# Properties

 Label 1960.2.q.a Level $1960$ Weight $2$ Character orbit 1960.q Analytic conductor $15.651$ 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] = [1960,2,Mod(361,1960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1960, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))

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

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

chi := DirichletCharacter("1960.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1960 = 2^{3} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1960.q (of order $$3$$, degree $$2$$, 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: $$15.6506787962$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (3 \zeta_{6} - 3) q^{3} + \zeta_{6} q^{5} - 6 \zeta_{6} q^{9} +O(q^{10})$$ q + (3*z - 3) * q^3 + z * q^5 - 6*z * q^9 $$q + (3 \zeta_{6} - 3) q^{3} + \zeta_{6} q^{5} - 6 \zeta_{6} q^{9} + ( - 5 \zeta_{6} + 5) q^{11} + 5 q^{13} - 3 q^{15} + (7 \zeta_{6} - 7) q^{17} - 2 \zeta_{6} q^{19} + 2 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + 9 q^{27} + 7 q^{29} + ( - 4 \zeta_{6} + 4) q^{31} + 15 \zeta_{6} q^{33} + 6 \zeta_{6} q^{37} + (15 \zeta_{6} - 15) q^{39} + 12 q^{41} - 2 q^{43} + ( - 6 \zeta_{6} + 6) q^{45} + \zeta_{6} q^{47} - 21 \zeta_{6} q^{51} + 5 q^{55} + 6 q^{57} + (4 \zeta_{6} - 4) q^{59} + 4 \zeta_{6} q^{61} + 5 \zeta_{6} q^{65} + (8 \zeta_{6} - 8) q^{67} - 6 q^{69} + ( - 6 \zeta_{6} + 6) q^{73} - 3 \zeta_{6} q^{75} + 3 \zeta_{6} q^{79} + (9 \zeta_{6} - 9) q^{81} + 4 q^{83} - 7 q^{85} + (21 \zeta_{6} - 21) q^{87} + 12 \zeta_{6} q^{93} + ( - 2 \zeta_{6} + 2) q^{95} - 13 q^{97} - 30 q^{99} +O(q^{100})$$ q + (3*z - 3) * q^3 + z * q^5 - 6*z * q^9 + (-5*z + 5) * q^11 + 5 * q^13 - 3 * q^15 + (7*z - 7) * q^17 - 2*z * q^19 + 2*z * q^23 + (z - 1) * q^25 + 9 * q^27 + 7 * q^29 + (-4*z + 4) * q^31 + 15*z * q^33 + 6*z * q^37 + (15*z - 15) * q^39 + 12 * q^41 - 2 * q^43 + (-6*z + 6) * q^45 + z * q^47 - 21*z * q^51 + 5 * q^55 + 6 * q^57 + (4*z - 4) * q^59 + 4*z * q^61 + 5*z * q^65 + (8*z - 8) * q^67 - 6 * q^69 + (-6*z + 6) * q^73 - 3*z * q^75 + 3*z * q^79 + (9*z - 9) * q^81 + 4 * q^83 - 7 * q^85 + (21*z - 21) * q^87 + 12*z * q^93 + (-2*z + 2) * q^95 - 13 * q^97 - 30 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{3} + q^{5} - 6 q^{9}+O(q^{10})$$ 2 * q - 3 * q^3 + q^5 - 6 * q^9 $$2 q - 3 q^{3} + q^{5} - 6 q^{9} + 5 q^{11} + 10 q^{13} - 6 q^{15} - 7 q^{17} - 2 q^{19} + 2 q^{23} - q^{25} + 18 q^{27} + 14 q^{29} + 4 q^{31} + 15 q^{33} + 6 q^{37} - 15 q^{39} + 24 q^{41} - 4 q^{43} + 6 q^{45} + q^{47} - 21 q^{51} + 10 q^{55} + 12 q^{57} - 4 q^{59} + 4 q^{61} + 5 q^{65} - 8 q^{67} - 12 q^{69} + 6 q^{73} - 3 q^{75} + 3 q^{79} - 9 q^{81} + 8 q^{83} - 14 q^{85} - 21 q^{87} + 12 q^{93} + 2 q^{95} - 26 q^{97} - 60 q^{99}+O(q^{100})$$ 2 * q - 3 * q^3 + q^5 - 6 * q^9 + 5 * q^11 + 10 * q^13 - 6 * q^15 - 7 * q^17 - 2 * q^19 + 2 * q^23 - q^25 + 18 * q^27 + 14 * q^29 + 4 * q^31 + 15 * q^33 + 6 * q^37 - 15 * q^39 + 24 * q^41 - 4 * q^43 + 6 * q^45 + q^47 - 21 * q^51 + 10 * q^55 + 12 * q^57 - 4 * q^59 + 4 * q^61 + 5 * q^65 - 8 * q^67 - 12 * q^69 + 6 * q^73 - 3 * q^75 + 3 * q^79 - 9 * q^81 + 8 * q^83 - 14 * q^85 - 21 * q^87 + 12 * q^93 + 2 * q^95 - 26 * q^97 - 60 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1960\mathbb{Z}\right)^\times$$.

 $$n$$ $$981$$ $$1081$$ $$1177$$ $$1471$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 −1.50000 + 2.59808i 0 0.500000 + 0.866025i 0 0 0 −3.00000 5.19615i 0
961.1 0 −1.50000 2.59808i 0 0.500000 0.866025i 0 0 0 −3.00000 + 5.19615i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1960.2.q.a 2
7.b odd 2 1 1960.2.q.o 2
7.c even 3 1 1960.2.a.o 1
7.c even 3 1 inner 1960.2.q.a 2
7.d odd 6 1 280.2.a.a 1
7.d odd 6 1 1960.2.q.o 2
21.g even 6 1 2520.2.a.i 1
28.f even 6 1 560.2.a.f 1
28.g odd 6 1 3920.2.a.c 1
35.i odd 6 1 1400.2.a.n 1
35.j even 6 1 9800.2.a.a 1
35.k even 12 2 1400.2.g.a 2
56.j odd 6 1 2240.2.a.z 1
56.m even 6 1 2240.2.a.a 1
84.j odd 6 1 5040.2.a.a 1
140.s even 6 1 2800.2.a.c 1
140.x odd 12 2 2800.2.g.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.a 1 7.d odd 6 1
560.2.a.f 1 28.f even 6 1
1400.2.a.n 1 35.i odd 6 1
1400.2.g.a 2 35.k even 12 2
1960.2.a.o 1 7.c even 3 1
1960.2.q.a 2 1.a even 1 1 trivial
1960.2.q.a 2 7.c even 3 1 inner
1960.2.q.o 2 7.b odd 2 1
1960.2.q.o 2 7.d odd 6 1
2240.2.a.a 1 56.m even 6 1
2240.2.a.z 1 56.j odd 6 1
2520.2.a.i 1 21.g even 6 1
2800.2.a.c 1 140.s even 6 1
2800.2.g.b 2 140.x odd 12 2
3920.2.a.c 1 28.g odd 6 1
5040.2.a.a 1 84.j odd 6 1
9800.2.a.a 1 35.j even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1960, [\chi])$$:

 $$T_{3}^{2} + 3T_{3} + 9$$ T3^2 + 3*T3 + 9 $$T_{11}^{2} - 5T_{11} + 25$$ T11^2 - 5*T11 + 25 $$T_{13} - 5$$ T13 - 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 3T + 9$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 5T + 25$$
$13$ $$(T - 5)^{2}$$
$17$ $$T^{2} + 7T + 49$$
$19$ $$T^{2} + 2T + 4$$
$23$ $$T^{2} - 2T + 4$$
$29$ $$(T - 7)^{2}$$
$31$ $$T^{2} - 4T + 16$$
$37$ $$T^{2} - 6T + 36$$
$41$ $$(T - 12)^{2}$$
$43$ $$(T + 2)^{2}$$
$47$ $$T^{2} - T + 1$$
$53$ $$T^{2}$$
$59$ $$T^{2} + 4T + 16$$
$61$ $$T^{2} - 4T + 16$$
$67$ $$T^{2} + 8T + 64$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 6T + 36$$
$79$ $$T^{2} - 3T + 9$$
$83$ $$(T - 4)^{2}$$
$89$ $$T^{2}$$
$97$ $$(T + 13)^{2}$$