# Properties

 Label 2704.2.f.b Level $2704$ Weight $2$ Character orbit 2704.f Analytic conductor $21.592$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2704,2,Mod(337,2704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2704, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2704.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2704 = 2^{4} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2704.f (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$21.5915487066$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 13) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 q^{3} - \beta q^{5} + q^{9} +O(q^{10})$$ q - 2 * q^3 - b * q^5 + q^9 $$q - 2 q^{3} - \beta q^{5} + q^{9} + 2 \beta q^{15} - 3 q^{17} + 2 \beta q^{19} + 6 q^{23} + 2 q^{25} + 4 q^{27} + 3 q^{29} - 2 \beta q^{31} - 5 \beta q^{37} + 3 \beta q^{41} - 8 q^{43} - \beta q^{45} + 2 \beta q^{47} + 7 q^{49} + 6 q^{51} - 3 q^{53} - 4 \beta q^{57} - 4 \beta q^{59} + q^{61} + 2 \beta q^{67} - 12 q^{69} - 2 \beta q^{71} + \beta q^{73} - 4 q^{75} - 4 q^{79} - 11 q^{81} - 8 \beta q^{83} + 3 \beta q^{85} - 6 q^{87} + 4 \beta q^{89} + 4 \beta q^{93} + 6 q^{95} + 4 \beta q^{97} +O(q^{100})$$ q - 2 * q^3 - b * q^5 + q^9 + 2*b * q^15 - 3 * q^17 + 2*b * q^19 + 6 * q^23 + 2 * q^25 + 4 * q^27 + 3 * q^29 - 2*b * q^31 - 5*b * q^37 + 3*b * q^41 - 8 * q^43 - b * q^45 + 2*b * q^47 + 7 * q^49 + 6 * q^51 - 3 * q^53 - 4*b * q^57 - 4*b * q^59 + q^61 + 2*b * q^67 - 12 * q^69 - 2*b * q^71 + b * q^73 - 4 * q^75 - 4 * q^79 - 11 * q^81 - 8*b * q^83 + 3*b * q^85 - 6 * q^87 + 4*b * q^89 + 4*b * q^93 + 6 * q^95 + 4*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{3} + 2 q^{9}+O(q^{10})$$ 2 * q - 4 * q^3 + 2 * q^9 $$2 q - 4 q^{3} + 2 q^{9} - 6 q^{17} + 12 q^{23} + 4 q^{25} + 8 q^{27} + 6 q^{29} - 16 q^{43} + 14 q^{49} + 12 q^{51} - 6 q^{53} + 2 q^{61} - 24 q^{69} - 8 q^{75} - 8 q^{79} - 22 q^{81} - 12 q^{87} + 12 q^{95}+O(q^{100})$$ 2 * q - 4 * q^3 + 2 * q^9 - 6 * q^17 + 12 * q^23 + 4 * q^25 + 8 * q^27 + 6 * q^29 - 16 * q^43 + 14 * q^49 + 12 * q^51 - 6 * q^53 + 2 * q^61 - 24 * q^69 - 8 * q^75 - 8 * q^79 - 22 * q^81 - 12 * q^87 + 12 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2704\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$1185$$ $$2367$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 −2.00000 0 1.73205i 0 0 0 1.00000 0
337.2 0 −2.00000 0 1.73205i 0 0 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2704.2.f.b 2
4.b odd 2 1 169.2.b.a 2
12.b even 2 1 1521.2.b.a 2
13.b even 2 1 inner 2704.2.f.b 2
13.c even 3 1 208.2.w.b 2
13.d odd 4 2 2704.2.a.o 2
13.e even 6 1 208.2.w.b 2
39.h odd 6 1 1872.2.by.d 2
39.i odd 6 1 1872.2.by.d 2
52.b odd 2 1 169.2.b.a 2
52.f even 4 2 169.2.a.a 2
52.i odd 6 1 13.2.e.a 2
52.i odd 6 1 169.2.e.a 2
52.j odd 6 1 13.2.e.a 2
52.j odd 6 1 169.2.e.a 2
52.l even 12 4 169.2.c.a 4
104.n odd 6 1 832.2.w.d 2
104.p odd 6 1 832.2.w.d 2
104.r even 6 1 832.2.w.a 2
104.s even 6 1 832.2.w.a 2
156.h even 2 1 1521.2.b.a 2
156.l odd 4 2 1521.2.a.k 2
156.p even 6 1 117.2.q.c 2
156.r even 6 1 117.2.q.c 2
260.u even 4 2 4225.2.a.v 2
260.v odd 6 1 325.2.n.a 2
260.w odd 6 1 325.2.n.a 2
260.bg even 12 2 325.2.m.a 4
260.bj even 12 2 325.2.m.a 4
364.p odd 4 2 8281.2.a.q 2
364.q odd 6 1 637.2.u.c 2
364.s odd 6 1 637.2.u.c 2
364.v even 6 1 637.2.q.a 2
364.w even 6 1 637.2.k.c 2
364.ba even 6 1 637.2.k.c 2
364.bc even 6 1 637.2.q.a 2
364.bi odd 6 1 637.2.k.a 2
364.bk odd 6 1 637.2.k.a 2
364.bp even 6 1 637.2.u.b 2
364.br even 6 1 637.2.u.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.2.e.a 2 52.i odd 6 1
13.2.e.a 2 52.j odd 6 1
117.2.q.c 2 156.p even 6 1
117.2.q.c 2 156.r even 6 1
169.2.a.a 2 52.f even 4 2
169.2.b.a 2 4.b odd 2 1
169.2.b.a 2 52.b odd 2 1
169.2.c.a 4 52.l even 12 4
169.2.e.a 2 52.i odd 6 1
169.2.e.a 2 52.j odd 6 1
208.2.w.b 2 13.c even 3 1
208.2.w.b 2 13.e even 6 1
325.2.m.a 4 260.bg even 12 2
325.2.m.a 4 260.bj even 12 2
325.2.n.a 2 260.v odd 6 1
325.2.n.a 2 260.w odd 6 1
637.2.k.a 2 364.bi odd 6 1
637.2.k.a 2 364.bk odd 6 1
637.2.k.c 2 364.w even 6 1
637.2.k.c 2 364.ba even 6 1
637.2.q.a 2 364.v even 6 1
637.2.q.a 2 364.bc even 6 1
637.2.u.b 2 364.bp even 6 1
637.2.u.b 2 364.br even 6 1
637.2.u.c 2 364.q odd 6 1
637.2.u.c 2 364.s odd 6 1
832.2.w.a 2 104.r even 6 1
832.2.w.a 2 104.s even 6 1
832.2.w.d 2 104.n odd 6 1
832.2.w.d 2 104.p odd 6 1
1521.2.a.k 2 156.l odd 4 2
1521.2.b.a 2 12.b even 2 1
1521.2.b.a 2 156.h even 2 1
1872.2.by.d 2 39.h odd 6 1
1872.2.by.d 2 39.i odd 6 1
2704.2.a.o 2 13.d odd 4 2
2704.2.f.b 2 1.a even 1 1 trivial
2704.2.f.b 2 13.b even 2 1 inner
4225.2.a.v 2 260.u even 4 2
8281.2.a.q 2 364.p odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2704, [\chi])$$:

 $$T_{3} + 2$$ T3 + 2 $$T_{5}^{2} + 3$$ T5^2 + 3 $$T_{11}$$ T11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 2)^{2}$$
$5$ $$T^{2} + 3$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$(T + 3)^{2}$$
$19$ $$T^{2} + 12$$
$23$ $$(T - 6)^{2}$$
$29$ $$(T - 3)^{2}$$
$31$ $$T^{2} + 12$$
$37$ $$T^{2} + 75$$
$41$ $$T^{2} + 27$$
$43$ $$(T + 8)^{2}$$
$47$ $$T^{2} + 12$$
$53$ $$(T + 3)^{2}$$
$59$ $$T^{2} + 48$$
$61$ $$(T - 1)^{2}$$
$67$ $$T^{2} + 12$$
$71$ $$T^{2} + 12$$
$73$ $$T^{2} + 3$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2} + 192$$
$89$ $$T^{2} + 48$$
$97$ $$T^{2} + 48$$