Properties

 Label 1216.2.a.v Level $1216$ Weight $2$ Character orbit 1216.a Self dual yes Analytic conductor $9.710$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1216.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$9.70980888579$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.961.1 Defining polynomial: $$x^{3} - x^{2} - 10 x + 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 152) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + \beta_{2} q^{5} + ( -2 + \beta_{1} - \beta_{2} ) q^{7} + ( 5 - \beta_{1} + 2 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + \beta_{2} q^{5} + ( -2 + \beta_{1} - \beta_{2} ) q^{7} + ( 5 - \beta_{1} + 2 \beta_{2} ) q^{9} + ( -2 - \beta_{2} ) q^{11} + ( -2 + \beta_{1} ) q^{13} + ( 4 + 2 \beta_{2} ) q^{15} + ( \beta_{1} - \beta_{2} ) q^{17} - q^{19} + ( 4 - 3 \beta_{1} ) q^{21} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{23} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{25} + ( 3 \beta_{1} + 2 \beta_{2} ) q^{27} + ( 2 + \beta_{1} - 2 \beta_{2} ) q^{29} + ( -4 - 2 \beta_{1} - 2 \beta_{2} ) q^{33} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{35} + 2 q^{37} + ( 8 - 3 \beta_{1} + 2 \beta_{2} ) q^{39} + ( 2 + 2 \beta_{1} ) q^{41} + ( 6 - 2 \beta_{1} - \beta_{2} ) q^{43} + ( 8 + 4 \beta_{1} + \beta_{2} ) q^{45} + ( 2 - 2 \beta_{1} + 3 \beta_{2} ) q^{47} + ( 3 - 3 \beta_{1} + \beta_{2} ) q^{49} + ( 4 - \beta_{1} ) q^{51} + ( -2 + \beta_{1} - 4 \beta_{2} ) q^{53} + ( -6 - 2 \beta_{1} - \beta_{2} ) q^{55} -\beta_{1} q^{57} + ( -8 + \beta_{1} ) q^{59} + ( -2 \beta_{1} + \beta_{2} ) q^{61} + ( -18 + 4 \beta_{1} - 3 \beta_{2} ) q^{63} + 4 q^{65} + ( 4 + \beta_{1} - 2 \beta_{2} ) q^{67} + ( 16 - 3 \beta_{1} + 2 \beta_{2} ) q^{69} + ( 4 - 2 \beta_{1} - 2 \beta_{2} ) q^{71} + ( 3 \beta_{1} - \beta_{2} ) q^{73} + ( 12 - \beta_{1} + 2 \beta_{2} ) q^{75} + ( 6 + \beta_{2} ) q^{77} + ( -8 - 2 \beta_{1} ) q^{79} + ( 17 + 4 \beta_{2} ) q^{81} + ( -4 + 2 \beta_{1} - 4 \beta_{2} ) q^{83} + ( -2 - 2 \beta_{1} + 3 \beta_{2} ) q^{85} + ( \beta_{1} - 2 \beta_{2} ) q^{87} + ( 6 - 2 \beta_{1} - 2 \beta_{2} ) q^{89} + ( 8 - 5 \beta_{1} + 2 \beta_{2} ) q^{91} -\beta_{2} q^{95} + ( -2 + 2 \beta_{2} ) q^{97} + ( -18 - 2 \beta_{1} - 5 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + q^{3} - q^{5} - 4q^{7} + 12q^{9} + O(q^{10})$$ $$3q + q^{3} - q^{5} - 4q^{7} + 12q^{9} - 5q^{11} - 5q^{13} + 10q^{15} + 2q^{17} - 3q^{19} + 9q^{21} + 5q^{23} + 6q^{25} + q^{27} + 9q^{29} - 12q^{33} - 9q^{35} + 6q^{37} + 19q^{39} + 8q^{41} + 17q^{43} + 27q^{45} + q^{47} + 5q^{49} + 11q^{51} - q^{53} - 19q^{55} - q^{57} - 23q^{59} - 3q^{61} - 47q^{63} + 12q^{65} + 15q^{67} + 43q^{69} + 12q^{71} + 4q^{73} + 33q^{75} + 17q^{77} - 26q^{79} + 47q^{81} - 6q^{83} - 11q^{85} + 3q^{87} + 18q^{89} + 17q^{91} + q^{95} - 8q^{97} - 51q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 10 x + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{2} + \nu - 8$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2} - \beta_{1} + 8$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −3.08387 0.786802 3.29707
0 −3.08387 0 −0.786802 0 −4.29707 0 6.51027 0
1.2 0 0.786802 0 −3.29707 0 2.08387 0 −2.38094 0
1.3 0 3.29707 0 3.08387 0 −1.78680 0 7.87067 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$19$$ $$1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.a.v 3
4.b odd 2 1 1216.2.a.u 3
8.b even 2 1 304.2.a.g 3
8.d odd 2 1 152.2.a.c 3
24.f even 2 1 1368.2.a.n 3
24.h odd 2 1 2736.2.a.bd 3
40.e odd 2 1 3800.2.a.r 3
40.f even 2 1 7600.2.a.bv 3
40.k even 4 2 3800.2.d.j 6
56.e even 2 1 7448.2.a.bf 3
152.b even 2 1 2888.2.a.o 3
152.g odd 2 1 5776.2.a.bp 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.a.c 3 8.d odd 2 1
304.2.a.g 3 8.b even 2 1
1216.2.a.u 3 4.b odd 2 1
1216.2.a.v 3 1.a even 1 1 trivial
1368.2.a.n 3 24.f even 2 1
2736.2.a.bd 3 24.h odd 2 1
2888.2.a.o 3 152.b even 2 1
3800.2.a.r 3 40.e odd 2 1
3800.2.d.j 6 40.k even 4 2
5776.2.a.bp 3 152.g odd 2 1
7448.2.a.bf 3 56.e even 2 1
7600.2.a.bv 3 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1216))$$:

 $$T_{3}^{3} - T_{3}^{2} - 10 T_{3} + 8$$ $$T_{5}^{3} + T_{5}^{2} - 10 T_{5} - 8$$ $$T_{7}^{3} + 4 T_{7}^{2} - 5 T_{7} - 16$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$8 - 10 T - T^{2} + T^{3}$$
$5$ $$-8 - 10 T + T^{2} + T^{3}$$
$7$ $$-16 - 5 T + 4 T^{2} + T^{3}$$
$11$ $$-8 - 2 T + 5 T^{2} + T^{3}$$
$13$ $$-8 - 2 T + 5 T^{2} + T^{3}$$
$17$ $$2 - 9 T - 2 T^{2} + T^{3}$$
$19$ $$( 1 + T )^{3}$$
$23$ $$256 - 64 T - 5 T^{2} + T^{3}$$
$29$ $$4 - 4 T - 9 T^{2} + T^{3}$$
$31$ $$T^{3}$$
$37$ $$( -2 + T )^{3}$$
$41$ $$128 - 20 T - 8 T^{2} + T^{3}$$
$43$ $$368 + 24 T - 17 T^{2} + T^{3}$$
$47$ $$256 - 72 T - T^{2} + T^{3}$$
$53$ $$-256 - 134 T + T^{2} + T^{3}$$
$59$ $$376 + 166 T + 23 T^{2} + T^{3}$$
$61$ $$-92 - 28 T + 3 T^{2} + T^{3}$$
$67$ $$-32 + 44 T - 15 T^{2} + T^{3}$$
$71$ $$928 - 76 T - 12 T^{2} + T^{3}$$
$73$ $$326 - 67 T - 4 T^{2} + T^{3}$$
$79$ $$256 + 184 T + 26 T^{2} + T^{3}$$
$83$ $$-736 - 112 T + 6 T^{2} + T^{3}$$
$89$ $$1024 - 16 T - 18 T^{2} + T^{3}$$
$97$ $$-128 - 20 T + 8 T^{2} + T^{3}$$