Embedded newform 37.5.g.a.14.6

• Label: 37.5.g.a.14.6
• Level: $37$
• Weight: $5$
• Character: 37.14
• Analytic conductor: $3.825$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	37.g (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.82468863410\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(11\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			14.6
Character	\(\chi\)	\(=\)	37.14
Dual form			37.5.g.a.8.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.112513 + 0.0301478i) q^{2} +(1.29148 - 0.745634i) q^{3} +(-13.8447 + 7.99322i) q^{4} +(-28.7245 - 7.69670i) q^{5} +(-0.122829 + 0.122829i) q^{6} +(-12.4901 - 21.6335i) q^{7} +(2.63458 - 2.63458i) q^{8} +(-39.3881 + 68.2221i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 10 q^{2} - 6 q^{3} - 24 q^{4} + 98 q^{5} - 36 q^{6} - 2 q^{7} + 240 q^{8} + 306 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{11}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.112513	+	0.0301478i	−0.0281283	+	0.00753696i	−0.272856	−	0.962055i	\(-0.587968\pi\)
							0.244727	+	0.969592i	\(0.421302\pi\)
\(3\)	1.29148	−	0.745634i	0.143497	−	0.0828483i	−0.426532	−	0.904472i	\(-0.640265\pi\)
							0.570030	+	0.821624i	\(0.306932\pi\)
\(5\)	−28.7245	−	7.69670i	−1.14898	−	0.307868i	−0.366423	−	0.930448i	\(-0.619418\pi\)
							−0.782556	+	0.622580i	\(0.786084\pi\)
\(7\)	−12.4901	−	21.6335i	−0.254900	−	0.441499i	0.709969	−	0.704233i	\(-0.248709\pi\)
							−0.964868	+	0.262734i	\(0.915376\pi\)
\(11\)		−	18.5940i		−	0.153669i	−0.997044	−	0.0768347i	\(-0.975519\pi\)
							0.997044	−	0.0768347i	\(-0.0244814\pi\)
\(13\)	−175.674	−	47.0716i	−1.03949	−	0.278530i	−0.301587	−	0.953439i	\(-0.597516\pi\)
							−0.737902	+	0.674908i	\(0.764183\pi\)
\(17\)	58.6244	+	218.789i	0.202853	+	0.757057i	0.990093	+	0.140411i	\(0.0448423\pi\)
							−0.787241	+	0.616646i	\(0.788491\pi\)
\(19\)	338.801	+	90.7814i	0.938507	+	0.251472i	0.695478	−	0.718547i	\(-0.255193\pi\)
							0.243028	+	0.970019i	\(0.421859\pi\)
\(23\)	28.2565	−	28.2565i	0.0534149	−	0.0534149i	−0.679895	−	0.733310i	\(-0.737975\pi\)
							0.733310	+	0.679895i	\(0.237975\pi\)
\(29\)	−183.666	−	183.666i	−0.218390	−	0.218390i	0.589430	−	0.807820i	\(-0.299353\pi\)
							−0.807820	+	0.589430i	\(0.799353\pi\)
\(31\)	−1032.26	−	1032.26i	−1.07416	−	1.07416i	−0.997021	−	0.0771362i	\(-0.975422\pi\)
							−0.0771362	−	0.997021i	\(-0.524578\pi\)
\(37\)	−503.544	+	1273.03i	−0.367819	+	0.929898i
							\(41\)	−2221.94	+	1282.84i	−1.32179	+	0.763139i	−0.984015	−	0.178086i	\(-0.943009\pi\)
−0.337780	+	0.941225i	\(0.609676\pi\)
\(43\)	−258.411	+	258.411i	−0.139757	+	0.139757i	−0.773524	−	0.633767i	\(-0.781508\pi\)
							0.633767	+	0.773524i	\(0.281508\pi\)
\(47\)	−2931.90			−1.32725			−0.663625	−	0.748065i	\(-0.730983\pi\)
							−0.663625	+	0.748065i	\(0.730983\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.5.g.a.14.6	yes	44
37.8	odd	12	inner	37.5.g.a.8.6	✓	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.5.g.a.8.6	✓	44	37.8	odd	12	inner
37.5.g.a.14.6	yes	44	1.1	even	1	trivial