Embedded newform 441.2.z.a.142.52

• Label: 441.2.z.a.142.52
• Level: $441$
• Weight: $2$
• Character: 441.142
• 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.z (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			142.52
Character	\(\chi\)	\(=\)	441.142
Dual form			441.2.z.a.382.52

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.969985 - 2.47148i) q^{2} +(-1.62598 + 0.596818i) q^{3} +(-3.70125 - 3.43426i) q^{4} +(-2.70476 + 1.30254i) q^{5} +(-0.102150 + 4.59748i) q^{6} +(2.23130 + 1.42172i) q^{7} +(-7.29370 + 3.51246i) q^{8} +(2.28762 - 1.94083i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 8 q^{2} - 13 q^{3} + 44 q^{4} + 3 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 39 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{4}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.969985	−	2.47148i	0.685883	−	1.74760i	0.0220091	−	0.999758i	\(-0.492994\pi\)
							0.663874	−	0.747844i	\(-0.268911\pi\)
\(3\)	−1.62598	+	0.596818i	−0.938759	+	0.344573i
							\(5\)	−2.70476	+	1.30254i	−1.20961	+	0.582516i	−0.926399	−	0.376543i	\(-0.877113\pi\)
−0.283207	+	0.959059i	\(0.591398\pi\)
\(7\)	2.23130	+	1.42172i	0.843352	+	0.537361i
							\(11\)	0.141496	+	0.177430i	0.0426626	+	0.0534972i	0.802705	−	0.596376i	\(-0.203393\pi\)
−0.760042	+	0.649873i	\(0.774822\pi\)
\(13\)	−1.85705	+	4.73168i	−0.515052	+	1.31233i	0.402756	+	0.915307i	\(0.368052\pi\)
							−0.917808	+	0.397024i	\(0.870043\pi\)
\(17\)	−3.43055	+	3.18309i	−0.832032	+	0.772013i	−0.976046	−	0.217563i	\(-0.930189\pi\)
							0.144015	+	0.989576i	\(0.453999\pi\)
\(19\)	2.97385	−	5.15086i	0.682248	−	1.18169i	−0.292046	−	0.956404i	\(-0.594336\pi\)
							0.974293	−	0.225283i	\(-0.0723308\pi\)
\(23\)	−1.24461	+	5.45301i	−0.259520	+	1.13703i	0.662247	+	0.749286i	\(0.269603\pi\)
							−0.921767	+	0.387745i	\(0.873254\pi\)
\(29\)	−0.773284	−	0.238526i	−0.143595	−	0.0442932i	0.222124	−	0.975018i	\(-0.428701\pi\)
							−0.365720	+	0.930725i	\(0.619177\pi\)
\(31\)	0.361489	−	0.626117i	0.0649253	−	0.112454i	−0.831736	−	0.555172i	\(-0.812652\pi\)
							0.896661	+	0.442718i	\(0.145986\pi\)
\(37\)	1.12677	+	0.347564i	0.185240	+	0.0571391i	0.385988	−	0.922504i	\(-0.373861\pi\)
							−0.200747	+	0.979643i	\(0.564337\pi\)
\(41\)	1.34357	+	0.916030i	0.209830	+	0.143060i	0.663675	−	0.748021i	\(-0.268996\pi\)
							−0.453845	+	0.891081i	\(0.649948\pi\)
\(43\)	−6.33403	+	4.31847i	−0.965931	+	0.658561i	−0.940026	−	0.341104i	\(-0.889199\pi\)
							−0.0259054	+	0.999664i	\(0.508247\pi\)
\(47\)	−2.96124	+	7.54512i	−0.431942	+	1.10057i	0.534837	+	0.844955i	\(0.320373\pi\)
							−0.966779	+	0.255614i	\(0.917722\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.z.a.142.52	yes	648
9.4	even	3		441.2.y.a.436.52	yes	648
49.39	even	21		441.2.y.a.88.52	✓	648
441.382	even	21	inner	441.2.z.a.382.52	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.88.52	✓	648	49.39	even	21
441.2.y.a.436.52	yes	648	9.4	even	3
441.2.z.a.142.52	yes	648	1.1	even	1	trivial
441.2.z.a.382.52	yes	648	441.382	even	21	inner