Embedded newform 441.2.y.a.25.5

• Label: 441.2.y.a.25.5
• Level: $441$
• Weight: $2$
• Character: 441.25
• 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.y (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.52140272914\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(54\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			25.5
Character	\(\chi\)	\(=\)	441.25
Dual form			441.2.y.a.247.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.524991 + 2.30014i) q^{2} +(-1.18605 - 1.26225i) q^{3} +(-3.21307 - 1.54733i) q^{4} +(1.00605 + 2.56336i) q^{5} +(3.52602 - 2.06541i) q^{6} +(-1.26982 - 2.32111i) q^{7} +(2.30393 - 2.88904i) q^{8} +(-0.186564 + 2.99419i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{8}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.524991	+	2.30014i	−0.371225	+	1.62644i	0.352120	+	0.935955i	\(0.385461\pi\)
							−0.723345	+	0.690487i	\(0.757396\pi\)
\(3\)	−1.18605	−	1.26225i	−0.684767	−	0.728762i
							\(5\)	1.00605	+	2.56336i	0.449918	+	1.14637i	0.958433	+	0.285318i	\(0.0920993\pi\)
−0.508515	+	0.861053i	\(0.669805\pi\)
\(7\)	−1.26982	−	2.32111i	−0.479948	−	0.877297i
							\(11\)	−2.21604	+	0.683558i	−0.668161	+	0.206100i	−0.610235	−	0.792221i	\(-0.708925\pi\)
−0.0579260	+	0.998321i	\(0.518449\pi\)
\(13\)	−2.11529	+	0.652480i	−0.586675	+	0.180965i	−0.573861	−	0.818953i	\(-0.694555\pi\)
							−0.0128139	+	0.999918i	\(0.504079\pi\)
\(17\)	0.332952	−	4.44293i	0.0807527	−	1.07757i	−0.798746	−	0.601668i	\(-0.794503\pi\)
							0.879499	−	0.475901i	\(-0.157878\pi\)
\(19\)	−2.68692	−	4.65389i	−0.616423	−	1.06768i	−0.990133	−	0.140130i	\(-0.955248\pi\)
							0.373710	−	0.927545i	\(-0.378085\pi\)
\(23\)	−0.735719	+	0.501605i	−0.153408	+	0.104592i	−0.637591	−	0.770375i	\(-0.720069\pi\)
							0.484183	+	0.874967i	\(0.339117\pi\)
\(29\)	−0.0140351	+	0.187285i	−0.00260625	+	0.0347780i	−0.998355	−	0.0573298i	\(-0.981741\pi\)
							0.995749	+	0.0921077i	\(0.0293604\pi\)
\(31\)	−0.153013			−0.0274819			−0.0137409	−	0.999906i	\(-0.504374\pi\)
							−0.0137409	+	0.999906i	\(0.504374\pi\)
\(37\)	−4.68812	−	3.19630i	−0.770722	−	0.525469i	0.112936	−	0.993602i	\(-0.463974\pi\)
							−0.883658	+	0.468133i	\(0.844927\pi\)
\(41\)	−9.19151	−	1.38540i	−1.43547	−	0.216363i	−0.615216	−	0.788359i	\(-0.710931\pi\)
							−0.820256	+	0.571996i	\(0.806169\pi\)
\(43\)	7.86054	−	1.18479i	1.19872	−	0.180678i	0.480818	−	0.876821i	\(-0.340340\pi\)
							0.717904	+	0.696143i	\(0.245102\pi\)
\(47\)	2.17566	−	9.53218i	0.317352	−	1.39041i	−0.524825	−	0.851210i	\(-0.675869\pi\)
							0.842177	−	0.539201i	\(-0.181274\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.25.5	✓	648
9.4	even	3		441.2.z.a.319.5	yes	648
49.2	even	21		441.2.z.a.394.5	yes	648
441.247	even	21	inner	441.2.y.a.247.5	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.25.5	✓	648	1.1	even	1	trivial
441.2.y.a.247.5	yes	648	441.247	even	21	inner
441.2.z.a.319.5	yes	648	9.4	even	3
441.2.z.a.394.5	yes	648	49.2	even	21