Newform orbit 900.2.bf.d

• Label: 900.2.bf.d
• Level: $900$
• Weight: $2$
• Character orbit: 900.bf
• Analytic conductor: $7.187$
• Analytic rank: $0$
• Dimension: $64$
• Inner twists: $16$

Newspace parameters

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	900.bf (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.18653618192\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 16 q^{6} - 36 q^{16} + 80 q^{21} - 128 q^{26} + 24 q^{36} - 64 q^{41} + 8 q^{46} - 20 q^{56} - 16 q^{61} - 8 q^{66} + 40 q^{76} - 64 q^{81} - 80 q^{86} + 92 q^{96}+O(q^{100}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
7.1	−1.40331	+	0.175242i	−1.64832	+	0.532003i	1.93858	−	0.491839i		0		2.21989	−	1.03542i	0.408602	+	1.52492i	−2.63425	+	1.02992i	2.43395	−	1.75383i		0
7.2	−1.30293	+	0.549893i	1.64832	−	0.532003i	1.39524	−	1.43294i		0		−1.85510	+	1.59956i	−0.408602	−	1.52492i	−1.02992	+	2.63425i	2.43395	−	1.75383i		0
7.3	−1.29795	−	0.561531i	−0.0328891	+	1.73174i	1.36937	+	1.45768i		0		1.01511	−	2.22925i	−0.621822	−	2.32067i	−0.958842	−	2.66094i	−2.99784	−	0.113911i		0
7.4	−1.06468	−	0.930841i	−1.40780	−	1.00901i	0.267070	+	1.98209i		0		0.559630	+	2.38470i	0.884613	+	3.30142i	1.56067	−	2.35888i	0.963816	+	2.84096i		0
7.5	−0.963334	−	1.03537i	1.14019	−	1.30383i	−0.143974	+	1.99481i		0		−2.44833	+	0.0755025i	0.0598944	+	0.223529i	2.20406	−	1.77260i	−0.399926	−	2.97322i		0
7.6	−0.843296	+	1.13528i	0.0328891	−	1.73174i	−0.577705	−	1.91475i		0		1.93827	+	1.49771i	0.621822	+	2.32067i	2.66094	+	0.958842i	−2.99784	−	0.113911i		0
7.7	−0.456616	+	1.33847i	1.40780	+	1.00901i	−1.58300	−	1.22233i		0		−1.99335	+	1.42357i	−0.884613	−	3.30142i	2.35888	−	1.56067i	0.963816	+	2.84096i		0
7.8	−0.316588	+	1.37832i	−1.14019	+	1.30383i	−1.79954	−	0.872720i		0		−1.43612	−	1.98433i	−0.0598944	−	0.223529i	1.77260	−	2.20406i	−0.399926	−	2.97322i		0
7.9	0.316588	−	1.37832i	1.14019	−	1.30383i	−1.79954	−	0.872720i		0		−1.43612	−	1.98433i	0.0598944	+	0.223529i	−1.77260	+	2.20406i	−0.399926	−	2.97322i		0
7.10	0.456616	−	1.33847i	−1.40780	−	1.00901i	−1.58300	−	1.22233i		0		−1.99335	+	1.42357i	0.884613	+	3.30142i	−2.35888	+	1.56067i	0.963816	+	2.84096i		0
7.11	0.843296	−	1.13528i	−0.0328891	+	1.73174i	−0.577705	−	1.91475i		0		1.93827	+	1.49771i	−0.621822	−	2.32067i	−2.66094	−	0.958842i	−2.99784	−	0.113911i		0
7.12	0.963334	+	1.03537i	−1.14019	+	1.30383i	−0.143974	+	1.99481i		0		−2.44833	+	0.0755025i	−0.0598944	−	0.223529i	−2.20406	+	1.77260i	−0.399926	−	2.97322i		0
7.13	1.06468	+	0.930841i	1.40780	+	1.00901i	0.267070	+	1.98209i		0		0.559630	+	2.38470i	−0.884613	−	3.30142i	−1.56067	+	2.35888i	0.963816	+	2.84096i		0
7.14	1.29795	+	0.561531i	0.0328891	−	1.73174i	1.36937	+	1.45768i		0		1.01511	−	2.22925i	0.621822	+	2.32067i	0.958842	+	2.66094i	−2.99784	−	0.113911i		0
7.15	1.30293	−	0.549893i	−1.64832	+	0.532003i	1.39524	−	1.43294i		0		−1.85510	+	1.59956i	0.408602	+	1.52492i	1.02992	−	2.63425i	2.43395	−	1.75383i		0
7.16	1.40331	−	0.175242i	1.64832	−	0.532003i	1.93858	−	0.491839i		0		2.21989	−	1.03542i	−0.408602	−	1.52492i	2.63425	−	1.02992i	2.43395	−	1.75383i		0
43.1	−1.37832	−	0.316588i	1.30383	+	1.14019i	1.79954	+	0.872720i		0		−1.43612	−	1.98433i	0.223529	−	0.0598944i	−2.20406	−	1.77260i	0.399926	+	2.97322i		0
43.2	−1.33847	−	0.456616i	1.00901	−	1.40780i	1.58300	+	1.22233i		0		−1.99335	+	1.42357i	3.30142	−	0.884613i	−1.56067	−	2.35888i	−0.963816	−	2.84096i		0
43.3	−1.13528	−	0.843296i	−1.73174	−	0.0328891i	0.577705	+	1.91475i		0		1.93827	+	1.49771i	−2.32067	+	0.621822i	0.958842	−	2.66094i	2.99784	+	0.113911i		0
43.4	−1.03537	+	0.963334i	1.30383	+	1.14019i	0.143974	−	1.99481i		0		−2.44833	+	0.0755025i	0.223529	−	0.0598944i	1.77260	+	2.20406i	0.399926	+	2.97322i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
5.b	even	2	1	inner
5.c	odd	4	2	inner
9.c	even	3	1	inner
20.d	odd	2	1	inner
20.e	even	4	2	inner
36.f	odd	6	1	inner
45.j	even	6	1	inner
45.k	odd	12	2	inner
180.p	odd	6	1	inner
180.x	even	12	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	900.2.bf.d	✓	64
4.b	odd	2	1	inner	900.2.bf.d	✓	64
5.b	even	2	1	inner	900.2.bf.d	✓	64
5.c	odd	4	2	inner	900.2.bf.d	✓	64
9.c	even	3	1	inner	900.2.bf.d	✓	64
20.d	odd	2	1	inner	900.2.bf.d	✓	64
20.e	even	4	2	inner	900.2.bf.d	✓	64
36.f	odd	6	1	inner	900.2.bf.d	✓	64
45.j	even	6	1	inner	900.2.bf.d	✓	64
45.k	odd	12	2	inner	900.2.bf.d	✓	64
180.p	odd	6	1	inner	900.2.bf.d	✓	64
180.x	even	12	2	inner	900.2.bf.d	✓	64

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
900.2.bf.d	✓	64	1.a	even	1	1	trivial
900.2.bf.d	✓	64	4.b	odd	2	1	inner
900.2.bf.d	✓	64	5.b	even	2	1	inner
900.2.bf.d	✓	64	5.c	odd	4	2	inner
900.2.bf.d	✓	64	9.c	even	3	1	inner
900.2.bf.d	✓	64	20.d	odd	2	1	inner
900.2.bf.d	✓	64	20.e	even	4	2	inner
900.2.bf.d	✓	64	36.f	odd	6	1	inner
900.2.bf.d	✓	64	45.j	even	6	1	inner
900.2.bf.d	✓	64	45.k	odd	12	2	inner
900.2.bf.d	✓	64	180.p	odd	6	1	inner
900.2.bf.d	✓	64	180.x	even	12	2	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T7^32 - 176*T7^28 + 25374*T7^24 - 929432*T7^20 + 26408563*T7^16 - 158284008*T7^12 + 798173838*T7^8 - 2289060*T7^4 + 6561