Embedded newform 37.9.g.a.8.20

• Label: 37.9.g.a.8.20
• Level: $37$
• Weight: $9$
• Character: 37.8
• Analytic conductor: $15.073$
• Analytic rank: $0$
• Dimension: $100$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	37.g (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(15.0730085723\)
Analytic rank:	\(0\)
Dimension:	\(100\)
Relative dimension:	\(25\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			8.20
Character	\(\chi\)	\(=\)	37.8
Dual form			37.9.g.a.14.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(21.1757 + 5.67401i) q^{2} +(-53.5012 - 30.8889i) q^{3} +(194.514 + 112.303i) q^{4} +(280.857 - 75.2555i) q^{5} +(-957.661 - 957.661i) q^{6} +(727.522 - 1260.11i) q^{7} +(-486.675 - 486.675i) q^{8} +(-1372.25 - 2376.80i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 100 q - 10 q^{2} - 6 q^{3} + 552 q^{4} - 1642 q^{5} - 516 q^{6} - 2 q^{7} - 7440 q^{8} + 117258 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	21.1757	+	5.67401i	1.32348	+	0.354626i	0.850281	−	0.526329i	\(-0.176432\pi\)
							0.473201	+	0.880955i	\(0.343099\pi\)
\(3\)	−53.5012	−	30.8889i	−0.660508	−	0.381345i	0.131962	−	0.991255i	\(-0.457872\pi\)
							−0.792471	+	0.609910i	\(0.791206\pi\)
\(5\)	280.857	−	75.2555i	0.449372	−	0.120409i	−0.0270335	−	0.999635i	\(-0.508606\pi\)
							0.476406	+	0.879226i	\(0.341939\pi\)
\(7\)	727.522	−	1260.11i	0.303008	−	0.524825i	−0.673808	−	0.738907i	\(-0.735342\pi\)
							0.976816	+	0.214081i	\(0.0686758\pi\)
\(11\)		−	3152.83i		−	0.215342i	−0.994187	−	0.107671i	\(-0.965661\pi\)
							0.994187	−	0.107671i	\(-0.0343393\pi\)
\(13\)	22965.2	−	6153.52i	0.804077	−	0.215452i	0.166704	−	0.986007i	\(-0.446688\pi\)
							0.637373	+	0.770555i	\(0.280021\pi\)
\(17\)	36645.2	−	136762.i	0.438754	−	1.63745i	−0.293165	−	0.956062i	\(-0.594708\pi\)
							0.731919	−	0.681392i	\(-0.238625\pi\)
\(19\)	47644.1	−	12766.2i	0.365590	−	0.0979597i	−0.0713470	−	0.997452i	\(-0.522730\pi\)
							0.436937	+	0.899492i	\(0.356063\pi\)
\(23\)	111733.	+	111733.i	0.399274	+	0.399274i	0.877977	−	0.478703i	\(-0.158893\pi\)
							−0.478703	+	0.877977i	\(0.658893\pi\)
\(29\)	−141261.	+	141261.i	−0.199724	+	0.199724i	−0.799882	−	0.600158i	\(-0.795104\pi\)
							0.600158	+	0.799882i	\(0.295104\pi\)
\(31\)	107150.	−	107150.i	0.116023	−	0.116023i	−0.646711	−	0.762735i	\(-0.723856\pi\)
							0.762735	+	0.646711i	\(0.223856\pi\)
\(37\)	−1.41643e6	+	1.22727e6i	−0.755768	+	0.654839i
							\(41\)	517278.	+	298651.i	0.183058	+	0.105689i	0.588729	−	0.808331i	\(-0.299629\pi\)
−0.405671	+	0.914019i	\(0.632962\pi\)
\(43\)	747639.	+	747639.i	0.218684	+	0.218684i	0.807944	−	0.589259i	\(-0.200580\pi\)
							−0.589259	+	0.807944i	\(0.700580\pi\)
\(47\)	116242.			0.0238216			0.0119108	−	0.999929i	\(-0.496209\pi\)
							0.0119108	+	0.999929i	\(0.496209\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.9.g.a.8.20	✓	100
37.14	odd	12	inner	37.9.g.a.14.20	yes	100

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.9.g.a.8.20	✓	100	1.1	even	1	trivial
37.9.g.a.14.20	yes	100	37.14	odd	12	inner