Embedded newform 378.2.bf.a.5.9

• Label: 378.2.bf.a.5.9
• Level: $378$
• Weight: $2$
• Character: 378.5
• Analytic conductor: $3.018$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	378.bf (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.01834519640\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(24\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			5.9
Character	\(\chi\)	\(=\)	378.5
Dual form			378.2.bf.a.227.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.984808 + 0.173648i) q^{2} +(0.891925 + 1.48475i) q^{3} +(0.939693 - 0.342020i) q^{4} +(0.217432 - 1.23312i) q^{5} +(-1.13620 - 1.30731i) q^{6} +(-2.23659 + 1.41339i) q^{7} +(-0.866025 + 0.500000i) q^{8} +(-1.40894 + 2.64856i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q + 18 q^{6} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.984808	+	0.173648i	−0.696364	+	0.122788i
							\(3\)	0.891925	+	1.48475i	0.514953	+	0.857218i
\(5\)	0.217432	−	1.23312i	0.0972385	−	0.551467i								−0.896800	−	0.442436i	\(-0.854114\pi\)
							0.994038	−	0.109031i	\(-0.0347746\pi\)
\(7\)	−2.23659	+	1.41339i	−0.845352	+	0.534210i
							\(11\)	−2.76303	+	0.487197i	−0.833086	+	0.146896i	−0.573893	−	0.818930i	\(-0.694568\pi\)
−0.259193	+	0.965826i	\(0.583456\pi\)
\(13\)	−4.01380	+	4.78346i	−1.11323	+	1.32669i	−0.173474	+	0.984838i	\(0.555499\pi\)
							−0.939753	+	0.341854i	\(0.888945\pi\)
\(17\)	6.08286			1.47531			0.737656	−	0.675177i	\(-0.235933\pi\)
							0.737656	+	0.675177i	\(0.235933\pi\)
\(19\)			5.71089i			1.31017i	0.755556	+	0.655084i	\(0.227367\pi\)
							−0.755556	+	0.655084i	\(0.772633\pi\)
\(23\)	−0.212471	+	0.253213i	−0.0443032	+	0.0527985i	−0.787741	−	0.616007i	\(-0.788749\pi\)
							0.743438	+	0.668805i	\(0.233194\pi\)
\(29\)	−0.547037	−	0.651934i	−0.101582	−	0.121061i	0.712858	−	0.701308i	\(-0.247400\pi\)
							−0.814440	+	0.580247i	\(0.802956\pi\)
\(31\)	−1.14922	−	3.15747i	−0.206407	−	0.567098i	0.792689	−	0.609627i	\(-0.208681\pi\)
							−0.999095	+	0.0425291i	\(0.986458\pi\)
\(37\)	2.93407	+	5.08197i	0.482359	+	0.835470i	0.999795	−	0.0202517i	\(-0.00644676\pi\)
							−0.517436	+	0.855722i	\(0.673113\pi\)
\(41\)	−3.05897	−	2.56678i	−0.477731	−	0.400864i	0.371874	−	0.928283i	\(-0.378715\pi\)
							−0.849605	+	0.527419i	\(0.823160\pi\)
\(43\)	−4.15313	−	1.51162i	−0.633347	−	0.230519i	0.00534054	−	0.999986i	\(-0.498300\pi\)
							−0.638687	+	0.769466i	\(0.720522\pi\)
\(47\)	−1.10121	−	0.400806i	−0.160627	−	0.0584636i	0.260455	−	0.965486i	\(-0.416127\pi\)
							−0.421082	+	0.907022i	\(0.638350\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.2.bf.a.5.9	yes	144
7.3	odd	6		378.2.ba.a.59.16	✓	144
27.11	odd	18		378.2.ba.a.173.16	yes	144
189.38	even	18	inner	378.2.bf.a.227.9	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.2.ba.a.59.16	✓	144	7.3	odd	6
378.2.ba.a.173.16	yes	144	27.11	odd	18
378.2.bf.a.5.9	yes	144	1.1	even	1	trivial
378.2.bf.a.227.9	yes	144	189.38	even	18	inner