Embedded newform 451.2.i.a.92.17

• Label: 451.2.i.a.92.17
• Level: $451$
• Weight: $2$
• Character: 451.92
• Analytic conductor: $3.601$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 451 = 11 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	451.i (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.60125313116\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(40\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			92.17
Character	\(\chi\)	\(=\)	451.92
Dual form			451.2.i.a.201.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.156942 - 0.483018i) q^{2} +(-0.858828 + 2.64320i) q^{3} +(1.40936 - 1.02396i) q^{4} +(-2.04169 - 1.48337i) q^{5} +1.41150 q^{6} +(3.85897 + 2.80370i) q^{7} +(-1.53754 - 1.11709i) q^{8} +(-3.82188 - 2.77676i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + q^{2} - 7 q^{3} - 39 q^{4} - q^{5} - 4 q^{6} - q^{7} + 13 q^{8} - 45 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/451\mathbb{Z}\right)^\times\).

\(n\)	\(288\)	\(375\)
\(\chi(n)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.156942	−	0.483018i	−0.110975	−	0.341545i	0.880111	−	0.474767i	\(-0.157468\pi\)
							−0.991086	+	0.133222i	\(0.957468\pi\)
\(3\)	−0.858828	+	2.64320i	−0.495845	+	1.52605i	0.319791	+	0.947488i	\(0.396387\pi\)
							−0.815636	+	0.578565i	\(0.803613\pi\)
\(5\)	−2.04169	−	1.48337i	−0.913071	−	0.663385i	0.0287185	−	0.999588i	\(-0.490857\pi\)
							−0.941790	+	0.336202i	\(0.890857\pi\)
\(7\)	3.85897	+	2.80370i	1.45855	+	1.05970i	0.983737	+	0.179614i	\(0.0574850\pi\)
							0.474815	+	0.880086i	\(0.342515\pi\)
\(11\)	1.94005	−	2.69002i	0.584946	−	0.811072i
							\(13\)	1.41713	+	4.36149i	0.393042	+	1.20966i	0.930476	+	0.366353i	\(0.119394\pi\)
−0.537434	+	0.843306i	\(0.680606\pi\)
\(17\)	0.312221	+	0.960919i	0.0757248	+	0.233057i	0.981753	−	0.190160i	\(-0.0609007\pi\)
							−0.906028	+	0.423217i	\(0.860901\pi\)
\(19\)	0.906056	+	2.78855i	0.207864	+	0.639738i	0.999584	+	0.0288541i	\(0.00918583\pi\)
							−0.791720	+	0.610884i	\(0.790814\pi\)
\(23\)	5.57520	+	4.05062i	1.16251	+	0.844613i	0.990093	−	0.140411i	\(-0.0448425\pi\)
							0.172417	+	0.985024i	\(0.444842\pi\)
\(29\)	−4.45096	+	3.23382i	−0.826523	+	0.600504i	−0.918574	−	0.395250i	\(-0.870658\pi\)
							0.0920502	+	0.995754i	\(0.470658\pi\)
\(31\)	1.38184	−	1.00397i	0.248186	−	0.180318i	−0.456736	−	0.889602i	\(-0.650982\pi\)
							0.704923	+	0.709284i	\(0.250982\pi\)
\(37\)	−4.04763	+	2.94078i	−0.665426	+	0.483461i	−0.868491	−	0.495705i	\(-0.834910\pi\)
							0.203065	+	0.979165i	\(0.434910\pi\)
\(41\)	6.31888	+	1.03528i	0.986843	+	0.161684i
							\(43\)	5.57293	+	4.04897i	0.849864	+	0.617462i	0.925109	−	0.379703i	\(-0.123974\pi\)
−0.0752444	+	0.997165i	\(0.523974\pi\)
\(47\)	1.16793	−	0.848553i	0.170361	−	0.123774i	−0.499338	−	0.866407i	\(-0.666423\pi\)
							0.669698	+	0.742633i	\(0.266423\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	451.2.i.a.92.17	✓	160
11.3	even	5		451.2.l.a.256.17	yes	160
41.37	even	5		451.2.l.a.37.17	yes	160
451.201	even	5	inner	451.2.i.a.201.17	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
451.2.i.a.92.17	✓	160	1.1	even	1	trivial
451.2.i.a.201.17	yes	160	451.201	even	5	inner
451.2.l.a.37.17	yes	160	41.37	even	5
451.2.l.a.256.17	yes	160	11.3	even	5