# 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

# Related objects

## 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: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ 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
 0.5 − 0.866025i 0.5 + 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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