Properties

Label 8470.2.a.bx
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: \( 1 \)
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.

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

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 −1.61803 1.00000 −1.00000 −1.61803 −1.00000 1.00000 −0.381966 −1.00000
1.2 1.00000 0.618034 1.00000 −1.00000 0.618034 −1.00000 1.00000 −2.61803 −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.bx 2
11.b odd 2 1 8470.2.a.bk 2
11.d odd 10 2 770.2.n.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.n.c 4 11.d odd 10 2
8470.2.a.bk 2 11.b odd 2 1
8470.2.a.bx 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( -1 + T + T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( 4 - 6 T + T^{2} \)
$17$ \( -1 - T + T^{2} \)
$19$ \( 1 - 3 T + T^{2} \)
$23$ \( -4 + 8 T + T^{2} \)
$29$ \( ( 4 + T )^{2} \)
$31$ \( -20 + T^{2} \)
$37$ \( -36 - 6 T + T^{2} \)
$41$ \( 55 + 15 T + T^{2} \)
$43$ \( 11 - 7 T + T^{2} \)
$47$ \( 76 + 18 T + T^{2} \)
$53$ \( -16 + 4 T + T^{2} \)
$59$ \( 5 + 5 T + T^{2} \)
$61$ \( ( -8 + T )^{2} \)
$67$ \( -31 + T + T^{2} \)
$71$ \( -16 + 4 T + T^{2} \)
$73$ \( -49 - 7 T + T^{2} \)
$79$ \( -20 - 10 T + T^{2} \)
$83$ \( -1 + 11 T + T^{2} \)
$89$ \( 31 - 13 T + T^{2} \)
$97$ \( -149 - 3 T + T^{2} \)
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