Embedded newform 69.5.h.a.2.19

• Label: 69.5.h.a.2.19
• 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			2.19
Character	\(\chi\)	\(=\)	69.2
Dual form			69.5.h.a.35.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.16876 - 1.81863i) q^{2} +(-1.06097 - 8.93724i) q^{3} +(4.70524 + 10.3030i) q^{4} +(4.27625 - 14.5636i) q^{5} +(-17.4935 - 8.51599i) q^{6} +(33.8030 - 39.0108i) q^{7} +(58.4735 + 8.40722i) q^{8} +(-78.7487 + 18.9643i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 300 q - 19 q^{3} + 234 q^{4} - 10 q^{6} - 18 q^{7} - 111 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.16876	−	1.81863i	0.292190	−	0.454657i	−0.663861	−	0.747856i	\(-0.731083\pi\)
							0.956051	+	0.293199i	\(0.0947198\pi\)
\(3\)	−1.06097	−	8.93724i	−0.117886	−	0.993027i
							\(5\)	4.27625	−	14.5636i	0.171050	−	0.582543i	−0.828689	−	0.559709i	\(-0.810913\pi\)
0.999739	−	0.0228338i	\(-0.00726886\pi\)
\(7\)	33.8030	−	39.0108i	0.689858	−	0.796139i	−0.297487	−	0.954726i	\(-0.596148\pi\)
							0.987345	+	0.158587i	\(0.0506939\pi\)
\(11\)	−124.642	−	193.947i	−1.03010	−	1.60287i	−0.770808	−	0.637067i	\(-0.780147\pi\)
							−0.259292	−	0.965799i	\(-0.583489\pi\)
\(13\)	83.2695	+	96.0981i	0.492719	+	0.568628i	0.946590	−	0.322440i	\(-0.104503\pi\)
							−0.453871	+	0.891067i	\(0.649957\pi\)
\(17\)	−274.845	−	125.518i	−0.951021	−	0.434317i	−0.121375	−	0.992607i	\(-0.538730\pi\)
							−0.829646	+	0.558290i	\(0.811458\pi\)
\(19\)	70.6661	+	154.737i	0.195751	+	0.428635i	0.981899	−	0.189403i	\(-0.0606552\pi\)
							−0.786148	+	0.618038i	\(0.787928\pi\)
\(23\)	81.8618	−	522.628i	0.154748	−	0.987954i
							\(29\)	1463.27	+	668.254i	1.73992	+	0.794595i	0.991304	+	0.131590i	\(0.0420083\pi\)
0.748616	+	0.663004i	\(0.230719\pi\)
\(31\)	216.291	−	1504.34i	0.225069	−	1.56539i	−0.493384	−	0.869812i	\(-0.664240\pi\)
							0.718453	−	0.695576i	\(-0.244851\pi\)
\(37\)	348.307	−	102.272i	0.254424	−	0.0747057i	−0.152033	−	0.988375i	\(-0.548582\pi\)
							0.406457	+	0.913670i	\(0.366764\pi\)
\(41\)	−500.478	+	1704.47i	−0.297726	+	1.01396i	0.665752	+	0.746173i	\(0.268111\pi\)
							−0.963478	+	0.267789i	\(0.913707\pi\)
\(43\)	440.159	+	3061.38i	0.238053	+	1.65569i	0.661628	+	0.749832i	\(0.269866\pi\)
							−0.423576	+	0.905861i	\(0.639225\pi\)
\(47\)			1470.99i			0.665907i	0.942943	+	0.332954i	\(0.108045\pi\)
							−0.942943	+	0.332954i	\(0.891955\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.5.h.a.2.19	yes	300
3.2	odd	2	inner	69.5.h.a.2.12	✓	300
23.12	even	11	inner	69.5.h.a.35.12	yes	300
69.35	odd	22	inner	69.5.h.a.35.19	yes	300

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.5.h.a.2.12	✓	300	3.2	odd	2	inner
69.5.h.a.2.19	yes	300	1.1	even	1	trivial
69.5.h.a.35.12	yes	300	23.12	even	11	inner
69.5.h.a.35.19	yes	300	69.35	odd	22	inner