orbit 4018.a Self dual yes Analytic conductor 32.084 Analytic rank 0 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + \beta_{2} q^{3} + q^{4} -\beta_{1} q^{5} + \beta_{2} q^{6} + q^{8} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{9} +O(q^{10})$$ $$q + q^{2} + \beta_{2} q^{3} + q^{4} -\beta_{1} q^{5} + \beta_{2} q^{6} + q^{8} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{9} -\beta_{1} q^{10} + ( 2 + \beta_{1} + \beta_{2} ) q^{11} + \beta_{2} q^{12} + ( 2 + \beta_{2} ) q^{15} + q^{16} + ( 2 - \beta_{2} ) q^{17} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{18} + ( -2 - 2 \beta_{2} ) q^{19} -\beta_{1} q^{20} + ( 2 + \beta_{1} + \beta_{2} ) q^{22} + ( 2 - 2 \beta_{1} ) q^{23} + \beta_{2} q^{24} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{25} + ( -2 + 2 \beta_{1} + 3 \beta_{2} ) q^{27} + ( 2 - \beta_{1} ) q^{29} + ( 2 + \beta_{2} ) q^{30} + ( -2 + \beta_{2} ) q^{31} + q^{32} + ( 4 - 2 \beta_{1} ) q^{33} + ( 2 - \beta_{2} ) q^{34} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{36} + 2 q^{37} + ( -2 - 2 \beta_{2} ) q^{38} -\beta_{1} q^{40} + q^{41} + ( -2 + 3 \beta_{2} ) q^{43} + ( 2 + \beta_{1} + \beta_{2} ) q^{44} + ( 6 + \beta_{1} + \beta_{2} ) q^{45} + ( 2 - 2 \beta_{1} ) q^{46} -2 q^{47} + \beta_{2} q^{48} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{50} + ( -6 + 2 \beta_{1} + 3 \beta_{2} ) q^{51} + ( -2 - 3 \beta_{1} - 2 \beta_{2} ) q^{53} + ( -2 + 2 \beta_{1} + 3 \beta_{2} ) q^{54} + ( -2 - 4 \beta_{1} ) q^{55} + ( -12 + 4 \beta_{1} ) q^{57} + ( 2 - \beta_{1} ) q^{58} + ( 3 \beta_{1} + \beta_{2} ) q^{59} + ( 2 + \beta_{2} ) q^{60} + ( 4 - \beta_{1} - 2 \beta_{2} ) q^{61} + ( -2 + \beta_{2} ) q^{62} + q^{64} + ( 4 - 2 \beta_{1} ) q^{66} + ( 6 + \beta_{1} - 3 \beta_{2} ) q^{67} + ( 2 - \beta_{2} ) q^{68} + ( 4 + 4 \beta_{2} ) q^{69} + ( 2 + 4 \beta_{1} + 3 \beta_{2} ) q^{71} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{72} + ( 10 + 2 \beta_{2} ) q^{73} + 2 q^{74} + ( 2 - 2 \beta_{1} - 4 \beta_{2} ) q^{75} + ( -2 - 2 \beta_{2} ) q^{76} + ( 6 + 2 \beta_{1} + \beta_{2} ) q^{79} -\beta_{1} q^{80} + ( 5 - 4 \beta_{2} ) q^{81} + q^{82} + ( 4 + \beta_{1} - \beta_{2} ) q^{83} + ( -2 - 2 \beta_{1} - \beta_{2} ) q^{85} + ( -2 + 3 \beta_{2} ) q^{86} + ( 2 + 3 \beta_{2} ) q^{87} + ( 2 + \beta_{1} + \beta_{2} ) q^{88} + ( -8 + 2 \beta_{1} + \beta_{2} ) q^{89} + ( 6 + \beta_{1} + \beta_{2} ) q^{90} + ( 2 - 2 \beta_{1} ) q^{92} + ( 6 - 2 \beta_{1} - 3 \beta_{2} ) q^{93} -2 q^{94} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{95} + \beta_{2} q^{96} + ( 8 + 2 \beta_{1} + \beta_{2} ) q^{97} + ( -2 - 3 \beta_{1} + 3 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{2} - q^{3} + 3q^{4} - q^{5} - q^{6} + 3q^{8} + 8q^{9} + O(q^{10})$$ $$3q + 3q^{2} - q^{3} + 3q^{4} - q^{5} - q^{6} + 3q^{8} + 8q^{9} - q^{10} + 6q^{11} - q^{12} + 5q^{15} + 3q^{16} + 7q^{17} + 8q^{18} - 4q^{19} - q^{20} + 6q^{22} + 4q^{23} - q^{24} - 2q^{25} - 7q^{27} + 5q^{29} + 5q^{30} - 7q^{31} + 3q^{32} + 10q^{33} + 7q^{34} + 8q^{36} + 6q^{37} - 4q^{38} - q^{40} + 3q^{41} - 9q^{43} + 6q^{44} + 18q^{45} + 4q^{46} - 6q^{47} - q^{48} - 2q^{50} - 19q^{51} - 7q^{53} - 7q^{54} - 10q^{55} - 32q^{57} + 5q^{58} + 2q^{59} + 5q^{60} + 13q^{61} - 7q^{62} + 3q^{64} + 10q^{66} + 22q^{67} + 7q^{68} + 8q^{69} + 7q^{71} + 8q^{72} + 28q^{73} + 6q^{74} + 8q^{75} - 4q^{76} + 19q^{79} - q^{80} + 19q^{81} + 3q^{82} + 14q^{83} - 7q^{85} - 9q^{86} + 3q^{87} + 6q^{88} - 23q^{89} + 18q^{90} + 4q^{92} + 19q^{93} - 6q^{94} - 8q^{95} - q^{96} + 25q^{97} - 12q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 6 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2 \nu - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2 \beta_{1} + 4$$

## 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
 −0.363328 3.12489 −1.76156
1.00000 −3.14134 1.00000 0.363328 −3.14134 0 1.00000 6.86799 0.363328
1.2 1.00000 −0.484862 1.00000 −3.12489 −0.484862 0 1.00000 −2.76491 −3.12489
1.3 1.00000 2.62620 1.00000 1.76156 2.62620 0 1.00000 3.89692 1.76156
 $$n$$: 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.bg 3
7.b odd 2 1 574.2.a.l 3
21.c even 2 1 5166.2.a.bt 3
28.d even 2 1 4592.2.a.s 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.a.l 3 7.b odd 2 1
4018.2.a.bg 3 1.a even 1 1 trivial
4592.2.a.s 3 28.d even 2 1
5166.2.a.bt 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4018))$$:

 $$T_{3}^{3} + T_{3}^{2} - 8 T_{3} - 4$$ $$T_{5}^{3} + T_{5}^{2} - 6 T_{5} + 2$$ $$T_{11}^{3} - 6 T_{11}^{2} + 2 T_{11} + 20$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T )^{3}$$
$3$ $$1 + T + T^{2} + 2 T^{3} + 3 T^{4} + 9 T^{5} + 27 T^{6}$$
$5$ $$1 + T + 9 T^{2} + 12 T^{3} + 45 T^{4} + 25 T^{5} + 125 T^{6}$$
$7$ 1
$11$ $$1 - 6 T + 35 T^{2} - 112 T^{3} + 385 T^{4} - 726 T^{5} + 1331 T^{6}$$
$13$ $$( 1 + 13 T^{2} )^{3}$$
$17$ $$1 - 7 T + 59 T^{2} - 230 T^{3} + 1003 T^{4} - 2023 T^{5} + 4913 T^{6}$$
$19$ $$1 + 4 T + 29 T^{2} + 120 T^{3} + 551 T^{4} + 1444 T^{5} + 6859 T^{6}$$
$23$ $$1 - 4 T + 49 T^{2} - 120 T^{3} + 1127 T^{4} - 2116 T^{5} + 12167 T^{6}$$
$29$ $$1 - 5 T + 89 T^{2} - 280 T^{3} + 2581 T^{4} - 4205 T^{5} + 24389 T^{6}$$
$31$ $$1 + 7 T + 101 T^{2} + 426 T^{3} + 3131 T^{4} + 6727 T^{5} + 29791 T^{6}$$
$37$ $$( 1 - 2 T + 37 T^{2} )^{3}$$
$41$ $$( 1 - T )^{3}$$
$43$ $$1 + 9 T + 81 T^{2} + 542 T^{3} + 3483 T^{4} + 16641 T^{5} + 79507 T^{6}$$
$47$ $$( 1 + 2 T + 47 T^{2} )^{3}$$
$53$ $$1 + 7 T + 113 T^{2} + 632 T^{3} + 5989 T^{4} + 19663 T^{5} + 148877 T^{6}$$
$59$ $$1 - 2 T + 127 T^{2} - 336 T^{3} + 7493 T^{4} - 6962 T^{5} + 205379 T^{6}$$
$61$ $$1 - 13 T + 209 T^{2} - 1596 T^{3} + 12749 T^{4} - 48373 T^{5} + 226981 T^{6}$$
$67$ $$1 - 22 T + 267 T^{2} - 2368 T^{3} + 17889 T^{4} - 98758 T^{5} + 300763 T^{6}$$
$71$ $$1 - 7 T + 109 T^{2} - 666 T^{3} + 7739 T^{4} - 35287 T^{5} + 357911 T^{6}$$
$73$ $$1 - 28 T + 447 T^{2} - 4600 T^{3} + 32631 T^{4} - 149212 T^{5} + 389017 T^{6}$$
$79$ $$1 - 19 T + 333 T^{2} - 3130 T^{3} + 26307 T^{4} - 118579 T^{5} + 493039 T^{6}$$
$83$ $$1 - 14 T + 295 T^{2} - 2304 T^{3} + 24485 T^{4} - 96446 T^{5} + 571787 T^{6}$$
$89$ $$1 + 23 T + 419 T^{2} + 4330 T^{3} + 37291 T^{4} + 182183 T^{5} + 704969 T^{6}$$
$97$ $$1 - 25 T + 475 T^{2} - 5254 T^{3} + 46075 T^{4} - 235225 T^{5} + 912673 T^{6}$$