Properties

Label 2563.1.b.d
Level 2563
Weight 1
Character orbit 2563.b
Analytic conductor 1.279
Analytic rank 0
Dimension 2
Projective image \(S_{4}\)
CM/RM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 2563 = 11 \cdot 233 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 2563.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(1.27910362738\)
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 4.2.2563.1
Artin image size \(48\)
Artin image $\GL(2,3)$
Artin field Galois closure of 8.2.16836267547.1

$q$-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.

\(f(q)\) \(=\) \( q -\beta q^{2} + \beta q^{3} - q^{4} -\beta q^{5} + 2 q^{6} - q^{9} +O(q^{10})\) \( q -\beta q^{2} + \beta q^{3} - q^{4} -\beta q^{5} + 2 q^{6} - q^{9} -2 q^{10} - q^{11} -\beta q^{12} + 2 q^{15} - q^{16} - q^{17} + \beta q^{18} -\beta q^{19} + \beta q^{20} + \beta q^{22} - q^{23} - q^{25} -\beta q^{29} -2 \beta q^{30} - q^{31} + \beta q^{32} -\beta q^{33} + \beta q^{34} + q^{36} + q^{37} -2 q^{38} - q^{41} + q^{43} + q^{44} + \beta q^{45} + \beta q^{46} + \beta q^{47} -\beta q^{48} + q^{49} + \beta q^{50} -\beta q^{51} -\beta q^{53} + \beta q^{55} + 2 q^{57} -2 q^{58} -2 q^{60} + \beta q^{62} + q^{64} -2 q^{66} + q^{68} -\beta q^{69} - q^{71} + q^{73} -\beta q^{74} -\beta q^{75} + \beta q^{76} - q^{79} + \beta q^{80} - q^{81} + \beta q^{82} + \beta q^{85} -\beta q^{86} + 2 q^{87} + q^{89} + 2 q^{90} + q^{92} -\beta q^{93} + 2 q^{94} -2 q^{95} -2 q^{96} -\beta q^{97} -\beta q^{98} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 4q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 4q^{6} - 2q^{9} - 4q^{10} - 2q^{11} + 4q^{15} - 2q^{16} - 2q^{17} - 2q^{23} - 2q^{25} - 2q^{31} + 2q^{36} + 2q^{37} - 4q^{38} - 2q^{41} + 2q^{43} + 2q^{44} + 2q^{49} + 4q^{57} - 4q^{58} - 4q^{60} + 2q^{64} - 4q^{66} + 2q^{68} - 2q^{71} + 2q^{73} - 2q^{79} - 2q^{81} + 4q^{87} + 2q^{89} + 4q^{90} + 2q^{92} + 4q^{94} - 4q^{95} - 4q^{96} + 2q^{99} + O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2563\mathbb{Z}\right)^\times\).

\(n\) \(1399\) \(2333\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2562.1
1.41421i
1.41421i
1.41421i 1.41421i −1.00000 1.41421i 2.00000 0 0 −1.00000 −2.00000
2562.2 1.41421i 1.41421i −1.00000 1.41421i 2.00000 0 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
2563.b Odd 1 yes

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2563, [\chi])\):

\( T_{2}^{2} + 2 \)
\( T_{17} + 1 \)