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Properties

Label 800.6.a.w
Level $800$
Weight $6$
Character orbit 800.a
Self dual yes
Analytic conductor $128.307$
Analytic rank $0$
Dimension $6$
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] = [800,6,Mod(1,800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 800.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: \(128.307055850\)
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: \( x^{6} - 383x^{4} + 27543x^{2} - 289161 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{15}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 160)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 3) q^{3} + (\beta_{4} - 5 \beta_1 + 77) q^{7} + (3 \beta_{4} + 4 \beta_1 + 21) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 3) q^{3} + (\beta_{4} - 5 \beta_1 + 77) q^{7} + (3 \beta_{4} + 4 \beta_1 + 21) q^{9} - \beta_{2} q^{11} - \beta_{5} q^{13} + (\beta_{5} - \beta_{3} - 4 \beta_{2}) q^{17} + ( - 2 \beta_{5} - 2 \beta_{3} - \beta_{2}) q^{19} + (9 \beta_{4} - 112 \beta_1 + 1044) q^{21} + (57 \beta_{4} + 61 \beta_1 + 211) q^{23} + ( - 30 \beta_{4} + 98 \beta_1 - 354) q^{27} + ( - 12 \beta_{4} - 248 \beta_1 + 738) q^{29} + ( - 2 \beta_{5} + 14 \beta_{3} + 8 \beta_{2}) q^{31} + (7 \beta_{5} + 3 \beta_{3} - 4 \beta_{2}) q^{33} + ( - 6 \beta_{5} + 9 \beta_{3} + 8 \beta_{2}) q^{37} + (8 \beta_{5} - 24 \beta_{3} + 10 \beta_{2}) q^{39} + (113 \beta_{4} - 380 \beta_1 + 5590) q^{41} + (146 \beta_{4} - 77 \beta_1 - 2543) q^{43} + ( - 135 \beta_{4} - 393 \beta_1 + 10177) q^{47} + (159 \beta_{4} - 1180 \beta_1 - 1103) q^{49} + (10 \beta_{5} + 42 \beta_{3} - 16 \beta_{2}) q^{51} + ( - 11 \beta_{5} - 28 \beta_{3} + 8 \beta_{2}) q^{53} + (3 \beta_{5} - 33 \beta_{3} + 36 \beta_{2}) q^{57} + (10 \beta_{5} - 54 \beta_{3} - 35 \beta_{2}) q^{59} + ( - 87 \beta_{4} - 1064 \beta_1 - 9498) q^{61} + (39 \beta_{4} - 77 \beta_1 + 6717) q^{63} + ( - 622 \beta_{4} + 617 \beta_1 - 6989) q^{67} + ( - 525 \beta_{4} - 2552 \beta_1 - 16188) q^{69} + (2 \beta_{5} + 18 \beta_{3} - 30 \beta_{2}) q^{71} + ( - 29 \beta_{5} + 99 \beta_{3} + 44 \beta_{2}) q^{73} + (45 \beta_{5} + 73 \beta_{3} - 72 \beta_{2}) q^{77} + ( - 70 \beta_{5} + 10 \beta_{3} + 28 \beta_{2}) q^{79} + ( - 843 \beta_{4} + 484 \beta_1 - 29031) q^{81} + ( - 594 \beta_{4} - 1473 \beta_1 - 11195) q^{83} + (816 \beta_{4} - 10 \beta_1 + 61026) q^{87} + (756 \beta_{4} - 3576 \beta_1 - 28386) q^{89} + ( - 82 \beta_{5} + \cdots + 130 \beta_{2}) q^{91}+ \cdots + (2 \beta_{5} + 162 \beta_{3} + 127 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 20 q^{3} + 452 q^{7} + 134 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 20 q^{3} + 452 q^{7} + 134 q^{9} + 6040 q^{21} + 1388 q^{23} - 1928 q^{27} + 3932 q^{29} + 32780 q^{41} - 15412 q^{43} + 60276 q^{47} - 8978 q^{49} - 59116 q^{61} + 40148 q^{63} - 40700 q^{67} - 102232 q^{69} - 173218 q^{81} - 70116 q^{83} + 366136 q^{87} - 177468 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 383x^{4} + 27543x^{2} - 289161 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{4} - 272\nu^{2} + 4371 ) / 540 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -34\nu^{5} + 15188\nu^{3} - 1433274\nu ) / 7695 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{5} + 1630\nu^{3} - 65325\nu ) / 513 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{4} + 380\nu^{2} - 17979 ) / 162 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -109\nu^{5} + 39638\nu^{3} - 1797549\nu ) / 7695 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{3} - \beta_{2} ) / 80 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{4} + 10\beta _1 + 252 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 241\beta_{5} - 309\beta_{3} - 91\beta_{2} ) / 80 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 408\beta_{4} + 1900\beta _1 + 29901 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 65501\beta_{5} - 95877\beta_{3} - 16601\beta_{2} ) / 80 \) Copy content Toggle raw display

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

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
17.0908
−17.0908
−3.55673
3.55673
−8.84621
8.84621
0 −21.9637 0 0 0 29.6983 0 239.406 0
1.2 0 −21.9637 0 0 0 29.6983 0 239.406 0
1.3 0 −5.01877 0 0 0 −15.3895 0 −217.812 0
1.4 0 −5.01877 0 0 0 −15.3895 0 −217.812 0
1.5 0 16.9825 0 0 0 211.691 0 45.4059 0
1.6 0 16.9825 0 0 0 211.691 0 45.4059 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
\(2\) \(1\)
\(5\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.6.a.w 6
4.b odd 2 1 800.6.a.bb 6
5.b even 2 1 800.6.a.bb 6
5.c odd 4 2 160.6.c.d ✓ 12
20.d odd 2 1 inner 800.6.a.w 6
20.e even 4 2 160.6.c.d ✓ 12
40.i odd 4 2 320.6.c.l 12
40.k even 4 2 320.6.c.l 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.c.d ✓ 12 5.c odd 4 2
160.6.c.d ✓ 12 20.e even 4 2
320.6.c.l 12 40.i odd 4 2
320.6.c.l 12 40.k even 4 2
800.6.a.w 6 1.a even 1 1 trivial
800.6.a.w 6 20.d odd 2 1 inner
800.6.a.bb 6 4.b odd 2 1
800.6.a.bb 6 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(800))\):

\( T_{3}^{3} + 10T_{3}^{2} - 348T_{3} - 1872 \) Copy content Toggle raw display
\( T_{11}^{6} - 654656T_{11}^{4} + 121093324800T_{11}^{2} - 4638399528960000 \) Copy content Toggle raw display
\( T_{13}^{6} - 2044496T_{13}^{4} + 1228724736000T_{13}^{2} - 225213825024000000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{3} + 10 T^{2} + \cdots - 1872)^{2} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( (T^{3} - 226 T^{2} + \cdots + 96752)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots - 46\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots - 22\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots - 98\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots - 41\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{3} - 694 T^{2} + \cdots + 17305310224)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 1966 T^{2} + \cdots + 29310748632)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{3} - 16390 T^{2} + \cdots + 287527662200)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 7706 T^{2} + \cdots + 175861521008)^{2} \) Copy content Toggle raw display
$47$ \( (T^{3} + \cdots + 1008559697072)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots - 20\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots - 1325305098360)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + \cdots - 46234016389424)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots - 78\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{3} + \cdots - 5656685262160)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} + \cdots - 408997402152984)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots - 31\!\cdots\!00 \) Copy content Toggle raw display
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