Embedded newform 287.2.w.a.3.15

• Label: 287.2.w.a.3.15
• Level: $287$
• Weight: $2$
• Character: 287.3
• Analytic conductor: $2.292$
• Analytic rank: $0$
• Dimension: $208$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	287.w (of order \(24\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29170653801\)
Analytic rank:	\(0\)
Dimension:	\(208\)
Relative dimension:	\(26\) over \(\Q(\zeta_{24})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{24}]$

Embedding invariants

Embedding label			3.15
Character	\(\chi\)	\(=\)	287.3
Dual form			287.2.w.a.96.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0492328 + 0.0131919i) q^{2} +(-1.37958 + 1.05859i) q^{3} +(-1.72980 - 0.998701i) q^{4} +(2.44382 + 0.654820i) q^{5} +(-0.0818854 + 0.0339181i) q^{6} +(0.674922 - 2.55822i) q^{7} +(-0.144070 - 0.144070i) q^{8} +(0.00617438 - 0.0230431i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 4 q^{2} - 12 q^{3} - 12 q^{5} - 8 q^{7} - 32 q^{8} + 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/287\mathbb{Z}\right)^\times\).

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{3}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0492328	+	0.0131919i	0.0348128	+	0.00932807i	0.276183	−	0.961105i	\(-0.410930\pi\)
							−0.241371	+	0.970433i	\(0.577597\pi\)
\(3\)	−1.37958	+	1.05859i	−0.796501	+	0.611177i	−0.924754	−	0.380564i	\(-0.875730\pi\)
							0.128253	+	0.991741i	\(0.459063\pi\)
\(5\)	2.44382	+	0.654820i	1.09291	+	0.292844i	0.759875	−	0.650070i	\(-0.225260\pi\)
							0.333036	+	0.942914i	\(0.391927\pi\)
\(7\)	0.674922	−	2.55822i	0.255097	−	0.966916i
							\(11\)	1.12636	−	0.864287i	0.339610	−	0.260592i	−0.424849	−	0.905264i	\(-0.639673\pi\)
0.764459	+	0.644672i	\(0.223006\pi\)
\(13\)	4.77223	+	1.97672i	1.32358	+	0.548244i	0.928817	−	0.370540i	\(-0.120827\pi\)
							0.394762	+	0.918784i	\(0.370827\pi\)
\(17\)	−0.157819	+	1.19875i	−0.0382767	+	0.290740i	0.961591	+	0.274486i	\(0.0885075\pi\)
							−0.999868	+	0.0162546i	\(0.994826\pi\)
\(19\)	4.18135	+	3.20846i	0.959268	+	0.736072i	0.964546	−	0.263914i	\(-0.0850135\pi\)
							−0.00527835	+	0.999986i	\(0.501680\pi\)
\(23\)	4.74479	−	2.73940i	0.989357	−	0.571205i	0.0842747	−	0.996443i	\(-0.473143\pi\)
							0.905082	+	0.425237i	\(0.139809\pi\)
\(29\)	−0.982733	+	2.37253i	−0.182489	+	0.440567i	−0.988478	−	0.151363i	\(-0.951634\pi\)
							0.805989	+	0.591930i	\(0.201634\pi\)
\(31\)	2.81243	−	4.87127i	0.505127	−	0.874906i	−0.494855	−	0.868975i	\(-0.664779\pi\)
							0.999982	−	0.00593043i	\(-0.00188773\pi\)
\(37\)	2.77108	+	4.79965i	0.455562	+	0.789057i	0.998720	−	0.0505735i	\(-0.0161049\pi\)
							−0.543158	+	0.839630i	\(0.682772\pi\)
\(41\)	−1.36755	−	6.25538i	−0.213575	−	0.976927i
							\(43\)	−2.13434	+	2.13434i	−0.325485	+	0.325485i	−0.850866	−	0.525382i	\(-0.823922\pi\)
0.525382	+	0.850866i	\(0.323922\pi\)
\(47\)	−5.27012	−	4.04391i	−0.768726	−	0.589864i	0.148183	−	0.988960i	\(-0.452657\pi\)
							−0.916909	+	0.399096i	\(0.869324\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.w.a.3.15	✓	208
7.5	odd	6	inner	287.2.w.a.208.12	yes	208
41.14	odd	8	inner	287.2.w.a.178.12	yes	208
287.96	even	24	inner	287.2.w.a.96.15	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.w.a.3.15	✓	208	1.1	even	1	trivial
287.2.w.a.96.15	yes	208	287.96	even	24	inner
287.2.w.a.178.12	yes	208	41.14	odd	8	inner
287.2.w.a.208.12	yes	208	7.5	odd	6	inner