# Properties

 Label 8281.2.a.l 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 2 q^{4} - 3 q^{5} - 3 q^{9}+O(q^{10})$$ q + 2 * q^2 + 2 * q^4 - 3 * q^5 - 3 * q^9 $$q + 2 q^{2} + 2 q^{4} - 3 q^{5} - 3 q^{9} - 6 q^{10} + 6 q^{11} - 4 q^{16} - 4 q^{17} - 6 q^{18} + 5 q^{19} - 6 q^{20} + 12 q^{22} + 3 q^{23} + 4 q^{25} - 5 q^{29} - 3 q^{31} - 8 q^{32} - 8 q^{34} - 6 q^{36} + 4 q^{37} + 10 q^{38} - 6 q^{41} - q^{43} + 12 q^{44} + 9 q^{45} + 6 q^{46} + 7 q^{47} + 8 q^{50} - 9 q^{53} - 18 q^{55} - 10 q^{58} + 8 q^{59} + 10 q^{61} - 6 q^{62} - 8 q^{64} + 6 q^{67} - 8 q^{68} + 8 q^{71} - 13 q^{73} + 8 q^{74} + 10 q^{76} + 3 q^{79} + 12 q^{80} + 9 q^{81} - 12 q^{82} + 15 q^{83} + 12 q^{85} - 2 q^{86} + 3 q^{89} + 18 q^{90} + 6 q^{92} + 14 q^{94} - 15 q^{95} + 7 q^{97} - 18 q^{99}+O(q^{100})$$ q + 2 * q^2 + 2 * q^4 - 3 * q^5 - 3 * q^9 - 6 * q^10 + 6 * q^11 - 4 * q^16 - 4 * q^17 - 6 * q^18 + 5 * q^19 - 6 * q^20 + 12 * q^22 + 3 * q^23 + 4 * q^25 - 5 * q^29 - 3 * q^31 - 8 * q^32 - 8 * q^34 - 6 * q^36 + 4 * q^37 + 10 * q^38 - 6 * q^41 - q^43 + 12 * q^44 + 9 * q^45 + 6 * q^46 + 7 * q^47 + 8 * q^50 - 9 * q^53 - 18 * q^55 - 10 * q^58 + 8 * q^59 + 10 * q^61 - 6 * q^62 - 8 * q^64 + 6 * q^67 - 8 * q^68 + 8 * q^71 - 13 * q^73 + 8 * q^74 + 10 * q^76 + 3 * q^79 + 12 * q^80 + 9 * q^81 - 12 * q^82 + 15 * q^83 + 12 * q^85 - 2 * q^86 + 3 * q^89 + 18 * q^90 + 6 * q^92 + 14 * q^94 - 15 * q^95 + 7 * q^97 - 18 * 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 0 2.00000 −3.00000 0 0 0 −3.00000 −6.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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.l 1
7.b odd 2 1 1183.2.a.b 1
13.b even 2 1 637.2.a.a 1
39.d odd 2 1 5733.2.a.l 1
91.b odd 2 1 91.2.a.a 1
91.i even 4 2 1183.2.c.b 2
91.r even 6 2 637.2.e.d 2
91.s odd 6 2 637.2.e.e 2
273.g even 2 1 819.2.a.f 1
364.h even 2 1 1456.2.a.g 1
455.h odd 2 1 2275.2.a.h 1
728.b even 2 1 5824.2.a.t 1
728.l odd 2 1 5824.2.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.a 1 91.b odd 2 1
637.2.a.a 1 13.b even 2 1
637.2.e.d 2 91.r even 6 2
637.2.e.e 2 91.s odd 6 2
819.2.a.f 1 273.g even 2 1
1183.2.a.b 1 7.b odd 2 1
1183.2.c.b 2 91.i even 4 2
1456.2.a.g 1 364.h even 2 1
2275.2.a.h 1 455.h odd 2 1
5733.2.a.l 1 39.d odd 2 1
5824.2.a.s 1 728.l odd 2 1
5824.2.a.t 1 728.b even 2 1
8281.2.a.l 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8281))$$:

 $$T_{2} - 2$$ T2 - 2 $$T_{3}$$ T3 $$T_{5} + 3$$ T5 + 3 $$T_{11} - 6$$ T11 - 6 $$T_{17} + 4$$ T17 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T$$
$5$ $$T + 3$$
$7$ $$T$$
$11$ $$T - 6$$
$13$ $$T$$
$17$ $$T + 4$$
$19$ $$T - 5$$
$23$ $$T - 3$$
$29$ $$T + 5$$
$31$ $$T + 3$$
$37$ $$T - 4$$
$41$ $$T + 6$$
$43$ $$T + 1$$
$47$ $$T - 7$$
$53$ $$T + 9$$
$59$ $$T - 8$$
$61$ $$T - 10$$
$67$ $$T - 6$$
$71$ $$T - 8$$
$73$ $$T + 13$$
$79$ $$T - 3$$
$83$ $$T - 15$$
$89$ $$T - 3$$
$97$ $$T - 7$$