# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$9.70980888579$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.49787136.1 Defining polynomial: $$x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{12}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + \beta_{3} q^{5} + \beta_{2} q^{7} -4 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + \beta_{3} q^{5} + \beta_{2} q^{7} -4 q^{9} -\beta_{7} q^{13} -\beta_{4} q^{15} -3 q^{17} + ( \beta_{1} + \beta_{5} ) q^{19} + \beta_{7} q^{21} + 3 \beta_{2} q^{23} -7 q^{25} -\beta_{1} q^{27} -3 \beta_{7} q^{29} + \beta_{5} q^{35} -4 \beta_{7} q^{37} + 7 \beta_{2} q^{39} -\beta_{6} q^{41} + 3 \beta_{5} q^{43} -4 \beta_{3} q^{45} + 6 q^{49} -3 \beta_{1} q^{51} + 3 \beta_{7} q^{53} + ( -7 - \beta_{6} ) q^{57} + 3 \beta_{1} q^{59} -2 \beta_{3} q^{61} -4 \beta_{2} q^{63} + \beta_{6} q^{65} -5 \beta_{1} q^{67} + 3 \beta_{7} q^{69} + \beta_{4} q^{71} + 7 q^{73} -7 \beta_{1} q^{75} -\beta_{4} q^{79} -5 q^{81} -5 \beta_{5} q^{83} -3 \beta_{3} q^{85} + 21 \beta_{2} q^{87} + \beta_{6} q^{89} + \beta_{1} q^{91} + ( -12 \beta_{2} - \beta_{4} ) q^{95} -\beta_{6} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 32q^{9} + O(q^{10})$$ $$8q - 32q^{9} - 24q^{17} - 56q^{25} + 48q^{49} - 56q^{57} + 56q^{73} - 40q^{81} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2 \nu^{6} - 9$$$$)/5$$ $$\beta_{2}$$ $$=$$ $$($$$$-3 \nu^{7} - 5 \nu^{5} + 5 \nu^{3} - 16 \nu$$$$)/40$$ $$\beta_{3}$$ $$=$$ $$($$$$3 \nu^{6} + 5 \nu^{4} + 15 \nu^{2} + 26$$$$)/5$$ $$\beta_{4}$$ $$=$$ $$-\nu^{6} - 3 \nu^{4} - \nu^{2} - 6$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{5} - \nu^{3} - 8 \nu$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$7 \nu^{7} + 25 \nu^{5} + 55 \nu^{3} + 184 \nu$$$$)/20$$ $$\beta_{7}$$ $$=$$ $$($$$$3 \nu^{7} + 5 \nu^{5} + 11 \nu^{3} + 16 \nu$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-2 \beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{2}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{4} + 3 \beta_{3} + 2 \beta_{1} - 6$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{7} + 5 \beta_{2}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-3 \beta_{4} - \beta_{3} + 6 \beta_{1} - 2$$$$)/8$$ $$\nu^{5}$$ $$=$$ $$($$$$2 \beta_{7} + \beta_{6} + 11 \beta_{5} - 22 \beta_{2}$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$-5 \beta_{1} - 9$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$14 \beta_{7} - 7 \beta_{6} - 13 \beta_{5} - 26 \beta_{2}$$$$)/8$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
607.1
 1.09445 − 0.895644i −1.09445 + 0.895644i 0.228425 + 1.39564i −0.228425 − 1.39564i −0.228425 + 1.39564i 0.228425 − 1.39564i −1.09445 − 0.895644i 1.09445 + 0.895644i
0 2.64575i 0 3.46410i 0 1.00000i 0 −4.00000 0
607.2 0 2.64575i 0 3.46410i 0 1.00000i 0 −4.00000 0
607.3 0 2.64575i 0 3.46410i 0 1.00000i 0 −4.00000 0
607.4 0 2.64575i 0 3.46410i 0 1.00000i 0 −4.00000 0
607.5 0 2.64575i 0 3.46410i 0 1.00000i 0 −4.00000 0
607.6 0 2.64575i 0 3.46410i 0 1.00000i 0 −4.00000 0
607.7 0 2.64575i 0 3.46410i 0 1.00000i 0 −4.00000 0
607.8 0 2.64575i 0 3.46410i 0 1.00000i 0 −4.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 607.8 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
19.b odd 2 1 inner
76.d even 2 1 inner
152.b even 2 1 inner
152.g 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.b.d 8
4.b odd 2 1 inner 1216.2.b.d 8
8.b even 2 1 inner 1216.2.b.d 8
8.d odd 2 1 inner 1216.2.b.d 8
19.b odd 2 1 inner 1216.2.b.d 8
76.d even 2 1 inner 1216.2.b.d 8
152.b even 2 1 inner 1216.2.b.d 8
152.g odd 2 1 inner 1216.2.b.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.2.b.d 8 1.a even 1 1 trivial
1216.2.b.d 8 4.b odd 2 1 inner
1216.2.b.d 8 8.b even 2 1 inner
1216.2.b.d 8 8.d odd 2 1 inner
1216.2.b.d 8 19.b odd 2 1 inner
1216.2.b.d 8 76.d even 2 1 inner
1216.2.b.d 8 152.b even 2 1 inner
1216.2.b.d 8 152.g odd 2 1 inner

## Hecke kernels

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

 $$T_{3}^{2} + 7$$ $$T_{5}^{2} + 12$$ $$T_{13}^{2} - 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 7 + T^{2} )^{4}$$
$5$ $$( 12 + T^{2} )^{4}$$
$7$ $$( 1 + T^{2} )^{4}$$
$11$ $$T^{8}$$
$13$ $$( -7 + T^{2} )^{4}$$
$17$ $$( 3 + T )^{8}$$
$19$ $$( 361 - 10 T^{2} + T^{4} )^{2}$$
$23$ $$( 9 + T^{2} )^{4}$$
$29$ $$( -63 + T^{2} )^{4}$$
$31$ $$T^{8}$$
$37$ $$( -112 + T^{2} )^{4}$$
$41$ $$( 84 + T^{2} )^{4}$$
$43$ $$( -108 + T^{2} )^{4}$$
$47$ $$T^{8}$$
$53$ $$( -63 + T^{2} )^{4}$$
$59$ $$( 63 + T^{2} )^{4}$$
$61$ $$( 48 + T^{2} )^{4}$$
$67$ $$( 175 + T^{2} )^{4}$$
$71$ $$( -84 + T^{2} )^{4}$$
$73$ $$( -7 + T )^{8}$$
$79$ $$( -84 + T^{2} )^{4}$$
$83$ $$( -300 + T^{2} )^{4}$$
$89$ $$( 84 + T^{2} )^{4}$$
$97$ $$( 84 + T^{2} )^{4}$$