Embedded newform 441.2.ba.a.106.9

• Label: 441.2.ba.a.106.9
• Level: $441$
• Weight: $2$
• Character: 441.106
• 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.ba (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			106.9
Character	\(\chi\)	\(=\)	441.106
Dual form			441.2.ba.a.337.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.12192 + 0.319828i) q^{2} +(1.21433 - 1.23508i) q^{3} +(2.48910 - 0.767787i) q^{4} +(3.12995 + 2.13396i) q^{5} +(-2.18169 + 3.00910i) q^{6} +(1.89973 - 1.84147i) q^{7} +(-1.16936 + 0.563136i) q^{8} +(-0.0508217 - 2.99957i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 10 q^{3} + 47 q^{4} - 9 q^{5} - 34 q^{6} - 7 q^{7} - 28 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.12192	+	0.319828i	−1.50042	+	0.226152i	−0.847306	−	0.531105i	\(-0.821777\pi\)
							−0.653117	+	0.757257i	\(0.726539\pi\)
\(3\)	1.21433	−	1.23508i	0.701092	−	0.713071i
							\(5\)	3.12995	+	2.13396i	1.39975	+	0.954336i	0.999277	+	0.0380299i	\(0.0121082\pi\)
0.400478	+	0.916306i	\(0.368844\pi\)
\(7\)	1.89973	−	1.84147i	0.718031	−	0.696011i
							\(11\)	3.89230	−	0.586669i	1.17357	−	0.176888i	0.466822	−	0.884351i	\(-0.345399\pi\)
0.706750	+	0.707464i	\(0.250161\pi\)
\(13\)	0.269707	−	0.687203i	0.0748033	−	0.190596i	−0.888581	−	0.458719i	\(-0.848308\pi\)
							0.963385	+	0.268124i	\(0.0864036\pi\)
\(17\)	0.713315	+	3.12524i	0.173004	+	0.757981i	0.984751	+	0.173971i	\(0.0556600\pi\)
							−0.811747	+	0.584010i	\(0.801483\pi\)
\(19\)	−5.18224			−1.18889			−0.594443	−	0.804137i	\(-0.702628\pi\)
							−0.594443	+	0.804137i	\(0.702628\pi\)
\(23\)	−6.58264	+	2.03048i	−1.37257	+	0.423383i	−0.891391	−	0.453236i	\(-0.850269\pi\)
							−0.481184	+	0.876619i	\(0.659793\pi\)
\(29\)	2.61953	+	0.808019i	0.486435	+	0.150045i	0.528265	−	0.849079i	\(-0.322843\pi\)
							−0.0418302	+	0.999125i	\(0.513319\pi\)
\(31\)	−2.97170	−	5.14713i	−0.533732	−	0.924452i	−0.999224	−	0.0393992i	\(-0.987456\pi\)
							0.465491	−	0.885053i	\(-0.345878\pi\)
\(37\)	−0.166896	−	0.731218i	−0.0274375	−	0.120211i	0.959355	−	0.282203i	\(-0.0910653\pi\)
							−0.986792	+	0.161992i	\(0.948208\pi\)
\(41\)	5.11994	+	3.49072i	0.799600	+	0.545158i	0.892781	−	0.450491i	\(-0.148751\pi\)
							−0.0931808	+	0.995649i	\(0.529703\pi\)
\(43\)	−3.33483	+	2.27365i	−0.508556	+	0.346728i	−0.790258	−	0.612774i	\(-0.790053\pi\)
							0.281702	+	0.959502i	\(0.409101\pi\)
\(47\)	−9.47503	+	1.42813i	−1.38208	+	0.208314i	−0.797644	−	0.603129i	\(-0.793921\pi\)
							−0.584431	+	0.811443i	\(0.698682\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.ba.a.106.9	yes	648
9.4	even	3	inner	441.2.ba.a.400.46	yes	648
49.43	even	7	inner	441.2.ba.a.43.46	✓	648
441.337	even	21	inner	441.2.ba.a.337.9	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.ba.a.43.46	✓	648	49.43	even	7	inner
441.2.ba.a.106.9	yes	648	1.1	even	1	trivial
441.2.ba.a.337.9	yes	648	441.337	even	21	inner
441.2.ba.a.400.46	yes	648	9.4	even	3	inner