Properties

 Label 847.2.a.i Level $847$ Weight $2$ Character orbit 847.a Self dual yes Analytic conductor $6.763$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$847 = 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 847.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$6.76332905120$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 Defining polynomial: $$x^{3} - x^{2} - 6 x - 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{2} -\beta_{1} q^{3} + ( 3 + \beta_{2} ) q^{4} -\beta_{2} q^{5} + ( -4 - \beta_{1} - \beta_{2} ) q^{6} - q^{7} + ( -3 + \beta_{1} - 2 \beta_{2} ) q^{8} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{2} -\beta_{1} q^{3} + ( 3 + \beta_{2} ) q^{4} -\beta_{2} q^{5} + ( -4 - \beta_{1} - \beta_{2} ) q^{6} - q^{7} + ( -3 + \beta_{1} - 2 \beta_{2} ) q^{8} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{9} + ( 2 + 2 \beta_{2} ) q^{10} + ( 2 - 3 \beta_{1} + \beta_{2} ) q^{12} + ( -2 - \beta_{1} + \beta_{2} ) q^{13} + ( 1 - \beta_{1} ) q^{14} + ( -2 - \beta_{2} ) q^{15} + ( 5 - 2 \beta_{1} + 3 \beta_{2} ) q^{16} + ( -2 - \beta_{1} + \beta_{2} ) q^{17} + ( 5 + 3 \beta_{1} ) q^{18} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{19} + ( -6 + 2 \beta_{1} - 2 \beta_{2} ) q^{20} + \beta_{1} q^{21} + ( 2 - \beta_{2} ) q^{23} + ( -8 + \beta_{1} - 3 \beta_{2} ) q^{24} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{25} + ( -4 - 3 \beta_{1} - 3 \beta_{2} ) q^{26} + ( -6 - 2 \beta_{1} - \beta_{2} ) q^{27} + ( -3 - \beta_{2} ) q^{28} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{29} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{30} + ( -4 - \beta_{1} ) q^{31} + ( -13 + \beta_{1} - 4 \beta_{2} ) q^{32} + ( -4 - 3 \beta_{1} - 3 \beta_{2} ) q^{34} + \beta_{2} q^{35} + ( 5 + 4 \beta_{1} + \beta_{2} ) q^{36} + ( -6 + 2 \beta_{1} + \beta_{2} ) q^{37} + ( 4 + 2 \beta_{1} - 2 \beta_{2} ) q^{38} + ( 6 + 4 \beta_{1} + 2 \beta_{2} ) q^{39} + ( 14 - 4 \beta_{1} + 2 \beta_{2} ) q^{40} + ( -6 + \beta_{1} - \beta_{2} ) q^{41} + ( 4 + \beta_{1} + \beta_{2} ) q^{42} + ( -2 - 2 \beta_{2} ) q^{43} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{45} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{46} + ( 2 - \beta_{1} + \beta_{2} ) q^{47} + ( 14 - \beta_{1} + 5 \beta_{2} ) q^{48} + q^{49} + ( -7 - \beta_{1} ) q^{50} + ( 6 + 4 \beta_{1} + 2 \beta_{2} ) q^{51} + ( 2 - 5 \beta_{1} + \beta_{2} ) q^{52} + ( -4 - 2 \beta_{2} ) q^{53} -8 \beta_{1} q^{54} + ( 3 - \beta_{1} + 2 \beta_{2} ) q^{56} + ( -4 - 4 \beta_{1} ) q^{57} + ( -4 - 2 \beta_{1} + 2 \beta_{2} ) q^{58} + \beta_{1} q^{59} + ( -12 + 2 \beta_{1} - 4 \beta_{2} ) q^{60} + ( 6 - \beta_{1} + \beta_{2} ) q^{61} + ( -5 \beta_{1} - \beta_{2} ) q^{62} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{63} + ( 15 - 8 \beta_{1} + 3 \beta_{2} ) q^{64} + ( -8 + 2 \beta_{1} + 2 \beta_{2} ) q^{65} + ( -2 + 4 \beta_{1} + \beta_{2} ) q^{67} + ( 2 - 5 \beta_{1} + \beta_{2} ) q^{68} + ( -2 - 2 \beta_{1} - \beta_{2} ) q^{69} + ( -2 - 2 \beta_{2} ) q^{70} + ( 2 + 4 \beta_{1} + 5 \beta_{2} ) q^{71} + ( -1 + 3 \beta_{1} + 2 \beta_{2} ) q^{72} + ( -6 + \beta_{1} - \beta_{2} ) q^{73} + ( 12 - 4 \beta_{1} ) q^{74} + ( 6 + 3 \beta_{1} + \beta_{2} ) q^{75} + ( 8 + 2 \beta_{1} + 2 \beta_{2} ) q^{76} + ( 6 + 10 \beta_{1} ) q^{78} + ( -10 - 2 \beta_{2} ) q^{79} + ( -22 + 6 \beta_{1} - 4 \beta_{2} ) q^{80} + ( 3 + 4 \beta_{1} - 2 \beta_{2} ) q^{81} + ( 12 - 5 \beta_{1} + 3 \beta_{2} ) q^{82} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{83} + ( -2 + 3 \beta_{1} - \beta_{2} ) q^{84} + ( -8 + 2 \beta_{1} + 2 \beta_{2} ) q^{85} + ( 6 - 2 \beta_{1} + 4 \beta_{2} ) q^{86} + ( 4 + 4 \beta_{1} ) q^{87} + ( -8 + 2 \beta_{1} - \beta_{2} ) q^{89} + ( 6 - 2 \beta_{2} ) q^{90} + ( 2 + \beta_{1} - \beta_{2} ) q^{91} + 2 \beta_{1} q^{92} + ( 4 + 6 \beta_{1} + \beta_{2} ) q^{93} + ( -8 + \beta_{1} - 3 \beta_{2} ) q^{94} + ( -8 + 4 \beta_{1} + 4 \beta_{2} ) q^{95} + ( -12 + 11 \beta_{1} - 5 \beta_{2} ) q^{96} + ( -4 - 2 \beta_{1} - 3 \beta_{2} ) q^{97} + ( -1 + \beta_{1} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} - q^{3} + 8 q^{4} + q^{5} - 12 q^{6} - 3 q^{7} - 6 q^{8} + 4 q^{9} + O(q^{10})$$ $$3 q - 2 q^{2} - q^{3} + 8 q^{4} + q^{5} - 12 q^{6} - 3 q^{7} - 6 q^{8} + 4 q^{9} + 4 q^{10} + 2 q^{12} - 8 q^{13} + 2 q^{14} - 5 q^{15} + 10 q^{16} - 8 q^{17} + 18 q^{18} - 14 q^{20} + q^{21} + 7 q^{23} - 20 q^{24} + 2 q^{25} - 12 q^{26} - 19 q^{27} - 8 q^{28} + 8 q^{30} - 13 q^{31} - 34 q^{32} - 12 q^{34} - q^{35} + 18 q^{36} - 17 q^{37} + 16 q^{38} + 20 q^{39} + 36 q^{40} - 16 q^{41} + 12 q^{42} - 4 q^{43} - 6 q^{45} + 4 q^{47} + 36 q^{48} + 3 q^{49} - 22 q^{50} + 20 q^{51} - 10 q^{53} - 8 q^{54} + 6 q^{56} - 16 q^{57} - 16 q^{58} + q^{59} - 30 q^{60} + 16 q^{61} - 4 q^{62} - 4 q^{63} + 34 q^{64} - 24 q^{65} - 3 q^{67} - 7 q^{69} - 4 q^{70} + 5 q^{71} - 2 q^{72} - 16 q^{73} + 32 q^{74} + 20 q^{75} + 24 q^{76} + 28 q^{78} - 28 q^{79} - 56 q^{80} + 15 q^{81} + 28 q^{82} - 8 q^{83} - 2 q^{84} - 24 q^{85} + 12 q^{86} + 16 q^{87} - 21 q^{89} + 20 q^{90} + 8 q^{91} + 2 q^{92} + 17 q^{93} - 20 q^{94} - 24 q^{95} - 20 q^{96} - 11 q^{97} - 2 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 6 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2 \nu - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2 \beta_{1} + 4$$

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
 −1.76156 −0.363328 3.12489
−2.76156 1.76156 5.62620 −2.62620 −4.86464 −1.00000 −10.0140 0.103084 7.25240
1.2 −1.36333 0.363328 −0.141336 3.14134 −0.495336 −1.00000 2.91934 −2.86799 −4.28267
1.3 2.12489 −3.12489 2.51514 0.484862 −6.64002 −1.00000 1.09461 6.76491 1.03028
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$1$$
$$11$$ $$1$$

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 847.2.a.i 3
3.b odd 2 1 7623.2.a.ce 3
7.b odd 2 1 5929.2.a.t 3
11.b odd 2 1 847.2.a.j yes 3
11.c even 5 4 847.2.f.u 12
11.d odd 10 4 847.2.f.t 12
33.d even 2 1 7623.2.a.bz 3
77.b even 2 1 5929.2.a.y 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.a.i 3 1.a even 1 1 trivial
847.2.a.j yes 3 11.b odd 2 1
847.2.f.t 12 11.d odd 10 4
847.2.f.u 12 11.c even 5 4
5929.2.a.t 3 7.b odd 2 1
5929.2.a.y 3 77.b even 2 1
7623.2.a.bz 3 33.d even 2 1
7623.2.a.ce 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(847))$$:

 $$T_{2}^{3} + 2 T_{2}^{2} - 5 T_{2} - 8$$ $$T_{3}^{3} + T_{3}^{2} - 6 T_{3} + 2$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-8 - 5 T + 2 T^{2} + T^{3}$$
$3$ $$2 - 6 T + T^{2} + T^{3}$$
$5$ $$4 - 8 T - T^{2} + T^{3}$$
$7$ $$( 1 + T )^{3}$$
$11$ $$T^{3}$$
$13$ $$-64 + 2 T + 8 T^{2} + T^{3}$$
$17$ $$-64 + 2 T + 8 T^{2} + T^{3}$$
$19$ $$64 - 40 T + T^{3}$$
$23$ $$8 + 8 T - 7 T^{2} + T^{3}$$
$29$ $$-64 - 40 T + T^{3}$$
$31$ $$58 + 50 T + 13 T^{2} + T^{3}$$
$37$ $$16 + 72 T + 17 T^{2} + T^{3}$$
$41$ $$80 + 66 T + 16 T^{2} + T^{3}$$
$43$ $$-32 - 28 T + 4 T^{2} + T^{3}$$
$47$ $$-8 - 14 T - 4 T^{2} + T^{3}$$
$53$ $$-64 + 10 T^{2} + T^{3}$$
$59$ $$-2 - 6 T - T^{2} + T^{3}$$
$61$ $$-80 + 66 T - 16 T^{2} + T^{3}$$
$67$ $$-424 - 88 T + 3 T^{2} + T^{3}$$
$71$ $$1480 - 208 T - 5 T^{2} + T^{3}$$
$73$ $$80 + 66 T + 16 T^{2} + T^{3}$$
$79$ $$512 + 228 T + 28 T^{2} + T^{3}$$
$83$ $$64 - 56 T + 8 T^{2} + T^{3}$$
$89$ $$100 + 104 T + 21 T^{2} + T^{3}$$
$97$ $$-452 - 32 T + 11 T^{2} + T^{3}$$