# Properties

 Label 847.2.a.h Level $847$ Weight $2$ Character orbit 847.a Self dual yes Analytic conductor $6.763$ Analytic rank $0$ Dimension $2$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −0.618034 1.61803
0.381966 −1.61803 −1.85410 1.00000 −0.618034 −1.00000 −1.47214 −0.381966 0.381966
1.2 2.61803 0.618034 4.85410 1.00000 1.61803 −1.00000 7.47214 −2.61803 2.61803
 $$n$$: 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.h yes 2
3.b odd 2 1 7623.2.a.t 2
7.b odd 2 1 5929.2.a.s 2
11.b odd 2 1 847.2.a.d 2
11.c even 5 2 847.2.f.c 4
11.c even 5 2 847.2.f.j 4
11.d odd 10 2 847.2.f.d 4
11.d odd 10 2 847.2.f.l 4
33.d even 2 1 7623.2.a.bx 2
77.b even 2 1 5929.2.a.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.a.d 2 11.b odd 2 1
847.2.a.h yes 2 1.a even 1 1 trivial
847.2.f.c 4 11.c even 5 2
847.2.f.d 4 11.d odd 10 2
847.2.f.j 4 11.c even 5 2
847.2.f.l 4 11.d odd 10 2
5929.2.a.i 2 77.b even 2 1
5929.2.a.s 2 7.b odd 2 1
7623.2.a.t 2 3.b odd 2 1
7623.2.a.bx 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(847))$$:

 $$T_{2}^{2} - 3T_{2} + 1$$ T2^2 - 3*T2 + 1 $$T_{3}^{2} + T_{3} - 1$$ T3^2 + T3 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3T + 1$$
$3$ $$T^{2} + T - 1$$
$5$ $$(T - 1)^{2}$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 2T - 4$$
$17$ $$T^{2} - 5T - 25$$
$19$ $$T^{2} - 8T + 11$$
$23$ $$T^{2} + T - 31$$
$29$ $$T^{2} - 7T + 11$$
$31$ $$T^{2} + 4T - 1$$
$37$ $$T^{2} - 4T - 16$$
$41$ $$T^{2} - 125$$
$43$ $$T^{2} - 5T - 95$$
$47$ $$T^{2} + 11T + 29$$
$53$ $$T^{2} + 7T + 11$$
$59$ $$T^{2} - 11T - 1$$
$61$ $$T^{2} - 13T + 41$$
$67$ $$T^{2} + T - 61$$
$71$ $$T^{2} + 21T + 79$$
$73$ $$T^{2} - 24T + 139$$
$79$ $$T^{2} + 15T + 55$$
$83$ $$T^{2} - 8T - 29$$
$89$ $$T^{2} - 7T + 1$$
$97$ $$(T - 7)^{2}$$