Embedded newform 37.5.g.a.8.1

• Label: 37.5.g.a.8.1
• Level: $37$
• Weight: $5$
• Character: 37.8
• 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			8.1
Character	\(\chi\)	\(=\)	37.8
Dual form			37.5.g.a.14.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.08279 - 1.89783i) q^{2} +(-0.547763 - 0.316251i) q^{3} +(32.7077 + 18.8838i) q^{4} +(12.4417 - 3.33374i) q^{5} +(3.27950 + 3.27950i) q^{6} +(-4.56428 + 7.90557i) q^{7} +(-112.864 - 112.864i) q^{8} +(-40.3000 - 69.8016i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 10 q^{2} - 6 q^{3} - 24 q^{4} + 98 q^{5} - 36 q^{6} - 2 q^{7} + 240 q^{8} + 306 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.08279	−	1.89783i	−1.77070	−	0.474457i	−0.781860	−	0.623454i	\(-0.785729\pi\)
							−0.988838	+	0.148997i	\(0.952395\pi\)
\(3\)	−0.547763	−	0.316251i	−0.0608625	−	0.0351390i	0.469260	−	0.883060i	\(-0.344521\pi\)
							−0.530122	+	0.847921i	\(0.677854\pi\)
\(5\)	12.4417	−	3.33374i	0.497667	−	0.133350i	−0.00124982	−	0.999999i	\(-0.500398\pi\)
							0.498917	+	0.866650i	\(0.333731\pi\)
\(7\)	−4.56428	+	7.90557i	−0.0931486	+	0.161338i	−0.908834	−	0.417157i	\(-0.863026\pi\)
							0.815686	+	0.578495i	\(0.196360\pi\)
\(11\)		−	111.675i		−	0.922930i	−0.887158	−	0.461465i	\(-0.847324\pi\)
							0.887158	−	0.461465i	\(-0.152676\pi\)
\(13\)	−107.041	+	28.6815i	−0.633378	+	0.169713i	−0.561202	−	0.827679i	\(-0.689661\pi\)
							−0.0721761	+	0.997392i	\(0.522994\pi\)
\(17\)	104.523	−	390.085i	0.361671	−	1.34977i	−0.510207	−	0.860052i	\(-0.670431\pi\)
							0.871878	−	0.489723i	\(-0.162902\pi\)
\(19\)	228.955	−	61.3483i	0.634224	−	0.169940i	0.0726383	−	0.997358i	\(-0.476858\pi\)
							0.561586	+	0.827418i	\(0.310191\pi\)
\(23\)	−595.125	−	595.125i	−1.12500	−	1.12500i	−0.990978	−	0.134022i	\(-0.957211\pi\)
							−0.134022	−	0.990978i	\(-0.542789\pi\)
\(29\)	24.0431	−	24.0431i	0.0285887	−	0.0285887i	−0.692668	−	0.721257i	\(-0.743565\pi\)
							0.721257	+	0.692668i	\(0.243565\pi\)
\(31\)	1192.52	−	1192.52i	1.24092	−	1.24092i	0.281299	−	0.959620i	\(-0.409235\pi\)
							0.959620	−	0.281299i	\(-0.0907652\pi\)
\(37\)	−1322.08	−	355.345i	−0.965726	−	0.259565i
							\(41\)	1811.74	+	1046.01i	1.07778	+	0.622254i	0.930295	−	0.366811i	\(-0.119550\pi\)
0.147480	+	0.989065i	\(0.452884\pi\)
\(43\)	1481.93	+	1481.93i	0.801479	+	0.801479i	0.983327	−	0.181848i	\(-0.0582078\pi\)
							−0.181848	+	0.983327i	\(0.558208\pi\)
\(47\)	−2361.58			−1.06907			−0.534536	−	0.845146i	\(-0.679514\pi\)
							−0.534536	+	0.845146i	\(0.679514\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.5.g.a.8.1	✓	44
37.14	odd	12	inner	37.5.g.a.14.1	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.5.g.a.8.1	✓	44	1.1	even	1	trivial
37.5.g.a.14.1	yes	44	37.14	odd	12	inner