# Properties

 Label 7360.2.a.bd 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: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 1) q^{3} + q^{5} + (\beta - 2) q^{7} + (3 \beta + 1) q^{9}+O(q^{10})$$ q + (-b - 1) * q^3 + q^5 + (b - 2) * q^7 + (3*b + 1) * q^9 $$q + ( - \beta - 1) q^{3} + q^{5} + (\beta - 2) q^{7} + (3 \beta + 1) q^{9} + (\beta + 1) q^{11} + \beta q^{13} + ( - \beta - 1) q^{15} + (\beta + 5) q^{17} + ( - 3 \beta + 4) q^{19} - q^{21} + q^{23} + q^{25} + ( - 4 \beta - 7) q^{27} + ( - 2 \beta - 4) q^{29} + ( - \beta - 2) q^{31} + ( - 3 \beta - 4) q^{33} + (\beta - 2) q^{35} + ( - 4 \beta + 4) q^{37} + ( - 2 \beta - 3) q^{39} + (\beta - 7) q^{41} - 4 q^{43} + (3 \beta + 1) q^{45} + (2 \beta - 8) q^{47} - 3 \beta q^{49} + ( - 7 \beta - 8) q^{51} + (4 \beta - 2) q^{53} + (\beta + 1) q^{55} + (2 \beta + 5) q^{57} + (2 \beta - 6) q^{59} + ( - \beta + 1) q^{61} + ( - 2 \beta + 7) q^{63} + \beta q^{65} - 4 \beta q^{67} + ( - \beta - 1) q^{69} + ( - \beta - 3) q^{71} + ( - 2 \beta + 10) q^{73} + ( - \beta - 1) q^{75} + q^{77} + 4 q^{79} + (6 \beta + 16) q^{81} + ( - 4 \beta - 2) q^{83} + (\beta + 5) q^{85} + (8 \beta + 10) q^{87} + ( - 4 \beta + 12) q^{89} + ( - \beta + 3) q^{91} + (4 \beta + 5) q^{93} + ( - 3 \beta + 4) q^{95} + ( - 9 \beta + 7) q^{97} + (7 \beta + 10) q^{99} +O(q^{100})$$ q + (-b - 1) * q^3 + q^5 + (b - 2) * q^7 + (3*b + 1) * q^9 + (b + 1) * q^11 + b * q^13 + (-b - 1) * q^15 + (b + 5) * q^17 + (-3*b + 4) * q^19 - q^21 + q^23 + q^25 + (-4*b - 7) * q^27 + (-2*b - 4) * q^29 + (-b - 2) * q^31 + (-3*b - 4) * q^33 + (b - 2) * q^35 + (-4*b + 4) * q^37 + (-2*b - 3) * q^39 + (b - 7) * q^41 - 4 * q^43 + (3*b + 1) * q^45 + (2*b - 8) * q^47 - 3*b * q^49 + (-7*b - 8) * q^51 + (4*b - 2) * q^53 + (b + 1) * q^55 + (2*b + 5) * q^57 + (2*b - 6) * q^59 + (-b + 1) * q^61 + (-2*b + 7) * q^63 + b * q^65 - 4*b * q^67 + (-b - 1) * q^69 + (-b - 3) * q^71 + (-2*b + 10) * q^73 + (-b - 1) * q^75 + q^77 + 4 * q^79 + (6*b + 16) * q^81 + (-4*b - 2) * q^83 + (b + 5) * q^85 + (8*b + 10) * q^87 + (-4*b + 12) * q^89 + (-b + 3) * q^91 + (4*b + 5) * q^93 + (-3*b + 4) * q^95 + (-9*b + 7) * q^97 + (7*b + 10) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{3} + 2 q^{5} - 3 q^{7} + 5 q^{9}+O(q^{10})$$ 2 * q - 3 * q^3 + 2 * q^5 - 3 * q^7 + 5 * q^9 $$2 q - 3 q^{3} + 2 q^{5} - 3 q^{7} + 5 q^{9} + 3 q^{11} + q^{13} - 3 q^{15} + 11 q^{17} + 5 q^{19} - 2 q^{21} + 2 q^{23} + 2 q^{25} - 18 q^{27} - 10 q^{29} - 5 q^{31} - 11 q^{33} - 3 q^{35} + 4 q^{37} - 8 q^{39} - 13 q^{41} - 8 q^{43} + 5 q^{45} - 14 q^{47} - 3 q^{49} - 23 q^{51} + 3 q^{55} + 12 q^{57} - 10 q^{59} + q^{61} + 12 q^{63} + q^{65} - 4 q^{67} - 3 q^{69} - 7 q^{71} + 18 q^{73} - 3 q^{75} + 2 q^{77} + 8 q^{79} + 38 q^{81} - 8 q^{83} + 11 q^{85} + 28 q^{87} + 20 q^{89} + 5 q^{91} + 14 q^{93} + 5 q^{95} + 5 q^{97} + 27 q^{99}+O(q^{100})$$ 2 * q - 3 * q^3 + 2 * q^5 - 3 * q^7 + 5 * q^9 + 3 * q^11 + q^13 - 3 * q^15 + 11 * q^17 + 5 * q^19 - 2 * q^21 + 2 * q^23 + 2 * q^25 - 18 * q^27 - 10 * q^29 - 5 * q^31 - 11 * q^33 - 3 * q^35 + 4 * q^37 - 8 * q^39 - 13 * q^41 - 8 * q^43 + 5 * q^45 - 14 * q^47 - 3 * q^49 - 23 * q^51 + 3 * q^55 + 12 * q^57 - 10 * q^59 + q^61 + 12 * q^63 + q^65 - 4 * q^67 - 3 * q^69 - 7 * q^71 + 18 * q^73 - 3 * q^75 + 2 * q^77 + 8 * q^79 + 38 * q^81 - 8 * q^83 + 11 * q^85 + 28 * q^87 + 20 * q^89 + 5 * q^91 + 14 * q^93 + 5 * q^95 + 5 * q^97 + 27 * 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
 2.30278 −1.30278
0 −3.30278 0 1.00000 0 0.302776 0 7.90833 0
1.2 0 0.302776 0 1.00000 0 −3.30278 0 −2.90833 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.bd 2
4.b odd 2 1 7360.2.a.bv 2
8.b even 2 1 3680.2.a.p yes 2
8.d odd 2 1 3680.2.a.k 2

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

 $$T_{3}^{2} + 3T_{3} - 1$$ T3^2 + 3*T3 - 1 $$T_{7}^{2} + 3T_{7} - 1$$ T7^2 + 3*T7 - 1 $$T_{11}^{2} - 3T_{11} - 1$$ T11^2 - 3*T11 - 1 $$T_{13}^{2} - T_{13} - 3$$ T13^2 - T13 - 3 $$T_{17}^{2} - 11T_{17} + 27$$ T17^2 - 11*T17 + 27

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 3T - 1$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} + 3T - 1$$
$11$ $$T^{2} - 3T - 1$$
$13$ $$T^{2} - T - 3$$
$17$ $$T^{2} - 11T + 27$$
$19$ $$T^{2} - 5T - 23$$
$23$ $$(T - 1)^{2}$$
$29$ $$T^{2} + 10T + 12$$
$31$ $$T^{2} + 5T + 3$$
$37$ $$T^{2} - 4T - 48$$
$41$ $$T^{2} + 13T + 39$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2} + 14T + 36$$
$53$ $$T^{2} - 52$$
$59$ $$T^{2} + 10T + 12$$
$61$ $$T^{2} - T - 3$$
$67$ $$T^{2} + 4T - 48$$
$71$ $$T^{2} + 7T + 9$$
$73$ $$T^{2} - 18T + 68$$
$79$ $$(T - 4)^{2}$$
$83$ $$T^{2} + 8T - 36$$
$89$ $$T^{2} - 20T + 48$$
$97$ $$T^{2} - 5T - 257$$