Embedded newform 76.2.f.a.31.8

• Label: 76.2.f.a.31.8
• Level: $76$
• Weight: $2$
• Character: 76.31
• Analytic conductor: $0.607$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	76.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.606863055362\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	x^16 - 3*x^15 + 6*x^14 - 9*x^13 + 12*x^12 - 9*x^11 + 3*x^10 + 6*x^9 - 10*x^8 + 12*x^7 + 12*x^6 - 72*x^5 + 192*x^4 - 288*x^3 + 384*x^2 - 384*x + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			31.8
Root			\(-0.327894 - 1.37568i\) of defining polynomial
Character	\(\chi\)	\(=\)	76.31
Dual form			76.2.f.a.27.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.35532 + 0.403874i) q^{2} +(-1.42689 - 2.47144i) q^{3} +(1.67377 + 1.09475i) q^{4} +(-0.139977 - 0.242447i) q^{5} +(-0.935735 - 3.92587i) q^{6} +1.55280i q^{7} +(1.82635 + 2.15973i) q^{8} +(-2.57201 + 4.45486i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 3 q^{2} - 3 q^{4} - 2 q^{5} + 5 q^{6} - 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\)	\(21\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.35532	+	0.403874i	0.958354	+	0.285582i
							\(3\)	−1.42689	−	2.47144i	−0.823814	−	1.42689i	−0.902823	−	0.430013i	\(-0.858509\pi\)
0.0790092	−	0.996874i	\(-0.474824\pi\)
\(5\)	−0.139977	−	0.242447i	−0.0625997	−	0.108426i	0.833027	−	0.553232i	\(-0.186606\pi\)
							−0.895627	+	0.444806i	\(0.853272\pi\)
\(7\)			1.55280i			0.586902i	0.955974	+	0.293451i	\(0.0948038\pi\)
							−0.955974	+	0.293451i	\(0.905196\pi\)
\(11\)			2.44102i			0.735994i	0.929827	+	0.367997i	\(0.119956\pi\)
							−0.929827	+	0.367997i	\(0.880044\pi\)
\(13\)	−5.47527	−	3.16115i	−1.51857	−	0.876745i	−0.999761	−	0.0218524i	\(-0.993044\pi\)
							−0.518805	−	0.854892i	\(-0.673623\pi\)
\(17\)	2.18321	+	3.78143i	0.529507	+	0.917132i	0.999408	+	0.0344132i	\(0.0109562\pi\)
							−0.469901	+	0.882719i	\(0.655710\pi\)
\(19\)	3.17111	−	2.99066i	0.727503	−	0.686105i
							\(23\)	−5.71910	−	3.30192i	−1.19251	−	0.688498i	−0.233638	−	0.972324i	\(-0.575063\pi\)
−0.958876	+	0.283825i	\(0.908396\pi\)
\(29\)	0.695180	+	0.401362i	0.129092	+	0.0745311i	0.563155	−	0.826351i	\(-0.309587\pi\)
							−0.434064	+	0.900882i	\(0.642921\pi\)
\(31\)	1.42017			0.255070			0.127535	−	0.991834i	\(-0.459293\pi\)
							0.127535	+	0.991834i	\(0.459293\pi\)
\(37\)		−	4.74916i		−	0.780757i	−0.920654	−	0.390379i	\(-0.872344\pi\)
							0.920654	−	0.390379i	\(-0.127656\pi\)
\(41\)	−5.84961	+	3.37728i	−0.913556	+	0.527442i	−0.881574	−	0.472047i	\(-0.843515\pi\)
							−0.0319825	+	0.999488i	\(0.510182\pi\)
\(43\)	−0.146372	+	0.0845082i	−0.0223216	+	0.0128874i	−0.511119	−	0.859510i	\(-0.670769\pi\)
							0.488798	+	0.872397i	\(0.337436\pi\)
\(47\)	−1.19839	−	0.691890i	−0.174803	−	0.100923i	0.410046	−	0.912065i	\(-0.365513\pi\)
							−0.584849	+	0.811142i	\(0.698846\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.2.f.a.31.8	yes	16
3.2	odd	2		684.2.r.a.487.1		16
4.3	odd	2	inner	76.2.f.a.31.6	yes	16
8.3	odd	2		1216.2.n.f.639.1		16
8.5	even	2		1216.2.n.f.639.8		16
12.11	even	2		684.2.r.a.487.3		16
19.8	odd	6	inner	76.2.f.a.27.6	✓	16
57.8	even	6		684.2.r.a.559.3		16
76.27	even	6	inner	76.2.f.a.27.8	yes	16
152.27	even	6		1216.2.n.f.255.8		16
152.141	odd	6		1216.2.n.f.255.1		16
228.179	odd	6		684.2.r.a.559.1		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.2.f.a.27.6	✓	16	19.8	odd	6	inner
76.2.f.a.27.8	yes	16	76.27	even	6	inner
76.2.f.a.31.6	yes	16	4.3	odd	2	inner
76.2.f.a.31.8	yes	16	1.1	even	1	trivial
684.2.r.a.487.1		16	3.2	odd	2
684.2.r.a.487.3		16	12.11	even	2
684.2.r.a.559.1		16	228.179	odd	6
684.2.r.a.559.3		16	57.8	even	6
1216.2.n.f.255.1		16	152.141	odd	6
1216.2.n.f.255.8		16	152.27	even	6
1216.2.n.f.639.1		16	8.3	odd	2
1216.2.n.f.639.8		16	8.5	even	2