Embedded newform 441.2.bn.a.5.4

• Label: 441.2.bn.a.5.4
• Level: $441$
• Weight: $2$
• Character: 441.5
• 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			5.4
Character	\(\chi\)	\(=\)	441.5
Dual form			441.2.bn.a.353.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.98600 + 1.58378i) q^{2} +(1.73017 + 0.0806248i) q^{3} +(0.990793 - 4.34095i) q^{4} +(-3.40470 - 2.32129i) q^{5} +(-3.56382 + 2.58010i) q^{6} +(-2.11613 - 1.58809i) q^{7} +(2.70311 + 5.61307i) q^{8} +(2.98700 + 0.278990i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 21 q^{2} - 11 q^{3} + 99 q^{4} - 18 q^{5} - 34 q^{6} - 5 q^{7} - 23 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{29}{42}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.98600	+	1.58378i	−1.40432	+	1.11991i	−0.427945	+	0.903805i	\(0.640762\pi\)
							−0.976371	+	0.216101i	\(0.930666\pi\)
\(3\)	1.73017	+	0.0806248i	0.998916	+	0.0465488i
							\(5\)	−3.40470	−	2.32129i	−1.52263	−	1.03811i	−0.980200	−	0.198008i	\(-0.936553\pi\)
−0.542428	−	0.840102i	\(-0.682495\pi\)
\(7\)	−2.11613	−	1.58809i	−0.799820	−	0.600240i
							\(11\)	0.472238	+	3.13309i	0.142385	+	0.944663i	0.939921	+	0.341392i	\(0.110898\pi\)
−0.797536	+	0.603271i	\(0.793864\pi\)
\(13\)	0.572102	+	3.79565i	0.158673	+	1.05272i	0.916432	+	0.400190i	\(0.131056\pi\)
							−0.757760	+	0.652534i	\(0.773706\pi\)
\(17\)	−5.38118	+	4.99301i	−1.30513	+	1.21098i	−0.342701	+	0.939445i	\(0.611342\pi\)
							−0.962428	+	0.271538i	\(0.912468\pi\)
\(19\)	−1.72748	−	0.997363i	−0.396312	−	0.228811i	0.288580	−	0.957456i	\(-0.406817\pi\)
							−0.684891	+	0.728645i	\(0.740150\pi\)
\(23\)	0.468287	+	1.51815i	0.0976445	+	0.316556i	0.991196	−	0.132406i	\(-0.0422702\pi\)
							−0.893551	+	0.448962i	\(0.851794\pi\)
\(29\)	2.35536	+	2.53848i	0.437380	+	0.471383i	0.912818	−	0.408366i	\(-0.133901\pi\)
							−0.475439	+	0.879749i	\(0.657711\pi\)
\(31\)			1.35826i			0.243951i	0.992533	+	0.121976i	\(0.0389230\pi\)
							−0.992533	+	0.121976i	\(0.961077\pi\)
\(37\)	−3.36382	−	1.03760i	−0.553009	−	0.170581i	0.00564110	−	0.999984i	\(-0.498204\pi\)
							−0.558650	+	0.829403i	\(0.688681\pi\)
\(41\)	−0.118309	+	1.57872i	−0.0184768	+	0.246555i	0.980316	+	0.197434i	\(0.0632610\pi\)
							−0.998793	+	0.0491208i	\(0.984358\pi\)
\(43\)	0.419251	+	5.59451i	0.0639352	+	0.853156i	0.933326	+	0.359030i	\(0.116893\pi\)
							−0.869391	+	0.494125i	\(0.835488\pi\)
\(47\)	−5.09449	−	6.38829i	−0.743108	−	0.931828i	0.256287	−	0.966601i	\(-0.417501\pi\)
							−0.999395	+	0.0347729i	\(0.988929\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bn.a.5.4	yes	648
9.2	odd	6		441.2.bd.a.299.4	yes	648
49.10	odd	42		441.2.bd.a.59.4	✓	648
441.353	even	42	inner	441.2.bn.a.353.4	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.59.4	✓	648	49.10	odd	42
441.2.bd.a.299.4	yes	648	9.2	odd	6
441.2.bn.a.5.4	yes	648	1.1	even	1	trivial
441.2.bn.a.353.4	yes	648	441.353	even	42	inner