# Properties

 Label 7360.2.a.c 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 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{3} - q^{5} - q^{7} + q^{9}+O(q^{10})$$ q - 2 * q^3 - q^5 - q^7 + q^9 $$q - 2 q^{3} - q^{5} - q^{7} + q^{9} + 2 q^{11} + 4 q^{13} + 2 q^{15} - 3 q^{17} + 4 q^{19} + 2 q^{21} + q^{23} + q^{25} + 4 q^{27} - 3 q^{29} + 3 q^{31} - 4 q^{33} + q^{35} + 3 q^{37} - 8 q^{39} - 3 q^{41} - 4 q^{43} - q^{45} - 6 q^{49} + 6 q^{51} + 11 q^{53} - 2 q^{55} - 8 q^{57} + 5 q^{59} - 6 q^{61} - q^{63} - 4 q^{65} + 5 q^{67} - 2 q^{69} + 5 q^{71} - 4 q^{73} - 2 q^{75} - 2 q^{77} - 8 q^{79} - 11 q^{81} + 9 q^{83} + 3 q^{85} + 6 q^{87} - 10 q^{89} - 4 q^{91} - 6 q^{93} - 4 q^{95} - 2 q^{97} + 2 q^{99}+O(q^{100})$$ q - 2 * q^3 - q^5 - q^7 + q^9 + 2 * q^11 + 4 * q^13 + 2 * q^15 - 3 * q^17 + 4 * q^19 + 2 * q^21 + q^23 + q^25 + 4 * q^27 - 3 * q^29 + 3 * q^31 - 4 * q^33 + q^35 + 3 * q^37 - 8 * q^39 - 3 * q^41 - 4 * q^43 - q^45 - 6 * q^49 + 6 * q^51 + 11 * q^53 - 2 * q^55 - 8 * q^57 + 5 * q^59 - 6 * q^61 - q^63 - 4 * q^65 + 5 * q^67 - 2 * q^69 + 5 * q^71 - 4 * q^73 - 2 * q^75 - 2 * q^77 - 8 * q^79 - 11 * q^81 + 9 * q^83 + 3 * q^85 + 6 * q^87 - 10 * q^89 - 4 * q^91 - 6 * q^93 - 4 * q^95 - 2 * 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 −1.00000 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.c 1
4.b odd 2 1 7360.2.a.w 1
8.b even 2 1 3680.2.a.j yes 1
8.d odd 2 1 3680.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3680.2.a.d 1 8.d odd 2 1
3680.2.a.j yes 1 8.b even 2 1
7360.2.a.c 1 1.a even 1 1 trivial
7360.2.a.w 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 $$T_{7} + 1$$ T7 + 1 $$T_{11} - 2$$ T11 - 2 $$T_{13} - 4$$ T13 - 4 $$T_{17} + 3$$ T17 + 3

## Hecke characteristic polynomials

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