# Properties

 Label 1216.2.c.h Level $1216$ Weight $2$ Character orbit 1216.c Analytic conductor $9.710$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1216.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.70980888579$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.303595776.1 Defining polynomial: $$x^{8} + 5 x^{6} + 16 x^{4} + 45 x^{2} + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + \beta_{1} q^{5} + ( -\beta_{3} + 2 \beta_{6} ) q^{7} + ( -5 + \beta_{7} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} + \beta_{1} q^{5} + ( -\beta_{3} + 2 \beta_{6} ) q^{7} + ( -5 + \beta_{7} ) q^{9} + ( -\beta_{2} - \beta_{4} ) q^{11} + ( 3 \beta_{1} - \beta_{5} ) q^{13} + ( 2 \beta_{3} + 2 \beta_{6} ) q^{15} + 5 q^{17} + \beta_{4} q^{19} + ( -\beta_{1} - 5 \beta_{5} ) q^{21} + ( -\beta_{3} + \beta_{6} ) q^{23} + ( 2 + \beta_{7} ) q^{25} + ( -3 \beta_{2} + 8 \beta_{4} ) q^{27} + ( \beta_{1} - 3 \beta_{5} ) q^{29} + 2 \beta_{3} q^{31} + 8 q^{33} + ( -\beta_{2} + 5 \beta_{4} ) q^{35} + 4 \beta_{1} q^{37} + ( 7 \beta_{3} + 3 \beta_{6} ) q^{39} + ( -\beta_{2} - 7 \beta_{4} ) q^{43} + ( -7 \beta_{1} - 2 \beta_{5} ) q^{45} + ( 4 \beta_{3} - 3 \beta_{6} ) q^{47} + 4 q^{49} + 5 \beta_{2} q^{51} + ( -\beta_{1} + 5 \beta_{5} ) q^{53} + ( -2 \beta_{3} - \beta_{6} ) q^{55} -\beta_{7} q^{57} + ( -\beta_{2} - 2 \beta_{4} ) q^{59} + ( -3 \beta_{1} - 4 \beta_{5} ) q^{61} -11 \beta_{6} q^{63} + ( -8 + 2 \beta_{7} ) q^{65} + ( \beta_{2} + 6 \beta_{4} ) q^{67} + ( \beta_{1} - 3 \beta_{5} ) q^{69} + ( 2 \beta_{3} - 4 \beta_{6} ) q^{71} + ( -5 - 4 \beta_{7} ) q^{73} + ( \beta_{2} + 8 \beta_{4} ) q^{75} + ( -\beta_{1} + 6 \beta_{5} ) q^{77} + ( -6 \beta_{3} + 4 \beta_{6} ) q^{79} + ( 9 - 8 \beta_{7} ) q^{81} -8 \beta_{4} q^{83} + 5 \beta_{1} q^{85} + ( 5 \beta_{3} - 7 \beta_{6} ) q^{87} + ( -2 + 4 \beta_{7} ) q^{89} + ( -\beta_{2} + 16 \beta_{4} ) q^{91} + ( -6 \beta_{1} + 2 \beta_{5} ) q^{93} -\beta_{6} q^{95} + ( 4 + 2 \beta_{7} ) q^{97} + ( 5 \beta_{2} - 3 \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 44q^{9} + O(q^{10})$$ $$8q - 44q^{9} + 40q^{17} + 12q^{25} + 64q^{33} + 32q^{49} + 4q^{57} - 72q^{65} - 24q^{73} + 104q^{81} - 32q^{89} + 24q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 5 x^{6} + 16 x^{4} + 45 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{6} + 4 \nu^{4} + 20 \nu^{2} + 27$$$$)/36$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} + 8 \nu^{5} + 40 \nu^{3} + 165 \nu$$$$)/72$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} + 4 \nu^{5} + 2 \nu^{3} - 9 \nu$$$$)/54$$ $$\beta_{4}$$ $$=$$ $$($$$$5 \nu^{7} + 16 \nu^{5} + 8 \nu^{3} + 81 \nu$$$$)/216$$ $$\beta_{5}$$ $$=$$ $$($$$$5 \nu^{6} + 16 \nu^{4} + 80 \nu^{2} + 153$$$$)/72$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} + 5 \nu^{5} + 16 \nu^{3} + 18 \nu$$$$)/27$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{6} + 5 \nu^{4} + 7 \nu^{2} + 18$$$$)/9$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{6} + \beta_{4} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{7} + 2 \beta_{5} + \beta_{1} - 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{6} - 4 \beta_{4} - \beta_{3}$$ $$\nu^{4}$$ $$=$$ $$($$$$5 \beta_{7} - 6 \beta_{5} + 5 \beta_{1} - 1$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{6} + 15 \beta_{4} + 16 \beta_{3} - \beta_{2}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$8 \beta_{5} - 16 \beta_{1} - 5$$ $$\nu^{7}$$ $$=$$ $$($$$$13 \beta_{6} + 35 \beta_{4} - 48 \beta_{3} - 13 \beta_{2}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$705$$ $$837$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
609.1
 1.26217 − 1.18614i −1.26217 − 1.18614i −0.396143 − 1.68614i 0.396143 − 1.68614i 0.396143 + 1.68614i −0.396143 + 1.68614i −1.26217 + 1.18614i 1.26217 + 1.18614i
0 3.37228i 0 2.52434i 0 −3.31662 0 −8.37228 0
609.2 0 3.37228i 0 2.52434i 0 3.31662 0 −8.37228 0
609.3 0 2.37228i 0 0.792287i 0 3.31662 0 −2.62772 0
609.4 0 2.37228i 0 0.792287i 0 −3.31662 0 −2.62772 0
609.5 0 2.37228i 0 0.792287i 0 −3.31662 0 −2.62772 0
609.6 0 2.37228i 0 0.792287i 0 3.31662 0 −2.62772 0
609.7 0 3.37228i 0 2.52434i 0 3.31662 0 −8.37228 0
609.8 0 3.37228i 0 2.52434i 0 −3.31662 0 −8.37228 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 609.8 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
8.b even 2 1 inner
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.c.h 8
4.b odd 2 1 inner 1216.2.c.h 8
8.b even 2 1 inner 1216.2.c.h 8
8.d odd 2 1 inner 1216.2.c.h 8
16.e even 4 1 4864.2.a.bi 4
16.e even 4 1 4864.2.a.bl 4
16.f odd 4 1 4864.2.a.bi 4
16.f odd 4 1 4864.2.a.bl 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.2.c.h 8 1.a even 1 1 trivial
1216.2.c.h 8 4.b odd 2 1 inner
1216.2.c.h 8 8.b even 2 1 inner
1216.2.c.h 8 8.d odd 2 1 inner
4864.2.a.bi 4 16.e even 4 1
4864.2.a.bi 4 16.f odd 4 1
4864.2.a.bl 4 16.e even 4 1
4864.2.a.bl 4 16.f odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1216, [\chi])$$:

 $$T_{3}^{4} + 17 T_{3}^{2} + 64$$ $$T_{5}^{4} + 7 T_{5}^{2} + 4$$ $$T_{7}^{2} - 11$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 64 + 17 T^{2} + T^{4} )^{2}$$
$5$ $$( 4 + 7 T^{2} + T^{4} )^{2}$$
$7$ $$( -11 + T^{2} )^{4}$$
$11$ $$( 64 + 17 T^{2} + T^{4} )^{2}$$
$13$ $$( 576 + 51 T^{2} + T^{4} )^{2}$$
$17$ $$( -5 + T )^{8}$$
$19$ $$( 1 + T^{2} )^{4}$$
$23$ $$( 4 - 7 T^{2} + T^{4} )^{2}$$
$29$ $$( 256 + 43 T^{2} + T^{4} )^{2}$$
$31$ $$( -12 + T^{2} )^{4}$$
$37$ $$( 1024 + 112 T^{2} + T^{4} )^{2}$$
$41$ $$T^{8}$$
$43$ $$( 1156 + 101 T^{2} + T^{4} )^{2}$$
$47$ $$( 36 - 87 T^{2} + T^{4} )^{2}$$
$53$ $$( 3364 + 127 T^{2} + T^{4} )^{2}$$
$59$ $$( 36 + 21 T^{2} + T^{4} )^{2}$$
$61$ $$( 4356 + 231 T^{2} + T^{4} )^{2}$$
$67$ $$( 484 + 77 T^{2} + T^{4} )^{2}$$
$71$ $$( -44 + T^{2} )^{4}$$
$73$ $$( -123 + 6 T + T^{2} )^{4}$$
$79$ $$( 16 - 184 T^{2} + T^{4} )^{2}$$
$83$ $$( 64 + T^{2} )^{4}$$
$89$ $$( -116 + 8 T + T^{2} )^{4}$$
$97$ $$( -24 - 6 T + T^{2} )^{4}$$