# Properties

 Label 847.2.a.n Level $847$ Weight $2$ Character orbit 847.a Self dual yes Analytic conductor $6.763$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.6.7674048.1 Defining polynomial: $$x^{6} - 2 x^{5} - 5 x^{4} + 8 x^{3} + 7 x^{2} - 6 x - 2$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} ) q^{2} + \beta_{5} q^{3} + ( 1 - \beta_{1} + \beta_{2} ) q^{4} + ( -1 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{5} + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{6} + q^{7} + ( 2 + 2 \beta_{2} - \beta_{3} ) q^{8} + ( 2 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{1} ) q^{2} + \beta_{5} q^{3} + ( 1 - \beta_{1} + \beta_{2} ) q^{4} + ( -1 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{5} + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{6} + q^{7} + ( 2 + 2 \beta_{2} - \beta_{3} ) q^{8} + ( 2 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{9} + ( -1 + \beta_{2} + \beta_{5} ) q^{10} + ( -3 + 2 \beta_{1} - \beta_{2} + 3 \beta_{3} - 2 \beta_{4} ) q^{12} + ( 1 + \beta_{2} - \beta_{4} + \beta_{5} ) q^{13} + ( 1 - \beta_{1} ) q^{14} + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{5} ) q^{15} + ( 2 - 2 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{16} + ( 4 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{17} + ( 4 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{18} + ( 1 + \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{19} + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} ) q^{20} + \beta_{5} q^{21} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{23} + ( -4 + 2 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} - \beta_{4} ) q^{24} + ( -4 \beta_{2} - 2 \beta_{5} ) q^{25} + ( 2 - \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{26} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{27} + ( 1 - \beta_{1} + \beta_{2} ) q^{28} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{29} + ( 3 - \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{5} ) q^{30} + ( -1 + \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{31} + ( 2 - 3 \beta_{1} + \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{32} + ( 5 - 3 \beta_{1} + \beta_{2} + \beta_{4} ) q^{34} + ( -1 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{35} + ( 3 - 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{36} + ( 4 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{37} + ( -4 - 2 \beta_{2} - 2 \beta_{3} - \beta_{5} ) q^{38} + ( 4 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{39} + ( 2 + \beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{40} + ( 4 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{41} + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{42} + ( -2 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{43} + ( -6 + \beta_{1} + 4 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{45} + ( 1 - \beta_{2} - 3 \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{46} + ( -3 + 2 \beta_{1} + 3 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{47} + ( -5 + 4 \beta_{1} - 7 \beta_{2} + 3 \beta_{3} + \beta_{5} ) q^{48} + q^{49} + ( -2 + 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{50} + ( 1 - \beta_{2} - \beta_{3} + \beta_{4} + 5 \beta_{5} ) q^{51} + ( 3 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{52} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{4} ) q^{53} + ( -6 + 2 \beta_{1} - 2 \beta_{2} + \beta_{4} ) q^{54} + ( 2 + 2 \beta_{2} - \beta_{3} ) q^{56} + ( 4 - \beta_{1} - 3 \beta_{2} - 4 \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{57} + ( 1 - 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{58} + ( -3 + 4 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{5} ) q^{59} + ( 5 - \beta_{1} - 3 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{60} + ( -4 + 6 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{61} + ( 4 - 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} ) q^{62} + ( 2 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{63} + ( 3 + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} ) q^{64} + ( 3 - 3 \beta_{2} + 2 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{65} + ( -1 + 4 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{67} + ( 3 - 4 \beta_{1} + 5 \beta_{2} - \beta_{5} ) q^{68} + ( -3 + 4 \beta_{1} - \beta_{2} - 3 \beta_{3} + 2 \beta_{5} ) q^{69} + ( -1 + \beta_{2} + \beta_{5} ) q^{70} + ( 4 + \beta_{1} - \beta_{2} + 2 \beta_{5} ) q^{71} + ( 5 - 7 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} ) q^{72} + ( 1 + \beta_{1} - 4 \beta_{2} - \beta_{3} - 2 \beta_{4} - 3 \beta_{5} ) q^{73} + ( 9 - 4 \beta_{1} + 3 \beta_{2} - 3 \beta_{4} + \beta_{5} ) q^{74} + ( -2 + 6 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} ) q^{75} + ( -7 + 3 \beta_{1} + \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{76} + ( 6 - 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} ) q^{78} + ( -6 + 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{79} + ( -2 + \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{80} + ( -2 + 3 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{81} + ( -3 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{82} + ( 3 + 3 \beta_{1} + 3 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{83} + ( -3 + 2 \beta_{1} - \beta_{2} + 3 \beta_{3} - 2 \beta_{4} ) q^{84} + ( -6 + 5 \beta_{1} + 5 \beta_{2} + \beta_{3} - 5 \beta_{4} - \beta_{5} ) q^{85} + ( -5 + 2 \beta_{1} - 3 \beta_{2} + 5 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{86} + ( 5 - 3 \beta_{1} + \beta_{3} + \beta_{5} ) q^{87} + ( -\beta_{1} - 4 \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{89} + ( -4 + 3 \beta_{1} + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{90} + ( 1 + \beta_{2} - \beta_{4} + \beta_{5} ) q^{91} + ( \beta_{1} + 3 \beta_{2} - 4 \beta_{3} - 4 \beta_{5} ) q^{92} + ( -10 + 3 \beta_{1} - \beta_{2} + 6 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{93} + ( -7 + \beta_{1} - 5 \beta_{3} - \beta_{5} ) q^{94} + ( -5 + \beta_{2} + 3 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{95} + ( -13 + 9 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{96} + ( -1 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{97} + ( 1 - \beta_{1} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} - 6 q^{6} + 6 q^{7} + 12 q^{8} + 8 q^{9} + O(q^{10})$$ $$6 q + 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} - 6 q^{6} + 6 q^{7} + 12 q^{8} + 8 q^{9} - 8 q^{10} - 14 q^{12} + 4 q^{13} + 4 q^{14} + 2 q^{15} + 8 q^{16} + 22 q^{17} + 24 q^{18} + 6 q^{19} + 2 q^{20} - 2 q^{21} + 2 q^{23} - 20 q^{24} + 4 q^{25} + 6 q^{26} - 2 q^{27} + 4 q^{28} + 12 q^{29} + 20 q^{30} - 2 q^{31} + 8 q^{32} + 24 q^{34} - 4 q^{35} + 18 q^{36} + 14 q^{37} - 22 q^{38} + 20 q^{39} + 18 q^{40} + 26 q^{41} - 6 q^{42} - 4 q^{43} - 36 q^{45} + 12 q^{46} - 16 q^{47} - 24 q^{48} + 6 q^{49} - 4 q^{50} - 4 q^{51} + 12 q^{52} + 4 q^{53} - 32 q^{54} + 12 q^{56} + 20 q^{57} - 2 q^{58} - 4 q^{59} + 24 q^{60} - 8 q^{61} + 20 q^{62} + 8 q^{63} + 26 q^{64} + 24 q^{65} + 6 q^{67} + 12 q^{68} - 14 q^{69} - 8 q^{70} + 22 q^{71} + 16 q^{72} + 14 q^{73} + 44 q^{74} - 20 q^{75} - 30 q^{76} + 32 q^{78} - 28 q^{79} - 4 q^{80} - 6 q^{81} - 4 q^{82} + 22 q^{83} - 14 q^{84} - 24 q^{85} - 30 q^{86} + 22 q^{87} - 22 q^{90} + 4 q^{91} + 10 q^{92} - 50 q^{93} - 38 q^{94} - 24 q^{95} - 62 q^{96} - 4 q^{97} + 4 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} - 5 x^{4} + 8 x^{3} + 7 x^{2} - 6 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 3 \nu + 1$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - \nu^{3} - 4 \nu^{2} + 2 \nu + 2$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} - 2 \nu^{4} - 4 \nu^{3} + 6 \nu^{2} + 4 \nu - 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} + 5 \beta_{2} + 6 \beta_{1} + 7$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} + 2 \beta_{4} + 6 \beta_{3} + 8 \beta_{2} + 18 \beta_{1} + 8$$

## 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
 2.38595 1.82356 0.879640 −0.276564 −1.10939 −1.70320
−1.38595 −0.122479 −0.0791355 −0.133004 0.169750 1.00000 2.88158 −2.98500 0.184338
1.2 −0.823556 −0.960649 −1.32176 2.98565 0.791148 1.00000 2.73565 −2.07715 −2.45885
1.3 0.120360 2.76784 −1.98551 −2.80853 0.333137 1.00000 −0.479696 4.66094 −0.338034
1.4 1.27656 −2.57603 −0.370384 −4.09144 −3.28847 1.00000 −3.02595 3.63595 −5.22298
1.5 2.10939 1.69851 2.44952 0.492391 3.58282 1.00000 0.948212 −0.115054 1.03864
1.6 2.70320 −2.80719 5.30727 −0.445072 −7.58839 1.00000 8.94020 4.88032 −1.20312
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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.n yes 6
3.b odd 2 1 7623.2.a.cp 6
7.b odd 2 1 5929.2.a.bm 6
11.b odd 2 1 847.2.a.m 6
11.c even 5 4 847.2.f.y 24
11.d odd 10 4 847.2.f.z 24
33.d even 2 1 7623.2.a.cs 6
77.b even 2 1 5929.2.a.bj 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.a.m 6 11.b odd 2 1
847.2.a.n yes 6 1.a even 1 1 trivial
847.2.f.y 24 11.c even 5 4
847.2.f.z 24 11.d odd 10 4
5929.2.a.bj 6 77.b even 2 1
5929.2.a.bm 6 7.b odd 2 1
7623.2.a.cp 6 3.b odd 2 1
7623.2.a.cs 6 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(847))$$:

 $$T_{2}^{6} - 4 T_{2}^{5} + 12 T_{2}^{3} - 4 T_{2}^{2} - 8 T_{2} + 1$$ $$T_{3}^{6} + 2 T_{3}^{5} - 11 T_{3}^{4} - 20 T_{3}^{3} + 25 T_{3}^{2} + 36 T_{3} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 8 T - 4 T^{2} + 12 T^{3} - 4 T^{5} + T^{6}$$
$3$ $$4 + 36 T + 25 T^{2} - 20 T^{3} - 11 T^{4} + 2 T^{5} + T^{6}$$
$5$ $$1 + 8 T - T^{2} - 36 T^{3} - 9 T^{4} + 4 T^{5} + T^{6}$$
$7$ $$( -1 + T )^{6}$$
$11$ $$T^{6}$$
$13$ $$16 - 32 T - 36 T^{2} + 72 T^{3} - 19 T^{4} - 4 T^{5} + T^{6}$$
$17$ $$-2687 + 1192 T + 774 T^{2} - 666 T^{3} + 182 T^{4} - 22 T^{5} + T^{6}$$
$19$ $$592 + 1328 T + 925 T^{2} + 142 T^{3} - 50 T^{4} - 6 T^{5} + T^{6}$$
$23$ $$8656 - 8056 T + 1425 T^{2} + 318 T^{3} - 85 T^{4} - 2 T^{5} + T^{6}$$
$29$ $$481 - 224 T - 272 T^{2} + 92 T^{3} + 28 T^{4} - 12 T^{5} + T^{6}$$
$31$ $$6736 + 9736 T + 3321 T^{2} - 282 T^{3} - 118 T^{4} + 2 T^{5} + T^{6}$$
$37$ $$2896 + 1056 T - 3404 T^{2} + 1052 T^{3} - 35 T^{4} - 14 T^{5} + T^{6}$$
$41$ $$-3803 + 2034 T + 1195 T^{2} - 1000 T^{3} + 247 T^{4} - 26 T^{5} + T^{6}$$
$43$ $$-2288 - 112 T + 1881 T^{2} - 252 T^{3} - 121 T^{4} + 4 T^{5} + T^{6}$$
$47$ $$9088 - 1456 T - 5803 T^{2} - 1422 T^{3} - 27 T^{4} + 16 T^{5} + T^{6}$$
$53$ $$2257 + 4596 T + 1864 T^{2} + 22 T^{3} - 80 T^{4} - 4 T^{5} + T^{6}$$
$59$ $$-313856 + 45024 T + 19477 T^{2} - 946 T^{3} - 287 T^{4} + 4 T^{5} + T^{6}$$
$61$ $$-971552 + 122232 T + 31669 T^{2} - 2066 T^{3} - 323 T^{4} + 8 T^{5} + T^{6}$$
$67$ $$-48896 - 14848 T + 6301 T^{2} + 670 T^{3} - 161 T^{4} - 6 T^{5} + T^{6}$$
$71$ $$-2048 + 10432 T - 4923 T^{2} + 474 T^{3} + 107 T^{4} - 22 T^{5} + T^{6}$$
$73$ $$-23024 - 9960 T + 4885 T^{2} + 1358 T^{3} - 122 T^{4} - 14 T^{5} + T^{6}$$
$79$ $$-183488 - 133440 T - 33023 T^{2} - 2668 T^{3} + 127 T^{4} + 28 T^{5} + T^{6}$$
$83$ $$-44012 + 49696 T - 18163 T^{2} + 2142 T^{3} + 42 T^{4} - 22 T^{5} + T^{6}$$
$89$ $$62137 - 38150 T + 3850 T^{2} + 1400 T^{3} - 254 T^{4} + T^{6}$$
$97$ $$-62759 + 10904 T + 6167 T^{2} - 420 T^{3} - 153 T^{4} + 4 T^{5} + T^{6}$$