Embedded newform 451.2.h.a.59.12

• Label: 451.2.h.a.59.12
• Level: $451$
• Weight: $2$
• Character: 451.59
• 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.h (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			59.12
Character	\(\chi\)	\(=\)	451.59
Dual form			451.2.h.a.344.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.472524 + 1.45428i) q^{2} +(-1.00332 + 3.08789i) q^{3} +(-0.273616 - 0.198794i) q^{4} +(-0.450992 + 1.38801i) q^{5} +(-4.01657 - 2.91821i) q^{6} +1.39505 q^{7} +(-2.05578 + 1.49361i) q^{8} +(-6.10139 - 4.43292i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + q^{2} - 7 q^{3} - 39 q^{4} - q^{5} - 4 q^{6} - 6 q^{7} + 3 q^{8} - 45 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(288\)	\(375\)
\(\chi(n)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.472524	+	1.45428i	−0.334125	+	1.02833i	0.633026	+	0.774130i	\(0.281812\pi\)
							−0.967151	+	0.254201i	\(0.918188\pi\)
\(3\)	−1.00332	+	3.08789i	−0.579265	+	1.78280i	0.0419066	+	0.999122i	\(0.486657\pi\)
							−0.621172	+	0.783674i	\(0.713343\pi\)
\(5\)	−0.450992	+	1.38801i	−0.201690	+	0.620737i	0.798144	+	0.602467i	\(0.205816\pi\)
							−0.999833	+	0.0182693i	\(0.994184\pi\)
\(7\)	1.39505			0.527280			0.263640	−	0.964621i	\(-0.415077\pi\)
							0.263640	+	0.964621i	\(0.415077\pi\)
\(11\)	3.03624	−	1.33463i	0.915461	−	0.402407i
							\(13\)	−0.634783	+	1.95366i	−0.176057	+	0.541848i	−0.999680	−	0.0252896i	\(-0.991949\pi\)
0.823623	+	0.567138i	\(0.191949\pi\)
\(17\)	−6.12768			−1.48618			−0.743090	−	0.669192i	\(-0.766641\pi\)
							−0.743090	+	0.669192i	\(0.766641\pi\)
\(19\)	3.71979	+	2.70258i	0.853377	+	0.620015i	0.926075	−	0.377339i	\(-0.123161\pi\)
							−0.0726978	+	0.997354i	\(0.523161\pi\)
\(23\)	−0.781179	−	2.40422i	−0.162887	−	0.501315i	0.835987	−	0.548749i	\(-0.184896\pi\)
							−0.998874	+	0.0474339i	\(0.984896\pi\)
\(29\)	0.892565	+	0.648487i	0.165745	+	0.120421i	0.667566	−	0.744551i	\(-0.267336\pi\)
							−0.501821	+	0.864972i	\(0.667336\pi\)
\(31\)	1.60003	+	4.92440i	0.287375	+	0.884448i	0.985677	+	0.168645i	\(0.0539391\pi\)
							−0.698302	+	0.715803i	\(0.746061\pi\)
\(37\)	8.50888	+	6.18207i	1.39885	+	1.01633i	0.994828	+	0.101577i	\(0.0323889\pi\)
							0.404024	+	0.914748i	\(0.367611\pi\)
\(41\)	0.614070	−	6.37361i	0.0959016	−	0.995391i
							\(43\)	0.218856	+	0.673569i	0.0333752	+	0.102718i	0.966356	−	0.257207i	\(-0.0828022\pi\)
−0.932981	+	0.359925i	\(0.882802\pi\)
\(47\)	−4.96998			−0.724946			−0.362473	−	0.931994i	\(-0.618067\pi\)
							−0.362473	+	0.931994i	\(0.618067\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	451.2.h.a.59.12	✓	160
11.3	even	5		451.2.j.a.223.29	yes	160
41.16	even	5		451.2.j.a.180.29	yes	160
451.344	even	5	inner	451.2.h.a.344.12	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
451.2.h.a.59.12	✓	160	1.1	even	1	trivial
451.2.h.a.344.12	yes	160	451.344	even	5	inner
451.2.j.a.180.29	yes	160	41.16	even	5
451.2.j.a.223.29	yes	160	11.3	even	5