Embedded newform 69.8.g.a.5.19

• Label: 69.8.g.a.5.19
• Level: $69$
• Weight: $8$
• Character: 69.5
• Analytic conductor: $21.555$
• Analytic rank: $0$
• Dimension: $540$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	69.g (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(21.5545667584\)
Analytic rank:	\(0\)
Dimension:	\(540\)
Relative dimension:	\(54\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.19
Character	\(\chi\)	\(=\)	69.5
Dual form			69.8.g.a.14.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.50658 + 8.53663i) q^{2} +(-3.74386 - 46.6153i) q^{3} +(41.0893 + 26.4065i) q^{4} +(-49.8381 + 346.631i) q^{5} +(407.322 + 84.8851i) q^{6} +(206.284 - 94.2068i) q^{7} +(-1189.08 + 1030.34i) q^{8} +(-2158.97 + 349.042i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 540 q - 35 q^{3} + 3362 q^{4} - 1280 q^{6} - 22 q^{7} - 659 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.50658	+	8.53663i	−0.221553	+	0.754539i	0.771434	+	0.636310i	\(0.219540\pi\)
							−0.992986	+	0.118229i	\(0.962278\pi\)
\(3\)	−3.74386	−	46.6153i	−0.0800562	−	0.996790i
							\(5\)	−49.8381	+	346.631i	−0.178306	+	1.24015i	0.682375	+	0.731002i	\(0.260947\pi\)
−0.860681	+	0.509144i	\(0.829962\pi\)
\(7\)	206.284	−	94.2068i	0.227312	−	0.103810i	−0.298503	−	0.954409i	\(-0.596487\pi\)
							0.525816	+	0.850599i	\(0.323760\pi\)
\(11\)	5475.73	−	1607.82i	1.24042	−	0.364220i	0.405246	−	0.914208i	\(-0.367186\pi\)
							0.835172	+	0.549988i	\(0.185368\pi\)
\(13\)	1518.30	−	3324.61i	0.191671	−	0.419700i	−0.789260	−	0.614060i	\(-0.789535\pi\)
							0.980930	+	0.194359i	\(0.0622627\pi\)
\(17\)	−17717.5	+	11386.3i	−0.874643	+	0.562099i	−0.899170	−	0.437600i	\(-0.855829\pi\)
							0.0245264	+	0.999699i	\(0.492192\pi\)
\(19\)	−19996.7	+	31115.4i	−0.668837	+	1.04073i	0.326587	+	0.945167i	\(0.394101\pi\)
							−0.995423	+	0.0955629i	\(0.969535\pi\)
\(23\)	−55897.9	−	16740.8i	−0.957961	−	0.286898i
							\(29\)	57860.8	+	90033.2i	0.440546	+	0.685503i	0.988537	−	0.150981i	\(-0.0482434\pi\)
−0.547990	+	0.836485i	\(0.684607\pi\)
\(31\)	−109203.	−	126027.i	−0.658368	−	0.759797i	0.324142	−	0.946008i	\(-0.394924\pi\)
							−0.982510	+	0.186212i	\(0.940379\pi\)
\(37\)	−568841.	+	81787.0i	−1.84623	+	0.265447i	−0.974474	−	0.224500i	\(-0.927925\pi\)
							−0.871752	+	0.489947i	\(0.837016\pi\)
\(41\)	−146152.	−	21013.5i	−0.331177	−	0.0476161i	−0.0252795	−	0.999680i	\(-0.508048\pi\)
							−0.305898	+	0.952064i	\(0.598957\pi\)
\(43\)	327652.	+	283912.i	0.628454	+	0.544559i	0.909800	−	0.415048i	\(-0.136235\pi\)
							−0.281345	+	0.959607i	\(0.590781\pi\)
\(47\)			1.05626e6i			1.48398i	0.670411	+	0.741990i	\(0.266118\pi\)
							−0.670411	+	0.741990i	\(0.733882\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.g.a.5.19	✓	540
3.2	odd	2	inner	69.8.g.a.5.36	yes	540
23.14	odd	22	inner	69.8.g.a.14.36	yes	540
69.14	even	22	inner	69.8.g.a.14.19	yes	540

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.g.a.5.19	✓	540	1.1	even	1	trivial
69.8.g.a.5.36	yes	540	3.2	odd	2	inner
69.8.g.a.14.19	yes	540	69.14	even	22	inner
69.8.g.a.14.36	yes	540	23.14	odd	22	inner