Embedded newform 451.2.t.a.81.12

• Label: 451.2.t.a.81.12
• 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			81.12
Character	\(\chi\)	\(=\)	451.81
Dual form			451.2.t.a.245.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.27627 - 0.927264i) q^{2} +(0.470503 + 0.152876i) q^{3} +(0.151011 + 0.464764i) q^{4} +(-1.14393 + 0.831112i) q^{5} +(-0.458732 - 0.631391i) q^{6} +(2.29572 - 0.745926i) q^{7} +(-0.736755 + 2.26750i) q^{8} +(-2.22905 - 1.61950i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 8 q^{2} - 44 q^{4} - 6 q^{5} - 4 q^{8} + 42 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.27627	−	0.927264i	−0.902459	−	0.655675i	0.0366376	−	0.999329i	\(-0.488335\pi\)
							−0.939096	+	0.343654i	\(0.888335\pi\)
\(3\)	0.470503	+	0.152876i	0.271645	+	0.0882628i	0.441672	−	0.897177i	\(-0.354385\pi\)
							−0.170027	+	0.985439i	\(0.554385\pi\)
\(5\)	−1.14393	+	0.831112i	−0.511580	+	0.371685i	−0.813423	−	0.581673i	\(-0.802398\pi\)
							0.301843	+	0.953358i	\(0.402398\pi\)
\(7\)	2.29572	−	0.745926i	0.867702	−	0.281934i	0.158860	−	0.987301i	\(-0.449218\pi\)
							0.708842	+	0.705368i	\(0.249218\pi\)
\(11\)	−3.24674	+	0.677267i	−0.978928	+	0.204204i
							\(13\)	−0.605103	+	0.832852i	−0.167825	+	0.230992i	−0.884643	−	0.466269i	\(-0.845598\pi\)
0.716818	+	0.697261i	\(0.245598\pi\)
\(17\)	−2.86017	−	3.93668i	−0.693692	−	0.954785i	−0.999996	−	0.00285133i	\(-0.999092\pi\)
							0.306304	−	0.951934i	\(-0.400908\pi\)
\(19\)	−2.53609	−	0.824027i	−0.581820	−	0.189045i	0.00329581	−	0.999995i	\(-0.498951\pi\)
							−0.585116	+	0.810950i	\(0.698951\pi\)
\(23\)	0.274434			0.0572234			0.0286117	−	0.999591i	\(-0.490891\pi\)
							0.0286117	+	0.999591i	\(0.490891\pi\)
\(29\)	−7.99830	+	2.59881i	−1.48525	+	0.482586i	−0.935676	−	0.352860i	\(-0.885209\pi\)
							−0.549572	+	0.835447i	\(0.685209\pi\)
\(31\)	−5.58500	−	4.05774i	−1.00310	−	0.728792i	−0.0403458	−	0.999186i	\(-0.512846\pi\)
							−0.962750	+	0.270394i	\(0.912846\pi\)
\(37\)	2.98128	+	9.17542i	0.490119	+	1.50843i	0.824428	+	0.565967i	\(0.191497\pi\)
							−0.334309	+	0.942463i	\(0.608503\pi\)
\(41\)	3.15943	−	5.56938i	0.493419	−	0.869792i
							\(43\)	5.34596			0.815252			0.407626	−	0.913149i	\(-0.366357\pi\)
0.407626	+	0.913149i	\(0.366357\pi\)
\(47\)	−10.4223	−	3.38641i	−1.52025	−	0.493959i	−0.574404	−	0.818572i	\(-0.694766\pi\)
							−0.945847	+	0.324613i	\(0.894766\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	451.2.t.a.81.12	yes	160
11.3	even	5	inner	451.2.t.a.245.11	yes	160
41.40	even	2	inner	451.2.t.a.81.11	✓	160
451.245	even	10	inner	451.2.t.a.245.12	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
451.2.t.a.81.11	✓	160	41.40	even	2	inner
451.2.t.a.81.12	yes	160	1.1	even	1	trivial
451.2.t.a.245.11	yes	160	11.3	even	5	inner
451.2.t.a.245.12	yes	160	451.245	even	10	inner