Embedded newform 37.5.i.a.2.6

• Label: 37.5.i.a.2.6
• Level: $37$
• Weight: $5$
• Character: 37.2
• 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			2.6
Character	\(\chi\)	\(=\)	37.2
Dual form			37.5.i.a.19.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.13246 - 0.0990776i) q^{2} +(9.68265 - 11.5393i) q^{3} +(-14.4843 - 2.55397i) q^{4} +(6.08797 - 13.0557i) q^{5} +(-12.1085 + 12.1085i) q^{6} +(-66.6341 + 24.2528i) q^{7} +(33.7187 + 9.03489i) q^{8} +(-25.3370 - 143.693i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 12 q^{3} + 6 q^{4} - 120 q^{5} - 12 q^{6} - 12 q^{7} - 282 q^{8} + 198 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{36}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.13246	−	0.0990776i	−0.283115	−	0.0247694i	−0.0552864	−	0.998471i	\(-0.517607\pi\)
							−0.227829	+	0.973701i	\(0.573163\pi\)
\(3\)	9.68265	−	11.5393i	1.07585	−	1.28215i	0.118582	−	0.992944i	\(-0.462165\pi\)
							0.957268	−	0.289203i	\(-0.0933904\pi\)
\(5\)	6.08797	−	13.0557i	0.243519	−	0.522228i	−0.746119	−	0.665812i	\(-0.768085\pi\)
							0.989638	+	0.143584i	\(0.0458629\pi\)
\(7\)	−66.6341	+	24.2528i	−1.35988	+	0.494955i	−0.916017	−	0.401140i	\(-0.868614\pi\)
							−0.443862	+	0.896095i	\(0.646392\pi\)
\(11\)	166.836	−	96.3228i	1.37881	−	0.796056i	0.386793	−	0.922167i	\(-0.373583\pi\)
							0.992016	+	0.126111i	\(0.0402495\pi\)
\(13\)	−68.1829	−	97.3753i	−0.403449	−	0.576185i	0.565095	−	0.825026i	\(-0.308839\pi\)
							−0.968544	+	0.248841i	\(0.919950\pi\)
\(17\)	−37.8374	−	26.4940i	−0.130925	−	0.0916748i	0.506279	−	0.862370i	\(-0.331020\pi\)
							−0.637205	+	0.770695i	\(0.719909\pi\)
\(19\)	436.952	−	38.2283i	1.21039	−	0.105896i	0.535967	−	0.844239i	\(-0.319947\pi\)
							0.674425	+	0.738343i	\(0.264391\pi\)
\(23\)	154.310	+	575.891i	0.291700	+	1.08864i	0.943803	+	0.330510i	\(0.107221\pi\)
							−0.652102	+	0.758131i	\(0.726113\pi\)
\(29\)	191.117	−	713.257i	0.227249	−	0.848106i	−0.754242	−	0.656597i	\(-0.771995\pi\)
							0.981491	−	0.191509i	\(-0.0613380\pi\)
\(31\)	−912.146	−	912.146i	−0.949164	−	0.949164i	0.0496051	−	0.998769i	\(-0.484204\pi\)
							−0.998769	+	0.0496051i	\(0.984204\pi\)
\(37\)	748.997	+	1145.93i	0.547112	+	0.837059i
							\(41\)	753.226	+	132.814i	0.448082	+	0.0790089i	0.393136	−	0.919480i	\(-0.371390\pi\)
0.0549461	+	0.998489i	\(0.482501\pi\)
\(43\)	708.183	−	708.183i	0.383009	−	0.383009i	−0.489176	−	0.872185i	\(-0.662703\pi\)
							0.872185	+	0.489176i	\(0.162703\pi\)
\(47\)	−727.360	+	1259.82i	−0.329271	+	0.570315i	−0.982367	−	0.186960i	\(-0.940136\pi\)
							0.653096	+	0.757275i	\(0.273470\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.5.i.a.2.6	✓	144
37.19	odd	36	inner	37.5.i.a.19.6	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.5.i.a.2.6	✓	144	1.1	even	1	trivial
37.5.i.a.19.6	yes	144	37.19	odd	36	inner