# Properties

 Label 684.2.a.b Level $684$ Weight $2$ Character orbit 684.a Self dual yes Analytic conductor $5.462$ 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] = [684,2,Mod(1,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, 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("684.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$684 = 2^{2} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 684.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: $$5.46176749826$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 76) 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)

 $$f(q)$$ $$=$$ $$q + q^{5} - 3 q^{7}+O(q^{10})$$ q + q^5 - 3 * q^7 $$q + q^{5} - 3 q^{7} - 5 q^{11} - 4 q^{13} + 3 q^{17} - q^{19} - 8 q^{23} - 4 q^{25} + 2 q^{29} + 4 q^{31} - 3 q^{35} + 10 q^{37} - 10 q^{41} + q^{43} + q^{47} + 2 q^{49} + 4 q^{53} - 5 q^{55} - 6 q^{59} - 13 q^{61} - 4 q^{65} - 12 q^{67} - 2 q^{71} + 9 q^{73} + 15 q^{77} + 8 q^{79} + 12 q^{83} + 3 q^{85} - 12 q^{89} + 12 q^{91} - q^{95} - 8 q^{97}+O(q^{100})$$ q + q^5 - 3 * q^7 - 5 * q^11 - 4 * q^13 + 3 * q^17 - q^19 - 8 * q^23 - 4 * q^25 + 2 * q^29 + 4 * q^31 - 3 * q^35 + 10 * q^37 - 10 * q^41 + q^43 + q^47 + 2 * q^49 + 4 * q^53 - 5 * q^55 - 6 * q^59 - 13 * q^61 - 4 * q^65 - 12 * q^67 - 2 * q^71 + 9 * q^73 + 15 * q^77 + 8 * q^79 + 12 * q^83 + 3 * q^85 - 12 * q^89 + 12 * q^91 - q^95 - 8 * q^97

## 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 0 0 1.00000 0 −3.00000 0 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$19$$ $$1$$

## 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 684.2.a.b 1
3.b odd 2 1 76.2.a.a 1
4.b odd 2 1 2736.2.a.q 1
12.b even 2 1 304.2.a.a 1
15.d odd 2 1 1900.2.a.b 1
15.e even 4 2 1900.2.c.b 2
21.c even 2 1 3724.2.a.a 1
24.f even 2 1 1216.2.a.q 1
24.h odd 2 1 1216.2.a.c 1
33.d even 2 1 9196.2.a.f 1
57.d even 2 1 1444.2.a.a 1
57.f even 6 2 1444.2.e.c 2
57.h odd 6 2 1444.2.e.a 2
60.h even 2 1 7600.2.a.p 1
228.b odd 2 1 5776.2.a.p 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.a.a 1 3.b odd 2 1
304.2.a.a 1 12.b even 2 1
684.2.a.b 1 1.a even 1 1 trivial
1216.2.a.c 1 24.h odd 2 1
1216.2.a.q 1 24.f even 2 1
1444.2.a.a 1 57.d even 2 1
1444.2.e.a 2 57.h odd 6 2
1444.2.e.c 2 57.f even 6 2
1900.2.a.b 1 15.d odd 2 1
1900.2.c.b 2 15.e even 4 2
2736.2.a.q 1 4.b odd 2 1
3724.2.a.a 1 21.c even 2 1
5776.2.a.p 1 228.b odd 2 1
7600.2.a.p 1 60.h even 2 1
9196.2.a.f 1 33.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(684))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T + 3$$
$11$ $$T + 5$$
$13$ $$T + 4$$
$17$ $$T - 3$$
$19$ $$T + 1$$
$23$ $$T + 8$$
$29$ $$T - 2$$
$31$ $$T - 4$$
$37$ $$T - 10$$
$41$ $$T + 10$$
$43$ $$T - 1$$
$47$ $$T - 1$$
$53$ $$T - 4$$
$59$ $$T + 6$$
$61$ $$T + 13$$
$67$ $$T + 12$$
$71$ $$T + 2$$
$73$ $$T - 9$$
$79$ $$T - 8$$
$83$ $$T - 12$$
$89$ $$T + 12$$
$97$ $$T + 8$$