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 disc. -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 \(10\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(0.0464130461749\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{5}\)
Projective field Galois closure of 5.1.8311689.1

$q$-expansion

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

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 CM by \(\Q(\sqrt{-3}) \) yes
31.d Even 1 yes
93.l Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(93, [\chi])\).