# Properties

 Label 3380.2.a.n Level $3380$ Weight $2$ Character orbit 3380.a Self dual yes Analytic conductor $26.989$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(1,3380)]

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, 0]))

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

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

chi := DirichletCharacter("3380.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## 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: yes Analytic conductor: $$26.9894358832$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.756.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 6x - 2$$ x^3 - 6*x - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 260) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + q^{5} + \beta_{2} q^{7} + (\beta_{2} + 1) q^{9}+O(q^{10})$$ q + b1 * q^3 + q^5 + b2 * q^7 + (b2 + 1) * q^9 $$q + \beta_1 q^{3} + q^{5} + \beta_{2} q^{7} + (\beta_{2} + 1) q^{9} + (\beta_{2} - \beta_1 + 2) q^{11} + \beta_1 q^{15} + ( - \beta_{2} + 3 \beta_1) q^{19} + (2 \beta_1 + 2) q^{21} + (2 \beta_{2} - \beta_1 + 2) q^{23} + q^{25} + 2 q^{27} + (\beta_{2} - 2 \beta_1 - 2) q^{29} + ( - \beta_{2} - \beta_1 + 2) q^{31} + ( - \beta_{2} + 4 \beta_1 - 2) q^{33} + \beta_{2} q^{35} + ( - \beta_{2} - 2 \beta_1 + 4) q^{37} + (2 \beta_{2} - 2 \beta_1 - 2) q^{41} + ( - 3 \beta_1 - 2) q^{43} + (\beta_{2} + 1) q^{45} + ( - \beta_{2} + 4 \beta_1 + 4) q^{47} + ( - 2 \beta_{2} + 2 \beta_1 + 1) q^{49} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{53} + (\beta_{2} - \beta_1 + 2) q^{55} + (3 \beta_{2} - 2 \beta_1 + 10) q^{57} + ( - \beta_{2} + \beta_1 - 8) q^{59} + ( - 3 \beta_{2} - 2) q^{61} + ( - \beta_{2} + 2 \beta_1 + 8) q^{63} + ( - 3 \beta_{2} + 2 \beta_1 - 4) q^{67} + ( - \beta_{2} + 6 \beta_1) q^{69} + (3 \beta_{2} + 3 \beta_1) q^{71} + (\beta_{2} + 2 \beta_1 + 8) q^{73} + \beta_1 q^{75} + 6 q^{77} + (2 \beta_{2} + 4 \beta_1 - 4) q^{79} + ( - 3 \beta_{2} + 2 \beta_1 - 3) q^{81} + ( - \beta_{2} - 2 \beta_1 + 4) q^{83} + ( - 2 \beta_{2} - 6) q^{87} + ( - 2 \beta_{2} - 4 \beta_1 + 8) q^{89} + ( - \beta_{2} - 6) q^{93} + ( - \beta_{2} + 3 \beta_1) q^{95} + ( - 2 \beta_{2} + 6) q^{97} + (\beta_{2} - \beta_1 + 8) q^{99}+O(q^{100})$$ q + b1 * q^3 + q^5 + b2 * q^7 + (b2 + 1) * q^9 + (b2 - b1 + 2) * q^11 + b1 * q^15 + (-b2 + 3*b1) * q^19 + (2*b1 + 2) * q^21 + (2*b2 - b1 + 2) * q^23 + q^25 + 2 * q^27 + (b2 - 2*b1 - 2) * q^29 + (-b2 - b1 + 2) * q^31 + (-b2 + 4*b1 - 2) * q^33 + b2 * q^35 + (-b2 - 2*b1 + 4) * q^37 + (2*b2 - 2*b1 - 2) * q^41 + (-3*b1 - 2) * q^43 + (b2 + 1) * q^45 + (-b2 + 4*b1 + 4) * q^47 + (-2*b2 + 2*b1 + 1) * q^49 + (-2*b2 - 2*b1 - 2) * q^53 + (b2 - b1 + 2) * q^55 + (3*b2 - 2*b1 + 10) * q^57 + (-b2 + b1 - 8) * q^59 + (-3*b2 - 2) * q^61 + (-b2 + 2*b1 + 8) * q^63 + (-3*b2 + 2*b1 - 4) * q^67 + (-b2 + 6*b1) * q^69 + (3*b2 + 3*b1) * q^71 + (b2 + 2*b1 + 8) * q^73 + b1 * q^75 + 6 * q^77 + (2*b2 + 4*b1 - 4) * q^79 + (-3*b2 + 2*b1 - 3) * q^81 + (-b2 - 2*b1 + 4) * q^83 + (-2*b2 - 6) * q^87 + (-2*b2 - 4*b1 + 8) * q^89 + (-b2 - 6) * q^93 + (-b2 + 3*b1) * q^95 + (-2*b2 + 6) * q^97 + (b2 - b1 + 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{5} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^5 + 3 * q^9 $$3 q + 3 q^{5} + 3 q^{9} + 6 q^{11} + 6 q^{21} + 6 q^{23} + 3 q^{25} + 6 q^{27} - 6 q^{29} + 6 q^{31} - 6 q^{33} + 12 q^{37} - 6 q^{41} - 6 q^{43} + 3 q^{45} + 12 q^{47} + 3 q^{49} - 6 q^{53} + 6 q^{55} + 30 q^{57} - 24 q^{59} - 6 q^{61} + 24 q^{63} - 12 q^{67} + 24 q^{73} + 18 q^{77} - 12 q^{79} - 9 q^{81} + 12 q^{83} - 18 q^{87} + 24 q^{89} - 18 q^{93} + 18 q^{97} + 24 q^{99}+O(q^{100})$$ 3 * q + 3 * q^5 + 3 * q^9 + 6 * q^11 + 6 * q^21 + 6 * q^23 + 3 * q^25 + 6 * q^27 - 6 * q^29 + 6 * q^31 - 6 * q^33 + 12 * q^37 - 6 * q^41 - 6 * q^43 + 3 * q^45 + 12 * q^47 + 3 * q^49 - 6 * q^53 + 6 * q^55 + 30 * q^57 - 24 * q^59 - 6 * q^61 + 24 * q^63 - 12 * q^67 + 24 * q^73 + 18 * q^77 - 12 * q^79 - 9 * q^81 + 12 * q^83 - 18 * q^87 + 24 * q^89 - 18 * q^93 + 18 * q^97 + 24 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 6x - 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ b2 + 4

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.26180 −0.339877 2.60168
0 −2.26180 0 1.00000 0 1.11575 0 2.11575 0
1.2 0 −0.339877 0 1.00000 0 −3.88448 0 −2.88448 0
1.3 0 2.60168 0 1.00000 0 2.76873 0 3.76873 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$13$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3380.2.a.n 3
13.b even 2 1 3380.2.a.m 3
13.d odd 4 2 260.2.f.a 6
39.f even 4 2 2340.2.c.d 6
52.f even 4 2 1040.2.k.c 6
65.f even 4 2 1300.2.d.d 6
65.g odd 4 2 1300.2.f.e 6
65.k even 4 2 1300.2.d.c 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.f.a 6 13.d odd 4 2
1040.2.k.c 6 52.f even 4 2
1300.2.d.c 6 65.k even 4 2
1300.2.d.d 6 65.f even 4 2
1300.2.f.e 6 65.g odd 4 2
2340.2.c.d 6 39.f even 4 2
3380.2.a.m 3 13.b even 2 1
3380.2.a.n 3 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3380))$$:

 $$T_{3}^{3} - 6T_{3} - 2$$ T3^3 - 6*T3 - 2 $$T_{7}^{3} - 12T_{7} + 12$$ T7^3 - 12*T7 + 12 $$T_{19}^{3} - 48T_{19} + 114$$ T19^3 - 48*T19 + 114

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - 6T - 2$$
$5$ $$(T - 1)^{3}$$
$7$ $$T^{3} - 12T + 12$$
$11$ $$T^{3} - 6T^{2} + 18$$
$13$ $$T^{3}$$
$17$ $$T^{3}$$
$19$ $$T^{3} - 48T + 114$$
$23$ $$T^{3} - 6 T^{2} - 30 T + 174$$
$29$ $$T^{3} + 6 T^{2} - 12 T - 84$$
$31$ $$T^{3} - 6 T^{2} - 12 T + 66$$
$37$ $$T^{3} - 12T^{2} + 252$$
$41$ $$T^{3} + 6 T^{2} - 36 T - 72$$
$43$ $$T^{3} + 6 T^{2} - 42 T - 46$$
$47$ $$T^{3} - 12 T^{2} - 36 T + 468$$
$53$ $$T^{3} + 6 T^{2} - 84 T + 24$$
$59$ $$T^{3} + 24 T^{2} + 180 T + 414$$
$61$ $$T^{3} + 6 T^{2} - 96 T - 532$$
$67$ $$T^{3} + 12 T^{2} - 48 T - 588$$
$71$ $$T^{3} - 216T - 702$$
$73$ $$T^{3} - 24 T^{2} + 144 T - 252$$
$79$ $$T^{3} + 12 T^{2} - 144 T - 1696$$
$83$ $$T^{3} - 12T^{2} + 252$$
$89$ $$T^{3} - 24T^{2} + 2016$$
$97$ $$T^{3} - 18 T^{2} + 60 T - 24$$