Properties

Label 1560.4.a.l
Level $1560$
Weight $4$
Character orbit 1560.a
Self dual yes
Analytic conductor $92.043$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1560.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(92.0429796090\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - x^{3} - 112x^{2} - 28x + 1648 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q - 3 q^{3} - 5 q^{5} + ( - \beta_{3} - 3) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} - 5 q^{5} + ( - \beta_{3} - 3) q^{7} + 9 q^{9} + ( - \beta_{2} - \beta_1 + 8) q^{11} - 13 q^{13} + 15 q^{15} + (\beta_{2} - 2) q^{17} + (2 \beta_{3} + 4 \beta_{2} + \beta_1 + 10) q^{19} + (3 \beta_{3} + 9) q^{21} + (4 \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 36) q^{23} + 25 q^{25} - 27 q^{27} + ( - \beta_{3} - 5 \beta_{2} + 5 \beta_1 + 43) q^{29} + ( - \beta_{3} + \beta_{2} + 6 \beta_1 + 73) q^{31} + (3 \beta_{2} + 3 \beta_1 - 24) q^{33} + (5 \beta_{3} + 15) q^{35} + (3 \beta_{3} + 2 \beta_{2} - 7 \beta_1 + 7) q^{37} + 39 q^{39} + ( - 3 \beta_{3} + 2 \beta_{2} - \beta_1 + 61) q^{41} + (8 \beta_{3} + 8 \beta_{2} - 2 \beta_1 + 36) q^{43} - 45 q^{45} + (11 \beta_{3} + 5 \beta_{2} - 14 \beta_1 - 3) q^{47} + (7 \beta_{3} - 6 \beta_{2} - 9 \beta_1 + 50) q^{49} + ( - 3 \beta_{2} + 6) q^{51} + (5 \beta_{3} + 6 \beta_{2} - 7 \beta_1 + 61) q^{53} + (5 \beta_{2} + 5 \beta_1 - 40) q^{55} + ( - 6 \beta_{3} - 12 \beta_{2} - 3 \beta_1 - 30) q^{57} + (3 \beta_{3} - 21 \beta_{2} + 149) q^{59} + ( - 21 \beta_{3} + 8 \beta_{2} + 5 \beta_1 + 179) q^{61} + ( - 9 \beta_{3} - 27) q^{63} + 65 q^{65} + ( - 29 \beta_{3} + 3 \beta_{2} + 22 \beta_1 + 169) q^{67} + ( - 12 \beta_{3} + 9 \beta_{2} - 6 \beta_1 + 108) q^{69} + (28 \beta_{3} + 31 \beta_{2} - 11 \beta_1 + 244) q^{71} + ( - 9 \beta_{3} - 25 \beta_{2} + 13 \beta_1 - 365) q^{73} - 75 q^{75} + ( - 39 \beta_{3} - 22 \beta_{2} + 9 \beta_1 + 39) q^{77} + (9 \beta_{3} - 24 \beta_{2} - 11 \beta_1 + 63) q^{79} + 81 q^{81} + ( - 21 \beta_{3} - 45 \beta_{2} + 30 \beta_1 - 31) q^{83} + ( - 5 \beta_{2} + 10) q^{85} + (3 \beta_{3} + 15 \beta_{2} - 15 \beta_1 - 129) q^{87} + (55 \beta_{3} + 18 \beta_{2} - 27 \beta_1 - 325) q^{89} + (13 \beta_{3} + 39) q^{91} + (3 \beta_{3} - 3 \beta_{2} - 18 \beta_1 - 219) q^{93} + ( - 10 \beta_{3} - 20 \beta_{2} - 5 \beta_1 - 50) q^{95} + ( - 18 \beta_{3} - 25 \beta_{2} + 30 \beta_1 - 244) q^{97} + ( - 9 \beta_{2} - 9 \beta_1 + 72) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{3} - 20 q^{5} - 11 q^{7} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{3} - 20 q^{5} - 11 q^{7} + 36 q^{9} + 35 q^{11} - 52 q^{13} + 60 q^{15} - 9 q^{17} + 32 q^{19} + 33 q^{21} - 149 q^{23} + 100 q^{25} - 108 q^{27} + 168 q^{29} + 280 q^{31} - 105 q^{33} + 55 q^{35} + 37 q^{37} + 156 q^{39} + 247 q^{41} + 132 q^{43} - 180 q^{45} + 217 q^{49} + 27 q^{51} + 247 q^{53} - 175 q^{55} - 96 q^{57} + 614 q^{59} + 719 q^{61} - 99 q^{63} + 260 q^{65} + 658 q^{67} + 447 q^{69} + 939 q^{71} - 1452 q^{73} - 300 q^{75} + 199 q^{77} + 289 q^{79} + 324 q^{81} - 118 q^{83} + 45 q^{85} - 504 q^{87} - 1319 q^{89} + 143 q^{91} - 840 q^{93} - 160 q^{95} - 993 q^{97} + 315 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 112x^{2} - 28x + 1648 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + \nu^{2} - 62\nu - 152 ) / 12 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 5\nu^{2} - 92\nu + 196 ) / 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{2} - 3\nu - 56 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{3} - \beta_{2} + \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{3} - 3\beta_{2} + 3\beta _1 + 227 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -67\beta_{3} - 59\beta_{2} + 107\beta _1 + 443 ) / 4 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−8.89553
10.4982
−4.51296
3.91028
0 −3.00000 0 −5.00000 0 −27.9085 0 9.00000 0
1.2 0 −3.00000 0 −5.00000 0 −14.3589 0 9.00000 0
1.3 0 −3.00000 0 −5.00000 0 8.04714 0 9.00000 0
1.4 0 −3.00000 0 −5.00000 0 23.2203 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)
\(13\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1560.4.a.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.4.a.l 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 11T_{7}^{3} - 734T_{7}^{2} - 4632T_{7} + 74880 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1560))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T + 3)^{4} \) Copy content Toggle raw display
$5$ \( (T + 5)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 11 T^{3} - 734 T^{2} + \cdots + 74880 \) Copy content Toggle raw display
$11$ \( T^{4} - 35 T^{3} - 1834 T^{2} + \cdots + 901888 \) Copy content Toggle raw display
$13$ \( (T + 13)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 9 T^{3} - 806 T^{2} + \cdots - 79544 \) Copy content Toggle raw display
$19$ \( T^{4} - 32 T^{3} - 16732 T^{2} + \cdots - 3617024 \) Copy content Toggle raw display
$23$ \( T^{4} + 149 T^{3} + \cdots + 158058368 \) Copy content Toggle raw display
$29$ \( T^{4} - 168 T^{3} + \cdots + 12907088 \) Copy content Toggle raw display
$31$ \( T^{4} - 280 T^{3} + \cdots + 222656512 \) Copy content Toggle raw display
$37$ \( T^{4} - 37 T^{3} - 55786 T^{2} + \cdots - 45918024 \) Copy content Toggle raw display
$41$ \( T^{4} - 247 T^{3} + 7778 T^{2} + \cdots + 7745736 \) Copy content Toggle raw display
$43$ \( T^{4} - 132 T^{3} + \cdots + 673244672 \) Copy content Toggle raw display
$47$ \( T^{4} - 243412 T^{2} + \cdots + 23627264 \) Copy content Toggle raw display
$53$ \( T^{4} - 247 T^{3} + \cdots - 304394568 \) Copy content Toggle raw display
$59$ \( T^{4} - 614 T^{3} + \cdots - 41560807424 \) Copy content Toggle raw display
$61$ \( T^{4} - 719 T^{3} + \cdots - 4148421320 \) Copy content Toggle raw display
$67$ \( T^{4} - 658 T^{3} + \cdots - 133654471808 \) Copy content Toggle raw display
$71$ \( T^{4} - 939 T^{3} + \cdots + 169844545536 \) Copy content Toggle raw display
$73$ \( T^{4} + 1452 T^{3} + \cdots - 26344765584 \) Copy content Toggle raw display
$79$ \( T^{4} - 289 T^{3} + \cdots - 19819318272 \) Copy content Toggle raw display
$83$ \( T^{4} + 118 T^{3} + \cdots + 584811051136 \) Copy content Toggle raw display
$89$ \( T^{4} + 1319 T^{3} + \cdots - 805979945160 \) Copy content Toggle raw display
$97$ \( T^{4} + 993 T^{3} + \cdots - 171137263400 \) Copy content Toggle raw display
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