Embedded newform 177.4.f.a.89.2

• Label: 177.4.f.a.89.2
• Level: $177$
• Weight: $4$
• Character: 177.89
• 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			89.2
Character	\(\chi\)	\(=\)	177.89
Dual form			177.4.f.a.2.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.285117 - 5.25867i) q^{2} +(5.18186 - 0.385093i) q^{3} +(-19.6192 + 2.13372i) q^{4} +(6.02049 + 17.8682i) q^{5} +(-3.50252 - 27.1399i) q^{6} +(18.4360 - 27.1910i) q^{7} +(9.99825 + 60.9866i) q^{8} +(26.7034 - 3.99100i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1624 q - 21 q^{3} - 278 q^{4} - 29 q^{6} - 42 q^{7} - 25 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{57}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.285117	−	5.25867i	−0.100804	−	1.85922i	−0.414947	−	0.909846i	\(-0.636200\pi\)
							0.314143	−	0.949376i	\(-0.398283\pi\)
\(3\)	5.18186	−	0.385093i	0.997250	−	0.0741112i
							\(5\)	6.02049	+	17.8682i	0.538489	+	1.59818i	0.778659	+	0.627447i	\(0.215900\pi\)
−0.240171	+	0.970731i	\(0.577203\pi\)
\(7\)	18.4360	−	27.1910i	0.995449	−	1.46818i	0.115386	−	0.993321i	\(-0.463189\pi\)
							0.880063	−	0.474857i	\(-0.157500\pi\)
\(11\)	33.6603	+	7.40920i	0.922633	+	0.203087i	0.650793	−	0.759255i	\(-0.274437\pi\)
							0.271840	+	0.962342i	\(0.412368\pi\)
\(13\)	−30.7484	−	26.1179i	−0.656005	−	0.557215i	0.256366	−	0.966580i	\(-0.417475\pi\)
							−0.912371	+	0.409364i	\(0.865750\pi\)
\(17\)	21.0742	−	14.2886i	0.300661	−	0.203853i	−0.401523	−	0.915849i	\(-0.631519\pi\)
							0.702184	+	0.711996i	\(0.252209\pi\)
\(19\)	−27.1221	−	51.1576i	−0.327486	−	0.617703i	0.664028	−	0.747708i	\(-0.268846\pi\)
							−0.991513	+	0.130005i	\(0.958501\pi\)
\(23\)	40.8133	+	38.6605i	0.370007	+	0.350490i	0.850000	−	0.526783i	\(-0.176602\pi\)
							−0.479992	+	0.877273i	\(0.659361\pi\)
\(29\)	95.8565	+	5.19719i	0.613797	+	0.0332791i	0.358417	−	0.933562i	\(-0.383317\pi\)
							0.255380	+	0.966841i	\(0.417799\pi\)
\(31\)	−172.470	−	91.4376i	−0.999241	−	0.529764i	−0.113299	−	0.993561i	\(-0.536142\pi\)
							−0.885941	+	0.463797i	\(0.846487\pi\)
\(37\)	273.967	+	44.9147i	1.21730	+	0.199566i	0.735993	−	0.676989i	\(-0.236715\pi\)
							0.481303	+	0.876554i	\(0.340164\pi\)
\(41\)	−102.169	−	107.858i	−0.389172	−	0.410844i	0.501741	−	0.865018i	\(-0.332693\pi\)
							−0.890913	+	0.454174i	\(0.849935\pi\)
\(43\)	89.6907	+	407.469i	0.318086	+	1.44508i	0.816100	+	0.577911i	\(0.196132\pi\)
							−0.498014	+	0.867169i	\(0.665937\pi\)
\(47\)	−253.882	−	85.5429i	−0.787927	−	0.265483i	−0.103562	−	0.994623i	\(-0.533024\pi\)
							−0.684365	+	0.729140i	\(0.739920\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.f.a.89.2	yes	1624
3.2	odd	2	inner	177.4.f.a.89.57	yes	1624
59.2	odd	58	inner	177.4.f.a.2.57	yes	1624
177.2	even	58	inner	177.4.f.a.2.2	✓	1624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.f.a.2.2	✓	1624	177.2	even	58	inner
177.4.f.a.2.57	yes	1624	59.2	odd	58	inner
177.4.f.a.89.2	yes	1624	1.1	even	1	trivial
177.4.f.a.89.57	yes	1624	3.2	odd	2	inner