Embedded newform 441.2.bn.a.101.23

• Label: 441.2.bn.a.101.23
• Level: $441$
• Weight: $2$
• Character: 441.101
• 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			101.23
Character	\(\chi\)	\(=\)	441.101
Dual form			441.2.bn.a.131.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.419561 + 0.334589i) q^{2} +(-1.08122 - 1.35313i) q^{3} +(-0.380960 + 1.66910i) q^{4} +(0.121548 - 1.62194i) q^{5} +(0.906381 + 0.205955i) q^{6} +(-2.56323 + 0.655614i) q^{7} +(-0.864302 - 1.79474i) q^{8} +(-0.661914 + 2.92607i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 21 q^{2} - 11 q^{3} + 99 q^{4} - 18 q^{5} - 34 q^{6} - 5 q^{7} - 23 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.419561	+	0.334589i	−0.296674	+	0.236590i	−0.760502	−	0.649335i	\(-0.775047\pi\)
							0.463828	+	0.885925i	\(0.346476\pi\)
\(3\)	−1.08122	−	1.35313i	−0.624244	−	0.781229i
							\(5\)	0.121548	−	1.62194i	0.0543578	−	0.725355i	−0.901735	−	0.432290i	\(-0.857706\pi\)
0.956093	−	0.293065i	\(-0.0946752\pi\)
\(7\)	−2.56323	+	0.655614i	−0.968812	+	0.247799i
							\(11\)	1.54228	+	0.605300i	0.465015	+	0.182505i	0.586269	−	0.810117i	\(-0.300596\pi\)
−0.121254	+	0.992622i	\(0.538692\pi\)
\(13\)	5.17322	+	2.03034i	1.43479	+	0.563114i	0.950204	−	0.311628i	\(-0.100874\pi\)
							0.484588	+	0.874743i	\(0.338970\pi\)
\(17\)	4.27109	+	1.31746i	1.03589	+	0.319530i	0.765647	−	0.643261i	\(-0.222419\pi\)
							0.270245	+	0.962791i	\(0.412895\pi\)
\(19\)	−0.895437	+	0.516981i	−0.205427	+	0.118604i	−0.599184	−	0.800611i	\(-0.704508\pi\)
							0.393757	+	0.919214i	\(0.371175\pi\)
\(23\)	2.69770	−	2.90743i	0.562510	−	0.606242i	−0.386190	−	0.922419i	\(-0.626209\pi\)
							0.948700	+	0.316177i	\(0.102399\pi\)
\(29\)	−1.15420	+	3.74183i	−0.214330	+	0.694841i	0.783014	+	0.622004i	\(0.213681\pi\)
							−0.997344	+	0.0728368i	\(0.976795\pi\)
\(31\)			2.17271i			0.390229i	0.980780	+	0.195115i	\(0.0625079\pi\)
							−0.980780	+	0.195115i	\(0.937492\pi\)
\(37\)	−4.26592	+	3.95820i	−0.701313	+	0.650724i	−0.947963	−	0.318381i	\(-0.896861\pi\)
							0.246650	+	0.969105i	\(0.420670\pi\)
\(41\)	9.00665	+	6.14063i	1.40660	+	0.959005i	0.998947	+	0.0458684i	\(0.0146055\pi\)
							0.407654	+	0.913136i	\(0.366347\pi\)
\(43\)	3.00057	−	2.04575i	0.457583	−	0.311975i	−0.312501	−	0.949917i	\(-0.601167\pi\)
							0.770084	+	0.637943i	\(0.220214\pi\)
\(47\)	2.85790	+	3.58369i	0.416867	+	0.522735i	0.945283	−	0.326251i	\(-0.105785\pi\)
							−0.528416	+	0.848985i	\(0.677214\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bn.a.101.23	yes	648
9.5	odd	6		441.2.bd.a.248.32	✓	648
49.33	odd	42		441.2.bd.a.425.32	yes	648
441.131	even	42	inner	441.2.bn.a.131.23	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.248.32	✓	648	9.5	odd	6
441.2.bd.a.425.32	yes	648	49.33	odd	42
441.2.bn.a.101.23	yes	648	1.1	even	1	trivial
441.2.bn.a.131.23	yes	648	441.131	even	42	inner