Embedded newform 451.2.t.a.81.5

• Label: 451.2.t.a.81.5
• Level: $451$
• Weight: $2$
• Character: 451.81
• Analytic conductor: $3.601$
• Analytic rank: $0$
• Dimension: $160$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 451 = 11 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	451.t (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			81.5
Character	\(\chi\)	\(=\)	451.81
Dual form			451.2.t.a.245.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.89175 - 1.37444i) q^{2} +(-3.11579 - 1.01238i) q^{3} +(1.07161 + 3.29809i) q^{4} +(0.541307 - 0.393283i) q^{5} +(4.50286 + 6.19765i) q^{6} +(-2.87335 + 0.933607i) q^{7} +(1.06062 - 3.26426i) q^{8} +(6.25620 + 4.54540i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 8 q^{2} - 44 q^{4} - 6 q^{5} - 4 q^{8} + 42 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.89175	−	1.37444i	−1.33767	−	0.971875i	−0.999526	−	0.0307814i	\(-0.990200\pi\)
							−0.338145	−	0.941094i	\(-0.609800\pi\)
\(3\)	−3.11579	−	1.01238i	−1.79890	−	0.584500i	−0.799045	−	0.601271i	\(-0.794661\pi\)
							−0.999859	+	0.0167714i	\(0.994661\pi\)
\(5\)	0.541307	−	0.393283i	0.242080	−	0.175881i	−0.460130	−	0.887852i	\(-0.652197\pi\)
							0.702209	+	0.711970i	\(0.252197\pi\)
\(7\)	−2.87335	+	0.933607i	−1.08602	+	0.352870i	−0.796709	−	0.604364i	\(-0.793427\pi\)
							−0.289315	+	0.957234i	\(0.593427\pi\)
\(11\)	−2.53185	+	2.14236i	−0.763382	+	0.645947i
							\(13\)	3.62765	−	4.99304i	1.00613	−	1.38482i	0.0846429	−	0.996411i	\(-0.473025\pi\)
0.921487	−	0.388408i	\(-0.126975\pi\)
\(17\)	0.750668	+	1.03321i	0.182064	+	0.250589i	0.890288	−	0.455399i	\(-0.150503\pi\)
							−0.708224	+	0.705988i	\(0.750503\pi\)
\(19\)	6.59146	+	2.14170i	1.51219	+	0.491339i	0.943544	−	0.331247i	\(-0.107469\pi\)
							0.568641	+	0.822586i	\(0.307469\pi\)
\(23\)	−1.53896			−0.320896			−0.160448	−	0.987044i	\(-0.551294\pi\)
							−0.160448	+	0.987044i	\(0.551294\pi\)
\(29\)	−2.86786	+	0.931824i	−0.532548	+	0.173035i	−0.562932	−	0.826503i	\(-0.690327\pi\)
							0.0303843	+	0.999538i	\(0.490327\pi\)
\(31\)	−4.82267	−	3.50387i	−0.866177	−	0.629314i	0.0633816	−	0.997989i	\(-0.479811\pi\)
							−0.929558	+	0.368675i	\(0.879811\pi\)
\(37\)	0.657816	+	2.02455i	0.108144	+	0.332834i	0.990456	−	0.137832i	\(-0.0440136\pi\)
							−0.882311	+	0.470666i	\(0.844014\pi\)
\(41\)	−1.85474	−	6.12862i	−0.289662	−	0.957129i
							\(43\)	5.66934			0.864566			0.432283	−	0.901738i	\(-0.357708\pi\)
0.432283	+	0.901738i	\(0.357708\pi\)
\(47\)	4.61584	+	1.49978i	0.673289	+	0.218765i	0.625655	−	0.780100i	\(-0.284832\pi\)
							0.0476340	+	0.998865i	\(0.484832\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	451.2.t.a.81.5	✓	160
11.3	even	5	inner	451.2.t.a.245.6	yes	160
41.40	even	2	inner	451.2.t.a.81.6	yes	160
451.245	even	10	inner	451.2.t.a.245.5	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
451.2.t.a.81.5	✓	160	1.1	even	1	trivial
451.2.t.a.81.6	yes	160	41.40	even	2	inner
451.2.t.a.245.5	yes	160	451.245	even	10	inner
451.2.t.a.245.6	yes	160	11.3	even	5	inner