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

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
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.

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

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
−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:

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

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(847))\):

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T + 5 T^{2} - 6 T^{3} + 4 T^{4} \)
$3$ \( 1 + T + 5 T^{2} + 3 T^{3} + 9 T^{4} \)
$5$ \( ( 1 - T + 5 T^{2} )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( \)
$13$ \( 1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4} \)
$17$ \( 1 - 5 T + 9 T^{2} - 85 T^{3} + 289 T^{4} \)
$19$ \( 1 - 8 T + 49 T^{2} - 152 T^{3} + 361 T^{4} \)
$23$ \( 1 + T + 15 T^{2} + 23 T^{3} + 529 T^{4} \)
$29$ \( 1 - 7 T + 69 T^{2} - 203 T^{3} + 841 T^{4} \)
$31$ \( 1 + 4 T + 61 T^{2} + 124 T^{3} + 961 T^{4} \)
$37$ \( 1 - 4 T + 58 T^{2} - 148 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 43 T^{2} + 1681 T^{4} \)
$43$ \( 1 - 5 T - 9 T^{2} - 215 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 11 T + 123 T^{2} + 517 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 7 T + 117 T^{2} + 371 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 11 T + 117 T^{2} - 649 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 13 T + 163 T^{2} - 793 T^{3} + 3721 T^{4} \)
$67$ \( 1 + T + 73 T^{2} + 67 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 21 T + 221 T^{2} + 1491 T^{3} + 5041 T^{4} \)
$73$ \( 1 - 24 T + 285 T^{2} - 1752 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 15 T + 213 T^{2} + 1185 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 8 T + 137 T^{2} - 664 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 7 T + 179 T^{2} - 623 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 - 7 T + 97 T^{2} )^{2} \)
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