Embedded newform 289.2.g.a.152.14

• Label: 289.2.g.a.152.14
• Level: $289$
• Weight: $2$
• Character: 289.152
• Analytic conductor: $2.308$
• Analytic rank: $0$
• Dimension: $384$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 289 = 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	289.g (of order \(34\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.30767661842\)
Analytic rank:	\(0\)
Dimension:	\(384\)
Relative dimension:	\(24\) over \(\Q(\zeta_{34})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{34}]$

Embedding invariants

Embedding label			152.14
Character	\(\chi\)	\(=\)	289.152
Dual form			289.2.g.a.135.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.205042 - 0.0383290i) q^{2} +(2.62408 + 1.30664i) q^{3} +(-1.82437 + 0.706765i) q^{4} +(1.01942 - 1.64642i) q^{5} +(0.588130 + 0.167338i) q^{6} +(3.47020 - 2.62057i) q^{7} +(-0.701684 + 0.434464i) q^{8} +(3.37061 + 4.46341i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q - 17 q^{2} - 17 q^{3} - 41 q^{4} + 17 q^{5} - 17 q^{6} - 17 q^{7} - 17 q^{8} + 7 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{34}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.205042	−	0.0383290i	0.144987	−	0.0271027i	−0.110756	−	0.993848i	\(-0.535327\pi\)
							0.255743	+	0.966745i	\(0.417680\pi\)
\(3\)	2.62408	+	1.30664i	1.51502	+	0.754388i	0.995010	−	0.0997745i	\(-0.0318121\pi\)
							0.520006	+	0.854163i	\(0.325930\pi\)
\(5\)	1.01942	−	1.64642i	0.455899	−	0.736303i	−0.538427	−	0.842672i	\(-0.680981\pi\)
							0.994327	+	0.106369i	\(0.0339226\pi\)
\(7\)	3.47020	−	2.62057i	1.31161	−	0.990483i	0.312433	−	0.949940i	\(-0.398856\pi\)
							0.999178	−	0.0405431i	\(-0.0129088\pi\)
\(11\)	−0.880071	−	2.27172i	−0.265351	−	0.684951i	−0.999979	−	0.00648015i	\(-0.997937\pi\)
							0.734628	−	0.678470i	\(-0.237357\pi\)
\(13\)	−5.09830	+	3.15674i	−1.41401	+	0.875521i	−0.999405	−	0.0344787i	\(-0.989023\pi\)
							−0.414609	+	0.910000i	\(0.636082\pi\)
\(17\)	2.23353	+	3.46574i	0.541710	+	0.840565i
							\(19\)	−3.36291	−	0.628637i	−0.771505	−	0.144219i	−0.216743	−	0.976229i	\(-0.569543\pi\)
−0.554762	+	0.832009i	\(0.687191\pi\)
\(23\)	−5.28305	+	3.98958i	−1.10159	+	0.831884i	−0.986871	−	0.161513i	\(-0.948363\pi\)
							−0.114722	+	0.993398i	\(0.536598\pi\)
\(29\)	−0.556760	−	1.43716i	−0.103388	−	0.266875i	0.871181	−	0.490961i	\(-0.163354\pi\)
							−0.974569	+	0.224087i	\(0.928060\pi\)
\(31\)	−2.39521	−	3.86840i	−0.430193	−	0.694785i	0.560911	−	0.827876i	\(-0.310451\pi\)
							−0.991104	+	0.133091i	\(0.957510\pi\)
\(37\)	−6.78399	+	0.628630i	−1.11528	+	0.103346i	−0.634083	−	0.773265i	\(-0.718622\pi\)
							−0.481199	+	0.876611i	\(0.659799\pi\)
\(41\)	4.96233	+	2.47094i	0.774985	+	0.385897i	0.789305	−	0.614001i	\(-0.210441\pi\)
							−0.0143204	+	0.999897i	\(0.504558\pi\)
\(43\)	1.20083	+	1.09470i	0.183124	+	0.166940i	0.760067	−	0.649845i	\(-0.225166\pi\)
							−0.576943	+	0.816785i	\(0.695754\pi\)
\(47\)	4.39998	−	5.82651i	0.641802	−	0.849884i	−0.354566	−	0.935031i	\(-0.615371\pi\)
							0.996368	+	0.0851472i	\(0.0271360\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	289.2.g.a.152.14	yes	384
289.135	even	34	inner	289.2.g.a.135.14	✓	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
289.2.g.a.135.14	✓	384	289.135	even	34	inner
289.2.g.a.152.14	yes	384	1.1	even	1	trivial