Embedded newform 108.4.l.a.59.45

• Label: 108.4.l.a.59.45
• Level: $108$
• Weight: $4$
• Character: 108.59
• Analytic conductor: $6.372$
• Analytic rank: $0$
• Dimension: $312$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	108.l (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.37220628062\)
Analytic rank:	\(0\)
Dimension:	\(312\)
Relative dimension:	\(52\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			59.45
Character	\(\chi\)	\(=\)	108.59
Dual form			108.4.l.a.11.45

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.40490 - 1.48878i) q^{2} +(4.43306 + 2.71072i) q^{3} +(3.56707 - 7.16072i) q^{4} +(5.32629 + 14.6339i) q^{5} +(14.6967 - 0.0808436i) q^{6} +(-6.67869 + 1.17763i) q^{7} +(-2.08229 - 22.5314i) q^{8} +(12.3040 + 24.0335i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 6 q^{6} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/108\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(55\)
\(\chi(n)\)	\(e\left(\frac{5}{18}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.40490	−	1.48878i	0.850260	−	0.526363i
							\(3\)	4.43306	+	2.71072i	0.853143	+	0.521678i
\(5\)	5.32629	+	14.6339i	0.476398	+	1.30889i								0.912530	+	0.409010i	\(0.134126\pi\)
							−0.436132	+	0.899883i	\(0.643652\pi\)
\(7\)	−6.67869	+	1.17763i	−0.360616	+	0.0635863i	−0.351021	−	0.936368i	\(-0.614165\pi\)
							−0.00959492	+	0.999954i	\(0.503054\pi\)
\(11\)	32.8435	+	11.9540i	0.900243	+	0.327662i	0.750350	−	0.661041i	\(-0.229885\pi\)
							0.149893	+	0.988702i	\(0.452107\pi\)
\(13\)	−58.0957	−	48.7481i	−1.23945	−	1.04002i	−0.997567	−	0.0697169i	\(-0.977790\pi\)
							−0.241883	−	0.970305i	\(-0.577765\pi\)
\(17\)	−26.5003	−	15.3000i	−0.378075	−	0.218282i	0.298905	−	0.954283i	\(-0.403378\pi\)
							−0.676980	+	0.736001i	\(0.736712\pi\)
\(19\)	47.8487	−	27.6255i	0.577750	−	0.333564i	−0.182489	−	0.983208i	\(-0.558415\pi\)
							0.760239	+	0.649644i	\(0.225082\pi\)
\(23\)	16.2062	−	91.9101i	0.146923	−	0.833243i	−0.818880	−	0.573965i	\(-0.805404\pi\)
							0.965803	−	0.259277i	\(-0.0834844\pi\)
\(29\)	−146.505	−	174.598i	−0.938115	−	1.11800i	−0.992834	−	0.119501i	\(-0.961871\pi\)
							0.0547186	−	0.998502i	\(-0.482574\pi\)
\(31\)	134.207	+	23.6644i	0.777560	+	0.137105i	0.548324	−	0.836266i	\(-0.315266\pi\)
							0.229236	+	0.973371i	\(0.426377\pi\)
\(37\)	−109.255	+	189.235i	−0.485442	+	0.840810i	−0.999860	−	0.0167296i	\(-0.994675\pi\)
							0.514418	+	0.857539i	\(0.328008\pi\)
\(41\)	42.2042	−	50.2970i	0.160761	−	0.191587i	−0.679651	−	0.733536i	\(-0.737869\pi\)
							0.840412	+	0.541948i	\(0.182313\pi\)
\(43\)	−37.6216	+	103.364i	−0.133424	+	0.366580i	−0.988356	−	0.152161i	\(-0.951377\pi\)
							0.854932	+	0.518741i	\(0.173599\pi\)
\(47\)	1.93936	+	10.9987i	0.00601883	+	0.0341345i	0.987669	−	0.156555i	\(-0.0500388\pi\)
							−0.981650	+	0.190689i	\(0.938928\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.4.l.a.59.45	yes	312
4.3	odd	2	inner	108.4.l.a.59.15	yes	312
27.11	odd	18	inner	108.4.l.a.11.15	✓	312
108.11	even	18	inner	108.4.l.a.11.45	yes	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.l.a.11.15	✓	312	27.11	odd	18	inner
108.4.l.a.11.45	yes	312	108.11	even	18	inner
108.4.l.a.59.15	yes	312	4.3	odd	2	inner
108.4.l.a.59.45	yes	312	1.1	even	1	trivial