Embedded newform 158.5.f.a.15.17

• Label: 158.5.f.a.15.17
• Level: $158$
• Weight: $5$
• Character: 158.15
• Analytic conductor: $16.332$
• Analytic rank: $0$
• Dimension: $336$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 158 = 2 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	158.f (of order \(26\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(16.3324541672\)
Analytic rank:	\(0\)
Dimension:	\(336\)
Relative dimension:	\(28\) over \(\Q(\zeta_{26})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{26}]$

Embedding invariants

Embedding label			15.17
Character	\(\chi\)	\(=\)	158.15
Dual form			158.5.f.a.137.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.11711 + 1.87559i) q^{2} +(-9.44597 + 6.52008i) q^{3} +(0.964293 + 7.94167i) q^{4} +(-38.2121 - 9.41843i) q^{5} +(-32.2272 - 3.91308i) q^{6} +(-63.8026 + 44.0398i) q^{7} +(-12.8538 + 18.6220i) q^{8} +(17.9918 - 47.4406i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 224 q^{4} + 728 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/158\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{21}{26}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.11711	+	1.87559i	0.529277	+	0.468899i
							\(3\)	−9.44597	+	6.52008i	−1.04955	+	0.724453i	−0.962371	−	0.271739i	\(-0.912401\pi\)
−0.0871809	+	0.996193i	\(0.527786\pi\)
\(5\)	−38.2121	−	9.41843i	−1.52848	−	0.376737i	−0.616630	−	0.787253i	\(-0.711502\pi\)
							−0.911853	+	0.410516i	\(0.865349\pi\)
\(7\)	−63.8026	+	44.0398i	−1.30209	+	0.898771i	−0.998539	−	0.0540429i	\(-0.982789\pi\)
							−0.303556	+	0.952814i	\(0.598174\pi\)
\(11\)	159.041	−	39.2000i	1.31439	−	0.323967i	0.481031	−	0.876704i	\(-0.340262\pi\)
							0.833356	+	0.552736i	\(0.186416\pi\)
\(13\)	2.42612	+	19.9809i	0.0143557	+	0.118230i	0.998125	−	0.0612077i	\(-0.0194952\pi\)
							−0.983769	+	0.179438i	\(0.942572\pi\)
\(17\)	496.713	−	60.3119i	1.71873	−	0.208692i	0.799025	−	0.601298i	\(-0.205350\pi\)
							0.919707	+	0.392606i	\(0.128427\pi\)
\(19\)	−356.971	+	187.353i	−0.988838	+	0.518982i	−0.879947	−	0.475071i	\(-0.842422\pi\)
							−0.108891	+	0.994054i	\(0.534730\pi\)
\(23\)	−673.860			−1.27384			−0.636919	−	0.770931i	\(-0.719791\pi\)
							−0.636919	+	0.770931i	\(0.719791\pi\)
\(29\)	−31.1267	+	11.8048i	−0.0370116	+	0.0140366i	−0.373041	−	0.927815i	\(-0.621685\pi\)
							0.336030	+	0.941851i	\(0.390916\pi\)
\(31\)	−571.759	−	506.534i	−0.594962	−	0.527091i	0.311022	−	0.950403i	\(-0.399329\pi\)
							−0.905984	+	0.423312i	\(0.860867\pi\)
\(37\)	1040.61	+	1982.72i	0.760125	+	1.44830i	0.889612	+	0.456717i	\(0.150975\pi\)
							−0.129487	+	0.991581i	\(0.541333\pi\)
\(41\)	580.034	−	2353.29i	0.345053	−	1.39993i	−0.500359	−	0.865818i	\(-0.666799\pi\)
							0.845411	−	0.534116i	\(-0.179355\pi\)
\(43\)	367.498	−	1491.00i	0.198755	−	0.806381i	−0.783892	−	0.620897i	\(-0.786769\pi\)
							0.982647	−	0.185484i	\(-0.0593853\pi\)
\(47\)	−436.135	+	830.986i	−0.197436	+	0.376182i	−0.963966	−	0.266026i	\(-0.914289\pi\)
							0.766530	+	0.642208i	\(0.221982\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	158.5.f.a.15.17	✓	336
79.58	odd	26	inner	158.5.f.a.137.17	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
158.5.f.a.15.17	✓	336	1.1	even	1	trivial
158.5.f.a.137.17	yes	336	79.58	odd	26	inner