# Properties

 Label 1170.2.a.j Level $1170$ Weight $2$ Character orbit 1170.a Self dual yes Analytic conductor $9.342$ 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] = [1170,2,Mod(1,1170)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1170, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1170.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1170.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: $$9.34249703649$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - q^{5} - 2 q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 - q^5 - 2 * q^7 + q^8 $$q + q^{2} + q^{4} - q^{5} - 2 q^{7} + q^{8} - q^{10} - 4 q^{11} - q^{13} - 2 q^{14} + q^{16} - 4 q^{17} - 2 q^{19} - q^{20} - 4 q^{22} - 2 q^{23} + q^{25} - q^{26} - 2 q^{28} - 8 q^{29} + 4 q^{31} + q^{32} - 4 q^{34} + 2 q^{35} + 6 q^{37} - 2 q^{38} - q^{40} - 10 q^{41} + 4 q^{43} - 4 q^{44} - 2 q^{46} - 3 q^{49} + q^{50} - q^{52} - 6 q^{53} + 4 q^{55} - 2 q^{56} - 8 q^{58} + 12 q^{59} - 2 q^{61} + 4 q^{62} + q^{64} + q^{65} - 8 q^{67} - 4 q^{68} + 2 q^{70} + 6 q^{74} - 2 q^{76} + 8 q^{77} - 8 q^{79} - q^{80} - 10 q^{82} + 12 q^{83} + 4 q^{85} + 4 q^{86} - 4 q^{88} + 10 q^{89} + 2 q^{91} - 2 q^{92} + 2 q^{95} - 8 q^{97} - 3 q^{98}+O(q^{100})$$ q + q^2 + q^4 - q^5 - 2 * q^7 + q^8 - q^10 - 4 * q^11 - q^13 - 2 * q^14 + q^16 - 4 * q^17 - 2 * q^19 - q^20 - 4 * q^22 - 2 * q^23 + q^25 - q^26 - 2 * q^28 - 8 * q^29 + 4 * q^31 + q^32 - 4 * q^34 + 2 * q^35 + 6 * q^37 - 2 * q^38 - q^40 - 10 * q^41 + 4 * q^43 - 4 * q^44 - 2 * q^46 - 3 * q^49 + q^50 - q^52 - 6 * q^53 + 4 * q^55 - 2 * q^56 - 8 * q^58 + 12 * q^59 - 2 * q^61 + 4 * q^62 + q^64 + q^65 - 8 * q^67 - 4 * q^68 + 2 * q^70 + 6 * q^74 - 2 * q^76 + 8 * q^77 - 8 * q^79 - q^80 - 10 * q^82 + 12 * q^83 + 4 * q^85 + 4 * q^86 - 4 * q^88 + 10 * q^89 + 2 * q^91 - 2 * q^92 + 2 * q^95 - 8 * q^97 - 3 * 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
1.00000 0 1.00000 −1.00000 0 −2.00000 1.00000 0 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$13$$ $$1$$

## 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 1170.2.a.j 1
3.b odd 2 1 390.2.a.b 1
4.b odd 2 1 9360.2.a.v 1
5.b even 2 1 5850.2.a.s 1
5.c odd 4 2 5850.2.e.h 2
12.b even 2 1 3120.2.a.y 1
15.d odd 2 1 1950.2.a.ba 1
15.e even 4 2 1950.2.e.m 2
39.d odd 2 1 5070.2.a.n 1
39.f even 4 2 5070.2.b.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.a.b 1 3.b odd 2 1
1170.2.a.j 1 1.a even 1 1 trivial
1950.2.a.ba 1 15.d odd 2 1
1950.2.e.m 2 15.e even 4 2
3120.2.a.y 1 12.b even 2 1
5070.2.a.n 1 39.d odd 2 1
5070.2.b.f 2 39.f even 4 2
5850.2.a.s 1 5.b even 2 1
5850.2.e.h 2 5.c odd 4 2
9360.2.a.v 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1170))$$:

 $$T_{7} + 2$$ T7 + 2 $$T_{11} + 4$$ T11 + 4 $$T_{17} + 4$$ T17 + 4 $$T_{31} - 4$$ T31 - 4

## Hecke characteristic polynomials

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