Embedded newform 108.4.l.a.11.12

• Label: 108.4.l.a.11.12
• 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			11.12
Character	\(\chi\)	\(=\)	108.11
Dual form			108.4.l.a.59.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.10574 - 1.88834i) q^{2} +(-3.74172 - 3.60549i) q^{3} +(0.868318 + 7.95274i) q^{4} +(-1.60571 + 4.41165i) q^{5} +(1.07071 + 14.6579i) q^{6} +(23.1356 + 4.07942i) q^{7} +(13.1890 - 18.3861i) q^{8} +(1.00093 + 26.9814i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 6 q^{6} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.10574	−	1.88834i	−0.744493	−	0.667630i
							\(3\)	−3.74172	−	3.60549i	−0.720094	−	0.693876i
\(5\)	−1.60571	+	4.41165i	−0.143619	+	0.394590i								−0.990557	−	0.137102i	\(-0.956221\pi\)
							0.846938	+	0.531692i	\(0.178443\pi\)
\(7\)	23.1356	+	4.07942i	1.24920	+	0.220268i	0.758856	−	0.651259i	\(-0.225759\pi\)
							0.490347	+	0.871527i	\(0.336870\pi\)
\(11\)	−26.1319	+	9.51124i	−0.716279	+	0.260704i	−0.674345	−	0.738416i	\(-0.735574\pi\)
							−0.0419338	+	0.999120i	\(0.513352\pi\)
\(13\)	58.1938	−	48.8304i	1.24154	−	1.04178i	0.244141	−	0.969740i	\(-0.421494\pi\)
							0.997402	−	0.0720380i	\(-0.0229503\pi\)
\(17\)	21.6675	−	12.5097i	0.309126	−	0.178474i	−0.337409	−	0.941358i	\(-0.609551\pi\)
							0.646535	+	0.762884i	\(0.276217\pi\)
\(19\)	−39.2385	−	22.6543i	−0.473785	−	0.273540i	0.244038	−	0.969766i	\(-0.421528\pi\)
							−0.717823	+	0.696226i	\(0.754861\pi\)
\(23\)	−13.9803	−	79.2863i	−0.126743	−	0.718797i	−0.980257	−	0.197727i	\(-0.936644\pi\)
							0.853514	−	0.521070i	\(-0.174467\pi\)
\(29\)	167.711	−	199.870i	1.07390	−	1.27983i	0.115841	−	0.993268i	\(-0.463044\pi\)
							0.958062	−	0.286560i	\(-0.0925119\pi\)
\(31\)	246.562	−	43.4756i	1.42851	−	0.251886i	0.594707	−	0.803942i	\(-0.297268\pi\)
							0.833807	+	0.552057i	\(0.186157\pi\)
\(37\)	70.2583	+	121.691i	0.312173	+	0.540699i	0.978832	−	0.204663i	\(-0.0656098\pi\)
							−0.666660	+	0.745362i	\(0.732277\pi\)
\(41\)	19.4231	+	23.1475i	0.0739848	+	0.0881716i	0.801767	−	0.597636i	\(-0.203893\pi\)
							−0.727782	+	0.685808i	\(0.759449\pi\)
\(43\)	41.0421	+	112.762i	0.145555	+	0.399909i	0.990950	−	0.134233i	\(-0.0428570\pi\)
							−0.845395	+	0.534142i	\(0.820635\pi\)
\(47\)	−81.7043	+	463.368i	−0.253570	+	1.43807i	0.546147	+	0.837690i	\(0.316094\pi\)
							−0.799717	+	0.600378i	\(0.795017\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.4.l.a.11.12	✓	312
4.3	odd	2	inner	108.4.l.a.11.40	yes	312
27.5	odd	18	inner	108.4.l.a.59.40	yes	312
108.59	even	18	inner	108.4.l.a.59.12	yes	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.l.a.11.12	✓	312	1.1	even	1	trivial
108.4.l.a.11.40	yes	312	4.3	odd	2	inner
108.4.l.a.59.12	yes	312	108.59	even	18	inner
108.4.l.a.59.40	yes	312	27.5	odd	18	inner