Properties

Label 1470.2.g.j
Level $1470$
Weight $2$
Character orbit 1470.g
Analytic conductor $11.738$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: 8.0.1698758656.6
Defining polynomial: \(x^{8} + 18 x^{6} + 97 x^{4} + 176 x^{2} + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{5} q^{2} -\beta_{5} q^{3} - q^{4} + ( -1 + \beta_{7} ) q^{5} - q^{6} + \beta_{5} q^{8} - q^{9} +O(q^{10})\) \( q -\beta_{5} q^{2} -\beta_{5} q^{3} - q^{4} + ( -1 + \beta_{7} ) q^{5} - q^{6} + \beta_{5} q^{8} - q^{9} + \beta_{4} q^{10} + ( \beta_{1} + \beta_{3} - \beta_{4} ) q^{11} + \beta_{5} q^{12} + ( 2 \beta_{2} + \beta_{6} - \beta_{7} ) q^{13} + \beta_{4} q^{15} + q^{16} + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{17} + \beta_{5} q^{18} + ( 2 + \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} ) q^{19} + ( 1 - \beta_{7} ) q^{20} + ( -\beta_{2} - \beta_{6} + \beta_{7} ) q^{22} + ( \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{23} + q^{24} + ( 1 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{7} ) q^{25} + ( \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{26} + \beta_{5} q^{27} + ( 4 + \beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{29} + ( 1 - \beta_{7} ) q^{30} + ( -4 - \beta_{1} - \beta_{3} + \beta_{4} ) q^{31} -\beta_{5} q^{32} + ( -\beta_{2} - \beta_{6} + \beta_{7} ) q^{33} + ( 2 + \beta_{1} + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{34} + q^{36} + ( 3 \beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{37} + ( \beta_{1} - 2 \beta_{2} + \beta_{4} - 4 \beta_{5} - \beta_{6} + \beta_{7} ) q^{38} + ( \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{39} -\beta_{4} q^{40} + ( -2 - 2 \beta_{3} - \beta_{6} - \beta_{7} ) q^{41} + ( \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{43} + ( -\beta_{1} - \beta_{3} + \beta_{4} ) q^{44} + ( 1 - \beta_{7} ) q^{45} + ( 2 - \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} ) q^{46} + ( -\beta_{1} - \beta_{2} - \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{47} -\beta_{5} q^{48} + ( 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{6} ) q^{50} + ( 2 + \beta_{1} + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{51} + ( -2 \beta_{2} - \beta_{6} + \beta_{7} ) q^{52} + ( \beta_{1} - 2 \beta_{2} + \beta_{4} + 4 \beta_{5} - \beta_{6} + \beta_{7} ) q^{53} + q^{54} + ( -\beta_{1} - 2 \beta_{2} - 3 \beta_{3} + \beta_{4} - 4 \beta_{5} ) q^{55} + ( \beta_{1} - 2 \beta_{2} + \beta_{4} - 4 \beta_{5} - \beta_{6} + \beta_{7} ) q^{57} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{6} + \beta_{7} ) q^{58} + ( 6 + \beta_{1} - 4 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{59} -\beta_{4} q^{60} + ( -6 + 3 \beta_{6} + 3 \beta_{7} ) q^{61} + ( \beta_{2} + 4 \beta_{5} + \beta_{6} - \beta_{7} ) q^{62} - q^{64} + ( 4 + \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{65} + ( -\beta_{1} - \beta_{3} + \beta_{4} ) q^{66} + ( \beta_{1} - \beta_{2} + \beta_{4} ) q^{67} + ( -\beta_{1} - \beta_{2} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{68} + ( 2 - \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} ) q^{69} + ( -2 + \beta_{1} - 2 \beta_{3} - \beta_{4} - 3 \beta_{6} - 3 \beta_{7} ) q^{71} -\beta_{5} q^{72} + ( -\beta_{1} - 2 \beta_{2} - \beta_{4} + 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{73} + ( 2 - \beta_{1} + 3 \beta_{3} + \beta_{4} ) q^{74} + ( 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{6} ) q^{75} + ( -2 - \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} ) q^{76} + ( -2 \beta_{2} - \beta_{6} + \beta_{7} ) q^{78} + ( 4 + 2 \beta_{1} - 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{79} + ( -1 + \beta_{7} ) q^{80} + q^{81} + ( -\beta_{1} + 2 \beta_{2} - \beta_{4} + 4 \beta_{5} ) q^{82} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - 4 \beta_{5} ) q^{83} + ( 4 - \beta_{2} + 4 \beta_{3} - \beta_{4} - 5 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{85} + ( 2 - 2 \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{86} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{6} + \beta_{7} ) q^{87} + ( \beta_{2} + \beta_{6} - \beta_{7} ) q^{88} + ( 10 + 4 \beta_{3} + \beta_{6} + \beta_{7} ) q^{89} -\beta_{4} q^{90} + ( -\beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{92} + ( \beta_{2} + 4 \beta_{5} + \beta_{6} - \beta_{7} ) q^{93} + ( 2 - 2 \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{6} + \beta_{7} ) q^{94} + ( 2 - \beta_{1} - 6 \beta_{3} + \beta_{4} - 4 \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{95} - q^{96} + ( \beta_{1} - 4 \beta_{2} + \beta_{4} + 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{97} + ( -\beta_{1} - \beta_{3} + \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{4} - 4q^{5} - 8q^{6} - 8q^{9} + O(q^{10}) \) \( 8q - 8q^{4} - 4q^{5} - 8q^{6} - 8q^{9} + 8q^{16} + 24q^{19} + 4q^{20} + 8q^{24} + 4q^{25} + 16q^{29} + 4q^{30} - 32q^{31} + 8q^{34} + 8q^{36} - 24q^{41} + 4q^{45} + 8q^{46} - 4q^{50} + 8q^{51} + 8q^{54} + 40q^{59} - 24q^{61} - 8q^{64} + 28q^{65} + 8q^{69} - 40q^{71} + 16q^{74} - 4q^{75} - 24q^{76} + 16q^{79} - 4q^{80} + 8q^{81} + 28q^{85} + 8q^{86} + 88q^{89} + 24q^{94} + 24q^{95} - 8q^{96} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 18 x^{6} + 97 x^{4} + 176 x^{2} + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{7} - 2 \nu^{6} + 10 \nu^{5} - 20 \nu^{4} - 15 \nu^{3} - 2 \nu^{2} - 184 \nu + 80 \)\()/64\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} - 18 \nu^{5} - 89 \nu^{3} - 104 \nu \)\()/32\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} + 14 \nu^{4} + 37 \nu^{2} - 8 \)\()/16\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} + 2 \nu^{6} + 10 \nu^{5} + 20 \nu^{4} - 15 \nu^{3} + 2 \nu^{2} - 184 \nu - 80 \)\()/64\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{7} + 46 \nu^{5} + 179 \nu^{3} + 168 \nu \)\()/64\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{7} + 6 \nu^{6} - 10 \nu^{5} + 92 \nu^{4} + 15 \nu^{3} + 358 \nu^{2} + 120 \nu + 336 \)\()/64\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} + 6 \nu^{6} + 10 \nu^{5} + 92 \nu^{4} - 15 \nu^{3} + 358 \nu^{2} - 120 \nu + 336 \)\()/64\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} - \beta_{6} - \beta_{4} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} + \beta_{6} + \beta_{4} - 4 \beta_{3} - \beta_{1} - 10\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-9 \beta_{7} + 9 \beta_{6} + 8 \beta_{5} + 5 \beta_{4} + 8 \beta_{2} + 5 \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-9 \beta_{7} - 9 \beta_{6} - 17 \beta_{4} + 44 \beta_{3} + 17 \beta_{1} + 74\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(81 \beta_{7} - 81 \beta_{6} - 104 \beta_{5} - 37 \beta_{4} - 112 \beta_{2} - 37 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(89 \beta_{7} + 89 \beta_{6} + 201 \beta_{4} - 436 \beta_{3} - 201 \beta_{1} - 650\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-761 \beta_{7} + 761 \beta_{6} + 1160 \beta_{5} + 325 \beta_{4} + 1240 \beta_{2} + 325 \beta_{1}\)\()/2\)

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
1.69230i
3.16053i
2.16053i
0.692297i
1.69230i
3.16053i
2.16053i
0.692297i
1.00000i 1.00000i −1.00000 −2.18183 + 0.489528i −1.00000 0 1.00000i −1.00000 0.489528 + 2.18183i
589.2 1.00000i 1.00000i −1.00000 −1.63280 1.52773i −1.00000 0 1.00000i −1.00000 −1.52773 + 1.63280i
589.3 1.00000i 1.00000i −1.00000 −0.0743018 + 2.23483i −1.00000 0 1.00000i −1.00000 2.23483 + 0.0743018i
589.4 1.00000i 1.00000i −1.00000 1.88893 1.19663i −1.00000 0 1.00000i −1.00000 −1.19663 1.88893i
589.5 1.00000i 1.00000i −1.00000 −2.18183 0.489528i −1.00000 0 1.00000i −1.00000 0.489528 2.18183i
589.6 1.00000i 1.00000i −1.00000 −1.63280 + 1.52773i −1.00000 0 1.00000i −1.00000 −1.52773 1.63280i
589.7 1.00000i 1.00000i −1.00000 −0.0743018 2.23483i −1.00000 0 1.00000i −1.00000 2.23483 0.0743018i
589.8 1.00000i 1.00000i −1.00000 1.88893 + 1.19663i −1.00000 0 1.00000i −1.00000 −1.19663 + 1.88893i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 589.8
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.j 8
5.b even 2 1 inner 1470.2.g.j 8
5.c odd 4 1 7350.2.a.ds 4
5.c odd 4 1 7350.2.a.dt 4
7.b odd 2 1 1470.2.g.k yes 8
7.c even 3 2 1470.2.n.l 16
7.d odd 6 2 1470.2.n.k 16
35.c odd 2 1 1470.2.g.k yes 8
35.f even 4 1 7350.2.a.dr 4
35.f even 4 1 7350.2.a.du 4
35.i odd 6 2 1470.2.n.k 16
35.j even 6 2 1470.2.n.l 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.g.j 8 1.a even 1 1 trivial
1470.2.g.j 8 5.b even 2 1 inner
1470.2.g.k yes 8 7.b odd 2 1
1470.2.g.k yes 8 35.c odd 2 1
1470.2.n.k 16 7.d odd 6 2
1470.2.n.k 16 35.i odd 6 2
1470.2.n.l 16 7.c even 3 2
1470.2.n.l 16 35.j even 6 2
7350.2.a.dr 4 35.f even 4 1
7350.2.a.ds 4 5.c odd 4 1
7350.2.a.dt 4 5.c odd 4 1
7350.2.a.du 4 35.f even 4 1

Hecke kernels

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

\( T_{11}^{4} - 18 T_{11}^{2} - 16 T_{11} + 32 \)
\( T_{17}^{8} + 80 T_{17}^{6} + 1336 T_{17}^{4} + 8000 T_{17}^{2} + 15376 \)
\( T_{19}^{4} - 12 T_{19}^{3} + 12 T_{19}^{2} + 224 T_{19} - 448 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{4} \)
$3$ \( ( 1 + T^{2} )^{4} \)
$5$ \( 625 + 500 T + 150 T^{2} + 20 T^{3} + 2 T^{4} + 4 T^{5} + 6 T^{6} + 4 T^{7} + T^{8} \)
$7$ \( T^{8} \)
$11$ \( ( 32 - 16 T - 18 T^{2} + T^{4} )^{2} \)
$13$ \( 256 + 1472 T^{2} + 644 T^{4} + 52 T^{6} + T^{8} \)
$17$ \( 15376 + 8000 T^{2} + 1336 T^{4} + 80 T^{6} + T^{8} \)
$19$ \( ( -448 + 224 T + 12 T^{2} - 12 T^{3} + T^{4} )^{2} \)
$23$ \( 16384 + 9216 T^{2} + 1552 T^{4} + 88 T^{6} + T^{8} \)
$29$ \( ( 1568 + 336 T - 66 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$31$ \( ( 64 + 128 T + 78 T^{2} + 16 T^{3} + T^{4} )^{2} \)
$37$ \( 40000 + 31200 T^{2} + 4084 T^{4} + 148 T^{6} + T^{8} \)
$41$ \( ( -376 - 120 T + 26 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$43$ \( 541696 + 297344 T^{2} + 16516 T^{4} + 236 T^{6} + T^{8} \)
$47$ \( 1048576 + 200704 T^{2} + 11140 T^{4} + 204 T^{6} + T^{8} \)
$53$ \( 4096 + 11776 T^{2} + 5392 T^{4} + 184 T^{6} + T^{8} \)
$59$ \( ( -3008 + 736 T + 52 T^{2} - 20 T^{3} + T^{4} )^{2} \)
$61$ \( ( -648 - 1512 T - 126 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$67$ \( 1024 + 1152 T^{2} + 388 T^{4} + 44 T^{6} + T^{8} \)
$71$ \( ( 2944 - 1472 T - 36 T^{2} + 20 T^{3} + T^{4} )^{2} \)
$73$ \( 61504 + 244640 T^{2} + 19764 T^{4} + 316 T^{6} + T^{8} \)
$79$ \( ( -3584 + 1792 T - 176 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$83$ \( 16384 + 26624 T^{2} + 6976 T^{4} + 240 T^{6} + T^{8} \)
$89$ \( ( 5048 - 3688 T + 658 T^{2} - 44 T^{3} + T^{4} )^{2} \)
$97$ \( 107661376 + 4479328 T^{2} + 66804 T^{4} + 428 T^{6} + T^{8} \)
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