Embedded newform 287.3.m.a.85.12

• Label: 287.3.m.a.85.12
• Level: $287$
• Weight: $3$
• Character: 287.85
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $168$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.m (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(168\)
Relative dimension:	\(42\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			85.12
Character	\(\chi\)	\(=\)	287.85
Dual form			287.3.m.a.260.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.53733 + 1.53733i) q^{2} +(-1.13549 + 2.74132i) q^{3} -0.726768i q^{4} +(4.22119 + 4.22119i) q^{5} +(-2.46869 - 5.95995i) q^{6} +(-1.01249 + 2.44436i) q^{7} +(-5.03204 - 5.03204i) q^{8} +(0.138450 + 0.138450i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q + 8 q^{2} - 16 q^{3} + 24 q^{6} - 48 q^{8} - 48 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.53733	+	1.53733i	−0.768665	+	0.768665i	−0.977872	−	0.209206i	\(-0.932912\pi\)
							0.209206	+	0.977872i	\(0.432912\pi\)
\(3\)	−1.13549	+	2.74132i	−0.378498	+	0.913775i	0.613750	+	0.789500i	\(0.289660\pi\)
							−0.992248	+	0.124274i	\(0.960340\pi\)
\(5\)	4.22119	+	4.22119i	0.844239	+	0.844239i	0.989407	−	0.145168i	\(-0.0463723\pi\)
							−0.145168	+	0.989407i	\(0.546372\pi\)
\(7\)	−1.01249	+	2.44436i	−0.144641	+	0.349194i
							\(11\)	−10.7743	−	4.46287i	−0.979483	−	0.405715i	−0.165249	−	0.986252i	\(-0.552843\pi\)
−0.814234	+	0.580537i	\(0.802843\pi\)
\(13\)	−5.55675	+	13.4152i	−0.427442	+	1.03194i	0.552653	+	0.833411i	\(0.313615\pi\)
							−0.980096	+	0.198526i	\(0.936385\pi\)
\(17\)	11.9922	+	28.9517i	0.705423	+	1.70304i	0.711133	+	0.703057i	\(0.248182\pi\)
							−0.00571052	+	0.999984i	\(0.501818\pi\)
\(19\)	−4.84236	−	11.6905i	−0.254861	−	0.615289i	0.743723	−	0.668488i	\(-0.233058\pi\)
							−0.998584	+	0.0531989i	\(0.983058\pi\)
\(23\)		−	15.1735i		−	0.659719i	−0.944030	−	0.329860i	\(-0.892999\pi\)
							0.944030	−	0.329860i	\(-0.107001\pi\)
\(29\)	7.54774	−	18.2219i	0.260267	−	0.628340i	−0.738688	−	0.674048i	\(-0.764554\pi\)
							0.998955	+	0.0457075i	\(0.0145542\pi\)
\(31\)			5.57092i			0.179707i	0.995955	+	0.0898536i	\(0.0286399\pi\)
							−0.995955	+	0.0898536i	\(0.971360\pi\)
\(37\)	−23.2796			−0.629177			−0.314589	−	0.949228i	\(-0.601867\pi\)
							−0.314589	+	0.949228i	\(0.601867\pi\)
\(41\)	30.0484	−	27.8943i	0.732889	−	0.680349i
							\(43\)	39.0239	−	39.0239i	0.907532	−	0.907532i	−0.0885407	−	0.996073i	\(-0.528220\pi\)
0.996073	+	0.0885407i	\(0.0282203\pi\)
\(47\)	8.95869	+	21.6282i	0.190610	+	0.460174i	0.990075	−	0.140539i	\(-0.0448834\pi\)
							−0.799465	+	0.600713i	\(0.794883\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.m.a.85.12	✓	168
41.14	odd	8	inner	287.3.m.a.260.12	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.m.a.85.12	✓	168	1.1	even	1	trivial
287.3.m.a.260.12	yes	168	41.14	odd	8	inner