Embedded newform 36.6.h.a.11.9

• Label: 36.6.h.a.11.9
• 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			11.9
Character	\(\chi\)	\(=\)	36.11
Dual form			36.6.h.a.23.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.75408 + 4.23165i) q^{2} +(-1.51698 - 15.5145i) q^{3} +(-3.81374 - 31.7719i) q^{4} +(-52.1793 + 30.1257i) q^{5} +(71.3467 + 51.8233i) q^{6} +(185.316 + 106.992i) q^{7} +(148.765 + 103.136i) q^{8} +(-238.398 + 47.0702i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 3 q^{2} - q^{4} - 6 q^{5} - 27 q^{6} + 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.75408	+	4.23165i	−0.663634	+	0.748057i
							\(3\)	−1.51698	−	15.5145i	−0.0973142	−	0.995254i
\(5\)	−52.1793	+	30.1257i	−0.933412	+	0.538906i								−0.887889	−	0.460058i	\(-0.847829\pi\)
							−0.0455230	+	0.998963i	\(0.514495\pi\)
\(7\)	185.316	+	106.992i	1.42944	+	0.825290i	0.997077	−	0.0764081i	\(-0.0243452\pi\)
							0.432367	+	0.901698i	\(0.357679\pi\)
\(11\)	−185.570	+	321.417i	−0.462410	+	0.800917i	−0.999080	−	0.0428745i	\(-0.986348\pi\)
							0.536671	+	0.843792i	\(0.319682\pi\)
\(13\)	402.759	+	697.599i	0.660978	+	1.14485i	0.980359	+	0.197221i	\(0.0631917\pi\)
							−0.319381	+	0.947626i	\(0.603475\pi\)
\(17\)		−	34.0704i		−	0.0285926i	−0.999898	−	0.0142963i	\(-0.995449\pi\)
							0.999898	−	0.0142963i	\(-0.00455082\pi\)
\(19\)		−	1173.48i		−	0.745747i	−0.927882	−	0.372874i	\(-0.878373\pi\)
							0.927882	−	0.372874i	\(-0.121627\pi\)
\(23\)	1980.40	+	3430.16i	0.780610	+	1.35206i	0.931587	+	0.363519i	\(0.118425\pi\)
							−0.150977	+	0.988537i	\(0.548242\pi\)
\(29\)	−382.894	−	221.064i	−0.0845441	−	0.0488115i	0.457132	−	0.889399i	\(-0.348877\pi\)
							−0.541676	+	0.840587i	\(0.682210\pi\)
\(31\)	1381.28	−	797.481i	0.258153	−	0.149045i	−0.365339	−	0.930875i	\(-0.619047\pi\)
							0.623492	+	0.781830i	\(0.285714\pi\)
\(37\)	−11057.6			−1.32787			−0.663937	−	0.747788i	\(-0.731116\pi\)
							−0.663937	+	0.747788i	\(0.731116\pi\)
\(41\)	−7969.76	+	4601.35i	−0.740433	+	0.427489i	−0.822227	−	0.569160i	\(-0.807268\pi\)
							0.0817937	+	0.996649i	\(0.473935\pi\)
\(43\)	−1061.38	−	612.787i	−0.0875384	−	0.0505403i	0.455592	−	0.890189i	\(-0.349428\pi\)
							−0.543130	+	0.839648i	\(0.682761\pi\)
\(47\)	1222.66	−	2117.71i	0.0807348	−	0.139837i	−0.822831	−	0.568286i	\(-0.807607\pi\)
							0.903566	+	0.428449i	\(0.140940\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	36.6.h.a.11.9	✓	56
3.2	odd	2		108.6.h.a.35.20		56
4.3	odd	2	inner	36.6.h.a.11.18	yes	56
9.4	even	3		108.6.h.a.71.11		56
9.5	odd	6	inner	36.6.h.a.23.18	yes	56
12.11	even	2		108.6.h.a.35.11		56
36.23	even	6	inner	36.6.h.a.23.9	yes	56
36.31	odd	6		108.6.h.a.71.20		56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.6.h.a.11.9	✓	56	1.1	even	1	trivial
36.6.h.a.11.18	yes	56	4.3	odd	2	inner
36.6.h.a.23.9	yes	56	36.23	even	6	inner
36.6.h.a.23.18	yes	56	9.5	odd	6	inner
108.6.h.a.35.11		56	12.11	even	2
108.6.h.a.35.20		56	3.2	odd	2
108.6.h.a.71.11		56	9.4	even	3
108.6.h.a.71.20		56	36.31	odd	6