Embedded newform 152.2.o.c.107.3

• Label: 152.2.o.c.107.3
• Level: $152$
• Weight: $2$
• Character: 152.107
• Analytic conductor: $1.214$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.21372611072\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(14\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			107.3
Character	\(\chi\)	\(=\)	152.107
Dual form			152.2.o.c.27.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.34230 - 0.445227i) q^{2} +(0.705625 - 0.407393i) q^{3} +(1.60355 + 1.19526i) q^{4} +(-3.59735 + 2.07693i) q^{5} +(-1.12854 + 0.232681i) q^{6} +3.24695i q^{7} +(-1.62028 - 2.31834i) q^{8} +(-1.16806 + 2.02314i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 3 q^{2} - 6 q^{3} + q^{4} - 3 q^{6} + 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(39\)	\(77\)	\(97\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.34230	−	0.445227i	−0.949150	−	0.314823i
							\(3\)	0.705625	−	0.407393i	0.407393	−	0.235208i	−0.282276	−	0.959333i	\(-0.591089\pi\)
0.689669	+	0.724125i	\(0.257756\pi\)
\(5\)	−3.59735	+	2.07693i	−1.60878	+	0.928831i	−0.619139	+	0.785282i	\(0.712518\pi\)
							−0.989643	+	0.143549i	\(0.954149\pi\)
\(7\)			3.24695i			1.22723i	0.789604	+	0.613617i	\(0.210286\pi\)
							−0.789604	+	0.613617i	\(0.789714\pi\)
\(11\)	−1.21833			−0.367340			−0.183670	−	0.982988i	\(-0.558798\pi\)
							−0.183670	+	0.982988i	\(0.558798\pi\)
\(13\)	2.01425	−	3.48878i	0.558652	−	0.967614i	−0.438957	−	0.898508i	\(-0.644652\pi\)
							0.997609	−	0.0691058i	\(-0.0220146\pi\)
\(17\)	1.50282	+	2.60296i	0.364487	+	0.631310i	0.988694	−	0.149949i	\(-0.0479110\pi\)
							−0.624207	+	0.781259i	\(0.714578\pi\)
\(19\)	−2.16355	+	3.78405i	−0.496352	+	0.868121i
							\(23\)	3.01328	+	1.73972i	0.628313	+	0.362756i	0.780098	−	0.625657i	\(-0.215169\pi\)
−0.151786	+	0.988413i	\(0.548502\pi\)
\(29\)	2.35755	−	4.08339i	0.437785	−	0.758266i	−0.559733	−	0.828673i	\(-0.689096\pi\)
							0.997518	+	0.0704067i	\(0.0224297\pi\)
\(31\)	−5.99801			−1.07728			−0.538638	−	0.842538i	\(-0.681061\pi\)
							−0.538638	+	0.842538i	\(0.681061\pi\)
\(37\)	−1.34860			−0.221708			−0.110854	−	0.993837i	\(-0.535359\pi\)
							−0.110854	+	0.993837i	\(0.535359\pi\)
\(41\)	2.97093	−	1.71527i	0.463981	−	0.267880i	−0.249736	−	0.968314i	\(-0.580344\pi\)
							0.713717	+	0.700434i	\(0.247010\pi\)
\(43\)	−1.01615	−	1.76003i	−0.154962	−	0.268402i	0.778083	−	0.628161i	\(-0.216192\pi\)
							−0.933045	+	0.359759i	\(0.882859\pi\)
\(47\)	8.37318	+	4.83426i	1.22135	+	0.705149i	0.965207	−	0.261489i	\(-0.0842134\pi\)
							0.256147	+	0.966638i	\(0.417547\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	152.2.o.c.107.3	yes	28
4.3	odd	2		608.2.s.c.335.5		28
8.3	odd	2	inner	152.2.o.c.107.7	yes	28
8.5	even	2		608.2.s.c.335.6		28
19.8	odd	6	inner	152.2.o.c.27.7	yes	28
76.27	even	6		608.2.s.c.559.6		28
152.27	even	6	inner	152.2.o.c.27.3	✓	28
152.141	odd	6		608.2.s.c.559.5		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.o.c.27.3	✓	28	152.27	even	6	inner
152.2.o.c.27.7	yes	28	19.8	odd	6	inner
152.2.o.c.107.3	yes	28	1.1	even	1	trivial
152.2.o.c.107.7	yes	28	8.3	odd	2	inner
608.2.s.c.335.5		28	4.3	odd	2
608.2.s.c.335.6		28	8.5	even	2
608.2.s.c.559.5		28	152.141	odd	6
608.2.s.c.559.6		28	76.27	even	6