# Properties

 Label 8281.2.a.cf Level $8281$ Weight $2$ Character orbit 8281.a Self dual yes Analytic conductor $66.124$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.6.6995813.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} - 6x^{4} + 4x^{3} + 7x^{2} - x - 1$$ x^6 - x^5 - 6*x^4 + 4*x^3 + 7*x^2 - x - 1 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 a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{4} + \beta_1) q^{2} - \beta_{4} q^{3} + (\beta_{5} + 1) q^{4} - \beta_{3} q^{5} + (\beta_{5} - \beta_{2} + 2) q^{6} + (\beta_{5} - \beta_1 + 1) q^{8} + (\beta_{5} - \beta_{2}) q^{9}+O(q^{10})$$ q + (-b4 + b1) * q^2 - b4 * q^3 + (b5 + 1) * q^4 - b3 * q^5 + (b5 - b2 + 2) * q^6 + (b5 - b1 + 1) * q^8 + (b5 - b2) * q^9 $$q + ( - \beta_{4} + \beta_1) q^{2} - \beta_{4} q^{3} + (\beta_{5} + 1) q^{4} - \beta_{3} q^{5} + (\beta_{5} - \beta_{2} + 2) q^{6} + (\beta_{5} - \beta_1 + 1) q^{8} + (\beta_{5} - \beta_{2}) q^{9} + ( - \beta_{3} - \beta_1 + 1) q^{10} + (\beta_{5} + \beta_{4} + \beta_1 + 1) q^{11} + (\beta_{5} - 2 \beta_{4} - \beta_{3} + 1) q^{12} + \beta_{5} q^{15} + ( - \beta_{5} - 2 \beta_{4} - \beta_{2} + \cdots - 2) q^{16}+ \cdots + (2 \beta_{4} - 2 \beta_{2} - 4 \beta_1 + 3) q^{99}+O(q^{100})$$ q + (-b4 + b1) * q^2 - b4 * q^3 + (b5 + 1) * q^4 - b3 * q^5 + (b5 - b2 + 2) * q^6 + (b5 - b1 + 1) * q^8 + (b5 - b2) * q^9 + (-b3 - b1 + 1) * q^10 + (b5 + b4 + b1 + 1) * q^11 + (b5 - 2*b4 - b3 + 1) * q^12 + b5 * q^15 + (-b5 - 2*b4 - b2 + b1 - 2) * q^16 + (b5 - b4 + 1) * q^17 + (b5 - 2*b4 - b3 - 2*b1 + 1) * q^18 + (b5 + 2*b4 - b3 + 2*b2 - 2*b1 + 1) * q^19 + (-b4 + b3 - b2) * q^20 + (-2*b4 + 2*b2 + b1) * q^22 + (b5 - b4 - 3*b3 + b2 - 2*b1 + 1) * q^23 + (b5 - 2*b4 - b3 + 2) * q^24 + (-2*b5 - b4 - b3 + b1 - 2) * q^25 + (b5 - b3 + b1 + 1) * q^27 + (-b5 - b4 + 2*b2 - 2*b1 - 1) * q^29 + (b5 - b4 + 1) * q^30 + (b5 + 2*b3 + 2*b1 - 3) * q^31 + (-b5 + 2*b4 - b3 - b2 - 2*b1 + 2) * q^32 + (-2*b4 - b3 + b2 - 3) * q^33 + (2*b5 - 2*b4 - b2 + b1 + 3) * q^34 + (b5 - 2*b4 - b3 - 2*b2 + 4) * q^36 + (-b5 - 3*b3 - 2) * q^37 + (-b5 + b3 + 4*b1 - 4) * q^38 + (b5 - b4 + 2*b3 - b2 + b1 - 1) * q^40 + (2*b5 + b4 + b3 + 2) * q^41 + (2*b5 + b4 - 2*b1 + 3) * q^43 + (2*b3 - b2 + 2*b1 + 3) * q^44 + (b5 - b4 + 2*b3 + 1) * q^45 + (2*b5 - b4 - 2*b3 - 3*b2 + 4) * q^46 + (-2*b5 - 3*b4 + 2*b2 + 4*b1 - 2) * q^47 + (b5 + b4 + b3 - 2*b2 + b1 + 4) * q^48 + (-b5 + 4*b4 - b3 - 3*b1 + 2) * q^50 + (2*b5 - 2*b4 - b3 - b2 + 4) * q^51 + (3*b3 - 2*b2 + b1) * q^53 + (b5 - 2*b4 - b3 + b2 + 3) * q^54 + (-2*b5 - b4 - b2 - b1 + 1) * q^55 + (2*b4 - b3 + 2*b2 - 2*b1 - 3) * q^57 + (4*b4 + 2*b3 - 3*b2 + 3*b1 - 1) * q^58 + (2*b5 + b4 - b3 - 2*b2 + b1 - 1) * q^59 + (-2*b4 - b2 + b1 + 3) * q^60 + (2*b5 + 5*b4 - 2*b3 - 6*b1 + 2) * q^61 + (b5 + 2*b4 + 2*b3 + 2*b2 - b1 + 1) * q^62 + (-b5 + 2*b4 - 2*b3 + 2*b2 - 3*b1 - 2) * q^64 + (2*b5 + 4*b4 - 2*b2 - 2*b1 + 5) * q^66 + (-3*b5 - 2*b4 - 2*b3 + 2*b2 - b1 - 3) * q^67 + (2*b5 - 4*b4 - b3 - b2 + b1 + 5) * q^68 + (5*b5 - b3 - b2 - b1 + 6) * q^69 + (2*b5 + 3*b4 - b3 + 2*b2 + 2) * q^71 + (b5 - 3*b4 - b3 - 2*b2 + 3*b1 + 4) * q^72 + (-b4 + 5*b1 + 4) * q^73 + (-b5 + 3*b4 - 3*b3 - 5*b1 + 2) * q^74 + (4*b4 + 2*b3 - b2) * q^75 + (-3*b5 + b4 + 3*b3 + b1) * q^76 + (-3*b4 + 2*b3 + b2 - b1 - 2) * q^79 + (2*b5 + b4 - b3 + 2*b2 - b1 + 2) * q^80 + (-b5 - 2*b4 - b3 + 3*b2) * q^81 + (b5 - 4*b4 + b3 + b2 + 3*b1 - 1) * q^82 + (-b5 + b4 - b3 - 3*b2 - 4) * q^83 + (b5 - b4 + b3 - b2) * q^85 + (b5 - 5*b4 - b2 + 3*b1 - 2) * q^86 + (6*b4 + b3 - b2 - 2*b1 + 4) * q^87 + (b3 - 2*b2 + b1) * q^88 + (-4*b5 - 5*b4 - b2 + 8*b1 - 4) * q^89 + (2*b5 - 2*b4 + 2*b3 - b2 + 3*b1 + 1) * q^90 + (b5 - 7*b4 + b3 - 3*b2 + 4) * q^92 + (-b5 + 2*b4 - b3 - 1) * q^93 + (b5 + 6*b4 + 2*b3 + b2 + 2*b1 + 8) * q^94 + (-4*b5 - 2*b4 + 2*b3 - 3*b2 + 3*b1 - 1) * q^95 + (-2*b5 - 3*b4 + b3 + 2*b2 + b1 - 5) * q^96 + (3*b5 + 4*b4 - b2 - 2*b1 + 8) * q^97 + (2*b4 - 2*b2 - 4*b1 + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 2 q^{2} + q^{3} + 4 q^{4} - q^{5} + 9 q^{6} + 3 q^{8} - 3 q^{9}+O(q^{10})$$ 6 * q + 2 * q^2 + q^3 + 4 * q^4 - q^5 + 9 * q^6 + 3 * q^8 - 3 * q^9 $$6 q + 2 q^{2} + q^{3} + 4 q^{4} - q^{5} + 9 q^{6} + 3 q^{8} - 3 q^{9} + 4 q^{10} + 4 q^{11} + 5 q^{12} - 2 q^{15} - 8 q^{16} + 5 q^{17} + 3 q^{18} + q^{19} + q^{20} + 5 q^{22} + q^{23} + 11 q^{24} - 7 q^{25} + 4 q^{27} - 3 q^{29} + 5 q^{30} - 16 q^{31} + 8 q^{32} - 16 q^{33} + 16 q^{34} + 21 q^{36} - 13 q^{37} - 17 q^{38} - 5 q^{40} + 8 q^{41} + 11 q^{43} + 21 q^{44} + 7 q^{45} + 16 q^{46} + q^{47} + 21 q^{48} + 6 q^{50} + 20 q^{51} + 2 q^{53} + 18 q^{54} + 9 q^{55} - 21 q^{57} - 8 q^{58} - 13 q^{59} + 20 q^{60} - 5 q^{61} + 5 q^{62} - 15 q^{64} + 18 q^{66} - 11 q^{67} + 29 q^{68} + 23 q^{69} + 6 q^{71} + 25 q^{72} + 30 q^{73} + 3 q^{74} - 3 q^{75} + 9 q^{76} - 7 q^{79} + 7 q^{80} + 6 q^{81} + q^{82} - 27 q^{83} - q^{85} - 7 q^{86} + 16 q^{87} - 4 q^{89} + 8 q^{90} + 27 q^{92} - 7 q^{93} + 45 q^{94} + 6 q^{95} - 19 q^{96} + 35 q^{97} + 10 q^{99}+O(q^{100})$$ 6 * q + 2 * q^2 + q^3 + 4 * q^4 - q^5 + 9 * q^6 + 3 * q^8 - 3 * q^9 + 4 * q^10 + 4 * q^11 + 5 * q^12 - 2 * q^15 - 8 * q^16 + 5 * q^17 + 3 * q^18 + q^19 + q^20 + 5 * q^22 + q^23 + 11 * q^24 - 7 * q^25 + 4 * q^27 - 3 * q^29 + 5 * q^30 - 16 * q^31 + 8 * q^32 - 16 * q^33 + 16 * q^34 + 21 * q^36 - 13 * q^37 - 17 * q^38 - 5 * q^40 + 8 * q^41 + 11 * q^43 + 21 * q^44 + 7 * q^45 + 16 * q^46 + q^47 + 21 * q^48 + 6 * q^50 + 20 * q^51 + 2 * q^53 + 18 * q^54 + 9 * q^55 - 21 * q^57 - 8 * q^58 - 13 * q^59 + 20 * q^60 - 5 * q^61 + 5 * q^62 - 15 * q^64 + 18 * q^66 - 11 * q^67 + 29 * q^68 + 23 * q^69 + 6 * q^71 + 25 * q^72 + 30 * q^73 + 3 * q^74 - 3 * q^75 + 9 * q^76 - 7 * q^79 + 7 * q^80 + 6 * q^81 + q^82 - 27 * q^83 - q^85 - 7 * q^86 + 16 * q^87 - 4 * q^89 + 8 * q^90 + 27 * q^92 - 7 * q^93 + 45 * q^94 + 6 * q^95 - 19 * q^96 + 35 * q^97 + 10 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 6x^{4} + 4x^{3} + 7x^{2} - x - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{3}$$ $$=$$ $$\nu^{5} - \nu^{4} - 5\nu^{3} + 4\nu^{2} + 2\nu - 1$$ v^5 - v^4 - 5*v^3 + 4*v^2 + 2*v - 1 $$\beta_{4}$$ $$=$$ $$\nu^{5} - \nu^{4} - 6\nu^{3} + 4\nu^{2} + 7\nu - 1$$ v^5 - v^4 - 6*v^3 + 4*v^2 + 7*v - 1 $$\beta_{5}$$ $$=$$ $$-\nu^{5} + 2\nu^{4} + 5\nu^{3} - 9\nu^{2} - 3\nu + 3$$ -v^5 + 2*v^4 + 5*v^3 - 9*v^2 - 3*v + 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2 $$\nu^{3}$$ $$=$$ $$-\beta_{4} + \beta_{3} + 5\beta_1$$ -b4 + b3 + 5*b1 $$\nu^{4}$$ $$=$$ $$\beta_{5} + \beta_{3} + 5\beta_{2} + \beta _1 + 8$$ b5 + b3 + 5*b2 + b1 + 8 $$\nu^{5}$$ $$=$$ $$\beta_{5} - 5\beta_{4} + 7\beta_{3} + \beta_{2} + 24\beta _1 + 1$$ b5 - 5*b4 + 7*b3 + b2 + 24*b1 + 1

## 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
 0.435907 −2.04394 −0.874884 1.51235 2.33401 −0.363441
−1.85816 −2.29407 1.45276 −0.197362 4.26275 0 1.01686 2.26275 0.366731
1.2 −1.55469 0.489252 0.417051 −1.19151 −0.760633 0 2.46099 −2.76063 1.85243
1.3 0.268125 1.14301 −1.92811 −2.56175 0.306470 0 −1.05323 −1.69353 −0.686871
1.4 0.851125 −0.661223 −1.27559 3.44148 −0.562784 0 −2.78793 −2.56278 2.92913
1.5 1.90556 −0.428448 1.63116 −1.47313 −0.816433 0 −0.702849 −2.81643 −2.80714
1.6 2.38804 2.75148 3.70272 0.982280 6.57063 0 4.06616 4.57063 2.34572
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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.cf 6
7.b odd 2 1 8281.2.a.ce 6
7.d odd 6 2 1183.2.e.g 12
13.b even 2 1 8281.2.a.ca 6
13.e even 6 2 637.2.f.j 12
91.b odd 2 1 8281.2.a.bz 6
91.k even 6 2 637.2.h.l 12
91.l odd 6 2 91.2.h.b yes 12
91.p odd 6 2 91.2.g.b 12
91.s odd 6 2 1183.2.e.h 12
91.t odd 6 2 637.2.f.k 12
91.u even 6 2 637.2.g.l 12
273.y even 6 2 819.2.n.d 12
273.br even 6 2 819.2.s.d 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.g.b 12 91.p odd 6 2
91.2.h.b yes 12 91.l odd 6 2
637.2.f.j 12 13.e even 6 2
637.2.f.k 12 91.t odd 6 2
637.2.g.l 12 91.u even 6 2
637.2.h.l 12 91.k even 6 2
819.2.n.d 12 273.y even 6 2
819.2.s.d 12 273.br even 6 2
1183.2.e.g 12 7.d odd 6 2
1183.2.e.h 12 91.s odd 6 2
8281.2.a.bz 6 91.b odd 2 1
8281.2.a.ca 6 13.b even 2 1
8281.2.a.ce 6 7.b odd 2 1
8281.2.a.cf 6 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8281))$$:

 $$T_{2}^{6} - 2T_{2}^{5} - 6T_{2}^{4} + 11T_{2}^{3} + 8T_{2}^{2} - 14T_{2} + 3$$ T2^6 - 2*T2^5 - 6*T2^4 + 11*T2^3 + 8*T2^2 - 14*T2 + 3 $$T_{3}^{6} - T_{3}^{5} - 7T_{3}^{4} + 4T_{3}^{3} + 6T_{3}^{2} - T_{3} - 1$$ T3^6 - T3^5 - 7*T3^4 + 4*T3^3 + 6*T3^2 - T3 - 1 $$T_{5}^{6} + T_{5}^{5} - 11T_{5}^{4} - 18T_{5}^{3} + 6T_{5}^{2} + 17T_{5} + 3$$ T5^6 + T5^5 - 11*T5^4 - 18*T5^3 + 6*T5^2 + 17*T5 + 3 $$T_{11}^{6} - 4T_{11}^{5} - 21T_{11}^{4} + 76T_{11}^{3} + 81T_{11}^{2} - 207T_{11} + 81$$ T11^6 - 4*T11^5 - 21*T11^4 + 76*T11^3 + 81*T11^2 - 207*T11 + 81 $$T_{17}^{6} - 5T_{17}^{5} - 12T_{17}^{4} + 14T_{17}^{3} + 20T_{17}^{2} - 8T_{17} - 9$$ T17^6 - 5*T17^5 - 12*T17^4 + 14*T17^3 + 20*T17^2 - 8*T17 - 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 2 T^{5} + \cdots + 3$$
$3$ $$T^{6} - T^{5} - 7 T^{4} + \cdots - 1$$
$5$ $$T^{6} + T^{5} - 11 T^{4} + \cdots + 3$$
$7$ $$T^{6}$$
$11$ $$T^{6} - 4 T^{5} + \cdots + 81$$
$13$ $$T^{6}$$
$17$ $$T^{6} - 5 T^{5} + \cdots - 9$$
$19$ $$T^{6} - T^{5} + \cdots + 873$$
$23$ $$T^{6} - T^{5} + \cdots - 24387$$
$29$ $$T^{6} + 3 T^{5} + \cdots - 201$$
$31$ $$T^{6} + 16 T^{5} + \cdots - 2477$$
$37$ $$T^{6} + 13 T^{5} + \cdots - 13477$$
$41$ $$T^{6} - 8 T^{5} + \cdots + 2043$$
$43$ $$T^{6} - 11 T^{5} + \cdots + 37$$
$47$ $$T^{6} - T^{5} + \cdots - 17847$$
$53$ $$T^{6} - 2 T^{5} + \cdots - 69$$
$59$ $$T^{6} + 13 T^{5} + \cdots + 9123$$
$61$ $$T^{6} + 5 T^{5} + \cdots + 32481$$
$67$ $$T^{6} + 11 T^{5} + \cdots - 16623$$
$71$ $$T^{6} - 6 T^{5} + \cdots + 23043$$
$73$ $$T^{6} - 30 T^{5} + \cdots - 14029$$
$79$ $$T^{6} + 7 T^{5} + \cdots + 10529$$
$83$ $$T^{6} + 27 T^{5} + \cdots + 2673$$
$89$ $$T^{6} + 4 T^{5} + \cdots - 304479$$
$97$ $$T^{6} - 35 T^{5} + \cdots - 3899$$