Embedded newform 451.2.bf.a.4.15

• Label: 451.2.bf.a.4.15
• Level: $451$
• Weight: $2$
• Character: 451.4
• 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.bf (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.60125313116\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(40\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			4.15
Character	\(\chi\)	\(=\)	451.4
Dual form			451.2.bf.a.113.15

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.02865 q^{2} +(-1.58915 - 0.516347i) q^{3} -0.941883 q^{4} +(-1.13371 - 3.48921i) q^{5} +(1.63468 + 0.531140i) q^{6} +(-1.91178 - 0.621174i) q^{7} +3.02616 q^{8} +(-0.168258 - 0.122246i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 2 q^{2} + 5 q^{3} + 146 q^{4} - q^{5} - 20 q^{6} - 5 q^{7} + 6 q^{8} + 27 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(288\)	\(375\)
\(\chi(n)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{3}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.02865			−0.727364			−0.363682	−	0.931523i	\(-0.618481\pi\)
							−0.363682	+	0.931523i	\(0.618481\pi\)
\(3\)	−1.58915	−	0.516347i	−0.917498	−	0.298113i	−0.188058	−	0.982158i	\(-0.560219\pi\)
							−0.729440	+	0.684045i	\(0.760219\pi\)
\(5\)	−1.13371	−	3.48921i	−0.507012	−	1.56042i	−0.797362	−	0.603501i	\(-0.793772\pi\)
							0.290351	−	0.956920i	\(-0.406228\pi\)
\(7\)	−1.91178	−	0.621174i	−0.722584	−	0.234782i	−0.0754410	−	0.997150i	\(-0.524036\pi\)
							−0.647143	+	0.762368i	\(0.724036\pi\)
\(11\)	0.206852	−	3.31017i	0.0623682	−	0.998053i
							\(13\)		−	5.27351i		−	1.46261i	−0.682052	−	0.731304i	\(-0.738912\pi\)
0.682052	−	0.731304i	\(-0.261088\pi\)
\(17\)	−3.63977	+	5.00972i	−0.882775	+	1.21504i	0.0928700	+	0.995678i	\(0.470396\pi\)
							−0.975645	+	0.219357i	\(0.929604\pi\)
\(19\)	−1.49970	−	2.06417i	−0.344056	−	0.473552i	0.601565	−	0.798824i	\(-0.294544\pi\)
							−0.945621	+	0.325272i	\(0.894544\pi\)
\(23\)	3.19236	−	2.31939i	0.665654	−	0.483626i	−0.202914	−	0.979197i	\(-0.565041\pi\)
							0.868567	+	0.495571i	\(0.165041\pi\)
\(29\)			9.45609i			1.75595i	0.478704	+	0.877976i	\(0.341107\pi\)
							−0.478704	+	0.877976i	\(0.658893\pi\)
\(31\)	2.02120	−	6.22062i	0.363018	−	1.11726i	−0.588194	−	0.808720i	\(-0.700161\pi\)
							0.951213	−	0.308536i	\(-0.0998391\pi\)
\(37\)	5.54208			0.911112			0.455556	−	0.890207i	\(-0.349441\pi\)
							0.455556	+	0.890207i	\(0.349441\pi\)
\(41\)	−4.65457	−	4.39715i	−0.726922	−	0.686720i
							\(43\)	−6.12615	+	4.45091i	−0.934230	+	0.678757i	−0.947025	−	0.321161i	\(-0.895927\pi\)
0.0127952	+	0.999918i	\(0.495927\pi\)
\(47\)	4.66465	−	1.51564i	0.680409	−	0.221078i	0.0516343	−	0.998666i	\(-0.483557\pi\)
							0.628774	+	0.777588i	\(0.283557\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	451.2.bf.a.4.15	yes	160
11.3	even	5		451.2.r.a.168.15	✓	160
41.31	even	10		451.2.r.a.400.15	yes	160
451.113	even	10	inner	451.2.bf.a.113.15	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
451.2.r.a.168.15	✓	160	11.3	even	5
451.2.r.a.400.15	yes	160	41.31	even	10
451.2.bf.a.4.15	yes	160	1.1	even	1	trivial
451.2.bf.a.113.15	yes	160	451.113	even	10	inner