# Properties

 Label 1200.2.v.f Level $1200$ Weight $2$ Character orbit 1200.v Analytic conductor $9.582$ Analytic rank $0$ Dimension $4$ CM discriminant -3 Inner twists $8$

# Learn more

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: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 75) Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} + \beta_1 q^{7} - 3 \beta_{2} q^{9}+O(q^{10})$$ q + b3 * q^3 + b1 * q^7 - 3*b2 * q^9 $$q + \beta_{3} q^{3} + \beta_1 q^{7} - 3 \beta_{2} q^{9} + 3 \beta_{3} q^{13} + \beta_{2} q^{19} - 3 q^{21} + 3 \beta_1 q^{27} - 7 q^{31} + 4 \beta_1 q^{37} - 9 \beta_{2} q^{39} + 7 \beta_{3} q^{43} - 4 \beta_{2} q^{49} - \beta_1 q^{57} - 13 q^{61} - 3 \beta_{3} q^{63} - 9 \beta_1 q^{67} + 8 \beta_{3} q^{73} - 4 \beta_{2} q^{79} - 9 q^{81} - 9 q^{91} - 7 \beta_{3} q^{93} - 11 \beta_1 q^{97}+O(q^{100})$$ q + b3 * q^3 + b1 * q^7 - 3*b2 * q^9 + 3*b3 * q^13 + b2 * q^19 - 3 * q^21 + 3*b1 * q^27 - 7 * q^31 + 4*b1 * q^37 - 9*b2 * q^39 + 7*b3 * q^43 - 4*b2 * q^49 - b1 * q^57 - 13 * q^61 - 3*b3 * q^63 - 9*b1 * q^67 + 8*b3 * q^73 - 4*b2 * q^79 - 9 * q^81 - 9 * q^91 - 7*b3 * q^93 - 11*b1 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 12 q^{21} - 28 q^{31} - 52 q^{61} - 36 q^{81} - 36 q^{91}+O(q^{100})$$ 4 * q - 12 * q^21 - 28 * q^31 - 52 * q^61 - 36 * q^81 - 36 * q^91

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 3$$ (v^2) / 3 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 3$$ (v^3) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$3\beta_{2}$$ 3*b2 $$\nu^{3}$$ $$=$$ $$3\beta_{3}$$ 3*b3

## Character values

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

 $$n$$ $$401$$ $$577$$ $$751$$ $$901$$ $$\chi(n)$$ $$-1$$ $$\beta_{2}$$ $$1$$ $$1$$

## 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
 1.22474 + 1.22474i −1.22474 − 1.22474i 1.22474 − 1.22474i −1.22474 + 1.22474i
0 −1.22474 + 1.22474i 0 0 0 1.22474 + 1.22474i 0 3.00000i 0
257.2 0 1.22474 1.22474i 0 0 0 −1.22474 1.22474i 0 3.00000i 0
593.1 0 −1.22474 1.22474i 0 0 0 1.22474 1.22474i 0 3.00000i 0
593.2 0 1.22474 + 1.22474i 0 0 0 −1.22474 + 1.22474i 0 3.00000i 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 CM by $$\Q(\sqrt{-3})$$
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.2.v.f 4
3.b odd 2 1 CM 1200.2.v.f 4
4.b odd 2 1 75.2.e.b 4
5.b even 2 1 inner 1200.2.v.f 4
5.c odd 4 2 inner 1200.2.v.f 4
12.b even 2 1 75.2.e.b 4
15.d odd 2 1 inner 1200.2.v.f 4
15.e even 4 2 inner 1200.2.v.f 4
20.d odd 2 1 75.2.e.b 4
20.e even 4 2 75.2.e.b 4
60.h even 2 1 75.2.e.b 4
60.l odd 4 2 75.2.e.b 4

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

## Hecke kernels

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

 $$T_{7}^{4} + 9$$ T7^4 + 9 $$T_{11}$$ T11 $$T_{17}$$ T17

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 9$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 9$$
$11$ $$T^{4}$$
$13$ $$T^{4} + 729$$
$17$ $$T^{4}$$
$19$ $$(T^{2} + 1)^{2}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$(T + 7)^{4}$$
$37$ $$T^{4} + 2304$$
$41$ $$T^{4}$$
$43$ $$T^{4} + 21609$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$(T + 13)^{4}$$
$67$ $$T^{4} + 59049$$
$71$ $$T^{4}$$
$73$ $$T^{4} + 36864$$
$79$ $$(T^{2} + 16)^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4} + 131769$$
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