Embedded newform 69.6.g.a.5.2

• Label: 69.6.g.a.5.2
• Level: $69$
• Weight: $6$
• Character: 69.5
• Analytic conductor: $11.066$
• Analytic rank: $0$
• Dimension: $380$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	69.g (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.0664835671\)
Analytic rank:	\(0\)
Dimension:	\(380\)
Relative dimension:	\(38\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.2
Character	\(\chi\)	\(=\)	69.5
Dual form			69.6.g.a.14.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.97006 + 10.1151i) q^{2} +(-14.6309 - 5.37922i) q^{3} +(-66.5741 - 42.7846i) q^{4} +(8.40244 - 58.4402i) q^{5} +(97.8662 - 132.017i) q^{6} +(-141.246 + 64.5047i) q^{7} +(375.549 - 325.415i) q^{8} +(185.128 + 157.406i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 380 q - 11 q^{3} + 578 q^{4} + 64 q^{6} - 22 q^{7} + 433 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{1}{22}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.97006	+	10.1151i	−0.525038	+	1.78812i	0.0857440	+	0.996317i	\(0.472673\pi\)
							−0.610782	+	0.791799i	\(0.709145\pi\)
\(3\)	−14.6309	−	5.37922i	−0.938574	−	0.345077i
							\(5\)	8.40244	−	58.4402i	0.150307	−	1.04541i	−0.765397	−	0.643558i	\(-0.777457\pi\)
0.915705	−	0.401852i	\(-0.131633\pi\)
\(7\)	−141.246	+	64.5047i	−1.08951	+	0.497561i	−0.877436	−	0.479693i	\(-0.840748\pi\)
							−0.212071	+	0.977254i	\(0.568021\pi\)
\(11\)	−342.218	+	100.484i	−0.852749	+	0.250390i	−0.678762	−	0.734358i	\(-0.737483\pi\)
							−0.173987	+	0.984748i	\(0.555665\pi\)
\(13\)	−149.661	+	327.713i	−0.245613	+	0.537818i	−0.991782	−	0.127939i	\(-0.959164\pi\)
							0.746169	+	0.665757i	\(0.231891\pi\)
\(17\)	1826.55	−	1173.85i	1.53288	−	0.985123i	0.543560	−	0.839371i	\(-0.317076\pi\)
							0.989321	−	0.145752i	\(-0.0465601\pi\)
\(19\)	280.771	−	436.888i	0.178430	−	0.277643i	−0.740505	−	0.672050i	\(-0.765414\pi\)
							0.918936	+	0.394408i	\(0.129050\pi\)
\(23\)	−2362.21	+	925.379i	−0.931104	+	0.364754i
							\(29\)	3609.41	+	5616.35i	0.796968	+	1.24011i	0.967027	+	0.254673i	\(0.0819677\pi\)
−0.170059	+	0.985434i	\(0.554396\pi\)
\(31\)	679.891	+	784.636i	0.127068	+	0.146644i	0.815718	−	0.578450i	\(-0.196342\pi\)
							−0.688651	+	0.725093i	\(0.741797\pi\)
\(37\)	−1655.18	+	237.979i	−0.198765	+	0.0285781i	−0.240978	−	0.970530i	\(-0.577468\pi\)
							0.0422130	+	0.999109i	\(0.486559\pi\)
\(41\)	11867.9	+	1706.35i	1.10259	+	0.158529i	0.669495	−	0.742817i	\(-0.266511\pi\)
							0.433100	+	0.901346i	\(0.357420\pi\)
\(43\)	13434.4	+	11640.9i	1.10802	+	0.960101i	0.999422	−	0.0339857i	\(-0.0108201\pi\)
							0.108593	+	0.994086i	\(0.465366\pi\)
\(47\)		−	28170.4i		−	1.86015i	−0.367370	−	0.930075i	\(-0.619742\pi\)
							0.367370	−	0.930075i	\(-0.380258\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.6.g.a.5.2	✓	380
3.2	odd	2	inner	69.6.g.a.5.37	yes	380
23.14	odd	22	inner	69.6.g.a.14.37	yes	380
69.14	even	22	inner	69.6.g.a.14.2	yes	380

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.g.a.5.2	✓	380	1.1	even	1	trivial
69.6.g.a.5.37	yes	380	3.2	odd	2	inner
69.6.g.a.14.2	yes	380	69.14	even	22	inner
69.6.g.a.14.37	yes	380	23.14	odd	22	inner