Embedded newform 756.2.cf.a.155.5

• Label: 756.2.cf.a.155.5
• 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			155.5
Character	\(\chi\)	\(=\)	756.155
Dual form			756.2.cf.a.239.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37502 - 0.330626i) q^{2} +(0.740762 + 1.56565i) q^{3} +(1.78137 + 0.909235i) q^{4} +(3.54867 - 0.625726i) q^{5} +(-0.500919 - 2.39772i) q^{6} +(0.642788 - 0.766044i) q^{7} +(-2.14881 - 1.83919i) q^{8} +(-1.90254 + 2.31955i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37502	−	0.330626i	−0.972288	−	0.233788i
							\(3\)	0.740762	+	1.56565i	0.427679	+	0.903931i
\(5\)	3.54867	−	0.625726i	1.58701	−	0.279833i								0.690662	−	0.723178i	\(-0.257319\pi\)
							0.896351	+	0.443345i	\(0.146208\pi\)
\(7\)	0.642788	−	0.766044i	0.242951	−	0.289538i
							\(11\)	−0.571956	+	3.24372i	−0.172451	+	0.978019i	0.768594	+	0.639737i	\(0.220957\pi\)
−0.941045	+	0.338282i	\(0.890154\pi\)
\(13\)	−0.926577	−	0.337246i	−0.256986	−	0.0935353i	0.210314	−	0.977634i	\(-0.432551\pi\)
							−0.467301	+	0.884098i	\(0.654773\pi\)
\(17\)	−0.192637	+	0.111219i	−0.0467215	+	0.0269746i	−0.523179	−	0.852223i	\(-0.675254\pi\)
							0.476457	+	0.879198i	\(0.341921\pi\)
\(19\)	6.44241	+	3.71952i	1.47799	+	0.853317i	0.999690	−	0.0248798i	\(-0.00792031\pi\)
							0.478299	+	0.878197i	\(0.341254\pi\)
\(23\)	−4.63223	+	3.88691i	−0.965887	+	0.810476i	−0.981901	−	0.189396i	\(-0.939347\pi\)
							0.0160134	+	0.999872i	\(0.494903\pi\)
\(29\)	−0.179401	−	0.492900i	−0.0333139	−	0.0915292i	0.921921	−	0.387379i	\(-0.126619\pi\)
							−0.955235	+	0.295850i	\(0.904397\pi\)
\(31\)	−2.50675	−	2.98743i	−0.450226	−	0.536558i	0.492418	−	0.870359i	\(-0.336113\pi\)
							−0.942644	+	0.333801i	\(0.891669\pi\)
\(37\)	2.26147	+	3.91698i	0.371783	+	0.643947i	0.989840	−	0.142186i	\(-0.0454132\pi\)
							−0.618057	+	0.786133i	\(0.712080\pi\)
\(41\)	3.26279	−	8.96443i	0.509562	−	1.40001i	−0.372129	−	0.928181i	\(-0.621372\pi\)
							0.881691	−	0.471828i	\(-0.156406\pi\)
\(43\)	7.14795	+	1.26038i	1.09005	+	0.192206i	0.689656	−	0.724137i	\(-0.257762\pi\)
							0.400396	+	0.916342i	\(0.368873\pi\)
\(47\)	−2.47403	−	2.07596i	−0.360875	−	0.302810i	0.444265	−	0.895896i	\(-0.353465\pi\)
							−0.805139	+	0.593086i	\(0.797909\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.2.cf.a.155.5	✓	648
4.3	odd	2	inner	756.2.cf.a.155.77	yes	648
27.23	odd	18	inner	756.2.cf.a.239.77	yes	648
108.23	even	18	inner	756.2.cf.a.239.5	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.cf.a.155.5	✓	648	1.1	even	1	trivial
756.2.cf.a.155.77	yes	648	4.3	odd	2	inner
756.2.cf.a.239.5	yes	648	108.23	even	18	inner
756.2.cf.a.239.77	yes	648	27.23	odd	18	inner