# Properties

 Label 1960.2.q.h Level $1960$ Weight $2$ Character orbit 1960.q Analytic conductor $15.651$ Analytic rank $1$ 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: $$1$$ 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 40) 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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{5} + 3 \zeta_{6} q^{9} +O(q^{10})$$ q - z * q^5 + 3*z * q^9 $$q - \zeta_{6} q^{5} + 3 \zeta_{6} q^{9} + (4 \zeta_{6} - 4) q^{11} - 2 q^{13} + (2 \zeta_{6} - 2) q^{17} - 4 \zeta_{6} q^{19} - 4 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} - 2 q^{29} + ( - 8 \zeta_{6} + 8) q^{31} - 6 \zeta_{6} q^{37} - 6 q^{41} - 8 q^{43} + ( - 3 \zeta_{6} + 3) q^{45} - 4 \zeta_{6} q^{47} + (6 \zeta_{6} - 6) q^{53} + 4 q^{55} + ( - 4 \zeta_{6} + 4) q^{59} + 2 \zeta_{6} q^{61} + 2 \zeta_{6} q^{65} + (8 \zeta_{6} - 8) q^{67} + ( - 6 \zeta_{6} + 6) q^{73} + (9 \zeta_{6} - 9) q^{81} - 16 q^{83} + 2 q^{85} + 6 \zeta_{6} q^{89} + (4 \zeta_{6} - 4) q^{95} - 14 q^{97} - 12 q^{99} +O(q^{100})$$ q - z * q^5 + 3*z * q^9 + (4*z - 4) * q^11 - 2 * q^13 + (2*z - 2) * q^17 - 4*z * q^19 - 4*z * q^23 + (z - 1) * q^25 - 2 * q^29 + (-8*z + 8) * q^31 - 6*z * q^37 - 6 * q^41 - 8 * q^43 + (-3*z + 3) * q^45 - 4*z * q^47 + (6*z - 6) * q^53 + 4 * q^55 + (-4*z + 4) * q^59 + 2*z * q^61 + 2*z * q^65 + (8*z - 8) * q^67 + (-6*z + 6) * q^73 + (9*z - 9) * q^81 - 16 * q^83 + 2 * q^85 + 6*z * q^89 + (4*z - 4) * q^95 - 14 * q^97 - 12 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{5} + 3 q^{9}+O(q^{10})$$ 2 * q - q^5 + 3 * q^9 $$2 q - q^{5} + 3 q^{9} - 4 q^{11} - 4 q^{13} - 2 q^{17} - 4 q^{19} - 4 q^{23} - q^{25} - 4 q^{29} + 8 q^{31} - 6 q^{37} - 12 q^{41} - 16 q^{43} + 3 q^{45} - 4 q^{47} - 6 q^{53} + 8 q^{55} + 4 q^{59} + 2 q^{61} + 2 q^{65} - 8 q^{67} + 6 q^{73} - 9 q^{81} - 32 q^{83} + 4 q^{85} + 6 q^{89} - 4 q^{95} - 28 q^{97} - 24 q^{99}+O(q^{100})$$ 2 * q - q^5 + 3 * q^9 - 4 * q^11 - 4 * q^13 - 2 * q^17 - 4 * q^19 - 4 * q^23 - q^25 - 4 * q^29 + 8 * q^31 - 6 * q^37 - 12 * q^41 - 16 * q^43 + 3 * q^45 - 4 * q^47 - 6 * q^53 + 8 * q^55 + 4 * q^59 + 2 * q^61 + 2 * q^65 - 8 * q^67 + 6 * q^73 - 9 * q^81 - 32 * q^83 + 4 * q^85 + 6 * q^89 - 4 * q^95 - 28 * q^97 - 24 * 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 0 0 −0.500000 0.866025i 0 0 0 1.50000 + 2.59808i 0
961.1 0 0 0 −0.500000 + 0.866025i 0 0 0 1.50000 2.59808i 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.h 2
7.b odd 2 1 1960.2.q.i 2
7.c even 3 1 40.2.a.a 1
7.c even 3 1 inner 1960.2.q.h 2
7.d odd 6 1 1960.2.a.g 1
7.d odd 6 1 1960.2.q.i 2
21.h odd 6 1 360.2.a.a 1
28.f even 6 1 3920.2.a.s 1
28.g odd 6 1 80.2.a.a 1
35.i odd 6 1 9800.2.a.x 1
35.j even 6 1 200.2.a.c 1
35.l odd 12 2 200.2.c.b 2
56.k odd 6 1 320.2.a.d 1
56.p even 6 1 320.2.a.c 1
63.g even 3 1 3240.2.q.k 2
63.h even 3 1 3240.2.q.k 2
63.j odd 6 1 3240.2.q.x 2
63.n odd 6 1 3240.2.q.x 2
77.h odd 6 1 4840.2.a.f 1
84.n even 6 1 720.2.a.e 1
91.r even 6 1 6760.2.a.i 1
105.o odd 6 1 1800.2.a.v 1
105.x even 12 2 1800.2.f.a 2
112.u odd 12 2 1280.2.d.a 2
112.w even 12 2 1280.2.d.j 2
140.p odd 6 1 400.2.a.e 1
140.w even 12 2 400.2.c.d 2
168.s odd 6 1 2880.2.a.t 1
168.v even 6 1 2880.2.a.bg 1
280.bf even 6 1 1600.2.a.o 1
280.bi odd 6 1 1600.2.a.k 1
280.br even 12 2 1600.2.c.m 2
280.bt odd 12 2 1600.2.c.k 2
308.n even 6 1 9680.2.a.q 1
420.ba even 6 1 3600.2.a.h 1
420.bp odd 12 2 3600.2.f.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.a.a 1 7.c even 3 1
80.2.a.a 1 28.g odd 6 1
200.2.a.c 1 35.j even 6 1
200.2.c.b 2 35.l odd 12 2
320.2.a.c 1 56.p even 6 1
320.2.a.d 1 56.k odd 6 1
360.2.a.a 1 21.h odd 6 1
400.2.a.e 1 140.p odd 6 1
400.2.c.d 2 140.w even 12 2
720.2.a.e 1 84.n even 6 1
1280.2.d.a 2 112.u odd 12 2
1280.2.d.j 2 112.w even 12 2
1600.2.a.k 1 280.bi odd 6 1
1600.2.a.o 1 280.bf even 6 1
1600.2.c.k 2 280.bt odd 12 2
1600.2.c.m 2 280.br even 12 2
1800.2.a.v 1 105.o odd 6 1
1800.2.f.a 2 105.x even 12 2
1960.2.a.g 1 7.d odd 6 1
1960.2.q.h 2 1.a even 1 1 trivial
1960.2.q.h 2 7.c even 3 1 inner
1960.2.q.i 2 7.b odd 2 1
1960.2.q.i 2 7.d odd 6 1
2880.2.a.t 1 168.s odd 6 1
2880.2.a.bg 1 168.v even 6 1
3240.2.q.k 2 63.g even 3 1
3240.2.q.k 2 63.h even 3 1
3240.2.q.x 2 63.j odd 6 1
3240.2.q.x 2 63.n odd 6 1
3600.2.a.h 1 420.ba even 6 1
3600.2.f.t 2 420.bp odd 12 2
3920.2.a.s 1 28.f even 6 1
4840.2.a.f 1 77.h odd 6 1
6760.2.a.i 1 91.r even 6 1
9680.2.a.q 1 308.n even 6 1
9800.2.a.x 1 35.i odd 6 1

## Hecke kernels

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

 $$T_{3}$$ T3 $$T_{11}^{2} + 4T_{11} + 16$$ T11^2 + 4*T11 + 16 $$T_{13} + 2$$ T13 + 2

## Hecke characteristic polynomials

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