Newform orbit 65.3.s.a

• Label: 65.3.s.a
• Level: $65$
• Weight: $3$
• Character orbit: 65.s
• Analytic conductor: $1.771$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 65 = 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	65.s (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.77112171834\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 12 q^{4} - 2 q^{5} - 24 q^{6} + 44 q^{9}+O(q^{10}) \)

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	−3.14494	+	0.842684i	0.104365	+	0.0602551i	5.71643	−	3.30038i	−1.81196	+	4.66013i	−0.378997	−	0.101552i	−5.82488	−	1.56077i	−5.98764	+	5.98764i	−4.49274	−	7.78165i	1.77149	−	16.1827i
19.2	−2.81828	+	0.755156i	4.93366	+	2.84845i	3.90835	−	2.25648i	4.35399	−	2.45820i	−16.0555	−	4.30205i	−3.75529	−	1.00623i	−1.05832	+	1.05832i	11.7273	+	20.3123i	−10.4145	+	10.2158i
19.3	−2.69925	+	0.723261i	0.0293708	+	0.0169572i	3.29872	−	1.90452i	−3.01936	−	3.98541i	−0.0915435	−	0.0245290i	9.18466	+	2.46102i	0.377336	−	0.377336i	−4.49942	−	7.79323i	11.0325	+	8.57381i
19.4	−1.99539	+	0.534664i	−3.91301	−	2.25918i	0.231624	−	0.133728i	4.11651	+	2.83802i	9.01590	+	2.41580i	7.29556	+	1.95484i	5.45223	−	5.45223i	5.70777	+	9.88616i	−9.73144	−	3.46201i
19.5	−0.921230	+	0.246843i	−1.41409	−	0.816426i	−2.67637	+	1.54520i	2.37278	−	4.40113i	1.50423	+	0.403058i	−11.6987	−	3.13466i	4.78168	−	4.78168i	−3.16690	−	5.48523i	−1.09949	+	4.64015i
19.6	−0.396050	+	0.106121i	2.89568	+	1.67182i	−3.31851	+	1.91594i	−0.559860	+	4.96856i	−1.32425	−	0.354832i	0.834189	+	0.223520i	2.27069	−	2.27069i	1.08998	+	1.88791i	−0.305537	−	2.02721i
19.7	0.396050	−	0.106121i	−2.89568	−	1.67182i	−3.31851	+	1.91594i	−4.96856	+	0.559860i	−1.32425	−	0.354832i	−0.834189	−	0.223520i	−2.27069	+	2.27069i	1.08998	+	1.88791i	−1.90838	+	0.749002i
19.8	0.921230	−	0.246843i	1.41409	+	0.816426i	−2.67637	+	1.54520i	4.40113	−	2.37278i	1.50423	+	0.403058i	11.6987	+	3.13466i	−4.78168	+	4.78168i	−3.16690	−	5.48523i	3.46875	−	3.27226i
19.9	1.99539	−	0.534664i	3.91301	+	2.25918i	0.231624	−	0.133728i	−2.83802	−	4.11651i	9.01590	+	2.41580i	−7.29556	−	1.95484i	−5.45223	+	5.45223i	5.70777	+	9.88616i	−7.86391	−	6.69667i
19.10	2.69925	−	0.723261i	−0.0293708	−	0.0169572i	3.29872	−	1.90452i	3.98541	+	3.01936i	−0.0915435	−	0.0245290i	−9.18466	−	2.46102i	−0.377336	+	0.377336i	−4.49942	−	7.79323i	12.9414	+	5.26751i
19.11	2.81828	−	0.755156i	−4.93366	−	2.84845i	3.90835	−	2.25648i	2.45820	−	4.35399i	−16.0555	−	4.30205i	3.75529	+	1.00623i	1.05832	−	1.05832i	11.7273	+	20.3123i	3.63995	−	14.1271i
19.12	3.14494	−	0.842684i	−0.104365	−	0.0602551i	5.71643	−	3.30038i	−4.66013	+	1.81196i	−0.378997	−	0.101552i	5.82488	+	1.56077i	5.98764	−	5.98764i	−4.49274	−	7.78165i	−13.1289	+	9.62553i
24.1	−3.14494	−	0.842684i	0.104365	−	0.0602551i	5.71643	+	3.30038i	−1.81196	−	4.66013i	−0.378997	+	0.101552i	−5.82488	+	1.56077i	−5.98764	−	5.98764i	−4.49274	+	7.78165i	1.77149	+	16.1827i
24.2	−2.81828	−	0.755156i	4.93366	−	2.84845i	3.90835	+	2.25648i	4.35399	+	2.45820i	−16.0555	+	4.30205i	−3.75529	+	1.00623i	−1.05832	−	1.05832i	11.7273	−	20.3123i	−10.4145	−	10.2158i
24.3	−2.69925	−	0.723261i	0.0293708	−	0.0169572i	3.29872	+	1.90452i	−3.01936	+	3.98541i	−0.0915435	+	0.0245290i	9.18466	−	2.46102i	0.377336	+	0.377336i	−4.49942	+	7.79323i	11.0325	−	8.57381i
24.4	−1.99539	−	0.534664i	−3.91301	+	2.25918i	0.231624	+	0.133728i	4.11651	−	2.83802i	9.01590	−	2.41580i	7.29556	−	1.95484i	5.45223	+	5.45223i	5.70777	−	9.88616i	−9.73144	+	3.46201i
24.5	−0.921230	−	0.246843i	−1.41409	+	0.816426i	−2.67637	−	1.54520i	2.37278	+	4.40113i	1.50423	−	0.403058i	−11.6987	+	3.13466i	4.78168	+	4.78168i	−3.16690	+	5.48523i	−1.09949	−	4.64015i
24.6	−0.396050	−	0.106121i	2.89568	−	1.67182i	−3.31851	−	1.91594i	−0.559860	−	4.96856i	−1.32425	+	0.354832i	0.834189	−	0.223520i	2.27069	+	2.27069i	1.08998	−	1.88791i	−0.305537	+	2.02721i
24.7	0.396050	+	0.106121i	−2.89568	+	1.67182i	−3.31851	−	1.91594i	−4.96856	−	0.559860i	−1.32425	+	0.354832i	−0.834189	+	0.223520i	−2.27069	−	2.27069i	1.08998	−	1.88791i	−1.90838	−	0.749002i
24.8	0.921230	+	0.246843i	1.41409	−	0.816426i	−2.67637	−	1.54520i	4.40113	+	2.37278i	1.50423	−	0.403058i	11.6987	−	3.13466i	−4.78168	−	4.78168i	−3.16690	+	5.48523i	3.46875	+	3.27226i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
13.f	odd	12	1	inner
65.s	odd	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	65.3.s.a	✓	48
5.b	even	2	1	inner	65.3.s.a	✓	48
5.c	odd	4	2		325.3.t.e		48
13.f	odd	12	1	inner	65.3.s.a	✓	48
65.o	even	12	1		325.3.t.e		48
65.s	odd	12	1	inner	65.3.s.a	✓	48
65.t	even	12	1		325.3.t.e		48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
65.3.s.a	✓	48	1.a	even	1	1	trivial
65.3.s.a	✓	48	5.b	even	2	1	inner
65.3.s.a	✓	48	13.f	odd	12	1	inner
65.3.s.a	✓	48	65.s	odd	12	1	inner
325.3.t.e		48	5.c	odd	4	2
325.3.t.e		48	65.o	even	12	1
325.3.t.e		48	65.t	even	12	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(65, [\chi])\).