Embedded newform 37.6.h.a.3.6

• Label: 37.6.h.a.3.6
• Level: $37$
• Weight: $6$
• Character: 37.3
• Analytic conductor: $5.934$
• Analytic rank: $0$
• Dimension: $90$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	37.h (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.93420133308\)
Analytic rank:	\(0\)
Dimension:	\(90\)
Relative dimension:	\(15\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			3.6
Character	\(\chi\)	\(=\)	37.3
Dual form			37.6.h.a.25.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.40929 + 2.87128i) q^{2} +(21.9339 - 18.4047i) q^{3} +(3.11718 + 17.6784i) q^{4} +(22.1706 - 60.9132i) q^{5} +107.320i q^{6} +(-26.8299 - 9.76528i) q^{7} +(-162.142 - 93.6130i) q^{8} +(100.165 - 568.064i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 90 q - 3 q^{2} - 9 q^{3} - 39 q^{4} - 153 q^{5} + 201 q^{7} + 792 q^{8} + 87 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.40929	+	2.87128i	−0.425906	+	0.507575i	−0.935736	−	0.352700i	\(-0.885264\pi\)
							0.509831	+	0.860275i	\(0.329708\pi\)
\(3\)	21.9339	−	18.4047i	1.40706	−	1.18066i	0.449196	−	0.893433i	\(-0.351711\pi\)
							0.957862	−	0.287229i	\(-0.0927339\pi\)
\(5\)	22.1706	−	60.9132i	0.396599	−	1.08965i	−0.567330	−	0.823491i	\(-0.692024\pi\)
							0.963930	−	0.266158i	\(-0.0857542\pi\)
\(7\)	−26.8299	−	9.76528i	−0.206954	−	0.0753251i	0.236463	−	0.971640i	\(-0.424012\pi\)
							−0.443417	+	0.896315i	\(0.646234\pi\)
\(11\)	−153.575	+	266.000i	−0.382684	+	0.662827i	−0.991445	−	0.130526i	\(-0.958333\pi\)
							0.608761	+	0.793353i	\(0.291667\pi\)
\(13\)	1148.40	−	202.494i	1.88467	−	0.332319i	0.891890	−	0.452252i	\(-0.149379\pi\)
							0.992782	+	0.119934i	\(0.0382681\pi\)
\(17\)	477.169	+	84.1377i	0.400451	+	0.0706104i	0.370247	−	0.928933i	\(-0.379273\pi\)
							0.0302043	+	0.999544i	\(0.490384\pi\)
\(19\)	156.764	+	186.824i	0.0996234	+	0.118727i	0.813553	−	0.581491i	\(-0.197531\pi\)
							−0.713929	+	0.700218i	\(0.753086\pi\)
\(23\)	−3248.07	+	1875.27i	−1.28028	+	0.739171i	−0.976900	−	0.213698i	\(-0.931449\pi\)
							−0.303382	+	0.952869i	\(0.598116\pi\)
\(29\)	−432.571	−	249.745i	−0.0955130	−	0.0551445i	0.451483	−	0.892280i	\(-0.350895\pi\)
							−0.546996	+	0.837135i	\(0.684229\pi\)
\(31\)			6660.88i			1.24488i	0.782667	+	0.622440i	\(0.213859\pi\)
							−0.782667	+	0.622440i	\(0.786141\pi\)
\(37\)	772.547	+	8291.39i	0.0927727	+	0.995687i
							\(41\)	−669.897	−	3799.18i	−0.0622370	−	0.352963i	−0.999984	−	0.00564460i	\(-0.998203\pi\)
0.937747	−	0.347319i	\(-0.112908\pi\)
\(43\)			20104.3i			1.65813i	0.559156	+	0.829063i	\(0.311125\pi\)
							−0.559156	+	0.829063i	\(0.688875\pi\)
\(47\)	−79.8740	−	138.346i	−0.00527425	−	0.00913527i	0.863376	−	0.504561i	\(-0.168346\pi\)
							−0.868650	+	0.495425i	\(0.835012\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.6.h.a.3.6	✓	90
37.25	even	18	inner	37.6.h.a.25.6	yes	90

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.6.h.a.3.6	✓	90	1.1	even	1	trivial
37.6.h.a.25.6	yes	90	37.25	even	18	inner