# Properties

 Label 9.16.a.a Level $9$ Weight $16$ Character orbit 9.a Self dual yes Analytic conductor $12.842$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9,16,Mod(1,9)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 16, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9.1");

S:= CuspForms(chi, 16);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9 = 3^{2}$$ Weight: $$k$$ $$=$$ $$16$$ Character orbit: $$[\chi]$$ $$=$$ 9.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$12.8424154590$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 216 q^{2} + 13888 q^{4} - 52110 q^{5} + 2822456 q^{7} + 4078080 q^{8}+O(q^{10})$$ q - 216 * q^2 + 13888 * q^4 - 52110 * q^5 + 2822456 * q^7 + 4078080 * q^8 $$q - 216 q^{2} + 13888 q^{4} - 52110 q^{5} + 2822456 q^{7} + 4078080 q^{8} + 11255760 q^{10} - 20586852 q^{11} - 190073338 q^{13} - 609650496 q^{14} - 1335947264 q^{16} - 1646527986 q^{17} + 1563257180 q^{19} - 723703680 q^{20} + 4446760032 q^{22} - 9451116072 q^{23} - 27802126025 q^{25} + 41055841008 q^{26} + 39198268928 q^{28} + 36902568330 q^{29} + 71588483552 q^{31} + 154934083584 q^{32} + 355650044976 q^{34} - 147078182160 q^{35} - 1033652081554 q^{37} - 337663550880 q^{38} - 212508748800 q^{40} - 1641974018202 q^{41} - 492403109308 q^{43} - 285910200576 q^{44} + 2041441071552 q^{46} + 3410684952624 q^{47} + 3218696361993 q^{49} + 6005259221400 q^{50} - 2639738518144 q^{52} - 6797151655902 q^{53} + 1072780857720 q^{55} + 11510201364480 q^{56} - 7970954759280 q^{58} - 9858856815540 q^{59} + 4931842626902 q^{61} - 15463112447232 q^{62} + 10310557892608 q^{64} + 9904721643180 q^{65} - 28837826625364 q^{67} - 22866980669568 q^{68} + 31768887346560 q^{70} - 125050114914552 q^{71} - 82171455513478 q^{73} + 223268849615664 q^{74} + 21710515715840 q^{76} - 58105483948512 q^{77} - 25413078694480 q^{79} + 69616211927040 q^{80} + 354666387931632 q^{82} + 281736730890468 q^{83} + 85800573350460 q^{85} + 106359071610528 q^{86} - 83954829404160 q^{88} - 715618564776810 q^{89} - 536473633278128 q^{91} - 131257100007936 q^{92} - 736707949766784 q^{94} - 81461331649800 q^{95} + 612786136081826 q^{97} - 695238414190488 q^{98}+O(q^{100})$$ q - 216 * q^2 + 13888 * q^4 - 52110 * q^5 + 2822456 * q^7 + 4078080 * q^8 + 11255760 * q^10 - 20586852 * q^11 - 190073338 * q^13 - 609650496 * q^14 - 1335947264 * q^16 - 1646527986 * q^17 + 1563257180 * q^19 - 723703680 * q^20 + 4446760032 * q^22 - 9451116072 * q^23 - 27802126025 * q^25 + 41055841008 * q^26 + 39198268928 * q^28 + 36902568330 * q^29 + 71588483552 * q^31 + 154934083584 * q^32 + 355650044976 * q^34 - 147078182160 * q^35 - 1033652081554 * q^37 - 337663550880 * q^38 - 212508748800 * q^40 - 1641974018202 * q^41 - 492403109308 * q^43 - 285910200576 * q^44 + 2041441071552 * q^46 + 3410684952624 * q^47 + 3218696361993 * q^49 + 6005259221400 * q^50 - 2639738518144 * q^52 - 6797151655902 * q^53 + 1072780857720 * q^55 + 11510201364480 * q^56 - 7970954759280 * q^58 - 9858856815540 * q^59 + 4931842626902 * q^61 - 15463112447232 * q^62 + 10310557892608 * q^64 + 9904721643180 * q^65 - 28837826625364 * q^67 - 22866980669568 * q^68 + 31768887346560 * q^70 - 125050114914552 * q^71 - 82171455513478 * q^73 + 223268849615664 * q^74 + 21710515715840 * q^76 - 58105483948512 * q^77 - 25413078694480 * q^79 + 69616211927040 * q^80 + 354666387931632 * q^82 + 281736730890468 * q^83 + 85800573350460 * q^85 + 106359071610528 * q^86 - 83954829404160 * q^88 - 715618564776810 * q^89 - 536473633278128 * q^91 - 131257100007936 * q^92 - 736707949766784 * q^94 - 81461331649800 * q^95 + 612786136081826 * q^97 - 695238414190488 * q^98

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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
−216.000 0 13888.0 −52110.0 0 2.82246e6 4.07808e6 0 1.12558e7
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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 9.16.a.a 1
3.b odd 2 1 1.16.a.a 1
4.b odd 2 1 144.16.a.f 1
12.b even 2 1 16.16.a.d 1
15.d odd 2 1 25.16.a.a 1
15.e even 4 2 25.16.b.a 2
21.c even 2 1 49.16.a.a 1
21.g even 6 2 49.16.c.b 2
21.h odd 6 2 49.16.c.c 2
24.f even 2 1 64.16.a.c 1
24.h odd 2 1 64.16.a.i 1
33.d even 2 1 121.16.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.16.a.a 1 3.b odd 2 1
9.16.a.a 1 1.a even 1 1 trivial
16.16.a.d 1 12.b even 2 1
25.16.a.a 1 15.d odd 2 1
25.16.b.a 2 15.e even 4 2
49.16.a.a 1 21.c even 2 1
49.16.c.b 2 21.g even 6 2
49.16.c.c 2 21.h odd 6 2
64.16.a.c 1 24.f even 2 1
64.16.a.i 1 24.h odd 2 1
121.16.a.a 1 33.d even 2 1
144.16.a.f 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 216$$ acting on $$S_{16}^{\mathrm{new}}(\Gamma_0(9))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 216$$
$3$ $$T$$
$5$ $$T + 52110$$
$7$ $$T - 2822456$$
$11$ $$T + 20586852$$
$13$ $$T + 190073338$$
$17$ $$T + 1646527986$$
$19$ $$T - 1563257180$$
$23$ $$T + 9451116072$$
$29$ $$T - 36902568330$$
$31$ $$T - 71588483552$$
$37$ $$T + 1033652081554$$
$41$ $$T + 1641974018202$$
$43$ $$T + 492403109308$$
$47$ $$T - 3410684952624$$
$53$ $$T + 6797151655902$$
$59$ $$T + 9858856815540$$
$61$ $$T - 4931842626902$$
$67$ $$T + 28837826625364$$
$71$ $$T + 125050114914552$$
$73$ $$T + 82171455513478$$
$79$ $$T + 25413078694480$$
$83$ $$T - 281736730890468$$
$89$ $$T + 715618564776810$$
$97$ $$T - 612786136081826$$