Embedded newform 959.2.o.b.50.17

• Label: 959.2.o.b.50.17
• Level: $959$
• Weight: $2$
• Character: 959.50
• Analytic conductor: $7.658$
• Analytic rank: $0$
• Dimension: $576$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 959 = 7 \cdot 137 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	959.o (of order \(17\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.65765355384\)
Analytic rank:	\(0\)
Dimension:	\(576\)
Relative dimension:	\(36\) over \(\Q(\zeta_{17})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{17}]$

Embedding invariants

Embedding label			50.17
Character	\(\chi\)	\(=\)	959.50
Dual form			959.2.o.b.211.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0391285 - 0.0356703i) q^{2} +(1.47775 + 0.914985i) q^{3} +(-0.184278 + 1.98868i) q^{4} +(2.44859 + 2.23219i) q^{5} +(0.0904599 - 0.0169099i) q^{6} +(-0.273663 - 0.961826i) q^{7} +(0.127542 + 0.168893i) q^{8} +(0.00933609 + 0.0187494i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 576 q - 6 q^{2} - 4 q^{3} - 14 q^{4} - 6 q^{5} - 16 q^{6} - 36 q^{7} - 24 q^{8} - 54 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/959\mathbb{Z}\right)^\times\).

\(n\)	\(414\)	\(549\)
\(\chi(n)\)	\(e\left(\frac{3}{17}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0391285	−	0.0356703i	0.0276680	−	0.0252227i	−0.659744	−	0.751491i	\(-0.729335\pi\)
							0.687412	+	0.726268i	\(0.258747\pi\)
\(3\)	1.47775	+	0.914985i	0.853180	+	0.528267i	0.881963	−	0.471319i	\(-0.156222\pi\)
							−0.0287828	+	0.999586i	\(0.509163\pi\)
\(5\)	2.44859	+	2.23219i	1.09504	+	0.998265i	0.999996	+	0.00279199i	\(0.000888718\pi\)
							0.0950481	+	0.995473i	\(0.469700\pi\)
\(7\)	−0.273663	−	0.961826i	−0.103435	−	0.363536i
							\(11\)	−0.0683731	+	0.737863i	−0.0206153	+	0.222474i	0.979194	+	0.202928i	\(0.0650457\pi\)
−0.999809	+	0.0195463i	\(0.993778\pi\)
\(13\)	0.931623	+	3.27432i	0.258386	+	0.908132i	0.977092	+	0.212818i	\(0.0682641\pi\)
							−0.718706	+	0.695314i	\(0.755265\pi\)
\(17\)	1.38435	−	1.83317i	0.335753	−	0.444609i	−0.598551	−	0.801084i	\(-0.704257\pi\)
							0.934305	+	0.356475i	\(0.116022\pi\)
\(19\)	1.42174	+	0.550784i	0.326169	+	0.126359i	0.518758	−	0.854921i	\(-0.326394\pi\)
							−0.192589	+	0.981279i	\(0.561689\pi\)
\(23\)	2.25188	+	0.420950i	0.469550	+	0.0877742i	0.413207	−	0.910637i	\(-0.364409\pi\)
							0.0563437	+	0.998411i	\(0.482056\pi\)
\(29\)	−3.39311	−	0.634282i	−0.630085	−	0.117783i	−0.140986	−	0.990012i	\(-0.545027\pi\)
							−0.489099	+	0.872228i	\(0.662674\pi\)
\(31\)	3.87466	−	1.50105i	0.695910	−	0.269597i	0.0128058	−	0.999918i	\(-0.495924\pi\)
							0.683104	+	0.730321i	\(0.260630\pi\)
\(37\)	−1.17007			−0.192358			−0.0961790	−	0.995364i	\(-0.530662\pi\)
							−0.0961790	+	0.995364i	\(0.530662\pi\)
\(41\)	−1.84243			−0.287740			−0.143870	−	0.989597i	\(-0.545955\pi\)
							−0.143870	+	0.989597i	\(0.545955\pi\)
\(43\)	−0.884569	−	0.342684i	−0.134896	−	0.0522588i	0.292848	−	0.956159i	\(-0.405397\pi\)
							−0.427743	+	0.903900i	\(0.640691\pi\)
\(47\)	1.01935	+	2.04713i	0.148688	+	0.298605i	0.956968	−	0.290192i	\(-0.0937193\pi\)
							−0.808281	+	0.588797i	\(0.799602\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	959.2.o.b.50.17	✓	576
137.74	even	17	inner	959.2.o.b.211.17	yes	576

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
959.2.o.b.50.17	✓	576	1.1	even	1	trivial
959.2.o.b.211.17	yes	576	137.74	even	17	inner