Properties

Label 1470.2.i.g
Level $1470$
Weight $2$
Character orbit 1470.i
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.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.7380090971\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(491\) \(1081\) \(1177\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −1.00000 0 1.00000 −0.500000 0.866025i −0.500000 + 0.866025i
961.1 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −1.00000 0 1.00000 −0.500000 + 0.866025i −0.500000 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.2.i.g 2
7.b odd 2 1 1470.2.i.c 2
7.c even 3 1 1470.2.a.n 1
7.c even 3 1 inner 1470.2.i.g 2
7.d odd 6 1 1470.2.a.p yes 1
7.d odd 6 1 1470.2.i.c 2
21.g even 6 1 4410.2.a.n 1
21.h odd 6 1 4410.2.a.e 1
35.i odd 6 1 7350.2.a.o 1
35.j even 6 1 7350.2.a.bh 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.a.n 1 7.c even 3 1
1470.2.a.p yes 1 7.d odd 6 1
1470.2.i.c 2 7.b odd 2 1
1470.2.i.c 2 7.d odd 6 1
1470.2.i.g 2 1.a even 1 1 trivial
1470.2.i.g 2 7.c even 3 1 inner
4410.2.a.e 1 21.h odd 6 1
4410.2.a.n 1 21.g even 6 1
7350.2.a.o 1 35.i odd 6 1
7350.2.a.bh 1 35.j even 6 1

Hecke kernels

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

\( T_{11}^{2} + 2 T_{11} + 4 \)
\( T_{13} - 2 \)
\( T_{17}^{2} - 4 T_{17} + 16 \)
\( T_{19} \)
\( T_{31}^{2} - 2 T_{31} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( 1 + T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 4 + 2 T + T^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( 16 - 4 T + T^{2} \)
$19$ \( T^{2} \)
$23$ \( 64 + 8 T + T^{2} \)
$29$ \( T^{2} \)
$31$ \( 4 - 2 T + T^{2} \)
$37$ \( 64 + 8 T + T^{2} \)
$41$ \( ( 2 + T )^{2} \)
$43$ \( ( 2 + T )^{2} \)
$47$ \( 100 + 10 T + T^{2} \)
$53$ \( 4 - 2 T + T^{2} \)
$59$ \( 16 + 4 T + T^{2} \)
$61$ \( 100 - 10 T + T^{2} \)
$67$ \( 4 + 2 T + T^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( 100 + 10 T + T^{2} \)
$79$ \( 256 + 16 T + T^{2} \)
$83$ \( ( -16 + T )^{2} \)
$89$ \( 196 + 14 T + T^{2} \)
$97$ \( ( -6 + T )^{2} \)
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