# Properties

 Label 2156.4.a.b Level $2156$ Weight $4$ Character orbit 2156.a Self dual yes Analytic conductor $127.208$ 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] = [2156,4,Mod(1,2156)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2156, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2156.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2156 = 2^{2} \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2156.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: $$127.208117972$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 44) 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)

 $$f(q)$$ $$=$$ $$q + 5 q^{3} + 7 q^{5} - 2 q^{9}+O(q^{10})$$ q + 5 * q^3 + 7 * q^5 - 2 * q^9 $$q + 5 q^{3} + 7 q^{5} - 2 q^{9} - 11 q^{11} - 52 q^{13} + 35 q^{15} - 46 q^{17} + 96 q^{19} + 27 q^{23} - 76 q^{25} - 145 q^{27} + 16 q^{29} + 293 q^{31} - 55 q^{33} - 29 q^{37} - 260 q^{39} + 472 q^{41} - 110 q^{43} - 14 q^{45} + 224 q^{47} - 230 q^{51} + 754 q^{53} - 77 q^{55} + 480 q^{57} - 825 q^{59} + 548 q^{61} - 364 q^{65} - 123 q^{67} + 135 q^{69} + 1001 q^{71} + 1020 q^{73} - 380 q^{75} + 526 q^{79} - 671 q^{81} + 158 q^{83} - 322 q^{85} + 80 q^{87} + 1217 q^{89} + 1465 q^{93} + 672 q^{95} + 263 q^{97} + 22 q^{99}+O(q^{100})$$ q + 5 * q^3 + 7 * q^5 - 2 * q^9 - 11 * q^11 - 52 * q^13 + 35 * q^15 - 46 * q^17 + 96 * q^19 + 27 * q^23 - 76 * q^25 - 145 * q^27 + 16 * q^29 + 293 * q^31 - 55 * q^33 - 29 * q^37 - 260 * q^39 + 472 * q^41 - 110 * q^43 - 14 * q^45 + 224 * q^47 - 230 * q^51 + 754 * q^53 - 77 * q^55 + 480 * q^57 - 825 * q^59 + 548 * q^61 - 364 * q^65 - 123 * q^67 + 135 * q^69 + 1001 * q^71 + 1020 * q^73 - 380 * q^75 + 526 * q^79 - 671 * q^81 + 158 * q^83 - 322 * q^85 + 80 * q^87 + 1217 * q^89 + 1465 * q^93 + 672 * q^95 + 263 * q^97 + 22 * 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 5.00000 0 7.00000 0 0 0 −2.00000 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
$$2$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$+1$$

## 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 2156.4.a.b 1
7.b odd 2 1 44.4.a.a 1
21.c even 2 1 396.4.a.e 1
28.d even 2 1 176.4.a.e 1
35.c odd 2 1 1100.4.a.d 1
35.f even 4 2 1100.4.b.c 2
56.e even 2 1 704.4.a.c 1
56.h odd 2 1 704.4.a.j 1
77.b even 2 1 484.4.a.a 1
84.h odd 2 1 1584.4.a.p 1
308.g odd 2 1 1936.4.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.4.a.a 1 7.b odd 2 1
176.4.a.e 1 28.d even 2 1
396.4.a.e 1 21.c even 2 1
484.4.a.a 1 77.b even 2 1
704.4.a.c 1 56.e even 2 1
704.4.a.j 1 56.h odd 2 1
1100.4.a.d 1 35.c odd 2 1
1100.4.b.c 2 35.f even 4 2
1584.4.a.p 1 84.h odd 2 1
1936.4.a.m 1 308.g odd 2 1
2156.4.a.b 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 5$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2156))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 5$$
$5$ $$T - 7$$
$7$ $$T$$
$11$ $$T + 11$$
$13$ $$T + 52$$
$17$ $$T + 46$$
$19$ $$T - 96$$
$23$ $$T - 27$$
$29$ $$T - 16$$
$31$ $$T - 293$$
$37$ $$T + 29$$
$41$ $$T - 472$$
$43$ $$T + 110$$
$47$ $$T - 224$$
$53$ $$T - 754$$
$59$ $$T + 825$$
$61$ $$T - 548$$
$67$ $$T + 123$$
$71$ $$T - 1001$$
$73$ $$T - 1020$$
$79$ $$T - 526$$
$83$ $$T - 158$$
$89$ $$T - 1217$$
$97$ $$T - 263$$