# Properties

 Label 9.10.a.a Level 9 Weight 10 Character orbit 9.a Self dual yes Analytic conductor 4.635 Analytic rank 0 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$9 = 3^{2}$$ Weight: $$k$$ = $$10$$ Character orbit: $$[\chi]$$ = 9.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.63532252547$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 18q^{2} - 188q^{4} + 1530q^{5} + 9128q^{7} + 12600q^{8} + O(q^{10})$$ $$q - 18q^{2} - 188q^{4} + 1530q^{5} + 9128q^{7} + 12600q^{8} - 27540q^{10} - 21132q^{11} + 31214q^{13} - 164304q^{14} - 130544q^{16} + 279342q^{17} + 144020q^{19} - 287640q^{20} + 380376q^{22} + 1763496q^{23} + 387775q^{25} - 561852q^{26} - 1716064q^{28} - 4692510q^{29} - 369088q^{31} - 4101408q^{32} - 5028156q^{34} + 13965840q^{35} + 9347078q^{37} - 2592360q^{38} + 19278000q^{40} + 7226838q^{41} - 23147476q^{43} + 3972816q^{44} - 31742928q^{46} - 22971888q^{47} + 42966777q^{49} - 6979950q^{50} - 5868232q^{52} - 78477174q^{53} - 32331960q^{55} + 115012800q^{56} + 84465180q^{58} + 20310660q^{59} - 179339938q^{61} + 6643584q^{62} + 140663872q^{64} + 47757420q^{65} + 274528388q^{67} - 52516296q^{68} - 251385120q^{70} + 36342648q^{71} - 247089526q^{73} - 168247404q^{74} - 27075760q^{76} - 192892896q^{77} + 191874800q^{79} - 199732320q^{80} - 130083084q^{82} + 276159276q^{83} + 427393260q^{85} + 416654568q^{86} - 266263200q^{88} + 678997350q^{89} + 284921392q^{91} - 331537248q^{92} + 413493984q^{94} + 220350600q^{95} - 567657502q^{97} - 773401986q^{98} + O(q^{100})$$

## 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
−18.0000 0 −188.000 1530.00 0 9128.00 12600.0 0 −27540.0
 $$n$$: 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 9.10.a.a 1
3.b odd 2 1 3.10.a.b 1
4.b odd 2 1 144.10.a.m 1
5.b even 2 1 225.10.a.e 1
5.c odd 4 2 225.10.b.c 2
9.c even 3 2 81.10.c.d 2
9.d odd 6 2 81.10.c.b 2
12.b even 2 1 48.10.a.a 1
15.d odd 2 1 75.10.a.b 1
15.e even 4 2 75.10.b.c 2
21.c even 2 1 147.10.a.c 1
24.f even 2 1 192.10.a.n 1
24.h odd 2 1 192.10.a.g 1
33.d even 2 1 363.10.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.10.a.b 1 3.b odd 2 1
9.10.a.a 1 1.a even 1 1 trivial
48.10.a.a 1 12.b even 2 1
75.10.a.b 1 15.d odd 2 1
75.10.b.c 2 15.e even 4 2
81.10.c.b 2 9.d odd 6 2
81.10.c.d 2 9.c even 3 2
144.10.a.m 1 4.b odd 2 1
147.10.a.c 1 21.c even 2 1
192.10.a.g 1 24.h odd 2 1
192.10.a.n 1 24.f even 2 1
225.10.a.e 1 5.b even 2 1
225.10.b.c 2 5.c odd 4 2
363.10.a.a 1 33.d even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 18$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(9))$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 18 T + 512 T^{2}$$
$3$ 
$5$ $$1 - 1530 T + 1953125 T^{2}$$
$7$ $$1 - 9128 T + 40353607 T^{2}$$
$11$ $$1 + 21132 T + 2357947691 T^{2}$$
$13$ $$1 - 31214 T + 10604499373 T^{2}$$
$17$ $$1 - 279342 T + 118587876497 T^{2}$$
$19$ $$1 - 144020 T + 322687697779 T^{2}$$
$23$ $$1 - 1763496 T + 1801152661463 T^{2}$$
$29$ $$1 + 4692510 T + 14507145975869 T^{2}$$
$31$ $$1 + 369088 T + 26439622160671 T^{2}$$
$37$ $$1 - 9347078 T + 129961739795077 T^{2}$$
$41$ $$1 - 7226838 T + 327381934393961 T^{2}$$
$43$ $$1 + 23147476 T + 502592611936843 T^{2}$$
$47$ $$1 + 22971888 T + 1119130473102767 T^{2}$$
$53$ $$1 + 78477174 T + 3299763591802133 T^{2}$$
$59$ $$1 - 20310660 T + 8662995818654939 T^{2}$$
$61$ $$1 + 179339938 T + 11694146092834141 T^{2}$$
$67$ $$1 - 274528388 T + 27206534396294947 T^{2}$$
$71$ $$1 - 36342648 T + 45848500718449031 T^{2}$$
$73$ $$1 + 247089526 T + 58871586708267913 T^{2}$$
$79$ $$1 - 191874800 T + 119851595982618319 T^{2}$$
$83$ $$1 - 276159276 T + 186940255267540403 T^{2}$$
$89$ $$1 - 678997350 T + 350356403707485209 T^{2}$$
$97$ $$1 + 567657502 T + 760231058654565217 T^{2}$$