Properties

Label 1560.4.a.t
Level $1560$
Weight $4$
Character orbit 1560.a
Self dual yes
Analytic conductor $92.043$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1560,4,Mod(1,1560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1560, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1560.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1560.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(92.0429796090\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 1422x^{4} + 15963x^{3} + 172073x^{2} - 1167798x - 2107216 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} - 5 q^{5} + (\beta_1 - 1) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} - 5 q^{5} + (\beta_1 - 1) q^{7} + 9 q^{9} + (\beta_{4} - 1) q^{11} - 13 q^{13} - 15 q^{15} + (\beta_{5} + \beta_{4} + \beta_1 + 15) q^{17} + (\beta_{3} + \beta_1 - 3) q^{19} + (3 \beta_1 - 3) q^{21} + ( - \beta_{2} + 2 \beta_1 + 2) q^{23} + 25 q^{25} + 27 q^{27} + ( - 2 \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 + 20) q^{29} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} + 3 \beta_1 - 18) q^{31} + (3 \beta_{4} - 3) q^{33} + ( - 5 \beta_1 + 5) q^{35} + ( - 2 \beta_{5} - 3 \beta_{4} + \beta_{2} - 5 \beta_1 - 52) q^{37} - 39 q^{39} + (\beta_{4} - 3 \beta_{2} + 3 \beta_1 + 46) q^{41} + ( - \beta_{5} - \beta_{4} + \beta_{2} + 9 \beta_1 + 125) q^{43} - 45 q^{45} + ( - 3 \beta_{5} - \beta_{3} - \beta_{2} + \beta_1 + 60) q^{47} + (\beta_{5} - 4 \beta_{4} - 2 \beta_{2} + 148) q^{49} + (3 \beta_{5} + 3 \beta_{4} + 3 \beta_1 + 45) q^{51} + (\beta_{5} - 2 \beta_{3} + 6 \beta_{2} - 4 \beta_1 + 65) q^{53} + ( - 5 \beta_{4} + 5) q^{55} + (3 \beta_{3} + 3 \beta_1 - 9) q^{57} + (2 \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 139) q^{59} + ( - 2 \beta_{5} - 10 \beta_{4} - \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 30) q^{61} + (9 \beta_1 - 9) q^{63} + 65 q^{65} + ( - 5 \beta_{4} + \beta_{3} - 2 \beta_{2} + 8 \beta_1 + 247) q^{67} + ( - 3 \beta_{2} + 6 \beta_1 + 6) q^{69} + (9 \beta_{4} - 2 \beta_{3} + 2 \beta_1 + 65) q^{71} + (5 \beta_{5} + 5 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} - 7 \beta_1 + 109) q^{73} + 75 q^{75} + ( - 2 \beta_{5} - 2 \beta_{4} + 7 \beta_{3} - 4 \beta_{2} - 20 \beta_1 - 102) q^{77} + (7 \beta_{5} - 3 \beta_{4} + \beta_{3} - \beta_{2} + 19 \beta_1 + 303) q^{79} + 81 q^{81} + (9 \beta_{4} + 3 \beta_{3} - 6 \beta_1 + 179) q^{83} + ( - 5 \beta_{5} - 5 \beta_{4} - 5 \beta_1 - 75) q^{85} + ( - 6 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} - 6 \beta_1 + 60) q^{87} + (\beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_{2} - 15 \beta_1 + 1) q^{89} + ( - 13 \beta_1 + 13) q^{91} + ( - 3 \beta_{5} + 6 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} + 9 \beta_1 - 54) q^{93} + ( - 5 \beta_{3} - 5 \beta_1 + 15) q^{95} + ( - 5 \beta_{5} + 10 \beta_{4} - 5 \beta_{3} + 5 \beta_{2} - 32 \beta_1 + 251) q^{97} + (9 \beta_{4} - 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 18 q^{3} - 30 q^{5} - 6 q^{7} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 18 q^{3} - 30 q^{5} - 6 q^{7} + 54 q^{9} - 8 q^{11} - 78 q^{13} - 90 q^{15} + 88 q^{17} - 20 q^{19} - 18 q^{21} + 12 q^{23} + 150 q^{25} + 162 q^{27} + 126 q^{29} - 110 q^{31} - 24 q^{33} + 30 q^{35} - 306 q^{37} - 234 q^{39} + 274 q^{41} + 752 q^{43} - 270 q^{45} + 362 q^{47} + 896 q^{49} + 264 q^{51} + 394 q^{53} + 40 q^{55} - 60 q^{57} + 834 q^{59} + 202 q^{61} - 54 q^{63} + 390 q^{65} + 1490 q^{67} + 36 q^{69} + 376 q^{71} + 638 q^{73} + 450 q^{75} - 622 q^{77} + 1822 q^{79} + 486 q^{81} + 1050 q^{83} - 440 q^{85} + 378 q^{87} + 10 q^{89} + 78 q^{91} - 330 q^{93} + 100 q^{95} + 1496 q^{97} - 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 1422x^{4} + 15963x^{3} + 172073x^{2} - 1167798x - 2107216 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 866\nu^{5} + 11672\nu^{4} - 947750\nu^{3} + 2163569\nu^{2} + 71893199\nu - 129452261 ) / 7926255 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3017\nu^{5} - 115919\nu^{4} + 1977710\nu^{3} + 69649402\nu^{2} - 108567308\nu - 4436776588 ) / 23778765 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3916\nu^{5} - 114205\nu^{4} + 2647942\nu^{3} + 40471409\nu^{2} + 326129426\nu - 1506832166 ) / 23778765 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4321\nu^{5} + 94036\nu^{4} - 4141906\nu^{3} - 23242790\nu^{2} + 372648259\nu + 76365605 ) / 23778765 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4186\nu^{5} - 100759\nu^{4} + 3643918\nu^{3} + 28985663\nu^{2} - 203132404\nu - 545261537 ) / 7926255 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{5} - 2\beta_{4} + \beta_{3} + 1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 11\beta_{5} + 74\beta_{4} - 15\beta_{3} + 28\beta_{2} - 60\beta _1 + 3813 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -1213\beta_{5} - 3616\beta_{4} + 963\beta_{3} - 362\beta_{2} + 1182\beta _1 - 59047 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 30739\beta_{5} + 135750\beta_{4} - 30443\beta_{3} + 32560\beta_{2} - 85272\beta _1 + 4432053 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -1686273\beta_{5} - 5805838\beta_{4} + 1418677\beta_{3} - 904972\beta_{2} + 2666004\beta _1 - 132769839 ) / 8 \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.51887
22.6779
−40.6602
23.1984
6.81694
−9.51414
0 3.00000 0 −5.00000 0 −30.0530 0 9.00000 0
1.2 0 3.00000 0 −5.00000 0 −20.9894 0 9.00000 0
1.3 0 3.00000 0 −5.00000 0 −14.4091 0 9.00000 0
1.4 0 3.00000 0 −5.00000 0 7.75305 0 9.00000 0
1.5 0 3.00000 0 −5.00000 0 24.0939 0 9.00000 0
1.6 0 3.00000 0 −5.00000 0 27.6046 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.t 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.4.a.t 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} + 6T_{7}^{5} - 1459T_{7}^{4} - 7528T_{7}^{3} + 578468T_{7}^{2} + 2643120T_{7} - 46869120 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1560))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T - 3)^{6} \) Copy content Toggle raw display
$5$ \( (T + 5)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 6 T^{5} - 1459 T^{4} + \cdots - 46869120 \) Copy content Toggle raw display
$11$ \( T^{6} + 8 T^{5} + \cdots - 2813910016 \) Copy content Toggle raw display
$13$ \( (T + 13)^{6} \) Copy content Toggle raw display
$17$ \( T^{6} - 88 T^{5} + \cdots + 45378853184 \) Copy content Toggle raw display
$19$ \( T^{6} + 20 T^{5} + \cdots - 3101487910400 \) Copy content Toggle raw display
$23$ \( T^{6} - 12 T^{5} + \cdots - 187866479360 \) Copy content Toggle raw display
$29$ \( T^{6} - 126 T^{5} + \cdots + 779991110528 \) Copy content Toggle raw display
$31$ \( T^{6} + 110 T^{5} + \cdots - 22819448422400 \) Copy content Toggle raw display
$37$ \( T^{6} + 306 T^{5} + \cdots - 11230108888080 \) Copy content Toggle raw display
$41$ \( T^{6} - 274 T^{5} + \cdots - 40785543888 \) Copy content Toggle raw display
$43$ \( T^{6} - 752 T^{5} + \cdots + 63173993136128 \) Copy content Toggle raw display
$47$ \( T^{6} - 362 T^{5} + \cdots - 31354450927616 \) Copy content Toggle raw display
$53$ \( T^{6} - 394 T^{5} + \cdots + 55\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{6} - 834 T^{5} + \cdots + 98545717477376 \) Copy content Toggle raw display
$61$ \( T^{6} - 202 T^{5} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{6} - 1490 T^{5} + \cdots - 10\!\cdots\!40 \) Copy content Toggle raw display
$71$ \( T^{6} - 376 T^{5} + \cdots - 26845782835200 \) Copy content Toggle raw display
$73$ \( T^{6} - 638 T^{5} + \cdots + 18\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{6} - 1822 T^{5} + \cdots + 19\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{6} - 1050 T^{5} + \cdots + 85\!\cdots\!80 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 257876154228240 \) Copy content Toggle raw display
$97$ \( T^{6} - 1496 T^{5} + \cdots + 55\!\cdots\!20 \) Copy content Toggle raw display
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