Embedded newform 76.2.f.a.31.5

• Label: 76.2.f.a.31.5
• 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			31.5
Root			\(1.16486 - 0.801943i\) of defining polynomial
Character	\(\chi\)	\(=\)	76.31
Dual form			76.2.f.a.27.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.112075 + 1.40977i) q^{2} +(-0.305055 - 0.528371i) q^{3} +(-1.97488 + 0.316000i) q^{4} +(1.59295 + 2.75907i) q^{5} +(0.710690 - 0.489273i) q^{6} +2.36291i q^{7} +(-0.666820 - 2.74870i) q^{8} +(1.31388 - 2.27571i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 3 q^{2} - 3 q^{4} - 2 q^{5} + 5 q^{6} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.112075	+	1.40977i	0.0792492	+	0.996855i
							\(3\)	−0.305055	−	0.528371i	−0.176124	−	0.305055i	0.764426	−	0.644712i	\(-0.223023\pi\)
−0.940550	+	0.339657i	\(0.889689\pi\)
\(5\)	1.59295	+	2.75907i	0.712389	+	1.23389i	0.963958	+	0.266055i	\(0.0857203\pi\)
							−0.251569	+	0.967839i	\(0.580946\pi\)
\(7\)			2.36291i			0.893097i	0.894759	+	0.446548i	\(0.147347\pi\)
							−0.894759	+	0.446548i	\(0.852653\pi\)
\(11\)		−	5.46750i		−	1.64851i	−0.566216	−	0.824257i	\(-0.691593\pi\)
							0.566216	−	0.824257i	\(-0.308407\pi\)
\(13\)	−2.31924	−	1.33901i	−0.643241	−	0.371375i	0.142621	−	0.989777i	\(-0.454447\pi\)
							−0.785862	+	0.618402i	\(0.787780\pi\)
\(17\)	−0.552780	−	0.957443i	−0.134069	−	0.232214i	0.791173	−	0.611593i	\(-0.209471\pi\)
							−0.925241	+	0.379379i	\(0.876138\pi\)
\(19\)	1.37952	+	4.13484i	0.316484	+	0.948598i
							\(23\)	2.46168	+	1.42125i	0.513296	+	0.296352i	0.734187	−	0.678947i	\(-0.237563\pi\)
−0.220891	+	0.975298i	\(0.570897\pi\)
\(29\)	−5.63736	−	3.25473i	−1.04683	−	0.604389i	−0.125071	−	0.992148i	\(-0.539916\pi\)
							−0.921761	+	0.387759i	\(0.873249\pi\)
\(31\)	1.01504			0.182306			0.0911531	−	0.995837i	\(-0.470945\pi\)
							0.0911531	+	0.995837i	\(0.470945\pi\)
\(37\)		−	0.450315i		−	0.0740314i	−0.999315	−	0.0370157i	\(-0.988215\pi\)
							0.999315	−	0.0370157i	\(-0.0117851\pi\)
\(41\)	0.336089	−	0.194041i	0.0524882	−	0.0303041i	−0.473526	−	0.880780i	\(-0.657019\pi\)
							0.526014	+	0.850476i	\(0.323686\pi\)
\(43\)	−4.96197	+	2.86479i	−0.756693	+	0.436877i	−0.828107	−	0.560570i	\(-0.810582\pi\)
							0.0714141	+	0.997447i	\(0.477249\pi\)
\(47\)	2.91563	+	1.68334i	0.425288	+	0.245540i	0.697337	−	0.716743i	\(-0.254368\pi\)
							−0.272049	+	0.962283i	\(0.587701\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.2.f.a.31.5	yes	16
3.2	odd	2		684.2.r.a.487.4		16
4.3	odd	2	inner	76.2.f.a.31.2	yes	16
8.3	odd	2		1216.2.n.f.639.4		16
8.5	even	2		1216.2.n.f.639.5		16
12.11	even	2		684.2.r.a.487.7		16
19.8	odd	6	inner	76.2.f.a.27.2	✓	16
57.8	even	6		684.2.r.a.559.7		16
76.27	even	6	inner	76.2.f.a.27.5	yes	16
152.27	even	6		1216.2.n.f.255.5		16
152.141	odd	6		1216.2.n.f.255.4		16
228.179	odd	6		684.2.r.a.559.4		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.2.f.a.27.2	✓	16	19.8	odd	6	inner
76.2.f.a.27.5	yes	16	76.27	even	6	inner
76.2.f.a.31.2	yes	16	4.3	odd	2	inner
76.2.f.a.31.5	yes	16	1.1	even	1	trivial
684.2.r.a.487.4		16	3.2	odd	2
684.2.r.a.487.7		16	12.11	even	2
684.2.r.a.559.4		16	228.179	odd	6
684.2.r.a.559.7		16	57.8	even	6
1216.2.n.f.255.4		16	152.141	odd	6
1216.2.n.f.255.5		16	152.27	even	6
1216.2.n.f.639.4		16	8.3	odd	2
1216.2.n.f.639.5		16	8.5	even	2