Embedded newform 49.4.g.a.11.2

• Label: 49.4.g.a.11.2
• Level: $49$
• Weight: $4$
• Character: 49.11
• 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			11.2
Character	\(\chi\)	\(=\)	49.11
Dual form			49.4.g.a.9.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.58962 + 3.33068i) q^{2} +(2.54362 - 0.383389i) q^{3} +(1.19410 - 15.9341i) q^{4} +(4.69056 - 11.9514i) q^{5} +(-7.85369 + 9.84821i) q^{6} +(11.1756 - 14.7684i) q^{7} +(24.3601 + 30.5466i) q^{8} +(-19.4774 + 6.00799i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 13 q^{2} - 7 q^{3} + 35 q^{4} - 7 q^{5} - 70 q^{6} - 42 q^{7} + 16 q^{8} + 198 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{20}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.58962	+	3.33068i	−1.26912	+	1.17757i	−0.294023	+	0.955798i	\(0.594994\pi\)
							−0.975098	+	0.221774i	\(0.928815\pi\)
\(3\)	2.54362	−	0.383389i	0.489520	−	0.0737833i	0.100356	−	0.994952i	\(-0.468002\pi\)
							0.389165	+	0.921168i	\(0.372764\pi\)
\(5\)	4.69056	−	11.9514i	0.419536	−	1.06896i	−0.552371	−	0.833598i	\(-0.686277\pi\)
							0.971908	−	0.235363i	\(-0.0756279\pi\)
\(7\)	11.1756	−	14.7684i	0.603425	−	0.797420i
							\(11\)	55.7683	+	17.2022i	1.52862	+	0.471515i	0.941223	−	0.337787i	\(-0.109678\pi\)
0.587393	+	0.809302i	\(0.300154\pi\)
\(13\)	−6.73362	−	29.5019i	−0.143659	−	0.629412i	−0.994567	−	0.104097i	\(-0.966805\pi\)
							0.850908	−	0.525315i	\(-0.176052\pi\)
\(17\)	3.05888	−	2.08551i	0.0436404	−	0.0297535i	−0.541303	−	0.840827i	\(-0.682069\pi\)
							0.584944	+	0.811074i	\(0.301117\pi\)
\(19\)	−58.6116	−	101.518i	−0.707706	−	1.22578i	−0.965706	−	0.259638i	\(-0.916397\pi\)
							0.258000	−	0.966145i	\(-0.416937\pi\)
\(23\)	90.4557	+	61.6717i	0.820058	+	0.559106i	0.899089	−	0.437766i	\(-0.144230\pi\)
							−0.0790311	+	0.996872i	\(0.525183\pi\)
\(29\)	80.2504	+	38.6465i	0.513866	+	0.247465i	0.672800	−	0.739824i	\(-0.265091\pi\)
							−0.158934	+	0.987289i	\(0.550806\pi\)
\(31\)	99.8138	−	172.883i	0.578293	−	1.00163i	−0.417382	−	0.908731i	\(-0.637052\pi\)
							0.995675	−	0.0929021i	\(-0.0296144\pi\)
\(37\)	7.83331	+	104.528i	0.0348051	+	0.464442i	0.987191	+	0.159545i	\(0.0510028\pi\)
							−0.952386	+	0.304896i	\(0.901378\pi\)
\(41\)	−8.27637	−	10.3782i	−0.0315256	−	0.0395319i	0.765819	−	0.643056i	\(-0.222334\pi\)
							−0.797344	+	0.603525i	\(0.793763\pi\)
\(43\)	−299.930	+	376.100i	−1.06369	+	1.33383i	−0.123822	+	0.992304i	\(0.539515\pi\)
							−0.939872	+	0.341526i	\(0.889056\pi\)
\(47\)	−352.253	+	326.843i	−1.09322	+	1.01436i	−0.0933973	+	0.995629i	\(0.529773\pi\)
							−0.999825	+	0.0187327i	\(0.994037\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.g.a.11.2	yes	156
49.3	odd	42		2401.4.a.f.1.72		78
49.9	even	21	inner	49.4.g.a.9.2	✓	156
49.46	even	21		2401.4.a.g.1.72		78

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.g.a.9.2	✓	156	49.9	even	21	inner
49.4.g.a.11.2	yes	156	1.1	even	1	trivial
2401.4.a.f.1.72		78	49.3	odd	42
2401.4.a.g.1.72		78	49.46	even	21