# Properties

 Label 40.6.a.c Level 40 Weight 6 Character orbit 40.a Self dual yes Analytic conductor 6.415 Analytic rank 1 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: $$1$$ 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

 $$f(q)$$ $$=$$ $$q - 2q^{3} - 25q^{5} - 62q^{7} - 239q^{9} + O(q^{10})$$ $$q - 2q^{3} - 25q^{5} - 62q^{7} - 239q^{9} - 144q^{11} - 654q^{13} + 50q^{15} - 1190q^{17} + 556q^{19} + 124q^{21} + 2182q^{23} + 625q^{25} + 964q^{27} - 1578q^{29} + 9660q^{31} + 288q^{33} + 1550q^{35} - 3534q^{37} + 1308q^{39} + 7462q^{41} - 7114q^{43} + 5975q^{45} - 28294q^{47} - 12963q^{49} + 2380q^{51} - 13046q^{53} + 3600q^{55} - 1112q^{57} - 37092q^{59} + 39570q^{61} + 14818q^{63} + 16350q^{65} - 56734q^{67} - 4364q^{69} + 45588q^{71} + 11842q^{73} - 1250q^{75} + 8928q^{77} + 94216q^{79} + 56149q^{81} - 31482q^{83} + 29750q^{85} + 3156q^{87} - 94054q^{89} + 40548q^{91} - 19320q^{93} - 13900q^{95} + 23714q^{97} + 34416q^{99} + O(q^{100})$$

## 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 −2.00000 0 −25.0000 0 −62.0000 0 −239.000 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.c 1
3.b odd 2 1 360.6.a.f 1
4.b odd 2 1 80.6.a.d 1
5.b even 2 1 200.6.a.b 1
5.c odd 4 2 200.6.c.d 2
8.b even 2 1 320.6.a.i 1
8.d odd 2 1 320.6.a.h 1
12.b even 2 1 720.6.a.t 1
20.d odd 2 1 400.6.a.h 1
20.e even 4 2 400.6.c.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.a.c 1 1.a even 1 1 trivial
80.6.a.d 1 4.b odd 2 1
200.6.a.b 1 5.b even 2 1
200.6.c.d 2 5.c odd 4 2
320.6.a.h 1 8.d odd 2 1
320.6.a.i 1 8.b even 2 1
360.6.a.f 1 3.b odd 2 1
400.6.a.h 1 20.d odd 2 1
400.6.c.k 2 20.e even 4 2
720.6.a.t 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 $$T_{3} + 2$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(40))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 2 T + 243 T^{2}$$
$5$ $$1 + 25 T$$
$7$ $$1 + 62 T + 16807 T^{2}$$
$11$ $$1 + 144 T + 161051 T^{2}$$
$13$ $$1 + 654 T + 371293 T^{2}$$
$17$ $$1 + 1190 T + 1419857 T^{2}$$
$19$ $$1 - 556 T + 2476099 T^{2}$$
$23$ $$1 - 2182 T + 6436343 T^{2}$$
$29$ $$1 + 1578 T + 20511149 T^{2}$$
$31$ $$1 - 9660 T + 28629151 T^{2}$$
$37$ $$1 + 3534 T + 69343957 T^{2}$$
$41$ $$1 - 7462 T + 115856201 T^{2}$$
$43$ $$1 + 7114 T + 147008443 T^{2}$$
$47$ $$1 + 28294 T + 229345007 T^{2}$$
$53$ $$1 + 13046 T + 418195493 T^{2}$$
$59$ $$1 + 37092 T + 714924299 T^{2}$$
$61$ $$1 - 39570 T + 844596301 T^{2}$$
$67$ $$1 + 56734 T + 1350125107 T^{2}$$
$71$ $$1 - 45588 T + 1804229351 T^{2}$$
$73$ $$1 - 11842 T + 2073071593 T^{2}$$
$79$ $$1 - 94216 T + 3077056399 T^{2}$$
$83$ $$1 + 31482 T + 3939040643 T^{2}$$
$89$ $$1 + 94054 T + 5584059449 T^{2}$$
$97$ $$1 - 23714 T + 8587340257 T^{2}$$