Embedded newform 26.5.f.b.19.2

• Label: 26.5.f.b.19.2
• Level: $26$
• Weight: $5$
• Character: 26.19
• Analytic conductor: $2.688$
• Analytic rank: $0$
• Dimension: $8$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 26 = 2 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	26.f (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.68761904018\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{12})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial:	\( x^{8} - 494x^{6} + 90747x^{4} - 7343258x^{2} + 221087161 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			19.2
Root			\(12.0941 - 0.500000i\) of defining polynomial
Character	\(\chi\)	\(=\)	26.19
Dual form			26.5.f.b.11.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.732051 - 2.73205i) q^{2} +(7.34608 - 12.7238i) q^{3} +(-6.92820 + 4.00000i) q^{4} +(-10.3806 + 10.3806i) q^{5} +(-40.1397 - 10.7554i) q^{6} +(6.98005 - 26.0499i) q^{7} +(16.0000 + 16.0000i) q^{8} +(-67.4297 - 116.792i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 18 q^{5} - 36 q^{6} + 4 q^{7} + 128 q^{8} - 198 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)
\(\chi(n)\)	\(e\left(\frac{5}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.732051	−	2.73205i	−0.183013	−	0.683013i
							\(3\)	7.34608	−	12.7238i	0.816231	−	1.41375i	−0.0922102	−	0.995740i	\(-0.529393\pi\)
0.908441	−	0.418013i	\(-0.137273\pi\)
\(5\)	−10.3806	+	10.3806i	−0.415225	+	0.415225i	−0.883554	−	0.468329i	\(-0.844856\pi\)
							0.468329	+	0.883554i	\(0.344856\pi\)
\(7\)	6.98005	−	26.0499i	0.142450	−	0.531631i	−0.857406	−	0.514641i	\(-0.827925\pi\)
							0.999856	−	0.0169895i	\(-0.00540819\pi\)
\(11\)	143.137	−	38.3535i	1.18295	−	0.316971i	0.386855	−	0.922141i	\(-0.373562\pi\)
							0.796096	+	0.605170i	\(0.206895\pi\)
\(13\)	−26.7410	+	166.871i	−0.158231	+	0.987402i
							\(17\)	122.438	−	70.6894i	0.423660	−	0.244600i	−0.272982	−	0.962019i	\(-0.588010\pi\)
0.696642	+	0.717419i	\(0.254677\pi\)
\(19\)	662.337	+	177.473i	1.83473	+	0.491614i	0.998396	−	0.0566093i	\(-0.0180289\pi\)
							0.836332	+	0.548223i	\(0.184696\pi\)
\(23\)	−732.944	−	423.165i	−1.38553	−	0.799935i	−0.392720	−	0.919658i	\(-0.628466\pi\)
							−0.992807	+	0.119723i	\(0.961799\pi\)
\(29\)	113.079	−	195.859i	0.134458	−	0.232888i	−0.790932	−	0.611904i	\(-0.790404\pi\)
							0.925390	+	0.379016i	\(0.123737\pi\)
\(31\)	−997.522	+	997.522i	−1.03800	+	1.03800i	−0.0387558	+	0.999249i	\(0.512339\pi\)
							−0.999249	+	0.0387558i	\(0.987661\pi\)
\(37\)	−26.7375	+	7.16429i	−0.0195307	+	0.00523323i	−0.268571	−	0.963260i	\(-0.586551\pi\)
							0.249040	+	0.968493i	\(0.419885\pi\)
\(41\)	32.0735	+	119.700i	0.0190800	+	0.0712075i	0.974809	−	0.223039i	\(-0.0715979\pi\)
							−0.955729	+	0.294247i	\(0.904931\pi\)
\(43\)	−2831.15	+	1634.57i	−1.53118	+	0.884027i	−0.531872	+	0.846824i	\(0.678511\pi\)
							−0.999308	+	0.0372028i	\(0.988155\pi\)
\(47\)	405.040	+	405.040i	0.183359	+	0.183359i	0.792818	−	0.609459i	\(-0.208613\pi\)
							−0.609459	+	0.792818i	\(0.708613\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	26.5.f.b.19.2	yes	8
3.2	odd	2		234.5.bb.a.19.1		8
13.4	even	6		338.5.d.g.239.1		8
13.6	odd	12		338.5.d.g.99.1		8
13.7	odd	12		338.5.d.f.99.1		8
13.9	even	3		338.5.d.f.239.1		8
13.11	odd	12	inner	26.5.f.b.11.2	✓	8
39.11	even	12		234.5.bb.a.37.1		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.5.f.b.11.2	✓	8	13.11	odd	12	inner
26.5.f.b.19.2	yes	8	1.1	even	1	trivial
234.5.bb.a.19.1		8	3.2	odd	2
234.5.bb.a.37.1		8	39.11	even	12
338.5.d.f.99.1		8	13.7	odd	12
338.5.d.f.239.1		8	13.9	even	3
338.5.d.g.99.1		8	13.6	odd	12
338.5.d.g.239.1		8	13.4	even	6