Embedded newform 69.4.e.b.13.2

• Label: 69.4.e.b.13.2
• Level: $69$
• Weight: $4$
• Character: 69.13
• 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			13.2
Character	\(\chi\)	\(=\)	69.13
Dual form			69.4.e.b.16.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.23546 - 2.57986i) q^{2} +(-2.52376 - 1.62192i) q^{3} +(-0.519874 + 3.61580i) q^{4} +(8.61254 - 18.8588i) q^{5} +(1.45744 + 10.1367i) q^{6} +(-2.20230 - 0.646654i) q^{7} +(-12.4835 + 8.02266i) q^{8} +(3.73874 + 8.18669i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q + 4 q^{2} + 18 q^{3} - 28 q^{4} - 6 q^{5} + 21 q^{6} - 4 q^{7} - 52 q^{8} - 54 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.23546	−	2.57986i	−0.790356	−	0.912119i	0.207456	−	0.978244i	\(-0.433482\pi\)
							−0.997811	+	0.0661254i	\(0.978936\pi\)
\(3\)	−2.52376	−	1.62192i	−0.485698	−	0.312139i
							\(5\)	8.61254	−	18.8588i	0.770329	−	1.68679i	0.0444025	−	0.999014i	\(-0.485862\pi\)
0.725927	−	0.687772i	\(-0.241411\pi\)
\(7\)	−2.20230	−	0.646654i	−0.118913	−	0.0349160i	0.221735	−	0.975107i	\(-0.428828\pi\)
							−0.340648	+	0.940191i	\(0.610646\pi\)
\(11\)	11.3756	−	13.1281i	0.311806	−	0.359844i	−0.578117	−	0.815954i	\(-0.696212\pi\)
							0.889923	+	0.456110i	\(0.150758\pi\)
\(13\)	−66.5253	+	19.5336i	−1.41929	+	0.416741i	−0.899263	−	0.437408i	\(-0.855897\pi\)
							−0.520028	+	0.854149i	\(0.674078\pi\)
\(17\)	7.14880	+	49.7210i	0.101991	+	0.709360i	0.975089	+	0.221815i	\(0.0711982\pi\)
							−0.873098	+	0.487545i	\(0.837893\pi\)
\(19\)	0.526607	−	3.66263i	0.00635853	−	0.0442245i	−0.986395	−	0.164392i	\(-0.947434\pi\)
							0.992754	+	0.120167i	\(0.0383430\pi\)
\(23\)	91.3733	−	61.7893i	0.828376	−	0.560172i
							\(29\)	−33.9413	−	236.067i	−0.217336	−	1.51161i	−0.747814	−	0.663908i	\(-0.768897\pi\)
0.530478	−	0.847699i	\(-0.322012\pi\)
\(31\)	10.5041	−	6.75056i	0.0608576	−	0.0391108i	−0.509858	−	0.860258i	\(-0.670302\pi\)
							0.570716	+	0.821148i	\(0.306666\pi\)
\(37\)	42.6763	+	93.4481i	0.189620	+	0.415210i	0.980434	−	0.196846i	\(-0.0630699\pi\)
							−0.790814	+	0.612056i	\(0.790343\pi\)
\(41\)	43.0459	−	94.2573i	0.163967	−	0.359037i	−0.809758	−	0.586763i	\(-0.800402\pi\)
							0.973725	+	0.227726i	\(0.0731291\pi\)
\(43\)	310.117	+	199.300i	1.09982	+	0.706814i	0.959053	−	0.283228i	\(-0.0914054\pi\)
							0.140771	+	0.990042i	\(0.455042\pi\)
\(47\)	−135.575			−0.420759			−0.210380	−	0.977620i	\(-0.567470\pi\)
							−0.210380	+	0.977620i	\(0.567470\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.4.e.b.13.2	✓	60
3.2	odd	2		207.4.i.b.82.5		60
23.4	even	11		1587.4.a.w.1.24		30
23.16	even	11	inner	69.4.e.b.16.2	yes	60
23.19	odd	22		1587.4.a.v.1.24		30
69.62	odd	22		207.4.i.b.154.5		60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.4.e.b.13.2	✓	60	1.1	even	1	trivial
69.4.e.b.16.2	yes	60	23.16	even	11	inner
207.4.i.b.82.5		60	3.2	odd	2
207.4.i.b.154.5		60	69.62	odd	22
1587.4.a.v.1.24		30	23.19	odd	22
1587.4.a.w.1.24		30	23.4	even	11