# Properties

 Label 7360.2.a.f 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 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{3} + q^{5} + 5 q^{7} + q^{9}+O(q^{10})$$ q - 2 * q^3 + q^5 + 5 * q^7 + q^9 $$q - 2 q^{3} + q^{5} + 5 q^{7} + q^{9} - 2 q^{11} - 4 q^{13} - 2 q^{15} + 3 q^{17} - 10 q^{21} - q^{23} + q^{25} + 4 q^{27} + q^{29} - q^{31} + 4 q^{33} + 5 q^{35} - 7 q^{37} + 8 q^{39} - 3 q^{41} - 4 q^{43} + q^{45} - 12 q^{47} + 18 q^{49} - 6 q^{51} + q^{53} - 2 q^{55} - 3 q^{59} + 14 q^{61} + 5 q^{63} - 4 q^{65} - 5 q^{67} + 2 q^{69} - 15 q^{71} + 8 q^{73} - 2 q^{75} - 10 q^{77} - 4 q^{79} - 11 q^{81} - 17 q^{83} + 3 q^{85} - 2 q^{87} - 14 q^{89} - 20 q^{91} + 2 q^{93} - 6 q^{97} - 2 q^{99}+O(q^{100})$$ q - 2 * q^3 + q^5 + 5 * q^7 + q^9 - 2 * q^11 - 4 * q^13 - 2 * q^15 + 3 * q^17 - 10 * q^21 - q^23 + q^25 + 4 * q^27 + q^29 - q^31 + 4 * q^33 + 5 * q^35 - 7 * q^37 + 8 * q^39 - 3 * q^41 - 4 * q^43 + q^45 - 12 * q^47 + 18 * q^49 - 6 * q^51 + q^53 - 2 * q^55 - 3 * q^59 + 14 * q^61 + 5 * q^63 - 4 * q^65 - 5 * q^67 + 2 * q^69 - 15 * q^71 + 8 * q^73 - 2 * q^75 - 10 * q^77 - 4 * q^79 - 11 * q^81 - 17 * q^83 + 3 * q^85 - 2 * q^87 - 14 * q^89 - 20 * q^91 + 2 * q^93 - 6 * q^97 - 2 * 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 −2.00000 0 1.00000 0 5.00000 0 1.00000 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.f 1
4.b odd 2 1 7360.2.a.x 1
8.b even 2 1 3680.2.a.i yes 1
8.d odd 2 1 3680.2.a.a 1

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

 $$T_{3} + 2$$ T3 + 2 $$T_{7} - 5$$ T7 - 5 $$T_{11} + 2$$ T11 + 2 $$T_{13} + 4$$ T13 + 4 $$T_{17} - 3$$ T17 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 2$$
$5$ $$T - 1$$
$7$ $$T - 5$$
$11$ $$T + 2$$
$13$ $$T + 4$$
$17$ $$T - 3$$
$19$ $$T$$
$23$ $$T + 1$$
$29$ $$T - 1$$
$31$ $$T + 1$$
$37$ $$T + 7$$
$41$ $$T + 3$$
$43$ $$T + 4$$
$47$ $$T + 12$$
$53$ $$T - 1$$
$59$ $$T + 3$$
$61$ $$T - 14$$
$67$ $$T + 5$$
$71$ $$T + 15$$
$73$ $$T - 8$$
$79$ $$T + 4$$
$83$ $$T + 17$$
$89$ $$T + 14$$
$97$ $$T + 6$$