Properties

Label 847.2.a.k
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

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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: \(4\)
Coefficient field: 4.4.2525.1
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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} - 2q^{3} + 4q^{4} - 6q^{5} + q^{6} + 4q^{7} - 9q^{8} - 6q^{9} + O(q^{10}) \) \( 4q - 2q^{2} - 2q^{3} + 4q^{4} - 6q^{5} + q^{6} + 4q^{7} - 9q^{8} - 6q^{9} + 14q^{10} - 7q^{12} - 2q^{14} + 3q^{15} - 4q^{16} + 3q^{17} + 3q^{18} - 3q^{19} - 17q^{20} - 2q^{21} - 8q^{23} + 12q^{24} - 12q^{26} + 4q^{27} + 4q^{28} - 3q^{29} - 12q^{30} - 3q^{31} + 10q^{32} - 12q^{34} - 6q^{35} - q^{36} - 7q^{37} - 20q^{38} - 5q^{39} + 13q^{40} - 4q^{41} + q^{42} - 8q^{43} + 9q^{45} + 3q^{46} - 14q^{47} - 3q^{48} + 4q^{49} - 33q^{50} + 11q^{51} + 17q^{52} - 9q^{53} - 2q^{54} - 9q^{56} - 6q^{57} + 3q^{58} - 25q^{59} + 21q^{60} + 19q^{61} - 10q^{62} - 6q^{63} + 3q^{64} - 12q^{65} - 15q^{67} + q^{68} + 14q^{69} + 14q^{70} - 7q^{71} + 6q^{72} + 11q^{73} - 8q^{74} - 5q^{75} + 26q^{76} - 9q^{78} - 8q^{79} - 4q^{80} - 4q^{81} + 3q^{82} + q^{83} - 7q^{84} - 15q^{85} + 4q^{86} - 6q^{87} - 17q^{89} - 16q^{90} - 17q^{92} - 11q^{93} + 20q^{94} - 17q^{95} + 10q^{96} - 15q^{97} - 2q^{98} + O(q^{100}) \)

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
2.46673
1.77748
−0.777484
−1.46673
−2.46673 −1.61803 4.08477 −3.46673 3.99126 1.00000 −5.14256 −0.381966 8.55150
1.2 −1.77748 0.618034 1.15945 −2.77748 −1.09855 1.00000 1.49406 −2.61803 4.93693
1.3 0.777484 0.618034 −1.39552 −0.222516 0.480512 1.00000 −2.63996 −2.61803 −0.173002
1.4 1.46673 −1.61803 0.151302 0.466732 −2.37322 1.00000 −2.71154 −0.381966 0.684570
\(n\): 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.k 4
3.b odd 2 1 7623.2.a.co 4
7.b odd 2 1 5929.2.a.bb 4
11.b odd 2 1 847.2.a.l 4
11.c even 5 2 847.2.f.q 8
11.c even 5 2 847.2.f.s 8
11.d odd 10 2 77.2.f.a 8
11.d odd 10 2 847.2.f.p 8
33.d even 2 1 7623.2.a.ch 4
33.f even 10 2 693.2.m.g 8
77.b even 2 1 5929.2.a.bi 4
77.l even 10 2 539.2.f.d 8
77.n even 30 4 539.2.q.b 16
77.o odd 30 4 539.2.q.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.f.a 8 11.d odd 10 2
539.2.f.d 8 77.l even 10 2
539.2.q.b 16 77.n even 30 4
539.2.q.c 16 77.o odd 30 4
693.2.m.g 8 33.f even 10 2
847.2.a.k 4 1.a even 1 1 trivial
847.2.a.l 4 11.b odd 2 1
847.2.f.p 8 11.d odd 10 2
847.2.f.q 8 11.c even 5 2
847.2.f.s 8 11.c even 5 2
5929.2.a.bb 4 7.b odd 2 1
5929.2.a.bi 4 77.b even 2 1
7623.2.a.ch 4 33.d even 2 1
7623.2.a.co 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(847))\):

\( T_{2}^{4} + 2 T_{2}^{3} - 4 T_{2}^{2} - 5 T_{2} + 5 \)
\( T_{3}^{2} + T_{3} - 1 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 4 T^{2} + 7 T^{3} + 13 T^{4} + 14 T^{5} + 16 T^{6} + 16 T^{7} + 16 T^{8} \)
$3$ \( ( 1 + T + 5 T^{2} + 3 T^{3} + 9 T^{4} )^{2} \)
$5$ \( 1 + 6 T + 28 T^{2} + 87 T^{3} + 229 T^{4} + 435 T^{5} + 700 T^{6} + 750 T^{7} + 625 T^{8} \)
$7$ \( ( 1 - T )^{4} \)
$11$ \( \)
$13$ \( 1 + 20 T^{2} + 65 T^{3} + 153 T^{4} + 845 T^{5} + 3380 T^{6} + 28561 T^{8} \)
$17$ \( 1 - 3 T + 45 T^{2} - 62 T^{3} + 881 T^{4} - 1054 T^{5} + 13005 T^{6} - 14739 T^{7} + 83521 T^{8} \)
$19$ \( 1 + 3 T + 47 T^{2} + 36 T^{3} + 919 T^{4} + 684 T^{5} + 16967 T^{6} + 20577 T^{7} + 130321 T^{8} \)
$23$ \( 1 + 8 T + 83 T^{2} + 402 T^{3} + 2555 T^{4} + 9246 T^{5} + 43907 T^{6} + 97336 T^{7} + 279841 T^{8} \)
$29$ \( 1 + 3 T + 62 T^{2} + 261 T^{3} + 2319 T^{4} + 7569 T^{5} + 52142 T^{6} + 73167 T^{7} + 707281 T^{8} \)
$31$ \( 1 + 3 T + 100 T^{2} + 299 T^{3} + 4283 T^{4} + 9269 T^{5} + 96100 T^{6} + 89373 T^{7} + 923521 T^{8} \)
$37$ \( 1 + 7 T + 154 T^{2} + 767 T^{3} + 8653 T^{4} + 28379 T^{5} + 210826 T^{6} + 354571 T^{7} + 1874161 T^{8} \)
$41$ \( 1 + 4 T + 107 T^{2} + 470 T^{3} + 5491 T^{4} + 19270 T^{5} + 179867 T^{6} + 275684 T^{7} + 2825761 T^{8} \)
$43$ \( ( 1 + 4 T + 45 T^{2} + 172 T^{3} + 1849 T^{4} )^{2} \)
$47$ \( 1 + 14 T + 162 T^{2} + 1449 T^{3} + 10505 T^{4} + 68103 T^{5} + 357858 T^{6} + 1453522 T^{7} + 4879681 T^{8} \)
$53$ \( 1 + 9 T + 108 T^{2} + 725 T^{3} + 4961 T^{4} + 38425 T^{5} + 303372 T^{6} + 1339893 T^{7} + 7890481 T^{8} \)
$59$ \( 1 + 25 T + 428 T^{2} + 4725 T^{3} + 42353 T^{4} + 278775 T^{5} + 1489868 T^{6} + 5134475 T^{7} + 12117361 T^{8} \)
$61$ \( 1 - 19 T + 238 T^{2} - 2447 T^{3} + 20599 T^{4} - 149267 T^{5} + 885598 T^{6} - 4312639 T^{7} + 13845841 T^{8} \)
$67$ \( 1 + 15 T + 335 T^{2} + 3060 T^{3} + 35713 T^{4} + 205020 T^{5} + 1503815 T^{6} + 4511445 T^{7} + 20151121 T^{8} \)
$71$ \( 1 + 7 T + 201 T^{2} + 812 T^{3} + 17469 T^{4} + 57652 T^{5} + 1013241 T^{6} + 2505377 T^{7} + 25411681 T^{8} \)
$73$ \( 1 - 11 T + 176 T^{2} - 1649 T^{3} + 20013 T^{4} - 120377 T^{5} + 937904 T^{6} - 4279187 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 8 T + 308 T^{2} + 1735 T^{3} + 35911 T^{4} + 137065 T^{5} + 1922228 T^{6} + 3944312 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - T + 96 T^{2} - 1149 T^{3} + 4403 T^{4} - 95367 T^{5} + 661344 T^{6} - 571787 T^{7} + 47458321 T^{8} \)
$89$ \( 1 + 17 T + 312 T^{2} + 3419 T^{3} + 38939 T^{4} + 304291 T^{5} + 2471352 T^{6} + 11984473 T^{7} + 62742241 T^{8} \)
$97$ \( 1 + 15 T + 278 T^{2} + 2415 T^{3} + 30889 T^{4} + 234255 T^{5} + 2615702 T^{6} + 13690095 T^{7} + 88529281 T^{8} \)
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