Embedded newform 169.4.h.a.12.3

• Label: 169.4.h.a.12.3
• 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			12.3
Character	\(\chi\)	\(=\)	169.12
Dual form			169.4.h.a.155.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.22925 + 2.91924i) q^{2} +(-8.43671 - 4.42793i) q^{3} +(6.52775 - 17.2123i) q^{4} +(-3.39733 - 3.83479i) q^{5} +(48.6071 - 5.90198i) q^{6} +(0.351475 + 1.42599i) q^{7} +(12.8006 + 51.9342i) q^{8} +(36.2338 + 52.4938i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 528 q - 13 q^{2} - 11 q^{3} + 157 q^{4} - 13 q^{5} - 13 q^{6} - 13 q^{7} - 13 q^{8} - 339 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.22925	+	2.91924i	−1.49527	+	1.03211i	−0.510175	+	0.860071i	\(0.670419\pi\)
							−0.985091	+	0.172036i	\(0.944965\pi\)
\(3\)	−8.43671	−	4.42793i	−1.62365	−	0.852155i	−0.997364	−	0.0725588i	\(-0.976884\pi\)
							−0.626282	−	0.779597i	\(-0.715424\pi\)
\(5\)	−3.39733	−	3.83479i	−0.303866	−	0.342994i	0.576666	−	0.816980i	\(-0.304353\pi\)
							−0.880533	+	0.473986i	\(0.842815\pi\)
\(7\)	0.351475	+	1.42599i	0.0189779	+	0.0769963i	0.979667	−	0.200629i	\(-0.0642987\pi\)
							−0.960689	+	0.277626i	\(0.910453\pi\)
\(11\)	11.1929	+	7.72593i	0.306800	+	0.211769i	0.711509	−	0.702677i	\(-0.248012\pi\)
							−0.404709	+	0.914445i	\(0.632627\pi\)
\(13\)	−14.1440	−	44.6872i	−0.301757	−	0.953385i
							\(17\)	71.2499	−	17.5615i	1.01651	−	0.250547i	0.304346	−	0.952562i	\(-0.401562\pi\)
0.712162	+	0.702015i	\(0.247716\pi\)
\(19\)			83.8135i			1.01201i	0.862531	+	0.506004i	\(0.168878\pi\)
							−0.862531	+	0.506004i	\(0.831122\pi\)
\(23\)	128.703			1.16681			0.583403	−	0.812183i	\(-0.301721\pi\)
							0.583403	+	0.812183i	\(0.301721\pi\)
\(29\)	85.8220	+	124.335i	0.549543	+	0.796151i	0.995076	−	0.0991149i	\(-0.0316011\pi\)
							−0.445533	+	0.895266i	\(0.646986\pi\)
\(31\)	−89.0606	+	10.8139i	−0.515992	+	0.0626528i	−0.374391	−	0.927271i	\(-0.622148\pi\)
							−0.141601	+	0.989924i	\(0.545225\pi\)
\(37\)	−230.155	+	27.9458i	−1.02263	+	0.124169i	−0.614631	−	0.788815i	\(-0.710695\pi\)
							−0.407996	+	0.912984i	\(0.633772\pi\)
\(41\)	164.619	−	313.656i	0.627054	−	1.19475i	−0.340059	−	0.940404i	\(-0.610447\pi\)
							0.967113	−	0.254347i	\(-0.0818606\pi\)
\(43\)	62.0191	−	510.773i	0.219950	−	1.81145i	−0.296394	−	0.955066i	\(-0.595784\pi\)
							0.516344	−	0.856381i	\(-0.327293\pi\)
\(47\)	−187.081	+	70.9503i	−0.580607	+	0.220195i	−0.627384	−	0.778710i	\(-0.715874\pi\)
							0.0467769	+	0.998905i	\(0.485105\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.h.a.12.3	✓	528
169.155	even	26	inner	169.4.h.a.155.3	yes	528

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.4.h.a.12.3	✓	528	1.1	even	1	trivial
169.4.h.a.155.3	yes	528	169.155	even	26	inner