Properties

 Label 2704.2.f.j 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: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + i q^{5} - i q^{7} + 6 q^{9} +O(q^{10})$$ q + 3 * q^3 + i * q^5 - i * q^7 + 6 * q^9 $$q + 3 q^{3} + i q^{5} - i q^{7} + 6 q^{9} + 2 i q^{11} + 3 i q^{15} + 3 q^{17} + 6 i q^{19} - 3 i q^{21} - 4 q^{23} + 4 q^{25} + 9 q^{27} + 2 q^{29} + 4 i q^{31} + 6 i q^{33} + q^{35} + 3 i q^{37} - 5 q^{43} + 6 i q^{45} - 13 i q^{47} + 6 q^{49} + 9 q^{51} + 12 q^{53} - 2 q^{55} + 18 i q^{57} + 10 i q^{59} - 8 q^{61} - 6 i q^{63} - 2 i q^{67} - 12 q^{69} - 5 i q^{71} - 10 i q^{73} + 12 q^{75} + 2 q^{77} + 4 q^{79} + 9 q^{81} + 3 i q^{85} + 6 q^{87} + 6 i q^{89} + 12 i q^{93} - 6 q^{95} - 14 i q^{97} + 12 i q^{99} +O(q^{100})$$ q + 3 * q^3 + i * q^5 - i * q^7 + 6 * q^9 + 2*i * q^11 + 3*i * q^15 + 3 * q^17 + 6*i * q^19 - 3*i * q^21 - 4 * q^23 + 4 * q^25 + 9 * q^27 + 2 * q^29 + 4*i * q^31 + 6*i * q^33 + q^35 + 3*i * q^37 - 5 * q^43 + 6*i * q^45 - 13*i * q^47 + 6 * q^49 + 9 * q^51 + 12 * q^53 - 2 * q^55 + 18*i * q^57 + 10*i * q^59 - 8 * q^61 - 6*i * q^63 - 2*i * q^67 - 12 * q^69 - 5*i * q^71 - 10*i * q^73 + 12 * q^75 + 2 * q^77 + 4 * q^79 + 9 * q^81 + 3*i * q^85 + 6 * q^87 + 6*i * q^89 + 12*i * q^93 - 6 * q^95 - 14*i * q^97 + 12*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} + 12 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 + 12 * q^9 $$2 q + 6 q^{3} + 12 q^{9} + 6 q^{17} - 8 q^{23} + 8 q^{25} + 18 q^{27} + 4 q^{29} + 2 q^{35} - 10 q^{43} + 12 q^{49} + 18 q^{51} + 24 q^{53} - 4 q^{55} - 16 q^{61} - 24 q^{69} + 24 q^{75} + 4 q^{77} + 8 q^{79} + 18 q^{81} + 12 q^{87} - 12 q^{95}+O(q^{100})$$ 2 * q + 6 * q^3 + 12 * q^9 + 6 * q^17 - 8 * q^23 + 8 * q^25 + 18 * q^27 + 4 * q^29 + 2 * q^35 - 10 * q^43 + 12 * q^49 + 18 * q^51 + 24 * q^53 - 4 * q^55 - 16 * q^61 - 24 * q^69 + 24 * q^75 + 4 * q^77 + 8 * q^79 + 18 * 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
 − 1.00000i 1.00000i
0 3.00000 0 1.00000i 0 1.00000i 0 6.00000 0
337.2 0 3.00000 0 1.00000i 0 1.00000i 0 6.00000 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.j 2
4.b odd 2 1 338.2.b.a 2
12.b even 2 1 3042.2.b.f 2
13.b even 2 1 inner 2704.2.f.j 2
13.d odd 4 1 208.2.a.d 1
13.d odd 4 1 2704.2.a.n 1
39.f even 4 1 1872.2.a.m 1
52.b odd 2 1 338.2.b.a 2
52.f even 4 1 26.2.a.b 1
52.f even 4 1 338.2.a.a 1
52.i odd 6 2 338.2.e.d 4
52.j odd 6 2 338.2.e.d 4
52.l even 12 2 338.2.c.c 2
52.l even 12 2 338.2.c.g 2
65.g odd 4 1 5200.2.a.c 1
104.j odd 4 1 832.2.a.a 1
104.m even 4 1 832.2.a.j 1
156.h even 2 1 3042.2.b.f 2
156.l odd 4 1 234.2.a.b 1
156.l odd 4 1 3042.2.a.l 1
208.l even 4 1 3328.2.b.g 2
208.m odd 4 1 3328.2.b.k 2
208.r odd 4 1 3328.2.b.k 2
208.s even 4 1 3328.2.b.g 2
260.l odd 4 1 650.2.b.a 2
260.s odd 4 1 650.2.b.a 2
260.u even 4 1 650.2.a.g 1
260.u even 4 1 8450.2.a.y 1
312.w odd 4 1 7488.2.a.w 1
312.y even 4 1 7488.2.a.v 1
364.p odd 4 1 1274.2.a.o 1
364.bw odd 12 2 1274.2.f.a 2
364.ce even 12 2 1274.2.f.l 2
468.bs even 12 2 2106.2.e.h 2
468.ch odd 12 2 2106.2.e.t 2
572.k odd 4 1 3146.2.a.a 1
780.u even 4 1 5850.2.e.v 2
780.bb odd 4 1 5850.2.a.bn 1
780.bn even 4 1 5850.2.e.v 2
884.t even 4 1 7514.2.a.i 1
988.p odd 4 1 9386.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.b 1 52.f even 4 1
208.2.a.d 1 13.d odd 4 1
234.2.a.b 1 156.l odd 4 1
338.2.a.a 1 52.f even 4 1
338.2.b.a 2 4.b odd 2 1
338.2.b.a 2 52.b odd 2 1
338.2.c.c 2 52.l even 12 2
338.2.c.g 2 52.l even 12 2
338.2.e.d 4 52.i odd 6 2
338.2.e.d 4 52.j odd 6 2
650.2.a.g 1 260.u even 4 1
650.2.b.a 2 260.l odd 4 1
650.2.b.a 2 260.s odd 4 1
832.2.a.a 1 104.j odd 4 1
832.2.a.j 1 104.m even 4 1
1274.2.a.o 1 364.p odd 4 1
1274.2.f.a 2 364.bw odd 12 2
1274.2.f.l 2 364.ce even 12 2
1872.2.a.m 1 39.f even 4 1
2106.2.e.h 2 468.bs even 12 2
2106.2.e.t 2 468.ch odd 12 2
2704.2.a.n 1 13.d odd 4 1
2704.2.f.j 2 1.a even 1 1 trivial
2704.2.f.j 2 13.b even 2 1 inner
3042.2.a.l 1 156.l odd 4 1
3042.2.b.f 2 12.b even 2 1
3042.2.b.f 2 156.h even 2 1
3146.2.a.a 1 572.k odd 4 1
3328.2.b.g 2 208.l even 4 1
3328.2.b.g 2 208.s even 4 1
3328.2.b.k 2 208.m odd 4 1
3328.2.b.k 2 208.r odd 4 1
5200.2.a.c 1 65.g odd 4 1
5850.2.a.bn 1 780.bb odd 4 1
5850.2.e.v 2 780.u even 4 1
5850.2.e.v 2 780.bn even 4 1
7488.2.a.v 1 312.y even 4 1
7488.2.a.w 1 312.w odd 4 1
7514.2.a.i 1 884.t even 4 1
8450.2.a.y 1 260.u even 4 1
9386.2.a.f 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 $$S_{2}^{\mathrm{new}}(2704, [\chi])$$:

 $$T_{3} - 3$$ T3 - 3 $$T_{5}^{2} + 1$$ T5^2 + 1 $$T_{11}^{2} + 4$$ T11^2 + 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 3)^{2}$$
$5$ $$T^{2} + 1$$
$7$ $$T^{2} + 1$$
$11$ $$T^{2} + 4$$
$13$ $$T^{2}$$
$17$ $$(T - 3)^{2}$$
$19$ $$T^{2} + 36$$
$23$ $$(T + 4)^{2}$$
$29$ $$(T - 2)^{2}$$
$31$ $$T^{2} + 16$$
$37$ $$T^{2} + 9$$
$41$ $$T^{2}$$
$43$ $$(T + 5)^{2}$$
$47$ $$T^{2} + 169$$
$53$ $$(T - 12)^{2}$$
$59$ $$T^{2} + 100$$
$61$ $$(T + 8)^{2}$$
$67$ $$T^{2} + 4$$
$71$ $$T^{2} + 25$$
$73$ $$T^{2} + 100$$
$79$ $$(T - 4)^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2} + 36$$
$97$ $$T^{2} + 196$$