# Properties

 Label 99.2.e.c 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: $$\Z[a_1, \ldots, a_{11}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 2 - 2 \zeta_{6} ) q^{2} + ( 1 - 2 \zeta_{6} ) q^{3} -2 \zeta_{6} q^{4} + 2 \zeta_{6} q^{5} + ( -2 - 2 \zeta_{6} ) q^{6} + ( -4 + 4 \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})$$ $$q + ( 2 - 2 \zeta_{6} ) q^{2} + ( 1 - 2 \zeta_{6} ) q^{3} -2 \zeta_{6} q^{4} + 2 \zeta_{6} q^{5} + ( -2 - 2 \zeta_{6} ) q^{6} + ( -4 + 4 \zeta_{6} ) q^{7} -3 q^{9} + 4 q^{10} + ( -1 + \zeta_{6} ) q^{11} + ( -4 + 2 \zeta_{6} ) q^{12} -4 \zeta_{6} q^{13} + 8 \zeta_{6} q^{14} + ( 4 - 2 \zeta_{6} ) q^{15} + ( 4 - 4 \zeta_{6} ) q^{16} + 4 q^{17} + ( -6 + 6 \zeta_{6} ) q^{18} -6 q^{19} + ( 4 - 4 \zeta_{6} ) q^{20} + ( 4 + 4 \zeta_{6} ) q^{21} + 2 \zeta_{6} q^{22} + \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{25} -8 q^{26} + ( -3 + 6 \zeta_{6} ) q^{27} + 8 q^{28} + ( 4 - 8 \zeta_{6} ) q^{30} -\zeta_{6} q^{31} -8 \zeta_{6} q^{32} + ( 1 + \zeta_{6} ) q^{33} + ( 8 - 8 \zeta_{6} ) q^{34} -8 q^{35} + 6 \zeta_{6} q^{36} + 3 q^{37} + ( -12 + 12 \zeta_{6} ) q^{38} + ( -8 + 4 \zeta_{6} ) q^{39} + 2 \zeta_{6} q^{41} + ( 16 - 8 \zeta_{6} ) q^{42} + ( -12 + 12 \zeta_{6} ) q^{43} + 2 q^{44} -6 \zeta_{6} q^{45} + 2 q^{46} + ( 7 - 7 \zeta_{6} ) q^{47} + ( -4 - 4 \zeta_{6} ) q^{48} -9 \zeta_{6} q^{49} -2 \zeta_{6} q^{50} + ( 4 - 8 \zeta_{6} ) q^{51} + ( -8 + 8 \zeta_{6} ) q^{52} + 3 q^{53} + ( 6 + 6 \zeta_{6} ) q^{54} -2 q^{55} + ( -6 + 12 \zeta_{6} ) q^{57} -11 \zeta_{6} q^{59} + ( -4 - 4 \zeta_{6} ) q^{60} -2 q^{62} + ( 12 - 12 \zeta_{6} ) q^{63} -8 q^{64} + ( 8 - 8 \zeta_{6} ) q^{65} + ( 4 - 2 \zeta_{6} ) q^{66} + 4 \zeta_{6} q^{67} -8 \zeta_{6} q^{68} + ( 2 - \zeta_{6} ) q^{69} + ( -16 + 16 \zeta_{6} ) q^{70} + 15 q^{71} -8 q^{73} + ( 6 - 6 \zeta_{6} ) q^{74} + ( -1 - \zeta_{6} ) q^{75} + 12 \zeta_{6} q^{76} -4 \zeta_{6} q^{77} + ( -8 + 16 \zeta_{6} ) q^{78} + ( 10 - 10 \zeta_{6} ) q^{79} + 8 q^{80} + 9 q^{81} + 4 q^{82} + ( -12 + 12 \zeta_{6} ) q^{83} + ( 8 - 16 \zeta_{6} ) q^{84} + 8 \zeta_{6} q^{85} + 24 \zeta_{6} q^{86} + 3 q^{89} -12 q^{90} + 16 q^{91} + ( 2 - 2 \zeta_{6} ) q^{92} + ( -2 + \zeta_{6} ) q^{93} -14 \zeta_{6} q^{94} -12 \zeta_{6} q^{95} + ( -16 + 8 \zeta_{6} ) q^{96} + ( -17 + 17 \zeta_{6} ) q^{97} -18 q^{98} + ( 3 - 3 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 2q^{4} + 2q^{5} - 6q^{6} - 4q^{7} - 6q^{9} + O(q^{10})$$ $$2q + 2q^{2} - 2q^{4} + 2q^{5} - 6q^{6} - 4q^{7} - 6q^{9} + 8q^{10} - q^{11} - 6q^{12} - 4q^{13} + 8q^{14} + 6q^{15} + 4q^{16} + 8q^{17} - 6q^{18} - 12q^{19} + 4q^{20} + 12q^{21} + 2q^{22} + q^{23} + q^{25} - 16q^{26} + 16q^{28} - q^{31} - 8q^{32} + 3q^{33} + 8q^{34} - 16q^{35} + 6q^{36} + 6q^{37} - 12q^{38} - 12q^{39} + 2q^{41} + 24q^{42} - 12q^{43} + 4q^{44} - 6q^{45} + 4q^{46} + 7q^{47} - 12q^{48} - 9q^{49} - 2q^{50} - 8q^{52} + 6q^{53} + 18q^{54} - 4q^{55} - 11q^{59} - 12q^{60} - 4q^{62} + 12q^{63} - 16q^{64} + 8q^{65} + 6q^{66} + 4q^{67} - 8q^{68} + 3q^{69} - 16q^{70} + 30q^{71} - 16q^{73} + 6q^{74} - 3q^{75} + 12q^{76} - 4q^{77} + 10q^{79} + 16q^{80} + 18q^{81} + 8q^{82} - 12q^{83} + 8q^{85} + 24q^{86} + 6q^{89} - 24q^{90} + 32q^{91} + 2q^{92} - 3q^{93} - 14q^{94} - 12q^{95} - 24q^{96} - 17q^{97} - 36q^{98} + 3q^{99} + O(q^{100})$$

## 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
1.00000 1.73205i 1.73205i −1.00000 1.73205i 1.00000 + 1.73205i −3.00000 1.73205i −2.00000 + 3.46410i 0 −3.00000 4.00000
67.1 1.00000 + 1.73205i 1.73205i −1.00000 + 1.73205i 1.00000 1.73205i −3.00000 + 1.73205i −2.00000 3.46410i 0 −3.00000 4.00000
 $$n$$: 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.c 2
3.b odd 2 1 297.2.e.a 2
9.c even 3 1 inner 99.2.e.c 2
9.c even 3 1 891.2.a.a 1
9.d odd 6 1 297.2.e.a 2
9.d odd 6 1 891.2.a.h 1
11.b odd 2 1 1089.2.e.a 2
99.g even 6 1 9801.2.a.a 1
99.h odd 6 1 1089.2.e.a 2
99.h odd 6 1 9801.2.a.l 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 2 T_{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(99, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4}$$
$3$ $$1 + 3 T^{2}$$
$5$ $$1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4}$$
$7$ $$( 1 - T + 7 T^{2} )( 1 + 5 T + 7 T^{2} )$$
$11$ $$1 + T + T^{2}$$
$13$ $$1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4}$$
$17$ $$( 1 - 4 T + 17 T^{2} )^{2}$$
$19$ $$( 1 + 6 T + 19 T^{2} )^{2}$$
$23$ $$1 - T - 22 T^{2} - 23 T^{3} + 529 T^{4}$$
$29$ $$1 - 29 T^{2} + 841 T^{4}$$
$31$ $$1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4}$$
$37$ $$( 1 - 3 T + 37 T^{2} )^{2}$$
$41$ $$1 - 2 T - 37 T^{2} - 82 T^{3} + 1681 T^{4}$$
$43$ $$1 + 12 T + 101 T^{2} + 516 T^{3} + 1849 T^{4}$$
$47$ $$1 - 7 T + 2 T^{2} - 329 T^{3} + 2209 T^{4}$$
$53$ $$( 1 - 3 T + 53 T^{2} )^{2}$$
$59$ $$1 + 11 T + 62 T^{2} + 649 T^{3} + 3481 T^{4}$$
$61$ $$1 - 61 T^{2} + 3721 T^{4}$$
$67$ $$1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4}$$
$71$ $$( 1 - 15 T + 71 T^{2} )^{2}$$
$73$ $$( 1 + 8 T + 73 T^{2} )^{2}$$
$79$ $$1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4}$$
$83$ $$1 + 12 T + 61 T^{2} + 996 T^{3} + 6889 T^{4}$$
$89$ $$( 1 - 3 T + 89 T^{2} )^{2}$$
$97$ $$1 + 17 T + 192 T^{2} + 1649 T^{3} + 9409 T^{4}$$