Properties

Label 8470.2.a.cp
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: \(\Q(\zeta_{24})^+\)
Defining polynomial: \(x^{4} - 4 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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.

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

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
0.517638
−1.93185
1.93185
−0.517638
−1.00000 −2.41421 1.00000 1.00000 2.41421 1.00000 −1.00000 2.82843 −1.00000
1.2 −1.00000 −2.41421 1.00000 1.00000 2.41421 1.00000 −1.00000 2.82843 −1.00000
1.3 −1.00000 0.414214 1.00000 1.00000 −0.414214 1.00000 −1.00000 −2.82843 −1.00000
1.4 −1.00000 0.414214 1.00000 1.00000 −0.414214 1.00000 −1.00000 −2.82843 −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.cp 4
11.b odd 2 1 8470.2.a.cr yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8470.2.a.cp 4 1.a even 1 1 trivial
8470.2.a.cr yes 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\):

\( T_{3}^{2} + 2 T_{3} - 1 \)
\( T_{13}^{4} - 22 T_{13}^{2} - 48 T_{13} - 23 \)
\( T_{17}^{2} + 4 T_{17} - 2 \)
\( T_{19}^{4} - 12 T_{19}^{3} + 26 T_{19}^{2} + 60 T_{19} - 167 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{4} \)
$3$ \( ( -1 + 2 T + T^{2} )^{2} \)
$5$ \( ( -1 + T )^{4} \)
$7$ \( ( -1 + T )^{4} \)
$11$ \( T^{4} \)
$13$ \( -23 - 48 T - 22 T^{2} + T^{4} \)
$17$ \( ( -2 + 4 T + T^{2} )^{2} \)
$19$ \( -167 + 60 T + 26 T^{2} - 12 T^{3} + T^{4} \)
$23$ \( 1 - 8 T - 10 T^{2} + 8 T^{3} + T^{4} \)
$29$ \( 16 + 32 T - 16 T^{2} - 8 T^{3} + T^{4} \)
$31$ \( 2116 - 124 T^{2} + T^{4} \)
$37$ \( -800 - 160 T + 56 T^{2} + 16 T^{3} + T^{4} \)
$41$ \( -2396 - 976 T - 84 T^{2} + 8 T^{3} + T^{4} \)
$43$ \( -284 + 304 T - 60 T^{2} - 8 T^{3} + T^{4} \)
$47$ \( 772 - 544 T - 4 T^{2} + 16 T^{3} + T^{4} \)
$53$ \( 4 - 192 T - 76 T^{2} + T^{4} \)
$59$ \( 73 - 136 T - 18 T^{2} + 8 T^{3} + T^{4} \)
$61$ \( -128 + 128 T - 16 T^{2} - 8 T^{3} + T^{4} \)
$67$ \( 900 - 132 T^{2} + T^{4} \)
$71$ \( -3008 - 1984 T - 192 T^{2} + 8 T^{3} + T^{4} \)
$73$ \( ( 2 + 4 T + T^{2} )^{2} \)
$79$ \( -1031 - 72 T + 158 T^{2} - 24 T^{3} + T^{4} \)
$83$ \( -5807 - 2632 T - 282 T^{2} + 8 T^{3} + T^{4} \)
$89$ \( 676 - 148 T^{2} + T^{4} \)
$97$ \( -2672 - 1504 T - 144 T^{2} + 8 T^{3} + T^{4} \)
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