# Properties

 Label 1200.2.h.a Level $1200$ Weight $2$ Character orbit 1200.h Analytic conductor $9.582$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,2,Mod(1151,1200)]

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, 0]))

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

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

chi := DirichletCharacter("1200.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1200.h (of order $$2$$, degree $$1$$, 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} - 1) q^{3} + 3 \zeta_{6} q^{9}+O(q^{10})$$ q + (-z - 1) * q^3 + 3*z * q^9 $$q + ( - \zeta_{6} - 1) q^{3} + 3 \zeta_{6} q^{9} - 3 q^{11} + 2 q^{13} + ( - 6 \zeta_{6} + 3) q^{17} + (6 \zeta_{6} - 3) q^{19} - 6 q^{23} + ( - 6 \zeta_{6} + 3) q^{27} + (12 \zeta_{6} - 6) q^{29} + ( - 4 \zeta_{6} + 2) q^{31} + (3 \zeta_{6} + 3) q^{33} - 8 q^{37} + ( - 2 \zeta_{6} - 2) q^{39} + (6 \zeta_{6} - 3) q^{41} + (4 \zeta_{6} - 2) q^{43} + 6 q^{47} + 7 q^{49} + (9 \zeta_{6} - 9) q^{51} + (12 \zeta_{6} - 6) q^{53} + ( - 9 \zeta_{6} + 9) q^{57} - 12 q^{59} + 8 q^{61} + (14 \zeta_{6} - 7) q^{67} + (6 \zeta_{6} + 6) q^{69} + 6 q^{71} - q^{73} + (8 \zeta_{6} - 4) q^{79} + (9 \zeta_{6} - 9) q^{81} + 9 q^{83} + ( - 18 \zeta_{6} + 18) q^{87} + (6 \zeta_{6} - 3) q^{89} + (6 \zeta_{6} - 6) q^{93} - 10 q^{97} - 9 \zeta_{6} q^{99} +O(q^{100})$$ q + (-z - 1) * q^3 + 3*z * q^9 - 3 * q^11 + 2 * q^13 + (-6*z + 3) * q^17 + (6*z - 3) * q^19 - 6 * q^23 + (-6*z + 3) * q^27 + (12*z - 6) * q^29 + (-4*z + 2) * q^31 + (3*z + 3) * q^33 - 8 * q^37 + (-2*z - 2) * q^39 + (6*z - 3) * q^41 + (4*z - 2) * q^43 + 6 * q^47 + 7 * q^49 + (9*z - 9) * q^51 + (12*z - 6) * q^53 + (-9*z + 9) * q^57 - 12 * q^59 + 8 * q^61 + (14*z - 7) * q^67 + (6*z + 6) * q^69 + 6 * q^71 - q^73 + (8*z - 4) * q^79 + (9*z - 9) * q^81 + 9 * q^83 + (-18*z + 18) * q^87 + (6*z - 3) * q^89 + (6*z - 6) * q^93 - 10 * q^97 - 9*z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{3} + 3 q^{9}+O(q^{10})$$ 2 * q - 3 * q^3 + 3 * q^9 $$2 q - 3 q^{3} + 3 q^{9} - 6 q^{11} + 4 q^{13} - 12 q^{23} + 9 q^{33} - 16 q^{37} - 6 q^{39} + 12 q^{47} + 14 q^{49} - 9 q^{51} + 9 q^{57} - 24 q^{59} + 16 q^{61} + 18 q^{69} + 12 q^{71} - 2 q^{73} - 9 q^{81} + 18 q^{83} + 18 q^{87} - 6 q^{93} - 20 q^{97} - 9 q^{99}+O(q^{100})$$ 2 * q - 3 * q^3 + 3 * q^9 - 6 * q^11 + 4 * q^13 - 12 * q^23 + 9 * q^33 - 16 * q^37 - 6 * q^39 + 12 * q^47 + 14 * q^49 - 9 * q^51 + 9 * q^57 - 24 * q^59 + 16 * q^61 + 18 * q^69 + 12 * q^71 - 2 * q^73 - 9 * q^81 + 18 * q^83 + 18 * q^87 - 6 * q^93 - 20 * q^97 - 9 * q^99

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1151.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 −1.50000 0.866025i 0 0 0 0 0 1.50000 + 2.59808i 0
1151.2 0 −1.50000 + 0.866025i 0 0 0 0 0 1.50000 2.59808i 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
12.b 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.h.a 2
3.b odd 2 1 1200.2.h.i yes 2
4.b odd 2 1 1200.2.h.i yes 2
5.b even 2 1 1200.2.h.h yes 2
5.c odd 4 2 1200.2.o.c 4
12.b even 2 1 inner 1200.2.h.a 2
15.d odd 2 1 1200.2.h.b yes 2
15.e even 4 2 1200.2.o.d 4
20.d odd 2 1 1200.2.h.b yes 2
20.e even 4 2 1200.2.o.d 4
60.h even 2 1 1200.2.h.h yes 2
60.l odd 4 2 1200.2.o.c 4

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

## Hecke kernels

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

 $$T_{7}$$ T7 $$T_{11} + 3$$ T11 + 3 $$T_{13} - 2$$ T13 - 2 $$T_{23} + 6$$ T23 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 3T + 3$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$(T + 3)^{2}$$
$13$ $$(T - 2)^{2}$$
$17$ $$T^{2} + 27$$
$19$ $$T^{2} + 27$$
$23$ $$(T + 6)^{2}$$
$29$ $$T^{2} + 108$$
$31$ $$T^{2} + 12$$
$37$ $$(T + 8)^{2}$$
$41$ $$T^{2} + 27$$
$43$ $$T^{2} + 12$$
$47$ $$(T - 6)^{2}$$
$53$ $$T^{2} + 108$$
$59$ $$(T + 12)^{2}$$
$61$ $$(T - 8)^{2}$$
$67$ $$T^{2} + 147$$
$71$ $$(T - 6)^{2}$$
$73$ $$(T + 1)^{2}$$
$79$ $$T^{2} + 48$$
$83$ $$(T - 9)^{2}$$
$89$ $$T^{2} + 27$$
$97$ $$(T + 10)^{2}$$