Properties

Label 283.1.b.b
Level 283
Weight 1
Character orbit 283.b
Analytic conductor 0.141
Analytic rank 0
Dimension 2
Projective image \(S_{4}\)
CM/RM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 283 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 283.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.141235398575\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(S_{4}\)
Projective field Galois closure of 4.2.283.1
Artin image size \(48\)
Artin image $\GL(2,3)$
Artin field Galois closure of 8.2.22665187.3

$q$-expansion

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

\(f(q)\) \(=\) \( q -\beta q^{2} + \beta q^{3} - q^{4} + \beta q^{5} + 2 q^{6} - q^{7} - q^{9} +O(q^{10})\) \( q -\beta q^{2} + \beta q^{3} - q^{4} + \beta q^{5} + 2 q^{6} - q^{7} - q^{9} + 2 q^{10} + q^{11} -\beta q^{12} + q^{13} + \beta q^{14} -2 q^{15} - q^{16} + \beta q^{18} -\beta q^{19} -\beta q^{20} -\beta q^{21} -\beta q^{22} - q^{23} - q^{25} -\beta q^{26} + q^{28} - q^{29} + 2 \beta q^{30} -\beta q^{31} + \beta q^{32} + \beta q^{33} -\beta q^{35} + q^{36} -2 q^{38} + \beta q^{39} + q^{41} -2 q^{42} -\beta q^{43} - q^{44} -\beta q^{45} + \beta q^{46} + \beta q^{47} -\beta q^{48} + \beta q^{50} - q^{52} + \beta q^{55} + 2 q^{57} + \beta q^{58} + q^{59} + 2 q^{60} + q^{61} -2 q^{62} + q^{63} + q^{64} + \beta q^{65} + 2 q^{66} -\beta q^{69} -2 q^{70} -\beta q^{75} + \beta q^{76} - q^{77} + 2 q^{78} -\beta q^{80} - q^{81} -\beta q^{82} -2 q^{83} + \beta q^{84} -2 q^{86} -\beta q^{87} - q^{89} -2 q^{90} - q^{91} + q^{92} + 2 q^{93} + 2 q^{94} + 2 q^{95} -2 q^{96} + q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 4q^{6} - 2q^{7} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 4q^{6} - 2q^{7} - 2q^{9} + 4q^{10} + 2q^{11} + 2q^{13} - 4q^{15} - 2q^{16} - 2q^{23} - 2q^{25} + 2q^{28} - 2q^{29} + 2q^{36} - 4q^{38} + 2q^{41} - 4q^{42} - 2q^{44} - 2q^{52} + 4q^{57} + 2q^{59} + 4q^{60} + 2q^{61} - 4q^{62} + 2q^{63} + 2q^{64} + 4q^{66} - 4q^{70} - 2q^{77} + 4q^{78} - 2q^{81} - 4q^{83} - 4q^{86} - 2q^{89} - 4q^{90} - 2q^{91} + 2q^{92} + 4q^{93} + 4q^{94} + 4q^{95} - 4q^{96} + 2q^{97} - 2q^{99} + O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/283\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
282.1
1.41421i
1.41421i
1.41421i 1.41421i −1.00000 1.41421i 2.00000 −1.00000 0 −1.00000 2.00000
282.2 1.41421i 1.41421i −1.00000 1.41421i 2.00000 −1.00000 0 −1.00000 2.00000
\(n\): 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
283.b Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2 \) acting on \(S_{1}^{\mathrm{new}}(283, [\chi])\).