# Properties

 Label 1200.2.o.b Level $1200$ Weight $2$ Character orbit 1200.o Analytic conductor $9.582$ Analytic rank $0$ 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(1199,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1200.1199");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1200.o (of order $$2$$, degree $$1$$, 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: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 240) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} + \beta_{3} q^{7} - 3 q^{9}+O(q^{10})$$ q - b2 * q^3 + b3 * q^7 - 3 * q^9 $$q - \beta_{2} q^{3} + \beta_{3} q^{7} - 3 q^{9} + \beta_{3} q^{11} + 2 \beta_1 q^{13} + 6 q^{17} + 2 \beta_{2} q^{19} - 3 \beta_1 q^{21} + 2 \beta_{2} q^{23} + 3 \beta_{2} q^{27} - 3 \beta_1 q^{29} + 2 \beta_{2} q^{31} - 3 \beta_1 q^{33} + 2 \beta_1 q^{37} + 2 \beta_{3} q^{39} - 6 \beta_1 q^{41} + 2 \beta_{3} q^{43} + 2 \beta_{2} q^{47} + 5 q^{49} - 6 \beta_{2} q^{51} - 6 q^{53} + 6 q^{57} - \beta_{3} q^{59} - 10 q^{61} - 3 \beta_{3} q^{63} - 2 \beta_{3} q^{67} + 6 q^{69} + 4 \beta_{3} q^{71} - \beta_1 q^{73} + 12 q^{77} - 6 \beta_{2} q^{79} + 9 q^{81} - 6 \beta_{2} q^{83} - 3 \beta_{3} q^{87} + 8 \beta_{2} q^{91} + 6 q^{93} - 5 \beta_1 q^{97} - 3 \beta_{3} q^{99}+O(q^{100})$$ q - b2 * q^3 + b3 * q^7 - 3 * q^9 + b3 * q^11 + 2*b1 * q^13 + 6 * q^17 + 2*b2 * q^19 - 3*b1 * q^21 + 2*b2 * q^23 + 3*b2 * q^27 - 3*b1 * q^29 + 2*b2 * q^31 - 3*b1 * q^33 + 2*b1 * q^37 + 2*b3 * q^39 - 6*b1 * q^41 + 2*b3 * q^43 + 2*b2 * q^47 + 5 * q^49 - 6*b2 * q^51 - 6 * q^53 + 6 * q^57 - b3 * q^59 - 10 * q^61 - 3*b3 * q^63 - 2*b3 * q^67 + 6 * q^69 + 4*b3 * q^71 - b1 * q^73 + 12 * q^77 - 6*b2 * q^79 + 9 * q^81 - 6*b2 * q^83 - 3*b3 * q^87 + 8*b2 * q^91 + 6 * q^93 - 5*b1 * q^97 - 3*b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{9}+O(q^{10})$$ 4 * q - 12 * q^9 $$4 q - 12 q^{9} + 24 q^{17} + 20 q^{49} - 24 q^{53} + 24 q^{57} - 40 q^{61} + 24 q^{69} + 48 q^{77} + 36 q^{81} + 24 q^{93}+O(q^{100})$$ 4 * q - 12 * q^9 + 24 * q^17 + 20 * q^49 - 24 * q^53 + 24 * q^57 - 40 * q^61 + 24 * q^69 + 48 * q^77 + 36 * q^81 + 24 * q^93

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{12}^{3}$$ 2*v^3 $$\beta_{2}$$ $$=$$ $$2\zeta_{12}^{2} - 1$$ 2*v^2 - 1 $$\beta_{3}$$ $$=$$ $$-2\zeta_{12}^{3} + 4\zeta_{12}$$ -2*v^3 + 4*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 4$$ (b3 + b1) / 4 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta_{2} + 1 ) / 2$$ (b2 + 1) / 2 $$\zeta_{12}^{3}$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1199.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0 1.73205i 0 0 0 −3.46410 0 −3.00000 0
1199.2 0 1.73205i 0 0 0 3.46410 0 −3.00000 0
1199.3 0 1.73205i 0 0 0 −3.46410 0 −3.00000 0
1199.4 0 1.73205i 0 0 0 3.46410 0 −3.00000 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
4.b odd 2 1 inner
15.d odd 2 1 inner
60.h even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.2.o.b 4
3.b odd 2 1 1200.2.o.a 4
4.b odd 2 1 inner 1200.2.o.b 4
5.b even 2 1 1200.2.o.a 4
5.c odd 4 1 240.2.h.b 4
5.c odd 4 1 1200.2.h.m 4
12.b even 2 1 1200.2.o.a 4
15.d odd 2 1 inner 1200.2.o.b 4
15.e even 4 1 240.2.h.b 4
15.e even 4 1 1200.2.h.m 4
20.d odd 2 1 1200.2.o.a 4
20.e even 4 1 240.2.h.b 4
20.e even 4 1 1200.2.h.m 4
40.i odd 4 1 960.2.h.d 4
40.k even 4 1 960.2.h.d 4
60.h even 2 1 inner 1200.2.o.b 4
60.l odd 4 1 240.2.h.b 4
60.l odd 4 1 1200.2.h.m 4
120.q odd 4 1 960.2.h.d 4
120.w even 4 1 960.2.h.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.h.b 4 5.c odd 4 1
240.2.h.b 4 15.e even 4 1
240.2.h.b 4 20.e even 4 1
240.2.h.b 4 60.l odd 4 1
960.2.h.d 4 40.i odd 4 1
960.2.h.d 4 40.k even 4 1
960.2.h.d 4 120.q odd 4 1
960.2.h.d 4 120.w even 4 1
1200.2.h.m 4 5.c odd 4 1
1200.2.h.m 4 15.e even 4 1
1200.2.h.m 4 20.e even 4 1
1200.2.h.m 4 60.l odd 4 1
1200.2.o.a 4 3.b odd 2 1
1200.2.o.a 4 5.b even 2 1
1200.2.o.a 4 12.b even 2 1
1200.2.o.a 4 20.d odd 2 1
1200.2.o.b 4 1.a even 1 1 trivial
1200.2.o.b 4 4.b odd 2 1 inner
1200.2.o.b 4 15.d odd 2 1 inner
1200.2.o.b 4 60.h even 2 1 inner

## Hecke kernels

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

 $$T_{7}^{2} - 12$$ T7^2 - 12 $$T_{11}^{2} - 12$$ T11^2 - 12 $$T_{17} - 6$$ T17 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 3)^{2}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} - 12)^{2}$$
$11$ $$(T^{2} - 12)^{2}$$
$13$ $$(T^{2} + 16)^{2}$$
$17$ $$(T - 6)^{4}$$
$19$ $$(T^{2} + 12)^{2}$$
$23$ $$(T^{2} + 12)^{2}$$
$29$ $$(T^{2} + 36)^{2}$$
$31$ $$(T^{2} + 12)^{2}$$
$37$ $$(T^{2} + 16)^{2}$$
$41$ $$(T^{2} + 144)^{2}$$
$43$ $$(T^{2} - 48)^{2}$$
$47$ $$(T^{2} + 12)^{2}$$
$53$ $$(T + 6)^{4}$$
$59$ $$(T^{2} - 12)^{2}$$
$61$ $$(T + 10)^{4}$$
$67$ $$(T^{2} - 48)^{2}$$
$71$ $$(T^{2} - 192)^{2}$$
$73$ $$(T^{2} + 4)^{2}$$
$79$ $$(T^{2} + 108)^{2}$$
$83$ $$(T^{2} + 108)^{2}$$
$89$ $$T^{4}$$
$97$ $$(T^{2} + 100)^{2}$$