Embedded newform 152.4.v.b.3.7

• Label: 152.4.v.b.3.7
• Level: $152$
• Weight: $4$
• Character: 152.3
• Analytic conductor: $8.968$
• Analytic rank: $0$
• Dimension: $336$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 152 = 2^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	152.v (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.96829032087\)
Analytic rank:	\(0\)
Dimension:	\(336\)
Relative dimension:	\(56\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			3.7
Character	\(\chi\)	\(=\)	152.3
Dual form			152.4.v.b.51.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.59925 - 1.11530i) q^{2} +(0.142296 + 0.390954i) q^{3} +(5.51222 + 5.79788i) q^{4} +(-20.2800 - 3.57591i) q^{5} +(0.0661683 - 1.17489i) q^{6} +(-16.1140 + 9.30340i) q^{7} +(-7.86127 - 21.2179i) q^{8} +(20.5506 - 17.2440i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 6 q^{2} - 42 q^{3} + 12 q^{4} + 36 q^{6} - 9 q^{8} + 126 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(39\)	\(77\)	\(97\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{13}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.59925	−	1.11530i	−0.918974	−	0.394318i
							\(3\)	0.142296	+	0.390954i	0.0273848	+	0.0752391i	0.952632	−	0.304126i	\(-0.0983645\pi\)
−0.925247	+	0.379366i	\(0.876142\pi\)
\(5\)	−20.2800	−	3.57591i	−1.81390	−	0.319839i	−0.839277	−	0.543704i	\(-0.817022\pi\)
							−0.974620	+	0.223864i	\(0.928133\pi\)
\(7\)	−16.1140	+	9.30340i	−0.870072	+	0.502336i	−0.867372	−	0.497660i	\(-0.834193\pi\)
							−0.00269980	+	0.999996i	\(0.500859\pi\)
\(11\)	4.07367	−	7.05581i	0.111660	−	0.193401i	−0.804780	−	0.593574i	\(-0.797717\pi\)
							0.916440	+	0.400173i	\(0.131050\pi\)
\(13\)	26.9604	+	9.81279i	0.575191	+	0.209352i	0.613203	−	0.789925i	\(-0.289881\pi\)
							−0.0380129	+	0.999277i	\(0.512103\pi\)
\(17\)	32.2805	+	27.0866i	0.460540	+	0.386439i	0.843330	−	0.537397i	\(-0.180592\pi\)
							−0.382790	+	0.923835i	\(0.625037\pi\)
\(19\)	−57.1414	+	59.9488i	−0.689955	+	0.723853i
							\(23\)	179.881	−	31.7178i	1.63077	−	0.287549i	0.718006	−	0.696037i	\(-0.245055\pi\)
0.912764	+	0.408488i	\(0.133944\pi\)
\(29\)	15.3848	−	12.9094i	0.0985133	−	0.0826624i	−0.592201	−	0.805790i	\(-0.701741\pi\)
							0.690714	+	0.723128i	\(0.257296\pi\)
\(31\)	8.46371	+	14.6596i	0.0490364	+	0.0849335i	0.889502	−	0.456932i	\(-0.151052\pi\)
							−0.840465	+	0.541865i	\(0.817718\pi\)
\(37\)	235.284			1.04542			0.522710	−	0.852511i	\(-0.324921\pi\)
							0.522710	+	0.852511i	\(0.324921\pi\)
\(41\)	−110.560	−	303.760i	−0.421134	−	1.15706i	−0.951058	−	0.309012i	\(-0.900002\pi\)
							0.529924	−	0.848045i	\(-0.322221\pi\)
\(43\)	12.2311	−	69.3658i	0.0433772	−	0.246004i	−0.955407	−	0.295291i	\(-0.904584\pi\)
							0.998785	+	0.0492862i	\(0.0156946\pi\)
\(47\)	338.396	+	403.285i	1.05022	+	1.25160i	0.966922	+	0.255073i	\(0.0820994\pi\)
							0.0832934	+	0.996525i	\(0.473456\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	152.4.v.b.3.7	✓	336
8.3	odd	2	inner	152.4.v.b.3.39	yes	336
19.13	odd	18	inner	152.4.v.b.51.39	yes	336
152.51	even	18	inner	152.4.v.b.51.7	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.4.v.b.3.7	✓	336	1.1	even	1	trivial
152.4.v.b.3.39	yes	336	8.3	odd	2	inner
152.4.v.b.51.7	yes	336	152.51	even	18	inner
152.4.v.b.51.39	yes	336	19.13	odd	18	inner