↑

Properties

Label 800.3.b.d
Level $800$
Weight $3$
Character orbit 800.b
Analytic conductor $21.798$
Analytic rank $0$
Dimension $4$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [800,3,Mod(351,800)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("800.351"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 800.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-20,0,0,0,-64,0,0,0,40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(21.7984211488\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (4 \beta_{2} - \beta_1) q^{7} + (3 \beta_{3} - 5) q^{9} + ( - \beta_{2} - 3 \beta_1) q^{11} + ( - \beta_{3} - 16) q^{13} + ( - 2 \beta_{3} + 10) q^{17} + (3 \beta_{2} + 3 \beta_1) q^{19}+ \cdots + ( - 25 \beta_{2} + 45 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 20 q^{9} - 64 q^{13} + 40 q^{17} - 88 q^{21} - 72 q^{29} - 160 q^{33} - 112 q^{37} + 160 q^{41} - 308 q^{49} - 128 q^{53} + 144 q^{57} + 64 q^{61} - 136 q^{69} + 8 q^{73} - 160 q^{77} + 316 q^{81}+ \cdots - 264 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu^{3} + 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{3} + 6\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{2} + 6 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{2} + 3\beta_1 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
351.1
− 1.61803i
0.618034i
− 0.618034i
1.61803i
0 − 5.23607i 0 0 0 − 10.1803i 0 −18.4164 0
351.2 0 − 0.763932i 0 0 0 12.1803i 0 8.41641 0
351.3 0 0.763932i 0 0 0 − 12.1803i 0 8.41641 0
351.4 0 5.23607i 0 0 0 10.1803i 0 −18.4164 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.3.b.d 4
4.b odd 2 1 inner 800.3.b.d 4
5.b even 2 1 160.3.b.a ✓ 4
5.c odd 4 1 800.3.h.c 4
5.c odd 4 1 800.3.h.j 4
8.b even 2 1 1600.3.b.n 4
8.d odd 2 1 1600.3.b.n 4
15.d odd 2 1 1440.3.e.b 4
20.d odd 2 1 160.3.b.a ✓ 4
20.e even 4 1 800.3.h.c 4
20.e even 4 1 800.3.h.j 4
40.e odd 2 1 320.3.b.b 4
40.f even 2 1 320.3.b.b 4
40.i odd 4 1 1600.3.h.d 4
40.i odd 4 1 1600.3.h.m 4
40.k even 4 1 1600.3.h.d 4
40.k even 4 1 1600.3.h.m 4
60.h even 2 1 1440.3.e.b 4
80.k odd 4 1 1280.3.g.a 4
80.k odd 4 1 1280.3.g.d 4
80.q even 4 1 1280.3.g.a 4
80.q even 4 1 1280.3.g.d 4
120.i odd 2 1 2880.3.e.a 4
120.m even 2 1 2880.3.e.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.3.b.a ✓ 4 5.b even 2 1
160.3.b.a ✓ 4 20.d odd 2 1
320.3.b.b 4 40.e odd 2 1
320.3.b.b 4 40.f even 2 1
800.3.b.d 4 1.a even 1 1 trivial
800.3.b.d 4 4.b odd 2 1 inner
800.3.h.c 4 5.c odd 4 1
800.3.h.c 4 20.e even 4 1
800.3.h.j 4 5.c odd 4 1
800.3.h.j 4 20.e even 4 1
1280.3.g.a 4 80.k odd 4 1
1280.3.g.a 4 80.q even 4 1
1280.3.g.d 4 80.k odd 4 1
1280.3.g.d 4 80.q even 4 1
1440.3.e.b 4 15.d odd 2 1
1440.3.e.b 4 60.h even 2 1
1600.3.b.n 4 8.b even 2 1
1600.3.b.n 4 8.d odd 2 1
1600.3.h.d 4 40.i odd 4 1
1600.3.h.d 4 40.k even 4 1
1600.3.h.m 4 40.i odd 4 1
1600.3.h.m 4 40.k even 4 1
2880.3.e.a 4 120.i odd 2 1
2880.3.e.a 4 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(800, [\chi])\):

\( T_{3}^{4} + 28T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{13}^{2} + 32T_{13} + 236 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 28T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 252 T^{2} + 15376 \) Copy content Toggle raw display
$11$ \( T^{4} + 240T^{2} + 6400 \) Copy content Toggle raw display
$13$ \( (T^{2} + 32 T + 236)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 20 T + 20)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 1308 T^{2} + 309136 \) Copy content Toggle raw display
$29$ \( (T^{2} + 36 T - 396)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 752T^{2} + 256 \) Copy content Toggle raw display
$37$ \( (T^{2} + 56 T + 764)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 80 T + 620)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 732 T^{2} + 15376 \) Copy content Toggle raw display
$47$ \( T^{4} + 2492 T^{2} + 1008016 \) Copy content Toggle raw display
$53$ \( (T^{2} + 64 T + 44)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 9888 T^{2} + 8386816 \) Copy content Toggle raw display
$61$ \( (T^{2} - 32 T - 2164)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 15260 T^{2} + 57456400 \) Copy content Toggle raw display
$71$ \( T^{4} + 16688 T^{2} + 26050816 \) Copy content Toggle raw display
$73$ \( (T^{2} - 4 T - 17996)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 24768 T^{2} + 119771136 \) Copy content Toggle raw display
$83$ \( T^{4} + 7868 T^{2} + 2835856 \) Copy content Toggle raw display
$89$ \( (T + 30)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 132 T + 1476)^{2} \) Copy content Toggle raw display
show more
show less