Embedded newform 102.3.e.b.47.5

• Label: 102.3.e.b.47.5
• Level: $102$
• Weight: $3$
• Character: 102.47
• Analytic conductor: $2.779$
• Analytic rank: $0$
• Dimension: $20$
• 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			47.5
Root			\(-2.30685 + 1.91792i\) of defining polynomial
Character	\(\chi\)	\(=\)	102.47
Dual form			102.3.e.b.89.5

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +(2.98737 + 0.275015i) q^{3} +2.00000 q^{4} +(-7.00867 + 7.00867i) q^{5} +(-4.22478 - 0.388930i) q^{6} +(-5.33829 + 5.33829i) q^{7} -2.82843 q^{8} +(8.84873 + 1.64314i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{3} + 40 q^{4} + 4 q^{6} + 20 q^{7}+O(q^{10}) \)

Character values

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

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

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
							\(3\)	2.98737	+	0.275015i	0.995789	+	0.0916716i
\(5\)	−7.00867	+	7.00867i	−1.40173	+	1.40173i								−0.607134	+	0.794599i	\(0.707681\pi\)
							−0.794599	+	0.607134i	\(0.792319\pi\)
\(7\)	−5.33829	+	5.33829i	−0.762613	+	0.762613i	−0.976794	−	0.214181i	\(-0.931292\pi\)
							0.214181	+	0.976794i	\(0.431292\pi\)
\(11\)	−3.04728	−	3.04728i	−0.277026	−	0.277026i	0.554895	−	0.831920i	\(-0.312758\pi\)
							−0.831920	+	0.554895i	\(0.812758\pi\)
\(13\)	0.439668			0.0338206			0.0169103	−	0.999857i	\(-0.494617\pi\)
							0.0169103	+	0.999857i	\(0.494617\pi\)
\(17\)	14.3861	+	9.05766i	0.846239	+	0.532804i
							\(19\)			13.8300i			0.727893i	0.931420	+	0.363946i	\(0.118571\pi\)
−0.931420	+	0.363946i	\(0.881429\pi\)
\(23\)	14.8394	+	14.8394i	0.645191	+	0.645191i	0.951827	−	0.306636i	\(-0.0992033\pi\)
							−0.306636	+	0.951827i	\(0.599203\pi\)
\(29\)	11.2152	−	11.2152i	0.386730	−	0.386730i	−0.486789	−	0.873519i	\(-0.661832\pi\)
							0.873519	+	0.486789i	\(0.161832\pi\)
\(31\)	−20.9916	−	20.9916i	−0.677150	−	0.677150i	0.282205	−	0.959354i	\(-0.408934\pi\)
							−0.959354	+	0.282205i	\(0.908934\pi\)
\(37\)	44.8591	+	44.8591i	1.21241	+	1.21241i	0.970233	+	0.242175i	\(0.0778608\pi\)
							0.242175	+	0.970233i	\(0.422139\pi\)
\(41\)	14.5266	+	14.5266i	0.354307	+	0.354307i	0.861709	−	0.507402i	\(-0.169394\pi\)
							−0.507402	+	0.861709i	\(0.669394\pi\)
\(43\)		−	35.3283i		−	0.821589i	−0.911728	−	0.410794i	\(-0.865251\pi\)
							0.911728	−	0.410794i	\(-0.134749\pi\)
\(47\)		−	20.6719i		−	0.439827i	−0.975519	−	0.219913i	\(-0.929423\pi\)
							0.975519	−	0.219913i	\(-0.0705775\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	102.3.e.b.47.5	✓	20
3.2	odd	2	inner	102.3.e.b.47.8	yes	20
17.4	even	4	inner	102.3.e.b.89.8	yes	20
51.38	odd	4	inner	102.3.e.b.89.5	yes	20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
102.3.e.b.47.5	✓	20	1.1	even	1	trivial
102.3.e.b.47.8	yes	20	3.2	odd	2	inner
102.3.e.b.89.5	yes	20	51.38	odd	4	inner
102.3.e.b.89.8	yes	20	17.4	even	4	inner