Properties

Label 8470.2.a.cs
Level $8470$
Weight $2$
Character orbit 8470.a
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.4400.1
Defining polynomial: \(x^{4} - 7 x^{2} + 11\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 7 x^{2} + 11\):

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

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
−2.14896
−1.54336
1.54336
2.14896
1.00000 −3.14896 1.00000 1.00000 −3.14896 1.00000 1.00000 6.91596 1.00000
1.2 1.00000 −2.54336 1.00000 1.00000 −2.54336 1.00000 1.00000 3.46869 1.00000
1.3 1.00000 0.543362 1.00000 1.00000 0.543362 1.00000 1.00000 −2.70476 1.00000
1.4 1.00000 1.14896 1.00000 1.00000 1.14896 1.00000 1.00000 −1.67989 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.cs 4
11.b odd 2 1 8470.2.a.co 4
11.d odd 10 2 770.2.n.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.n.f 8 11.d odd 10 2
8470.2.a.co 4 11.b odd 2 1
8470.2.a.cs 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\):

\( T_{3}^{4} + 4 T_{3}^{3} - T_{3}^{2} - 10 T_{3} + 5 \)
\( T_{13}^{4} + 14 T_{13}^{3} + 54 T_{13}^{2} - 220 \)
\( T_{17}^{4} + 12 T_{17}^{3} + 41 T_{17}^{2} + 30 T_{17} - 25 \)
\( T_{19}^{4} - 2 T_{19}^{3} - 29 T_{19}^{2} + 10 T_{19} + 145 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{4} \)
$3$ \( 5 - 10 T - T^{2} + 4 T^{3} + T^{4} \)
$5$ \( ( -1 + T )^{4} \)
$7$ \( ( -1 + T )^{4} \)
$11$ \( T^{4} \)
$13$ \( -220 + 54 T^{2} + 14 T^{3} + T^{4} \)
$17$ \( -25 + 30 T + 41 T^{2} + 12 T^{3} + T^{4} \)
$19$ \( 145 + 10 T - 29 T^{2} - 2 T^{3} + T^{4} \)
$23$ \( 176 - 32 T^{2} + T^{4} \)
$29$ \( -164 - 440 T - 38 T^{2} + 10 T^{3} + T^{4} \)
$31$ \( -4684 - 1128 T + 14 T^{2} + 18 T^{3} + T^{4} \)
$37$ \( -1100 + 440 T + 46 T^{2} - 18 T^{3} + T^{4} \)
$41$ \( 839 + 708 T + 203 T^{2} + 24 T^{3} + T^{4} \)
$43$ \( 971 - 280 T - 53 T^{2} + 10 T^{3} + T^{4} \)
$47$ \( -1616 - 736 T - 68 T^{2} + 8 T^{3} + T^{4} \)
$53$ \( -2804 + 500 T + 78 T^{2} - 20 T^{3} + T^{4} \)
$59$ \( -155 - 290 T - 21 T^{2} + 14 T^{3} + T^{4} \)
$61$ \( -284 - 156 T + 16 T^{2} + 14 T^{3} + T^{4} \)
$67$ \( -29 - 50 T - 23 T^{2} + T^{4} \)
$71$ \( -556 - 152 T + 38 T^{2} + 14 T^{3} + T^{4} \)
$73$ \( -26249 - 3580 T + 103 T^{2} + 30 T^{3} + T^{4} \)
$79$ \( 1420 - 300 T - 74 T^{2} + 8 T^{3} + T^{4} \)
$83$ \( -1055 - 850 T - 129 T^{2} + 8 T^{3} + T^{4} \)
$89$ \( -2749 - 1396 T - 169 T^{2} + 4 T^{3} + T^{4} \)
$97$ \( -1429 - 760 T - 77 T^{2} + 10 T^{3} + T^{4} \)
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