# Properties

 Label 3380.2.a.m 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

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$26.9894358832$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.756.1 Defining polynomial: $$x^{3} - 6 x - 2$$ 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

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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$

## 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.

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.m 3
13.b even 2 1 3380.2.a.n 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.c 6
65.g odd 4 2 1300.2.f.e 6
65.k even 4 2 1300.2.d.d 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.f even 4 2
1300.2.d.d 6 65.k 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 1.a even 1 1 trivial
3380.2.a.n 3 13.b even 2 1

## Hecke kernels

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

 $$T_{3}^{3} - 6 T_{3} - 2$$ $$T_{7}^{3} - 12 T_{7} - 12$$ $$T_{19}^{3} - 48 T_{19} - 114$$

## Hecke characteristic polynomials

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