Embedded newform 999.1.dq.a.163.1

• Label: 999.1.dq.a.163.1
• Level: $999$
• Weight: $1$
• Character: 999.163
• Analytic conductor: $0.499$
• Analytic rank: $0$
• Dimension: $12$
• Projective image: $D_{36}$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 999 = 3^{3} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	999.dq (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.498565947620\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	\(\Q(\zeta_{36})\)
Defining polynomial:	\( x^{12} - x^{6} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{36}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{36} - \cdots)\)

Embedding invariants

Embedding label			163.1
Root			\(0.642788 + 0.766044i\) of defining polynomial
Character	\(\chi\)	\(=\)	999.163
Dual form			999.1.dq.a.190.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.642788 - 0.766044i) q^{4} +(-0.342020 + 1.93969i) q^{7} +(1.63207 - 0.142788i) q^{13} +(-0.173648 - 0.984808i) q^{16} +(-0.816436 + 1.75085i) q^{19} +(0.342020 - 0.939693i) q^{25} +(1.26604 + 1.50881i) q^{28} +(-0.811160 - 0.811160i) q^{31} +(0.500000 - 0.866025i) q^{37} +(0.597672 - 0.597672i) q^{43} +(-2.70574 - 0.984808i) q^{49} +(0.939693 - 1.34202i) q^{52} +(0.0451151 + 0.515668i) q^{61} +(-0.866025 - 0.500000i) q^{64} +(-1.50881 - 0.266044i) q^{67} -0.347296i q^{73} +(0.816436 + 1.75085i) q^{76} +(0.0999810 + 0.142788i) q^{79} +(-0.281237 + 3.21455i) q^{91} +(0.515668 + 1.92450i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{28} + 6 q^{37} - 12 q^{49} - 6 q^{91}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/999\mathbb{Z}\right)^\times\).

\(n\)	\(298\)	\(704\)
\(\chi(n)\)	\(e\left(\frac{13}{36}\right)\)	\(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		0			0		0.906308	−	0.422618i	\(-0.138889\pi\)
							−0.906308	+	0.422618i	\(0.861111\pi\)
\(3\)		0			0
							\(5\)		0			0		0.819152	−	0.573576i	\(-0.194444\pi\)
−0.819152	+	0.573576i	\(0.805556\pi\)
\(7\)	−0.342020	+	1.93969i	−0.342020	+	1.93969i			1.00000i	\(0.5\pi\)
							−0.342020	+	0.939693i	\(0.611111\pi\)
\(11\)		0			0		−0.500000	−	0.866025i	\(-0.666667\pi\)
							0.500000	+	0.866025i	\(0.333333\pi\)
\(13\)	1.63207	−	0.142788i	1.63207	−	0.142788i	0.766044	−	0.642788i	\(-0.222222\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)
\(17\)		0			0		−0.996195	−	0.0871557i	\(-0.972222\pi\)
							0.996195	+	0.0871557i	\(0.0277778\pi\)
\(19\)	−0.816436	+	1.75085i	−0.816436	+	1.75085i	−0.173648	+	0.984808i	\(0.555556\pi\)
							−0.642788	+	0.766044i	\(0.722222\pi\)
\(23\)		0			0		0.965926	−	0.258819i	\(-0.0833333\pi\)
							−0.965926	+	0.258819i	\(0.916667\pi\)
\(29\)		0			0		−0.965926	−	0.258819i	\(-0.916667\pi\)
							0.965926	+	0.258819i	\(0.0833333\pi\)
\(31\)	−0.811160	−	0.811160i	−0.811160	−	0.811160i	0.173648	−	0.984808i	\(-0.444444\pi\)
							−0.984808	+	0.173648i	\(0.944444\pi\)
\(37\)	0.500000	−	0.866025i	0.500000	−	0.866025i
							\(41\)		0			0		−0.766044	−	0.642788i	\(-0.777778\pi\)
0.766044	+	0.642788i	\(0.222222\pi\)
\(43\)	0.597672	−	0.597672i	0.597672	−	0.597672i	−0.342020	−	0.939693i	\(-0.611111\pi\)
							0.939693	+	0.342020i	\(0.111111\pi\)
\(47\)		0			0		−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)
\(53\)		0			0		0.984808	−	0.173648i	\(-0.0555556\pi\)
							−0.984808	+	0.173648i	\(0.944444\pi\)
\(59\)		0			0		0.573576	−	0.819152i	\(-0.305556\pi\)
							−0.573576	+	0.819152i	\(0.694444\pi\)
\(61\)	0.0451151	+	0.515668i	0.0451151	+	0.515668i	0.984808	+	0.173648i	\(0.0555556\pi\)
							−0.939693	+	0.342020i	\(0.888889\pi\)
\(67\)	−1.50881	−	0.266044i	−1.50881	−	0.266044i	−0.642788	−	0.766044i	\(-0.722222\pi\)
							−0.866025	+	0.500000i	\(0.833333\pi\)
\(71\)		0			0		−0.342020	−	0.939693i	\(-0.611111\pi\)
							0.342020	+	0.939693i	\(0.388889\pi\)
\(73\)		−	0.347296i		−	0.347296i	−0.984808	−	0.173648i	\(-0.944444\pi\)
							0.984808	−	0.173648i	\(-0.0555556\pi\)
\(79\)	0.0999810	+	0.142788i	0.0999810	+	0.142788i	0.866025	−	0.500000i	\(-0.166667\pi\)
							−0.766044	+	0.642788i	\(0.777778\pi\)
\(83\)		0			0		−0.642788	−	0.766044i	\(-0.722222\pi\)
							0.642788	+	0.766044i	\(0.277778\pi\)
\(89\)		0			0		−0.819152	−	0.573576i	\(-0.805556\pi\)
							0.819152	+	0.573576i	\(0.194444\pi\)
\(97\)	0.515668	+	1.92450i	0.515668	+	1.92450i	0.342020	+	0.939693i	\(0.388889\pi\)
							0.173648	+	0.984808i	\(0.444444\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	999.1.dq.a.163.1	✓	12
3.2	odd	2	CM	999.1.dq.a.163.1	✓	12
9.2	odd	6		2997.1.dy.a.1162.1		12
9.4	even	3		2997.1.dp.a.2161.1		12
9.5	odd	6		2997.1.dp.a.2161.1		12
9.7	even	3		2997.1.dy.a.1162.1		12
37.5	odd	36	inner	999.1.dq.a.190.1	yes	12
111.5	even	36	inner	999.1.dq.a.190.1	yes	12
333.5	even	36		2997.1.dy.a.1189.1		12
333.79	odd	36		2997.1.dp.a.190.1		12
333.227	even	36		2997.1.dp.a.190.1		12
333.301	odd	36		2997.1.dy.a.1189.1		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
999.1.dq.a.163.1	✓	12	1.1	even	1	trivial
999.1.dq.a.163.1	✓	12	3.2	odd	2	CM
999.1.dq.a.190.1	yes	12	37.5	odd	36	inner
999.1.dq.a.190.1	yes	12	111.5	even	36	inner
2997.1.dp.a.190.1		12	333.79	odd	36
2997.1.dp.a.190.1		12	333.227	even	36
2997.1.dp.a.2161.1		12	9.4	even	3
2997.1.dp.a.2161.1		12	9.5	odd	6
2997.1.dy.a.1162.1		12	9.2	odd	6
2997.1.dy.a.1162.1		12	9.7	even	3
2997.1.dy.a.1189.1		12	333.5	even	36
2997.1.dy.a.1189.1		12	333.301	odd	36