Properties

 Label 1200.1.bd.b Level $1200$ Weight $1$ Character orbit 1200.bd 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(443,1200)]

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, 3]))

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

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

chi := DirichletCharacter("1200.443");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1200.bd (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: $$\Z[a_1, a_2]$$ 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.

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

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
443.1
 − 1.00000i 1.00000i
1.00000i 1.00000 −1.00000 0 1.00000i 0 1.00000i 1.00000 0
707.1 1.00000i 1.00000 −1.00000 0 1.00000i 0 1.00000i 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.s even 4 1 inner
240.bd 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.bd.b yes 2
3.b odd 2 1 1200.1.bd.a yes 2
5.b even 2 1 1200.1.bd.a yes 2
5.c odd 4 1 1200.1.z.a 2
5.c odd 4 1 1200.1.z.b yes 2
15.d odd 2 1 CM 1200.1.bd.b yes 2
15.e even 4 1 1200.1.z.a 2
15.e even 4 1 1200.1.z.b yes 2
16.f odd 4 1 1200.1.z.a 2
48.k even 4 1 1200.1.z.b yes 2
80.j even 4 1 1200.1.bd.a yes 2
80.k odd 4 1 1200.1.z.b yes 2
80.s even 4 1 inner 1200.1.bd.b yes 2
240.t even 4 1 1200.1.z.a 2
240.z odd 4 1 1200.1.bd.a yes 2
240.bd odd 4 1 inner 1200.1.bd.b yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1200.1.z.a 2 5.c odd 4 1
1200.1.z.a 2 15.e even 4 1
1200.1.z.a 2 16.f odd 4 1
1200.1.z.a 2 240.t even 4 1
1200.1.z.b yes 2 5.c odd 4 1
1200.1.z.b yes 2 15.e even 4 1
1200.1.z.b yes 2 48.k even 4 1
1200.1.z.b yes 2 80.k odd 4 1
1200.1.bd.a yes 2 3.b odd 2 1
1200.1.bd.a yes 2 5.b even 2 1
1200.1.bd.a yes 2 80.j even 4 1
1200.1.bd.a yes 2 240.z odd 4 1
1200.1.bd.b yes 2 1.a even 1 1 trivial
1200.1.bd.b yes 2 15.d odd 2 1 CM
1200.1.bd.b yes 2 80.s even 4 1 inner
1200.1.bd.b yes 2 240.bd 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 $$S_{1}^{\mathrm{new}}(1200, [\chi])$$.

Hecke characteristic polynomials

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