Embedded newform 36.6.h.a.23.12

• Label: 36.6.h.a.23.12
• 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			23.12
Character	\(\chi\)	\(=\)	36.23
Dual form			36.6.h.a.11.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.78216 - 5.36879i) q^{2} +(-15.4584 - 2.00950i) q^{3} +(-25.6478 + 19.1360i) q^{4} +(12.8631 + 7.42651i) q^{5} +(16.7607 + 86.5741i) q^{6} +(-15.2989 + 8.83282i) q^{7} +(148.446 + 103.595i) q^{8} +(234.924 + 62.1273i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 3 q^{2} - q^{4} - 6 q^{5} - 27 q^{6} + 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.78216	−	5.36879i	−0.315044	−	0.949077i
							\(3\)	−15.4584	−	2.00950i	−0.991656	−	0.128910i
\(5\)	12.8631	+	7.42651i	0.230102	+	0.132849i								0.610619	−	0.791924i	\(-0.290921\pi\)
							−0.380517	+	0.924774i	\(0.624254\pi\)
\(7\)	−15.2989	+	8.83282i	−0.118009	+	0.0681325i	−0.557843	−	0.829947i	\(-0.688371\pi\)
							0.439834	+	0.898079i	\(0.355037\pi\)
\(11\)	143.245	+	248.107i	0.356941	+	0.618240i	0.987448	−	0.157943i	\(-0.0504863\pi\)
							−0.630507	+	0.776184i	\(0.717153\pi\)
\(13\)	42.2460	−	73.1722i	0.0693309	−	0.120085i	−0.829276	−	0.558839i	\(-0.811247\pi\)
							0.898607	+	0.438755i	\(0.144580\pi\)
\(17\)			1609.65i			1.35085i	0.737427	+	0.675427i	\(0.236040\pi\)
							−0.737427	+	0.675427i	\(0.763960\pi\)
\(19\)			1669.01i			1.06065i	0.847793	+	0.530327i	\(0.177931\pi\)
							−0.847793	+	0.530327i	\(0.822069\pi\)
\(23\)	−2324.29	+	4025.79i	−0.916159	+	1.58683i	−0.110963	+	0.993825i	\(0.535394\pi\)
							−0.805196	+	0.593009i	\(0.797940\pi\)
\(29\)	2414.87	−	1394.23i	0.533211	−	0.307850i	−0.209112	−	0.977892i	\(-0.567057\pi\)
							0.742323	+	0.670042i	\(0.233724\pi\)
\(31\)	2781.39	+	1605.84i	0.519825	+	0.300121i	0.736863	−	0.676042i	\(-0.236306\pi\)
							−0.217038	+	0.976163i	\(0.569640\pi\)
\(37\)	−4624.19			−0.555304			−0.277652	−	0.960682i	\(-0.589556\pi\)
							−0.277652	+	0.960682i	\(0.589556\pi\)
\(41\)	9810.20	+	5663.92i	0.911420	+	0.526208i	0.880888	−	0.473325i	\(-0.156947\pi\)
							0.0305320	+	0.999534i	\(0.490280\pi\)
\(43\)	12396.0	−	7156.83i	1.02237	−	0.590268i	0.107584	−	0.994196i	\(-0.465689\pi\)
							0.914791	+	0.403928i	\(0.132355\pi\)
\(47\)	−1695.63	−	2936.92i	−0.111966	−	0.193931i	0.804597	−	0.593822i	\(-0.202382\pi\)
							−0.916563	+	0.399890i	\(0.869048\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	36.6.h.a.23.12	yes	56
3.2	odd	2		108.6.h.a.71.17		56
4.3	odd	2	inner	36.6.h.a.23.21	yes	56
9.2	odd	6	inner	36.6.h.a.11.21	yes	56
9.7	even	3		108.6.h.a.35.8		56
12.11	even	2		108.6.h.a.71.8		56
36.7	odd	6		108.6.h.a.35.17		56
36.11	even	6	inner	36.6.h.a.11.12	✓	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.6.h.a.11.12	✓	56	36.11	even	6	inner
36.6.h.a.11.21	yes	56	9.2	odd	6	inner
36.6.h.a.23.12	yes	56	1.1	even	1	trivial
36.6.h.a.23.21	yes	56	4.3	odd	2	inner
108.6.h.a.35.8		56	9.7	even	3
108.6.h.a.35.17		56	36.7	odd	6
108.6.h.a.71.8		56	12.11	even	2
108.6.h.a.71.17		56	3.2	odd	2