# Properties

 Label 1260.1.u.b Level $1260$ Weight $1$ Character orbit 1260.u Analytic conductor $0.629$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ CM discriminant -84 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1260,1,Mod(307,1260)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1260, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 1, 2]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("1260.307");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1260.u (of order $$4$$, degree $$2$$, 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: $$0.628821915918$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.220500.3

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - \zeta_{8}^{3} q^{2} - \zeta_{8}^{2} q^{4} + q^{5} - \zeta_{8} q^{7} - \zeta_{8} q^{8} +O(q^{10})$$ q - z^3 * q^2 - z^2 * q^4 + q^5 - z * q^7 - z * q^8 $$q - \zeta_{8}^{3} q^{2} - \zeta_{8}^{2} q^{4} + q^{5} - \zeta_{8} q^{7} - \zeta_{8} q^{8} - \zeta_{8}^{3} q^{10} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{11} - q^{14} - q^{16} + (\zeta_{8}^{2} - 1) q^{17} + (\zeta_{8}^{3} + \zeta_{8}) q^{19} - \zeta_{8}^{2} q^{20} + ( - \zeta_{8}^{2} - 1) q^{22} + q^{25} + \zeta_{8}^{3} q^{28} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{31} + \zeta_{8}^{3} q^{32} + (\zeta_{8}^{3} + \zeta_{8}) q^{34} - \zeta_{8} q^{35} + (\zeta_{8}^{2} + 1) q^{37} + (\zeta_{8}^{2} + 1) q^{38} - \zeta_{8} q^{40} - \zeta_{8}^{2} q^{41} + (\zeta_{8}^{3} - \zeta_{8}) q^{44} + \zeta_{8}^{2} q^{49} - \zeta_{8}^{3} q^{50} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{55} + \zeta_{8}^{2} q^{56} + ( - \zeta_{8}^{2} + 1) q^{62} + \zeta_{8}^{2} q^{64} + (\zeta_{8}^{2} + 1) q^{68} - q^{70} + (\zeta_{8}^{3} + \zeta_{8}) q^{71} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{74} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{76} + (\zeta_{8}^{2} - 1) q^{77} - q^{80} - 2 \zeta_{8} q^{82} + (\zeta_{8}^{2} - 1) q^{85} + (\zeta_{8}^{2} - 1) q^{88} + (\zeta_{8}^{3} + \zeta_{8}) q^{95} + \zeta_{8} q^{98} +O(q^{100})$$ q - z^3 * q^2 - z^2 * q^4 + q^5 - z * q^7 - z * q^8 - z^3 * q^10 + (-z^3 - z) * q^11 - q^14 - q^16 + (z^2 - 1) * q^17 + (z^3 + z) * q^19 - z^2 * q^20 + (-z^2 - 1) * q^22 + q^25 + z^3 * q^28 + (-z^3 + z) * q^31 + z^3 * q^32 + (z^3 + z) * q^34 - z * q^35 + (z^2 + 1) * q^37 + (z^2 + 1) * q^38 - z * q^40 - z^2 * q^41 + (z^3 - z) * q^44 + z^2 * q^49 - z^3 * q^50 + (-z^3 - z) * q^55 + z^2 * q^56 + (-z^2 + 1) * q^62 + z^2 * q^64 + (z^2 + 1) * q^68 - q^70 + (z^3 + z) * q^71 + (-z^3 + z) * q^74 + (-z^3 + z) * q^76 + (z^2 - 1) * q^77 - q^80 - 2*z * q^82 + (z^2 - 1) * q^85 + (z^2 - 1) * q^88 + (z^3 + z) * q^95 + z * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{5}+O(q^{10})$$ 4 * q + 4 * q^5 $$4 q + 4 q^{5} - 4 q^{14} - 4 q^{16} - 4 q^{17} - 4 q^{22} + 4 q^{25} + 4 q^{37} + 4 q^{38} + 4 q^{62} + 4 q^{68} - 4 q^{70} - 4 q^{77} - 4 q^{80} - 4 q^{85} - 4 q^{88}+O(q^{100})$$ 4 * q + 4 * q^5 - 4 * q^14 - 4 * q^16 - 4 * q^17 - 4 * q^22 + 4 * q^25 + 4 * q^37 + 4 * q^38 + 4 * q^62 + 4 * q^68 - 4 * q^70 - 4 * q^77 - 4 * q^80 - 4 * q^85 - 4 * q^88

## Character values

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

 $$n$$ $$281$$ $$631$$ $$757$$ $$1081$$ $$\chi(n)$$ $$1$$ $$-1$$ $$\zeta_{8}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
307.1
 −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
−0.707107 + 0.707107i 0 1.00000i 1.00000 0 0.707107 + 0.707107i 0.707107 + 0.707107i 0 −0.707107 + 0.707107i
307.2 0.707107 0.707107i 0 1.00000i 1.00000 0 −0.707107 0.707107i −0.707107 0.707107i 0 0.707107 0.707107i
1063.1 −0.707107 0.707107i 0 1.00000i 1.00000 0 0.707107 0.707107i 0.707107 0.707107i 0 −0.707107 0.707107i
1063.2 0.707107 + 0.707107i 0 1.00000i 1.00000 0 −0.707107 + 0.707107i −0.707107 + 0.707107i 0 0.707107 + 0.707107i
 $$n$$: 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
84.h odd 2 1 CM by $$\Q(\sqrt{-21})$$
4.b odd 2 1 inner
15.e even 4 1 inner
21.c even 2 1 inner
35.f even 4 1 inner
60.l odd 4 1 inner
140.j odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.1.u.b yes 4
3.b odd 2 1 1260.1.u.a 4
4.b odd 2 1 inner 1260.1.u.b yes 4
5.c odd 4 1 1260.1.u.a 4
7.b odd 2 1 1260.1.u.a 4
12.b even 2 1 1260.1.u.a 4
15.e even 4 1 inner 1260.1.u.b yes 4
20.e even 4 1 1260.1.u.a 4
21.c even 2 1 inner 1260.1.u.b yes 4
28.d even 2 1 1260.1.u.a 4
35.f even 4 1 inner 1260.1.u.b yes 4
60.l odd 4 1 inner 1260.1.u.b yes 4
84.h odd 2 1 CM 1260.1.u.b yes 4
105.k odd 4 1 1260.1.u.a 4
140.j odd 4 1 inner 1260.1.u.b yes 4
420.w even 4 1 1260.1.u.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.1.u.a 4 3.b odd 2 1
1260.1.u.a 4 5.c odd 4 1
1260.1.u.a 4 7.b odd 2 1
1260.1.u.a 4 12.b even 2 1
1260.1.u.a 4 20.e even 4 1
1260.1.u.a 4 28.d even 2 1
1260.1.u.a 4 105.k odd 4 1
1260.1.u.a 4 420.w even 4 1
1260.1.u.b yes 4 1.a even 1 1 trivial
1260.1.u.b yes 4 4.b odd 2 1 inner
1260.1.u.b yes 4 15.e even 4 1 inner
1260.1.u.b yes 4 21.c even 2 1 inner
1260.1.u.b yes 4 35.f even 4 1 inner
1260.1.u.b yes 4 60.l odd 4 1 inner
1260.1.u.b yes 4 84.h odd 2 1 CM
1260.1.u.b yes 4 140.j odd 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{17}^{2} + 2T_{17} + 2$$ acting on $$S_{1}^{\mathrm{new}}(1260, [\chi])$$.

## Hecke characteristic polynomials

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