Embedded newform 451.2.h.a.324.38

• Label: 451.2.h.a.324.38
• Level: $451$
• Weight: $2$
• Character: 451.324
• 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			324.38
Character	\(\chi\)	\(=\)	451.324
Dual form			451.2.h.a.174.38

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.10625 + 1.53028i) q^{2} +(0.229091 + 0.166445i) q^{3} +(1.47650 + 4.54418i) q^{4} +(0.946320 + 0.687542i) q^{5} +(0.227816 + 0.701147i) q^{6} -1.70079 q^{7} +(-2.23497 + 6.87854i) q^{8} +(-0.902272 - 2.77691i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + q^{2} - 7 q^{3} - 39 q^{4} - q^{5} - 4 q^{6} - 6 q^{7} + 3 q^{8} - 45 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(288\)	\(375\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.10625	+	1.53028i	1.48934	+	1.08207i	0.974393	+	0.224850i	\(0.0721893\pi\)
							0.514949	+	0.857221i	\(0.327811\pi\)
\(3\)	0.229091	+	0.166445i	0.132266	+	0.0960968i	0.651951	−	0.758261i	\(-0.273951\pi\)
							−0.519685	+	0.854358i	\(0.673951\pi\)
\(5\)	0.946320	+	0.687542i	0.423207	+	0.307478i	0.778927	−	0.627115i	\(-0.215764\pi\)
							−0.355720	+	0.934593i	\(0.615764\pi\)
\(7\)	−1.70079			−0.642838			−0.321419	−	0.946937i	\(-0.604160\pi\)
							−0.321419	+	0.946937i	\(0.604160\pi\)
\(11\)	−0.410374	+	3.29114i	−0.123733	+	0.992316i
							\(13\)	−2.62678	−	1.90847i	−0.728539	−	0.529314i	0.160562	−	0.987026i	\(-0.448669\pi\)
−0.889101	+	0.457711i	\(0.848669\pi\)
\(17\)	7.95346			1.92900			0.964499	−	0.264085i	\(-0.0850701\pi\)
							0.964499	+	0.264085i	\(0.0850701\pi\)
\(19\)	−0.198449	−	0.610763i	−0.0455273	−	0.140119i	0.925709	−	0.378237i	\(-0.123469\pi\)
							−0.971236	+	0.238118i	\(0.923469\pi\)
\(23\)	7.42550	−	5.39494i	1.54832	−	1.12492i	0.603491	−	0.797370i	\(-0.293776\pi\)
							0.944833	−	0.327553i	\(-0.106224\pi\)
\(29\)	2.01193	+	6.19207i	0.373605	+	1.14984i	0.944415	+	0.328756i	\(0.106629\pi\)
							−0.570810	+	0.821082i	\(0.693371\pi\)
\(31\)	−3.67583	+	2.67065i	−0.660198	+	0.479662i	−0.866730	−	0.498778i	\(-0.833782\pi\)
							0.206531	+	0.978440i	\(0.433782\pi\)
\(37\)	−3.04442	−	9.36977i	−0.500500	−	1.54038i	−0.808206	−	0.588900i	\(-0.799561\pi\)
							0.307706	−	0.951481i	\(-0.400439\pi\)
\(41\)	−6.08983	+	1.97839i	−0.951071	+	0.308972i
							\(43\)	−0.459789	+	0.334056i	−0.0701172	+	0.0509431i	−0.622292	−	0.782785i	\(-0.713798\pi\)
0.552174	+	0.833729i	\(0.313798\pi\)
\(47\)	−11.0381			−1.61007			−0.805037	−	0.593225i	\(-0.797854\pi\)
							−0.805037	+	0.593225i	\(0.797854\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	451.2.h.a.324.38	yes	160
11.9	even	5		451.2.j.a.119.3	yes	160
41.10	even	5		451.2.j.a.379.3	yes	160
451.174	even	5	inner	451.2.h.a.174.38	✓	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
451.2.h.a.174.38	✓	160	451.174	even	5	inner
451.2.h.a.324.38	yes	160	1.1	even	1	trivial
451.2.j.a.119.3	yes	160	11.9	even	5
451.2.j.a.379.3	yes	160	41.10	even	5