Embedded newform 26.8.c.b.9.1

• Label: 26.8.c.b.9.1
• Level: $26$
• Weight: $8$
• Character: 26.9
• Analytic conductor: $8.122$
• Analytic rank: $0$
• Dimension: $8$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 26 = 2 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	26.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.12201066259\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial:	\( x^{8} + 4654x^{6} + 7012369x^{4} + 3763719168x^{2} + 637953638400 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 2^{6}\cdot 3 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			9.1
Root			\(45.7672i\) of defining polynomial
Character	\(\chi\)	\(=\)	26.9
Dual form			26.8.c.b.3.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.00000 - 6.92820i) q^{2} +(-39.6356 + 68.6509i) q^{3} +(-32.0000 - 55.4256i) q^{4} +132.638 q^{5} +(317.085 + 549.207i) q^{6} +(-754.459 - 1306.76i) q^{7} -512.000 q^{8} +(-2048.46 - 3548.04i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 32 q^{2} - 256 q^{4} + 556 q^{5} - 548 q^{7} - 4096 q^{8} - 5214 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/26\mathbb{Z}\right)^\times\).

\(n\)	\(15\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.00000	−	6.92820i	0.353553	−	0.612372i
							\(3\)	−39.6356	+	68.6509i	−0.847542	+	1.46799i	0.0358539	+	0.999357i	\(0.488585\pi\)
−0.883396	+	0.468628i	\(0.844748\pi\)
\(5\)	132.638			0.474539			0.237270	−	0.971444i	\(-0.423748\pi\)
							0.237270	+	0.971444i	\(0.423748\pi\)
\(7\)	−754.459	−	1306.76i	−0.831366	−	1.43997i	−0.896955	−	0.442122i	\(-0.854226\pi\)
							0.0655888	−	0.997847i	\(-0.479107\pi\)
\(11\)	1463.11	−	2534.19i	0.331439	−	0.574070i	−0.651355	−	0.758773i	\(-0.725799\pi\)
							0.982794	+	0.184703i	\(0.0591325\pi\)
\(13\)	−7917.74	−	240.503i	−0.999539	−	0.0303612i
							\(17\)	10869.5	+	18826.5i	0.536583	+	0.929389i	0.999085	+	0.0427706i	\(0.0136185\pi\)
−0.462502	+	0.886618i	\(0.653048\pi\)
\(19\)	−26739.1	−	46313.4i	−0.894352	−	1.54906i	−0.834604	−	0.550850i	\(-0.814303\pi\)
							−0.0597485	−	0.998213i	\(-0.519030\pi\)
\(23\)	14224.0	−	24636.6i	0.243766	−	0.422215i	−0.718018	−	0.696025i	\(-0.754950\pi\)
							0.961784	+	0.273809i	\(0.0882837\pi\)
\(29\)	−76324.1	+	132197.i	−0.581124	+	1.00654i	0.414222	+	0.910176i	\(0.364054\pi\)
							−0.995346	+	0.0963608i	\(0.969280\pi\)
\(31\)	83326.6			0.502363			0.251182	−	0.967940i	\(-0.419181\pi\)
							0.251182	+	0.967940i	\(0.419181\pi\)
\(37\)	−56229.4	+	97392.2i	−0.182498	+	0.316095i	−0.942730	−	0.333555i	\(-0.891752\pi\)
							0.760233	+	0.649651i	\(0.225085\pi\)
\(41\)	−55065.1	+	95375.6i	−0.124777	+	0.216119i	−0.921646	−	0.388033i	\(-0.873155\pi\)
							0.796869	+	0.604152i	\(0.206488\pi\)
\(43\)	−52967.4	−	91742.2i	−0.101594	−	0.175966i	0.810747	−	0.585396i	\(-0.199061\pi\)
							−0.912342	+	0.409430i	\(0.865728\pi\)
\(47\)	413100.			0.580381			0.290190	−	0.956969i	\(-0.406281\pi\)
							0.290190	+	0.956969i	\(0.406281\pi\)