Embedded newform 49.6.g.a.2.7

• Label: 49.6.g.a.2.7
• 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			2.7
Character	\(\chi\)	\(=\)	49.2
Dual form			49.6.g.a.25.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.24707 - 1.61851i) q^{2} +(7.63255 - 19.4474i) q^{3} +(-1.52748 - 1.04142i) q^{4} +(-57.8178 + 8.71463i) q^{5} +(-71.5243 + 89.6886i) q^{6} +(-83.4438 - 99.2176i) q^{7} +(115.884 + 145.314i) q^{8} +(-141.815 - 131.585i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 13 q^{2} - 22 q^{3} + 307 q^{4} - 52 q^{5} - 130 q^{6} + 154 q^{7} - 320 q^{8} + 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.24707	−	1.61851i	−0.927559	−	0.286114i	−0.206068	−	0.978538i	\(-0.566067\pi\)
							−0.721491	+	0.692424i	\(0.756543\pi\)
\(3\)	7.63255	−	19.4474i	0.489628	−	1.24755i	−0.446149	−	0.894959i	\(-0.647205\pi\)
							0.935777	−	0.352593i	\(-0.114700\pi\)
\(5\)	−57.8178	+	8.71463i	−1.03428	+	0.155892i	−0.644174	−	0.764879i	\(-0.722799\pi\)
							−0.390103	+	0.920771i	\(0.627560\pi\)
\(7\)	−83.4438	−	99.2176i	−0.643649	−	0.765321i
							\(11\)	213.985	−	198.549i	0.533214	−	0.494750i	−0.366987	−	0.930226i	\(-0.619610\pi\)
0.900200	+	0.435476i	\(0.143420\pi\)
\(13\)	48.9961	+	214.666i	0.0804087	+	0.352294i	0.999087	−	0.0427120i	\(-0.0135998\pi\)
							−0.918679	+	0.395006i	\(0.870743\pi\)
\(17\)	73.8039	+	984.844i	0.0619380	+	0.826505i	0.938514	+	0.345241i	\(0.112203\pi\)
							−0.876576	+	0.481263i	\(0.840178\pi\)
\(19\)	−1101.74	+	1908.27i	−0.700155	+	1.21270i	0.268257	+	0.963347i	\(0.413552\pi\)
							−0.968412	+	0.249357i	\(0.919781\pi\)
\(23\)	62.6302	−	835.742i	0.0246868	−	0.329422i	−0.971163	−	0.238415i	\(-0.923372\pi\)
							0.995850	−	0.0910074i	\(-0.0290087\pi\)
\(29\)	−1991.44	−	959.026i	−0.439715	−	0.211756i	0.200909	−	0.979610i	\(-0.435610\pi\)
							−0.640625	+	0.767854i	\(0.721325\pi\)
\(31\)	−895.140	−	1550.43i	−0.167296	−	0.289766i	0.770172	−	0.637836i	\(-0.220170\pi\)
							−0.937468	+	0.348070i	\(0.886837\pi\)
\(37\)	−3862.30	+	2633.27i	−0.463812	+	0.316222i	−0.772582	−	0.634915i	\(-0.781035\pi\)
							0.308770	+	0.951137i	\(0.400083\pi\)
\(41\)	−10634.9	−	13335.8i	−0.988042	−	1.23897i	−0.970992	−	0.239111i	\(-0.923144\pi\)
							−0.0170502	−	0.999855i	\(-0.505428\pi\)
\(43\)	−12418.7	+	15572.6i	−1.02425	+	1.28437i	−0.0661863	+	0.997807i	\(0.521083\pi\)
							−0.958062	+	0.286560i	\(0.907488\pi\)
\(47\)	222.748	+	68.7088i	0.0147086	+	0.00453699i	0.302101	−	0.953276i	\(-0.402312\pi\)
							−0.287392	+	0.957813i	\(0.592788\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.g.a.2.7	✓	264
49.25	even	21	inner	49.6.g.a.25.7	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.g.a.2.7	✓	264	1.1	even	1	trivial
49.6.g.a.25.7	yes	264	49.25	even	21	inner