Embedded newform 49.4.e.a.43.12

• Label: 49.4.e.a.43.12
• Level: $49$
• Weight: $4$
• Character: 49.43
• 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			43.12
Character	\(\chi\)	\(=\)	49.43
Dual form			49.4.e.a.8.12

$q$-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 \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{7}\right)\)

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	−0.0483758	−	0.998829i	\(-0.515405\pi\)
							0.984551	−	0.175097i	\(-0.0560241\pi\)
\(3\)	6.62695	+	3.19137i	1.27536	+	0.614180i	0.944193	−	0.329393i	\(-0.106844\pi\)
							0.331165	+	0.943573i	\(0.392558\pi\)
\(5\)	−17.1881	−	8.27737i	−1.53735	−	0.740350i	−0.542346	−	0.840155i	\(-0.682464\pi\)
							−0.995007	+	0.0998054i	\(0.968178\pi\)
\(7\)	16.3388	+	8.72030i	0.882212	+	0.470852i
							\(11\)	−8.30653	+	10.4161i	−0.227683	+	0.285505i	−0.882530	−	0.470256i	\(-0.844162\pi\)
0.654847	+	0.755761i	\(0.272733\pi\)
\(13\)	−27.7617	+	34.8121i	−0.592286	+	0.742703i	−0.984154	−	0.177318i	\(-0.943258\pi\)
							0.391867	+	0.920022i	\(0.371829\pi\)
\(17\)	−9.98101	+	43.7297i	−0.142397	+	0.623882i	0.852477	+	0.522764i	\(0.175099\pi\)
							−0.994874	+	0.101118i	\(0.967758\pi\)
\(19\)	−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	−0.983655	−	0.180063i	\(-0.942370\pi\)
							0.808116	−	0.589023i	\(-0.200487\pi\)
\(29\)	33.7748	−	147.977i	0.216269	−	0.947539i	−0.743937	−	0.668249i	\(-0.767044\pi\)
							0.960207	−	0.279289i	\(-0.0900989\pi\)
\(31\)	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	−0.929277	−	0.369383i	\(-0.879569\pi\)
							0.997519	+	0.0703960i	\(0.0224263\pi\)
\(41\)	−167.948	−	80.8796i	−0.639734	−	0.308080i	0.0857519	−	0.996317i	\(-0.472671\pi\)
							−0.725486	+	0.688237i	\(0.758385\pi\)
\(43\)	−107.322	+	51.6834i	−0.380614	+	0.183294i	−0.614401	−	0.788994i	\(-0.710602\pi\)
							0.233787	+	0.972288i	\(0.424888\pi\)
\(47\)	359.776	−	451.145i	1.11657	−	1.40013i	0.210195	−	0.977660i	\(-0.432590\pi\)
							0.906375	−	0.422474i	\(-0.138838\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.e.a.8.12	✓	78	49.8	even	7	inner
49.4.e.a.43.12	yes	78	1.1	even	1	trivial
2401.4.a.c.1.34		39	49.20	odd	14
2401.4.a.d.1.34		39	49.29	even	7