Embedded newform 49.6.g.a.2.15

• Label: 49.6.g.a.2.15
• 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			2.15
Character	\(\chi\)	\(=\)	49.2
Dual form			49.6.g.a.25.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.24738 + 1.31014i) q^{2} +(-9.79667 + 24.9615i) q^{3} +(-10.1159 - 6.89687i) q^{4} +(-27.7351 + 4.18039i) q^{5} +(-74.3134 + 93.1860i) q^{6} +(129.357 - 8.58801i) q^{7} +(-122.612 - 153.751i) q^{8} +(-348.971 - 323.797i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 13 q^{2} - 22 q^{3} + 307 q^{4} - 52 q^{5} - 130 q^{6} + 154 q^{7} - 320 q^{8} + 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.24738	+	1.31014i	0.750838	+	0.231603i	0.646474	−	0.762936i	\(-0.276243\pi\)
							0.104365	+	0.994539i	\(0.466719\pi\)
\(3\)	−9.79667	+	24.9615i	−0.628457	+	1.60128i	0.160640	+	0.987013i	\(0.448644\pi\)
							−0.789097	+	0.614269i	\(0.789451\pi\)
\(5\)	−27.7351	+	4.18039i	−0.496141	+	0.0747812i	−0.392346	−	0.919818i	\(-0.628336\pi\)
							−0.103794	+	0.994599i	\(0.533098\pi\)
\(7\)	129.357	−	8.58801i	0.997803	−	0.0662441i
							\(11\)	−488.818	+	453.557i	−1.21805	+	1.13019i	−0.230470	+	0.973079i	\(0.574026\pi\)
−0.987581	+	0.157108i	\(0.949783\pi\)
\(13\)	−152.165	−	666.680i	−0.249722	−	1.09411i	−0.931842	−	0.362865i	\(-0.881799\pi\)
							0.682119	−	0.731241i	\(-0.261058\pi\)
\(17\)	46.2115	+	616.649i	0.0387818	+	0.517507i	0.982439	+	0.186583i	\(0.0597413\pi\)
							−0.943657	+	0.330924i	\(0.892640\pi\)
\(19\)	−125.282	+	216.995i	−0.0796169	+	0.137900i	−0.903085	−	0.429463i	\(-0.858703\pi\)
							0.823468	+	0.567363i	\(0.192036\pi\)
\(23\)	−320.725	+	4279.78i	−0.126419	+	1.68695i	0.468736	+	0.883338i	\(0.344710\pi\)
							−0.595155	+	0.803611i	\(0.702909\pi\)
\(29\)	−4174.16	−	2010.17i	−0.921667	−	0.443851i	−0.0880010	−	0.996120i	\(-0.528048\pi\)
							−0.833666	+	0.552269i	\(0.813762\pi\)
\(31\)	3784.94	+	6555.70i	0.707382	+	1.22522i	0.965825	+	0.259196i	\(0.0834574\pi\)
							−0.258442	+	0.966027i	\(0.583209\pi\)
\(37\)	5149.15	−	3510.63i	0.618345	−	0.421581i	−0.213223	−	0.977004i	\(-0.568396\pi\)
							0.831568	+	0.555423i	\(0.187444\pi\)
\(41\)	2473.50	+	3101.67i	0.229801	+	0.288162i	0.883341	−	0.468730i	\(-0.155288\pi\)
							−0.653540	+	0.756892i	\(0.726717\pi\)
\(43\)	1549.91	−	1943.53i	0.127831	−	0.160295i	−0.713797	−	0.700352i	\(-0.753026\pi\)
							0.841628	+	0.540058i	\(0.181598\pi\)
\(47\)	−20755.7	−	6402.29i	−1.37054	−	0.422757i	−0.479847	−	0.877352i	\(-0.659308\pi\)
							−0.890698	+	0.454595i	\(0.849784\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.g.a.2.15	✓	264
49.25	even	21	inner	49.6.g.a.25.15	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.g.a.2.15	✓	264	1.1	even	1	trivial
49.6.g.a.25.15	yes	264	49.25	even	21	inner