Properties

Label 8470.2.a.cy
Level $8470$
Weight $2$
Character orbit 8470.a
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
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: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.13298000.1
Defining polynomial: \(x^{6} - x^{5} - 10 x^{4} + 3 x^{3} + 26 x^{2} + 13 x - 1\)
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.

\(f(q)\) \(=\) \( q - q^{2} + ( -\beta_{2} - \beta_{5} ) q^{3} + q^{4} + q^{5} + ( \beta_{2} + \beta_{5} ) q^{6} - q^{7} - q^{8} + ( 4 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} ) q^{9} +O(q^{10})\) \( q - q^{2} + ( -\beta_{2} - \beta_{5} ) q^{3} + q^{4} + q^{5} + ( \beta_{2} + \beta_{5} ) q^{6} - q^{7} - q^{8} + ( 4 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} ) q^{9} - q^{10} + ( -\beta_{2} - \beta_{5} ) q^{12} + ( -1 - \beta_{1} - \beta_{3} + \beta_{5} ) q^{13} + q^{14} + ( -\beta_{2} - \beta_{5} ) q^{15} + q^{16} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{17} + ( -4 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} ) q^{18} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} ) q^{19} + q^{20} + ( \beta_{2} + \beta_{5} ) q^{21} + ( -2 + 2 \beta_{1} + 2 \beta_{5} ) q^{23} + ( \beta_{2} + \beta_{5} ) q^{24} + q^{25} + ( 1 + \beta_{1} + \beta_{3} - \beta_{5} ) q^{26} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{27} - q^{28} + ( 2 + \beta_{2} + 4 \beta_{3} - \beta_{4} + \beta_{5} ) q^{29} + ( \beta_{2} + \beta_{5} ) q^{30} + ( \beta_{2} + \beta_{4} + \beta_{5} ) q^{31} - q^{32} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{34} - q^{35} + ( 4 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} ) q^{36} + ( -4 - \beta_{2} - 3 \beta_{3} - 2 \beta_{4} ) q^{37} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} ) q^{38} + ( -5 + 2 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{5} ) q^{39} - q^{40} + ( -4 + \beta_{1} - 3 \beta_{3} + \beta_{4} ) q^{41} + ( -\beta_{2} - \beta_{5} ) q^{42} + ( -4 - 2 \beta_{1} - 4 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{43} + ( 4 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} ) q^{45} + ( 2 - 2 \beta_{1} - 2 \beta_{5} ) q^{46} + ( 4 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{47} + ( -\beta_{2} - \beta_{5} ) q^{48} + q^{49} - q^{50} + ( 5 - \beta_{1} + 3 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{51} + ( -1 - \beta_{1} - \beta_{3} + \beta_{5} ) q^{52} + ( \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{53} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 3 \beta_{5} ) q^{54} + q^{56} + ( -8 + 3 \beta_{1} - 3 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{57} + ( -2 - \beta_{2} - 4 \beta_{3} + \beta_{4} - \beta_{5} ) q^{58} + ( -2 \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{59} + ( -\beta_{2} - \beta_{5} ) q^{60} + ( -3 - \beta_{2} + \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{61} + ( -\beta_{2} - \beta_{4} - \beta_{5} ) q^{62} + ( -4 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} ) q^{63} + q^{64} + ( -1 - \beta_{1} - \beta_{3} + \beta_{5} ) q^{65} + ( -4 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} ) q^{67} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{68} + ( -6 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 6 \beta_{5} ) q^{69} + q^{70} + ( 3 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{71} + ( -4 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} ) q^{72} + ( 1 - 5 \beta_{1} + 3 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{73} + ( 4 + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{74} + ( -\beta_{2} - \beta_{5} ) q^{75} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} ) q^{76} + ( 5 - 2 \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{5} ) q^{78} + ( \beta_{1} - \beta_{2} - 3 \beta_{3} + 5 \beta_{4} - \beta_{5} ) q^{79} + q^{80} + ( 3 - \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - 3 \beta_{5} ) q^{81} + ( 4 - \beta_{1} + 3 \beta_{3} - \beta_{4} ) q^{82} + ( 2 + 4 \beta_{1} - \beta_{2} + 5 \beta_{3} + \beta_{4} ) q^{83} + ( \beta_{2} + \beta_{5} ) q^{84} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{85} + ( 4 + 2 \beta_{1} + 4 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{86} + ( -6 + 5 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} ) q^{87} + ( -5 + 4 \beta_{1} - \beta_{2} - 6 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{89} + ( -4 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} ) q^{90} + ( 1 + \beta_{1} + \beta_{3} - \beta_{5} ) q^{91} + ( -2 + 2 \beta_{1} + 2 \beta_{5} ) q^{92} + ( -8 + \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{93} + ( -4 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{94} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} ) q^{95} + ( \beta_{2} + \beta_{5} ) q^{96} + ( -5 + \beta_{1} - 3 \beta_{2} - 4 \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{97} - q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{2} + q^{3} + 6q^{4} + 6q^{5} - q^{6} - 6q^{7} - 6q^{8} + 15q^{9} + O(q^{10}) \) \( 6q - 6q^{2} + q^{3} + 6q^{4} + 6q^{5} - q^{6} - 6q^{7} - 6q^{8} + 15q^{9} - 6q^{10} + q^{12} - 2q^{13} + 6q^{14} + q^{15} + 6q^{16} - 7q^{17} - 15q^{18} - 11q^{19} + 6q^{20} - q^{21} - 6q^{23} - q^{24} + 6q^{25} + 2q^{26} + 4q^{27} - 6q^{28} - 2q^{29} - q^{30} - 6q^{32} + 7q^{34} - 6q^{35} + 15q^{36} - 14q^{37} + 11q^{38} - 20q^{39} - 6q^{40} - 13q^{41} + q^{42} - 19q^{43} + 15q^{45} + 6q^{46} + 22q^{47} + q^{48} + 6q^{49} - 6q^{50} + 14q^{51} - 2q^{52} - 10q^{53} - 4q^{54} + 6q^{56} - 32q^{57} + 2q^{58} - 7q^{59} + q^{60} - 22q^{61} - 15q^{63} + 6q^{64} - 2q^{65} + 5q^{67} - 7q^{68} - 36q^{69} + 6q^{70} + 8q^{71} - 15q^{72} - 13q^{73} + 14q^{74} + q^{75} - 11q^{76} + 20q^{78} + 16q^{79} + 6q^{80} + 18q^{81} + 13q^{82} + 5q^{83} - q^{84} - 7q^{85} + 19q^{86} - 14q^{87} + q^{89} - 15q^{90} + 2q^{91} - 6q^{92} - 42q^{93} - 22q^{94} - 11q^{95} - q^{96} - 3q^{97} - 6q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} - 10 x^{4} + 3 x^{3} + 26 x^{2} + 13 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} + 4 \nu^{4} + 8 \nu^{3} - 22 \nu^{2} - 20 \nu - 3 \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{5} + 3 \nu^{4} + 16 \nu^{3} - 14 \nu^{2} - 30 \nu - 6 \)\()/5\)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{5} + 3 \nu^{4} + 21 \nu^{3} - 14 \nu^{2} - 60 \nu - 16 \)\()/5\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{5} - 7 \nu^{4} - 24 \nu^{3} + 41 \nu^{2} + 45 \nu - 11 \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} + \beta_{3} + \beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{4} - \beta_{3} + 6 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(6 \beta_{5} + 5 \beta_{3} + 8 \beta_{2} + 8 \beta_{1} + 24\)
\(\nu^{5}\)\(=\)\(2 \beta_{5} + 8 \beta_{4} - 10 \beta_{3} + 5 \beta_{2} + 38 \beta_{1} + 21\)

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
−2.27063
2.79700
−1.24667
2.42266
0.0677009
−0.770059
−1.00000 −2.80837 1.00000 1.00000 2.80837 −1.00000 −1.00000 4.88695 −1.00000
1.2 −1.00000 −2.64424 1.00000 1.00000 2.64424 −1.00000 −1.00000 3.99201 −1.00000
1.3 −1.00000 −0.418877 1.00000 1.00000 0.418877 −1.00000 −1.00000 −2.82454 −1.00000
1.4 −1.00000 1.17142 1.00000 1.00000 −1.17142 −1.00000 −1.00000 −1.62777 −1.00000
1.5 −1.00000 2.44508 1.00000 1.00000 −2.44508 −1.00000 −1.00000 2.97843 −1.00000
1.6 −1.00000 3.25498 1.00000 1.00000 −3.25498 −1.00000 −1.00000 7.59492 −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.cy 6
11.b odd 2 1 8470.2.a.de 6
11.d odd 10 2 770.2.n.i 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.n.i 12 11.d odd 10 2
8470.2.a.cy 6 1.a even 1 1 trivial
8470.2.a.de 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\):

\( T_{3}^{6} - T_{3}^{5} - 16 T_{3}^{4} + 13 T_{3}^{3} + 66 T_{3}^{2} - 45 T_{3} - 29 \)
\( T_{13}^{6} + 2 T_{13}^{5} - 32 T_{13}^{4} - 94 T_{13}^{3} + 160 T_{13}^{2} + 704 T_{13} + 484 \)
\( T_{17}^{6} + 7 T_{17}^{5} - 20 T_{17}^{4} - 183 T_{17}^{3} - 308 T_{17}^{2} - 139 T_{17} + 11 \)
\( T_{19}^{6} + 11 T_{19}^{5} + 20 T_{19}^{4} - 177 T_{19}^{3} - 896 T_{19}^{2} - 1445 T_{19} - 725 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{6} \)
$3$ \( -29 - 45 T + 66 T^{2} + 13 T^{3} - 16 T^{4} - T^{5} + T^{6} \)
$5$ \( ( -1 + T )^{6} \)
$7$ \( ( 1 + T )^{6} \)
$11$ \( T^{6} \)
$13$ \( 484 + 704 T + 160 T^{2} - 94 T^{3} - 32 T^{4} + 2 T^{5} + T^{6} \)
$17$ \( 11 - 139 T - 308 T^{2} - 183 T^{3} - 20 T^{4} + 7 T^{5} + T^{6} \)
$19$ \( -725 - 1445 T - 896 T^{2} - 177 T^{3} + 20 T^{4} + 11 T^{5} + T^{6} \)
$23$ \( -1856 + 480 T + 768 T^{2} - 184 T^{3} - 56 T^{4} + 6 T^{5} + T^{6} \)
$29$ \( 3020 + 3340 T + 776 T^{2} - 178 T^{3} - 62 T^{4} + 2 T^{5} + T^{6} \)
$31$ \( -284 - 468 T + 316 T^{2} + 58 T^{3} - 42 T^{4} + T^{6} \)
$37$ \( -4 + 276 T - 4320 T^{2} - 1334 T^{3} - 52 T^{4} + 14 T^{5} + T^{6} \)
$41$ \( 3649 + 225 T - 1254 T^{2} - 351 T^{3} + 16 T^{4} + 13 T^{5} + T^{6} \)
$43$ \( -20719 - 24251 T - 9780 T^{2} - 1415 T^{3} + 20 T^{4} + 19 T^{5} + T^{6} \)
$47$ \( 49856 + 9824 T - 12768 T^{2} + 1928 T^{3} + 40 T^{4} - 22 T^{5} + T^{6} \)
$53$ \( -244 + 1880 T + 800 T^{2} - 338 T^{3} - 50 T^{4} + 10 T^{5} + T^{6} \)
$59$ \( 95 + 175 T - 124 T^{2} - 293 T^{3} - 72 T^{4} + 7 T^{5} + T^{6} \)
$61$ \( -112156 - 92484 T - 25280 T^{2} - 2478 T^{3} + 44 T^{4} + 22 T^{5} + T^{6} \)
$67$ \( -5821 + 2633 T + 6014 T^{2} + 1179 T^{3} - 226 T^{4} - 5 T^{5} + T^{6} \)
$71$ \( -13036 - 5160 T + 3516 T^{2} + 1086 T^{3} - 174 T^{4} - 8 T^{5} + T^{6} \)
$73$ \( 71779 + 46377 T - 1552 T^{2} - 2443 T^{3} - 170 T^{4} + 13 T^{5} + T^{6} \)
$79$ \( -848020 - 198380 T + 19424 T^{2} + 3822 T^{3} - 250 T^{4} - 16 T^{5} + T^{6} \)
$83$ \( -53629 + 2471 T + 6724 T^{2} + 27 T^{3} - 194 T^{4} - 5 T^{5} + T^{6} \)
$89$ \( -362975 + 55835 T + 31116 T^{2} - 179 T^{3} - 338 T^{4} - T^{5} + T^{6} \)
$97$ \( -4649 - 10511 T - 8088 T^{2} - 2437 T^{3} - 240 T^{4} + 3 T^{5} + T^{6} \)
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