Properties

Label 8470.2.a.cg
Level $8470$
Weight $2$
Character orbit 8470.a
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $3$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 2\)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 4 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{2}\)\(=\)\( -\nu^{2} + 2 \nu + 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{1} + 3\)

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
0.470683
−1.81361
2.34292
−1.00000 −2.77846 1.00000 −1.00000 2.77846 −1.00000 −1.00000 4.71982 1.00000
1.2 −1.00000 0.289169 1.00000 −1.00000 −0.289169 −1.00000 −1.00000 −2.91638 1.00000
1.3 −1.00000 2.48929 1.00000 −1.00000 −2.48929 −1.00000 −1.00000 3.19656 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.cg 3
11.b odd 2 1 8470.2.a.ck yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8470.2.a.cg 3 1.a even 1 1 trivial
8470.2.a.ck yes 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\):

\( T_{3}^{3} - 7 T_{3} + 2 \)
\( T_{13}^{3} - 7 T_{13} - 2 \)
\( T_{17}^{3} + 11 T_{17}^{2} + 24 T_{17} - 32 \)
\( T_{19}^{3} + 5 T_{19}^{2} - 21 T_{19} - 17 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{3} \)
$3$ \( 2 - 7 T + T^{3} \)
$5$ \( ( 1 + T )^{3} \)
$7$ \( ( 1 + T )^{3} \)
$11$ \( T^{3} \)
$13$ \( -2 - 7 T + T^{3} \)
$17$ \( -32 + 24 T + 11 T^{2} + T^{3} \)
$19$ \( -17 - 21 T + 5 T^{2} + T^{3} \)
$23$ \( -106 - 27 T + 4 T^{2} + T^{3} \)
$29$ \( 128 - 64 T + 2 T^{2} + T^{3} \)
$31$ \( T^{3} \)
$37$ \( -8 - 68 T + 2 T^{2} + T^{3} \)
$41$ \( 16 - 28 T + T^{3} \)
$43$ \( -548 - 88 T + 7 T^{2} + T^{3} \)
$47$ \( 128 - 48 T - 8 T^{2} + T^{3} \)
$53$ \( -16 - 16 T - T^{2} + T^{3} \)
$59$ \( 83 - 13 T - 7 T^{2} + T^{3} \)
$61$ \( -316 + 13 T^{2} + T^{3} \)
$67$ \( -992 - 120 T + 9 T^{2} + T^{3} \)
$71$ \( -64 + 64 T - 17 T^{2} + T^{3} \)
$73$ \( -16 + 88 T + 19 T^{2} + T^{3} \)
$79$ \( 41 - T - 9 T^{2} + T^{3} \)
$83$ \( -8 + 5 T + 6 T^{2} + T^{3} \)
$89$ \( 32 + 20 T - 12 T^{2} + T^{3} \)
$97$ \( -128 - 16 T + 7 T^{2} + T^{3} \)
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