Properties

Label 1470.2.i.v
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: \(\Z[a_1, \ldots, a_{11}]\)
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.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{2} ) q^{2} -\beta_{2} q^{3} + \beta_{2} q^{4} + ( 1 + \beta_{2} ) q^{5} - q^{6} + q^{8} + ( -1 - \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{2} ) q^{2} -\beta_{2} q^{3} + \beta_{2} q^{4} + ( 1 + \beta_{2} ) q^{5} - q^{6} + q^{8} + ( -1 - \beta_{2} ) q^{9} -\beta_{2} q^{10} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{11} + ( 1 + \beta_{2} ) q^{12} + q^{15} + ( -1 - \beta_{2} ) q^{16} + ( -\beta_{1} - \beta_{3} ) q^{17} + \beta_{2} q^{18} -2 \beta_{1} q^{19} - q^{20} + ( 2 + \beta_{3} ) q^{22} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{23} -\beta_{2} q^{24} + \beta_{2} q^{25} - q^{27} + ( 4 - 3 \beta_{3} ) q^{29} + ( -1 - \beta_{2} ) q^{30} + ( -5 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} ) q^{31} + \beta_{2} q^{32} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{33} + \beta_{3} q^{34} + q^{36} + \beta_{1} q^{37} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{38} + ( 1 + \beta_{2} ) q^{40} + ( 6 - 2 \beta_{3} ) q^{41} + ( 6 + \beta_{3} ) q^{43} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{44} -\beta_{2} q^{45} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{46} + ( -2 - 5 \beta_{1} - 2 \beta_{2} ) q^{47} - q^{48} + q^{50} -\beta_{1} q^{51} + ( -8 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} ) q^{53} + ( 1 + \beta_{2} ) q^{54} + ( -2 - \beta_{3} ) q^{55} + 2 \beta_{3} q^{57} + ( -4 - 3 \beta_{1} - 4 \beta_{2} ) q^{58} + ( -6 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{59} + \beta_{2} q^{60} + ( -6 - 4 \beta_{1} - 6 \beta_{2} ) q^{61} + ( 2 + 5 \beta_{3} ) q^{62} + q^{64} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{66} + ( 7 \beta_{1} - 2 \beta_{2} + 7 \beta_{3} ) q^{67} + \beta_{1} q^{68} + ( -2 + 2 \beta_{3} ) q^{69} + ( 8 - 2 \beta_{3} ) q^{71} + ( -1 - \beta_{2} ) q^{72} + ( -4 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{73} + ( -\beta_{1} - \beta_{3} ) q^{74} + ( 1 + \beta_{2} ) q^{75} -2 \beta_{3} q^{76} + 8 \beta_{1} q^{79} -\beta_{2} q^{80} + \beta_{2} q^{81} + ( -6 - 2 \beta_{1} - 6 \beta_{2} ) q^{82} + ( 8 + 2 \beta_{3} ) q^{83} -\beta_{3} q^{85} + ( -6 + \beta_{1} - 6 \beta_{2} ) q^{86} + ( -3 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{87} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{88} + ( 2 - 6 \beta_{1} + 2 \beta_{2} ) q^{89} - q^{90} + ( 2 - 2 \beta_{3} ) q^{92} + ( 2 - 5 \beta_{1} + 2 \beta_{2} ) q^{93} + ( 5 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} ) q^{94} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{95} + ( 1 + \beta_{2} ) q^{96} + ( -6 + 2 \beta_{3} ) q^{97} + ( 2 + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} + 2q^{3} - 2q^{4} + 2q^{5} - 4q^{6} + 4q^{8} - 2q^{9} + O(q^{10}) \) \( 4q - 2q^{2} + 2q^{3} - 2q^{4} + 2q^{5} - 4q^{6} + 4q^{8} - 2q^{9} + 2q^{10} - 4q^{11} + 2q^{12} + 4q^{15} - 2q^{16} - 2q^{18} - 4q^{20} + 8q^{22} - 4q^{23} + 2q^{24} - 2q^{25} - 4q^{27} + 16q^{29} - 2q^{30} - 4q^{31} - 2q^{32} + 4q^{33} + 4q^{36} + 2q^{40} + 24q^{41} + 24q^{43} - 4q^{44} + 2q^{45} - 4q^{46} - 4q^{47} - 4q^{48} + 4q^{50} - 4q^{53} + 2q^{54} - 8q^{55} - 8q^{58} - 12q^{59} - 2q^{60} - 12q^{61} + 8q^{62} + 4q^{64} + 4q^{66} + 4q^{67} - 8q^{69} + 32q^{71} - 2q^{72} + 4q^{73} + 2q^{75} + 2q^{80} - 2q^{81} - 12q^{82} + 32q^{83} - 12q^{86} + 8q^{87} - 4q^{88} + 4q^{89} - 4q^{90} + 8q^{92} + 4q^{93} - 4q^{94} + 2q^{96} - 24q^{97} + 8q^{99} + O(q^{100}) \)

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 \(\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times\).

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

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.v 4
7.b odd 2 1 1470.2.i.u 4
7.c even 3 1 1470.2.a.u 2
7.c even 3 1 inner 1470.2.i.v 4
7.d odd 6 1 1470.2.a.v yes 2
7.d odd 6 1 1470.2.i.u 4
21.g even 6 1 4410.2.a.bn 2
21.h odd 6 1 4410.2.a.br 2
35.i odd 6 1 7350.2.a.dd 2
35.j even 6 1 7350.2.a.df 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.a.u 2 7.c even 3 1
1470.2.a.v yes 2 7.d odd 6 1
1470.2.i.u 4 7.b odd 2 1
1470.2.i.u 4 7.d odd 6 1
1470.2.i.v 4 1.a even 1 1 trivial
1470.2.i.v 4 7.c even 3 1 inner
4410.2.a.bn 2 21.g even 6 1
4410.2.a.br 2 21.h odd 6 1
7350.2.a.dd 2 35.i odd 6 1
7350.2.a.df 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 \(S_{2}^{\mathrm{new}}(1470, [\chi])\):

\( T_{11}^{4} + 4 T_{11}^{3} + 14 T_{11}^{2} + 8 T_{11} + 4 \)
\( T_{13} \)
\( T_{17}^{4} + 2 T_{17}^{2} + 4 \)
\( T_{19}^{4} + 8 T_{19}^{2} + 64 \)
\( T_{31}^{4} + 4 T_{31}^{3} + 62 T_{31}^{2} - 184 T_{31} + 2116 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( ( 1 - T + T^{2} )^{2} \)
$5$ \( ( 1 - T + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( 4 + 8 T + 14 T^{2} + 4 T^{3} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( 4 + 2 T^{2} + T^{4} \)
$19$ \( 64 + 8 T^{2} + T^{4} \)
$23$ \( 16 - 16 T + 20 T^{2} + 4 T^{3} + T^{4} \)
$29$ \( ( -2 - 8 T + T^{2} )^{2} \)
$31$ \( 2116 - 184 T + 62 T^{2} + 4 T^{3} + T^{4} \)
$37$ \( 4 + 2 T^{2} + T^{4} \)
$41$ \( ( 28 - 12 T + T^{2} )^{2} \)
$43$ \( ( 34 - 12 T + T^{2} )^{2} \)
$47$ \( 2116 - 184 T + 62 T^{2} + 4 T^{3} + T^{4} \)
$53$ \( 15376 - 496 T + 140 T^{2} + 4 T^{3} + T^{4} \)
$59$ \( 1296 - 432 T + 180 T^{2} + 12 T^{3} + T^{4} \)
$61$ \( 16 + 48 T + 140 T^{2} + 12 T^{3} + T^{4} \)
$67$ \( 8836 + 376 T + 110 T^{2} - 4 T^{3} + T^{4} \)
$71$ \( ( 56 - 16 T + T^{2} )^{2} \)
$73$ \( 784 + 112 T + 44 T^{2} - 4 T^{3} + T^{4} \)
$79$ \( 16384 + 128 T^{2} + T^{4} \)
$83$ \( ( 56 - 16 T + T^{2} )^{2} \)
$89$ \( 4624 + 272 T + 84 T^{2} - 4 T^{3} + T^{4} \)
$97$ \( ( 28 + 12 T + T^{2} )^{2} \)
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