# Properties

 Label 847.2.a.e Level $847$ Weight $2$ Character orbit 847.a Self dual yes Analytic conductor $6.763$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 2.30278 −1.30278
−2.30278 −2.30278 3.30278 3.60555 5.30278 1.00000 −3.00000 2.30278 −8.30278
1.2 1.30278 1.30278 −0.302776 −3.60555 1.69722 1.00000 −3.00000 −1.30278 −4.69722
 $$n$$: 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.e 2
3.b odd 2 1 7623.2.a.bs 2
7.b odd 2 1 5929.2.a.k 2
11.b odd 2 1 847.2.a.g yes 2
11.c even 5 4 847.2.f.r 8
11.d odd 10 4 847.2.f.o 8
33.d even 2 1 7623.2.a.bc 2
77.b even 2 1 5929.2.a.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.a.e 2 1.a even 1 1 trivial
847.2.a.g yes 2 11.b odd 2 1
847.2.f.o 8 11.d odd 10 4
847.2.f.r 8 11.c even 5 4
5929.2.a.k 2 7.b odd 2 1
5929.2.a.p 2 77.b even 2 1
7623.2.a.bc 2 33.d even 2 1
7623.2.a.bs 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(847))$$:

 $$T_{2}^{2} + T_{2} - 3$$ $$T_{3}^{2} + T_{3} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-3 + T + T^{2}$$
$3$ $$-3 + T + T^{2}$$
$5$ $$-13 + T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$T^{2}$$
$13$ $$-4 + 6 T + T^{2}$$
$17$ $$17 + 9 T + T^{2}$$
$19$ $$( -3 + T )^{2}$$
$23$ $$17 + 9 T + T^{2}$$
$29$ $$39 + 13 T + T^{2}$$
$31$ $$( -1 + T )^{2}$$
$37$ $$-48 - 4 T + T^{2}$$
$41$ $$( 7 + T )^{2}$$
$43$ $$9 + 7 T + T^{2}$$
$47$ $$-17 - 7 T + T^{2}$$
$53$ $$27 + 15 T + T^{2}$$
$59$ $$69 - 17 T + T^{2}$$
$61$ $$3 - 5 T + T^{2}$$
$67$ $$-81 + T + T^{2}$$
$71$ $$3 + 5 T + T^{2}$$
$73$ $$( -5 + T )^{2}$$
$79$ $$39 - 13 T + T^{2}$$
$83$ $$( 3 + T )^{2}$$
$89$ $$-261 - 3 T + T^{2}$$
$97$ $$-13 + T^{2}$$