Properties

Label 8470.2.a.cn
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.

\(f(q)\) \(=\) \( q + q^{2} + ( 1 + \beta_{2} ) q^{3} + q^{4} + q^{5} + ( 1 + \beta_{2} ) q^{6} + q^{7} + q^{8} + ( 3 - \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + q^{2} + ( 1 + \beta_{2} ) q^{3} + q^{4} + q^{5} + ( 1 + \beta_{2} ) q^{6} + q^{7} + q^{8} + ( 3 - \beta_{1} + \beta_{2} ) q^{9} + q^{10} + ( 1 + \beta_{2} ) q^{12} + q^{14} + ( 1 + \beta_{2} ) q^{15} + q^{16} + ( 2 - \beta_{1} + \beta_{2} ) q^{17} + ( 3 - \beta_{1} + \beta_{2} ) q^{18} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{19} + q^{20} + ( 1 + \beta_{2} ) q^{21} + ( 3 - \beta_{1} ) q^{23} + ( 1 + \beta_{2} ) q^{24} + q^{25} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{27} + q^{28} + ( -1 + \beta_{1} ) q^{29} + ( 1 + \beta_{2} ) q^{30} + 2 \beta_{1} q^{31} + q^{32} + ( 2 - \beta_{1} + \beta_{2} ) q^{34} + q^{35} + ( 3 - \beta_{1} + \beta_{2} ) q^{36} + ( 3 - \beta_{1} ) q^{37} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{38} + q^{40} + ( -1 - 3 \beta_{2} ) q^{41} + ( 1 + \beta_{2} ) q^{42} + ( \beta_{1} + \beta_{2} ) q^{43} + ( 3 - \beta_{1} + \beta_{2} ) q^{45} + ( 3 - \beta_{1} ) q^{46} + ( 4 - \beta_{1} + 3 \beta_{2} ) q^{47} + ( 1 + \beta_{2} ) q^{48} + q^{49} + q^{50} + ( 6 - 2 \beta_{1} + 4 \beta_{2} ) q^{51} + ( -1 + 3 \beta_{1} ) q^{53} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{54} + q^{56} + ( -8 + 3 \beta_{1} - \beta_{2} ) q^{57} + ( -1 + \beta_{1} ) q^{58} + ( -4 + \beta_{1} - \beta_{2} ) q^{59} + ( 1 + \beta_{2} ) q^{60} + ( -2 + \beta_{1} - 3 \beta_{2} ) q^{61} + 2 \beta_{1} q^{62} + ( 3 - \beta_{1} + \beta_{2} ) q^{63} + q^{64} + ( -6 + 2 \beta_{1} - 2 \beta_{2} ) q^{67} + ( 2 - \beta_{1} + \beta_{2} ) q^{68} + ( 2 - \beta_{1} + 5 \beta_{2} ) q^{69} + q^{70} -4 \beta_{2} q^{71} + ( 3 - \beta_{1} + \beta_{2} ) q^{72} + ( -4 - \beta_{1} - \beta_{2} ) q^{73} + ( 3 - \beta_{1} ) q^{74} + ( 1 + \beta_{2} ) q^{75} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{76} + ( 1 - 3 \beta_{1} + 2 \beta_{2} ) q^{79} + q^{80} + ( 3 - \beta_{1} + 5 \beta_{2} ) q^{81} + ( -1 - 3 \beta_{2} ) q^{82} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{83} + ( 1 + \beta_{2} ) q^{84} + ( 2 - \beta_{1} + \beta_{2} ) q^{85} + ( \beta_{1} + \beta_{2} ) q^{86} + ( \beta_{1} - 3 \beta_{2} ) q^{87} + ( 6 + 4 \beta_{2} ) q^{89} + ( 3 - \beta_{1} + \beta_{2} ) q^{90} + ( 3 - \beta_{1} ) q^{92} + ( 2 + 2 \beta_{1} - 4 \beta_{2} ) q^{93} + ( 4 - \beta_{1} + 3 \beta_{2} ) q^{94} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{95} + ( 1 + \beta_{2} ) q^{96} + ( 15 - \beta_{1} ) q^{97} + q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 2q^{3} + 3q^{4} + 3q^{5} + 2q^{6} + 3q^{7} + 3q^{8} + 7q^{9} + O(q^{10}) \) \( 3q + 3q^{2} + 2q^{3} + 3q^{4} + 3q^{5} + 2q^{6} + 3q^{7} + 3q^{8} + 7q^{9} + 3q^{10} + 2q^{12} + 3q^{14} + 2q^{15} + 3q^{16} + 4q^{17} + 7q^{18} + 6q^{19} + 3q^{20} + 2q^{21} + 8q^{23} + 2q^{24} + 3q^{25} + 8q^{27} + 3q^{28} - 2q^{29} + 2q^{30} + 2q^{31} + 3q^{32} + 4q^{34} + 3q^{35} + 7q^{36} + 8q^{37} + 6q^{38} + 3q^{40} + 2q^{42} + 7q^{45} + 8q^{46} + 8q^{47} + 2q^{48} + 3q^{49} + 3q^{50} + 12q^{51} + 8q^{54} + 3q^{56} - 20q^{57} - 2q^{58} - 10q^{59} + 2q^{60} - 2q^{61} + 2q^{62} + 7q^{63} + 3q^{64} - 14q^{67} + 4q^{68} + 3q^{70} + 4q^{71} + 7q^{72} - 12q^{73} + 8q^{74} + 2q^{75} + 6q^{76} - 2q^{79} + 3q^{80} + 3q^{81} + 4q^{83} + 2q^{84} + 4q^{85} + 4q^{87} + 14q^{89} + 7q^{90} + 8q^{92} + 12q^{93} + 8q^{94} + 6q^{95} + 2q^{96} + 44q^{97} + 3q^{98} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} + \nu - 3 \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 6\)\()/2\)

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
2.34292
−1.81361
1.00000 −2.24914 1.00000 1.00000 −2.24914 1.00000 1.00000 2.05863 1.00000
1.2 1.00000 1.14637 1.00000 1.00000 1.14637 1.00000 1.00000 −1.68585 1.00000
1.3 1.00000 3.10278 1.00000 1.00000 3.10278 1.00000 1.00000 6.62721 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.cn yes 3
11.b odd 2 1 8470.2.a.ch 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8470.2.a.ch 3 11.b odd 2 1
8470.2.a.cn yes 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\):

\( T_{3}^{3} - 2 T_{3}^{2} - 6 T_{3} + 8 \)
\( T_{13} \)
\( T_{17}^{3} - 4 T_{17}^{2} - 12 T_{17} + 16 \)
\( T_{19}^{3} - 6 T_{19}^{2} - 22 T_{19} + 136 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{3} \)
$3$ \( 8 - 6 T - 2 T^{2} + T^{3} \)
$5$ \( ( -1 + T )^{3} \)
$7$ \( ( -1 + T )^{3} \)
$11$ \( T^{3} \)
$13$ \( T^{3} \)
$17$ \( 16 - 12 T - 4 T^{2} + T^{3} \)
$19$ \( 136 - 22 T - 6 T^{2} + T^{3} \)
$23$ \( 44 + 6 T - 8 T^{2} + T^{3} \)
$29$ \( -32 - 14 T + 2 T^{2} + T^{3} \)
$31$ \( -136 - 60 T - 2 T^{2} + T^{3} \)
$37$ \( 44 + 6 T - 8 T^{2} + T^{3} \)
$41$ \( -92 - 66 T + T^{3} \)
$43$ \( 16 - 28 T + T^{3} \)
$47$ \( -16 - 44 T - 8 T^{2} + T^{3} \)
$53$ \( -596 - 138 T + T^{3} \)
$59$ \( -16 + 16 T + 10 T^{2} + T^{3} \)
$61$ \( 128 - 64 T + 2 T^{2} + T^{3} \)
$67$ \( -184 - 4 T + 14 T^{2} + T^{3} \)
$71$ \( -64 - 112 T - 4 T^{2} + T^{3} \)
$73$ \( -64 + 20 T + 12 T^{2} + T^{3} \)
$79$ \( 184 - 134 T + 2 T^{2} + T^{3} \)
$83$ \( 128 - 64 T - 4 T^{2} + T^{3} \)
$89$ \( 664 - 52 T - 14 T^{2} + T^{3} \)
$97$ \( -2908 + 630 T - 44 T^{2} + T^{3} \)
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