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 $1$

Related objects

Downloads

Learn more

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: \(\Q(\sqrt{5}) \)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
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.

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

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:

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.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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(847))\):

\( T_{2}^{2} - 3T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{2} + T_{3} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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