# Properties

 Label 8281.2.a.i Level $8281$ Weight $2$ Character orbit 8281.a Self dual yes Analytic conductor $66.124$ 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] = [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: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - 3 q^{3} - q^{4} + 3 q^{5} - 3 q^{6} - 3 q^{8} + 6 q^{9}+O(q^{10})$$ q + q^2 - 3 * q^3 - q^4 + 3 * q^5 - 3 * q^6 - 3 * q^8 + 6 * q^9 $$q + q^{2} - 3 q^{3} - q^{4} + 3 q^{5} - 3 q^{6} - 3 q^{8} + 6 q^{9} + 3 q^{10} - 3 q^{11} + 3 q^{12} - 9 q^{15} - q^{16} - 2 q^{17} + 6 q^{18} - q^{19} - 3 q^{20} - 3 q^{22} + 9 q^{24} + 4 q^{25} - 9 q^{27} + 7 q^{29} - 9 q^{30} + 3 q^{31} + 5 q^{32} + 9 q^{33} - 2 q^{34} - 6 q^{36} + 2 q^{37} - q^{38} - 9 q^{40} + 3 q^{41} - 7 q^{43} + 3 q^{44} + 18 q^{45} + q^{47} + 3 q^{48} + 4 q^{50} + 6 q^{51} + 3 q^{53} - 9 q^{54} - 9 q^{55} + 3 q^{57} + 7 q^{58} - 4 q^{59} + 9 q^{60} - 13 q^{61} + 3 q^{62} + 7 q^{64} + 9 q^{66} - 3 q^{67} + 2 q^{68} + 13 q^{71} - 18 q^{72} - 13 q^{73} + 2 q^{74} - 12 q^{75} + q^{76} - 3 q^{79} - 3 q^{80} + 9 q^{81} + 3 q^{82} - 6 q^{85} - 7 q^{86} - 21 q^{87} + 9 q^{88} + 6 q^{89} + 18 q^{90} - 9 q^{93} + q^{94} - 3 q^{95} - 15 q^{96} - 5 q^{97} - 18 q^{99}+O(q^{100})$$ q + q^2 - 3 * q^3 - q^4 + 3 * q^5 - 3 * q^6 - 3 * q^8 + 6 * q^9 + 3 * q^10 - 3 * q^11 + 3 * q^12 - 9 * q^15 - q^16 - 2 * q^17 + 6 * q^18 - q^19 - 3 * q^20 - 3 * q^22 + 9 * q^24 + 4 * q^25 - 9 * q^27 + 7 * q^29 - 9 * q^30 + 3 * q^31 + 5 * q^32 + 9 * q^33 - 2 * q^34 - 6 * q^36 + 2 * q^37 - q^38 - 9 * q^40 + 3 * q^41 - 7 * q^43 + 3 * q^44 + 18 * q^45 + q^47 + 3 * q^48 + 4 * q^50 + 6 * q^51 + 3 * q^53 - 9 * q^54 - 9 * q^55 + 3 * q^57 + 7 * q^58 - 4 * q^59 + 9 * q^60 - 13 * q^61 + 3 * q^62 + 7 * q^64 + 9 * q^66 - 3 * q^67 + 2 * q^68 + 13 * q^71 - 18 * q^72 - 13 * q^73 + 2 * q^74 - 12 * q^75 + q^76 - 3 * q^79 - 3 * q^80 + 9 * q^81 + 3 * q^82 - 6 * q^85 - 7 * q^86 - 21 * q^87 + 9 * q^88 + 6 * q^89 + 18 * q^90 - 9 * q^93 + q^94 - 3 * q^95 - 15 * q^96 - 5 * 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
1.00000 −3.00000 −1.00000 3.00000 −3.00000 0 −3.00000 6.00000 3.00000
 $$n$$: 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.i 1
7.b odd 2 1 8281.2.a.j 1
7.c even 3 2 1183.2.e.a 2
13.b even 2 1 8281.2.a.c 1
13.c even 3 2 637.2.f.b 2
91.b odd 2 1 8281.2.a.g 1
91.g even 3 2 91.2.g.a 2
91.h even 3 2 91.2.h.a yes 2
91.m odd 6 2 637.2.g.a 2
91.n odd 6 2 637.2.f.a 2
91.r even 6 2 1183.2.e.c 2
91.v odd 6 2 637.2.h.a 2
273.s odd 6 2 819.2.s.a 2
273.bm odd 6 2 819.2.n.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.g.a 2 91.g even 3 2
91.2.h.a yes 2 91.h even 3 2
637.2.f.a 2 91.n odd 6 2
637.2.f.b 2 13.c even 3 2
637.2.g.a 2 91.m odd 6 2
637.2.h.a 2 91.v odd 6 2
819.2.n.c 2 273.bm odd 6 2
819.2.s.a 2 273.s odd 6 2
1183.2.e.a 2 7.c even 3 2
1183.2.e.c 2 91.r even 6 2
8281.2.a.c 1 13.b even 2 1
8281.2.a.g 1 91.b odd 2 1
8281.2.a.i 1 1.a even 1 1 trivial
8281.2.a.j 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 $$T_{3} + 3$$ T3 + 3 $$T_{5} - 3$$ T5 - 3 $$T_{11} + 3$$ T11 + 3 $$T_{17} + 2$$ T17 + 2

## Hecke characteristic polynomials

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