Properties

Label 1560.4.a.s
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: \( x^{6} - x^{5} - 834x^{4} + 965x^{3} + 200177x^{2} - 261462x - 11549822 \) 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

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.

\(f(q)\) \(=\) \( q - 3 q^{3} + 5 q^{5} + (\beta_{2} - 3) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + 5 q^{5} + (\beta_{2} - 3) q^{7} + 9 q^{9} + (\beta_{5} + \beta_{2}) q^{11} - 13 q^{13} - 15 q^{15} + (\beta_{3} + 6) q^{17} + (\beta_{5} + \beta_{4} + \beta_{2} - 11) q^{19} + ( - 3 \beta_{2} + 9) q^{21} + ( - \beta_{3} + 4 \beta_{2} - \beta_1 - 5) q^{23} + 25 q^{25} - 27 q^{27} + (\beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1 + 11) q^{29} + (2 \beta_{5} + \beta_{3} + 5 \beta_{2} + \beta_1 + 36) q^{31} + ( - 3 \beta_{5} - 3 \beta_{2}) q^{33} + (5 \beta_{2} - 15) q^{35} + (\beta_{5} - 2 \beta_{4} + \beta_{3} + 6 \beta_{2} + \beta_1 + 74) q^{37} + 39 q^{39} + ( - 3 \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1 + 56) q^{41} + ( - 2 \beta_{5} - 5 \beta_{4} - 3 \beta_{3} + 4 \beta_{2} - 3 \beta_1 - 10) q^{43} + 45 q^{45} + ( - 2 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + \beta_{2} + 5 \beta_1 - 114) q^{47} + ( - \beta_{5} - \beta_{4} - 2 \beta_{2} - 6 \beta_1 + 185) q^{49} + ( - 3 \beta_{3} - 18) q^{51} + ( - \beta_{5} + 3 \beta_{4} - 4 \beta_{3} - 6 \beta_{2} + 2 \beta_1 - 70) q^{53} + (5 \beta_{5} + 5 \beta_{2}) q^{55} + ( - 3 \beta_{5} - 3 \beta_{4} - 3 \beta_{2} + 33) q^{57} + (4 \beta_{5} + 4 \beta_{4} - \beta_{3} - 7 \beta_{2} + 2 \beta_1 - 121) q^{59} + (7 \beta_{5} + 4 \beta_{4} - \beta_{3} - 8 \beta_{2} + 8 \beta_1 + 111) q^{61} + (9 \beta_{2} - 27) q^{63} - 65 q^{65} + (2 \beta_{5} + 5 \beta_{4} + 4 \beta_{3} - 13 \beta_{2} - 2 \beta_1 - 52) q^{67} + (3 \beta_{3} - 12 \beta_{2} + 3 \beta_1 + 15) q^{69} + ( - 13 \beta_{5} - 5 \beta_{4} - \beta_{3} + 3 \beta_{2} - 8 \beta_1 + 47) q^{71} + ( - 9 \beta_{5} + 8 \beta_{4} + 4 \beta_{3} - 12 \beta_{2} - 4 \beta_1 + 151) q^{73} - 75 q^{75} + ( - 11 \beta_{5} - 8 \beta_{4} - 9 \beta_{3} + 18 \beta_{2} - 22 \beta_1 + 307) q^{77} + ( - 7 \beta_{5} - 11 \beta_{4} - 2 \beta_{3} - 12 \beta_{2} + 3 \beta_1 - 261) q^{79} + 81 q^{81} + ( - 2 \beta_{5} - 3 \beta_{4} + 12 \beta_{3} - \beta_{2} + 8 \beta_1 - 262) q^{83} + (5 \beta_{3} + 30) q^{85} + ( - 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} - 6 \beta_{2} - 3 \beta_1 - 33) q^{87} + (17 \beta_{5} + 5 \beta_{4} + 2 \beta_{2} + 5 \beta_1 + 249) q^{89} + ( - 13 \beta_{2} + 39) q^{91} + ( - 6 \beta_{5} - 3 \beta_{3} - 15 \beta_{2} - 3 \beta_1 - 108) q^{93} + (5 \beta_{5} + 5 \beta_{4} + 5 \beta_{2} - 55) q^{95} + (12 \beta_{5} + 4 \beta_{4} + \beta_{3} - 6 \beta_{2} + 23 \beta_1 + 139) q^{97} + (9 \beta_{5} + 9 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 18 q^{3} + 30 q^{5} - 18 q^{7} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 18 q^{3} + 30 q^{5} - 18 q^{7} + 54 q^{9} - 78 q^{13} - 90 q^{15} + 36 q^{17} - 64 q^{19} + 54 q^{21} - 28 q^{23} + 150 q^{25} - 162 q^{27} + 62 q^{29} + 214 q^{31} - 90 q^{35} + 438 q^{37} + 234 q^{39} + 342 q^{41} - 64 q^{43} + 270 q^{45} - 690 q^{47} + 1120 q^{49} - 108 q^{51} - 418 q^{53} + 192 q^{57} - 722 q^{59} + 658 q^{61} - 162 q^{63} - 390 q^{65} - 298 q^{67} + 84 q^{69} + 288 q^{71} + 930 q^{73} - 450 q^{75} + 1870 q^{77} - 1594 q^{79} + 486 q^{81} - 1594 q^{83} + 180 q^{85} - 186 q^{87} + 1494 q^{89} + 234 q^{91} - 642 q^{93} - 320 q^{95} + 796 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 834x^{4} + 965x^{3} + 200177x^{2} - 261462x - 11549822 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5230\nu^{5} + 274679\nu^{4} + 873670\nu^{3} - 125665941\nu^{2} + 440629942\nu + 7725428543 ) / 136354419 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -20341\nu^{5} + 1216046\nu^{4} + 15130174\nu^{3} - 623741613\nu^{2} - 2923649840\nu + 54610834910 ) / 272708838 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 17317\nu^{5} + 55162\nu^{4} - 11844046\nu^{3} - 58924467\nu^{2} + 1778065976\nu + 10621919128 ) / 90902946 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -22169\nu^{5} + 60616\nu^{4} + 12306950\nu^{3} - 26859687\nu^{2} - 1421495416\nu + 1664464732 ) / 90902946 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -8\beta_{5} - 9\beta_{4} - 3\beta_{3} + 12\beta_{2} - 5\beta _1 + 1114 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -38\beta_{5} - 47\beta_{4} + 69\beta_{3} - 126\beta_{2} + 368\beta _1 - 123 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4292\beta_{5} - 4491\beta_{4} - 837\beta_{3} + 6612\beta_{2} - 2403\beta _1 + 393950 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -39540\beta_{5} - 27467\beta_{4} + 39651\beta_{3} - 66408\beta_{2} + 139659\beta _1 - 104624 ) / 4 \) 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.

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
−18.8313
−8.14751
16.6225
21.1549
10.5304
−20.3291
0 −3.00000 0 5.00000 0 −32.6504 0 9.00000 0
1.2 0 −3.00000 0 5.00000 0 −27.0616 0 9.00000 0
1.3 0 −3.00000 0 5.00000 0 −12.7285 0 9.00000 0
1.4 0 −3.00000 0 5.00000 0 11.1788 0 9.00000 0
1.5 0 −3.00000 0 5.00000 0 12.7751 0 9.00000 0
1.6 0 −3.00000 0 5.00000 0 30.4866 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.s 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.4.a.s 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} + 18T_{7}^{5} - 1427T_{7}^{4} - 19340T_{7}^{3} + 507356T_{7}^{2} + 2663200T_{7} - 48965280 \) 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
$7$ \( T^{6} + 18 T^{5} - 1427 T^{4} + \cdots - 48965280 \) Copy content Toggle raw display
$11$ \( T^{6} - 7443 T^{4} + \cdots - 7758842112 \) Copy content Toggle raw display
$13$ \( (T + 13)^{6} \) Copy content Toggle raw display
$17$ \( T^{6} - 36 T^{5} + \cdots - 48858044256 \) Copy content Toggle raw display
$19$ \( T^{6} + 64 T^{5} + \cdots - 9663362176 \) Copy content Toggle raw display
$23$ \( T^{6} + 28 T^{5} - 41903 T^{4} + \cdots + 91055424 \) Copy content Toggle raw display
$29$ \( T^{6} - 62 T^{5} + \cdots - 170223974656 \) Copy content Toggle raw display
$31$ \( T^{6} - 214 T^{5} + \cdots - 278767937024 \) Copy content Toggle raw display
$37$ \( T^{6} - 438 T^{5} + \cdots + 59868044062128 \) Copy content Toggle raw display
$41$ \( T^{6} - 342 T^{5} + \cdots + 15167720117616 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 252983393722368 \) Copy content Toggle raw display
$47$ \( T^{6} + 690 T^{5} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{6} + 418 T^{5} + \cdots - 55\!\cdots\!12 \) Copy content Toggle raw display
$59$ \( T^{6} + 722 T^{5} + \cdots - 39102420828160 \) Copy content Toggle raw display
$61$ \( T^{6} - 658 T^{5} + \cdots - 31\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{6} + 298 T^{5} + \cdots - 29120342679552 \) Copy content Toggle raw display
$71$ \( T^{6} - 288 T^{5} + \cdots - 19\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{6} - 930 T^{5} + \cdots - 10\!\cdots\!28 \) Copy content Toggle raw display
$79$ \( T^{6} + 1594 T^{5} + \cdots - 14\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{6} + 1594 T^{5} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{6} - 1494 T^{5} + \cdots + 94\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{6} - 796 T^{5} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
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