# Properties

 Label 1200.1.z.b Level $1200$ Weight $1$ Character orbit 1200.z Analytic conductor $0.599$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -15 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,1,Mod(107,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("1200.107");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1200.z (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: $$0.598878015160$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.153600.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q + q^{2} - i q^{3} + q^{4} - i q^{6} + q^{8} - q^{9} +O(q^{10})$$ q + q^2 - z * q^3 + q^4 - z * q^6 + q^8 - q^9 $$q + q^{2} - i q^{3} + q^{4} - i q^{6} + q^{8} - q^{9} - i q^{12} + q^{16} + ( - i + 1) q^{17} - q^{18} + (i - 1) q^{19} + ( - i - 1) q^{23} - i q^{24} + i q^{27} + 2 i q^{31} + q^{32} + ( - i + 1) q^{34} - q^{36} + (i - 1) q^{38} + ( - i - 1) q^{46} + ( - i - 1) q^{47} - i q^{48} - i q^{49} + ( - i - 1) q^{51} + 2 i q^{53} + i q^{54} + (i + 1) q^{57} + (i - 1) q^{61} + 2 i q^{62} + q^{64} + ( - i + 1) q^{68} + (i - 1) q^{69} - q^{72} + (i - 1) q^{76} + q^{81} + ( - i - 1) q^{92} + 2 q^{93} + ( - i - 1) q^{94} - i q^{96} - i q^{98} +O(q^{100})$$ q + q^2 - z * q^3 + q^4 - z * q^6 + q^8 - q^9 - z * q^12 + q^16 + (-z + 1) * q^17 - q^18 + (z - 1) * q^19 + (-z - 1) * q^23 - z * q^24 + z * q^27 + 2*z * q^31 + q^32 + (-z + 1) * q^34 - q^36 + (z - 1) * q^38 + (-z - 1) * q^46 + (-z - 1) * q^47 - z * q^48 - z * q^49 + (-z - 1) * q^51 + 2*z * q^53 + z * q^54 + (z + 1) * q^57 + (z - 1) * q^61 + 2*z * q^62 + q^64 + (-z + 1) * q^68 + (z - 1) * q^69 - q^72 + (z - 1) * q^76 + q^81 + (-z - 1) * q^92 + 2 * q^93 + (-z - 1) * q^94 - z * q^96 - z * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9} + 2 q^{16} + 2 q^{17} - 2 q^{18} - 2 q^{19} - 2 q^{23} + 2 q^{32} + 2 q^{34} - 2 q^{36} - 2 q^{38} - 2 q^{46} - 2 q^{47} - 2 q^{51} + 2 q^{57} - 2 q^{61} + 2 q^{64} + 2 q^{68} - 2 q^{69} - 2 q^{72} - 2 q^{76} + 2 q^{81} - 2 q^{92} + 4 q^{93} - 2 q^{94}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 + 2 * q^16 + 2 * q^17 - 2 * q^18 - 2 * q^19 - 2 * q^23 + 2 * q^32 + 2 * q^34 - 2 * q^36 - 2 * q^38 - 2 * q^46 - 2 * q^47 - 2 * q^51 + 2 * q^57 - 2 * q^61 + 2 * q^64 + 2 * q^68 - 2 * q^69 - 2 * q^72 - 2 * q^76 + 2 * q^81 - 2 * q^92 + 4 * q^93 - 2 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$577$$ $$751$$ $$901$$ $$\chi(n)$$ $$-1$$ $$i$$ $$-1$$ $$i$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1
 1.00000i − 1.00000i
1.00000 1.00000i 1.00000 0 1.00000i 0 1.00000 −1.00000 0
1043.1 1.00000 1.00000i 1.00000 0 1.00000i 0 1.00000 −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
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
80.j even 4 1 inner
240.z odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.1.z.b yes 2
3.b odd 2 1 1200.1.z.a 2
5.b even 2 1 1200.1.z.a 2
5.c odd 4 1 1200.1.bd.a yes 2
5.c odd 4 1 1200.1.bd.b yes 2
15.d odd 2 1 CM 1200.1.z.b yes 2
15.e even 4 1 1200.1.bd.a yes 2
15.e even 4 1 1200.1.bd.b yes 2
16.f odd 4 1 1200.1.bd.a yes 2
48.k even 4 1 1200.1.bd.b yes 2
80.j even 4 1 inner 1200.1.z.b yes 2
80.k odd 4 1 1200.1.bd.b yes 2
80.s even 4 1 1200.1.z.a 2
240.t even 4 1 1200.1.bd.a yes 2
240.z odd 4 1 inner 1200.1.z.b yes 2
240.bd odd 4 1 1200.1.z.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1200.1.z.a 2 3.b odd 2 1
1200.1.z.a 2 5.b even 2 1
1200.1.z.a 2 80.s even 4 1
1200.1.z.a 2 240.bd odd 4 1
1200.1.z.b yes 2 1.a even 1 1 trivial
1200.1.z.b yes 2 15.d odd 2 1 CM
1200.1.z.b yes 2 80.j even 4 1 inner
1200.1.z.b yes 2 240.z odd 4 1 inner
1200.1.bd.a yes 2 5.c odd 4 1
1200.1.bd.a yes 2 15.e even 4 1
1200.1.bd.a yes 2 16.f odd 4 1
1200.1.bd.a yes 2 240.t even 4 1
1200.1.bd.b yes 2 5.c odd 4 1
1200.1.bd.b yes 2 15.e even 4 1
1200.1.bd.b yes 2 48.k even 4 1
1200.1.bd.b yes 2 80.k odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{17}^{2} - 2T_{17} + 2$$ acting on $$S_{1}^{\mathrm{new}}(1200, [\chi])$$.

## Hecke characteristic polynomials

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