Properties

Label 1470.2.a.d
Level $1470$
Weight $2$
Character orbit 1470.a
Self dual yes
Analytic conductor $11.738$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.7380090971\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{8} + q^{9} - q^{10} - q^{12} - 2q^{13} - q^{15} + q^{16} - 6q^{17} - q^{18} + 4q^{19} + q^{20} + q^{24} + q^{25} + 2q^{26} - q^{27} - 6q^{29} + q^{30} - 8q^{31} - q^{32} + 6q^{34} + q^{36} + 2q^{37} - 4q^{38} + 2q^{39} - q^{40} + 6q^{41} - 4q^{43} + q^{45} - q^{48} - q^{50} + 6q^{51} - 2q^{52} - 6q^{53} + q^{54} - 4q^{57} + 6q^{58} - q^{60} + 10q^{61} + 8q^{62} + q^{64} - 2q^{65} - 4q^{67} - 6q^{68} - q^{72} - 2q^{73} - 2q^{74} - q^{75} + 4q^{76} - 2q^{78} + 8q^{79} + q^{80} + q^{81} - 6q^{82} - 12q^{83} - 6q^{85} + 4q^{86} + 6q^{87} - 18q^{89} - q^{90} + 8q^{93} + 4q^{95} + q^{96} - 2q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−1.00000 −1.00000 1.00000 1.00000 1.00000 0 −1.00000 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(5\) \(-1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.2.a.d 1
3.b odd 2 1 4410.2.a.z 1
5.b even 2 1 7350.2.a.ct 1
7.b odd 2 1 30.2.a.a 1
7.c even 3 2 1470.2.i.q 2
7.d odd 6 2 1470.2.i.o 2
21.c even 2 1 90.2.a.c 1
28.d even 2 1 240.2.a.b 1
35.c odd 2 1 150.2.a.b 1
35.f even 4 2 150.2.c.a 2
56.e even 2 1 960.2.a.p 1
56.h odd 2 1 960.2.a.e 1
63.l odd 6 2 810.2.e.l 2
63.o even 6 2 810.2.e.b 2
77.b even 2 1 3630.2.a.w 1
84.h odd 2 1 720.2.a.j 1
91.b odd 2 1 5070.2.a.w 1
91.i even 4 2 5070.2.b.k 2
105.g even 2 1 450.2.a.d 1
105.k odd 4 2 450.2.c.b 2
112.j even 4 2 3840.2.k.f 2
112.l odd 4 2 3840.2.k.y 2
119.d odd 2 1 8670.2.a.g 1
140.c even 2 1 1200.2.a.k 1
140.j odd 4 2 1200.2.f.e 2
168.e odd 2 1 2880.2.a.q 1
168.i even 2 1 2880.2.a.a 1
280.c odd 2 1 4800.2.a.cq 1
280.n even 2 1 4800.2.a.d 1
280.s even 4 2 4800.2.f.p 2
280.y odd 4 2 4800.2.f.w 2
420.o odd 2 1 3600.2.a.f 1
420.w even 4 2 3600.2.f.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.a.a 1 7.b odd 2 1
90.2.a.c 1 21.c even 2 1
150.2.a.b 1 35.c odd 2 1
150.2.c.a 2 35.f even 4 2
240.2.a.b 1 28.d even 2 1
450.2.a.d 1 105.g even 2 1
450.2.c.b 2 105.k odd 4 2
720.2.a.j 1 84.h odd 2 1
810.2.e.b 2 63.o even 6 2
810.2.e.l 2 63.l odd 6 2
960.2.a.e 1 56.h odd 2 1
960.2.a.p 1 56.e even 2 1
1200.2.a.k 1 140.c even 2 1
1200.2.f.e 2 140.j odd 4 2
1470.2.a.d 1 1.a even 1 1 trivial
1470.2.i.o 2 7.d odd 6 2
1470.2.i.q 2 7.c even 3 2
2880.2.a.a 1 168.i even 2 1
2880.2.a.q 1 168.e odd 2 1
3600.2.a.f 1 420.o odd 2 1
3600.2.f.i 2 420.w even 4 2
3630.2.a.w 1 77.b even 2 1
3840.2.k.f 2 112.j even 4 2
3840.2.k.y 2 112.l odd 4 2
4410.2.a.z 1 3.b odd 2 1
4800.2.a.d 1 280.n even 2 1
4800.2.a.cq 1 280.c odd 2 1
4800.2.f.p 2 280.s even 4 2
4800.2.f.w 2 280.y odd 4 2
5070.2.a.w 1 91.b odd 2 1
5070.2.b.k 2 91.i even 4 2
7350.2.a.ct 1 5.b even 2 1
8670.2.a.g 1 119.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1470))\):

\( T_{11} \)
\( T_{13} + 2 \)
\( T_{17} + 6 \)
\( T_{19} - 4 \)
\( T_{31} + 8 \)