Embedded newform 108.4.l.a.59.12

• Label: 108.4.l.a.59.12
• 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			59.12
Character	\(\chi\)	\(=\)	108.59
Dual form			108.4.l.a.11.12

$q$-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 \(\left(\mathbb{Z}/108\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.l.a.11.12	✓	312	108.11	even	18	inner
108.4.l.a.11.40	yes	312	27.11	odd	18	inner
108.4.l.a.59.12	yes	312	1.1	even	1	trivial
108.4.l.a.59.40	yes	312	4.3	odd	2	inner