Embedded newform 49.6.e.a.8.12

• Label: 49.6.e.a.8.12
• 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			8.12
Character	\(\chi\)	\(=\)	49.8
Dual form			49.6.e.a.43.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.469190 - 0.588346i) q^{2} +(2.32193 - 1.11818i) q^{3} +(6.99466 - 30.6456i) q^{4} +(-89.5344 + 43.1175i) q^{5} +(-1.74731 - 0.841459i) q^{6} +(102.032 + 79.9786i) q^{7} +(-43.0080 + 20.7116i) q^{8} +(-147.367 + 184.792i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 138 q - 5 q^{2} + 13 q^{3} - 389 q^{4} + 67 q^{5} + 331 q^{6} - 56 q^{7} + 305 q^{8} - 918 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.469190	−	0.588346i	−0.0829419	−	0.104006i	0.738629	−	0.674112i	\(-0.235474\pi\)
							−0.821571	+	0.570106i	\(0.806902\pi\)
\(3\)	2.32193	−	1.11818i	0.148952	−	0.0717315i	−0.357923	−	0.933751i	\(-0.616515\pi\)
							0.506875	+	0.862020i	\(0.330801\pi\)
\(5\)	−89.5344	+	43.1175i	−1.60164	+	0.771309i	−0.999631	−	0.0271798i	\(-0.991347\pi\)
							−0.602009	+	0.798489i	\(0.705633\pi\)
\(7\)	102.032	+	79.9786i	0.787026	+	0.616920i
							\(11\)	128.392	+	160.998i	0.319931	+	0.401180i	0.915627	−	0.402030i	\(-0.131695\pi\)
−0.595696	+	0.803210i	\(0.703124\pi\)
\(13\)	70.1729	+	87.9940i	0.115163	+	0.144409i	0.836071	−	0.548621i	\(-0.184847\pi\)
							−0.720909	+	0.693030i	\(0.756275\pi\)
\(17\)	407.534	+	1785.52i	0.342012	+	1.49845i	0.794820	+	0.606846i	\(0.207565\pi\)
							−0.452808	+	0.891608i	\(0.649578\pi\)
\(19\)	−2881.41			−1.83114			−0.915570	−	0.402158i	\(-0.868260\pi\)
							−0.915570	+	0.402158i	\(0.868260\pi\)
\(23\)	−254.878	+	1116.69i	−0.100465	+	0.440164i	0.899530	+	0.436859i	\(0.143909\pi\)
							−0.999995	+	0.00330486i	\(0.998948\pi\)
\(29\)	−974.125	−	4267.92i	−0.215090	−	0.942369i	−0.961049	−	0.276378i	\(-0.910866\pi\)
							0.745959	−	0.665991i	\(-0.231991\pi\)
\(31\)	−1837.76			−0.343466			−0.171733	−	0.985144i	\(-0.554937\pi\)
							−0.171733	+	0.985144i	\(0.554937\pi\)
\(37\)	−2174.10	−	9525.33i	−0.261080	−	1.14387i	−0.920082	−	0.391726i	\(-0.871878\pi\)
							0.659001	−	0.752142i	\(-0.270979\pi\)
\(41\)	6167.34	−	2970.03i	0.572978	−	0.275932i	−0.124875	−	0.992172i	\(-0.539853\pi\)
							0.697853	+	0.716241i	\(0.254139\pi\)
\(43\)	−1210.85	−	583.116i	−0.0998666	−	0.0480932i	0.383284	−	0.923631i	\(-0.374793\pi\)
							−0.483150	+	0.875538i	\(0.660507\pi\)
\(47\)	8905.59	+	11167.3i	0.588055	+	0.737398i	0.983463	−	0.181108i	\(-0.0579683\pi\)
							−0.395408	+	0.918506i	\(0.629397\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.e.a.8.12	✓	138
49.43	even	7	inner	49.6.e.a.43.12	yes	138

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.e.a.8.12	✓	138	1.1	even	1	trivial
49.6.e.a.43.12	yes	138	49.43	even	7	inner