Properties

Label 8470.2.a.by
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.

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

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 −2.23607 1.00000 −1.00000 −2.23607 −1.00000 1.00000 2.00000 −1.00000
1.2 1.00000 2.23607 1.00000 −1.00000 2.23607 −1.00000 1.00000 2.00000 −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.by yes 2
11.b odd 2 1 8470.2.a.bn 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8470.2.a.bn 2 11.b odd 2 1
8470.2.a.by yes 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\):

\( T_{3}^{2} - 5 \)
\( T_{13} + 1 \)
\( T_{17}^{2} + 2 T_{17} - 4 \)
\( T_{19}^{2} - 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( -5 + T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( -4 + 2 T + T^{2} \)
$19$ \( -5 + T^{2} \)
$23$ \( 31 + 12 T + T^{2} \)
$29$ \( -20 + T^{2} \)
$31$ \( 4 + 6 T + T^{2} \)
$37$ \( -80 + T^{2} \)
$41$ \( 20 - 10 T + T^{2} \)
$43$ \( -4 + 2 T + T^{2} \)
$47$ \( -44 - 2 T + T^{2} \)
$53$ \( -44 - 2 T + T^{2} \)
$59$ \( -41 + 4 T + T^{2} \)
$61$ \( 44 - 16 T + T^{2} \)
$67$ \( 20 + 10 T + T^{2} \)
$71$ \( 16 + 12 T + T^{2} \)
$73$ \( 20 - 10 T + T^{2} \)
$79$ \( 59 + 16 T + T^{2} \)
$83$ \( -1 + 4 T + T^{2} \)
$89$ \( 220 + 30 T + T^{2} \)
$97$ \( 44 - 16 T + T^{2} \)
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