Embedded newform 441.2.y.a.25.18

• Label: 441.2.y.a.25.18
• 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			25.18
Character	\(\chi\)	\(=\)	441.25
Dual form			441.2.y.a.247.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.263181 + 1.15307i) q^{2} +(1.58469 - 0.699101i) q^{3} +(0.541631 + 0.260836i) q^{4} +(-0.0198537 - 0.0505863i) q^{5} +(0.389051 + 2.01125i) q^{6} +(1.10457 + 2.40415i) q^{7} +(-1.91814 + 2.40527i) q^{8} +(2.02252 - 2.21572i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.263181	+	1.15307i	−0.186097	+	0.815344i	0.792552	+	0.609804i	\(0.208752\pi\)
							−0.978649	+	0.205539i	\(0.934105\pi\)
\(3\)	1.58469	−	0.699101i	0.914924	−	0.403626i
							\(5\)	−0.0198537	−	0.0505863i	−0.00887883	−	0.0226229i	0.926366	−	0.376625i	\(-0.122916\pi\)
−0.935244	+	0.354003i	\(0.884820\pi\)
\(7\)	1.10457	+	2.40415i	0.417488	+	0.908683i
							\(11\)	0.990935	−	0.305663i	0.298778	−	0.0921609i	−0.141742	−	0.989904i	\(-0.545270\pi\)
0.440520	+	0.897743i	\(0.354794\pi\)
\(13\)	−0.807733	+	0.249153i	−0.224025	+	0.0691025i	−0.404735	−	0.914434i	\(-0.632636\pi\)
							0.180711	+	0.983536i	\(0.442160\pi\)
\(17\)	−0.119230	+	1.59102i	−0.0289176	+	0.385879i	0.963902	+	0.266257i	\(0.0857871\pi\)
							−0.992820	+	0.119621i	\(0.961832\pi\)
\(19\)	−1.50088	−	2.59959i	−0.344324	−	0.596387i	0.640906	−	0.767619i	\(-0.278559\pi\)
							−0.985231	+	0.171232i	\(0.945225\pi\)
\(23\)	−5.65103	+	3.85280i	−1.17832	+	0.803365i	−0.984104	−	0.177593i	\(-0.943169\pi\)
							−0.194216	+	0.980959i	\(0.562216\pi\)
\(29\)	0.452142	−	6.03341i	0.0839606	−	1.12038i	−0.783007	−	0.622013i	\(-0.786315\pi\)
							0.866968	−	0.498364i	\(-0.166066\pi\)
\(31\)	8.31503			1.49342			0.746711	−	0.665148i	\(-0.231632\pi\)
							0.746711	+	0.665148i	\(0.231632\pi\)
\(37\)	8.39311	+	5.72232i	1.37982	+	0.940744i	0.999876	+	0.0157341i	\(0.00500851\pi\)
							0.379942	+	0.925010i	\(0.375944\pi\)
\(41\)	−4.40134	−	0.663395i	−0.687373	−	0.103605i	−0.203936	−	0.978984i	\(-0.565374\pi\)
							−0.483437	+	0.875379i	\(0.660612\pi\)
\(43\)	−7.84409	+	1.18231i	−1.19621	+	0.180300i	−0.716793	−	0.697286i	\(-0.754391\pi\)
							−0.479419	+	0.877586i	\(0.659153\pi\)
\(47\)	2.38207	−	10.4365i	0.347460	−	1.52232i	−0.435464	−	0.900206i	\(-0.643416\pi\)
							0.782924	−	0.622117i	\(-0.213727\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.25.18	✓	648
9.4	even	3		441.2.z.a.319.18	yes	648
49.2	even	21		441.2.z.a.394.18	yes	648
441.247	even	21	inner	441.2.y.a.247.18	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.25.18	✓	648	1.1	even	1	trivial
441.2.y.a.247.18	yes	648	441.247	even	21	inner
441.2.z.a.319.18	yes	648	9.4	even	3
441.2.z.a.394.18	yes	648	49.2	even	21