Embedded newform 69.6.e.b.13.5

• Label: 69.6.e.b.13.5
• Level: $69$
• Weight: $6$
• Character: 69.13
• Analytic conductor: $11.066$
• Analytic rank: $0$
• Dimension: $100$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	69.e (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.0664835671\)
Analytic rank:	\(0\)
Dimension:	\(100\)
Relative dimension:	\(10\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			13.5
Character	\(\chi\)	\(=\)	69.13
Dual form			69.6.e.b.16.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.220291 - 0.254230i) q^{2} +(7.57128 + 4.86577i) q^{3} +(4.53797 - 31.5623i) q^{4} +(21.3588 - 46.7692i) q^{5} +(-0.430865 - 2.99673i) q^{6} +(-219.257 - 64.3798i) q^{7} +(-18.0795 + 11.6190i) q^{8} +(33.6486 + 73.6802i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 100 q + 12 q^{2} - 90 q^{3} - 148 q^{4} - 50 q^{5} + 9 q^{6} - 78 q^{7} - 1844 q^{8} - 810 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{7}{11}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.220291	−	0.254230i	−0.0389424	−	0.0449419i	0.735944	−	0.677042i	\(-0.236739\pi\)
							−0.774887	+	0.632100i	\(0.782193\pi\)
\(3\)	7.57128	+	4.86577i	0.485698	+	0.312139i
							\(5\)	21.3588	−	46.7692i	0.382078	−	0.836633i	−0.616700	−	0.787199i	\(-0.711531\pi\)
0.998777	−	0.0494349i	\(-0.0157420\pi\)
\(7\)	−219.257	−	64.3798i	−1.69125	−	0.496597i	−0.712507	−	0.701665i	\(-0.752440\pi\)
							−0.978748	+	0.205068i	\(0.934259\pi\)
\(11\)	−314.566	+	363.028i	−0.783844	+	0.904604i	−0.997381	−	0.0723311i	\(-0.976956\pi\)
							0.213537	+	0.976935i	\(0.431502\pi\)
\(13\)	−42.3295	+	12.4291i	−0.0694680	+	0.0203976i	−0.316282	−	0.948665i	\(-0.602435\pi\)
							0.246814	+	0.969063i	\(0.420616\pi\)
\(17\)	−95.5236	−	664.381i	−0.0801657	−	0.557564i	−0.989834	−	0.142229i	\(-0.954573\pi\)
							0.909668	−	0.415336i	\(-0.136336\pi\)
\(19\)	−6.76833	+	47.0747i	−0.00430128	+	0.0299160i	−0.991859	−	0.127341i	\(-0.959356\pi\)
							0.987558	+	0.157257i	\(0.0502650\pi\)
\(23\)	−65.8003	−	2536.14i	−0.0259363	−	0.999664i
							\(29\)	−660.892	−	4596.61i	−0.145927	−	1.01494i	−0.922798	−	0.385285i	\(-0.874103\pi\)
0.776871	−	0.629660i	\(-0.216806\pi\)
\(31\)	2965.82	−	1906.01i	0.554294	−	0.356223i	−0.233313	−	0.972402i	\(-0.574957\pi\)
							0.787606	+	0.616179i	\(0.211320\pi\)
\(37\)	−5157.02	−	11292.3i	−0.619290	−	1.35606i	−0.916034	−	0.401100i	\(-0.868628\pi\)
							0.296744	−	0.954957i	\(-0.404099\pi\)
\(41\)	5914.40	−	12950.7i	0.549479	−	1.20319i	−0.407546	−	0.913185i	\(-0.633615\pi\)
							0.957025	−	0.290006i	\(-0.0936574\pi\)
\(43\)	18465.2	+	11866.9i	1.52294	+	0.978736i	0.991279	+	0.131781i	\(0.0420697\pi\)
							0.531665	+	0.846955i	\(0.321567\pi\)
\(47\)	7823.34			0.516592			0.258296	−	0.966066i	\(-0.416839\pi\)
							0.258296	+	0.966066i	\(0.416839\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.6.e.b.13.5	✓	100
23.16	even	11	inner	69.6.e.b.16.5	yes	100

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.e.b.13.5	✓	100	1.1	even	1	trivial
69.6.e.b.16.5	yes	100	23.16	even	11	inner