# Properties

 Label 3380.2.f.e Level $3380$ Weight $2$ Character orbit 3380.f Analytic conductor $26.989$ 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] = [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 260) 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.

 $$f(q)$$ $$=$$ $$q + q^{3} - i q^{5} - i q^{7} - 2 q^{9} +O(q^{10})$$ q + q^3 - i * q^5 - i * q^7 - 2 * q^9 $$q + q^{3} - i q^{5} - i q^{7} - 2 q^{9} + 3 i q^{11} - i q^{15} + 3 q^{17} + 7 i q^{19} - i q^{21} + 3 q^{23} - q^{25} - 5 q^{27} + 3 q^{29} + 4 i q^{31} + 3 i q^{33} - q^{35} - 7 i q^{37} + 9 i q^{41} - 11 q^{43} + 2 i q^{45} + 6 q^{49} + 3 q^{51} - 6 q^{53} + 3 q^{55} + 7 i q^{57} - 3 i q^{59} + 11 q^{61} + 2 i q^{63} + 7 i q^{67} + 3 q^{69} + 3 i q^{71} + 2 i q^{73} - q^{75} + 3 q^{77} + 8 q^{79} + q^{81} + 12 i q^{83} - 3 i q^{85} + 3 q^{87} + 15 i q^{89} + 4 i q^{93} + 7 q^{95} + 7 i q^{97} - 6 i q^{99} +O(q^{100})$$ q + q^3 - i * q^5 - i * q^7 - 2 * q^9 + 3*i * q^11 - i * q^15 + 3 * q^17 + 7*i * q^19 - i * q^21 + 3 * q^23 - q^25 - 5 * q^27 + 3 * q^29 + 4*i * q^31 + 3*i * q^33 - q^35 - 7*i * q^37 + 9*i * q^41 - 11 * q^43 + 2*i * q^45 + 6 * q^49 + 3 * q^51 - 6 * q^53 + 3 * q^55 + 7*i * q^57 - 3*i * q^59 + 11 * q^61 + 2*i * q^63 + 7*i * q^67 + 3 * q^69 + 3*i * q^71 + 2*i * q^73 - q^75 + 3 * q^77 + 8 * q^79 + q^81 + 12*i * q^83 - 3*i * q^85 + 3 * q^87 + 15*i * q^89 + 4*i * q^93 + 7 * q^95 + 7*i * q^97 - 6*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 $$2 q + 2 q^{3} - 4 q^{9} + 6 q^{17} + 6 q^{23} - 2 q^{25} - 10 q^{27} + 6 q^{29} - 2 q^{35} - 22 q^{43} + 12 q^{49} + 6 q^{51} - 12 q^{53} + 6 q^{55} + 22 q^{61} + 6 q^{69} - 2 q^{75} + 6 q^{77} + 16 q^{79} + 2 q^{81} + 6 q^{87} + 14 q^{95}+O(q^{100})$$ 2 * q + 2 * q^3 - 4 * q^9 + 6 * q^17 + 6 * q^23 - 2 * q^25 - 10 * q^27 + 6 * q^29 - 2 * q^35 - 22 * q^43 + 12 * q^49 + 6 * q^51 - 12 * q^53 + 6 * q^55 + 22 * q^61 + 6 * q^69 - 2 * q^75 + 6 * q^77 + 16 * q^79 + 2 * q^81 + 6 * q^87 + 14 * 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 1.00000 0 1.00000i 0 1.00000i 0 −2.00000 0
3041.2 0 1.00000 0 1.00000i 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 3380.2.f.e 2
13.b even 2 1 inner 3380.2.f.e 2
13.d odd 4 1 3380.2.a.g 1
13.d odd 4 1 3380.2.a.h 1
13.f odd 12 2 260.2.i.b 2
39.k even 12 2 2340.2.q.b 2
52.l even 12 2 1040.2.q.j 2
65.o even 12 2 1300.2.bb.a 4
65.s odd 12 2 1300.2.i.e 2
65.t even 12 2 1300.2.bb.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.i.b 2 13.f odd 12 2
1040.2.q.j 2 52.l even 12 2
1300.2.i.e 2 65.s odd 12 2
1300.2.bb.a 4 65.o even 12 2
1300.2.bb.a 4 65.t even 12 2
2340.2.q.b 2 39.k even 12 2
3380.2.a.g 1 13.d odd 4 1
3380.2.a.h 1 13.d odd 4 1
3380.2.f.e 2 1.a even 1 1 trivial
3380.2.f.e 2 13.b even 2 1 inner

## Hecke kernels

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

 $$T_{3} - 1$$ T3 - 1 $$T_{19}^{2} + 49$$ T19^2 + 49

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} + 1$$
$7$ $$T^{2} + 1$$
$11$ $$T^{2} + 9$$
$13$ $$T^{2}$$
$17$ $$(T - 3)^{2}$$
$19$ $$T^{2} + 49$$
$23$ $$(T - 3)^{2}$$
$29$ $$(T - 3)^{2}$$
$31$ $$T^{2} + 16$$
$37$ $$T^{2} + 49$$
$41$ $$T^{2} + 81$$
$43$ $$(T + 11)^{2}$$
$47$ $$T^{2}$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} + 9$$
$61$ $$(T - 11)^{2}$$
$67$ $$T^{2} + 49$$
$71$ $$T^{2} + 9$$
$73$ $$T^{2} + 4$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$T^{2} + 225$$
$97$ $$T^{2} + 49$$