Embedded newform 49.4.e.a.8.12

• Label: 49.4.e.a.8.12
• 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			8.12
Character	\(\chi\)	\(=\)	49.8
Dual form			49.4.e.a.43.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.64790 + 3.32037i) q^{2} +(6.62695 - 3.19137i) q^{3} +(-2.23327 + 9.78459i) q^{4} +(-17.1881 + 8.27737i) q^{5} +(28.1441 + 13.5535i) q^{6} +(16.3388 - 8.72030i) q^{7} +(-7.79127 + 3.75208i) q^{8} +(16.8974 - 21.1887i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q - 5 q^{2} - 5 q^{3} - 53 q^{4} - 23 q^{5} + 19 q^{6} - 31 q^{8} - 174 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.64790	+	3.32037i	0.936175	+	1.17393i	0.984551	+	0.175097i	\(0.0560241\pi\)
							−0.0483758	+	0.998829i	\(0.515405\pi\)
\(3\)	6.62695	−	3.19137i	1.27536	−	0.614180i	0.331165	−	0.943573i	\(-0.392558\pi\)
							0.944193	+	0.329393i	\(0.106844\pi\)
\(5\)	−17.1881	+	8.27737i	−1.53735	+	0.740350i	−0.995007	−	0.0998054i	\(-0.968178\pi\)
							−0.542346	+	0.840155i	\(0.682464\pi\)
\(7\)	16.3388	−	8.72030i	0.882212	−	0.470852i
							\(11\)	−8.30653	−	10.4161i	−0.227683	−	0.285505i	0.654847	−	0.755761i	\(-0.272733\pi\)
−0.882530	+	0.470256i	\(0.844162\pi\)
\(13\)	−27.7617	−	34.8121i	−0.592286	−	0.742703i	0.391867	−	0.920022i	\(-0.371829\pi\)
							−0.984154	+	0.177318i	\(0.943258\pi\)
\(17\)	−9.98101	−	43.7297i	−0.142397	−	0.623882i	−0.994874	−	0.101118i	\(-0.967758\pi\)
							0.852477	−	0.522764i	\(-0.175099\pi\)
\(19\)	−112.743			−1.36131			−0.680657	−	0.732602i	\(-0.738306\pi\)
							−0.680657	+	0.732602i	\(0.738306\pi\)
\(23\)	−19.3627	+	84.8335i	−0.175539	+	0.769087i	0.808116	+	0.589023i	\(0.200487\pi\)
							−0.983655	+	0.180063i	\(0.942370\pi\)
\(29\)	33.7748	+	147.977i	0.216269	+	0.947539i	0.960207	+	0.279289i	\(0.0900989\pi\)
							−0.743937	+	0.668249i	\(0.767044\pi\)
\(31\)	256.908			1.48846			0.744228	−	0.667926i	\(-0.232818\pi\)
							0.744228	+	0.667926i	\(0.232818\pi\)
\(37\)	15.3586	+	67.2906i	0.0682418	+	0.298987i	0.997519	−	0.0703960i	\(-0.0224263\pi\)
							−0.929277	+	0.369383i	\(0.879569\pi\)
\(41\)	−167.948	+	80.8796i	−0.639734	+	0.308080i	−0.725486	−	0.688237i	\(-0.758385\pi\)
							0.0857519	+	0.996317i	\(0.472671\pi\)
\(43\)	−107.322	−	51.6834i	−0.380614	−	0.183294i	0.233787	−	0.972288i	\(-0.424888\pi\)
							−0.614401	+	0.788994i	\(0.710602\pi\)
\(47\)	359.776	+	451.145i	1.11657	+	1.40013i	0.906375	+	0.422474i	\(0.138838\pi\)
							0.210195	+	0.977660i	\(0.432590\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.e.a.8.12	✓	78
49.22	even	7		2401.4.a.d.1.34		39
49.27	odd	14		2401.4.a.c.1.34		39
49.43	even	7	inner	49.4.e.a.43.12	yes	78

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.e.a.8.12	✓	78	1.1	even	1	trivial
49.4.e.a.43.12	yes	78	49.43	even	7	inner
2401.4.a.c.1.34		39	49.27	odd	14
2401.4.a.d.1.34		39	49.22	even	7