Embedded newform 49.4.g.a.11.1

• Label: 49.4.g.a.11.1
• 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			11.1
Character	\(\chi\)	\(=\)	49.11
Dual form			49.4.g.a.9.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.86677 + 3.58784i) q^{2} +(-8.83156 + 1.33114i) q^{3} +(1.48149 - 19.7691i) q^{4} +(-3.38608 + 8.62758i) q^{5} +(29.3737 - 36.8335i) q^{6} +(-14.1685 + 11.9270i) q^{7} +(38.8891 + 48.7653i) q^{8} +(50.4240 - 15.5538i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 13 q^{2} - 7 q^{3} + 35 q^{4} - 7 q^{5} - 70 q^{6} - 42 q^{7} + 16 q^{8} + 198 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.86677	+	3.58784i	−1.36711	+	1.26849i	−0.437860	+	0.899043i	\(0.644264\pi\)
							−0.929250	+	0.369450i	\(0.879546\pi\)
\(3\)	−8.83156	+	1.33114i	−1.69963	+	0.256179i	−0.926019	−	0.377477i	\(-0.876792\pi\)
							−0.773615	+	0.633655i	\(0.781554\pi\)
\(5\)	−3.38608	+	8.62758i	−0.302860	+	0.771675i	0.695716	+	0.718317i	\(0.255087\pi\)
							−0.998575	+	0.0533572i	\(0.983008\pi\)
\(7\)	−14.1685	+	11.9270i	−0.765027	+	0.643998i
							\(11\)	11.8174	+	3.64520i	0.323918	+	0.0999154i	0.452450	−	0.891790i	\(-0.350550\pi\)
−0.128532	+	0.991705i	\(0.541026\pi\)
\(13\)	−6.06718	−	26.5821i	−0.129441	−	0.567118i	−0.997501	−	0.0706582i	\(-0.977490\pi\)
							0.868060	−	0.496460i	\(-0.165367\pi\)
\(17\)	14.9733	−	10.2086i	0.213621	−	0.145644i	−0.451783	−	0.892128i	\(-0.649212\pi\)
							0.665404	+	0.746484i	\(0.268259\pi\)
\(19\)	−50.4392	−	87.3632i	−0.609028	−	1.05487i	−0.991401	−	0.130860i	\(-0.958226\pi\)
							0.382372	−	0.924008i	\(-0.375107\pi\)
\(23\)	−57.2439	−	39.0282i	−0.518964	−	0.353824i	0.275337	−	0.961348i	\(-0.411211\pi\)
							−0.794301	+	0.607524i	\(0.792163\pi\)
\(29\)	140.470	+	67.6469i	0.899472	+	0.433163i	0.825698	−	0.564112i	\(-0.190782\pi\)
							0.0737735	+	0.997275i	\(0.476496\pi\)
\(31\)	−112.473	+	194.808i	−0.651635	+	1.12867i	0.331091	+	0.943599i	\(0.392583\pi\)
							−0.982726	+	0.185066i	\(0.940750\pi\)
\(37\)	−25.9918	−	346.836i	−0.115487	−	1.54107i	−0.690746	−	0.723097i	\(-0.742718\pi\)
							0.575259	−	0.817971i	\(-0.304901\pi\)
\(41\)	−114.065	−	143.033i	−0.434486	−	0.544828i	0.515595	−	0.856832i	\(-0.327571\pi\)
							−0.950081	+	0.312005i	\(0.899000\pi\)
\(43\)	265.504	−	332.932i	0.941606	−	1.18074i	−0.0417654	−	0.999127i	\(-0.513298\pi\)
							0.983371	−	0.181609i	\(-0.0581304\pi\)
\(47\)	−274.074	+	254.303i	−0.850590	+	0.789233i	−0.979292	−	0.202451i	\(-0.935109\pi\)
							0.128702	+	0.991683i	\(0.458919\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.g.a.11.1	yes	156
49.3	odd	42		2401.4.a.f.1.75		78
49.9	even	21	inner	49.4.g.a.9.1	✓	156
49.46	even	21		2401.4.a.g.1.75		78

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.g.a.9.1	✓	156	49.9	even	21	inner
49.4.g.a.11.1	yes	156	1.1	even	1	trivial
2401.4.a.f.1.75		78	49.3	odd	42
2401.4.a.g.1.75		78	49.46	even	21