# Properties

 Label 2352.2.a.w Level $2352$ Weight $2$ Character orbit 2352.a Self dual yes Analytic conductor $18.781$ Analytic rank $0$ 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] = [2352,2,Mod(1,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, 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("2352.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2352.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: $$18.7808145554$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) 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)

 $$f(q)$$ $$=$$ $$q + q^{3} + 2 q^{5} + q^{9}+O(q^{10})$$ q + q^3 + 2 * q^5 + q^9 $$q + q^{3} + 2 q^{5} + q^{9} + 2 q^{11} - q^{13} + 2 q^{15} + q^{19} - q^{25} + q^{27} + 4 q^{29} + 9 q^{31} + 2 q^{33} + 3 q^{37} - q^{39} + 10 q^{41} - 5 q^{43} + 2 q^{45} - 6 q^{47} + 12 q^{53} + 4 q^{55} + q^{57} - 12 q^{59} - 10 q^{61} - 2 q^{65} + 5 q^{67} + 6 q^{71} + 3 q^{73} - q^{75} + q^{79} + q^{81} + 6 q^{83} + 4 q^{87} - 16 q^{89} + 9 q^{93} + 2 q^{95} + 6 q^{97} + 2 q^{99}+O(q^{100})$$ q + q^3 + 2 * q^5 + q^9 + 2 * q^11 - q^13 + 2 * q^15 + q^19 - q^25 + q^27 + 4 * q^29 + 9 * q^31 + 2 * q^33 + 3 * q^37 - q^39 + 10 * q^41 - 5 * q^43 + 2 * q^45 - 6 * q^47 + 12 * q^53 + 4 * q^55 + q^57 - 12 * q^59 - 10 * q^61 - 2 * q^65 + 5 * q^67 + 6 * q^71 + 3 * q^73 - q^75 + q^79 + q^81 + 6 * q^83 + 4 * q^87 - 16 * q^89 + 9 * q^93 + 2 * q^95 + 6 * q^97 + 2 * 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 1.00000 0 2.00000 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

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$-1$$

## 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 2352.2.a.w 1
3.b odd 2 1 7056.2.a.m 1
4.b odd 2 1 147.2.a.b 1
7.b odd 2 1 2352.2.a.d 1
7.c even 3 2 2352.2.q.c 2
7.d odd 6 2 336.2.q.f 2
8.b even 2 1 9408.2.a.k 1
8.d odd 2 1 9408.2.a.bz 1
12.b even 2 1 441.2.a.a 1
20.d odd 2 1 3675.2.a.c 1
21.c even 2 1 7056.2.a.bp 1
21.g even 6 2 1008.2.s.d 2
28.d even 2 1 147.2.a.c 1
28.f even 6 2 21.2.e.a 2
28.g odd 6 2 147.2.e.a 2
56.e even 2 1 9408.2.a.bg 1
56.h odd 2 1 9408.2.a.cv 1
56.j odd 6 2 1344.2.q.c 2
56.m even 6 2 1344.2.q.m 2
84.h odd 2 1 441.2.a.b 1
84.j odd 6 2 63.2.e.b 2
84.n even 6 2 441.2.e.e 2
140.c even 2 1 3675.2.a.a 1
140.s even 6 2 525.2.i.e 2
140.x odd 12 4 525.2.r.e 4
252.n even 6 2 567.2.g.a 2
252.r odd 6 2 567.2.h.a 2
252.bj even 6 2 567.2.h.f 2
252.bn odd 6 2 567.2.g.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.e.a 2 28.f even 6 2
63.2.e.b 2 84.j odd 6 2
147.2.a.b 1 4.b odd 2 1
147.2.a.c 1 28.d even 2 1
147.2.e.a 2 28.g odd 6 2
336.2.q.f 2 7.d odd 6 2
441.2.a.a 1 12.b even 2 1
441.2.a.b 1 84.h odd 2 1
441.2.e.e 2 84.n even 6 2
525.2.i.e 2 140.s even 6 2
525.2.r.e 4 140.x odd 12 4
567.2.g.a 2 252.n even 6 2
567.2.g.f 2 252.bn odd 6 2
567.2.h.a 2 252.r odd 6 2
567.2.h.f 2 252.bj even 6 2
1008.2.s.d 2 21.g even 6 2
1344.2.q.c 2 56.j odd 6 2
1344.2.q.m 2 56.m even 6 2
2352.2.a.d 1 7.b odd 2 1
2352.2.a.w 1 1.a even 1 1 trivial
2352.2.q.c 2 7.c even 3 2
3675.2.a.a 1 140.c even 2 1
3675.2.a.c 1 20.d odd 2 1
7056.2.a.m 1 3.b odd 2 1
7056.2.a.bp 1 21.c even 2 1
9408.2.a.k 1 8.b even 2 1
9408.2.a.bg 1 56.e even 2 1
9408.2.a.bz 1 8.d odd 2 1
9408.2.a.cv 1 56.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2352))$$:

 $$T_{5} - 2$$ T5 - 2 $$T_{11} - 2$$ T11 - 2 $$T_{13} + 1$$ T13 + 1 $$T_{17}$$ T17

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T - 2$$
$7$ $$T$$
$11$ $$T - 2$$
$13$ $$T + 1$$
$17$ $$T$$
$19$ $$T - 1$$
$23$ $$T$$
$29$ $$T - 4$$
$31$ $$T - 9$$
$37$ $$T - 3$$
$41$ $$T - 10$$
$43$ $$T + 5$$
$47$ $$T + 6$$
$53$ $$T - 12$$
$59$ $$T + 12$$
$61$ $$T + 10$$
$67$ $$T - 5$$
$71$ $$T - 6$$
$73$ $$T - 3$$
$79$ $$T - 1$$
$83$ $$T - 6$$
$89$ $$T + 16$$
$97$ $$T - 6$$
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