Embedded newform 289.2.i.a.117.6

• Label: 289.2.i.a.117.6
• Level: $289$
• Weight: $2$
• Character: 289.117
• 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			117.6
Character	\(\chi\)	\(=\)	289.117
Dual form			289.2.i.a.42.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00585 - 1.46836i) q^{2} +(1.48570 - 1.70733i) q^{3} +(-0.421856 + 1.08894i) q^{4} +(2.18820 - 3.04124i) q^{5} +(-4.00135 - 0.464221i) q^{6} +(-3.35105 + 1.96929i) q^{7} +(-1.44184 + 0.339118i) q^{8} +(-0.293205 - 2.10192i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1600 q - 64 q^{2} - 64 q^{3} - 68 q^{4} - 68 q^{5} - 72 q^{6} - 64 q^{7} - 72 q^{8} - 76 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{99}{136}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00585	−	1.46836i	−0.711241	−	1.03828i	−0.996801	−	0.0799232i	\(-0.974533\pi\)
							0.285560	−	0.958361i	\(-0.407820\pi\)
\(3\)	1.48570	−	1.70733i	0.857768	−	0.985727i	−0.142224	−	0.989835i	\(-0.545425\pi\)
							0.999992	+	0.00410788i	\(0.00130758\pi\)
\(5\)	2.18820	−	3.04124i	0.978591	−	1.36009i	0.0460827	−	0.998938i	\(-0.485326\pi\)
							0.932509	−	0.361148i	\(-0.117615\pi\)
\(7\)	−3.35105	+	1.96929i	−1.26658	+	0.744320i	−0.977083	−	0.212860i	\(-0.931722\pi\)
							−0.289494	+	0.957180i	\(0.593487\pi\)
\(11\)	1.64320	−	1.72092i	0.495443	−	0.518878i	−0.427396	−	0.904065i	\(-0.640569\pi\)
							0.922839	+	0.385187i	\(0.125863\pi\)
\(13\)	−1.21206	+	1.95754i	−0.336164	+	0.542923i	−0.973798	−	0.227416i	\(-0.926972\pi\)
							0.637634	+	0.770340i	\(0.279913\pi\)
\(17\)	1.43203	−	3.86643i	0.347319	−	0.937747i
							\(19\)	5.21384	+	3.57156i	1.19614	+	0.819373i	0.987345	−	0.158588i	\(-0.0506940\pi\)
0.208792	+	0.977960i	\(0.433047\pi\)
\(23\)	1.03228	+	1.75659i	0.215246	+	0.366274i	0.950097	−	0.311954i	\(-0.100983\pi\)
							−0.734851	+	0.678228i	\(0.762748\pi\)
\(29\)	−2.09498	+	0.0484025i	−0.389028	+	0.00898812i	−0.217170	−	0.976134i	\(-0.569683\pi\)
							−0.171858	+	0.985122i	\(0.554977\pi\)
\(31\)	1.54340	−	9.46150i	0.277202	−	1.69934i	−0.363539	−	0.931579i	\(-0.618432\pi\)
							0.640741	−	0.767757i	\(-0.278627\pi\)
\(37\)	−6.51179	+	2.01647i	−1.07053	+	0.331505i	−0.780271	−	0.625442i	\(-0.784919\pi\)
							−0.290261	+	0.956947i	\(0.593742\pi\)
\(41\)	0.0539120	+	0.776707i	0.00841964	+	0.121301i	0.999964	−	0.00846566i	\(-0.00269473\pi\)
							−0.991545	+	0.129767i	\(0.958577\pi\)
\(43\)	−0.432307	+	9.35066i	−0.0659262	+	1.42596i	0.666698	+	0.745328i	\(0.267707\pi\)
							−0.732624	+	0.680634i	\(0.761705\pi\)
\(47\)	−1.17278	+	0.885640i	−0.171067	+	0.129184i	−0.684697	−	0.728828i	\(-0.740065\pi\)
							0.513630	+	0.858012i	\(0.328301\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	289.2.i.a.117.6	yes	1600
289.42	even	136	inner	289.2.i.a.42.6	✓	1600

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
289.2.i.a.42.6	✓	1600	289.42	even	136	inner
289.2.i.a.117.6	yes	1600	1.1	even	1	trivial