Embedded newform 102.3.e.b.89.7

• Label: 102.3.e.b.89.7
• Level: $102$
• Weight: $3$
• Character: 102.89
• Analytic conductor: $2.779$
• Analytic rank: $0$
• Dimension: $20$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 102 = 2 \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	102.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.77929869648\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(10\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial:	x^20 - 10*x^18 + 149*x^16 - 800*x^14 - 1986*x^12 + 2844*x^10 - 160866*x^8 - 5248800*x^6 + 79184709*x^4 - 430467210*x^2 + 3486784401
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{8}\cdot 3^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			89.7
Root			\(-2.40171 + 1.79772i\) of defining polynomial
Character	\(\chi\)	\(=\)	102.89
Dual form			102.3.e.b.47.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +(-0.427083 + 2.96944i) q^{3} +2.00000 q^{4} +(1.26169 + 1.26169i) q^{5} +(-0.603986 + 4.19943i) q^{6} +(2.01679 + 2.01679i) q^{7} +2.82843 q^{8} +(-8.63520 - 2.53640i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{3} + 40 q^{4} + 4 q^{6} + 20 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/102\mathbb{Z}\right)^\times\).

\(n\)	\(35\)	\(37\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41421			0.707107
							\(3\)	−0.427083	+	2.96944i	−0.142361	+	0.989815i
\(5\)	1.26169	+	1.26169i	0.252338	+	0.252338i								0.821929	−	0.569591i	\(-0.192898\pi\)
							−0.569591	+	0.821929i	\(0.692898\pi\)
\(7\)	2.01679	+	2.01679i	0.288112	+	0.288112i	0.836333	−	0.548221i	\(-0.184695\pi\)
							−0.548221	+	0.836333i	\(0.684695\pi\)
\(11\)	−8.23314	+	8.23314i	−0.748467	+	0.748467i	−0.974191	−	0.225724i	\(-0.927525\pi\)
							0.225724	+	0.974191i	\(0.427525\pi\)
\(13\)	22.6733			1.74410			0.872050	−	0.489416i	\(-0.162790\pi\)
							0.872050	+	0.489416i	\(0.162790\pi\)
\(17\)	−8.81003	−	14.5390i	−0.518237	−	0.855237i
							\(19\)		−	19.5669i		−	1.02984i	−0.857239	−	0.514920i	\(-0.827822\pi\)
0.857239	−	0.514920i	\(-0.172178\pi\)
\(23\)	6.84267	−	6.84267i	0.297507	−	0.297507i	−0.542529	−	0.840037i	\(-0.682533\pi\)
							0.840037	+	0.542529i	\(0.182533\pi\)
\(29\)	−8.85961	−	8.85961i	−0.305504	−	0.305504i	0.537659	−	0.843162i	\(-0.319309\pi\)
							−0.843162	+	0.537659i	\(0.819309\pi\)
\(31\)	−16.6659	+	16.6659i	−0.537608	+	0.537608i	−0.922826	−	0.385218i	\(-0.874126\pi\)
							0.385218	+	0.922826i	\(0.374126\pi\)
\(37\)	23.1242	−	23.1242i	0.624978	−	0.624978i	−0.321822	−	0.946800i	\(-0.604295\pi\)
							0.946800	+	0.321822i	\(0.104295\pi\)
\(41\)	−17.3200	+	17.3200i	−0.422438	+	0.422438i	−0.886042	−	0.463604i	\(-0.846556\pi\)
							0.463604	+	0.886042i	\(0.346556\pi\)
\(43\)			53.6267i			1.24713i	0.781770	+	0.623567i	\(0.214317\pi\)
							−0.781770	+	0.623567i	\(0.785683\pi\)
\(47\)			12.1592i			0.258706i	0.991599	+	0.129353i	\(0.0412900\pi\)
							−0.991599	+	0.129353i	\(0.958710\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	102.3.e.b.89.7	yes	20
3.2	odd	2	inner	102.3.e.b.89.1	yes	20
17.13	even	4	inner	102.3.e.b.47.1	✓	20
51.47	odd	4	inner	102.3.e.b.47.7	yes	20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
102.3.e.b.47.1	✓	20	17.13	even	4	inner
102.3.e.b.47.7	yes	20	51.47	odd	4	inner
102.3.e.b.89.1	yes	20	3.2	odd	2	inner
102.3.e.b.89.7	yes	20	1.1	even	1	trivial