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.

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

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 \(S_{1}^{\mathrm{new}}(88, [\chi])\).