# Properties

 Label 4368.2.a.u Level $4368$ Weight $2$ Character orbit 4368.a Self dual yes Analytic conductor $34.879$ 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] = [4368,2,Mod(1,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4368.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: $$34.8786556029$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1092) 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)

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{7} + q^{9}+O(q^{10})$$ q + q^3 + q^7 + q^9 $$q + q^{3} + q^{7} + q^{9} + 2 q^{11} - q^{13} + 4 q^{17} - 4 q^{19} + q^{21} - 2 q^{23} - 5 q^{25} + q^{27} + 6 q^{29} + 2 q^{33} + 10 q^{37} - q^{39} + 4 q^{41} + 4 q^{43} + q^{49} + 4 q^{51} + 10 q^{53} - 4 q^{57} - 4 q^{59} - 2 q^{61} + q^{63} - 2 q^{69} + 6 q^{71} + 14 q^{73} - 5 q^{75} + 2 q^{77} + 4 q^{79} + q^{81} - 16 q^{83} + 6 q^{87} + 12 q^{89} - q^{91} + 6 q^{97} + 2 q^{99}+O(q^{100})$$ q + q^3 + q^7 + q^9 + 2 * q^11 - q^13 + 4 * q^17 - 4 * q^19 + q^21 - 2 * q^23 - 5 * q^25 + q^27 + 6 * q^29 + 2 * q^33 + 10 * q^37 - q^39 + 4 * q^41 + 4 * q^43 + q^49 + 4 * q^51 + 10 * q^53 - 4 * q^57 - 4 * q^59 - 2 * q^61 + q^63 - 2 * q^69 + 6 * q^71 + 14 * q^73 - 5 * q^75 + 2 * q^77 + 4 * q^79 + q^81 - 16 * q^83 + 6 * q^87 + 12 * q^89 - q^91 + 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 0 0 1.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$-1$$
$$13$$ $$1$$

## 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 4368.2.a.u 1
4.b odd 2 1 1092.2.a.c 1
12.b even 2 1 3276.2.a.d 1
28.d even 2 1 7644.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1092.2.a.c 1 4.b odd 2 1
3276.2.a.d 1 12.b even 2 1
4368.2.a.u 1 1.a even 1 1 trivial
7644.2.a.h 1 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4368))$$:

 $$T_{5}$$ T5 $$T_{11} - 2$$ T11 - 2 $$T_{17} - 4$$ T17 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T - 1$$
$11$ $$T - 2$$
$13$ $$T + 1$$
$17$ $$T - 4$$
$19$ $$T + 4$$
$23$ $$T + 2$$
$29$ $$T - 6$$
$31$ $$T$$
$37$ $$T - 10$$
$41$ $$T - 4$$
$43$ $$T - 4$$
$47$ $$T$$
$53$ $$T - 10$$
$59$ $$T + 4$$
$61$ $$T + 2$$
$67$ $$T$$
$71$ $$T - 6$$
$73$ $$T - 14$$
$79$ $$T - 4$$
$83$ $$T + 16$$
$89$ $$T - 12$$
$97$ $$T - 6$$