Properties

Label 8470.2.a.cl
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: 3.3.892.1
Defining polynomial: \(x^{3} - x^{2} - 8 x + 10\)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 8 x + 10\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} + \nu - 6 \)
\(\beta_{2}\)\(=\)\( -\nu^{2} + \nu + 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{2} + \beta_{1} + 11\)\()/2\)

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
1.31955
−2.91729
2.59774
1.00000 −2.93923 1.00000 −1.00000 −2.93923 1.00000 1.00000 5.63910 −1.00000
1.2 1.00000 −0.406728 1.00000 −1.00000 −0.406728 1.00000 1.00000 −2.83457 −1.00000
1.3 1.00000 3.34596 1.00000 −1.00000 3.34596 1.00000 1.00000 8.19547 −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.cl 3
11.b odd 2 1 770.2.a.l 3
33.d even 2 1 6930.2.a.cl 3
44.c even 2 1 6160.2.a.bi 3
55.d odd 2 1 3850.2.a.bu 3
55.e even 4 2 3850.2.c.z 6
77.b even 2 1 5390.2.a.bz 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.a.l 3 11.b odd 2 1
3850.2.a.bu 3 55.d odd 2 1
3850.2.c.z 6 55.e even 4 2
5390.2.a.bz 3 77.b even 2 1
6160.2.a.bi 3 44.c even 2 1
6930.2.a.cl 3 33.d even 2 1
8470.2.a.cl 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\):

\( T_{3}^{3} - 10 T_{3} - 4 \)
\( T_{13}^{3} - 40 T_{13} + 32 \)
\( T_{17}^{3} - 8 T_{17}^{2} - 12 T_{17} + 128 \)
\( T_{19}^{3} - 2 T_{19}^{2} - 30 T_{19} + 56 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{3} \)
$3$ \( -4 - 10 T + T^{3} \)
$5$ \( ( 1 + T )^{3} \)
$7$ \( ( -1 + T )^{3} \)
$11$ \( T^{3} \)
$13$ \( 32 - 40 T + T^{3} \)
$17$ \( 128 - 12 T - 8 T^{2} + T^{3} \)
$19$ \( 56 - 30 T - 2 T^{2} + T^{3} \)
$23$ \( 56 - 30 T - 2 T^{2} + T^{3} \)
$29$ \( -428 - 82 T + 4 T^{2} + T^{3} \)
$31$ \( ( -6 + T )^{3} \)
$37$ \( 124 - 50 T - 4 T^{2} + T^{3} \)
$41$ \( -160 + 98 T - 18 T^{2} + T^{3} \)
$43$ \( -16 - 28 T + 8 T^{2} + T^{3} \)
$47$ \( -64 - 48 T - 2 T^{2} + T^{3} \)
$53$ \( 428 - 82 T - 4 T^{2} + T^{3} \)
$59$ \( 608 - 64 T - 10 T^{2} + T^{3} \)
$61$ \( 64 + 84 T + 20 T^{2} + T^{3} \)
$67$ \( ( -8 + T )^{3} \)
$71$ \( 32 - 104 T - 8 T^{2} + T^{3} \)
$73$ \( 784 - 140 T - 4 T^{2} + T^{3} \)
$79$ \( 2272 - 262 T - 6 T^{2} + T^{3} \)
$83$ \( -160 + 152 T - 24 T^{2} + T^{3} \)
$89$ \( -40 - 28 T + 6 T^{2} + T^{3} \)
$97$ \( 100 - 10 T - 8 T^{2} + T^{3} \)
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