Embedded newform 288.2.bf.a.11.7

• Label: 288.2.bf.a.11.7
• Level: $288$
• Weight: $2$
• Character: 288.11
• Analytic conductor: $2.300$
• Analytic rank: $0$
• Dimension: $368$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	288.bf (of order \(24\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29969157821\)
Analytic rank:	\(0\)
Dimension:	\(368\)
Relative dimension:	\(46\) over \(\Q(\zeta_{24})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{24}]$

Embedding invariants

Embedding label			11.7
Character	\(\chi\)	\(=\)	288.11
Dual form			288.2.bf.a.131.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28187 + 0.597341i) q^{2} +(-1.23148 + 1.21798i) q^{3} +(1.28637 - 1.53142i) q^{4} +(-0.451380 - 3.42857i) q^{5} +(0.851043 - 2.29689i) q^{6} +(0.762324 + 2.84503i) q^{7} +(-0.734171 + 2.73148i) q^{8} +(0.0330693 - 2.99982i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 12 q^{2} - 8 q^{3} - 4 q^{4} - 12 q^{5} - 8 q^{6} - 4 q^{7} - 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\)	\(37\)	\(65\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.28187	+	0.597341i	−0.906417	+	0.422384i
							\(3\)	−1.23148	+	1.21798i	−0.710993	+	0.703199i
\(5\)	−0.451380	−	3.42857i	−0.201863	−	1.53330i								−0.729588	−	0.683887i	\(-0.760288\pi\)
							0.527725	−	0.849415i	\(-0.323045\pi\)
\(7\)	0.762324	+	2.84503i	0.288131	+	1.07532i	0.946521	+	0.322643i	\(0.104571\pi\)
							−0.658389	+	0.752677i	\(0.728762\pi\)
\(11\)	−2.71791	+	2.08552i	−0.819480	+	0.628809i	−0.931027	−	0.364950i	\(-0.881086\pi\)
							0.111547	+	0.993759i	\(0.464419\pi\)
\(13\)	−3.74750	+	4.88384i	−1.03937	+	1.35453i	−0.106504	+	0.994312i	\(0.533966\pi\)
							−0.932867	+	0.360222i	\(0.882701\pi\)
\(17\)	3.38416			0.820780			0.410390	−	0.911910i	\(-0.365393\pi\)
							0.410390	+	0.911910i	\(0.365393\pi\)
\(19\)	−2.96770	+	1.22926i	−0.680838	+	0.282012i	−0.696177	−	0.717870i	\(-0.745117\pi\)
							0.0153393	+	0.999882i	\(0.495117\pi\)
\(23\)	−5.41028	−	1.44968i	−1.12812	−	0.302279i	−0.353956	−	0.935262i	\(-0.615164\pi\)
							−0.774166	+	0.632983i	\(0.781830\pi\)
\(29\)	−6.01017	−	0.791254i	−1.11606	−	0.146932i	−0.450142	−	0.892957i	\(-0.648627\pi\)
							−0.665918	+	0.746025i	\(0.731960\pi\)
\(31\)	−0.785209	+	0.453340i	−0.141028	+	0.0814224i	−0.568854	−	0.822439i	\(-0.692613\pi\)
							0.427826	+	0.903861i	\(0.359280\pi\)
\(37\)	0.444182	−	1.07235i	0.0730230	−	0.176293i	−0.883153	−	0.469084i	\(-0.844584\pi\)
							0.956176	+	0.292791i	\(0.0945841\pi\)
\(41\)	−3.23920	+	12.0889i	−0.505878	+	1.88796i	−0.0482122	+	0.998837i	\(0.515352\pi\)
							−0.457666	+	0.889124i	\(0.651314\pi\)
\(43\)	2.21400	+	2.88534i	0.337631	+	0.440010i	0.931054	−	0.364881i	\(-0.118890\pi\)
							−0.593423	+	0.804891i	\(0.702224\pi\)
\(47\)	−4.40315	−	2.54216i	−0.642265	−	0.370812i	0.143222	−	0.989691i	\(-0.454254\pi\)
							−0.785486	+	0.618879i	\(0.787587\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	288.2.bf.a.11.7	✓	368
3.2	odd	2		864.2.bn.a.683.40		368
9.4	even	3		864.2.bn.a.395.25		368
9.5	odd	6	inner	288.2.bf.a.203.22	yes	368
32.3	odd	8	inner	288.2.bf.a.227.22	yes	368
96.35	even	8		864.2.bn.a.35.25		368
288.67	odd	24		864.2.bn.a.611.40		368
288.131	even	24	inner	288.2.bf.a.131.7	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.2.bf.a.11.7	✓	368	1.1	even	1	trivial
288.2.bf.a.131.7	yes	368	288.131	even	24	inner
288.2.bf.a.203.22	yes	368	9.5	odd	6	inner
288.2.bf.a.227.22	yes	368	32.3	odd	8	inner
864.2.bn.a.35.25		368	96.35	even	8
864.2.bn.a.395.25		368	9.4	even	3
864.2.bn.a.611.40		368	288.67	odd	24
864.2.bn.a.683.40		368	3.2	odd	2