Embedded newform 49.6.g.a.2.6

• Label: 49.6.g.a.2.6
• 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			2.6
Character	\(\chi\)	\(=\)	49.2
Dual form			49.6.g.a.25.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.81543 - 2.10228i) q^{2} +(3.52552 - 8.98289i) q^{3} +(15.5909 + 10.6297i) q^{4} +(80.2537 - 12.0963i) q^{5} +(-42.9125 + 53.8106i) q^{6} +(-85.3330 + 97.5975i) q^{7} +(58.3893 + 73.2178i) q^{8} +(109.869 + 101.943i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 13 q^{2} - 22 q^{3} + 307 q^{4} - 52 q^{5} - 130 q^{6} + 154 q^{7} - 320 q^{8} + 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.81543	−	2.10228i	−1.20481	−	0.371635i	−0.373598	−	0.927591i	\(-0.621876\pi\)
							−0.831212	+	0.555956i	\(0.812352\pi\)
\(3\)	3.52552	−	8.98289i	0.226162	−	0.576253i	−0.772079	−	0.635526i	\(-0.780783\pi\)
							0.998242	+	0.0592736i	\(0.0188784\pi\)
\(5\)	80.2537	−	12.0963i	1.43562	−	0.216385i	0.615302	−	0.788291i	\(-0.289034\pi\)
							0.820319	+	0.571906i	\(0.193796\pi\)
\(7\)	−85.3330	+	97.5975i	−0.658221	+	0.752824i
							\(11\)	415.840	−	385.843i	1.03620	−	0.961454i	0.0369184	−	0.999318i	\(-0.488246\pi\)
0.999283	+	0.0378640i	\(0.0120554\pi\)
\(13\)	20.6664	+	90.5452i	0.0339161	+	0.148596i	0.989051	−	0.147576i	\(-0.0471470\pi\)
							−0.955135	+	0.296172i	\(0.904290\pi\)
\(17\)	−5.71997	−	76.3277i	−0.00480034	−	0.0640560i	0.994289	−	0.106720i	\(-0.0340347\pi\)
							−0.999089	+	0.0426635i	\(0.986416\pi\)
\(19\)	1254.84	−	2173.45i	0.797452	−	1.38123i	−0.123818	−	0.992305i	\(-0.539514\pi\)
							0.921270	−	0.388923i	\(-0.127153\pi\)
\(23\)	−51.6075	+	688.654i	−0.0203420	+	0.271445i	0.977754	+	0.209757i	\(0.0672673\pi\)
							−0.998096	+	0.0616876i	\(0.980352\pi\)
\(29\)	−5080.93	−	2446.85i	−1.12188	−	0.540271i	−0.221410	−	0.975181i	\(-0.571066\pi\)
							−0.900474	+	0.434910i	\(0.856780\pi\)
\(31\)	3354.97	+	5810.98i	0.627025	+	1.08604i	0.988146	+	0.153520i	\(0.0490608\pi\)
							−0.361121	+	0.932519i	\(0.617606\pi\)
\(37\)	−5429.12	+	3701.51i	−0.651966	+	0.444503i	−0.843606	−	0.536963i	\(-0.819572\pi\)
							0.191640	+	0.981465i	\(0.438619\pi\)
\(41\)	−6181.61	−	7751.49i	−0.574304	−	0.720154i	0.406826	−	0.913506i	\(-0.366636\pi\)
							−0.981130	+	0.193351i	\(0.938064\pi\)
\(43\)	12687.6	−	15909.8i	1.04643	−	1.31218i	0.0979984	−	0.995187i	\(-0.468756\pi\)
							0.948428	−	0.316991i	\(-0.102673\pi\)
\(47\)	26412.8	+	8147.27i	1.74409	+	0.537981i	0.992565	−	0.121719i	\(-0.0388408\pi\)
							0.751528	+	0.659701i	\(0.229317\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.g.a.2.6	✓	264
49.25	even	21	inner	49.6.g.a.25.6	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.g.a.2.6	✓	264	1.1	even	1	trivial
49.6.g.a.25.6	yes	264	49.25	even	21	inner