Embedded newform 441.2.bn.a.353.20

• Label: 441.2.bn.a.353.20
• Level: $441$
• Weight: $2$
• Character: 441.353
• 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.bn (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			353.20
Character	\(\chi\)	\(=\)	441.353
Dual form			441.2.bn.a.5.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.744514 - 0.593730i) q^{2} +(-1.04233 - 1.38331i) q^{3} +(-0.243256 - 1.06577i) q^{4} +(0.397639 - 0.271106i) q^{5} +(-0.0452820 + 1.64876i) q^{6} +(-2.38053 + 1.15459i) q^{7} +(-1.27802 + 2.65384i) q^{8} +(-0.827090 + 2.88373i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 21 q^{2} - 11 q^{3} + 99 q^{4} - 18 q^{5} - 34 q^{6} - 5 q^{7} - 23 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{13}{42}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.744514	−	0.593730i	−0.526451	−	0.419831i	0.323865	−	0.946103i	\(-0.395018\pi\)
							−0.850316	+	0.526273i	\(0.823589\pi\)
\(3\)	−1.04233	−	1.38331i	−0.601790	−	0.798654i
							\(5\)	0.397639	−	0.271106i	0.177830	−	0.121242i	−0.471134	−	0.882062i	\(-0.656155\pi\)
0.648964	+	0.760820i	\(0.275203\pi\)
\(7\)	−2.38053	+	1.15459i	−0.899756	+	0.436394i
							\(11\)	−0.227802	+	1.51137i	−0.0686850	+	0.455695i	0.927880	+	0.372879i	\(0.121629\pi\)
−0.996565	+	0.0828158i	\(0.973609\pi\)
\(13\)	0.172566	−	1.14490i	0.0478611	−	0.317538i	−0.952071	−	0.305877i	\(-0.901050\pi\)
							0.999932	−	0.0116602i	\(-0.00371164\pi\)
\(17\)	1.74339	+	1.61763i	0.422835	+	0.392334i	0.862688	−	0.505736i	\(-0.168779\pi\)
							−0.439853	+	0.898070i	\(0.644969\pi\)
\(19\)	−5.11573	+	2.95357i	−1.17363	+	0.677595i	−0.954532	−	0.298109i	\(-0.903644\pi\)
							−0.219096	+	0.975703i	\(0.570311\pi\)
\(23\)	0.309968	−	1.00489i	0.0646327	−	0.209534i	−0.917496	−	0.397745i	\(-0.869793\pi\)
							0.982129	+	0.188211i	\(0.0602689\pi\)
\(29\)	6.27208	−	6.75969i	1.16470	−	1.25524i	0.203592	−	0.979056i	\(-0.434738\pi\)
							0.961104	−	0.276188i	\(-0.0890711\pi\)
\(31\)		−	7.44428i		−	1.33703i	−0.743698	−	0.668516i	\(-0.766930\pi\)
							0.743698	−	0.668516i	\(-0.233070\pi\)
\(37\)	−8.67490	+	2.67585i	−1.42614	+	0.439907i	−0.909289	−	0.416166i	\(-0.863374\pi\)
							−0.516855	+	0.856073i	\(0.672897\pi\)
\(41\)	0.620827	+	8.28436i	0.0969569	+	1.29380i	0.807383	+	0.590028i	\(0.200883\pi\)
							−0.710426	+	0.703772i	\(0.751498\pi\)
\(43\)	−0.683300	+	9.11801i	−0.104202	+	1.39048i	0.662779	+	0.748815i	\(0.269377\pi\)
							−0.766982	+	0.641669i	\(0.778242\pi\)
\(47\)	−6.24642	+	7.83276i	−0.911134	+	1.14253i	0.0782114	+	0.996937i	\(0.475079\pi\)
							−0.989345	+	0.145589i	\(0.953492\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bn.a.353.20	yes	648
9.5	odd	6		441.2.bd.a.59.20	✓	648
49.5	odd	42		441.2.bd.a.299.20	yes	648
441.5	even	42	inner	441.2.bn.a.5.20	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.59.20	✓	648	9.5	odd	6
441.2.bd.a.299.20	yes	648	49.5	odd	42
441.2.bn.a.5.20	yes	648	441.5	even	42	inner
441.2.bn.a.353.20	yes	648	1.1	even	1	trivial