# Properties

 Label 7360.2.a.bx Level $7360$ Weight $2$ Character orbit 7360.a Self dual yes Analytic conductor $58.770$ 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] = [7360,2,Mod(1,7360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7360, 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("7360.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7360 = 2^{6} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7360.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: $$58.7698958877$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.1573.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 7x + 2$$ x^3 - x^2 - 7*x + 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3680) 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.

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

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 5$$ v^2 - v - 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 5$$ b2 + b1 + 5

## 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
 3.06871 0.277754 −2.34646
0 −3.06871 0 −1.00000 0 −4.06871 0 6.41697 0
1.2 0 −0.277754 0 −1.00000 0 −1.27775 0 −2.92285 0
1.3 0 2.34646 0 −1.00000 0 1.34646 0 2.50588 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
$$2$$ $$1$$
$$5$$ $$1$$
$$23$$ $$1$$

## 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 7360.2.a.bx 3
4.b odd 2 1 7360.2.a.cd 3
8.b even 2 1 3680.2.a.r yes 3
8.d odd 2 1 3680.2.a.q 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3680.2.a.q 3 8.d odd 2 1
3680.2.a.r yes 3 8.b even 2 1
7360.2.a.bx 3 1.a even 1 1 trivial
7360.2.a.cd 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7360))$$:

 $$T_{3}^{3} + T_{3}^{2} - 7T_{3} - 2$$ T3^3 + T3^2 - 7*T3 - 2 $$T_{7}^{3} + 4T_{7}^{2} - 2T_{7} - 7$$ T7^3 + 4*T7^2 - 2*T7 - 7 $$T_{11}^{3} + T_{11}^{2} - 7T_{11} - 2$$ T11^3 + T11^2 - 7*T11 - 2 $$T_{13}^{3} + 3T_{13}^{2} - 19T_{13} - 32$$ T13^3 + 3*T13^2 - 19*T13 - 32 $$T_{17}^{3} - 2T_{17}^{2} - 28T_{17} + 49$$ T17^3 - 2*T17^2 - 28*T17 + 49

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} + T^{2} - 7T - 2$$
$5$ $$(T + 1)^{3}$$
$7$ $$T^{3} + 4 T^{2} - 2 T - 7$$
$11$ $$T^{3} + T^{2} - 7T - 2$$
$13$ $$T^{3} + 3 T^{2} - 19 T - 32$$
$17$ $$T^{3} - 2 T^{2} - 28 T + 49$$
$19$ $$T^{3} - 3 T^{2} - 19 T + 32$$
$23$ $$(T + 1)^{3}$$
$29$ $$T^{3} + 17 T^{2} + 78 T + 52$$
$31$ $$T^{3} + 16 T^{2} + 78 T + 113$$
$37$ $$T^{3} + 9 T^{2} - 28 T + 16$$
$41$ $$T^{3} - 16 T^{2} - 10 T + 701$$
$43$ $$(T + 4)^{3}$$
$47$ $$T^{3} + 2 T^{2} - 116 T + 160$$
$53$ $$T^{3} + 17 T^{2} + 56 T - 124$$
$59$ $$T^{3} - 9 T^{2} - 94 T + 820$$
$61$ $$T^{3} + 15 T^{2} - 145 T - 2174$$
$67$ $$T^{3} - 23 T^{2} + 136 T - 64$$
$71$ $$T^{3} - 16 T^{2} - 76 T + 1493$$
$73$ $$T^{3} - 14 T^{2} + 36 T + 32$$
$79$ $$(T - 8)^{3}$$
$83$ $$T^{3} - 7 T^{2} - 24 T + 4$$
$89$ $$T^{3} - 10 T^{2} - 128 T + 1120$$
$97$ $$T^{3} - 9 T^{2} - 39 T + 182$$