Newform orbit 420.2.i.a

• Label: 420.2.i.a
• Level: $420$
• Weight: $2$
• Character orbit: 420.i
• Analytic conductor: $3.354$
• Analytic rank: $0$
• Dimension: $48$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	420.i (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.35371688489\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 48 q^{9} + 20 q^{14} - 16 q^{16} + 8 q^{25} - 16 q^{30} - 40 q^{44} + 16 q^{46} - 16 q^{49} + 48 q^{50} + 28 q^{56} - 32 q^{60} - 112 q^{74} + 48 q^{81} - 28 q^{84} + 56 q^{85} + 8 q^{86}+O(q^{100}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
139.1	−1.36843	−	0.356944i		−	1.00000i	1.74518	+	0.976904i	−1.65234	+	1.50657i	−0.356944	+	1.36843i	−2.07310	+	1.64386i	−2.03945	−	1.95975i	−1.00000			2.79887	−	1.47184i
139.2	−1.36843	−	0.356944i			1.00000i	1.74518	+	0.976904i	1.65234	−	1.50657i	0.356944	−	1.36843i	−2.07310	−	1.64386i	−2.03945	−	1.95975i	−1.00000			−2.79887	+	1.47184i
139.3	−1.36843	+	0.356944i		−	1.00000i	1.74518	−	0.976904i	1.65234	+	1.50657i	0.356944	+	1.36843i	−2.07310	+	1.64386i	−2.03945	+	1.95975i	−1.00000			−2.79887	−	1.47184i
139.4	−1.36843	+	0.356944i			1.00000i	1.74518	−	0.976904i	−1.65234	−	1.50657i	−0.356944	−	1.36843i	−2.07310	−	1.64386i	−2.03945	+	1.95975i	−1.00000			2.79887	+	1.47184i
139.5	−1.33186	−	0.475542i		−	1.00000i	1.54772	+	1.26671i	2.22087	+	0.260295i	−0.475542	+	1.33186i	1.17807	−	2.36900i	−1.45898	−	2.42309i	−1.00000			−2.83411	−	1.40279i
139.6	−1.33186	−	0.475542i			1.00000i	1.54772	+	1.26671i	−2.22087	−	0.260295i	0.475542	−	1.33186i	1.17807	+	2.36900i	−1.45898	−	2.42309i	−1.00000			2.83411	+	1.40279i
139.7	−1.33186	+	0.475542i		−	1.00000i	1.54772	−	1.26671i	−2.22087	+	0.260295i	0.475542	+	1.33186i	1.17807	−	2.36900i	−1.45898	+	2.42309i	−1.00000			2.83411	−	1.40279i
139.8	−1.33186	+	0.475542i			1.00000i	1.54772	−	1.26671i	2.22087	−	0.260295i	−0.475542	−	1.33186i	1.17807	+	2.36900i	−1.45898	+	2.42309i	−1.00000			−2.83411	+	1.40279i
139.9	−1.08769	−	0.903842i		−	1.00000i	0.366139	+	1.96620i	−0.660150	−	2.13640i	−0.903842	+	1.08769i	−2.64346	−	0.110159i	1.37889	−	2.46955i	−1.00000			−1.21293	+	2.92041i
139.10	−1.08769	−	0.903842i			1.00000i	0.366139	+	1.96620i	0.660150	+	2.13640i	0.903842	−	1.08769i	−2.64346	+	0.110159i	1.37889	−	2.46955i	−1.00000			1.21293	−	2.92041i
139.11	−1.08769	+	0.903842i		−	1.00000i	0.366139	−	1.96620i	0.660150	−	2.13640i	0.903842	+	1.08769i	−2.64346	−	0.110159i	1.37889	+	2.46955i	−1.00000			1.21293	+	2.92041i
139.12	−1.08769	+	0.903842i			1.00000i	0.366139	−	1.96620i	−0.660150	+	2.13640i	−0.903842	−	1.08769i	−2.64346	+	0.110159i	1.37889	+	2.46955i	−1.00000			−1.21293	−	2.92041i
139.13	−0.846354	−	1.13300i		−	1.00000i	−0.567368	+	1.91784i	−2.22361	+	0.235752i	−1.13300	+	0.846354i	2.54904	+	0.708793i	2.65310	−	0.980342i	−1.00000			2.14906	+	2.31981i
139.14	−0.846354	−	1.13300i			1.00000i	−0.567368	+	1.91784i	2.22361	−	0.235752i	1.13300	−	0.846354i	2.54904	−	0.708793i	2.65310	−	0.980342i	−1.00000			−2.14906	−	2.31981i
139.15	−0.846354	+	1.13300i		−	1.00000i	−0.567368	−	1.91784i	2.22361	+	0.235752i	1.13300	+	0.846354i	2.54904	+	0.708793i	2.65310	+	0.980342i	−1.00000			−2.14906	+	2.31981i
139.16	−0.846354	+	1.13300i			1.00000i	−0.567368	−	1.91784i	−2.22361	−	0.235752i	−1.13300	−	0.846354i	2.54904	−	0.708793i	2.65310	+	0.980342i	−1.00000			2.14906	−	2.31981i
139.17	−0.654401	−	1.25370i		−	1.00000i	−1.14352	+	1.64084i	−0.206931	+	2.22647i	−1.25370	+	0.654401i	−0.859661	−	2.50220i	2.80544	+	0.359858i	−1.00000			2.92674	−	1.19758i
139.18	−0.654401	−	1.25370i			1.00000i	−1.14352	+	1.64084i	0.206931	−	2.22647i	1.25370	−	0.654401i	−0.859661	+	2.50220i	2.80544	+	0.359858i	−1.00000			−2.92674	+	1.19758i
139.19	−0.654401	+	1.25370i		−	1.00000i	−1.14352	−	1.64084i	0.206931	+	2.22647i	1.25370	+	0.654401i	−0.859661	−	2.50220i	2.80544	−	0.359858i	−1.00000			−2.92674	−	1.19758i
139.20	−0.654401	+	1.25370i			1.00000i	−1.14352	−	1.64084i	−0.206931	−	2.22647i	−1.25370	−	0.654401i	−0.859661	+	2.50220i	2.80544	−	0.359858i	−1.00000			2.92674	+	1.19758i

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
7.b	odd	2	1	inner
20.d	odd	2	1	inner
28.d	even	2	1	inner
35.c	odd	2	1	inner
140.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	420.2.i.a	✓	48
4.b	odd	2	1	inner	420.2.i.a	✓	48
5.b	even	2	1	inner	420.2.i.a	✓	48
7.b	odd	2	1	inner	420.2.i.a	✓	48
20.d	odd	2	1	inner	420.2.i.a	✓	48
28.d	even	2	1	inner	420.2.i.a	✓	48
35.c	odd	2	1	inner	420.2.i.a	✓	48
140.c	even	2	1	inner	420.2.i.a	✓	48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
420.2.i.a	✓	48	1.a	even	1	1	trivial
420.2.i.a	✓	48	4.b	odd	2	1	inner
420.2.i.a	✓	48	5.b	even	2	1	inner
420.2.i.a	✓	48	7.b	odd	2	1	inner
420.2.i.a	✓	48	20.d	odd	2	1	inner
420.2.i.a	✓	48	28.d	even	2	1	inner
420.2.i.a	✓	48	35.c	odd	2	1	inner
420.2.i.a	✓	48	140.c	even	2	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(420, [\chi])\).