# Properties

 Label 40.6.a.a Level 40 Weight 6 Character orbit 40.a Self dual yes Analytic conductor 6.415 Analytic rank 0 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$40 = 2^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 40.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.41535279252$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 18q^{3} - 25q^{5} + 242q^{7} + 81q^{9} + O(q^{10})$$ $$q - 18q^{3} - 25q^{5} + 242q^{7} + 81q^{9} + 656q^{11} - 206q^{13} + 450q^{15} + 1690q^{17} - 1364q^{19} - 4356q^{21} + 2198q^{23} + 625q^{25} + 2916q^{27} - 2218q^{29} - 1700q^{31} - 11808q^{33} - 6050q^{35} - 846q^{37} + 3708q^{39} - 1818q^{41} + 10534q^{43} - 2025q^{45} + 12074q^{47} + 41757q^{49} - 30420q^{51} + 32586q^{53} - 16400q^{55} + 24552q^{57} + 8668q^{59} - 34670q^{61} + 19602q^{63} + 5150q^{65} - 47566q^{67} - 39564q^{69} + 948q^{71} - 63102q^{73} - 11250q^{75} + 158752q^{77} + 46536q^{79} - 72171q^{81} - 88778q^{83} - 42250q^{85} + 39924q^{87} - 104934q^{89} - 49852q^{91} + 30600q^{93} + 34100q^{95} - 36254q^{97} + 53136q^{99} + O(q^{100})$$

## 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
 0
0 −18.0000 0 −25.0000 0 242.000 0 81.0000 0
 $$n$$: 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 40.6.a.a 1
3.b odd 2 1 360.6.a.i 1
4.b odd 2 1 80.6.a.g 1
5.b even 2 1 200.6.a.d 1
5.c odd 4 2 200.6.c.b 2
8.b even 2 1 320.6.a.m 1
8.d odd 2 1 320.6.a.d 1
12.b even 2 1 720.6.a.k 1
20.d odd 2 1 400.6.a.b 1
20.e even 4 2 400.6.c.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.a.a 1 1.a even 1 1 trivial
80.6.a.g 1 4.b odd 2 1
200.6.a.d 1 5.b even 2 1
200.6.c.b 2 5.c odd 4 2
320.6.a.d 1 8.d odd 2 1
320.6.a.m 1 8.b even 2 1
360.6.a.i 1 3.b odd 2 1
400.6.a.b 1 20.d odd 2 1
400.6.c.e 2 20.e even 4 2
720.6.a.k 1 12.b even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 18$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(40))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 18 T + 243 T^{2}$$
$5$ $$1 + 25 T$$
$7$ $$1 - 242 T + 16807 T^{2}$$
$11$ $$1 - 656 T + 161051 T^{2}$$
$13$ $$1 + 206 T + 371293 T^{2}$$
$17$ $$1 - 1690 T + 1419857 T^{2}$$
$19$ $$1 + 1364 T + 2476099 T^{2}$$
$23$ $$1 - 2198 T + 6436343 T^{2}$$
$29$ $$1 + 2218 T + 20511149 T^{2}$$
$31$ $$1 + 1700 T + 28629151 T^{2}$$
$37$ $$1 + 846 T + 69343957 T^{2}$$
$41$ $$1 + 1818 T + 115856201 T^{2}$$
$43$ $$1 - 10534 T + 147008443 T^{2}$$
$47$ $$1 - 12074 T + 229345007 T^{2}$$
$53$ $$1 - 32586 T + 418195493 T^{2}$$
$59$ $$1 - 8668 T + 714924299 T^{2}$$
$61$ $$1 + 34670 T + 844596301 T^{2}$$
$67$ $$1 + 47566 T + 1350125107 T^{2}$$
$71$ $$1 - 948 T + 1804229351 T^{2}$$
$73$ $$1 + 63102 T + 2073071593 T^{2}$$
$79$ $$1 - 46536 T + 3077056399 T^{2}$$
$83$ $$1 + 88778 T + 3939040643 T^{2}$$
$89$ $$1 + 104934 T + 5584059449 T^{2}$$
$97$ $$1 + 36254 T + 8587340257 T^{2}$$