# Properties

 Label 2016.2.s.o Level $2016$ Weight $2$ Character orbit 2016.s Analytic conductor $16.098$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 7x^{2} + 49$$ x^4 + 7*x^2 + 49 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 224) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 \beta_{2} q^{5} + \beta_{3} q^{7}+O(q^{10})$$ q + 3*b2 * q^5 + b3 * q^7 $$q + 3 \beta_{2} q^{5} + \beta_{3} q^{7} - \beta_1 q^{11} - 4 q^{13} + (\beta_{2} + 1) q^{17} + ( - 3 \beta_{3} - 3 \beta_1) q^{19} + (\beta_{3} + \beta_1) q^{23} + ( - 4 \beta_{2} - 4) q^{25} + 4 q^{29} + \beta_1 q^{31} + ( - 3 \beta_{3} - 3 \beta_1) q^{35} - 5 \beta_{2} q^{37} - 8 q^{41} - 4 \beta_{3} q^{43} + ( - \beta_{3} - \beta_1) q^{47} + 7 q^{49} + (7 \beta_{2} + 7) q^{53} - 3 \beta_{3} q^{55} + \beta_1 q^{59} - 5 \beta_{2} q^{61} - 12 \beta_{2} q^{65} - \beta_1 q^{67} + (9 \beta_{2} + 9) q^{73} + (7 \beta_{2} + 7) q^{77} + ( - \beta_{3} - \beta_1) q^{79} - 4 \beta_{3} q^{83} - 3 q^{85} + 9 \beta_{2} q^{89} - 4 \beta_{3} q^{91} + 9 \beta_1 q^{95} - 8 q^{97}+O(q^{100})$$ q + 3*b2 * q^5 + b3 * q^7 - b1 * q^11 - 4 * q^13 + (b2 + 1) * q^17 + (-3*b3 - 3*b1) * q^19 + (b3 + b1) * q^23 + (-4*b2 - 4) * q^25 + 4 * q^29 + b1 * q^31 + (-3*b3 - 3*b1) * q^35 - 5*b2 * q^37 - 8 * q^41 - 4*b3 * q^43 + (-b3 - b1) * q^47 + 7 * q^49 + (7*b2 + 7) * q^53 - 3*b3 * q^55 + b1 * q^59 - 5*b2 * q^61 - 12*b2 * q^65 - b1 * q^67 + (9*b2 + 9) * q^73 + (7*b2 + 7) * q^77 + (-b3 - b1) * q^79 - 4*b3 * q^83 - 3 * q^85 + 9*b2 * q^89 - 4*b3 * q^91 + 9*b1 * q^95 - 8 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{5}+O(q^{10})$$ 4 * q - 6 * q^5 $$4 q - 6 q^{5} - 16 q^{13} + 2 q^{17} - 8 q^{25} + 16 q^{29} + 10 q^{37} - 32 q^{41} + 28 q^{49} + 14 q^{53} + 10 q^{61} + 24 q^{65} + 18 q^{73} + 14 q^{77} - 12 q^{85} - 18 q^{89} - 32 q^{97}+O(q^{100})$$ 4 * q - 6 * q^5 - 16 * q^13 + 2 * q^17 - 8 * q^25 + 16 * q^29 + 10 * q^37 - 32 * q^41 + 28 * q^49 + 14 * q^53 + 10 * q^61 + 24 * q^65 + 18 * q^73 + 14 * q^77 - 12 * q^85 - 18 * q^89 - 32 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7x^{2} + 49$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 7$$ (v^2) / 7 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 7$$ (v^3) / 7
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$7\beta_{2}$$ 7*b2 $$\nu^{3}$$ $$=$$ $$7\beta_{3}$$ 7*b3

## Character values

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

 $$n$$ $$127$$ $$577$$ $$1765$$ $$1793$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$ $$1$$ $$1$$

## 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
 1.32288 + 2.29129i −1.32288 − 2.29129i 1.32288 − 2.29129i −1.32288 + 2.29129i
0 0 0 −1.50000 + 2.59808i 0 −2.64575 0 0 0
289.2 0 0 0 −1.50000 + 2.59808i 0 2.64575 0 0 0
865.1 0 0 0 −1.50000 2.59808i 0 −2.64575 0 0 0
865.2 0 0 0 −1.50000 2.59808i 0 2.64575 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.c even 3 1 inner
28.g odd 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.o 4
3.b odd 2 1 224.2.i.c 4
4.b odd 2 1 inner 2016.2.s.o 4
7.c even 3 1 inner 2016.2.s.o 4
12.b even 2 1 224.2.i.c 4
21.c even 2 1 1568.2.i.p 4
21.g even 6 1 1568.2.a.t 2
21.g even 6 1 1568.2.i.p 4
21.h odd 6 1 224.2.i.c 4
21.h odd 6 1 1568.2.a.m 2
24.f even 2 1 448.2.i.h 4
24.h odd 2 1 448.2.i.h 4
28.g odd 6 1 inner 2016.2.s.o 4
84.h odd 2 1 1568.2.i.p 4
84.j odd 6 1 1568.2.a.t 2
84.j odd 6 1 1568.2.i.p 4
84.n even 6 1 224.2.i.c 4
84.n even 6 1 1568.2.a.m 2
168.s odd 6 1 448.2.i.h 4
168.s odd 6 1 3136.2.a.bv 2
168.v even 6 1 448.2.i.h 4
168.v even 6 1 3136.2.a.bv 2
168.ba even 6 1 3136.2.a.bg 2
168.be odd 6 1 3136.2.a.bg 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.i.c 4 3.b odd 2 1
224.2.i.c 4 12.b even 2 1
224.2.i.c 4 21.h odd 6 1
224.2.i.c 4 84.n even 6 1
448.2.i.h 4 24.f even 2 1
448.2.i.h 4 24.h odd 2 1
448.2.i.h 4 168.s odd 6 1
448.2.i.h 4 168.v even 6 1
1568.2.a.m 2 21.h odd 6 1
1568.2.a.m 2 84.n even 6 1
1568.2.a.t 2 21.g even 6 1
1568.2.a.t 2 84.j odd 6 1
1568.2.i.p 4 21.c even 2 1
1568.2.i.p 4 21.g even 6 1
1568.2.i.p 4 84.h odd 2 1
1568.2.i.p 4 84.j odd 6 1
2016.2.s.o 4 1.a even 1 1 trivial
2016.2.s.o 4 4.b odd 2 1 inner
2016.2.s.o 4 7.c even 3 1 inner
2016.2.s.o 4 28.g odd 6 1 inner
3136.2.a.bg 2 168.ba even 6 1
3136.2.a.bg 2 168.be odd 6 1
3136.2.a.bv 2 168.s odd 6 1
3136.2.a.bv 2 168.v even 6 1

## Hecke kernels

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

 $$T_{5}^{2} + 3T_{5} + 9$$ T5^2 + 3*T5 + 9 $$T_{11}^{4} + 7T_{11}^{2} + 49$$ T11^4 + 7*T11^2 + 49 $$T_{13} + 4$$ T13 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 3 T + 9)^{2}$$
$7$ $$(T^{2} - 7)^{2}$$
$11$ $$T^{4} + 7T^{2} + 49$$
$13$ $$(T + 4)^{4}$$
$17$ $$(T^{2} - T + 1)^{2}$$
$19$ $$T^{4} + 63T^{2} + 3969$$
$23$ $$T^{4} + 7T^{2} + 49$$
$29$ $$(T - 4)^{4}$$
$31$ $$T^{4} + 7T^{2} + 49$$
$37$ $$(T^{2} - 5 T + 25)^{2}$$
$41$ $$(T + 8)^{4}$$
$43$ $$(T^{2} - 112)^{2}$$
$47$ $$T^{4} + 7T^{2} + 49$$
$53$ $$(T^{2} - 7 T + 49)^{2}$$
$59$ $$T^{4} + 7T^{2} + 49$$
$61$ $$(T^{2} - 5 T + 25)^{2}$$
$67$ $$T^{4} + 7T^{2} + 49$$
$71$ $$T^{4}$$
$73$ $$(T^{2} - 9 T + 81)^{2}$$
$79$ $$T^{4} + 7T^{2} + 49$$
$83$ $$(T^{2} - 112)^{2}$$
$89$ $$(T^{2} + 9 T + 81)^{2}$$
$97$ $$(T + 8)^{4}$$