# Properties

 Label 8281.2.a.r Level $8281$ Weight $2$ Character orbit 8281.a Self dual yes Analytic conductor $66.124$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + ( - \beta - 1) q^{3} + q^{4} - \beta q^{5} + ( - \beta - 3) q^{6} - \beta q^{8} + (2 \beta + 1) q^{9} +O(q^{10})$$ q + b * q^2 + (-b - 1) * q^3 + q^4 - b * q^5 + (-b - 3) * q^6 - b * q^8 + (2*b + 1) * q^9 $$q + \beta q^{2} + ( - \beta - 1) q^{3} + q^{4} - \beta q^{5} + ( - \beta - 3) q^{6} - \beta q^{8} + (2 \beta + 1) q^{9} - 3 q^{10} + ( - \beta + 3) q^{11} + ( - \beta - 1) q^{12} + (\beta + 3) q^{15} - 5 q^{16} + (\beta + 6) q^{17} + (\beta + 6) q^{18} - 2 q^{19} - \beta q^{20} + (3 \beta - 3) q^{22} + (\beta + 3) q^{23} + (\beta + 3) q^{24} - 2 q^{25} - 4 q^{27} - 3 q^{29} + (3 \beta + 3) q^{30} + ( - 3 \beta + 1) q^{31} - 3 \beta q^{32} - 2 \beta q^{33} + (6 \beta + 3) q^{34} + (2 \beta + 1) q^{36} - 7 q^{37} - 2 \beta q^{38} + 3 q^{40} + 3 \beta q^{41} + ( - 3 \beta + 5) q^{43} + ( - \beta + 3) q^{44} + ( - \beta - 6) q^{45} + (3 \beta + 3) q^{46} + ( - 4 \beta - 6) q^{47} + (5 \beta + 5) q^{48} - 2 \beta q^{50} + ( - 7 \beta - 9) q^{51} + ( - 4 \beta - 3) q^{53} - 4 \beta q^{54} + ( - 3 \beta + 3) q^{55} + (2 \beta + 2) q^{57} - 3 \beta q^{58} + (\beta - 9) q^{59} + (\beta + 3) q^{60} + ( - 3 \beta + 10) q^{61} + (\beta - 9) q^{62} + q^{64} - 6 q^{66} + ( - 3 \beta - 1) q^{67} + (\beta + 6) q^{68} + ( - 4 \beta - 6) q^{69} + 6 q^{71} + ( - \beta - 6) q^{72} + (3 \beta - 2) q^{73} - 7 \beta q^{74} + (2 \beta + 2) q^{75} - 2 q^{76} + (3 \beta + 11) q^{79} + 5 \beta q^{80} + ( - 2 \beta + 1) q^{81} + 9 q^{82} + (3 \beta - 3) q^{83} + ( - 6 \beta - 3) q^{85} + (5 \beta - 9) q^{86} + (3 \beta + 3) q^{87} + ( - 3 \beta + 3) q^{88} + ( - 4 \beta - 6) q^{89} + ( - 6 \beta - 3) q^{90} + (\beta + 3) q^{92} + (2 \beta + 8) q^{93} + ( - 6 \beta - 12) q^{94} + 2 \beta q^{95} + (3 \beta + 9) q^{96} + ( - 6 \beta + 4) q^{97} + (5 \beta - 3) q^{99} +O(q^{100})$$ q + b * q^2 + (-b - 1) * q^3 + q^4 - b * q^5 + (-b - 3) * q^6 - b * q^8 + (2*b + 1) * q^9 - 3 * q^10 + (-b + 3) * q^11 + (-b - 1) * q^12 + (b + 3) * q^15 - 5 * q^16 + (b + 6) * q^17 + (b + 6) * q^18 - 2 * q^19 - b * q^20 + (3*b - 3) * q^22 + (b + 3) * q^23 + (b + 3) * q^24 - 2 * q^25 - 4 * q^27 - 3 * q^29 + (3*b + 3) * q^30 + (-3*b + 1) * q^31 - 3*b * q^32 - 2*b * q^33 + (6*b + 3) * q^34 + (2*b + 1) * q^36 - 7 * q^37 - 2*b * q^38 + 3 * q^40 + 3*b * q^41 + (-3*b + 5) * q^43 + (-b + 3) * q^44 + (-b - 6) * q^45 + (3*b + 3) * q^46 + (-4*b - 6) * q^47 + (5*b + 5) * q^48 - 2*b * q^50 + (-7*b - 9) * q^51 + (-4*b - 3) * q^53 - 4*b * q^54 + (-3*b + 3) * q^55 + (2*b + 2) * q^57 - 3*b * q^58 + (b - 9) * q^59 + (b + 3) * q^60 + (-3*b + 10) * q^61 + (b - 9) * q^62 + q^64 - 6 * q^66 + (-3*b - 1) * q^67 + (b + 6) * q^68 + (-4*b - 6) * q^69 + 6 * q^71 + (-b - 6) * q^72 + (3*b - 2) * q^73 - 7*b * q^74 + (2*b + 2) * q^75 - 2 * q^76 + (3*b + 11) * q^79 + 5*b * q^80 + (-2*b + 1) * q^81 + 9 * q^82 + (3*b - 3) * q^83 + (-6*b - 3) * q^85 + (5*b - 9) * q^86 + (3*b + 3) * q^87 + (-3*b + 3) * q^88 + (-4*b - 6) * q^89 + (-6*b - 3) * q^90 + (b + 3) * q^92 + (2*b + 8) * q^93 + (-6*b - 12) * q^94 + 2*b * q^95 + (3*b + 9) * q^96 + (-6*b + 4) * q^97 + (5*b - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{4} - 6 q^{6} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^4 - 6 * q^6 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{4} - 6 q^{6} + 2 q^{9} - 6 q^{10} + 6 q^{11} - 2 q^{12} + 6 q^{15} - 10 q^{16} + 12 q^{17} + 12 q^{18} - 4 q^{19} - 6 q^{22} + 6 q^{23} + 6 q^{24} - 4 q^{25} - 8 q^{27} - 6 q^{29} + 6 q^{30} + 2 q^{31} + 6 q^{34} + 2 q^{36} - 14 q^{37} + 6 q^{40} + 10 q^{43} + 6 q^{44} - 12 q^{45} + 6 q^{46} - 12 q^{47} + 10 q^{48} - 18 q^{51} - 6 q^{53} + 6 q^{55} + 4 q^{57} - 18 q^{59} + 6 q^{60} + 20 q^{61} - 18 q^{62} + 2 q^{64} - 12 q^{66} - 2 q^{67} + 12 q^{68} - 12 q^{69} + 12 q^{71} - 12 q^{72} - 4 q^{73} + 4 q^{75} - 4 q^{76} + 22 q^{79} + 2 q^{81} + 18 q^{82} - 6 q^{83} - 6 q^{85} - 18 q^{86} + 6 q^{87} + 6 q^{88} - 12 q^{89} - 6 q^{90} + 6 q^{92} + 16 q^{93} - 24 q^{94} + 18 q^{96} + 8 q^{97} - 6 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^4 - 6 * q^6 + 2 * q^9 - 6 * q^10 + 6 * q^11 - 2 * q^12 + 6 * q^15 - 10 * q^16 + 12 * q^17 + 12 * q^18 - 4 * q^19 - 6 * q^22 + 6 * q^23 + 6 * q^24 - 4 * q^25 - 8 * q^27 - 6 * q^29 + 6 * q^30 + 2 * q^31 + 6 * q^34 + 2 * q^36 - 14 * q^37 + 6 * q^40 + 10 * q^43 + 6 * q^44 - 12 * q^45 + 6 * q^46 - 12 * q^47 + 10 * q^48 - 18 * q^51 - 6 * q^53 + 6 * q^55 + 4 * q^57 - 18 * q^59 + 6 * q^60 + 20 * q^61 - 18 * q^62 + 2 * q^64 - 12 * q^66 - 2 * q^67 + 12 * q^68 - 12 * q^69 + 12 * q^71 - 12 * q^72 - 4 * q^73 + 4 * q^75 - 4 * q^76 + 22 * q^79 + 2 * q^81 + 18 * q^82 - 6 * q^83 - 6 * q^85 - 18 * q^86 + 6 * q^87 + 6 * q^88 - 12 * q^89 - 6 * q^90 + 6 * q^92 + 16 * q^93 - 24 * q^94 + 18 * q^96 + 8 * q^97 - 6 * 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
 −1.73205 1.73205
−1.73205 0.732051 1.00000 1.73205 −1.26795 0 1.73205 −2.46410 −3.00000
1.2 1.73205 −2.73205 1.00000 −1.73205 −4.73205 0 −1.73205 4.46410 −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.r 2
7.b odd 2 1 1183.2.a.f 2
13.b even 2 1 8281.2.a.t 2
13.c even 3 2 637.2.f.d 4
91.b odd 2 1 1183.2.a.e 2
91.g even 3 2 637.2.g.d 4
91.h even 3 2 637.2.h.e 4
91.i even 4 2 1183.2.c.e 4
91.m odd 6 2 637.2.g.e 4
91.n odd 6 2 91.2.f.b 4
91.v odd 6 2 637.2.h.d 4
273.bn even 6 2 819.2.o.b 4
364.v even 6 2 1456.2.s.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.b 4 91.n odd 6 2
637.2.f.d 4 13.c even 3 2
637.2.g.d 4 91.g even 3 2
637.2.g.e 4 91.m odd 6 2
637.2.h.d 4 91.v odd 6 2
637.2.h.e 4 91.h even 3 2
819.2.o.b 4 273.bn even 6 2
1183.2.a.e 2 91.b odd 2 1
1183.2.a.f 2 7.b odd 2 1
1183.2.c.e 4 91.i even 4 2
1456.2.s.o 4 364.v even 6 2
8281.2.a.r 2 1.a even 1 1 trivial
8281.2.a.t 2 13.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8281))$$:

 $$T_{2}^{2} - 3$$ T2^2 - 3 $$T_{3}^{2} + 2T_{3} - 2$$ T3^2 + 2*T3 - 2 $$T_{5}^{2} - 3$$ T5^2 - 3 $$T_{11}^{2} - 6T_{11} + 6$$ T11^2 - 6*T11 + 6 $$T_{17}^{2} - 12T_{17} + 33$$ T17^2 - 12*T17 + 33

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3$$
$3$ $$T^{2} + 2T - 2$$
$5$ $$T^{2} - 3$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 6T + 6$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 12T + 33$$
$19$ $$(T + 2)^{2}$$
$23$ $$T^{2} - 6T + 6$$
$29$ $$(T + 3)^{2}$$
$31$ $$T^{2} - 2T - 26$$
$37$ $$(T + 7)^{2}$$
$41$ $$T^{2} - 27$$
$43$ $$T^{2} - 10T - 2$$
$47$ $$T^{2} + 12T - 12$$
$53$ $$T^{2} + 6T - 39$$
$59$ $$T^{2} + 18T + 78$$
$61$ $$T^{2} - 20T + 73$$
$67$ $$T^{2} + 2T - 26$$
$71$ $$(T - 6)^{2}$$
$73$ $$T^{2} + 4T - 23$$
$79$ $$T^{2} - 22T + 94$$
$83$ $$T^{2} + 6T - 18$$
$89$ $$T^{2} + 12T - 12$$
$97$ $$T^{2} - 8T - 92$$