Embedded newform 287.3.x.a.250.18

• Label: 287.3.x.a.250.18
• Level: $287$
• Weight: $3$
• Character: 287.250
• 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.x (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			250.18
Character	\(\chi\)	\(=\)	287.250
Dual form			287.3.x.a.31.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.149486 - 1.42226i) q^{2} +(-2.44356 - 4.23237i) q^{3} +(1.91210 - 0.406429i) q^{4} +(-3.74322 + 3.37041i) q^{5} +(-5.65427 + 4.10807i) q^{6} +(0.725236 + 6.96233i) q^{7} +(-2.63158 - 8.09918i) q^{8} +(-7.44197 + 12.8899i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 432 q - 3 q^{2} + 101 q^{4} - 9 q^{5} - 10 q^{7} - 40 q^{8} - 624 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{3}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.149486	−	1.42226i	−0.0747430	−	0.711132i	−0.966159	−	0.257947i	\(-0.916954\pi\)
							0.891416	−	0.453186i	\(-0.149713\pi\)
\(3\)	−2.44356	−	4.23237i	−0.814520	−	1.41079i	−0.909672	−	0.415327i	\(-0.863667\pi\)
							0.0951524	−	0.995463i	\(-0.469666\pi\)
\(5\)	−3.74322	+	3.37041i	−0.748643	+	0.674081i	−0.952370	−	0.304946i	\(-0.901361\pi\)
							0.203726	+	0.979028i	\(0.434695\pi\)
\(7\)	0.725236	+	6.96233i	0.103605	+	0.994619i
							\(11\)	13.4176	+	12.0813i	1.21978	+	1.09830i	0.992215	+	0.124533i	\(0.0397434\pi\)
0.227566	+	0.973763i	\(0.426923\pi\)
\(13\)	−15.0111	+	10.9062i	−1.15470	+	0.838939i	−0.989099	−	0.147253i	\(-0.952957\pi\)
							−0.165602	+	0.986193i	\(0.552957\pi\)
\(17\)	−13.4826	+	14.9739i	−0.793094	+	0.880820i	−0.995131	−	0.0985575i	\(-0.968577\pi\)
							0.202037	+	0.979378i	\(0.435244\pi\)
\(19\)	−4.12533	+	1.83671i	−0.217123	+	0.0966692i	−0.512416	−	0.858737i	\(-0.671249\pi\)
							0.295293	+	0.955407i	\(0.404583\pi\)
\(23\)	−2.84273	−	27.0467i	−0.123597	−	1.17594i	−0.863897	−	0.503668i	\(-0.831983\pi\)
							0.740300	−	0.672276i	\(-0.234683\pi\)
\(29\)	−6.00647	−	1.95162i	−0.207120	−	0.0672973i	0.203620	−	0.979050i	\(-0.434729\pi\)
							−0.410739	+	0.911753i	\(0.634729\pi\)
\(31\)	20.3436	+	18.3174i	0.656244	+	0.590885i	0.928497	−	0.371341i	\(-0.121102\pi\)
							−0.272252	+	0.962226i	\(0.587768\pi\)
\(37\)	−8.85841	−	9.83826i	−0.239417	−	0.265899i	0.611447	−	0.791285i	\(-0.290588\pi\)
							−0.850864	+	0.525386i	\(0.823921\pi\)
\(41\)	10.3071	−	39.6833i	0.251393	−	0.967885i
							\(43\)	−24.0663	+	17.4852i	−0.559683	+	0.406633i	−0.831343	−	0.555760i	\(-0.812427\pi\)
0.271660	+	0.962393i	\(0.412427\pi\)
\(47\)	7.28144	+	69.2782i	0.154924	+	1.47400i	0.745220	+	0.666818i	\(0.232344\pi\)
							−0.590296	+	0.807187i	\(0.700989\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.x.a.250.18	yes	432
7.3	odd	6	inner	287.3.x.a.45.37	yes	432
41.31	even	10	inner	287.3.x.a.236.37	yes	432
287.31	odd	30	inner	287.3.x.a.31.18	✓	432

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.x.a.31.18	✓	432	287.31	odd	30	inner
287.3.x.a.45.37	yes	432	7.3	odd	6	inner
287.3.x.a.236.37	yes	432	41.31	even	10	inner
287.3.x.a.250.18	yes	432	1.1	even	1	trivial