# Properties

 Label 1200.2.v.a Level $1200$ Weight $2$ Character orbit 1200.v Analytic conductor $9.582$ Analytic rank $1$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,2,Mod(257,1200)]

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([0, 0, 2, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1200.257");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1200.v (of order $$4$$, 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: $$9.58204824255$$ Analytic rank: $$1$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 600) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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 $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{8}^{3} - \zeta_{8}^{2} - 1) q^{3} + (2 \zeta_{8}^{2} - 2) q^{7} + ( - 2 \zeta_{8}^{3} + \cdots + 2 \zeta_{8}) q^{9}+O(q^{10})$$ q + (z^3 - z^2 - 1) * q^3 + (2*z^2 - 2) * q^7 + (-2*z^3 + z^2 + 2*z) * q^9 $$q + (\zeta_{8}^{3} - \zeta_{8}^{2} - 1) q^{3} + (2 \zeta_{8}^{2} - 2) q^{7} + ( - 2 \zeta_{8}^{3} + \cdots + 2 \zeta_{8}) q^{9}+ \cdots + (4 \zeta_{8}^{3} + 16 \zeta_{8}^{2} - 4 \zeta_{8}) q^{99}+O(q^{100})$$ q + (z^3 - z^2 - 1) * q^3 + (2*z^2 - 2) * q^7 + (-2*z^3 + z^2 + 2*z) * q^9 + (4*z^3 + 4*z) * q^11 - 4*z * q^17 - 4*z^2 * q^19 + (-2*z^3 - 2*z + 4) * q^21 - 6*z^3 * q^23 + (z^2 - 5*z - 1) * q^27 + (-4*z^3 + 4*z) * q^29 - 8 * q^31 + (-8*z^3 - 4*z^2 - 4) * q^33 + (8*z^2 - 8) * q^37 + (-4*z^3 - 4*z) * q^41 + (-2*z^2 - 2) * q^43 - 2*z * q^47 - z^2 * q^49 + (4*z^3 + 4*z + 4) * q^51 + 8*z^3 * q^53 + (4*z^2 + 4*z - 4) * q^57 + (4*z^3 - 4*z) * q^59 - 6 * q^61 + (8*z^3 - 2*z^2 - 2) * q^63 + (6*z^2 - 6) * q^67 + (6*z^3 + 6*z^2 - 6*z) * q^69 + (-8*z^3 - 8*z) * q^71 + (-8*z^2 - 8) * q^73 - 16*z * q^77 + (4*z^3 + 4*z + 7) * q^81 - 14*z^3 * q^83 + (4*z^2 - 8*z - 4) * q^87 + (8*z^3 - 8*z) * q^89 + (-8*z^3 + 8*z^2 + 8) * q^93 + (-8*z^2 + 8) * q^97 + (4*z^3 + 16*z^2 - 4*z) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} - 8 q^{7}+O(q^{10})$$ 4 * q - 4 * q^3 - 8 * q^7 $$4 q - 4 q^{3} - 8 q^{7} + 16 q^{21} - 4 q^{27} - 32 q^{31} - 16 q^{33} - 32 q^{37} - 8 q^{43} + 16 q^{51} - 16 q^{57} - 24 q^{61} - 8 q^{63} - 24 q^{67} - 32 q^{73} + 28 q^{81} - 16 q^{87} + 32 q^{93} + 32 q^{97}+O(q^{100})$$ 4 * q - 4 * q^3 - 8 * q^7 + 16 * q^21 - 4 * q^27 - 32 * q^31 - 16 * q^33 - 32 * q^37 - 8 * q^43 + 16 * q^51 - 16 * q^57 - 24 * q^61 - 8 * q^63 - 24 * q^67 - 32 * q^73 + 28 * q^81 - 16 * q^87 + 32 * q^93 + 32 * q^97

## Character values

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

 $$n$$ $$401$$ $$577$$ $$751$$ $$901$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{8}^{2}$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1
 0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i −0.707107 − 0.707107i
0 −1.70711 + 0.292893i 0 0 0 −2.00000 2.00000i 0 2.82843 1.00000i 0
257.2 0 −0.292893 + 1.70711i 0 0 0 −2.00000 2.00000i 0 −2.82843 1.00000i 0
593.1 0 −1.70711 0.292893i 0 0 0 −2.00000 + 2.00000i 0 2.82843 + 1.00000i 0
593.2 0 −0.292893 1.70711i 0 0 0 −2.00000 + 2.00000i 0 −2.82843 + 1.00000i 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
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.2.v.a 4
3.b odd 2 1 inner 1200.2.v.a 4
4.b odd 2 1 600.2.r.e yes 4
5.b even 2 1 1200.2.v.k 4
5.c odd 4 1 inner 1200.2.v.a 4
5.c odd 4 1 1200.2.v.k 4
12.b even 2 1 600.2.r.e yes 4
15.d odd 2 1 1200.2.v.k 4
15.e even 4 1 inner 1200.2.v.a 4
15.e even 4 1 1200.2.v.k 4
20.d odd 2 1 600.2.r.a 4
20.e even 4 1 600.2.r.a 4
20.e even 4 1 600.2.r.e yes 4
60.h even 2 1 600.2.r.a 4
60.l odd 4 1 600.2.r.a 4
60.l odd 4 1 600.2.r.e yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.r.a 4 20.d odd 2 1
600.2.r.a 4 20.e even 4 1
600.2.r.a 4 60.h even 2 1
600.2.r.a 4 60.l odd 4 1
600.2.r.e yes 4 4.b odd 2 1
600.2.r.e yes 4 12.b even 2 1
600.2.r.e yes 4 20.e even 4 1
600.2.r.e yes 4 60.l odd 4 1
1200.2.v.a 4 1.a even 1 1 trivial
1200.2.v.a 4 3.b odd 2 1 inner
1200.2.v.a 4 5.c odd 4 1 inner
1200.2.v.a 4 15.e even 4 1 inner
1200.2.v.k 4 5.b even 2 1
1200.2.v.k 4 5.c odd 4 1
1200.2.v.k 4 15.d odd 2 1
1200.2.v.k 4 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1200, [\chi])$$:

 $$T_{7}^{2} + 4T_{7} + 8$$ T7^2 + 4*T7 + 8 $$T_{11}^{2} + 32$$ T11^2 + 32 $$T_{17}^{4} + 256$$ T17^4 + 256

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 4 T^{3} + \cdots + 9$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 4 T + 8)^{2}$$
$11$ $$(T^{2} + 32)^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4} + 256$$
$19$ $$(T^{2} + 16)^{2}$$
$23$ $$T^{4} + 1296$$
$29$ $$(T^{2} - 32)^{2}$$
$31$ $$(T + 8)^{4}$$
$37$ $$(T^{2} + 16 T + 128)^{2}$$
$41$ $$(T^{2} + 32)^{2}$$
$43$ $$(T^{2} + 4 T + 8)^{2}$$
$47$ $$T^{4} + 16$$
$53$ $$T^{4} + 4096$$
$59$ $$(T^{2} - 32)^{2}$$
$61$ $$(T + 6)^{4}$$
$67$ $$(T^{2} + 12 T + 72)^{2}$$
$71$ $$(T^{2} + 128)^{2}$$
$73$ $$(T^{2} + 16 T + 128)^{2}$$
$79$ $$T^{4}$$
$83$ $$T^{4} + 38416$$
$89$ $$(T^{2} - 128)^{2}$$
$97$ $$(T^{2} - 16 T + 128)^{2}$$