# Properties

 Label 1764.2.a.h Level $1764$ Weight $2$ Character orbit 1764.a Self dual yes Analytic conductor $14.086$ 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] = [1764,2,Mod(1,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, 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("1764.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1764 = 2^{2} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1764.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: $$14.0856109166$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{5}+O(q^{10})$$ q + 2 * q^5 $$q + 2 q^{5} - 2 q^{11} - 3 q^{13} - 8 q^{17} - q^{19} - 8 q^{23} - q^{25} - 4 q^{29} + 3 q^{31} - q^{37} - 6 q^{41} + 11 q^{43} - 6 q^{47} + 12 q^{53} - 4 q^{55} - 4 q^{59} - 6 q^{61} - 6 q^{65} + 13 q^{67} + 10 q^{71} - 11 q^{73} - 3 q^{79} - 2 q^{83} - 16 q^{85} - 2 q^{95} + 10 q^{97}+O(q^{100})$$ q + 2 * q^5 - 2 * q^11 - 3 * q^13 - 8 * q^17 - q^19 - 8 * q^23 - q^25 - 4 * q^29 + 3 * q^31 - q^37 - 6 * q^41 + 11 * q^43 - 6 * q^47 + 12 * q^53 - 4 * q^55 - 4 * q^59 - 6 * q^61 - 6 * q^65 + 13 * q^67 + 10 * q^71 - 11 * q^73 - 3 * q^79 - 2 * q^83 - 16 * q^85 - 2 * q^95 + 10 * 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 2.00000 0 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$+1$$

## 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 1764.2.a.h 1
3.b odd 2 1 588.2.a.a 1
4.b odd 2 1 7056.2.a.bs 1
7.b odd 2 1 1764.2.a.c 1
7.c even 3 2 252.2.k.a 2
7.d odd 6 2 1764.2.k.j 2
12.b even 2 1 2352.2.a.o 1
21.c even 2 1 588.2.a.f 1
21.g even 6 2 588.2.i.b 2
21.h odd 6 2 84.2.i.a 2
24.f even 2 1 9408.2.a.bi 1
24.h odd 2 1 9408.2.a.cx 1
28.d even 2 1 7056.2.a.o 1
28.g odd 6 2 1008.2.s.c 2
63.g even 3 2 2268.2.l.g 2
63.h even 3 2 2268.2.i.b 2
63.j odd 6 2 2268.2.i.g 2
63.n odd 6 2 2268.2.l.b 2
84.h odd 2 1 2352.2.a.k 1
84.j odd 6 2 2352.2.q.q 2
84.n even 6 2 336.2.q.c 2
105.o odd 6 2 2100.2.q.b 2
105.x even 12 4 2100.2.bc.a 4
168.e odd 2 1 9408.2.a.bx 1
168.i even 2 1 9408.2.a.i 1
168.s odd 6 2 1344.2.q.b 2
168.v even 6 2 1344.2.q.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.i.a 2 21.h odd 6 2
252.2.k.a 2 7.c even 3 2
336.2.q.c 2 84.n even 6 2
588.2.a.a 1 3.b odd 2 1
588.2.a.f 1 21.c even 2 1
588.2.i.b 2 21.g even 6 2
1008.2.s.c 2 28.g odd 6 2
1344.2.q.b 2 168.s odd 6 2
1344.2.q.n 2 168.v even 6 2
1764.2.a.c 1 7.b odd 2 1
1764.2.a.h 1 1.a even 1 1 trivial
1764.2.k.j 2 7.d odd 6 2
2100.2.q.b 2 105.o odd 6 2
2100.2.bc.a 4 105.x even 12 4
2268.2.i.b 2 63.h even 3 2
2268.2.i.g 2 63.j odd 6 2
2268.2.l.b 2 63.n odd 6 2
2268.2.l.g 2 63.g even 3 2
2352.2.a.k 1 84.h odd 2 1
2352.2.a.o 1 12.b even 2 1
2352.2.q.q 2 84.j odd 6 2
7056.2.a.o 1 28.d even 2 1
7056.2.a.bs 1 4.b odd 2 1
9408.2.a.i 1 168.i even 2 1
9408.2.a.bi 1 24.f even 2 1
9408.2.a.bx 1 168.e odd 2 1
9408.2.a.cx 1 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1764))$$:

 $$T_{5} - 2$$ T5 - 2 $$T_{11} + 2$$ T11 + 2 $$T_{13} + 3$$ T13 + 3

## Hecke characteristic polynomials

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