Embedded newform 675.2.l.c.76.1

• Label: 675.2.l.c.76.1
• Level: $675$
• Weight: $2$
• Character: 675.76
• Analytic conductor: $5.390$
• Analytic rank: $0$
• Dimension: $12$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	675.l (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.38990213644\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(2\) over \(\Q(\zeta_{9})\)
Coefficient field:	12.0.1952986685049.1
Defining polynomial:	x^12 - 6*x^11 + 27*x^10 - 80*x^9 + 186*x^8 - 330*x^7 + 463*x^6 - 504*x^5 + 420*x^4 - 258*x^3 + 108*x^2 - 27*x + 3
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			76.1
Root			\(0.500000 + 2.22827i\) of defining polynomial
Character	\(\chi\)	\(=\)	675.76
Dual form			675.2.l.c.151.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.318266 - 0.267057i) q^{2} +(-0.159815 + 1.72466i) q^{3} +(-0.317323 + 1.79963i) q^{4} +(0.409719 + 0.591580i) q^{6} +(0.229151 + 1.29958i) q^{7} +(0.795075 + 1.37711i) q^{8} +(-2.94892 - 0.551252i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{2} + 6 q^{3} - 6 q^{4} + 6 q^{7} - 6 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).

\(n\)	\(326\)	\(352\)
\(\chi(n)\)	\(e\left(\frac{7}{9}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.318266	−	0.267057i	0.225048	−	0.188837i	−0.523291	−	0.852154i	\(-0.675296\pi\)
							0.748339	+	0.663316i	\(0.230852\pi\)
\(3\)	−0.159815	+	1.72466i	−0.0922690	+	0.995734i
							\(5\)		0			0
\(7\)	0.229151	+	1.29958i	0.0866110	+	0.491195i								0.996997	+	0.0774361i	\(0.0246734\pi\)
							−0.910386	+	0.413759i	\(0.864216\pi\)
\(11\)	4.90067	+	1.78370i	1.47761	+	0.537805i	0.950155	−	0.311778i	\(-0.100925\pi\)
							0.527454	+	0.849584i	\(0.323147\pi\)
\(13\)	0.0138336	+	0.0116078i	0.00383676	+	0.00321942i	0.644704	−	0.764432i	\(-0.276981\pi\)
							−0.640867	+	0.767652i	\(0.721425\pi\)
\(17\)	−1.56640	+	2.71308i	−0.379907	+	0.658019i	−0.991048	−	0.133503i	\(-0.957377\pi\)
							0.611141	+	0.791522i	\(0.290711\pi\)
\(19\)	−0.208676	−	0.361438i	−0.0478736	−	0.0829195i	0.841096	−	0.540886i	\(-0.181911\pi\)
							−0.888969	+	0.457967i	\(0.848578\pi\)
\(23\)	0.179619	−	1.01867i	0.0374532	−	0.212408i	−0.960338	−	0.278839i	\(-0.910050\pi\)
							0.997791	+	0.0664316i	\(0.0211614\pi\)
\(29\)	−5.98068	+	5.01839i	−1.11058	+	0.931891i	−0.998091	−	0.0617615i	\(-0.980328\pi\)
							−0.112493	+	0.993652i	\(0.535884\pi\)
\(31\)	0.647649	−	3.67300i	0.116321	−	0.659691i	−0.869766	−	0.493464i	\(-0.835730\pi\)
							0.986088	−	0.166227i	\(-0.0531584\pi\)
\(37\)	2.21238	−	3.83195i	0.363713	−	0.629969i	−0.624856	−	0.780740i	\(-0.714842\pi\)
							0.988569	+	0.150771i	\(0.0481756\pi\)
\(41\)	−2.81517	−	2.36221i	−0.439655	−	0.368915i	0.395925	−	0.918283i	\(-0.370424\pi\)
							−0.835580	+	0.549368i	\(0.814868\pi\)
\(43\)	−7.80685	−	2.84146i	−1.19053	−	0.433319i	−0.330622	−	0.943763i	\(-0.607259\pi\)
							−0.859911	+	0.510445i	\(0.829481\pi\)
\(47\)	1.23254	+	6.99008i	0.179784	+	1.01961i	0.932475	+	0.361234i	\(0.117644\pi\)
							−0.752691	+	0.658374i	\(0.771245\pi\)