Properties

Label 1470.2.g.g
Level $1470$
Weight $2$
Character orbit 1470.g
Analytic conductor $11.738$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1470.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.7380090971\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} + i q^{3} - q^{4} + ( 2 - i ) q^{5} - q^{6} -i q^{8} - q^{9} +O(q^{10})\) \( q + i q^{2} + i q^{3} - q^{4} + ( 2 - i ) q^{5} - q^{6} -i q^{8} - q^{9} + ( 1 + 2 i ) q^{10} + 2 q^{11} -i q^{12} -6 i q^{13} + ( 1 + 2 i ) q^{15} + q^{16} + 2 i q^{17} -i q^{18} + ( -2 + i ) q^{20} + 2 i q^{22} -4 i q^{23} + q^{24} + ( 3 - 4 i ) q^{25} + 6 q^{26} -i q^{27} + ( -2 + i ) q^{30} + 8 q^{31} + i q^{32} + 2 i q^{33} -2 q^{34} + q^{36} -2 i q^{37} + 6 q^{39} + ( -1 - 2 i ) q^{40} -2 q^{41} -4 i q^{43} -2 q^{44} + ( -2 + i ) q^{45} + 4 q^{46} -8 i q^{47} + i q^{48} + ( 4 + 3 i ) q^{50} -2 q^{51} + 6 i q^{52} + 6 i q^{53} + q^{54} + ( 4 - 2 i ) q^{55} + 10 q^{59} + ( -1 - 2 i ) q^{60} -2 q^{61} + 8 i q^{62} - q^{64} + ( -6 - 12 i ) q^{65} -2 q^{66} + 8 i q^{67} -2 i q^{68} + 4 q^{69} + 12 q^{71} + i q^{72} + 4 i q^{73} + 2 q^{74} + ( 4 + 3 i ) q^{75} + 6 i q^{78} + ( 2 - i ) q^{80} + q^{81} -2 i q^{82} + 4 i q^{83} + ( 2 + 4 i ) q^{85} + 4 q^{86} -2 i q^{88} -10 q^{89} + ( -1 - 2 i ) q^{90} + 4 i q^{92} + 8 i q^{93} + 8 q^{94} - q^{96} -8 i q^{97} -2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 4q^{5} - 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 4q^{5} - 2q^{6} - 2q^{9} + 2q^{10} + 4q^{11} + 2q^{15} + 2q^{16} - 4q^{20} + 2q^{24} + 6q^{25} + 12q^{26} - 4q^{30} + 16q^{31} - 4q^{34} + 2q^{36} + 12q^{39} - 2q^{40} - 4q^{41} - 4q^{44} - 4q^{45} + 8q^{46} + 8q^{50} - 4q^{51} + 2q^{54} + 8q^{55} + 20q^{59} - 2q^{60} - 4q^{61} - 2q^{64} - 12q^{65} - 4q^{66} + 8q^{69} + 24q^{71} + 4q^{74} + 8q^{75} + 4q^{80} + 2q^{81} + 4q^{85} + 8q^{86} - 20q^{89} - 2q^{90} + 16q^{94} - 2q^{96} - 4q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times\).

\(n\) \(491\) \(1081\) \(1177\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
589.1
1.00000i
1.00000i
1.00000i 1.00000i −1.00000 2.00000 + 1.00000i −1.00000 0 1.00000i −1.00000 1.00000 2.00000i
589.2 1.00000i 1.00000i −1.00000 2.00000 1.00000i −1.00000 0 1.00000i −1.00000 1.00000 + 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.2.g.g 2
5.b even 2 1 inner 1470.2.g.g 2
5.c odd 4 1 7350.2.a.bg 1
5.c odd 4 1 7350.2.a.cc 1
7.b odd 2 1 30.2.c.a 2
7.c even 3 2 1470.2.n.a 4
7.d odd 6 2 1470.2.n.h 4
21.c even 2 1 90.2.c.a 2
28.d even 2 1 240.2.f.a 2
35.c odd 2 1 30.2.c.a 2
35.f even 4 1 150.2.a.a 1
35.f even 4 1 150.2.a.c 1
35.i odd 6 2 1470.2.n.h 4
35.j even 6 2 1470.2.n.a 4
56.e even 2 1 960.2.f.i 2
56.h odd 2 1 960.2.f.h 2
63.l odd 6 2 810.2.i.e 4
63.o even 6 2 810.2.i.b 4
84.h odd 2 1 720.2.f.f 2
105.g even 2 1 90.2.c.a 2
105.k odd 4 1 450.2.a.b 1
105.k odd 4 1 450.2.a.f 1
112.j even 4 1 3840.2.d.j 2
112.j even 4 1 3840.2.d.x 2
112.l odd 4 1 3840.2.d.g 2
112.l odd 4 1 3840.2.d.y 2
140.c even 2 1 240.2.f.a 2
140.j odd 4 1 1200.2.a.g 1
140.j odd 4 1 1200.2.a.m 1
168.e odd 2 1 2880.2.f.c 2
168.i even 2 1 2880.2.f.e 2
280.c odd 2 1 960.2.f.h 2
280.n even 2 1 960.2.f.i 2
280.s even 4 1 4800.2.a.l 1
280.s even 4 1 4800.2.a.cg 1
280.y odd 4 1 4800.2.a.m 1
280.y odd 4 1 4800.2.a.cj 1
315.z even 6 2 810.2.i.b 4
315.bg odd 6 2 810.2.i.e 4
420.o odd 2 1 720.2.f.f 2
420.w even 4 1 3600.2.a.o 1
420.w even 4 1 3600.2.a.bg 1
560.be even 4 1 3840.2.d.j 2
560.be even 4 1 3840.2.d.x 2
560.bf odd 4 1 3840.2.d.g 2
560.bf odd 4 1 3840.2.d.y 2
840.b odd 2 1 2880.2.f.c 2
840.u even 2 1 2880.2.f.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.c.a 2 7.b odd 2 1
30.2.c.a 2 35.c odd 2 1
90.2.c.a 2 21.c even 2 1
90.2.c.a 2 105.g even 2 1
150.2.a.a 1 35.f even 4 1
150.2.a.c 1 35.f even 4 1
240.2.f.a 2 28.d even 2 1
240.2.f.a 2 140.c even 2 1
450.2.a.b 1 105.k odd 4 1
450.2.a.f 1 105.k odd 4 1
720.2.f.f 2 84.h odd 2 1
720.2.f.f 2 420.o odd 2 1
810.2.i.b 4 63.o even 6 2
810.2.i.b 4 315.z even 6 2
810.2.i.e 4 63.l odd 6 2
810.2.i.e 4 315.bg odd 6 2
960.2.f.h 2 56.h odd 2 1
960.2.f.h 2 280.c odd 2 1
960.2.f.i 2 56.e even 2 1
960.2.f.i 2 280.n even 2 1
1200.2.a.g 1 140.j odd 4 1
1200.2.a.m 1 140.j odd 4 1
1470.2.g.g 2 1.a even 1 1 trivial
1470.2.g.g 2 5.b even 2 1 inner
1470.2.n.a 4 7.c even 3 2
1470.2.n.a 4 35.j even 6 2
1470.2.n.h 4 7.d odd 6 2
1470.2.n.h 4 35.i odd 6 2
2880.2.f.c 2 168.e odd 2 1
2880.2.f.c 2 840.b odd 2 1
2880.2.f.e 2 168.i even 2 1
2880.2.f.e 2 840.u even 2 1
3600.2.a.o 1 420.w even 4 1
3600.2.a.bg 1 420.w even 4 1
3840.2.d.g 2 112.l odd 4 1
3840.2.d.g 2 560.bf odd 4 1
3840.2.d.j 2 112.j even 4 1
3840.2.d.j 2 560.be even 4 1
3840.2.d.x 2 112.j even 4 1
3840.2.d.x 2 560.be even 4 1
3840.2.d.y 2 112.l odd 4 1
3840.2.d.y 2 560.bf odd 4 1
4800.2.a.l 1 280.s even 4 1
4800.2.a.m 1 280.y odd 4 1
4800.2.a.cg 1 280.s even 4 1
4800.2.a.cj 1 280.y odd 4 1
7350.2.a.bg 1 5.c odd 4 1
7350.2.a.cc 1 5.c odd 4 1

Hecke kernels

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

\( T_{11} - 2 \)
\( T_{17}^{2} + 4 \)
\( T_{19} \)