Embedded newform 36.6.h.a.23.14

• Label: 36.6.h.a.23.14
• Level: $36$
• Weight: $6$
• Character: 36.23
• Analytic conductor: $5.774$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 36 = 2^{2} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	36.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.77381751327\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(28\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			23.14
Character	\(\chi\)	\(=\)	36.23
Dual form			36.6.h.a.11.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0499164 - 5.65663i) q^{2} +(14.1974 + 6.43687i) q^{3} +(-31.9950 + 0.564717i) q^{4} +(70.1957 + 40.5275i) q^{5} +(35.7024 - 80.6309i) q^{6} +(89.9985 - 51.9607i) q^{7} +(4.79147 + 180.956i) q^{8} +(160.133 + 182.774i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 3 q^{2} - q^{4} - 6 q^{5} - 27 q^{6} + 18 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).

\(n\)	\(19\)	\(29\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0499164	−	5.65663i	−0.00882405	−	0.999961i
							\(3\)	14.1974	+	6.43687i	0.910765	+	0.412926i
\(5\)	70.1957	+	40.5275i	1.25570	+	0.724978i								0.972235	−	0.234005i	\(-0.0751833\pi\)
							0.283463	+	0.958983i	\(0.408517\pi\)
\(7\)	89.9985	−	51.9607i	0.694209	−	0.400802i	−0.110978	−	0.993823i	\(-0.535398\pi\)
							0.805187	+	0.593021i	\(0.202065\pi\)
\(11\)	−214.271	−	371.129i	−0.533927	−	0.924789i	−0.999214	−	0.0396293i	\(-0.987382\pi\)
							0.465287	−	0.885160i	\(-0.345951\pi\)
\(13\)	42.9574	−	74.4044i	0.0704985	−	0.122107i	−0.828621	−	0.559809i	\(-0.810874\pi\)
							0.899120	+	0.437702i	\(0.144208\pi\)
\(17\)		−	1509.75i		−	1.26701i	−0.773737	−	0.633507i	\(-0.781615\pi\)
							0.773737	−	0.633507i	\(-0.218385\pi\)
\(19\)			1224.53i			0.778189i	0.921198	+	0.389094i	\(0.127212\pi\)
							−0.921198	+	0.389094i	\(0.872788\pi\)
\(23\)	−1530.14	+	2650.27i	−0.603129	+	1.04465i	0.389215	+	0.921147i	\(0.372746\pi\)
							−0.992344	+	0.123503i	\(0.960587\pi\)
\(29\)	−4347.44	+	2510.00i	−0.959929	+	0.554215i	−0.896151	−	0.443749i	\(-0.853648\pi\)
							−0.0637776	+	0.997964i	\(0.520315\pi\)
\(31\)	−3400.78	−	1963.44i	−0.635587	−	0.366956i	0.147326	−	0.989088i	\(-0.452933\pi\)
							−0.782913	+	0.622132i	\(0.786267\pi\)
\(37\)	13035.3			1.56537			0.782684	−	0.622419i	\(-0.213850\pi\)
							0.782684	+	0.622419i	\(0.213850\pi\)
\(41\)	−15071.6	−	8701.62i	−1.40024	−	0.808426i	−0.405819	−	0.913954i	\(-0.633014\pi\)
							−0.994416	+	0.105527i	\(0.966347\pi\)
\(43\)	−5754.33	+	3322.27i	−0.474596	+	0.274008i	−0.718162	−	0.695876i	\(-0.755016\pi\)
							0.243566	+	0.969884i	\(0.421683\pi\)
\(47\)	−10068.4	−	17438.9i	−0.664836	−	1.15153i	−0.979330	−	0.202270i	\(-0.935168\pi\)
							0.314494	−	0.949260i	\(-0.398165\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	36.6.h.a.23.14	yes	56
3.2	odd	2		108.6.h.a.71.15		56
4.3	odd	2	inner	36.6.h.a.23.24	yes	56
9.2	odd	6	inner	36.6.h.a.11.24	yes	56
9.7	even	3		108.6.h.a.35.5		56
12.11	even	2		108.6.h.a.71.5		56
36.7	odd	6		108.6.h.a.35.15		56
36.11	even	6	inner	36.6.h.a.11.14	✓	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.6.h.a.11.14	✓	56	36.11	even	6	inner
36.6.h.a.11.24	yes	56	9.2	odd	6	inner
36.6.h.a.23.14	yes	56	1.1	even	1	trivial
36.6.h.a.23.24	yes	56	4.3	odd	2	inner
108.6.h.a.35.5		56	9.7	even	3
108.6.h.a.35.15		56	36.7	odd	6
108.6.h.a.71.5		56	12.11	even	2
108.6.h.a.71.15		56	3.2	odd	2