Embedded newform 49.6.g.a.2.8

• Label: 49.6.g.a.2.8
• Level: $49$
• Weight: $6$
• Character: 49.2
• 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			2.8
Character	\(\chi\)	\(=\)	49.2
Dual form			49.6.g.a.25.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.14587 - 0.970373i) q^{2} +(-3.81108 + 9.71048i) q^{3} +(-17.4847 - 11.9209i) q^{4} +(67.5754 - 10.1854i) q^{5} +(21.4120 - 26.8498i) q^{6} +(-81.7407 - 100.625i) q^{7} +(109.121 + 136.833i) q^{8} +(98.3625 + 91.2671i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 13 q^{2} - 22 q^{3} + 307 q^{4} - 52 q^{5} - 130 q^{6} + 154 q^{7} - 320 q^{8} + 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{13}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.14587	−	0.970373i	−0.556117	−	0.171539i	0.00394177	−	0.999992i	\(-0.498745\pi\)
							−0.560059	+	0.828453i	\(0.689221\pi\)
\(3\)	−3.81108	+	9.71048i	−0.244481	+	0.622928i	−0.999461	−	0.0328200i	\(-0.989551\pi\)
							0.754980	+	0.655748i	\(0.227646\pi\)
\(5\)	67.5754	−	10.1854i	1.20883	−	0.182201i	0.486454	−	0.873706i	\(-0.338290\pi\)
							0.722372	+	0.691505i	\(0.243052\pi\)
\(7\)	−81.7407	−	100.625i	−0.630512	−	0.776179i
							\(11\)	−557.865	+	517.623i	−1.39010	+	1.28983i	−0.478903	+	0.877868i	\(0.658965\pi\)
−0.911202	+	0.411961i	\(0.864844\pi\)
\(13\)	36.0988	+	158.159i	0.0592427	+	0.259559i	0.995873	−	0.0907609i	\(-0.0289299\pi\)
							−0.936630	+	0.350320i	\(0.886073\pi\)
\(17\)	121.941	+	1627.19i	0.102336	+	1.36557i	0.777954	+	0.628322i	\(0.216258\pi\)
							−0.675618	+	0.737252i	\(0.736123\pi\)
\(19\)	257.655	−	446.272i	0.163740	−	0.283606i	−0.772467	−	0.635055i	\(-0.780977\pi\)
							0.936207	+	0.351449i	\(0.114311\pi\)
\(23\)	−249.543	+	3329.92i	−0.0983616	+	1.31254i	0.701634	+	0.712538i	\(0.252454\pi\)
							−0.799995	+	0.600006i	\(0.795165\pi\)
\(29\)	3843.54	+	1850.95i	0.848666	+	0.408696i	0.807082	−	0.590439i	\(-0.201045\pi\)
							0.0415836	+	0.999135i	\(0.486760\pi\)
\(31\)	−2374.95	−	4113.53i	−0.443864	−	0.768795i	0.554108	−	0.832445i	\(-0.313059\pi\)
							−0.997972	+	0.0636493i	\(0.979726\pi\)
\(37\)	5055.85	−	3447.02i	0.607142	−	0.413942i	−0.220349	−	0.975421i	\(-0.570720\pi\)
							0.827491	+	0.561479i	\(0.189767\pi\)
\(41\)	−838.874	−	1051.91i	−0.0779358	−	0.0977284i	0.741335	−	0.671135i	\(-0.234193\pi\)
							−0.819271	+	0.573407i	\(0.805622\pi\)
\(43\)	−6600.57	+	8276.85i	−0.544390	+	0.682643i	−0.975587	−	0.219615i	\(-0.929520\pi\)
							0.431197	+	0.902258i	\(0.358091\pi\)
\(47\)	10594.5	+	3267.96i	0.699575	+	0.215790i	0.624085	−	0.781356i	\(-0.285472\pi\)
							0.0754895	+	0.997147i	\(0.475948\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.g.a.2.8	✓	264
49.25	even	21	inner	49.6.g.a.25.8	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.g.a.2.8	✓	264	1.1	even	1	trivial
49.6.g.a.25.8	yes	264	49.25	even	21	inner