Embedded newform 49.6.e.a.15.11

• Label: 49.6.e.a.15.11
• Level: $49$
• Weight: $6$
• Character: 49.15
• 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			15.11
Character	\(\chi\)	\(=\)	49.15
Dual form			49.6.e.a.36.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.342340 + 1.49989i) q^{2} +(9.55577 - 11.9826i) q^{3} +(26.6985 + 12.8573i) q^{4} +(41.6644 - 52.2455i) q^{5} +(14.7012 + 18.4347i) q^{6} +(-96.6695 - 86.3830i) q^{7} +(-59.1194 + 74.1334i) q^{8} +(1.80370 + 7.90254i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 138 q - 5 q^{2} + 13 q^{3} - 389 q^{4} + 67 q^{5} + 331 q^{6} - 56 q^{7} + 305 q^{8} - 918 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.342340	+	1.49989i	−0.0605177	+	0.265145i	−0.996131	−	0.0878813i	\(-0.971990\pi\)
							0.935613	+	0.353027i	\(0.114848\pi\)
\(3\)	9.55577	−	11.9826i	0.613003	−	0.768681i	−0.374338	−	0.927292i	\(-0.622130\pi\)
							0.987341	+	0.158611i	\(0.0507016\pi\)
\(5\)	41.6644	−	52.2455i	0.745315	−	0.934596i	−0.254154	−	0.967164i	\(-0.581797\pi\)
							0.999470	+	0.0325678i	\(0.0103685\pi\)
\(7\)	−96.6695	−	86.3830i	−0.745666	−	0.666320i
							\(11\)	14.4165	−	63.1627i	0.0359234	−	0.157391i	−0.953785	−	0.300490i	\(-0.902850\pi\)
0.989708	+	0.143099i	\(0.0457069\pi\)
\(13\)	133.441	−	584.643i	0.218993	−	0.959472i	−0.739230	−	0.673453i	\(-0.764811\pi\)
							0.958224	−	0.286020i	\(-0.0923323\pi\)
\(17\)	296.538	−	142.805i	0.248861	−	0.119845i	−0.305295	−	0.952258i	\(-0.598755\pi\)
							0.554157	+	0.832412i	\(0.313041\pi\)
\(19\)	949.207			0.603222			0.301611	−	0.953431i	\(-0.402476\pi\)
							0.301611	+	0.953431i	\(0.402476\pi\)
\(23\)	986.273	+	474.964i	0.388756	+	0.187215i	0.618044	−	0.786144i	\(-0.287925\pi\)
							−0.229288	+	0.973359i	\(0.573640\pi\)
\(29\)	−6597.08	+	3176.98i	−1.45665	+	0.701488i	−0.983736	−	0.179619i	\(-0.942514\pi\)
							−0.472918	+	0.881106i	\(0.656799\pi\)
\(31\)	−2074.22			−0.387660			−0.193830	−	0.981035i	\(-0.562091\pi\)
							−0.193830	+	0.981035i	\(0.562091\pi\)
\(37\)	−2078.28	+	1000.85i	−0.249575	+	0.120189i	−0.554489	−	0.832191i	\(-0.687086\pi\)
							0.304915	+	0.952380i	\(0.401372\pi\)
\(41\)	498.685	−	625.332i	0.0463305	−	0.0580966i	−0.758126	−	0.652108i	\(-0.773885\pi\)
							0.804457	+	0.594011i	\(0.202456\pi\)
\(43\)	10086.7	+	12648.4i	0.831917	+	1.04319i	0.998366	+	0.0571405i	\(0.0181983\pi\)
							−0.166449	+	0.986050i	\(0.553230\pi\)
\(47\)	−2538.33	+	11121.2i	−0.167612	+	0.734354i	0.819336	+	0.573313i	\(0.194342\pi\)
							−0.986948	+	0.161041i	\(0.948515\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.e.a.15.11	✓	138
49.36	even	7	inner	49.6.e.a.36.11	yes	138

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.e.a.15.11	✓	138	1.1	even	1	trivial
49.6.e.a.36.11	yes	138	49.36	even	7	inner