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Properties

Label 1560.4.g.c
Level $1560$
Weight $4$
Character orbit 1560.g
Analytic conductor $92.043$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1560,4,Mod(961,1560)]
 
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, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1560.961");
 
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.g (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(92.0429796090\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 2831 x^{18} + 3052845 x^{16} + 1573840355 x^{14} + 402678072963 x^{12} + 50151013456349 x^{10} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{25} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 q^{3} - 5 \beta_{7} q^{5} + ( - \beta_{17} - \beta_{7}) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} - 5 \beta_{7} q^{5} + ( - \beta_{17} - \beta_{7}) q^{7} + 9 q^{9} + (\beta_{17} + \beta_{7} + \beta_1) q^{11} + ( - \beta_{13} - 2 \beta_{7} - 2) q^{13} + 15 \beta_{7} q^{15} + (\beta_{4} - \beta_{3} - \beta_{2} - 3) q^{17} + (\beta_{13} - \beta_{12} + \cdots + 2 \beta_1) q^{19}+ \cdots + (9 \beta_{17} + 9 \beta_{7} + 9 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 60 q^{3} + 180 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 60 q^{3} + 180 q^{9} - 38 q^{13} - 58 q^{17} + 86 q^{23} - 500 q^{25} - 540 q^{27} - 220 q^{29} - 90 q^{35} + 114 q^{39} - 340 q^{43} - 482 q^{49} + 174 q^{51} - 106 q^{53} + 110 q^{55} - 70 q^{61} - 150 q^{65} - 258 q^{69} + 1500 q^{75} + 4058 q^{77} + 2054 q^{79} + 1620 q^{81} + 660 q^{87} - 2366 q^{91} - 20 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 2831 x^{18} + 3052845 x^{16} + 1573840355 x^{14} + 402678072963 x^{12} + 50151013456349 x^{10} + \cdots + 17\!\cdots\!36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 11\!\cdots\!29 \nu^{18} + \cdots - 13\!\cdots\!52 ) / 63\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 31\!\cdots\!35 \nu^{18} + \cdots + 60\!\cdots\!72 ) / 45\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 14\!\cdots\!71 \nu^{18} + \cdots - 32\!\cdots\!76 ) / 14\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 37\!\cdots\!73 \nu^{18} + \cdots - 27\!\cdots\!88 ) / 24\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 52\!\cdots\!07 \nu^{18} + \cdots + 50\!\cdots\!48 ) / 18\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 41\!\cdots\!99 \nu^{19} + \cdots - 51\!\cdots\!88 \nu ) / 65\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 49\!\cdots\!63 \nu^{19} + \cdots - 39\!\cdots\!04 ) / 18\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 49\!\cdots\!63 \nu^{19} + \cdots + 39\!\cdots\!44 ) / 18\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 15\!\cdots\!47 \nu^{19} + \cdots - 31\!\cdots\!84 \nu ) / 39\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 37\!\cdots\!51 \nu^{19} + \cdots + 87\!\cdots\!00 \nu ) / 48\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 32\!\cdots\!01 \nu^{19} + \cdots + 60\!\cdots\!08 ) / 12\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 32\!\cdots\!01 \nu^{19} + \cdots + 60\!\cdots\!08 ) / 12\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 97\!\cdots\!03 \nu^{19} + \cdots + 29\!\cdots\!68 ) / 37\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 97\!\cdots\!03 \nu^{19} + \cdots + 10\!\cdots\!52 ) / 37\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 10\!\cdots\!35 \nu^{19} + \cdots + 79\!\cdots\!08 ) / 37\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 15\!\cdots\!73 \nu^{19} + \cdots + 46\!\cdots\!08 \nu ) / 47\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 44\!\cdots\!85 \nu^{19} + \cdots + 15\!\cdots\!04 ) / 72\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 11\!\cdots\!01 \nu^{19} + \cdots + 55\!\cdots\!40 \nu ) / 94\!\cdots\!20 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 5 \beta_{15} + \beta_{14} - 7 \beta_{13} - 13 \beta_{12} - 10 \beta_{9} + 10 \beta_{8} + \beta_{6} + \cdots - 1133 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 4 \beta_{19} - 21 \beta_{18} + 110 \beta_{17} - 59 \beta_{16} - 29 \beta_{13} + 29 \beta_{12} + \cdots + 111 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3066 \beta_{15} - 1606 \beta_{14} + 5401 \beta_{13} + 10073 \beta_{12} + 8163 \beta_{9} - 8163 \beta_{8} + \cdots + 762243 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 18435 \beta_{19} - 1823 \beta_{18} - 42393 \beta_{17} + 36947 \beta_{16} - 18759 \beta_{13} + \cdots - 56756 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 2090911 \beta_{15} + 1965147 \beta_{14} - 3856659 \beta_{13} - 7912717 \beta_{12} - 6498536 \beta_{9} + \cdots - 591117941 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 56302384 \beta_{19} + 16371373 \beta_{18} + 75388830 \beta_{17} - 79442037 \beta_{16} + \cdots + 98052161 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1550693756 \beta_{15} - 2218636832 \beta_{14} + 2693607001 \beta_{13} + 6462937589 \beta_{12} + \cdots + 483977241763 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 33412649721 \beta_{19} - 10784695544 \beta_{18} - 38755239753 \beta_{17} + 40687756016 \beta_{16} + \cdots - 40121024245 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1225742384425 \beta_{15} + 2369139606829 \beta_{14} - 1841050584059 \beta_{13} + \cdots - 409432462138381 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 71986714653788 \beta_{19} + 23103858577775 \beta_{18} + 83636254661934 \beta_{17} + \cdots + 63775793035003 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 10\!\cdots\!78 \beta_{15} + \cdots + 35\!\cdots\!47 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 36\!\cdots\!95 \beta_{19} + \cdots - 24\!\cdots\!54 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 86\!\cdots\!47 \beta_{15} + \cdots - 31\!\cdots\!57 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 73\!\cdots\!92 \beta_{19} + \cdots + 38\!\cdots\!29 ) / 4 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 76\!\cdots\!40 \beta_{15} + \cdots + 27\!\cdots\!67 ) / 4 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 36\!\cdots\!97 \beta_{19} + \cdots - 14\!\cdots\!71 ) / 2 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 68\!\cdots\!49 \beta_{15} + \cdots - 24\!\cdots\!45 ) / 4 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 69\!\cdots\!24 \beta_{19} + \cdots + 21\!\cdots\!31 ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1560\mathbb{Z}\right)^\times\).

\(n\) \(391\) \(521\) \(781\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1
− 2.38827i
− 6.10182i
− 23.8633i
27.6662i
− 18.5307i
9.03772i
− 5.44159i
30.5218i
− 1.95654i
− 7.94352i
7.94352i
1.95654i
− 30.5218i
5.44159i
− 9.03772i
18.5307i
− 27.6662i
23.8633i
6.10182i
2.38827i
0 −3.00000 0 − 5.00000i 0 − 29.6657i 0 9.00000 0
961.2 0 −3.00000 0 − 5.00000i 0 − 21.3688i 0 9.00000 0
961.3 0 −3.00000 0 − 5.00000i 0 − 19.3824i 0 9.00000 0
961.4 0 −3.00000 0 − 5.00000i 0 − 12.5225i 0 9.00000 0
961.5 0 −3.00000 0 − 5.00000i 0 − 9.77575i 0 9.00000 0
961.6 0 −3.00000 0 − 5.00000i 0 2.41605i 0 9.00000 0
961.7 0 −3.00000 0 − 5.00000i 0 15.1033i 0 9.00000 0
961.8 0 −3.00000 0 − 5.00000i 0 20.3539i 0 9.00000 0
961.9 0 −3.00000 0 − 5.00000i 0 21.0107i 0 9.00000 0
961.10 0 −3.00000 0 − 5.00000i 0 24.8312i 0 9.00000 0
961.11 0 −3.00000 0 5.00000i 0 − 24.8312i 0 9.00000 0
961.12 0 −3.00000 0 5.00000i 0 − 21.0107i 0 9.00000 0
961.13 0 −3.00000 0 5.00000i 0 − 20.3539i 0 9.00000 0
961.14 0 −3.00000 0 5.00000i 0 − 15.1033i 0 9.00000 0
961.15 0 −3.00000 0 5.00000i 0 − 2.41605i 0 9.00000 0
961.16 0 −3.00000 0 5.00000i 0 9.77575i 0 9.00000 0
961.17 0 −3.00000 0 5.00000i 0 12.5225i 0 9.00000 0
961.18 0 −3.00000 0 5.00000i 0 19.3824i 0 9.00000 0
961.19 0 −3.00000 0 5.00000i 0 21.3688i 0 9.00000 0
961.20 0 −3.00000 0 5.00000i 0 29.6657i 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 961.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1560.4.g.c ✓ 20
13.b even 2 1 inner 1560.4.g.c ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.4.g.c ✓ 20 1.a even 1 1 trivial
1560.4.g.c ✓ 20 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{20} + 3671 T_{7}^{18} + 5759815 T_{7}^{16} + 5066509885 T_{7}^{14} + 2748855683068 T_{7}^{12} + \cdots + 33\!\cdots\!96 \) acting on \(S_{4}^{\mathrm{new}}(1560, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T + 3)^{20} \) Copy content Toggle raw display
$5$ \( (T^{2} + 25)^{10} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 33\!\cdots\!96 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 26\!\cdots\!49 \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots + 24\!\cdots\!60)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots - 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 35\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 22\!\cdots\!52)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 49\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots - 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 29\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 52\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 70\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
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