# Properties

 Label 847.2.a.l Level $847$ Weight $2$ Character orbit 847.a Self dual yes Analytic conductor $6.763$ Analytic rank $1$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.2525.1 Defining polynomial: $$x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 5$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 77) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -1 - \beta_{2} ) q^{3} + ( 1 + \beta_{1} + \beta_{2} ) q^{4} + ( -1 - \beta_{1} ) q^{5} + ( -\beta_{1} - \beta_{3} ) q^{6} - q^{7} + ( 3 + \beta_{2} + \beta_{3} ) q^{8} + ( -1 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -1 - \beta_{2} ) q^{3} + ( 1 + \beta_{1} + \beta_{2} ) q^{4} + ( -1 - \beta_{1} ) q^{5} + ( -\beta_{1} - \beta_{3} ) q^{6} - q^{7} + ( 3 + \beta_{2} + \beta_{3} ) q^{8} + ( -1 + \beta_{2} ) q^{9} + ( -3 - 2 \beta_{1} - \beta_{2} ) q^{10} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{12} + ( -\beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{13} -\beta_{1} q^{14} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{15} + ( -1 + \beta_{1} + 2 \beta_{3} ) q^{16} + ( 1 - \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{17} + ( -\beta_{1} + \beta_{3} ) q^{18} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{19} + ( -4 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{20} + ( 1 + \beta_{2} ) q^{21} + ( -3 - \beta_{2} - 2 \beta_{3} ) q^{23} + ( -4 - \beta_{1} - 3 \beta_{2} ) q^{24} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{25} + ( -1 - \beta_{1} + 3 \beta_{2} ) q^{26} + ( 3 + 4 \beta_{2} ) q^{27} + ( -1 - \beta_{1} - \beta_{2} ) q^{28} -3 \beta_{3} q^{29} + ( 4 + 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{30} + ( \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{31} + ( -1 + 3 \beta_{2} ) q^{32} + ( -4 - 3 \beta_{2} + 2 \beta_{3} ) q^{34} + ( 1 + \beta_{1} ) q^{35} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{36} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{37} + ( -7 - \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{38} + ( 2 - \beta_{1} + \beta_{3} ) q^{39} + ( -4 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{40} + ( -1 - 3 \beta_{2} - 2 \beta_{3} ) q^{41} + ( \beta_{1} + \beta_{3} ) q^{42} + ( 5 + 6 \beta_{2} ) q^{43} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{45} + ( -2 - 3 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{46} + ( -3 - \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{47} + ( 1 - 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{48} + q^{49} + ( 9 + 2 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{50} + ( -4 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{51} + ( -3 + 3 \beta_{2} - \beta_{3} ) q^{52} + ( -4 + 4 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{53} + ( 3 \beta_{1} + 4 \beta_{3} ) q^{54} + ( -3 - \beta_{2} - \beta_{3} ) q^{56} + ( 3 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{57} + ( -3 - 6 \beta_{2} - 3 \beta_{3} ) q^{58} + ( -6 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{59} + ( 6 + 4 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} ) q^{60} + ( -7 + 5 \beta_{1} + \beta_{3} ) q^{61} + ( 2 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{62} + ( 1 - \beta_{2} ) q^{63} + ( 2 - 3 \beta_{1} - \beta_{3} ) q^{64} + ( 1 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{65} + ( -4 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{67} + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{68} + ( 4 + 2 \beta_{1} + 3 \beta_{2} ) q^{69} + ( 3 + 2 \beta_{1} + \beta_{2} ) q^{70} + ( 1 - 2 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} ) q^{71} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{72} + ( -4 + 2 \beta_{1} - 3 \beta_{2} + 5 \beta_{3} ) q^{73} + ( 2 - \beta_{1} - \beta_{2} ) q^{74} + ( -3 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{75} + ( -7 - 4 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} ) q^{76} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{78} + ( 2 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{79} + ( -4 - \beta_{1} - 5 \beta_{2} - 4 \beta_{3} ) q^{80} + ( -4 - 6 \beta_{2} ) q^{81} + ( -2 - \beta_{1} - 4 \beta_{2} - 5 \beta_{3} ) q^{82} + ( -4 + 3 \beta_{1} - 7 \beta_{2} + 5 \beta_{3} ) q^{83} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{84} + ( 3 + \beta_{1} - \beta_{3} ) q^{85} + ( 5 \beta_{1} + 6 \beta_{3} ) q^{86} + 3 \beta_{1} q^{87} + ( -4 - \beta_{1} - 3 \beta_{2} + 5 \beta_{3} ) q^{89} + ( 2 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{90} + ( \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{91} + ( -6 - 5 \beta_{1} - 7 \beta_{2} - 3 \beta_{3} ) q^{92} + ( -3 - \beta_{3} ) q^{93} + ( 1 - 4 \beta_{1} + 7 \beta_{2} + 2 \beta_{3} ) q^{94} + ( 6 + 3 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} ) q^{95} + ( -2 + \beta_{2} ) q^{96} + ( -6 - 2 \beta_{2} - 5 \beta_{3} ) q^{97} + \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 2 q^{3} + 4 q^{4} - 6 q^{5} - q^{6} - 4 q^{7} + 9 q^{8} - 6 q^{9} + O(q^{10})$$ $$4 q + 2 q^{2} - 2 q^{3} + 4 q^{4} - 6 q^{5} - q^{6} - 4 q^{7} + 9 q^{8} - 6 q^{9} - 14 q^{10} - 7 q^{12} - 2 q^{14} + 3 q^{15} - 4 q^{16} - 3 q^{17} - 3 q^{18} + 3 q^{19} - 17 q^{20} + 2 q^{21} - 8 q^{23} - 12 q^{24} - 12 q^{26} + 4 q^{27} - 4 q^{28} + 3 q^{29} + 12 q^{30} - 3 q^{31} - 10 q^{32} - 12 q^{34} + 6 q^{35} - q^{36} - 7 q^{37} - 20 q^{38} + 5 q^{39} - 13 q^{40} + 4 q^{41} + q^{42} + 8 q^{43} + 9 q^{45} - 3 q^{46} - 14 q^{47} - 3 q^{48} + 4 q^{49} + 33 q^{50} - 11 q^{51} - 17 q^{52} - 9 q^{53} + 2 q^{54} - 9 q^{56} + 6 q^{57} + 3 q^{58} - 25 q^{59} + 21 q^{60} - 19 q^{61} + 10 q^{62} + 6 q^{63} + 3 q^{64} + 12 q^{65} - 15 q^{67} - q^{68} + 14 q^{69} + 14 q^{70} - 7 q^{71} - 6 q^{72} - 11 q^{73} + 8 q^{74} - 5 q^{75} - 26 q^{76} - 9 q^{78} + 8 q^{79} - 4 q^{80} - 4 q^{81} + 3 q^{82} - q^{83} + 7 q^{84} + 15 q^{85} + 4 q^{86} + 6 q^{87} - 17 q^{89} + 16 q^{90} - 17 q^{92} - 11 q^{93} - 20 q^{94} + 17 q^{95} - 10 q^{96} - 15 q^{97} + 2 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 5$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 3 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4 \beta_{1} + 3$$

## 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
 −1.46673 −0.777484 1.77748 2.46673
−1.46673 −1.61803 0.151302 0.466732 2.37322 −1.00000 2.71154 −0.381966 −0.684570
1.2 −0.777484 0.618034 −1.39552 −0.222516 −0.480512 −1.00000 2.63996 −2.61803 0.173002
1.3 1.77748 0.618034 1.15945 −2.77748 1.09855 −1.00000 −1.49406 −2.61803 −4.93693
1.4 2.46673 −1.61803 4.08477 −3.46673 −3.99126 −1.00000 5.14256 −0.381966 −8.55150
 $$n$$: 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.l 4
3.b odd 2 1 7623.2.a.ch 4
7.b odd 2 1 5929.2.a.bi 4
11.b odd 2 1 847.2.a.k 4
11.c even 5 2 77.2.f.a 8
11.c even 5 2 847.2.f.p 8
11.d odd 10 2 847.2.f.q 8
11.d odd 10 2 847.2.f.s 8
33.d even 2 1 7623.2.a.co 4
33.h odd 10 2 693.2.m.g 8
77.b even 2 1 5929.2.a.bb 4
77.j odd 10 2 539.2.f.d 8
77.m even 15 4 539.2.q.c 16
77.p odd 30 4 539.2.q.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.f.a 8 11.c even 5 2
539.2.f.d 8 77.j odd 10 2
539.2.q.b 16 77.p odd 30 4
539.2.q.c 16 77.m even 15 4
693.2.m.g 8 33.h odd 10 2
847.2.a.k 4 11.b odd 2 1
847.2.a.l 4 1.a even 1 1 trivial
847.2.f.p 8 11.c even 5 2
847.2.f.q 8 11.d odd 10 2
847.2.f.s 8 11.d odd 10 2
5929.2.a.bb 4 77.b even 2 1
5929.2.a.bi 4 7.b odd 2 1
7623.2.a.ch 4 3.b odd 2 1
7623.2.a.co 4 33.d even 2 1

## Hecke kernels

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

 $$T_{2}^{4} - 2 T_{2}^{3} - 4 T_{2}^{2} + 5 T_{2} + 5$$ $$T_{3}^{2} + T_{3} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$5 + 5 T - 4 T^{2} - 2 T^{3} + T^{4}$$
$3$ $$( -1 + T + T^{2} )^{2}$$
$5$ $$-1 - 3 T + 8 T^{2} + 6 T^{3} + T^{4}$$
$7$ $$( 1 + T )^{4}$$
$11$ $$T^{4}$$
$13$ $$-29 - 65 T - 32 T^{2} + T^{4}$$
$17$ $$-71 - 91 T - 23 T^{2} + 3 T^{3} + T^{4}$$
$19$ $$-145 + 135 T - 29 T^{2} - 3 T^{3} + T^{4}$$
$23$ $$-205 - 150 T - 9 T^{2} + 8 T^{3} + T^{4}$$
$29$ $$405 - 54 T^{2} - 3 T^{3} + T^{4}$$
$31$ $$5 + 20 T - 24 T^{2} + 3 T^{3} + T^{4}$$
$37$ $$-5 - 10 T + 6 T^{2} + 7 T^{3} + T^{4}$$
$41$ $$79 + 22 T - 57 T^{2} - 4 T^{3} + T^{4}$$
$43$ $$( -41 - 4 T + T^{2} )^{2}$$
$47$ $$-305 - 525 T - 26 T^{2} + 14 T^{3} + T^{4}$$
$53$ $$-869 - 706 T - 104 T^{2} + 9 T^{3} + T^{4}$$
$59$ $$-1189 + 300 T + 192 T^{2} + 25 T^{3} + T^{4}$$
$61$ $$-995 - 1030 T - 6 T^{2} + 19 T^{3} + T^{4}$$
$67$ $$-199 + 45 T + 67 T^{2} + 15 T^{3} + T^{4}$$
$71$ $$-991 - 679 T - 83 T^{2} + 7 T^{3} + T^{4}$$
$73$ $$4975 - 760 T - 116 T^{2} + 11 T^{3} + T^{4}$$
$79$ $$-271 + 161 T - 8 T^{2} - 8 T^{3} + T^{4}$$
$83$ $$2245 + 900 T - 236 T^{2} + T^{3} + T^{4}$$
$89$ $$-755 - 1120 T - 44 T^{2} + 17 T^{3} + T^{4}$$
$97$ $$-4225 - 1950 T - 110 T^{2} + 15 T^{3} + T^{4}$$