Newform orbit 169.3.d.e

• Label: 169.3.d.e
• Level: $169$
• Weight: $3$
• Character orbit: 169.d
• Analytic conductor: $4.605$
• Analytic rank: $0$
• Dimension: $24$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	169.d (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.60491646769\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{3} + 84 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
70.1	−2.28029	+	2.28029i	−2.04607				−	6.39942i	−5.50472	+	5.50472i	4.66563	−	4.66563i	−2.31365	−	2.31365i	5.47136	+	5.47136i	−4.81360				−	25.1047i
70.2	−2.02149	+	2.02149i	4.64845				−	4.17281i	2.43155	−	2.43155i	−9.39676	+	9.39676i	0.759383	+	0.759383i	0.349324	+	0.349324i	12.6080					9.83067i
70.3	−1.66446	+	1.66446i	−5.02350				−	1.54085i	1.10964	−	1.10964i	8.36141	−	8.36141i	−4.64797	−	4.64797i	−4.09315	−	4.09315i	16.2355					3.69391i
70.4	−1.40788	+	1.40788i	−0.599528					0.0357421i	5.65111	−	5.65111i	0.844064	−	0.844064i	−1.43861	−	1.43861i	−5.68184	−	5.68184i	−8.64057					15.9122i
70.5	−1.17461	+	1.17461i	1.35405					1.24058i	−1.57408	+	1.57408i	−1.59048	+	1.59048i	−8.90433	−	8.90433i	−6.15564	−	6.15564i	−7.16655				−	3.69787i
70.6	−0.285703	+	0.285703i	4.66660					3.83675i	4.14024	−	4.14024i	−1.33326	+	1.33326i	1.61555	+	1.61555i	−2.23898	−	2.23898i	12.7772					2.36576i
70.7	0.285703	−	0.285703i	4.66660					3.83675i	−4.14024	+	4.14024i	1.33326	−	1.33326i	−1.61555	−	1.61555i	2.23898	+	2.23898i	12.7772					2.36576i
70.8	1.17461	−	1.17461i	1.35405					1.24058i	1.57408	−	1.57408i	1.59048	−	1.59048i	8.90433	+	8.90433i	6.15564	+	6.15564i	−7.16655				−	3.69787i
70.9	1.40788	−	1.40788i	−0.599528					0.0357421i	−5.65111	+	5.65111i	−0.844064	+	0.844064i	1.43861	+	1.43861i	5.68184	+	5.68184i	−8.64057					15.9122i
70.10	1.66446	−	1.66446i	−5.02350				−	1.54085i	−1.10964	+	1.10964i	−8.36141	+	8.36141i	4.64797	+	4.64797i	4.09315	+	4.09315i	16.2355					3.69391i
70.11	2.02149	−	2.02149i	4.64845				−	4.17281i	−2.43155	+	2.43155i	9.39676	−	9.39676i	−0.759383	−	0.759383i	−0.349324	−	0.349324i	12.6080					9.83067i
70.12	2.28029	−	2.28029i	−2.04607				−	6.39942i	5.50472	−	5.50472i	−4.66563	+	4.66563i	2.31365	+	2.31365i	−5.47136	−	5.47136i	−4.81360				−	25.1047i
99.1	−2.28029	−	2.28029i	−2.04607					6.39942i	−5.50472	−	5.50472i	4.66563	+	4.66563i	−2.31365	+	2.31365i	5.47136	−	5.47136i	−4.81360					25.1047i
99.2	−2.02149	−	2.02149i	4.64845					4.17281i	2.43155	+	2.43155i	−9.39676	−	9.39676i	0.759383	−	0.759383i	0.349324	−	0.349324i	12.6080				−	9.83067i
99.3	−1.66446	−	1.66446i	−5.02350					1.54085i	1.10964	+	1.10964i	8.36141	+	8.36141i	−4.64797	+	4.64797i	−4.09315	+	4.09315i	16.2355				−	3.69391i
99.4	−1.40788	−	1.40788i	−0.599528				−	0.0357421i	5.65111	+	5.65111i	0.844064	+	0.844064i	−1.43861	+	1.43861i	−5.68184	+	5.68184i	−8.64057				−	15.9122i
99.5	−1.17461	−	1.17461i	1.35405				−	1.24058i	−1.57408	−	1.57408i	−1.59048	−	1.59048i	−8.90433	+	8.90433i	−6.15564	+	6.15564i	−7.16655					3.69787i
99.6	−0.285703	−	0.285703i	4.66660				−	3.83675i	4.14024	+	4.14024i	−1.33326	−	1.33326i	1.61555	−	1.61555i	−2.23898	+	2.23898i	12.7772				−	2.36576i
99.7	0.285703	+	0.285703i	4.66660				−	3.83675i	−4.14024	−	4.14024i	1.33326	+	1.33326i	−1.61555	+	1.61555i	2.23898	−	2.23898i	12.7772				−	2.36576i
99.8	1.17461	+	1.17461i	1.35405				−	1.24058i	1.57408	+	1.57408i	1.59048	+	1.59048i	8.90433	−	8.90433i	6.15564	−	6.15564i	−7.16655					3.69787i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.b	even	2	1	inner
13.d	odd	4	2	inner

Twists

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

    

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 229T_{2}^{20} + 17518T_{2}^{16} + 540679T_{2}^{12} + 6695460T_{2}^{8} + 26716280T_{2}^{4} + 707281 \) acting on \(S_{3}^{\mathrm{new}}(169, [\chi])\).