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 discriminant -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 \(6\), degree \(2\), minimal)

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: \( 1 \)
Twist minimal: yes
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.891.1
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.

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
9.c even 3 1 inner
99.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.1.h.a 2
3.b odd 2 1 297.1.h.a 2
4.b odd 2 1 1584.1.bf.b 2
5.b even 2 1 2475.1.y.a 2
5.c odd 4 2 2475.1.t.a 4
9.c even 3 1 inner 99.1.h.a 2
9.c even 3 1 891.1.c.a 1
9.d odd 6 1 297.1.h.a 2
9.d odd 6 1 891.1.c.b 1
11.b odd 2 1 CM 99.1.h.a 2
11.c even 5 4 1089.1.s.a 8
11.d odd 10 4 1089.1.s.a 8
33.d even 2 1 297.1.h.a 2
33.f even 10 4 3267.1.w.a 8
33.h odd 10 4 3267.1.w.a 8
36.f odd 6 1 1584.1.bf.b 2
44.c even 2 1 1584.1.bf.b 2
45.j even 6 1 2475.1.y.a 2
45.k odd 12 2 2475.1.t.a 4
55.d odd 2 1 2475.1.y.a 2
55.e even 4 2 2475.1.t.a 4
99.g even 6 1 297.1.h.a 2
99.g even 6 1 891.1.c.b 1
99.h odd 6 1 inner 99.1.h.a 2
99.h odd 6 1 891.1.c.a 1
99.m even 15 4 1089.1.s.a 8
99.n odd 30 4 3267.1.w.a 8
99.o odd 30 4 1089.1.s.a 8
99.p even 30 4 3267.1.w.a 8
396.k even 6 1 1584.1.bf.b 2
495.o odd 6 1 2475.1.y.a 2
495.bf even 12 2 2475.1.t.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.1.h.a 2 1.a even 1 1 trivial
99.1.h.a 2 9.c even 3 1 inner
99.1.h.a 2 11.b odd 2 1 CM
99.1.h.a 2 99.h odd 6 1 inner
297.1.h.a 2 3.b odd 2 1
297.1.h.a 2 9.d odd 6 1
297.1.h.a 2 33.d even 2 1
297.1.h.a 2 99.g even 6 1
891.1.c.a 1 9.c even 3 1
891.1.c.a 1 99.h odd 6 1
891.1.c.b 1 9.d odd 6 1
891.1.c.b 1 99.g even 6 1
1089.1.s.a 8 11.c even 5 4
1089.1.s.a 8 11.d odd 10 4
1089.1.s.a 8 99.m even 15 4
1089.1.s.a 8 99.o odd 30 4
1584.1.bf.b 2 4.b odd 2 1
1584.1.bf.b 2 36.f odd 6 1
1584.1.bf.b 2 44.c even 2 1
1584.1.bf.b 2 396.k even 6 1
2475.1.t.a 4 5.c odd 4 2
2475.1.t.a 4 45.k odd 12 2
2475.1.t.a 4 55.e even 4 2
2475.1.t.a 4 495.bf even 12 2
2475.1.y.a 2 5.b even 2 1
2475.1.y.a 2 45.j even 6 1
2475.1.y.a 2 55.d odd 2 1
2475.1.y.a 2 495.o odd 6 1
3267.1.w.a 8 33.f even 10 4
3267.1.w.a 8 33.h odd 10 4
3267.1.w.a 8 99.n odd 30 4
3267.1.w.a 8 99.p even 30 4

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(99, [\chi])\).

Hecke characteristic polynomials

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