Embedded newform 49.6.g.a.11.19

• Label: 49.6.g.a.11.19
• Level: $49$
• Weight: $6$
• Character: 49.11
• 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			11.19
Character	\(\chi\)	\(=\)	49.11
Dual form			49.6.g.a.9.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(6.60436 - 6.12796i) q^{2} +(-7.36828 + 1.11059i) q^{3} +(3.67444 - 49.0320i) q^{4} +(14.6588 - 37.3501i) q^{5} +(-41.8572 + 52.4872i) q^{6} +(44.9121 - 121.614i) q^{7} +(-96.4456 - 120.939i) q^{8} +(-179.146 + 55.2592i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 13 q^{2} - 22 q^{3} + 307 q^{4} - 52 q^{5} - 130 q^{6} + 154 q^{7} - 320 q^{8} + 6 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	6.60436	−	6.12796i	1.16750	−	1.08328i	0.172335	−	0.985038i	\(-0.444869\pi\)
							0.995163	−	0.0982413i	\(-0.0313217\pi\)
\(3\)	−7.36828	+	1.11059i	−0.472675	+	0.0712444i	−0.381060	−	0.924550i	\(-0.624441\pi\)
							−0.0916150	+	0.995795i	\(0.529203\pi\)
\(5\)	14.6588	−	37.3501i	0.262225	−	0.668138i	−0.737751	−	0.675073i	\(-0.764112\pi\)
							0.999976	+	0.00693477i	\(0.00220742\pi\)
\(7\)	44.9121	−	121.614i	0.346432	−	0.938075i
							\(11\)	−322.459	−	99.4653i	−0.803512	−	0.247850i	−0.134318	−	0.990938i	\(-0.542884\pi\)
−0.669194	+	0.743088i	\(0.733360\pi\)
\(13\)	−36.1863	−	158.543i	−0.0593863	−	0.260188i	0.936516	−	0.350625i	\(-0.114031\pi\)
							−0.995902	+	0.0904367i	\(0.971174\pi\)
\(17\)	1251.89	−	853.525i	1.05062	−	0.716299i	0.0904999	−	0.995896i	\(-0.471154\pi\)
							0.960117	+	0.279598i	\(0.0902011\pi\)
\(19\)	384.884	+	666.639i	0.244594	+	0.423650i	0.962017	−	0.272988i	\(-0.0880119\pi\)
							−0.717423	+	0.696638i	\(0.754679\pi\)
\(23\)	2683.27	+	1829.42i	1.05766	+	0.721099i	0.961649	−	0.274281i	\(-0.0884400\pi\)
							0.0960087	+	0.995380i	\(0.469392\pi\)
\(29\)	−2487.17	−	1197.76i	−0.549175	−	0.264469i	0.138655	−	0.990341i	\(-0.455722\pi\)
							−0.687830	+	0.725872i	\(0.741436\pi\)
\(31\)	3254.52	−	5637.00i	0.608252	−	1.05352i	−0.383277	−	0.923634i	\(-0.625204\pi\)
							0.991529	−	0.129889i	\(-0.0414622\pi\)
\(37\)	545.631	+	7280.94i	0.0655231	+	0.874345i	0.929029	+	0.370007i	\(0.120645\pi\)
							−0.863506	+	0.504339i	\(0.831736\pi\)
\(41\)	2642.72	+	3313.86i	0.245522	+	0.307875i	0.889288	−	0.457347i	\(-0.151200\pi\)
							−0.643766	+	0.765223i	\(0.722629\pi\)
\(43\)	−10379.4	+	13015.4i	−0.856055	+	1.07346i	0.140464	+	0.990086i	\(0.455141\pi\)
							−0.996518	+	0.0833727i	\(0.973431\pi\)
\(47\)	3902.85	−	3621.32i	0.257714	−	0.239123i	−0.540725	−	0.841199i	\(-0.681850\pi\)
							0.798439	+	0.602076i	\(0.205660\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.g.a.11.19	yes	264
49.9	even	21	inner	49.6.g.a.9.19	✓	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.g.a.9.19	✓	264	49.9	even	21	inner
49.6.g.a.11.19	yes	264	1.1	even	1	trivial