# Properties

 Label 8281.2.a.e Level $8281$ Weight $2$ Character orbit 8281.a Self dual yes Analytic conductor $66.124$ 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] = [8281,2,Mod(1,8281)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8281, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8281.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8281 = 7^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8281.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: $$66.1241179138$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{4} + 3 q^{8} - 3 q^{9}+O(q^{10})$$ q - q^2 - q^4 + 3 * q^8 - 3 * q^9 $$q - q^{2} - q^{4} + 3 q^{8} - 3 q^{9} + 3 q^{11} - q^{16} - 7 q^{17} + 3 q^{18} - 7 q^{19} - 3 q^{22} - 6 q^{23} - 5 q^{25} - 5 q^{29} - 5 q^{32} + 7 q^{34} + 3 q^{36} - 8 q^{37} + 7 q^{38} + 2 q^{43} - 3 q^{44} + 6 q^{46} + 7 q^{47} + 5 q^{50} - 3 q^{53} + 5 q^{58} - 7 q^{59} + 7 q^{61} + 7 q^{64} + 3 q^{67} + 7 q^{68} + 5 q^{71} - 9 q^{72} + 14 q^{73} + 8 q^{74} + 7 q^{76} - 6 q^{79} + 9 q^{81} - 2 q^{86} + 9 q^{88} + 6 q^{92} - 7 q^{94} - 14 q^{97} - 9 q^{99}+O(q^{100})$$ q - q^2 - q^4 + 3 * q^8 - 3 * q^9 + 3 * q^11 - q^16 - 7 * q^17 + 3 * q^18 - 7 * q^19 - 3 * q^22 - 6 * q^23 - 5 * q^25 - 5 * q^29 - 5 * q^32 + 7 * q^34 + 3 * q^36 - 8 * q^37 + 7 * q^38 + 2 * q^43 - 3 * q^44 + 6 * q^46 + 7 * q^47 + 5 * q^50 - 3 * q^53 + 5 * q^58 - 7 * q^59 + 7 * q^61 + 7 * q^64 + 3 * q^67 + 7 * q^68 + 5 * q^71 - 9 * q^72 + 14 * q^73 + 8 * q^74 + 7 * q^76 - 6 * q^79 + 9 * q^81 - 2 * q^86 + 9 * q^88 + 6 * q^92 - 7 * q^94 - 14 * q^97 - 9 * 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 0 −1.00000 0 0 0 3.00000 −3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$-1$$
$$13$$ $$+1$$

## 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 8281.2.a.e 1
7.b odd 2 1 8281.2.a.f 1
7.d odd 6 2 1183.2.e.b 2
13.b even 2 1 637.2.a.d 1
39.d odd 2 1 5733.2.a.d 1
91.b odd 2 1 637.2.a.c 1
91.r even 6 2 637.2.e.a 2
91.s odd 6 2 91.2.e.a 2
273.g even 2 1 5733.2.a.c 1
273.ba even 6 2 819.2.j.b 2
364.x even 6 2 1456.2.r.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.e.a 2 91.s odd 6 2
637.2.a.c 1 91.b odd 2 1
637.2.a.d 1 13.b even 2 1
637.2.e.a 2 91.r even 6 2
819.2.j.b 2 273.ba even 6 2
1183.2.e.b 2 7.d odd 6 2
1456.2.r.g 2 364.x even 6 2
5733.2.a.c 1 273.g even 2 1
5733.2.a.d 1 39.d odd 2 1
8281.2.a.e 1 1.a even 1 1 trivial
8281.2.a.f 1 7.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8281))$$:

 $$T_{2} + 1$$ T2 + 1 $$T_{3}$$ T3 $$T_{5}$$ T5 $$T_{11} - 3$$ T11 - 3 $$T_{17} + 7$$ T17 + 7

## Hecke characteristic polynomials

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