Properties

 Label 4680.2.a.y Level $4680$ Weight $2$ Character orbit 4680.a Self dual yes Analytic conductor $37.370$ 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] = [4680,2,Mod(1,4680)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4680.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: $$37.3699881460$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1560) 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{5} - \beta q^{7} +O(q^{10})$$ q - q^5 - b * q^7 $$q - q^{5} - \beta q^{7} + ( - \beta + 4) q^{11} + q^{13} + ( - 3 \beta + 2) q^{17} - 2 \beta q^{19} + ( - \beta + 4) q^{23} + q^{25} + 2 q^{29} + (2 \beta - 4) q^{31} + \beta q^{35} + ( - 5 \beta + 2) q^{37} + (3 \beta + 2) q^{41} + (4 \beta - 4) q^{43} + (2 \beta - 4) q^{47} + (\beta - 3) q^{49} + (\beta - 2) q^{53} + (\beta - 4) q^{55} + 8 q^{59} + (3 \beta - 6) q^{61} - q^{65} + (4 \beta - 8) q^{67} + 3 \beta q^{71} + ( - 4 \beta + 10) q^{73} + ( - 3 \beta + 4) q^{77} + ( - 5 \beta + 4) q^{79} + 8 q^{83} + (3 \beta - 2) q^{85} + ( - 3 \beta + 10) q^{89} - \beta q^{91} + 2 \beta q^{95} + (5 \beta - 10) q^{97} +O(q^{100})$$ q - q^5 - b * q^7 + (-b + 4) * q^11 + q^13 + (-3*b + 2) * q^17 - 2*b * q^19 + (-b + 4) * q^23 + q^25 + 2 * q^29 + (2*b - 4) * q^31 + b * q^35 + (-5*b + 2) * q^37 + (3*b + 2) * q^41 + (4*b - 4) * q^43 + (2*b - 4) * q^47 + (b - 3) * q^49 + (b - 2) * q^53 + (b - 4) * q^55 + 8 * q^59 + (3*b - 6) * q^61 - q^65 + (4*b - 8) * q^67 + 3*b * q^71 + (-4*b + 10) * q^73 + (-3*b + 4) * q^77 + (-5*b + 4) * q^79 + 8 * q^83 + (3*b - 2) * q^85 + (-3*b + 10) * q^89 - b * q^91 + 2*b * q^95 + (5*b - 10) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} - q^{7}+O(q^{10})$$ 2 * q - 2 * q^5 - q^7 $$2 q - 2 q^{5} - q^{7} + 7 q^{11} + 2 q^{13} + q^{17} - 2 q^{19} + 7 q^{23} + 2 q^{25} + 4 q^{29} - 6 q^{31} + q^{35} - q^{37} + 7 q^{41} - 4 q^{43} - 6 q^{47} - 5 q^{49} - 3 q^{53} - 7 q^{55} + 16 q^{59} - 9 q^{61} - 2 q^{65} - 12 q^{67} + 3 q^{71} + 16 q^{73} + 5 q^{77} + 3 q^{79} + 16 q^{83} - q^{85} + 17 q^{89} - q^{91} + 2 q^{95} - 15 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 - q^7 + 7 * q^11 + 2 * q^13 + q^17 - 2 * q^19 + 7 * q^23 + 2 * q^25 + 4 * q^29 - 6 * q^31 + q^35 - q^37 + 7 * q^41 - 4 * q^43 - 6 * q^47 - 5 * q^49 - 3 * q^53 - 7 * q^55 + 16 * q^59 - 9 * q^61 - 2 * q^65 - 12 * q^67 + 3 * q^71 + 16 * q^73 + 5 * q^77 + 3 * q^79 + 16 * q^83 - q^85 + 17 * q^89 - q^91 + 2 * q^95 - 15 * 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
 2.56155 −1.56155
0 0 0 −1.00000 0 −2.56155 0 0 0
1.2 0 0 0 −1.00000 0 1.56155 0 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$13$$ $$-1$$

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 4680.2.a.y 2
3.b odd 2 1 1560.2.a.o 2
4.b odd 2 1 9360.2.a.ci 2
12.b even 2 1 3120.2.a.bg 2
15.d odd 2 1 7800.2.a.bd 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.a.o 2 3.b odd 2 1
3120.2.a.bg 2 12.b even 2 1
4680.2.a.y 2 1.a even 1 1 trivial
7800.2.a.bd 2 15.d odd 2 1
9360.2.a.ci 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4680))$$:

 $$T_{7}^{2} + T_{7} - 4$$ T7^2 + T7 - 4 $$T_{11}^{2} - 7T_{11} + 8$$ T11^2 - 7*T11 + 8 $$T_{17}^{2} - T_{17} - 38$$ T17^2 - T17 - 38 $$T_{19}^{2} + 2T_{19} - 16$$ T19^2 + 2*T19 - 16

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} + T - 4$$
$11$ $$T^{2} - 7T + 8$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} - T - 38$$
$19$ $$T^{2} + 2T - 16$$
$23$ $$T^{2} - 7T + 8$$
$29$ $$(T - 2)^{2}$$
$31$ $$T^{2} + 6T - 8$$
$37$ $$T^{2} + T - 106$$
$41$ $$T^{2} - 7T - 26$$
$43$ $$T^{2} + 4T - 64$$
$47$ $$T^{2} + 6T - 8$$
$53$ $$T^{2} + 3T - 2$$
$59$ $$(T - 8)^{2}$$
$61$ $$T^{2} + 9T - 18$$
$67$ $$T^{2} + 12T - 32$$
$71$ $$T^{2} - 3T - 36$$
$73$ $$T^{2} - 16T - 4$$
$79$ $$T^{2} - 3T - 104$$
$83$ $$(T - 8)^{2}$$
$89$ $$T^{2} - 17T + 34$$
$97$ $$T^{2} + 15T - 50$$