# Properties

 Label 600.4.a.o Level $600$ Weight $4$ Character orbit 600.a Self dual yes Analytic conductor $35.401$ 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] = [600,4,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, 4, names="a")

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

chi := DirichletCharacter("600.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$35.4011460034$$ 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + 10 q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 + 10 * q^7 + 9 * q^9 $$q + 3 q^{3} + 10 q^{7} + 9 q^{9} - 46 q^{11} - 34 q^{13} + 66 q^{17} + 104 q^{19} + 30 q^{21} + 164 q^{23} + 27 q^{27} + 224 q^{29} - 72 q^{31} - 138 q^{33} - 22 q^{37} - 102 q^{39} + 194 q^{41} + 108 q^{43} - 480 q^{47} - 243 q^{49} + 198 q^{51} + 286 q^{53} + 312 q^{57} + 426 q^{59} + 698 q^{61} + 90 q^{63} + 328 q^{67} + 492 q^{69} + 188 q^{71} - 740 q^{73} - 460 q^{77} + 1168 q^{79} + 81 q^{81} + 412 q^{83} + 672 q^{87} + 1206 q^{89} - 340 q^{91} - 216 q^{93} - 1384 q^{97} - 414 q^{99}+O(q^{100})$$ q + 3 * q^3 + 10 * q^7 + 9 * q^9 - 46 * q^11 - 34 * q^13 + 66 * q^17 + 104 * q^19 + 30 * q^21 + 164 * q^23 + 27 * q^27 + 224 * q^29 - 72 * q^31 - 138 * q^33 - 22 * q^37 - 102 * q^39 + 194 * q^41 + 108 * q^43 - 480 * q^47 - 243 * q^49 + 198 * q^51 + 286 * q^53 + 312 * q^57 + 426 * q^59 + 698 * q^61 + 90 * q^63 + 328 * q^67 + 492 * q^69 + 188 * q^71 - 740 * q^73 - 460 * q^77 + 1168 * q^79 + 81 * q^81 + 412 * q^83 + 672 * q^87 + 1206 * q^89 - 340 * q^91 - 216 * q^93 - 1384 * q^97 - 414 * q^99

## 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 3.00000 0 0 0 10.0000 0 9.00000 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 600.4.a.o 1
3.b odd 2 1 1800.4.a.y 1
4.b odd 2 1 1200.4.a.f 1
5.b even 2 1 600.4.a.b 1
5.c odd 4 2 120.4.f.a 2
15.d odd 2 1 1800.4.a.j 1
15.e even 4 2 360.4.f.c 2
20.d odd 2 1 1200.4.a.bh 1
20.e even 4 2 240.4.f.b 2
40.i odd 4 2 960.4.f.l 2
40.k even 4 2 960.4.f.g 2
60.l odd 4 2 720.4.f.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.f.a 2 5.c odd 4 2
240.4.f.b 2 20.e even 4 2
360.4.f.c 2 15.e even 4 2
600.4.a.b 1 5.b even 2 1
600.4.a.o 1 1.a even 1 1 trivial
720.4.f.g 2 60.l odd 4 2
960.4.f.g 2 40.k even 4 2
960.4.f.l 2 40.i odd 4 2
1200.4.a.f 1 4.b odd 2 1
1200.4.a.bh 1 20.d odd 2 1
1800.4.a.j 1 15.d odd 2 1
1800.4.a.y 1 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(600))$$:

 $$T_{7} - 10$$ T7 - 10 $$T_{11} + 46$$ T11 + 46

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T$$
$7$ $$T - 10$$
$11$ $$T + 46$$
$13$ $$T + 34$$
$17$ $$T - 66$$
$19$ $$T - 104$$
$23$ $$T - 164$$
$29$ $$T - 224$$
$31$ $$T + 72$$
$37$ $$T + 22$$
$41$ $$T - 194$$
$43$ $$T - 108$$
$47$ $$T + 480$$
$53$ $$T - 286$$
$59$ $$T - 426$$
$61$ $$T - 698$$
$67$ $$T - 328$$
$71$ $$T - 188$$
$73$ $$T + 740$$
$79$ $$T - 1168$$
$83$ $$T - 412$$
$89$ $$T - 1206$$
$97$ $$T + 1384$$