Newform orbit 375.2.e.c

• Label: 375.2.e.c
• Level: $375$
• Weight: $2$
• Character orbit: 375.e
• Analytic conductor: $2.994$
• Analytic rank: $0$
• Dimension: $32$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 375 = 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	375.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.99439007580\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 12 q^{6} - 24 q^{16} + 32 q^{21} - 56 q^{31} - 116 q^{36} - 96 q^{46} - 48 q^{51} + 144 q^{61} + 120 q^{66} + 168 q^{76} + 132 q^{81} - 56 q^{91} + 76 q^{96}+O(q^{100}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
68.1	−1.73446	−	1.73446i	−0.992284	−	1.41964i			4.01670i		0		−0.741229	+	4.18338i	−2.24842	+	2.24842i	3.49789	−	3.49789i	−1.03075	+	2.81737i		0
68.2	−1.73446	−	1.73446i	1.41964	+	0.992284i			4.01670i		0		−0.741229	−	4.18338i	2.24842	−	2.24842i	3.49789	−	3.49789i	1.03075	+	2.81737i		0
68.3	−1.42125	−	1.42125i	−1.71967	−	0.206704i			2.03989i		0		2.15030	+	2.73786i	1.62063	−	1.62063i	0.0566872	−	0.0566872i	2.91455	+	0.710926i		0
68.4	−1.42125	−	1.42125i	0.206704	+	1.71967i			2.03989i		0		2.15030	−	2.73786i	−1.62063	+	1.62063i	0.0566872	−	0.0566872i	−2.91455	+	0.710926i		0
68.5	−0.548330	−	0.548330i	−1.73176	−	0.0316680i		−	1.39867i		0		0.932212	+	0.966941i	1.30711	−	1.30711i	−1.86359	+	1.86359i	2.99799	+	0.109683i		0
68.6	−0.548330	−	0.548330i	0.0316680	+	1.73176i		−	1.39867i		0		0.932212	−	0.966941i	−1.30711	+	1.30711i	−1.86359	+	1.86359i	−2.99799	+	0.109683i		0
68.7	−0.413570	−	0.413570i	0.334821	−	1.69938i		−	1.65792i		0		−0.841285	+	0.564341i	−2.93422	+	2.93422i	−1.51281	+	1.51281i	−2.77579	−	1.13798i		0
68.8	−0.413570	−	0.413570i	1.69938	−	0.334821i		−	1.65792i		0		−0.841285	−	0.564341i	2.93422	−	2.93422i	−1.51281	+	1.51281i	2.77579	−	1.13798i		0
68.9	0.413570	+	0.413570i	−1.69938	+	0.334821i		−	1.65792i		0		−0.841285	−	0.564341i	−2.93422	+	2.93422i	1.51281	−	1.51281i	2.77579	−	1.13798i		0
68.10	0.413570	+	0.413570i	−0.334821	+	1.69938i		−	1.65792i		0		−0.841285	+	0.564341i	2.93422	−	2.93422i	1.51281	−	1.51281i	−2.77579	−	1.13798i		0
68.11	0.548330	+	0.548330i	−0.0316680	−	1.73176i		−	1.39867i		0		0.932212	−	0.966941i	1.30711	−	1.30711i	1.86359	−	1.86359i	−2.99799	+	0.109683i		0
68.12	0.548330	+	0.548330i	1.73176	+	0.0316680i		−	1.39867i		0		0.932212	+	0.966941i	−1.30711	+	1.30711i	1.86359	−	1.86359i	2.99799	+	0.109683i		0
68.13	1.42125	+	1.42125i	−0.206704	−	1.71967i			2.03989i		0		2.15030	−	2.73786i	1.62063	−	1.62063i	−0.0566872	+	0.0566872i	−2.91455	+	0.710926i		0
68.14	1.42125	+	1.42125i	1.71967	+	0.206704i			2.03989i		0		2.15030	+	2.73786i	−1.62063	+	1.62063i	−0.0566872	+	0.0566872i	2.91455	+	0.710926i		0
68.15	1.73446	+	1.73446i	−1.41964	−	0.992284i			4.01670i		0		−0.741229	−	4.18338i	−2.24842	+	2.24842i	−3.49789	+	3.49789i	1.03075	+	2.81737i		0
68.16	1.73446	+	1.73446i	0.992284	+	1.41964i			4.01670i		0		−0.741229	+	4.18338i	2.24842	−	2.24842i	−3.49789	+	3.49789i	−1.03075	+	2.81737i		0
182.1	−1.73446	+	1.73446i	−0.992284	+	1.41964i		−	4.01670i		0		−0.741229	−	4.18338i	−2.24842	−	2.24842i	3.49789	+	3.49789i	−1.03075	−	2.81737i		0
182.2	−1.73446	+	1.73446i	1.41964	−	0.992284i		−	4.01670i		0		−0.741229	+	4.18338i	2.24842	+	2.24842i	3.49789	+	3.49789i	1.03075	−	2.81737i		0
182.3	−1.42125	+	1.42125i	−1.71967	+	0.206704i		−	2.03989i		0		2.15030	−	2.73786i	1.62063	+	1.62063i	0.0566872	+	0.0566872i	2.91455	−	0.710926i		0
182.4	−1.42125	+	1.42125i	0.206704	−	1.71967i		−	2.03989i		0		2.15030	+	2.73786i	−1.62063	−	1.62063i	0.0566872	+	0.0566872i	−2.91455	−	0.710926i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.b	even	2	1	inner
5.c	odd	4	2	inner
15.d	odd	2	1	inner
15.e	even	4	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	375.2.e.c	✓	32
3.b	odd	2	1	inner	375.2.e.c	✓	32
5.b	even	2	1	inner	375.2.e.c	✓	32
5.c	odd	4	2	inner	375.2.e.c	✓	32
15.d	odd	2	1	inner	375.2.e.c	✓	32
15.e	even	4	2	inner	375.2.e.c	✓	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
375.2.e.c	✓	32	1.a	even	1	1	trivial
375.2.e.c	✓	32	3.b	odd	2	1	inner
375.2.e.c	✓	32	5.b	even	2	1	inner
375.2.e.c	✓	32	5.c	odd	4	2	inner
375.2.e.c	✓	32	15.d	odd	2	1	inner
375.2.e.c	✓	32	15.e	even	4	2	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 53T_{2}^{12} + 616T_{2}^{8} + 285T_{2}^{4} + 25 \) acting on \(S_{2}^{\mathrm{new}}(375, [\chi])\).