Embedded newform 287.3.y.a.10.12

• Label: 287.3.y.a.10.12
• Level: $287$
• Weight: $3$
• Character: 287.10
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $432$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.y (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(432\)
Relative dimension:	\(54\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			10.12
Character	\(\chi\)	\(=\)	287.10
Dual form			287.3.y.a.201.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.262361 + 2.49620i) q^{2} +(3.99634 + 2.30729i) q^{3} +(-2.24959 - 0.478164i) q^{4} +(-1.09445 - 0.985445i) q^{5} +(-6.80793 + 9.37031i) q^{6} +(6.92637 + 1.01262i) q^{7} +(-1.31867 + 4.05845i) q^{8} +(6.14714 + 10.6472i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 432 q - 3 q^{2} - 24 q^{3} + 101 q^{4} - 9 q^{5} - 6 q^{7} - 16 q^{8} + 576 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.262361	+	2.49620i	−0.131181	+	1.24810i	0.708771	+	0.705438i	\(0.249250\pi\)
							−0.839952	+	0.542661i	\(0.817417\pi\)
\(3\)	3.99634	+	2.30729i	1.33211	+	0.769095i	0.985623	−	0.168959i	\(-0.0540405\pi\)
							0.346489	+	0.938054i	\(0.387374\pi\)
\(5\)	−1.09445	−	0.985445i	−0.218890	−	0.197089i	0.552390	−	0.833586i	\(-0.313716\pi\)
							−0.771279	+	0.636497i	\(0.780383\pi\)
\(7\)	6.92637	+	1.01262i	0.989482	+	0.144659i
							\(11\)	1.30503	+	1.44938i	0.118639	+	0.131762i	0.799538	−	0.600615i	\(-0.205078\pi\)
−0.680899	+	0.732377i	\(0.738411\pi\)
\(13\)	7.05428	−	9.70939i	0.542637	−	0.746876i	−0.446353	−	0.894857i	\(-0.647277\pi\)
							0.988990	+	0.147981i	\(0.0472775\pi\)
\(17\)	−9.18838	+	8.27325i	−0.540493	+	0.486662i	−0.893565	−	0.448934i	\(-0.851804\pi\)
							0.353072	+	0.935596i	\(0.385137\pi\)
\(19\)	−3.27957	+	7.36603i	−0.172609	+	0.387686i	−0.979047	−	0.203632i	\(-0.934725\pi\)
							0.806439	+	0.591318i	\(0.201392\pi\)
\(23\)	−1.99161	+	18.9489i	−0.0865918	+	0.823866i	0.861903	+	0.507073i	\(0.169273\pi\)
							−0.948495	+	0.316793i	\(0.897394\pi\)
\(29\)	−4.17582	−	12.8519i	−0.143994	−	0.443167i	0.852886	−	0.522096i	\(-0.174850\pi\)
							−0.996880	+	0.0789291i	\(0.974850\pi\)
\(31\)	18.9357	−	17.0497i	0.610828	−	0.549992i	−0.304597	−	0.952481i	\(-0.598522\pi\)
							0.915425	+	0.402489i	\(0.131855\pi\)
\(37\)	−12.9355	+	14.3664i	−0.349609	+	0.388281i	−0.892143	−	0.451754i	\(-0.850799\pi\)
							0.542533	+	0.840034i	\(0.317465\pi\)
\(41\)	4.74275	−	40.7248i	0.115677	−	0.993287i
							\(43\)	−11.6327	−	8.45163i	−0.270527	−	0.196549i	0.444248	−	0.895904i	\(-0.353471\pi\)
−0.714775	+	0.699354i	\(0.753471\pi\)
\(47\)	−20.3382	−	2.13763i	−0.432727	−	0.0454814i	−0.114339	−	0.993442i	\(-0.536475\pi\)
							−0.318388	+	0.947960i	\(0.603142\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.y.a.10.12	✓	432
7.5	odd	6	inner	287.3.y.a.215.43	yes	432
41.37	even	5	inner	287.3.y.a.283.43	yes	432
287.201	odd	30	inner	287.3.y.a.201.12	yes	432

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.y.a.10.12	✓	432	1.1	even	1	trivial
287.3.y.a.201.12	yes	432	287.201	odd	30	inner
287.3.y.a.215.43	yes	432	7.5	odd	6	inner
287.3.y.a.283.43	yes	432	41.37	even	5	inner