Embedded newform 451.2.i.a.201.17

• Label: 451.2.i.a.201.17
• Level: $451$
• Weight: $2$
• Character: 451.201
• Analytic conductor: $3.601$
• Analytic rank: $0$
• Dimension: $160$
• 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			201.17
Character	\(\chi\)	\(=\)	451.201
Dual form			451.2.i.a.92.17

$q$-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 \(\left(\mathbb{Z}/451\mathbb{Z}\right)^\times\).

\(n\)	\(288\)	\(375\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
451.2.i.a.92.17	✓	160	451.92	even	5	inner
451.2.i.a.201.17	yes	160	1.1	even	1	trivial
451.2.l.a.37.17	yes	160	11.4	even	5
451.2.l.a.256.17	yes	160	41.10	even	5