# Properties

 Label 3120.2.l.k Level $3120$ Weight $2$ Character orbit 3120.l Analytic conductor $24.913$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3120,2,Mod(1249,3120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3120, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3120.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3120.l (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$24.9133254306$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 390) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} - \beta_{2} q^{5} + (\beta_{2} - \beta_1) q^{7} - q^{9}+O(q^{10})$$ q + b1 * q^3 - b2 * q^5 + (b2 - b1) * q^7 - q^9 $$q + \beta_1 q^{3} - \beta_{2} q^{5} + (\beta_{2} - \beta_1) q^{7} - q^{9} + 2 \beta_{3} q^{11} - \beta_1 q^{13} + \beta_{3} q^{15} + ( - \beta_{2} + 5 \beta_1) q^{17} + ( - \beta_{3} - 5) q^{19} + ( - \beta_{3} + 1) q^{21} + (\beta_{2} + 5 \beta_1) q^{23} - 5 q^{25} - \beta_1 q^{27} + ( - 3 \beta_{3} - 3) q^{29} + 4 q^{31} + 2 \beta_{2} q^{33} + ( - \beta_{3} + 5) q^{35} + (4 \beta_{2} - 2 \beta_1) q^{37} + q^{39} + (2 \beta_{3} + 8) q^{41} + (2 \beta_{2} + 2 \beta_1) q^{43} + \beta_{2} q^{45} + (4 \beta_{2} - 4 \beta_1) q^{47} + (2 \beta_{3} + 1) q^{49} + (\beta_{3} - 5) q^{51} + ( - 2 \beta_{2} - 4 \beta_1) q^{53} - 10 \beta_1 q^{55} + ( - \beta_{2} - 5 \beta_1) q^{57} + (2 \beta_{3} - 4) q^{59} + (4 \beta_{3} - 2) q^{61} + ( - \beta_{2} + \beta_1) q^{63} - \beta_{3} q^{65} + ( - \beta_{3} - 5) q^{69} + ( - 2 \beta_{3} - 2) q^{71} + (\beta_{2} - 11 \beta_1) q^{73} - 5 \beta_1 q^{75} + ( - 2 \beta_{2} + 10 \beta_1) q^{77} - 4 q^{79} + q^{81} + (4 \beta_{2} + 4 \beta_1) q^{83} + (5 \beta_{3} - 5) q^{85} + ( - 3 \beta_{2} - 3 \beta_1) q^{87} + (2 \beta_{3} + 4) q^{89} + (\beta_{3} - 1) q^{91} + 4 \beta_1 q^{93} + (5 \beta_{2} + 5 \beta_1) q^{95} + (3 \beta_{2} + 3 \beta_1) q^{97} - 2 \beta_{3} q^{99}+O(q^{100})$$ q + b1 * q^3 - b2 * q^5 + (b2 - b1) * q^7 - q^9 + 2*b3 * q^11 - b1 * q^13 + b3 * q^15 + (-b2 + 5*b1) * q^17 + (-b3 - 5) * q^19 + (-b3 + 1) * q^21 + (b2 + 5*b1) * q^23 - 5 * q^25 - b1 * q^27 + (-3*b3 - 3) * q^29 + 4 * q^31 + 2*b2 * q^33 + (-b3 + 5) * q^35 + (4*b2 - 2*b1) * q^37 + q^39 + (2*b3 + 8) * q^41 + (2*b2 + 2*b1) * q^43 + b2 * q^45 + (4*b2 - 4*b1) * q^47 + (2*b3 + 1) * q^49 + (b3 - 5) * q^51 + (-2*b2 - 4*b1) * q^53 - 10*b1 * q^55 + (-b2 - 5*b1) * q^57 + (2*b3 - 4) * q^59 + (4*b3 - 2) * q^61 + (-b2 + b1) * q^63 - b3 * q^65 + (-b3 - 5) * q^69 + (-2*b3 - 2) * q^71 + (b2 - 11*b1) * q^73 - 5*b1 * q^75 + (-2*b2 + 10*b1) * q^77 - 4 * q^79 + q^81 + (4*b2 + 4*b1) * q^83 + (5*b3 - 5) * q^85 + (-3*b2 - 3*b1) * q^87 + (2*b3 + 4) * q^89 + (b3 - 1) * q^91 + 4*b1 * q^93 + (5*b2 + 5*b1) * q^95 + (3*b2 + 3*b1) * q^97 - 2*b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^9 $$4 q - 4 q^{9} - 20 q^{19} + 4 q^{21} - 20 q^{25} - 12 q^{29} + 16 q^{31} + 20 q^{35} + 4 q^{39} + 32 q^{41} + 4 q^{49} - 20 q^{51} - 16 q^{59} - 8 q^{61} - 20 q^{69} - 8 q^{71} - 16 q^{79} + 4 q^{81} - 20 q^{85} + 16 q^{89} - 4 q^{91}+O(q^{100})$$ 4 * q - 4 * q^9 - 20 * q^19 + 4 * q^21 - 20 * q^25 - 12 * q^29 + 16 * q^31 + 20 * q^35 + 4 * q^39 + 32 * q^41 + 4 * q^49 - 20 * q^51 - 16 * q^59 - 8 * q^61 - 20 * q^69 - 8 * q^71 - 16 * q^79 + 4 * q^81 - 20 * q^85 + 16 * q^89 - 4 * q^91

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{3} + 2\nu$$ v^3 + 2*v $$\beta_{2}$$ $$=$$ $$\nu^{3} + 4\nu$$ v^3 + 4*v $$\beta_{3}$$ $$=$$ $$2\nu^{2} + 3$$ 2*v^2 + 3
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 2$$ (b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 3 ) / 2$$ (b3 - 3) / 2 $$\nu^{3}$$ $$=$$ $$-\beta_{2} + 2\beta_1$$ -b2 + 2*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3120\mathbb{Z}\right)^\times$$.

 $$n$$ $$1951$$ $$2081$$ $$2341$$ $$2497$$ $$2641$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1249.1
 1.61803i − 0.618034i 0.618034i − 1.61803i
0 1.00000i 0 2.23607i 0 3.23607i 0 −1.00000 0
1249.2 0 1.00000i 0 2.23607i 0 1.23607i 0 −1.00000 0
1249.3 0 1.00000i 0 2.23607i 0 1.23607i 0 −1.00000 0
1249.4 0 1.00000i 0 2.23607i 0 3.23607i 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3120.2.l.k 4
4.b odd 2 1 390.2.e.e 4
5.b even 2 1 inner 3120.2.l.k 4
12.b even 2 1 1170.2.e.e 4
20.d odd 2 1 390.2.e.e 4
20.e even 4 1 1950.2.a.be 2
20.e even 4 1 1950.2.a.bf 2
60.h even 2 1 1170.2.e.e 4
60.l odd 4 1 5850.2.a.cf 2
60.l odd 4 1 5850.2.a.cm 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.e.e 4 4.b odd 2 1
390.2.e.e 4 20.d odd 2 1
1170.2.e.e 4 12.b even 2 1
1170.2.e.e 4 60.h even 2 1
1950.2.a.be 2 20.e even 4 1
1950.2.a.bf 2 20.e even 4 1
3120.2.l.k 4 1.a even 1 1 trivial
3120.2.l.k 4 5.b even 2 1 inner
5850.2.a.cf 2 60.l odd 4 1
5850.2.a.cm 2 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3120, [\chi])$$:

 $$T_{7}^{4} + 12T_{7}^{2} + 16$$ T7^4 + 12*T7^2 + 16 $$T_{11}^{2} - 20$$ T11^2 - 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 1)^{2}$$
$5$ $$(T^{2} + 5)^{2}$$
$7$ $$T^{4} + 12T^{2} + 16$$
$11$ $$(T^{2} - 20)^{2}$$
$13$ $$(T^{2} + 1)^{2}$$
$17$ $$T^{4} + 60T^{2} + 400$$
$19$ $$(T^{2} + 10 T + 20)^{2}$$
$23$ $$T^{4} + 60T^{2} + 400$$
$29$ $$(T^{2} + 6 T - 36)^{2}$$
$31$ $$(T - 4)^{4}$$
$37$ $$T^{4} + 168T^{2} + 5776$$
$41$ $$(T^{2} - 16 T + 44)^{2}$$
$43$ $$T^{4} + 48T^{2} + 256$$
$47$ $$T^{4} + 192T^{2} + 4096$$
$53$ $$T^{4} + 72T^{2} + 16$$
$59$ $$(T^{2} + 8 T - 4)^{2}$$
$61$ $$(T^{2} + 4 T - 76)^{2}$$
$67$ $$T^{4}$$
$71$ $$(T^{2} + 4 T - 16)^{2}$$
$73$ $$T^{4} + 252 T^{2} + 13456$$
$79$ $$(T + 4)^{4}$$
$83$ $$T^{4} + 192T^{2} + 4096$$
$89$ $$(T^{2} - 8 T - 4)^{2}$$
$97$ $$T^{4} + 108T^{2} + 1296$$