# Properties

 Label 4416.2.a.bp Level $4416$ Weight $2$ Character orbit 4416.a Self dual yes Analytic conductor $35.262$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4416 = 2^{6} \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4416.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$35.2619375326$$ Analytic rank: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{3} -\beta_{2} q^{5} + ( 1 + \beta_{1} ) q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} -\beta_{2} q^{5} + ( 1 + \beta_{1} ) q^{7} + q^{9} + ( -1 + \beta_{1} + \beta_{2} ) q^{11} -2 q^{13} + \beta_{2} q^{15} + ( 1 - \beta_{1} ) q^{17} + ( -2 + \beta_{2} ) q^{19} + ( -1 - \beta_{1} ) q^{21} - q^{23} + ( 5 - 2 \beta_{1} - 2 \beta_{2} ) q^{25} - q^{27} -2 \beta_{1} q^{29} + ( -2 - 2 \beta_{1} ) q^{31} + ( 1 - \beta_{1} - \beta_{2} ) q^{33} + ( 2 - 2 \beta_{1} ) q^{35} + ( -5 - \beta_{1} + \beta_{2} ) q^{37} + 2 q^{39} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{41} + ( -2 \beta_{1} - \beta_{2} ) q^{43} -\beta_{2} q^{45} + ( -6 - 2 \beta_{1} ) q^{47} + ( 3 + 2 \beta_{1} + 2 \beta_{2} ) q^{49} + ( -1 + \beta_{1} ) q^{51} + ( 2 + 2 \beta_{1} + 3 \beta_{2} ) q^{53} + ( -8 + 4 \beta_{2} ) q^{55} + ( 2 - \beta_{2} ) q^{57} + ( 4 - 2 \beta_{2} ) q^{59} + ( -7 + \beta_{1} - \beta_{2} ) q^{61} + ( 1 + \beta_{1} ) q^{63} + 2 \beta_{2} q^{65} + ( 2 \beta_{1} + \beta_{2} ) q^{67} + q^{69} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{71} + 2 q^{73} + ( -5 + 2 \beta_{1} + 2 \beta_{2} ) q^{75} + ( 6 + 2 \beta_{1} + 2 \beta_{2} ) q^{77} + ( -9 - \beta_{1} ) q^{79} + q^{81} + ( 1 - \beta_{1} + 3 \beta_{2} ) q^{83} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{85} + 2 \beta_{1} q^{87} + ( -3 - \beta_{1} - 2 \beta_{2} ) q^{89} + ( -2 - 2 \beta_{1} ) q^{91} + ( 2 + 2 \beta_{1} ) q^{93} + ( -10 + 2 \beta_{1} + 4 \beta_{2} ) q^{95} + ( 6 + 2 \beta_{2} ) q^{97} + ( -1 + \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{3} + 2q^{7} + 3q^{9} + O(q^{10})$$ $$3q - 3q^{3} + 2q^{7} + 3q^{9} - 4q^{11} - 6q^{13} + 4q^{17} - 6q^{19} - 2q^{21} - 3q^{23} + 17q^{25} - 3q^{27} + 2q^{29} - 4q^{31} + 4q^{33} + 8q^{35} - 14q^{37} + 6q^{39} - 2q^{41} + 2q^{43} - 16q^{47} + 7q^{49} - 4q^{51} + 4q^{53} - 24q^{55} + 6q^{57} + 12q^{59} - 22q^{61} + 2q^{63} - 2q^{67} + 3q^{69} - 8q^{71} + 6q^{73} - 17q^{75} + 16q^{77} - 26q^{79} + 3q^{81} + 4q^{83} - 8q^{85} - 2q^{87} - 8q^{89} - 4q^{91} + 4q^{93} - 32q^{95} + 18q^{97} - 4q^{99} + O(q^{100})$$

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 −1.00000 0 −3.35026 0 −2.96239 0 1.00000 0
1.2 0 −1.00000 0 −1.07838 0 4.34017 0 1.00000 0
1.3 0 −1.00000 0 4.42864 0 0.622216 0 1.00000 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 4416.2.a.bp 3
4.b odd 2 1 4416.2.a.bs 3
8.b even 2 1 552.2.a.g 3
8.d odd 2 1 1104.2.a.o 3
24.f even 2 1 3312.2.a.bf 3
24.h odd 2 1 1656.2.a.n 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.2.a.g 3 8.b even 2 1
1104.2.a.o 3 8.d odd 2 1
1656.2.a.n 3 24.h odd 2 1
3312.2.a.bf 3 24.f even 2 1
4416.2.a.bp 3 1.a even 1 1 trivial
4416.2.a.bs 3 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4416))$$:

 $$T_{5}^{3} - 16 T_{5} - 16$$ $$T_{7}^{3} - 2 T_{7}^{2} - 12 T_{7} + 8$$ $$T_{11}^{3} + 4 T_{11}^{2} - 16 T_{11} - 32$$ $$T_{17}^{3} - 4 T_{17}^{2} - 8 T_{17} + 16$$ $$T_{19}^{3} + 6 T_{19}^{2} - 4 T_{19} - 8$$

## Hecke characteristic polynomials

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