# Properties

 Label 99.2.e.a 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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{2} + ( -2 + \zeta_{6} ) q^{3} + \zeta_{6} q^{4} -\zeta_{6} q^{5} + ( 1 - 2 \zeta_{6} ) q^{6} + ( -4 + 4 \zeta_{6} ) q^{7} -3 q^{8} + ( 3 - 3 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{2} + ( -2 + \zeta_{6} ) q^{3} + \zeta_{6} q^{4} -\zeta_{6} q^{5} + ( 1 - 2 \zeta_{6} ) q^{6} + ( -4 + 4 \zeta_{6} ) q^{7} -3 q^{8} + ( 3 - 3 \zeta_{6} ) q^{9} + q^{10} + ( -1 + \zeta_{6} ) q^{11} + ( -1 - \zeta_{6} ) q^{12} + 2 \zeta_{6} q^{13} -4 \zeta_{6} q^{14} + ( 1 + \zeta_{6} ) q^{15} + ( 1 - \zeta_{6} ) q^{16} + 4 q^{17} + 3 \zeta_{6} q^{18} + 6 q^{19} + ( 1 - \zeta_{6} ) q^{20} + ( 4 - 8 \zeta_{6} ) q^{21} -\zeta_{6} q^{22} + 4 \zeta_{6} q^{23} + ( 6 - 3 \zeta_{6} ) q^{24} + ( 4 - 4 \zeta_{6} ) q^{25} -2 q^{26} + ( -3 + 6 \zeta_{6} ) q^{27} -4 q^{28} + ( -6 + 6 \zeta_{6} ) q^{29} + ( -2 + \zeta_{6} ) q^{30} -7 \zeta_{6} q^{31} -5 \zeta_{6} q^{32} + ( 1 - 2 \zeta_{6} ) q^{33} + ( -4 + 4 \zeta_{6} ) q^{34} + 4 q^{35} + 3 q^{36} + 3 q^{37} + ( -6 + 6 \zeta_{6} ) q^{38} + ( -2 - 2 \zeta_{6} ) q^{39} + 3 \zeta_{6} q^{40} + 2 \zeta_{6} q^{41} + ( 4 + 4 \zeta_{6} ) q^{42} + ( -6 + 6 \zeta_{6} ) q^{43} - q^{44} -3 q^{45} -4 q^{46} + ( 7 - 7 \zeta_{6} ) q^{47} + ( -1 + 2 \zeta_{6} ) q^{48} -9 \zeta_{6} q^{49} + 4 \zeta_{6} q^{50} + ( -8 + 4 \zeta_{6} ) q^{51} + ( -2 + 2 \zeta_{6} ) q^{52} -9 q^{53} + ( -3 - 3 \zeta_{6} ) q^{54} + q^{55} + ( 12 - 12 \zeta_{6} ) q^{56} + ( -12 + 6 \zeta_{6} ) q^{57} -6 \zeta_{6} q^{58} + 7 \zeta_{6} q^{59} + ( -1 + 2 \zeta_{6} ) q^{60} + 7 q^{62} + 12 \zeta_{6} q^{63} + 7 q^{64} + ( 2 - 2 \zeta_{6} ) q^{65} + ( 1 + \zeta_{6} ) q^{66} -11 \zeta_{6} q^{67} + 4 \zeta_{6} q^{68} + ( -4 - 4 \zeta_{6} ) q^{69} + ( -4 + 4 \zeta_{6} ) q^{70} + 9 q^{71} + ( -9 + 9 \zeta_{6} ) q^{72} + 4 q^{73} + ( -3 + 3 \zeta_{6} ) q^{74} + ( -4 + 8 \zeta_{6} ) q^{75} + 6 \zeta_{6} q^{76} -4 \zeta_{6} q^{77} + ( 4 - 2 \zeta_{6} ) q^{78} + ( -8 + 8 \zeta_{6} ) q^{79} - q^{80} -9 \zeta_{6} q^{81} -2 q^{82} + ( 12 - 12 \zeta_{6} ) q^{83} + ( 8 - 4 \zeta_{6} ) q^{84} -4 \zeta_{6} q^{85} -6 \zeta_{6} q^{86} + ( 6 - 12 \zeta_{6} ) q^{87} + ( 3 - 3 \zeta_{6} ) q^{88} + 6 q^{89} + ( 3 - 3 \zeta_{6} ) q^{90} -8 q^{91} + ( -4 + 4 \zeta_{6} ) q^{92} + ( 7 + 7 \zeta_{6} ) q^{93} + 7 \zeta_{6} q^{94} -6 \zeta_{6} q^{95} + ( 5 + 5 \zeta_{6} ) q^{96} + ( 19 - 19 \zeta_{6} ) q^{97} + 9 q^{98} + 3 \zeta_{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - 3q^{3} + q^{4} - q^{5} - 4q^{7} - 6q^{8} + 3q^{9} + O(q^{10})$$ $$2q - q^{2} - 3q^{3} + q^{4} - q^{5} - 4q^{7} - 6q^{8} + 3q^{9} + 2q^{10} - q^{11} - 3q^{12} + 2q^{13} - 4q^{14} + 3q^{15} + q^{16} + 8q^{17} + 3q^{18} + 12q^{19} + q^{20} - q^{22} + 4q^{23} + 9q^{24} + 4q^{25} - 4q^{26} - 8q^{28} - 6q^{29} - 3q^{30} - 7q^{31} - 5q^{32} - 4q^{34} + 8q^{35} + 6q^{36} + 6q^{37} - 6q^{38} - 6q^{39} + 3q^{40} + 2q^{41} + 12q^{42} - 6q^{43} - 2q^{44} - 6q^{45} - 8q^{46} + 7q^{47} - 9q^{49} + 4q^{50} - 12q^{51} - 2q^{52} - 18q^{53} - 9q^{54} + 2q^{55} + 12q^{56} - 18q^{57} - 6q^{58} + 7q^{59} + 14q^{62} + 12q^{63} + 14q^{64} + 2q^{65} + 3q^{66} - 11q^{67} + 4q^{68} - 12q^{69} - 4q^{70} + 18q^{71} - 9q^{72} + 8q^{73} - 3q^{74} + 6q^{76} - 4q^{77} + 6q^{78} - 8q^{79} - 2q^{80} - 9q^{81} - 4q^{82} + 12q^{83} + 12q^{84} - 4q^{85} - 6q^{86} + 3q^{88} + 12q^{89} + 3q^{90} - 16q^{91} - 4q^{92} + 21q^{93} + 7q^{94} - 6q^{95} + 15q^{96} + 19q^{97} + 18q^{98} + 3q^{99} + O(q^{100})$$

## 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
−0.500000 + 0.866025i −1.50000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 1.73205i −2.00000 + 3.46410i −3.00000 1.50000 2.59808i 1.00000
67.1 −0.500000 0.866025i −1.50000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 1.73205i −2.00000 3.46410i −3.00000 1.50000 + 2.59808i 1.00000
 $$n$$: 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.a 2
3.b odd 2 1 297.2.e.c 2
9.c even 3 1 inner 99.2.e.a 2
9.c even 3 1 891.2.a.g 1
9.d odd 6 1 297.2.e.c 2
9.d odd 6 1 891.2.a.c 1
11.b odd 2 1 1089.2.e.c 2
99.g even 6 1 9801.2.a.i 1
99.h odd 6 1 1089.2.e.c 2
99.h odd 6 1 9801.2.a.c 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(99, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T - T^{2} + 2 T^{3} + 4 T^{4}$$
$3$ $$1 + 3 T + 3 T^{2}$$
$5$ $$1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4}$$
$7$ $$( 1 - T + 7 T^{2} )( 1 + 5 T + 7 T^{2} )$$
$11$ $$1 + T + T^{2}$$
$13$ $$( 1 - 7 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} )$$
$17$ $$( 1 - 4 T + 17 T^{2} )^{2}$$
$19$ $$( 1 - 6 T + 19 T^{2} )^{2}$$
$23$ $$1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4}$$
$29$ $$1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4}$$
$31$ $$( 1 - 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} )$$
$37$ $$( 1 - 3 T + 37 T^{2} )^{2}$$
$41$ $$1 - 2 T - 37 T^{2} - 82 T^{3} + 1681 T^{4}$$
$43$ $$1 + 6 T - 7 T^{2} + 258 T^{3} + 1849 T^{4}$$
$47$ $$1 - 7 T + 2 T^{2} - 329 T^{3} + 2209 T^{4}$$
$53$ $$( 1 + 9 T + 53 T^{2} )^{2}$$
$59$ $$1 - 7 T - 10 T^{2} - 413 T^{3} + 3481 T^{4}$$
$61$ $$1 - 61 T^{2} + 3721 T^{4}$$
$67$ $$( 1 - 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} )$$
$71$ $$( 1 - 9 T + 71 T^{2} )^{2}$$
$73$ $$( 1 - 4 T + 73 T^{2} )^{2}$$
$79$ $$1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4}$$
$83$ $$1 - 12 T + 61 T^{2} - 996 T^{3} + 6889 T^{4}$$
$89$ $$( 1 - 6 T + 89 T^{2} )^{2}$$
$97$ $$( 1 - 14 T + 97 T^{2} )( 1 - 5 T + 97 T^{2} )$$