Properties

Label 8470.2.a.ct
Level $8470$
Weight $2$
Character orbit 8470.a
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.5225.1
Defining polynomial: \(x^{4} - x^{3} - 8 x^{2} + x + 11\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 8 x^{2} + x + 11\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 2 \nu^{2} - 4 \nu + 3 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 4 \nu^{2} + 2 \nu - 11 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(2 \beta_{3} + 4 \beta_{2} + 6 \beta_{1} + 5\)

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
1.26498
−1.48718
3.10522
−1.88301
1.00000 −2.04678 1.00000 −1.00000 −2.04678 1.00000 1.00000 1.18929 −1.00000
1.2 1.00000 −0.919131 1.00000 −1.00000 −0.919131 1.00000 1.00000 −2.15520 −1.00000
1.3 1.00000 1.91913 1.00000 −1.00000 1.91913 1.00000 1.00000 0.683063 −1.00000
1.4 1.00000 3.04678 1.00000 −1.00000 3.04678 1.00000 1.00000 6.28284 −1.00000
\(n\): e.g. 2-40 or 990-1000
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.ct 4
11.b odd 2 1 8470.2.a.cq 4
11.d odd 10 2 770.2.n.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.n.e 8 11.d odd 10 2
8470.2.a.cq 4 11.b odd 2 1
8470.2.a.ct 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\):

\( T_{3}^{4} - 2 T_{3}^{3} - 7 T_{3}^{2} + 8 T_{3} + 11 \)
\( T_{13}^{4} - T_{13}^{3} - 13 T_{13}^{2} + 16 T_{13} - 4 \)
\( T_{17}^{4} - 2 T_{17}^{3} - 17 T_{17}^{2} + 38 T_{17} - 19 \)
\( T_{19}^{4} + 3 T_{19}^{3} - 41 T_{19}^{2} - 18 T_{19} + 236 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{4} \)
$3$ \( 11 + 8 T - 7 T^{2} - 2 T^{3} + T^{4} \)
$5$ \( ( 1 + T )^{4} \)
$7$ \( ( -1 + T )^{4} \)
$11$ \( T^{4} \)
$13$ \( -4 + 16 T - 13 T^{2} - T^{3} + T^{4} \)
$17$ \( -19 + 38 T - 17 T^{2} - 2 T^{3} + T^{4} \)
$19$ \( 236 - 18 T - 41 T^{2} + 3 T^{3} + T^{4} \)
$23$ \( ( 2 + T )^{4} \)
$29$ \( 44 - 90 T + 61 T^{2} - 15 T^{3} + T^{4} \)
$31$ \( ( -44 - 2 T + T^{2} )^{2} \)
$37$ \( 64 + 176 T - 60 T^{2} - 2 T^{3} + T^{4} \)
$41$ \( 1024 + 352 T - 35 T^{2} - 11 T^{3} + T^{4} \)
$43$ \( 1076 + 298 T - 67 T^{2} - 7 T^{3} + T^{4} \)
$47$ \( 10604 + 510 T - 201 T^{2} - 5 T^{3} + T^{4} \)
$53$ \( 64 - 44 T^{2} - 10 T^{3} + T^{4} \)
$59$ \( 2804 + 874 T - 85 T^{2} - 13 T^{3} + T^{4} \)
$61$ \( -976 - 328 T + 200 T^{2} - 26 T^{3} + T^{4} \)
$67$ \( -284 - 198 T + 19 T^{2} + 13 T^{3} + T^{4} \)
$71$ \( -1364 - 778 T - 51 T^{2} + 13 T^{3} + T^{4} \)
$73$ \( 869 - 654 T - 25 T^{2} + 18 T^{3} + T^{4} \)
$79$ \( 44 - 120 T + 71 T^{2} - 15 T^{3} + T^{4} \)
$83$ \( 79 + 244 T - 115 T^{2} + 2 T^{3} + T^{4} \)
$89$ \( 1084 - 646 T - 105 T^{2} + 7 T^{3} + T^{4} \)
$97$ \( ( -145 - 5 T + T^{2} )^{2} \)
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