Properties

Label 731.2.d.a
Level 731
Weight 2
Character orbit 731.d
Analytic conductor 5.837
Analytic rank 0
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{4} + \beta q^{5} -3 q^{8} + 3 q^{9} +O(q^{10})\) \( q + q^{2} - q^{4} + \beta q^{5} -3 q^{8} + 3 q^{9} + \beta q^{10} + \beta q^{11} -6 q^{13} - q^{16} + ( -3 - \beta ) q^{17} + 3 q^{18} -4 q^{19} -\beta q^{20} + \beta q^{22} + \beta q^{23} -3 q^{25} -6 q^{26} -\beta q^{29} + 3 \beta q^{31} + 5 q^{32} + ( -3 - \beta ) q^{34} -3 q^{36} + 3 \beta q^{37} -4 q^{38} -3 \beta q^{40} + 2 \beta q^{41} + q^{43} -\beta q^{44} + 3 \beta q^{45} + \beta q^{46} + 7 q^{49} -3 q^{50} + 6 q^{52} -10 q^{53} -8 q^{55} -\beta q^{58} -4 q^{59} -3 \beta q^{61} + 3 \beta q^{62} + 7 q^{64} -6 \beta q^{65} + 12 q^{67} + ( 3 + \beta ) q^{68} -4 \beta q^{71} -9 q^{72} + 5 \beta q^{73} + 3 \beta q^{74} + 4 q^{76} -3 \beta q^{79} -\beta q^{80} + 9 q^{81} + 2 \beta q^{82} + 4 q^{83} + ( 8 - 3 \beta ) q^{85} + q^{86} -3 \beta q^{88} + 6 q^{89} + 3 \beta q^{90} -\beta q^{92} -4 \beta q^{95} + 4 \beta q^{97} + 7 q^{98} + 3 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{4} - 6q^{8} + 6q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{4} - 6q^{8} + 6q^{9} - 12q^{13} - 2q^{16} - 6q^{17} + 6q^{18} - 8q^{19} - 6q^{25} - 12q^{26} + 10q^{32} - 6q^{34} - 6q^{36} - 8q^{38} + 2q^{43} + 14q^{49} - 6q^{50} + 12q^{52} - 20q^{53} - 16q^{55} - 8q^{59} + 14q^{64} + 24q^{67} + 6q^{68} - 18q^{72} + 8q^{76} + 18q^{81} + 8q^{83} + 16q^{85} + 2q^{86} + 12q^{89} + 14q^{98} + O(q^{100}) \)

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
1.41421i
1.41421i
1.00000 0 −1.00000 2.82843i 0 0 −3.00000 3.00000 2.82843i
560.2 1.00000 0 −1.00000 2.82843i 0 0 −3.00000 3.00000 2.82843i
\(n\): e.g. 2-40 or 990-1000
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 \( T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(731, [\chi])\).