Properties

Label 8470.2.a.bz
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

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 −1.73205 1.00000 −1.00000 −1.73205 1.00000 1.00000 0 −1.00000
1.2 1.00000 1.73205 1.00000 −1.00000 1.73205 1.00000 1.00000 0 −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.bz yes 2
11.b odd 2 1 8470.2.a.bm 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8470.2.a.bm 2 11.b odd 2 1
8470.2.a.bz 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( -3 + T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( 13 + 8 T + T^{2} \)
$17$ \( -2 - 2 T + T^{2} \)
$19$ \( 1 + 4 T + T^{2} \)
$23$ \( ( 5 + T )^{2} \)
$29$ \( ( -8 + T )^{2} \)
$31$ \( -2 + 2 T + T^{2} \)
$37$ \( ( -6 + T )^{2} \)
$41$ \( -26 + 2 T + T^{2} \)
$43$ \( 46 + 14 T + T^{2} \)
$47$ \( 6 + 6 T + T^{2} \)
$53$ \( -26 + 2 T + T^{2} \)
$59$ \( 69 + 18 T + T^{2} \)
$61$ \( -32 - 8 T + T^{2} \)
$67$ \( -74 + 2 T + T^{2} \)
$71$ \( -48 + T^{2} \)
$73$ \( -138 + 6 T + T^{2} \)
$79$ \( ( 7 + T )^{2} \)
$83$ \( ( -5 + T )^{2} \)
$89$ \( -50 + 10 T + T^{2} \)
$97$ \( -32 - 8 T + T^{2} \)
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