# Properties

 Label 9280.2.a.bk Level $9280$ Weight $2$ Character orbit 9280.a Self dual yes Analytic conductor $74.101$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9280,2,Mod(1,9280)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("9280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9280 = 2^{6} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9280.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: $$74.1011730757$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.564.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 5x + 3$$ x^3 - x^2 - 5*x + 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 580) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - 1) q^{3} + q^{5} + (\beta_{2} - 1) q^{7} + (\beta_1 + 4) q^{9}+O(q^{10})$$ q + (-b2 - 1) * q^3 + q^5 + (b2 - 1) * q^7 + (b1 + 4) * q^9 $$q + ( - \beta_{2} - 1) q^{3} + q^{5} + (\beta_{2} - 1) q^{7} + (\beta_1 + 4) q^{9} + ( - \beta_{2} - 3) q^{11} + (\beta_1 + 1) q^{13} + ( - \beta_{2} - 1) q^{15} + ( - \beta_{2} - \beta_1) q^{17} + (\beta_{2} + \beta_1 - 2) q^{19} + (2 \beta_{2} - \beta_1 - 5) q^{21} + (\beta_{2} - \beta_1) q^{23} + q^{25} + ( - 2 \beta_{2} - 2 \beta_1 - 4) q^{27} + q^{29} + (\beta_{2} + \beta_1 + 2) q^{31} + (2 \beta_{2} + \beta_1 + 9) q^{33} + (\beta_{2} - 1) q^{35} + (3 \beta_{2} - \beta_1 + 4) q^{37} + ( - 2 \beta_{2} - 2 \beta_1 - 4) q^{39} + ( - \beta_1 + 3) q^{41} + (\beta_{2} - 2 \beta_1 - 5) q^{43} + (\beta_1 + 4) q^{45} + (\beta_{2} - 3) q^{47} + ( - 4 \beta_{2} + \beta_1) q^{49} + (3 \beta_1 + 9) q^{51} + (2 \beta_{2} - \beta_1 - 3) q^{53} + ( - \beta_{2} - 3) q^{55} + (2 \beta_{2} - 3 \beta_1 - 7) q^{57} + (2 \beta_{2} - 3 \beta_1 - 3) q^{59} + ( - 2 \beta_{2} + \beta_1 + 7) q^{61} + (5 \beta_{2} - 1) q^{63} + (\beta_1 + 1) q^{65} + ( - \beta_{2} + \beta_1 + 4) q^{67} + (2 \beta_{2} + \beta_1 - 3) q^{69} + ( - 2 \beta_{2} + \beta_1 - 3) q^{71} + (\beta_{2} - 2 \beta_1 - 1) q^{73} + ( - \beta_{2} - 1) q^{75} + ( - \beta_1 - 3) q^{77} + ( - \beta_{2} - 2 \beta_1 - 1) q^{79} + (4 \beta_{2} + 3 \beta_1 + 10) q^{81} + (3 \beta_{2} - 3) q^{83} + ( - \beta_{2} - \beta_1) q^{85} + ( - \beta_{2} - 1) q^{87} + (2 \beta_{2} - 2 \beta_1 + 6) q^{89} + (2 \beta_{2} + 2) q^{91} + ( - 2 \beta_{2} - 3 \beta_1 - 11) q^{93} + (\beta_{2} + \beta_1 - 2) q^{95} + ( - \beta_{2} - 13) q^{97} + ( - 5 \beta_{2} - 4 \beta_1 - 15) q^{99}+O(q^{100})$$ q + (-b2 - 1) * q^3 + q^5 + (b2 - 1) * q^7 + (b1 + 4) * q^9 + (-b2 - 3) * q^11 + (b1 + 1) * q^13 + (-b2 - 1) * q^15 + (-b2 - b1) * q^17 + (b2 + b1 - 2) * q^19 + (2*b2 - b1 - 5) * q^21 + (b2 - b1) * q^23 + q^25 + (-2*b2 - 2*b1 - 4) * q^27 + q^29 + (b2 + b1 + 2) * q^31 + (2*b2 + b1 + 9) * q^33 + (b2 - 1) * q^35 + (3*b2 - b1 + 4) * q^37 + (-2*b2 - 2*b1 - 4) * q^39 + (-b1 + 3) * q^41 + (b2 - 2*b1 - 5) * q^43 + (b1 + 4) * q^45 + (b2 - 3) * q^47 + (-4*b2 + b1) * q^49 + (3*b1 + 9) * q^51 + (2*b2 - b1 - 3) * q^53 + (-b2 - 3) * q^55 + (2*b2 - 3*b1 - 7) * q^57 + (2*b2 - 3*b1 - 3) * q^59 + (-2*b2 + b1 + 7) * q^61 + (5*b2 - 1) * q^63 + (b1 + 1) * q^65 + (-b2 + b1 + 4) * q^67 + (2*b2 + b1 - 3) * q^69 + (-2*b2 + b1 - 3) * q^71 + (b2 - 2*b1 - 1) * q^73 + (-b2 - 1) * q^75 + (-b1 - 3) * q^77 + (-b2 - 2*b1 - 1) * q^79 + (4*b2 + 3*b1 + 10) * q^81 + (3*b2 - 3) * q^83 + (-b2 - b1) * q^85 + (-b2 - 1) * q^87 + (2*b2 - 2*b1 + 6) * q^89 + (2*b2 + 2) * q^91 + (-2*b2 - 3*b1 - 11) * q^93 + (b2 + b1 - 2) * q^95 + (-b2 - 13) * q^97 + (-5*b2 - 4*b1 - 15) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{3} + 3 q^{5} - 4 q^{7} + 11 q^{9}+O(q^{10})$$ 3 * q - 2 * q^3 + 3 * q^5 - 4 * q^7 + 11 * q^9 $$3 q - 2 q^{3} + 3 q^{5} - 4 q^{7} + 11 q^{9} - 8 q^{11} + 2 q^{13} - 2 q^{15} + 2 q^{17} - 8 q^{19} - 16 q^{21} + 3 q^{25} - 8 q^{27} + 3 q^{29} + 4 q^{31} + 24 q^{33} - 4 q^{35} + 10 q^{37} - 8 q^{39} + 10 q^{41} - 14 q^{43} + 11 q^{45} - 10 q^{47} + 3 q^{49} + 24 q^{51} - 10 q^{53} - 8 q^{55} - 20 q^{57} - 8 q^{59} + 22 q^{61} - 8 q^{63} + 2 q^{65} + 12 q^{67} - 12 q^{69} - 8 q^{71} - 2 q^{73} - 2 q^{75} - 8 q^{77} + 23 q^{81} - 12 q^{83} + 2 q^{85} - 2 q^{87} + 18 q^{89} + 4 q^{91} - 28 q^{93} - 8 q^{95} - 38 q^{97} - 36 q^{99}+O(q^{100})$$ 3 * q - 2 * q^3 + 3 * q^5 - 4 * q^7 + 11 * q^9 - 8 * q^11 + 2 * q^13 - 2 * q^15 + 2 * q^17 - 8 * q^19 - 16 * q^21 + 3 * q^25 - 8 * q^27 + 3 * q^29 + 4 * q^31 + 24 * q^33 - 4 * q^35 + 10 * q^37 - 8 * q^39 + 10 * q^41 - 14 * q^43 + 11 * q^45 - 10 * q^47 + 3 * q^49 + 24 * q^51 - 10 * q^53 - 8 * q^55 - 20 * q^57 - 8 * q^59 + 22 * q^61 - 8 * q^63 + 2 * q^65 + 12 * q^67 - 12 * q^69 - 8 * q^71 - 2 * q^73 - 2 * q^75 - 8 * q^77 + 23 * q^81 - 12 * q^83 + 2 * q^85 - 2 * q^87 + 18 * q^89 + 4 * q^91 - 28 * q^93 - 8 * q^95 - 38 * q^97 - 36 * q^99

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

 $$\beta_{1}$$ $$=$$ $$2\nu - 1$$ 2*v - 1 $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4
 $$\nu$$ $$=$$ $$( \beta _1 + 1 ) / 2$$ (b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ b2 + 4

## 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
 2.51414 −2.08613 0.571993
0 −3.32088 0 1.00000 0 1.32088 0 8.02827 0
1.2 0 −1.35194 0 1.00000 0 −0.648061 0 −1.17226 0
1.3 0 2.67282 0 1.00000 0 −4.67282 0 4.14399 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$-1$$
$$29$$ $$-1$$

## 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 9280.2.a.bk 3
4.b odd 2 1 9280.2.a.bw 3
8.b even 2 1 580.2.a.c 3
8.d odd 2 1 2320.2.a.m 3
24.h odd 2 1 5220.2.a.x 3
40.f even 2 1 2900.2.a.g 3
40.i odd 4 2 2900.2.c.f 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
580.2.a.c 3 8.b even 2 1
2320.2.a.m 3 8.d odd 2 1
2900.2.a.g 3 40.f even 2 1
2900.2.c.f 6 40.i odd 4 2
5220.2.a.x 3 24.h odd 2 1
9280.2.a.bk 3 1.a even 1 1 trivial
9280.2.a.bw 3 4.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9280))$$:

 $$T_{3}^{3} + 2T_{3}^{2} - 8T_{3} - 12$$ T3^3 + 2*T3^2 - 8*T3 - 12 $$T_{7}^{3} + 4T_{7}^{2} - 4T_{7} - 4$$ T7^3 + 4*T7^2 - 4*T7 - 4 $$T_{11}^{3} + 8T_{11}^{2} + 12T_{11} - 12$$ T11^3 + 8*T11^2 + 12*T11 - 12 $$T_{13}^{3} - 2T_{13}^{2} - 20T_{13} + 24$$ T13^3 - 2*T13^2 - 20*T13 + 24 $$T_{19}^{3} + 8T_{19}^{2} - 16T_{19} - 164$$ T19^3 + 8*T19^2 - 16*T19 - 164

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} + 2 T^{2} + \cdots - 12$$
$5$ $$(T - 1)^{3}$$
$7$ $$T^{3} + 4 T^{2} + \cdots - 4$$
$11$ $$T^{3} + 8 T^{2} + \cdots - 12$$
$13$ $$T^{3} - 2 T^{2} + \cdots + 24$$
$17$ $$T^{3} - 2 T^{2} + \cdots + 108$$
$19$ $$T^{3} + 8 T^{2} + \cdots - 164$$
$23$ $$T^{3} - 24T - 36$$
$29$ $$(T - 1)^{3}$$
$31$ $$T^{3} - 4 T^{2} + \cdots - 36$$
$37$ $$T^{3} - 10 T^{2} + \cdots + 508$$
$41$ $$T^{3} - 10 T^{2} + \cdots + 24$$
$43$ $$T^{3} + 14 T^{2} + \cdots - 548$$
$47$ $$T^{3} + 10 T^{2} + \cdots + 12$$
$53$ $$T^{3} + 10 T^{2} + \cdots - 72$$
$59$ $$T^{3} + 8 T^{2} + \cdots - 1488$$
$61$ $$T^{3} - 22 T^{2} + \cdots - 104$$
$67$ $$T^{3} - 12 T^{2} + \cdots + 68$$
$71$ $$T^{3} + 8 T^{2} + \cdots - 144$$
$73$ $$T^{3} + 2 T^{2} + \cdots - 324$$
$79$ $$T^{3} - 108T + 244$$
$83$ $$T^{3} + 12 T^{2} + \cdots - 108$$
$89$ $$T^{3} - 18 T^{2} + \cdots + 72$$
$97$ $$T^{3} + 38 T^{2} + \cdots + 1908$$