Embedded newform 19.4.e.a.5.2

• Label: 19.4.e.a.5.2
• Level: $19$
• Weight: $4$
• Character: 19.5
• Analytic conductor: $1.121$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	19.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.12103629011\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(4\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			5.2
Character	\(\chi\)	\(=\)	19.5
Dual form			19.4.e.a.4.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.16911 + 0.789492i) q^{2} +(-0.930433 + 5.27675i) q^{3} +(-2.04660 + 1.71731i) q^{4} +(6.28846 + 5.27664i) q^{5} +(-2.14774 - 12.1804i) q^{6} +(-1.44586 - 2.50430i) q^{7} +(12.3168 - 21.3333i) q^{8} +(-1.60664 - 0.584768i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 6 q^{2} - 3 q^{3} - 24 q^{4} - 6 q^{5} + 42 q^{6} + 3 q^{7} - 75 q^{8} - 51 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/19\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.16911	+	0.789492i	−0.766897	+	0.279128i	−0.695698	−	0.718334i	\(-0.744905\pi\)
							−0.0711991	+	0.997462i	\(0.522683\pi\)
\(3\)	−0.930433	+	5.27675i	−0.179062	+	1.01551i	0.754288	+	0.656543i	\(0.227982\pi\)
							−0.933350	+	0.358967i	\(0.883129\pi\)
\(5\)	6.28846	+	5.27664i	0.562457	+	0.471957i	0.879133	−	0.476576i	\(-0.158122\pi\)
							−0.316676	+	0.948534i	\(0.602567\pi\)
\(7\)	−1.44586	−	2.50430i	−0.0780691	−	0.135220i	0.824348	−	0.566084i	\(-0.191542\pi\)
							−0.902417	+	0.430864i	\(0.858209\pi\)
\(11\)	14.1179	−	24.4529i	0.386973	−	0.670257i	−0.605068	−	0.796174i	\(-0.706854\pi\)
							0.992041	+	0.125917i	\(0.0401872\pi\)
\(13\)	14.7989	+	83.9287i	0.315729	+	1.79059i	0.568103	+	0.822957i	\(0.307677\pi\)
							−0.252374	+	0.967630i	\(0.581211\pi\)
\(17\)	33.2517	−	12.1026i	0.474396	−	0.172666i	−0.0937467	−	0.995596i	\(-0.529884\pi\)
							0.568143	+	0.822930i	\(0.307662\pi\)
\(19\)	−70.3411	−	43.7165i	−0.849335	−	0.527855i
							\(23\)	131.166	−	110.062i	1.18913	−	0.997801i	0.189259	−	0.981927i	\(-0.439391\pi\)
0.999874	−	0.0158739i	\(-0.00505302\pi\)
\(29\)	−62.1167	−	22.6086i	−0.397751	−	0.144770i	0.135397	−	0.990791i	\(-0.456769\pi\)
							−0.533148	+	0.846022i	\(0.678991\pi\)
\(31\)	−76.2433	−	132.057i	−0.441732	−	0.765102i	0.556086	−	0.831125i	\(-0.312302\pi\)
							−0.997818	+	0.0660223i	\(0.978969\pi\)
\(37\)	129.616			0.575912			0.287956	−	0.957644i	\(-0.407024\pi\)
							0.287956	+	0.957644i	\(0.407024\pi\)
\(41\)	−27.0433	+	153.370i	−0.103011	+	0.584206i	0.888985	+	0.457937i	\(0.151411\pi\)
							−0.991996	+	0.126269i	\(0.959700\pi\)
\(43\)	242.696	+	203.646i	0.860715	+	0.722226i	0.962122	−	0.272619i	\(-0.0878899\pi\)
							−0.101407	+	0.994845i	\(0.532334\pi\)
\(47\)	−450.304	−	163.897i	−1.39752	−	0.508657i	−0.470082	−	0.882623i	\(-0.655776\pi\)
							−0.927442	+	0.373966i	\(0.877998\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.4.e.a.5.2	yes	24
3.2	odd	2		171.4.u.b.100.3		24
19.2	odd	18		361.4.a.m.1.5		12
19.4	even	9	inner	19.4.e.a.4.2	✓	24
19.17	even	9		361.4.a.n.1.8		12
57.23	odd	18		171.4.u.b.118.3		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.4.e.a.4.2	✓	24	19.4	even	9	inner
19.4.e.a.5.2	yes	24	1.1	even	1	trivial
171.4.u.b.100.3		24	3.2	odd	2
171.4.u.b.118.3		24	57.23	odd	18
361.4.a.m.1.5		12	19.2	odd	18
361.4.a.n.1.8		12	19.17	even	9