# Properties

 Label 8281.2.a.be Level $8281$ Weight $2$ Character orbit 8281.a Self dual yes Analytic conductor $66.124$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

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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: $$3$$ Coefficient field: 3.3.148.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} - \beta_{2} q^{3} + (\beta_{2} - \beta_1 + 1) q^{4} + ( - \beta_{2} + 1) q^{5} + (\beta_{2} - \beta_1 + 1) q^{6} + ( - 2 \beta_{2} - 2) q^{8} + ( - \beta_{2} - \beta_1) q^{9}+O(q^{10})$$ q + (b1 - 1) * q^2 - b2 * q^3 + (b2 - b1 + 1) * q^4 + (-b2 + 1) * q^5 + (b2 - b1 + 1) * q^6 + (-2*b2 - 2) * q^8 + (-b2 - b1) * q^9 $$q + (\beta_1 - 1) q^{2} - \beta_{2} q^{3} + (\beta_{2} - \beta_1 + 1) q^{4} + ( - \beta_{2} + 1) q^{5} + (\beta_{2} - \beta_1 + 1) q^{6} + ( - 2 \beta_{2} - 2) q^{8} + ( - \beta_{2} - \beta_1) q^{9} + \beta_{2} q^{10} + (\beta_1 - 3) q^{11} + (2 \beta_1 - 4) q^{12} + ( - 2 \beta_{2} - \beta_1 + 3) q^{15} + ( - 2 \beta_1 + 2) q^{16} + (\beta_{2} + 2 \beta_1 - 2) q^{17} + ( - \beta_1 - 1) q^{18} + (4 \beta_{2} + \beta_1) q^{19} + (\beta_{2} + \beta_1 - 3) q^{20} + (\beta_{2} - 3 \beta_1 + 5) q^{22} + (\beta_{2} + 3 \beta_1 - 2) q^{23} + ( - 2 \beta_1 + 6) q^{24} + ( - 3 \beta_{2} - \beta_1 - 1) q^{25} + (2 \beta_{2} + 2) q^{27} + ( - 2 \beta_{2} + 2 \beta_1 - 3) q^{29} + (\beta_{2} + \beta_1 - 3) q^{30} + ( - \beta_{2} + 2 \beta_1 + 3) q^{31} + (2 \beta_{2} + 2 \beta_1 - 2) q^{32} + (3 \beta_{2} - \beta_1 + 1) q^{33} + (\beta_{2} - \beta_1 + 5) q^{34} + (\beta_{2} + \beta_1 - 1) q^{36} + ( - 3 \beta_1 - 3) q^{37} + ( - 3 \beta_{2} + 4 \beta_1 - 2) q^{38} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{40} + ( - 4 \beta_1 + 2) q^{41} + ( - 2 \beta_{2} - 2 \beta_1 + 5) q^{43} + ( - 4 \beta_{2} + 4 \beta_1 - 6) q^{44} + ( - 2 \beta_{2} - \beta_1 + 2) q^{45} + (2 \beta_{2} - \beta_1 + 7) q^{46} + (2 \beta_{2} + \beta_1 + 6) q^{47} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{48} + (2 \beta_{2} - 4 \beta_1 + 2) q^{50} + (3 \beta_{2} - \beta_1 - 1) q^{51} + ( - \beta_{2} + \beta_1) q^{53} + ( - 2 \beta_{2} + 4 \beta_1 - 4) q^{54} + (3 \beta_{2} - 2) q^{55} + (4 \beta_{2} + 3 \beta_1 - 11) q^{57} + (4 \beta_{2} - 5 \beta_1 + 9) q^{58} + ( - 3 \beta_{2} - \beta_1 + 1) q^{59} + (4 \beta_{2} - 2) q^{60} + (2 \beta_{2} - 2 \beta_1 - 4) q^{61} + (3 \beta_{2} + 2 \beta_1 + 2) q^{62} + 4 \beta_1 q^{64} + ( - 4 \beta_{2} + 4 \beta_1 - 6) q^{66} + (2 \beta_{2} + 4) q^{67} + ( - 4 \beta_{2} + 2 \beta_1 - 4) q^{68} + (3 \beta_{2} - 2 \beta_1) q^{69} + (2 \beta_{2} - 5 \beta_1 - 3) q^{71} + (2 \beta_1 + 4) q^{72} + ( - 4 \beta_{2} + 3 \beta_1 - 4) q^{73} + ( - 3 \beta_{2} - 3 \beta_1 - 3) q^{74} + ( - 2 \beta_{2} - 2 \beta_1 + 8) q^{75} + ( - \beta_{2} - 7 \beta_1 + 13) q^{76} + (4 \beta_{2} - 2 \beta_1 + 5) q^{79} - 2 \beta_{2} q^{80} + (3 \beta_{2} + 5 \beta_1 - 6) q^{81} + ( - 4 \beta_{2} + 2 \beta_1 - 10) q^{82} + (5 \beta_{2} + 4 \beta_1 - 1) q^{83} + (4 \beta_{2} + \beta_1 - 3) q^{85} + (3 \beta_1 - 7) q^{86} + (\beta_{2} - 4 \beta_1 + 8) q^{87} + (6 \beta_{2} - 4 \beta_1 + 8) q^{88} + ( - 2 \beta_{2} + 3 \beta_1) q^{89} + (\beta_{2} - 2) q^{90} + ( - 5 \beta_{2} + 3 \beta_1 - 7) q^{92} + ( - 4 \beta_{2} - 3 \beta_1 + 5) q^{93} + ( - \beta_{2} + 8 \beta_1 - 6) q^{94} + (8 \beta_{2} + 4 \beta_1 - 11) q^{95} + (4 \beta_{2} - 4) q^{96} + ( - 2 \beta_{2} - 9 \beta_1 + 8) q^{97} + (2 \beta_{2} + \beta_1 - 1) q^{99}+O(q^{100})$$ q + (b1 - 1) * q^2 - b2 * q^3 + (b2 - b1 + 1) * q^4 + (-b2 + 1) * q^5 + (b2 - b1 + 1) * q^6 + (-2*b2 - 2) * q^8 + (-b2 - b1) * q^9 + b2 * q^10 + (b1 - 3) * q^11 + (2*b1 - 4) * q^12 + (-2*b2 - b1 + 3) * q^15 + (-2*b1 + 2) * q^16 + (b2 + 2*b1 - 2) * q^17 + (-b1 - 1) * q^18 + (4*b2 + b1) * q^19 + (b2 + b1 - 3) * q^20 + (b2 - 3*b1 + 5) * q^22 + (b2 + 3*b1 - 2) * q^23 + (-2*b1 + 6) * q^24 + (-3*b2 - b1 - 1) * q^25 + (2*b2 + 2) * q^27 + (-2*b2 + 2*b1 - 3) * q^29 + (b2 + b1 - 3) * q^30 + (-b2 + 2*b1 + 3) * q^31 + (2*b2 + 2*b1 - 2) * q^32 + (3*b2 - b1 + 1) * q^33 + (b2 - b1 + 5) * q^34 + (b2 + b1 - 1) * q^36 + (-3*b1 - 3) * q^37 + (-3*b2 + 4*b1 - 2) * q^38 + (-2*b2 - 2*b1 + 4) * q^40 + (-4*b1 + 2) * q^41 + (-2*b2 - 2*b1 + 5) * q^43 + (-4*b2 + 4*b1 - 6) * q^44 + (-2*b2 - b1 + 2) * q^45 + (2*b2 - b1 + 7) * q^46 + (2*b2 + b1 + 6) * q^47 + (-2*b2 + 2*b1 - 2) * q^48 + (2*b2 - 4*b1 + 2) * q^50 + (3*b2 - b1 - 1) * q^51 + (-b2 + b1) * q^53 + (-2*b2 + 4*b1 - 4) * q^54 + (3*b2 - 2) * q^55 + (4*b2 + 3*b1 - 11) * q^57 + (4*b2 - 5*b1 + 9) * q^58 + (-3*b2 - b1 + 1) * q^59 + (4*b2 - 2) * q^60 + (2*b2 - 2*b1 - 4) * q^61 + (3*b2 + 2*b1 + 2) * q^62 + 4*b1 * q^64 + (-4*b2 + 4*b1 - 6) * q^66 + (2*b2 + 4) * q^67 + (-4*b2 + 2*b1 - 4) * q^68 + (3*b2 - 2*b1) * q^69 + (2*b2 - 5*b1 - 3) * q^71 + (2*b1 + 4) * q^72 + (-4*b2 + 3*b1 - 4) * q^73 + (-3*b2 - 3*b1 - 3) * q^74 + (-2*b2 - 2*b1 + 8) * q^75 + (-b2 - 7*b1 + 13) * q^76 + (4*b2 - 2*b1 + 5) * q^79 - 2*b2 * q^80 + (3*b2 + 5*b1 - 6) * q^81 + (-4*b2 + 2*b1 - 10) * q^82 + (5*b2 + 4*b1 - 1) * q^83 + (4*b2 + b1 - 3) * q^85 + (3*b1 - 7) * q^86 + (b2 - 4*b1 + 8) * q^87 + (6*b2 - 4*b1 + 8) * q^88 + (-2*b2 + 3*b1) * q^89 + (b2 - 2) * q^90 + (-5*b2 + 3*b1 - 7) * q^92 + (-4*b2 - 3*b1 + 5) * q^93 + (-b2 + 8*b1 - 6) * q^94 + (8*b2 + 4*b1 - 11) * q^95 + (4*b2 - 4) * q^96 + (-2*b2 - 9*b1 + 8) * q^97 + (2*b2 + b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} + 2 q^{4} + 3 q^{5} + 2 q^{6} - 6 q^{8} - q^{9}+O(q^{10})$$ 3 * q - 2 * q^2 + 2 * q^4 + 3 * q^5 + 2 * q^6 - 6 * q^8 - q^9 $$3 q - 2 q^{2} + 2 q^{4} + 3 q^{5} + 2 q^{6} - 6 q^{8} - q^{9} - 8 q^{11} - 10 q^{12} + 8 q^{15} + 4 q^{16} - 4 q^{17} - 4 q^{18} + q^{19} - 8 q^{20} + 12 q^{22} - 3 q^{23} + 16 q^{24} - 4 q^{25} + 6 q^{27} - 7 q^{29} - 8 q^{30} + 11 q^{31} - 4 q^{32} + 2 q^{33} + 14 q^{34} - 2 q^{36} - 12 q^{37} - 2 q^{38} + 10 q^{40} + 2 q^{41} + 13 q^{43} - 14 q^{44} + 5 q^{45} + 20 q^{46} + 19 q^{47} - 4 q^{48} + 2 q^{50} - 4 q^{51} + q^{53} - 8 q^{54} - 6 q^{55} - 30 q^{57} + 22 q^{58} + 2 q^{59} - 6 q^{60} - 14 q^{61} + 8 q^{62} + 4 q^{64} - 14 q^{66} + 12 q^{67} - 10 q^{68} - 2 q^{69} - 14 q^{71} + 14 q^{72} - 9 q^{73} - 12 q^{74} + 22 q^{75} + 32 q^{76} + 13 q^{79} - 13 q^{81} - 28 q^{82} + q^{83} - 8 q^{85} - 18 q^{86} + 20 q^{87} + 20 q^{88} + 3 q^{89} - 6 q^{90} - 18 q^{92} + 12 q^{93} - 10 q^{94} - 29 q^{95} - 12 q^{96} + 15 q^{97} - 2 q^{99}+O(q^{100})$$ 3 * q - 2 * q^2 + 2 * q^4 + 3 * q^5 + 2 * q^6 - 6 * q^8 - q^9 - 8 * q^11 - 10 * q^12 + 8 * q^15 + 4 * q^16 - 4 * q^17 - 4 * q^18 + q^19 - 8 * q^20 + 12 * q^22 - 3 * q^23 + 16 * q^24 - 4 * q^25 + 6 * q^27 - 7 * q^29 - 8 * q^30 + 11 * q^31 - 4 * q^32 + 2 * q^33 + 14 * q^34 - 2 * q^36 - 12 * q^37 - 2 * q^38 + 10 * q^40 + 2 * q^41 + 13 * q^43 - 14 * q^44 + 5 * q^45 + 20 * q^46 + 19 * q^47 - 4 * q^48 + 2 * q^50 - 4 * q^51 + q^53 - 8 * q^54 - 6 * q^55 - 30 * q^57 + 22 * q^58 + 2 * q^59 - 6 * q^60 - 14 * q^61 + 8 * q^62 + 4 * q^64 - 14 * q^66 + 12 * q^67 - 10 * q^68 - 2 * q^69 - 14 * q^71 + 14 * q^72 - 9 * q^73 - 12 * q^74 + 22 * q^75 + 32 * q^76 + 13 * q^79 - 13 * q^81 - 28 * q^82 + q^83 - 8 * q^85 - 18 * q^86 + 20 * q^87 + 20 * q^88 + 3 * q^89 - 6 * q^90 - 18 * q^92 + 12 * q^93 - 10 * q^94 - 29 * q^95 - 12 * q^96 + 15 * q^97 - 2 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ v^2 - v - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 2$$ b2 + b1 + 2

## 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
 −1.48119 0.311108 2.17009
−2.48119 −1.67513 4.15633 −0.675131 4.15633 0 −5.35026 −0.193937 1.67513
1.2 −0.688892 2.21432 −1.52543 3.21432 −1.52543 0 2.42864 1.90321 −2.21432
1.3 1.17009 −0.539189 −0.630898 0.460811 −0.630898 0 −3.07838 −2.70928 0.539189
 $$n$$: 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.be 3
7.b odd 2 1 1183.2.a.h 3
13.b even 2 1 8281.2.a.bi 3
13.d odd 4 2 637.2.c.d 6
91.b odd 2 1 1183.2.a.j 3
91.i even 4 2 91.2.c.a 6
91.z odd 12 4 637.2.r.d 12
91.bb even 12 4 637.2.r.e 12
273.o odd 4 2 819.2.c.b 6
364.p odd 4 2 1456.2.k.c 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.c.a 6 91.i even 4 2
637.2.c.d 6 13.d odd 4 2
637.2.r.d 12 91.z odd 12 4
637.2.r.e 12 91.bb even 12 4
819.2.c.b 6 273.o odd 4 2
1183.2.a.h 3 7.b odd 2 1
1183.2.a.j 3 91.b odd 2 1
1456.2.k.c 6 364.p odd 4 2
8281.2.a.be 3 1.a even 1 1 trivial
8281.2.a.bi 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8281))$$:

 $$T_{2}^{3} + 2T_{2}^{2} - 2T_{2} - 2$$ T2^3 + 2*T2^2 - 2*T2 - 2 $$T_{3}^{3} - 4T_{3} - 2$$ T3^3 - 4*T3 - 2 $$T_{5}^{3} - 3T_{5}^{2} - T_{5} + 1$$ T5^3 - 3*T5^2 - T5 + 1 $$T_{11}^{3} + 8T_{11}^{2} + 18T_{11} + 10$$ T11^3 + 8*T11^2 + 18*T11 + 10 $$T_{17}^{3} + 4T_{17}^{2} - 8T_{17} - 34$$ T17^3 + 4*T17^2 - 8*T17 - 34

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + 2 T^{2} - 2 T - 2$$
$3$ $$T^{3} - 4T - 2$$
$5$ $$T^{3} - 3T^{2} - T + 1$$
$7$ $$T^{3}$$
$11$ $$T^{3} + 8 T^{2} + 18 T + 10$$
$13$ $$T^{3}$$
$17$ $$T^{3} + 4 T^{2} - 8 T - 34$$
$19$ $$T^{3} - T^{2} - 59 T + 193$$
$23$ $$T^{3} + 3 T^{2} - 25 T - 79$$
$29$ $$T^{3} + 7 T^{2} - 21 T + 5$$
$31$ $$T^{3} - 11 T^{2} + 19 T + 65$$
$37$ $$T^{3} + 12 T^{2} + 18 T - 54$$
$41$ $$T^{3} - 2 T^{2} - 52 T + 40$$
$43$ $$T^{3} - 13 T^{2} + 35 T + 17$$
$47$ $$T^{3} - 19 T^{2} + 105 T - 137$$
$53$ $$T^{3} - T^{2} - 9T + 13$$
$59$ $$T^{3} - 2 T^{2} - 32 T - 52$$
$61$ $$T^{3} + 14 T^{2} + 28 T - 152$$
$67$ $$T^{3} - 12 T^{2} + 32 T + 16$$
$71$ $$T^{3} + 14 T^{2} - 54 T - 890$$
$73$ $$T^{3} + 9 T^{2} - 91 T + 31$$
$79$ $$T^{3} - 13 T^{2} - 37 T + 185$$
$83$ $$T^{3} - T^{2} - 113 T + 163$$
$89$ $$T^{3} - 3 T^{2} - 55 T + 227$$
$97$ $$T^{3} - 15 T^{2} - 175 T + 2183$$
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