# Properties

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

# 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: 8.0.41342275584.5 Defining polynomial: $$x^{8} - 2 x^{7} + 13 x^{6} + 2 x^{5} + 81 x^{4} - 8 x^{3} + 208 x^{2} + 128 x + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{2} + \beta_{4} - \beta_{6} ) q^{3} + ( -1 - \beta_{1} + \beta_{4} + \beta_{5} ) q^{5} + ( -1 + \beta_{3} ) q^{7} + 2 \beta_{2} q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{2} + \beta_{4} - \beta_{6} ) q^{3} + ( -1 - \beta_{1} + \beta_{4} + \beta_{5} ) q^{5} + ( -1 + \beta_{3} ) q^{7} + 2 \beta_{2} q^{9} + \beta_{5} q^{11} + ( -\beta_{1} - 2 \beta_{2} - \beta_{7} ) q^{13} + ( \beta_{1} + \beta_{2} + \beta_{7} ) q^{15} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{17} + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{5} + \beta_{6} ) q^{19} + ( 1 - 2 \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{21} + ( \beta_{1} - 3 \beta_{2} - \beta_{7} ) q^{23} + ( \beta_{1} - \beta_{4} - \beta_{7} ) q^{25} + ( 1 - \beta_{6} ) q^{27} + ( -\beta_{1} - 2 \beta_{2} + 3 \beta_{4} + 2 \beta_{7} ) q^{29} + ( 3 - \beta_{3} - 4 \beta_{6} ) q^{31} + ( -1 + \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{33} + ( 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{5} - 4 \beta_{6} ) q^{35} + ( 1 - \beta_{3} + 2 \beta_{6} ) q^{37} + ( -3 - 2 \beta_{3} + 3 \beta_{5} - 3 \beta_{6} ) q^{39} + ( 3 + 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{41} + ( 5 + \beta_{1} - \beta_{2} - 5 \beta_{4} - \beta_{5} - \beta_{6} ) q^{43} + ( -2 + 2 \beta_{3} + 2 \beta_{6} ) q^{45} + ( \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{7} ) q^{47} + ( 3 - 2 \beta_{3} + 2 \beta_{5} ) q^{49} + ( \beta_{1} + 3 \beta_{2} - 5 \beta_{4} ) q^{51} + ( \beta_{1} - 4 \beta_{2} + 2 \beta_{4} - \beta_{7} ) q^{53} + ( 5 - \beta_{3} - 5 \beta_{4} - \beta_{7} ) q^{55} + ( 1 + \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{57} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{59} + ( \beta_{1} + 4 \beta_{2} - 5 \beta_{4} ) q^{61} + 4 \beta_{1} q^{63} + ( -4 - \beta_{3} - \beta_{5} - 6 \beta_{6} ) q^{65} + ( 4 \beta_{1} + 3 \beta_{2} - 3 \beta_{4} ) q^{67} + ( -7 + \beta_{5} - 4 \beta_{6} ) q^{69} + ( -3 - 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{71} + ( -1 - 2 \beta_{1} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{73} + \beta_{5} q^{75} + ( -1 + \beta_{3} - 2 \beta_{5} - 4 \beta_{6} ) q^{77} + ( 5 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - 5 \beta_{4} - 3 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{79} + ( 1 + 6 \beta_{2} - \beta_{4} + 6 \beta_{6} ) q^{81} + ( 2 \beta_{3} + \beta_{5} ) q^{83} + ( -\beta_{1} + 6 \beta_{2} + 6 \beta_{4} + 3 \beta_{7} ) q^{85} + ( -6 + \beta_{3} - 3 \beta_{5} - 3 \beta_{6} ) q^{87} + ( -\beta_{1} - 2 \beta_{2} + \beta_{7} ) q^{89} + ( -4 \beta_{2} + 10 \beta_{4} + 2 \beta_{7} ) q^{91} + ( 5 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - 5 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{93} + ( 10 + \beta_{1} - \beta_{2} - \beta_{3} - 7 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{95} + ( -5 - 4 \beta_{1} - 6 \beta_{2} + 5 \beta_{4} + 4 \beta_{5} - 6 \beta_{6} ) q^{97} + ( -2 \beta_{4} - 2 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{3} - 2q^{5} - 8q^{7} + O(q^{10})$$ $$8q - 4q^{3} - 2q^{5} - 8q^{7} + 4q^{11} - 2q^{13} + 2q^{15} - 2q^{17} - 2q^{19} + 8q^{21} + 2q^{23} - 2q^{25} + 8q^{27} + 10q^{29} + 24q^{31} - 6q^{33} - 4q^{35} + 8q^{37} - 12q^{39} + 8q^{41} + 18q^{43} - 16q^{45} - 6q^{47} + 32q^{49} - 18q^{51} + 10q^{53} + 20q^{55} + 10q^{57} + 8q^{59} - 18q^{61} + 8q^{63} - 36q^{65} - 4q^{67} - 52q^{69} - 6q^{71} + 4q^{75} - 16q^{77} + 14q^{79} + 4q^{81} + 4q^{83} + 22q^{85} - 60q^{87} - 2q^{89} + 40q^{91} + 16q^{93} + 50q^{95} - 12q^{97} - 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} + 13 x^{6} + 2 x^{5} + 81 x^{4} - 8 x^{3} + 208 x^{2} + 128 x + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-3 \nu^{7} - 212 \nu^{6} + 901 \nu^{5} - 3304 \nu^{4} + 4081 \nu^{3} - 9010 \nu^{2} + 23992 \nu - 15264$$$$)/22304$$ $$\beta_{3}$$ $$=$$ $$($$$$27 \nu^{7} - 142 \nu^{6} + 255 \nu^{5} + 462 \nu^{4} - 1797 \nu^{3} + 1632 \nu^{2} + 1536 \nu + 34384$$$$)/11152$$ $$\beta_{4}$$ $$=$$ $$($$$$-77 \nu^{7} + 244 \nu^{6} - 1037 \nu^{5} + 696 \nu^{4} - 4697 \nu^{3} + 10370 \nu^{2} - 10576 \nu + 17568$$$$)/22304$$ $$\beta_{5}$$ $$=$$ $$($$$$-45 \nu^{7} + 18 \nu^{6} - 425 \nu^{5} - 770 \nu^{4} - 4877 \nu^{3} - 2720 \nu^{2} - 2560 \nu - 9856$$$$)/11152$$ $$\beta_{6}$$ $$=$$ $$($$$$9 \nu^{7} - 10 \nu^{6} + 85 \nu^{5} + 154 \nu^{4} + 473 \nu^{3} + 544 \nu^{2} + 512 \nu + 1728$$$$)/1088$$ $$\beta_{7}$$ $$=$$ $$($$$$241 \nu^{7} - 900 \nu^{6} + 3825 \nu^{5} - 5944 \nu^{4} + 17325 \nu^{3} - 38250 \nu^{2} + 22384 \nu - 64800$$$$)/22304$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} - \beta_{5} + 5 \beta_{4} + \beta_{3} + \beta_{1} - 5$$ $$\nu^{3}$$ $$=$$ $$-4 \beta_{6} - 7 \beta_{5} + 2 \beta_{3} - 6$$ $$\nu^{4}$$ $$=$$ $$-13 \beta_{7} - 41 \beta_{4} + 8 \beta_{2} - 15 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$-36 \beta_{7} + 52 \beta_{6} + 69 \beta_{5} - 96 \beta_{4} - 36 \beta_{3} + 52 \beta_{2} - 69 \beta_{1} + 96$$ $$\nu^{6}$$ $$=$$ $$144 \beta_{6} + 201 \beta_{5} - 157 \beta_{3} + 433$$ $$\nu^{7}$$ $$=$$ $$502 \beta_{7} + 1306 \beta_{4} - 628 \beta_{2} + 791 \beta_{1}$$

## Character values

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

 $$n$$ $$191$$ $$705$$ $$837$$ $$\chi(n)$$ $$1$$ $$-\beta_{4}$$ $$1$$

## 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
 −0.564469 + 0.977689i 1.77158 − 3.06846i −1.10890 + 1.92067i 0.901794 − 1.56195i −0.564469 − 0.977689i 1.77158 + 3.06846i −1.10890 − 1.92067i 0.901794 + 1.56195i
0 −1.20711 2.09077i 0 −1.06447 1.84371i 0 1.59656 0 −1.41421 + 2.44949i 0
577.2 0 −1.20711 2.09077i 0 1.27158 + 2.20243i 0 −5.01077 0 −1.41421 + 2.44949i 0
577.3 0 0.207107 + 0.358719i 0 −1.60890 2.78670i 0 −3.13644 0 1.41421 2.44949i 0
577.4 0 0.207107 + 0.358719i 0 0.401794 + 0.695928i 0 2.55066 0 1.41421 2.44949i 0
961.1 0 −1.20711 + 2.09077i 0 −1.06447 + 1.84371i 0 1.59656 0 −1.41421 2.44949i 0
961.2 0 −1.20711 + 2.09077i 0 1.27158 2.20243i 0 −5.01077 0 −1.41421 2.44949i 0
961.3 0 0.207107 0.358719i 0 −1.60890 + 2.78670i 0 −3.13644 0 1.41421 + 2.44949i 0
961.4 0 0.207107 0.358719i 0 0.401794 0.695928i 0 2.55066 0 1.41421 + 2.44949i 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
19.c even 3 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{3}^{4} + 2 T_{3}^{3} + 5 T_{3}^{2} - 2 T_{3} + 1$$ $$T_{5}^{8} + \cdots$$ $$T_{7}^{4} + 4 T_{7}^{3} - 14 T_{7}^{2} - 32 T_{7} + 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 1 - 2 T + 5 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$5$ $$196 - 168 T + 270 T^{2} + 52 T^{3} + 91 T^{4} + 6 T^{5} + 13 T^{6} + 2 T^{7} + T^{8}$$
$7$ $$( 64 - 32 T - 14 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$11$ $$( 16 + 8 T - 9 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$13$ $$8464 + 11408 T + 12156 T^{2} + 4708 T^{3} + 1565 T^{4} + 178 T^{5} + 39 T^{6} + 2 T^{7} + T^{8}$$
$17$ $$33856 + 30912 T + 21048 T^{2} + 7288 T^{3} + 2041 T^{4} + 258 T^{5} + 43 T^{6} + 2 T^{7} + T^{8}$$
$19$ $$130321 + 13718 T - 6137 T^{2} - 646 T^{3} + 4 T^{4} - 34 T^{5} - 17 T^{6} + 2 T^{7} + T^{8}$$
$23$ $$42436 - 51912 T + 51350 T^{2} - 15692 T^{3} + 4191 T^{4} - 386 T^{5} + 63 T^{6} - 2 T^{7} + T^{8}$$
$29$ $$1119364 - 782920 T + 478830 T^{2} - 69260 T^{3} + 12683 T^{4} - 830 T^{5} + 165 T^{6} - 10 T^{7} + T^{8}$$
$31$ $$( -392 + 328 T - 14 T^{2} - 12 T^{3} + T^{4} )^{2}$$
$37$ $$( -16 + 64 T - 38 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$41$ $$25921 + 19320 T + 17298 T^{2} + 416 T^{3} + 1123 T^{4} - 96 T^{5} + 82 T^{6} - 8 T^{7} + T^{8}$$
$43$ $$64 - 1536 T + 36040 T^{2} - 19488 T^{3} + 7145 T^{4} - 1470 T^{5} + 221 T^{6} - 18 T^{7} + T^{8}$$
$47$ $$31684 + 24208 T + 15826 T^{2} + 4176 T^{3} + 1219 T^{4} + 182 T^{5} + 51 T^{6} + 6 T^{7} + T^{8}$$
$53$ $$795664 - 567312 T + 359004 T^{2} - 50276 T^{3} + 9853 T^{4} - 762 T^{5} + 151 T^{6} - 10 T^{7} + T^{8}$$
$59$ $$47089 + 26040 T + 17438 T^{2} + 1792 T^{3} + 939 T^{4} - 128 T^{5} + 78 T^{6} - 8 T^{7} + T^{8}$$
$61$ $$103684 + 47656 T + 42190 T^{2} + 2268 T^{3} + 6955 T^{4} + 1430 T^{5} + 261 T^{6} + 18 T^{7} + T^{8}$$
$67$ $$27636049 - 1282708 T + 848086 T^{2} - 5456 T^{3} + 18219 T^{4} - 112 T^{5} + 166 T^{6} + 4 T^{7} + T^{8}$$
$71$ $$11262736 + 4846064 T + 1585092 T^{2} + 255428 T^{3} + 34221 T^{4} + 1994 T^{5} + 185 T^{6} + 6 T^{7} + T^{8}$$
$73$ $$124609 + 34594 T^{2} + 9251 T^{4} + 98 T^{6} + T^{8}$$
$79$ $$33362176 - 11413376 T + 3205680 T^{2} - 400824 T^{3} + 48081 T^{4} - 2258 T^{5} + 317 T^{6} - 14 T^{7} + T^{8}$$
$83$ $$( 1052 + 20 T - 77 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$89$ $$3136 - 4928 T + 5560 T^{2} - 3208 T^{3} + 1401 T^{4} - 254 T^{5} + 43 T^{6} + 2 T^{7} + T^{8}$$
$97$ $$508369 - 664516 T + 984130 T^{2} + 133872 T^{3} + 36715 T^{4} - 80 T^{5} + 306 T^{6} + 12 T^{7} + T^{8}$$