Properties

Label 93.1.l.a
Level 93
Weight 1
Character orbit 93.l
Analytic conductor 0.046
Analytic rank 0
Dimension 4
Projective image \(D_{5}\)
CM discriminant -3
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 93 = 3 \cdot 31 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 93.l (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.0464130461749\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \(x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{5}\)
Projective field Galois closure of 5.1.8311689.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{10} q^{3} -\zeta_{10}^{3} q^{4} + ( \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{7} + \zeta_{10}^{2} q^{9} +O(q^{10})\) \( q -\zeta_{10} q^{3} -\zeta_{10}^{3} q^{4} + ( \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{7} + \zeta_{10}^{2} q^{9} + \zeta_{10}^{4} q^{12} + ( -\zeta_{10}^{3} + \zeta_{10}^{4} ) q^{13} -\zeta_{10} q^{16} + ( -\zeta_{10} + \zeta_{10}^{2} ) q^{19} + ( 1 - \zeta_{10}^{3} ) q^{21} + q^{25} -\zeta_{10}^{3} q^{27} + ( 1 + \zeta_{10}^{2} ) q^{28} -\zeta_{10} q^{31} + q^{36} + ( -\zeta_{10} + \zeta_{10}^{4} ) q^{37} + ( 1 + \zeta_{10}^{4} ) q^{39} + ( 1 - \zeta_{10}^{3} ) q^{43} + \zeta_{10}^{2} q^{48} + ( -\zeta_{10} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{49} + ( -\zeta_{10} + \zeta_{10}^{2} ) q^{52} + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{57} + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{61} + ( -\zeta_{10} + \zeta_{10}^{4} ) q^{63} + \zeta_{10}^{4} q^{64} + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{67} + ( \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{73} -\zeta_{10} q^{75} + ( 1 + \zeta_{10}^{4} ) q^{76} + ( 1 + \zeta_{10}^{4} ) q^{79} + \zeta_{10}^{4} q^{81} + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{84} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{91} + \zeta_{10}^{2} q^{93} + ( 1 - \zeta_{10} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - q^{3} - q^{4} - 2q^{7} - q^{9} + O(q^{10}) \) \( 4q - q^{3} - q^{4} - 2q^{7} - q^{9} - q^{12} - 2q^{13} - q^{16} - 2q^{19} + 3q^{21} + 4q^{25} - q^{27} + 3q^{28} - q^{31} + 4q^{36} - 2q^{37} + 3q^{39} + 3q^{43} - q^{48} - 3q^{49} - 2q^{52} - 2q^{57} - 2q^{61} - 2q^{63} - q^{64} - 2q^{67} - 2q^{73} - q^{75} + 3q^{76} + 3q^{79} - q^{81} - 2q^{84} + q^{91} - q^{93} + 3q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/93\mathbb{Z}\right)^\times\).

\(n\) \(32\) \(34\)
\(\chi(n)\) \(-1\) \(\zeta_{10}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
0.809017 0.587785i
−0.309017 0.951057i
−0.309017 + 0.951057i
0.809017 + 0.587785i
0 −0.809017 + 0.587785i 0.309017 + 0.951057i 0 0 −0.500000 1.53884i 0 0.309017 0.951057i 0
8.1 0 0.309017 + 0.951057i −0.809017 0.587785i 0 0 −0.500000 0.363271i 0 −0.809017 + 0.587785i 0
35.1 0 0.309017 0.951057i −0.809017 + 0.587785i 0 0 −0.500000 + 0.363271i 0 −0.809017 0.587785i 0
47.1 0 −0.809017 0.587785i 0.309017 0.951057i 0 0 −0.500000 + 1.53884i 0 0.309017 + 0.951057i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
31.d even 5 1 inner
93.l odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 93.1.l.a 4
3.b odd 2 1 CM 93.1.l.a 4
4.b odd 2 1 1488.1.br.a 4
5.b even 2 1 2325.1.ca.a 4
5.c odd 4 2 2325.1.bq.a 8
9.c even 3 2 2511.1.bu.a 8
9.d odd 6 2 2511.1.bu.a 8
12.b even 2 1 1488.1.br.a 4
15.d odd 2 1 2325.1.ca.a 4
15.e even 4 2 2325.1.bq.a 8
31.b odd 2 1 2883.1.l.b 4
31.c even 3 2 2883.1.o.d 8
31.d even 5 1 inner 93.1.l.a 4
31.d even 5 1 2883.1.b.b 2
31.d even 5 2 2883.1.l.a 4
31.e odd 6 2 2883.1.o.b 8
31.f odd 10 1 2883.1.b.a 2
31.f odd 10 1 2883.1.l.b 4
31.f odd 10 2 2883.1.l.c 4
31.g even 15 2 2883.1.h.a 4
31.g even 15 4 2883.1.o.c 8
31.g even 15 2 2883.1.o.d 8
31.h odd 30 2 2883.1.h.b 4
31.h odd 30 4 2883.1.o.a 8
31.h odd 30 2 2883.1.o.b 8
93.c even 2 1 2883.1.l.b 4
93.g even 6 2 2883.1.o.b 8
93.h odd 6 2 2883.1.o.d 8
93.k even 10 1 2883.1.b.a 2
93.k even 10 1 2883.1.l.b 4
93.k even 10 2 2883.1.l.c 4
93.l odd 10 1 inner 93.1.l.a 4
93.l odd 10 1 2883.1.b.b 2
93.l odd 10 2 2883.1.l.a 4
93.o odd 30 2 2883.1.h.a 4
93.o odd 30 4 2883.1.o.c 8
93.o odd 30 2 2883.1.o.d 8
93.p even 30 2 2883.1.h.b 4
93.p even 30 4 2883.1.o.a 8
93.p even 30 2 2883.1.o.b 8
124.l odd 10 1 1488.1.br.a 4
155.n even 10 1 2325.1.ca.a 4
155.s odd 20 2 2325.1.bq.a 8
279.z even 15 2 2511.1.bu.a 8
279.bf odd 30 2 2511.1.bu.a 8
372.t even 10 1 1488.1.br.a 4
465.x odd 10 1 2325.1.ca.a 4
465.bj even 20 2 2325.1.bq.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
93.1.l.a 4 1.a even 1 1 trivial
93.1.l.a 4 3.b odd 2 1 CM
93.1.l.a 4 31.d even 5 1 inner
93.1.l.a 4 93.l odd 10 1 inner
1488.1.br.a 4 4.b odd 2 1
1488.1.br.a 4 12.b even 2 1
1488.1.br.a 4 124.l odd 10 1
1488.1.br.a 4 372.t even 10 1
2325.1.bq.a 8 5.c odd 4 2
2325.1.bq.a 8 15.e even 4 2
2325.1.bq.a 8 155.s odd 20 2
2325.1.bq.a 8 465.bj even 20 2
2325.1.ca.a 4 5.b even 2 1
2325.1.ca.a 4 15.d odd 2 1
2325.1.ca.a 4 155.n even 10 1
2325.1.ca.a 4 465.x odd 10 1
2511.1.bu.a 8 9.c even 3 2
2511.1.bu.a 8 9.d odd 6 2
2511.1.bu.a 8 279.z even 15 2
2511.1.bu.a 8 279.bf odd 30 2
2883.1.b.a 2 31.f odd 10 1
2883.1.b.a 2 93.k even 10 1
2883.1.b.b 2 31.d even 5 1
2883.1.b.b 2 93.l odd 10 1
2883.1.h.a 4 31.g even 15 2
2883.1.h.a 4 93.o odd 30 2
2883.1.h.b 4 31.h odd 30 2
2883.1.h.b 4 93.p even 30 2
2883.1.l.a 4 31.d even 5 2
2883.1.l.a 4 93.l odd 10 2
2883.1.l.b 4 31.b odd 2 1
2883.1.l.b 4 31.f odd 10 1
2883.1.l.b 4 93.c even 2 1
2883.1.l.b 4 93.k even 10 1
2883.1.l.c 4 31.f odd 10 2
2883.1.l.c 4 93.k even 10 2
2883.1.o.a 8 31.h odd 30 4
2883.1.o.a 8 93.p even 30 4
2883.1.o.b 8 31.e odd 6 2
2883.1.o.b 8 31.h odd 30 2
2883.1.o.b 8 93.g even 6 2
2883.1.o.b 8 93.p even 30 2
2883.1.o.c 8 31.g even 15 4
2883.1.o.c 8 93.o odd 30 4
2883.1.o.d 8 31.c even 3 2
2883.1.o.d 8 31.g even 15 2
2883.1.o.d 8 93.h odd 6 2
2883.1.o.d 8 93.o odd 30 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(93, [\chi])\).

Hecke characteristic polynomials

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