Properties

Label 8470.2.a.da
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: 6.6.38498000.1
Defining polynomial: \(x^{6} - x^{5} - 12 x^{4} + 5 x^{3} + 38 x^{2} - x - 19\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} - 12 x^{4} + 5 x^{3} + 38 x^{2} - x - 19\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} - 2 \nu^{3} - 6 \nu^{2} + 8 \nu + 1 \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} - 2 \nu^{4} - 6 \nu^{3} + 8 \nu^{2} + \nu \)\()/5\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{4} + 2 \nu^{3} + 11 \nu^{2} - 13 \nu - 21 \)\()/5\)
\(\beta_{5}\)\(=\)\((\)\( -2 \nu^{5} + 4 \nu^{4} + 17 \nu^{3} - 21 \nu^{2} - 32 \nu + 10 \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} + 7 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(2 \beta_{5} + 8 \beta_{4} + 4 \beta_{3} + 13 \beta_{2} + 12 \beta_{1} + 27\)
\(\nu^{5}\)\(=\)\(10 \beta_{5} + 14 \beta_{4} + 25 \beta_{3} + 24 \beta_{2} + 57 \beta_{1} + 34\)

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
−2.29803
−1.95171
−0.824369
0.752765
2.15804
3.16330
1.00000 −3.29803 1.00000 −1.00000 −3.29803 1.00000 1.00000 7.87703 −1.00000
1.2 1.00000 −2.95171 1.00000 −1.00000 −2.95171 1.00000 1.00000 5.71258 −1.00000
1.3 1.00000 −1.82437 1.00000 −1.00000 −1.82437 1.00000 1.00000 0.328323 −1.00000
1.4 1.00000 −0.247235 1.00000 −1.00000 −0.247235 1.00000 1.00000 −2.93888 −1.00000
1.5 1.00000 1.15804 1.00000 −1.00000 1.15804 1.00000 1.00000 −1.65894 −1.00000
1.6 1.00000 2.16330 1.00000 −1.00000 2.16330 1.00000 1.00000 1.67988 −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.da 6
11.b odd 2 1 8470.2.a.cu 6
11.c even 5 2 770.2.n.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.n.h 12 11.c even 5 2
8470.2.a.cu 6 11.b odd 2 1
8470.2.a.da 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\):

\( T_{3}^{6} + 5 T_{3}^{5} - 2 T_{3}^{4} - 33 T_{3}^{3} - 14 T_{3}^{2} + 43 T_{3} + 11 \)
\( T_{13}^{6} - 38 T_{13}^{4} - 34 T_{13}^{3} + 316 T_{13}^{2} + 256 T_{13} - 556 \)
\( T_{17}^{6} + 15 T_{17}^{5} + 36 T_{17}^{4} - 305 T_{17}^{3} - 1076 T_{17}^{2} + 725 T_{17} + 545 \)
\( T_{19}^{6} + 13 T_{19}^{5} + 40 T_{19}^{4} - 103 T_{19}^{3} - 692 T_{19}^{2} - 963 T_{19} - 401 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{6} \)
$3$ \( 11 + 43 T - 14 T^{2} - 33 T^{3} - 2 T^{4} + 5 T^{5} + T^{6} \)
$5$ \( ( 1 + T )^{6} \)
$7$ \( ( -1 + T )^{6} \)
$11$ \( T^{6} \)
$13$ \( -556 + 256 T + 316 T^{2} - 34 T^{3} - 38 T^{4} + T^{6} \)
$17$ \( 545 + 725 T - 1076 T^{2} - 305 T^{3} + 36 T^{4} + 15 T^{5} + T^{6} \)
$19$ \( -401 - 963 T - 692 T^{2} - 103 T^{3} + 40 T^{4} + 13 T^{5} + T^{6} \)
$23$ \( 3520 - 4640 T + 1664 T^{2} + 24 T^{3} - 80 T^{4} + 2 T^{5} + T^{6} \)
$29$ \( -484 - 196 T + 568 T^{2} - 70 T^{3} - 52 T^{4} + 4 T^{5} + T^{6} \)
$31$ \( 2420 + 7700 T + 2476 T^{2} - 262 T^{3} - 116 T^{4} + 2 T^{5} + T^{6} \)
$37$ \( -6844 + 412 T + 1176 T^{2} - 22 T^{3} - 62 T^{4} + T^{6} \)
$41$ \( -3509 + 3795 T - 642 T^{2} - 357 T^{3} + 46 T^{4} + 17 T^{5} + T^{6} \)
$43$ \( -29 - 449 T - 396 T^{2} + 197 T^{3} + 36 T^{4} - 15 T^{5} + T^{6} \)
$47$ \( -17216 + 10272 T + 608 T^{2} - 888 T^{3} + 18 T^{5} + T^{6} \)
$53$ \( 1804 + 2956 T - 7460 T^{2} + 1974 T^{3} + 8 T^{4} - 22 T^{5} + T^{6} \)
$59$ \( -54121 + 32603 T + 11228 T^{2} - 1197 T^{3} - 220 T^{4} + 7 T^{5} + T^{6} \)
$61$ \( -22084 - 52680 T - 22760 T^{2} - 3162 T^{3} - 40 T^{4} + 20 T^{5} + T^{6} \)
$67$ \( -15679 - 28301 T - 14630 T^{2} - 1465 T^{3} + 170 T^{4} + 29 T^{5} + T^{6} \)
$71$ \( 44 - 2248 T + 3648 T^{2} - 150 T^{3} - 168 T^{4} + 2 T^{5} + T^{6} \)
$73$ \( -292471 + 40435 T + 20468 T^{2} - 211 T^{3} - 266 T^{4} - T^{5} + T^{6} \)
$79$ \( -49484 + 14280 T + 3368 T^{2} - 882 T^{3} - 84 T^{4} + 12 T^{5} + T^{6} \)
$83$ \( 1711 + 2057 T - 6264 T^{2} + 323 T^{3} + 348 T^{4} + 35 T^{5} + T^{6} \)
$89$ \( 370271 + 15771 T - 25312 T^{2} - 3011 T^{3} + 120 T^{4} + 29 T^{5} + T^{6} \)
$97$ \( 2383105 + 626365 T + 11284 T^{2} - 7043 T^{3} - 320 T^{4} + 19 T^{5} + T^{6} \)
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