Embedded newform 37.5.i.a.5.6

• Label: 37.5.i.a.5.6
• Level: $37$
• Weight: $5$
• Character: 37.5
• Analytic conductor: $3.825$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	37.i (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.82468863410\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(12\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			5.6
Character	\(\chi\)	\(=\)	37.5
Dual form			37.5.i.a.15.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.02938 + 2.20752i) q^{2} +(0.471556 - 1.29559i) q^{3} +(6.47109 + 7.71195i) q^{4} +(15.7198 - 22.4501i) q^{5} +(2.37463 + 2.37463i) q^{6} +(13.1089 + 74.3445i) q^{7} +(-61.3292 + 16.4331i) q^{8} +(60.5934 + 50.8439i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 12 q^{3} + 6 q^{4} - 120 q^{5} - 12 q^{6} - 12 q^{7} - 282 q^{8} + 198 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{23}{36}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.02938	+	2.20752i	−0.257346	+	0.551880i	−0.991952	−	0.126614i	\(-0.959589\pi\)
							0.734606	+	0.678494i	\(0.237367\pi\)
\(3\)	0.471556	−	1.29559i	0.0523951	−	0.143954i	−0.910734	−	0.412993i	\(-0.864484\pi\)
							0.963130	+	0.269038i	\(0.0867058\pi\)
\(5\)	15.7198	−	22.4501i	0.628790	−	0.898005i	−0.370737	−	0.928738i	\(-0.620895\pi\)
							0.999528	+	0.0307323i	\(0.00978393\pi\)
\(7\)	13.1089	+	74.3445i	0.267529	+	1.51723i	0.761734	+	0.647889i	\(0.224348\pi\)
							−0.494205	+	0.869345i	\(0.664541\pi\)
\(11\)	−42.0635	−	24.2854i	−0.347632	−	0.200706i	0.316010	−	0.948756i	\(-0.397657\pi\)
							−0.663642	+	0.748050i	\(0.730990\pi\)
\(13\)	155.957	+	13.6445i	0.922824	+	0.0807367i	0.538655	−	0.842526i	\(-0.318932\pi\)
							0.384169	+	0.923263i	\(0.374488\pi\)
\(17\)	10.5240	+	120.290i	0.0364153	+	0.416228i	0.992421	+	0.122882i	\(0.0392135\pi\)
							−0.956006	+	0.293347i	\(0.905231\pi\)
\(19\)	−286.071	−	613.481i	−0.792440	−	1.69939i	−0.711514	−	0.702672i	\(-0.751990\pi\)
							−0.0809262	−	0.996720i	\(-0.525788\pi\)
\(23\)	238.304	−	889.361i	0.450480	−	1.68121i	−0.250569	−	0.968099i	\(-0.580618\pi\)
							0.701048	−	0.713114i	\(-0.252716\pi\)
\(29\)	−78.9652	−	294.702i	−0.0938945	−	0.350419i	0.902955	−	0.429735i	\(-0.141393\pi\)
							−0.996850	+	0.0793161i	\(0.974726\pi\)
\(31\)	−986.880	+	986.880i	−1.02693	+	1.02693i	−0.0273035	+	0.999627i	\(0.508692\pi\)
							−0.999627	+	0.0273035i	\(0.991308\pi\)
\(37\)	1337.63	+	291.366i	0.977089	+	0.212831i
							\(41\)	−245.062	−	292.053i	−0.145783	−	0.173738i	0.688211	−	0.725510i	\(-0.258396\pi\)
−0.833995	+	0.551772i	\(0.813952\pi\)
\(43\)	−2366.99	−	2366.99i	−1.28014	−	1.28014i	−0.940584	−	0.339560i	\(-0.889722\pi\)
							−0.339560	−	0.940584i	\(-0.610278\pi\)
\(47\)	735.876	+	1274.57i	0.333126	+	0.576991i	0.983123	−	0.182945i	\(-0.0585632\pi\)
							−0.649997	+	0.759937i	\(0.725230\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.5.i.a.5.6	✓	144
37.15	odd	36	inner	37.5.i.a.15.6	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.5.i.a.5.6	✓	144	1.1	even	1	trivial
37.5.i.a.15.6	yes	144	37.15	odd	36	inner