Embedded newform 177.4.e.b.4.5

• Label: 177.4.e.b.4.5
• Level: $177$
• Weight: $4$
• Character: 177.4
• Analytic conductor: $10.443$
• Analytic rank: $0$
• Dimension: $420$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.e (of order \(29\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.4433380710\)
Analytic rank:	\(0\)
Dimension:	\(420\)
Relative dimension:	\(15\) over \(\Q(\zeta_{29})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{29}]$

Embedding invariants

Embedding label			4.5
Character	\(\chi\)	\(=\)	177.4
Dual form			177.4.e.b.133.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.96757 - 0.322743i) q^{2} +(1.94216 - 2.28649i) q^{3} +(0.889340 + 0.195759i) q^{4} +(-11.1726 - 8.49315i) q^{5} +(-6.50144 + 6.15849i) q^{6} +(12.7555 - 32.0138i) q^{7} +(20.0545 + 6.75714i) q^{8} +(-1.45604 - 8.88144i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q + 2 q^{2} - 45 q^{3} - 56 q^{4} - 14 q^{5} + 6 q^{6} + 6 q^{7} + 66 q^{8} - 135 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{1}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.96757	−	0.322743i	−1.04919	−	0.114107i	−0.432757	−	0.901511i	\(-0.642459\pi\)
							−0.616437	+	0.787404i	\(0.711425\pi\)
\(3\)	1.94216	−	2.28649i	0.373769	−	0.440034i
							\(5\)	−11.1726	−	8.49315i	−0.999303	−	0.759651i	−0.0285163	−	0.999593i	\(-0.509078\pi\)
−0.970787	+	0.239943i	\(0.922871\pi\)
\(7\)	12.7555	−	32.0138i	0.688731	−	1.72859i	0.00378384	−	0.999993i	\(-0.498796\pi\)
							0.684947	−	0.728592i	\(-0.259825\pi\)
\(11\)	1.79779	−	0.831748i	0.0492778	−	0.0227983i	−0.395095	−	0.918640i	\(-0.629288\pi\)
							0.444373	+	0.895842i	\(0.353426\pi\)
\(13\)	4.43627	−	27.0601i	0.0946462	−	0.577316i	−0.896342	−	0.443364i	\(-0.853785\pi\)
							0.990988	−	0.133952i	\(-0.0427669\pi\)
\(17\)	9.85155	+	24.7255i	0.140550	+	0.352754i	0.982354	−	0.187028i	\(-0.0598857\pi\)
							−0.841804	+	0.539783i	\(0.818506\pi\)
\(19\)	−28.1583	−	41.5304i	−0.339998	−	0.501459i	0.618800	−	0.785548i	\(-0.287619\pi\)
							−0.958798	+	0.284090i	\(0.908309\pi\)
\(23\)	−2.70992	+	49.9816i	−0.0245678	+	0.453125i	0.960207	+	0.279290i	\(0.0900990\pi\)
							−0.984775	+	0.173836i	\(0.944384\pi\)
\(29\)	−58.3702	+	6.34814i	−0.373761	+	0.0406490i	−0.293072	−	0.956090i	\(-0.594678\pi\)
							−0.0806891	+	0.996739i	\(0.525712\pi\)
\(31\)	48.9599	−	72.2105i	0.283660	−	0.418367i	−0.658989	−	0.752152i	\(-0.729016\pi\)
							0.942649	+	0.333785i	\(0.108326\pi\)
\(37\)	−360.797	+	121.567i	−1.60310	+	0.540146i	−0.971883	−	0.235463i	\(-0.924339\pi\)
							−0.631214	+	0.775609i	\(0.717443\pi\)
\(41\)	3.19233	+	58.8790i	0.0121599	+	0.224277i	0.998297	+	0.0583420i	\(0.0185814\pi\)
							−0.986137	+	0.165935i	\(0.946936\pi\)
\(43\)	353.761	+	163.667i	1.25461	+	0.580443i	0.930748	−	0.365662i	\(-0.119157\pi\)
							0.323859	+	0.946105i	\(0.395020\pi\)
\(47\)	−353.521	+	268.740i	−1.09716	+	0.834037i	−0.987135	−	0.159892i	\(-0.948885\pi\)
							−0.110023	+	0.993929i	\(0.535092\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.e.b.4.5	✓	420
59.15	even	29	inner	177.4.e.b.133.5	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.e.b.4.5	✓	420	1.1	even	1	trivial
177.4.e.b.133.5	yes	420	59.15	even	29	inner