# Properties

 Label 1776.2.a.c 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)

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{7} + q^{9}+O(q^{10})$$ q - q^3 + q^7 + q^9 $$q - q^{3} + q^{7} + q^{9} - 3 q^{11} - q^{13} - 3 q^{17} + 7 q^{19} - q^{21} - 3 q^{23} - 5 q^{25} - q^{27} - 2 q^{31} + 3 q^{33} + q^{37} + q^{39} - 6 q^{41} + 4 q^{43} - 6 q^{47} - 6 q^{49} + 3 q^{51} + 9 q^{53} - 7 q^{57} - 10 q^{61} + q^{63} - 2 q^{67} + 3 q^{69} - 12 q^{71} + 5 q^{73} + 5 q^{75} - 3 q^{77} - 2 q^{79} + q^{81} - 3 q^{83} - 3 q^{89} - q^{91} + 2 q^{93} + 2 q^{97} - 3 q^{99}+O(q^{100})$$ q - q^3 + q^7 + q^9 - 3 * q^11 - q^13 - 3 * q^17 + 7 * q^19 - q^21 - 3 * q^23 - 5 * q^25 - q^27 - 2 * q^31 + 3 * q^33 + q^37 + q^39 - 6 * q^41 + 4 * q^43 - 6 * q^47 - 6 * q^49 + 3 * q^51 + 9 * q^53 - 7 * q^57 - 10 * q^61 + q^63 - 2 * q^67 + 3 * q^69 - 12 * q^71 + 5 * q^73 + 5 * q^75 - 3 * q^77 - 2 * q^79 + q^81 - 3 * q^83 - 3 * q^89 - q^91 + 2 * q^93 + 2 * q^97 - 3 * 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$37$$ $$-1$$

## 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.c 1
3.b odd 2 1 5328.2.a.l 1
4.b odd 2 1 222.2.a.e 1
8.b even 2 1 7104.2.a.u 1
8.d odd 2 1 7104.2.a.g 1
12.b even 2 1 666.2.a.a 1
20.d odd 2 1 5550.2.a.h 1
148.b odd 2 1 8214.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
222.2.a.e 1 4.b odd 2 1
666.2.a.a 1 12.b even 2 1
1776.2.a.c 1 1.a even 1 1 trivial
5328.2.a.l 1 3.b odd 2 1
5550.2.a.h 1 20.d odd 2 1
7104.2.a.g 1 8.d odd 2 1
7104.2.a.u 1 8.b even 2 1
8214.2.a.d 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 $$T_{7} - 1$$ T7 - 1 $$T_{11} + 3$$ T11 + 3

## Hecke characteristic polynomials

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