Properties

Label 8470.2.a.cx
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: 6.6.745749504.1
Defining polynomial: \(x^{6} - 18 x^{4} - 4 x^{3} + 81 x^{2} + 36 x - 44\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 18 x^{4} - 4 x^{3} + 81 x^{2} + 36 x - 44\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 9 \nu - 2 \)\()/4\)
\(\beta_{3}\)\(=\)\( \nu^{2} - 6 \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} - 13 \nu^{2} - 2 \nu + 24 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{5} - \nu^{4} - 15 \nu^{3} + 11 \nu^{2} + 48 \nu - 12 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 6\)
\(\nu^{3}\)\(=\)\(4 \beta_{2} + 9 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(4 \beta_{4} + 13 \beta_{3} + 2 \beta_{1} + 54\)
\(\nu^{5}\)\(=\)\(4 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + 60 \beta_{2} + 89 \beta_{1} + 30\)

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
3.40870
2.67544
0.567932
−1.16996
−2.23874
−3.24337
−1.00000 −3.40870 1.00000 −1.00000 3.40870 1.00000 −1.00000 8.61924 1.00000
1.2 −1.00000 −2.67544 1.00000 −1.00000 2.67544 1.00000 −1.00000 4.15798 1.00000
1.3 −1.00000 −0.567932 1.00000 −1.00000 0.567932 1.00000 −1.00000 −2.67745 1.00000
1.4 −1.00000 1.16996 1.00000 −1.00000 −1.16996 1.00000 −1.00000 −1.63119 1.00000
1.5 −1.00000 2.23874 1.00000 −1.00000 −2.23874 1.00000 −1.00000 2.01195 1.00000
1.6 −1.00000 3.24337 1.00000 −1.00000 −3.24337 1.00000 −1.00000 7.51947 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.cx 6
11.b odd 2 1 8470.2.a.dd yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8470.2.a.cx 6 1.a even 1 1 trivial
8470.2.a.dd yes 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\):

\( T_{3}^{6} - 18 T_{3}^{4} + 4 T_{3}^{3} + 81 T_{3}^{2} - 36 T_{3} - 44 \)
\( T_{13}^{6} - 42 T_{13}^{4} - 8 T_{13}^{3} + 441 T_{13}^{2} + 168 T_{13} - 956 \)
\( T_{17}^{6} - 6 T_{17}^{5} - 51 T_{17}^{4} + 352 T_{17}^{3} - 48 T_{17}^{2} - 1416 T_{17} - 716 \)
\( T_{19}^{6} - 45 T_{19}^{4} - 104 T_{19}^{3} + 99 T_{19}^{2} + 216 T_{19} - 143 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{6} \)
$3$ \( -44 - 36 T + 81 T^{2} + 4 T^{3} - 18 T^{4} + T^{6} \)
$5$ \( ( 1 + T )^{6} \)
$7$ \( ( -1 + T )^{6} \)
$11$ \( T^{6} \)
$13$ \( -956 + 168 T + 441 T^{2} - 8 T^{3} - 42 T^{4} + T^{6} \)
$17$ \( -716 - 1416 T - 48 T^{2} + 352 T^{3} - 51 T^{4} - 6 T^{5} + T^{6} \)
$19$ \( -143 + 216 T + 99 T^{2} - 104 T^{3} - 45 T^{4} + T^{6} \)
$23$ \( 1188 - 1944 T + 729 T^{2} + 72 T^{3} - 54 T^{4} + T^{6} \)
$29$ \( 1600 - 960 T - 816 T^{2} + 448 T^{3} - 12 T^{4} - 12 T^{5} + T^{6} \)
$31$ \( 13312 - 11520 T + 2484 T^{2} + 256 T^{3} - 108 T^{4} + T^{6} \)
$37$ \( 1024 + 3840 T - 1824 T^{2} - 160 T^{3} + 168 T^{4} - 24 T^{5} + T^{6} \)
$41$ \( -34848 - 23760 T + 180 T^{2} + 1536 T^{3} - 108 T^{4} - 12 T^{5} + T^{6} \)
$43$ \( -1052 - 2376 T - 1368 T^{2} - 112 T^{3} + 81 T^{4} + 18 T^{5} + T^{6} \)
$47$ \( 15696 + 9504 T - 6300 T^{2} + 480 T^{3} + 144 T^{4} - 24 T^{5} + T^{6} \)
$53$ \( 62452 + 42768 T - 11880 T^{2} - 752 T^{3} + 405 T^{4} - 36 T^{5} + T^{6} \)
$59$ \( 126949 + 67122 T - 30273 T^{2} + 2524 T^{3} + 171 T^{4} - 30 T^{5} + T^{6} \)
$61$ \( -198272 + 107520 T - 15696 T^{2} - 872 T^{3} + 417 T^{4} - 36 T^{5} + T^{6} \)
$67$ \( -22412 - 30000 T - 14208 T^{2} - 2752 T^{3} - 147 T^{4} + 12 T^{5} + T^{6} \)
$71$ \( -3008 + 5568 T + 3648 T^{2} + 200 T^{3} - 111 T^{4} - 6 T^{5} + T^{6} \)
$73$ \( -28908 + 17928 T + 1800 T^{2} - 1344 T^{3} - 171 T^{4} + 6 T^{5} + T^{6} \)
$79$ \( 12469 - 6504 T - 7761 T^{2} - 1360 T^{3} + 75 T^{4} + 24 T^{5} + T^{6} \)
$83$ \( -132896 + 105000 T - 27303 T^{2} + 2432 T^{3} + 78 T^{4} - 24 T^{5} + T^{6} \)
$89$ \( 216928 - 3024 T - 17388 T^{2} + 976 T^{3} + 324 T^{4} - 36 T^{5} + T^{6} \)
$97$ \( 34192 + 96480 T + 30384 T^{2} - 536 T^{3} - 387 T^{4} + T^{6} \)
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