# Properties

 Label 7360.2.a.b Level $7360$ Weight $2$ Character orbit 7360.a Self dual yes Analytic conductor $58.770$ Analytic rank $0$ 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: $$0$$ 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)

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

## 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 −3.00000 0 1.00000 0 2.00000 0 6.00000 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.b 1
4.b odd 2 1 7360.2.a.bb 1
8.b even 2 1 460.2.a.d 1
8.d odd 2 1 1840.2.a.a 1
24.h odd 2 1 4140.2.a.k 1
40.e odd 2 1 9200.2.a.bk 1
40.f even 2 1 2300.2.a.a 1
40.i odd 4 2 2300.2.c.a 2

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T - 1$$
$7$ $$T - 2$$
$11$ $$T$$
$13$ $$T - 3$$
$17$ $$T - 4$$
$19$ $$T - 4$$
$23$ $$T + 1$$
$29$ $$T + 1$$
$31$ $$T - 1$$
$37$ $$T - 8$$
$41$ $$T - 11$$
$43$ $$T - 10$$
$47$ $$T + 1$$
$53$ $$T - 6$$
$59$ $$T - 8$$
$61$ $$T - 8$$
$67$ $$T + 12$$
$71$ $$T - 13$$
$73$ $$T - 7$$
$79$ $$T + 12$$
$83$ $$T + 16$$
$89$ $$T + 6$$
$97$ $$T - 2$$