# Properties

 Label 7360.2.a.i Level $7360$ Weight $2$ Character orbit 7360.a Self dual yes Analytic conductor $58.770$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 460) 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)

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{5} - 4 q^{7} - 2 q^{9}+O(q^{10})$$ q - q^3 + q^5 - 4 * q^7 - 2 * q^9 $$q - q^{3} + q^{5} - 4 q^{7} - 2 q^{9} + 6 q^{11} + q^{13} - q^{15} - 2 q^{19} + 4 q^{21} + q^{23} + q^{25} + 5 q^{27} - 9 q^{29} + 5 q^{31} - 6 q^{33} - 4 q^{35} - 2 q^{37} - q^{39} - 9 q^{41} + 4 q^{43} - 2 q^{45} - 3 q^{47} + 9 q^{49} + 6 q^{53} + 6 q^{55} + 2 q^{57} - 2 q^{61} + 8 q^{63} + q^{65} + 10 q^{67} - q^{69} - 3 q^{71} - 7 q^{73} - q^{75} - 24 q^{77} - 10 q^{79} + q^{81} + 12 q^{83} + 9 q^{87} - 4 q^{91} - 5 q^{93} - 2 q^{95} + 8 q^{97} - 12 q^{99}+O(q^{100})$$ q - q^3 + q^5 - 4 * q^7 - 2 * q^9 + 6 * q^11 + q^13 - q^15 - 2 * q^19 + 4 * q^21 + q^23 + q^25 + 5 * q^27 - 9 * q^29 + 5 * q^31 - 6 * q^33 - 4 * q^35 - 2 * q^37 - q^39 - 9 * q^41 + 4 * q^43 - 2 * q^45 - 3 * q^47 + 9 * q^49 + 6 * q^53 + 6 * q^55 + 2 * q^57 - 2 * q^61 + 8 * q^63 + q^65 + 10 * q^67 - q^69 - 3 * q^71 - 7 * q^73 - q^75 - 24 * q^77 - 10 * q^79 + q^81 + 12 * q^83 + 9 * q^87 - 4 * q^91 - 5 * q^93 - 2 * q^95 + 8 * q^97 - 12 * 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
 0
0 −1.00000 0 1.00000 0 −4.00000 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.i 1
4.b odd 2 1 7360.2.a.v 1
8.b even 2 1 460.2.a.c 1
8.d odd 2 1 1840.2.a.c 1
24.h odd 2 1 4140.2.a.f 1
40.e odd 2 1 9200.2.a.y 1
40.f even 2 1 2300.2.a.d 1
40.i odd 4 2 2300.2.c.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.a.c 1 8.b even 2 1
1840.2.a.c 1 8.d odd 2 1
2300.2.a.d 1 40.f even 2 1
2300.2.c.d 2 40.i odd 4 2
4140.2.a.f 1 24.h odd 2 1
7360.2.a.i 1 1.a even 1 1 trivial
7360.2.a.v 1 4.b odd 2 1
9200.2.a.y 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7360))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T - 1$$
$7$ $$T + 4$$
$11$ $$T - 6$$
$13$ $$T - 1$$
$17$ $$T$$
$19$ $$T + 2$$
$23$ $$T - 1$$
$29$ $$T + 9$$
$31$ $$T - 5$$
$37$ $$T + 2$$
$41$ $$T + 9$$
$43$ $$T - 4$$
$47$ $$T + 3$$
$53$ $$T - 6$$
$59$ $$T$$
$61$ $$T + 2$$
$67$ $$T - 10$$
$71$ $$T + 3$$
$73$ $$T + 7$$
$79$ $$T + 10$$
$83$ $$T - 12$$
$89$ $$T$$
$97$ $$T - 8$$