Properties

 Label 7360.2.a.bt Level $7360$ Weight $2$ Character orbit 7360.a Self dual yes Analytic conductor $58.770$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 115) 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 $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{5} + ( - \beta - 1) q^{7} - 2 q^{9}+O(q^{10})$$ q + q^3 + q^5 + (-b - 1) * q^7 - 2 * q^9 $$q + q^{3} + q^{5} + ( - \beta - 1) q^{7} - 2 q^{9} + ( - \beta + 1) q^{11} + ( - \beta + 4) q^{13} + q^{15} + ( - 2 \beta - 2) q^{17} + ( - 3 \beta - 1) q^{19} + ( - \beta - 1) q^{21} - q^{23} + q^{25} - 5 q^{27} + (2 \beta + 5) q^{29} + ( - \beta + 2) q^{31} + ( - \beta + 1) q^{33} + ( - \beta - 1) q^{35} + ( - 3 \beta + 3) q^{37} + ( - \beta + 4) q^{39} + (2 \beta - 3) q^{41} + (3 \beta + 3) q^{43} - 2 q^{45} + ( - 2 \beta + 5) q^{47} + (2 \beta - 1) q^{49} + ( - 2 \beta - 2) q^{51} + 6 q^{53} + ( - \beta + 1) q^{55} + ( - 3 \beta - 1) q^{57} + 4 \beta q^{59} + (5 \beta - 1) q^{61} + (2 \beta + 2) q^{63} + ( - \beta + 4) q^{65} + (3 \beta - 3) q^{67} - q^{69} + (\beta - 4) q^{71} + 3 \beta q^{73} + q^{75} + 4 q^{77} + (\beta + 11) q^{79} + q^{81} + ( - 2 \beta + 2) q^{83} + ( - 2 \beta - 2) q^{85} + (2 \beta + 5) q^{87} + (\beta + 5) q^{89} + ( - 3 \beta + 1) q^{91} + ( - \beta + 2) q^{93} + ( - 3 \beta - 1) q^{95} + (5 \beta + 5) q^{97} + (2 \beta - 2) q^{99}+O(q^{100})$$ q + q^3 + q^5 + (-b - 1) * q^7 - 2 * q^9 + (-b + 1) * q^11 + (-b + 4) * q^13 + q^15 + (-2*b - 2) * q^17 + (-3*b - 1) * q^19 + (-b - 1) * q^21 - q^23 + q^25 - 5 * q^27 + (2*b + 5) * q^29 + (-b + 2) * q^31 + (-b + 1) * q^33 + (-b - 1) * q^35 + (-3*b + 3) * q^37 + (-b + 4) * q^39 + (2*b - 3) * q^41 + (3*b + 3) * q^43 - 2 * q^45 + (-2*b + 5) * q^47 + (2*b - 1) * q^49 + (-2*b - 2) * q^51 + 6 * q^53 + (-b + 1) * q^55 + (-3*b - 1) * q^57 + 4*b * q^59 + (5*b - 1) * q^61 + (2*b + 2) * q^63 + (-b + 4) * q^65 + (3*b - 3) * q^67 - q^69 + (b - 4) * q^71 + 3*b * q^73 + q^75 + 4 * q^77 + (b + 11) * q^79 + q^81 + (-2*b + 2) * q^83 + (-2*b - 2) * q^85 + (2*b + 5) * q^87 + (b + 5) * q^89 + (-3*b + 1) * q^91 + (-b + 2) * q^93 + (-3*b - 1) * q^95 + (5*b + 5) * q^97 + (2*b - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} - 4 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 - 4 * q^9 $$2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} - 4 q^{9} + 2 q^{11} + 8 q^{13} + 2 q^{15} - 4 q^{17} - 2 q^{19} - 2 q^{21} - 2 q^{23} + 2 q^{25} - 10 q^{27} + 10 q^{29} + 4 q^{31} + 2 q^{33} - 2 q^{35} + 6 q^{37} + 8 q^{39} - 6 q^{41} + 6 q^{43} - 4 q^{45} + 10 q^{47} - 2 q^{49} - 4 q^{51} + 12 q^{53} + 2 q^{55} - 2 q^{57} - 2 q^{61} + 4 q^{63} + 8 q^{65} - 6 q^{67} - 2 q^{69} - 8 q^{71} + 2 q^{75} + 8 q^{77} + 22 q^{79} + 2 q^{81} + 4 q^{83} - 4 q^{85} + 10 q^{87} + 10 q^{89} + 2 q^{91} + 4 q^{93} - 2 q^{95} + 10 q^{97} - 4 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 - 4 * q^9 + 2 * q^11 + 8 * q^13 + 2 * q^15 - 4 * q^17 - 2 * q^19 - 2 * q^21 - 2 * q^23 + 2 * q^25 - 10 * q^27 + 10 * q^29 + 4 * q^31 + 2 * q^33 - 2 * q^35 + 6 * q^37 + 8 * q^39 - 6 * q^41 + 6 * q^43 - 4 * q^45 + 10 * q^47 - 2 * q^49 - 4 * q^51 + 12 * q^53 + 2 * q^55 - 2 * q^57 - 2 * q^61 + 4 * q^63 + 8 * q^65 - 6 * q^67 - 2 * q^69 - 8 * q^71 + 2 * q^75 + 8 * q^77 + 22 * q^79 + 2 * q^81 + 4 * q^83 - 4 * q^85 + 10 * q^87 + 10 * q^89 + 2 * q^91 + 4 * q^93 - 2 * q^95 + 10 * q^97 - 4 * 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
 1.61803 −0.618034
0 1.00000 0 1.00000 0 −3.23607 0 −2.00000 0
1.2 0 1.00000 0 1.00000 0 1.23607 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.bt 2
4.b odd 2 1 7360.2.a.bf 2
8.b even 2 1 115.2.a.b 2
8.d odd 2 1 1840.2.a.p 2
24.h odd 2 1 1035.2.a.m 2
40.e odd 2 1 9200.2.a.bm 2
40.f even 2 1 575.2.a.g 2
40.i odd 4 2 575.2.b.c 4
56.h odd 2 1 5635.2.a.n 2
120.i odd 2 1 5175.2.a.bb 2
184.e odd 2 1 2645.2.a.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.a.b 2 8.b even 2 1
575.2.a.g 2 40.f even 2 1
575.2.b.c 4 40.i odd 4 2
1035.2.a.m 2 24.h odd 2 1
1840.2.a.p 2 8.d odd 2 1
2645.2.a.d 2 184.e odd 2 1
5175.2.a.bb 2 120.i odd 2 1
5635.2.a.n 2 56.h odd 2 1
7360.2.a.bf 2 4.b odd 2 1
7360.2.a.bt 2 1.a even 1 1 trivial
9200.2.a.bm 2 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 $$T_{7}^{2} + 2T_{7} - 4$$ T7^2 + 2*T7 - 4 $$T_{11}^{2} - 2T_{11} - 4$$ T11^2 - 2*T11 - 4 $$T_{13}^{2} - 8T_{13} + 11$$ T13^2 - 8*T13 + 11 $$T_{17}^{2} + 4T_{17} - 16$$ T17^2 + 4*T17 - 16

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} + 2T - 4$$
$11$ $$T^{2} - 2T - 4$$
$13$ $$T^{2} - 8T + 11$$
$17$ $$T^{2} + 4T - 16$$
$19$ $$T^{2} + 2T - 44$$
$23$ $$(T + 1)^{2}$$
$29$ $$T^{2} - 10T + 5$$
$31$ $$T^{2} - 4T - 1$$
$37$ $$T^{2} - 6T - 36$$
$41$ $$T^{2} + 6T - 11$$
$43$ $$T^{2} - 6T - 36$$
$47$ $$T^{2} - 10T + 5$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} - 80$$
$61$ $$T^{2} + 2T - 124$$
$67$ $$T^{2} + 6T - 36$$
$71$ $$T^{2} + 8T + 11$$
$73$ $$T^{2} - 45$$
$79$ $$T^{2} - 22T + 116$$
$83$ $$T^{2} - 4T - 16$$
$89$ $$T^{2} - 10T + 20$$
$97$ $$T^{2} - 10T - 100$$