# Properties

 Label 3380.2.a.o 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: 3.3.564.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 5x + 3$$ x^3 - x^2 - 5*x + 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + 1) q^{3} - q^{5} + (\beta_1 + 1) q^{7} + (\beta_1 + 4) q^{9}+O(q^{10})$$ q + (b2 + 1) * q^3 - q^5 + (b1 + 1) * q^7 + (b1 + 4) * q^9 $$q + (\beta_{2} + 1) q^{3} - q^{5} + (\beta_1 + 1) q^{7} + (\beta_1 + 4) q^{9} + (\beta_{2} - \beta_1) q^{11} + ( - \beta_{2} - 1) q^{15} - 2 \beta_{2} q^{17} + (\beta_{2} + \beta_1 - 2) q^{19} + (2 \beta_{2} + 2 \beta_1 + 4) q^{21} + (\beta_{2} - 3) q^{23} + q^{25} + (2 \beta_{2} + 2 \beta_1 + 4) q^{27} + ( - \beta_1 + 3) q^{29} + (\beta_{2} - \beta_1 + 4) q^{31} + ( - 2 \beta_{2} - \beta_1 + 3) q^{33} + ( - \beta_1 - 1) q^{35} + (2 \beta_{2} - \beta_1 + 1) q^{37} - 2 \beta_{2} q^{41} + ( - \beta_{2} - 1) q^{43} + ( - \beta_1 - 4) q^{45} + ( - \beta_1 + 3) q^{47} + (4 \beta_{2} + 9) q^{49} + (2 \beta_{2} - 2 \beta_1 - 12) q^{51} + (2 \beta_{2} - 2 \beta_1 - 6) q^{53} + ( - \beta_{2} + \beta_1) q^{55} + ( - 2 \beta_{2} + 3 \beta_1 + 7) q^{57} + (3 \beta_{2} - \beta_1 + 6) q^{59} + (\beta_1 + 5) q^{61} + (4 \beta_{2} + 3 \beta_1 + 19) q^{63} + ( - 2 \beta_{2} + \beta_1 - 5) q^{67} + ( - 4 \beta_{2} + \beta_1 + 3) q^{69} + (\beta_{2} - \beta_1) q^{71} + (2 \beta_{2} - 3 \beta_1 - 5) q^{73} + (\beta_{2} + 1) q^{75} + ( - 2 \beta_{2} + 2 \beta_1 - 12) q^{77} + ( - 2 \beta_{2} + 2) q^{79} + (4 \beta_{2} + 3 \beta_1 + 10) q^{81} + (4 \beta_{2} - \beta_1 + 3) q^{83} + 2 \beta_{2} q^{85} + (2 \beta_{2} - 2 \beta_1) q^{87} + ( - 4 \beta_{2} + 2 \beta_1) q^{89} + (2 \beta_{2} - \beta_1 + 7) q^{93} + ( - \beta_{2} - \beta_1 + 2) q^{95} + (2 \beta_1 - 8) q^{97} + (\beta_{2} - \beta_1 - 12) q^{99}+O(q^{100})$$ q + (b2 + 1) * q^3 - q^5 + (b1 + 1) * q^7 + (b1 + 4) * q^9 + (b2 - b1) * q^11 + (-b2 - 1) * q^15 - 2*b2 * q^17 + (b2 + b1 - 2) * q^19 + (2*b2 + 2*b1 + 4) * q^21 + (b2 - 3) * q^23 + q^25 + (2*b2 + 2*b1 + 4) * q^27 + (-b1 + 3) * q^29 + (b2 - b1 + 4) * q^31 + (-2*b2 - b1 + 3) * q^33 + (-b1 - 1) * q^35 + (2*b2 - b1 + 1) * q^37 - 2*b2 * q^41 + (-b2 - 1) * q^43 + (-b1 - 4) * q^45 + (-b1 + 3) * q^47 + (4*b2 + 9) * q^49 + (2*b2 - 2*b1 - 12) * q^51 + (2*b2 - 2*b1 - 6) * q^53 + (-b2 + b1) * q^55 + (-2*b2 + 3*b1 + 7) * q^57 + (3*b2 - b1 + 6) * q^59 + (b1 + 5) * q^61 + (4*b2 + 3*b1 + 19) * q^63 + (-2*b2 + b1 - 5) * q^67 + (-4*b2 + b1 + 3) * q^69 + (b2 - b1) * q^71 + (2*b2 - 3*b1 - 5) * q^73 + (b2 + 1) * q^75 + (-2*b2 + 2*b1 - 12) * q^77 + (-2*b2 + 2) * q^79 + (4*b2 + 3*b1 + 10) * q^81 + (4*b2 - b1 + 3) * q^83 + 2*b2 * q^85 + (2*b2 - 2*b1) * q^87 + (-4*b2 + 2*b1) * q^89 + (2*b2 - b1 + 7) * q^93 + (-b2 - b1 + 2) * q^95 + (2*b1 - 8) * q^97 + (b2 - b1 - 12) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{3} - 3 q^{5} + 2 q^{7} + 11 q^{9}+O(q^{10})$$ 3 * q + 2 * q^3 - 3 * q^5 + 2 * q^7 + 11 * q^9 $$3 q + 2 q^{3} - 3 q^{5} + 2 q^{7} + 11 q^{9} - 2 q^{15} + 2 q^{17} - 8 q^{19} + 8 q^{21} - 10 q^{23} + 3 q^{25} + 8 q^{27} + 10 q^{29} + 12 q^{31} + 12 q^{33} - 2 q^{35} + 2 q^{37} + 2 q^{41} - 2 q^{43} - 11 q^{45} + 10 q^{47} + 23 q^{49} - 36 q^{51} - 18 q^{53} + 20 q^{57} + 16 q^{59} + 14 q^{61} + 50 q^{63} - 14 q^{67} + 12 q^{69} - 14 q^{73} + 2 q^{75} - 36 q^{77} + 8 q^{79} + 23 q^{81} + 6 q^{83} - 2 q^{85} + 2 q^{89} + 20 q^{93} + 8 q^{95} - 26 q^{97} - 36 q^{99}+O(q^{100})$$ 3 * q + 2 * q^3 - 3 * q^5 + 2 * q^7 + 11 * q^9 - 2 * q^15 + 2 * q^17 - 8 * q^19 + 8 * q^21 - 10 * q^23 + 3 * q^25 + 8 * q^27 + 10 * q^29 + 12 * q^31 + 12 * q^33 - 2 * q^35 + 2 * q^37 + 2 * q^41 - 2 * q^43 - 11 * q^45 + 10 * q^47 + 23 * q^49 - 36 * q^51 - 18 * q^53 + 20 * q^57 + 16 * q^59 + 14 * q^61 + 50 * q^63 - 14 * q^67 + 12 * q^69 - 14 * q^73 + 2 * q^75 - 36 * q^77 + 8 * q^79 + 23 * q^81 + 6 * q^83 - 2 * q^85 + 2 * q^89 + 20 * q^93 + 8 * q^95 - 26 * q^97 - 36 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 5x + 3$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu - 1$$ 2*v - 1 $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4
 $$\nu$$ $$=$$ $$( \beta _1 + 1 ) / 2$$ (b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ 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
 0.571993 −2.08613 2.51414
0 −2.67282 0 −1.00000 0 1.14399 0 4.14399 0
1.2 0 1.35194 0 −1.00000 0 −4.17226 0 −1.17226 0
1.3 0 3.32088 0 −1.00000 0 5.02827 0 8.02827 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.o 3
13.b even 2 1 260.2.a.b 3
13.d odd 4 2 3380.2.f.h 6
39.d odd 2 1 2340.2.a.n 3
52.b odd 2 1 1040.2.a.o 3
65.d even 2 1 1300.2.a.i 3
65.h odd 4 2 1300.2.c.f 6
104.e even 2 1 4160.2.a.bo 3
104.h odd 2 1 4160.2.a.br 3
156.h even 2 1 9360.2.a.da 3
260.g odd 2 1 5200.2.a.ci 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.a.b 3 13.b even 2 1
1040.2.a.o 3 52.b odd 2 1
1300.2.a.i 3 65.d even 2 1
1300.2.c.f 6 65.h odd 4 2
2340.2.a.n 3 39.d odd 2 1
3380.2.a.o 3 1.a even 1 1 trivial
3380.2.f.h 6 13.d odd 4 2
4160.2.a.bo 3 104.e even 2 1
4160.2.a.br 3 104.h odd 2 1
5200.2.a.ci 3 260.g odd 2 1
9360.2.a.da 3 156.h even 2 1

## Hecke kernels

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

 $$T_{3}^{3} - 2T_{3}^{2} - 8T_{3} + 12$$ T3^3 - 2*T3^2 - 8*T3 + 12 $$T_{7}^{3} - 2T_{7}^{2} - 20T_{7} + 24$$ T7^3 - 2*T7^2 - 20*T7 + 24 $$T_{19}^{3} + 8T_{19}^{2} - 16T_{19} - 164$$ T19^3 + 8*T19^2 - 16*T19 - 164

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - 2 T^{2} + \cdots + 12$$
$5$ $$(T + 1)^{3}$$
$7$ $$T^{3} - 2 T^{2} + \cdots + 24$$
$11$ $$T^{3} - 24T - 36$$
$13$ $$T^{3}$$
$17$ $$T^{3} - 2 T^{2} + \cdots - 24$$
$19$ $$T^{3} + 8 T^{2} + \cdots - 164$$
$23$ $$T^{3} + 10 T^{2} + \cdots + 12$$
$29$ $$T^{3} - 10 T^{2} + \cdots + 24$$
$31$ $$T^{3} - 12 T^{2} + \cdots - 4$$
$37$ $$T^{3} - 2 T^{2} + \cdots + 72$$
$41$ $$T^{3} - 2 T^{2} + \cdots - 24$$
$43$ $$T^{3} + 2 T^{2} + \cdots - 12$$
$47$ $$T^{3} - 10 T^{2} + \cdots + 24$$
$53$ $$T^{3} + 18 T^{2} + \cdots - 648$$
$59$ $$T^{3} - 16T^{2} + 564$$
$61$ $$T^{3} - 14 T^{2} + \cdots + 8$$
$67$ $$T^{3} + 14 T^{2} + \cdots - 152$$
$71$ $$T^{3} - 24T - 36$$
$73$ $$T^{3} + 14 T^{2} + \cdots - 1784$$
$79$ $$T^{3} - 8 T^{2} + \cdots + 32$$
$83$ $$T^{3} - 6 T^{2} + \cdots + 936$$
$89$ $$T^{3} - 2 T^{2} + \cdots - 216$$
$97$ $$T^{3} + 26 T^{2} + \cdots - 8$$