Embedded newform 77.3.p.a.59.1

• Label: 77.3.p.a.59.1
• Level: $77$
• Weight: $3$
• Character: 77.59
• Analytic conductor: $2.098$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 77 = 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	77.p (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.09809803557\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(14\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			59.1
Character	\(\chi\)	\(=\)	77.59
Dual form			77.3.p.a.47.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.351467 + 3.34399i) q^{2} +(-2.40951 + 2.16953i) q^{3} +(-7.14614 - 1.51896i) q^{4} +(-0.677037 + 1.52065i) q^{5} +(-6.40802 - 8.81989i) q^{6} +(5.36948 - 4.49096i) q^{7} +(3.43485 - 10.5714i) q^{8} +(0.158110 - 1.50431i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 5 q^{2} - 9 q^{3} + 27 q^{4} - 15 q^{5} - 23 q^{7} - 72 q^{8} - 27 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/77\mathbb{Z}\right)^\times\).

\(n\)	\(45\)	\(57\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.351467	+	3.34399i	−0.175734	+	1.67199i	0.450819	+	0.892615i	\(0.351132\pi\)
							−0.626553	+	0.779379i	\(0.715535\pi\)
\(3\)	−2.40951	+	2.16953i	−0.803170	+	0.723177i	−0.964603	−	0.263706i	\(-0.915055\pi\)
							0.161433	+	0.986884i	\(0.448388\pi\)
\(5\)	−0.677037	+	1.52065i	−0.135407	+	0.304130i	−0.968504	−	0.248997i	\(-0.919899\pi\)
							0.833097	+	0.553127i	\(0.186566\pi\)
\(7\)	5.36948	−	4.49096i	0.767068	−	0.641566i
							\(11\)	−5.98557	+	9.22892i	−0.544143	+	0.838993i
\(13\)	−0.691564	+	0.951857i	−0.0531973	+	0.0732197i								−0.834787	−	0.550573i	\(-0.814409\pi\)
							0.781590	+	0.623793i	\(0.214409\pi\)
\(17\)	−15.5416	+	1.63349i	−0.914214	+	0.0960878i	−0.549940	−	0.835204i	\(-0.685349\pi\)
							−0.364274	+	0.931292i	\(0.618683\pi\)
\(19\)	−1.82520	−	8.58688i	−0.0960631	−	0.451941i	−0.999720	−	0.0236830i	\(-0.992461\pi\)
							0.903656	−	0.428258i	\(-0.140873\pi\)
\(23\)	12.4979	+	21.6469i	0.543385	+	0.941170i	0.998707	+	0.0508432i	\(0.0161909\pi\)
							−0.455322	+	0.890327i	\(0.650476\pi\)
\(29\)	14.8495	+	45.7021i	0.512052	+	1.57593i	0.788582	+	0.614930i	\(0.210816\pi\)
							−0.276530	+	0.961005i	\(0.589184\pi\)
\(31\)	5.69291	+	12.7865i	0.183642	+	0.412467i	0.981783	−	0.190003i	\(-0.0608498\pi\)
							−0.798141	+	0.602470i	\(0.794183\pi\)
\(37\)	−9.79975	+	10.8837i	−0.264858	+	0.294155i	−0.860874	−	0.508818i	\(-0.830083\pi\)
							0.596016	+	0.802972i	\(0.296749\pi\)
\(41\)	75.8452	+	24.6436i	1.84988	+	0.601063i	0.996860	+	0.0791903i	\(0.0252335\pi\)
							0.853023	+	0.521873i	\(0.174767\pi\)
\(43\)	27.3716			0.636549			0.318275	−	0.947999i	\(-0.396897\pi\)
							0.318275	+	0.947999i	\(0.396897\pi\)
\(47\)	−15.5024	−	72.9329i	−0.329838	−	1.55176i	−0.760553	−	0.649276i	\(-0.775072\pi\)
							0.430715	−	0.902488i	\(-0.358261\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	77.3.p.a.59.1	yes	112
7.5	odd	6	inner	77.3.p.a.26.14	yes	112
11.3	even	5	inner	77.3.p.a.3.14	✓	112
77.47	odd	30	inner	77.3.p.a.47.1	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.3.p.a.3.14	✓	112	11.3	even	5	inner
77.3.p.a.26.14	yes	112	7.5	odd	6	inner
77.3.p.a.47.1	yes	112	77.47	odd	30	inner
77.3.p.a.59.1	yes	112	1.1	even	1	trivial