Properties

Label 1560.4.a.n
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: \( x^{4} - x^{3} - 41x^{2} - 30x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{6} \)
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.

\(f(q)\) \(=\) \( q + 3 q^{3} + 5 q^{5} + ( - \beta_{3} - 8) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} + 5 q^{5} + ( - \beta_{3} - 8) q^{7} + 9 q^{9} + (\beta_{2} - 10) q^{11} + 13 q^{13} + 15 q^{15} + (3 \beta_{3} - 2 \beta_1 - 14) q^{17} + (3 \beta_{3} - \beta_{2} - 46) q^{19} + ( - 3 \beta_{3} - 24) q^{21} + (2 \beta_{3} - 3 \beta_{2} + 5 \beta_1 - 42) q^{23} + 25 q^{25} + 27 q^{27} + ( - 6 \beta_{2} - \beta_1 - 18) q^{29} + (\beta_{3} - \beta_{2} + 3 \beta_1 - 102) q^{31} + (3 \beta_{2} - 30) q^{33} + ( - 5 \beta_{3} - 40) q^{35} + (7 \beta_{3} - 6 \beta_{2} + 5 \beta_1 - 10) q^{37} + 39 q^{39} + (9 \beta_{3} + 6 \beta_{2} - \beta_1 - 30) q^{41} + (9 \beta_{3} + 5 \beta_{2} - 13 \beta_1 - 190) q^{43} + 45 q^{45} + ( - 5 \beta_{3} + 5 \beta_{2} + \beta_1 - 130) q^{47} + (24 \beta_{3} + 5 \beta_{2} - 8 \beta_1 + 43) q^{49} + (9 \beta_{3} - 6 \beta_1 - 42) q^{51} + ( - 10 \beta_{3} + \beta_{2} - 10 \beta_1 - 80) q^{53} + (5 \beta_{2} - 50) q^{55} + (9 \beta_{3} - 3 \beta_{2} - 138) q^{57} + ( - 12 \beta_{3} - 52) q^{59} + ( - 14 \beta_{3} + \beta_{2} + 14 \beta_1 + 88) q^{61} + ( - 9 \beta_{3} - 72) q^{63} + 65 q^{65} + ( - 4 \beta_{3} + 4 \beta_{2} + 8 \beta_1 - 260) q^{67} + (6 \beta_{3} - 9 \beta_{2} + 15 \beta_1 - 126) q^{69} + ( - 20 \beta_{3} + \beta_{2} + 16 \beta_1 - 166) q^{71} + ( - 7 \beta_{3} + 15 \beta_{2} + 2 \beta_1 - 96) q^{73} + 75 q^{75} + ( - 2 \beta_{3} - 11 \beta_{2} + 18 \beta_1 + 10) q^{77} + ( - 31 \beta_{3} + 4 \beta_{2} - 21 \beta_1 - 156) q^{79} + 81 q^{81} + ( - 26 \beta_{3} + 10 \beta_{2} - 26 \beta_1 - 464) q^{83} + (15 \beta_{3} - 10 \beta_1 - 70) q^{85} + ( - 18 \beta_{2} - 3 \beta_1 - 54) q^{87} + (15 \beta_{3} - 30 \beta_{2} + 13 \beta_1 - 510) q^{89} + ( - 13 \beta_{3} - 104) q^{91} + (3 \beta_{3} - 3 \beta_{2} + 9 \beta_1 - 306) q^{93} + (15 \beta_{3} - 5 \beta_{2} - 230) q^{95} + ( - 24 \beta_{3} + 19 \beta_{2} + \beta_1 + 284) q^{97} + (9 \beta_{2} - 90) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{3} + 20 q^{5} - 34 q^{7} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{3} + 20 q^{5} - 34 q^{7} + 36 q^{9} - 38 q^{11} + 52 q^{13} + 60 q^{15} - 50 q^{17} - 180 q^{19} - 102 q^{21} - 170 q^{23} + 100 q^{25} + 108 q^{27} - 84 q^{29} - 408 q^{31} - 114 q^{33} - 170 q^{35} - 38 q^{37} + 156 q^{39} - 90 q^{41} - 732 q^{43} + 180 q^{45} - 520 q^{47} + 230 q^{49} - 150 q^{51} - 338 q^{53} - 190 q^{55} - 540 q^{57} - 232 q^{59} + 326 q^{61} - 306 q^{63} + 260 q^{65} - 1040 q^{67} - 510 q^{69} - 702 q^{71} - 368 q^{73} + 300 q^{75} + 14 q^{77} - 678 q^{79} + 324 q^{81} - 1888 q^{83} - 250 q^{85} - 252 q^{87} - 2070 q^{89} - 442 q^{91} - 1224 q^{93} - 900 q^{95} + 1126 q^{97} - 342 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 41x^{2} - 30x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -2\nu^{3} + 4\nu^{2} + 80\nu + 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -3\nu^{3} + 4\nu^{2} + 128\nu + 46 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -3\nu^{3} + 4\nu^{2} + 120\nu + 48 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} + 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{3} + 3\beta _1 + 84 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -6\beta_{3} + 5\beta_{2} + \beta _1 + 54 \) 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
−0.174957
−5.49345
7.24301
−0.574597
0 3.00000 0 5.00000 0 −35.1436 0 9.00000 0
1.2 0 3.00000 0 5.00000 0 −14.8428 0 9.00000 0
1.3 0 3.00000 0 5.00000 0 4.92462 0 9.00000 0
1.4 0 3.00000 0 5.00000 0 11.0618 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.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.4.a.n 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 34T_{7}^{3} - 223T_{7}^{2} - 5616T_{7} + 28416 \) 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
$7$ \( T^{4} + 34 T^{3} - 223 T^{2} + \cdots + 28416 \) Copy content Toggle raw display
$11$ \( T^{4} + 38 T^{3} - 1759 T^{2} + \cdots + 794640 \) Copy content Toggle raw display
$13$ \( (T - 13)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 50 T^{3} - 10091 T^{2} + \cdots + 2634756 \) Copy content Toggle raw display
$19$ \( T^{4} + 180 T^{3} + 4868 T^{2} + \cdots + 1578560 \) Copy content Toggle raw display
$23$ \( T^{4} + 170 T^{3} + \cdots - 115845696 \) Copy content Toggle raw display
$29$ \( T^{4} + 84 T^{3} + \cdots + 1641072688 \) Copy content Toggle raw display
$31$ \( T^{4} + 408 T^{3} + 49948 T^{2} + \cdots + 9081600 \) Copy content Toggle raw display
$37$ \( T^{4} + 38 T^{3} + \cdots - 153758476 \) Copy content Toggle raw display
$41$ \( T^{4} + 90 T^{3} + \cdots + 1993594900 \) Copy content Toggle raw display
$43$ \( T^{4} + 732 T^{3} + \cdots + 2881893632 \) Copy content Toggle raw display
$47$ \( T^{4} + 520 T^{3} + \cdots - 2001440000 \) Copy content Toggle raw display
$53$ \( T^{4} + 338 T^{3} + \cdots - 9572486780 \) Copy content Toggle raw display
$59$ \( T^{4} + 232 T^{3} + \cdots + 923054080 \) Copy content Toggle raw display
$61$ \( T^{4} - 326 T^{3} + \cdots + 13972040964 \) Copy content Toggle raw display
$67$ \( T^{4} + 1040 T^{3} + \cdots - 15293380352 \) Copy content Toggle raw display
$71$ \( T^{4} + 702 T^{3} + \cdots - 18758062080 \) Copy content Toggle raw display
$73$ \( T^{4} + 368 T^{3} + \cdots + 10624585104 \) Copy content Toggle raw display
$79$ \( T^{4} + 678 T^{3} + \cdots - 203520914688 \) Copy content Toggle raw display
$83$ \( T^{4} + 1888 T^{3} + \cdots - 460820907008 \) Copy content Toggle raw display
$89$ \( T^{4} + 2070 T^{3} + \cdots - 796693724300 \) Copy content Toggle raw display
$97$ \( T^{4} - 1126 T^{3} + \cdots - 13823634476 \) Copy content Toggle raw display
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