Properties

Label 8470.2.a.bo
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: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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 \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.41421
1.41421
−1.00000 −2.82843 1.00000 −1.00000 2.82843 1.00000 −1.00000 5.00000 1.00000
1.2 −1.00000 2.82843 1.00000 −1.00000 −2.82843 1.00000 −1.00000 5.00000 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.bo 2
11.b odd 2 1 770.2.a.i 2
33.d even 2 1 6930.2.a.br 2
44.c even 2 1 6160.2.a.ba 2
55.d odd 2 1 3850.2.a.bi 2
55.e even 4 2 3850.2.c.u 4
77.b even 2 1 5390.2.a.bt 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.a.i 2 11.b odd 2 1
3850.2.a.bi 2 55.d odd 2 1
3850.2.c.u 4 55.e even 4 2
5390.2.a.bt 2 77.b even 2 1
6160.2.a.ba 2 44.c even 2 1
6930.2.a.br 2 33.d even 2 1
8470.2.a.bo 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\):

\( T_{3}^{2} - 8 \)
\( T_{13} + 2 \)
\( T_{17}^{2} + 4 T_{17} - 28 \)
\( T_{19}^{2} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( -8 + T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( ( 2 + T )^{2} \)
$17$ \( -28 + 4 T + T^{2} \)
$19$ \( -8 + T^{2} \)
$23$ \( -8 + T^{2} \)
$29$ \( 28 + 12 T + T^{2} \)
$31$ \( -32 + T^{2} \)
$37$ \( 28 - 12 T + T^{2} \)
$41$ \( -4 + 4 T + T^{2} \)
$43$ \( -16 - 8 T + T^{2} \)
$47$ \( ( 8 + T )^{2} \)
$53$ \( -68 + 4 T + T^{2} \)
$59$ \( ( 8 + T )^{2} \)
$61$ \( -124 + 4 T + T^{2} \)
$67$ \( -112 - 8 T + T^{2} \)
$71$ \( -32 + T^{2} \)
$73$ \( -28 + 4 T + T^{2} \)
$79$ \( 56 - 16 T + T^{2} \)
$83$ \( 32 - 16 T + T^{2} \)
$89$ \( 4 + 12 T + T^{2} \)
$97$ \( -4 - 4 T + T^{2} \)
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