Embedded newform 177.4.f.a.2.18

• Label: 177.4.f.a.2.18
• Level: $177$
• Weight: $4$
• Character: 177.2
• Analytic conductor: $10.443$
• Analytic rank: $0$
• Dimension: $1624$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.f (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.4433380710\)
Analytic rank:	\(0\)
Dimension:	\(1624\)
Relative dimension:	\(58\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			2.18
Character	\(\chi\)	\(=\)	177.2
Dual form			177.4.f.a.89.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.142776 + 2.63336i) q^{2} +(0.601836 - 5.16118i) q^{3} +(1.03892 + 0.112990i) q^{4} +(-5.43994 + 16.1452i) q^{5} +(13.5053 + 2.32174i) q^{6} +(-11.5153 - 16.9838i) q^{7} +(-3.85912 + 23.5396i) q^{8} +(-26.2756 - 6.21237i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1624 q - 21 q^{3} - 278 q^{4} - 29 q^{6} - 42 q^{7} - 25 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/177\mathbb{Z}\right)^\times\).

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{1}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.142776	+	2.63336i	−0.0504791	+	0.931032i	0.858143	+	0.513412i	\(0.171619\pi\)
							−0.908622	+	0.417620i	\(0.862864\pi\)
\(3\)	0.601836	−	5.16118i	0.115823	−	0.993270i
							\(5\)	−5.43994	+	16.1452i	−0.486563	+	1.44407i	0.370488	+	0.928837i	\(0.379190\pi\)
−0.857051	+	0.515231i	\(0.827706\pi\)
\(7\)	−11.5153	−	16.9838i	−0.621768	−	0.917040i	0.378194	−	0.925726i	\(-0.376545\pi\)
							−0.999962	+	0.00868670i	\(0.997235\pi\)
\(11\)	6.66264	−	1.46656i	0.182624	−	0.0401985i	−0.122717	−	0.992442i	\(-0.539161\pi\)
							0.305341	+	0.952243i	\(0.401230\pi\)
\(13\)	−30.7242	+	26.0973i	−0.655488	+	0.556777i	−0.912217	−	0.409707i	\(-0.865631\pi\)
							0.256729	+	0.966483i	\(0.417355\pi\)
\(17\)	−16.3466	−	11.0833i	−0.233214	−	0.158123i	0.438983	−	0.898495i	\(-0.355339\pi\)
							−0.672197	+	0.740372i	\(0.734649\pi\)
\(19\)	−15.8766	+	29.9464i	−0.191702	+	0.361588i	−0.960667	−	0.277703i	\(-0.910427\pi\)
							0.768965	+	0.639291i	\(0.220772\pi\)
\(23\)	−83.1149	+	78.7306i	−0.753507	+	0.713759i	−0.964603	−	0.263706i	\(-0.915055\pi\)
							0.211097	+	0.977465i	\(0.432297\pi\)
\(29\)	−224.319	+	12.1622i	−1.43638	+	0.0778781i	−0.755802	−	0.654801i	\(-0.772753\pi\)
							−0.680575	+	0.732679i	\(0.738270\pi\)
\(31\)	−30.4126	+	16.1237i	−0.176202	+	0.0934164i	−0.554175	−	0.832400i	\(-0.686966\pi\)
							0.377973	+	0.925817i	\(0.376621\pi\)
\(37\)	225.272	−	36.9314i	1.00093	−	0.164094i	0.361032	−	0.932554i	\(-0.382425\pi\)
							0.639898	+	0.768459i	\(0.278976\pi\)
\(41\)	177.296	−	187.169i	0.675339	−	0.712947i	−0.294783	−	0.955564i	\(-0.595247\pi\)
							0.970122	+	0.242617i	\(0.0780059\pi\)
\(43\)	−35.3679	+	160.678i	−0.125432	+	0.569841i	0.871419	+	0.490539i	\(0.163200\pi\)
							−0.996851	+	0.0793020i	\(0.974731\pi\)
\(47\)	−281.620	+	94.8888i	−0.874010	+	0.294488i	−0.720329	−	0.693632i	\(-0.756009\pi\)
							−0.153681	+	0.988121i	\(0.549113\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.f.a.2.18	✓	1624
3.2	odd	2	inner	177.4.f.a.2.41	yes	1624
59.30	odd	58	inner	177.4.f.a.89.41	yes	1624
177.89	even	58	inner	177.4.f.a.89.18	yes	1624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.f.a.2.18	✓	1624	1.1	even	1	trivial
177.4.f.a.2.41	yes	1624	3.2	odd	2	inner
177.4.f.a.89.18	yes	1624	177.89	even	58	inner
177.4.f.a.89.41	yes	1624	59.30	odd	58	inner