# Properties

 Label 7360.2.a.bs Level $7360$ Weight $2$ Character orbit 7360.a Self dual yes Analytic conductor $58.770$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.73205 1.73205
0 1.00000 0 1.00000 0 −2.73205 0 −2.00000 0
1.2 0 1.00000 0 1.00000 0 0.732051 0 −2.00000 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.bs 2
4.b odd 2 1 7360.2.a.bg 2
8.b even 2 1 3680.2.a.l 2
8.d odd 2 1 3680.2.a.n yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3680.2.a.l 2 8.b even 2 1
3680.2.a.n yes 2 8.d odd 2 1
7360.2.a.bg 2 4.b odd 2 1
7360.2.a.bs 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7360))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} + 2T - 2$$
$11$ $$T^{2} + 2T - 2$$
$13$ $$T^{2} - 3$$
$17$ $$T^{2} - 4T - 8$$
$19$ $$T^{2} + 2T - 2$$
$23$ $$(T - 1)^{2}$$
$29$ $$T^{2} + 2T - 47$$
$31$ $$T^{2} - 4T + 1$$
$37$ $$T^{2} - 2T - 2$$
$41$ $$T^{2} + 2T - 11$$
$43$ $$T^{2} + 10T + 22$$
$47$ $$T^{2} + 2T - 47$$
$53$ $$T^{2} - 8T + 4$$
$59$ $$T^{2} - 108$$
$61$ $$T^{2} - 2T - 74$$
$67$ $$T^{2} + 6T - 66$$
$71$ $$T^{2} - 147$$
$73$ $$T^{2} + 28T + 193$$
$79$ $$T^{2} + 14T - 26$$
$83$ $$(T + 4)^{2}$$
$89$ $$T^{2} + 18T + 6$$
$97$ $$T^{2} - 2T - 74$$