Embedded newform 441.3.bm.a.11.10

• Label: 441.3.bm.a.11.10
• Level: $441$
• Weight: $3$
• Character: 441.11
• Analytic conductor: $12.016$
• Analytic rank: $0$
• Dimension: $1320$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(12.0163796583\)
Analytic rank:	\(0\)
Dimension:	\(1320\)
Relative dimension:	\(110\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			11.10
Character	\(\chi\)	\(=\)	441.11
Dual form			441.3.bm.a.401.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.37360 + 0.770003i) q^{2} +(-0.382520 - 2.97551i) q^{3} +(7.18442 - 3.45983i) q^{4} +(-0.509050 - 3.37733i) q^{5} +(3.58162 + 9.74366i) q^{6} +(4.00587 - 5.74047i) q^{7} +(-10.7516 + 8.57412i) q^{8} +(-8.70736 + 2.27638i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1320 q - 21 q^{2} - 13 q^{3} + 427 q^{4} - 18 q^{5} - 34 q^{6} - 5 q^{7} + 27 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{20}{21}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.37360	+	0.770003i	−1.68680	+	0.385001i	−0.955021	−	0.296539i	\(-0.904168\pi\)
							−0.731781	+	0.681540i	\(0.761310\pi\)
\(3\)	−0.382520	−	2.97551i	−0.127507	−	0.991838i
							\(5\)	−0.509050	−	3.37733i	−0.101810	−	0.675466i	−0.980272	−	0.197653i	\(-0.936668\pi\)
0.878462	−	0.477812i	\(-0.158570\pi\)
\(7\)	4.00587	−	5.74047i	0.572267	−	0.820067i
							\(11\)	−9.69576	−	10.4495i	−0.881433	−	0.949959i	0.117638	−	0.993057i	\(-0.462468\pi\)
−0.999071	+	0.0430978i	\(0.986277\pi\)
\(13\)	4.41578	−	4.09724i	0.339675	−	0.315172i	−0.491828	−	0.870692i	\(-0.663671\pi\)
							0.831503	+	0.555520i	\(0.187481\pi\)
\(17\)	−5.27285	−	7.73385i	−0.310167	−	0.454932i	0.639227	−	0.769018i	\(-0.279254\pi\)
							−0.949395	+	0.314086i	\(0.898302\pi\)
\(19\)	−12.2660	−	21.2453i	−0.645578	−	1.11817i	−0.984168	−	0.177239i	\(-0.943283\pi\)
							0.338590	−	0.940934i	\(-0.390050\pi\)
\(23\)	4.41753	+	0.331048i	0.192067	+	0.0143934i	0.170416	−	0.985372i	\(-0.445489\pi\)
							0.0216505	+	0.999766i	\(0.493108\pi\)
\(29\)	−2.80339	−	4.11181i	−0.0966685	−	0.141787i	0.774858	−	0.632136i	\(-0.217822\pi\)
							−0.871526	+	0.490349i	\(0.836869\pi\)
\(31\)	34.7590			1.12126			0.560629	−	0.828067i	\(-0.310560\pi\)
							0.560629	+	0.828067i	\(0.310560\pi\)
\(37\)	−0.446675	−	5.96047i	−0.0120723	−	0.161094i	−0.999982	−	0.00606098i	\(-0.998071\pi\)
							0.987909	−	0.155033i	\(-0.0495483\pi\)
\(41\)	−6.99125	−	2.74387i	−0.170518	−	0.0669235i	0.278546	−	0.960423i	\(-0.410147\pi\)
							−0.449065	+	0.893499i	\(0.648243\pi\)
\(43\)	13.4750	+	34.3338i	0.313373	+	0.798461i	0.997603	+	0.0691951i	\(0.0220431\pi\)
							−0.684230	+	0.729266i	\(0.739862\pi\)
\(47\)	10.2212	−	2.33291i	0.217471	−	0.0496364i	−0.112397	−	0.993663i	\(-0.535853\pi\)
							0.329868	+	0.944027i	\(0.392996\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.3.bm.a.11.10	yes	1320
9.5	odd	6		441.3.bi.a.158.10	✓	1320
49.9	even	21		441.3.bi.a.254.10	yes	1320
441.401	odd	42	inner	441.3.bm.a.401.10	yes	1320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.3.bi.a.158.10	✓	1320	9.5	odd	6
441.3.bi.a.254.10	yes	1320	49.9	even	21
441.3.bm.a.11.10	yes	1320	1.1	even	1	trivial
441.3.bm.a.401.10	yes	1320	441.401	odd	42	inner