Embedded newform 451.2.j.a.119.17

• Label: 451.2.j.a.119.17
• Level: $451$
• Weight: $2$
• Character: 451.119
• 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.j (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			119.17
Character	\(\chi\)	\(=\)	451.119
Dual form			451.2.j.a.379.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.163820 - 0.504186i) q^{2} +(-1.74772 + 1.26980i) q^{3} +(1.39067 - 1.01038i) q^{4} -0.466136 q^{5} +(0.926526 + 0.673161i) q^{6} +(0.0711135 + 0.218865i) q^{7} +(-1.59501 - 1.15884i) q^{8} +(0.515107 - 1.58534i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + q^{2} - 7 q^{3} - 39 q^{4} - 6 q^{5} + 6 q^{6} - q^{7} + 3 q^{8} - 45 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(288\)	\(375\)
\(\chi(n)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.163820	−	0.504186i	−0.115838	−	0.356514i	0.876283	−	0.481797i	\(-0.160016\pi\)
							−0.992121	+	0.125284i	\(0.960016\pi\)
\(3\)	−1.74772	+	1.26980i	−1.00905	+	0.733117i	−0.964009	−	0.265868i	\(-0.914341\pi\)
							−0.0450397	+	0.998985i	\(0.514341\pi\)
\(5\)	−0.466136			−0.208463			−0.104231	−	0.994553i	\(-0.533238\pi\)
							−0.104231	+	0.994553i	\(0.533238\pi\)
\(7\)	0.0711135	+	0.218865i	0.0268784	+	0.0827231i	0.963596	−	0.267363i	\(-0.0861523\pi\)
							−0.936717	+	0.350086i	\(0.886152\pi\)
\(11\)	1.00343	−	3.16119i	0.302544	−	0.953135i
							\(13\)	−0.0635451	−	0.195572i	−0.0176242	−	0.0542418i	0.941858	−	0.336012i	\(-0.109078\pi\)
−0.959482	+	0.281770i	\(0.909078\pi\)
\(17\)	2.84731	−	2.06869i	0.690574	−	0.501732i	−0.186275	−	0.982498i	\(-0.559641\pi\)
							0.876849	+	0.480766i	\(0.159641\pi\)
\(19\)	5.06170			1.16123			0.580616	−	0.814177i	\(-0.302812\pi\)
							0.580616	+	0.814177i	\(0.302812\pi\)
\(23\)	−1.52124	+	1.10525i	−0.317201	+	0.230460i	−0.734980	−	0.678089i	\(-0.762808\pi\)
							0.417779	+	0.908548i	\(0.362808\pi\)
\(29\)	4.45342	−	3.23560i	0.826979	−	0.600836i	−0.0917237	−	0.995784i	\(-0.529238\pi\)
							0.918703	+	0.394949i	\(0.129238\pi\)
\(31\)	8.15782			1.46519			0.732594	−	0.680666i	\(-0.238309\pi\)
							0.732594	+	0.680666i	\(0.238309\pi\)
\(37\)	7.44399	−	5.40838i	1.22378	−	0.889132i	0.227376	−	0.973807i	\(-0.426985\pi\)
							0.996409	+	0.0846754i	\(0.0269853\pi\)
\(41\)	−1.47785	−	6.23025i	−0.230801	−	0.973001i
							\(43\)	−1.49527	+	1.08638i	−0.228027	+	0.165671i	−0.695932	−	0.718107i	\(-0.745009\pi\)
0.467906	+	0.883778i	\(0.345009\pi\)
\(47\)	0.956579	−	2.94405i	0.139531	−	0.429434i	−0.856736	−	0.515755i	\(-0.827511\pi\)
							0.996267	+	0.0863218i	\(0.0275113\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	451.2.j.a.119.17	yes	160
11.5	even	5		451.2.h.a.324.24	yes	160
41.10	even	5		451.2.h.a.174.24	✓	160
451.379	even	5	inner	451.2.j.a.379.17	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
451.2.h.a.174.24	✓	160	41.10	even	5
451.2.h.a.324.24	yes	160	11.5	even	5
451.2.j.a.119.17	yes	160	1.1	even	1	trivial
451.2.j.a.379.17	yes	160	451.379	even	5	inner