# Properties

 Label 2704.2.f.f Level $2704$ Weight $2$ Character orbit 2704.f Analytic conductor $21.592$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 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

 Self dual: no Analytic conductor: $$21.5915487066$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 52) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{5} + \beta q^{7} - 3 q^{9} +O(q^{10})$$ q - b * q^5 + b * q^7 - 3 * q^9 $$q - \beta q^{5} + \beta q^{7} - 3 q^{9} + \beta q^{11} - 6 q^{17} - 3 \beta q^{19} + 8 q^{23} + q^{25} + 2 q^{29} + 5 \beta q^{31} + 4 q^{35} - 3 \beta q^{37} + 3 \beta q^{41} + 4 q^{43} + 3 \beta q^{45} + \beta q^{47} + 3 q^{49} + 6 q^{53} + 4 q^{55} + 5 \beta q^{59} - 2 q^{61} - 3 \beta q^{63} + 5 \beta q^{67} + 5 \beta q^{71} + \beta q^{73} - 4 q^{77} + 4 q^{79} + 9 q^{81} - 3 \beta q^{83} + 6 \beta q^{85} - 3 \beta q^{89} - 12 q^{95} - \beta q^{97} - 3 \beta q^{99} +O(q^{100})$$ q - b * q^5 + b * q^7 - 3 * q^9 + b * q^11 - 6 * q^17 - 3*b * q^19 + 8 * q^23 + q^25 + 2 * q^29 + 5*b * q^31 + 4 * q^35 - 3*b * q^37 + 3*b * q^41 + 4 * q^43 + 3*b * q^45 + b * q^47 + 3 * q^49 + 6 * q^53 + 4 * q^55 + 5*b * q^59 - 2 * q^61 - 3*b * q^63 + 5*b * q^67 + 5*b * q^71 + b * q^73 - 4 * q^77 + 4 * q^79 + 9 * q^81 - 3*b * q^83 + 6*b * q^85 - 3*b * q^89 - 12 * q^95 - b * q^97 - 3*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{9}+O(q^{10})$$ 2 * q - 6 * q^9 $$2 q - 6 q^{9} - 12 q^{17} + 16 q^{23} + 2 q^{25} + 4 q^{29} + 8 q^{35} + 8 q^{43} + 6 q^{49} + 12 q^{53} + 8 q^{55} - 4 q^{61} - 8 q^{77} + 8 q^{79} + 18 q^{81} - 24 q^{95}+O(q^{100})$$ 2 * q - 6 * q^9 - 12 * q^17 + 16 * q^23 + 2 * q^25 + 4 * q^29 + 8 * q^35 + 8 * q^43 + 6 * q^49 + 12 * q^53 + 8 * q^55 - 4 * q^61 - 8 * q^77 + 8 * q^79 + 18 * q^81 - 24 * 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 1.00000i − 1.00000i
0 0 0 2.00000i 0 2.00000i 0 −3.00000 0
337.2 0 0 0 2.00000i 0 2.00000i 0 −3.00000 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.f 2
4.b odd 2 1 676.2.d.c 2
12.b even 2 1 6084.2.b.m 2
13.b even 2 1 inner 2704.2.f.f 2
13.d odd 4 1 208.2.a.c 1
13.d odd 4 1 2704.2.a.g 1
39.f even 4 1 1872.2.a.f 1
52.b odd 2 1 676.2.d.c 2
52.f even 4 1 52.2.a.a 1
52.f even 4 1 676.2.a.c 1
52.i odd 6 2 676.2.h.c 4
52.j odd 6 2 676.2.h.c 4
52.l even 12 2 676.2.e.b 2
52.l even 12 2 676.2.e.c 2
65.g odd 4 1 5200.2.a.q 1
104.j odd 4 1 832.2.a.f 1
104.m even 4 1 832.2.a.e 1
156.h even 2 1 6084.2.b.m 2
156.l odd 4 1 468.2.a.b 1
156.l odd 4 1 6084.2.a.m 1
208.l even 4 1 3328.2.b.q 2
208.m odd 4 1 3328.2.b.e 2
208.r odd 4 1 3328.2.b.e 2
208.s even 4 1 3328.2.b.q 2
260.l odd 4 1 1300.2.c.c 2
260.s odd 4 1 1300.2.c.c 2
260.u even 4 1 1300.2.a.d 1
312.w odd 4 1 7488.2.a.bn 1
312.y even 4 1 7488.2.a.bw 1
364.p odd 4 1 2548.2.a.e 1
364.bw odd 12 2 2548.2.j.f 2
364.ce even 12 2 2548.2.j.e 2
468.bs even 12 2 4212.2.i.d 2
468.ch odd 12 2 4212.2.i.i 2
572.k odd 4 1 6292.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.2.a.a 1 52.f even 4 1
208.2.a.c 1 13.d odd 4 1
468.2.a.b 1 156.l odd 4 1
676.2.a.c 1 52.f even 4 1
676.2.d.c 2 4.b odd 2 1
676.2.d.c 2 52.b odd 2 1
676.2.e.b 2 52.l even 12 2
676.2.e.c 2 52.l even 12 2
676.2.h.c 4 52.i odd 6 2
676.2.h.c 4 52.j odd 6 2
832.2.a.e 1 104.m even 4 1
832.2.a.f 1 104.j odd 4 1
1300.2.a.d 1 260.u even 4 1
1300.2.c.c 2 260.l odd 4 1
1300.2.c.c 2 260.s odd 4 1
1872.2.a.f 1 39.f even 4 1
2548.2.a.e 1 364.p odd 4 1
2548.2.j.e 2 364.ce even 12 2
2548.2.j.f 2 364.bw odd 12 2
2704.2.a.g 1 13.d odd 4 1
2704.2.f.f 2 1.a even 1 1 trivial
2704.2.f.f 2 13.b even 2 1 inner
3328.2.b.e 2 208.m odd 4 1
3328.2.b.e 2 208.r odd 4 1
3328.2.b.q 2 208.l even 4 1
3328.2.b.q 2 208.s even 4 1
4212.2.i.d 2 468.bs even 12 2
4212.2.i.i 2 468.ch odd 12 2
5200.2.a.q 1 65.g odd 4 1
6084.2.a.m 1 156.l odd 4 1
6084.2.b.m 2 12.b even 2 1
6084.2.b.m 2 156.h even 2 1
6292.2.a.g 1 572.k odd 4 1
7488.2.a.bn 1 312.w odd 4 1
7488.2.a.bw 1 312.y even 4 1

## Hecke kernels

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

 $$T_{3}$$ T3 $$T_{5}^{2} + 4$$ T5^2 + 4 $$T_{11}^{2} + 4$$ T11^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 4$$
$7$ $$T^{2} + 4$$
$11$ $$T^{2} + 4$$
$13$ $$T^{2}$$
$17$ $$(T + 6)^{2}$$
$19$ $$T^{2} + 36$$
$23$ $$(T - 8)^{2}$$
$29$ $$(T - 2)^{2}$$
$31$ $$T^{2} + 100$$
$37$ $$T^{2} + 36$$
$41$ $$T^{2} + 36$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2} + 4$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} + 100$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} + 100$$
$71$ $$T^{2} + 100$$
$73$ $$T^{2} + 4$$
$79$ $$(T - 4)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$T^{2} + 36$$
$97$ $$T^{2} + 4$$