Embedded newform 49.6.e.a.8.18

• Label: 49.6.e.a.8.18
• Level: $49$
• Weight: $6$
• Character: 49.8
• Analytic conductor: $7.859$
• Analytic rank: $0$
• Dimension: $138$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(7.85880717084\)
Analytic rank:	\(0\)
Dimension:	\(138\)
Relative dimension:	\(23\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			8.18
Character	\(\chi\)	\(=\)	49.8
Dual form			49.6.e.a.43.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.97480 + 4.98424i) q^{2} +(-2.41848 + 1.16468i) q^{3} +(-1.92294 + 8.42495i) q^{4} +(98.2899 - 47.3339i) q^{5} +(-15.4180 - 7.42491i) q^{6} +(-15.6636 - 128.692i) q^{7} +(134.165 - 64.6103i) q^{8} +(-147.015 + 184.352i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 138 q - 5 q^{2} + 13 q^{3} - 389 q^{4} + 67 q^{5} + 331 q^{6} - 56 q^{7} + 305 q^{8} - 918 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{6}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.97480	+	4.98424i	0.702651	+	0.881097i	0.997218	−	0.0745339i	\(-0.0237469\pi\)
							−0.294567	+	0.955631i	\(0.595175\pi\)
\(3\)	−2.41848	+	1.16468i	−0.155145	+	0.0747141i	−0.509845	−	0.860266i	\(-0.670297\pi\)
							0.354700	+	0.934980i	\(0.384583\pi\)
\(5\)	98.2899	−	47.3339i	1.75826	−	0.846735i	0.784184	−	0.620528i	\(-0.213082\pi\)
							0.974079	−	0.226207i	\(-0.0726325\pi\)
\(7\)	−15.6636	−	128.692i	−0.120822	−	0.992674i
							\(11\)	−162.407	−	203.652i	−0.404691	−	0.507466i	0.537168	−	0.843475i	\(-0.319494\pi\)
−0.941859	+	0.336009i	\(0.890923\pi\)
\(13\)	627.239	+	786.533i	1.02938	+	1.29080i	0.955960	+	0.293497i	\(0.0948190\pi\)
							0.0734171	+	0.997301i	\(0.476610\pi\)
\(17\)	−26.5386	−	116.273i	−0.0222718	−	0.0975793i	0.962570	−	0.271032i	\(-0.0873648\pi\)
							−0.984842	+	0.173452i	\(0.944508\pi\)
\(19\)	−2049.60			−1.30252			−0.651260	−	0.758855i	\(-0.725759\pi\)
							−0.651260	+	0.758855i	\(0.725759\pi\)
\(23\)	−636.359	+	2788.07i	−0.250832	+	1.09897i	0.679912	+	0.733294i	\(0.262018\pi\)
							−0.930744	+	0.365672i	\(0.880839\pi\)
\(29\)	−340.986	−	1493.96i	−0.0752908	−	0.329870i	0.923230	−	0.384249i	\(-0.125539\pi\)
							−0.998520	+	0.0543782i	\(0.982682\pi\)
\(31\)	507.547			0.0948576			0.0474288	−	0.998875i	\(-0.484897\pi\)
							0.0474288	+	0.998875i	\(0.484897\pi\)
\(37\)	1151.11	+	5043.36i	0.138234	+	0.605641i	0.995823	+	0.0913059i	\(0.0291041\pi\)
							−0.857589	+	0.514335i	\(0.828039\pi\)
\(41\)	−12716.9	+	6124.13i	−1.18147	+	0.568964i	−0.918338	−	0.395797i	\(-0.870468\pi\)
							−0.263128	+	0.964761i	\(0.584754\pi\)
\(43\)	43.2581	+	20.8320i	0.00356776	+	0.00171815i	0.435667	−	0.900108i	\(-0.356513\pi\)
							−0.432099	+	0.901826i	\(0.642227\pi\)
\(47\)	3286.88	+	4121.62i	0.217040	+	0.272160i	0.878418	−	0.477893i	\(-0.158599\pi\)
							−0.661378	+	0.750053i	\(0.730028\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.e.a.8.18	✓	138
49.43	even	7	inner	49.6.e.a.43.18	yes	138

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.e.a.8.18	✓	138	1.1	even	1	trivial
49.6.e.a.43.18	yes	138	49.43	even	7	inner