Properties

Label 847.2.a.d
Level $847$
Weight $2$
Character orbit 847.a
Self dual yes
Analytic conductor $6.763$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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
1.61803
−0.618034
−2.61803 0.618034 4.85410 1.00000 −1.61803 1.00000 −7.47214 −2.61803 −2.61803
1.2 −0.381966 −1.61803 −1.85410 1.00000 0.618034 1.00000 1.47214 −0.381966 −0.381966
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.d 2
3.b odd 2 1 7623.2.a.bx 2
7.b odd 2 1 5929.2.a.i 2
11.b odd 2 1 847.2.a.h yes 2
11.c even 5 2 847.2.f.d 4
11.c even 5 2 847.2.f.l 4
11.d odd 10 2 847.2.f.c 4
11.d odd 10 2 847.2.f.j 4
33.d even 2 1 7623.2.a.t 2
77.b even 2 1 5929.2.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.a.d 2 1.a even 1 1 trivial
847.2.a.h yes 2 11.b odd 2 1
847.2.f.c 4 11.d odd 10 2
847.2.f.d 4 11.c even 5 2
847.2.f.j 4 11.d odd 10 2
847.2.f.l 4 11.c even 5 2
5929.2.a.i 2 7.b odd 2 1
5929.2.a.s 2 77.b even 2 1
7623.2.a.t 2 33.d even 2 1
7623.2.a.bx 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(847))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + T^{2} \)
$3$ \( -1 + T + T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( -4 - 2 T + T^{2} \)
$17$ \( -25 + 5 T + T^{2} \)
$19$ \( 11 + 8 T + T^{2} \)
$23$ \( -31 + T + T^{2} \)
$29$ \( 11 + 7 T + T^{2} \)
$31$ \( -1 + 4 T + T^{2} \)
$37$ \( -16 - 4 T + T^{2} \)
$41$ \( -125 + T^{2} \)
$43$ \( -95 + 5 T + T^{2} \)
$47$ \( 29 + 11 T + T^{2} \)
$53$ \( 11 + 7 T + T^{2} \)
$59$ \( -1 - 11 T + T^{2} \)
$61$ \( 41 + 13 T + T^{2} \)
$67$ \( -61 + T + T^{2} \)
$71$ \( 79 + 21 T + T^{2} \)
$73$ \( 139 + 24 T + T^{2} \)
$79$ \( 55 - 15 T + T^{2} \)
$83$ \( -29 + 8 T + T^{2} \)
$89$ \( 1 - 7 T + T^{2} \)
$97$ \( ( -7 + T )^{2} \)
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