# Properties

 Label 8281.2.a.bq Level $8281$ Weight $2$ Character orbit 8281.a Self dual yes Analytic conductor $66.124$ Analytic rank $1$ Dimension $4$ Inner twists $2$

# 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: $$4$$ Coefficient field: $$\Q(\sqrt{3}, \sqrt{7})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 5x^{2} + 1$$ x^4 - 5*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 637) 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)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + ( - \beta_{2} - 1) q^{3} - \beta_{2} q^{4} - \beta_{3} q^{5} + (2 \beta_{3} + \beta_1) q^{6} + (\beta_{3} + \beta_1) q^{8} + (\beta_{2} + 3) q^{9}+O(q^{10})$$ q + b3 * q^2 + (-b2 - 1) * q^3 - b2 * q^4 - b3 * q^5 + (2*b3 + b1) * q^6 + (b3 + b1) * q^8 + (b2 + 3) * q^9 $$q + \beta_{3} q^{2} + ( - \beta_{2} - 1) q^{3} - \beta_{2} q^{4} - \beta_{3} q^{5} + (2 \beta_{3} + \beta_1) q^{6} + (\beta_{3} + \beta_1) q^{8} + (\beta_{2} + 3) q^{9} + (\beta_{2} - 2) q^{10} + (\beta_{3} + 2 \beta_1) q^{11} + 5 q^{12} + ( - 2 \beta_{3} - \beta_1) q^{15} + (\beta_{2} + 1) q^{16} + 3 q^{17} - \beta_1 q^{18} - 3 \beta_{3} q^{19} + ( - 3 \beta_{3} - \beta_1) q^{20} - \beta_{2} q^{22} + (2 \beta_{2} - 2) q^{23} + (\beta_{3} - 2 \beta_1) q^{24} + ( - \beta_{2} - 3) q^{25} - 5 q^{27} + ( - \beta_{2} - 5) q^{29} + (2 \beta_{2} - 3) q^{30} + (5 \beta_{3} + 5 \beta_1) q^{31} + ( - 4 \beta_{3} - 3 \beta_1) q^{32} - 5 \beta_1 q^{33} + 3 \beta_{3} q^{34} + ( - 2 \beta_{2} - 5) q^{36} + ( - 4 \beta_{3} - 4 \beta_1) q^{37} + (3 \beta_{2} - 6) q^{38} + (\beta_{2} - 1) q^{40} + ( - 2 \beta_{3} - 4 \beta_1) q^{41} + ( - 3 \beta_{2} - 4) q^{43} + (\beta_{3} - 3 \beta_1) q^{44} + \beta_1 q^{45} + ( - 8 \beta_{3} - 2 \beta_1) q^{46} + (3 \beta_{3} + 5 \beta_1) q^{47} + ( - \beta_{2} - 6) q^{48} + \beta_1 q^{50} + ( - 3 \beta_{2} - 3) q^{51} + (4 \beta_{2} - 1) q^{53} - 5 \beta_{3} q^{54} + \beta_{2} q^{55} + ( - 6 \beta_{3} - 3 \beta_1) q^{57} + ( - 2 \beta_{3} + \beta_1) q^{58} + (3 \beta_{3} - 5 \beta_1) q^{59} - 5 \beta_{3} q^{60} + (6 \beta_{2} + 4) q^{61} + ( - 5 \beta_{2} + 5) q^{62} + (2 \beta_{2} - 7) q^{64} + 5 q^{66} + (5 \beta_{3} - \beta_1) q^{67} - 3 \beta_{2} q^{68} + (2 \beta_{2} - 8) q^{69} - 2 \beta_1 q^{71} + (\beta_{3} + 4 \beta_1) q^{72} + (2 \beta_{3} + 2 \beta_1) q^{73} + (4 \beta_{2} - 4) q^{74} + (3 \beta_{2} + 8) q^{75} + ( - 9 \beta_{3} - 3 \beta_1) q^{76} - 6 q^{79} + (2 \beta_{3} + \beta_1) q^{80} + (2 \beta_{2} - 4) q^{81} + 2 \beta_{2} q^{82} + (\beta_{3} - 3 \beta_1) q^{83} - 3 \beta_{3} q^{85} + (5 \beta_{3} + 3 \beta_1) q^{86} + (5 \beta_{2} + 10) q^{87} + (\beta_{2} + 5) q^{88} + ( - 3 \beta_{3} - 8 \beta_1) q^{89} - q^{90} + (4 \beta_{2} - 10) q^{92} + (5 \beta_{3} - 10 \beta_1) q^{93} + ( - 3 \beta_{2} + 1) q^{94} + ( - 3 \beta_{2} + 6) q^{95} + ( - 5 \beta_{3} + 5 \beta_1) q^{96} + ( - 8 \beta_{3} - 5 \beta_1) q^{97} + (2 \beta_{3} + 9 \beta_1) q^{99}+O(q^{100})$$ q + b3 * q^2 + (-b2 - 1) * q^3 - b2 * q^4 - b3 * q^5 + (2*b3 + b1) * q^6 + (b3 + b1) * q^8 + (b2 + 3) * q^9 + (b2 - 2) * q^10 + (b3 + 2*b1) * q^11 + 5 * q^12 + (-2*b3 - b1) * q^15 + (b2 + 1) * q^16 + 3 * q^17 - b1 * q^18 - 3*b3 * q^19 + (-3*b3 - b1) * q^20 - b2 * q^22 + (2*b2 - 2) * q^23 + (b3 - 2*b1) * q^24 + (-b2 - 3) * q^25 - 5 * q^27 + (-b2 - 5) * q^29 + (2*b2 - 3) * q^30 + (5*b3 + 5*b1) * q^31 + (-4*b3 - 3*b1) * q^32 - 5*b1 * q^33 + 3*b3 * q^34 + (-2*b2 - 5) * q^36 + (-4*b3 - 4*b1) * q^37 + (3*b2 - 6) * q^38 + (b2 - 1) * q^40 + (-2*b3 - 4*b1) * q^41 + (-3*b2 - 4) * q^43 + (b3 - 3*b1) * q^44 + b1 * q^45 + (-8*b3 - 2*b1) * q^46 + (3*b3 + 5*b1) * q^47 + (-b2 - 6) * q^48 + b1 * q^50 + (-3*b2 - 3) * q^51 + (4*b2 - 1) * q^53 - 5*b3 * q^54 + b2 * q^55 + (-6*b3 - 3*b1) * q^57 + (-2*b3 + b1) * q^58 + (3*b3 - 5*b1) * q^59 - 5*b3 * q^60 + (6*b2 + 4) * q^61 + (-5*b2 + 5) * q^62 + (2*b2 - 7) * q^64 + 5 * q^66 + (5*b3 - b1) * q^67 - 3*b2 * q^68 + (2*b2 - 8) * q^69 - 2*b1 * q^71 + (b3 + 4*b1) * q^72 + (2*b3 + 2*b1) * q^73 + (4*b2 - 4) * q^74 + (3*b2 + 8) * q^75 + (-9*b3 - 3*b1) * q^76 - 6 * q^79 + (2*b3 + b1) * q^80 + (2*b2 - 4) * q^81 + 2*b2 * q^82 + (b3 - 3*b1) * q^83 - 3*b3 * q^85 + (5*b3 + 3*b1) * q^86 + (5*b2 + 10) * q^87 + (b2 + 5) * q^88 + (-3*b3 - 8*b1) * q^89 - q^90 + (4*b2 - 10) * q^92 + (5*b3 - 10*b1) * q^93 + (-3*b2 + 1) * q^94 + (-3*b2 + 6) * q^95 + (-5*b3 + 5*b1) * q^96 + (-8*b3 - 5*b1) * q^97 + (2*b3 + 9*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} + 2 q^{4} + 10 q^{9}+O(q^{10})$$ 4 * q - 2 * q^3 + 2 * q^4 + 10 * q^9 $$4 q - 2 q^{3} + 2 q^{4} + 10 q^{9} - 10 q^{10} + 20 q^{12} + 2 q^{16} + 12 q^{17} + 2 q^{22} - 12 q^{23} - 10 q^{25} - 20 q^{27} - 18 q^{29} - 16 q^{30} - 16 q^{36} - 30 q^{38} - 6 q^{40} - 10 q^{43} - 22 q^{48} - 6 q^{51} - 12 q^{53} - 2 q^{55} + 4 q^{61} + 30 q^{62} - 32 q^{64} + 20 q^{66} + 6 q^{68} - 36 q^{69} - 24 q^{74} + 26 q^{75} - 24 q^{79} - 20 q^{81} - 4 q^{82} + 30 q^{87} + 18 q^{88} - 4 q^{90} - 48 q^{92} + 10 q^{94} + 30 q^{95}+O(q^{100})$$ 4 * q - 2 * q^3 + 2 * q^4 + 10 * q^9 - 10 * q^10 + 20 * q^12 + 2 * q^16 + 12 * q^17 + 2 * q^22 - 12 * q^23 - 10 * q^25 - 20 * q^27 - 18 * q^29 - 16 * q^30 - 16 * q^36 - 30 * q^38 - 6 * q^40 - 10 * q^43 - 22 * q^48 - 6 * q^51 - 12 * q^53 - 2 * q^55 + 4 * q^61 + 30 * q^62 - 32 * q^64 + 20 * q^66 + 6 * q^68 - 36 * q^69 - 24 * q^74 + 26 * q^75 - 24 * q^79 - 20 * q^81 - 4 * q^82 + 30 * q^87 + 18 * q^88 - 4 * q^90 - 48 * q^92 + 10 * q^94 + 30 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 5\nu$$ v^3 - 5*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 5\beta_1$$ b3 + 5*b1

## 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
 0.456850 2.18890 −2.18890 −0.456850
−2.18890 1.79129 2.79129 2.18890 −3.92095 0 −1.73205 0.208712 −4.79129
1.2 −0.456850 −2.79129 −1.79129 0.456850 1.27520 0 1.73205 4.79129 −0.208712
1.3 0.456850 −2.79129 −1.79129 −0.456850 −1.27520 0 −1.73205 4.79129 −0.208712
1.4 2.18890 1.79129 2.79129 −2.18890 3.92095 0 1.73205 0.208712 −4.79129
 $$n$$: 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8281.2.a.bq 4
7.b odd 2 1 8281.2.a.bs 4
13.b even 2 1 inner 8281.2.a.bq 4
13.f odd 12 2 637.2.q.f yes 4
91.b odd 2 1 8281.2.a.bs 4
91.w even 12 2 637.2.u.e 4
91.x odd 12 2 637.2.k.f 4
91.ba even 12 2 637.2.k.d 4
91.bc even 12 2 637.2.q.e 4
91.bd odd 12 2 637.2.u.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.k.d 4 91.ba even 12 2
637.2.k.f 4 91.x odd 12 2
637.2.q.e 4 91.bc even 12 2
637.2.q.f yes 4 13.f odd 12 2
637.2.u.d 4 91.bd odd 12 2
637.2.u.e 4 91.w even 12 2
8281.2.a.bq 4 1.a even 1 1 trivial
8281.2.a.bq 4 13.b even 2 1 inner
8281.2.a.bs 4 7.b odd 2 1
8281.2.a.bs 4 91.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8281))$$:

 $$T_{2}^{4} - 5T_{2}^{2} + 1$$ T2^4 - 5*T2^2 + 1 $$T_{3}^{2} + T_{3} - 5$$ T3^2 + T3 - 5 $$T_{5}^{4} - 5T_{5}^{2} + 1$$ T5^4 - 5*T5^2 + 1 $$T_{11}^{4} - 17T_{11}^{2} + 25$$ T11^4 - 17*T11^2 + 25 $$T_{17} - 3$$ T17 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 5T^{2} + 1$$
$3$ $$(T^{2} + T - 5)^{2}$$
$5$ $$T^{4} - 5T^{2} + 1$$
$7$ $$T^{4}$$
$11$ $$T^{4} - 17T^{2} + 25$$
$13$ $$T^{4}$$
$17$ $$(T - 3)^{4}$$
$19$ $$T^{4} - 45T^{2} + 81$$
$23$ $$(T^{2} + 6 T - 12)^{2}$$
$29$ $$(T^{2} + 9 T + 15)^{2}$$
$31$ $$(T^{2} - 75)^{2}$$
$37$ $$(T^{2} - 48)^{2}$$
$41$ $$T^{4} - 68T^{2} + 400$$
$43$ $$(T^{2} + 5 T - 41)^{2}$$
$47$ $$T^{4} - 110T^{2} + 1681$$
$53$ $$(T^{2} + 6 T - 75)^{2}$$
$59$ $$T^{4} - 230 T^{2} + 11881$$
$61$ $$(T^{2} - 2 T - 188)^{2}$$
$67$ $$T^{4} - 150T^{2} + 2601$$
$71$ $$T^{4} - 20T^{2} + 16$$
$73$ $$(T^{2} - 12)^{2}$$
$79$ $$(T + 6)^{4}$$
$83$ $$T^{4} - 62T^{2} + 625$$
$89$ $$T^{4} - 269T^{2} + 2209$$
$97$ $$T^{4} - 285 T^{2} + 12321$$