# Properties

 Label 2352.4.a.e Level $2352$ Weight $4$ Character orbit 2352.a Self dual yes Analytic conductor $138.772$ 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] = [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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} - 6 q^{5} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 - 6 * q^5 + 9 * q^9 $$q - 3 q^{3} - 6 q^{5} + 9 q^{9} - 12 q^{11} - 38 q^{13} + 18 q^{15} + 126 q^{17} + 20 q^{19} - 168 q^{23} - 89 q^{25} - 27 q^{27} + 30 q^{29} - 88 q^{31} + 36 q^{33} + 254 q^{37} + 114 q^{39} - 42 q^{41} + 52 q^{43} - 54 q^{45} - 96 q^{47} - 378 q^{51} + 198 q^{53} + 72 q^{55} - 60 q^{57} - 660 q^{59} + 538 q^{61} + 228 q^{65} - 884 q^{67} + 504 q^{69} - 792 q^{71} - 218 q^{73} + 267 q^{75} + 520 q^{79} + 81 q^{81} - 492 q^{83} - 756 q^{85} - 90 q^{87} - 810 q^{89} + 264 q^{93} - 120 q^{95} - 1154 q^{97} - 108 q^{99}+O(q^{100})$$ q - 3 * q^3 - 6 * q^5 + 9 * q^9 - 12 * q^11 - 38 * q^13 + 18 * q^15 + 126 * q^17 + 20 * q^19 - 168 * q^23 - 89 * q^25 - 27 * q^27 + 30 * q^29 - 88 * q^31 + 36 * q^33 + 254 * q^37 + 114 * q^39 - 42 * q^41 + 52 * q^43 - 54 * q^45 - 96 * q^47 - 378 * q^51 + 198 * q^53 + 72 * q^55 - 60 * q^57 - 660 * q^59 + 538 * q^61 + 228 * q^65 - 884 * q^67 + 504 * q^69 - 792 * q^71 - 218 * q^73 + 267 * q^75 + 520 * q^79 + 81 * q^81 - 492 * q^83 - 756 * q^85 - 90 * q^87 - 810 * q^89 + 264 * q^93 - 120 * q^95 - 1154 * q^97 - 108 * 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 −6.00000 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.e 1
4.b odd 2 1 294.4.a.e 1
7.b odd 2 1 48.4.a.c 1
12.b even 2 1 882.4.a.n 1
21.c even 2 1 144.4.a.c 1
28.d even 2 1 6.4.a.a 1
28.f even 6 2 294.4.e.h 2
28.g odd 6 2 294.4.e.g 2
35.c odd 2 1 1200.4.a.b 1
35.f even 4 2 1200.4.f.j 2
56.e even 2 1 192.4.a.i 1
56.h odd 2 1 192.4.a.c 1
84.h odd 2 1 18.4.a.a 1
84.j odd 6 2 882.4.g.i 2
84.n even 6 2 882.4.g.f 2
112.j even 4 2 768.4.d.n 2
112.l odd 4 2 768.4.d.c 2
140.c even 2 1 150.4.a.i 1
140.j odd 4 2 150.4.c.d 2
168.e odd 2 1 576.4.a.q 1
168.i even 2 1 576.4.a.r 1
252.s odd 6 2 162.4.c.c 2
252.bi even 6 2 162.4.c.f 2
308.g odd 2 1 726.4.a.f 1
364.h even 2 1 1014.4.a.g 1
364.p odd 4 2 1014.4.b.d 2
420.o odd 2 1 450.4.a.h 1
420.w even 4 2 450.4.c.e 2
476.e even 2 1 1734.4.a.d 1
532.b odd 2 1 2166.4.a.i 1
924.n even 2 1 2178.4.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 28.d even 2 1
18.4.a.a 1 84.h odd 2 1
48.4.a.c 1 7.b odd 2 1
144.4.a.c 1 21.c even 2 1
150.4.a.i 1 140.c even 2 1
150.4.c.d 2 140.j odd 4 2
162.4.c.c 2 252.s odd 6 2
162.4.c.f 2 252.bi even 6 2
192.4.a.c 1 56.h odd 2 1
192.4.a.i 1 56.e even 2 1
294.4.a.e 1 4.b odd 2 1
294.4.e.g 2 28.g odd 6 2
294.4.e.h 2 28.f even 6 2
450.4.a.h 1 420.o odd 2 1
450.4.c.e 2 420.w even 4 2
576.4.a.q 1 168.e odd 2 1
576.4.a.r 1 168.i even 2 1
726.4.a.f 1 308.g odd 2 1
768.4.d.c 2 112.l odd 4 2
768.4.d.n 2 112.j even 4 2
882.4.a.n 1 12.b even 2 1
882.4.g.f 2 84.n even 6 2
882.4.g.i 2 84.j odd 6 2
1014.4.a.g 1 364.h even 2 1
1014.4.b.d 2 364.p odd 4 2
1200.4.a.b 1 35.c odd 2 1
1200.4.f.j 2 35.f even 4 2
1734.4.a.d 1 476.e even 2 1
2166.4.a.i 1 532.b odd 2 1
2178.4.a.e 1 924.n even 2 1
2352.4.a.e 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2352))$$:

 $$T_{5} + 6$$ T5 + 6 $$T_{11} + 12$$ T11 + 12

## Hecke characteristic polynomials

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