Properties

Label 8470.2.a.cf
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.

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

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 0.381966 1.00000 −1.00000 0.381966 1.00000 1.00000 −2.85410 −1.00000
1.2 1.00000 2.61803 1.00000 −1.00000 2.61803 1.00000 1.00000 3.85410 −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.cf 2
11.b odd 2 1 8470.2.a.bt 2
11.d odd 10 2 770.2.n.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.n.b 4 11.d odd 10 2
8470.2.a.bt 2 11.b odd 2 1
8470.2.a.cf 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\):

\( T_{3}^{2} - 3 T_{3} + 1 \)
\( T_{13} - 2 \)
\( T_{17}^{2} + T_{17} - 1 \)
\( T_{19}^{2} - 7 T_{19} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( 1 - 3 T + T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( -1 + T + T^{2} \)
$19$ \( 1 - 7 T + T^{2} \)
$23$ \( ( -6 + T )^{2} \)
$29$ \( -4 - 2 T + T^{2} \)
$31$ \( -4 - 2 T + T^{2} \)
$37$ \( -16 + 4 T + T^{2} \)
$41$ \( -55 + 5 T + T^{2} \)
$43$ \( -9 + 3 T + T^{2} \)
$47$ \( 44 - 14 T + T^{2} \)
$53$ \( -4 + 2 T + T^{2} \)
$59$ \( 41 + 13 T + T^{2} \)
$61$ \( 44 + 16 T + T^{2} \)
$67$ \( -31 - T + T^{2} \)
$71$ \( -36 + 6 T + T^{2} \)
$73$ \( 99 - 21 T + T^{2} \)
$79$ \( -4 + 8 T + T^{2} \)
$83$ \( -55 - 5 T + T^{2} \)
$89$ \( 25 - 15 T + T^{2} \)
$97$ \( -101 - T + T^{2} \)
show more
show less