# Properties

 Label 10.6.a.c Level 10 Weight 6 Character orbit 10.a Self dual yes Analytic conductor 1.604 Analytic rank 0 Dimension 1 CM no Inner twists 1

# Learn more about

## 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: $$0$$ 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

 $$f(q)$$ $$=$$ $$q + 4q^{2} + 6q^{3} + 16q^{4} - 25q^{5} + 24q^{6} - 118q^{7} + 64q^{8} - 207q^{9} + O(q^{10})$$ $$q + 4q^{2} + 6q^{3} + 16q^{4} - 25q^{5} + 24q^{6} - 118q^{7} + 64q^{8} - 207q^{9} - 100q^{10} + 192q^{11} + 96q^{12} + 1106q^{13} - 472q^{14} - 150q^{15} + 256q^{16} + 762q^{17} - 828q^{18} - 2740q^{19} - 400q^{20} - 708q^{21} + 768q^{22} + 1566q^{23} + 384q^{24} + 625q^{25} + 4424q^{26} - 2700q^{27} - 1888q^{28} + 5910q^{29} - 600q^{30} - 6868q^{31} + 1024q^{32} + 1152q^{33} + 3048q^{34} + 2950q^{35} - 3312q^{36} - 5518q^{37} - 10960q^{38} + 6636q^{39} - 1600q^{40} - 378q^{41} - 2832q^{42} - 2434q^{43} + 3072q^{44} + 5175q^{45} + 6264q^{46} + 13122q^{47} + 1536q^{48} - 2883q^{49} + 2500q^{50} + 4572q^{51} + 17696q^{52} - 9174q^{53} - 10800q^{54} - 4800q^{55} - 7552q^{56} - 16440q^{57} + 23640q^{58} - 34980q^{59} - 2400q^{60} - 9838q^{61} - 27472q^{62} + 24426q^{63} + 4096q^{64} - 27650q^{65} + 4608q^{66} + 33722q^{67} + 12192q^{68} + 9396q^{69} + 11800q^{70} + 70212q^{71} - 13248q^{72} + 21986q^{73} - 22072q^{74} + 3750q^{75} - 43840q^{76} - 22656q^{77} + 26544q^{78} + 4520q^{79} - 6400q^{80} + 34101q^{81} - 1512q^{82} - 109074q^{83} - 11328q^{84} - 19050q^{85} - 9736q^{86} + 35460q^{87} + 12288q^{88} + 38490q^{89} + 20700q^{90} - 130508q^{91} + 25056q^{92} - 41208q^{93} + 52488q^{94} + 68500q^{95} + 6144q^{96} - 1918q^{97} - 11532q^{98} - 39744q^{99} + O(q^{100})$$

## 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 6.00000 16.0000 −25.0000 24.0000 −118.000 64.0000 −207.000 −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.c 1
3.b odd 2 1 90.6.a.b 1
4.b odd 2 1 80.6.a.c 1
5.b even 2 1 50.6.a.b 1
5.c odd 4 2 50.6.b.b 2
7.b odd 2 1 490.6.a.k 1
8.b even 2 1 320.6.a.f 1
8.d odd 2 1 320.6.a.k 1
12.b even 2 1 720.6.a.v 1
15.d odd 2 1 450.6.a.u 1
15.e even 4 2 450.6.c.f 2
20.d odd 2 1 400.6.a.i 1
20.e even 4 2 400.6.c.i 2

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 4 T$$
$3$ $$1 - 6 T + 243 T^{2}$$
$5$ $$1 + 25 T$$
$7$ $$1 + 118 T + 16807 T^{2}$$
$11$ $$1 - 192 T + 161051 T^{2}$$
$13$ $$1 - 1106 T + 371293 T^{2}$$
$17$ $$1 - 762 T + 1419857 T^{2}$$
$19$ $$1 + 2740 T + 2476099 T^{2}$$
$23$ $$1 - 1566 T + 6436343 T^{2}$$
$29$ $$1 - 5910 T + 20511149 T^{2}$$
$31$ $$1 + 6868 T + 28629151 T^{2}$$
$37$ $$1 + 5518 T + 69343957 T^{2}$$
$41$ $$1 + 378 T + 115856201 T^{2}$$
$43$ $$1 + 2434 T + 147008443 T^{2}$$
$47$ $$1 - 13122 T + 229345007 T^{2}$$
$53$ $$1 + 9174 T + 418195493 T^{2}$$
$59$ $$1 + 34980 T + 714924299 T^{2}$$
$61$ $$1 + 9838 T + 844596301 T^{2}$$
$67$ $$1 - 33722 T + 1350125107 T^{2}$$
$71$ $$1 - 70212 T + 1804229351 T^{2}$$
$73$ $$1 - 21986 T + 2073071593 T^{2}$$
$79$ $$1 - 4520 T + 3077056399 T^{2}$$
$83$ $$1 + 109074 T + 3939040643 T^{2}$$
$89$ $$1 - 38490 T + 5584059449 T^{2}$$
$97$ $$1 + 1918 T + 8587340257 T^{2}$$
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