Embedded newform 570.2.bi.a.23.14

• Label: 570.2.bi.a.23.14
• Level: $570$
• Weight: $2$
• Character: 570.23
• Analytic conductor: $4.551$
• Analytic rank: $0$
• Dimension: $480$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	570.bi (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.55147291521\)
Analytic rank:	\(0\)
Dimension:	\(480\)
Relative dimension:	\(40\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			23.14
Character	\(\chi\)	\(=\)	570.23
Dual form			570.2.bi.a.347.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.422618 - 0.906308i) q^{2} +(1.13741 - 1.30625i) q^{3} +(-0.642788 + 0.766044i) q^{4} +(2.03932 - 0.917164i) q^{5} +(-1.66456 - 0.478800i) q^{6} +(-1.13371 - 4.23108i) q^{7} +(0.965926 + 0.258819i) q^{8} +(-0.412587 - 2.97149i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/570\mathbb{Z}\right)^\times\).

\(n\)	\(191\)	\(211\)	\(457\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.422618	−	0.906308i	−0.298836	−	0.640856i
							\(3\)	1.13741	−	1.30625i	0.656685	−	0.754165i
\(5\)	2.03932	−	0.917164i	0.912010	−	0.410168i
							\(7\)	−1.13371	−	4.23108i	−0.428504	−	1.59920i	−0.756150	−	0.654398i	\(-0.772922\pi\)
0.327646	−	0.944800i	\(-0.393745\pi\)
\(11\)	1.81764	−	1.04941i	0.548039	−	0.316410i	−0.200292	−	0.979736i	\(-0.564189\pi\)
							0.748331	+	0.663326i	\(0.230856\pi\)
\(13\)	−3.84366	+	2.69136i	−1.06604	+	0.746449i	−0.968558	−	0.248789i	\(-0.919967\pi\)
							−0.0974812	+	0.995237i	\(0.531079\pi\)
\(17\)	1.56811	+	3.36282i	0.380322	+	0.815602i	0.999538	+	0.0303823i	\(0.00967248\pi\)
							−0.619217	+	0.785220i	\(0.712550\pi\)
\(19\)	−1.31641	+	4.15537i	−0.302006	+	0.953306i
							\(23\)	6.58540	−	0.576148i	1.37315	−	0.120135i	0.623425	−	0.781884i	\(-0.285741\pi\)
0.749726	+	0.661748i	\(0.230185\pi\)
\(29\)	−7.59687	+	2.76504i	−1.41070	+	0.513454i	−0.931336	−	0.364160i	\(-0.881356\pi\)
							−0.479367	+	0.877614i	\(0.659134\pi\)
\(31\)	1.67251	−	2.89688i	0.300392	−	0.520294i	−0.675833	−	0.737055i	\(-0.736216\pi\)
							0.976225	+	0.216761i	\(0.0695492\pi\)
\(37\)	7.16856	−	7.16856i	1.17850	−	1.17850i	0.198378	−	0.980126i	\(-0.436433\pi\)
							0.980126	−	0.198378i	\(-0.0635674\pi\)
\(41\)	−7.18278	+	1.26652i	−1.12176	+	0.197797i	−0.703614	−	0.710583i	\(-0.748431\pi\)
							−0.418147	+	0.908379i	\(0.637320\pi\)
\(43\)	−0.953329	−	0.0834055i	−0.145381	−	0.0127192i	0.0142327	−	0.999899i	\(-0.495469\pi\)
							−0.159614	+	0.987179i	\(0.551025\pi\)
\(47\)	11.1066	+	5.17910i	1.62007	+	0.755450i	0.999586	−	0.0287713i	\(-0.00915946\pi\)
							0.620481	+	0.784221i	\(0.286937\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	570.2.bi.a.23.14	✓	480
3.2	odd	2	inner	570.2.bi.a.23.34	yes	480
5.2	odd	4	inner	570.2.bi.a.137.24	yes	480
15.2	even	4	inner	570.2.bi.a.137.5	yes	480
19.5	even	9	inner	570.2.bi.a.233.5	yes	480
57.5	odd	18	inner	570.2.bi.a.233.24	yes	480
95.62	odd	36	inner	570.2.bi.a.347.34	yes	480
285.62	even	36	inner	570.2.bi.a.347.14	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
570.2.bi.a.23.14	✓	480	1.1	even	1	trivial
570.2.bi.a.23.34	yes	480	3.2	odd	2	inner
570.2.bi.a.137.5	yes	480	15.2	even	4	inner
570.2.bi.a.137.24	yes	480	5.2	odd	4	inner
570.2.bi.a.233.5	yes	480	19.5	even	9	inner
570.2.bi.a.233.24	yes	480	57.5	odd	18	inner
570.2.bi.a.347.14	yes	480	285.62	even	36	inner
570.2.bi.a.347.34	yes	480	95.62	odd	36	inner