# Properties

 Label 1776.2.a.h Level $1776$ Weight $2$ Character orbit 1776.a Self dual yes Analytic conductor $14.181$ 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] = [1776,2,Mod(1,1776)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1776, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1776.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1776 = 2^{4} \cdot 3 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1776.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: $$14.1814313990$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 222) 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)

 $$f(q)$$ $$=$$ $$q + q^{3} - 3 q^{7} + q^{9}+O(q^{10})$$ q + q^3 - 3 * q^7 + q^9 $$q + q^{3} - 3 q^{7} + q^{9} - q^{11} + q^{13} - 3 q^{17} - 3 q^{19} - 3 q^{21} + q^{23} - 5 q^{25} + q^{27} - 4 q^{29} + 6 q^{31} - q^{33} - q^{37} + q^{39} - 10 q^{41} - 12 q^{43} + 6 q^{47} + 2 q^{49} - 3 q^{51} - q^{53} - 3 q^{57} + 2 q^{61} - 3 q^{63} - 2 q^{67} + q^{69} - 3 q^{73} - 5 q^{75} + 3 q^{77} - 14 q^{79} + q^{81} - 9 q^{83} - 4 q^{87} - 3 q^{89} - 3 q^{91} + 6 q^{93} - 10 q^{97} - q^{99}+O(q^{100})$$ q + q^3 - 3 * q^7 + q^9 - q^11 + q^13 - 3 * q^17 - 3 * q^19 - 3 * q^21 + q^23 - 5 * q^25 + q^27 - 4 * q^29 + 6 * q^31 - q^33 - q^37 + q^39 - 10 * q^41 - 12 * q^43 + 6 * q^47 + 2 * q^49 - 3 * q^51 - q^53 - 3 * q^57 + 2 * q^61 - 3 * q^63 - 2 * q^67 + q^69 - 3 * q^73 - 5 * q^75 + 3 * q^77 - 14 * q^79 + q^81 - 9 * q^83 - 4 * q^87 - 3 * q^89 - 3 * q^91 + 6 * q^93 - 10 * q^97 - 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 −3.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$37$$ $$+1$$

## 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 1776.2.a.h 1
3.b odd 2 1 5328.2.a.h 1
4.b odd 2 1 222.2.a.d 1
8.b even 2 1 7104.2.a.f 1
8.d odd 2 1 7104.2.a.v 1
12.b even 2 1 666.2.a.b 1
20.d odd 2 1 5550.2.a.n 1
148.b odd 2 1 8214.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
222.2.a.d 1 4.b odd 2 1
666.2.a.b 1 12.b even 2 1
1776.2.a.h 1 1.a even 1 1 trivial
5328.2.a.h 1 3.b odd 2 1
5550.2.a.n 1 20.d odd 2 1
7104.2.a.f 1 8.b even 2 1
7104.2.a.v 1 8.d odd 2 1
8214.2.a.b 1 148.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1776))$$:

 $$T_{5}$$ T5 $$T_{7} + 3$$ T7 + 3 $$T_{11} + 1$$ T11 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T + 3$$
$11$ $$T + 1$$
$13$ $$T - 1$$
$17$ $$T + 3$$
$19$ $$T + 3$$
$23$ $$T - 1$$
$29$ $$T + 4$$
$31$ $$T - 6$$
$37$ $$T + 1$$
$41$ $$T + 10$$
$43$ $$T + 12$$
$47$ $$T - 6$$
$53$ $$T + 1$$
$59$ $$T$$
$61$ $$T - 2$$
$67$ $$T + 2$$
$71$ $$T$$
$73$ $$T + 3$$
$79$ $$T + 14$$
$83$ $$T + 9$$
$89$ $$T + 3$$
$97$ $$T + 10$$