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

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

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:

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

Atkin-Lehner signs

\( p \) Sign
\(7\) \(1\)
\(11\) \(1\)

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

$p$ $F_p(T)$
$2$ \( 1 + 2 T + T^{2} + 2 T^{4} + 8 T^{5} + 8 T^{6} \)
$3$ \( 1 + T + 3 T^{2} + 8 T^{3} + 9 T^{4} + 9 T^{5} + 27 T^{6} \)
$5$ \( 1 - T + 7 T^{2} - 6 T^{3} + 35 T^{4} - 25 T^{5} + 125 T^{6} \)
$7$ \( ( 1 + T )^{3} \)
$11$ \( \)
$13$ \( 1 + 8 T + 41 T^{2} + 144 T^{3} + 533 T^{4} + 1352 T^{5} + 2197 T^{6} \)
$17$ \( 1 + 8 T + 53 T^{2} + 208 T^{3} + 901 T^{4} + 2312 T^{5} + 4913 T^{6} \)
$19$ \( 1 + 17 T^{2} + 64 T^{3} + 323 T^{4} + 6859 T^{6} \)
$23$ \( 1 - 7 T + 77 T^{2} - 314 T^{3} + 1771 T^{4} - 3703 T^{5} + 12167 T^{6} \)
$29$ \( 1 + 47 T^{2} - 64 T^{3} + 1363 T^{4} + 24389 T^{6} \)
$31$ \( 1 + 13 T + 143 T^{2} + 864 T^{3} + 4433 T^{4} + 12493 T^{5} + 29791 T^{6} \)
$37$ \( 1 + 17 T + 183 T^{2} + 1274 T^{3} + 6771 T^{4} + 23273 T^{5} + 50653 T^{6} \)
$41$ \( 1 + 16 T + 189 T^{2} + 1392 T^{3} + 7749 T^{4} + 26896 T^{5} + 68921 T^{6} \)
$43$ \( 1 + 4 T + 101 T^{2} + 312 T^{3} + 4343 T^{4} + 7396 T^{5} + 79507 T^{6} \)
$47$ \( 1 - 4 T + 127 T^{2} - 384 T^{3} + 5969 T^{4} - 8836 T^{5} + 103823 T^{6} \)
$53$ \( 1 + 10 T + 159 T^{2} + 996 T^{3} + 8427 T^{4} + 28090 T^{5} + 148877 T^{6} \)
$59$ \( 1 - T + 171 T^{2} - 120 T^{3} + 10089 T^{4} - 3481 T^{5} + 205379 T^{6} \)
$61$ \( 1 - 16 T + 249 T^{2} - 2032 T^{3} + 15189 T^{4} - 59536 T^{5} + 226981 T^{6} \)
$67$ \( 1 + 3 T + 113 T^{2} - 22 T^{3} + 7571 T^{4} + 13467 T^{5} + 300763 T^{6} \)
$71$ \( 1 - 5 T + 5 T^{2} + 770 T^{3} + 355 T^{4} - 25205 T^{5} + 357911 T^{6} \)
$73$ \( 1 + 16 T + 285 T^{2} + 2416 T^{3} + 20805 T^{4} + 85264 T^{5} + 389017 T^{6} \)
$79$ \( 1 + 28 T + 465 T^{2} + 4936 T^{3} + 36735 T^{4} + 174748 T^{5} + 493039 T^{6} \)
$83$ \( 1 + 8 T + 193 T^{2} + 1392 T^{3} + 16019 T^{4} + 55112 T^{5} + 571787 T^{6} \)
$89$ \( 1 + 21 T + 371 T^{2} + 3838 T^{3} + 33019 T^{4} + 166341 T^{5} + 704969 T^{6} \)
$97$ \( 1 + 11 T + 259 T^{2} + 1682 T^{3} + 25123 T^{4} + 103499 T^{5} + 912673 T^{6} \)
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