Embedded newform 287.3.m.a.85.1

• Label: 287.3.m.a.85.1
• 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			85.1
Character	\(\chi\)	\(=\)	287.85
Dual form			287.3.m.a.260.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.66518 + 2.66518i) q^{2} +(-0.402189 + 0.970969i) q^{3} -10.2064i q^{4} +(-1.88272 - 1.88272i) q^{5} +(-1.51590 - 3.65972i) q^{6} +(-1.01249 + 2.44436i) q^{7} +(16.5412 + 16.5412i) q^{8} +(5.58294 + 5.58294i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q + 8 q^{2} - 16 q^{3} + 24 q^{6} - 48 q^{8} - 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.66518	+	2.66518i	−1.33259	+	1.33259i	−0.429548	+	0.903044i	\(0.641327\pi\)
							−0.903044	+	0.429548i	\(0.858673\pi\)
\(3\)	−0.402189	+	0.970969i	−0.134063	+	0.323656i	−0.976628	−	0.214939i	\(-0.931045\pi\)
							0.842565	+	0.538595i	\(0.181045\pi\)
\(5\)	−1.88272	−	1.88272i	−0.376545	−	0.376545i	0.493309	−	0.869854i	\(-0.335787\pi\)
							−0.869854	+	0.493309i	\(0.835787\pi\)
\(7\)	−1.01249	+	2.44436i	−0.144641	+	0.349194i
							\(11\)	6.67629	+	2.76541i	0.606936	+	0.251401i	0.664918	−	0.746917i	\(-0.268467\pi\)
−0.0579819	+	0.998318i	\(0.518467\pi\)
\(13\)	−5.41661	+	13.0769i	−0.416662	+	1.00591i	0.566646	+	0.823962i	\(0.308241\pi\)
							−0.983308	+	0.181950i	\(0.941759\pi\)
\(17\)	−1.99895	−	4.82590i	−0.117585	−	0.283876i	0.854119	−	0.520078i	\(-0.174097\pi\)
							−0.971704	+	0.236202i	\(0.924097\pi\)
\(19\)	−10.1317	−	24.4600i	−0.533246	−	1.28737i	−0.929362	−	0.369169i	\(-0.879643\pi\)
							0.396117	−	0.918200i	\(-0.370357\pi\)
\(23\)			33.0028i			1.43491i	0.696607	+	0.717453i	\(0.254692\pi\)
							−0.696607	+	0.717453i	\(0.745308\pi\)
\(29\)	−7.86827	+	18.9957i	−0.271320	+	0.655024i	−0.999540	−	0.0303190i	\(-0.990348\pi\)
							0.728220	+	0.685343i	\(0.240348\pi\)
\(31\)			7.58225i			0.244589i	0.992494	+	0.122294i	\(0.0390252\pi\)
							−0.992494	+	0.122294i	\(0.960975\pi\)
\(37\)	−59.2044			−1.60012			−0.800059	−	0.599921i	\(-0.795198\pi\)
							−0.800059	+	0.599921i	\(0.795198\pi\)
\(41\)	34.9853	−	21.3782i	0.853301	−	0.521419i
							\(43\)	−45.0505	+	45.0505i	−1.04769	+	1.04769i	−0.0488818	+	0.998805i	\(0.515566\pi\)
−0.998805	+	0.0488818i	\(0.984434\pi\)
\(47\)	−24.7186	−	59.6760i	−0.525928	−	1.26970i	−0.934170	−	0.356829i	\(-0.883858\pi\)
							0.408242	−	0.912874i	\(-0.366142\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.m.a.85.1	✓	168
41.14	odd	8	inner	287.3.m.a.260.1	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.m.a.85.1	✓	168	1.1	even	1	trivial
287.3.m.a.260.1	yes	168	41.14	odd	8	inner