# Properties

 Label 2352.4.a.u 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 42) 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} - 15 q^{5} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 - 15 * q^5 + 9 * q^9 $$q + 3 q^{3} - 15 q^{5} + 9 q^{9} + 9 q^{11} - 88 q^{13} - 45 q^{15} - 84 q^{17} - 104 q^{19} + 84 q^{23} + 100 q^{25} + 27 q^{27} + 51 q^{29} - 185 q^{31} + 27 q^{33} + 44 q^{37} - 264 q^{39} - 168 q^{41} - 326 q^{43} - 135 q^{45} + 138 q^{47} - 252 q^{51} + 639 q^{53} - 135 q^{55} - 312 q^{57} - 159 q^{59} + 722 q^{61} + 1320 q^{65} + 166 q^{67} + 252 q^{69} - 1086 q^{71} + 218 q^{73} + 300 q^{75} + 583 q^{79} + 81 q^{81} + 597 q^{83} + 1260 q^{85} + 153 q^{87} - 1038 q^{89} - 555 q^{93} + 1560 q^{95} - 169 q^{97} + 81 q^{99}+O(q^{100})$$ q + 3 * q^3 - 15 * q^5 + 9 * q^9 + 9 * q^11 - 88 * q^13 - 45 * q^15 - 84 * q^17 - 104 * q^19 + 84 * q^23 + 100 * q^25 + 27 * q^27 + 51 * q^29 - 185 * q^31 + 27 * q^33 + 44 * q^37 - 264 * q^39 - 168 * q^41 - 326 * q^43 - 135 * q^45 + 138 * q^47 - 252 * q^51 + 639 * q^53 - 135 * q^55 - 312 * q^57 - 159 * q^59 + 722 * q^61 + 1320 * q^65 + 166 * q^67 + 252 * q^69 - 1086 * q^71 + 218 * q^73 + 300 * q^75 + 583 * q^79 + 81 * q^81 + 597 * q^83 + 1260 * q^85 + 153 * q^87 - 1038 * q^89 - 555 * q^93 + 1560 * q^95 - 169 * q^97 + 81 * 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 −15.0000 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.u 1
4.b odd 2 1 294.4.a.a 1
7.b odd 2 1 2352.4.a.q 1
7.c even 3 2 336.4.q.d 2
12.b even 2 1 882.4.a.r 1
28.d even 2 1 294.4.a.g 1
28.f even 6 2 294.4.e.e 2
28.g odd 6 2 42.4.e.b 2
84.h odd 2 1 882.4.a.h 1
84.j odd 6 2 882.4.g.l 2
84.n even 6 2 126.4.g.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.b 2 28.g odd 6 2
126.4.g.a 2 84.n even 6 2
294.4.a.a 1 4.b odd 2 1
294.4.a.g 1 28.d even 2 1
294.4.e.e 2 28.f even 6 2
336.4.q.d 2 7.c even 3 2
882.4.a.h 1 84.h odd 2 1
882.4.a.r 1 12.b even 2 1
882.4.g.l 2 84.j odd 6 2
2352.4.a.q 1 7.b odd 2 1
2352.4.a.u 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2352))$$:

 $$T_{5} + 15$$ T5 + 15 $$T_{11} - 9$$ T11 - 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T + 15$$
$7$ $$T$$
$11$ $$T - 9$$
$13$ $$T + 88$$
$17$ $$T + 84$$
$19$ $$T + 104$$
$23$ $$T - 84$$
$29$ $$T - 51$$
$31$ $$T + 185$$
$37$ $$T - 44$$
$41$ $$T + 168$$
$43$ $$T + 326$$
$47$ $$T - 138$$
$53$ $$T - 639$$
$59$ $$T + 159$$
$61$ $$T - 722$$
$67$ $$T - 166$$
$71$ $$T + 1086$$
$73$ $$T - 218$$
$79$ $$T - 583$$
$83$ $$T - 597$$
$89$ $$T + 1038$$
$97$ $$T + 169$$