Embedded newform 756.2.bs.a.11.1

• Label: 756.2.bs.a.11.1
• Level: $756$
• Weight: $2$
• Character: 756.11
• Analytic conductor: $6.037$
• Analytic rank: $0$
• Dimension: $840$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.03669039281\)
Analytic rank:	\(0\)
Dimension:	\(840\)
Relative dimension:	\(140\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			11.1
Character	\(\chi\)	\(=\)	756.11
Dual form			756.2.bs.a.275.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41413 + 0.0149120i) q^{2} +(0.154673 + 1.72513i) q^{3} +(1.99956 - 0.0421753i) q^{4} +(3.70389 - 0.653095i) q^{5} +(-0.244453 - 2.43726i) q^{6} +(-2.38269 - 1.15013i) q^{7} +(-2.82701 + 0.0894589i) q^{8} +(-2.95215 + 0.533661i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 3 q^{2} - 3 q^{4} - 6 q^{5} - 12 q^{6} - 18 q^{8} - 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(325\)	\(379\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41413	+	0.0149120i	−0.999944	+	0.0105444i
							\(3\)	0.154673	+	1.72513i	0.0893003	+	0.996005i
\(5\)	3.70389	−	0.653095i	1.65643	−	0.292073i								0.734262	−	0.678867i	\(-0.237529\pi\)
							0.922166	+	0.386794i	\(0.126417\pi\)
\(7\)	−2.38269	−	1.15013i	−0.900572	−	0.434706i
							\(11\)	0.518126	−	2.93844i	0.156221	−	0.885973i	−0.801440	−	0.598075i	\(-0.795932\pi\)
0.957661	−	0.287898i	\(-0.0929565\pi\)
\(13\)	3.90281	−	3.27484i	1.08244	−	0.908279i	0.0863228	−	0.996267i	\(-0.472488\pi\)
							0.996121	+	0.0879887i	\(0.0280439\pi\)
\(17\)		−	5.14410i		−	1.24763i	−0.781573	−	0.623814i	\(-0.785582\pi\)
							0.781573	−	0.623814i	\(-0.214418\pi\)
\(19\)		−	5.48072i		−	1.25736i	−0.777662	−	0.628682i	\(-0.783595\pi\)
							0.777662	−	0.628682i	\(-0.216405\pi\)
\(23\)	−2.08467	+	1.74925i	−0.434685	+	0.364744i	−0.833716	−	0.552194i	\(-0.813791\pi\)
							0.399031	+	0.916937i	\(0.369346\pi\)
\(29\)	−2.48873	+	2.96595i	−0.462146	+	0.550764i	−0.945908	−	0.324436i	\(-0.894826\pi\)
							0.483762	+	0.875200i	\(0.339270\pi\)
\(31\)	1.76191	−	4.84079i	0.316447	−	0.869432i	−0.674869	−	0.737937i	\(-0.735800\pi\)
							0.991317	−	0.131495i	\(-0.0419778\pi\)
\(37\)	−0.868980	+	1.50512i	−0.142859	+	0.247440i	−0.928572	−	0.371152i	\(-0.878963\pi\)
							0.785713	+	0.618591i	\(0.212296\pi\)
\(41\)	3.26572	+	3.89193i	0.510020	+	0.607818i	0.958191	−	0.286130i	\(-0.0923689\pi\)
							−0.448171	+	0.893948i	\(0.647924\pi\)
\(43\)	0.808573	+	2.22154i	0.123306	+	0.338781i	0.985952	−	0.167027i	\(-0.0534167\pi\)
							−0.862646	+	0.505808i	\(0.831194\pi\)
\(47\)	−5.29209	+	1.92616i	−0.771930	+	0.280960i	−0.697804	−	0.716289i	\(-0.745839\pi\)
							−0.0741267	+	0.997249i	\(0.523617\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.2.bs.a.11.1	✓	840
4.3	odd	2	inner	756.2.bs.a.11.15	yes	840
7.2	even	3		756.2.ci.a.443.93	yes	840
27.5	odd	18		756.2.ci.a.599.79	yes	840
28.23	odd	6		756.2.ci.a.443.79	yes	840
108.59	even	18		756.2.ci.a.599.93	yes	840
189.86	odd	18	inner	756.2.bs.a.275.15	yes	840
756.275	even	18	inner	756.2.bs.a.275.1	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.bs.a.11.1	✓	840	1.1	even	1	trivial
756.2.bs.a.11.15	yes	840	4.3	odd	2	inner
756.2.bs.a.275.1	yes	840	756.275	even	18	inner
756.2.bs.a.275.15	yes	840	189.86	odd	18	inner
756.2.ci.a.443.79	yes	840	28.23	odd	6
756.2.ci.a.443.93	yes	840	7.2	even	3
756.2.ci.a.599.79	yes	840	27.5	odd	18
756.2.ci.a.599.93	yes	840	108.59	even	18