# Properties

 Label 1200.2.f.i Level $1200$ Weight $2$ Character orbit 1200.f 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(49,1200)]

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

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

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

chi := DirichletCharacter("1200.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1200.f (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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 600) 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{3} + 5 i q^{7} - q^{9}+O(q^{10})$$ q + i * q^3 + 5*i * q^7 - q^9 $$q + i q^{3} + 5 i q^{7} - q^{9} + 6 q^{11} + 3 i q^{13} - 2 i q^{17} + q^{19} - 5 q^{21} - 2 i q^{23} - i q^{27} - 6 q^{29} - 3 q^{31} + 6 i q^{33} - 6 i q^{37} - 3 q^{39} + 4 q^{41} + 11 i q^{43} + 10 i q^{47} - 18 q^{49} + 2 q^{51} + 8 i q^{53} + i q^{57} - 6 q^{59} + 3 q^{61} - 5 i q^{63} + i q^{67} + 2 q^{69} + 12 q^{71} - 10 i q^{73} + 30 i q^{77} - 8 q^{79} + q^{81} - 6 i q^{83} - 6 i q^{87} + 16 q^{89} - 15 q^{91} - 3 i q^{93} - 7 i q^{97} - 6 q^{99} +O(q^{100})$$ q + i * q^3 + 5*i * q^7 - q^9 + 6 * q^11 + 3*i * q^13 - 2*i * q^17 + q^19 - 5 * q^21 - 2*i * q^23 - i * q^27 - 6 * q^29 - 3 * q^31 + 6*i * q^33 - 6*i * q^37 - 3 * q^39 + 4 * q^41 + 11*i * q^43 + 10*i * q^47 - 18 * q^49 + 2 * q^51 + 8*i * q^53 + i * q^57 - 6 * q^59 + 3 * q^61 - 5*i * q^63 + i * q^67 + 2 * q^69 + 12 * q^71 - 10*i * q^73 + 30*i * q^77 - 8 * q^79 + q^81 - 6*i * q^83 - 6*i * q^87 + 16 * q^89 - 15 * q^91 - 3*i * q^93 - 7*i * q^97 - 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} + 12 q^{11} + 2 q^{19} - 10 q^{21} - 12 q^{29} - 6 q^{31} - 6 q^{39} + 8 q^{41} - 36 q^{49} + 4 q^{51} - 12 q^{59} + 6 q^{61} + 4 q^{69} + 24 q^{71} - 16 q^{79} + 2 q^{81} + 32 q^{89} - 30 q^{91} - 12 q^{99}+O(q^{100})$$ 2 * q - 2 * q^9 + 12 * q^11 + 2 * q^19 - 10 * q^21 - 12 * q^29 - 6 * q^31 - 6 * q^39 + 8 * q^41 - 36 * q^49 + 4 * q^51 - 12 * q^59 + 6 * q^61 + 4 * q^69 + 24 * q^71 - 16 * q^79 + 2 * q^81 + 32 * q^89 - 30 * q^91 - 12 * 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 − 1.00000i 1.00000i
0 1.00000i 0 0 0 5.00000i 0 −1.00000 0
49.2 0 1.00000i 0 0 0 5.00000i 0 −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
5.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.f.i 2
3.b odd 2 1 3600.2.f.b 2
4.b odd 2 1 600.2.f.a 2
5.b even 2 1 inner 1200.2.f.i 2
5.c odd 4 1 1200.2.a.i 1
5.c odd 4 1 1200.2.a.j 1
8.b even 2 1 4800.2.f.a 2
8.d odd 2 1 4800.2.f.bj 2
12.b even 2 1 1800.2.f.k 2
15.d odd 2 1 3600.2.f.b 2
15.e even 4 1 3600.2.a.a 1
15.e even 4 1 3600.2.a.bq 1
20.d odd 2 1 600.2.f.a 2
20.e even 4 1 600.2.a.e 1
20.e even 4 1 600.2.a.f yes 1
40.e odd 2 1 4800.2.f.bj 2
40.f even 2 1 4800.2.f.a 2
40.i odd 4 1 4800.2.a.a 1
40.i odd 4 1 4800.2.a.cs 1
40.k even 4 1 4800.2.a.b 1
40.k even 4 1 4800.2.a.ct 1
60.h even 2 1 1800.2.f.k 2
60.l odd 4 1 1800.2.a.a 1
60.l odd 4 1 1800.2.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.a.e 1 20.e even 4 1
600.2.a.f yes 1 20.e even 4 1
600.2.f.a 2 4.b odd 2 1
600.2.f.a 2 20.d odd 2 1
1200.2.a.i 1 5.c odd 4 1
1200.2.a.j 1 5.c odd 4 1
1200.2.f.i 2 1.a even 1 1 trivial
1200.2.f.i 2 5.b even 2 1 inner
1800.2.a.a 1 60.l odd 4 1
1800.2.a.x 1 60.l odd 4 1
1800.2.f.k 2 12.b even 2 1
1800.2.f.k 2 60.h even 2 1
3600.2.a.a 1 15.e even 4 1
3600.2.a.bq 1 15.e even 4 1
3600.2.f.b 2 3.b odd 2 1
3600.2.f.b 2 15.d odd 2 1
4800.2.a.a 1 40.i odd 4 1
4800.2.a.b 1 40.k even 4 1
4800.2.a.cs 1 40.i odd 4 1
4800.2.a.ct 1 40.k even 4 1
4800.2.f.a 2 8.b even 2 1
4800.2.f.a 2 40.f even 2 1
4800.2.f.bj 2 8.d odd 2 1
4800.2.f.bj 2 40.e odd 2 1

## Hecke kernels

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

 $$T_{7}^{2} + 25$$ T7^2 + 25 $$T_{11} - 6$$ T11 - 6 $$T_{13}^{2} + 9$$ T13^2 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 25$$
$11$ $$(T - 6)^{2}$$
$13$ $$T^{2} + 9$$
$17$ $$T^{2} + 4$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} + 4$$
$29$ $$(T + 6)^{2}$$
$31$ $$(T + 3)^{2}$$
$37$ $$T^{2} + 36$$
$41$ $$(T - 4)^{2}$$
$43$ $$T^{2} + 121$$
$47$ $$T^{2} + 100$$
$53$ $$T^{2} + 64$$
$59$ $$(T + 6)^{2}$$
$61$ $$(T - 3)^{2}$$
$67$ $$T^{2} + 1$$
$71$ $$(T - 12)^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T - 16)^{2}$$
$97$ $$T^{2} + 49$$