# Properties

 Label 99.2.e.b Level 99 Weight 2 Character orbit 99.e Analytic conductor 0.791 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 99.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.790518980011$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{4} + 3 \zeta_{6} q^{5} + ( 4 - 4 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -1 - \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{4} + 3 \zeta_{6} q^{5} + ( 4 - 4 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} + ( -1 + \zeta_{6} ) q^{11} + ( 2 - 4 \zeta_{6} ) q^{12} -2 \zeta_{6} q^{13} + ( 3 - 6 \zeta_{6} ) q^{15} + ( -4 + 4 \zeta_{6} ) q^{16} -6 q^{17} + 2 q^{19} + ( -6 + 6 \zeta_{6} ) q^{20} + ( -8 + 4 \zeta_{6} ) q^{21} -3 \zeta_{6} q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} + 8 q^{28} + ( 6 - 6 \zeta_{6} ) q^{29} -8 \zeta_{6} q^{31} + ( 2 - \zeta_{6} ) q^{33} + 12 q^{35} + ( -6 + 6 \zeta_{6} ) q^{36} + 2 q^{37} + ( -2 + 4 \zeta_{6} ) q^{39} + ( -8 + 8 \zeta_{6} ) q^{43} -2 q^{44} + ( -9 + 9 \zeta_{6} ) q^{45} + ( -3 + 3 \zeta_{6} ) q^{47} + ( 8 - 4 \zeta_{6} ) q^{48} -9 \zeta_{6} q^{49} + ( 6 + 6 \zeta_{6} ) q^{51} + ( 4 - 4 \zeta_{6} ) q^{52} + 3 q^{53} -3 q^{55} + ( -2 - 2 \zeta_{6} ) q^{57} + ( 12 - 6 \zeta_{6} ) q^{60} + ( -8 + 8 \zeta_{6} ) q^{61} + 12 q^{63} -8 q^{64} + ( 6 - 6 \zeta_{6} ) q^{65} + 13 \zeta_{6} q^{67} -12 \zeta_{6} q^{68} + ( -3 + 6 \zeta_{6} ) q^{69} + 2 q^{73} + ( 8 - 4 \zeta_{6} ) q^{75} + 4 \zeta_{6} q^{76} + 4 \zeta_{6} q^{77} + ( -2 + 2 \zeta_{6} ) q^{79} -12 q^{80} + ( -9 + 9 \zeta_{6} ) q^{81} + ( 18 - 18 \zeta_{6} ) q^{83} + ( -8 - 8 \zeta_{6} ) q^{84} -18 \zeta_{6} q^{85} + ( -12 + 6 \zeta_{6} ) q^{87} + 3 q^{89} -8 q^{91} + ( 6 - 6 \zeta_{6} ) q^{92} + ( -8 + 16 \zeta_{6} ) q^{93} + 6 \zeta_{6} q^{95} + ( -2 + 2 \zeta_{6} ) q^{97} -3 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{3} + 2q^{4} + 3q^{5} + 4q^{7} + 3q^{9} + O(q^{10})$$ $$2q - 3q^{3} + 2q^{4} + 3q^{5} + 4q^{7} + 3q^{9} - q^{11} - 2q^{13} - 4q^{16} - 12q^{17} + 4q^{19} - 6q^{20} - 12q^{21} - 3q^{23} - 4q^{25} + 16q^{28} + 6q^{29} - 8q^{31} + 3q^{33} + 24q^{35} - 6q^{36} + 4q^{37} - 8q^{43} - 4q^{44} - 9q^{45} - 3q^{47} + 12q^{48} - 9q^{49} + 18q^{51} + 4q^{52} + 6q^{53} - 6q^{55} - 6q^{57} + 18q^{60} - 8q^{61} + 24q^{63} - 16q^{64} + 6q^{65} + 13q^{67} - 12q^{68} + 4q^{73} + 12q^{75} + 4q^{76} + 4q^{77} - 2q^{79} - 24q^{80} - 9q^{81} + 18q^{83} - 24q^{84} - 18q^{85} - 18q^{87} + 6q^{89} - 16q^{91} + 6q^{92} + 6q^{95} - 2q^{97} - 6q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
34.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 −1.50000 0.866025i 1.00000 + 1.73205i 1.50000 + 2.59808i 0 2.00000 3.46410i 0 1.50000 + 2.59808i 0
67.1 0 −1.50000 + 0.866025i 1.00000 1.73205i 1.50000 2.59808i 0 2.00000 + 3.46410i 0 1.50000 2.59808i 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.2.e.b 2
3.b odd 2 1 297.2.e.b 2
9.c even 3 1 inner 99.2.e.b 2
9.c even 3 1 891.2.a.d 1
9.d odd 6 1 297.2.e.b 2
9.d odd 6 1 891.2.a.e 1
11.b odd 2 1 1089.2.e.b 2
99.g even 6 1 9801.2.a.g 1
99.h odd 6 1 1089.2.e.b 2
99.h odd 6 1 9801.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.e.b 2 1.a even 1 1 trivial
99.2.e.b 2 9.c even 3 1 inner
297.2.e.b 2 3.b odd 2 1
297.2.e.b 2 9.d odd 6 1
891.2.a.d 1 9.c even 3 1
891.2.a.e 1 9.d odd 6 1
1089.2.e.b 2 11.b odd 2 1
1089.2.e.b 2 99.h odd 6 1
9801.2.a.f 1 99.h odd 6 1
9801.2.a.g 1 99.g even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{2}^{\mathrm{new}}(99, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T^{2} + 4 T^{4}$$
$3$ $$1 + 3 T + 3 T^{2}$$
$5$ $$1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4}$$
$7$ $$( 1 - 5 T + 7 T^{2} )( 1 + T + 7 T^{2} )$$
$11$ $$1 + T + T^{2}$$
$13$ $$( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$
$17$ $$( 1 + 6 T + 17 T^{2} )^{2}$$
$19$ $$( 1 - 2 T + 19 T^{2} )^{2}$$
$23$ $$1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4}$$
$29$ $$1 - 6 T + 7 T^{2} - 174 T^{3} + 841 T^{4}$$
$31$ $$1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4}$$
$37$ $$( 1 - 2 T + 37 T^{2} )^{2}$$
$41$ $$1 - 41 T^{2} + 1681 T^{4}$$
$43$ $$( 1 - 5 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} )$$
$47$ $$1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4}$$
$53$ $$( 1 - 3 T + 53 T^{2} )^{2}$$
$59$ $$1 - 59 T^{2} + 3481 T^{4}$$
$61$ $$1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4}$$
$67$ $$1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 71 T^{2} )^{2}$$
$73$ $$( 1 - 2 T + 73 T^{2} )^{2}$$
$79$ $$1 + 2 T - 75 T^{2} + 158 T^{3} + 6241 T^{4}$$
$83$ $$1 - 18 T + 241 T^{2} - 1494 T^{3} + 6889 T^{4}$$
$89$ $$( 1 - 3 T + 89 T^{2} )^{2}$$
$97$ $$1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4}$$