Embedded newform 77.4.f.a.36.4

• Label: 77.4.f.a.36.4
• Level: $77$
• Weight: $4$
• Character: 77.36
• Analytic conductor: $4.543$
• Analytic rank: $0$
• Dimension: $32$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 77 = 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	77.f (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.54314707044\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			36.4
Character	\(\chi\)	\(=\)	77.36
Dual form			77.4.f.a.15.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.756117 + 0.549351i) q^{2} +(2.59084 + 7.97378i) q^{3} +(-2.20221 + 6.77770i) q^{4} +(6.79916 + 4.93988i) q^{5} +(-6.33938 - 4.60583i) q^{6} +(2.16312 - 6.65740i) q^{7} +(-4.36870 - 13.4455i) q^{8} +(-35.0253 + 25.4474i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 2 q^{2} + 18 q^{3} - 48 q^{4} + 16 q^{5} + 30 q^{6} - 56 q^{7} + 62 q^{8} - 46 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.756117	+	0.549351i	−0.267328	+	0.194225i	−0.713371	−	0.700786i	\(-0.752833\pi\)
							0.446044	+	0.895011i	\(0.352833\pi\)
\(3\)	2.59084	+	7.97378i	0.498607	+	1.53456i	0.811259	+	0.584687i	\(0.198783\pi\)
							−0.312652	+	0.949868i	\(0.601217\pi\)
\(5\)	6.79916	+	4.93988i	0.608136	+	0.441836i	0.848757	−	0.528783i	\(-0.177351\pi\)
							−0.240622	+	0.970619i	\(0.577351\pi\)
\(7\)	2.16312	−	6.65740i	0.116797	−	0.359466i
							\(11\)	12.3128	−	34.3423i	0.337495	−	0.941327i
\(13\)	0.809303	−	0.587993i	0.0172662	−	0.0125446i								−0.579119	−	0.815243i	\(-0.696603\pi\)
							0.596385	+	0.802699i	\(0.296603\pi\)
\(17\)	74.7169	+	54.2850i	1.06597	+	0.774474i	0.975184	−	0.221396i	\(-0.0710615\pi\)
							0.0907879	+	0.995870i	\(0.471061\pi\)
\(19\)	19.6477	+	60.4693i	0.237236	+	0.730138i	0.996817	+	0.0797246i	\(0.0254041\pi\)
							−0.759581	+	0.650413i	\(0.774596\pi\)
\(23\)	−76.1736			−0.690578			−0.345289	−	0.938496i	\(-0.612219\pi\)
							−0.345289	+	0.938496i	\(0.612219\pi\)
\(29\)	−31.6157	+	97.3030i	−0.202444	+	0.623059i	0.797364	+	0.603498i	\(0.206227\pi\)
							−0.999809	+	0.0195611i	\(0.993773\pi\)
\(31\)	141.283	−	102.648i	0.818551	−	0.594712i	−0.0977459	−	0.995211i	\(-0.531163\pi\)
							0.916297	+	0.400499i	\(0.131163\pi\)
\(37\)	−32.1096	+	98.8231i	−0.142670	+	0.439092i	−0.996704	−	0.0811246i	\(-0.974149\pi\)
							0.854034	+	0.520217i	\(0.174149\pi\)
\(41\)	−111.774	−	344.004i	−0.425759	−	1.31035i	−0.902266	−	0.431180i	\(-0.858097\pi\)
							0.476507	−	0.879170i	\(-0.341903\pi\)
\(43\)	290.229			1.02929			0.514646	−	0.857403i	\(-0.327923\pi\)
							0.514646	+	0.857403i	\(0.327923\pi\)
\(47\)	119.865	+	368.905i	0.372001	+	1.14490i	0.945480	+	0.325680i	\(0.105593\pi\)
							−0.573479	+	0.819220i	\(0.694407\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	77.4.f.a.36.4	yes	32
11.2	odd	10		847.4.a.p.1.7		16
11.4	even	5	inner	77.4.f.a.15.4	✓	32
11.9	even	5		847.4.a.o.1.10		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.f.a.15.4	✓	32	11.4	even	5	inner
77.4.f.a.36.4	yes	32	1.1	even	1	trivial
847.4.a.o.1.10		16	11.9	even	5
847.4.a.p.1.7		16	11.2	odd	10