Embedded newform 37.10.h.a.3.9

• Label: 37.10.h.a.3.9
• Level: $37$
• Weight: $10$
• Character: 37.3
• Analytic conductor: $19.056$
• Analytic rank: $0$
• Dimension: $168$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	37.h (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.0563259381\)
Analytic rank:	\(0\)
Dimension:	\(168\)
Relative dimension:	\(28\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			3.9
Character	\(\chi\)	\(=\)	37.3
Dual form			37.10.h.a.25.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-13.9806 + 16.6614i) q^{2} +(158.307 - 132.835i) q^{3} +(6.76230 + 38.3509i) q^{4} +(-726.809 + 1996.89i) q^{5} +4494.73i q^{6} +(3057.98 + 1113.01i) q^{7} +(-10377.5 - 5991.47i) q^{8} +(3997.96 - 22673.6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q + 45 q^{2} + 216 q^{3} + 249 q^{4} + 177 q^{5} - 5820 q^{7} - 74520 q^{8} - 30198 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−13.9806	+	16.6614i	−0.617859	+	0.736336i	−0.980701	−	0.195515i	\(-0.937362\pi\)
							0.362842	+	0.931851i	\(0.381807\pi\)
\(3\)	158.307	−	132.835i	1.12838	−	0.946822i	0.129381	−	0.991595i	\(-0.458701\pi\)
							0.998997	+	0.0447734i	\(0.0142566\pi\)
\(5\)	−726.809	+	1996.89i	−0.520062	+	1.42886i	0.350390	+	0.936604i	\(0.386049\pi\)
							−0.870452	+	0.492254i	\(0.836173\pi\)
\(7\)	3057.98	+	1113.01i	0.481386	+	0.175210i	0.571303	−	0.820739i	\(-0.306438\pi\)
							−0.0899174	+	0.995949i	\(0.528660\pi\)
\(11\)	−10829.1	+	18756.5i	−0.223010	+	0.386265i	−0.955721	−	0.294276i	\(-0.904922\pi\)
							0.732711	+	0.680540i	\(0.238255\pi\)
\(13\)	−28611.2	+	5044.92i	−0.277837	+	0.0489902i	−0.310830	−	0.950465i	\(-0.600607\pi\)
							0.0329932	+	0.999456i	\(0.489496\pi\)
\(17\)	−179307.	−	31616.7i	−0.520688	−	0.0918114i	−0.0928737	−	0.995678i	\(-0.529605\pi\)
							−0.427815	+	0.903867i	\(0.640716\pi\)
\(19\)	78482.1	+	93531.3i	0.138159	+	0.164652i	0.830687	−	0.556739i	\(-0.187948\pi\)
							−0.692528	+	0.721391i	\(0.743503\pi\)
\(23\)	−1.46722e6	+	847102.i	−1.09325	+	0.631191i	−0.934441	−	0.356118i	\(-0.884100\pi\)
							−0.158813	+	0.987309i	\(0.550767\pi\)
\(29\)	−5.01165e6	−	2.89348e6i	−1.31580	−	0.759677i	−0.332749	−	0.943015i	\(-0.607976\pi\)
							−0.983050	+	0.183338i	\(0.941310\pi\)
\(31\)			3.97227e6i			0.772523i	0.922389	+	0.386262i	\(0.126234\pi\)
							−0.922389	+	0.386262i	\(0.873766\pi\)
\(37\)	−6.67116e6	−	9.24432e6i	−0.585185	−	0.810900i
							\(41\)	2.72065e6	+	1.54296e7i	0.150365	+	0.852760i	0.962902	+	0.269851i	\(0.0869744\pi\)
−0.812538	+	0.582909i	\(0.801914\pi\)
\(43\)			3.55972e7i			1.58784i	0.608020	+	0.793922i	\(0.291964\pi\)
							−0.608020	+	0.793922i	\(0.708036\pi\)
\(47\)	1.31313e7	+	2.27441e7i	0.392525	+	0.679874i	0.992782	−	0.119934i	\(-0.0382682\pi\)
							−0.600257	+	0.799808i	\(0.704935\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.h.a.3.9	✓	168
37.25	even	18	inner	37.10.h.a.25.9	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.h.a.3.9	✓	168	1.1	even	1	trivial
37.10.h.a.25.9	yes	168	37.25	even	18	inner