Embedded newform 69.4.e.a.16.2

• Label: 69.4.e.a.16.2
• Level: $69$
• Weight: $4$
• Character: 69.16
• Analytic conductor: $4.071$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	69.e (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.07113179040\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(6\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			16.2
Character	\(\chi\)	\(=\)	69.16
Dual form			69.4.e.a.13.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.24664 + 2.59277i) q^{2} +(2.52376 - 1.62192i) q^{3} +(-0.536504 - 3.73147i) q^{4} +(-5.18705 - 11.3581i) q^{5} +(-1.46473 + 10.1874i) q^{6} +(-25.5228 + 7.49416i) q^{7} +(-12.2087 - 7.84605i) q^{8} +(3.73874 - 8.18669i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 18 q^{3} - 28 q^{4} + 22 q^{5} - 33 q^{6} + 24 q^{7} + 16 q^{8} - 54 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/69\mathbb{Z}\right)^\times\).

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{4}{11}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.24664	+	2.59277i	−0.794309	+	0.916681i	−0.998055	−	0.0623428i	\(-0.980143\pi\)
							0.203746	+	0.979024i	\(0.434688\pi\)
\(3\)	2.52376	−	1.62192i	0.485698	−	0.312139i
							\(5\)	−5.18705	−	11.3581i	−0.463944	−	1.01590i	−0.986571	−	0.163334i	\(-0.947775\pi\)
0.522627	−	0.852562i	\(-0.324952\pi\)
\(7\)	−25.5228	+	7.49416i	−1.37810	+	0.404646i	−0.885107	−	0.465387i	\(-0.845915\pi\)
							−0.492992	+	0.870034i	\(0.664097\pi\)
\(11\)	−10.0086	−	11.5505i	−0.274337	−	0.316602i	0.601816	−	0.798635i	\(-0.294444\pi\)
							−0.876153	+	0.482033i	\(0.839899\pi\)
\(13\)	−27.1213	−	7.96353i	−0.578622	−	0.169899i	−0.0206909	−	0.999786i	\(-0.506587\pi\)
							−0.557931	+	0.829887i	\(0.688405\pi\)
\(17\)	18.9991	−	132.142i	0.271056	−	1.88524i	−0.166491	−	0.986043i	\(-0.553244\pi\)
							0.437547	−	0.899196i	\(-0.355847\pi\)
\(19\)	16.0342	+	111.520i	0.193605	+	1.34655i	0.822368	+	0.568956i	\(0.192653\pi\)
							−0.628762	+	0.777597i	\(0.716438\pi\)
\(23\)	−106.395	+	29.1037i	−0.964564	+	0.263849i
							\(29\)	−36.0806	+	250.946i	−0.231035	+	1.60688i	0.462608	+	0.886563i	\(0.346914\pi\)
−0.693643	+	0.720319i	\(0.743995\pi\)
\(31\)	−108.957	−	70.0224i	−0.631266	−	0.405690i	0.185512	−	0.982642i	\(-0.440606\pi\)
							−0.816778	+	0.576952i	\(0.804242\pi\)
\(37\)	2.65059	−	5.80399i	0.0117772	−	0.0257884i	−0.903652	−	0.428267i	\(-0.859124\pi\)
							0.915429	+	0.402479i	\(0.131851\pi\)
\(41\)	−57.4434	−	125.784i	−0.218809	−	0.479124i	0.768115	−	0.640312i	\(-0.221195\pi\)
							−0.986924	+	0.161188i	\(0.948468\pi\)
\(43\)	84.8845	−	54.5520i	0.301041	−	0.193467i	−0.381396	−	0.924412i	\(-0.624557\pi\)
							0.682437	+	0.730944i	\(0.260920\pi\)
\(47\)	−97.8777			−0.303764			−0.151882	−	0.988399i	\(-0.548533\pi\)
							−0.151882	+	0.988399i	\(0.548533\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.4.e.a.16.2	yes	60
3.2	odd	2		207.4.i.c.154.5		60
23.6	even	11		1587.4.a.t.1.24		30
23.13	even	11	inner	69.4.e.a.13.2	✓	60
23.17	odd	22		1587.4.a.u.1.24		30
69.59	odd	22		207.4.i.c.82.5		60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.4.e.a.13.2	✓	60	23.13	even	11	inner
69.4.e.a.16.2	yes	60	1.1	even	1	trivial
207.4.i.c.82.5		60	69.59	odd	22
207.4.i.c.154.5		60	3.2	odd	2
1587.4.a.t.1.24		30	23.6	even	11
1587.4.a.u.1.24		30	23.17	odd	22