Properties

Label 8470.2.a.bp
Level $8470$
Weight $2$
Character orbit 8470.a
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
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.

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

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(7\) \(-1\)
\(11\) \(1\)

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 8470.2.a.bp 2
11.b odd 2 1 8470.2.a.cb 2
11.c even 5 2 770.2.n.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.n.d 4 11.c even 5 2
8470.2.a.bp 2 1.a even 1 1 trivial
8470.2.a.cb 2 11.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\):

\( T_{3}^{2} - T_{3} - 1 \)
\( T_{13}^{2} + 6 T_{13} + 4 \)
\( T_{17}^{2} - 3 T_{17} + 1 \)
\( T_{19}^{2} - 9 T_{19} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( -1 - T + T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( 4 + 6 T + T^{2} \)
$17$ \( 1 - 3 T + T^{2} \)
$19$ \( 9 - 9 T + T^{2} \)
$23$ \( -4 - 8 T + T^{2} \)
$29$ \( -20 + T^{2} \)
$31$ \( -16 + 4 T + T^{2} \)
$37$ \( 76 + 18 T + T^{2} \)
$41$ \( -11 + T + T^{2} \)
$43$ \( -19 - 7 T + T^{2} \)
$47$ \( 44 - 14 T + T^{2} \)
$53$ \( 20 + 10 T + T^{2} \)
$59$ \( 45 - 15 T + T^{2} \)
$61$ \( 4 - 14 T + T^{2} \)
$67$ \( 19 + 9 T + T^{2} \)
$71$ \( -76 - 4 T + T^{2} \)
$73$ \( -11 + T + T^{2} \)
$79$ \( ( -4 + T )^{2} \)
$83$ \( 19 + 11 T + T^{2} \)
$89$ \( 121 + 23 T + T^{2} \)
$97$ \( 89 + 19 T + T^{2} \)
show more
show less