# Properties

 Label 9555.2.a.v Level $9555$ Weight $2$ Character orbit 9555.a Self dual yes Analytic conductor $76.297$ Analytic rank $0$ 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] = [9555,2,Mod(1,9555)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9555, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("9555.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9555 = 3 \cdot 5 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9555.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: $$76.2970591313$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 195) 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + q^{3} + 2 q^{4} - q^{5} + 2 q^{6} + q^{9}+O(q^{10})$$ q + 2 * q^2 + q^3 + 2 * q^4 - q^5 + 2 * q^6 + q^9 $$q + 2 q^{2} + q^{3} + 2 q^{4} - q^{5} + 2 q^{6} + q^{9} - 2 q^{10} - q^{11} + 2 q^{12} + q^{13} - q^{15} - 4 q^{16} + q^{17} + 2 q^{18} + 2 q^{19} - 2 q^{20} - 2 q^{22} - 3 q^{23} + q^{25} + 2 q^{26} + q^{27} - 2 q^{29} - 2 q^{30} + 6 q^{31} - 8 q^{32} - q^{33} + 2 q^{34} + 2 q^{36} + 11 q^{37} + 4 q^{38} + q^{39} + 5 q^{41} + 4 q^{43} - 2 q^{44} - q^{45} - 6 q^{46} + 10 q^{47} - 4 q^{48} + 2 q^{50} + q^{51} + 2 q^{52} + 11 q^{53} + 2 q^{54} + q^{55} + 2 q^{57} - 4 q^{58} - 8 q^{59} - 2 q^{60} - 13 q^{61} + 12 q^{62} - 8 q^{64} - q^{65} - 2 q^{66} + 12 q^{67} + 2 q^{68} - 3 q^{69} - 5 q^{71} - 10 q^{73} + 22 q^{74} + q^{75} + 4 q^{76} + 2 q^{78} - 3 q^{79} + 4 q^{80} + q^{81} + 10 q^{82} + 12 q^{83} - q^{85} + 8 q^{86} - 2 q^{87} + 15 q^{89} - 2 q^{90} - 6 q^{92} + 6 q^{93} + 20 q^{94} - 2 q^{95} - 8 q^{96} - 17 q^{97} - q^{99}+O(q^{100})$$ q + 2 * q^2 + q^3 + 2 * q^4 - q^5 + 2 * q^6 + q^9 - 2 * q^10 - q^11 + 2 * q^12 + q^13 - q^15 - 4 * q^16 + q^17 + 2 * q^18 + 2 * q^19 - 2 * q^20 - 2 * q^22 - 3 * q^23 + q^25 + 2 * q^26 + q^27 - 2 * q^29 - 2 * q^30 + 6 * q^31 - 8 * q^32 - q^33 + 2 * q^34 + 2 * q^36 + 11 * q^37 + 4 * q^38 + q^39 + 5 * q^41 + 4 * q^43 - 2 * q^44 - q^45 - 6 * q^46 + 10 * q^47 - 4 * q^48 + 2 * q^50 + q^51 + 2 * q^52 + 11 * q^53 + 2 * q^54 + q^55 + 2 * q^57 - 4 * q^58 - 8 * q^59 - 2 * q^60 - 13 * q^61 + 12 * q^62 - 8 * q^64 - q^65 - 2 * q^66 + 12 * q^67 + 2 * q^68 - 3 * q^69 - 5 * q^71 - 10 * q^73 + 22 * q^74 + q^75 + 4 * q^76 + 2 * q^78 - 3 * q^79 + 4 * q^80 + q^81 + 10 * q^82 + 12 * q^83 - q^85 + 8 * q^86 - 2 * q^87 + 15 * q^89 - 2 * q^90 - 6 * q^92 + 6 * q^93 + 20 * q^94 - 2 * q^95 - 8 * q^96 - 17 * q^97 - q^99

## 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
2.00000 1.00000 2.00000 −1.00000 2.00000 0 0 1.00000 −2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$-1$$
$$13$$ $$-1$$

## 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 9555.2.a.v 1
7.b odd 2 1 195.2.a.b 1
21.c even 2 1 585.2.a.b 1
28.d even 2 1 3120.2.a.u 1
35.c odd 2 1 975.2.a.c 1
35.f even 4 2 975.2.c.a 2
84.h odd 2 1 9360.2.a.d 1
91.b odd 2 1 2535.2.a.a 1
105.g even 2 1 2925.2.a.q 1
105.k odd 4 2 2925.2.c.c 2
273.g even 2 1 7605.2.a.u 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.a.b 1 7.b odd 2 1
585.2.a.b 1 21.c even 2 1
975.2.a.c 1 35.c odd 2 1
975.2.c.a 2 35.f even 4 2
2535.2.a.a 1 91.b odd 2 1
2925.2.a.q 1 105.g even 2 1
2925.2.c.c 2 105.k odd 4 2
3120.2.a.u 1 28.d even 2 1
7605.2.a.u 1 273.g even 2 1
9360.2.a.d 1 84.h odd 2 1
9555.2.a.v 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9555))$$:

 $$T_{2} - 2$$ T2 - 2 $$T_{11} + 1$$ T11 + 1 $$T_{17} - 1$$ T17 - 1 $$T_{19} - 2$$ T19 - 2 $$T_{23} + 3$$ T23 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T - 1$$
$5$ $$T + 1$$
$7$ $$T$$
$11$ $$T + 1$$
$13$ $$T - 1$$
$17$ $$T - 1$$
$19$ $$T - 2$$
$23$ $$T + 3$$
$29$ $$T + 2$$
$31$ $$T - 6$$
$37$ $$T - 11$$
$41$ $$T - 5$$
$43$ $$T - 4$$
$47$ $$T - 10$$
$53$ $$T - 11$$
$59$ $$T + 8$$
$61$ $$T + 13$$
$67$ $$T - 12$$
$71$ $$T + 5$$
$73$ $$T + 10$$
$79$ $$T + 3$$
$83$ $$T - 12$$
$89$ $$T - 15$$
$97$ $$T + 17$$