Embedded newform 756.2.cf.a.155.12

• Label: 756.2.cf.a.155.12
• Level: $756$
• Weight: $2$
• Character: 756.155
• Analytic conductor: $6.037$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	756.cf (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.03669039281\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(108\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			155.12
Character	\(\chi\)	\(=\)	756.155
Dual form			756.2.cf.a.239.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.35441 - 0.406892i) q^{2} +(1.16480 - 1.28189i) q^{3} +(1.66888 + 1.10220i) q^{4} +(-2.59226 + 0.457086i) q^{5} +(-2.09921 + 1.26227i) q^{6} +(-0.642788 + 0.766044i) q^{7} +(-1.81188 - 2.17189i) q^{8} +(-0.286497 - 2.98629i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/756\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(325\)	\(379\)
\(\chi(n)\)	\(e\left(\frac{7}{18}\right)\)	\(1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.35441	−	0.406892i	−0.957716	−	0.287716i
							\(3\)	1.16480	−	1.28189i	0.672496	−	0.740101i
\(5\)	−2.59226	+	0.457086i	−1.15930	+	0.204415i								−0.720031	−	0.693942i	\(-0.755872\pi\)
							−0.439265	+	0.898358i	\(0.644761\pi\)
\(7\)	−0.642788	+	0.766044i	−0.242951	+	0.289538i
							\(11\)	0.685645	−	3.88849i	0.206730	−	1.17242i	−0.687964	−	0.725744i	\(-0.741496\pi\)
0.894694	−	0.446679i	\(-0.147393\pi\)
\(13\)	4.65910	+	1.69578i	1.29220	+	0.470323i	0.894450	−	0.447169i	\(-0.147568\pi\)
							0.397754	+	0.917492i	\(0.369790\pi\)
\(17\)	2.57233	−	1.48513i	0.623881	−	0.360198i	−0.154498	−	0.987993i	\(-0.549376\pi\)
							0.778378	+	0.627796i	\(0.216043\pi\)
\(19\)	−3.25745	−	1.88069i	−0.747311	−	0.431460i	0.0774104	−	0.996999i	\(-0.475335\pi\)
							−0.824722	+	0.565539i	\(0.808668\pi\)
\(23\)	−3.43648	+	2.88355i	−0.716555	+	0.601261i	−0.926430	−	0.376467i	\(-0.877139\pi\)
							0.209875	+	0.977728i	\(0.432694\pi\)
\(29\)	−2.12828	−	5.84740i	−0.395212	−	1.08584i	−0.964589	−	0.263759i	\(-0.915038\pi\)
							0.569377	−	0.822077i	\(-0.307185\pi\)
\(31\)	−4.71843	−	5.62321i	−0.847456	−	1.00996i	−0.999766	−	0.0216257i	\(-0.993116\pi\)
							0.152310	−	0.988333i	\(-0.451329\pi\)
\(37\)	0.0385505	+	0.0667714i	0.00633767	+	0.0109772i	0.869177	−	0.494501i	\(-0.164649\pi\)
							−0.862839	+	0.505479i	\(0.831316\pi\)
\(41\)	−1.72922	+	4.75099i	−0.270059	+	0.741980i	0.728329	+	0.685227i	\(0.240297\pi\)
							−0.998388	+	0.0567533i	\(0.981925\pi\)
\(43\)	−11.1839	−	1.97203i	−1.70553	−	0.300732i	−0.765912	−	0.642945i	\(-0.777712\pi\)
							−0.939622	+	0.342213i	\(0.888824\pi\)
\(47\)	−4.28007	−	3.59141i	−0.624313	−	0.523861i	0.274843	−	0.961489i	\(-0.411374\pi\)
							−0.899156	+	0.437629i	\(0.855819\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.2.cf.a.155.12	✓	648
4.3	odd	2	inner	756.2.cf.a.155.74	yes	648
27.23	odd	18	inner	756.2.cf.a.239.74	yes	648
108.23	even	18	inner	756.2.cf.a.239.12	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.cf.a.155.12	✓	648	1.1	even	1	trivial
756.2.cf.a.155.74	yes	648	4.3	odd	2	inner
756.2.cf.a.239.12	yes	648	108.23	even	18	inner
756.2.cf.a.239.74	yes	648	27.23	odd	18	inner