# Properties

 Label 2352.4.a.t Level $2352$ Weight $4$ Character orbit 2352.a Self dual yes Analytic conductor $138.772$ Analytic rank $1$ 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] = [2352,4,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, 4, names="a")

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

chi := DirichletCharacter("2352.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$138.772492334$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 147) 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} - 18 q^{5} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 - 18 * q^5 + 9 * q^9 $$q + 3 q^{3} - 18 q^{5} + 9 q^{9} + 50 q^{11} + 36 q^{13} - 54 q^{15} - 126 q^{17} - 72 q^{19} - 14 q^{23} + 199 q^{25} + 27 q^{27} + 158 q^{29} - 36 q^{31} + 150 q^{33} - 162 q^{37} + 108 q^{39} + 270 q^{41} + 324 q^{43} - 162 q^{45} - 72 q^{47} - 378 q^{51} - 22 q^{53} - 900 q^{55} - 216 q^{57} + 468 q^{59} - 792 q^{61} - 648 q^{65} - 232 q^{67} - 42 q^{69} + 734 q^{71} - 180 q^{73} + 597 q^{75} - 236 q^{79} + 81 q^{81} + 36 q^{83} + 2268 q^{85} + 474 q^{87} - 234 q^{89} - 108 q^{93} + 1296 q^{95} - 468 q^{97} + 450 q^{99}+O(q^{100})$$ q + 3 * q^3 - 18 * q^5 + 9 * q^9 + 50 * q^11 + 36 * q^13 - 54 * q^15 - 126 * q^17 - 72 * q^19 - 14 * q^23 + 199 * q^25 + 27 * q^27 + 158 * q^29 - 36 * q^31 + 150 * q^33 - 162 * q^37 + 108 * q^39 + 270 * q^41 + 324 * q^43 - 162 * q^45 - 72 * q^47 - 378 * q^51 - 22 * q^53 - 900 * q^55 - 216 * q^57 + 468 * q^59 - 792 * q^61 - 648 * q^65 - 232 * q^67 - 42 * q^69 + 734 * q^71 - 180 * q^73 + 597 * q^75 - 236 * q^79 + 81 * q^81 + 36 * q^83 + 2268 * q^85 + 474 * q^87 - 234 * q^89 - 108 * q^93 + 1296 * q^95 - 468 * q^97 + 450 * 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 −18.0000 0 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$-1$$

## 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 2352.4.a.t 1
4.b odd 2 1 147.4.a.f 1
7.b odd 2 1 2352.4.a.s 1
12.b even 2 1 441.4.a.c 1
28.d even 2 1 147.4.a.h yes 1
28.f even 6 2 147.4.e.a 2
28.g odd 6 2 147.4.e.d 2
84.h odd 2 1 441.4.a.a 1
84.j odd 6 2 441.4.e.o 2
84.n even 6 2 441.4.e.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.4.a.f 1 4.b odd 2 1
147.4.a.h yes 1 28.d even 2 1
147.4.e.a 2 28.f even 6 2
147.4.e.d 2 28.g odd 6 2
441.4.a.a 1 84.h odd 2 1
441.4.a.c 1 12.b even 2 1
441.4.e.l 2 84.n even 6 2
441.4.e.o 2 84.j odd 6 2
2352.4.a.s 1 7.b odd 2 1
2352.4.a.t 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2352))$$:

 $$T_{5} + 18$$ T5 + 18 $$T_{11} - 50$$ T11 - 50

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T + 18$$
$7$ $$T$$
$11$ $$T - 50$$
$13$ $$T - 36$$
$17$ $$T + 126$$
$19$ $$T + 72$$
$23$ $$T + 14$$
$29$ $$T - 158$$
$31$ $$T + 36$$
$37$ $$T + 162$$
$41$ $$T - 270$$
$43$ $$T - 324$$
$47$ $$T + 72$$
$53$ $$T + 22$$
$59$ $$T - 468$$
$61$ $$T + 792$$
$67$ $$T + 232$$
$71$ $$T - 734$$
$73$ $$T + 180$$
$79$ $$T + 236$$
$83$ $$T - 36$$
$89$ $$T + 234$$
$97$ $$T + 468$$