Embedded newform 78.5.l.b.7.2

• Label: 78.5.l.b.7.2
• Level: $78$
• Weight: $5$
• Character: 78.7
• Analytic conductor: $8.063$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 78 = 2 \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	78.l (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.06285712054\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{12})\)
Coefficient field:	8.0.1579585536.10
Defining polynomial:	\( x^{8} - 2x^{7} - 4x^{6} + 28x^{5} - 38x^{4} + 8x^{3} + 200x^{2} - 352x + 484 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 3^{2}\cdot 13^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			7.2
Root			\(-2.59436 + 0.0368949i\) of defining polynomial
Character	\(\chi\)	\(=\)	78.7
Dual form			78.5.l.b.67.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.73205 - 0.732051i) q^{2} +(-2.59808 + 4.50000i) q^{3} +(6.92820 - 4.00000i) q^{4} +(1.16932 + 1.16932i) q^{5} +(-3.80385 + 14.1962i) q^{6} +(50.3452 + 13.4900i) q^{7} +(16.0000 - 16.0000i) q^{8} +(-13.5000 - 23.3827i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 24 q^{5} - 72 q^{6} + 22 q^{7} + 128 q^{8} - 108 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/78\mathbb{Z}\right)^\times\).

\(n\)	\(53\)	\(67\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{11}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.73205	−	0.732051i	0.683013	−	0.183013i
							\(3\)	−2.59808	+	4.50000i	−0.288675	+	0.500000i
\(5\)	1.16932	+	1.16932i	0.0467730	+	0.0467730i								0.730106	−	0.683333i	\(-0.239470\pi\)
							−0.683333	+	0.730106i	\(0.739470\pi\)
\(7\)	50.3452	+	13.4900i	1.02745	+	0.275305i	0.732905	−	0.680331i	\(-0.238164\pi\)
							0.294549	+	0.955637i	\(0.404831\pi\)
\(11\)	46.0771	+	171.962i	0.380803	+	1.42117i	0.844678	+	0.535274i	\(0.179792\pi\)
							−0.463876	+	0.885900i	\(0.653542\pi\)
\(13\)	161.899	+	48.4728i	0.957984	+	0.286821i
							\(17\)	133.137	−	76.8669i	0.460683	−	0.265975i	−0.251648	−	0.967819i	\(-0.580973\pi\)
0.712331	+	0.701843i	\(0.247639\pi\)
\(19\)	−61.0751	+	227.935i	−0.169183	+	0.631400i	0.828286	+	0.560305i	\(0.189316\pi\)
							−0.997470	+	0.0710952i	\(0.977351\pi\)
\(23\)	−183.439	−	105.909i	−0.346766	−	0.200206i	0.316494	−	0.948595i	\(-0.397494\pi\)
							−0.663260	+	0.748389i	\(0.730828\pi\)
\(29\)	42.5668	−	73.7279i	0.0506146	−	0.0876670i	−0.839608	−	0.543193i	\(-0.817215\pi\)
							0.890223	+	0.455526i	\(0.150549\pi\)
\(31\)	−871.003	−	871.003i	−0.906351	−	0.906351i	0.0896251	−	0.995976i	\(-0.471433\pi\)
							−0.995976	+	0.0896251i	\(0.971433\pi\)
\(37\)	87.0800	+	324.987i	0.0636085	+	0.237390i	0.990410	−	0.138163i	\(-0.0441196\pi\)
							−0.926801	+	0.375553i	\(0.877453\pi\)
\(41\)	−2430.30	+	651.198i	−1.44575	+	0.387387i	−0.894543	−	0.446983i	\(-0.852499\pi\)
							−0.551205	+	0.834370i	\(0.685832\pi\)
\(43\)	2150.81	−	1241.77i	1.16323	−	0.671590i	0.211153	−	0.977453i	\(-0.432278\pi\)
							0.952076	+	0.305863i	\(0.0989450\pi\)
\(47\)	307.843	−	307.843i	0.139359	−	0.139359i	−0.633986	−	0.773345i	\(-0.718582\pi\)
							0.773345	+	0.633986i	\(0.218582\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	78.5.l.b.7.2	✓	8
3.2	odd	2		234.5.bb.b.163.1		8
13.2	odd	12	inner	78.5.l.b.67.2	yes	8
39.2	even	12		234.5.bb.b.145.1		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.5.l.b.7.2	✓	8	1.1	even	1	trivial
78.5.l.b.67.2	yes	8	13.2	odd	12	inner
234.5.bb.b.145.1		8	39.2	even	12
234.5.bb.b.163.1		8	3.2	odd	2