Embedded newform 154.4.e.b.67.5

• Label: 154.4.e.b.67.5
• Level: $154$
• Weight: $4$
• Character: 154.67
• Analytic conductor: $9.086$
• Analytic rank: $0$
• Dimension: $10$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 154 = 2 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	154.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.08629414088\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Relative dimension:	\(5\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial:	x^10 - 2*x^9 + 4*x^8 - 99*x^7 + 51*x^6 + 738*x^5 + 1071*x^4 - 43659*x^3 + 37044*x^2 - 388962*x + 4084101
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{4}\cdot 3^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			67.5
Root			\(4.57567 - 0.251569i\) of defining polynomial
Character	\(\chi\)	\(=\)	154.67
Dual form			154.4.e.b.23.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.73205i) q^{2} +(5.07567 - 8.79131i) q^{3} +(-2.00000 + 3.46410i) q^{4} +(-6.95600 - 12.0482i) q^{5} -20.3027 q^{6} +(13.9227 + 12.2131i) q^{7} +8.00000 q^{8} +(-38.0248 - 65.8608i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{2} + 7 q^{3} - 20 q^{4} + 10 q^{5} - 28 q^{6} + 62 q^{7} + 80 q^{8} - 80 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/154\mathbb{Z}\right)^\times\).

\(n\)	\(45\)	\(57\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00000	−	1.73205i	−0.353553	−	0.612372i
							\(3\)	5.07567	−	8.79131i	0.976812	−	1.69189i	0.302992	−	0.952993i	\(-0.402014\pi\)
0.673820	−	0.738895i	\(-0.264652\pi\)
\(5\)	−6.95600	−	12.0482i	−0.622164	−	1.07762i	−0.989082	−	0.147366i	\(-0.952920\pi\)
							0.366918	−	0.930253i	\(-0.380413\pi\)
\(7\)	13.9227	+	12.2131i	0.751754	+	0.659444i
							\(11\)	5.50000	−	9.52628i	0.150756	−	0.261116i
\(13\)	−29.0380			−0.619514										−0.309757	−	0.950816i	\(-0.600248\pi\)
							−0.309757	+	0.950816i	\(0.600248\pi\)
\(17\)	22.6566	−	39.2424i	0.323237	−	0.559863i	−0.657917	−	0.753091i	\(-0.728562\pi\)
							0.981154	+	0.193228i	\(0.0618955\pi\)
\(19\)	34.4109	+	59.6015i	0.415495	+	0.719659i	0.995480	−	0.0949682i	\(-0.0302749\pi\)
							−0.579985	+	0.814627i	\(0.696942\pi\)
\(23\)	60.6419	+	105.035i	0.549770	+	0.952230i	0.998290	+	0.0584569i	\(0.0186180\pi\)
							−0.448520	+	0.893773i	\(0.648049\pi\)
\(29\)	249.820			1.59967			0.799836	−	0.600219i	\(-0.204920\pi\)
							0.799836	+	0.600219i	\(0.204920\pi\)
\(31\)	−63.8963	+	110.672i	−0.370197	+	0.641201i	−0.989596	−	0.143876i	\(-0.954043\pi\)
							0.619398	+	0.785077i	\(0.287377\pi\)
\(37\)	−152.670	−	264.431i	−0.678344	−	1.17493i	−0.975479	−	0.220091i	\(-0.929365\pi\)
							0.297136	−	0.954835i	\(-0.403969\pi\)
\(41\)	−330.681			−1.25960			−0.629801	−	0.776756i	\(-0.716864\pi\)
							−0.629801	+	0.776756i	\(0.716864\pi\)
\(43\)	74.3080			0.263532			0.131766	−	0.991281i	\(-0.457935\pi\)
							0.131766	+	0.991281i	\(0.457935\pi\)
\(47\)	5.99262	+	10.3795i	0.0185982	+	0.0322130i	0.875175	−	0.483807i	\(-0.160746\pi\)
							−0.856577	+	0.516020i	\(0.827413\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	154.4.e.b.67.5	yes	10
7.2	even	3	inner	154.4.e.b.23.5	✓	10
7.3	odd	6		1078.4.a.z.1.5		5
7.4	even	3		1078.4.a.y.1.1		5

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.4.e.b.23.5	✓	10	7.2	even	3	inner
154.4.e.b.67.5	yes	10	1.1	even	1	trivial
1078.4.a.y.1.1		5	7.4	even	3
1078.4.a.z.1.5		5	7.3	odd	6