Newform orbit 171.4.y.a

• Label: 171.4.y.a
• Level: $171$
• Weight: $4$
• Character orbit: 171.y
• Analytic conductor: $10.089$
• Analytic rank: $0$
• Dimension: $120$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	171.y (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.0893266110\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(20\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 36 q^{4}+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
53.1	−4.90666	−	1.78588i		0		14.7576	+	12.3831i	−3.22388	−	3.84207i		0		−0.347063	+	0.601131i	−29.4095	−	50.9388i		0		8.95701	+	24.6092i
53.2	−4.79619	−	1.74567i		0		13.8277	+	11.6028i	0.869736	+	1.03651i		0		−17.0305	+	29.4978i	−25.6497	−	44.4266i		0		−2.36201	−	6.48958i
53.3	−4.20988	−	1.53227i		0		9.24691	+	7.75908i	−12.4048	−	14.7835i		0		14.3815	−	24.9094i	−9.11914	−	15.7948i		0		29.5705	+	81.2444i
53.4	−3.41758	−	1.24390i		0		4.00419	+	3.35992i	8.24027	+	9.82037i		0		8.78095	−	15.2090i	5.04239	+	8.73367i		0		−15.9462	−	43.8119i
53.5	−3.36331	−	1.22414i		0		3.68495	+	3.09204i	13.3019	+	15.8526i		0		−7.77429	+	13.4655i	5.70811	+	9.88673i		0		−25.3325	−	69.6004i
53.6	−3.13604	−	1.14142i		0		2.40352	+	2.01679i	−0.417489	−	0.497545i		0		11.8643	−	20.5495i	8.11369	+	14.0533i		0		0.741353	+	2.03685i
53.7	−2.30065	−	0.837369i		0		−1.53654	−	1.28931i	−10.9536	−	13.0540i		0		−16.1646	+	27.9979i	12.2486	+	21.2153i		0		14.2695	+	39.2050i
53.8	−1.91636	−	0.697498i		0		−2.94242	−	2.46899i	0.792095	+	0.943982i		0		−7.99477	+	13.8473i	12.0740	+	20.9128i		0		−0.859514	−	2.36149i
53.9	−1.02676	−	0.373709i		0		−5.21378	−	4.37488i	−4.49187	−	5.35320i		0		9.55306	−	16.5464i	8.08896	+	14.0105i		0		2.61152	+	7.17509i
53.10	−0.393398	−	0.143185i		0		−5.99410	−	5.02964i	−6.91842	−	8.24506i		0		−0.354982	+	0.614847i	3.31248	+	5.73738i		0		1.54113	+	4.23421i
53.11	0.393398	+	0.143185i		0		−5.99410	−	5.02964i	6.91842	+	8.24506i		0		−0.354982	+	0.614847i	−3.31248	−	5.73738i		0		1.54113	+	4.23421i
53.12	1.02676	+	0.373709i		0		−5.21378	−	4.37488i	4.49187	+	5.35320i		0		9.55306	−	16.5464i	−8.08896	−	14.0105i		0		2.61152	+	7.17509i
53.13	1.91636	+	0.697498i		0		−2.94242	−	2.46899i	−0.792095	−	0.943982i		0		−7.99477	+	13.8473i	−12.0740	−	20.9128i		0		−0.859514	−	2.36149i
53.14	2.30065	+	0.837369i		0		−1.53654	−	1.28931i	10.9536	+	13.0540i		0		−16.1646	+	27.9979i	−12.2486	−	21.2153i		0		14.2695	+	39.2050i
53.15	3.13604	+	1.14142i		0		2.40352	+	2.01679i	0.417489	+	0.497545i		0		11.8643	−	20.5495i	−8.11369	−	14.0533i		0		0.741353	+	2.03685i
53.16	3.36331	+	1.22414i		0		3.68495	+	3.09204i	−13.3019	−	15.8526i		0		−7.77429	+	13.4655i	−5.70811	−	9.88673i		0		−25.3325	−	69.6004i
53.17	3.41758	+	1.24390i		0		4.00419	+	3.35992i	−8.24027	−	9.82037i		0		8.78095	−	15.2090i	−5.04239	−	8.73367i		0		−15.9462	−	43.8119i
53.18	4.20988	+	1.53227i		0		9.24691	+	7.75908i	12.4048	+	14.7835i		0		14.3815	−	24.9094i	9.11914	+	15.7948i		0		29.5705	+	81.2444i
53.19	4.79619	+	1.74567i		0		13.8277	+	11.6028i	−0.869736	−	1.03651i		0		−17.0305	+	29.4978i	25.6497	+	44.4266i		0		−2.36201	−	6.48958i
53.20	4.90666	+	1.78588i		0		14.7576	+	12.3831i	3.22388	+	3.84207i		0		−0.347063	+	0.601131i	29.4095	+	50.9388i		0		8.95701	+	24.6092i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
19.f	odd	18	1	inner
57.j	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	171.4.y.a	✓	120
3.b	odd	2	1	inner	171.4.y.a	✓	120
19.f	odd	18	1	inner	171.4.y.a	✓	120
57.j	even	18	1	inner	171.4.y.a	✓	120

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
171.4.y.a	✓	120	1.a	even	1	1	trivial
171.4.y.a	✓	120	3.b	odd	2	1	inner
171.4.y.a	✓	120	19.f	odd	18	1	inner
171.4.y.a	✓	120	57.j	even	18	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(171, [\chi])\).