# Properties

 Label 2704.2.a.j Level $2704$ Weight $2$ Character orbit 2704.a Self dual yes Analytic conductor $21.592$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2704 = 2^{4} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2704.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$21.5915487066$$ Analytic rank: $$1$$ 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

 $$f(q)$$ $$=$$ $$q + q^{3} - 3 q^{5} + 3 q^{7} - 2 q^{9}+O(q^{10})$$ q + q^3 - 3 * q^5 + 3 * q^7 - 2 * q^9 $$q + q^{3} - 3 q^{5} + 3 q^{7} - 2 q^{9} - 3 q^{15} - 3 q^{17} + 6 q^{19} + 3 q^{21} - 6 q^{23} + 4 q^{25} - 5 q^{27} - 9 q^{35} - 3 q^{37} - q^{43} + 6 q^{45} + 3 q^{47} + 2 q^{49} - 3 q^{51} - 6 q^{53} + 6 q^{57} - 6 q^{59} - 8 q^{61} - 6 q^{63} + 12 q^{67} - 6 q^{69} - 15 q^{71} - 6 q^{73} + 4 q^{75} - 10 q^{79} + q^{81} - 6 q^{83} + 9 q^{85} + 6 q^{89} - 18 q^{95} - 12 q^{97}+O(q^{100})$$ q + q^3 - 3 * q^5 + 3 * q^7 - 2 * q^9 - 3 * q^15 - 3 * q^17 + 6 * q^19 + 3 * q^21 - 6 * q^23 + 4 * q^25 - 5 * q^27 - 9 * q^35 - 3 * q^37 - q^43 + 6 * q^45 + 3 * q^47 + 2 * q^49 - 3 * q^51 - 6 * q^53 + 6 * q^57 - 6 * q^59 - 8 * q^61 - 6 * q^63 + 12 * q^67 - 6 * q^69 - 15 * q^71 - 6 * q^73 + 4 * q^75 - 10 * q^79 + q^81 - 6 * q^83 + 9 * q^85 + 6 * q^89 - 18 * q^95 - 12 * 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.

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 −3.00000 0 3.00000 0 −2.00000 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.j 1
4.b odd 2 1 338.2.a.d 1
12.b even 2 1 3042.2.a.g 1
13.b even 2 1 2704.2.a.k 1
13.d odd 4 2 208.2.f.a 2
20.d odd 2 1 8450.2.a.h 1
39.f even 4 2 1872.2.c.f 2
52.b odd 2 1 338.2.a.b 1
52.f even 4 2 26.2.b.a 2
52.i odd 6 2 338.2.c.f 2
52.j odd 6 2 338.2.c.b 2
52.l even 12 4 338.2.e.c 4
104.j odd 4 2 832.2.f.b 2
104.m even 4 2 832.2.f.d 2
156.h even 2 1 3042.2.a.j 1
156.l odd 4 2 234.2.b.b 2
260.g odd 2 1 8450.2.a.u 1
260.l odd 4 2 650.2.c.a 2
260.s odd 4 2 650.2.c.d 2
260.u even 4 2 650.2.d.b 2
364.p odd 4 2 1274.2.d.c 2
364.bw odd 12 4 1274.2.n.c 4
364.ce even 12 4 1274.2.n.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.b.a 2 52.f even 4 2
208.2.f.a 2 13.d odd 4 2
234.2.b.b 2 156.l odd 4 2
338.2.a.b 1 52.b odd 2 1
338.2.a.d 1 4.b odd 2 1
338.2.c.b 2 52.j odd 6 2
338.2.c.f 2 52.i odd 6 2
338.2.e.c 4 52.l even 12 4
650.2.c.a 2 260.l odd 4 2
650.2.c.d 2 260.s odd 4 2
650.2.d.b 2 260.u even 4 2
832.2.f.b 2 104.j odd 4 2
832.2.f.d 2 104.m even 4 2
1274.2.d.c 2 364.p odd 4 2
1274.2.n.c 4 364.bw odd 12 4
1274.2.n.d 4 364.ce even 12 4
1872.2.c.f 2 39.f even 4 2
2704.2.a.j 1 1.a even 1 1 trivial
2704.2.a.k 1 13.b even 2 1
3042.2.a.g 1 12.b even 2 1
3042.2.a.j 1 156.h even 2 1
8450.2.a.h 1 20.d odd 2 1
8450.2.a.u 1 260.g odd 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T + 3$$
$7$ $$T - 3$$
$11$ $$T$$
$13$ $$T$$
$17$ $$T + 3$$
$19$ $$T - 6$$
$23$ $$T + 6$$
$29$ $$T$$
$31$ $$T$$
$37$ $$T + 3$$
$41$ $$T$$
$43$ $$T + 1$$
$47$ $$T - 3$$
$53$ $$T + 6$$
$59$ $$T + 6$$
$61$ $$T + 8$$
$67$ $$T - 12$$
$71$ $$T + 15$$
$73$ $$T + 6$$
$79$ $$T + 10$$
$83$ $$T + 6$$
$89$ $$T - 6$$
$97$ $$T + 12$$