Properties

Label 1560.2.l.d
Level $1560$
Weight $2$
Character orbit 1560.l
Analytic conductor $12.457$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1560.l (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.4566627153\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.57815240704.2
Defining polynomial: \(x^{8} - 2 x^{7} + 2 x^{6} + 89 x^{4} - 170 x^{3} + 162 x^{2} - 72 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} -\beta_{7} q^{5} + ( -\beta_{1} + 2 \beta_{2} ) q^{7} - q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} -\beta_{7} q^{5} + ( -\beta_{1} + 2 \beta_{2} ) q^{7} - q^{9} + ( \beta_{3} + \beta_{4} + \beta_{5} ) q^{11} + \beta_{2} q^{13} -\beta_{5} q^{15} + ( -\beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{17} + ( 2 + \beta_{3} ) q^{21} + ( -\beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{23} + ( -\beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{6} ) q^{25} + \beta_{2} q^{27} + ( -2 + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{29} + ( \beta_{1} + \beta_{6} - \beta_{7} ) q^{33} + ( -2 + \beta_{1} + 2 \beta_{4} + 2 \beta_{5} ) q^{35} + ( \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{37} + q^{39} + ( 2 - \beta_{3} - \beta_{6} - \beta_{7} ) q^{41} + ( -2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{43} + \beta_{7} q^{45} + ( 2 \beta_{2} - \beta_{6} + \beta_{7} ) q^{47} + ( -3 - 3 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{49} + ( \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{51} + ( \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{53} + ( 1 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{55} + ( -2 + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{59} + ( 2 + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{61} + ( \beta_{1} - 2 \beta_{2} ) q^{63} + \beta_{5} q^{65} + ( 4 \beta_{2} + 2 \beta_{6} - 2 \beta_{7} ) q^{67} + ( -2 + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{69} + ( 4 + 3 \beta_{3} + \beta_{6} + \beta_{7} ) q^{71} + ( -2 \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{73} + ( -1 - 2 \beta_{1} + \beta_{4} + \beta_{7} ) q^{75} + ( -\beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{5} - 3 \beta_{6} + 3 \beta_{7} ) q^{77} + ( -\beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{79} + q^{81} + ( 4 \beta_{1} - 6 \beta_{2} + \beta_{6} - \beta_{7} ) q^{83} + ( 2 - \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{85} + ( 2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{87} + ( -2 + 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{89} + ( -2 - \beta_{3} ) q^{91} + ( 5 \beta_{1} + 2 \beta_{6} - 2 \beta_{7} ) q^{97} + ( -\beta_{3} - \beta_{4} - \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 2q^{5} - 8q^{9} + O(q^{10}) \) \( 8q - 2q^{5} - 8q^{9} + 2q^{11} - 2q^{15} + 14q^{21} - 16q^{29} - 8q^{35} + 8q^{39} + 14q^{41} + 2q^{45} - 18q^{49} + 6q^{51} + 10q^{55} - 4q^{59} + 22q^{61} + 2q^{65} - 10q^{69} + 30q^{71} - 4q^{75} + 2q^{79} + 8q^{81} + 24q^{85} - 18q^{89} - 14q^{91} - 2q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} + 2 x^{6} + 89 x^{4} - 170 x^{3} + 162 x^{2} - 72 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 72 \nu^{7} + 747 \nu^{6} + 76 \nu^{5} + 32 \nu^{4} + 6804 \nu^{3} + 63783 \nu^{2} + 9448 \nu - 2736 \)\()/18170\)
\(\beta_{2}\)\(=\)\((\)\( -1881 \nu^{7} + 2970 \nu^{6} - 2894 \nu^{5} - 836 \nu^{4} - 167761 \nu^{3} + 244926 \nu^{2} - 234110 \nu + 67844 \)\()/36340\)
\(\beta_{3}\)\(=\)\((\)\( -1611 \nu^{7} + 1683 \nu^{6} - 792 \nu^{5} - 716 \nu^{4} - 144063 \nu^{3} + 136611 \nu^{2} - 60588 \nu + 28512 \)\()/18170\)
\(\beta_{4}\)\(=\)\((\)\( 358 \nu^{7} - 374 \nu^{6} + 176 \nu^{5} + 361 \nu^{4} + 32014 \nu^{3} - 30358 \nu^{2} + 11647 \nu + 2749 \)\()/1817\)
\(\beta_{5}\)\(=\)\((\)\( 1988 \nu^{7} - 2087 \nu^{6} + 1089 \nu^{5} + 1893 \nu^{4} + 178781 \nu^{3} - 169443 \nu^{2} + 84217 \nu + 6221 \)\()/9085\)
\(\beta_{6}\)\(=\)\((\)\( 1665 \nu^{7} - 3394 \nu^{6} + 2666 \nu^{5} + 740 \nu^{4} + 147349 \nu^{3} - 289098 \nu^{2} + 213034 \nu - 59636 \)\()/7268\)
\(\beta_{7}\)\(=\)\((\)\( -9117 \nu^{7} + 17838 \nu^{6} - 14166 \nu^{5} - 4052 \nu^{4} - 811589 \nu^{3} + 1516102 \nu^{2} - 1096418 \nu + 328276 \)\()/36340\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - 10 \beta_{2} - 4 \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-9 \beta_{7} - 9 \beta_{6} + 9 \beta_{5} - 9 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} - 2 \beta_{1} + 4\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(9 \beta_{7} + 9 \beta_{6} + 9 \beta_{5} + 9 \beta_{4} + 40 \beta_{3} - 90\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-85 \beta_{7} - 85 \beta_{6} - 85 \beta_{5} + 85 \beta_{4} + 22 \beta_{3} - 36 \beta_{2} + 22 \beta_{1} + 36\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-85 \beta_{7} + 85 \beta_{6} + 77 \beta_{5} - 77 \beta_{4} + 850 \beta_{2} + 384 \beta_{1}\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(801 \beta_{7} + 809 \beta_{6} - 809 \beta_{5} + 801 \beta_{4} - 230 \beta_{3} - 300 \beta_{2} + 230 \beta_{1} - 300\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1560\mathbb{Z}\right)^\times\).

\(n\) \(391\) \(521\) \(781\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
2.20793 2.20793i
0.594137 0.594137i
0.353624 0.353624i
−2.15569 + 2.15569i
2.20793 + 2.20793i
0.594137 + 0.594137i
0.353624 + 0.353624i
−2.15569 2.15569i
0 1.00000i 0 −2.20793 0.353624i 0 1.65573i 0 −1.00000 0
1249.2 0 1.00000i 0 −0.594137 + 2.15569i 0 4.92778i 0 −1.00000 0
1249.3 0 1.00000i 0 −0.353624 2.20793i 0 3.09417i 0 −1.00000 0
1249.4 0 1.00000i 0 2.15569 0.594137i 0 0.633776i 0 −1.00000 0
1249.5 0 1.00000i 0 −2.20793 + 0.353624i 0 1.65573i 0 −1.00000 0
1249.6 0 1.00000i 0 −0.594137 2.15569i 0 4.92778i 0 −1.00000 0
1249.7 0 1.00000i 0 −0.353624 + 2.20793i 0 3.09417i 0 −1.00000 0
1249.8 0 1.00000i 0 2.15569 + 0.594137i 0 0.633776i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1249.8
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 1560.2.l.d 8
3.b odd 2 1 4680.2.l.g 8
4.b odd 2 1 3120.2.l.n 8
5.b even 2 1 inner 1560.2.l.d 8
5.c odd 4 1 7800.2.a.bt 4
5.c odd 4 1 7800.2.a.by 4
15.d odd 2 1 4680.2.l.g 8
20.d odd 2 1 3120.2.l.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.l.d 8 1.a even 1 1 trivial
1560.2.l.d 8 5.b even 2 1 inner
3120.2.l.n 8 4.b odd 2 1
3120.2.l.n 8 20.d odd 2 1
4680.2.l.g 8 3.b odd 2 1
4680.2.l.g 8 15.d odd 2 1
7800.2.a.bt 4 5.c odd 4 1
7800.2.a.by 4 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 37 T_{7}^{6} + 340 T_{7}^{4} + 768 T_{7}^{2} + 256 \) acting on \(S_{2}^{\mathrm{new}}(1560, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 1 + T^{2} )^{4} \)
$5$ \( 625 + 250 T + 50 T^{2} - 30 T^{3} - 46 T^{4} - 6 T^{5} + 2 T^{6} + 2 T^{7} + T^{8} \)
$7$ \( 256 + 768 T^{2} + 340 T^{4} + 37 T^{6} + T^{8} \)
$11$ \( ( -4 - 26 T - 20 T^{2} - T^{3} + T^{4} )^{2} \)
$13$ \( ( 1 + T^{2} )^{4} \)
$17$ \( 64 + 1712 T^{2} + 908 T^{4} + 61 T^{6} + T^{8} \)
$19$ \( T^{8} \)
$23$ \( 1024 + 3904 T^{2} + 1056 T^{4} + 65 T^{6} + T^{8} \)
$29$ \( ( 256 - 144 T - 20 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$31$ \( T^{8} \)
$37$ \( 141376 + 60208 T^{2} + 5068 T^{4} + 133 T^{6} + T^{8} \)
$41$ \( ( -4 + 22 T - 8 T^{2} - 7 T^{3} + T^{4} )^{2} \)
$43$ \( 65536 + 30976 T^{2} + 3216 T^{4} + 104 T^{6} + T^{8} \)
$47$ \( 16384 + 7312 T^{2} + 1108 T^{4} + 64 T^{6} + T^{8} \)
$53$ \( 141376 + 60208 T^{2} + 5068 T^{4} + 133 T^{6} + T^{8} \)
$59$ \( ( 1072 - 36 T - 86 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$61$ \( ( 32 + 128 T - 22 T^{2} - 11 T^{3} + T^{4} )^{2} \)
$67$ \( 65536 + 37888 T^{2} + 5440 T^{4} + 192 T^{6} + T^{8} \)
$71$ \( ( 1256 + 530 T - 30 T^{2} - 15 T^{3} + T^{4} )^{2} \)
$73$ \( 1024 + 28992 T^{2} + 3568 T^{4} + 124 T^{6} + T^{8} \)
$79$ \( ( 7328 + 52 T - 176 T^{2} - T^{3} + T^{4} )^{2} \)
$83$ \( 7311616 + 2652624 T^{2} + 65620 T^{4} + 464 T^{6} + T^{8} \)
$89$ \( ( 268 - 378 T - 136 T^{2} + 9 T^{3} + T^{4} )^{2} \)
$97$ \( 41783296 + 3611280 T^{2} + 86296 T^{4} + 545 T^{6} + T^{8} \)
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