Embedded newform 114.4.i.d.73.1

• Label: 114.4.i.d.73.1
• Level: $114$
• Weight: $4$
• Character: 114.73
• Analytic conductor: $6.726$
• Analytic rank: $0$
• Dimension: $18$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 114 = 2 \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	114.i (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.72621774065\)
Analytic rank:	\(0\)
Dimension:	\(18\)
Relative dimension:	\(3\) over \(\Q(\zeta_{9})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{18} + \cdots)\)
Defining polynomial:	x^18 + 603*x^16 - 1528*x^15 + 240417*x^14 - 786198*x^13 + 54822410*x^12 - 261322128*x^11 + 9122210535*x^10 - 45012474508*x^9 + 1000170346788*x^8 - 5035551052410*x^7 + 79008124941737*x^6 - 336274219954836*x^5 + 3621639622047612*x^4 - 11991799127581216*x^3 + 107004711561222960*x^2 - 278928948783924960*x + 1161091951621493824
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 3^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			73.1
Root			\(8.23264 - 14.2593i\) of defining polynomial
Character	\(\chi\)	\(=\)	114.73
Dual form			114.4.i.d.25.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.53209 + 1.28558i) q^{2} +(-2.81908 + 1.02606i) q^{3} +(0.694593 + 3.93923i) q^{4} +(-2.27791 + 12.9187i) q^{5} +(-5.63816 - 2.05212i) q^{6} +(-8.34510 - 14.4541i) q^{7} +(-4.00000 + 6.92820i) q^{8} +(6.89440 - 5.78509i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 15 q^{5} - 72 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/114\mathbb{Z}\right)^\times\).

\(n\)	\(77\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.53209	+	1.28558i	0.541675	+	0.454519i
							\(3\)	−2.81908	+	1.02606i	−0.542532	+	0.197465i
\(5\)	−2.27791	+	12.9187i	−0.203743	+	1.15548i								0.695663	+	0.718368i	\(0.255111\pi\)
							−0.899406	+	0.437115i	\(0.856000\pi\)
\(7\)	−8.34510	−	14.4541i	−0.450593	−	0.780450i	0.547830	−	0.836590i	\(-0.315454\pi\)
							−0.998423	+	0.0561400i	\(0.982121\pi\)
\(11\)	−18.8408	+	32.6332i	−0.516428	+	0.894479i	0.483390	+	0.875405i	\(0.339405\pi\)
							−0.999818	+	0.0190742i	\(0.993928\pi\)
\(13\)	−60.4374	−	21.9974i	−1.28941	−	0.469306i	−0.395876	−	0.918304i	\(-0.629559\pi\)
							−0.893533	+	0.448998i	\(0.851781\pi\)
\(17\)	−43.8862	−	36.8249i	−0.626116	−	0.525374i	0.273603	−	0.961843i	\(-0.411784\pi\)
							−0.899719	+	0.436469i	\(0.856229\pi\)
\(19\)	33.3236	+	75.8191i	0.402367	+	0.915479i
							\(23\)	30.3565	+	172.161i	0.275208	+	1.56078i	0.738301	+	0.674471i	\(0.235628\pi\)
−0.463094	+	0.886309i	\(0.653261\pi\)
\(29\)	45.0267	−	37.7819i	0.288319	−	0.241928i	−0.487144	−	0.873322i	\(-0.661961\pi\)
							0.775463	+	0.631394i	\(0.217517\pi\)
\(31\)	4.64175	+	8.03976i	0.0268930	+	0.0465801i	0.879159	−	0.476529i	\(-0.158105\pi\)
							−0.852266	+	0.523109i	\(0.824772\pi\)
\(37\)	−21.9872			−0.0976939			−0.0488469	−	0.998806i	\(-0.515555\pi\)
							−0.0488469	+	0.998806i	\(0.515555\pi\)
\(41\)	380.584	−	138.521i	1.44969	−	0.527643i	0.507184	−	0.861838i	\(-0.330686\pi\)
							0.942504	+	0.334194i	\(0.108464\pi\)
\(43\)	−46.1258	+	261.593i	−0.163584	+	0.927732i	0.786928	+	0.617045i	\(0.211670\pi\)
							−0.950512	+	0.310687i	\(0.899441\pi\)
\(47\)	−11.3292	+	9.50632i	−0.0351603	+	0.0295030i	−0.660198	−	0.751091i	\(-0.729528\pi\)
							0.625038	+	0.780594i	\(0.285083\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	114.4.i.d.73.1	yes	18
19.5	even	9		2166.4.a.bm.1.1		9
19.6	even	9	inner	114.4.i.d.25.1	✓	18
19.14	odd	18		2166.4.a.bj.1.1		9

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.i.d.25.1	✓	18	19.6	even	9	inner
114.4.i.d.73.1	yes	18	1.1	even	1	trivial
2166.4.a.bj.1.1		9	19.14	odd	18
2166.4.a.bm.1.1		9	19.5	even	9