Embedded newform 36.6.h.a.11.12

• Label: 36.6.h.a.11.12
• Level: $36$
• Weight: $6$
• Character: 36.11
• 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			11.12
Character	\(\chi\)	\(=\)	36.11
Dual form			36.6.h.a.23.12

$q$-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 \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).

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

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								−0.380517	−	0.924774i	\(-0.624254\pi\)
							0.610619	+	0.791924i	\(0.290921\pi\)
\(7\)	−15.2989	−	8.83282i	−0.118009	−	0.0681325i	0.439834	−	0.898079i	\(-0.355037\pi\)
							−0.557843	+	0.829947i	\(0.688371\pi\)
\(11\)	143.245	−	248.107i	0.356941	−	0.618240i	−0.630507	−	0.776184i	\(-0.717153\pi\)
							0.987448	+	0.157943i	\(0.0504863\pi\)
\(13\)	42.2460	+	73.1722i	0.0693309	+	0.120085i	0.898607	−	0.438755i	\(-0.144580\pi\)
							−0.829276	+	0.558839i	\(0.811247\pi\)
\(17\)		−	1609.65i		−	1.35085i	−0.737427	−	0.675427i	\(-0.763960\pi\)
							0.737427	−	0.675427i	\(-0.236040\pi\)
\(19\)		−	1669.01i		−	1.06065i	−0.847793	−	0.530327i	\(-0.822069\pi\)
							0.847793	−	0.530327i	\(-0.177931\pi\)
\(23\)	−2324.29	−	4025.79i	−0.916159	−	1.58683i	−0.805196	−	0.593009i	\(-0.797940\pi\)
							−0.110963	−	0.993825i	\(-0.535394\pi\)
\(29\)	2414.87	+	1394.23i	0.533211	+	0.307850i	0.742323	−	0.670042i	\(-0.233724\pi\)
							−0.209112	+	0.977892i	\(0.567057\pi\)
\(31\)	2781.39	−	1605.84i	0.519825	−	0.300121i	−0.217038	−	0.976163i	\(-0.569640\pi\)
							0.736863	+	0.676042i	\(0.236306\pi\)
\(37\)	−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	0.0305320	−	0.999534i	\(-0.490280\pi\)
							0.880888	+	0.473325i	\(0.156947\pi\)
\(43\)	12396.0	+	7156.83i	1.02237	+	0.590268i	0.914791	−	0.403928i	\(-0.132355\pi\)
							0.107584	+	0.994196i	\(0.465689\pi\)
\(47\)	−1695.63	+	2936.92i	−0.111966	+	0.193931i	−0.916563	−	0.399890i	\(-0.869048\pi\)
							0.804597	+	0.593822i	\(0.202382\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.6.h.a.11.12	✓	56	1.1	even	1	trivial
36.6.h.a.11.21	yes	56	4.3	odd	2	inner
36.6.h.a.23.12	yes	56	36.23	even	6	inner
36.6.h.a.23.21	yes	56	9.5	odd	6	inner
108.6.h.a.35.8		56	12.11	even	2
108.6.h.a.35.17		56	3.2	odd	2
108.6.h.a.71.8		56	9.4	even	3
108.6.h.a.71.17		56	36.31	odd	6