Properties

Label 800.4.a.x
Level $800$
Weight $4$
Character orbit 800.a
Self dual yes
Analytic conductor $47.202$
Analytic rank $0$
Dimension $3$
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] = [800,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(47.2015280046\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.5685.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 11x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 2) q^{3} + ( - \beta_{2} - \beta_1 - 1) q^{7} + (\beta_{2} + 3 \beta_1 + 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 2) q^{3} + ( - \beta_{2} - \beta_1 - 1) q^{7} + (\beta_{2} + 3 \beta_1 + 7) q^{9} + (\beta_{2} + 2 \beta_1 + 11) q^{11} + ( - 3 \beta_{2} - \beta_1 - 3) q^{13} + ( - \beta_{2} + 13 \beta_1 + 34) q^{17} + (2 \beta_{2} + 9 \beta_1 + 32) q^{19} + ( - 3 \beta_{2} - 17 \beta_1 - 29) q^{21} + (2 \beta_{2} - 4 \beta_1 - 42) q^{23} + (5 \beta_{2} - 2 \beta_1 + 47) q^{27} + (\beta_{2} + 27 \beta_1 + 37) q^{29} + ( - 6 \beta_{2} - 12 \beta_1 + 94) q^{31} + (4 \beta_{2} + 28 \beta_1 + 79) q^{33} + ( - \beta_{2} + 53 \beta_1 + 49) q^{37} + ( - 7 \beta_{2} - 49 \beta_1 - 27) q^{39} + (4 \beta_{2} - 36 \beta_1 - 127) q^{41} + ( - 9 \beta_{2} + 13 \beta_1 - 133) q^{43} + ( - 5 \beta_{2} + 37 \beta_1 + 287) q^{47} + ( - 12 \beta_{2} + 28 \beta_1 + 129) q^{49} + (11 \beta_{2} + 32 \beta_1 + 461) q^{51} + (16 \beta_{2} + 32 \beta_1 - 14) q^{53} + (13 \beta_{2} + 71 \beta_1 + 328) q^{57} + ( - 5 \beta_{2} + 85 \beta_1 + 375) q^{59} + (19 \beta_{2} + \beta_1 + 161) q^{61} + (4 \beta_{2} - 64 \beta_1 - 532) q^{63} + ( - 13 \beta_{2} - 96 \beta_1 + 213) q^{67} + ( - 16 \beta_1 - 210) q^{69} + (3 \beta_{2} - 59 \beta_1 + 663) q^{71} + ( - 15 \beta_{2} - 45 \beta_1 - 10) q^{73} + (\beta_{2} - 53 \beta_1 - 509) q^{77} + (33 \beta_{2} - 59 \beta_1 + 313) q^{79} + ( - 19 \beta_{2} + 39 \beta_1 - 170) q^{81} + (31 \beta_{2} + 144 \beta_1 - 359) q^{83} + (29 \beta_{2} + 79 \beta_1 + 881) q^{87} + (37 \beta_{2} - \beta_1 + 396) q^{89} + ( - 38 \beta_{2} + 54 \beta_1 + 1362) q^{91} + ( - 24 \beta_{2} - 8 \beta_1 - 154) q^{93} + ( - 50 \beta_{2} - 230 \beta_1 + 260) q^{97} + (9 \beta_{2} + 113 \beta_1 + 689) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 5 q^{3} - 2 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 5 q^{3} - 2 q^{7} + 18 q^{9} + 31 q^{11} - 8 q^{13} + 89 q^{17} + 87 q^{19} - 70 q^{21} - 122 q^{23} + 143 q^{27} + 84 q^{29} + 294 q^{31} + 209 q^{33} + 94 q^{37} - 32 q^{39} - 345 q^{41} - 412 q^{43} + 824 q^{47} + 359 q^{49} + 1351 q^{51} - 74 q^{53} + 913 q^{57} + 1040 q^{59} + 482 q^{61} - 1532 q^{63} + 735 q^{67} - 614 q^{69} + 2048 q^{71} + 15 q^{73} - 1474 q^{77} + 998 q^{79} - 549 q^{81} - 1221 q^{83} + 2564 q^{87} + 1189 q^{89} + 4032 q^{91} - 454 q^{93} + 1010 q^{97} + 1954 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 11x + 6 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\nu^{2} - 2\nu - 30 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + \beta _1 + 31 ) / 4 \) 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
−3.12873
0.533386
3.59535
0 −5.25747 0 0 0 −9.15590 0 0.640965 0
1.2 0 2.06677 0 0 0 28.8620 0 −22.7285 0
1.3 0 8.19070 0 0 0 −21.7061 0 40.0875 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( +1 \)

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 800.4.a.x yes 3
4.b odd 2 1 800.4.a.u 3
5.b even 2 1 800.4.a.v yes 3
5.c odd 4 2 800.4.c.n 6
8.b even 2 1 1600.4.a.cq 3
8.d odd 2 1 1600.4.a.ct 3
20.d odd 2 1 800.4.a.w yes 3
20.e even 4 2 800.4.c.m 6
40.e odd 2 1 1600.4.a.cr 3
40.f even 2 1 1600.4.a.cs 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.4.a.u 3 4.b odd 2 1
800.4.a.v yes 3 5.b even 2 1
800.4.a.w yes 3 20.d odd 2 1
800.4.a.x yes 3 1.a even 1 1 trivial
800.4.c.m 6 20.e even 4 2
800.4.c.n 6 5.c odd 4 2
1600.4.a.cq 3 8.b even 2 1
1600.4.a.cr 3 40.e odd 2 1
1600.4.a.cs 3 40.f even 2 1
1600.4.a.ct 3 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(800))\):

\( T_{3}^{3} - 5T_{3}^{2} - 37T_{3} + 89 \) Copy content Toggle raw display
\( T_{11}^{3} - 31T_{11}^{2} - 485T_{11} + 8475 \) Copy content Toggle raw display
\( T_{13}^{3} + 8T_{13}^{2} - 6000T_{13} - 192000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 5 T^{2} + \cdots + 89 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 2 T^{2} + \cdots - 5736 \) Copy content Toggle raw display
$11$ \( T^{3} - 31 T^{2} + \cdots + 8475 \) Copy content Toggle raw display
$13$ \( T^{3} + 8 T^{2} + \cdots - 192000 \) Copy content Toggle raw display
$17$ \( T^{3} - 89 T^{2} + \cdots + 490725 \) Copy content Toggle raw display
$19$ \( T^{3} - 87 T^{2} + \cdots - 7925 \) Copy content Toggle raw display
$23$ \( T^{3} + 122 T^{2} + \cdots - 68808 \) Copy content Toggle raw display
$29$ \( T^{3} - 84 T^{2} + \cdots + 278528 \) Copy content Toggle raw display
$31$ \( T^{3} - 294 T^{2} + \cdots + 1628600 \) Copy content Toggle raw display
$37$ \( T^{3} - 94 T^{2} + \cdots + 10497400 \) Copy content Toggle raw display
$41$ \( T^{3} + 345 T^{2} + \cdots - 14242325 \) Copy content Toggle raw display
$43$ \( T^{3} + 412 T^{2} + \cdots - 9200896 \) Copy content Toggle raw display
$47$ \( T^{3} - 824 T^{2} + \cdots + 11410304 \) Copy content Toggle raw display
$53$ \( T^{3} + 74 T^{2} + \cdots + 76600 \) Copy content Toggle raw display
$59$ \( T^{3} - 1040 T^{2} + \cdots + 140160000 \) Copy content Toggle raw display
$61$ \( T^{3} - 482 T^{2} + \cdots + 80633480 \) Copy content Toggle raw display
$67$ \( T^{3} - 735 T^{2} + \cdots + 240845067 \) Copy content Toggle raw display
$71$ \( T^{3} - 2048 T^{2} + \cdots - 220992000 \) Copy content Toggle raw display
$73$ \( T^{3} - 15 T^{2} + \cdots + 18845875 \) Copy content Toggle raw display
$79$ \( T^{3} - 998 T^{2} + \cdots + 361942200 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots - 1162222465 \) Copy content Toggle raw display
$89$ \( T^{3} - 1189 T^{2} + \cdots + 641970729 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots + 3811625000 \) Copy content Toggle raw display
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