Properties

Label 88.1.l.a
Level 88
Weight 1
Character orbit 88.l
Analytic conductor 0.044
Analytic rank 0
Dimension 4
Projective image \(D_{5}\)
CM disc. -8
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 88 = 2^{3} \cdot 11 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 88.l (of order \(10\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(0.0439177211117\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{5}\)
Projective field Galois closure of 5.1.937024.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{10} q^{2} + ( \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{3} + \zeta_{10}^{2} q^{4} + ( 1 - \zeta_{10}^{3} ) q^{6} -\zeta_{10}^{3} q^{8} + ( -\zeta_{10} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{9} +O(q^{10})\) \( q -\zeta_{10} q^{2} + ( \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{3} + \zeta_{10}^{2} q^{4} + ( 1 - \zeta_{10}^{3} ) q^{6} -\zeta_{10}^{3} q^{8} + ( -\zeta_{10} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{9} -\zeta_{10}^{3} q^{11} + ( -\zeta_{10} + \zeta_{10}^{4} ) q^{12} + \zeta_{10}^{4} q^{16} + ( -\zeta_{10} + \zeta_{10}^{2} ) q^{17} + ( 1 + \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{18} + ( 1 - \zeta_{10} ) q^{19} + \zeta_{10}^{4} q^{22} + ( 1 + \zeta_{10}^{2} ) q^{24} -\zeta_{10}^{3} q^{25} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{27} + q^{32} + ( 1 + \zeta_{10}^{2} ) q^{33} + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{34} + ( 1 - \zeta_{10} - \zeta_{10}^{3} ) q^{36} + ( -\zeta_{10} + \zeta_{10}^{2} ) q^{38} + ( \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{41} + ( -\zeta_{10} + \zeta_{10}^{4} ) q^{43} + q^{44} + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{48} + \zeta_{10}^{4} q^{49} + \zeta_{10}^{4} q^{50} + ( 1 - \zeta_{10} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{51} + ( -\zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{54} + ( 1 + \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{57} + ( 1 + \zeta_{10}^{4} ) q^{59} -\zeta_{10} q^{64} + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{66} + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{67} + ( -\zeta_{10}^{3} + \zeta_{10}^{4} ) q^{68} + ( -\zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{72} + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{73} + ( 1 + \zeta_{10}^{2} ) q^{75} + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{76} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{81} + ( 1 - \zeta_{10}^{3} ) q^{82} + ( 1 - \zeta_{10}^{3} ) q^{83} + ( 1 + \zeta_{10}^{2} ) q^{86} -\zeta_{10} q^{88} + ( -\zeta_{10} + \zeta_{10}^{4} ) q^{89} + ( \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{96} + ( 1 + \zeta_{10}^{2} ) q^{97} + q^{98} + ( -\zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - q^{2} - 2q^{3} - q^{4} + 3q^{6} - q^{8} - 3q^{9} + O(q^{10}) \) \( 4q - q^{2} - 2q^{3} - q^{4} + 3q^{6} - q^{8} - 3q^{9} - q^{11} - 2q^{12} - q^{16} - 2q^{17} + 2q^{18} + 3q^{19} - q^{22} + 3q^{24} - q^{25} + q^{27} + 4q^{32} + 3q^{33} - 2q^{34} + 2q^{36} - 2q^{38} - 2q^{41} - 2q^{43} + 4q^{44} - 2q^{48} - q^{49} - q^{50} + q^{51} - 4q^{54} + q^{57} + 3q^{59} - q^{64} - 2q^{66} - 2q^{67} - 2q^{68} - 3q^{72} - 2q^{73} + 3q^{75} - 2q^{76} + 3q^{82} + 3q^{83} + 3q^{86} - q^{88} - 2q^{89} - 2q^{96} + 3q^{97} + 4q^{98} - 3q^{99} + O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/88\mathbb{Z}\right)^\times\).

\(n\) \(23\) \(45\) \(57\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{10}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0.809017 0.587785i
−0.309017 0.951057i
0.809017 + 0.587785i
−0.309017 + 0.951057i
−0.809017 + 0.587785i −0.500000 1.53884i 0.309017 0.951057i 0 1.30902 + 0.951057i 0 0.309017 + 0.951057i −1.30902 + 0.951057i 0
27.1 0.309017 + 0.951057i −0.500000 0.363271i −0.809017 + 0.587785i 0 0.190983 0.587785i 0 −0.809017 0.587785i −0.190983 0.587785i 0
59.1 −0.809017 0.587785i −0.500000 + 1.53884i 0.309017 + 0.951057i 0 1.30902 0.951057i 0 0.309017 0.951057i −1.30902 0.951057i 0
75.1 0.309017 0.951057i −0.500000 + 0.363271i −0.809017 0.587785i 0 0.190983 + 0.587785i 0 −0.809017 + 0.587785i −0.190983 + 0.587785i 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
8.d Odd 1 CM by \(\Q(\sqrt{-2}) \) yes
11.c Even 1 yes
88.l Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(88, [\chi])\).