Embedded newform 49.6.g.a.11.3

• Label: 49.6.g.a.11.3
• Level: $49$
• Weight: $6$
• Character: 49.11
• Analytic conductor: $7.859$
• Analytic rank: $0$
• Dimension: $264$
• 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			11.3
Character	\(\chi\)	\(=\)	49.11
Dual form			49.6.g.a.9.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.71729 + 6.23273i) q^{2} +(-0.814088 + 0.122704i) q^{3} +(3.88364 - 51.8236i) q^{4} +(-38.8287 + 98.9340i) q^{5} +(4.70368 - 5.89823i) q^{6} +(-59.4337 - 115.216i) q^{7} +(114.089 + 143.063i) q^{8} +(-231.557 + 71.4257i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 13 q^{2} - 22 q^{3} + 307 q^{4} - 52 q^{5} - 130 q^{6} + 154 q^{7} - 320 q^{8} + 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.71729	+	6.23273i	−1.18746	+	1.10180i	−0.194802	+	0.980843i	\(0.562407\pi\)
							−0.992657	+	0.120959i	\(0.961403\pi\)
\(3\)	−0.814088	+	0.122704i	−0.0522237	+	0.00787146i	−0.175102	−	0.984550i	\(-0.556026\pi\)
							0.122878	+	0.992422i	\(0.460787\pi\)
\(5\)	−38.8287	+	98.9340i	−0.694590	+	1.76979i	−0.0583466	+	0.998296i	\(0.518583\pi\)
							−0.636243	+	0.771489i	\(0.719512\pi\)
\(7\)	−59.4337	−	115.216i	−0.458445	−	0.888723i
							\(11\)	214.800	+	66.2572i	0.535246	+	0.165102i	0.550582	−	0.834781i	\(-0.314406\pi\)
−0.0153363	+	0.999882i	\(0.504882\pi\)
\(13\)	5.08225	+	22.2668i	0.00834061	+	0.0365426i	0.978927	−	0.204211i	\(-0.0654627\pi\)
							−0.970586	+	0.240753i	\(0.922606\pi\)
\(17\)	1505.64	−	1026.53i	1.26357	−	0.861489i	0.268758	−	0.963208i	\(-0.413387\pi\)
							0.994813	+	0.101719i	\(0.0324343\pi\)
\(19\)	−10.9654	−	18.9926i	−0.00696851	−	0.0120698i	0.862520	−	0.506023i	\(-0.168885\pi\)
							−0.869489	+	0.493953i	\(0.835551\pi\)
\(23\)	600.703	+	409.553i	0.236778	+	0.161432i	0.675896	−	0.736997i	\(-0.263757\pi\)
							−0.439119	+	0.898429i	\(0.644709\pi\)
\(29\)	−5066.16	−	2439.74i	−1.11862	−	0.538701i	−0.219157	−	0.975690i	\(-0.570331\pi\)
							−0.899467	+	0.436989i	\(0.856045\pi\)
\(31\)	1887.51	−	3269.26i	0.352765	−	0.611006i	−0.633968	−	0.773359i	\(-0.718575\pi\)
							0.986733	+	0.162353i	\(0.0519083\pi\)
\(37\)	199.811	+	2666.29i	0.0239947	+	0.320187i	0.996266	+	0.0863328i	\(0.0275148\pi\)
							−0.972272	+	0.233854i	\(0.924866\pi\)
\(41\)	−8930.14	−	11198.0i	−0.829657	−	1.04036i	−0.998502	−	0.0547143i	\(-0.982575\pi\)
							0.168845	−	0.985643i	\(-0.445996\pi\)
\(43\)	6354.38	−	7968.14i	0.524086	−	0.657182i	−0.447385	−	0.894341i	\(-0.647645\pi\)
							0.971471	+	0.237159i	\(0.0762162\pi\)
\(47\)	−18662.5	+	17316.3i	−1.23233	+	1.14343i	−0.247685	+	0.968841i	\(0.579670\pi\)
							−0.984641	+	0.174591i	\(0.944140\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.g.a.11.3	yes	264
49.9	even	21	inner	49.6.g.a.9.3	✓	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.g.a.9.3	✓	264	49.9	even	21	inner
49.6.g.a.11.3	yes	264	1.1	even	1	trivial