Properties

 Label 3380.2.f.b Level $3380$ Weight $2$ Character orbit 3380.f Analytic conductor $26.989$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(3041,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, 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("3380.3041");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3380 = 2^{2} \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3380.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: $$26.9894358832$$ 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 20) 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.

 $$f(q)$$ $$=$$ $$q - 2 q^{3} + i q^{5} + 2 i q^{7} + q^{9}+O(q^{10})$$ q - 2 * q^3 + i * q^5 + 2*i * q^7 + q^9 $$q - 2 q^{3} + i q^{5} + 2 i q^{7} + q^{9} - 2 i q^{15} + 6 q^{17} + 4 i q^{19} - 4 i q^{21} - 6 q^{23} - q^{25} + 4 q^{27} + 6 q^{29} + 4 i q^{31} - 2 q^{35} + 2 i q^{37} - 6 i q^{41} + 10 q^{43} + i q^{45} - 6 i q^{47} + 3 q^{49} - 12 q^{51} - 6 q^{53} - 8 i q^{57} + 12 i q^{59} + 2 q^{61} + 2 i q^{63} - 2 i q^{67} + 12 q^{69} + 12 i q^{71} + 2 i q^{73} + 2 q^{75} + 8 q^{79} - 11 q^{81} - 6 i q^{83} + 6 i q^{85} - 12 q^{87} - 6 i q^{89} - 8 i q^{93} - 4 q^{95} - 2 i q^{97} +O(q^{100})$$ q - 2 * q^3 + i * q^5 + 2*i * q^7 + q^9 - 2*i * q^15 + 6 * q^17 + 4*i * q^19 - 4*i * q^21 - 6 * q^23 - q^25 + 4 * q^27 + 6 * q^29 + 4*i * q^31 - 2 * q^35 + 2*i * q^37 - 6*i * q^41 + 10 * q^43 + i * q^45 - 6*i * q^47 + 3 * q^49 - 12 * q^51 - 6 * q^53 - 8*i * q^57 + 12*i * q^59 + 2 * q^61 + 2*i * q^63 - 2*i * q^67 + 12 * q^69 + 12*i * q^71 + 2*i * q^73 + 2 * q^75 + 8 * q^79 - 11 * q^81 - 6*i * q^83 + 6*i * q^85 - 12 * q^87 - 6*i * q^89 - 8*i * q^93 - 4 * q^95 - 2*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{3} + 2 q^{9}+O(q^{10})$$ 2 * q - 4 * q^3 + 2 * q^9 $$2 q - 4 q^{3} + 2 q^{9} + 12 q^{17} - 12 q^{23} - 2 q^{25} + 8 q^{27} + 12 q^{29} - 4 q^{35} + 20 q^{43} + 6 q^{49} - 24 q^{51} - 12 q^{53} + 4 q^{61} + 24 q^{69} + 4 q^{75} + 16 q^{79} - 22 q^{81} - 24 q^{87} - 8 q^{95}+O(q^{100})$$ 2 * q - 4 * q^3 + 2 * q^9 + 12 * q^17 - 12 * q^23 - 2 * q^25 + 8 * q^27 + 12 * q^29 - 4 * q^35 + 20 * q^43 + 6 * q^49 - 24 * q^51 - 12 * q^53 + 4 * q^61 + 24 * q^69 + 4 * q^75 + 16 * q^79 - 22 * q^81 - 24 * q^87 - 8 * q^95

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$1691$$ $$1861$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3041.1
 − 1.00000i 1.00000i
0 −2.00000 0 1.00000i 0 2.00000i 0 1.00000 0
3041.2 0 −2.00000 0 1.00000i 0 2.00000i 0 1.00000 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 3380.2.f.b 2
13.b even 2 1 inner 3380.2.f.b 2
13.d odd 4 1 20.2.a.a 1
13.d odd 4 1 3380.2.a.c 1
39.f even 4 1 180.2.a.a 1
52.f even 4 1 80.2.a.b 1
65.f even 4 1 100.2.c.a 2
65.g odd 4 1 100.2.a.a 1
65.k even 4 1 100.2.c.a 2
91.i even 4 1 980.2.a.h 1
91.z odd 12 2 980.2.i.i 2
91.bb even 12 2 980.2.i.c 2
104.j odd 4 1 320.2.a.f 1
104.m even 4 1 320.2.a.a 1
117.y odd 12 2 1620.2.i.h 2
117.z even 12 2 1620.2.i.b 2
143.g even 4 1 2420.2.a.a 1
156.l odd 4 1 720.2.a.h 1
195.j odd 4 1 900.2.d.c 2
195.n even 4 1 900.2.a.b 1
195.u odd 4 1 900.2.d.c 2
208.l even 4 1 1280.2.d.g 2
208.m odd 4 1 1280.2.d.c 2
208.r odd 4 1 1280.2.d.c 2
208.s even 4 1 1280.2.d.g 2
221.g odd 4 1 5780.2.a.f 1
221.h odd 4 1 5780.2.c.a 2
221.i odd 4 1 5780.2.c.a 2
247.i even 4 1 7220.2.a.f 1
260.l odd 4 1 400.2.c.b 2
260.s odd 4 1 400.2.c.b 2
260.u even 4 1 400.2.a.c 1
273.o odd 4 1 8820.2.a.g 1
312.w odd 4 1 2880.2.a.f 1
312.y even 4 1 2880.2.a.m 1
364.p odd 4 1 3920.2.a.h 1
455.n odd 4 1 4900.2.e.f 2
455.u even 4 1 4900.2.a.e 1
455.w odd 4 1 4900.2.e.f 2
520.t even 4 1 1600.2.a.w 1
520.x odd 4 1 1600.2.c.e 2
520.y even 4 1 1600.2.c.d 2
520.bj even 4 1 1600.2.c.d 2
520.bk odd 4 1 1600.2.c.e 2
520.bo odd 4 1 1600.2.a.c 1
572.k odd 4 1 9680.2.a.ba 1
780.u even 4 1 3600.2.f.j 2
780.bb odd 4 1 3600.2.a.be 1
780.bn even 4 1 3600.2.f.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.a.a 1 13.d odd 4 1
80.2.a.b 1 52.f even 4 1
100.2.a.a 1 65.g odd 4 1
100.2.c.a 2 65.f even 4 1
100.2.c.a 2 65.k even 4 1
180.2.a.a 1 39.f even 4 1
320.2.a.a 1 104.m even 4 1
320.2.a.f 1 104.j odd 4 1
400.2.a.c 1 260.u even 4 1
400.2.c.b 2 260.l odd 4 1
400.2.c.b 2 260.s odd 4 1
720.2.a.h 1 156.l odd 4 1
900.2.a.b 1 195.n even 4 1
900.2.d.c 2 195.j odd 4 1
900.2.d.c 2 195.u odd 4 1
980.2.a.h 1 91.i even 4 1
980.2.i.c 2 91.bb even 12 2
980.2.i.i 2 91.z odd 12 2
1280.2.d.c 2 208.m odd 4 1
1280.2.d.c 2 208.r odd 4 1
1280.2.d.g 2 208.l even 4 1
1280.2.d.g 2 208.s even 4 1
1600.2.a.c 1 520.bo odd 4 1
1600.2.a.w 1 520.t even 4 1
1600.2.c.d 2 520.y even 4 1
1600.2.c.d 2 520.bj even 4 1
1600.2.c.e 2 520.x odd 4 1
1600.2.c.e 2 520.bk odd 4 1
1620.2.i.b 2 117.z even 12 2
1620.2.i.h 2 117.y odd 12 2
2420.2.a.a 1 143.g even 4 1
2880.2.a.f 1 312.w odd 4 1
2880.2.a.m 1 312.y even 4 1
3380.2.a.c 1 13.d odd 4 1
3380.2.f.b 2 1.a even 1 1 trivial
3380.2.f.b 2 13.b even 2 1 inner
3600.2.a.be 1 780.bb odd 4 1
3600.2.f.j 2 780.u even 4 1
3600.2.f.j 2 780.bn even 4 1
3920.2.a.h 1 364.p odd 4 1
4900.2.a.e 1 455.u even 4 1
4900.2.e.f 2 455.n odd 4 1
4900.2.e.f 2 455.w odd 4 1
5780.2.a.f 1 221.g odd 4 1
5780.2.c.a 2 221.h odd 4 1
5780.2.c.a 2 221.i odd 4 1
7220.2.a.f 1 247.i even 4 1
8820.2.a.g 1 273.o odd 4 1
9680.2.a.ba 1 572.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3380, [\chi])$$:

 $$T_{3} + 2$$ T3 + 2 $$T_{19}^{2} + 16$$ T19^2 + 16

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 2)^{2}$$
$5$ $$T^{2} + 1$$
$7$ $$T^{2} + 4$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$(T - 6)^{2}$$
$19$ $$T^{2} + 16$$
$23$ $$(T + 6)^{2}$$
$29$ $$(T - 6)^{2}$$
$31$ $$T^{2} + 16$$
$37$ $$T^{2} + 4$$
$41$ $$T^{2} + 36$$
$43$ $$(T - 10)^{2}$$
$47$ $$T^{2} + 36$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} + 144$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2} + 4$$
$71$ $$T^{2} + 144$$
$73$ $$T^{2} + 4$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$T^{2} + 36$$
$97$ $$T^{2} + 4$$