# Properties

 Label 99.2.d.a Level 99 Weight 2 Character orbit 99.d Analytic conductor 0.791 Analytic rank 0 Dimension 4 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 99.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.790518980011$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + q^{4} -\beta_{1} q^{5} -\beta_{3} q^{7} + \beta_{2} q^{8} +O(q^{10})$$ $$q -\beta_{2} q^{2} + q^{4} -\beta_{1} q^{5} -\beta_{3} q^{7} + \beta_{2} q^{8} + \beta_{3} q^{10} + ( -2 \beta_{1} + \beta_{2} ) q^{11} -2 \beta_{3} q^{13} + 3 \beta_{1} q^{14} -5 q^{16} + 3 \beta_{3} q^{19} -\beta_{1} q^{20} + ( -3 + 2 \beta_{3} ) q^{22} + 2 \beta_{1} q^{23} + 3 q^{25} + 6 \beta_{1} q^{26} -\beta_{3} q^{28} -4 \beta_{2} q^{29} -4 q^{31} + 3 \beta_{2} q^{32} -2 \beta_{2} q^{35} + 8 q^{37} -9 \beta_{1} q^{38} -\beta_{3} q^{40} + 4 \beta_{2} q^{41} -\beta_{3} q^{43} + ( -2 \beta_{1} + \beta_{2} ) q^{44} -2 \beta_{3} q^{46} + 2 \beta_{1} q^{47} + q^{49} -3 \beta_{2} q^{50} -2 \beta_{3} q^{52} -7 \beta_{1} q^{53} + ( -4 - \beta_{3} ) q^{55} -3 \beta_{1} q^{56} + 12 q^{58} + 8 \beta_{1} q^{59} + 2 \beta_{3} q^{61} + 4 \beta_{2} q^{62} + q^{64} -4 \beta_{2} q^{65} -4 q^{67} + 6 q^{70} + 2 \beta_{1} q^{71} -8 \beta_{2} q^{74} + 3 \beta_{3} q^{76} + ( -3 \beta_{1} - 4 \beta_{2} ) q^{77} + 5 \beta_{3} q^{79} + 5 \beta_{1} q^{80} -12 q^{82} + 8 \beta_{2} q^{83} + 3 \beta_{1} q^{86} + ( 3 - 2 \beta_{3} ) q^{88} + 5 \beta_{1} q^{89} -12 q^{91} + 2 \beta_{1} q^{92} -2 \beta_{3} q^{94} + 6 \beta_{2} q^{95} -10 q^{97} -\beta_{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{4} + O(q^{10})$$ $$4q + 4q^{4} - 20q^{16} - 12q^{22} + 12q^{25} - 16q^{31} + 32q^{37} + 4q^{49} - 16q^{55} + 48q^{58} + 4q^{64} - 16q^{67} + 24q^{70} - 48q^{82} + 12q^{88} - 48q^{91} - 40q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 4 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} + 3 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 5 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{3} + 5 \beta_{1}$$$$)/2$$

## Character values

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

 $$n$$ $$46$$ $$56$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
98.1
 0.517638i − 0.517638i − 1.93185i 1.93185i
−1.73205 0 1.00000 1.41421i 0 2.44949i 1.73205 0 2.44949i
98.2 −1.73205 0 1.00000 1.41421i 0 2.44949i 1.73205 0 2.44949i
98.3 1.73205 0 1.00000 1.41421i 0 2.44949i −1.73205 0 2.44949i
98.4 1.73205 0 1.00000 1.41421i 0 2.44949i −1.73205 0 2.44949i
 $$n$$: 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
3.b odd 2 1 inner
11.b odd 2 1 inner
33.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.2.d.a 4
3.b odd 2 1 inner 99.2.d.a 4
4.b odd 2 1 1584.2.b.e 4
5.b even 2 1 2475.2.f.e 4
5.c odd 4 2 2475.2.d.a 8
8.b even 2 1 6336.2.b.s 4
8.d odd 2 1 6336.2.b.t 4
9.c even 3 2 891.2.g.c 8
9.d odd 6 2 891.2.g.c 8
11.b odd 2 1 inner 99.2.d.a 4
12.b even 2 1 1584.2.b.e 4
15.d odd 2 1 2475.2.f.e 4
15.e even 4 2 2475.2.d.a 8
24.f even 2 1 6336.2.b.t 4
24.h odd 2 1 6336.2.b.s 4
33.d even 2 1 inner 99.2.d.a 4
44.c even 2 1 1584.2.b.e 4
55.d odd 2 1 2475.2.f.e 4
55.e even 4 2 2475.2.d.a 8
88.b odd 2 1 6336.2.b.s 4
88.g even 2 1 6336.2.b.t 4
99.g even 6 2 891.2.g.c 8
99.h odd 6 2 891.2.g.c 8
132.d odd 2 1 1584.2.b.e 4
165.d even 2 1 2475.2.f.e 4
165.l odd 4 2 2475.2.d.a 8
264.m even 2 1 6336.2.b.s 4
264.p odd 2 1 6336.2.b.t 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.d.a 4 1.a even 1 1 trivial
99.2.d.a 4 3.b odd 2 1 inner
99.2.d.a 4 11.b odd 2 1 inner
99.2.d.a 4 33.d even 2 1 inner
891.2.g.c 8 9.c even 3 2
891.2.g.c 8 9.d odd 6 2
891.2.g.c 8 99.g even 6 2
891.2.g.c 8 99.h odd 6 2
1584.2.b.e 4 4.b odd 2 1
1584.2.b.e 4 12.b even 2 1
1584.2.b.e 4 44.c even 2 1
1584.2.b.e 4 132.d odd 2 1
2475.2.d.a 8 5.c odd 4 2
2475.2.d.a 8 15.e even 4 2
2475.2.d.a 8 55.e even 4 2
2475.2.d.a 8 165.l odd 4 2
2475.2.f.e 4 5.b even 2 1
2475.2.f.e 4 15.d odd 2 1
2475.2.f.e 4 55.d odd 2 1
2475.2.f.e 4 165.d even 2 1
6336.2.b.s 4 8.b even 2 1
6336.2.b.s 4 24.h odd 2 1
6336.2.b.s 4 88.b odd 2 1
6336.2.b.s 4 264.m even 2 1
6336.2.b.t 4 8.d odd 2 1
6336.2.b.t 4 24.f even 2 1
6336.2.b.t 4 88.g even 2 1
6336.2.b.t 4 264.p odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(99, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} + 4 T^{4} )^{2}$$
$3$ 1
$5$ $$( 1 - 8 T^{2} + 25 T^{4} )^{2}$$
$7$ $$( 1 - 8 T^{2} + 49 T^{4} )^{2}$$
$11$ $$1 + 10 T^{2} + 121 T^{4}$$
$13$ $$( 1 - 2 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 + 17 T^{2} )^{4}$$
$19$ $$( 1 + 16 T^{2} + 361 T^{4} )^{2}$$
$23$ $$( 1 - 38 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 + 10 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 + 4 T + 31 T^{2} )^{4}$$
$37$ $$( 1 - 8 T + 37 T^{2} )^{4}$$
$41$ $$( 1 + 34 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 80 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$( 1 - 86 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 - 8 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$( 1 + 10 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - 98 T^{2} + 3721 T^{4} )^{2}$$
$67$ $$( 1 + 4 T + 67 T^{2} )^{4}$$
$71$ $$( 1 - 134 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 73 T^{2} )^{4}$$
$79$ $$( 1 - 8 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$( 1 - 26 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 - 128 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 + 10 T + 97 T^{2} )^{4}$$