Properties

Label 731.2.d.b
Level 731
Weight 2
Character orbit 731.d
Analytic conductor 5.837
Analytic rank 0
Dimension 8
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 731 = 17 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 731.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(5.83706438776\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} -2 q^{4} -\beta_{3} q^{5} + ( \beta_{1} - \beta_{7} ) q^{7} + ( -2 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} -2 q^{4} -\beta_{3} q^{5} + ( \beta_{1} - \beta_{7} ) q^{7} + ( -2 + \beta_{2} ) q^{9} + ( \beta_{4} + \beta_{5} ) q^{11} -2 \beta_{1} q^{12} + ( -2 + \beta_{2} - \beta_{4} + \beta_{6} ) q^{13} + ( \beta_{4} - \beta_{6} ) q^{15} + 4 q^{16} + ( -\beta_{3} - \beta_{6} ) q^{17} + 2 q^{19} + 2 \beta_{3} q^{20} + ( -4 + \beta_{2} + \beta_{5} - \beta_{6} ) q^{21} + ( -\beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{23} + ( -1 - \beta_{2} + \beta_{5} - \beta_{6} ) q^{25} + ( 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{27} + ( -2 \beta_{1} + 2 \beta_{7} ) q^{28} + 2 \beta_{3} q^{29} + ( 2 \beta_{1} + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{31} + ( 4 - 4 \beta_{2} + 2 \beta_{4} - 2 \beta_{6} ) q^{33} + ( 3 + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{35} + ( 4 - 2 \beta_{2} ) q^{36} + ( -\beta_{1} - 2 \beta_{7} ) q^{37} + ( -\beta_{1} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{39} + ( 2 \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{41} + q^{43} + ( -2 \beta_{4} - 2 \beta_{5} ) q^{44} + ( -2 \beta_{1} + \beta_{7} ) q^{45} + ( -1 + 2 \beta_{2} + \beta_{5} - \beta_{6} ) q^{47} + 4 \beta_{1} q^{48} + ( -4 - \beta_{4} + \beta_{5} ) q^{49} + ( -2 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{7} ) q^{51} + ( 4 - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{6} ) q^{52} + ( 5 + 2 \beta_{2} - \beta_{4} + \beta_{6} ) q^{53} + ( -2 \beta_{2} + 2 \beta_{4} - 2 \beta_{6} ) q^{55} + 2 \beta_{1} q^{57} + ( -2 + \beta_{2} - \beta_{5} + \beta_{6} ) q^{59} + ( -2 \beta_{4} + 2 \beta_{6} ) q^{60} + ( \beta_{1} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{61} + ( -2 \beta_{1} + 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{63} -8 q^{64} + ( 3 \beta_{1} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{65} + ( -1 + 2 \beta_{2} - \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{67} + ( 2 \beta_{3} + 2 \beta_{6} ) q^{68} + ( -2 + 4 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{69} + ( -3 \beta_{1} - \beta_{3} ) q^{71} + ( \beta_{1} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{73} + ( -\beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{75} -4 q^{76} + ( 2 - 4 \beta_{2} + 2 \beta_{4} - 2 \beta_{6} ) q^{77} + ( -2 \beta_{1} - 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{79} -4 \beta_{3} q^{80} + ( 1 - 5 \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{81} + ( 1 + \beta_{2} - \beta_{4} + \beta_{5} ) q^{83} + ( 8 - 2 \beta_{2} - 2 \beta_{5} + 2 \beta_{6} ) q^{84} + ( -6 - 2 \beta_{1} + \beta_{3} - 2 \beta_{4} ) q^{85} + ( -2 \beta_{4} + 2 \beta_{6} ) q^{87} + ( -2 - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{89} + ( -\beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{91} + ( 2 \beta_{4} + 2 \beta_{5} + 4 \beta_{7} ) q^{92} + ( -6 - 2 \beta_{2} ) q^{93} -2 \beta_{3} q^{95} + ( 2 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{97} + ( 4 \beta_{1} - 2 \beta_{3} - 5 \beta_{4} - 5 \beta_{5} - 2 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 16q^{4} - 16q^{9} + O(q^{10}) \) \( 8q - 16q^{4} - 16q^{9} - 10q^{13} - 6q^{15} + 32q^{16} - 3q^{17} + 16q^{19} - 32q^{21} - 8q^{25} + 20q^{33} + 12q^{35} + 32q^{36} + 8q^{43} - 8q^{47} - 26q^{49} - 16q^{51} + 20q^{52} + 46q^{53} - 12q^{55} - 16q^{59} + 12q^{60} - 64q^{64} - 2q^{67} + 6q^{68} - 4q^{69} - 32q^{76} + 4q^{77} - 4q^{81} + 14q^{83} + 64q^{84} - 42q^{85} + 12q^{87} - 28q^{89} - 48q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 20 x^{6} + 129 x^{4} + 323 x^{2} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 5 \)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} - 12 \nu^{5} - 9 \nu^{3} + 85 \nu \)\()/24\)
\(\beta_{4}\)\(=\)\((\)\( 7 \nu^{7} + 16 \nu^{6} + 108 \nu^{5} + 288 \nu^{4} + 423 \nu^{3} + 1488 \nu^{2} + 437 \nu + 2096 \)\()/96\)
\(\beta_{5}\)\(=\)\((\)\( 7 \nu^{7} - 16 \nu^{6} + 108 \nu^{5} - 288 \nu^{4} + 423 \nu^{3} - 1488 \nu^{2} + 437 \nu - 2096 \)\()/96\)
\(\beta_{6}\)\(=\)\((\)\( 7 \nu^{7} + 48 \nu^{6} + 108 \nu^{5} + 768 \nu^{4} + 423 \nu^{3} + 3120 \nu^{2} + 437 \nu + 3120 \)\()/96\)
\(\beta_{7}\)\(=\)\((\)\( -5 \nu^{7} - 84 \nu^{5} - 381 \nu^{3} - 463 \nu \)\()/24\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 5\)
\(\nu^{3}\)\(=\)\(\beta_{7} + 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 6 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{6} - \beta_{5} + 2 \beta_{4} - 14 \beta_{2} + 37\)
\(\nu^{5}\)\(=\)\(-15 \beta_{7} - 28 \beta_{5} - 28 \beta_{4} - 23 \beta_{3} + 47 \beta_{1}\)
\(\nu^{6}\)\(=\)\(18 \beta_{6} + 15 \beta_{5} - 33 \beta_{4} + 159 \beta_{2} - 332\)
\(\nu^{7}\)\(=\)\(171 \beta_{7} + 318 \beta_{5} + 318 \beta_{4} + 234 \beta_{3} - 425 \beta_{1}\)

Character Values

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

\(n\) \(173\) \(562\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
560.1
3.20222i
2.15629i
1.89948i
1.21991i
1.21991i
1.89948i
2.15629i
3.20222i
0 3.20222i −2.00000 0.837930i 0 2.87630i 0 −7.25421 0
560.2 0 2.15629i −2.00000 2.88045i 0 2.59528i 0 −1.64959 0
560.3 0 1.89948i −2.00000 0.651103i 0 2.29537i 0 −0.608027 0
560.4 0 1.21991i −2.00000 3.81798i 0 4.55222i 0 1.51183 0
560.5 0 1.21991i −2.00000 3.81798i 0 4.55222i 0 1.51183 0
560.6 0 1.89948i −2.00000 0.651103i 0 2.29537i 0 −0.608027 0
560.7 0 2.15629i −2.00000 2.88045i 0 2.59528i 0 −1.64959 0
560.8 0 3.20222i −2.00000 0.837930i 0 2.87630i 0 −7.25421 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 560.8
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
17.b Even 1 yes

Hecke kernels

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