Properties

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

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 731 = 17 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 731.f (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(5.83706438776\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} + ( 2 + 2 i ) q^{3} + q^{4} + ( -2 + 2 i ) q^{6} + 3 i q^{8} + 5 i q^{9} +O(q^{10})\) \( q + i q^{2} + ( 2 + 2 i ) q^{3} + q^{4} + ( -2 + 2 i ) q^{6} + 3 i q^{8} + 5 i q^{9} + ( -1 + i ) q^{11} + ( 2 + 2 i ) q^{12} + 2 q^{13} - q^{16} + ( -4 - i ) q^{17} -5 q^{18} -8 i q^{19} + ( -1 - i ) q^{22} + ( -1 + i ) q^{23} + ( -6 + 6 i ) q^{24} -5 i q^{25} + 2 i q^{26} + ( -4 + 4 i ) q^{27} + ( 6 + 6 i ) q^{29} + ( 1 + i ) q^{31} + 5 i q^{32} -4 q^{33} + ( 1 - 4 i ) q^{34} + 5 i q^{36} + ( -2 - 2 i ) q^{37} + 8 q^{38} + ( 4 + 4 i ) q^{39} + ( 1 - i ) q^{41} -i q^{43} + ( -1 + i ) q^{44} + ( -1 - i ) q^{46} + 2 q^{47} + ( -2 - 2 i ) q^{48} + 7 i q^{49} + 5 q^{50} + ( -6 - 10 i ) q^{51} + 2 q^{52} -12 i q^{53} + ( -4 - 4 i ) q^{54} + ( 16 - 16 i ) q^{57} + ( -6 + 6 i ) q^{58} -6 i q^{59} + ( -6 + 6 i ) q^{61} + ( -1 + i ) q^{62} -7 q^{64} -4 i q^{66} -2 q^{67} + ( -4 - i ) q^{68} -4 q^{69} + ( -4 - 4 i ) q^{71} -15 q^{72} + ( 6 + 6 i ) q^{73} + ( 2 - 2 i ) q^{74} + ( 10 - 10 i ) q^{75} -8 i q^{76} + ( -4 + 4 i ) q^{78} + ( 7 - 7 i ) q^{79} - q^{81} + ( 1 + i ) q^{82} + 12 i q^{83} + q^{86} + 24 i q^{87} + ( -3 - 3 i ) q^{88} + 6 q^{89} + ( -1 + i ) q^{92} + 4 i q^{93} + 2 i q^{94} + ( -10 + 10 i ) q^{96} + ( -13 - 13 i ) q^{97} -7 q^{98} + ( -5 - 5 i ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{3} + 2q^{4} - 4q^{6} + O(q^{10}) \) \( 2q + 4q^{3} + 2q^{4} - 4q^{6} - 2q^{11} + 4q^{12} + 4q^{13} - 2q^{16} - 8q^{17} - 10q^{18} - 2q^{22} - 2q^{23} - 12q^{24} - 8q^{27} + 12q^{29} + 2q^{31} - 8q^{33} + 2q^{34} - 4q^{37} + 16q^{38} + 8q^{39} + 2q^{41} - 2q^{44} - 2q^{46} + 4q^{47} - 4q^{48} + 10q^{50} - 12q^{51} + 4q^{52} - 8q^{54} + 32q^{57} - 12q^{58} - 12q^{61} - 2q^{62} - 14q^{64} - 4q^{67} - 8q^{68} - 8q^{69} - 8q^{71} - 30q^{72} + 12q^{73} + 4q^{74} + 20q^{75} - 8q^{78} + 14q^{79} - 2q^{81} + 2q^{82} + 2q^{86} - 6q^{88} + 12q^{89} - 2q^{92} - 20q^{96} - 26q^{97} - 14q^{98} - 10q^{99} + O(q^{100}) \)

Character Values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
259.1
1.00000i
1.00000i
1.00000i 2.00000 2.00000i 1.00000 0 −2.00000 2.00000i 0 3.00000i 5.00000i 0
302.1 1.00000i 2.00000 + 2.00000i 1.00000 0 −2.00000 + 2.00000i 0 3.00000i 5.00000i 0
\(n\): 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.c Even 1 yes

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(731, [\chi])\):

\( T_{2}^{2} + 1 \)
\( T_{3}^{2} - 4 T_{3} + 8 \)