Embedded newform 289.2.i.a.121.18

• Label: 289.2.i.a.121.18
• Level: $289$
• Weight: $2$
• Character: 289.121
• Analytic conductor: $2.308$
• Analytic rank: $0$
• Dimension: $1600$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 289 = 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	289.i (of order \(136\), degree \(64\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.30767661842\)
Analytic rank:	\(0\)
Dimension:	\(1600\)
Relative dimension:	\(25\) over \(\Q(\zeta_{136})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{136}]$

Embedding invariants

Embedding label			121.18
Character	\(\chi\)	\(=\)	289.121
Dual form			289.2.i.a.43.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.02159 + 0.451077i) q^{2} +(2.39569 + 1.40786i) q^{3} +(-0.507212 - 0.556385i) q^{4} +(-1.83107 + 2.10422i) q^{5} +(1.81236 + 2.51889i) q^{6} +(1.50172 + 1.18949i) q^{7} +(-0.976975 - 2.91490i) q^{8} +(2.29744 + 4.12469i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1600 q - 64 q^{2} - 64 q^{3} - 68 q^{4} - 68 q^{5} - 72 q^{6} - 64 q^{7} - 72 q^{8} - 76 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/289\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{23}{136}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.02159	+	0.451077i	0.722374	+	0.318959i	0.732291	−	0.680991i	\(-0.238451\pi\)
							−0.00991708	+	0.999951i	\(0.503157\pi\)
\(3\)	2.39569	+	1.40786i	1.38315	+	0.812826i	0.994108	−	0.108394i	\(-0.0345708\pi\)
							0.389042	+	0.921220i	\(0.372806\pi\)
\(5\)	−1.83107	+	2.10422i	−0.818878	+	0.941035i	−0.999093	−	0.0425919i	\(-0.986438\pi\)
							0.180214	+	0.983627i	\(0.442321\pi\)
\(7\)	1.50172	+	1.18949i	0.567596	+	0.449586i	0.862584	−	0.505915i	\(-0.168845\pi\)
							−0.294988	+	0.955501i	\(0.595316\pi\)
\(11\)	1.21730	−	0.439528i	0.367031	−	0.132523i	−0.149565	−	0.988752i	\(-0.547787\pi\)
							0.516596	+	0.856229i	\(0.327199\pi\)
\(13\)	−2.08613	−	1.03877i	−0.578589	−	0.288103i	0.132525	−	0.991180i	\(-0.457691\pi\)
							−0.711114	+	0.703077i	\(0.751809\pi\)
\(17\)	2.17977	−	3.49980i	0.528672	−	0.848826i
							\(19\)	1.06985	+	2.42297i	0.245439	+	0.555867i	0.994270	−	0.106894i	\(-0.0340904\pi\)
−0.748831	+	0.662761i	\(0.769385\pi\)
\(23\)	1.21603	−	1.53521i	0.253559	−	0.320114i	−0.640591	−	0.767882i	\(-0.721310\pi\)
							0.894150	+	0.447768i	\(0.147781\pi\)
\(29\)	−6.27582	−	2.94610i	−1.16539	−	0.547077i	−0.257498	−	0.966279i	\(-0.582898\pi\)
							−0.907893	+	0.419202i	\(0.862310\pi\)
\(31\)	0.759310	−	10.9393i	0.136376	−	1.96476i	−0.105396	−	0.994430i	\(-0.533611\pi\)
							0.241773	−	0.970333i	\(-0.422271\pi\)
\(37\)	1.37812	−	0.897980i	0.226561	−	0.147627i	−0.427636	−	0.903951i	\(-0.640654\pi\)
							0.654197	+	0.756324i	\(0.273007\pi\)
\(41\)	−6.89593	+	1.79096i	−1.07696	+	0.279701i	−0.747361	−	0.664418i	\(-0.768679\pi\)
							−0.329602	+	0.944120i	\(0.606915\pi\)
\(43\)	9.14305	+	7.59229i	1.39430	+	1.15781i	0.967322	+	0.253552i	\(0.0815990\pi\)
							0.426980	+	0.904261i	\(0.359578\pi\)
\(47\)	5.91710	−	1.68356i	0.863098	−	0.245573i	0.186978	−	0.982364i	\(-0.440131\pi\)
							0.676120	+	0.736792i	\(0.263660\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	289.2.i.a.121.18	yes	1600
289.43	even	136	inner	289.2.i.a.43.18	✓	1600

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
289.2.i.a.43.18	✓	1600	289.43	even	136	inner
289.2.i.a.121.18	yes	1600	1.1	even	1	trivial