# Properties

 Label 7360.2.a.bw 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: 3.3.229.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 4x - 1$$ x^3 - 4*x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 920) 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.

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

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3

## 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
 2.11491 −1.86081 −0.254102
0 −2.47283 0 −1.00000 0 0.527166 0 3.11491 0
1.2 0 −1.46260 0 −1.00000 0 1.53740 0 −0.860806 0
1.3 0 1.93543 0 −1.00000 0 4.93543 0 0.745898 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.bw 3
4.b odd 2 1 7360.2.a.cf 3
8.b even 2 1 920.2.a.i 3
8.d odd 2 1 1840.2.a.q 3
24.h odd 2 1 8280.2.a.bl 3
40.e odd 2 1 9200.2.a.ci 3
40.f even 2 1 4600.2.a.v 3
40.i odd 4 2 4600.2.e.q 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.i 3 8.b even 2 1
1840.2.a.q 3 8.d odd 2 1
4600.2.a.v 3 40.f even 2 1
4600.2.e.q 6 40.i odd 4 2
7360.2.a.bw 3 1.a even 1 1 trivial
7360.2.a.cf 3 4.b odd 2 1
8280.2.a.bl 3 24.h odd 2 1
9200.2.a.ci 3 40.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7360))$$:

 $$T_{3}^{3} + 2T_{3}^{2} - 4T_{3} - 7$$ T3^3 + 2*T3^2 - 4*T3 - 7 $$T_{7}^{3} - 7T_{7}^{2} + 11T_{7} - 4$$ T7^3 - 7*T7^2 + 11*T7 - 4 $$T_{11}^{3} + 3T_{11}^{2} - T_{11} - 2$$ T11^3 + 3*T11^2 - T11 - 2 $$T_{13}^{3} + 6T_{13}^{2} + 8T_{13} + 1$$ T13^3 + 6*T13^2 + 8*T13 + 1 $$T_{17}^{3} + 5T_{17}^{2} - 31T_{17} - 148$$ T17^3 + 5*T17^2 - 31*T17 - 148

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} + 2 T^{2} - 4 T - 7$$
$5$ $$(T + 1)^{3}$$
$7$ $$T^{3} - 7 T^{2} + 11 T - 4$$
$11$ $$T^{3} + 3T^{2} - T - 2$$
$13$ $$T^{3} + 6 T^{2} + 8 T + 1$$
$17$ $$T^{3} + 5 T^{2} - 31 T - 148$$
$19$ $$T^{3} + 7 T^{2} - 11 T - 106$$
$23$ $$(T + 1)^{3}$$
$29$ $$T^{3} - T^{2} - 90 T + 148$$
$31$ $$T^{3} - 10 T^{2} + 14 T + 53$$
$37$ $$T^{3} + 2 T^{2} - 128 T - 512$$
$41$ $$T^{3} + 10 T^{2} - 48 T - 481$$
$43$ $$T^{3} + 12 T^{2} - 16 T - 256$$
$47$ $$T^{3} - T^{2} - 6T + 4$$
$53$ $$T^{3} + 10 T^{2} - 52 T - 8$$
$59$ $$T^{3} + 10 T^{2} - 124 T - 1184$$
$61$ $$T^{3} - 13 T^{2} - 29 T + 214$$
$67$ $$T^{3} - 6 T^{2} - 160 T + 1184$$
$71$ $$T^{3} - 10 T^{2} - 100 T + 875$$
$73$ $$T^{3} - 7 T^{2} - 34 T + 236$$
$79$ $$T^{3} - 14 T^{2} + 16 T + 224$$
$83$ $$T^{3} + 16 T^{2} + 60 T + 32$$
$89$ $$T^{3} + 20 T^{2} + 32 T - 256$$
$97$ $$T^{3} - 5 T^{2} - 23 T + 56$$