# Properties

 Label 576.4.a.q Level $576$ Weight $4$ Character orbit 576.a Self dual yes Analytic conductor $33.985$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,4,Mod(1,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("576.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$576 = 2^{6} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 576.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: $$33.9851001633$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 6) 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)

 $$f(q)$$ $$=$$ $$q + 6 q^{5} - 16 q^{7}+O(q^{10})$$ q + 6 * q^5 - 16 * q^7 $$q + 6 q^{5} - 16 q^{7} + 12 q^{11} - 38 q^{13} + 126 q^{17} - 20 q^{19} - 168 q^{23} - 89 q^{25} + 30 q^{29} - 88 q^{31} - 96 q^{35} - 254 q^{37} - 42 q^{41} + 52 q^{43} + 96 q^{47} - 87 q^{49} + 198 q^{53} + 72 q^{55} - 660 q^{59} + 538 q^{61} - 228 q^{65} - 884 q^{67} - 792 q^{71} + 218 q^{73} - 192 q^{77} - 520 q^{79} - 492 q^{83} + 756 q^{85} - 810 q^{89} + 608 q^{91} - 120 q^{95} + 1154 q^{97}+O(q^{100})$$ q + 6 * q^5 - 16 * q^7 + 12 * q^11 - 38 * q^13 + 126 * q^17 - 20 * q^19 - 168 * q^23 - 89 * q^25 + 30 * q^29 - 88 * q^31 - 96 * q^35 - 254 * q^37 - 42 * q^41 + 52 * q^43 + 96 * q^47 - 87 * q^49 + 198 * q^53 + 72 * q^55 - 660 * q^59 + 538 * q^61 - 228 * q^65 - 884 * q^67 - 792 * q^71 + 218 * q^73 - 192 * q^77 - 520 * q^79 - 492 * q^83 + 756 * q^85 - 810 * q^89 + 608 * q^91 - 120 * q^95 + 1154 * 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 0 0 6.00000 0 −16.0000 0 0 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
$$2$$ $$1$$
$$3$$ $$-1$$

## 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 576.4.a.q 1
3.b odd 2 1 192.4.a.i 1
4.b odd 2 1 576.4.a.r 1
8.b even 2 1 18.4.a.a 1
8.d odd 2 1 144.4.a.c 1
12.b even 2 1 192.4.a.c 1
24.f even 2 1 48.4.a.c 1
24.h odd 2 1 6.4.a.a 1
40.f even 2 1 450.4.a.h 1
40.i odd 4 2 450.4.c.e 2
48.i odd 4 2 768.4.d.n 2
48.k even 4 2 768.4.d.c 2
56.h odd 2 1 882.4.a.n 1
56.j odd 6 2 882.4.g.f 2
56.p even 6 2 882.4.g.i 2
72.j odd 6 2 162.4.c.f 2
72.n even 6 2 162.4.c.c 2
88.b odd 2 1 2178.4.a.e 1
120.i odd 2 1 150.4.a.i 1
120.m even 2 1 1200.4.a.b 1
120.q odd 4 2 1200.4.f.j 2
120.w even 4 2 150.4.c.d 2
168.e odd 2 1 2352.4.a.e 1
168.i even 2 1 294.4.a.e 1
168.s odd 6 2 294.4.e.h 2
168.ba even 6 2 294.4.e.g 2
264.m even 2 1 726.4.a.f 1
312.b odd 2 1 1014.4.a.g 1
312.y even 4 2 1014.4.b.d 2
408.b odd 2 1 1734.4.a.d 1
456.p even 2 1 2166.4.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 24.h odd 2 1
18.4.a.a 1 8.b even 2 1
48.4.a.c 1 24.f even 2 1
144.4.a.c 1 8.d odd 2 1
150.4.a.i 1 120.i odd 2 1
150.4.c.d 2 120.w even 4 2
162.4.c.c 2 72.n even 6 2
162.4.c.f 2 72.j odd 6 2
192.4.a.c 1 12.b even 2 1
192.4.a.i 1 3.b odd 2 1
294.4.a.e 1 168.i even 2 1
294.4.e.g 2 168.ba even 6 2
294.4.e.h 2 168.s odd 6 2
450.4.a.h 1 40.f even 2 1
450.4.c.e 2 40.i odd 4 2
576.4.a.q 1 1.a even 1 1 trivial
576.4.a.r 1 4.b odd 2 1
726.4.a.f 1 264.m even 2 1
768.4.d.c 2 48.k even 4 2
768.4.d.n 2 48.i odd 4 2
882.4.a.n 1 56.h odd 2 1
882.4.g.f 2 56.j odd 6 2
882.4.g.i 2 56.p even 6 2
1014.4.a.g 1 312.b odd 2 1
1014.4.b.d 2 312.y even 4 2
1200.4.a.b 1 120.m even 2 1
1200.4.f.j 2 120.q odd 4 2
1734.4.a.d 1 408.b odd 2 1
2166.4.a.i 1 456.p even 2 1
2178.4.a.e 1 88.b odd 2 1
2352.4.a.e 1 168.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(576))$$:

 $$T_{5} - 6$$ T5 - 6 $$T_{7} + 16$$ T7 + 16 $$T_{11} - 12$$ T11 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 6$$
$7$ $$T + 16$$
$11$ $$T - 12$$
$13$ $$T + 38$$
$17$ $$T - 126$$
$19$ $$T + 20$$
$23$ $$T + 168$$
$29$ $$T - 30$$
$31$ $$T + 88$$
$37$ $$T + 254$$
$41$ $$T + 42$$
$43$ $$T - 52$$
$47$ $$T - 96$$
$53$ $$T - 198$$
$59$ $$T + 660$$
$61$ $$T - 538$$
$67$ $$T + 884$$
$71$ $$T + 792$$
$73$ $$T - 218$$
$79$ $$T + 520$$
$83$ $$T + 492$$
$89$ $$T + 810$$
$97$ $$T - 1154$$
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