# Properties

 Label 4018.2.a.bi Level 4018 Weight 2 Character orbit 4018.a Self dual yes Analytic conductor 32.084 Analytic rank 1 Dimension 4 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: $$4$$ Coefficient field: 4.4.113481.1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} + ( -1 + \beta_{3} ) q^{5} + q^{6} - q^{8} -2 q^{9} +O(q^{10})$$ $$q - q^{2} - q^{3} + q^{4} + ( -1 + \beta_{3} ) q^{5} + q^{6} - q^{8} -2 q^{9} + ( 1 - \beta_{3} ) q^{10} + ( \beta_{1} + \beta_{2} ) q^{11} - q^{12} + ( 1 + \beta_{2} - \beta_{3} ) q^{13} + ( 1 - \beta_{3} ) q^{15} + q^{16} + ( 2 - 2 \beta_{1} - \beta_{3} ) q^{17} + 2 q^{18} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{19} + ( -1 + \beta_{3} ) q^{20} + ( -\beta_{1} - \beta_{2} ) q^{22} + ( -2 - \beta_{1} + \beta_{3} ) q^{23} + q^{24} + ( 4 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{25} + ( -1 - \beta_{2} + \beta_{3} ) q^{26} + 5 q^{27} + ( -5 - \beta_{2} ) q^{29} + ( -1 + \beta_{3} ) q^{30} + ( 3 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{31} - q^{32} + ( -\beta_{1} - \beta_{2} ) q^{33} + ( -2 + 2 \beta_{1} + \beta_{3} ) q^{34} -2 q^{36} + ( 2 - 3 \beta_{1} ) q^{37} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{38} + ( -1 - \beta_{2} + \beta_{3} ) q^{39} + ( 1 - \beta_{3} ) q^{40} + q^{41} + ( 1 - \beta_{2} - \beta_{3} ) q^{43} + ( \beta_{1} + \beta_{2} ) q^{44} + ( 2 - 2 \beta_{3} ) q^{45} + ( 2 + \beta_{1} - \beta_{3} ) q^{46} + ( 1 - \beta_{1} + \beta_{2} ) q^{47} - q^{48} + ( -4 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{50} + ( -2 + 2 \beta_{1} + \beta_{3} ) q^{51} + ( 1 + \beta_{2} - \beta_{3} ) q^{52} + ( 2 - \beta_{1} ) q^{53} -5 q^{54} + ( 1 - 2 \beta_{2} - 2 \beta_{3} ) q^{55} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{57} + ( 5 + \beta_{2} ) q^{58} + ( -1 + 3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{59} + ( 1 - \beta_{3} ) q^{60} + ( 3 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{61} + ( -3 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{62} + q^{64} + ( -7 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{65} + ( \beta_{1} + \beta_{2} ) q^{66} + ( 3 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{67} + ( 2 - 2 \beta_{1} - \beta_{3} ) q^{68} + ( 2 + \beta_{1} - \beta_{3} ) q^{69} + ( -3 + 3 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{71} + 2 q^{72} + ( 7 - 2 \beta_{1} + \beta_{2} ) q^{73} + ( -2 + 3 \beta_{1} ) q^{74} + ( -4 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{75} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{76} + ( 1 + \beta_{2} - \beta_{3} ) q^{78} + ( -6 + \beta_{2} - 2 \beta_{3} ) q^{79} + ( -1 + \beta_{3} ) q^{80} + q^{81} - q^{82} + ( 6 - 5 \beta_{1} - 2 \beta_{3} ) q^{83} + ( -8 + 3 \beta_{1} + 2 \beta_{3} ) q^{85} + ( -1 + \beta_{2} + \beta_{3} ) q^{86} + ( 5 + \beta_{2} ) q^{87} + ( -\beta_{1} - \beta_{2} ) q^{88} + ( -3 + 2 \beta_{1} - \beta_{2} ) q^{89} + ( -2 + 2 \beta_{3} ) q^{90} + ( -2 - \beta_{1} + \beta_{3} ) q^{92} + ( -3 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{93} + ( -1 + \beta_{1} - \beta_{2} ) q^{94} + ( -2 - 3 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} ) q^{95} + q^{96} + ( -1 - \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{97} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} - 4q^{3} + 4q^{4} - 3q^{5} + 4q^{6} - 4q^{8} - 8q^{9} + O(q^{10})$$ $$4q - 4q^{2} - 4q^{3} + 4q^{4} - 3q^{5} + 4q^{6} - 4q^{8} - 8q^{9} + 3q^{10} - 4q^{12} + q^{13} + 3q^{15} + 4q^{16} + 3q^{17} + 8q^{18} - 2q^{19} - 3q^{20} - 9q^{23} + 4q^{24} + 19q^{25} - q^{26} + 20q^{27} - 18q^{29} - 3q^{30} + 19q^{31} - 4q^{32} - 3q^{34} - 8q^{36} + 2q^{37} + 2q^{38} - q^{39} + 3q^{40} + 4q^{41} + 5q^{43} + 6q^{45} + 9q^{46} - 4q^{48} - 19q^{50} - 3q^{51} + q^{52} + 6q^{53} - 20q^{54} + 6q^{55} + 2q^{57} + 18q^{58} + 3q^{59} + 3q^{60} + q^{61} - 19q^{62} + 4q^{64} - 24q^{65} + 11q^{67} + 3q^{68} + 9q^{69} - 9q^{71} + 8q^{72} + 22q^{73} - 2q^{74} - 19q^{75} - 2q^{76} + q^{78} - 28q^{79} - 3q^{80} + 4q^{81} - 4q^{82} + 12q^{83} - 24q^{85} - 5q^{86} + 18q^{87} - 6q^{89} - 6q^{90} - 9q^{92} - 19q^{93} - 27q^{95} + 4q^{96} - 11q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 9 x^{2} + 3 x + 12$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2 \nu - 5$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 3 \nu^{2} - 5 \nu + 6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2 \beta_{1} + 5$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 3 \beta_{2} + 11 \beta_{1} + 9$$

## 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
 1.26053 −1.84745 3.90611 −1.31920
−1.00000 −1.00000 1.00000 −4.06659 1.00000 0 −1.00000 −2.00000 4.06659
1.2 −1.00000 −1.00000 1.00000 −2.30741 1.00000 0 −1.00000 −2.00000 2.30741
1.3 −1.00000 −1.00000 1.00000 −0.705371 1.00000 0 −1.00000 −2.00000 0.705371
1.4 −1.00000 −1.00000 1.00000 4.07936 1.00000 0 −1.00000 −2.00000 −4.07936
 $$n$$: 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.bi 4
7.b odd 2 1 4018.2.a.bk 4
7.d odd 6 2 574.2.e.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.e.f 8 7.d odd 6 2
4018.2.a.bi 4 1.a even 1 1 trivial
4018.2.a.bk 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4018))$$:

 $$T_{3} + 1$$ $$T_{5}^{4} + 3 T_{5}^{3} - 15 T_{5}^{2} - 50 T_{5} - 27$$ $$T_{11}^{4} - 33 T_{11}^{2} - 49 T_{11} + 15$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{4}$$
$3$ $$( 1 + T + 3 T^{2} )^{4}$$
$5$ $$1 + 3 T + 5 T^{2} - 5 T^{3} - 27 T^{4} - 25 T^{5} + 125 T^{6} + 375 T^{7} + 625 T^{8}$$
$7$ 1
$11$ $$1 + 11 T^{2} - 49 T^{3} + 15 T^{4} - 539 T^{5} + 1331 T^{6} + 14641 T^{8}$$
$13$ $$1 - T + 25 T^{2} - 14 T^{3} + 434 T^{4} - 182 T^{5} + 4225 T^{6} - 2197 T^{7} + 28561 T^{8}$$
$17$ $$1 - 3 T + 26 T^{2} - 62 T^{3} + 354 T^{4} - 1054 T^{5} + 7514 T^{6} - 14739 T^{7} + 83521 T^{8}$$
$19$ $$1 + 2 T + 13 T^{2} + 63 T^{3} + 453 T^{4} + 1197 T^{5} + 4693 T^{6} + 13718 T^{7} + 130321 T^{8}$$
$23$ $$1 + 9 T + 86 T^{2} + 440 T^{3} + 2670 T^{4} + 10120 T^{5} + 45494 T^{6} + 109503 T^{7} + 279841 T^{8}$$
$29$ $$1 + 18 T + 215 T^{2} + 1689 T^{3} + 10572 T^{4} + 48981 T^{5} + 180815 T^{6} + 439002 T^{7} + 707281 T^{8}$$
$31$ $$1 - 19 T + 205 T^{2} - 1484 T^{3} + 8978 T^{4} - 46004 T^{5} + 197005 T^{6} - 566029 T^{7} + 923521 T^{8}$$
$37$ $$1 - 2 T + 55 T^{2} + 61 T^{3} + 2110 T^{4} + 2257 T^{5} + 75295 T^{6} - 101306 T^{7} + 1874161 T^{8}$$
$41$ $$( 1 - T )^{4}$$
$43$ $$1 - 5 T + 127 T^{2} - 696 T^{3} + 7236 T^{4} - 29928 T^{5} + 234823 T^{6} - 397535 T^{7} + 3418801 T^{8}$$
$47$ $$1 + 155 T^{2} + 37 T^{3} + 10176 T^{4} + 1739 T^{5} + 342395 T^{6} + 4879681 T^{8}$$
$53$ $$1 - 6 T + 215 T^{2} - 929 T^{3} + 17154 T^{4} - 49237 T^{5} + 603935 T^{6} - 893262 T^{7} + 7890481 T^{8}$$
$59$ $$1 - 3 T + 68 T^{2} - 322 T^{3} + 7560 T^{4} - 18998 T^{5} + 236708 T^{6} - 616137 T^{7} + 12117361 T^{8}$$
$61$ $$1 - T + 73 T^{2} - 671 T^{3} + 1181 T^{4} - 40931 T^{5} + 271633 T^{6} - 226981 T^{7} + 13845841 T^{8}$$
$67$ $$1 - 11 T + 268 T^{2} - 2026 T^{3} + 26702 T^{4} - 135742 T^{5} + 1203052 T^{6} - 3308393 T^{7} + 20151121 T^{8}$$
$71$ $$1 + 9 T + 107 T^{2} + 199 T^{3} + 1653 T^{4} + 14129 T^{5} + 539387 T^{6} + 3221199 T^{7} + 25411681 T^{8}$$
$73$ $$1 - 22 T + 409 T^{2} - 4787 T^{3} + 48782 T^{4} - 349451 T^{5} + 2179561 T^{6} - 8558374 T^{7} + 28398241 T^{8}$$
$79$ $$1 + 28 T + 541 T^{2} + 7283 T^{3} + 73511 T^{4} + 575357 T^{5} + 3376381 T^{6} + 13805092 T^{7} + 38950081 T^{8}$$
$83$ $$1 - 12 T + 125 T^{2} - 957 T^{3} + 3390 T^{4} - 79431 T^{5} + 861125 T^{6} - 6861444 T^{7} + 47458321 T^{8}$$
$89$ $$1 + 6 T + 305 T^{2} + 1435 T^{3} + 39018 T^{4} + 127715 T^{5} + 2415905 T^{6} + 4229814 T^{7} + 62742241 T^{8}$$
$97$ $$1 + 11 T + 298 T^{2} + 2922 T^{3} + 39354 T^{4} + 283434 T^{5} + 2803882 T^{6} + 10039403 T^{7} + 88529281 T^{8}$$