Embedded newform 441.2.bn.a.131.8

• Label: 441.2.bn.a.131.8
• 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			131.8
Character	\(\chi\)	\(=\)	441.131
Dual form			441.2.bn.a.101.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.71887 - 1.37075i) q^{2} +(0.978038 + 1.42949i) q^{3} +(0.630511 + 2.76245i) q^{4} +(0.322472 + 4.30309i) q^{5} +(0.278359 - 3.79776i) q^{6} +(-2.62405 - 0.338191i) q^{7} +(0.795066 - 1.65097i) q^{8} +(-1.08688 + 2.79619i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 21 q^{2} - 11 q^{3} + 99 q^{4} - 18 q^{5} - 34 q^{6} - 5 q^{7} - 23 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.71887	−	1.37075i	−1.21543	−	0.969270i	−0.215452	−	0.976514i	\(-0.569123\pi\)
							−0.999974	+	0.00724452i	\(0.997694\pi\)
\(3\)	0.978038	+	1.42949i	0.564670	+	0.825317i
							\(5\)	0.322472	+	4.30309i	0.144214	+	1.92440i	0.333998	+	0.942574i	\(0.391602\pi\)
−0.189784	+	0.981826i	\(0.560779\pi\)
\(7\)	−2.62405	−	0.338191i	−0.991797	−	0.127824i
							\(11\)	1.29551	−	0.508449i	0.390610	−	0.153303i	−0.161904	−	0.986806i	\(-0.551764\pi\)
0.552514	+	0.833503i	\(0.313668\pi\)
\(13\)	−1.64005	+	0.643671i	−0.454867	+	0.178522i	−0.581703	−	0.813402i	\(-0.697613\pi\)
							0.126835	+	0.991924i	\(0.459518\pi\)
\(17\)	3.40897	−	1.05153i	0.826797	−	0.255033i	0.147654	−	0.989039i	\(-0.452828\pi\)
							0.679143	+	0.734006i	\(0.262352\pi\)
\(19\)	−6.82324	−	3.93940i	−1.56536	−	0.903760i	−0.996699	−	0.0811877i	\(-0.974129\pi\)
							−0.568660	−	0.822573i	\(-0.692538\pi\)
\(23\)	0.105702	+	0.113919i	0.0220403	+	0.0237538i	0.743976	−	0.668207i	\(-0.232938\pi\)
							−0.721935	+	0.691960i	\(0.756747\pi\)
\(29\)	0.317460	+	1.02918i	0.0589509	+	0.191114i	0.980258	−	0.197721i	\(-0.0633539\pi\)
							−0.921308	+	0.388835i	\(0.872878\pi\)
\(31\)		−	6.23204i		−	1.11931i	−0.828726	−	0.559654i	\(-0.810934\pi\)
							0.828726	−	0.559654i	\(-0.189066\pi\)
\(37\)	5.31954	+	4.93581i	0.874527	+	0.811442i	0.983180	−	0.182638i	\(-0.0584635\pi\)
							−0.108654	+	0.994080i	\(0.534654\pi\)
\(41\)	−2.63157	+	1.79417i	−0.410982	+	0.280203i	−0.751111	−	0.660176i	\(-0.770482\pi\)
							0.340129	+	0.940379i	\(0.389529\pi\)
\(43\)	3.44060	+	2.34576i	0.524687	+	0.357726i	0.796514	−	0.604621i	\(-0.206675\pi\)
							−0.271826	+	0.962346i	\(0.587628\pi\)
\(47\)	−4.44060	+	5.56833i	−0.647728	+	0.812225i	−0.991946	−	0.126665i	\(-0.959573\pi\)
							0.344218	+	0.938890i	\(0.388144\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bn.a.131.8	yes	648
9.2	odd	6		441.2.bd.a.425.47	yes	648
49.3	odd	42		441.2.bd.a.248.47	✓	648
441.101	even	42	inner	441.2.bn.a.101.8	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.248.47	✓	648	49.3	odd	42
441.2.bd.a.425.47	yes	648	9.2	odd	6
441.2.bn.a.101.8	yes	648	441.101	even	42	inner
441.2.bn.a.131.8	yes	648	1.1	even	1	trivial