Embedded newform 574.2.bf.b.101.3

• Label: 574.2.bf.b.101.3
• Level: $574$
• Weight: $2$
• Character: 574.101
• Analytic conductor: $4.583$
• Analytic rank: $0$
• Dimension: $448$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 574 = 2 \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	574.bf (of order \(120\), degree \(32\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.58341307602\)
Analytic rank:	\(0\)
Dimension:	\(448\)
Relative dimension:	\(14\) over \(\Q(\zeta_{120})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{120}]$

Embedding invariants

Embedding label			101.3
Character	\(\chi\)	\(=\)	574.101
Dual form			574.2.bf.b.341.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.838671 + 0.544639i) q^{2} +(-0.309323 + 2.34954i) q^{3} +(0.406737 + 0.913545i) q^{4} +(-3.21496 + 2.60343i) q^{5} +(-1.53907 + 1.80202i) q^{6} +(2.64345 - 0.110429i) q^{7} +(-0.156434 + 0.987688i) q^{8} +(-2.52688 - 0.677075i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 448 q + 8 q^{7} + 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/574\mathbb{Z}\right)^\times\).

\(n\)	\(211\)	\(493\)
\(\chi(n)\)	\(e\left(\frac{9}{40}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.838671	+	0.544639i	0.593030	+	0.385118i
							\(3\)	−0.309323	+	2.34954i	−0.178588	+	1.35651i	0.633849	+	0.773457i	\(0.281474\pi\)
−0.812436	+	0.583050i	\(0.801859\pi\)
\(5\)	−3.21496	+	2.60343i	−1.43777	+	1.16429i	−0.479821	+	0.877367i	\(0.659298\pi\)
							−0.957954	+	0.286921i	\(0.907368\pi\)
\(7\)	2.64345	−	0.110429i	0.999129	−	0.0417383i
							\(11\)	2.97623	+	5.48154i	0.897368	+	1.65275i	0.747727	+	0.664006i	\(0.231145\pi\)
0.149641	+	0.988740i	\(0.452188\pi\)
\(13\)	−0.294925	−	3.74738i	−0.0817975	−	1.03934i	−0.891657	−	0.452712i	\(-0.850457\pi\)
							0.809859	−	0.586624i	\(-0.199543\pi\)
\(17\)	1.50543	−	5.08225i	0.365120	−	1.23263i	−0.551942	−	0.833882i	\(-0.686113\pi\)
							0.917063	−	0.398743i	\(-0.130554\pi\)
\(19\)	−3.28268	+	1.56576i	−0.753099	+	0.359210i	−0.767806	−	0.640682i	\(-0.778652\pi\)
							0.0147074	+	0.999892i	\(0.495318\pi\)
\(23\)	−0.198448	−	0.933624i	−0.0413793	−	0.194674i	0.952603	−	0.304217i	\(-0.0983947\pi\)
							−0.993982	+	0.109543i	\(0.965061\pi\)
\(29\)	0.518996	−	2.16177i	0.0963752	−	0.401432i	−0.903241	−	0.429134i	\(-0.858819\pi\)
							0.999616	+	0.0277020i	\(0.00881896\pi\)
\(31\)	0.189013	+	1.79834i	0.0339477	+	0.322991i	0.998296	+	0.0583542i	\(0.0185853\pi\)
							−0.964348	+	0.264636i	\(0.914748\pi\)
\(37\)	0.197727	−	1.88124i	0.0325061	−	0.309275i	−0.966173	−	0.257895i	\(-0.916971\pi\)
							0.998679	−	0.0513802i	\(-0.0163620\pi\)
\(41\)	−0.923333	+	6.33620i	−0.144200	+	0.989549i
							\(43\)	4.13337	−	2.10606i	0.630332	−	0.321170i	−0.109475	−	0.993990i	\(-0.534917\pi\)
0.739807	+	0.672819i	\(0.234917\pi\)
\(47\)	3.92981	+	0.728347i	0.573222	+	0.106240i	0.460932	−	0.887435i	\(-0.347515\pi\)
							0.112289	+	0.993676i	\(0.464182\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	574.2.bf.b.101.3	yes	448
7.5	odd	6	inner	574.2.bf.b.19.12	✓	448
41.13	odd	40	inner	574.2.bf.b.423.12	yes	448
287.54	even	120	inner	574.2.bf.b.341.3	yes	448

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
574.2.bf.b.19.12	✓	448	7.5	odd	6	inner
574.2.bf.b.101.3	yes	448	1.1	even	1	trivial
574.2.bf.b.341.3	yes	448	287.54	even	120	inner
574.2.bf.b.423.12	yes	448	41.13	odd	40	inner