# Properties

 Label 8085.2.a.w Level $8085$ Weight $2$ Character orbit 8085.a Self dual yes Analytic conductor $64.559$ 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] = [8085,2,Mod(1,8085)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8085, 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("8085.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8085 = 3 \cdot 5 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8085.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: $$64.5590500342$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1155) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} - q^{4} + q^{5} + q^{6} - 3 q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 - q^4 + q^5 + q^6 - 3 * q^8 + q^9 $$q + q^{2} + q^{3} - q^{4} + q^{5} + q^{6} - 3 q^{8} + q^{9} + q^{10} - q^{11} - q^{12} + 2 q^{13} + q^{15} - q^{16} - 6 q^{17} + q^{18} - 4 q^{19} - q^{20} - q^{22} - 4 q^{23} - 3 q^{24} + q^{25} + 2 q^{26} + q^{27} + 10 q^{29} + q^{30} + 4 q^{31} + 5 q^{32} - q^{33} - 6 q^{34} - q^{36} - 2 q^{37} - 4 q^{38} + 2 q^{39} - 3 q^{40} - 10 q^{41} + 12 q^{43} + q^{44} + q^{45} - 4 q^{46} - q^{48} + q^{50} - 6 q^{51} - 2 q^{52} + 10 q^{53} + q^{54} - q^{55} - 4 q^{57} + 10 q^{58} + 12 q^{59} - q^{60} + 2 q^{61} + 4 q^{62} + 7 q^{64} + 2 q^{65} - q^{66} + 4 q^{67} + 6 q^{68} - 4 q^{69} + 8 q^{71} - 3 q^{72} + 14 q^{73} - 2 q^{74} + q^{75} + 4 q^{76} + 2 q^{78} + 4 q^{79} - q^{80} + q^{81} - 10 q^{82} - 8 q^{83} - 6 q^{85} + 12 q^{86} + 10 q^{87} + 3 q^{88} - 6 q^{89} + q^{90} + 4 q^{92} + 4 q^{93} - 4 q^{95} + 5 q^{96} + 10 q^{97} - q^{99}+O(q^{100})$$ q + q^2 + q^3 - q^4 + q^5 + q^6 - 3 * q^8 + q^9 + q^10 - q^11 - q^12 + 2 * q^13 + q^15 - q^16 - 6 * q^17 + q^18 - 4 * q^19 - q^20 - q^22 - 4 * q^23 - 3 * q^24 + q^25 + 2 * q^26 + q^27 + 10 * q^29 + q^30 + 4 * q^31 + 5 * q^32 - q^33 - 6 * q^34 - q^36 - 2 * q^37 - 4 * q^38 + 2 * q^39 - 3 * q^40 - 10 * q^41 + 12 * q^43 + q^44 + q^45 - 4 * q^46 - q^48 + q^50 - 6 * q^51 - 2 * q^52 + 10 * q^53 + q^54 - q^55 - 4 * q^57 + 10 * q^58 + 12 * q^59 - q^60 + 2 * q^61 + 4 * q^62 + 7 * q^64 + 2 * q^65 - q^66 + 4 * q^67 + 6 * q^68 - 4 * q^69 + 8 * q^71 - 3 * q^72 + 14 * q^73 - 2 * q^74 + q^75 + 4 * q^76 + 2 * q^78 + 4 * q^79 - q^80 + q^81 - 10 * q^82 - 8 * q^83 - 6 * q^85 + 12 * q^86 + 10 * q^87 + 3 * q^88 - 6 * q^89 + q^90 + 4 * q^92 + 4 * q^93 - 4 * q^95 + 5 * q^96 + 10 * 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
1.00000 1.00000 −1.00000 1.00000 1.00000 0 −3.00000 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
$$3$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$1$$

## 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 8085.2.a.w 1
7.b odd 2 1 1155.2.a.j 1
21.c even 2 1 3465.2.a.g 1
35.c odd 2 1 5775.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.a.j 1 7.b odd 2 1
3465.2.a.g 1 21.c even 2 1
5775.2.a.f 1 35.c odd 2 1
8085.2.a.w 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8085))$$:

 $$T_{2} - 1$$ T2 - 1 $$T_{13} - 2$$ T13 - 2 $$T_{17} + 6$$ T17 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 1$$
$5$ $$T - 1$$
$7$ $$T$$
$11$ $$T + 1$$
$13$ $$T - 2$$
$17$ $$T + 6$$
$19$ $$T + 4$$
$23$ $$T + 4$$
$29$ $$T - 10$$
$31$ $$T - 4$$
$37$ $$T + 2$$
$41$ $$T + 10$$
$43$ $$T - 12$$
$47$ $$T$$
$53$ $$T - 10$$
$59$ $$T - 12$$
$61$ $$T - 2$$
$67$ $$T - 4$$
$71$ $$T - 8$$
$73$ $$T - 14$$
$79$ $$T - 4$$
$83$ $$T + 8$$
$89$ $$T + 6$$
$97$ $$T - 10$$