Embedded newform 37.5.i.a.5.1

• Label: 37.5.i.a.5.1
• 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			5.1
Character	\(\chi\)	\(=\)	37.5
Dual form			37.5.i.a.15.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.29658 + 7.06954i) q^{2} +(-3.57062 + 9.81021i) q^{3} +(-28.8263 - 34.3539i) q^{4} +(0.215509 - 0.307778i) q^{5} +(-57.5828 - 57.5828i) q^{6} +(11.8673 + 67.3028i) q^{7} +(217.341 - 58.2363i) q^{8} +(-21.4413 - 17.9914i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 12 q^{3} + 6 q^{4} - 120 q^{5} - 12 q^{6} - 12 q^{7} - 282 q^{8} + 198 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.29658	+	7.06954i	−0.824145	+	1.76738i	−0.210871	+	0.977514i	\(0.567630\pi\)
							−0.613274	+	0.789870i	\(0.710148\pi\)
\(3\)	−3.57062	+	9.81021i	−0.396736	+	1.09002i	0.567128	+	0.823630i	\(0.308054\pi\)
							−0.963864	+	0.266394i	\(0.914168\pi\)
\(5\)	0.215509	−	0.307778i	0.00862035	−	0.0123111i	−0.814819	−	0.579716i	\(-0.803164\pi\)
							0.823439	+	0.567405i	\(0.192052\pi\)
\(7\)	11.8673	+	67.3028i	0.242190	+	1.37353i	0.826930	+	0.562305i	\(0.190085\pi\)
							−0.584740	+	0.811221i	\(0.698803\pi\)
\(11\)	−109.303	−	63.1061i	−0.903330	−	0.521538i	−0.0250511	−	0.999686i	\(-0.507975\pi\)
							−0.878279	+	0.478148i	\(0.841308\pi\)
\(13\)	−16.8507	−	1.47424i	−0.0997082	−	0.00872333i	0.0371925	−	0.999308i	\(-0.488159\pi\)
							−0.136901	+	0.990585i	\(0.543714\pi\)
\(17\)	−35.3549	−	404.109i	−0.122335	−	1.39830i	−0.770810	−	0.637065i	\(-0.780148\pi\)
							0.648475	−	0.761236i	\(-0.275407\pi\)
\(19\)	−44.6604	−	95.7746i	−0.123713	−	0.265304i	0.834691	−	0.550719i	\(-0.185646\pi\)
							−0.958404	+	0.285415i	\(0.907869\pi\)
\(23\)	−210.401	+	785.227i	−0.397733	+	1.48436i	0.419341	+	0.907829i	\(0.362261\pi\)
							−0.817074	+	0.576532i	\(0.804406\pi\)
\(29\)	−189.332	−	706.595i	−0.225127	−	0.840184i	−0.982354	−	0.187033i	\(-0.940113\pi\)
							0.757227	−	0.653152i	\(-0.226554\pi\)
\(31\)	−364.480	+	364.480i	−0.379272	+	0.379272i	−0.870839	−	0.491568i	\(-0.836424\pi\)
							0.491568	+	0.870839i	\(0.336424\pi\)
\(37\)	−1319.21	+	365.851i	−0.963630	+	0.267239i
							\(41\)	521.120	+	621.047i	0.310006	+	0.369451i	0.898441	−	0.439093i	\(-0.144700\pi\)
−0.588435	+	0.808544i	\(0.700256\pi\)
\(43\)	1592.47	+	1592.47i	0.861260	+	0.861260i	0.991485	−	0.130224i	\(-0.0415697\pi\)
							−0.130224	+	0.991485i	\(0.541570\pi\)
\(47\)	430.013	+	744.805i	0.194664	+	0.337168i	0.946790	−	0.321851i	\(-0.104305\pi\)
							−0.752126	+	0.659019i	\(0.770972\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.5.i.a.5.1	✓	144
37.15	odd	36	inner	37.5.i.a.15.1	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.5.i.a.5.1	✓	144	1.1	even	1	trivial
37.5.i.a.15.1	yes	144	37.15	odd	36	inner