Embedded newform 441.2.bh.a.398.9

• Label: 441.2.bh.a.398.9
• Level: $441$
• Weight: $2$
• Character: 441.398
• Analytic conductor: $3.521$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.bh (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			398.9
Character	\(\chi\)	\(=\)	441.398
Dual form			441.2.bh.a.41.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.96663 + 0.771846i) q^{2} +(1.72620 + 0.142302i) q^{3} +(1.80579 - 1.67552i) q^{4} +(-0.0783338 - 1.04529i) q^{5} +(-3.50462 + 1.05250i) q^{6} +(-1.26281 + 2.32493i) q^{7} +(-0.424760 + 0.882023i) q^{8} +(2.95950 + 0.491281i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 15 q^{2} - 14 q^{3} - 57 q^{4} - 21 q^{5} + 14 q^{6} - 5 q^{7} - 20 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.96663	+	0.771846i	−1.39062	+	0.545777i	−0.938220	−	0.346039i	\(-0.887526\pi\)
							−0.452398	+	0.891816i	\(0.649431\pi\)
\(3\)	1.72620	+	0.142302i	0.996619	+	0.0821579i
							\(5\)	−0.0783338	−	1.04529i	−0.0350319	−	0.467469i	−0.986940	−	0.161086i	\(-0.948500\pi\)
0.951908	−	0.306383i	\(-0.0991187\pi\)
\(7\)	−1.26281	+	2.32493i	−0.477296	+	0.878743i
							\(11\)	−4.41399	+	1.73236i	−1.33087	+	0.522327i	−0.920909	−	0.389777i	\(-0.872552\pi\)
−0.409959	+	0.912104i	\(0.634457\pi\)
\(13\)	−0.288100	+	1.91142i	−0.0799045	+	0.530132i	0.912690	+	0.408654i	\(0.134002\pi\)
							−0.992594	+	0.121478i	\(0.961237\pi\)
\(17\)	0.344193	−	1.50801i	0.0834790	−	0.365745i	−0.915884	−	0.401444i	\(-0.868508\pi\)
							0.999363	+	0.0356986i	\(0.0113656\pi\)
\(19\)			7.78644i			1.78633i	0.449728	+	0.893165i	\(0.351521\pi\)
							−0.449728	+	0.893165i	\(0.648479\pi\)
\(23\)	3.82214	+	4.11929i	0.796971	+	0.858931i	0.992560	−	0.121755i	\(-0.0388522\pi\)
							−0.195589	+	0.980686i	\(0.562662\pi\)
\(29\)	−0.847787	+	0.913697i	−0.157430	+	0.169669i	−0.806842	−	0.590768i	\(-0.798825\pi\)
							0.649411	+	0.760437i	\(0.275015\pi\)
\(31\)	3.80980	−	2.19959i	0.684260	−	0.395058i	−0.117198	−	0.993109i	\(-0.537391\pi\)
							0.801458	+	0.598051i	\(0.204058\pi\)
\(37\)	0.460052	−	2.01562i	0.0756321	−	0.331366i	−0.922930	−	0.384967i	\(-0.874213\pi\)
							0.998562	+	0.0536011i	\(0.0170700\pi\)
\(41\)	0.0882905	+	1.17816i	0.0137887	+	0.183997i	0.999845	+	0.0175871i	\(0.00559843\pi\)
							−0.986057	+	0.166410i	\(0.946783\pi\)
\(43\)	−0.576134	+	7.68797i	−0.0878596	+	1.17240i	0.762781	+	0.646657i	\(0.223833\pi\)
							−0.850641	+	0.525748i	\(0.823786\pi\)
\(47\)	2.75006	+	7.00703i	0.401137	+	1.02208i	0.978620	+	0.205678i	\(0.0659398\pi\)
							−0.577483	+	0.816403i	\(0.695965\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bh.a.398.9	yes	648
9.5	odd	6	inner	441.2.bh.a.104.9	yes	648
49.41	odd	14	inner	441.2.bh.a.335.9	yes	648
441.41	even	42	inner	441.2.bh.a.41.9	✓	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bh.a.41.9	✓	648	441.41	even	42	inner
441.2.bh.a.104.9	yes	648	9.5	odd	6	inner
441.2.bh.a.335.9	yes	648	49.41	odd	14	inner
441.2.bh.a.398.9	yes	648	1.1	even	1	trivial