Embedded newform 102.3.e.b.89.2

• Label: 102.3.e.b.89.2
• 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			89.2
Root			\(-0.660460 + 2.92640i\) of defining polynomial
Character	\(\chi\)	\(=\)	102.89
Dual form			102.3.e.b.47.2

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +(-1.60226 - 2.53629i) q^{3} +2.00000 q^{4} +(-0.525641 - 0.525641i) q^{5} +(2.26594 + 3.58686i) q^{6} +(-3.53019 - 3.53019i) q^{7} -2.82843 q^{8} +(-3.86553 + 8.12759i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{3} + 40 q^{4} + 4 q^{6} + 20 q^{7}+O(q^{10}) \)

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
							\(3\)	−1.60226	−	2.53629i	−0.534086	−	0.845430i
\(5\)	−0.525641	−	0.525641i	−0.105128	−	0.105128i								0.652586	−	0.757714i	\(-0.273684\pi\)
							−0.757714	+	0.652586i	\(0.773684\pi\)
\(7\)	−3.53019	−	3.53019i	−0.504313	−	0.504313i	0.408462	−	0.912775i	\(-0.366065\pi\)
							−0.912775	+	0.408462i	\(0.866065\pi\)
\(11\)	−2.79550	+	2.79550i	−0.254136	+	0.254136i	−0.822664	−	0.568528i	\(-0.807513\pi\)
							0.568528	+	0.822664i	\(0.307513\pi\)
\(13\)	−22.7371			−1.74901			−0.874505	−	0.485016i	\(-0.838814\pi\)
							−0.874505	+	0.485016i	\(0.838814\pi\)
\(17\)	−16.3239	+	4.74669i	−0.960228	+	0.279217i
							\(19\)		−	6.57842i		−	0.346233i	−0.984901	−	0.173116i	\(-0.944616\pi\)
0.984901	−	0.173116i	\(-0.0553837\pi\)
\(23\)	12.0306	−	12.0306i	0.523069	−	0.523069i	−0.395428	−	0.918497i	\(-0.629404\pi\)
							0.918497	+	0.395428i	\(0.129404\pi\)
\(29\)	14.6240	+	14.6240i	0.504277	+	0.504277i	0.912764	−	0.408487i	\(-0.133943\pi\)
							−0.408487	+	0.912764i	\(0.633943\pi\)
\(31\)	3.61802	−	3.61802i	0.116710	−	0.116710i	−0.646340	−	0.763050i	\(-0.723701\pi\)
							0.763050	+	0.646340i	\(0.223701\pi\)
\(37\)	−19.3388	+	19.3388i	−0.522669	+	0.522669i	−0.918377	−	0.395707i	\(-0.870499\pi\)
							0.395707	+	0.918377i	\(0.370499\pi\)
\(41\)	23.2169	−	23.2169i	0.566265	−	0.566265i	−0.364815	−	0.931080i	\(-0.618868\pi\)
							0.931080	+	0.364815i	\(0.118868\pi\)
\(43\)		−	18.9890i		−	0.441604i	−0.975319	−	0.220802i	\(-0.929132\pi\)
							0.975319	−	0.220802i	\(-0.0708675\pi\)
\(47\)		−	70.0604i		−	1.49065i	−0.666703	−	0.745323i	\(-0.732295\pi\)
							0.666703	−	0.745323i	\(-0.267705\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	102.3.e.b.89.2	yes	20
3.2	odd	2	inner	102.3.e.b.89.10	yes	20
17.13	even	4	inner	102.3.e.b.47.10	yes	20
51.47	odd	4	inner	102.3.e.b.47.2	✓	20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
102.3.e.b.47.2	✓	20	51.47	odd	4	inner
102.3.e.b.47.10	yes	20	17.13	even	4	inner
102.3.e.b.89.2	yes	20	1.1	even	1	trivial
102.3.e.b.89.10	yes	20	3.2	odd	2	inner