Embedded newform 177.4.e.b.7.5

• Label: 177.4.e.b.7.5
• Level: $177$
• Weight: $4$
• Character: 177.7
• 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			7.5
Character	\(\chi\)	\(=\)	177.7
Dual form			177.4.e.b.76.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28887 - 1.90094i) q^{2} +(0.162417 - 2.99560i) q^{3} +(1.00873 - 2.53171i) q^{4} +(8.34074 - 3.85884i) q^{5} +(-5.90378 + 3.55218i) q^{6} +(3.61604 - 13.0238i) q^{7} +(-24.0566 + 5.29526i) q^{8} +(-8.94724 - 0.973071i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q + 2 q^{2} - 45 q^{3} - 56 q^{4} - 14 q^{5} + 6 q^{6} + 6 q^{7} + 66 q^{8} - 135 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{9}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.28887	−	1.90094i	−0.455683	−	0.672082i	0.528561	−	0.848895i	\(-0.322732\pi\)
							−0.984244	+	0.176813i	\(0.943421\pi\)
\(3\)	0.162417	−	2.99560i	0.0312571	−	0.576504i
							\(5\)	8.34074	−	3.85884i	0.746018	−	0.345145i	−0.00977758	−	0.999952i	\(-0.503112\pi\)
0.755796	+	0.654807i	\(0.227250\pi\)
\(7\)	3.61604	−	13.0238i	0.195248	−	0.703219i	−0.799482	−	0.600690i	\(-0.794893\pi\)
							0.994730	−	0.102529i	\(-0.0326935\pi\)
\(11\)	41.1733	−	39.0014i	1.12857	−	1.06903i	0.131591	−	0.991304i	\(-0.457991\pi\)
							0.996974	−	0.0777302i	\(-0.0247673\pi\)
\(13\)	1.22137	−	0.132832i	0.0260575	−	0.00283392i	−0.0950810	−	0.995470i	\(-0.530311\pi\)
							0.121138	+	0.992636i	\(0.461345\pi\)
\(17\)	4.41016	+	15.8839i	0.0629188	+	0.226613i	0.988324	−	0.152366i	\(-0.0486891\pi\)
							−0.925405	+	0.378979i	\(0.876275\pi\)
\(19\)	78.0681	−	59.3458i	0.942634	−	0.716572i	−0.0163990	−	0.999866i	\(-0.505220\pi\)
							0.959033	+	0.283294i	\(0.0914271\pi\)
\(23\)	−57.7795	+	108.984i	−0.523820	+	0.988029i	0.470581	+	0.882357i	\(0.344044\pi\)
							−0.994401	+	0.105673i	\(0.966300\pi\)
\(29\)	−28.6233	+	42.2163i	−0.183283	+	0.270323i	−0.908220	−	0.418493i	\(-0.862559\pi\)
							0.724937	+	0.688815i	\(0.241869\pi\)
\(31\)	−251.276	−	191.015i	−1.45582	−	1.10669i	−0.972955	−	0.230996i	\(-0.925801\pi\)
							−0.482870	−	0.875692i	\(-0.660405\pi\)
\(37\)	−103.863	−	22.8619i	−0.461484	−	0.101580i	−0.0218575	−	0.999761i	\(-0.506958\pi\)
							−0.439626	+	0.898181i	\(0.644889\pi\)
\(41\)	206.441	+	389.389i	0.786357	+	1.48323i	0.873390	+	0.487021i	\(0.161916\pi\)
							−0.0870334	+	0.996205i	\(0.527739\pi\)
\(43\)	−124.866	−	118.280i	−0.442835	−	0.419476i	0.433518	−	0.901145i	\(-0.357272\pi\)
							−0.876353	+	0.481669i	\(0.840031\pi\)
\(47\)	414.756	+	191.887i	1.28720	+	0.595522i	0.939649	−	0.342139i	\(-0.111151\pi\)
							0.347551	+	0.937661i	\(0.387013\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.e.b.7.5	✓	420
59.17	even	29	inner	177.4.e.b.76.5	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.e.b.7.5	✓	420	1.1	even	1	trivial
177.4.e.b.76.5	yes	420	59.17	even	29	inner