# Properties

 Label 3120.2.l.i Level $3120$ Weight $2$ Character orbit 3120.l Analytic conductor $24.913$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{3} + ( - i + 2) q^{5} + 4 i q^{7} - q^{9}+O(q^{10})$$ q + i * q^3 + (-i + 2) * q^5 + 4*i * q^7 - q^9 $$q + i q^{3} + ( - i + 2) q^{5} + 4 i q^{7} - q^{9} + 6 q^{11} - i q^{13} + (2 i + 1) q^{15} - 4 i q^{17} + 2 q^{19} - 4 q^{21} - 6 i q^{23} + ( - 4 i + 3) q^{25} - i q^{27} + 10 q^{29} - 4 q^{31} + 6 i q^{33} + (8 i + 4) q^{35} - 6 i q^{37} + q^{39} + 10 q^{41} + (i - 2) q^{45} - 8 i q^{47} - 9 q^{49} + 4 q^{51} + 6 i q^{53} + ( - 6 i + 12) q^{55} + 2 i q^{57} - 6 q^{59} - 6 q^{61} - 4 i q^{63} + ( - 2 i - 1) q^{65} + 12 i q^{67} + 6 q^{69} - 2 i q^{73} + (3 i + 4) q^{75} + 24 i q^{77} - 8 q^{79} + q^{81} - 4 i q^{83} + ( - 8 i - 4) q^{85} + 10 i q^{87} - 14 q^{89} + 4 q^{91} - 4 i q^{93} + ( - 2 i + 4) q^{95} + 14 i q^{97} - 6 q^{99} +O(q^{100})$$ q + i * q^3 + (-i + 2) * q^5 + 4*i * q^7 - q^9 + 6 * q^11 - i * q^13 + (2*i + 1) * q^15 - 4*i * q^17 + 2 * q^19 - 4 * q^21 - 6*i * q^23 + (-4*i + 3) * q^25 - i * q^27 + 10 * q^29 - 4 * q^31 + 6*i * q^33 + (8*i + 4) * q^35 - 6*i * q^37 + q^39 + 10 * q^41 + (i - 2) * q^45 - 8*i * q^47 - 9 * q^49 + 4 * q^51 + 6*i * q^53 + (-6*i + 12) * q^55 + 2*i * q^57 - 6 * q^59 - 6 * q^61 - 4*i * q^63 + (-2*i - 1) * q^65 + 12*i * q^67 + 6 * q^69 - 2*i * q^73 + (3*i + 4) * q^75 + 24*i * q^77 - 8 * q^79 + q^81 - 4*i * q^83 + (-8*i - 4) * q^85 + 10*i * q^87 - 14 * q^89 + 4 * q^91 - 4*i * q^93 + (-2*i + 4) * q^95 + 14*i * q^97 - 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{5} - 2 q^{9}+O(q^{10})$$ 2 * q + 4 * q^5 - 2 * q^9 $$2 q + 4 q^{5} - 2 q^{9} + 12 q^{11} + 2 q^{15} + 4 q^{19} - 8 q^{21} + 6 q^{25} + 20 q^{29} - 8 q^{31} + 8 q^{35} + 2 q^{39} + 20 q^{41} - 4 q^{45} - 18 q^{49} + 8 q^{51} + 24 q^{55} - 12 q^{59} - 12 q^{61} - 2 q^{65} + 12 q^{69} + 8 q^{75} - 16 q^{79} + 2 q^{81} - 8 q^{85} - 28 q^{89} + 8 q^{91} + 8 q^{95} - 12 q^{99}+O(q^{100})$$ 2 * q + 4 * q^5 - 2 * q^9 + 12 * q^11 + 2 * q^15 + 4 * q^19 - 8 * q^21 + 6 * q^25 + 20 * q^29 - 8 * q^31 + 8 * q^35 + 2 * q^39 + 20 * q^41 - 4 * q^45 - 18 * q^49 + 8 * q^51 + 24 * q^55 - 12 * q^59 - 12 * q^61 - 2 * q^65 + 12 * q^69 + 8 * q^75 - 16 * q^79 + 2 * q^81 - 8 * q^85 - 28 * q^89 + 8 * q^91 + 8 * q^95 - 12 * q^99

## 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
 − 1.00000i 1.00000i
0 1.00000i 0 2.00000 + 1.00000i 0 4.00000i 0 −1.00000 0
1249.2 0 1.00000i 0 2.00000 1.00000i 0 4.00000i 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.i 2
4.b odd 2 1 390.2.e.d 2
5.b even 2 1 inner 3120.2.l.i 2
12.b even 2 1 1170.2.e.c 2
20.d odd 2 1 390.2.e.d 2
20.e even 4 1 1950.2.a.d 1
20.e even 4 1 1950.2.a.v 1
60.h even 2 1 1170.2.e.c 2
60.l odd 4 1 5850.2.a.e 1
60.l odd 4 1 5850.2.a.cc 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.e.d 2 4.b odd 2 1
390.2.e.d 2 20.d odd 2 1
1170.2.e.c 2 12.b even 2 1
1170.2.e.c 2 60.h even 2 1
1950.2.a.d 1 20.e even 4 1
1950.2.a.v 1 20.e even 4 1
3120.2.l.i 2 1.a even 1 1 trivial
3120.2.l.i 2 5.b even 2 1 inner
5850.2.a.e 1 60.l odd 4 1
5850.2.a.cc 1 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 $$S_{2}^{\mathrm{new}}(3120, [\chi])$$:

 $$T_{7}^{2} + 16$$ T7^2 + 16 $$T_{11} - 6$$ T11 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2} - 4T + 5$$
$7$ $$T^{2} + 16$$
$11$ $$(T - 6)^{2}$$
$13$ $$T^{2} + 1$$
$17$ $$T^{2} + 16$$
$19$ $$(T - 2)^{2}$$
$23$ $$T^{2} + 36$$
$29$ $$(T - 10)^{2}$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} + 36$$
$41$ $$(T - 10)^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 36$$
$59$ $$(T + 6)^{2}$$
$61$ $$(T + 6)^{2}$$
$67$ $$T^{2} + 144$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 4$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$(T + 14)^{2}$$
$97$ $$T^{2} + 196$$