Embedded newform 169.4.h.a.12.19

• Label: 169.4.h.a.12.19
• Level: $169$
• Weight: $4$
• Character: 169.12
• Analytic conductor: $9.971$
• Analytic rank: $0$
• Dimension: $528$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	169.h (of order \(26\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.97132279097\)
Analytic rank:	\(0\)
Dimension:	\(528\)
Relative dimension:	\(44\) over \(\Q(\zeta_{26})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{26}]$

Embedding invariants

Embedding label			12.19
Character	\(\chi\)	\(=\)	169.12
Dual form			169.4.h.a.155.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.997032 + 0.688202i) q^{2} +(-6.71687 - 3.52528i) q^{3} +(-2.31639 + 6.10781i) q^{4} +(-8.17932 - 9.23255i) q^{5} +(9.12304 - 1.10774i) q^{6} +(-7.34833 - 29.8134i) q^{7} +(-4.21331 - 17.0941i) q^{8} +(17.3509 + 25.1372i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 528 q - 13 q^{2} - 11 q^{3} + 157 q^{4} - 13 q^{5} - 13 q^{6} - 13 q^{7} - 13 q^{8} - 339 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{21}{26}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.997032	+	0.688202i	−0.352504	+	0.243316i	−0.731137	−	0.682231i	\(-0.761010\pi\)
							0.378633	+	0.925547i	\(0.376394\pi\)
\(3\)	−6.71687	−	3.52528i	−1.29266	−	0.678441i	−0.328688	−	0.944439i	\(-0.606606\pi\)
							−0.963974	+	0.265997i	\(0.914299\pi\)
\(5\)	−8.17932	−	9.23255i	−0.731581	−	0.825784i	0.258540	−	0.966001i	\(-0.416759\pi\)
							−0.990121	+	0.140216i	\(0.955220\pi\)
\(7\)	−7.34833	−	29.8134i	−0.396773	−	1.60977i	−0.741422	−	0.671039i	\(-0.765849\pi\)
							0.344650	−	0.938731i	\(-0.387998\pi\)
\(11\)	0.0788444	+	0.0544223i	0.00216113	+	0.00149172i	0.569145	−	0.822237i	\(-0.307274\pi\)
							−0.566984	+	0.823729i	\(0.691890\pi\)
\(13\)	−45.0481	+	12.9488i	−0.961084	+	0.276257i
							\(17\)	25.0357	−	6.17075i	0.357180	−	0.0880369i	−0.0566427	−	0.998395i	\(-0.518040\pi\)
0.413822	+	0.910358i	\(0.364193\pi\)
\(19\)			37.9058i			0.457694i	0.973462	+	0.228847i	\(0.0734955\pi\)
							−0.973462	+	0.228847i	\(0.926504\pi\)
\(23\)	−130.535			−1.18341			−0.591703	−	0.806156i	\(-0.701544\pi\)
							−0.591703	+	0.806156i	\(0.701544\pi\)
\(29\)	102.966	+	149.172i	0.659323	+	0.955194i	0.999888	+	0.0149732i	\(0.00476628\pi\)
							−0.340565	+	0.940221i	\(0.610618\pi\)
\(31\)	−44.4743	+	5.40016i	−0.257672	+	0.0312870i	−0.248353	−	0.968670i	\(-0.579889\pi\)
							−0.00931855	+	0.999957i	\(0.502966\pi\)
\(37\)	408.688	−	49.6238i	1.81589	−	0.220489i	0.859209	−	0.511625i	\(-0.170956\pi\)
							0.956682	+	0.291136i	\(0.0940331\pi\)
\(41\)	0.289392	−	0.551390i	0.00110233	−	0.00210031i	−0.884904	−	0.465773i	\(-0.845776\pi\)
							0.886007	+	0.463673i	\(0.153469\pi\)
\(43\)	49.6688	−	409.059i	0.176149	−	1.45072i	−0.589660	−	0.807652i	\(-0.700738\pi\)
							0.765809	−	0.643068i	\(-0.222339\pi\)
\(47\)	−254.795	+	96.6309i	−0.790758	+	0.299895i	−0.716721	−	0.697360i	\(-0.754358\pi\)
							−0.0740377	+	0.997255i	\(0.523589\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.h.a.12.19	✓	528
169.155	even	26	inner	169.4.h.a.155.19	yes	528

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.4.h.a.12.19	✓	528	1.1	even	1	trivial
169.4.h.a.155.19	yes	528	169.155	even	26	inner