Newform orbit 289.3.e.p

• Label: 289.3.e.p
• Level: $289$
• Weight: $3$
• Character orbit: 289.e
• Analytic conductor: $7.875$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $16$

Newspace parameters

Level:	\( N \)	\(=\)	\( 289 = 17^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	289.e (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.87467964001\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(4\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
40.1	−1.65493	−	0.685496i	−0.448072	+	0.0891271i	−0.559525	−	0.559525i	0.708463	−	1.06029i	0.802626	+	0.159652i	−2.73896	−	4.09915i	3.28441	+	7.92927i	−8.12209	+	3.36428i	−1.89929	+	1.26906i
40.2	−1.65493	−	0.685496i	0.448072	−	0.0891271i	−0.559525	−	0.559525i	−0.708463	+	1.06029i	−0.802626	−	0.159652i	2.73896	+	4.09915i	3.28441	+	7.92927i	−8.12209	+	3.36428i	1.89929	−	1.26906i
40.3	2.57881	+	1.06818i	−2.14684	+	0.427033i	2.68085	+	2.68085i	−2.17836	+	3.26015i	−5.99245	−	1.19197i	−7.55034	−	11.2999i	−0.222942	−	0.538230i	−3.88834	+	1.61060i	−9.10002	+	6.08044i
40.4	2.57881	+	1.06818i	2.14684	−	0.427033i	2.68085	+	2.68085i	2.17836	−	3.26015i	5.99245	+	1.19197i	7.55034	+	11.2999i	−0.222942	−	0.538230i	−3.88834	+	1.61060i	9.10002	−	6.08044i
65.1	−2.57881	+	1.06818i	−0.427033	+	2.14684i	2.68085	−	2.68085i	−3.26015	+	2.17836i	−1.19197	−	5.99245i	−11.2999	−	7.55034i	0.222942	−	0.538230i	3.88834	+	1.61060i	6.08044	−	9.10002i
65.2	−2.57881	+	1.06818i	0.427033	−	2.14684i	2.68085	−	2.68085i	3.26015	−	2.17836i	1.19197	+	5.99245i	11.2999	+	7.55034i	0.222942	−	0.538230i	3.88834	+	1.61060i	−6.08044	+	9.10002i
65.3	1.65493	−	0.685496i	−0.0891271	+	0.448072i	−0.559525	+	0.559525i	1.06029	−	0.708463i	0.159652	+	0.802626i	−4.09915	−	2.73896i	−3.28441	+	7.92927i	8.12209	+	3.36428i	1.26906	−	1.89929i
65.4	1.65493	−	0.685496i	0.0891271	−	0.448072i	−0.559525	+	0.559525i	−1.06029	+	0.708463i	−0.159652	−	0.802626i	4.09915	+	2.73896i	−3.28441	+	7.92927i	8.12209	+	3.36428i	−1.26906	+	1.89929i
75.1	−1.06818	+	2.57881i	−1.21609	+	1.82000i	−2.68085	−	2.68085i	−0.764940	−	3.84561i	−3.39445	−	5.08016i	−2.65133	+	13.3291i	−0.538230	+	0.222942i	1.61060	+	3.88834i	10.7342	+	2.13517i
75.2	−1.06818	+	2.57881i	1.21609	−	1.82000i	−2.68085	−	2.68085i	0.764940	+	3.84561i	3.39445	+	5.08016i	2.65133	−	13.3291i	−0.538230	+	0.222942i	1.61060	+	3.88834i	−10.7342	−	2.13517i
75.3	0.685496	−	1.65493i	−0.253812	+	0.379857i	0.559525	+	0.559525i	0.248779	+	1.25070i	0.454651	+	0.680433i	−0.961796	+	4.83527i	7.92927	−	3.28441i	3.36428	+	8.12209i	2.24036	+	0.445635i
75.4	0.685496	−	1.65493i	0.253812	−	0.379857i	0.559525	+	0.559525i	−0.248779	−	1.25070i	−0.454651	−	0.680433i	0.961796	−	4.83527i	7.92927	−	3.28441i	3.36428	+	8.12209i	−2.24036	−	0.445635i
131.1	−0.685496	−	1.65493i	−0.379857	+	0.253812i	0.559525	−	0.559525i	−1.25070	−	0.248779i	0.680433	+	0.454651i	4.83527	−	0.961796i	−7.92927	−	3.28441i	−3.36428	+	8.12209i	0.445635	+	2.24036i
131.2	−0.685496	−	1.65493i	0.379857	−	0.253812i	0.559525	−	0.559525i	1.25070	+	0.248779i	−0.680433	−	0.454651i	−4.83527	+	0.961796i	−7.92927	−	3.28441i	−3.36428	+	8.12209i	−0.445635	−	2.24036i
131.3	1.06818	+	2.57881i	−1.82000	+	1.21609i	−2.68085	+	2.68085i	3.84561	+	0.764940i	−5.08016	−	3.39445i	13.3291	−	2.65133i	0.538230	+	0.222942i	−1.61060	+	3.88834i	2.13517	+	10.7342i
131.4	1.06818	+	2.57881i	1.82000	−	1.21609i	−2.68085	+	2.68085i	−3.84561	−	0.764940i	5.08016	+	3.39445i	−13.3291	+	2.65133i	0.538230	+	0.222942i	−1.61060	+	3.88834i	−2.13517	−	10.7342i
158.1	−1.06818	−	2.57881i	−1.21609	−	1.82000i	−2.68085	+	2.68085i	−0.764940	+	3.84561i	−3.39445	+	5.08016i	−2.65133	−	13.3291i	−0.538230	−	0.222942i	1.61060	−	3.88834i	10.7342	−	2.13517i
158.2	−1.06818	−	2.57881i	1.21609	+	1.82000i	−2.68085	+	2.68085i	0.764940	−	3.84561i	3.39445	−	5.08016i	2.65133	+	13.3291i	−0.538230	−	0.222942i	1.61060	−	3.88834i	−10.7342	+	2.13517i
158.3	0.685496	+	1.65493i	−0.253812	−	0.379857i	0.559525	−	0.559525i	0.248779	−	1.25070i	0.454651	−	0.680433i	−0.961796	−	4.83527i	7.92927	+	3.28441i	3.36428	−	8.12209i	2.24036	−	0.445635i
158.4	0.685496	+	1.65493i	0.253812	+	0.379857i	0.559525	−	0.559525i	−0.248779	+	1.25070i	−0.454651	+	0.680433i	0.961796	+	4.83527i	7.92927	+	3.28441i	3.36428	−	8.12209i	−2.24036	+	0.445635i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
17.b	even	2	1	inner
17.c	even	4	2	inner
17.d	even	8	4	inner
17.e	odd	16	8	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	289.3.e.p	✓	32
17.b	even	2	1	inner	289.3.e.p	✓	32
17.c	even	4	2	inner	289.3.e.p	✓	32
17.d	even	8	4	inner	289.3.e.p	✓	32
17.e	odd	16	8	inner	289.3.e.p	✓	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
289.3.e.p	✓	32	1.a	even	1	1	trivial
289.3.e.p	✓	32	17.b	even	2	1	inner
289.3.e.p	✓	32	17.c	even	4	2	inner
289.3.e.p	✓	32	17.d	even	8	4	inner
289.3.e.p	✓	32	17.e	odd	16	8	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(289, [\chi])\):

\( T_{2}^{16} + 3791T_{2}^{8} + 390625 \)

\( T_{3}^{32} + 277727T_{3}^{16} + 1 \)