Embedded newform 441.2.bn.a.5.2

• Label: 441.2.bn.a.5.2
• Level: $441$
• Weight: $2$
• Character: 441.5
• 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			5.2
Character	\(\chi\)	\(=\)	441.5
Dual form			441.2.bn.a.353.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.15589 + 1.71927i) q^{2} +(-1.44111 - 0.960829i) q^{3} +(1.24695 - 5.46324i) q^{4} +(-1.16954 - 0.797380i) q^{5} +(4.75880 - 0.406215i) q^{6} +(1.71430 - 2.01524i) q^{7} +(4.31162 + 8.95316i) q^{8} +(1.15362 + 2.76933i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 21 q^{2} - 11 q^{3} + 99 q^{4} - 18 q^{5} - 34 q^{6} - 5 q^{7} - 23 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{29}{42}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.15589	+	1.71927i	−1.52444	+	1.21570i	−0.623459	+	0.781856i	\(0.714273\pi\)
							−0.900986	+	0.433848i	\(0.857156\pi\)
\(3\)	−1.44111	−	0.960829i	−0.832027	−	0.554735i
							\(5\)	−1.16954	−	0.797380i	−0.523035	−	0.356599i	0.272841	−	0.962059i	\(-0.412037\pi\)
−0.795876	+	0.605460i	\(0.792989\pi\)
\(7\)	1.71430	−	2.01524i	0.647945	−	0.761688i
							\(11\)	0.0494234	+	0.327903i	0.0149017	+	0.0988663i	0.995045	−	0.0994227i	\(-0.0316996\pi\)
−0.980144	+	0.198289i	\(0.936462\pi\)
\(13\)	−0.0685054	−	0.454504i	−0.0190000	−	0.126057i	0.977320	−	0.211767i	\(-0.0679220\pi\)
							−0.996320	+	0.0857109i	\(0.972684\pi\)
\(17\)	1.77059	−	1.64287i	0.429432	−	0.398455i	−0.435625	−	0.900128i	\(-0.643473\pi\)
							0.865057	+	0.501674i	\(0.167282\pi\)
\(19\)	4.18608	+	2.41683i	0.960352	+	0.554460i	0.896281	−	0.443486i	\(-0.146258\pi\)
							0.0640709	+	0.997945i	\(0.479592\pi\)
\(23\)	−2.02294	−	6.55820i	−0.421811	−	1.36748i	−0.878453	−	0.477829i	\(-0.841424\pi\)
							0.456642	−	0.889651i	\(-0.349052\pi\)
\(29\)	−1.43127	−	1.54255i	−0.265781	−	0.286443i	0.586034	−	0.810286i	\(-0.300688\pi\)
							−0.851815	+	0.523843i	\(0.824498\pi\)
\(31\)			8.28618i			1.48824i	0.668045	+	0.744121i	\(0.267131\pi\)
							−0.668045	+	0.744121i	\(0.732869\pi\)
\(37\)	−8.11144	−	2.50205i	−1.33351	−	0.411334i	−0.455613	−	0.890178i	\(-0.650580\pi\)
							−0.877900	+	0.478844i	\(0.841056\pi\)
\(41\)	−0.100337	+	1.33890i	−0.0156700	+	0.209102i	0.983873	+	0.178869i	\(0.0572438\pi\)
							−0.999543	+	0.0302326i	\(0.990375\pi\)
\(43\)	−0.468804	−	6.25575i	−0.0714919	−	0.953993i	−0.911468	−	0.411370i	\(-0.865050\pi\)
							0.839976	−	0.542623i	\(-0.182569\pi\)
\(47\)	5.40906	+	6.78275i	0.788993	+	0.989366i	0.999930	+	0.0118662i	\(0.00377723\pi\)
							−0.210937	+	0.977500i	\(0.567651\pi\)