Properties

Label 99.1.h.a
Level 99
Weight 1
Character orbit 99.h
Analytic conductor 0.049
Analytic rank 0
Dimension 2
Projective image \(D_{3}\)
CM disc. -11
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 99.h (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.0494074362507\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.891.1
Artin image size \(18\)
Artin image $C_3\times S_3$
Artin field Galois closure of 6.0.107811.1

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + \zeta_{6}^{2} q^{3} \) \( + \zeta_{6}^{2} q^{4} \) \( -\zeta_{6}^{2} q^{5} \) \( -\zeta_{6} q^{9} \) \(+O(q^{10})\) \( q\) \( + \zeta_{6}^{2} q^{3} \) \( + \zeta_{6}^{2} q^{4} \) \( -\zeta_{6}^{2} q^{5} \) \( -\zeta_{6} q^{9} \) \( -\zeta_{6} q^{11} \) \( -\zeta_{6} q^{12} \) \( + \zeta_{6} q^{15} \) \( -\zeta_{6} q^{16} \) \( + \zeta_{6} q^{20} \) \( + 2 \zeta_{6}^{2} q^{23} \) \(+ q^{27}\) \( -\zeta_{6}^{2} q^{31} \) \(+ q^{33}\) \(+ q^{36}\) \(- q^{37}\) \(+ q^{44}\) \(- q^{45}\) \( + \zeta_{6} q^{47} \) \(+ q^{48}\) \( + \zeta_{6}^{2} q^{49} \) \(- q^{53}\) \(- q^{55}\) \( -\zeta_{6}^{2} q^{59} \) \(- q^{60}\) \(+ q^{64}\) \( -\zeta_{6}^{2} q^{67} \) \( -2 \zeta_{6} q^{69} \) \(- q^{71}\) \(- q^{80}\) \( + \zeta_{6}^{2} q^{81} \) \( + 2 q^{89} \) \( -2 \zeta_{6} q^{92} \) \( + \zeta_{6} q^{93} \) \( + \zeta_{6} q^{97} \) \( + \zeta_{6}^{2} q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut -\mathstrut q^{4} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut -\mathstrut q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut -\mathstrut q^{4} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut -\mathstrut q^{9} \) \(\mathstrut -\mathstrut q^{11} \) \(\mathstrut -\mathstrut q^{12} \) \(\mathstrut +\mathstrut q^{15} \) \(\mathstrut -\mathstrut q^{16} \) \(\mathstrut +\mathstrut q^{20} \) \(\mathstrut -\mathstrut 2q^{23} \) \(\mathstrut +\mathstrut 2q^{27} \) \(\mathstrut +\mathstrut q^{31} \) \(\mathstrut +\mathstrut 2q^{33} \) \(\mathstrut +\mathstrut 2q^{36} \) \(\mathstrut -\mathstrut 2q^{37} \) \(\mathstrut +\mathstrut 2q^{44} \) \(\mathstrut -\mathstrut 2q^{45} \) \(\mathstrut +\mathstrut q^{47} \) \(\mathstrut +\mathstrut 2q^{48} \) \(\mathstrut -\mathstrut q^{49} \) \(\mathstrut -\mathstrut 2q^{53} \) \(\mathstrut -\mathstrut 2q^{55} \) \(\mathstrut +\mathstrut q^{59} \) \(\mathstrut -\mathstrut 2q^{60} \) \(\mathstrut +\mathstrut 2q^{64} \) \(\mathstrut +\mathstrut q^{67} \) \(\mathstrut -\mathstrut 2q^{69} \) \(\mathstrut -\mathstrut 2q^{71} \) \(\mathstrut -\mathstrut 2q^{80} \) \(\mathstrut -\mathstrut q^{81} \) \(\mathstrut +\mathstrut 4q^{89} \) \(\mathstrut -\mathstrut 2q^{92} \) \(\mathstrut +\mathstrut q^{93} \) \(\mathstrut +\mathstrut q^{97} \) \(\mathstrut -\mathstrut q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\)
\(\chi(n)\) \(-1\) \(\zeta_{6}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −0.500000 0.866025i −0.500000 0.866025i 0.500000 + 0.866025i 0 0 0 −0.500000 + 0.866025i 0
76.1 0 −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 0.866025i 0 0 0 −0.500000 0.866025i 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
11.b Odd 1 CM by \(\Q(\sqrt{-11}) \) yes
9.c Even 1 yes
99.h Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(99, [\chi])\).