Embedded newform 49.4.e.a.8.1

• Label: 49.4.e.a.8.1
• Level: $49$
• Weight: $4$
• Character: 49.8
• Analytic conductor: $2.891$
• Analytic rank: $0$
• Dimension: $78$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	49.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.89109359028\)
Analytic rank:	\(0\)
Dimension:	\(78\)
Relative dimension:	\(13\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			8.1
Character	\(\chi\)	\(=\)	49.8
Dual form			49.4.e.a.43.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.44244 - 4.31668i) q^{2} +(3.65176 - 1.75859i) q^{3} +(-5.00318 + 21.9203i) q^{4} +(-12.6967 + 6.11441i) q^{5} +(-20.1622 - 9.70961i) q^{6} +(-15.1671 + 10.6282i) q^{7} +(72.0505 - 34.6977i) q^{8} +(-6.59154 + 8.26553i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q - 5 q^{2} - 5 q^{3} - 53 q^{4} - 23 q^{5} + 19 q^{6} - 31 q^{8} - 174 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.44244	−	4.31668i	−1.21708	−	1.52618i	−0.778711	−	0.627383i	\(-0.784126\pi\)
							−0.438374	−	0.898793i	\(-0.644446\pi\)
\(3\)	3.65176	−	1.75859i	0.702781	−	0.338442i	−0.0481335	−	0.998841i	\(-0.515327\pi\)
							0.750915	+	0.660399i	\(0.229613\pi\)
\(5\)	−12.6967	+	6.11441i	−1.13563	+	0.546890i	−0.904686	−	0.426078i	\(-0.859895\pi\)
							−0.230942	+	0.972968i	\(0.574181\pi\)
\(7\)	−15.1671	+	10.6282i	−0.818949	+	0.573866i
							\(11\)	−21.5735	−	27.0524i	−0.591333	−	0.741509i	0.392666	−	0.919681i	\(-0.371553\pi\)
−0.983999	+	0.178173i	\(0.942981\pi\)
\(13\)	11.5286	+	14.4565i	0.245959	+	0.308423i	0.889452	−	0.457029i	\(-0.151087\pi\)
							−0.643492	+	0.765453i	\(0.722515\pi\)
\(17\)	−12.0700	−	52.8822i	−0.172200	−	0.754459i	−0.985090	−	0.172041i	\(-0.944964\pi\)
							0.812889	−	0.582418i	\(-0.197893\pi\)
\(19\)	−61.4377			−0.741831			−0.370915	−	0.928667i	\(-0.620956\pi\)
							−0.370915	+	0.928667i	\(0.620956\pi\)
\(23\)	−20.4282	+	89.5019i	−0.185199	+	0.811410i	0.793904	+	0.608044i	\(0.208045\pi\)
							−0.979103	+	0.203366i	\(0.934812\pi\)
\(29\)	1.60090	+	7.01401i	0.0102510	+	0.0449127i	0.979795	−	0.200006i	\(-0.0640960\pi\)
							−0.969544	+	0.244918i	\(0.921239\pi\)
\(31\)	−58.2544			−0.337510			−0.168755	−	0.985658i	\(-0.553975\pi\)
							−0.168755	+	0.985658i	\(0.553975\pi\)
\(37\)	−4.07514	−	17.8543i	−0.0181067	−	0.0793307i	0.965067	−	0.262002i	\(-0.0843827\pi\)
							−0.983174	+	0.182671i	\(0.941526\pi\)
\(41\)	282.635	−	136.110i	1.07659	−	0.518458i	0.190364	−	0.981713i	\(-0.439033\pi\)
							0.886225	+	0.463255i	\(0.153319\pi\)
\(43\)	51.7021	+	24.8984i	0.183360	+	0.0883017i	0.523312	−	0.852141i	\(-0.324696\pi\)
							−0.339951	+	0.940443i	\(0.610411\pi\)
\(47\)	−208.341	−	261.251i	−0.646588	−	0.810796i	0.345221	−	0.938521i	\(-0.387804\pi\)
							−0.991810	+	0.127725i	\(0.959232\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.e.a.8.1	✓	78
49.22	even	7		2401.4.a.d.1.1		39
49.27	odd	14		2401.4.a.c.1.1		39
49.43	even	7	inner	49.4.e.a.43.1	yes	78

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.e.a.8.1	✓	78	1.1	even	1	trivial
49.4.e.a.43.1	yes	78	49.43	even	7	inner
2401.4.a.c.1.1		39	49.27	odd	14
2401.4.a.d.1.1		39	49.22	even	7