Embedded newform 441.2.bn.a.131.18

• Label: 441.2.bn.a.131.18
• Level: $441$
• Weight: $2$
• Character: 441.131
• 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			131.18
Character	\(\chi\)	\(=\)	441.131
Dual form			441.2.bn.a.101.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00554 - 0.801893i) q^{2} +(-1.06595 - 1.36519i) q^{3} +(-0.0769596 - 0.337182i) q^{4} +(0.271847 + 3.62755i) q^{5} +(-0.0228859 + 2.22754i) q^{6} +(1.01646 - 2.44270i) q^{7} +(-1.30907 + 2.71830i) q^{8} +(-0.727515 + 2.91045i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 21 q^{2} - 11 q^{3} + 99 q^{4} - 18 q^{5} - 34 q^{6} - 5 q^{7} - 23 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{41}{42}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00554	−	0.801893i	−0.711025	−	0.567024i	0.199789	−	0.979839i	\(-0.435974\pi\)
							−0.910815	+	0.412815i	\(0.864546\pi\)
\(3\)	−1.06595	−	1.36519i	−0.615425	−	0.788196i
							\(5\)	0.271847	+	3.62755i	0.121574	+	1.62229i	0.640839	+	0.767676i	\(0.278587\pi\)
−0.519265	+	0.854613i	\(0.673794\pi\)
\(7\)	1.01646	−	2.44270i	0.384187	−	0.923255i
							\(11\)	3.96212	−	1.55502i	1.19462	−	0.468855i	0.317102	−	0.948392i	\(-0.397290\pi\)
0.877522	+	0.479536i	\(0.159195\pi\)
\(13\)	−0.639715	+	0.251070i	−0.177425	+	0.0696342i	−0.452391	−	0.891820i	\(-0.649429\pi\)
							0.274966	+	0.961454i	\(0.411334\pi\)
\(17\)	2.80364	−	0.864808i	0.679982	−	0.209747i	0.0645237	−	0.997916i	\(-0.479447\pi\)
							0.615458	+	0.788170i	\(0.288971\pi\)
\(19\)	5.82672	+	3.36406i	1.33674	+	0.771768i	0.986323	−	0.164826i	\(-0.0527062\pi\)
							0.350418	+	0.936593i	\(0.386040\pi\)
\(23\)	−3.05503	−	3.29254i	−0.637017	−	0.686541i	0.329203	−	0.944259i	\(-0.393220\pi\)
							−0.966220	+	0.257718i	\(0.917030\pi\)
\(29\)	0.899882	+	2.91735i	0.167104	+	0.541738i	0.999930	−	0.0117966i	\(-0.00375505\pi\)
							−0.832827	+	0.553534i	\(0.813279\pi\)
\(31\)		−	7.99939i		−	1.43673i	−0.695665	−	0.718367i	\(-0.744890\pi\)
							0.695665	−	0.718367i	\(-0.255110\pi\)
\(37\)	−5.70667	−	5.29502i	−0.938171	−	0.870495i	0.0535948	−	0.998563i	\(-0.482932\pi\)
							−0.991765	+	0.128068i	\(0.959123\pi\)
\(41\)	7.09899	−	4.84001i	1.10868	−	0.755883i	0.136618	−	0.990624i	\(-0.456377\pi\)
							0.972058	+	0.234741i	\(0.0754242\pi\)
\(43\)	1.74253	+	1.18803i	0.265733	+	0.181173i	0.688855	−	0.724899i	\(-0.258114\pi\)
							−0.423122	+	0.906073i	\(0.639066\pi\)
\(47\)	−1.23674	+	1.55082i	−0.180397	+	0.226211i	−0.863805	−	0.503826i	\(-0.831925\pi\)
							0.683408	+	0.730036i	\(0.260497\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bn.a.131.18	yes	648
9.2	odd	6		441.2.bd.a.425.37	yes	648
49.3	odd	42		441.2.bd.a.248.37	✓	648
441.101	even	42	inner	441.2.bn.a.101.18	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.248.37	✓	648	49.3	odd	42
441.2.bd.a.425.37	yes	648	9.2	odd	6
441.2.bn.a.101.18	yes	648	441.101	even	42	inner
441.2.bn.a.131.18	yes	648	1.1	even	1	trivial