# Properties

 Label 3312.2.a.bf Level $3312$ Weight $2$ Character orbit 3312.a Self dual yes Analytic conductor $26.446$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3312 = 2^{4} \cdot 3^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3312.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$26.4464531494$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 552) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{5} + ( -1 - \beta_{1} ) q^{7} +O(q^{10})$$ $$q -\beta_{2} q^{5} + ( -1 - \beta_{1} ) q^{7} + ( 1 - \beta_{1} - \beta_{2} ) q^{11} + 2 q^{13} + ( -1 + \beta_{1} ) q^{17} + ( -2 + \beta_{2} ) q^{19} - q^{23} + ( 5 - 2 \beta_{1} - 2 \beta_{2} ) q^{25} -2 \beta_{1} q^{29} + ( 2 + 2 \beta_{1} ) q^{31} + ( -2 + 2 \beta_{1} ) q^{35} + ( 5 + \beta_{1} - \beta_{2} ) q^{37} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{41} + ( -2 \beta_{1} - \beta_{2} ) q^{43} + ( -6 - 2 \beta_{1} ) q^{47} + ( 3 + 2 \beta_{1} + 2 \beta_{2} ) q^{49} + ( 2 + 2 \beta_{1} + 3 \beta_{2} ) q^{53} + ( 8 - 4 \beta_{2} ) q^{55} + ( -4 + 2 \beta_{2} ) q^{59} + ( 7 - \beta_{1} + \beta_{2} ) q^{61} -2 \beta_{2} q^{65} + ( 2 \beta_{1} + \beta_{2} ) q^{67} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{71} + 2 q^{73} + ( 6 + 2 \beta_{1} + 2 \beta_{2} ) q^{77} + ( 9 + \beta_{1} ) q^{79} + ( -1 + \beta_{1} - 3 \beta_{2} ) q^{83} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{85} + ( 3 + \beta_{1} + 2 \beta_{2} ) q^{89} + ( -2 - 2 \beta_{1} ) q^{91} + ( -10 + 2 \beta_{1} + 4 \beta_{2} ) q^{95} + ( 6 + 2 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 2q^{7} + O(q^{10})$$ $$3q - 2q^{7} + 4q^{11} + 6q^{13} - 4q^{17} - 6q^{19} - 3q^{23} + 17q^{25} + 2q^{29} + 4q^{31} - 8q^{35} + 14q^{37} + 2q^{41} + 2q^{43} - 16q^{47} + 7q^{49} + 4q^{53} + 24q^{55} - 12q^{59} + 22q^{61} - 2q^{67} - 8q^{71} + 6q^{73} + 16q^{77} + 26q^{79} - 4q^{83} + 8q^{85} + 8q^{89} - 4q^{91} - 32q^{95} + 18q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu - 1$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{2} - 2 \nu - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 5$$$$)/2$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.48119 2.17009 0.311108
0 0 0 −3.35026 0 2.96239 0 0 0
1.2 0 0 0 −1.07838 0 −4.34017 0 0 0
1.3 0 0 0 4.42864 0 −0.622216 0 0 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$23$$ $$1$$

## 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 3312.2.a.bf 3
3.b odd 2 1 1104.2.a.o 3
4.b odd 2 1 1656.2.a.n 3
12.b even 2 1 552.2.a.g 3
24.f even 2 1 4416.2.a.bp 3
24.h odd 2 1 4416.2.a.bs 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.2.a.g 3 12.b even 2 1
1104.2.a.o 3 3.b odd 2 1
1656.2.a.n 3 4.b odd 2 1
3312.2.a.bf 3 1.a even 1 1 trivial
4416.2.a.bp 3 24.f even 2 1
4416.2.a.bs 3 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3312))$$:

 $$T_{5}^{3} - 16 T_{5} - 16$$ $$T_{7}^{3} + 2 T_{7}^{2} - 12 T_{7} - 8$$ $$T_{11}^{3} - 4 T_{11}^{2} - 16 T_{11} + 32$$ $$T_{13} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3}$$
$5$ $$-16 - 16 T + T^{3}$$
$7$ $$-8 - 12 T + 2 T^{2} + T^{3}$$
$11$ $$32 - 16 T - 4 T^{2} + T^{3}$$
$13$ $$( -2 + T )^{3}$$
$17$ $$-16 - 8 T + 4 T^{2} + T^{3}$$
$19$ $$-8 - 4 T + 6 T^{2} + T^{3}$$
$23$ $$( 1 + T )^{3}$$
$29$ $$40 - 52 T - 2 T^{2} + T^{3}$$
$31$ $$64 - 48 T - 4 T^{2} + T^{3}$$
$37$ $$152 + 28 T - 14 T^{2} + T^{3}$$
$41$ $$104 - 84 T - 2 T^{2} + T^{3}$$
$43$ $$184 - 52 T - 2 T^{2} + T^{3}$$
$47$ $$-128 + 32 T + 16 T^{2} + T^{3}$$
$53$ $$592 - 144 T - 4 T^{2} + T^{3}$$
$59$ $$-64 - 16 T + 12 T^{2} + T^{3}$$
$61$ $$-200 + 124 T - 22 T^{2} + T^{3}$$
$67$ $$-184 - 52 T + 2 T^{2} + T^{3}$$
$71$ $$256 - 128 T + 8 T^{2} + T^{3}$$
$73$ $$( -2 + T )^{3}$$
$79$ $$-536 + 212 T - 26 T^{2} + T^{3}$$
$83$ $$-160 - 176 T + 4 T^{2} + T^{3}$$
$89$ $$304 - 40 T - 8 T^{2} + T^{3}$$
$97$ $$296 + 44 T - 18 T^{2} + T^{3}$$