Properties

Label 8470.2.a.dg
Level $8470$
Weight $2$
Character orbit 8470.a
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 16 x^{6} + 69 x^{4} - 10 x^{3} - 70 x^{2} + 10 x + 5\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_{1} q^{3} + q^{4} - q^{5} -\beta_{1} q^{6} + q^{7} - q^{8} + ( 1 + \beta_{5} - \beta_{6} ) q^{9} +O(q^{10})\) \( q - q^{2} + \beta_{1} q^{3} + q^{4} - q^{5} -\beta_{1} q^{6} + q^{7} - q^{8} + ( 1 + \beta_{5} - \beta_{6} ) q^{9} + q^{10} + \beta_{1} q^{12} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} ) q^{13} - q^{14} -\beta_{1} q^{15} + q^{16} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} ) q^{17} + ( -1 - \beta_{5} + \beta_{6} ) q^{18} + ( 1 - 2 \beta_{1} + 3 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{19} - q^{20} + \beta_{1} q^{21} + ( 2 - 2 \beta_{2} ) q^{23} -\beta_{1} q^{24} + q^{25} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} ) q^{26} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{27} + q^{28} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{29} + \beta_{1} q^{30} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{31} - q^{32} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{7} ) q^{34} - q^{35} + ( 1 + \beta_{5} - \beta_{6} ) q^{36} + ( -2 + \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{37} + ( -1 + 2 \beta_{1} - 3 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{38} + ( -7 + \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{39} + q^{40} + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{41} -\beta_{1} q^{42} + ( 1 - \beta_{2} - 2 \beta_{4} - \beta_{6} ) q^{43} + ( -1 - \beta_{5} + \beta_{6} ) q^{45} + ( -2 + 2 \beta_{2} ) q^{46} + ( -1 + 3 \beta_{3} - \beta_{4} + 2 \beta_{6} ) q^{47} + \beta_{1} q^{48} + q^{49} - q^{50} + ( 2 \beta_{1} - 3 \beta_{2} - 5 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{51} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} ) q^{52} + ( -3 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{6} - \beta_{7} ) q^{53} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{54} - q^{56} + ( -4 - \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{57} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} ) q^{58} + ( -2 + 2 \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{59} -\beta_{1} q^{60} + ( -6 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{5} ) q^{61} + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{62} + ( 1 + \beta_{5} - \beta_{6} ) q^{63} + q^{64} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} ) q^{65} + ( 4 - 3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{6} ) q^{67} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} ) q^{68} + ( 8 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{69} + q^{70} + ( 2 + \beta_{1} + \beta_{2} - 2 \beta_{7} ) q^{71} + ( -1 - \beta_{5} + \beta_{6} ) q^{72} + ( 3 - \beta_{2} + \beta_{4} + \beta_{6} + 3 \beta_{7} ) q^{73} + ( 2 - \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{74} + \beta_{1} q^{75} + ( 1 - 2 \beta_{1} + 3 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{76} + ( 7 - \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{78} + ( -6 - 2 \beta_{1} + \beta_{2} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{79} - q^{80} + ( 5 + 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{6} ) q^{81} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{82} + ( -6 - \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{83} + \beta_{1} q^{84} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{7} ) q^{85} + ( -1 + \beta_{2} + 2 \beta_{4} + \beta_{6} ) q^{86} + ( 3 - \beta_{1} + 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{87} + ( -1 + \beta_{1} - \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{7} ) q^{89} + ( 1 + \beta_{5} - \beta_{6} ) q^{90} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} ) q^{91} + ( 2 - 2 \beta_{2} ) q^{92} + ( 1 - 3 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{93} + ( 1 - 3 \beta_{3} + \beta_{4} - 2 \beta_{6} ) q^{94} + ( -1 + 2 \beta_{1} - 3 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{95} -\beta_{1} q^{96} + ( 1 - 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{97} - q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{2} + 8q^{4} - 8q^{5} + 8q^{7} - 8q^{8} + 8q^{9} + O(q^{10}) \) \( 8q - 8q^{2} + 8q^{4} - 8q^{5} + 8q^{7} - 8q^{8} + 8q^{9} + 8q^{10} + q^{13} - 8q^{14} + 8q^{16} - 6q^{17} - 8q^{18} - 5q^{19} - 8q^{20} + 10q^{23} + 8q^{25} - q^{26} + 8q^{28} - 3q^{29} - 8q^{31} - 8q^{32} + 6q^{34} - 8q^{35} + 8q^{36} - 6q^{37} + 5q^{38} - 35q^{39} + 8q^{40} - 11q^{41} + 5q^{43} - 8q^{45} - 10q^{46} - 15q^{47} + 8q^{49} - 8q^{50} + 6q^{51} + q^{52} - 16q^{53} - 8q^{56} - 38q^{57} + 3q^{58} - 9q^{59} - 32q^{61} + 8q^{62} + 8q^{63} + 8q^{64} - q^{65} + 33q^{67} - 6q^{68} - 22q^{69} + 8q^{70} + 11q^{71} - 8q^{72} + 34q^{73} + 6q^{74} - 5q^{76} + 35q^{78} - 31q^{79} - 8q^{80} + 20q^{81} + 11q^{82} - 50q^{83} + 6q^{85} - 5q^{86} + 12q^{87} + q^{89} + 8q^{90} + q^{91} + 10q^{92} + 26q^{93} + 15q^{94} + 5q^{95} - 4q^{97} - 8q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 16 x^{6} + 69 x^{4} - 10 x^{3} - 70 x^{2} + 10 x + 5\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -3 \nu^{6} + \nu^{5} + 34 \nu^{4} - 3 \nu^{3} - 70 \nu^{2} + 17 \nu + 29 \)\()/34\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} + 19 \nu^{5} - \nu^{4} - 103 \nu^{3} + 13 \nu^{2} + 140 \nu - 27 \)\()/34\)
\(\beta_{4}\)\(=\)\((\)\( 4 \nu^{7} - 5 \nu^{6} - 63 \nu^{5} + 72 \nu^{4} + 271 \nu^{3} - 282 \nu^{2} - 237 \nu + 111 \)\()/34\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{7} - \nu^{6} + 46 \nu^{5} + 14 \nu^{4} - 174 \nu^{3} - 7 \nu^{2} + 97 \nu - 26 \)\()/17\)
\(\beta_{6}\)\(=\)\((\)\( -3 \nu^{7} - \nu^{6} + 46 \nu^{5} + 14 \nu^{4} - 174 \nu^{3} - 24 \nu^{2} + 97 \nu + 42 \)\()/17\)
\(\beta_{7}\)\(=\)\((\)\( 9 \nu^{7} + 6 \nu^{6} - 139 \nu^{5} - 93 \nu^{4} + 559 \nu^{3} + 261 \nu^{2} - 512 \nu - 87 \)\()/34\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{6} + \beta_{5} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 7 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-8 \beta_{6} + 9 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + 2 \beta_{1} + 32\)
\(\nu^{5}\)\(=\)\(12 \beta_{7} + 10 \beta_{6} + 13 \beta_{5} + 13 \beta_{4} + 22 \beta_{3} - 13 \beta_{2} + 56 \beta_{1} + 12\)
\(\nu^{6}\)\(=\)\(3 \beta_{7} - 65 \beta_{6} + 82 \beta_{5} + 26 \beta_{4} + 29 \beta_{3} - 60 \beta_{2} + 40 \beta_{1} + 283\)
\(\nu^{7}\)\(=\)\(125 \beta_{7} + 82 \beta_{6} + 148 \beta_{5} + 142 \beta_{4} + 279 \beta_{3} - 140 \beta_{2} + 481 \beta_{1} + 221\)

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
−2.90474
−2.53932
−1.12455
−0.211079
0.365778
1.22201
2.00431
3.18761
−1.00000 −2.90474 1.00000 −1.00000 2.90474 1.00000 −1.00000 5.43751 1.00000
1.2 −1.00000 −2.53932 1.00000 −1.00000 2.53932 1.00000 −1.00000 3.44815 1.00000
1.3 −1.00000 −1.12455 1.00000 −1.00000 1.12455 1.00000 −1.00000 −1.73538 1.00000
1.4 −1.00000 −0.211079 1.00000 −1.00000 0.211079 1.00000 −1.00000 −2.95545 1.00000
1.5 −1.00000 0.365778 1.00000 −1.00000 −0.365778 1.00000 −1.00000 −2.86621 1.00000
1.6 −1.00000 1.22201 1.00000 −1.00000 −1.22201 1.00000 −1.00000 −1.50670 1.00000
1.7 −1.00000 2.00431 1.00000 −1.00000 −2.00431 1.00000 −1.00000 1.01724 1.00000
1.8 −1.00000 3.18761 1.00000 −1.00000 −3.18761 1.00000 −1.00000 7.16083 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.dg 8
11.b odd 2 1 8470.2.a.dh 8
11.d odd 10 2 770.2.n.k 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.n.k 16 11.d odd 10 2
8470.2.a.dg 8 1.a even 1 1 trivial
8470.2.a.dh 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\):

\( T_{3}^{8} - 16 T_{3}^{6} + 69 T_{3}^{4} - 10 T_{3}^{3} - 70 T_{3}^{2} + 10 T_{3} + 5 \)
\(T_{13}^{8} - \cdots\)
\(T_{17}^{8} + \cdots\)
\(T_{19}^{8} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{8} \)
$3$ \( 5 + 10 T - 70 T^{2} - 10 T^{3} + 69 T^{4} - 16 T^{6} + T^{8} \)
$5$ \( ( 1 + T )^{8} \)
$7$ \( ( -1 + T )^{8} \)
$11$ \( T^{8} \)
$13$ \( -12764 + 39548 T - 14908 T^{2} - 4850 T^{3} + 2216 T^{4} + 140 T^{5} - 87 T^{6} - T^{7} + T^{8} \)
$17$ \( 109 - 556 T - 2152 T^{2} + 3264 T^{3} + 749 T^{4} - 350 T^{5} - 62 T^{6} + 6 T^{7} + T^{8} \)
$19$ \( -70900 - 158450 T - 33635 T^{2} + 15235 T^{3} + 3534 T^{4} - 485 T^{5} - 106 T^{6} + 5 T^{7} + T^{8} \)
$23$ \( 1280 - 5760 T + 8000 T^{2} - 3200 T^{3} - 576 T^{4} + 480 T^{5} - 36 T^{6} - 10 T^{7} + T^{8} \)
$29$ \( -16636 - 24912 T - 3908 T^{2} + 7922 T^{3} + 2734 T^{4} - 310 T^{5} - 113 T^{6} + 3 T^{7} + T^{8} \)
$31$ \( 135344 + 277528 T + 190172 T^{2} + 45924 T^{3} + 204 T^{4} - 1350 T^{5} - 120 T^{6} + 8 T^{7} + T^{8} \)
$37$ \( 3280 + 1080 T - 7140 T^{2} - 1020 T^{3} + 3584 T^{4} - 442 T^{5} - 130 T^{6} + 6 T^{7} + T^{8} \)
$41$ \( 21296 - 1026300 T - 180387 T^{2} + 101649 T^{3} + 9544 T^{4} - 2279 T^{5} - 204 T^{6} + 11 T^{7} + T^{8} \)
$43$ \( -591484 - 612198 T - 169261 T^{2} + 14663 T^{3} + 10984 T^{4} + 393 T^{5} - 188 T^{6} - 5 T^{7} + T^{8} \)
$47$ \( 614336 + 972960 T + 504544 T^{2} + 83560 T^{3} - 4944 T^{4} - 2410 T^{5} - 101 T^{6} + 15 T^{7} + T^{8} \)
$53$ \( 10229824 - 7182736 T - 563892 T^{2} + 301404 T^{3} + 17024 T^{4} - 3970 T^{5} - 232 T^{6} + 16 T^{7} + T^{8} \)
$59$ \( 10961956 - 1447614 T - 806503 T^{2} + 87723 T^{3} + 21090 T^{4} - 1613 T^{5} - 238 T^{6} + 9 T^{7} + T^{8} \)
$61$ \( 58256 + 131832 T - 2684 T^{2} - 28128 T^{3} - 4756 T^{4} + 962 T^{5} + 336 T^{6} + 32 T^{7} + T^{8} \)
$67$ \( 162964 - 333930 T - 13175 T^{2} + 80879 T^{3} - 19566 T^{4} + 435 T^{5} + 306 T^{6} - 33 T^{7} + T^{8} \)
$71$ \( 44 - 524 T - 3240 T^{2} - 750 T^{3} + 5786 T^{4} + 708 T^{5} - 143 T^{6} - 11 T^{7} + T^{8} \)
$73$ \( -7365731 - 3020566 T + 1777518 T^{2} + 32644 T^{3} - 64771 T^{4} + 5470 T^{5} + 168 T^{6} - 34 T^{7} + T^{8} \)
$79$ \( 15119876 + 8618540 T + 804008 T^{2} - 362726 T^{3} - 93746 T^{4} - 6944 T^{5} + 81 T^{6} + 31 T^{7} + T^{8} \)
$83$ \( -13602031 - 14467992 T - 5572566 T^{2} - 961370 T^{3} - 58159 T^{4} + 4336 T^{5} + 880 T^{6} + 50 T^{7} + T^{8} \)
$89$ \( 108020 - 22690 T - 62945 T^{2} + 14135 T^{3} + 6756 T^{4} - 539 T^{5} - 208 T^{6} - T^{7} + T^{8} \)
$97$ \( -968759 - 1541370 T - 87842 T^{2} + 187326 T^{3} + 31449 T^{4} - 1716 T^{5} - 384 T^{6} + 4 T^{7} + T^{8} \)
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