Embedded newform 152.3.u.b.35.14

• Label: 152.3.u.b.35.14
• Level: $152$
• Weight: $3$
• Character: 152.35
• Analytic conductor: $4.142$
• Analytic rank: $0$
• Dimension: $216$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 152 = 2^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	152.u (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.14170001828\)
Analytic rank:	\(0\)
Dimension:	\(216\)
Relative dimension:	\(36\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			35.14
Character	\(\chi\)	\(=\)	152.35
Dual form			152.3.u.b.139.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.484715 + 1.94037i) q^{2} +(0.839443 - 0.305532i) q^{3} +(-3.53010 - 1.88106i) q^{4} +(-7.22003 - 1.27309i) q^{5} +(0.185957 + 1.77693i) q^{6} +(5.32261 - 3.07301i) q^{7} +(5.36104 - 5.93795i) q^{8} +(-6.28309 + 5.27214i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 6 q^{2} - 18 q^{3} - 12 q^{4} - 12 q^{6} - 51 q^{8} - 54 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/152\mathbb{Z}\right)^\times\).

\(n\)	\(39\)	\(77\)	\(97\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{2}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.484715	+	1.94037i	−0.242357	+	0.970187i
							\(3\)	0.839443	−	0.305532i	0.279814	−	0.101844i	−0.198301	−	0.980141i	\(-0.563542\pi\)
0.478115	+	0.878297i	\(0.341320\pi\)
\(5\)	−7.22003	−	1.27309i	−1.44401	−	0.254617i	−0.603910	−	0.797053i	\(-0.706391\pi\)
							−0.840097	+	0.542436i	\(0.817502\pi\)
\(7\)	5.32261	−	3.07301i	0.760372	−	0.439001i	−0.0690570	−	0.997613i	\(-0.521999\pi\)
							0.829429	+	0.558611i	\(0.188666\pi\)
\(11\)	7.57294	−	13.1167i	0.688450	−	1.19243i	−0.283890	−	0.958857i	\(-0.591625\pi\)
							0.972339	−	0.233573i	\(-0.0750417\pi\)
\(13\)	6.86104	−	18.8505i	0.527772	−	1.45004i	−0.333916	−	0.942603i	\(-0.608370\pi\)
							0.861688	−	0.507439i	\(-0.169408\pi\)
\(17\)	−9.10122	−	7.63683i	−0.535366	−	0.449225i	0.334584	−	0.942366i	\(-0.391404\pi\)
							−0.869950	+	0.493141i	\(0.835849\pi\)
\(19\)	−16.7435	−	8.98079i	−0.881238	−	0.472673i
							\(23\)	−26.1726	+	4.61493i	−1.13794	+	0.200649i	−0.710703	−	0.703492i	\(-0.751623\pi\)
−0.427234	+	0.904141i	\(0.640512\pi\)
\(29\)	11.7295	+	13.9786i	0.404464	+	0.482022i	0.929376	−	0.369135i	\(-0.120346\pi\)
							−0.524911	+	0.851157i	\(0.675902\pi\)
\(31\)	12.8488	−	7.41827i	0.414478	−	0.239299i	−0.278234	−	0.960513i	\(-0.589749\pi\)
							0.692712	+	0.721214i	\(0.256416\pi\)
\(37\)		−	57.5033i		−	1.55414i	−0.629413	−	0.777071i	\(-0.716704\pi\)
							0.629413	−	0.777071i	\(-0.283296\pi\)
\(41\)	−2.83181	+	1.03070i	−0.0690686	+	0.0251389i	−0.376323	−	0.926488i	\(-0.622812\pi\)
							0.307255	+	0.951627i	\(0.400590\pi\)
\(43\)	−8.19985	+	46.5037i	−0.190694	+	1.08148i	0.727724	+	0.685870i	\(0.240578\pi\)
							−0.918418	+	0.395610i	\(0.870533\pi\)
\(47\)	13.9806	+	16.6614i	0.297460	+	0.354499i	0.893986	−	0.448095i	\(-0.147897\pi\)
							−0.596526	+	0.802594i	\(0.703453\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	152.3.u.b.35.14	✓	216
8.3	odd	2	inner	152.3.u.b.35.32	yes	216
19.6	even	9	inner	152.3.u.b.139.32	yes	216
152.139	odd	18	inner	152.3.u.b.139.14	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.3.u.b.35.14	✓	216	1.1	even	1	trivial
152.3.u.b.35.32	yes	216	8.3	odd	2	inner
152.3.u.b.139.14	yes	216	152.139	odd	18	inner
152.3.u.b.139.32	yes	216	19.6	even	9	inner