# Properties

 Label 7360.2.a.be 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: $$\Q(\sqrt{7})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 7$$ x^2 - 7 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 $$\beta = \sqrt{7}$$. 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 $$q - q^{3} - q^{5} + ( - \beta - 1) q^{7} - 2 q^{9} + ( - \beta - 1) q^{11} + \beta q^{13} + q^{15} + (2 \beta - 2) q^{17} + ( - \beta + 3) q^{19} + (\beta + 1) q^{21} - q^{23} + q^{25} + 5 q^{27} + 3 q^{29} + (\beta + 2) q^{31} + (\beta + 1) q^{33} + (\beta + 1) q^{35} + ( - \beta + 7) q^{37} - \beta q^{39} + ( - 2 \beta - 1) q^{41} + ( - \beta - 1) q^{43} + 2 q^{45} - 3 q^{47} + (2 \beta + 1) q^{49} + ( - 2 \beta + 2) q^{51} + 2 \beta q^{53} + (\beta + 1) q^{55} + (\beta - 3) q^{57} - 2 \beta q^{59} + (\beta + 11) q^{61} + (2 \beta + 2) q^{63} - \beta q^{65} + (3 \beta + 1) q^{67} + q^{69} + ( - \beta - 4) q^{71} + (\beta - 2) q^{73} - q^{75} + (2 \beta + 8) q^{77} + (\beta + 1) q^{79} + q^{81} + (4 \beta - 4) q^{83} + ( - 2 \beta + 2) q^{85} - 3 q^{87} + (\beta + 1) q^{89} + ( - \beta - 7) q^{91} + ( - \beta - 2) q^{93} + (\beta - 3) q^{95} + ( - 3 \beta - 5) q^{97} + (2 \beta + 2) q^{99} +O(q^{100})$$ 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 + 3) * q^19 + (b + 1) * q^21 - q^23 + q^25 + 5 * q^27 + 3 * q^29 + (b + 2) * q^31 + (b + 1) * q^33 + (b + 1) * q^35 + (-b + 7) * q^37 - b * q^39 + (-2*b - 1) * q^41 + (-b - 1) * q^43 + 2 * q^45 - 3 * q^47 + (2*b + 1) * q^49 + (-2*b + 2) * q^51 + 2*b * q^53 + (b + 1) * q^55 + (b - 3) * q^57 - 2*b * q^59 + (b + 11) * q^61 + (2*b + 2) * q^63 - b * q^65 + (3*b + 1) * q^67 + q^69 + (-b - 4) * q^71 + (b - 2) * q^73 - q^75 + (2*b + 8) * q^77 + (b + 1) * q^79 + q^81 + (4*b - 4) * q^83 + (-2*b + 2) * q^85 - 3 * q^87 + (b + 1) * q^89 + (-b - 7) * q^91 + (-b - 2) * q^93 + (b - 3) * q^95 + (-3*b - 5) * 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 $$2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} - 4 q^{9} - 2 q^{11} + 2 q^{15} - 4 q^{17} + 6 q^{19} + 2 q^{21} - 2 q^{23} + 2 q^{25} + 10 q^{27} + 6 q^{29} + 4 q^{31} + 2 q^{33} + 2 q^{35} + 14 q^{37} - 2 q^{41} - 2 q^{43} + 4 q^{45} - 6 q^{47} + 2 q^{49} + 4 q^{51} + 2 q^{55} - 6 q^{57} + 22 q^{61} + 4 q^{63} + 2 q^{67} + 2 q^{69} - 8 q^{71} - 4 q^{73} - 2 q^{75} + 16 q^{77} + 2 q^{79} + 2 q^{81} - 8 q^{83} + 4 q^{85} - 6 q^{87} + 2 q^{89} - 14 q^{91} - 4 q^{93} - 6 q^{95} - 10 q^{97} + 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 - 4 * q^9 - 2 * q^11 + 2 * q^15 - 4 * q^17 + 6 * q^19 + 2 * q^21 - 2 * q^23 + 2 * q^25 + 10 * q^27 + 6 * q^29 + 4 * q^31 + 2 * q^33 + 2 * q^35 + 14 * q^37 - 2 * q^41 - 2 * q^43 + 4 * q^45 - 6 * q^47 + 2 * q^49 + 4 * q^51 + 2 * q^55 - 6 * q^57 + 22 * q^61 + 4 * q^63 + 2 * q^67 + 2 * q^69 - 8 * q^71 - 4 * q^73 - 2 * q^75 + 16 * q^77 + 2 * q^79 + 2 * q^81 - 8 * q^83 + 4 * q^85 - 6 * q^87 + 2 * q^89 - 14 * q^91 - 4 * q^93 - 6 * q^95 - 10 * 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
 2.64575 −2.64575
0 −1.00000 0 −1.00000 0 −3.64575 0 −2.00000 0
1.2 0 −1.00000 0 −1.00000 0 1.64575 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.be 2
4.b odd 2 1 7360.2.a.br 2
8.b even 2 1 3680.2.a.o yes 2
8.d odd 2 1 3680.2.a.m 2

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

 $$T_{3} + 1$$ T3 + 1 $$T_{7}^{2} + 2T_{7} - 6$$ T7^2 + 2*T7 - 6 $$T_{11}^{2} + 2T_{11} - 6$$ T11^2 + 2*T11 - 6 $$T_{13}^{2} - 7$$ T13^2 - 7 $$T_{17}^{2} + 4T_{17} - 24$$ T17^2 + 4*T17 - 24

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} + 2T - 6$$
$11$ $$T^{2} + 2T - 6$$
$13$ $$T^{2} - 7$$
$17$ $$T^{2} + 4T - 24$$
$19$ $$T^{2} - 6T + 2$$
$23$ $$(T + 1)^{2}$$
$29$ $$(T - 3)^{2}$$
$31$ $$T^{2} - 4T - 3$$
$37$ $$T^{2} - 14T + 42$$
$41$ $$T^{2} + 2T - 27$$
$43$ $$T^{2} + 2T - 6$$
$47$ $$(T + 3)^{2}$$
$53$ $$T^{2} - 28$$
$59$ $$T^{2} - 28$$
$61$ $$T^{2} - 22T + 114$$
$67$ $$T^{2} - 2T - 62$$
$71$ $$T^{2} + 8T + 9$$
$73$ $$T^{2} + 4T - 3$$
$79$ $$T^{2} - 2T - 6$$
$83$ $$T^{2} + 8T - 96$$
$89$ $$T^{2} - 2T - 6$$
$97$ $$T^{2} + 10T - 38$$