Properties

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

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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: \(1\)
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: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + ( -1 - \beta ) q^{3} + q^{4} + q^{5} + ( 1 + \beta ) q^{6} + q^{7} - q^{8} + ( 3 + 2 \beta ) q^{9} +O(q^{10})\) \( q - q^{2} + ( -1 - \beta ) q^{3} + q^{4} + q^{5} + ( 1 + \beta ) q^{6} + q^{7} - q^{8} + ( 3 + 2 \beta ) q^{9} - q^{10} + ( -1 - \beta ) q^{12} -2 \beta q^{13} - q^{14} + ( -1 - \beta ) q^{15} + q^{16} -2 q^{17} + ( -3 - 2 \beta ) q^{18} + ( 3 + \beta ) q^{19} + q^{20} + ( -1 - \beta ) q^{21} + ( -5 - \beta ) q^{23} + ( 1 + \beta ) q^{24} + q^{25} + 2 \beta q^{26} + ( -10 - 2 \beta ) q^{27} + q^{28} + ( -1 + \beta ) q^{29} + ( 1 + \beta ) q^{30} + 4 \beta q^{31} - q^{32} + 2 q^{34} + q^{35} + ( 3 + 2 \beta ) q^{36} + ( -1 - \beta ) q^{37} + ( -3 - \beta ) q^{38} + ( 10 + 2 \beta ) q^{39} - q^{40} + ( 1 + 3 \beta ) q^{41} + ( 1 + \beta ) q^{42} + 8 q^{43} + ( 3 + 2 \beta ) q^{45} + ( 5 + \beta ) q^{46} + ( -4 + 2 \beta ) q^{47} + ( -1 - \beta ) q^{48} + q^{49} - q^{50} + ( 2 + 2 \beta ) q^{51} -2 \beta q^{52} + ( -3 + \beta ) q^{53} + ( 10 + 2 \beta ) q^{54} - q^{56} + ( -8 - 4 \beta ) q^{57} + ( 1 - \beta ) q^{58} -8 q^{59} + ( -1 - \beta ) q^{60} + ( -10 - 2 \beta ) q^{61} -4 \beta q^{62} + ( 3 + 2 \beta ) q^{63} + q^{64} -2 \beta q^{65} -6 q^{67} -2 q^{68} + ( 10 + 6 \beta ) q^{69} - q^{70} -8 q^{71} + ( -3 - 2 \beta ) q^{72} + ( 4 + 2 \beta ) q^{73} + ( 1 + \beta ) q^{74} + ( -1 - \beta ) q^{75} + ( 3 + \beta ) q^{76} + ( -10 - 2 \beta ) q^{78} + ( -3 - \beta ) q^{79} + q^{80} + ( 11 + 6 \beta ) q^{81} + ( -1 - 3 \beta ) q^{82} -8 q^{83} + ( -1 - \beta ) q^{84} -2 q^{85} -8 q^{86} -4 q^{87} + ( -6 - 4 \beta ) q^{89} + ( -3 - 2 \beta ) q^{90} -2 \beta q^{91} + ( -5 - \beta ) q^{92} + ( -20 - 4 \beta ) q^{93} + ( 4 - 2 \beta ) q^{94} + ( 3 + \beta ) q^{95} + ( 1 + \beta ) q^{96} + ( -5 + 3 \beta ) q^{97} - q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 2q^{7} - 2q^{8} + 6q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 2q^{7} - 2q^{8} + 6q^{9} - 2q^{10} - 2q^{12} - 2q^{14} - 2q^{15} + 2q^{16} - 4q^{17} - 6q^{18} + 6q^{19} + 2q^{20} - 2q^{21} - 10q^{23} + 2q^{24} + 2q^{25} - 20q^{27} + 2q^{28} - 2q^{29} + 2q^{30} - 2q^{32} + 4q^{34} + 2q^{35} + 6q^{36} - 2q^{37} - 6q^{38} + 20q^{39} - 2q^{40} + 2q^{41} + 2q^{42} + 16q^{43} + 6q^{45} + 10q^{46} - 8q^{47} - 2q^{48} + 2q^{49} - 2q^{50} + 4q^{51} - 6q^{53} + 20q^{54} - 2q^{56} - 16q^{57} + 2q^{58} - 16q^{59} - 2q^{60} - 20q^{61} + 6q^{63} + 2q^{64} - 12q^{67} - 4q^{68} + 20q^{69} - 2q^{70} - 16q^{71} - 6q^{72} + 8q^{73} + 2q^{74} - 2q^{75} + 6q^{76} - 20q^{78} - 6q^{79} + 2q^{80} + 22q^{81} - 2q^{82} - 16q^{83} - 2q^{84} - 4q^{85} - 16q^{86} - 8q^{87} - 12q^{89} - 6q^{90} - 10q^{92} - 40q^{93} + 8q^{94} + 6q^{95} + 2q^{96} - 10q^{97} - 2q^{98} + O(q^{100}) \)

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
1.61803
−0.618034
−1.00000 −3.23607 1.00000 1.00000 3.23607 1.00000 −1.00000 7.47214 −1.00000
1.2 −1.00000 1.23607 1.00000 1.00000 −1.23607 1.00000 −1.00000 −1.47214 −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.bj 2
11.b odd 2 1 8470.2.a.bv yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8470.2.a.bj 2 1.a even 1 1 trivial
8470.2.a.bv yes 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\):

\( T_{3}^{2} + 2 T_{3} - 4 \)
\( T_{13}^{2} - 20 \)
\( T_{17} + 2 \)
\( T_{19}^{2} - 6 T_{19} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( -4 + 2 T + T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( -20 + T^{2} \)
$17$ \( ( 2 + T )^{2} \)
$19$ \( 4 - 6 T + T^{2} \)
$23$ \( 20 + 10 T + T^{2} \)
$29$ \( -4 + 2 T + T^{2} \)
$31$ \( -80 + T^{2} \)
$37$ \( -4 + 2 T + T^{2} \)
$41$ \( -44 - 2 T + T^{2} \)
$43$ \( ( -8 + T )^{2} \)
$47$ \( -4 + 8 T + T^{2} \)
$53$ \( 4 + 6 T + T^{2} \)
$59$ \( ( 8 + T )^{2} \)
$61$ \( 80 + 20 T + T^{2} \)
$67$ \( ( 6 + T )^{2} \)
$71$ \( ( 8 + T )^{2} \)
$73$ \( -4 - 8 T + T^{2} \)
$79$ \( 4 + 6 T + T^{2} \)
$83$ \( ( 8 + T )^{2} \)
$89$ \( -44 + 12 T + T^{2} \)
$97$ \( -20 + 10 T + T^{2} \)
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