Newform orbit 289.3.e.q

• Label: 289.3.e.q
• Level: $289$
• Weight: $3$
• Character orbit: 289.e
• Analytic conductor: $7.875$
• Analytic rank: $0$
• Dimension: $48$
• 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:	\(48\)
Relative dimension:	\(6\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q+O(q^{10}) \)

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	−3.15179	−	1.30551i	−3.71146	+	0.738255i	5.40099	+	5.40099i	2.25875	−	3.38045i	12.6615	+	2.51854i	3.99701	+	5.98194i	−4.74967	−	11.4667i	4.91500	−	2.03586i	−11.5323	+	7.70566i
40.2	−3.15179	−	1.30551i	3.71146	−	0.738255i	5.40099	+	5.40099i	−2.25875	+	3.38045i	−12.6615	−	2.51854i	−3.99701	−	5.98194i	−4.74967	−	11.4667i	4.91500	−	2.03586i	11.5323	−	7.70566i
40.3	1.09461	+	0.453400i	−5.48139	+	1.09032i	−1.83584	−	1.83584i	0.116065	−	0.173703i	−6.49431	−	1.29180i	−2.23436	−	3.34395i	−2.99075	−	7.22031i	20.5419	−	8.50875i	0.205802	−	0.137513i
40.4	1.09461	+	0.453400i	5.48139	−	1.09032i	−1.83584	−	1.83584i	−0.116065	+	0.173703i	6.49431	+	1.29180i	2.23436	+	3.34395i	−2.99075	−	7.22031i	20.5419	−	8.50875i	−0.205802	+	0.137513i
40.5	2.05719	+	0.852114i	−4.09654	+	0.814852i	0.677487	+	0.677487i	−3.39884	+	5.08673i	−9.12169	−	1.81442i	1.76265	+	2.63799i	−2.59204	−	6.25773i	7.80274	−	3.23200i	−11.3265	+	7.56814i
40.6	2.05719	+	0.852114i	4.09654	−	0.814852i	0.677487	+	0.677487i	3.39884	−	5.08673i	9.12169	+	1.81442i	−1.76265	−	2.63799i	−2.59204	−	6.25773i	7.80274	−	3.23200i	11.3265	−	7.56814i
65.1	−2.05719	+	0.852114i	−0.814852	+	4.09654i	0.677487	−	0.677487i	−5.08673	+	3.39884i	−1.81442	−	9.12169i	2.63799	+	1.76265i	2.59204	−	6.25773i	−7.80274	−	3.23200i	7.56814	−	11.3265i
65.2	−2.05719	+	0.852114i	0.814852	−	4.09654i	0.677487	−	0.677487i	5.08673	−	3.39884i	1.81442	+	9.12169i	−2.63799	−	1.76265i	2.59204	−	6.25773i	−7.80274	−	3.23200i	−7.56814	+	11.3265i
65.3	−1.09461	+	0.453400i	−1.09032	+	5.48139i	−1.83584	+	1.83584i	0.173703	−	0.116065i	−1.29180	−	6.49431i	−3.34395	−	2.23436i	2.99075	−	7.22031i	−20.5419	−	8.50875i	−0.137513	+	0.205802i
65.4	−1.09461	+	0.453400i	1.09032	−	5.48139i	−1.83584	+	1.83584i	−0.173703	+	0.116065i	1.29180	+	6.49431i	3.34395	+	2.23436i	2.99075	−	7.22031i	−20.5419	−	8.50875i	0.137513	−	0.205802i
65.5	3.15179	−	1.30551i	−0.738255	+	3.71146i	5.40099	−	5.40099i	3.38045	−	2.25875i	2.51854	+	12.6615i	5.98194	+	3.99701i	4.74967	−	11.4667i	−4.91500	−	2.03586i	7.70566	−	11.5323i
65.6	3.15179	−	1.30551i	0.738255	−	3.71146i	5.40099	−	5.40099i	−3.38045	+	2.25875i	−2.51854	−	12.6615i	−5.98194	−	3.99701i	4.74967	−	11.4667i	−4.91500	−	2.03586i	−7.70566	+	11.5323i
75.1	−0.852114	+	2.05719i	−2.32050	+	3.47288i	−0.677487	−	0.677487i	−1.19352	−	6.00021i	−5.16702	−	7.73300i	0.618960	−	3.11172i	−6.25773	+	2.59204i	−3.23200	−	7.80274i	13.3605	+	2.65758i
75.2	−0.852114	+	2.05719i	2.32050	−	3.47288i	−0.677487	−	0.677487i	1.19352	+	6.00021i	5.16702	+	7.73300i	−0.618960	+	3.11172i	−6.25773	+	2.59204i	−3.23200	−	7.80274i	−13.3605	−	2.65758i
75.3	−0.453400	+	1.09461i	−3.10496	+	4.64690i	1.83584	+	1.83584i	0.0407565	+	0.204897i	−3.67873	−	5.50561i	−0.784602	+	3.94446i	−7.22031	+	2.99075i	−8.50875	−	20.5419i	−0.242760	−	0.0482880i
75.4	−0.453400	+	1.09461i	3.10496	−	4.64690i	1.83584	+	1.83584i	−0.0407565	−	0.204897i	3.67873	+	5.50561i	0.784602	−	3.94446i	−7.22031	+	2.99075i	−8.50875	−	20.5419i	0.242760	+	0.0482880i
75.5	1.30551	−	3.15179i	−2.10237	+	3.14642i	−5.40099	−	5.40099i	0.793166	+	3.98752i	7.17219	+	10.7339i	1.40356	−	7.05618i	−11.4667	+	4.74967i	−2.03586	−	4.91500i	13.6033	+	2.70587i
75.6	1.30551	−	3.15179i	2.10237	−	3.14642i	−5.40099	−	5.40099i	−0.793166	−	3.98752i	−7.17219	−	10.7339i	−1.40356	+	7.05618i	−11.4667	+	4.74967i	−2.03586	−	4.91500i	−13.6033	−	2.70587i
131.1	−1.30551	−	3.15179i	−3.14642	+	2.10237i	−5.40099	+	5.40099i	−3.98752	−	0.793166i	10.7339	+	7.17219i	−7.05618	+	1.40356i	11.4667	+	4.74967i	2.03586	−	4.91500i	2.70587	+	13.6033i
131.2	−1.30551	−	3.15179i	3.14642	−	2.10237i	−5.40099	+	5.40099i	3.98752	+	0.793166i	−10.7339	−	7.17219i	7.05618	−	1.40356i	11.4667	+	4.74967i	2.03586	−	4.91500i	−2.70587	−	13.6033i

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.q	✓	48
17.b	even	2	1	inner	289.3.e.q	✓	48
17.c	even	4	2	inner	289.3.e.q	✓	48
17.d	even	8	4	inner	289.3.e.q	✓	48
17.e	odd	16	8	inner	289.3.e.q	✓	48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
289.3.e.q	✓	48	1.a	even	1	1	trivial
289.3.e.q	✓	48	17.b	even	2	1	inner
289.3.e.q	✓	48	17.c	even	4	2	inner
289.3.e.q	✓	48	17.d	even	8	4	inner
289.3.e.q	✓	48	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 \(S_{3}^{\mathrm{new}}(289, [\chi])\):

\( T_{2}^{24} + 18954T_{2}^{16} + 11160261T_{2}^{8} + 43046721 \)

\( T_{3}^{48} + 916222372677T_{3}^{32} + 9389424276833804858970T_{3}^{16} + 13743357866057596647857618555361 \)