Properties

Label 1470.2.g.i
Level $1470$
Weight $2$
Character orbit 1470.g
Analytic conductor $11.738$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.0.29160000.2
Defining polynomial: \(x^{6} - 20 x^{3} + 125\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 20 x^{3} + 125\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} + 20 \nu^{2} \)\()/25\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} - 10 \)\()/5\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} - 10 \nu \)\()/5\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{5} - 15 \nu^{2} \)\()/25\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} + 2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(5 \beta_{3} + 10\)
\(\nu^{4}\)\(=\)\(5 \beta_{4} + 10 \beta_{1}\)
\(\nu^{5}\)\(=\)\(20 \beta_{5} + 15 \beta_{2}\)

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
−1.40280 + 1.74131i
−0.806615 2.08551i
2.20942 + 0.344208i
−1.40280 1.74131i
−0.806615 + 2.08551i
2.20942 0.344208i
1.00000i 1.00000i −1.00000 −1.40280 1.74131i 1.00000 0 1.00000i −1.00000 −1.74131 + 1.40280i
589.2 1.00000i 1.00000i −1.00000 −0.806615 + 2.08551i 1.00000 0 1.00000i −1.00000 2.08551 + 0.806615i
589.3 1.00000i 1.00000i −1.00000 2.20942 0.344208i 1.00000 0 1.00000i −1.00000 −0.344208 2.20942i
589.4 1.00000i 1.00000i −1.00000 −1.40280 + 1.74131i 1.00000 0 1.00000i −1.00000 −1.74131 1.40280i
589.5 1.00000i 1.00000i −1.00000 −0.806615 2.08551i 1.00000 0 1.00000i −1.00000 2.08551 0.806615i
589.6 1.00000i 1.00000i −1.00000 2.20942 + 0.344208i 1.00000 0 1.00000i −1.00000 −0.344208 + 2.20942i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 589.6
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.i 6
5.b even 2 1 inner 1470.2.g.i 6
5.c odd 4 1 7350.2.a.dn 3
5.c odd 4 1 7350.2.a.dq 3
7.b odd 2 1 1470.2.g.h 6
7.c even 3 2 210.2.n.b 12
7.d odd 6 2 1470.2.n.j 12
21.h odd 6 2 630.2.u.f 12
28.g odd 6 2 1680.2.di.c 12
35.c odd 2 1 1470.2.g.h 6
35.f even 4 1 7350.2.a.do 3
35.f even 4 1 7350.2.a.dp 3
35.i odd 6 2 1470.2.n.j 12
35.j even 6 2 210.2.n.b 12
35.l odd 12 2 1050.2.i.u 6
35.l odd 12 2 1050.2.i.v 6
105.o odd 6 2 630.2.u.f 12
140.p odd 6 2 1680.2.di.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.n.b 12 7.c even 3 2
210.2.n.b 12 35.j even 6 2
630.2.u.f 12 21.h odd 6 2
630.2.u.f 12 105.o odd 6 2
1050.2.i.u 6 35.l odd 12 2
1050.2.i.v 6 35.l odd 12 2
1470.2.g.h 6 7.b odd 2 1
1470.2.g.h 6 35.c odd 2 1
1470.2.g.i 6 1.a even 1 1 trivial
1470.2.g.i 6 5.b even 2 1 inner
1470.2.n.j 12 7.d odd 6 2
1470.2.n.j 12 35.i odd 6 2
1680.2.di.c 12 28.g odd 6 2
1680.2.di.c 12 140.p odd 6 2
7350.2.a.dn 3 5.c odd 4 1
7350.2.a.do 3 35.f even 4 1
7350.2.a.dp 3 35.f even 4 1
7350.2.a.dq 3 5.c odd 4 1

Hecke kernels

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

\( T_{11}^{3} - 3 T_{11}^{2} - 27 T_{11} + 49 \)
\( T_{17}^{6} + 132 T_{17}^{4} + 5568 T_{17}^{2} + 73984 \)
\( T_{19}^{3} + 3 T_{19}^{2} - 12 T_{19} - 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{3} \)
$3$ \( ( 1 + T^{2} )^{3} \)
$5$ \( 125 - 20 T^{3} + T^{6} \)
$7$ \( T^{6} \)
$11$ \( ( 49 - 27 T - 3 T^{2} + T^{3} )^{2} \)
$13$ \( 576 + 288 T^{2} + 33 T^{4} + T^{6} \)
$17$ \( 73984 + 5568 T^{2} + 132 T^{4} + T^{6} \)
$19$ \( ( -24 - 12 T + 3 T^{2} + T^{3} )^{2} \)
$23$ \( 64 + 288 T^{2} + 57 T^{4} + T^{6} \)
$29$ \( ( 24 + 33 T + 12 T^{2} + T^{3} )^{2} \)
$31$ \( ( -10 - 15 T + T^{3} )^{2} \)
$37$ \( 64 + 288 T^{2} + 57 T^{4} + T^{6} \)
$41$ \( ( -128 - 48 T + 9 T^{2} + T^{3} )^{2} \)
$43$ \( ( 4 + T^{2} )^{3} \)
$47$ \( 163216 + 19848 T^{2} + 273 T^{4} + T^{6} \)
$53$ \( 660969 + 23283 T^{2} + 267 T^{4} + T^{6} \)
$59$ \( ( 436 - 27 T - 12 T^{2} + T^{3} )^{2} \)
$61$ \( ( 712 - 108 T - 6 T^{2} + T^{3} )^{2} \)
$67$ \( 153664 + 16368 T^{2} + 252 T^{4} + T^{6} \)
$71$ \( ( -6 + T )^{6} \)
$73$ \( ( 16 + T^{2} )^{3} \)
$79$ \( ( 402 + 177 T + 24 T^{2} + T^{3} )^{2} \)
$83$ \( 7396 + 2793 T^{2} + 198 T^{4} + T^{6} \)
$89$ \( ( 392 - 108 T - 6 T^{2} + T^{3} )^{2} \)
$97$ \( 12544 + 8313 T^{2} + 342 T^{4} + T^{6} \)
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