# Properties

 Label 4018.2.a.bh 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: 3.3.321.1 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4 x + 1$$:

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

## 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
 0.239123 2.46050 −1.69963
1.00000 1.00000 1.00000 −4.18194 1.00000 0 1.00000 −2.00000 −4.18194
1.2 1.00000 1.00000 1.00000 −0.406421 1.00000 0 1.00000 −2.00000 −0.406421
1.3 1.00000 1.00000 1.00000 0.588364 1.00000 0 1.00000 −2.00000 0.588364
 $$n$$: 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.bh 3
7.b odd 2 1 4018.2.a.bf 3
7.d odd 6 2 574.2.e.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.e.e 6 7.d odd 6 2
4018.2.a.bf 3 7.b odd 2 1
4018.2.a.bh 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4018))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T )^{3}$$
$3$ $$( 1 - T + 3 T^{2} )^{3}$$
$5$ $$1 + 4 T + 14 T^{2} + 39 T^{3} + 70 T^{4} + 100 T^{5} + 125 T^{6}$$
$7$ 1
$11$ $$1 + 5 T + 37 T^{2} + 107 T^{3} + 407 T^{4} + 605 T^{5} + 1331 T^{6}$$
$13$ $$1 + T + 14 T^{2} - 23 T^{3} + 182 T^{4} + 169 T^{5} + 2197 T^{6}$$
$17$ $$1 - 5 T + 39 T^{2} - 107 T^{3} + 663 T^{4} - 1445 T^{5} + 4913 T^{6}$$
$19$ $$1 + 3 T + 33 T^{2} + 141 T^{3} + 627 T^{4} + 1083 T^{5} + 6859 T^{6}$$
$23$ $$1 + 9 T + 57 T^{2} + 335 T^{3} + 1311 T^{4} + 4761 T^{5} + 12167 T^{6}$$
$29$ $$1 + 12 T + 108 T^{2} + 599 T^{3} + 3132 T^{4} + 10092 T^{5} + 24389 T^{6}$$
$31$ $$1 - 11 T + 116 T^{2} - 671 T^{3} + 3596 T^{4} - 10571 T^{5} + 29791 T^{6}$$
$37$ $$1 + 18 T + 210 T^{2} + 1493 T^{3} + 7770 T^{4} + 24642 T^{5} + 50653 T^{6}$$
$41$ $$( 1 + T )^{3}$$
$43$ $$1 + 13 T + 104 T^{2} + 577 T^{3} + 4472 T^{4} + 24037 T^{5} + 79507 T^{6}$$
$47$ $$1 + 2 T + 34 T^{2} + 167 T^{3} + 1598 T^{4} + 4418 T^{5} + 103823 T^{6}$$
$53$ $$1 + 8 T + 124 T^{2} + 875 T^{3} + 6572 T^{4} + 22472 T^{5} + 148877 T^{6}$$
$59$ $$1 + 7 T + 143 T^{2} + 609 T^{3} + 8437 T^{4} + 24367 T^{5} + 205379 T^{6}$$
$61$ $$1 + 10 T + 64 T^{2} + 269 T^{3} + 3904 T^{4} + 37210 T^{5} + 226981 T^{6}$$
$67$ $$1 + 23 T + 365 T^{2} + 3425 T^{3} + 24455 T^{4} + 103247 T^{5} + 300763 T^{6}$$
$71$ $$1 + 180 T^{2} + 9 T^{3} + 12780 T^{4} + 357911 T^{6}$$
$73$ $$1 - 8 T + 176 T^{2} - 911 T^{3} + 12848 T^{4} - 42632 T^{5} + 389017 T^{6}$$
$79$ $$1 - T + 45 T^{2} - 167 T^{3} + 3555 T^{4} - 6241 T^{5} + 493039 T^{6}$$
$83$ $$1 + 4 T + 204 T^{2} + 487 T^{3} + 16932 T^{4} + 27556 T^{5} + 571787 T^{6}$$
$89$ $$1 - 10 T + 216 T^{2} - 1657 T^{3} + 19224 T^{4} - 79210 T^{5} + 704969 T^{6}$$
$97$ $$1 - 3 T + 213 T^{2} - 529 T^{3} + 20661 T^{4} - 28227 T^{5} + 912673 T^{6}$$