Embedded newform 451.2.h.a.59.13

• Label: 451.2.h.a.59.13
• 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			59.13
Character	\(\chi\)	\(=\)	451.59
Dual form			451.2.h.a.344.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.354023 + 1.08957i) q^{2} +(0.289073 - 0.889675i) q^{3} +(0.556200 + 0.404103i) q^{4} +(-0.219575 + 0.675783i) q^{5} +(0.867026 + 0.629932i) q^{6} -1.60325 q^{7} +(-2.49090 + 1.80974i) q^{8} +(1.71909 + 1.24899i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + q^{2} - 7 q^{3} - 39 q^{4} - q^{5} - 4 q^{6} - 6 q^{7} + 3 q^{8} - 45 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.354023	+	1.08957i	−0.250332	+	0.770444i	0.744381	+	0.667755i	\(0.232745\pi\)
							−0.994714	+	0.102689i	\(0.967255\pi\)
\(3\)	0.289073	−	0.889675i	0.166896	−	0.513654i	−0.832275	−	0.554364i	\(-0.812962\pi\)
							0.999171	+	0.0407093i	\(0.0129617\pi\)
\(5\)	−0.219575	+	0.675783i	−0.0981970	+	0.302219i	−0.988074	−	0.153982i	\(-0.950790\pi\)
							0.889877	+	0.456201i	\(0.150790\pi\)
\(7\)	−1.60325			−0.605973			−0.302986	−	0.952995i	\(-0.597984\pi\)
							−0.302986	+	0.952995i	\(0.597984\pi\)
\(11\)	2.88153	−	1.64219i	0.868814	−	0.495139i
							\(13\)	−2.08530	+	6.41791i	−0.578360	+	1.78001i	0.0460839	+	0.998938i	\(0.485326\pi\)
−0.624443	+	0.781070i	\(0.714674\pi\)
\(17\)	−1.31308			−0.318468			−0.159234	−	0.987241i	\(-0.550902\pi\)
							−0.159234	+	0.987241i	\(0.550902\pi\)
\(19\)	−1.43441	−	1.04216i	−0.329075	−	0.239087i	0.410963	−	0.911652i	\(-0.365193\pi\)
							−0.740038	+	0.672565i	\(0.765193\pi\)
\(23\)	0.662954	+	2.04036i	0.138235	+	0.425445i	0.996079	−	0.0884656i	\(-0.0281963\pi\)
							−0.857844	+	0.513910i	\(0.828196\pi\)
\(29\)	7.06821	+	5.13535i	1.31253	+	0.953611i	0.999993	+	0.00370658i	\(0.00117985\pi\)
							0.312540	+	0.949905i	\(0.398820\pi\)
\(31\)	−0.0185262	−	0.0570178i	−0.00332741	−	0.0102407i	0.949379	−	0.314133i	\(-0.101714\pi\)
							−0.952706	+	0.303892i	\(0.901714\pi\)
\(37\)	−6.42438	−	4.66759i	−1.05616	−	0.767347i	−0.0827875	−	0.996567i	\(-0.526382\pi\)
							−0.973374	+	0.229221i	\(0.926382\pi\)
\(41\)	1.84423	+	6.13179i	0.288020	+	0.957624i
							\(43\)	−2.16812	−	6.67279i	−0.330635	−	1.01759i	−0.968832	−	0.247717i	\(-0.920320\pi\)
0.638197	−	0.769873i	\(-0.279680\pi\)
\(47\)	0.686053			0.100071			0.0500356	−	0.998747i	\(-0.484067\pi\)
							0.0500356	+	0.998747i	\(0.484067\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	451.2.h.a.59.13	✓	160
11.3	even	5		451.2.j.a.223.28	yes	160
41.16	even	5		451.2.j.a.180.28	yes	160
451.344	even	5	inner	451.2.h.a.344.13	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
451.2.h.a.59.13	✓	160	1.1	even	1	trivial
451.2.h.a.344.13	yes	160	451.344	even	5	inner
451.2.j.a.180.28	yes	160	41.16	even	5
451.2.j.a.223.28	yes	160	11.3	even	5