# Properties

 Label 1216.3.e.n Level $1216$ Weight $3$ Character orbit 1216.e Analytic conductor $33.134$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1216.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$33.1336001462$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 34 x^{6} + 345 x^{4} + 1064 x^{2} + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: no (minimal twist has level 152) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( 2 + \beta_{4} ) q^{5} + ( 1 - \beta_{3} ) q^{7} + ( -\beta_{4} + \beta_{6} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( 2 + \beta_{4} ) q^{5} + ( 1 - \beta_{3} ) q^{7} + ( -\beta_{4} + \beta_{6} ) q^{9} + ( -4 - \beta_{4} + 2 \beta_{6} ) q^{11} + ( -2 \beta_{1} + \beta_{7} ) q^{13} + ( 4 \beta_{1} - \beta_{2} ) q^{15} + ( -3 - 2 \beta_{4} + \beta_{6} ) q^{17} + ( 2 + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{19} + ( 2 \beta_{1} + \beta_{2} + 6 \beta_{5} + \beta_{7} ) q^{21} + ( -1 - \beta_{3} - \beta_{4} - 2 \beta_{6} ) q^{23} + ( 5 + 2 \beta_{3} + \beta_{4} - 4 \beta_{6} ) q^{25} + ( 5 \beta_{1} + \beta_{2} + 2 \beta_{5} ) q^{27} + ( -6 \beta_{1} + \beta_{2} - 4 \beta_{5} + \beta_{7} ) q^{29} + ( -4 \beta_{1} + \beta_{2} - 2 \beta_{7} ) q^{31} + ( -10 \beta_{1} + \beta_{2} + 4 \beta_{5} ) q^{33} + ( -6 - 4 \beta_{3} - \beta_{4} + 2 \beta_{6} ) q^{35} + ( 2 \beta_{1} + 3 \beta_{2} + 6 \beta_{5} ) q^{37} + ( 13 + 3 \beta_{3} - \beta_{4} - 2 \beta_{6} ) q^{39} + ( 8 \beta_{1} - \beta_{2} + 4 \beta_{5} + 2 \beta_{7} ) q^{41} + ( 10 - 4 \beta_{3} + 5 \beta_{4} ) q^{43} + ( -22 - 2 \beta_{3} - 3 \beta_{4} + 6 \beta_{6} ) q^{45} + ( -4 - 6 \beta_{3} - 3 \beta_{4} + 8 \beta_{6} ) q^{47} + ( 6 - 8 \beta_{3} + 6 \beta_{4} + \beta_{6} ) q^{49} + ( -9 \beta_{1} + 2 \beta_{2} + 2 \beta_{5} ) q^{51} + ( 8 \beta_{1} - \beta_{2} - 4 \beta_{5} + \beta_{7} ) q^{53} + ( -26 - 2 \beta_{3} - 11 \beta_{4} + 8 \beta_{6} ) q^{55} + ( 1 + 2 \beta_{1} + \beta_{2} + 6 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} ) q^{57} + ( -13 \beta_{1} - 3 \beta_{2} - 20 \beta_{5} ) q^{59} + ( 18 + 6 \beta_{3} + 5 \beta_{4} + 6 \beta_{6} ) q^{61} + ( 2 + 2 \beta_{3} + 3 \beta_{4} - 6 \beta_{6} ) q^{63} + ( -2 \beta_{1} + 4 \beta_{2} - 4 \beta_{5} + 2 \beta_{7} ) q^{65} + ( -9 \beta_{1} + 16 \beta_{5} + 2 \beta_{7} ) q^{67} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{5} + \beta_{7} ) q^{69} + ( -16 \beta_{1} - 16 \beta_{5} + 2 \beta_{7} ) q^{71} + ( -21 - 6 \beta_{3} + 8 \beta_{4} + 13 \beta_{6} ) q^{73} + ( 13 \beta_{1} - 3 \beta_{2} - 20 \beta_{5} - 2 \beta_{7} ) q^{75} + ( -8 + 6 \beta_{3} + 5 \beta_{4} - 10 \beta_{6} ) q^{77} + ( 14 \beta_{1} + \beta_{2} - 16 \beta_{5} ) q^{79} + ( -37 + 4 \beta_{3} - 6 \beta_{4} + 10 \beta_{6} ) q^{81} + ( -12 + 10 \beta_{3} - 14 \beta_{4} - 8 \beta_{6} ) q^{83} + ( -54 - 4 \beta_{3} - 5 \beta_{4} + 10 \beta_{6} ) q^{85} + ( 45 + \beta_{3} + 11 \beta_{4} - 4 \beta_{6} ) q^{87} + ( 12 \beta_{1} + 5 \beta_{2} + 4 \beta_{7} ) q^{89} + ( 19 \beta_{1} - \beta_{2} + 18 \beta_{5} - 2 \beta_{7} ) q^{91} + ( 50 - 4 \beta_{3} + 18 \beta_{4} - 6 \beta_{6} ) q^{93} + ( -26 - 16 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 7 \beta_{4} + 18 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} ) q^{95} + ( 26 \beta_{1} - 20 \beta_{5} + 4 \beta_{7} ) q^{97} + ( 66 + 6 \beta_{3} + 9 \beta_{4} + 2 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 14q^{5} + 6q^{7} + 4q^{9} + O(q^{10})$$ $$8q + 14q^{5} + 6q^{7} + 4q^{9} - 26q^{11} - 18q^{17} + 16q^{19} - 12q^{23} + 34q^{25} - 50q^{35} + 108q^{39} + 62q^{43} - 162q^{45} - 22q^{47} + 22q^{49} - 174q^{55} + 4q^{57} + 158q^{61} + 2q^{63} - 170q^{73} - 82q^{77} - 256q^{81} - 64q^{83} - 410q^{85} + 332q^{87} + 344q^{93} - 222q^{95} + 526q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 34 x^{6} + 345 x^{4} + 1064 x^{2} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} - 30 \nu^{5} - 205 \nu^{3} + 16 \nu$$$$)/40$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{6} - 20 \nu^{4} - 35 \nu^{2} + 216$$$$)/40$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{6} + 30 \nu^{4} + 205 \nu^{2} + 64$$$$)/40$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + 30 \nu^{5} + 245 \nu^{3} + 504 \nu$$$$)/80$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{6} + 30 \nu^{4} + 245 \nu^{2} + 424$$$$)/40$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{7} - 40 \nu^{5} - 495 \nu^{3} - 1784 \nu$$$$)/40$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} - \beta_{4} - 9$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{5} + \beta_{2} - 13 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-17 \beta_{6} + 21 \beta_{4} + 4 \beta_{3} + 125$$ $$\nu^{5}$$ $$=$$ $$-4 \beta_{7} - 58 \beta_{5} - 25 \beta_{2} + 197 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$305 \beta_{6} - 385 \beta_{4} - 120 \beta_{3} - 1969$$ $$\nu^{7}$$ $$=$$ $$120 \beta_{7} + 1330 \beta_{5} + 505 \beta_{2} - 3229 \beta_{1}$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1025.1
 − 4.27138i − 3.20945i − 2.27869i − 0.512197i 0.512197i 2.27869i 3.20945i 4.27138i
0 4.27138i 0 7.92033 0 −5.75693 0 −9.24469 0
1025.2 0 3.20945i 0 3.06310 0 12.3151 0 −1.30054 0
1025.3 0 2.27869i 0 −6.29008 0 1.03740 0 3.80758 0
1025.4 0 0.512197i 0 2.30665 0 −4.59559 0 8.73765 0
1025.5 0 0.512197i 0 2.30665 0 −4.59559 0 8.73765 0
1025.6 0 2.27869i 0 −6.29008 0 1.03740 0 3.80758 0
1025.7 0 3.20945i 0 3.06310 0 12.3151 0 −1.30054 0
1025.8 0 4.27138i 0 7.92033 0 −5.75693 0 −9.24469 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1025.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.3.e.n 8
4.b odd 2 1 1216.3.e.m 8
8.b even 2 1 304.3.e.g 8
8.d odd 2 1 152.3.e.b 8
19.b odd 2 1 inner 1216.3.e.n 8
24.f even 2 1 1368.3.o.b 8
24.h odd 2 1 2736.3.o.p 8
76.d even 2 1 1216.3.e.m 8
152.b even 2 1 152.3.e.b 8
152.g odd 2 1 304.3.e.g 8
456.l odd 2 1 1368.3.o.b 8
456.p even 2 1 2736.3.o.p 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.3.e.b 8 8.d odd 2 1
152.3.e.b 8 152.b even 2 1
304.3.e.g 8 8.b even 2 1
304.3.e.g 8 152.g odd 2 1
1216.3.e.m 8 4.b odd 2 1
1216.3.e.m 8 76.d even 2 1
1216.3.e.n 8 1.a even 1 1 trivial
1216.3.e.n 8 19.b odd 2 1 inner
1368.3.o.b 8 24.f even 2 1
1368.3.o.b 8 456.l odd 2 1
2736.3.o.p 8 24.h odd 2 1
2736.3.o.p 8 456.p even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1216, [\chi])$$:

 $$T_{3}^{8} + 34 T_{3}^{6} + 345 T_{3}^{4} + 1064 T_{3}^{2} + 256$$ $$T_{5}^{4} - 7 T_{5}^{3} - 34 T_{5}^{2} + 256 T_{5} - 352$$ $$T_{7}^{4} - 3 T_{7}^{3} - 99 T_{7}^{2} - 221 T_{7} + 338$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$256 + 1064 T^{2} + 345 T^{4} + 34 T^{6} + T^{8}$$
$5$ $$( -352 + 256 T - 34 T^{2} - 7 T^{3} + T^{4} )^{2}$$
$7$ $$( 338 - 221 T - 99 T^{2} - 3 T^{3} + T^{4} )^{2}$$
$11$ $$( -5912 - 2260 T - 174 T^{2} + 13 T^{3} + T^{4} )^{2}$$
$13$ $$246866944 + 11290176 T^{2} + 161977 T^{4} + 778 T^{6} + T^{8}$$
$17$ $$( 4814 - 253 T - 195 T^{2} + 9 T^{3} + T^{4} )^{2}$$
$19$ $$16983563041 - 752734096 T + 133448704 T^{2} - 6059024 T^{3} + 479902 T^{4} - 16784 T^{5} + 1024 T^{6} - 16 T^{7} + T^{8}$$
$23$ $$( 27836 + 504 T - 499 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$29$ $$36014930176 + 474986408 T^{2} + 1927129 T^{4} + 2498 T^{6} + T^{8}$$
$31$ $$2723080830976 + 8628732032 T^{2} + 10137808 T^{4} + 5232 T^{6} + T^{8}$$
$37$ $$5146582257664 + 20060072576 T^{2} + 22018512 T^{4} + 8272 T^{6} + T^{8}$$
$41$ $$5382400000000 + 16332880000 T^{2} + 17109456 T^{4} + 7216 T^{6} + T^{8}$$
$43$ $$( 258656 + 7264 T - 1982 T^{2} - 31 T^{3} + T^{4} )^{2}$$
$47$ $$( 2613512 + 84772 T - 6302 T^{2} + 11 T^{3} + T^{4} )^{2}$$
$53$ $$13467138304 + 2071871016 T^{2} + 6302521 T^{4} + 5410 T^{6} + T^{8}$$
$59$ $$247119406081024 + 303682748480 T^{2} + 126378201 T^{4} + 20362 T^{6} + T^{8}$$
$61$ $$( 17033600 + 298720 T - 7306 T^{2} - 79 T^{3} + T^{4} )^{2}$$
$67$ $$273018790144 + 85706332456 T^{2} + 72017257 T^{4} + 15538 T^{6} + T^{8}$$
$71$ $$1596725395456 + 25682937856 T^{2} + 53362320 T^{4} + 14696 T^{6} + T^{8}$$
$73$ $$( 28567486 - 288325 T - 12111 T^{2} + 85 T^{3} + T^{4} )^{2}$$
$79$ $$787748552704 + 110072696960 T^{2} + 103889616 T^{4} + 19312 T^{6} + T^{8}$$
$83$ $$( -454016 - 644096 T - 19300 T^{2} + 32 T^{3} + T^{4} )^{2}$$
$89$ $$682745962430464 + 992323117056 T^{2} + 291139072 T^{4} + 29768 T^{6} + T^{8}$$
$97$ $$19669519455748096 + 7988311560704 T^{2} + 1099281040 T^{4} + 58184 T^{6} + T^{8}$$