# Properties

 Label 600.2.a.a Level $600$ Weight $2$ Character orbit 600.a Self dual yes Analytic conductor $4.791$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,2,Mod(1,600)]

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, 0]))

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

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

chi := DirichletCharacter("600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## 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: yes Analytic conductor: $$4.79102412128$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 120) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{3} - 4 q^{7} + q^{9}+O(q^{10})$$ q - q^3 - 4 * q^7 + q^9 $$q - q^{3} - 4 q^{7} + q^{9} + 6 q^{13} + 2 q^{17} + 4 q^{19} + 4 q^{21} + 8 q^{23} - q^{27} - 6 q^{29} + 6 q^{37} - 6 q^{39} + 10 q^{41} + 4 q^{43} - 8 q^{47} + 9 q^{49} - 2 q^{51} - 10 q^{53} - 4 q^{57} + 6 q^{61} - 4 q^{63} + 4 q^{67} - 8 q^{69} + 14 q^{73} + 16 q^{79} + q^{81} - 12 q^{83} + 6 q^{87} + 2 q^{89} - 24 q^{91} - 2 q^{97}+O(q^{100})$$ q - q^3 - 4 * q^7 + q^9 + 6 * q^13 + 2 * q^17 + 4 * q^19 + 4 * q^21 + 8 * q^23 - q^27 - 6 * q^29 + 6 * q^37 - 6 * q^39 + 10 * q^41 + 4 * q^43 - 8 * q^47 + 9 * q^49 - 2 * q^51 - 10 * q^53 - 4 * q^57 + 6 * q^61 - 4 * q^63 + 4 * q^67 - 8 * q^69 + 14 * q^73 + 16 * q^79 + q^81 - 12 * q^83 + 6 * q^87 + 2 * q^89 - 24 * q^91 - 2 * q^97

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 −1.00000 0 0 0 −4.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$+1$$
$$5$$ $$+1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.2.a.a 1
3.b odd 2 1 1800.2.a.c 1
4.b odd 2 1 1200.2.a.r 1
5.b even 2 1 120.2.a.a 1
5.c odd 4 2 600.2.f.c 2
8.b even 2 1 4800.2.a.bl 1
8.d odd 2 1 4800.2.a.bh 1
12.b even 2 1 3600.2.a.bo 1
15.d odd 2 1 360.2.a.e 1
15.e even 4 2 1800.2.f.g 2
20.d odd 2 1 240.2.a.a 1
20.e even 4 2 1200.2.f.f 2
35.c odd 2 1 5880.2.a.p 1
40.e odd 2 1 960.2.a.n 1
40.f even 2 1 960.2.a.g 1
40.i odd 4 2 4800.2.f.u 2
40.k even 4 2 4800.2.f.n 2
45.h odd 6 2 3240.2.q.a 2
45.j even 6 2 3240.2.q.m 2
60.h even 2 1 720.2.a.f 1
60.l odd 4 2 3600.2.f.l 2
80.k odd 4 2 3840.2.k.z 2
80.q even 4 2 3840.2.k.a 2
120.i odd 2 1 2880.2.a.r 1
120.m even 2 1 2880.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.a.a 1 5.b even 2 1
240.2.a.a 1 20.d odd 2 1
360.2.a.e 1 15.d odd 2 1
600.2.a.a 1 1.a even 1 1 trivial
600.2.f.c 2 5.c odd 4 2
720.2.a.f 1 60.h even 2 1
960.2.a.g 1 40.f even 2 1
960.2.a.n 1 40.e odd 2 1
1200.2.a.r 1 4.b odd 2 1
1200.2.f.f 2 20.e even 4 2
1800.2.a.c 1 3.b odd 2 1
1800.2.f.g 2 15.e even 4 2
2880.2.a.b 1 120.m even 2 1
2880.2.a.r 1 120.i odd 2 1
3240.2.q.a 2 45.h odd 6 2
3240.2.q.m 2 45.j even 6 2
3600.2.a.bo 1 12.b even 2 1
3600.2.f.l 2 60.l odd 4 2
3840.2.k.a 2 80.q even 4 2
3840.2.k.z 2 80.k odd 4 2
4800.2.a.bh 1 8.d odd 2 1
4800.2.a.bl 1 8.b even 2 1
4800.2.f.n 2 40.k even 4 2
4800.2.f.u 2 40.i odd 4 2
5880.2.a.p 1 35.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(600))$$:

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

## Hecke characteristic polynomials

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