Embedded newform 102.3.e.b.89.4

• Label: 102.3.e.b.89.4
• 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			89.4
Root			\(-2.98496 + 0.299984i\) of defining polynomial
Character	\(\chi\)	\(=\)	102.89
Dual form			102.3.e.b.47.4

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +(1.89857 - 2.32281i) q^{3} +2.00000 q^{4} +(3.05768 + 3.05768i) q^{5} +(-2.68498 + 3.28495i) q^{6} +(3.50287 + 3.50287i) q^{7} -2.82843 q^{8} +(-1.79088 - 8.82002i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{3} + 40 q^{4} + 4 q^{6} + 20 q^{7}+O(q^{10}) \)

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
							\(3\)	1.89857	−	2.32281i	0.632856	−	0.774270i
\(5\)	3.05768	+	3.05768i	0.611536	+	0.611536i								0.943346	−	0.331810i	\(-0.107659\pi\)
							−0.331810	+	0.943346i	\(0.607659\pi\)
\(7\)	3.50287	+	3.50287i	0.500410	+	0.500410i	0.911565	−	0.411156i	\(-0.134875\pi\)
							−0.411156	+	0.911565i	\(0.634875\pi\)
\(11\)	6.44580	−	6.44580i	0.585981	−	0.585981i	−0.350559	−	0.936541i	\(-0.614008\pi\)
							0.936541	+	0.350559i	\(0.114008\pi\)
\(13\)	2.26456			0.174197			0.0870985	−	0.996200i	\(-0.472241\pi\)
							0.0870985	+	0.996200i	\(0.472241\pi\)
\(17\)	6.83994	−	15.5633i	0.402349	−	0.915486i
							\(19\)			16.3812i			0.862169i	0.902312	+	0.431085i	\(0.141869\pi\)
−0.902312	+	0.431085i	\(0.858131\pi\)
\(23\)	−7.15696	+	7.15696i	−0.311172	+	0.311172i	−0.845364	−	0.534191i	\(-0.820616\pi\)
							0.534191	+	0.845364i	\(0.320616\pi\)
\(29\)	−6.56388	−	6.56388i	−0.226341	−	0.226341i	0.584821	−	0.811162i	\(-0.301165\pi\)
							−0.811162	+	0.584821i	\(0.801165\pi\)
\(31\)	−22.2472	+	22.2472i	−0.717651	+	0.717651i	−0.968124	−	0.250473i	\(-0.919414\pi\)
							0.250473	+	0.968124i	\(0.419414\pi\)
\(37\)	−49.6392	+	49.6392i	−1.34160	+	1.34160i	−0.447134	+	0.894467i	\(0.647555\pi\)
							−0.894467	+	0.447134i	\(0.852445\pi\)
\(41\)	−16.9856	+	16.9856i	−0.414283	+	0.414283i	−0.883227	−	0.468945i	\(-0.844634\pi\)
							0.468945	+	0.883227i	\(0.344634\pi\)
\(43\)		−	18.0836i		−	0.420548i	−0.977643	−	0.210274i	\(-0.932564\pi\)
							0.977643	−	0.210274i	\(-0.0674356\pi\)
\(47\)			79.1555i			1.68416i	0.539353	+	0.842080i	\(0.318669\pi\)
							−0.539353	+	0.842080i	\(0.681331\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	102.3.e.b.89.4	yes	20
3.2	odd	2	inner	102.3.e.b.89.9	yes	20
17.13	even	4	inner	102.3.e.b.47.9	yes	20
51.47	odd	4	inner	102.3.e.b.47.4	✓	20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
102.3.e.b.47.4	✓	20	51.47	odd	4	inner
102.3.e.b.47.9	yes	20	17.13	even	4	inner
102.3.e.b.89.4	yes	20	1.1	even	1	trivial
102.3.e.b.89.9	yes	20	3.2	odd	2	inner