# Properties

 Label 156.2.a.a Level $156$ Weight $2$ Character orbit 156.a Self dual yes Analytic conductor $1.246$ 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] = [156,2,Mod(1,156)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(156, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("156.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$156 = 2^{2} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 156.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: $$1.24566627153$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes 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)

 $$f(q)$$ $$=$$ $$q - q^{3} - 4 q^{5} - 2 q^{7} + q^{9}+O(q^{10})$$ q - q^3 - 4 * q^5 - 2 * q^7 + q^9 $$q - q^{3} - 4 q^{5} - 2 q^{7} + q^{9} - 4 q^{11} + q^{13} + 4 q^{15} + 2 q^{17} - 2 q^{19} + 2 q^{21} + 11 q^{25} - q^{27} - 6 q^{29} - 10 q^{31} + 4 q^{33} + 8 q^{35} + 10 q^{37} - q^{39} + 8 q^{41} + 4 q^{43} - 4 q^{45} - 4 q^{47} - 3 q^{49} - 2 q^{51} - 10 q^{53} + 16 q^{55} + 2 q^{57} - 8 q^{59} - 14 q^{61} - 2 q^{63} - 4 q^{65} + 2 q^{67} + 16 q^{71} - 10 q^{73} - 11 q^{75} + 8 q^{77} - 16 q^{79} + q^{81} - 8 q^{85} + 6 q^{87} - 4 q^{89} - 2 q^{91} + 10 q^{93} + 8 q^{95} - 2 q^{97} - 4 q^{99}+O(q^{100})$$ q - q^3 - 4 * q^5 - 2 * q^7 + q^9 - 4 * q^11 + q^13 + 4 * q^15 + 2 * q^17 - 2 * q^19 + 2 * q^21 + 11 * q^25 - q^27 - 6 * q^29 - 10 * q^31 + 4 * q^33 + 8 * q^35 + 10 * q^37 - q^39 + 8 * q^41 + 4 * q^43 - 4 * q^45 - 4 * q^47 - 3 * q^49 - 2 * q^51 - 10 * q^53 + 16 * q^55 + 2 * q^57 - 8 * q^59 - 14 * q^61 - 2 * q^63 - 4 * q^65 + 2 * q^67 + 16 * q^71 - 10 * q^73 - 11 * q^75 + 8 * q^77 - 16 * q^79 + q^81 - 8 * q^85 + 6 * q^87 - 4 * q^89 - 2 * q^91 + 10 * q^93 + 8 * q^95 - 2 * q^97 - 4 * 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 −4.00000 0 −2.00000 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$13$$ $$-1$$

## 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 156.2.a.a 1
3.b odd 2 1 468.2.a.d 1
4.b odd 2 1 624.2.a.e 1
5.b even 2 1 3900.2.a.m 1
5.c odd 4 2 3900.2.h.b 2
7.b odd 2 1 7644.2.a.k 1
8.b even 2 1 2496.2.a.bc 1
8.d odd 2 1 2496.2.a.o 1
9.c even 3 2 4212.2.i.l 2
9.d odd 6 2 4212.2.i.b 2
12.b even 2 1 1872.2.a.s 1
13.b even 2 1 2028.2.a.c 1
13.c even 3 2 2028.2.i.e 2
13.d odd 4 2 2028.2.b.a 2
13.e even 6 2 2028.2.i.g 2
13.f odd 12 4 2028.2.q.h 4
24.f even 2 1 7488.2.a.d 1
24.h odd 2 1 7488.2.a.c 1
39.d odd 2 1 6084.2.a.b 1
39.f even 4 2 6084.2.b.j 2
52.b odd 2 1 8112.2.a.bi 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.a.a 1 1.a even 1 1 trivial
468.2.a.d 1 3.b odd 2 1
624.2.a.e 1 4.b odd 2 1
1872.2.a.s 1 12.b even 2 1
2028.2.a.c 1 13.b even 2 1
2028.2.b.a 2 13.d odd 4 2
2028.2.i.e 2 13.c even 3 2
2028.2.i.g 2 13.e even 6 2
2028.2.q.h 4 13.f odd 12 4
2496.2.a.o 1 8.d odd 2 1
2496.2.a.bc 1 8.b even 2 1
3900.2.a.m 1 5.b even 2 1
3900.2.h.b 2 5.c odd 4 2
4212.2.i.b 2 9.d odd 6 2
4212.2.i.l 2 9.c even 3 2
6084.2.a.b 1 39.d odd 2 1
6084.2.b.j 2 39.f even 4 2
7488.2.a.c 1 24.h odd 2 1
7488.2.a.d 1 24.f even 2 1
7644.2.a.k 1 7.b odd 2 1
8112.2.a.bi 1 52.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 4$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(156))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T + 4$$
$7$ $$T + 2$$
$11$ $$T + 4$$
$13$ $$T - 1$$
$17$ $$T - 2$$
$19$ $$T + 2$$
$23$ $$T$$
$29$ $$T + 6$$
$31$ $$T + 10$$
$37$ $$T - 10$$
$41$ $$T - 8$$
$43$ $$T - 4$$
$47$ $$T + 4$$
$53$ $$T + 10$$
$59$ $$T + 8$$
$61$ $$T + 14$$
$67$ $$T - 2$$
$71$ $$T - 16$$
$73$ $$T + 10$$
$79$ $$T + 16$$
$83$ $$T$$
$89$ $$T + 4$$
$97$ $$T + 2$$