# Properties

 Label 8670.2.a.r Level $8670$ Weight $2$ Character orbit 8670.a Self dual yes Analytic conductor $69.230$ 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] = [8670,2,Mod(1,8670)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8670, 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("8670.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8670.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: $$69.2302985525$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 510) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 - q^3 + q^4 + q^5 - q^6 + q^8 + q^9 $$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{8} + q^{9} + q^{10} - 4 q^{11} - q^{12} + 2 q^{13} - q^{15} + q^{16} + q^{18} - 4 q^{19} + q^{20} - 4 q^{22} - 4 q^{23} - q^{24} + q^{25} + 2 q^{26} - q^{27} - 2 q^{29} - q^{30} + 4 q^{31} + q^{32} + 4 q^{33} + q^{36} + 6 q^{37} - 4 q^{38} - 2 q^{39} + q^{40} + 10 q^{41} - 8 q^{43} - 4 q^{44} + q^{45} - 4 q^{46} - q^{48} - 7 q^{49} + q^{50} + 2 q^{52} + 6 q^{53} - q^{54} - 4 q^{55} + 4 q^{57} - 2 q^{58} - 8 q^{59} - q^{60} - 10 q^{61} + 4 q^{62} + q^{64} + 2 q^{65} + 4 q^{66} - 8 q^{67} + 4 q^{69} - 8 q^{71} + q^{72} + 2 q^{73} + 6 q^{74} - q^{75} - 4 q^{76} - 2 q^{78} - 4 q^{79} + q^{80} + q^{81} + 10 q^{82} + 4 q^{83} - 8 q^{86} + 2 q^{87} - 4 q^{88} - 14 q^{89} + q^{90} - 4 q^{92} - 4 q^{93} - 4 q^{95} - q^{96} + 10 q^{97} - 7 q^{98} - 4 q^{99}+O(q^{100})$$ q + q^2 - q^3 + q^4 + q^5 - q^6 + q^8 + q^9 + q^10 - 4 * q^11 - q^12 + 2 * q^13 - q^15 + q^16 + q^18 - 4 * q^19 + q^20 - 4 * q^22 - 4 * q^23 - q^24 + q^25 + 2 * q^26 - q^27 - 2 * q^29 - q^30 + 4 * q^31 + q^32 + 4 * q^33 + q^36 + 6 * q^37 - 4 * q^38 - 2 * q^39 + q^40 + 10 * q^41 - 8 * q^43 - 4 * q^44 + q^45 - 4 * q^46 - q^48 - 7 * q^49 + q^50 + 2 * q^52 + 6 * q^53 - q^54 - 4 * q^55 + 4 * q^57 - 2 * q^58 - 8 * q^59 - q^60 - 10 * q^61 + 4 * q^62 + q^64 + 2 * q^65 + 4 * q^66 - 8 * q^67 + 4 * q^69 - 8 * q^71 + q^72 + 2 * q^73 + 6 * q^74 - q^75 - 4 * q^76 - 2 * q^78 - 4 * q^79 + q^80 + q^81 + 10 * q^82 + 4 * q^83 - 8 * q^86 + 2 * q^87 - 4 * q^88 - 14 * q^89 + q^90 - 4 * q^92 - 4 * q^93 - 4 * q^95 - q^96 + 10 * q^97 - 7 * q^98 - 4 * 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
1.00000 −1.00000 1.00000 1.00000 −1.00000 0 1.00000 1.00000 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$-1$$
$$17$$ $$1$$

## 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 8670.2.a.r 1
17.b even 2 1 510.2.a.f 1
51.c odd 2 1 1530.2.a.f 1
68.d odd 2 1 4080.2.a.c 1
85.c even 2 1 2550.2.a.d 1
85.g odd 4 2 2550.2.d.s 2
255.h odd 2 1 7650.2.a.bw 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.a.f 1 17.b even 2 1
1530.2.a.f 1 51.c odd 2 1
2550.2.a.d 1 85.c even 2 1
2550.2.d.s 2 85.g odd 4 2
4080.2.a.c 1 68.d odd 2 1
7650.2.a.bw 1 255.h odd 2 1
8670.2.a.r 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8670))$$:

 $$T_{7}$$ T7 $$T_{11} + 4$$ T11 + 4 $$T_{13} - 2$$ T13 - 2 $$T_{23} + 4$$ T23 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 1$$
$5$ $$T - 1$$
$7$ $$T$$
$11$ $$T + 4$$
$13$ $$T - 2$$
$17$ $$T$$
$19$ $$T + 4$$
$23$ $$T + 4$$
$29$ $$T + 2$$
$31$ $$T - 4$$
$37$ $$T - 6$$
$41$ $$T - 10$$
$43$ $$T + 8$$
$47$ $$T$$
$53$ $$T - 6$$
$59$ $$T + 8$$
$61$ $$T + 10$$
$67$ $$T + 8$$
$71$ $$T + 8$$
$73$ $$T - 2$$
$79$ $$T + 4$$
$83$ $$T - 4$$
$89$ $$T + 14$$
$97$ $$T - 10$$