Properties

Label 8470.2.a.dc
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: 6.6.19898000.1
Defining polynomial: \(x^{6} - x^{5} - 10 x^{4} + 7 x^{3} + 24 x^{2} - 15 x - 5\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} - 10 x^{4} + 7 x^{3} + 24 x^{2} - 15 x - 5\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} - 2 \nu^{3} - 5 \nu^{2} + 8 \nu - 1 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{4} + 2 \nu^{3} + 7 \nu^{2} - 8 \nu - 7 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} - 2 \nu^{4} - 5 \nu^{3} + 8 \nu^{2} - \nu \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{5} + 2 \nu^{4} + 7 \nu^{3} - 10 \nu^{2} - 9 \nu + 6 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(2 \beta_{5} + 2 \beta_{4} + 7 \beta_{3} + 9 \beta_{2} + 2 \beta_{1} + 23\)
\(\nu^{5}\)\(=\)\(9 \beta_{5} + 11 \beta_{4} + 11 \beta_{3} + 15 \beta_{2} + 30 \beta_{1} + 19\)

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
2.86564
1.68692
0.935683
−0.245893
−2.05906
−2.18328
1.00000 −2.86564 1.00000 1.00000 −2.86564 −1.00000 1.00000 5.21187 1.00000
1.2 1.00000 −1.68692 1.00000 1.00000 −1.68692 −1.00000 1.00000 −0.154300 1.00000
1.3 1.00000 −0.935683 1.00000 1.00000 −0.935683 −1.00000 1.00000 −2.12450 1.00000
1.4 1.00000 0.245893 1.00000 1.00000 0.245893 −1.00000 1.00000 −2.93954 1.00000
1.5 1.00000 2.05906 1.00000 1.00000 2.05906 −1.00000 1.00000 1.23973 1.00000
1.6 1.00000 2.18328 1.00000 1.00000 2.18328 −1.00000 1.00000 1.76673 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.dc 6
11.b odd 2 1 8470.2.a.cw 6
11.d odd 10 2 770.2.n.j 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.n.j 12 11.d odd 10 2
8470.2.a.cw 6 11.b odd 2 1
8470.2.a.dc 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\):

\( T_{3}^{6} + T_{3}^{5} - 10 T_{3}^{4} - 7 T_{3}^{3} + 24 T_{3}^{2} + 15 T_{3} - 5 \)
\( T_{13}^{6} - 9 T_{13}^{5} + 15 T_{13}^{4} + 44 T_{13}^{3} - 138 T_{13}^{2} + 88 T_{13} + 4 \)
\( T_{17}^{6} - 9 T_{17}^{5} - 50 T_{17}^{4} + 581 T_{17}^{3} - 42 T_{17}^{2} - 8421 T_{17} + 14431 \)
\( T_{19}^{6} - 12 T_{19}^{5} + 23 T_{19}^{4} + 144 T_{19}^{3} - 635 T_{19}^{2} + 846 T_{19} - 356 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{6} \)
$3$ \( -5 + 15 T + 24 T^{2} - 7 T^{3} - 10 T^{4} + T^{5} + T^{6} \)
$5$ \( ( -1 + T )^{6} \)
$7$ \( ( 1 + T )^{6} \)
$11$ \( T^{6} \)
$13$ \( 4 + 88 T - 138 T^{2} + 44 T^{3} + 15 T^{4} - 9 T^{5} + T^{6} \)
$17$ \( 14431 - 8421 T - 42 T^{2} + 581 T^{3} - 50 T^{4} - 9 T^{5} + T^{6} \)
$19$ \( -356 + 846 T - 635 T^{2} + 144 T^{3} + 23 T^{4} - 12 T^{5} + T^{6} \)
$23$ \( -2624 + 2496 T + 2288 T^{2} + 160 T^{3} - 92 T^{4} - 4 T^{5} + T^{6} \)
$29$ \( 1796 - 1040 T - 590 T^{2} + 252 T^{3} + 35 T^{4} - 15 T^{5} + T^{6} \)
$31$ \( 80 + 440 T - 724 T^{2} + 312 T^{3} - 22 T^{4} - 8 T^{5} + T^{6} \)
$37$ \( 2416 + 2056 T - 180 T^{2} - 384 T^{3} - 62 T^{4} + 4 T^{5} + T^{6} \)
$41$ \( 23056 + 23796 T + 6313 T^{2} - 10 T^{3} - 147 T^{4} - 4 T^{5} + T^{6} \)
$43$ \( -45004 + 42150 T - 11235 T^{2} + 202 T^{3} + 255 T^{4} - 30 T^{5} + T^{6} \)
$47$ \( -4400 + 2400 T + 580 T^{2} - 260 T^{3} - 39 T^{4} + 7 T^{5} + T^{6} \)
$53$ \( -1216 + 832 T + 2236 T^{2} - 68 T^{3} - 130 T^{4} + 6 T^{5} + T^{6} \)
$59$ \( -284 - 386 T + 713 T^{2} + 76 T^{3} - 145 T^{4} - 4 T^{5} + T^{6} \)
$61$ \( -6416 + 24024 T + 44 T^{2} - 1592 T^{3} - 104 T^{4} + 14 T^{5} + T^{6} \)
$67$ \( -44 - 994 T - 1783 T^{2} + 682 T^{3} + 15 T^{4} - 18 T^{5} + T^{6} \)
$71$ \( 12724 + 27672 T - 14562 T^{2} + 1800 T^{3} + 67 T^{4} - 23 T^{5} + T^{6} \)
$73$ \( 55 - 315 T + 594 T^{2} - 481 T^{3} + 170 T^{4} - 23 T^{5} + T^{6} \)
$79$ \( 26356 + 8016 T - 13730 T^{2} + 2336 T^{3} + 3 T^{4} - 21 T^{5} + T^{6} \)
$83$ \( 80351 + 45717 T - 23604 T^{2} + 2543 T^{3} + 78 T^{4} - 25 T^{5} + T^{6} \)
$89$ \( -2420 + 2310 T + 569 T^{2} - 444 T^{3} - 35 T^{4} + 18 T^{5} + T^{6} \)
$97$ \( -89129 - 42979 T + 15946 T^{2} + 2399 T^{3} - 350 T^{4} - 7 T^{5} + T^{6} \)
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