Embedded newform 108.5.j.a.7.7

• Label: 108.5.j.a.7.7
• Level: $108$
• Weight: $5$
• Character: 108.7
• Analytic conductor: $11.164$
• Analytic rank: $0$
• Dimension: $420$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	108.j (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.1639560131\)
Analytic rank:	\(0\)
Dimension:	\(420\)
Relative dimension:	\(70\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			7.7
Character	\(\chi\)	\(=\)	108.7
Dual form			108.5.j.a.31.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.88326 + 0.959324i) q^{2} +(-8.67232 - 2.40643i) q^{3} +(14.1594 - 7.45061i) q^{4} +(-2.07418 - 11.7633i) q^{5} +(35.9854 + 1.02521i) q^{6} +(23.6467 - 28.1810i) q^{7} +(-47.8370 + 42.5161i) q^{8} +(69.4182 + 41.7386i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 6 q^{6} - 3 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(55\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.88326	+	0.959324i	−0.970815	+	0.239831i
							\(3\)	−8.67232	−	2.40643i	−0.963591	−	0.267381i
\(5\)	−2.07418	−	11.7633i	−0.0829672	−	0.470531i								−0.997777	−	0.0666450i	\(-0.978770\pi\)
							0.914810	−	0.403886i	\(-0.132341\pi\)
\(7\)	23.6467	−	28.1810i	0.482585	−	0.575122i	−0.468731	−	0.883341i	\(-0.655288\pi\)
							0.951315	+	0.308219i	\(0.0997329\pi\)
\(11\)	0.316309	+	0.0557738i	0.00261412	+	0.000460941i	0.174955	−	0.984576i	\(-0.444022\pi\)
							−0.172341	+	0.985037i	\(0.555133\pi\)
\(13\)	188.669	+	68.6697i	1.11638	+	0.406330i	0.833331	−	0.552775i	\(-0.186431\pi\)
							0.283051	+	0.959105i	\(0.408653\pi\)
\(17\)	−47.5601	−	82.3765i	−0.164568	−	0.285040i	0.771934	−	0.635703i	\(-0.219290\pi\)
							−0.936502	+	0.350663i	\(0.885956\pi\)
\(19\)	−423.240	−	244.358i	−1.17241	−	0.676891i	−0.218163	−	0.975912i	\(-0.570006\pi\)
							−0.954246	+	0.299022i	\(0.903340\pi\)
\(23\)	−285.004	−	339.655i	−0.538761	−	0.642070i	0.426149	−	0.904653i	\(-0.359870\pi\)
							−0.964909	+	0.262583i	\(0.915426\pi\)
\(29\)	−870.471	+	316.826i	−1.03504	+	0.376725i	−0.802999	−	0.595981i	\(-0.796764\pi\)
							−0.232044	+	0.972705i	\(0.574541\pi\)
\(31\)	−876.614	−	1044.71i	−0.912190	−	1.08711i	−0.995886	−	0.0906139i	\(-0.971117\pi\)
							0.0836965	−	0.996491i	\(-0.473327\pi\)
\(37\)	152.321	+	263.828i	0.111265	+	0.192716i	0.916280	−	0.400537i	\(-0.131177\pi\)
							−0.805016	+	0.593253i	\(0.797843\pi\)
\(41\)	−1392.59	−	506.862i	−0.828430	−	0.301524i	−0.107216	−	0.994236i	\(-0.534194\pi\)
							−0.721214	+	0.692712i	\(0.756416\pi\)
\(43\)	886.638	+	156.338i	0.479523	+	0.0845529i	0.408185	−	0.912899i	\(-0.366162\pi\)
							0.0713384	+	0.997452i	\(0.477273\pi\)
\(47\)	−622.204	+	741.513i	−0.281668	+	0.335678i	−0.888265	−	0.459330i	\(-0.848089\pi\)
							0.606598	+	0.795009i	\(0.292534\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.5.j.a.7.7	✓	420
4.3	odd	2	inner	108.5.j.a.7.10	yes	420
27.4	even	9	inner	108.5.j.a.31.10	yes	420
108.31	odd	18	inner	108.5.j.a.31.7	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.j.a.7.7	✓	420	1.1	even	1	trivial
108.5.j.a.7.10	yes	420	4.3	odd	2	inner
108.5.j.a.31.7	yes	420	108.31	odd	18	inner
108.5.j.a.31.10	yes	420	27.4	even	9	inner