# Properties

 Label 1216.2.i.p Level $1216$ Weight $2$ Character orbit 1216.i 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.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.70980888579$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.39075800976.1 Defining polynomial: $$x^{8} - x^{7} + 12 x^{6} - 13 x^{5} + 125 x^{4} - 116 x^{3} + 232 x^{2} + 96 x + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 608) 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{2} - \beta_{4} ) q^{3} -\beta_{1} q^{5} + ( -2 + 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{5} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{2} - \beta_{4} ) q^{3} -\beta_{1} q^{5} + ( -2 + 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{5} ) q^{9} + ( -\beta_{2} + \beta_{6} ) q^{11} + ( -1 + \beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{13} + ( 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{15} + ( 2 + \beta_{1} - 2 \beta_{3} ) q^{17} + ( \beta_{4} - \beta_{7} ) q^{19} + ( -\beta_{6} - \beta_{7} ) q^{23} + ( -1 + \beta_{1} + \beta_{3} + \beta_{5} ) q^{25} + ( -3 \beta_{2} + 2 \beta_{6} ) q^{27} + ( -3 + 3 \beta_{1} + 3 \beta_{5} ) q^{29} -2 \beta_{2} q^{31} + ( -3 - 5 \beta_{1} + 3 \beta_{3} ) q^{33} + ( -4 + 2 \beta_{5} ) q^{37} + ( -4 \beta_{2} + \beta_{6} ) q^{39} + ( -1 + \beta_{3} ) q^{41} + ( 2 \beta_{2} - 2 \beta_{4} - \beta_{7} ) q^{43} + ( 12 - 4 \beta_{5} ) q^{45} + ( 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{47} -7 q^{49} + ( -4 \beta_{4} + \beta_{6} + \beta_{7} ) q^{51} + ( 1 - \beta_{1} + 2 \beta_{3} - \beta_{5} ) q^{53} + ( 4 \beta_{2} - 4 \beta_{4} ) q^{55} + ( 10 - 3 \beta_{1} - 2 \beta_{3} - 5 \beta_{5} ) q^{57} + ( -\beta_{2} + \beta_{4} ) q^{59} + ( -1 + \beta_{1} - 4 \beta_{3} + \beta_{5} ) q^{61} + ( 7 - 3 \beta_{5} ) q^{65} + ( -3 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{67} + ( 1 - 3 \beta_{5} ) q^{69} -\beta_{7} q^{71} + ( 5 + 4 \beta_{1} - 5 \beta_{3} ) q^{73} + ( -\beta_{2} + \beta_{6} ) q^{75} + ( 2 \beta_{2} - 2 \beta_{4} + \beta_{7} ) q^{79} + ( -5 - 6 \beta_{1} + 5 \beta_{3} ) q^{81} + ( -\beta_{2} - \beta_{6} ) q^{83} + ( 3 - 3 \beta_{1} + 4 \beta_{3} - 3 \beta_{5} ) q^{85} + ( -6 \beta_{2} + 3 \beta_{6} ) q^{87} + ( -3 + 3 \beta_{1} + 6 \beta_{3} + 3 \beta_{5} ) q^{89} + ( -10 - 4 \beta_{1} + 10 \beta_{3} ) q^{93} + ( -2 \beta_{2} - 2 \beta_{4} - \beta_{7} ) q^{95} + ( -5 + 4 \beta_{1} + 5 \beta_{3} ) q^{97} + ( 10 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 2q^{5} - 12q^{9} + O(q^{10})$$ $$8q - 2q^{5} - 12q^{9} - 10q^{13} + 10q^{17} + 2q^{25} - 6q^{29} - 22q^{33} - 24q^{37} - 4q^{41} + 80q^{45} - 56q^{49} + 10q^{53} + 46q^{57} - 18q^{61} + 44q^{65} - 4q^{69} + 28q^{73} - 32q^{81} + 22q^{85} + 18q^{89} - 48q^{93} - 12q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 12 x^{6} - 13 x^{5} + 125 x^{4} - 116 x^{3} + 232 x^{2} + 96 x + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$83 \nu^{7} - 325 \nu^{6} + 2470 \nu^{5} - 3543 \nu^{4} + 26065 \nu^{3} - 38870 \nu^{2} + 139144 \nu - 11440$$$$)/47664$$ $$\beta_{2}$$ $$=$$ $$($$$$-1067 \nu^{7} - 1445 \nu^{6} - 10864 \nu^{5} - 873 \nu^{4} - 106543 \nu^{3} - 31816 \nu^{2} - 15520 \nu + 330448$$$$)/238320$$ $$\beta_{3}$$ $$=$$ $$($$$$557 \nu^{7} - 865 \nu^{6} + 6574 \nu^{5} - 10377 \nu^{4} + 69373 \nu^{3} - 103454 \nu^{2} + 120040 \nu + 48992$$$$)/79440$$ $$\beta_{4}$$ $$=$$ $$($$$$1138 \nu^{7} - 3140 \nu^{6} + 12941 \nu^{5} - 35178 \nu^{4} + 140612 \nu^{3} - 305041 \nu^{2} + 204320 \nu + 80128$$$$)/119160$$ $$\beta_{5}$$ $$=$$ $$($$$$154 \nu^{7} + 55 \nu^{6} + 1568 \nu^{5} + 126 \nu^{4} + 14456 \nu^{3} + 4592 \nu^{2} + 2240 \nu + 37684$$$$)/14895$$ $$\beta_{6}$$ $$=$$ $$($$$$649 \nu^{7} + 19 \nu^{6} + 6608 \nu^{5} + 531 \nu^{4} + 71561 \nu^{3} + 19352 \nu^{2} + 9440 \nu + 229456$$$$)/47664$$ $$\beta_{7}$$ $$=$$ $$($$$$5429 \nu^{7} - 9055 \nu^{6} + 68818 \nu^{5} - 118329 \nu^{4} + 726211 \nu^{3} - 1082978 \nu^{2} + 1956640 \nu - 318736$$$$)/238320$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} + \beta_{5} + 3 \beta_{4} - 11 \beta_{3} + \beta_{1} - 1$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{6} - 7 \beta_{5} - 4 \beta_{2} + 4$$ $$\nu^{4}$$ $$=$$ $$($$$$-7 \beta_{7} - 37 \beta_{4} + 99 \beta_{3} + 37 \beta_{2} - 9 \beta_{1} - 99$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-75 \beta_{7} - 75 \beta_{6} + 149 \beta_{5} + 79 \beta_{4} + 25 \beta_{3} + 149 \beta_{1} - 149$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-24 \beta_{6} - 55 \beta_{5} - 200 \beta_{2} + 532$$ $$\nu^{7}$$ $$=$$ $$($$$$725 \beta_{7} - 849 \beta_{4} + 7 \beta_{3} + 849 \beta_{2} - 1525 \beta_{1} - 7$$$$)/2$$

## Character values

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

 $$n$$ $$191$$ $$705$$ $$837$$ $$\chi(n)$$ $$1$$ $$-\beta_{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
577.1
 −1.63248 − 2.82754i 1.51772 + 2.62877i −0.236942 − 0.410396i 0.851703 + 1.47519i −1.63248 + 2.82754i 1.51772 − 2.62877i −0.236942 + 0.410396i 0.851703 − 1.47519i
0 −1.59084 2.75542i 0 −1.28078 2.21837i 0 0 0 −3.56155 + 6.16879i 0
577.2 0 −0.684999 1.18645i 0 0.780776 + 1.35234i 0 0 0 0.561553 0.972638i 0
577.3 0 0.684999 + 1.18645i 0 0.780776 + 1.35234i 0 0 0 0.561553 0.972638i 0
577.4 0 1.59084 + 2.75542i 0 −1.28078 2.21837i 0 0 0 −3.56155 + 6.16879i 0
961.1 0 −1.59084 + 2.75542i 0 −1.28078 + 2.21837i 0 0 0 −3.56155 6.16879i 0
961.2 0 −0.684999 + 1.18645i 0 0.780776 1.35234i 0 0 0 0.561553 + 0.972638i 0
961.3 0 0.684999 1.18645i 0 0.780776 1.35234i 0 0 0 0.561553 + 0.972638i 0
961.4 0 1.59084 2.75542i 0 −1.28078 + 2.21837i 0 0 0 −3.56155 6.16879i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 961.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.c even 3 1 inner
76.g odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.i.p 8
4.b odd 2 1 inner 1216.2.i.p 8
8.b even 2 1 608.2.i.d 8
8.d odd 2 1 608.2.i.d 8
19.c even 3 1 inner 1216.2.i.p 8
76.g odd 6 1 inner 1216.2.i.p 8
152.k odd 6 1 608.2.i.d 8
152.p even 6 1 608.2.i.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.2.i.d 8 8.b even 2 1
608.2.i.d 8 8.d odd 2 1
608.2.i.d 8 152.k odd 6 1
608.2.i.d 8 152.p even 6 1
1216.2.i.p 8 1.a even 1 1 trivial
1216.2.i.p 8 4.b odd 2 1 inner
1216.2.i.p 8 19.c even 3 1 inner
1216.2.i.p 8 76.g odd 6 1 inner

## Hecke kernels

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

 $$T_{3}^{8} + 12 T_{3}^{6} + 125 T_{3}^{4} + 228 T_{3}^{2} + 361$$ $$T_{5}^{4} + T_{5}^{3} + 5 T_{5}^{2} - 4 T_{5} + 16$$ $$T_{7}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$361 + 228 T^{2} + 125 T^{4} + 12 T^{6} + T^{8}$$
$5$ $$( 16 - 4 T + 5 T^{2} + T^{3} + T^{4} )^{2}$$
$7$ $$T^{8}$$
$11$ $$( 304 - 37 T^{2} + T^{4} )^{2}$$
$13$ $$( 4 + 10 T + 23 T^{2} + 5 T^{3} + T^{4} )^{2}$$
$17$ $$( 4 - 10 T + 23 T^{2} - 5 T^{3} + T^{4} )^{2}$$
$19$ $$130321 + 12635 T^{2} + 684 T^{4} + 35 T^{6} + T^{8}$$
$23$ $$5776 + 2052 T^{2} + 653 T^{4} + 27 T^{6} + T^{8}$$
$29$ $$( 1296 - 108 T + 45 T^{2} + 3 T^{3} + T^{4} )^{2}$$
$31$ $$( 304 - 48 T^{2} + T^{4} )^{2}$$
$37$ $$( -8 + 6 T + T^{2} )^{4}$$
$41$ $$( 1 + T + T^{2} )^{4}$$
$43$ $$1478656 + 96064 T^{2} + 5025 T^{4} + 79 T^{6} + T^{8}$$
$47$ $$92416 + 21584 T^{2} + 4737 T^{4} + 71 T^{6} + T^{8}$$
$53$ $$( 4 - 10 T + 23 T^{2} - 5 T^{3} + T^{4} )^{2}$$
$59$ $$361 + 228 T^{2} + 125 T^{4} + 12 T^{6} + T^{8}$$
$61$ $$( 256 + 144 T + 65 T^{2} + 9 T^{3} + T^{4} )^{2}$$
$67$ $$47045881 + 1563852 T^{2} + 45125 T^{4} + 228 T^{6} + T^{8}$$
$71$ $$5776 + 2052 T^{2} + 653 T^{4} + 27 T^{6} + T^{8}$$
$73$ $$( 361 + 266 T + 215 T^{2} - 14 T^{3} + T^{4} )^{2}$$
$79$ $$92416 + 21584 T^{2} + 4737 T^{4} + 71 T^{6} + T^{8}$$
$83$ $$( 76 - 41 T^{2} + T^{4} )^{2}$$
$89$ $$( 324 + 162 T + 99 T^{2} - 9 T^{3} + T^{4} )^{2}$$
$97$ $$( 3481 - 354 T + 95 T^{2} + 6 T^{3} + T^{4} )^{2}$$