# Properties

 Label 1260.2.k.c Level $1260$ Weight $2$ Character orbit 1260.k Analytic conductor $10.061$ 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] = [1260,2,Mod(1009,1260)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1260, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1260.1009");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1260.k (of order $$2$$, degree $$1$$, 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: $$10.0611506547$$ 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 140) 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.

 $$f(q)$$ $$=$$ $$q + (i + 2) q^{5} + i q^{7}+O(q^{10})$$ q + (i + 2) * q^5 + i * q^7 $$q + (i + 2) q^{5} + i q^{7} - 3 q^{11} - i q^{13} + 5 i q^{17} + 8 q^{19} + 2 i q^{23} + (4 i + 3) q^{25} - q^{29} - 2 q^{31} + (2 i - 1) q^{35} + 10 i q^{37} + 6 q^{41} + 4 i q^{43} - 11 i q^{47} - q^{49} + 6 i q^{53} + ( - 3 i - 6) q^{55} - 10 q^{59} + ( - 2 i + 1) q^{65} - 10 i q^{67} + 10 i q^{73} - 3 i q^{77} + 7 q^{79} + 12 i q^{83} + (10 i - 5) q^{85} + 8 q^{89} + q^{91} + (8 i + 16) q^{95} + 3 i q^{97} +O(q^{100})$$ q + (i + 2) * q^5 + i * q^7 - 3 * q^11 - i * q^13 + 5*i * q^17 + 8 * q^19 + 2*i * q^23 + (4*i + 3) * q^25 - q^29 - 2 * q^31 + (2*i - 1) * q^35 + 10*i * q^37 + 6 * q^41 + 4*i * q^43 - 11*i * q^47 - q^49 + 6*i * q^53 + (-3*i - 6) * q^55 - 10 * q^59 + (-2*i + 1) * q^65 - 10*i * q^67 + 10*i * q^73 - 3*i * q^77 + 7 * q^79 + 12*i * q^83 + (10*i - 5) * q^85 + 8 * q^89 + q^91 + (8*i + 16) * q^95 + 3*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{5}+O(q^{10})$$ 2 * q + 4 * q^5 $$2 q + 4 q^{5} - 6 q^{11} + 16 q^{19} + 6 q^{25} - 2 q^{29} - 4 q^{31} - 2 q^{35} + 12 q^{41} - 2 q^{49} - 12 q^{55} - 20 q^{59} + 2 q^{65} + 14 q^{79} - 10 q^{85} + 16 q^{89} + 2 q^{91} + 32 q^{95}+O(q^{100})$$ 2 * q + 4 * q^5 - 6 * q^11 + 16 * q^19 + 6 * q^25 - 2 * q^29 - 4 * q^31 - 2 * q^35 + 12 * q^41 - 2 * q^49 - 12 * q^55 - 20 * q^59 + 2 * q^65 + 14 * q^79 - 10 * q^85 + 16 * q^89 + 2 * q^91 + 32 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times$$.

 $$n$$ $$281$$ $$631$$ $$757$$ $$1081$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1009.1
 − 1.00000i 1.00000i
0 0 0 2.00000 1.00000i 0 1.00000i 0 0 0
1009.2 0 0 0 2.00000 + 1.00000i 0 1.00000i 0 0 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 1260.2.k.c 2
3.b odd 2 1 140.2.e.a 2
4.b odd 2 1 5040.2.t.s 2
5.b even 2 1 inner 1260.2.k.c 2
5.c odd 4 1 6300.2.a.c 1
5.c odd 4 1 6300.2.a.t 1
12.b even 2 1 560.2.g.a 2
15.d odd 2 1 140.2.e.a 2
15.e even 4 1 700.2.a.a 1
15.e even 4 1 700.2.a.j 1
20.d odd 2 1 5040.2.t.s 2
21.c even 2 1 980.2.e.b 2
21.g even 6 2 980.2.q.c 4
21.h odd 6 2 980.2.q.f 4
24.f even 2 1 2240.2.g.f 2
24.h odd 2 1 2240.2.g.e 2
60.h even 2 1 560.2.g.a 2
60.l odd 4 1 2800.2.a.a 1
60.l odd 4 1 2800.2.a.bf 1
105.g even 2 1 980.2.e.b 2
105.k odd 4 1 4900.2.a.b 1
105.k odd 4 1 4900.2.a.w 1
105.o odd 6 2 980.2.q.f 4
105.p even 6 2 980.2.q.c 4
120.i odd 2 1 2240.2.g.e 2
120.m even 2 1 2240.2.g.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.e.a 2 3.b odd 2 1
140.2.e.a 2 15.d odd 2 1
560.2.g.a 2 12.b even 2 1
560.2.g.a 2 60.h even 2 1
700.2.a.a 1 15.e even 4 1
700.2.a.j 1 15.e even 4 1
980.2.e.b 2 21.c even 2 1
980.2.e.b 2 105.g even 2 1
980.2.q.c 4 21.g even 6 2
980.2.q.c 4 105.p even 6 2
980.2.q.f 4 21.h odd 6 2
980.2.q.f 4 105.o odd 6 2
1260.2.k.c 2 1.a even 1 1 trivial
1260.2.k.c 2 5.b even 2 1 inner
2240.2.g.e 2 24.h odd 2 1
2240.2.g.e 2 120.i odd 2 1
2240.2.g.f 2 24.f even 2 1
2240.2.g.f 2 120.m even 2 1
2800.2.a.a 1 60.l odd 4 1
2800.2.a.bf 1 60.l odd 4 1
4900.2.a.b 1 105.k odd 4 1
4900.2.a.w 1 105.k odd 4 1
5040.2.t.s 2 4.b odd 2 1
5040.2.t.s 2 20.d odd 2 1
6300.2.a.c 1 5.c odd 4 1
6300.2.a.t 1 5.c odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11} + 3$$ acting on $$S_{2}^{\mathrm{new}}(1260, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 4T + 5$$
$7$ $$T^{2} + 1$$
$11$ $$(T + 3)^{2}$$
$13$ $$T^{2} + 1$$
$17$ $$T^{2} + 25$$
$19$ $$(T - 8)^{2}$$
$23$ $$T^{2} + 4$$
$29$ $$(T + 1)^{2}$$
$31$ $$(T + 2)^{2}$$
$37$ $$T^{2} + 100$$
$41$ $$(T - 6)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 121$$
$53$ $$T^{2} + 36$$
$59$ $$(T + 10)^{2}$$
$61$ $$T^{2}$$
$67$ $$T^{2} + 100$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$(T - 7)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$(T - 8)^{2}$$
$97$ $$T^{2} + 9$$