# Properties

 Label 600.2.f.b Level $600$ Weight $2$ Character orbit 600.f Analytic conductor $4.791$ 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] = [600,2,Mod(49,600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(600, 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("600.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 600.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: $$4.79102412128$$ 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 120) 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.

 $$f(q)$$ $$=$$ $$q + i q^{3} - q^{9}+O(q^{10})$$ q + i * q^3 - q^9 $$q + i q^{3} - q^{9} - 4 q^{11} + 6 i q^{13} + 6 i q^{17} + 4 q^{19} - i q^{27} + 2 q^{29} - 8 q^{31} - 4 i q^{33} + 2 i q^{37} - 6 q^{39} - 6 q^{41} + 12 i q^{43} - 8 i q^{47} + 7 q^{49} - 6 q^{51} + 6 i q^{53} + 4 i q^{57} - 12 q^{59} + 14 q^{61} - 4 i q^{67} + 8 q^{71} - 6 i q^{73} + 8 q^{79} + q^{81} - 12 i q^{83} + 2 i q^{87} - 10 q^{89} - 8 i q^{93} - 2 i q^{97} + 4 q^{99} +O(q^{100})$$ q + i * q^3 - q^9 - 4 * q^11 + 6*i * q^13 + 6*i * q^17 + 4 * q^19 - i * q^27 + 2 * q^29 - 8 * q^31 - 4*i * q^33 + 2*i * q^37 - 6 * q^39 - 6 * q^41 + 12*i * q^43 - 8*i * q^47 + 7 * q^49 - 6 * q^51 + 6*i * q^53 + 4*i * q^57 - 12 * q^59 + 14 * q^61 - 4*i * q^67 + 8 * q^71 - 6*i * q^73 + 8 * q^79 + q^81 - 12*i * q^83 + 2*i * q^87 - 10 * q^89 - 8*i * q^93 - 2*i * q^97 + 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} - 8 q^{11} + 8 q^{19} + 4 q^{29} - 16 q^{31} - 12 q^{39} - 12 q^{41} + 14 q^{49} - 12 q^{51} - 24 q^{59} + 28 q^{61} + 16 q^{71} + 16 q^{79} + 2 q^{81} - 20 q^{89} + 8 q^{99}+O(q^{100})$$ 2 * q - 2 * q^9 - 8 * q^11 + 8 * q^19 + 4 * q^29 - 16 * q^31 - 12 * q^39 - 12 * q^41 + 14 * q^49 - 12 * q^51 - 24 * q^59 + 28 * q^61 + 16 * q^71 + 16 * q^79 + 2 * q^81 - 20 * q^89 + 8 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$$.

 $$n$$ $$151$$ $$301$$ $$401$$ $$577$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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 0 0 −1.00000 0
49.2 0 1.00000i 0 0 0 0 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 600.2.f.b 2
3.b odd 2 1 1800.2.f.j 2
4.b odd 2 1 1200.2.f.g 2
5.b even 2 1 inner 600.2.f.b 2
5.c odd 4 1 120.2.a.b 1
5.c odd 4 1 600.2.a.c 1
8.b even 2 1 4800.2.f.bc 2
8.d odd 2 1 4800.2.f.i 2
12.b even 2 1 3600.2.f.c 2
15.d odd 2 1 1800.2.f.j 2
15.e even 4 1 360.2.a.b 1
15.e even 4 1 1800.2.a.n 1
20.d odd 2 1 1200.2.f.g 2
20.e even 4 1 240.2.a.c 1
20.e even 4 1 1200.2.a.o 1
35.f even 4 1 5880.2.a.a 1
40.e odd 2 1 4800.2.f.i 2
40.f even 2 1 4800.2.f.bc 2
40.i odd 4 1 960.2.a.c 1
40.i odd 4 1 4800.2.a.cd 1
40.k even 4 1 960.2.a.j 1
40.k even 4 1 4800.2.a.r 1
45.k odd 12 2 3240.2.q.g 2
45.l even 12 2 3240.2.q.q 2
60.h even 2 1 3600.2.f.c 2
60.l odd 4 1 720.2.a.d 1
60.l odd 4 1 3600.2.a.t 1
80.i odd 4 1 3840.2.k.o 2
80.j even 4 1 3840.2.k.j 2
80.s even 4 1 3840.2.k.j 2
80.t odd 4 1 3840.2.k.o 2
120.q odd 4 1 2880.2.a.bb 1
120.w even 4 1 2880.2.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.a.b 1 5.c odd 4 1
240.2.a.c 1 20.e even 4 1
360.2.a.b 1 15.e even 4 1
600.2.a.c 1 5.c odd 4 1
600.2.f.b 2 1.a even 1 1 trivial
600.2.f.b 2 5.b even 2 1 inner
720.2.a.d 1 60.l odd 4 1
960.2.a.c 1 40.i odd 4 1
960.2.a.j 1 40.k even 4 1
1200.2.a.o 1 20.e even 4 1
1200.2.f.g 2 4.b odd 2 1
1200.2.f.g 2 20.d odd 2 1
1800.2.a.n 1 15.e even 4 1
1800.2.f.j 2 3.b odd 2 1
1800.2.f.j 2 15.d odd 2 1
2880.2.a.x 1 120.w even 4 1
2880.2.a.bb 1 120.q odd 4 1
3240.2.q.g 2 45.k odd 12 2
3240.2.q.q 2 45.l even 12 2
3600.2.a.t 1 60.l odd 4 1
3600.2.f.c 2 12.b even 2 1
3600.2.f.c 2 60.h even 2 1
3840.2.k.j 2 80.j even 4 1
3840.2.k.j 2 80.s even 4 1
3840.2.k.o 2 80.i odd 4 1
3840.2.k.o 2 80.t odd 4 1
4800.2.a.r 1 40.k even 4 1
4800.2.a.cd 1 40.i odd 4 1
4800.2.f.i 2 8.d odd 2 1
4800.2.f.i 2 40.e odd 2 1
4800.2.f.bc 2 8.b even 2 1
4800.2.f.bc 2 40.f even 2 1
5880.2.a.a 1 35.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(600, [\chi])$$:

 $$T_{7}$$ T7 $$T_{11} + 4$$ T11 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$(T + 4)^{2}$$
$13$ $$T^{2} + 36$$
$17$ $$T^{2} + 36$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2}$$
$29$ $$(T - 2)^{2}$$
$31$ $$(T + 8)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} + 144$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 36$$
$59$ $$(T + 12)^{2}$$
$61$ $$(T - 14)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$(T - 8)^{2}$$
$73$ $$T^{2} + 36$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$(T + 10)^{2}$$
$97$ $$T^{2} + 4$$