Embedded newform 77.3.p.a.3.14

• Label: 77.3.p.a.3.14
• Level: $77$
• Weight: $3$
• Character: 77.3
• 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			3.14
Character	\(\chi\)	\(=\)	77.3
Dual form			77.3.p.a.26.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.07171 + 1.36761i) q^{2} +(0.674115 - 3.17146i) q^{3} +(4.88852 + 5.42926i) q^{4} +(-1.65544 - 0.173994i) q^{5} +(6.40802 - 8.81989i) q^{6} +(-6.98372 + 0.477162i) q^{7} +(3.43485 + 10.5714i) q^{8} +(-1.38183 - 0.615230i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 5 q^{2} - 9 q^{3} + 27 q^{4} - 15 q^{5} - 23 q^{7} - 72 q^{8} - 27 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(45\)	\(57\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.07171	+	1.36761i	1.53586	+	0.683807i	0.988238	−	0.152921i	\(-0.0488681\pi\)
							0.547618	+	0.836729i	\(0.315535\pi\)
\(3\)	0.674115	−	3.17146i	0.224705	−	1.05715i	−0.710678	−	0.703517i	\(-0.751612\pi\)
							0.935383	−	0.353636i	\(-0.115055\pi\)
\(5\)	−1.65544	−	0.173994i	−0.331088	−	0.0347988i	−0.0624739	−	0.998047i	\(-0.519899\pi\)
							−0.268614	+	0.963248i	\(0.586566\pi\)
\(7\)	−6.98372	+	0.477162i	−0.997674	+	0.0681660i
							\(11\)	−4.99969	+	9.79812i	−0.454517	+	0.890738i
\(13\)	0.691564	+	0.951857i	0.0531973	+	0.0732197i								0.834787	−	0.550573i	\(-0.185591\pi\)
							−0.781590	+	0.623793i	\(0.785591\pi\)
\(17\)	−6.35617	−	14.2762i	−0.373893	−	0.839777i	−0.998278	−	0.0586601i	\(-0.981317\pi\)
							0.624385	−	0.781116i	\(-0.285349\pi\)
\(19\)	6.52386	+	5.87411i	0.343361	+	0.309164i	0.822711	−	0.568460i	\(-0.192461\pi\)
							−0.479350	+	0.877624i	\(0.659127\pi\)
\(23\)	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	−0.276530	−	0.961005i	\(-0.589184\pi\)
							0.788582	−	0.614930i	\(-0.210816\pi\)
\(31\)	13.9199	−	1.46304i	0.449028	−	0.0471948i	0.122684	−	0.992446i	\(-0.460850\pi\)
							0.326344	+	0.945251i	\(0.394183\pi\)
\(37\)	14.3255	−	3.04497i	0.387174	−	0.0822965i	−0.0102119	−	0.999948i	\(-0.503251\pi\)
							0.397386	+	0.917651i	\(0.369917\pi\)
\(41\)	−75.8452	+	24.6436i	−1.84988	+	0.601063i	−0.853023	+	0.521873i	\(0.825233\pi\)
							−0.996860	+	0.0791903i	\(0.974767\pi\)
\(43\)	27.3716			0.636549			0.318275	−	0.947999i	\(-0.396897\pi\)
							0.318275	+	0.947999i	\(0.396897\pi\)
\(47\)	55.4106	+	49.8919i	1.17895	+	1.06153i	0.996935	+	0.0782337i	\(0.0249280\pi\)
							0.182013	+	0.983296i	\(0.441739\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.3.p.a.3.14	✓	112	1.1	even	1	trivial
77.3.p.a.26.14	yes	112	77.26	odd	30	inner
77.3.p.a.47.1	yes	112	7.5	odd	6	inner
77.3.p.a.59.1	yes	112	11.4	even	5	inner