Embedded newform 360.2.bd.b.59.8

• Label: 360.2.bd.b.59.8
• Level: $360$
• Weight: $2$
• Character: 360.59
• Analytic conductor: $2.875$
• Analytic rank: $0$
• Dimension: $128$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	360.bd (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.87461447277\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(64\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			59.8
Character	\(\chi\)	\(=\)	360.59
Dual form			360.2.bd.b.299.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.29321 + 0.572373i) q^{2} +(1.41725 - 0.995694i) q^{3} +(1.34478 - 1.48040i) q^{4} +(-0.776540 - 2.09690i) q^{5} +(-1.26289 + 2.09884i) q^{6} +(-0.720108 - 1.24726i) q^{7} +(-0.891741 + 2.68418i) q^{8} +(1.01719 - 2.82229i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 10 q^{4} + 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).

\(n\)	\(181\)	\(217\)	\(271\)	\(281\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.29321	+	0.572373i	−0.914437	+	0.404729i
							\(3\)	1.41725	−	0.995694i	0.818249	−	0.574864i
\(5\)	−0.776540	−	2.09690i	−0.347279	−	0.937762i
							\(7\)	−0.720108	−	1.24726i	−0.272175	−	0.471421i	0.697243	−	0.716834i	\(-0.254410\pi\)
−0.969419	+	0.245413i	\(0.921076\pi\)
\(11\)	−1.12107	+	0.647253i	−0.338017	+	0.195154i	−0.659395	−	0.751797i	\(-0.729187\pi\)
							0.321378	+	0.946951i	\(0.395854\pi\)
\(13\)	0.850182	−	1.47256i	0.235798	−	0.408414i	−0.723706	−	0.690108i	\(-0.757563\pi\)
							0.959504	+	0.281694i	\(0.0908962\pi\)
\(17\)	−4.93422			−1.19672			−0.598362	−	0.801226i	\(-0.704182\pi\)
							−0.598362	+	0.801226i	\(0.704182\pi\)
\(19\)	1.68641			0.386889			0.193444	−	0.981111i	\(-0.438034\pi\)
							0.193444	+	0.981111i	\(0.438034\pi\)
\(23\)	0.349093	+	0.201549i	0.0727910	+	0.0420259i	0.535954	−	0.844247i	\(-0.319952\pi\)
							−0.463163	+	0.886273i	\(0.653285\pi\)
\(29\)	−0.332757	−	0.576352i	−0.0617914	−	0.107026i	0.833475	−	0.552557i	\(-0.186348\pi\)
							−0.895266	+	0.445532i	\(0.853015\pi\)
\(31\)	−2.31653	−	1.33745i	−0.416061	−	0.240213i	0.277329	−	0.960775i	\(-0.410551\pi\)
							−0.693391	+	0.720562i	\(0.743884\pi\)
\(37\)	9.59254			1.57700			0.788502	−	0.615033i	\(-0.210857\pi\)
							0.788502	+	0.615033i	\(0.210857\pi\)
\(41\)	−3.52205	−	2.03346i	−0.550052	−	0.317573i	0.199091	−	0.979981i	\(-0.436201\pi\)
							−0.749143	+	0.662408i	\(0.769534\pi\)
\(43\)	6.92616	−	3.99882i	1.05623	−	0.609814i	0.131843	−	0.991271i	\(-0.457911\pi\)
							0.924387	+	0.381456i	\(0.124577\pi\)
\(47\)	6.55826	−	3.78641i	0.956620	−	0.552305i	0.0614890	−	0.998108i	\(-0.480415\pi\)
							0.895131	+	0.445803i	\(0.147082\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	360.2.bd.b.59.8	✓	128
5.4	even	2	inner	360.2.bd.b.59.57	yes	128
8.3	odd	2	inner	360.2.bd.b.59.52	yes	128
9.2	odd	6	inner	360.2.bd.b.299.13	yes	128
40.19	odd	2	inner	360.2.bd.b.59.13	yes	128
45.29	odd	6	inner	360.2.bd.b.299.52	yes	128
72.11	even	6	inner	360.2.bd.b.299.57	yes	128
360.299	even	6	inner	360.2.bd.b.299.8	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.bd.b.59.8	✓	128	1.1	even	1	trivial
360.2.bd.b.59.13	yes	128	40.19	odd	2	inner
360.2.bd.b.59.52	yes	128	8.3	odd	2	inner
360.2.bd.b.59.57	yes	128	5.4	even	2	inner
360.2.bd.b.299.8	yes	128	360.299	even	6	inner
360.2.bd.b.299.13	yes	128	9.2	odd	6	inner
360.2.bd.b.299.52	yes	128	45.29	odd	6	inner
360.2.bd.b.299.57	yes	128	72.11	even	6	inner