Embedded newform 31.3.h.a.21.4

• Label: 31.3.h.a.21.4
• Level: $31$
• Weight: $3$
• Character: 31.21
• Analytic conductor: $0.845$
• Analytic rank: $0$
• Dimension: $32$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	31.h (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.844688819517\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(4\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			21.4
Character	\(\chi\)	\(=\)	31.21
Dual form			31.3.h.a.3.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.09451 + 1.52175i) q^{2} +(-1.49940 + 0.157594i) q^{3} +(0.835177 + 2.57041i) q^{4} +(-1.26763 - 2.19560i) q^{5} +(-3.38033 - 1.95163i) q^{6} +(-0.0160743 - 0.00341670i) q^{7} +(1.03789 - 3.19430i) q^{8} +(-6.57995 + 1.39861i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 6 q^{2} - 10 q^{3} - 18 q^{4} - 7 q^{5} - 9 q^{6} - 22 q^{7} + 43 q^{8} - 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/31\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{29}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.09451	+	1.52175i	1.04725	+	0.760875i	0.971688	−	0.236266i	\(-0.0759237\pi\)
							0.0755659	+	0.997141i	\(0.475924\pi\)
\(3\)	−1.49940	+	0.157594i	−0.499801	+	0.0525312i	−0.351075	−	0.936347i	\(-0.614184\pi\)
							−0.148726	+	0.988878i	\(0.547517\pi\)
\(5\)	−1.26763	−	2.19560i	−0.253526	−	0.439121i	0.710968	−	0.703225i	\(-0.248257\pi\)
							−0.964494	+	0.264104i	\(0.914924\pi\)
\(7\)	−0.0160743	−	0.00341670i	−0.00229633	−	0.000488099i	0.206763	−	0.978391i	\(-0.433707\pi\)
							−0.209060	+	0.977903i	\(0.567040\pi\)
\(11\)	7.27610	+	6.55143i	0.661464	+	0.595585i	0.929944	−	0.367700i	\(-0.119855\pi\)
							−0.268480	+	0.963285i	\(0.586521\pi\)
\(13\)	0.883525	−	1.98443i	0.0679635	−	0.152648i	−0.876370	−	0.481639i	\(-0.840042\pi\)
							0.944333	+	0.328990i	\(0.106708\pi\)
\(17\)	−21.5079	+	19.3658i	−1.26517	+	1.13917i	−0.281428	+	0.959582i	\(0.590808\pi\)
							−0.983743	+	0.179583i	\(0.942525\pi\)
\(19\)	−6.69921	+	2.98268i	−0.352590	+	0.156983i	−0.575386	−	0.817882i	\(-0.695148\pi\)
							0.222796	+	0.974865i	\(0.428482\pi\)
\(23\)	34.9558	+	11.3578i	1.51982	+	0.493819i	0.945724	−	0.324971i	\(-0.105355\pi\)
							0.574094	+	0.818790i	\(0.305355\pi\)
\(29\)	−8.01759	+	11.0353i	−0.276469	+	0.380526i	−0.924560	−	0.381036i	\(-0.875567\pi\)
							0.648092	+	0.761562i	\(0.275567\pi\)
\(31\)	−17.5393	−	25.5612i	−0.565783	−	0.824554i
							\(37\)	−23.0379	−	13.3009i	−0.622645	−	0.359484i	0.155253	−	0.987875i	\(-0.450381\pi\)
−0.777898	+	0.628391i	\(0.783714\pi\)
\(41\)	5.30222	−	50.4473i	0.129322	−	1.23042i	−0.716741	−	0.697339i	\(-0.754367\pi\)
							0.846064	−	0.533082i	\(-0.178966\pi\)
\(43\)	−4.42018	−	9.92788i	−0.102795	−	0.230881i	0.854825	−	0.518917i	\(-0.173664\pi\)
							−0.957620	+	0.288036i	\(0.906998\pi\)
\(47\)	30.5454	−	22.1926i	0.649903	−	0.472182i	−0.213335	−	0.976979i	\(-0.568433\pi\)
							0.863238	+	0.504797i	\(0.168433\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	31.3.h.a.21.4	yes	32
3.2	odd	2		279.3.bc.b.145.1		32
31.3	odd	30	inner	31.3.h.a.3.4	✓	32
93.65	even	30		279.3.bc.b.127.1		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.3.h.a.3.4	✓	32	31.3	odd	30	inner
31.3.h.a.21.4	yes	32	1.1	even	1	trivial
279.3.bc.b.127.1		32	93.65	even	30
279.3.bc.b.145.1		32	3.2	odd	2