Embedded newform 667.2.s.a.16.9

• Label: 667.2.s.a.16.9
• Level: $667$
• Weight: $2$
• Character: 667.16
• Analytic conductor: $5.326$
• Analytic rank: $0$
• Dimension: $3480$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 667 = 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	667.s (of order \(77\), degree \(60\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.32602181482\)
Analytic rank:	\(0\)
Dimension:	\(3480\)
Relative dimension:	\(58\) over \(\Q(\zeta_{77})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{77}]$

Embedding invariants

Embedding label			16.9
Character	\(\chi\)	\(=\)	667.16
Dual form			667.2.s.a.542.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.98648 + 0.858663i) q^{2} +(0.240879 - 2.35336i) q^{3} +(1.83850 - 1.95461i) q^{4} +(0.0694953 - 3.40617i) q^{5} +(1.54225 + 4.88173i) q^{6} +(-0.823769 + 2.01780i) q^{7} +(-0.502672 + 1.39088i) q^{8} +(-2.54251 - 0.525987i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3480 q - 49 q^{2} - 45 q^{3} + 7 q^{4} - 49 q^{5} - 39 q^{6} - 49 q^{7} - 50 q^{8} + q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/667\mathbb{Z}\right)^\times\).

\(n\)	\(465\)	\(553\)
\(\chi(n)\)	\(e\left(\frac{4}{11}\right)\)	\(e\left(\frac{1}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.98648	+	0.858663i	−1.40465	+	0.607166i	−0.957798	−	0.287442i	\(-0.907195\pi\)
							−0.446852	+	0.894608i	\(0.647455\pi\)
\(3\)	0.240879	−	2.35336i	0.139071	−	1.35872i	−0.657173	−	0.753740i	\(-0.728248\pi\)
							0.796244	−	0.604976i	\(-0.206817\pi\)
\(5\)	0.0694953	−	3.40617i	0.0310792	−	1.52328i	−0.632137	−	0.774857i	\(-0.717822\pi\)
							0.663216	−	0.748428i	\(-0.269191\pi\)
\(7\)	−0.823769	+	2.01780i	−0.311355	+	0.762657i	0.687963	+	0.725746i	\(0.258505\pi\)
							−0.999319	+	0.0369111i	\(0.988248\pi\)
\(11\)	−3.36160	+	1.79130i	−1.01356	+	0.540098i	−0.891785	−	0.452459i	\(-0.850547\pi\)
							−0.121774	+	0.992558i	\(0.538858\pi\)
\(13\)	−2.59040	+	4.63104i	−0.718449	+	1.28442i	0.231750	+	0.972775i	\(0.425555\pi\)
							−0.950199	+	0.311644i	\(0.899120\pi\)
\(17\)	0.277976	−	1.93337i	0.0674192	−	0.468911i	−0.927944	−	0.372720i	\(-0.878425\pi\)
							0.995363	−	0.0961904i	\(-0.0306658\pi\)
\(19\)	−2.18168	+	2.31946i	−0.500513	+	0.532121i	−0.928336	−	0.371741i	\(-0.878761\pi\)
							0.427824	+	0.903862i	\(0.359280\pi\)
\(23\)	−2.02733	+	4.34625i	−0.422728	+	0.906257i
							\(29\)	3.55163	+	4.04796i	0.659521	+	0.751686i
\(31\)	−1.37248	−	2.23456i	−0.246504	−	0.401339i								0.704889	−	0.709318i	\(-0.250997\pi\)
							−0.951393	+	0.307979i	\(0.900347\pi\)
\(37\)	1.78883	+	0.370068i	0.294081	+	0.0608387i	0.347331	−	0.937742i	\(-0.387088\pi\)
							−0.0532501	+	0.998581i	\(0.516958\pi\)
\(41\)	1.58174	+	3.46353i	0.247027	+	0.540913i	0.992008	−	0.126172i	\(-0.0402691\pi\)
							−0.744982	+	0.667085i	\(0.767542\pi\)
\(43\)	1.75905	−	2.86395i	0.268253	−	0.436749i	−0.689523	−	0.724264i	\(-0.742180\pi\)
							0.957776	+	0.287515i	\(0.0928291\pi\)
\(47\)	0.645720	−	2.82908i	0.0941879	−	0.412664i	−0.905750	−	0.423813i	\(-0.860692\pi\)
							0.999938	+	0.0111483i	\(0.00354867\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	667.2.s.a.16.9	✓	3480
23.13	even	11	inner	667.2.s.a.335.9	yes	3480
29.20	even	7	inner	667.2.s.a.223.9	yes	3480
667.542	even	77	inner	667.2.s.a.542.9	yes	3480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.s.a.16.9	✓	3480	1.1	even	1	trivial
667.2.s.a.223.9	yes	3480	29.20	even	7	inner
667.2.s.a.335.9	yes	3480	23.13	even	11	inner
667.2.s.a.542.9	yes	3480	667.542	even	77	inner