Properties

Label 8470.2.a.df
Level $8470$
Weight $2$
Character orbit 8470.a
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.6.10784448.1
Defining polynomial: \(x^{6} - 11 x^{4} - 4 x^{3} + 31 x^{2} + 22 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 11 x^{4} - 4 x^{3} + 31 x^{2} + 22 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} - 6 \nu - 2 \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{4} - \nu^{3} + 7 \nu^{2} + 6 \nu - 4 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} + \nu^{3} - 5 \nu^{2} - 8 \nu - 4 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{4} + \nu^{3} + 7 \nu^{2} - 2 \nu - 8 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{5} - \nu^{4} - 17 \nu^{3} + 3 \nu^{2} + 32 \nu + 6 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} - \beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_{1} + 8\)\()/2\)
\(\nu^{3}\)\(=\)\(3 \beta_{4} - 3 \beta_{2} - 2 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\((\)\(7 \beta_{4} + 14 \beta_{3} + 3 \beta_{2} - 9 \beta_{1} + 44\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(2 \beta_{5} + 37 \beta_{4} + 4 \beta_{3} - 35 \beta_{2} - 21 \beta_{1} + 38\)\()/2\)

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
0.0816388
2.40765
−1.84763
−0.930827
2.58124
−2.29207
1.00000 −1.34292 1.00000 1.00000 −1.34292 1.00000 1.00000 −1.19656 1.00000
1.2 1.00000 −1.34292 1.00000 1.00000 −1.34292 1.00000 1.00000 −1.19656 1.00000
1.3 1.00000 0.529317 1.00000 1.00000 0.529317 1.00000 1.00000 −2.71982 1.00000
1.4 1.00000 0.529317 1.00000 1.00000 0.529317 1.00000 1.00000 −2.71982 1.00000
1.5 1.00000 2.81361 1.00000 1.00000 2.81361 1.00000 1.00000 4.91638 1.00000
1.6 1.00000 2.81361 1.00000 1.00000 2.81361 1.00000 1.00000 4.91638 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.df yes 6
11.b odd 2 1 8470.2.a.cz 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8470.2.a.cz 6 11.b odd 2 1
8470.2.a.df yes 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\):

\( T_{3}^{3} - 2 T_{3}^{2} - 3 T_{3} + 2 \)
\( T_{13}^{6} - 42 T_{13}^{4} + 441 T_{13}^{2} - 108 \)
\( T_{17}^{3} - T_{17}^{2} - 24 T_{17} - 38 \)
\( T_{19}^{6} - 41 T_{19}^{4} + 80 T_{19}^{3} + 283 T_{19}^{2} - 944 T_{19} + 709 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{6} \)
$3$ \( ( 2 - 3 T - 2 T^{2} + T^{3} )^{2} \)
$5$ \( ( -1 + T )^{6} \)
$7$ \( ( -1 + T )^{6} \)
$11$ \( T^{6} \)
$13$ \( -108 + 441 T^{2} - 42 T^{4} + T^{6} \)
$17$ \( ( -38 - 24 T - T^{2} + T^{3} )^{2} \)
$19$ \( 709 - 944 T + 283 T^{2} + 80 T^{3} - 41 T^{4} + T^{6} \)
$23$ \( 2932 - 2280 T + 109 T^{2} + 268 T^{3} - 50 T^{4} - 4 T^{5} + T^{6} \)
$29$ \( 256 - 640 T - 352 T^{2} + 336 T^{3} - 36 T^{4} - 8 T^{5} + T^{6} \)
$31$ \( ( 8 - 6 T - 2 T^{2} + T^{3} )^{2} \)
$37$ \( -512 - 384 T + 400 T^{2} + 112 T^{3} - 56 T^{4} - 4 T^{5} + T^{6} \)
$41$ \( -32 + 432 T - 876 T^{2} + 320 T^{3} - 12 T^{5} + T^{6} \)
$43$ \( 1012 - 536 T - 1388 T^{2} - 568 T^{3} - 59 T^{4} + 6 T^{5} + T^{6} \)
$47$ \( -26864 - 23456 T - 124 T^{2} + 1488 T^{3} - 60 T^{4} - 16 T^{5} + T^{6} \)
$53$ \( 436 + 144 T - 660 T^{2} + 248 T^{3} + 9 T^{4} - 12 T^{5} + T^{6} \)
$59$ \( -19679 + 20650 T - 6961 T^{2} + 588 T^{3} + 111 T^{4} - 22 T^{5} + T^{6} \)
$61$ \( -5888 - 3328 T + 4464 T^{2} + 48 T^{3} - 175 T^{4} + 4 T^{5} + T^{6} \)
$67$ \( 45748 + 6272 T - 11028 T^{2} + 1824 T^{3} + 17 T^{4} - 20 T^{5} + T^{6} \)
$71$ \( -402176 - 113536 T + 12528 T^{2} + 3240 T^{3} - 247 T^{4} - 14 T^{5} + T^{6} \)
$73$ \( -3788 - 6040 T - 1604 T^{2} + 736 T^{3} + 25 T^{4} - 18 T^{5} + T^{6} \)
$79$ \( -197027 + 141536 T - 33241 T^{2} + 2112 T^{3} + 219 T^{4} - 32 T^{5} + T^{6} \)
$83$ \( -25136 + 13376 T + 6081 T^{2} - 1680 T^{3} - 130 T^{4} + 16 T^{5} + T^{6} \)
$89$ \( 529504 - 336944 T + 40452 T^{2} + 2352 T^{3} - 424 T^{4} - 4 T^{5} + T^{6} \)
$97$ \( 46912 + 55776 T + 14488 T^{2} - 188 T^{3} - 251 T^{4} - 4 T^{5} + T^{6} \)
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