# Properties

 Label 10.6.a.a Level 10 Weight 6 Character orbit 10.a Self dual yes Analytic conductor 1.604 Analytic rank 1 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$10 = 2 \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 10.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.60383819813$$ 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

 $$f(q)$$ $$=$$ $$q - 4q^{2} - 26q^{3} + 16q^{4} - 25q^{5} + 104q^{6} - 22q^{7} - 64q^{8} + 433q^{9} + O(q^{10})$$ $$q - 4q^{2} - 26q^{3} + 16q^{4} - 25q^{5} + 104q^{6} - 22q^{7} - 64q^{8} + 433q^{9} + 100q^{10} - 768q^{11} - 416q^{12} - 46q^{13} + 88q^{14} + 650q^{15} + 256q^{16} + 378q^{17} - 1732q^{18} + 1100q^{19} - 400q^{20} + 572q^{21} + 3072q^{22} - 1986q^{23} + 1664q^{24} + 625q^{25} + 184q^{26} - 4940q^{27} - 352q^{28} - 5610q^{29} - 2600q^{30} - 3988q^{31} - 1024q^{32} + 19968q^{33} - 1512q^{34} + 550q^{35} + 6928q^{36} - 142q^{37} - 4400q^{38} + 1196q^{39} + 1600q^{40} + 1542q^{41} - 2288q^{42} - 5026q^{43} - 12288q^{44} - 10825q^{45} + 7944q^{46} + 24738q^{47} - 6656q^{48} - 16323q^{49} - 2500q^{50} - 9828q^{51} - 736q^{52} - 14166q^{53} + 19760q^{54} + 19200q^{55} + 1408q^{56} - 28600q^{57} + 22440q^{58} + 28380q^{59} + 10400q^{60} + 5522q^{61} + 15952q^{62} - 9526q^{63} + 4096q^{64} + 1150q^{65} - 79872q^{66} - 24742q^{67} + 6048q^{68} + 51636q^{69} - 2200q^{70} + 42372q^{71} - 27712q^{72} - 52126q^{73} + 568q^{74} - 16250q^{75} + 17600q^{76} + 16896q^{77} - 4784q^{78} - 39640q^{79} - 6400q^{80} + 23221q^{81} - 6168q^{82} - 59826q^{83} + 9152q^{84} - 9450q^{85} + 20104q^{86} + 145860q^{87} + 49152q^{88} + 57690q^{89} + 43300q^{90} + 1012q^{91} - 31776q^{92} + 103688q^{93} - 98952q^{94} - 27500q^{95} + 26624q^{96} - 144382q^{97} + 65292q^{98} - 332544q^{99} + O(q^{100})$$

## 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
−4.00000 −26.0000 16.0000 −25.0000 104.000 −22.0000 −64.0000 433.000 100.000
 $$n$$: 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 10.6.a.a 1
3.b odd 2 1 90.6.a.f 1
4.b odd 2 1 80.6.a.h 1
5.b even 2 1 50.6.a.g 1
5.c odd 4 2 50.6.b.d 2
7.b odd 2 1 490.6.a.j 1
8.b even 2 1 320.6.a.p 1
8.d odd 2 1 320.6.a.a 1
12.b even 2 1 720.6.a.r 1
15.d odd 2 1 450.6.a.h 1
15.e even 4 2 450.6.c.o 2
20.d odd 2 1 400.6.a.a 1
20.e even 4 2 400.6.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.a.a 1 1.a even 1 1 trivial
50.6.a.g 1 5.b even 2 1
50.6.b.d 2 5.c odd 4 2
80.6.a.h 1 4.b odd 2 1
90.6.a.f 1 3.b odd 2 1
320.6.a.a 1 8.d odd 2 1
320.6.a.p 1 8.b even 2 1
400.6.a.a 1 20.d odd 2 1
400.6.c.a 2 20.e even 4 2
450.6.a.h 1 15.d odd 2 1
450.6.c.o 2 15.e even 4 2
490.6.a.j 1 7.b odd 2 1
720.6.a.r 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 $$T_{3} + 26$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(10))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T$$
$3$ $$1 + 26 T + 243 T^{2}$$
$5$ $$1 + 25 T$$
$7$ $$1 + 22 T + 16807 T^{2}$$
$11$ $$1 + 768 T + 161051 T^{2}$$
$13$ $$1 + 46 T + 371293 T^{2}$$
$17$ $$1 - 378 T + 1419857 T^{2}$$
$19$ $$1 - 1100 T + 2476099 T^{2}$$
$23$ $$1 + 1986 T + 6436343 T^{2}$$
$29$ $$1 + 5610 T + 20511149 T^{2}$$
$31$ $$1 + 3988 T + 28629151 T^{2}$$
$37$ $$1 + 142 T + 69343957 T^{2}$$
$41$ $$1 - 1542 T + 115856201 T^{2}$$
$43$ $$1 + 5026 T + 147008443 T^{2}$$
$47$ $$1 - 24738 T + 229345007 T^{2}$$
$53$ $$1 + 14166 T + 418195493 T^{2}$$
$59$ $$1 - 28380 T + 714924299 T^{2}$$
$61$ $$1 - 5522 T + 844596301 T^{2}$$
$67$ $$1 + 24742 T + 1350125107 T^{2}$$
$71$ $$1 - 42372 T + 1804229351 T^{2}$$
$73$ $$1 + 52126 T + 2073071593 T^{2}$$
$79$ $$1 + 39640 T + 3077056399 T^{2}$$
$83$ $$1 + 59826 T + 3939040643 T^{2}$$
$89$ $$1 - 57690 T + 5584059449 T^{2}$$
$97$ $$1 + 144382 T + 8587340257 T^{2}$$