# Properties

 Label 2016.2.s.x Level $2016$ Weight $2$ Character orbit 2016.s Analytic conductor $16.098$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2016 = 2^{5} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2016.s (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.0978410475$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.1445900625.1 Defining polynomial: $$x^{8} + 9 x^{6} + 77 x^{4} + 36 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{7} q^{5} + ( -\beta_{1} + \beta_{4} - \beta_{5} ) q^{7} +O(q^{10})$$ $$q + \beta_{7} q^{5} + ( -\beta_{1} + \beta_{4} - \beta_{5} ) q^{7} -\beta_{6} q^{11} + ( 2 - \beta_{2} ) q^{13} + ( -2 \beta_{1} + 2 \beta_{4} ) q^{17} + ( \beta_{1} + \beta_{5} ) q^{19} + ( 2 + 2 \beta_{3} ) q^{23} + ( -\beta_{3} + \beta_{6} ) q^{25} + ( -\beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{29} + ( \beta_{1} - 3 \beta_{4} ) q^{31} + ( -5 + \beta_{2} - 2 \beta_{3} ) q^{35} + ( 2 - \beta_{2} + \beta_{3} + \beta_{6} ) q^{37} + 4 \beta_{5} q^{41} + ( 2 \beta_{4} + 3 \beta_{5} - 2 \beta_{7} ) q^{43} + ( 4 - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{6} ) q^{47} + ( -1 + \beta_{2} + \beta_{3} - 2 \beta_{6} ) q^{49} + ( -4 \beta_{1} + 5 \beta_{4} ) q^{53} + ( -5 \beta_{4} - 2 \beta_{5} + 5 \beta_{7} ) q^{55} + ( -6 \beta_{3} + \beta_{6} ) q^{59} + ( -4 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{6} ) q^{61} + ( -2 \beta_{1} - 2 \beta_{5} + 6 \beta_{7} ) q^{65} + ( \beta_{1} + 4 \beta_{4} ) q^{67} + 2 \beta_{2} q^{71} + ( \beta_{3} + \beta_{6} ) q^{73} + ( -2 \beta_{1} + 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{7} ) q^{77} + ( -3 \beta_{1} - 3 \beta_{5} - 5 \beta_{7} ) q^{79} + ( -5 - \beta_{2} ) q^{83} + ( -6 + 2 \beta_{2} ) q^{85} + 6 \beta_{7} q^{89} + ( \beta_{1} + 6 \beta_{4} + 3 \beta_{5} - 2 \beta_{7} ) q^{91} + 2 \beta_{3} q^{95} + ( -5 - \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + O(q^{10})$$ $$8 q + 2 q^{11} + 12 q^{13} + 8 q^{23} + 2 q^{25} - 28 q^{35} + 6 q^{37} + 12 q^{47} - 4 q^{49} + 22 q^{59} - 12 q^{61} + 8 q^{71} - 6 q^{73} - 44 q^{83} - 40 q^{85} - 8 q^{95} - 44 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 9 x^{6} + 77 x^{4} + 36 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} - 272$$$$)/77$$ $$\beta_{3}$$ $$=$$ $$($$$$-9 \nu^{6} - 77 \nu^{4} - 693 \nu^{2} - 324$$$$)/308$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} + 657 \nu$$$$)/154$$ $$\beta_{5}$$ $$=$$ $$($$$$9 \nu^{7} + 77 \nu^{5} + 693 \nu^{3} + 16 \nu$$$$)/308$$ $$\beta_{6}$$ $$=$$ $$($$$$-41 \nu^{6} - 385 \nu^{4} - 3157 \nu^{2} - 1476$$$$)/308$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{7} + 9 \nu^{5} + 77 \nu^{3} + 36 \nu$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} - 5 \beta_{3} - \beta_{2} - 4$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{7} + 9 \beta_{5} + 2 \beta_{4}$$ $$\nu^{4}$$ $$=$$ $$-9 \beta_{6} + 41 \beta_{3}$$ $$\nu^{5}$$ $$=$$ $$18 \beta_{7} - 77 \beta_{5} - 77 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$77 \beta_{2} + 272$$ $$\nu^{7}$$ $$=$$ $$-154 \beta_{4} + 657 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$577$$ $$1765$$ $$1793$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{3}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 0.342371 + 0.593004i 1.46040 + 2.52950i −1.46040 − 2.52950i −0.342371 − 0.593004i 0.342371 − 0.593004i 1.46040 − 2.52950i −1.46040 + 2.52950i −0.342371 + 0.593004i
0 0 0 −1.46040 + 2.52950i 0 1.80278 + 1.93649i 0 0 0
289.2 0 0 0 −0.342371 + 0.593004i 0 1.80278 1.93649i 0 0 0
289.3 0 0 0 0.342371 0.593004i 0 −1.80278 + 1.93649i 0 0 0
289.4 0 0 0 1.46040 2.52950i 0 −1.80278 1.93649i 0 0 0
865.1 0 0 0 −1.46040 2.52950i 0 1.80278 1.93649i 0 0 0
865.2 0 0 0 −0.342371 0.593004i 0 1.80278 + 1.93649i 0 0 0
865.3 0 0 0 0.342371 + 0.593004i 0 −1.80278 1.93649i 0 0 0
865.4 0 0 0 1.46040 + 2.52950i 0 −1.80278 + 1.93649i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 865.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
12.b even 2 1 inner
84.n even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2016.2.s.x yes 8
3.b odd 2 1 2016.2.s.w 8
4.b odd 2 1 2016.2.s.w 8
7.c even 3 1 inner 2016.2.s.x yes 8
12.b even 2 1 inner 2016.2.s.x yes 8
21.h odd 6 1 2016.2.s.w 8
28.g odd 6 1 2016.2.s.w 8
84.n even 6 1 inner 2016.2.s.x yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2016.2.s.w 8 3.b odd 2 1
2016.2.s.w 8 4.b odd 2 1
2016.2.s.w 8 21.h odd 6 1
2016.2.s.w 8 28.g odd 6 1
2016.2.s.x yes 8 1.a even 1 1 trivial
2016.2.s.x yes 8 7.c even 3 1 inner
2016.2.s.x yes 8 12.b even 2 1 inner
2016.2.s.x yes 8 84.n even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2016, [\chi])$$:

 $$T_{5}^{8} + 9 T_{5}^{6} + 77 T_{5}^{4} + 36 T_{5}^{2} + 16$$ $$T_{11}^{4} - T_{11}^{3} + 17 T_{11}^{2} + 16 T_{11} + 256$$ $$T_{13}^{2} - 3 T_{13} - 14$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$16 + 36 T^{2} + 77 T^{4} + 9 T^{6} + T^{8}$$
$7$ $$( 49 + T^{2} + T^{4} )^{2}$$
$11$ $$( 256 + 16 T + 17 T^{2} - T^{3} + T^{4} )^{2}$$
$13$ $$( -14 - 3 T + T^{2} )^{4}$$
$17$ $$( 400 + 20 T^{2} + T^{4} )^{2}$$
$19$ $$16 + 36 T^{2} + 77 T^{4} + 9 T^{6} + T^{8}$$
$23$ $$( 4 - 2 T + T^{2} )^{4}$$
$29$ $$( 784 - 61 T^{2} + T^{4} )^{2}$$
$31$ $$2401 + 3234 T^{2} + 4307 T^{4} + 66 T^{6} + T^{8}$$
$37$ $$( 196 + 42 T + 23 T^{2} - 3 T^{3} + T^{4} )^{2}$$
$41$ $$( 1024 - 144 T^{2} + T^{4} )^{2}$$
$43$ $$( 784 - 69 T^{2} + T^{4} )^{2}$$
$47$ $$( 3136 + 336 T + 92 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$53$ $$92236816 + 2007236 T^{2} + 34077 T^{4} + 209 T^{6} + T^{8}$$
$59$ $$( 196 - 154 T + 107 T^{2} - 11 T^{3} + T^{4} )^{2}$$
$61$ $$( 3136 - 336 T + 92 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$67$ $$24010000 + 906500 T^{2} + 29325 T^{4} + 185 T^{6} + T^{8}$$
$71$ $$( -64 - 2 T + T^{2} )^{4}$$
$73$ $$( 196 - 42 T + 23 T^{2} + 3 T^{3} + T^{4} )^{2}$$
$79$ $$1698181681 + 17555034 T^{2} + 140267 T^{4} + 426 T^{6} + T^{8}$$
$83$ $$( 14 + 11 T + T^{2} )^{4}$$
$89$ $$26873856 + 1679616 T^{2} + 99792 T^{4} + 324 T^{6} + T^{8}$$
$97$ $$( 14 + 11 T + T^{2} )^{4}$$