Embedded newform 26.8.c.b.3.2

• Label: 26.8.c.b.3.2
• Level: $26$
• Weight: $8$
• Character: 26.3
• 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			3.2
Root			\(-18.4011i\) of defining polynomial
Character	\(\chi\)	\(=\)	26.3
Dual form			26.8.c.b.9.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.00000 + 6.92820i) q^{2} +(-15.9358 - 27.6017i) q^{3} +(-32.0000 + 55.4256i) q^{4} -54.4265 q^{5} +(127.487 - 220.813i) q^{6} +(556.354 - 963.633i) q^{7} -512.000 q^{8} +(585.599 - 1014.29i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 32 q^{2} - 256 q^{4} + 556 q^{5} - 548 q^{7} - 4096 q^{8} - 5214 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)

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
							\(3\)	−15.9358	−	27.6017i	−0.340761	−	0.590216i	0.643813	−	0.765183i	\(-0.277351\pi\)
−0.984574	+	0.174967i	\(0.944018\pi\)
\(5\)	−54.4265			−0.194722			−0.0973611	−	0.995249i	\(-0.531040\pi\)
							−0.0973611	+	0.995249i	\(0.531040\pi\)
\(7\)	556.354	−	963.633i	0.613067	−	1.06186i	−0.377653	−	0.925947i	\(-0.623269\pi\)
							0.990720	−	0.135916i	\(-0.0433978\pi\)
\(11\)	−3566.84	−	6177.94i	−0.807996	−	1.39949i	−0.914250	−	0.405150i	\(-0.867219\pi\)
							0.106255	−	0.994339i	\(-0.466114\pi\)
\(13\)	7565.86	+	2346.53i	0.955117	+	0.296227i
							\(17\)	10252.8	−	17758.4i	0.506142	−	0.876663i	−0.493833	−	0.869557i	\(-0.664405\pi\)
0.999975	−	0.00710642i	\(-0.00226206\pi\)
\(19\)	−14035.8	+	24310.8i	−0.469462	+	0.813133i	−0.999390	−	0.0349098i	\(-0.988886\pi\)
							0.529928	+	0.848043i	\(0.322219\pi\)
\(23\)	−16961.2	−	29377.7i	−0.290677	−	0.503467i	0.683293	−	0.730144i	\(-0.260547\pi\)
							−0.973970	+	0.226677i	\(0.927214\pi\)
\(29\)	87734.9	+	151961.i	0.668004	+	1.15702i	0.978461	+	0.206430i	\(0.0661845\pi\)
							−0.310457	+	0.950587i	\(0.600482\pi\)
\(31\)	15690.1			0.0945930			0.0472965	−	0.998881i	\(-0.484939\pi\)
							0.0472965	+	0.998881i	\(0.484939\pi\)
\(37\)	−998.478	−	1729.41i	−0.00324065	−	0.00561297i	0.864401	−	0.502804i	\(-0.167698\pi\)
							−0.867641	+	0.497191i	\(0.834365\pi\)
\(41\)	−111820.	−	193678.i	−0.253382	−	0.438870i	0.711073	−	0.703118i	\(-0.248210\pi\)
							−0.964455	+	0.264248i	\(0.914876\pi\)
\(43\)	268966.	−	465862.i	0.515890	−	0.893548i	−0.483940	−	0.875101i	\(-0.660795\pi\)
							0.999830	−	0.0184468i	\(-0.00587213\pi\)
\(47\)	542249.			0.761826			0.380913	−	0.924611i	\(-0.375610\pi\)
							0.380913	+	0.924611i	\(0.375610\pi\)