Embedded newform 37.6.e.a.11.12

• Label: 37.6.e.a.11.12
• Level: $37$
• Weight: $6$
• Character: 37.11
• Analytic conductor: $5.934$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	37.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.93420133308\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.12
Character	\(\chi\)	\(=\)	37.11
Dual form			37.6.e.a.27.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.18743 - 2.99497i) q^{2} +(9.51908 - 16.4875i) q^{3} +(1.93964 - 3.35955i) q^{4} +(-4.99548 - 2.88414i) q^{5} -114.037i q^{6} +(118.181 - 204.696i) q^{7} +168.441i q^{8} +(-59.7259 - 103.448i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 6 q^{2} + 18 q^{3} + 298 q^{4} + 144 q^{5} - 52 q^{7} - 1490 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/37\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.18743	−	2.99497i	0.917017	−	0.529440i	0.0343349	−	0.999410i	\(-0.489069\pi\)
							0.882682	+	0.469970i	\(0.155735\pi\)
\(3\)	9.51908	−	16.4875i	0.610650	−	1.05768i	−0.380482	−	0.924788i	\(-0.624242\pi\)
							0.991131	−	0.132887i	\(-0.0424249\pi\)
\(5\)	−4.99548	−	2.88414i	−0.0893618	−	0.0515931i	0.454653	−	0.890669i	\(-0.349763\pi\)
							−0.544015	+	0.839075i	\(0.683097\pi\)
\(7\)	118.181	−	204.696i	0.911599	−	1.57894i	0.0997942	−	0.995008i	\(-0.468182\pi\)
							0.811805	−	0.583928i	\(-0.198485\pi\)
\(11\)	−663.051			−1.65221			−0.826104	−	0.563517i	\(-0.809448\pi\)
							−0.826104	+	0.563517i	\(0.809448\pi\)
\(13\)	417.993	+	241.328i	0.685978	+	0.396050i	0.802104	−	0.597185i	\(-0.203714\pi\)
							−0.116125	+	0.993235i	\(0.537047\pi\)
\(17\)	598.601	−	345.603i	0.502360	−	0.290038i	−0.227327	−	0.973818i	\(-0.572999\pi\)
							0.729688	+	0.683781i	\(0.239665\pi\)
\(19\)	2030.72	+	1172.44i	1.29053	+	0.745085i	0.978747	−	0.205069i	\(-0.0657420\pi\)
							0.311778	+	0.950155i	\(0.399075\pi\)
\(23\)			3476.38i			1.37028i	0.728413	+	0.685138i	\(0.240258\pi\)
							−0.728413	+	0.685138i	\(0.759742\pi\)
\(29\)			5127.79i			1.13223i	0.824326	+	0.566115i	\(0.191554\pi\)
							−0.824326	+	0.566115i	\(0.808446\pi\)
\(31\)		−	2029.05i		−	0.379218i	−0.981860	−	0.189609i	\(-0.939278\pi\)
							0.981860	−	0.189609i	\(-0.0607221\pi\)
\(37\)	−8137.25	+	1768.94i	−0.977177	+	0.212426i
							\(41\)	512.705	−	888.032i	0.0476330	−	0.0825028i	−0.841226	−	0.540684i	\(-0.818166\pi\)
0.888859	+	0.458181i	\(0.151499\pi\)
\(43\)			2812.85i			0.231993i	0.993250	+	0.115997i	\(0.0370062\pi\)
							−0.993250	+	0.115997i	\(0.962994\pi\)
\(47\)	−3506.55			−0.231545			−0.115773	−	0.993276i	\(-0.536934\pi\)
							−0.115773	+	0.993276i	\(0.536934\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.6.e.a.11.12	✓	32
37.27	even	6	inner	37.6.e.a.27.12	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.6.e.a.11.12	✓	32	1.1	even	1	trivial
37.6.e.a.27.12	yes	32	37.27	even	6	inner