Embedded newform 69.5.h.a.2.4

• Label: 69.5.h.a.2.4
• Level: $69$
• Weight: $5$
• Character: 69.2
• Analytic conductor: $7.133$
• Analytic rank: $0$
• Dimension: $300$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	69.h (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.13252745279\)
Analytic rank:	\(0\)
Dimension:	\(300\)
Relative dimension:	\(30\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			2.4
Character	\(\chi\)	\(=\)	69.2
Dual form			69.5.h.a.35.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.39074 + 5.27609i) q^{2} +(3.68875 - 8.20933i) q^{3} +(-9.69338 - 21.2255i) q^{4} +(-7.37394 + 25.1133i) q^{5} +(30.8056 + 47.2979i) q^{6} +(-0.916617 + 1.05783i) q^{7} +(45.5297 + 6.54619i) q^{8} +(-53.7862 - 60.5644i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 300 q - 19 q^{3} + 234 q^{4} - 10 q^{6} - 18 q^{7} - 111 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/69\mathbb{Z}\right)^\times\).

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{1}{11}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.39074	+	5.27609i	−0.847684	+	1.31902i	0.0984152	+	0.995145i	\(0.468623\pi\)
							−0.946099	+	0.323877i	\(0.895014\pi\)
\(3\)	3.68875	−	8.20933i	0.409861	−	0.912148i
							\(5\)	−7.37394	+	25.1133i	−0.294958	+	1.00453i	0.670051	+	0.742315i	\(0.266272\pi\)
−0.965009	+	0.262218i	\(0.915546\pi\)
\(7\)	−0.916617	+	1.05783i	−0.0187065	+	0.0215884i	−0.765026	−	0.644000i	\(-0.777274\pi\)
							0.746319	+	0.665588i	\(0.231819\pi\)
\(11\)	−67.9489	−	105.731i	−0.561561	−	0.873806i	0.438124	−	0.898915i	\(-0.355643\pi\)
							−0.999685	+	0.0251087i	\(0.992007\pi\)
\(13\)	−82.3774	−	95.0686i	−0.487440	−	0.562536i	0.457739	−	0.889086i	\(-0.348659\pi\)
							−0.945180	+	0.326550i	\(0.894114\pi\)
\(17\)	134.699	+	61.5148i	0.466085	+	0.212854i	0.634593	−	0.772846i	\(-0.281168\pi\)
							−0.168508	+	0.985700i	\(0.553895\pi\)
\(19\)	−155.919	−	341.415i	−0.431909	−	0.945749i	−0.993013	−	0.118004i	\(-0.962351\pi\)
							0.561104	−	0.827745i	\(-0.310377\pi\)
\(23\)	−436.406	−	298.983i	−0.824964	−	0.565185i
							\(29\)	−334.730	−	152.866i	−0.398014	−	0.181767i	0.206344	−	0.978479i	\(-0.433843\pi\)
−0.604359	+	0.796712i	\(0.706571\pi\)
\(31\)	167.033	−	1161.74i	0.173812	−	1.20889i	−0.696927	−	0.717142i	\(-0.745450\pi\)
							0.870739	−	0.491745i	\(-0.163641\pi\)
\(37\)	−1301.72	+	382.218i	−0.950851	+	0.279195i	−0.720141	−	0.693827i	\(-0.755923\pi\)
							−0.230710	+	0.973023i	\(0.574105\pi\)
\(41\)	−544.536	+	1854.52i	−0.323936	+	1.10322i	0.623112	+	0.782133i	\(0.285868\pi\)
							−0.947048	+	0.321092i	\(0.895950\pi\)
\(43\)	−143.406	−	997.413i	−0.0775588	−	0.539433i	−0.991145	−	0.132787i	\(-0.957607\pi\)
							0.913586	−	0.406646i	\(-0.133302\pi\)
\(47\)			989.924i			0.448132i	0.974574	+	0.224066i	\(0.0719331\pi\)
							−0.974574	+	0.224066i	\(0.928067\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.5.h.a.2.4	✓	300
3.2	odd	2	inner	69.5.h.a.2.27	yes	300
23.12	even	11	inner	69.5.h.a.35.27	yes	300
69.35	odd	22	inner	69.5.h.a.35.4	yes	300

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.5.h.a.2.4	✓	300	1.1	even	1	trivial
69.5.h.a.2.27	yes	300	3.2	odd	2	inner
69.5.h.a.35.4	yes	300	69.35	odd	22	inner
69.5.h.a.35.27	yes	300	23.12	even	11	inner