Properties

Label 8470.2.a.bi
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: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(\beta = 2\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.73205
1.73205
−1.00000 −1.00000 1.00000 1.00000 1.00000 −1.00000 −1.00000 −2.00000 −1.00000
1.2 −1.00000 −1.00000 1.00000 1.00000 1.00000 −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.bi 2
11.b odd 2 1 8470.2.a.bw yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8470.2.a.bi 2 1.a even 1 1 trivial
8470.2.a.bw 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\):

\( T_{3} + 1 \)
\( T_{13}^{2} + 2 T_{13} - 11 \)
\( T_{17}^{2} - 8 T_{17} + 4 \)
\( T_{19}^{2} - 2 T_{19} - 47 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( -11 + 2 T + T^{2} \)
$17$ \( 4 - 8 T + T^{2} \)
$19$ \( -47 - 2 T + T^{2} \)
$23$ \( -11 + 2 T + T^{2} \)
$29$ \( -48 + T^{2} \)
$31$ \( 4 + 8 T + T^{2} \)
$37$ \( -44 + 4 T + T^{2} \)
$41$ \( -104 + 4 T + T^{2} \)
$43$ \( -104 - 4 T + T^{2} \)
$47$ \( 4 + 8 T + T^{2} \)
$53$ \( 4 + 8 T + T^{2} \)
$59$ \( -39 - 6 T + T^{2} \)
$61$ \( -44 - 4 T + T^{2} \)
$67$ \( -92 + 8 T + T^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( 4 - 8 T + T^{2} \)
$79$ \( 109 - 22 T + T^{2} \)
$83$ \( ( -1 + T )^{2} \)
$89$ \( 184 + 28 T + T^{2} \)
$97$ \( 16 + 16 T + T^{2} \)
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