# Properties

 Label 3380.2.a.l Level $3380$ Weight $2$ Character orbit 3380.a Self dual yes Analytic conductor $26.989$ Analytic rank $1$ 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: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} + q^{5} + \beta_1 q^{7} + (\beta_{2} - 1) q^{9}+O(q^{10})$$ q - b1 * q^3 + q^5 + b1 * q^7 + (b2 - 1) * q^9 $$q - \beta_1 q^{3} + q^{5} + \beta_1 q^{7} + (\beta_{2} - 1) q^{9} + (\beta_{2} - \beta_1 - 1) q^{11} - \beta_1 q^{15} + ( - 2 \beta_{2} + 3 \beta_1 - 2) q^{17} + ( - 2 \beta_{2} + \beta_1 - 1) q^{19} + ( - \beta_{2} - 2) q^{21} + (\beta_{2} - 1) q^{23} + q^{25} + ( - \beta_{2} + 4 \beta_1 - 1) q^{27} + ( - 2 \beta_{2} - \beta_1 - 1) q^{29} + (2 \beta_{2} - 3 \beta_1 + 4) q^{31} + (\beta_1 + 1) q^{33} + \beta_1 q^{35} - 3 q^{37} + (\beta_{2} - 4 \beta_1 - 1) q^{41} + ( - 3 \beta_{2} + 5 \beta_1 - 3) q^{43} + (\beta_{2} - 1) q^{45} + ( - 7 \beta_{2} + 4 \beta_1 - 7) q^{47} + (\beta_{2} - 5) q^{49} + ( - \beta_{2} + 2 \beta_1 - 4) q^{51} + (7 \beta_{2} - \beta_1) q^{53} + (\beta_{2} - \beta_1 - 1) q^{55} + (\beta_{2} + \beta_1) q^{57} + (2 \beta_{2} + 4 \beta_1 + 5) q^{59} + (\beta_{2} - \beta_1 - 9) q^{61} + (\beta_{2} - \beta_1 + 1) q^{63} + (4 \beta_{2} - 4 \beta_1 - 3) q^{67} + ( - \beta_{2} + \beta_1 - 1) q^{69} + (3 \beta_{2} + 2) q^{71} + ( - 4 \beta_{2} - 2 \beta_1 - 6) q^{73} - \beta_1 q^{75} + ( - \beta_1 - 1) q^{77} + (3 \beta_{2} + 5 \beta_1 - 5) q^{79} + ( - 6 \beta_{2} + \beta_1 - 4) q^{81} + ( - 3 \beta_{2} - 4 \beta_1 + 6) q^{83} + ( - 2 \beta_{2} + 3 \beta_1 - 2) q^{85} + (3 \beta_{2} + \beta_1 + 4) q^{87} + (2 \beta_{2} + 8 \beta_1 - 5) q^{89} + (\beta_{2} - 4 \beta_1 + 4) q^{93} + ( - 2 \beta_{2} + \beta_1 - 1) q^{95} + (8 \beta_{2} - \beta_1 + 1) q^{97} + ( - 4 \beta_{2} + 2 \beta_1 + 1) q^{99}+O(q^{100})$$ q - b1 * q^3 + q^5 + b1 * q^7 + (b2 - 1) * q^9 + (b2 - b1 - 1) * q^11 - b1 * q^15 + (-2*b2 + 3*b1 - 2) * q^17 + (-2*b2 + b1 - 1) * q^19 + (-b2 - 2) * q^21 + (b2 - 1) * q^23 + q^25 + (-b2 + 4*b1 - 1) * q^27 + (-2*b2 - b1 - 1) * q^29 + (2*b2 - 3*b1 + 4) * q^31 + (b1 + 1) * q^33 + b1 * q^35 - 3 * q^37 + (b2 - 4*b1 - 1) * q^41 + (-3*b2 + 5*b1 - 3) * q^43 + (b2 - 1) * q^45 + (-7*b2 + 4*b1 - 7) * q^47 + (b2 - 5) * q^49 + (-b2 + 2*b1 - 4) * q^51 + (7*b2 - b1) * q^53 + (b2 - b1 - 1) * q^55 + (b2 + b1) * q^57 + (2*b2 + 4*b1 + 5) * q^59 + (b2 - b1 - 9) * q^61 + (b2 - b1 + 1) * q^63 + (4*b2 - 4*b1 - 3) * q^67 + (-b2 + b1 - 1) * q^69 + (3*b2 + 2) * q^71 + (-4*b2 - 2*b1 - 6) * q^73 - b1 * q^75 + (-b1 - 1) * q^77 + (3*b2 + 5*b1 - 5) * q^79 + (-6*b2 + b1 - 4) * q^81 + (-3*b2 - 4*b1 + 6) * q^83 + (-2*b2 + 3*b1 - 2) * q^85 + (3*b2 + b1 + 4) * q^87 + (2*b2 + 8*b1 - 5) * q^89 + (b2 - 4*b1 + 4) * q^93 + (-2*b2 + b1 - 1) * q^95 + (8*b2 - b1 + 1) * q^97 + (-4*b2 + 2*b1 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{3} + 3 q^{5} + q^{7} - 4 q^{9}+O(q^{10})$$ 3 * q - q^3 + 3 * q^5 + q^7 - 4 * q^9 $$3 q - q^{3} + 3 q^{5} + q^{7} - 4 q^{9} - 5 q^{11} - q^{15} - q^{17} - 5 q^{21} - 4 q^{23} + 3 q^{25} + 2 q^{27} - 2 q^{29} + 7 q^{31} + 4 q^{33} + q^{35} - 9 q^{37} - 8 q^{41} - q^{43} - 4 q^{45} - 10 q^{47} - 16 q^{49} - 9 q^{51} - 8 q^{53} - 5 q^{55} + 17 q^{59} - 29 q^{61} + q^{63} - 17 q^{67} - q^{69} + 3 q^{71} - 16 q^{73} - q^{75} - 4 q^{77} - 13 q^{79} - 5 q^{81} + 17 q^{83} - q^{85} + 10 q^{87} - 9 q^{89} + 7 q^{93} - 6 q^{97} + 9 q^{99}+O(q^{100})$$ 3 * q - q^3 + 3 * q^5 + q^7 - 4 * q^9 - 5 * q^11 - q^15 - q^17 - 5 * q^21 - 4 * q^23 + 3 * q^25 + 2 * q^27 - 2 * q^29 + 7 * q^31 + 4 * q^33 + q^35 - 9 * q^37 - 8 * q^41 - q^43 - 4 * q^45 - 10 * q^47 - 16 * q^49 - 9 * q^51 - 8 * q^53 - 5 * q^55 + 17 * q^59 - 29 * q^61 + q^63 - 17 * q^67 - q^69 + 3 * q^71 - 16 * q^73 - q^75 - 4 * q^77 - 13 * q^79 - 5 * q^81 + 17 * q^83 - q^85 + 10 * q^87 - 9 * q^89 + 7 * q^93 - 6 * q^97 + 9 * q^99

Basis of coefficient ring in terms of $$\nu = \zeta_{14} + \zeta_{14}^{-1}$$:

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

## 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
 1.80194 0.445042 −1.24698
0 −1.80194 0 1.00000 0 1.80194 0 0.246980 0
1.2 0 −0.445042 0 1.00000 0 0.445042 0 −2.80194 0
1.3 0 1.24698 0 1.00000 0 −1.24698 0 −1.44504 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.l yes 3
13.b even 2 1 3380.2.a.k 3
13.d odd 4 2 3380.2.f.g 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3380.2.a.k 3 13.b even 2 1
3380.2.a.l yes 3 1.a even 1 1 trivial
3380.2.f.g 6 13.d odd 4 2

## Hecke kernels

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

 $$T_{3}^{3} + T_{3}^{2} - 2T_{3} - 1$$ T3^3 + T3^2 - 2*T3 - 1 $$T_{7}^{3} - T_{7}^{2} - 2T_{7} + 1$$ T7^3 - T7^2 - 2*T7 + 1 $$T_{19}^{3} - 7T_{19} - 7$$ T19^3 - 7*T19 - 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} + T^{2} - 2T - 1$$
$5$ $$(T - 1)^{3}$$
$7$ $$T^{3} - T^{2} - 2T + 1$$
$11$ $$T^{3} + 5 T^{2} + 6 T + 1$$
$13$ $$T^{3}$$
$17$ $$T^{3} + T^{2} - 16 T + 13$$
$19$ $$T^{3} - 7T - 7$$
$23$ $$T^{3} + 4 T^{2} + 3 T - 1$$
$29$ $$T^{3} + 2 T^{2} - 15 T + 13$$
$31$ $$T^{3} - 7T^{2} + 7$$
$37$ $$(T + 3)^{3}$$
$41$ $$T^{3} + 8 T^{2} - 9 T - 113$$
$43$ $$T^{3} + T^{2} - 44 T + 83$$
$47$ $$T^{3} + 10 T^{2} - 53 T - 559$$
$53$ $$T^{3} + 8 T^{2} - 79 T - 169$$
$59$ $$T^{3} - 17 T^{2} + 31 T + 41$$
$61$ $$T^{3} + 29 T^{2} + 278 T + 881$$
$67$ $$T^{3} + 17 T^{2} + 59 T - 13$$
$71$ $$T^{3} - 3 T^{2} - 18 T + 13$$
$73$ $$T^{3} + 16 T^{2} + 20 T - 8$$
$79$ $$T^{3} + 13 T^{2} - 58 T - 797$$
$83$ $$T^{3} - 17 T^{2} + 10 T + 587$$
$89$ $$T^{3} + 9 T^{2} - 169 T - 953$$
$97$ $$T^{3} + 6 T^{2} - 121 T - 167$$