Embedded newform 287.3.i.a.40.12

• Label: 287.3.i.a.40.12
• Level: $287$
• Weight: $3$
• Character: 287.40
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Relative dimension:	\(54\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			40.12
Character	\(\chi\)	\(=\)	287.40
Dual form			287.3.i.a.122.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.24731 + 2.16040i) q^{2} +(0.923338 + 1.59927i) q^{3} +(-1.11156 - 1.92528i) q^{4} +(-1.05182 - 0.607270i) q^{5} -4.60675 q^{6} +(-4.85778 - 5.04004i) q^{7} -4.43264 q^{8} +(2.79489 - 4.84090i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 2 q^{2} - 106 q^{4} - 6 q^{5} + 20 q^{8} - 136 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.24731	+	2.16040i	−0.623655	+	1.08020i	0.365145	+	0.930951i	\(0.381019\pi\)
							−0.988799	+	0.149251i	\(0.952314\pi\)
\(3\)	0.923338	+	1.59927i	0.307779	+	0.533090i	0.977876	−	0.209184i	\(-0.0670808\pi\)
							−0.670097	+	0.742274i	\(0.733747\pi\)
\(5\)	−1.05182	−	0.607270i	−0.210364	−	0.121454i	0.391116	−	0.920341i	\(-0.372089\pi\)
							−0.601481	+	0.798887i	\(0.705422\pi\)
\(7\)	−4.85778	−	5.04004i	−0.693968	−	0.720005i
							\(11\)	10.0837	−	5.82181i	0.916698	−	0.529256i	0.0341178	−	0.999418i	\(-0.489138\pi\)
0.882580	+	0.470162i	\(0.155805\pi\)
\(13\)	−16.4289			−1.26376			−0.631882	−	0.775065i	\(-0.717717\pi\)
							−0.631882	+	0.775065i	\(0.717717\pi\)
\(17\)	−6.92311	−	11.9912i	−0.407242	−	0.705363i	0.587338	−	0.809342i	\(-0.300176\pi\)
							−0.994580	+	0.103978i	\(0.966843\pi\)
\(19\)	12.3068	−	21.3159i	0.647724	−	1.12189i	−0.335941	−	0.941883i	\(-0.609054\pi\)
							0.983665	−	0.180008i	\(-0.0576123\pi\)
\(23\)	−18.8731	+	32.6892i	−0.820571	+	1.42127i	0.0846863	+	0.996408i	\(0.473011\pi\)
							−0.905258	+	0.424863i	\(0.860322\pi\)
\(29\)		−	14.1571i		−	0.488176i	−0.969753	−	0.244088i	\(-0.921511\pi\)
							0.969753	−	0.244088i	\(-0.0784886\pi\)
\(31\)	−39.7485	+	22.9488i	−1.28221	+	0.740283i	−0.977252	−	0.212083i	\(-0.931975\pi\)
							−0.304957	+	0.952366i	\(0.598642\pi\)
\(37\)	−5.97634	+	10.3513i	−0.161523	+	0.279766i	−0.935415	−	0.353552i	\(-0.884974\pi\)
							0.773892	+	0.633317i	\(0.218307\pi\)
\(41\)	4.76832	−	40.7218i	0.116301	−	0.993214i
							\(43\)	38.2160			0.888743			0.444372	−	0.895843i	\(-0.353427\pi\)
0.444372	+	0.895843i	\(0.353427\pi\)
\(47\)	2.49961	−	4.32944i	0.0531831	−	0.0921158i	−0.838208	−	0.545350i	\(-0.816397\pi\)
							0.891391	+	0.453234i	\(0.149730\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.i.a.40.12	yes	108
7.3	odd	6	inner	287.3.i.a.122.11	yes	108
41.40	even	2	inner	287.3.i.a.40.11	✓	108
287.122	odd	6	inner	287.3.i.a.122.12	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.i.a.40.11	✓	108	41.40	even	2	inner
287.3.i.a.40.12	yes	108	1.1	even	1	trivial
287.3.i.a.122.11	yes	108	7.3	odd	6	inner
287.3.i.a.122.12	yes	108	287.122	odd	6	inner