# Properties

 Label 3328.2.b.g Level $3328$ Weight $2$ Character orbit 3328.b Analytic conductor $26.574$ 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] = [3328,2,Mod(1665,3328)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3328, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("3328.1665");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3328 = 2^{8} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3328.b (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: $$26.5742137927$$ 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.

 $$f(q)$$ $$=$$ $$q + 3 i q^{3} - i q^{5} - q^{7} - 6 q^{9} +O(q^{10})$$ q + 3*i * q^3 - i * q^5 - q^7 - 6 * q^9 $$q + 3 i q^{3} - i q^{5} - q^{7} - 6 q^{9} - 2 i q^{11} + i q^{13} + 3 q^{15} - 3 q^{17} - 6 i q^{19} - 3 i q^{21} + 4 q^{23} + 4 q^{25} - 9 i q^{27} - 2 i q^{29} + 4 q^{31} + 6 q^{33} + i q^{35} + 3 i q^{37} - 3 q^{39} - 5 i q^{43} + 6 i q^{45} + 13 q^{47} - 6 q^{49} - 9 i q^{51} + 12 i q^{53} - 2 q^{55} + 18 q^{57} - 10 i q^{59} + 8 i q^{61} + 6 q^{63} + q^{65} + 2 i q^{67} + 12 i q^{69} + 5 q^{71} + 10 q^{73} + 12 i q^{75} + 2 i q^{77} - 4 q^{79} + 9 q^{81} + 3 i q^{85} + 6 q^{87} - 6 q^{89} - i q^{91} + 12 i q^{93} - 6 q^{95} + 14 q^{97} + 12 i q^{99} +O(q^{100})$$ q + 3*i * q^3 - i * q^5 - q^7 - 6 * q^9 - 2*i * q^11 + i * q^13 + 3 * q^15 - 3 * q^17 - 6*i * q^19 - 3*i * q^21 + 4 * q^23 + 4 * q^25 - 9*i * q^27 - 2*i * q^29 + 4 * q^31 + 6 * q^33 + i * q^35 + 3*i * q^37 - 3 * q^39 - 5*i * q^43 + 6*i * q^45 + 13 * q^47 - 6 * q^49 - 9*i * q^51 + 12*i * q^53 - 2 * q^55 + 18 * q^57 - 10*i * q^59 + 8*i * q^61 + 6 * q^63 + q^65 + 2*i * q^67 + 12*i * q^69 + 5 * q^71 + 10 * q^73 + 12*i * q^75 + 2*i * q^77 - 4 * q^79 + 9 * q^81 + 3*i * q^85 + 6 * q^87 - 6 * q^89 - i * q^91 + 12*i * q^93 - 6 * q^95 + 14 * q^97 + 12*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{7} - 12 q^{9}+O(q^{10})$$ 2 * q - 2 * q^7 - 12 * q^9 $$2 q - 2 q^{7} - 12 q^{9} + 6 q^{15} - 6 q^{17} + 8 q^{23} + 8 q^{25} + 8 q^{31} + 12 q^{33} - 6 q^{39} + 26 q^{47} - 12 q^{49} - 4 q^{55} + 36 q^{57} + 12 q^{63} + 2 q^{65} + 10 q^{71} + 20 q^{73} - 8 q^{79} + 18 q^{81} + 12 q^{87} - 12 q^{89} - 12 q^{95} + 28 q^{97}+O(q^{100})$$ 2 * q - 2 * q^7 - 12 * q^9 + 6 * q^15 - 6 * q^17 + 8 * q^23 + 8 * q^25 + 8 * q^31 + 12 * q^33 - 6 * q^39 + 26 * q^47 - 12 * q^49 - 4 * q^55 + 36 * q^57 + 12 * q^63 + 2 * q^65 + 10 * q^71 + 20 * q^73 - 8 * q^79 + 18 * q^81 + 12 * q^87 - 12 * q^89 - 12 * q^95 + 28 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3328\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$769$$ $$1535$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1665.1
 − 1.00000i 1.00000i
0 3.00000i 0 1.00000i 0 −1.00000 0 −6.00000 0
1665.2 0 3.00000i 0 1.00000i 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

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3328.2.b.g 2
4.b odd 2 1 3328.2.b.k 2
8.b even 2 1 inner 3328.2.b.g 2
8.d odd 2 1 3328.2.b.k 2
16.e even 4 1 26.2.a.b 1
16.e even 4 1 832.2.a.j 1
16.f odd 4 1 208.2.a.d 1
16.f odd 4 1 832.2.a.a 1
48.i odd 4 1 234.2.a.b 1
48.i odd 4 1 7488.2.a.w 1
48.k even 4 1 1872.2.a.m 1
48.k even 4 1 7488.2.a.v 1
80.i odd 4 1 650.2.b.a 2
80.k odd 4 1 5200.2.a.c 1
80.q even 4 1 650.2.a.g 1
80.t odd 4 1 650.2.b.a 2
112.l odd 4 1 1274.2.a.o 1
112.w even 12 2 1274.2.f.l 2
112.x odd 12 2 1274.2.f.a 2
144.w odd 12 2 2106.2.e.t 2
144.x even 12 2 2106.2.e.h 2
176.l odd 4 1 3146.2.a.a 1
208.l even 4 1 2704.2.f.j 2
208.m odd 4 1 338.2.b.a 2
208.o odd 4 1 2704.2.a.n 1
208.p even 4 1 338.2.a.a 1
208.r odd 4 1 338.2.b.a 2
208.s even 4 1 2704.2.f.j 2
208.be odd 12 2 338.2.e.d 4
208.bh even 12 2 338.2.c.g 2
208.bj even 12 2 338.2.c.c 2
208.bl odd 12 2 338.2.e.d 4
240.bb even 4 1 5850.2.e.v 2
240.bf even 4 1 5850.2.e.v 2
240.bm odd 4 1 5850.2.a.bn 1
272.r even 4 1 7514.2.a.i 1
304.j odd 4 1 9386.2.a.f 1
624.u even 4 1 3042.2.b.f 2
624.bi odd 4 1 3042.2.a.l 1
624.bm even 4 1 3042.2.b.f 2
1040.be even 4 1 8450.2.a.y 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.b 1 16.e even 4 1
208.2.a.d 1 16.f odd 4 1
234.2.a.b 1 48.i odd 4 1
338.2.a.a 1 208.p even 4 1
338.2.b.a 2 208.m odd 4 1
338.2.b.a 2 208.r odd 4 1
338.2.c.c 2 208.bj even 12 2
338.2.c.g 2 208.bh even 12 2
338.2.e.d 4 208.be odd 12 2
338.2.e.d 4 208.bl odd 12 2
650.2.a.g 1 80.q even 4 1
650.2.b.a 2 80.i odd 4 1
650.2.b.a 2 80.t odd 4 1
832.2.a.a 1 16.f odd 4 1
832.2.a.j 1 16.e even 4 1
1274.2.a.o 1 112.l odd 4 1
1274.2.f.a 2 112.x odd 12 2
1274.2.f.l 2 112.w even 12 2
1872.2.a.m 1 48.k even 4 1
2106.2.e.h 2 144.x even 12 2
2106.2.e.t 2 144.w odd 12 2
2704.2.a.n 1 208.o odd 4 1
2704.2.f.j 2 208.l even 4 1
2704.2.f.j 2 208.s even 4 1
3042.2.a.l 1 624.bi odd 4 1
3042.2.b.f 2 624.u even 4 1
3042.2.b.f 2 624.bm even 4 1
3146.2.a.a 1 176.l odd 4 1
3328.2.b.g 2 1.a even 1 1 trivial
3328.2.b.g 2 8.b even 2 1 inner
3328.2.b.k 2 4.b odd 2 1
3328.2.b.k 2 8.d odd 2 1
5200.2.a.c 1 80.k odd 4 1
5850.2.a.bn 1 240.bm odd 4 1
5850.2.e.v 2 240.bb even 4 1
5850.2.e.v 2 240.bf even 4 1
7488.2.a.v 1 48.k even 4 1
7488.2.a.w 1 48.i odd 4 1
7514.2.a.i 1 272.r even 4 1
8450.2.a.y 1 1040.be even 4 1
9386.2.a.f 1 304.j odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3328, [\chi])$$:

 $$T_{3}^{2} + 9$$ T3^2 + 9 $$T_{5}^{2} + 1$$ T5^2 + 1 $$T_{7} + 1$$ T7 + 1 $$T_{11}^{2} + 4$$ T11^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 9$$
$5$ $$T^{2} + 1$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} + 4$$
$13$ $$T^{2} + 1$$
$17$ $$(T + 3)^{2}$$
$19$ $$T^{2} + 36$$
$23$ $$(T - 4)^{2}$$
$29$ $$T^{2} + 4$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 9$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 25$$
$47$ $$(T - 13)^{2}$$
$53$ $$T^{2} + 144$$
$59$ $$T^{2} + 100$$
$61$ $$T^{2} + 64$$
$67$ $$T^{2} + 4$$
$71$ $$(T - 5)^{2}$$
$73$ $$(T - 10)^{2}$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2}$$
$89$ $$(T + 6)^{2}$$
$97$ $$(T - 14)^{2}$$