Embedded newform 49.6.g.a.11.9

• Label: 49.6.g.a.11.9
• Level: $49$
• Weight: $6$
• Character: 49.11
• 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			11.9
Character	\(\chi\)	\(=\)	49.11
Dual form			49.6.g.a.9.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.64945 + 1.53047i) q^{2} +(0.953524 - 0.143721i) q^{3} +(-2.01300 + 26.8617i) q^{4} +(28.0594 - 71.4941i) q^{5} +(-1.35283 + 1.69640i) q^{6} +(123.655 + 38.9405i) q^{7} +(-82.6840 - 103.683i) q^{8} +(-231.316 + 71.3514i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 13 q^{2} - 22 q^{3} + 307 q^{4} - 52 q^{5} - 130 q^{6} + 154 q^{7} - 320 q^{8} + 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.64945	+	1.53047i	−0.291584	+	0.270551i	−0.812377	−	0.583132i	\(-0.801827\pi\)
							0.520793	+	0.853683i	\(0.325636\pi\)
\(3\)	0.953524	−	0.143721i	0.0611686	−	0.00921969i	−0.118387	−	0.992968i	\(-0.537772\pi\)
							0.179555	+	0.983748i	\(0.442534\pi\)
\(5\)	28.0594	−	71.4941i	0.501941	−	1.27892i	−0.425501	−	0.904958i	\(-0.639902\pi\)
							0.927442	−	0.373967i	\(-0.122003\pi\)
\(7\)	123.655	+	38.9405i	0.953823	+	0.300370i
							\(11\)	724.611	+	223.513i	1.80561	+	0.556956i	0.999399	−	0.0346774i	\(-0.0110404\pi\)
0.806207	+	0.591633i	\(0.201517\pi\)
\(13\)	154.899	+	678.658i	0.254209	+	1.11376i	0.927334	+	0.374234i	\(0.122094\pi\)
							−0.673125	+	0.739529i	\(0.735048\pi\)
\(17\)	1041.04	−	709.772i	0.873668	−	0.595657i	−0.0413160	−	0.999146i	\(-0.513155\pi\)
							0.914984	+	0.403489i	\(0.132203\pi\)
\(19\)	1158.26	+	2006.17i	0.736075	+	1.27492i	0.954250	+	0.299010i	\(0.0966563\pi\)
							−0.218175	+	0.975910i	\(0.570010\pi\)
\(23\)	755.592	+	515.154i	0.297830	+	0.203057i	0.703010	−	0.711180i	\(-0.251839\pi\)
							−0.405180	+	0.914237i	\(0.632791\pi\)
\(29\)	−4663.14	−	2245.65i	−1.02963	−	0.495846i	−0.158743	−	0.987320i	\(-0.550744\pi\)
							−0.870892	+	0.491474i	\(0.836458\pi\)
\(31\)	430.094	−	744.945i	0.0803821	−	0.139226i	−0.823032	−	0.567995i	\(-0.807719\pi\)
							0.903414	+	0.428769i	\(0.141053\pi\)
\(37\)	−736.621	−	9829.53i	−0.0884586	−	1.18040i	−0.848021	−	0.529963i	\(-0.822206\pi\)
							0.759562	−	0.650435i	\(-0.225413\pi\)
\(41\)	−8121.25	−	10183.7i	−0.754506	−	0.946121i	0.245221	−	0.969467i	\(-0.421139\pi\)
							−0.999727	+	0.0233460i	\(0.992568\pi\)
\(43\)	1350.68	−	1693.70i	0.111399	−	0.139690i	−0.723006	−	0.690842i	\(-0.757240\pi\)
							0.834405	+	0.551152i	\(0.185811\pi\)
\(47\)	5624.57	−	5218.84i	0.371402	−	0.344611i	−0.472285	−	0.881446i	\(-0.656571\pi\)
							0.843687	+	0.536835i	\(0.180380\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.g.a.11.9	yes	264
49.9	even	21	inner	49.6.g.a.9.9	✓	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.g.a.9.9	✓	264	49.9	even	21	inner
49.6.g.a.11.9	yes	264	1.1	even	1	trivial