Newform orbit 169.3.f.g

• Label: 169.3.f.g
• Level: $169$
• Weight: $3$
• Character orbit: 169.f
• Analytic conductor: $4.605$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	169.f (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.60491646769\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(12\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 12 q^{3} - 84 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
19.1	−0.834643	−	3.11493i	1.02304	−	1.77195i	−5.54206	+	3.19971i	−5.50472	+	5.50472i	−6.37336	−	1.70774i	−0.846854	+	3.16050i	5.47136	+	5.47136i	2.40680	+	4.16870i	21.7413	+	12.5523i
19.2	−0.739915	−	2.76140i	−2.32422	+	4.02567i	−3.61376	+	2.08640i	2.43155	−	2.43155i	12.8362	+	3.43945i	0.277954	−	1.03734i	0.349324	+	0.349324i	−6.30402	−	10.9189i	−8.51361	−	4.91533i
19.3	−0.609234	−	2.27369i	2.51175	−	4.35048i	−1.33442	+	0.770425i	1.10964	−	1.10964i	−11.4219	−	3.06049i	−1.70128	+	6.34925i	−4.09315	−	4.09315i	−8.11776	−	14.0604i	−3.19902	−	1.84696i
19.4	−0.515320	−	1.92320i	0.299764	−	0.519206i	0.0309536	−	0.0178711i	5.65111	−	5.65111i	−1.15301	−	0.308949i	−0.526568	+	1.96518i	−5.68184	−	5.68184i	4.32028	+	7.48295i	−13.7804	−	7.95610i
19.5	−0.429937	−	1.60455i	−0.677024	+	1.17264i	1.07438	−	0.620291i	−1.57408	+	1.57408i	2.17264	+	0.582156i	−3.25921	+	12.1635i	−6.15564	−	6.15564i	3.58328	+	6.20642i	3.20245	+	1.84893i
19.6	−0.104575	−	0.390278i	−2.33330	+	4.04139i	3.32272	−	1.91837i	4.14024	−	4.14024i	1.82127	+	0.488008i	0.591332	−	2.20688i	−2.23898	−	2.23898i	−6.38858	−	11.0653i	−2.04881	−	1.18288i
19.7	0.104575	+	0.390278i	−2.33330	+	4.04139i	3.32272	−	1.91837i	−4.14024	+	4.14024i	−1.82127	−	0.488008i	−0.591332	+	2.20688i	2.23898	+	2.23898i	−6.38858	−	11.0653i	−2.04881	−	1.18288i
19.8	0.429937	+	1.60455i	−0.677024	+	1.17264i	1.07438	−	0.620291i	1.57408	−	1.57408i	−2.17264	−	0.582156i	3.25921	−	12.1635i	6.15564	+	6.15564i	3.58328	+	6.20642i	3.20245	+	1.84893i
19.9	0.515320	+	1.92320i	0.299764	−	0.519206i	0.0309536	−	0.0178711i	−5.65111	+	5.65111i	1.15301	+	0.308949i	0.526568	−	1.96518i	5.68184	+	5.68184i	4.32028	+	7.48295i	−13.7804	−	7.95610i
19.10	0.609234	+	2.27369i	2.51175	−	4.35048i	−1.33442	+	0.770425i	−1.10964	+	1.10964i	11.4219	+	3.06049i	1.70128	−	6.34925i	4.09315	+	4.09315i	−8.11776	−	14.0604i	−3.19902	−	1.84696i
19.11	0.739915	+	2.76140i	−2.32422	+	4.02567i	−3.61376	+	2.08640i	−2.43155	+	2.43155i	−12.8362	−	3.43945i	−0.277954	+	1.03734i	−0.349324	−	0.349324i	−6.30402	−	10.9189i	−8.51361	−	4.91533i
19.12	0.834643	+	3.11493i	1.02304	−	1.77195i	−5.54206	+	3.19971i	5.50472	−	5.50472i	6.37336	+	1.70774i	0.846854	−	3.16050i	−5.47136	−	5.47136i	2.40680	+	4.16870i	21.7413	+	12.5523i
80.1	−3.11493	−	0.834643i	1.02304	+	1.77195i	5.54206	+	3.19971i	5.50472	−	5.50472i	−1.70774	−	6.37336i	−3.16050	+	0.846854i	−5.47136	−	5.47136i	2.40680	−	4.16870i	−21.7413	+	12.5523i
80.2	−2.76140	−	0.739915i	−2.32422	−	4.02567i	3.61376	+	2.08640i	−2.43155	+	2.43155i	3.43945	+	12.8362i	1.03734	−	0.277954i	−0.349324	−	0.349324i	−6.30402	+	10.9189i	8.51361	−	4.91533i
80.3	−2.27369	−	0.609234i	2.51175	+	4.35048i	1.33442	+	0.770425i	−1.10964	+	1.10964i	−3.06049	−	11.4219i	−6.34925	+	1.70128i	4.09315	+	4.09315i	−8.11776	+	14.0604i	3.19902	−	1.84696i
80.4	−1.92320	−	0.515320i	0.299764	+	0.519206i	−0.0309536	−	0.0178711i	−5.65111	+	5.65111i	−0.308949	−	1.15301i	−1.96518	+	0.526568i	5.68184	+	5.68184i	4.32028	−	7.48295i	13.7804	−	7.95610i
80.5	−1.60455	−	0.429937i	−0.677024	−	1.17264i	−1.07438	−	0.620291i	1.57408	−	1.57408i	0.582156	+	2.17264i	−12.1635	+	3.25921i	6.15564	+	6.15564i	3.58328	−	6.20642i	−3.20245	+	1.84893i
80.6	−0.390278	−	0.104575i	−2.33330	−	4.04139i	−3.32272	−	1.91837i	−4.14024	+	4.14024i	0.488008	+	1.82127i	2.20688	−	0.591332i	2.23898	+	2.23898i	−6.38858	+	11.0653i	2.04881	−	1.18288i
80.7	0.390278	+	0.104575i	−2.33330	−	4.04139i	−3.32272	−	1.91837i	4.14024	−	4.14024i	−0.488008	−	1.82127i	−2.20688	+	0.591332i	−2.23898	−	2.23898i	−6.38858	+	11.0653i	2.04881	−	1.18288i
80.8	1.60455	+	0.429937i	−0.677024	−	1.17264i	−1.07438	−	0.620291i	−1.57408	+	1.57408i	−0.582156	−	2.17264i	12.1635	−	3.25921i	−6.15564	−	6.15564i	3.58328	−	6.20642i	−3.20245	+	1.84893i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.b	even	2	1	inner
13.c	even	3	1	inner
13.d	odd	4	2	inner
13.e	even	6	1	inner
13.f	odd	12	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	169.3.f.g		48
13.b	even	2	1	inner	169.3.f.g		48
13.c	even	3	1		169.3.d.e	✓	24
13.c	even	3	1	inner	169.3.f.g		48
13.d	odd	4	2	inner	169.3.f.g		48
13.e	even	6	1		169.3.d.e	✓	24
13.e	even	6	1	inner	169.3.f.g		48
13.f	odd	12	2		169.3.d.e	✓	24
13.f	odd	12	2	inner	169.3.f.g		48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
169.3.d.e	✓	24	13.c	even	3	1
169.3.d.e	✓	24	13.e	even	6	1
169.3.d.e	✓	24	13.f	odd	12	2
169.3.f.g		48	1.a	even	1	1	trivial
169.3.f.g		48	13.b	even	2	1	inner
169.3.f.g		48	13.c	even	3	1	inner
169.3.f.g		48	13.d	odd	4	2	inner
169.3.f.g		48	13.e	even	6	1	inner
169.3.f.g		48	13.f	odd	12	2	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^48 - 229*T2^44 + 34923*T2^40 - 2930264*T2^36 + 176369373*T2^32 - 6431810322*T2^28 + 168926099203*T2^24 - 2684224998609*T2^20 + 30371862908922*T2^16 - 178112960121202*T2^12 + 709024045394140*T2^8 - 18895917234680*T2^4 + 500246412961