Embedded newform 49.6.e.a.8.4

• Label: 49.6.e.a.8.4
• 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			8.4
Character	\(\chi\)	\(=\)	49.8
Dual form			49.6.e.a.43.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.98029 - 6.24508i) q^{2} +(6.83688 - 3.29247i) q^{3} +(-7.07712 + 31.0069i) q^{4} +(10.7661 - 5.18467i) q^{5} +(-54.6114 - 26.2994i) q^{6} +(-98.9047 - 83.8145i) q^{7} +(-1.40854 + 0.678317i) q^{8} +(-115.605 + 144.965i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 138 q - 5 q^{2} + 13 q^{3} - 389 q^{4} + 67 q^{5} + 331 q^{6} - 56 q^{7} + 305 q^{8} - 918 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.98029	−	6.24508i	−0.880398	−	1.10398i	−0.993882	−	0.110445i	\(-0.964772\pi\)
							0.113484	−	0.993540i	\(-0.463799\pi\)
\(3\)	6.83688	−	3.29247i	0.438586	−	0.211212i	−0.201542	−	0.979480i	\(-0.564595\pi\)
							0.640129	+	0.768268i	\(0.278881\pi\)
\(5\)	10.7661	−	5.18467i	0.192590	−	0.0927462i	−0.335104	−	0.942181i	\(-0.608771\pi\)
							0.527693	+	0.849435i	\(0.323057\pi\)
\(7\)	−98.9047	−	83.8145i	−0.762907	−	0.646508i
							\(11\)	−328.298	−	411.673i	−0.818062	−	1.02582i	−0.999103	−	0.0423423i	\(-0.986518\pi\)
0.181041	−	0.983476i	\(-0.442053\pi\)
\(13\)	261.893	+	328.403i	0.429798	+	0.538950i	0.948823	−	0.315809i	\(-0.102276\pi\)
							−0.519024	+	0.854759i	\(0.673705\pi\)
\(17\)	476.352	+	2087.03i	0.399766	+	1.75149i	0.628318	+	0.777957i	\(0.283744\pi\)
							−0.228552	+	0.973532i	\(0.573399\pi\)
\(19\)	−2224.93			−1.41394			−0.706972	−	0.707241i	\(-0.749939\pi\)
							−0.706972	+	0.707241i	\(0.749939\pi\)
\(23\)	1001.93	−	4389.74i	0.394928	−	1.73029i	−0.251989	−	0.967730i	\(-0.581085\pi\)
							0.646916	−	0.762561i	\(-0.276058\pi\)
\(29\)	−214.204	−	938.488i	−0.0472968	−	0.207221i	0.945758	−	0.324871i	\(-0.105321\pi\)
							−0.993055	+	0.117650i	\(0.962464\pi\)
\(31\)	−2794.31			−0.522240			−0.261120	−	0.965306i	\(-0.584092\pi\)
							−0.261120	+	0.965306i	\(0.584092\pi\)
\(37\)	−1969.12	−	8627.27i	−0.236465	−	1.03602i	−0.944156	−	0.329500i	\(-0.893120\pi\)
							0.707690	−	0.706523i	\(-0.249737\pi\)
\(41\)	2530.32	−	1218.54i	0.235080	−	0.113208i	−0.312636	−	0.949873i	\(-0.601212\pi\)
							0.547715	+	0.836665i	\(0.315498\pi\)
\(43\)	−16135.1	−	7770.28i	−1.33077	−	0.640863i	−0.372845	−	0.927894i	\(-0.621618\pi\)
							−0.957921	+	0.287030i	\(0.907332\pi\)
\(47\)	−3519.71	−	4413.57i	−0.232414	−	0.291438i	0.651925	−	0.758283i	\(-0.273962\pi\)
							−0.884339	+	0.466846i	\(0.845390\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.e.a.8.4	✓	138
49.43	even	7	inner	49.6.e.a.43.4	yes	138

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.e.a.8.4	✓	138	1.1	even	1	trivial
49.6.e.a.43.4	yes	138	49.43	even	7	inner