Embedded newform 287.2.w.a.3.9

• Label: 287.2.w.a.3.9
• 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			3.9
Character	\(\chi\)	\(=\)	287.3
Dual form			287.2.w.a.96.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.05419 - 0.282470i) q^{2} +(2.06157 - 1.58190i) q^{3} +(-0.700516 - 0.404443i) q^{4} +(1.89345 + 0.507348i) q^{5} +(-2.62013 + 1.08530i) q^{6} +(1.12649 - 2.39395i) q^{7} +(2.16768 + 2.16768i) q^{8} +(0.971213 - 3.62462i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 4 q^{2} - 12 q^{3} - 12 q^{5} - 8 q^{7} - 32 q^{8} + 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.05419	−	0.282470i	−0.745427	−	0.199737i	−0.133938	−	0.990990i	\(-0.542762\pi\)
							−0.611489	+	0.791253i	\(0.709429\pi\)
\(3\)	2.06157	−	1.58190i	1.19025	−	0.913310i	0.192561	−	0.981285i	\(-0.438321\pi\)
							0.997687	+	0.0679751i	\(0.0216539\pi\)
\(5\)	1.89345	+	0.507348i	0.846777	+	0.226893i	0.656019	−	0.754744i	\(-0.272239\pi\)
							0.190757	+	0.981637i	\(0.438906\pi\)
\(7\)	1.12649	−	2.39395i	0.425775	−	0.904829i
							\(11\)	−1.36431	+	1.04687i	−0.411356	+	0.315645i	−0.793626	−	0.608406i	\(-0.791809\pi\)
0.382270	+	0.924051i	\(0.375143\pi\)
\(13\)	1.59489	+	0.660624i	0.442342	+	0.183224i	0.592727	−	0.805403i	\(-0.298051\pi\)
							−0.150385	+	0.988628i	\(0.548051\pi\)
\(17\)	0.189288	−	1.43779i	0.0459091	−	0.348714i	−0.953039	−	0.302849i	\(-0.902062\pi\)
							0.998948	−	0.0458651i	\(-0.0146044\pi\)
\(19\)	−0.0450319	−	0.0345542i	−0.0103310	−	0.00792727i	0.603583	−	0.797300i	\(-0.293739\pi\)
							−0.613914	+	0.789373i	\(0.710406\pi\)
\(23\)	−5.99929	+	3.46369i	−1.25094	+	0.722230i	−0.971296	−	0.237874i	\(-0.923550\pi\)
							−0.279643	+	0.960104i	\(0.590216\pi\)
\(29\)	−0.254025	+	0.613271i	−0.0471713	+	0.113882i	−0.945709	−	0.325015i	\(-0.894631\pi\)
							0.898538	+	0.438897i	\(0.144631\pi\)
\(31\)	1.06787	−	1.84961i	0.191796	−	0.332200i	−0.754050	−	0.656818i	\(-0.771902\pi\)
							0.945845	+	0.324617i	\(0.105235\pi\)
\(37\)	−2.58186	−	4.47191i	−0.424455	−	0.735177i	0.571914	−	0.820313i	\(-0.306201\pi\)
							−0.996369	+	0.0851358i	\(0.972868\pi\)
\(41\)	6.14946	−	1.78441i	0.960385	−	0.278678i
							\(43\)	−6.80999	+	6.80999i	−1.03851	+	1.03851i	−0.0392858	+	0.999228i	\(0.512508\pi\)
−0.999228	+	0.0392858i	\(0.987492\pi\)
\(47\)	9.30225	+	7.13787i	1.35687	+	1.04117i	0.994006	+	0.109325i	\(0.0348688\pi\)
							0.362867	+	0.931841i	\(0.381798\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.w.a.3.9	✓	208
7.5	odd	6	inner	287.2.w.a.208.18	yes	208
41.14	odd	8	inner	287.2.w.a.178.18	yes	208
287.96	even	24	inner	287.2.w.a.96.9	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.w.a.3.9	✓	208	1.1	even	1	trivial
287.2.w.a.96.9	yes	208	287.96	even	24	inner
287.2.w.a.178.18	yes	208	41.14	odd	8	inner
287.2.w.a.208.18	yes	208	7.5	odd	6	inner