Properties

Label 1470.2.g.f
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 210)
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.

\(f(q)\) \(=\) \( q -i q^{2} -i q^{3} - q^{4} + ( 2 + i ) q^{5} - q^{6} + i q^{8} - q^{9} +O(q^{10})\) \( q -i q^{2} -i q^{3} - q^{4} + ( 2 + i ) q^{5} - q^{6} + i q^{8} - q^{9} + ( 1 - 2 i ) q^{10} -5 q^{11} + i q^{12} -i q^{13} + ( 1 - 2 i ) q^{15} + q^{16} -2 i q^{17} + i q^{18} -7 q^{19} + ( -2 - i ) q^{20} + 5 i q^{22} -3 i q^{23} + q^{24} + ( 3 + 4 i ) q^{25} - q^{26} + i q^{27} + ( -2 - i ) q^{30} -6 q^{31} -i q^{32} + 5 i q^{33} -2 q^{34} + q^{36} -5 i q^{37} + 7 i q^{38} - q^{39} + ( -1 + 2 i ) q^{40} -9 q^{41} -10 i q^{43} + 5 q^{44} + ( -2 - i ) q^{45} -3 q^{46} -13 i q^{47} -i q^{48} + ( 4 - 3 i ) q^{50} -2 q^{51} + i q^{52} + i q^{53} + q^{54} + ( -10 - 5 i ) q^{55} + 7 i q^{57} -4 q^{59} + ( -1 + 2 i ) q^{60} -2 q^{61} + 6 i q^{62} - q^{64} + ( 1 - 2 i ) q^{65} + 5 q^{66} + 6 i q^{67} + 2 i q^{68} -3 q^{69} -2 q^{71} -i q^{72} -4 i q^{73} -5 q^{74} + ( 4 - 3 i ) q^{75} + 7 q^{76} + i q^{78} + 14 q^{79} + ( 2 + i ) q^{80} + q^{81} + 9 i q^{82} + 10 i q^{83} + ( 2 - 4 i ) q^{85} -10 q^{86} -5 i q^{88} -10 q^{89} + ( -1 + 2 i ) q^{90} + 3 i q^{92} + 6 i q^{93} -13 q^{94} + ( -14 - 7 i ) q^{95} - q^{96} + 8 i q^{97} + 5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 4q^{5} - 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 4q^{5} - 2q^{6} - 2q^{9} + 2q^{10} - 10q^{11} + 2q^{15} + 2q^{16} - 14q^{19} - 4q^{20} + 2q^{24} + 6q^{25} - 2q^{26} - 4q^{30} - 12q^{31} - 4q^{34} + 2q^{36} - 2q^{39} - 2q^{40} - 18q^{41} + 10q^{44} - 4q^{45} - 6q^{46} + 8q^{50} - 4q^{51} + 2q^{54} - 20q^{55} - 8q^{59} - 2q^{60} - 4q^{61} - 2q^{64} + 2q^{65} + 10q^{66} - 6q^{69} - 4q^{71} - 10q^{74} + 8q^{75} + 14q^{76} + 28q^{79} + 4q^{80} + 2q^{81} + 4q^{85} - 20q^{86} - 20q^{89} - 2q^{90} - 26q^{94} - 28q^{95} - 2q^{96} + 10q^{99} + O(q^{100}) \)

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.f 2
5.b even 2 1 inner 1470.2.g.f 2
5.c odd 4 1 7350.2.a.t 1
5.c odd 4 1 7350.2.a.bn 1
7.b odd 2 1 1470.2.g.a 2
7.c even 3 2 210.2.n.a 4
7.d odd 6 2 1470.2.n.i 4
21.h odd 6 2 630.2.u.c 4
28.g odd 6 2 1680.2.di.a 4
35.c odd 2 1 1470.2.g.a 2
35.f even 4 1 7350.2.a.b 1
35.f even 4 1 7350.2.a.ch 1
35.i odd 6 2 1470.2.n.i 4
35.j even 6 2 210.2.n.a 4
35.l odd 12 2 1050.2.i.f 2
35.l odd 12 2 1050.2.i.o 2
105.o odd 6 2 630.2.u.c 4
140.p odd 6 2 1680.2.di.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.n.a 4 7.c even 3 2
210.2.n.a 4 35.j even 6 2
630.2.u.c 4 21.h odd 6 2
630.2.u.c 4 105.o odd 6 2
1050.2.i.f 2 35.l odd 12 2
1050.2.i.o 2 35.l odd 12 2
1470.2.g.a 2 7.b odd 2 1
1470.2.g.a 2 35.c odd 2 1
1470.2.g.f 2 1.a even 1 1 trivial
1470.2.g.f 2 5.b even 2 1 inner
1470.2.n.i 4 7.d odd 6 2
1470.2.n.i 4 35.i odd 6 2
1680.2.di.a 4 28.g odd 6 2
1680.2.di.a 4 140.p odd 6 2
7350.2.a.b 1 35.f even 4 1
7350.2.a.t 1 5.c odd 4 1
7350.2.a.bn 1 5.c odd 4 1
7350.2.a.ch 1 35.f even 4 1

Hecke kernels

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

\( T_{11} + 5 \)
\( T_{17}^{2} + 4 \)
\( T_{19} + 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( 1 - 4 T + 5 T^{2} \)
$7$ 1
$11$ \( ( 1 + 5 T + 11 T^{2} )^{2} \)
$13$ \( 1 - 25 T^{2} + 169 T^{4} \)
$17$ \( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \)
$19$ \( ( 1 + 7 T + 19 T^{2} )^{2} \)
$23$ \( 1 - 37 T^{2} + 529 T^{4} \)
$29$ \( ( 1 + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 6 T + 31 T^{2} )^{2} \)
$37$ \( 1 - 49 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 + 9 T + 41 T^{2} )^{2} \)
$43$ \( 1 + 14 T^{2} + 1849 T^{4} \)
$47$ \( 1 + 75 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 105 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 + 4 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 2 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 98 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 2 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 130 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 - 14 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 66 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 + 10 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} ) \)
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