# Properties

 Label 3120.2.g.a Level $3120$ Weight $2$ Character orbit 3120.g Analytic conductor $24.913$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3120,2,Mod(961,3120)]

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, 0, 1]))

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

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

chi := DirichletCharacter("3120.961");

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.g (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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 195) 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + i q^{5} + 2 i q^{7} + q^{9}+O(q^{10})$$ q - q^3 + i * q^5 + 2*i * q^7 + q^9 $$q - q^{3} + i q^{5} + 2 i q^{7} + q^{9} + ( - 2 i - 3) q^{13} - i q^{15} + 2 q^{17} - 2 i q^{19} - 2 i q^{21} + 8 q^{23} - q^{25} - q^{27} + 2 q^{29} + 2 i q^{31} - 2 q^{35} - 8 i q^{37} + (2 i + 3) q^{39} - 2 i q^{41} + 4 q^{43} + i q^{45} + 4 i q^{47} + 3 q^{49} - 2 q^{51} - 6 q^{53} + 2 i q^{57} + 12 i q^{59} + 10 q^{61} + 2 i q^{63} + ( - 3 i + 2) q^{65} + 6 i q^{67} - 8 q^{69} - 8 i q^{71} + 16 i q^{73} + q^{75} + 8 q^{79} + q^{81} - 12 i q^{83} + 2 i q^{85} - 2 q^{87} + 6 i q^{89} + ( - 6 i + 4) q^{91} - 2 i q^{93} + 2 q^{95} + 16 i q^{97} +O(q^{100})$$ q - q^3 + i * q^5 + 2*i * q^7 + q^9 + (-2*i - 3) * q^13 - i * q^15 + 2 * q^17 - 2*i * q^19 - 2*i * q^21 + 8 * q^23 - q^25 - q^27 + 2 * q^29 + 2*i * q^31 - 2 * q^35 - 8*i * q^37 + (2*i + 3) * q^39 - 2*i * q^41 + 4 * q^43 + i * q^45 + 4*i * q^47 + 3 * q^49 - 2 * q^51 - 6 * q^53 + 2*i * q^57 + 12*i * q^59 + 10 * q^61 + 2*i * q^63 + (-3*i + 2) * q^65 + 6*i * q^67 - 8 * q^69 - 8*i * q^71 + 16*i * q^73 + q^75 + 8 * q^79 + q^81 - 12*i * q^83 + 2*i * q^85 - 2 * q^87 + 6*i * q^89 + (-6*i + 4) * q^91 - 2*i * q^93 + 2 * q^95 + 16*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{9} - 6 q^{13} + 4 q^{17} + 16 q^{23} - 2 q^{25} - 2 q^{27} + 4 q^{29} - 4 q^{35} + 6 q^{39} + 8 q^{43} + 6 q^{49} - 4 q^{51} - 12 q^{53} + 20 q^{61} + 4 q^{65} - 16 q^{69} + 2 q^{75} + 16 q^{79} + 2 q^{81} - 4 q^{87} + 8 q^{91} + 4 q^{95}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^9 - 6 * q^13 + 4 * q^17 + 16 * q^23 - 2 * q^25 - 2 * q^27 + 4 * q^29 - 4 * q^35 + 6 * q^39 + 8 * q^43 + 6 * q^49 - 4 * q^51 - 12 * q^53 + 20 * q^61 + 4 * q^65 - 16 * q^69 + 2 * q^75 + 16 * q^79 + 2 * q^81 - 4 * q^87 + 8 * q^91 + 4 * q^95

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
961.1
 − 1.00000i 1.00000i
0 −1.00000 0 1.00000i 0 2.00000i 0 1.00000 0
961.2 0 −1.00000 0 1.00000i 0 2.00000i 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
13.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.g.a 2
4.b odd 2 1 195.2.b.b 2
12.b even 2 1 585.2.b.a 2
13.b even 2 1 inner 3120.2.g.a 2
20.d odd 2 1 975.2.b.b 2
20.e even 4 1 975.2.h.a 2
20.e even 4 1 975.2.h.d 2
52.b odd 2 1 195.2.b.b 2
52.f even 4 1 2535.2.a.e 1
52.f even 4 1 2535.2.a.l 1
156.h even 2 1 585.2.b.a 2
156.l odd 4 1 7605.2.a.d 1
156.l odd 4 1 7605.2.a.p 1
260.g odd 2 1 975.2.b.b 2
260.p even 4 1 975.2.h.a 2
260.p even 4 1 975.2.h.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.b.b 2 4.b odd 2 1
195.2.b.b 2 52.b odd 2 1
585.2.b.a 2 12.b even 2 1
585.2.b.a 2 156.h even 2 1
975.2.b.b 2 20.d odd 2 1
975.2.b.b 2 260.g odd 2 1
975.2.h.a 2 20.e even 4 1
975.2.h.a 2 260.p even 4 1
975.2.h.d 2 20.e even 4 1
975.2.h.d 2 260.p even 4 1
2535.2.a.e 1 52.f even 4 1
2535.2.a.l 1 52.f even 4 1
3120.2.g.a 2 1.a even 1 1 trivial
3120.2.g.a 2 13.b even 2 1 inner
7605.2.a.d 1 156.l odd 4 1
7605.2.a.p 1 156.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 $$S_{2}^{\mathrm{new}}(3120, [\chi])$$:

 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{11}$$ T11 $$T_{17} - 2$$ T17 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} + 1$$
$7$ $$T^{2} + 4$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 6T + 13$$
$17$ $$(T - 2)^{2}$$
$19$ $$T^{2} + 4$$
$23$ $$(T - 8)^{2}$$
$29$ $$(T - 2)^{2}$$
$31$ $$T^{2} + 4$$
$37$ $$T^{2} + 64$$
$41$ $$T^{2} + 4$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2} + 16$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} + 144$$
$61$ $$(T - 10)^{2}$$
$67$ $$T^{2} + 36$$
$71$ $$T^{2} + 64$$
$73$ $$T^{2} + 256$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$T^{2} + 36$$
$97$ $$T^{2} + 256$$