Embedded newform 49.6.g.a.11.15

• Label: 49.6.g.a.11.15
• Level: $49$
• Weight: $6$
• Character: 49.11
• Analytic conductor: $7.859$
• Analytic rank: $0$
• Dimension: $264$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	49.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.85880717084\)
Analytic rank:	\(0\)
Dimension:	\(264\)
Relative dimension:	\(22\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			11.15
Character	\(\chi\)	\(=\)	49.11
Dual form			49.6.g.a.9.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.32025 - 3.08075i) q^{2} +(18.9500 - 2.85626i) q^{3} +(-0.858269 + 11.4528i) q^{4} +(20.6981 - 52.7379i) q^{5} +(54.1195 - 67.8637i) q^{6} +(-50.0442 - 119.593i) q^{7} +(122.802 + 153.989i) q^{8} +(118.741 - 36.6267i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 13 q^{2} - 22 q^{3} + 307 q^{4} - 52 q^{5} - 130 q^{6} + 154 q^{7} - 320 q^{8} + 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{20}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.32025	−	3.08075i	0.586944	−	0.544604i	−0.329815	−	0.944046i	\(-0.606986\pi\)
							0.916759	+	0.399441i	\(0.130796\pi\)
\(3\)	18.9500	−	2.85626i	1.21564	−	0.183229i	0.490262	−	0.871575i	\(-0.336901\pi\)
							0.725382	+	0.688346i	\(0.241663\pi\)
\(5\)	20.6981	−	52.7379i	0.370259	−	0.943405i	−0.617381	−	0.786664i	\(-0.711806\pi\)
							0.987640	−	0.156740i	\(-0.0500986\pi\)
\(7\)	−50.0442	−	119.593i	−0.386019	−	0.922491i
							\(11\)	535.257	+	165.105i	1.33377	+	0.411414i	0.877990	−	0.478679i	\(-0.158884\pi\)
0.455780	+	0.890092i	\(0.349360\pi\)
\(13\)	−40.3682	−	176.865i	−0.0662492	−	0.290257i	0.930941	−	0.365170i	\(-0.118989\pi\)
							−0.997190	+	0.0749132i	\(0.976132\pi\)
\(17\)	−1252.75	+	854.109i	−1.05134	+	0.716789i	−0.960275	−	0.279057i	\(-0.909978\pi\)
							−0.0910611	+	0.995845i	\(0.529026\pi\)
\(19\)	−819.850	−	1420.02i	−0.521015	−	0.902425i	−0.999701	−	0.0244390i	\(-0.992220\pi\)
							0.478686	−	0.877986i	\(-0.341113\pi\)
\(23\)	1942.47	+	1324.35i	0.765656	+	0.522016i	0.882032	−	0.471190i	\(-0.156175\pi\)
							−0.116376	+	0.993205i	\(0.537128\pi\)
\(29\)	−2268.21	−	1092.31i	−0.500828	−	0.241186i	0.166374	−	0.986063i	\(-0.446794\pi\)
							−0.667202	+	0.744877i	\(0.732508\pi\)
\(31\)	−3204.49	+	5550.34i	−0.598901	+	1.03733i	0.394083	+	0.919075i	\(0.371062\pi\)
							−0.992984	+	0.118251i	\(0.962271\pi\)
\(37\)	1037.92	+	13850.1i	0.124641	+	1.66321i	0.612635	+	0.790366i	\(0.290109\pi\)
							−0.487995	+	0.872847i	\(0.662272\pi\)
\(41\)	−3169.93	−	3974.97i	−0.294503	−	0.369296i	0.612463	−	0.790500i	\(-0.290179\pi\)
							−0.906966	+	0.421204i	\(0.861608\pi\)
\(43\)	3807.22	−	4774.11i	0.314005	−	0.393750i	−0.599635	−	0.800274i	\(-0.704688\pi\)
							0.913640	+	0.406523i	\(0.133259\pi\)
\(47\)	3999.85	−	3711.32i	0.264119	−	0.245067i	−0.536985	−	0.843592i	\(-0.680437\pi\)
							0.801104	+	0.598525i	\(0.204246\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.g.a.11.15	yes	264
49.9	even	21	inner	49.6.g.a.9.15	✓	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.g.a.9.15	✓	264	49.9	even	21	inner
49.6.g.a.11.15	yes	264	1.1	even	1	trivial