# Properties

 Label 99.2.g.a Level 99 Weight 2 Character orbit 99.g Analytic conductor 0.791 Analytic rank 0 Dimension 4 CM discriminant -11 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 99.g (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.790518980011$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + ( 2 - 2 \beta_{2} ) q^{4} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{5} + ( 3 \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + ( 2 - 2 \beta_{2} ) q^{4} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{5} + ( 3 \beta_{2} + \beta_{3} ) q^{9} + ( \beta_{2} + 2 \beta_{3} ) q^{11} -2 \beta_{3} q^{12} + ( 3 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{15} -4 \beta_{2} q^{16} + ( -6 + 2 \beta_{2} - 2 \beta_{3} ) q^{20} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{23} + ( -3 + 6 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} ) q^{25} + ( -3 - 2 \beta_{1} + 2 \beta_{3} ) q^{27} + ( 4 - 3 \beta_{1} - \beta_{2} + 6 \beta_{3} ) q^{31} + ( -6 + \beta_{1} - \beta_{3} ) q^{33} + ( 6 - 2 \beta_{1} + 2 \beta_{3} ) q^{36} + ( 5 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{37} + ( 2 - 4 \beta_{1} + 4 \beta_{3} ) q^{44} + ( 3 - 4 \beta_{1} - 9 \beta_{2} + \beta_{3} ) q^{45} + ( 12 - 7 \beta_{2} - 2 \beta_{3} ) q^{47} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{48} + ( -7 + 7 \beta_{2} ) q^{49} + ( 1 + 4 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} ) q^{53} + ( -4 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{55} + ( 8 - \beta_{1} + 7 \beta_{2} ) q^{59} + ( 6 + 2 \beta_{1} + 4 \beta_{3} ) q^{60} -8 q^{64} + ( -5 - 3 \beta_{1} + 8 \beta_{2} + 6 \beta_{3} ) q^{67} + ( -6 \beta_{2} - \beta_{3} ) q^{69} + ( 1 - 5 \beta_{1} + 3 \beta_{2} + 5 \beta_{3} ) q^{71} + ( 9 - 6 \beta_{1} - 18 \beta_{2} + 3 \beta_{3} ) q^{75} + ( -8 + 4 \beta_{1} + 12 \beta_{2} - 4 \beta_{3} ) q^{80} + ( -6 + 5 \beta_{1} + 6 \beta_{2} ) q^{81} + ( -5 + 10 \beta_{1} - 10 \beta_{3} ) q^{89} + ( 2 \beta_{2} + 4 \beta_{3} ) q^{92} + ( -18 + 3 \beta_{1} + 9 \beta_{2} - 4 \beta_{3} ) q^{93} + ( -3 + 6 \beta_{1} + 10 \beta_{2} - 3 \beta_{3} ) q^{97} + ( 3 + 5 \beta_{1} - 3 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - q^{3} + 4q^{4} - 9q^{5} + 5q^{9} + O(q^{10})$$ $$4q - q^{3} + 4q^{4} - 9q^{5} + 5q^{9} + 2q^{12} + 10q^{15} - 8q^{16} - 18q^{20} + 9q^{25} - 16q^{27} + 5q^{31} - 22q^{33} + 20q^{36} + 14q^{37} - 11q^{45} + 36q^{47} + 8q^{48} - 14q^{49} - 22q^{55} + 45q^{59} + 22q^{60} - 32q^{64} - 13q^{67} - 11q^{69} - 9q^{75} - 7q^{81} - 47q^{93} + 17q^{97} + 11q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} - 2 \nu - 3$$$$)/6$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu + 3$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{3} + 2 \beta_{1} + 3$$

## Character values

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

 $$n$$ $$46$$ $$56$$ $$\chi(n)$$ $$-1$$ $$1 - \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
32.1
 1.68614 + 0.396143i −1.18614 − 1.26217i 1.68614 − 0.396143i −1.18614 + 1.26217i
0 −1.68614 0.396143i 1.00000 1.73205i −3.68614 2.12819i 0 0 0 2.68614 + 1.33591i 0
32.2 0 1.18614 + 1.26217i 1.00000 1.73205i −0.813859 0.469882i 0 0 0 −0.186141 + 2.99422i 0
65.1 0 −1.68614 + 0.396143i 1.00000 + 1.73205i −3.68614 + 2.12819i 0 0 0 2.68614 1.33591i 0
65.2 0 1.18614 1.26217i 1.00000 + 1.73205i −0.813859 + 0.469882i 0 0 0 −0.186141 2.99422i 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.d odd 6 1 inner
99.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.2.g.a 4
3.b odd 2 1 297.2.g.a 4
9.c even 3 1 297.2.g.a 4
9.c even 3 1 891.2.d.a 4
9.d odd 6 1 inner 99.2.g.a 4
9.d odd 6 1 891.2.d.a 4
11.b odd 2 1 CM 99.2.g.a 4
33.d even 2 1 297.2.g.a 4
99.g even 6 1 inner 99.2.g.a 4
99.g even 6 1 891.2.d.a 4
99.h odd 6 1 297.2.g.a 4
99.h odd 6 1 891.2.d.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.g.a 4 1.a even 1 1 trivial
99.2.g.a 4 9.d odd 6 1 inner
99.2.g.a 4 11.b odd 2 1 CM
99.2.g.a 4 99.g even 6 1 inner
297.2.g.a 4 3.b odd 2 1
297.2.g.a 4 9.c even 3 1
297.2.g.a 4 33.d even 2 1
297.2.g.a 4 99.h odd 6 1
891.2.d.a 4 9.c even 3 1
891.2.d.a 4 9.d odd 6 1
891.2.d.a 4 99.g even 6 1
891.2.d.a 4 99.h odd 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

$p$ $F_p(T)$
$2$ $$( 1 - 2 T^{2} + 4 T^{4} )^{2}$$
$3$ $$1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4}$$
$5$ $$( 1 + 3 T + 5 T^{2} )^{2}( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} )$$
$7$ $$( 1 + 7 T^{2} + 49 T^{4} )^{2}$$
$11$ $$1 - 11 T^{2} + 121 T^{4}$$
$13$ $$( 1 + 13 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 + 17 T^{2} )^{4}$$
$19$ $$( 1 - 19 T^{2} )^{4}$$
$23$ $$( 1 - 9 T + 58 T^{2} - 207 T^{3} + 529 T^{4} )( 1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4} )$$
$29$ $$( 1 - 29 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 5 T + 31 T^{2} )^{2}( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} )$$
$37$ $$( 1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$( 1 - 41 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 + 43 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$( 1 - 12 T + 47 T^{2} )^{2}( 1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4} )$$
$53$ $$( 1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4} )( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} )$$
$59$ $$( 1 - 15 T + 59 T^{2} )^{2}( 1 - 15 T + 166 T^{2} - 885 T^{3} + 3481 T^{4} )$$
$61$ $$( 1 + 61 T^{2} + 3721 T^{4} )^{2}$$
$67$ $$( 1 + 13 T + 67 T^{2} )^{2}( 1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4} )$$
$71$ $$( 1 - 3 T - 62 T^{2} - 213 T^{3} + 5041 T^{4} )( 1 + 3 T - 62 T^{2} + 213 T^{3} + 5041 T^{4} )$$
$73$ $$( 1 - 73 T^{2} )^{4}$$
$79$ $$( 1 + 79 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$( 1 - 83 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 - 9 T + 89 T^{2} )^{2}( 1 + 9 T + 89 T^{2} )^{2}$$
$97$ $$( 1 - 17 T + 97 T^{2} )^{2}( 1 + 17 T + 192 T^{2} + 1649 T^{3} + 9409 T^{4} )$$