Properties

Label 731.2.s.a
Level 731
Weight 2
Character orbit 731.s
Analytic conductor 5.837
Analytic rank 0
Dimension 16
CM discriminant -43
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 731 = 17 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 731.s (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.83706438776\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{16})\)
Coefficient field: 16.0.3289935900927224469054816256.1
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{16}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_{4} q^{4} -3 \beta_{5} q^{9} +O(q^{10})\) \( q + 2 \beta_{4} q^{4} -3 \beta_{5} q^{9} + ( -\beta_{4} - \beta_{6} - \beta_{7} ) q^{11} + ( -1 + \beta_{2} + \beta_{8} - \beta_{9} ) q^{13} + 4 \beta_{8} q^{16} + ( 3 \beta_{13} - \beta_{15} ) q^{17} + ( 3 - \beta_{2} - 4 \beta_{10} + \beta_{12} ) q^{23} + 5 \beta_{13} q^{25} + ( \beta_{1} - 4 \beta_{3} + 5 \beta_{4} + \beta_{6} ) q^{31} -6 \beta_{10} q^{36} + ( -6 \beta_{5} + \beta_{7} + 5 \beta_{11} + \beta_{14} ) q^{41} + ( \beta_{13} - 2 \beta_{15} ) q^{43} + ( -2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{12} ) q^{44} + ( 2 \beta_{1} - \beta_{3} + 3 \beta_{5} - 2 \beta_{7} ) q^{47} -7 \beta_{3} q^{49} + ( -4 \beta_{4} - 2 \beta_{6} + 2 \beta_{11} + 2 \beta_{14} ) q^{52} + ( 7 \beta_{8} + \beta_{9} - 6 \beta_{11} - \beta_{14} ) q^{53} + ( -3 + 2 \beta_{2} - 3 \beta_{11} - 2 \beta_{14} ) q^{59} + 8 \beta_{11} q^{64} + ( -\beta_{1} + 7 \beta_{3} + 7 \beta_{13} + \beta_{15} ) q^{67} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{68} + ( 5 - 2 \beta_{2} + 7 \beta_{10} - 2 \beta_{12} ) q^{79} -9 \beta_{11} q^{81} + ( \beta_{10} - 2 \beta_{12} ) q^{83} + ( 8 \beta_{4} + 2 \beta_{6} - 8 \beta_{13} + 2 \beta_{15} ) q^{92} + ( 3 \beta_{1} + 2 \beta_{3} - \beta_{11} + 3 \beta_{14} ) q^{97} + ( 3 \beta_{12} - 3 \beta_{14} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 24q^{13} + 56q^{23} - 64q^{59} + 96q^{79} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 10319 x^{8} + 214358881\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{8} + 3070 \)\()/4179\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{9} + 3070 \nu \)\()/45969\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{10} + 49039 \nu^{2} \)\()/505659\)
\(\beta_{5}\)\(=\)\((\)\( -10 \nu^{11} + 15269 \nu^{3} \)\()/5562249\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{10} - 3070 \nu^{2} \)\()/45969\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{11} + 49039 \nu^{3} \)\()/505659\)
\(\beta_{8}\)\(=\)\((\)\( \nu^{12} + 24960 \nu^{4} \)\()/2913559\)
\(\beta_{9}\)\(=\)\((\)\( -10 \nu^{12} + 15269 \nu^{4} \)\()/5562249\)
\(\beta_{10}\)\(=\)\((\)\( -89 \nu^{13} + 692119 \nu^{5} \)\()/ 673032129 \)
\(\beta_{11}\)\(=\)\((\)\( 320 \nu^{14} + 5073641 \nu^{6} \)\()/ 7403353419 \)
\(\beta_{12}\)\(=\)\((\)\( -320 \nu^{13} - 5073641 \nu^{5} \)\()/ 673032129 \)
\(\beta_{13}\)\(=\)\((\)\( -659 \nu^{15} + 12686950 \nu^{7} \)\()/ 81436887609 \)
\(\beta_{14}\)\(=\)\((\)\( 89 \nu^{14} - 692119 \nu^{6} \)\()/ 673032129 \)
\(\beta_{15}\)\(=\)\((\)\( -\nu^{15} - 10319 \nu^{7} \)\()/19487171\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} + 11 \beta_{4}\)
\(\nu^{3}\)\(=\)\(10 \beta_{7} + 11 \beta_{5}\)
\(\nu^{4}\)\(=\)\(21 \beta_{9} + 110 \beta_{8}\)
\(\nu^{5}\)\(=\)\(-89 \beta_{12} + 320 \beta_{10}\)
\(\nu^{6}\)\(=\)\(-320 \beta_{14} + 979 \beta_{11}\)
\(\nu^{7}\)\(=\)\(-659 \beta_{15} + 4179 \beta_{13}\)
\(\nu^{8}\)\(=\)\(4179 \beta_{2} - 3070\)
\(\nu^{9}\)\(=\)\(45969 \beta_{3} - 3070 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-49039 \beta_{6} - 33770 \beta_{4}\)
\(\nu^{11}\)\(=\)\(15269 \beta_{7} - 539429 \beta_{5}\)
\(\nu^{12}\)\(=\)\(-524160 \beta_{9} + 167959 \beta_{8}\)
\(\nu^{13}\)\(=\)\(-692119 \beta_{12} - 5073641 \beta_{10}\)
\(\nu^{14}\)\(=\)\(5073641 \beta_{14} + 7613309 \beta_{11}\)
\(\nu^{15}\)\(=\)\(-12686950 \beta_{15} - 43123101 \beta_{13}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/731\mathbb{Z}\right)^\times\).

\(n\) \(173\) \(562\)
\(\chi(n)\) \(-\beta_{5}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
214.1
−0.792772 + 3.22048i
1.71665 2.83780i
3.22048 + 0.792772i
−2.83780 1.71665i
−3.22048 + 0.792772i
2.83780 1.71665i
−0.792772 3.22048i
1.71665 + 2.83780i
−1.71665 2.83780i
0.792772 + 3.22048i
3.22048 0.792772i
−2.83780 + 1.71665i
−3.22048 0.792772i
2.83780 + 1.71665i
−1.71665 + 2.83780i
0.792772 3.22048i
0 0 −1.41421 1.41421i 0 0 0 0 1.14805 + 2.77164i 0
214.2 0 0 −1.41421 1.41421i 0 0 0 0 1.14805 + 2.77164i 0
300.1 0 0 1.41421 + 1.41421i 0 0 0 0 −2.77164 + 1.14805i 0
300.2 0 0 1.41421 + 1.41421i 0 0 0 0 −2.77164 + 1.14805i 0
343.1 0 0 1.41421 1.41421i 0 0 0 0 2.77164 + 1.14805i 0
343.2 0 0 1.41421 1.41421i 0 0 0 0 2.77164 + 1.14805i 0
386.1 0 0 −1.41421 + 1.41421i 0 0 0 0 1.14805 2.77164i 0
386.2 0 0 −1.41421 + 1.41421i 0 0 0 0 1.14805 2.77164i 0
515.1 0 0 −1.41421 + 1.41421i 0 0 0 0 −1.14805 + 2.77164i 0
515.2 0 0 −1.41421 + 1.41421i 0 0 0 0 −1.14805 + 2.77164i 0
558.1 0 0 1.41421 1.41421i 0 0 0 0 −2.77164 1.14805i 0
558.2 0 0 1.41421 1.41421i 0 0 0 0 −2.77164 1.14805i 0
601.1 0 0 1.41421 + 1.41421i 0 0 0 0 2.77164 1.14805i 0
601.2 0 0 1.41421 + 1.41421i 0 0 0 0 2.77164 1.14805i 0
687.1 0 0 −1.41421 1.41421i 0 0 0 0 −1.14805 2.77164i 0
687.2 0 0 −1.41421 1.41421i 0 0 0 0 −1.14805 2.77164i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 687.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.b odd 2 1 CM by \(\Q(\sqrt{-43}) \)
17.e odd 16 1 inner
731.s even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 731.2.s.a 16
17.e odd 16 1 inner 731.2.s.a 16
43.b odd 2 1 CM 731.2.s.a 16
731.s even 16 1 inner 731.2.s.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
731.2.s.a 16 1.a even 1 1 trivial
731.2.s.a 16 17.e odd 16 1 inner
731.2.s.a 16 43.b odd 2 1 CM
731.2.s.a 16 731.s even 16 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(731, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 16 T^{8} )^{4} \)
$3$ \( ( 1 + 6561 T^{16} )^{2} \)
$5$ \( ( 1 + 390625 T^{16} )^{2} \)
$7$ \( ( 1 + 5764801 T^{16} )^{2} \)
$11$ \( ( 1 + 199 T^{4} + 14641 T^{8} )^{2}( 1 + 10319 T^{8} + 214358881 T^{16} ) \)
$13$ \( ( 1 + 3 T + 13 T^{2} )^{8}( 1 - 17 T^{2} + 169 T^{4} )^{4} \)
$17$ \( 1 + 79967 T^{8} + 6975757441 T^{16} \)
$19$ \( ( 1 + 130321 T^{8} )^{4} \)
$23$ \( ( 1 - 7 T + 23 T^{2} )^{8}( 1 + 540719 T^{8} + 78310985281 T^{16} ) \)
$29$ \( ( 1 + 500246412961 T^{16} )^{2} \)
$31$ \( ( 1 - 1561 T^{4} + 923521 T^{8} )^{2}( 1 + 589679 T^{8} + 852891037441 T^{16} ) \)
$37$ \( ( 1 + 3512479453921 T^{16} )^{2} \)
$41$ \( ( 1 - 1841 T^{4} + 2825761 T^{8} )^{2}( 1 - 2262241 T^{8} + 7984925229121 T^{16} ) \)
$43$ \( ( 1 + 3418801 T^{8} )^{2} \)
$47$ \( ( 1 - 6983806 T^{8} + 23811286661761 T^{16} )^{2} \)
$53$ \( ( 1 + 63 T^{2} + 2809 T^{4} )^{4}( 1 - 1649 T^{4} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 + 8 T + 59 T^{2} )^{8}( 1 - 4046 T^{4} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 + 191707312997281 T^{16} )^{2} \)
$67$ \( ( 1 - 39816433 T^{8} + 406067677556641 T^{16} )^{2} \)
$71$ \( ( 1 + 645753531245761 T^{16} )^{2} \)
$73$ \( ( 1 + 806460091894081 T^{16} )^{2} \)
$79$ \( ( 1 - 12 T + 79 T^{2} )^{8}( 1 + 73045634 T^{8} + 1517108809906561 T^{16} ) \)
$83$ \( ( 1 - 93091441 T^{8} + 2252292232139041 T^{16} )^{2} \)
$89$ \( ( 1 + 7921 T^{4} )^{8} \)
$97$ \( ( 1 + 18431 T^{4} + 88529281 T^{8} )^{2}( 1 + 162643199 T^{8} + 7837433594376961 T^{16} ) \)
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