Embedded newform 441.2.bn.a.416.1

• Label: 441.2.bn.a.416.1
• Level: $441$
• Weight: $2$
• Character: 441.416
• Analytic conductor: $3.521$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.bn (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.52140272914\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(54\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			416.1
Character	\(\chi\)	\(=\)	441.416
Dual form			441.2.bn.a.194.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.63066 - 0.600431i) q^{2} +(-0.946922 - 1.45029i) q^{3} +(4.75791 + 2.29129i) q^{4} +(0.479238 + 1.22108i) q^{5} +(1.62023 + 4.38378i) q^{6} +(-2.31702 + 1.27726i) q^{7} +(-6.92143 - 5.51966i) q^{8} +(-1.20668 + 2.74662i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 21 q^{2} - 11 q^{3} + 99 q^{4} - 18 q^{5} - 34 q^{6} - 5 q^{7} - 23 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{37}{42}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.63066	−	0.600431i	−1.86016	−	0.424569i	−0.863363	−	0.504583i	\(-0.831646\pi\)
							−0.996794	+	0.0800148i	\(0.974503\pi\)
\(3\)	−0.946922	−	1.45029i	−0.546706	−	0.837325i
							\(5\)	0.479238	+	1.22108i	0.214322	+	0.546083i	0.997090	−	0.0762301i	\(-0.0242884\pi\)
−0.782769	+	0.622313i	\(0.786193\pi\)
\(7\)	−2.31702	+	1.27726i	−0.875753	+	0.482760i
							\(11\)	−1.52865	−	4.95575i	−0.460904	−	1.49421i	−0.826229	−	0.563334i	\(-0.809518\pi\)
0.365325	−	0.930880i	\(-0.380958\pi\)
\(13\)	0.879664	+	2.85180i	0.243975	+	0.790947i	0.991884	+	0.127146i	\(0.0405817\pi\)
							−0.747909	+	0.663801i	\(0.768942\pi\)
\(17\)	−0.284048	+	3.79036i	−0.0688918	+	0.919297i	0.850505	+	0.525966i	\(0.176296\pi\)
							−0.919397	+	0.393330i	\(0.871323\pi\)
\(19\)	3.82107	−	2.20610i	0.876614	−	0.506113i	0.00707349	−	0.999975i	\(-0.497748\pi\)
							0.869540	+	0.493862i	\(0.164415\pi\)
\(23\)	−1.55787	−	2.28498i	−0.324839	−	0.476451i	0.628731	−	0.777623i	\(-0.283575\pi\)
							−0.953570	+	0.301172i	\(0.902622\pi\)
\(29\)	7.96142	+	0.596626i	1.47840	+	0.110791i	0.789358	−	0.613933i	\(-0.210413\pi\)
							0.689040	+	0.724723i	\(0.258032\pi\)
\(31\)		−	3.44251i		−	0.618294i	−0.951014	−	0.309147i	\(-0.899957\pi\)
							0.951014	−	0.309147i	\(-0.100043\pi\)
\(37\)	−4.53317	−	3.09067i	−0.745249	−	0.508102i	0.130155	−	0.991494i	\(-0.458452\pi\)
							−0.875404	+	0.483391i	\(0.839405\pi\)
\(41\)	1.12654	+	0.169799i	0.175937	+	0.0265182i	0.236420	−	0.971651i	\(-0.424026\pi\)
							−0.0604830	+	0.998169i	\(0.519264\pi\)
\(43\)	7.72095	−	1.16375i	1.17743	−	0.177470i	0.468969	−	0.883215i	\(-0.344626\pi\)
							0.708466	+	0.705745i	\(0.249388\pi\)
\(47\)	1.80850	−	7.92354i	0.263796	−	1.15577i	−0.653299	−	0.757100i	\(-0.726616\pi\)
							0.917096	−	0.398667i	\(-0.130527\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bn.a.416.1	yes	648
9.5	odd	6		441.2.bd.a.122.1	yes	648
49.47	odd	42		441.2.bd.a.47.1	✓	648
441.194	even	42	inner	441.2.bn.a.194.1	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.47.1	✓	648	49.47	odd	42
441.2.bd.a.122.1	yes	648	9.5	odd	6
441.2.bn.a.194.1	yes	648	441.194	even	42	inner
441.2.bn.a.416.1	yes	648	1.1	even	1	trivial