Properties

Label 1560.4.a.m
Level $1560$
Weight $4$
Character orbit 1560.a
Self dual yes
Analytic conductor $92.043$
Analytic rank $0$
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: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - x^{3} - 76x^{2} + 50x + 1156 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + 3 q^{3} - 5 q^{5} + (\beta_1 + 4) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} - 5 q^{5} + (\beta_1 + 4) q^{7} + 9 q^{9} + ( - \beta_{3} + \beta_1 - 2) q^{11} + 13 q^{13} - 15 q^{15} + ( - \beta_{2} + 2 \beta_1 - 18) q^{17} + (\beta_{3} + \beta_{2} + \beta_1 + 18) q^{19} + (3 \beta_1 + 12) q^{21} + ( - \beta_{2} + 2 \beta_1 - 12) q^{23} + 25 q^{25} + 27 q^{27} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 16) q^{29} + ( - 3 \beta_{2} - 3 \beta_1 + 56) q^{31} + ( - 3 \beta_{3} + 3 \beta_1 - 6) q^{33} + ( - 5 \beta_1 - 20) q^{35} + ( - \beta_{3} + 3 \beta_{2} + 76) q^{37} + 39 q^{39} + ( - 5 \beta_{3} + \beta_{2} - 2 \beta_1 + 36) q^{41} + (4 \beta_{2} - 4 \beta_1 + 84) q^{43} - 45 q^{45} + (3 \beta_{2} - 5 \beta_1 + 44) q^{47} + (5 \beta_{3} + 3 \beta_{2} + 6 \beta_1 + 99) q^{49} + ( - 3 \beta_{2} + 6 \beta_1 - 54) q^{51} + (\beta_{3} - 7 \beta_{2} + 8 \beta_1 + 88) q^{53} + (5 \beta_{3} - 5 \beta_1 + 10) q^{55} + (3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 54) q^{57} + (8 \beta_{3} - 13 \beta_{2} + 5 \beta_1 - 88) q^{59} + (\beta_{3} - 13 \beta_{2} + 12 \beta_1 + 56) q^{61} + (9 \beta_1 + 36) q^{63} - 65 q^{65} + (6 \beta_{3} - 7 \beta_{2} - 17 \beta_1 + 144) q^{67} + ( - 3 \beta_{2} + 6 \beta_1 - 36) q^{69} + (5 \beta_{3} + 14 \beta_{2} - 13 \beta_1 - 18) q^{71} + (5 \beta_{3} + 14 \beta_{2} + 22 \beta_1 + 200) q^{73} + 75 q^{75} + ( - 13 \beta_{3} + 21 \beta_{2} - 28 \beta_1 + 342) q^{77} + ( - 5 \beta_{3} + 23 \beta_{2} - 26 \beta_1 + 214) q^{79} + 81 q^{81} + ( - 6 \beta_{3} - \beta_{2} - 15 \beta_1 - 268) q^{83} + (5 \beta_{2} - 10 \beta_1 + 90) q^{85} + ( - 3 \beta_{3} + 6 \beta_{2} + 6 \beta_1 - 48) q^{87} + (11 \beta_{3} - 27 \beta_{2} - 4 \beta_1 + 252) q^{89} + (13 \beta_1 + 52) q^{91} + ( - 9 \beta_{2} - 9 \beta_1 + 168) q^{93} + ( - 5 \beta_{3} - 5 \beta_{2} - 5 \beta_1 - 90) q^{95} + (16 \beta_{3} - 29 \beta_{2} - 2 \beta_1 + 342) q^{97} + ( - 9 \beta_{3} + 9 \beta_1 - 18) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{3} - 20 q^{5} + 15 q^{7} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{3} - 20 q^{5} + 15 q^{7} + 36 q^{9} - 7 q^{11} + 52 q^{13} - 60 q^{15} - 73 q^{17} + 68 q^{19} + 45 q^{21} - 49 q^{23} + 100 q^{25} + 108 q^{27} - 66 q^{29} + 230 q^{31} - 21 q^{33} - 75 q^{35} + 303 q^{37} + 156 q^{39} + 155 q^{41} + 336 q^{43} - 180 q^{45} + 178 q^{47} + 377 q^{49} - 219 q^{51} + 349 q^{53} + 35 q^{55} + 204 q^{57} - 360 q^{59} + 223 q^{61} + 135 q^{63} - 260 q^{65} + 588 q^{67} - 147 q^{69} - 83 q^{71} + 754 q^{73} + 300 q^{75} + 1401 q^{77} + 869 q^{79} + 324 q^{81} - 1044 q^{83} + 365 q^{85} - 198 q^{87} + 1017 q^{89} + 195 q^{91} + 690 q^{93} - 340 q^{95} + 1367 q^{97} - 63 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 76x^{2} + 50x + 1156 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 11\nu^{2} + 122\nu - 438 ) / 26 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{3} - 33\nu^{2} - 54\nu + 1210 ) / 26 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6\nu^{3} + 12\nu^{2} - 264\nu - 518 ) / 13 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 3\beta _1 + 4 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 3\beta_{2} + 3\beta _1 + 230 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 11\beta_{3} + 28\beta_{2} + 60\beta _1 + 146 ) / 6 \) 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
−4.17243
−7.35294
5.02115
7.50422
0 3.00000 0 −5.00000 0 −22.2653 0 9.00000 0
1.2 0 3.00000 0 −5.00000 0 −9.18441 0 9.00000 0
1.3 0 3.00000 0 −5.00000 0 16.5122 0 9.00000 0
1.4 0 3.00000 0 −5.00000 0 29.9374 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.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.4.a.m 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 15T_{7}^{3} - 762T_{7}^{2} + 6048T_{7} + 101088 \) 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
$7$ \( T^{4} - 15 T^{3} - 762 T^{2} + \cdots + 101088 \) Copy content Toggle raw display
$11$ \( T^{4} + 7 T^{3} - 4536 T^{2} + \cdots + 1081600 \) Copy content Toggle raw display
$13$ \( (T - 13)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 73 T^{3} - 2304 T^{2} + \cdots + 237376 \) Copy content Toggle raw display
$19$ \( T^{4} - 68 T^{3} - 7020 T^{2} + \cdots + 3677440 \) Copy content Toggle raw display
$23$ \( T^{4} + 49 T^{3} - 3402 T^{2} + \cdots + 230080 \) Copy content Toggle raw display
$29$ \( T^{4} + 66 T^{3} - 12228 T^{2} + \cdots - 1489536 \) Copy content Toggle raw display
$31$ \( T^{4} - 230 T^{3} + \cdots + 26955136 \) Copy content Toggle raw display
$37$ \( T^{4} - 303 T^{3} + \cdots - 62816472 \) Copy content Toggle raw display
$41$ \( T^{4} - 155 T^{3} + \cdots + 289265992 \) Copy content Toggle raw display
$43$ \( T^{4} - 336 T^{3} + \cdots - 30613248 \) Copy content Toggle raw display
$47$ \( T^{4} - 178 T^{3} + \cdots + 27520000 \) Copy content Toggle raw display
$53$ \( T^{4} - 349 T^{3} + \cdots + 77826040 \) Copy content Toggle raw display
$59$ \( T^{4} + 360 T^{3} + \cdots + 30371328000 \) Copy content Toggle raw display
$61$ \( T^{4} - 223 T^{3} + \cdots + 9467381272 \) Copy content Toggle raw display
$67$ \( T^{4} - 588 T^{3} + \cdots + 5005863936 \) Copy content Toggle raw display
$71$ \( T^{4} + 83 T^{3} + \cdots + 4244171776 \) Copy content Toggle raw display
$73$ \( T^{4} - 754 T^{3} + \cdots - 50664163520 \) Copy content Toggle raw display
$79$ \( T^{4} - 869 T^{3} + \cdots + 239698264576 \) Copy content Toggle raw display
$83$ \( T^{4} + 1044 T^{3} + \cdots - 909439488 \) Copy content Toggle raw display
$89$ \( T^{4} - 1017 T^{3} + \cdots + 6573954600 \) Copy content Toggle raw display
$97$ \( T^{4} - 1367 T^{3} + \cdots - 193647346784 \) Copy content Toggle raw display
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