Embedded newform 108.4.l.a.11.5

• Label: 108.4.l.a.11.5
• Level: $108$
• Weight: $4$
• Character: 108.11
• 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			11.5
Character	\(\chi\)	\(=\)	108.11
Dual form			108.4.l.a.59.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.73813 - 0.708980i) q^{2} +(-4.61237 - 2.39291i) q^{3} +(6.99469 + 3.88256i) q^{4} +(0.577330 - 1.58620i) q^{5} +(10.9327 + 9.82218i) q^{6} +(-27.0016 - 4.76111i) q^{7} +(-16.3997 - 15.5900i) q^{8} +(15.5480 + 22.0740i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 6 q^{6} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(55\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.73813	−	0.708980i	−0.968075	−	0.250662i
							\(3\)	−4.61237	−	2.39291i	−0.887651	−	0.460516i
\(5\)	0.577330	−	1.58620i	0.0516380	−	0.141874i								−0.911192	−	0.411981i	\(-0.864837\pi\)
							0.962830	+	0.270107i	\(0.0870591\pi\)
\(7\)	−27.0016	−	4.76111i	−1.45795	−	0.257076i	−0.612220	−	0.790687i	\(-0.709723\pi\)
							−0.845730	+	0.533612i	\(0.820834\pi\)
\(11\)	19.7116	−	7.17445i	0.540299	−	0.196653i	−0.0574323	−	0.998349i	\(-0.518291\pi\)
							0.597731	+	0.801697i	\(0.296069\pi\)
\(13\)	−48.6339	+	40.8087i	−1.03759	+	0.870637i	−0.991734	−	0.128311i	\(-0.959044\pi\)
							−0.0458513	+	0.998948i	\(0.514600\pi\)
\(17\)	103.873	−	59.9710i	1.48193	−	0.855595i	0.482143	−	0.876092i	\(-0.339858\pi\)
							0.999790	+	0.0204978i	\(0.00652512\pi\)
\(19\)	110.023	+	63.5217i	1.32847	+	0.766994i	0.985063	−	0.172192i	\(-0.0550850\pi\)
							0.343409	+	0.939186i	\(0.388418\pi\)
\(23\)	21.9463	+	124.463i	0.198961	+	1.12837i	0.906663	+	0.421855i	\(0.138621\pi\)
							−0.707702	+	0.706511i	\(0.750268\pi\)
\(29\)	−3.97135	+	4.73287i	−0.0254297	+	0.0303059i	−0.778609	−	0.627509i	\(-0.784075\pi\)
							0.753180	+	0.657815i	\(0.228519\pi\)
\(31\)	−182.057	+	32.1016i	−1.05479	+	0.185988i	−0.674043	−	0.738692i	\(-0.735444\pi\)
							−0.380746	+	0.924680i	\(0.624333\pi\)
\(37\)	34.7162	+	60.1302i	0.154252	+	0.267172i	0.932786	−	0.360430i	\(-0.117370\pi\)
							−0.778535	+	0.627602i	\(0.784037\pi\)
\(41\)	77.0664	+	91.8441i	0.293555	+	0.349845i	0.892583	−	0.450883i	\(-0.148891\pi\)
							−0.599028	+	0.800728i	\(0.704446\pi\)
\(43\)	−16.6589	−	45.7699i	−0.0590804	−	0.162322i	0.906641	−	0.421903i	\(-0.138638\pi\)
							−0.965721	+	0.259581i	\(0.916416\pi\)
\(47\)	−87.1016	+	493.978i	−0.270321	+	1.53307i	0.483123	+	0.875553i	\(0.339502\pi\)
							−0.753444	+	0.657512i	\(0.771609\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.4.l.a.11.5	✓	312
4.3	odd	2	inner	108.4.l.a.11.34	yes	312
27.5	odd	18	inner	108.4.l.a.59.34	yes	312
108.59	even	18	inner	108.4.l.a.59.5	yes	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.l.a.11.5	✓	312	1.1	even	1	trivial
108.4.l.a.11.34	yes	312	4.3	odd	2	inner
108.4.l.a.59.5	yes	312	108.59	even	18	inner
108.4.l.a.59.34	yes	312	27.5	odd	18	inner