Embedded newform 289.2.i.a.121.7

• Label: 289.2.i.a.121.7
• 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			121.7
Character	\(\chi\)	\(=\)	289.121
Dual form			289.2.i.a.43.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.34228 - 0.592674i) q^{2} +(-1.01371 - 0.595719i) q^{3} +(0.103056 + 0.113047i) q^{4} +(2.74945 - 3.15960i) q^{5} +(1.00761 + 1.40042i) q^{6} +(3.67769 + 2.91306i) q^{7} +(0.861261 + 2.56965i) q^{8} +(-0.787089 - 1.41310i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1600 q - 64 q^{2} - 64 q^{3} - 68 q^{4} - 68 q^{5} - 72 q^{6} - 64 q^{7} - 72 q^{8} - 76 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.34228	−	0.592674i	−0.949134	−	0.419083i	−0.129249	−	0.991612i	\(-0.541257\pi\)
							−0.819885	+	0.572529i	\(0.805963\pi\)
\(3\)	−1.01371	−	0.595719i	−0.585265	−	0.343939i	0.183943	−	0.982937i	\(-0.441114\pi\)
							−0.769208	+	0.638998i	\(0.779349\pi\)
\(5\)	2.74945	−	3.15960i	1.22959	−	1.41302i	0.350364	−	0.936614i	\(-0.386058\pi\)
							0.879227	−	0.476403i	\(-0.158060\pi\)
\(7\)	3.67769	+	2.91306i	1.39004	+	1.10103i	0.982171	+	0.187990i	\(0.0601973\pi\)
							0.407865	+	0.913042i	\(0.366273\pi\)
\(11\)	1.76619	−	0.637711i	0.532526	−	0.192277i	−0.0594522	−	0.998231i	\(-0.518935\pi\)
							0.591978	+	0.805954i	\(0.298347\pi\)
\(13\)	4.44694	+	2.21431i	1.23336	+	0.614139i	0.939775	−	0.341794i	\(-0.111035\pi\)
							0.293583	+	0.955934i	\(0.405152\pi\)
\(17\)	−3.82149	−	1.54796i	−0.926848	−	0.375436i
							\(19\)	0.790605	+	1.79055i	0.181377	+	0.410780i	0.982127	−	0.188218i	\(-0.0602713\pi\)
−0.800750	+	0.598999i	\(0.795565\pi\)
\(23\)	2.66058	−	3.35894i	0.554769	−	0.700387i	−0.423928	−	0.905696i	\(-0.639349\pi\)
							0.978697	+	0.205308i	\(0.0658198\pi\)
\(29\)	−0.978912	−	0.459537i	−0.181779	−	0.0853338i	0.331906	−	0.943312i	\(-0.392308\pi\)
							−0.513685	+	0.857979i	\(0.671720\pi\)
\(31\)	−0.0263788	+	0.380038i	−0.00473777	+	0.0682568i	−0.999385	−	0.0350754i	\(-0.988833\pi\)
							0.994647	+	0.103332i	\(0.0329505\pi\)
\(37\)	−7.22009	+	4.70461i	−1.18698	+	0.773433i	−0.978832	−	0.204666i	\(-0.934389\pi\)
							−0.208144	+	0.978098i	\(0.566742\pi\)
\(41\)	−5.94039	+	1.54280i	−0.927733	+	0.240945i	−0.684491	−	0.729021i	\(-0.739976\pi\)
							−0.243242	+	0.969966i	\(0.578211\pi\)
\(43\)	−4.04831	−	3.36167i	−0.617361	−	0.512650i	0.276489	−	0.961017i	\(-0.410829\pi\)
							−0.893849	+	0.448367i	\(0.852006\pi\)
\(47\)	−3.26855	+	0.929982i	−0.476767	+	0.135652i	−0.503508	−	0.863991i	\(-0.667958\pi\)
							0.0267413	+	0.999642i	\(0.491487\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	289.2.i.a.121.7	yes	1600
289.43	even	136	inner	289.2.i.a.43.7	✓	1600

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
289.2.i.a.43.7	✓	1600	289.43	even	136	inner
289.2.i.a.121.7	yes	1600	1.1	even	1	trivial