# Properties

 Label 2704.2.a.n Level $2704$ Weight $2$ Character orbit 2704.a Self dual yes Analytic conductor $21.592$ Analytic rank $0$ 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] = [2704,2,Mod(1,2704)]

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, 0]))

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

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

chi := DirichletCharacter("2704.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2704 = 2^{4} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2704.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: $$21.5915487066$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + q^{5} + q^{7} + 6 q^{9}+O(q^{10})$$ q + 3 * q^3 + q^5 + q^7 + 6 * q^9 $$q + 3 q^{3} + q^{5} + q^{7} + 6 q^{9} - 2 q^{11} + 3 q^{15} - 3 q^{17} + 6 q^{19} + 3 q^{21} + 4 q^{23} - 4 q^{25} + 9 q^{27} + 2 q^{29} + 4 q^{31} - 6 q^{33} + q^{35} - 3 q^{37} + 5 q^{43} + 6 q^{45} + 13 q^{47} - 6 q^{49} - 9 q^{51} + 12 q^{53} - 2 q^{55} + 18 q^{57} - 10 q^{59} - 8 q^{61} + 6 q^{63} - 2 q^{67} + 12 q^{69} - 5 q^{71} + 10 q^{73} - 12 q^{75} - 2 q^{77} + 4 q^{79} + 9 q^{81} - 3 q^{85} + 6 q^{87} - 6 q^{89} + 12 q^{93} + 6 q^{95} - 14 q^{97} - 12 q^{99}+O(q^{100})$$ q + 3 * q^3 + q^5 + q^7 + 6 * q^9 - 2 * q^11 + 3 * q^15 - 3 * q^17 + 6 * q^19 + 3 * q^21 + 4 * q^23 - 4 * q^25 + 9 * q^27 + 2 * q^29 + 4 * q^31 - 6 * q^33 + q^35 - 3 * q^37 + 5 * q^43 + 6 * q^45 + 13 * q^47 - 6 * q^49 - 9 * q^51 + 12 * q^53 - 2 * q^55 + 18 * q^57 - 10 * q^59 - 8 * q^61 + 6 * q^63 - 2 * q^67 + 12 * q^69 - 5 * q^71 + 10 * q^73 - 12 * q^75 - 2 * q^77 + 4 * q^79 + 9 * q^81 - 3 * q^85 + 6 * q^87 - 6 * q^89 + 12 * q^93 + 6 * q^95 - 14 * q^97 - 12 * 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 3.00000 0 1.00000 0 1.00000 0 6.00000 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
$$2$$ $$-1$$
$$13$$ $$+1$$

## 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 2704.2.a.n 1
4.b odd 2 1 338.2.a.a 1
12.b even 2 1 3042.2.a.l 1
13.b even 2 1 208.2.a.d 1
13.d odd 4 2 2704.2.f.j 2
20.d odd 2 1 8450.2.a.y 1
39.d odd 2 1 1872.2.a.m 1
52.b odd 2 1 26.2.a.b 1
52.f even 4 2 338.2.b.a 2
52.i odd 6 2 338.2.c.c 2
52.j odd 6 2 338.2.c.g 2
52.l even 12 4 338.2.e.d 4
65.d even 2 1 5200.2.a.c 1
104.e even 2 1 832.2.a.a 1
104.h odd 2 1 832.2.a.j 1
156.h even 2 1 234.2.a.b 1
156.l odd 4 2 3042.2.b.f 2
208.o odd 4 2 3328.2.b.g 2
208.p even 4 2 3328.2.b.k 2
260.g odd 2 1 650.2.a.g 1
260.p even 4 2 650.2.b.a 2
312.b odd 2 1 7488.2.a.v 1
312.h even 2 1 7488.2.a.w 1
364.h even 2 1 1274.2.a.o 1
364.x even 6 2 1274.2.f.a 2
364.bl odd 6 2 1274.2.f.l 2
468.x even 6 2 2106.2.e.t 2
468.bg odd 6 2 2106.2.e.h 2
572.b even 2 1 3146.2.a.a 1
780.d even 2 1 5850.2.a.bn 1
780.w odd 4 2 5850.2.e.v 2
884.h odd 2 1 7514.2.a.i 1
988.g even 2 1 9386.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.b 1 52.b odd 2 1
208.2.a.d 1 13.b even 2 1
234.2.a.b 1 156.h even 2 1
338.2.a.a 1 4.b odd 2 1
338.2.b.a 2 52.f even 4 2
338.2.c.c 2 52.i odd 6 2
338.2.c.g 2 52.j odd 6 2
338.2.e.d 4 52.l even 12 4
650.2.a.g 1 260.g odd 2 1
650.2.b.a 2 260.p even 4 2
832.2.a.a 1 104.e even 2 1
832.2.a.j 1 104.h odd 2 1
1274.2.a.o 1 364.h even 2 1
1274.2.f.a 2 364.x even 6 2
1274.2.f.l 2 364.bl odd 6 2
1872.2.a.m 1 39.d odd 2 1
2106.2.e.h 2 468.bg odd 6 2
2106.2.e.t 2 468.x even 6 2
2704.2.a.n 1 1.a even 1 1 trivial
2704.2.f.j 2 13.d odd 4 2
3042.2.a.l 1 12.b even 2 1
3042.2.b.f 2 156.l odd 4 2
3146.2.a.a 1 572.b even 2 1
3328.2.b.g 2 208.o odd 4 2
3328.2.b.k 2 208.p even 4 2
5200.2.a.c 1 65.d even 2 1
5850.2.a.bn 1 780.d even 2 1
5850.2.e.v 2 780.w odd 4 2
7488.2.a.v 1 312.b odd 2 1
7488.2.a.w 1 312.h even 2 1
7514.2.a.i 1 884.h odd 2 1
8450.2.a.y 1 20.d odd 2 1
9386.2.a.f 1 988.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2704))$$:

 $$T_{3} - 3$$ T3 - 3 $$T_{5} - 1$$ T5 - 1 $$T_{7} - 1$$ T7 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T - 1$$
$7$ $$T - 1$$
$11$ $$T + 2$$
$13$ $$T$$
$17$ $$T + 3$$
$19$ $$T - 6$$
$23$ $$T - 4$$
$29$ $$T - 2$$
$31$ $$T - 4$$
$37$ $$T + 3$$
$41$ $$T$$
$43$ $$T - 5$$
$47$ $$T - 13$$
$53$ $$T - 12$$
$59$ $$T + 10$$
$61$ $$T + 8$$
$67$ $$T + 2$$
$71$ $$T + 5$$
$73$ $$T - 10$$
$79$ $$T - 4$$
$83$ $$T$$
$89$ $$T + 6$$
$97$ $$T + 14$$