Embedded newform 441.2.bn.a.353.2

• Label: 441.2.bn.a.353.2
• Level: $441$
• Weight: $2$
• Character: 441.353
• 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			353.2
Character	\(\chi\)	\(=\)	441.353
Dual form			441.2.bn.a.5.2

$q$-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 \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{13}{42}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

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

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bn.a.353.2	yes	648
9.5	odd	6		441.2.bd.a.59.2	✓	648
49.5	odd	42		441.2.bd.a.299.2	yes	648
441.5	even	42	inner	441.2.bn.a.5.2	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.59.2	✓	648	9.5	odd	6
441.2.bd.a.299.2	yes	648	49.5	odd	42
441.2.bn.a.5.2	yes	648	441.5	even	42	inner
441.2.bn.a.353.2	yes	648	1.1	even	1	trivial