Embedded newform 49.4.g.a.16.6

• Label: 49.4.g.a.16.6
• Level: $49$
• Weight: $4$
• Character: 49.16
• Analytic conductor: $2.891$
• Analytic rank: $0$
• Dimension: $156$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	49.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.89109359028\)
Analytic rank:	\(0\)
Dimension:	\(156\)
Relative dimension:	\(13\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			16.6
Character	\(\chi\)	\(=\)	49.16
Dual form			49.4.g.a.46.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.182357 - 0.464637i) q^{2} +(0.156731 + 2.09142i) q^{3} +(5.68178 - 5.27192i) q^{4} +(-0.383119 + 0.261206i) q^{5} +(0.943172 - 0.454208i) q^{6} +(18.4688 + 1.37983i) q^{7} +(-7.08332 - 3.41115i) q^{8} +(22.3489 - 3.36856i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 13 q^{2} - 7 q^{3} + 35 q^{4} - 7 q^{5} - 70 q^{6} - 42 q^{7} + 16 q^{8} + 198 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{10}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.182357	−	0.464637i	−0.0644728	−	0.164274i	0.895007	−	0.446053i	\(-0.147171\pi\)
							−0.959479	+	0.281779i	\(0.909075\pi\)
\(3\)	0.156731	+	2.09142i	0.0301628	+	0.402495i	0.991767	+	0.128055i	\(0.0408734\pi\)
							−0.961604	+	0.274440i	\(0.911508\pi\)
\(5\)	−0.383119	+	0.261206i	−0.0342672	+	0.0233630i	−0.580333	−	0.814380i	\(-0.697077\pi\)
							0.546065	+	0.837743i	\(0.316125\pi\)
\(7\)	18.4688	+	1.37983i	0.997221	+	0.0745036i
							\(11\)	1.92141	+	0.289605i	0.0526660	+	0.00793812i	0.175322	−	0.984511i	\(-0.443903\pi\)
−0.122656	+	0.992449i	\(0.539141\pi\)
\(13\)	5.77161	−	7.23736i	0.123135	−	0.154406i	−0.716443	−	0.697646i	\(-0.754231\pi\)
							0.839578	+	0.543239i	\(0.182802\pi\)
\(17\)	−110.911	+	34.2115i	−1.58235	+	0.488089i	−0.956469	−	0.291832i	\(-0.905735\pi\)
							−0.625877	+	0.779922i	\(0.715259\pi\)
\(19\)	−34.0185	+	58.9218i	−0.410757	+	0.711452i	−0.994973	−	0.100146i	\(-0.968069\pi\)
							0.584216	+	0.811598i	\(0.301402\pi\)
\(23\)	−66.1367	−	20.4005i	−0.599585	−	0.184947i	−0.0199190	−	0.999802i	\(-0.506341\pi\)
							−0.579666	+	0.814854i	\(0.696817\pi\)
\(29\)	12.8600	−	56.3434i	0.0823463	−	0.360783i	−0.916920	−	0.399070i	\(-0.869333\pi\)
							0.999267	+	0.0382872i	\(0.0121902\pi\)
\(31\)	−56.6623	−	98.1421i	−0.328286	−	0.568608i	0.653886	−	0.756593i	\(-0.273138\pi\)
							−0.982172	+	0.187985i	\(0.939804\pi\)
\(37\)	−147.494	−	136.855i	−0.655350	−	0.608076i	0.280762	−	0.959777i	\(-0.409413\pi\)
							−0.936112	+	0.351702i	\(0.885603\pi\)
\(41\)	246.219	+	118.573i	0.937876	+	0.451658i	0.839420	−	0.543484i	\(-0.182895\pi\)
							0.0984568	+	0.995141i	\(0.468609\pi\)
\(43\)	319.217	−	153.727i	1.13210	−	0.545188i	0.228487	−	0.973547i	\(-0.426622\pi\)
							0.903609	+	0.428359i	\(0.140908\pi\)
\(47\)	93.8246	+	239.061i	0.291186	+	0.741929i	0.999356	+	0.0358871i	\(0.0114257\pi\)
							−0.708170	+	0.706042i	\(0.750479\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.g.a.16.6	✓	156
49.12	odd	42		2401.4.a.f.1.37		78
49.37	even	21		2401.4.a.g.1.37		78
49.46	even	21	inner	49.4.g.a.46.6	yes	156

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.g.a.16.6	✓	156	1.1	even	1	trivial
49.4.g.a.46.6	yes	156	49.46	even	21	inner
2401.4.a.f.1.37		78	49.12	odd	42
2401.4.a.g.1.37		78	49.37	even	21