Properties

Label 1470.2.i.x
Level $1470$
Weight $2$
Character orbit 1470.i
Analytic conductor $11.738$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} + ( 1 + \beta_{2} ) q^{3} + ( -1 - \beta_{2} ) q^{4} + \beta_{2} q^{5} + q^{6} - q^{8} + \beta_{2} q^{9} +O(q^{10})\) \( q -\beta_{2} q^{2} + ( 1 + \beta_{2} ) q^{3} + ( -1 - \beta_{2} ) q^{4} + \beta_{2} q^{5} + q^{6} - q^{8} + \beta_{2} q^{9} + ( 1 + \beta_{2} ) q^{10} + ( 2 + 3 \beta_{1} + 2 \beta_{2} ) q^{11} -\beta_{2} q^{12} + 4 \beta_{3} q^{13} - q^{15} + \beta_{2} q^{16} + ( -4 - \beta_{1} - 4 \beta_{2} ) q^{17} + ( 1 + \beta_{2} ) q^{18} + ( 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{19} + q^{20} + ( 2 - 3 \beta_{3} ) q^{22} + ( 2 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{23} + ( -1 - \beta_{2} ) q^{24} + ( -1 - \beta_{2} ) q^{25} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{26} - q^{27} + ( 4 - \beta_{3} ) q^{29} + \beta_{2} q^{30} + ( 6 - 3 \beta_{1} + 6 \beta_{2} ) q^{31} + ( 1 + \beta_{2} ) q^{32} + ( 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{33} + ( -4 + \beta_{3} ) q^{34} + q^{36} + ( 3 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} ) q^{37} + ( -4 + 2 \beta_{1} - 4 \beta_{2} ) q^{38} -4 \beta_{1} q^{39} -\beta_{2} q^{40} + ( 2 + 6 \beta_{3} ) q^{41} + ( 2 + 5 \beta_{3} ) q^{43} + ( -3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{44} + ( -1 - \beta_{2} ) q^{45} + ( 6 + 2 \beta_{1} + 6 \beta_{2} ) q^{46} + ( 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{47} - q^{48} - q^{50} + ( -\beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{51} + 4 \beta_{1} q^{52} + ( -6 - 4 \beta_{1} - 6 \beta_{2} ) q^{53} + \beta_{2} q^{54} + ( -2 + 3 \beta_{3} ) q^{55} + ( 4 + 2 \beta_{3} ) q^{57} + ( -\beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{58} + ( 6 + 2 \beta_{1} + 6 \beta_{2} ) q^{59} + ( 1 + \beta_{2} ) q^{60} + ( 4 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{61} + ( 6 + 3 \beta_{3} ) q^{62} + q^{64} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{65} + ( 2 + 3 \beta_{1} + 2 \beta_{2} ) q^{66} + ( -2 + 5 \beta_{1} - 2 \beta_{2} ) q^{67} + ( \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{68} + ( -6 + 2 \beta_{3} ) q^{69} + ( 4 - 6 \beta_{3} ) q^{71} -\beta_{2} q^{72} + ( 2 + 2 \beta_{2} ) q^{73} + ( 4 + 3 \beta_{1} + 4 \beta_{2} ) q^{74} -\beta_{2} q^{75} + ( -4 - 2 \beta_{3} ) q^{76} + 4 \beta_{3} q^{78} + 8 \beta_{2} q^{79} + ( -1 - \beta_{2} ) q^{80} + ( -1 - \beta_{2} ) q^{81} + ( 6 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} ) q^{82} + ( 8 - 2 \beta_{3} ) q^{83} + ( 4 - \beta_{3} ) q^{85} + ( 5 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} ) q^{86} + ( 4 + \beta_{1} + 4 \beta_{2} ) q^{87} + ( -2 - 3 \beta_{1} - 2 \beta_{2} ) q^{88} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{89} - q^{90} + ( 6 - 2 \beta_{3} ) q^{92} + ( -3 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} ) q^{93} + ( 2 + 3 \beta_{1} + 2 \beta_{2} ) q^{94} + ( 4 - 2 \beta_{1} + 4 \beta_{2} ) q^{95} + \beta_{2} q^{96} + ( 2 - 6 \beta_{3} ) q^{97} + ( -2 + 3 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} + 2q^{3} - 2q^{4} - 2q^{5} + 4q^{6} - 4q^{8} - 2q^{9} + O(q^{10}) \) \( 4q + 2q^{2} + 2q^{3} - 2q^{4} - 2q^{5} + 4q^{6} - 4q^{8} - 2q^{9} + 2q^{10} + 4q^{11} + 2q^{12} - 4q^{15} - 2q^{16} - 8q^{17} + 2q^{18} + 8q^{19} + 4q^{20} + 8q^{22} - 12q^{23} - 2q^{24} - 2q^{25} - 4q^{27} + 16q^{29} - 2q^{30} + 12q^{31} + 2q^{32} - 4q^{33} - 16q^{34} + 4q^{36} - 8q^{37} - 8q^{38} + 2q^{40} + 8q^{41} + 8q^{43} + 4q^{44} - 2q^{45} + 12q^{46} - 4q^{47} - 4q^{48} - 4q^{50} + 8q^{51} - 12q^{53} - 2q^{54} - 8q^{55} + 16q^{57} + 8q^{58} + 12q^{59} + 2q^{60} + 12q^{61} + 24q^{62} + 4q^{64} + 4q^{66} - 4q^{67} - 8q^{68} - 24q^{69} + 16q^{71} + 2q^{72} + 4q^{73} + 8q^{74} + 2q^{75} - 16q^{76} - 16q^{79} - 2q^{80} - 2q^{81} + 4q^{82} + 32q^{83} + 16q^{85} + 4q^{86} + 8q^{87} - 4q^{88} - 4q^{89} - 4q^{90} + 24q^{92} - 12q^{93} + 4q^{94} + 8q^{95} - 2q^{96} + 8q^{97} - 8q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

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

\(n\) \(491\) \(1081\) \(1177\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(1\)

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
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
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
361.2 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
961.2 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.x 4
7.b odd 2 1 1470.2.i.w 4
7.c even 3 1 1470.2.a.s 2
7.c even 3 1 inner 1470.2.i.x 4
7.d odd 6 1 1470.2.a.t yes 2
7.d odd 6 1 1470.2.i.w 4
21.g even 6 1 4410.2.a.bz 2
21.h odd 6 1 4410.2.a.bw 2
35.i odd 6 1 7350.2.a.dh 2
35.j even 6 1 7350.2.a.dl 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.a.s 2 7.c even 3 1
1470.2.a.t yes 2 7.d odd 6 1
1470.2.i.w 4 7.b odd 2 1
1470.2.i.w 4 7.d odd 6 1
1470.2.i.x 4 1.a even 1 1 trivial
1470.2.i.x 4 7.c even 3 1 inner
4410.2.a.bw 2 21.h odd 6 1
4410.2.a.bz 2 21.g even 6 1
7350.2.a.dh 2 35.i odd 6 1
7350.2.a.dl 2 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 \(S_{2}^{\mathrm{new}}(1470, [\chi])\):

\( T_{11}^{4} - 4 T_{11}^{3} + 30 T_{11}^{2} + 56 T_{11} + 196 \)
\( T_{13}^{2} - 32 \)
\( T_{17}^{4} + 8 T_{17}^{3} + 50 T_{17}^{2} + 112 T_{17} + 196 \)
\( T_{19}^{4} - 8 T_{19}^{3} + 56 T_{19}^{2} - 64 T_{19} + 64 \)
\( T_{31}^{4} - 12 T_{31}^{3} + 126 T_{31}^{2} - 216 T_{31} + 324 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{2} \)
$3$ \( ( 1 - T + T^{2} )^{2} \)
$5$ \( ( 1 + T + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( 196 + 56 T + 30 T^{2} - 4 T^{3} + T^{4} \)
$13$ \( ( -32 + T^{2} )^{2} \)
$17$ \( 196 + 112 T + 50 T^{2} + 8 T^{3} + T^{4} \)
$19$ \( 64 - 64 T + 56 T^{2} - 8 T^{3} + T^{4} \)
$23$ \( 784 + 336 T + 116 T^{2} + 12 T^{3} + T^{4} \)
$29$ \( ( 14 - 8 T + T^{2} )^{2} \)
$31$ \( 324 - 216 T + 126 T^{2} - 12 T^{3} + T^{4} \)
$37$ \( 4 - 16 T + 66 T^{2} + 8 T^{3} + T^{4} \)
$41$ \( ( -68 - 4 T + T^{2} )^{2} \)
$43$ \( ( -46 - 4 T + T^{2} )^{2} \)
$47$ \( 196 - 56 T + 30 T^{2} + 4 T^{3} + T^{4} \)
$53$ \( 16 + 48 T + 140 T^{2} + 12 T^{3} + T^{4} \)
$59$ \( 784 - 336 T + 116 T^{2} - 12 T^{3} + T^{4} \)
$61$ \( 16 - 48 T + 140 T^{2} - 12 T^{3} + T^{4} \)
$67$ \( 2116 - 184 T + 62 T^{2} + 4 T^{3} + T^{4} \)
$71$ \( ( -56 - 8 T + T^{2} )^{2} \)
$73$ \( ( 4 - 2 T + T^{2} )^{2} \)
$79$ \( ( 64 + 8 T + T^{2} )^{2} \)
$83$ \( ( 56 - 16 T + T^{2} )^{2} \)
$89$ \( 16 - 16 T + 20 T^{2} + 4 T^{3} + T^{4} \)
$97$ \( ( -68 - 4 T + T^{2} )^{2} \)
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