Properties

Label 1470.2.i.c
Level $1470$
Weight $2$
Character orbit 1470.i
Analytic conductor $11.738$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

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

Hecke kernels

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

\( T_{11}^{2} + 2T_{11} + 4 \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display
\( T_{17}^{2} + 4T_{17} + 16 \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display
\( T_{31}^{2} + 2T_{31} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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