Properties

Label 1200.3.l.s
Level $1200$
Weight $3$
Character orbit 1200.l
Analytic conductor $32.698$
Analytic rank $0$
Dimension $2$
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] = [1200,3,Mod(401,1200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1200.401"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.l (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,5,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.6976317232\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
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 \(\beta = \frac{1}{2}(1 + \sqrt{-11})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 3) q^{3} + ( - 5 \beta + 6) q^{9} + ( - 10 \beta + 5) q^{11} - 10 q^{13} + (2 \beta - 1) q^{17} - 7 q^{19} + ( - 12 \beta + 6) q^{23} + ( - 16 \beta + 3) q^{27} + (20 \beta - 10) q^{29}+ \cdots + ( - 35 \beta - 120) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{3} + 7 q^{9} - 20 q^{13} - 14 q^{19} - 10 q^{27} - 84 q^{31} - 55 q^{33} + 80 q^{37} - 50 q^{39} - 100 q^{43} - 98 q^{49} + 11 q^{51} - 35 q^{57} - 16 q^{61} + 90 q^{67} - 66 q^{69} + 70 q^{73}+ \cdots - 275 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
0.500000 + 1.65831i
0.500000 1.65831i
0 2.50000 1.65831i 0 0 0 0 0 3.50000 8.29156i 0
401.2 0 2.50000 + 1.65831i 0 0 0 0 0 3.50000 + 8.29156i 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
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.3.l.s 2
3.b odd 2 1 inner 1200.3.l.s 2
4.b odd 2 1 75.3.c.c 2
5.b even 2 1 1200.3.l.f 2
5.c odd 4 2 1200.3.c.d 4
12.b even 2 1 75.3.c.c 2
15.d odd 2 1 1200.3.l.f 2
15.e even 4 2 1200.3.c.d 4
20.d odd 2 1 75.3.c.f yes 2
20.e even 4 2 75.3.d.c 4
60.h even 2 1 75.3.c.f yes 2
60.l odd 4 2 75.3.d.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.3.c.c 2 4.b odd 2 1
75.3.c.c 2 12.b even 2 1
75.3.c.f yes 2 20.d odd 2 1
75.3.c.f yes 2 60.h even 2 1
75.3.d.c 4 20.e even 4 2
75.3.d.c 4 60.l odd 4 2
1200.3.c.d 4 5.c odd 4 2
1200.3.c.d 4 15.e even 4 2
1200.3.l.f 2 5.b even 2 1
1200.3.l.f 2 15.d odd 2 1
1200.3.l.s 2 1.a even 1 1 trivial
1200.3.l.s 2 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7} \) Copy content Toggle raw display
\( T_{11}^{2} + 275 \) Copy content Toggle raw display
\( T_{13} + 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 5T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 275 \) Copy content Toggle raw display
$13$ \( (T + 10)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 11 \) Copy content Toggle raw display
$19$ \( (T + 7)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 396 \) Copy content Toggle raw display
$29$ \( T^{2} + 1100 \) Copy content Toggle raw display
$31$ \( (T + 42)^{2} \) Copy content Toggle raw display
$37$ \( (T - 40)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 275 \) Copy content Toggle raw display
$43$ \( (T + 50)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 2156 \) Copy content Toggle raw display
$53$ \( T^{2} + 2156 \) Copy content Toggle raw display
$59$ \( T^{2} + 4400 \) Copy content Toggle raw display
$61$ \( (T + 8)^{2} \) Copy content Toggle raw display
$67$ \( (T - 45)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 1100 \) Copy content Toggle raw display
$73$ \( (T - 35)^{2} \) Copy content Toggle raw display
$79$ \( (T + 12)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 4851 \) Copy content Toggle raw display
$89$ \( T^{2} + 22275 \) Copy content Toggle raw display
$97$ \( (T - 70)^{2} \) Copy content Toggle raw display
show more
show less