Embedded newform 154.4.e.d.67.2

• Label: 154.4.e.d.67.2
• 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 - x^9 + 106*x^8 - 287*x^7 + 9065*x^6 - 19649*x^5 + 261415*x^4 - 287434*x^3 + 5311012*x^2 - 7289436*x + 11431161
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			67.2
Root			\(3.18908 + 5.52365i\) of defining polynomial
Character	\(\chi\)	\(=\)	154.67
Dual form			154.4.e.d.23.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.73205i) q^{2} +(-3.18908 + 5.52365i) q^{3} +(-2.00000 + 3.46410i) q^{4} +(8.62120 + 14.9324i) q^{5} -12.7563 q^{6} +(15.8089 + 9.64778i) q^{7} -8.00000 q^{8} +(-6.84047 - 11.8480i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 10 q^{2} - q^{3} - 20 q^{4} + 10 q^{5} - 4 q^{6} + 48 q^{7} - 80 q^{8} - 76 q^{9}+O(q^{10}) \)

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
							\(3\)	−3.18908	+	5.52365i	−0.613739	+	1.06303i	0.376865	+	0.926268i	\(0.377002\pi\)
−0.990604	+	0.136759i	\(0.956331\pi\)
\(5\)	8.62120	+	14.9324i	0.771104	+	1.33559i	0.936958	+	0.349441i	\(0.113628\pi\)
							−0.165855	+	0.986150i	\(0.553038\pi\)
\(7\)	15.8089	+	9.64778i	0.853599	+	0.520931i
							\(11\)	−5.50000	+	9.52628i	−0.150756	+	0.261116i
\(13\)	45.7422			0.975893										0.487947	−	0.872874i	\(-0.337746\pi\)
							0.487947	+	0.872874i	\(0.337746\pi\)
\(17\)	33.6310	−	58.2505i	0.479806	−	0.831049i	−0.519926	−	0.854212i	\(-0.674040\pi\)
							0.999732	+	0.0231630i	\(0.00737367\pi\)
\(19\)	−49.1355	−	85.1052i	−0.593287	−	1.02760i	−0.993786	−	0.111306i	\(-0.964497\pi\)
							0.400499	−	0.916297i	\(-0.368837\pi\)
\(23\)	−4.55545	−	7.89026i	−0.0412990	−	0.0715319i	0.844637	−	0.535339i	\(-0.179816\pi\)
							−0.885936	+	0.463807i	\(0.846483\pi\)
\(29\)	−8.89814			−0.0569773			−0.0284887	−	0.999594i	\(-0.509069\pi\)
							−0.0284887	+	0.999594i	\(0.509069\pi\)
\(31\)	109.655	−	189.928i	0.635310	−	1.10039i	−0.351139	−	0.936323i	\(-0.614206\pi\)
							0.986449	−	0.164066i	\(-0.0524611\pi\)
\(37\)	43.2710	+	74.9476i	0.192262	+	0.333008i	0.946000	−	0.324168i	\(-0.105084\pi\)
							−0.753737	+	0.657176i	\(0.771751\pi\)
\(41\)	21.1006			0.0803748			0.0401874	−	0.999192i	\(-0.487205\pi\)
							0.0401874	+	0.999192i	\(0.487205\pi\)
\(43\)	499.901			1.77289			0.886445	−	0.462835i	\(-0.153168\pi\)
							0.886445	+	0.462835i	\(0.153168\pi\)
\(47\)	−85.6446	−	148.341i	−0.265799	−	0.460377i	0.701974	−	0.712203i	\(-0.252302\pi\)
							−0.967773	+	0.251826i	\(0.918969\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	154.4.e.d.67.2	yes	10
7.2	even	3	inner	154.4.e.d.23.2	✓	10
7.3	odd	6		1078.4.a.u.1.2		5
7.4	even	3		1078.4.a.v.1.4		5

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.4.e.d.23.2	✓	10	7.2	even	3	inner
154.4.e.d.67.2	yes	10	1.1	even	1	trivial
1078.4.a.u.1.2		5	7.3	odd	6
1078.4.a.v.1.4		5	7.4	even	3