Embedded newform 49.6.g.a.2.16

• Label: 49.6.g.a.2.16
• Level: $49$
• Weight: $6$
• Character: 49.2
• Analytic conductor: $7.859$
• Analytic rank: $0$
• Dimension: $264$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	49.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.85880717084\)
Analytic rank:	\(0\)
Dimension:	\(264\)
Relative dimension:	\(22\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			2.16
Character	\(\chi\)	\(=\)	49.2
Dual form			49.6.g.a.25.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.22133 + 1.61057i) q^{2} +(8.72914 - 22.2415i) q^{3} +(-1.77125 - 1.20762i) q^{4} +(-104.732 + 15.7858i) q^{5} +(81.3992 - 102.071i) q^{6} +(122.487 - 42.4725i) q^{7} +(-116.321 - 145.862i) q^{8} +(-240.355 - 223.016i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 13 q^{2} - 22 q^{3} + 307 q^{4} - 52 q^{5} - 130 q^{6} + 154 q^{7} - 320 q^{8} + 6 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{13}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.22133	+	1.61057i	0.923010	+	0.284711i	0.719605	−	0.694384i	\(-0.244323\pi\)
							0.203405	+	0.979095i	\(0.434799\pi\)
\(3\)	8.72914	−	22.2415i	0.559975	−	1.42679i	−0.318002	−	0.948090i	\(-0.603012\pi\)
							0.877977	−	0.478703i	\(-0.158893\pi\)
\(5\)	−104.732	+	15.7858i	−1.87350	+	0.282384i	−0.984481	−	0.175489i	\(-0.943849\pi\)
							−0.889017	+	0.457874i	\(0.848611\pi\)
\(7\)	122.487	−	42.4725i	0.944812	−	0.327614i
							\(11\)	99.8611	−	92.6576i	0.248837	−	0.230887i	−0.545886	−	0.837860i	\(-0.683807\pi\)
0.794723	+	0.606973i	\(0.207616\pi\)
\(13\)	62.1361	+	272.236i	0.101973	+	0.446773i	0.999977	+	0.00678577i	\(0.00215999\pi\)
							−0.898004	+	0.439988i	\(0.854983\pi\)
\(17\)	−58.6898	−	783.161i	−0.0492538	−	0.657247i	−0.966127	−	0.258068i	\(-0.916914\pi\)
							0.916873	−	0.399179i	\(-0.130705\pi\)
\(19\)	1383.15	−	2395.69i	0.878993	−	1.52246i	0.0265447	−	0.999648i	\(-0.491550\pi\)
							0.852448	−	0.522812i	\(-0.175117\pi\)
\(23\)	−236.457	+	3155.30i	−0.0932036	+	1.24372i	0.732955	+	0.680277i	\(0.238141\pi\)
							−0.826158	+	0.563438i	\(0.809478\pi\)
\(29\)	4712.66	+	2269.50i	1.04057	+	0.501112i	0.874512	−	0.485004i	\(-0.161182\pi\)
							0.166058	+	0.986116i	\(0.446896\pi\)
\(31\)	−2555.00	−	4425.39i	−0.477515	−	0.827081i	0.522153	−	0.852852i	\(-0.325129\pi\)
							−0.999668	+	0.0257715i	\(0.991796\pi\)
\(37\)	−1889.02	+	1287.91i	−0.226846	+	0.154661i	−0.671410	−	0.741086i	\(-0.734311\pi\)
							0.444564	+	0.895747i	\(0.353359\pi\)
\(41\)	−5796.48	−	7268.56i	−0.538524	−	0.675287i	0.435903	−	0.899994i	\(-0.356429\pi\)
							−0.974427	+	0.224706i	\(0.927858\pi\)
\(43\)	−8404.18	+	10538.5i	−0.693145	+	0.869177i	−0.996491	−	0.0837038i	\(-0.973325\pi\)
							0.303345	+	0.952881i	\(0.401896\pi\)
\(47\)	−4899.74	−	1511.37i	−0.323540	−	0.0997990i	0.128731	−	0.991680i	\(-0.458910\pi\)
							−0.452271	+	0.891881i	\(0.649386\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.g.a.2.16	✓	264
49.25	even	21	inner	49.6.g.a.25.16	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.g.a.2.16	✓	264	1.1	even	1	trivial
49.6.g.a.25.16	yes	264	49.25	even	21	inner