# Properties

 Label 2704.2.f.d 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) 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.

 $$f(q)$$ $$=$$ $$q - q^{3} - 3 i q^{5} - i q^{7} - 2 q^{9} +O(q^{10})$$ q - q^3 - 3*i * q^5 - i * q^7 - 2 * q^9 $$q - q^{3} - 3 i q^{5} - i q^{7} - 2 q^{9} + 6 i q^{11} + 3 i q^{15} + 3 q^{17} - 2 i q^{19} + i q^{21} - 4 q^{25} + 5 q^{27} + 6 q^{29} + 4 i q^{31} - 6 i q^{33} - 3 q^{35} + 7 i q^{37} - q^{43} + 6 i q^{45} + 3 i q^{47} + 6 q^{49} - 3 q^{51} + 18 q^{55} + 2 i q^{57} - 6 i q^{59} + 8 q^{61} + 2 i q^{63} - 14 i q^{67} + 3 i q^{71} - 2 i q^{73} + 4 q^{75} + 6 q^{77} - 8 q^{79} + q^{81} - 12 i q^{83} - 9 i q^{85} - 6 q^{87} + 6 i q^{89} - 4 i q^{93} - 6 q^{95} - 10 i q^{97} - 12 i q^{99} +O(q^{100})$$ q - q^3 - 3*i * q^5 - i * q^7 - 2 * q^9 + 6*i * q^11 + 3*i * q^15 + 3 * q^17 - 2*i * q^19 + i * q^21 - 4 * q^25 + 5 * q^27 + 6 * q^29 + 4*i * q^31 - 6*i * q^33 - 3 * q^35 + 7*i * q^37 - q^43 + 6*i * q^45 + 3*i * q^47 + 6 * q^49 - 3 * q^51 + 18 * q^55 + 2*i * q^57 - 6*i * q^59 + 8 * q^61 + 2*i * q^63 - 14*i * q^67 + 3*i * q^71 - 2*i * q^73 + 4 * q^75 + 6 * q^77 - 8 * q^79 + q^81 - 12*i * q^83 - 9*i * q^85 - 6 * q^87 + 6*i * q^89 - 4*i * q^93 - 6 * q^95 - 10*i * q^97 - 12*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 4 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 4 * q^9 $$2 q - 2 q^{3} - 4 q^{9} + 6 q^{17} - 8 q^{25} + 10 q^{27} + 12 q^{29} - 6 q^{35} - 2 q^{43} + 12 q^{49} - 6 q^{51} + 36 q^{55} + 16 q^{61} + 8 q^{75} + 12 q^{77} - 16 q^{79} + 2 q^{81} - 12 q^{87} - 12 q^{95}+O(q^{100})$$ 2 * q - 2 * q^3 - 4 * q^9 + 6 * q^17 - 8 * q^25 + 10 * q^27 + 12 * q^29 - 6 * q^35 - 2 * q^43 + 12 * q^49 - 6 * q^51 + 36 * q^55 + 16 * q^61 + 8 * q^75 + 12 * q^77 - 16 * q^79 + 2 * 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.

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 −1.00000 0 3.00000i 0 1.00000i 0 −2.00000 0
337.2 0 −1.00000 0 3.00000i 0 1.00000i 0 −2.00000 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.d 2
4.b odd 2 1 338.2.b.c 2
12.b even 2 1 3042.2.b.a 2
13.b even 2 1 inner 2704.2.f.d 2
13.d odd 4 1 208.2.a.a 1
13.d odd 4 1 2704.2.a.f 1
39.f even 4 1 1872.2.a.q 1
52.b odd 2 1 338.2.b.c 2
52.f even 4 1 26.2.a.a 1
52.f even 4 1 338.2.a.f 1
52.i odd 6 2 338.2.e.a 4
52.j odd 6 2 338.2.e.a 4
52.l even 12 2 338.2.c.a 2
52.l even 12 2 338.2.c.d 2
65.g odd 4 1 5200.2.a.x 1
104.j odd 4 1 832.2.a.i 1
104.m even 4 1 832.2.a.d 1
156.h even 2 1 3042.2.b.a 2
156.l odd 4 1 234.2.a.e 1
156.l odd 4 1 3042.2.a.a 1
208.l even 4 1 3328.2.b.m 2
208.m odd 4 1 3328.2.b.j 2
208.r odd 4 1 3328.2.b.j 2
208.s even 4 1 3328.2.b.m 2
260.l odd 4 1 650.2.b.d 2
260.s odd 4 1 650.2.b.d 2
260.u even 4 1 650.2.a.j 1
260.u even 4 1 8450.2.a.c 1
312.w odd 4 1 7488.2.a.g 1
312.y even 4 1 7488.2.a.h 1
364.p odd 4 1 1274.2.a.d 1
364.bw odd 12 2 1274.2.f.r 2
364.ce even 12 2 1274.2.f.p 2
468.bs even 12 2 2106.2.e.ba 2
468.ch odd 12 2 2106.2.e.b 2
572.k odd 4 1 3146.2.a.n 1
780.u even 4 1 5850.2.e.a 2
780.bb odd 4 1 5850.2.a.p 1
780.bn even 4 1 5850.2.e.a 2
884.t even 4 1 7514.2.a.c 1
988.p odd 4 1 9386.2.a.j 1

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

## Hecke kernels

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

 $$T_{3} + 1$$ T3 + 1 $$T_{5}^{2} + 9$$ T5^2 + 9 $$T_{11}^{2} + 36$$ T11^2 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} + 9$$
$7$ $$T^{2} + 1$$
$11$ $$T^{2} + 36$$
$13$ $$T^{2}$$
$17$ $$(T - 3)^{2}$$
$19$ $$T^{2} + 4$$
$23$ $$T^{2}$$
$29$ $$(T - 6)^{2}$$
$31$ $$T^{2} + 16$$
$37$ $$T^{2} + 49$$
$41$ $$T^{2}$$
$43$ $$(T + 1)^{2}$$
$47$ $$T^{2} + 9$$
$53$ $$T^{2}$$
$59$ $$T^{2} + 36$$
$61$ $$(T - 8)^{2}$$
$67$ $$T^{2} + 196$$
$71$ $$T^{2} + 9$$
$73$ $$T^{2} + 4$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$T^{2} + 36$$
$97$ $$T^{2} + 100$$