# Properties

 Label 4018.2.a.ba 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: no (minimal twist has level 82) 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.

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

## 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 −1.41421 1.00000 2.82843 −1.41421 0 1.00000 −1.00000 2.82843
1.2 1.00000 1.41421 1.00000 −2.82843 1.41421 0 1.00000 −1.00000 −2.82843
 $$n$$: 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.ba 2
7.b odd 2 1 82.2.a.b 2
21.c even 2 1 738.2.a.k 2
28.d even 2 1 656.2.a.e 2
35.c odd 2 1 2050.2.a.h 2
35.f even 4 2 2050.2.c.l 4
56.e even 2 1 2624.2.a.l 2
56.h odd 2 1 2624.2.a.j 2
77.b even 2 1 9922.2.a.i 2
84.h odd 2 1 5904.2.a.z 2
287.d odd 2 1 3362.2.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
82.2.a.b 2 7.b odd 2 1
656.2.a.e 2 28.d even 2 1
738.2.a.k 2 21.c even 2 1
2050.2.a.h 2 35.c odd 2 1
2050.2.c.l 4 35.f even 4 2
2624.2.a.j 2 56.h odd 2 1
2624.2.a.l 2 56.e even 2 1
3362.2.a.m 2 287.d odd 2 1
4018.2.a.ba 2 1.a even 1 1 trivial
5904.2.a.z 2 84.h odd 2 1
9922.2.a.i 2 77.b 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4018))$$:

 $$T_{3}^{2} - 2$$ $$T_{5}^{2} - 8$$ $$T_{11}^{2} - 18$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T )^{2}$$
$3$ $$1 + 4 T^{2} + 9 T^{4}$$
$5$ $$1 + 2 T^{2} + 25 T^{4}$$
$7$ 1
$11$ $$1 + 4 T^{2} + 121 T^{4}$$
$13$ $$( 1 + 13 T^{2} )^{2}$$
$17$ $$1 + 4 T + 6 T^{2} + 68 T^{3} + 289 T^{4}$$
$19$ $$1 - 8 T + 52 T^{2} - 152 T^{3} + 361 T^{4}$$
$23$ $$1 - 8 T + 54 T^{2} - 184 T^{3} + 529 T^{4}$$
$29$ $$1 - 8 T + 42 T^{2} - 232 T^{3} + 841 T^{4}$$
$31$ $$1 - 8 T + 70 T^{2} - 248 T^{3} + 961 T^{4}$$
$37$ $$1 + 2 T^{2} + 1369 T^{4}$$
$41$ $$( 1 - T )^{2}$$
$43$ $$1 - 8 T + 70 T^{2} - 344 T^{3} + 1849 T^{4}$$
$47$ $$1 - 4 T + 48 T^{2} - 188 T^{3} + 2209 T^{4}$$
$53$ $$( 1 - 12 T + 53 T^{2} )^{2}$$
$59$ $$1 - 8 T + 126 T^{2} - 472 T^{3} + 3481 T^{4}$$
$61$ $$( 1 + 6 T + 61 T^{2} )^{2}$$
$67$ $$1 + 8 T + 132 T^{2} + 536 T^{3} + 4489 T^{4}$$
$71$ $$1 + 4 T + 144 T^{2} + 284 T^{3} + 5041 T^{4}$$
$73$ $$1 - 16 T + 178 T^{2} - 1168 T^{3} + 5329 T^{4}$$
$79$ $$1 + 12 T + 176 T^{2} + 948 T^{3} + 6241 T^{4}$$
$83$ $$1 + 24 T + 278 T^{2} + 1992 T^{3} + 6889 T^{4}$$
$89$ $$1 - 12 T + 182 T^{2} - 1068 T^{3} + 7921 T^{4}$$
$97$ $$1 - 4 T + 166 T^{2} - 388 T^{3} + 9409 T^{4}$$