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 discriminant -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 \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.0439177211117\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \(x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
11.c even 5 1 inner
88.l odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 88.1.l.a 4
3.b odd 2 1 792.1.bu.a 4
4.b odd 2 1 352.1.t.a 4
5.b even 2 1 2200.1.cl.a 4
5.c odd 4 2 2200.1.dd.a 8
8.b even 2 1 352.1.t.a 4
8.d odd 2 1 CM 88.1.l.a 4
11.b odd 2 1 968.1.l.b 4
11.c even 5 1 inner 88.1.l.a 4
11.c even 5 1 968.1.f.b 2
11.c even 5 2 968.1.l.a 4
11.d odd 10 1 968.1.f.a 2
11.d odd 10 1 968.1.l.b 4
11.d odd 10 2 968.1.l.c 4
12.b even 2 1 3168.1.ck.a 4
16.e even 4 2 2816.1.v.c 8
16.f odd 4 2 2816.1.v.c 8
24.f even 2 1 792.1.bu.a 4
24.h odd 2 1 3168.1.ck.a 4
33.h odd 10 1 792.1.bu.a 4
40.e odd 2 1 2200.1.cl.a 4
40.k even 4 2 2200.1.dd.a 8
44.c even 2 1 3872.1.t.c 4
44.g even 10 1 3872.1.f.b 2
44.g even 10 2 3872.1.t.a 4
44.g even 10 1 3872.1.t.c 4
44.h odd 10 1 352.1.t.a 4
44.h odd 10 1 3872.1.f.a 2
44.h odd 10 2 3872.1.t.b 4
55.j even 10 1 2200.1.cl.a 4
55.k odd 20 2 2200.1.dd.a 8
88.b odd 2 1 3872.1.t.c 4
88.g even 2 1 968.1.l.b 4
88.k even 10 1 968.1.f.a 2
88.k even 10 1 968.1.l.b 4
88.k even 10 2 968.1.l.c 4
88.l odd 10 1 inner 88.1.l.a 4
88.l odd 10 1 968.1.f.b 2
88.l odd 10 2 968.1.l.a 4
88.o even 10 1 352.1.t.a 4
88.o even 10 1 3872.1.f.a 2
88.o even 10 2 3872.1.t.b 4
88.p odd 10 1 3872.1.f.b 2
88.p odd 10 2 3872.1.t.a 4
88.p odd 10 1 3872.1.t.c 4
132.o even 10 1 3168.1.ck.a 4
176.v odd 20 2 2816.1.v.c 8
176.w even 20 2 2816.1.v.c 8
264.t odd 10 1 3168.1.ck.a 4
264.w even 10 1 792.1.bu.a 4
440.bh odd 10 1 2200.1.cl.a 4
440.bs even 20 2 2200.1.dd.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.1.l.a 4 1.a even 1 1 trivial
88.1.l.a 4 8.d odd 2 1 CM
88.1.l.a 4 11.c even 5 1 inner
88.1.l.a 4 88.l odd 10 1 inner
352.1.t.a 4 4.b odd 2 1
352.1.t.a 4 8.b even 2 1
352.1.t.a 4 44.h odd 10 1
352.1.t.a 4 88.o even 10 1
792.1.bu.a 4 3.b odd 2 1
792.1.bu.a 4 24.f even 2 1
792.1.bu.a 4 33.h odd 10 1
792.1.bu.a 4 264.w even 10 1
968.1.f.a 2 11.d odd 10 1
968.1.f.a 2 88.k even 10 1
968.1.f.b 2 11.c even 5 1
968.1.f.b 2 88.l odd 10 1
968.1.l.a 4 11.c even 5 2
968.1.l.a 4 88.l odd 10 2
968.1.l.b 4 11.b odd 2 1
968.1.l.b 4 11.d odd 10 1
968.1.l.b 4 88.g even 2 1
968.1.l.b 4 88.k even 10 1
968.1.l.c 4 11.d odd 10 2
968.1.l.c 4 88.k even 10 2
2200.1.cl.a 4 5.b even 2 1
2200.1.cl.a 4 40.e odd 2 1
2200.1.cl.a 4 55.j even 10 1
2200.1.cl.a 4 440.bh odd 10 1
2200.1.dd.a 8 5.c odd 4 2
2200.1.dd.a 8 40.k even 4 2
2200.1.dd.a 8 55.k odd 20 2
2200.1.dd.a 8 440.bs even 20 2
2816.1.v.c 8 16.e even 4 2
2816.1.v.c 8 16.f odd 4 2
2816.1.v.c 8 176.v odd 20 2
2816.1.v.c 8 176.w even 20 2
3168.1.ck.a 4 12.b even 2 1
3168.1.ck.a 4 24.h odd 2 1
3168.1.ck.a 4 132.o even 10 1
3168.1.ck.a 4 264.t odd 10 1
3872.1.f.a 2 44.h odd 10 1
3872.1.f.a 2 88.o even 10 1
3872.1.f.b 2 44.g even 10 1
3872.1.f.b 2 88.p odd 10 1
3872.1.t.a 4 44.g even 10 2
3872.1.t.a 4 88.p odd 10 2
3872.1.t.b 4 44.h odd 10 2
3872.1.t.b 4 88.o even 10 2
3872.1.t.c 4 44.c even 2 1
3872.1.t.c 4 44.g even 10 1
3872.1.t.c 4 88.b odd 2 1
3872.1.t.c 4 88.p odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(88, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$3$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$5$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$7$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$11$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$13$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$17$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$19$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$23$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$29$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$31$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$37$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$41$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$43$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$47$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$53$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$59$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$61$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$67$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$71$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$73$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$79$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$83$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$89$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$97$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
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