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Properties

Label 200.3.l.a
Level $200$
Weight $3$
Character orbit 200.l
Analytic conductor $5.450$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [200,3,Mod(57,200)] 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("200.57"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(200, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 200.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-2,0,0,0,-14] 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: \(5.44960528721\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (i - 1) q^{3} + ( - 7 i - 7) q^{7} + 7 i q^{9} - 4 q^{11} + (12 i - 12) q^{13} + ( - 20 i - 20) q^{17} - 28 i q^{19} + 14 q^{21} + (9 i - 9) q^{23} + ( - 16 i - 16) q^{27} + 34 i q^{29} - 40 q^{31} + \cdots - 28 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 14 q^{7} - 8 q^{11} - 24 q^{13} - 40 q^{17} + 28 q^{21} - 18 q^{23} - 32 q^{27} - 80 q^{31} + 8 q^{33} + 32 q^{37} + 64 q^{41} - 14 q^{43} + 62 q^{47} + 80 q^{51} + 104 q^{53} + 56 q^{57}+ \cdots + 88 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/200\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(177\)
\(\chi(n)\) \(1\) \(1\) \(i\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
57.1
1.00000i
− 1.00000i
0 −1.00000 + 1.00000i 0 0 0 −7.00000 − 7.00000i 0 7.00000i 0
193.1 0 −1.00000 − 1.00000i 0 0 0 −7.00000 + 7.00000i 0 − 7.00000i 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
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 200.3.l.a ✓ 2
3.b odd 2 1 1800.3.v.a 2
4.b odd 2 1 400.3.p.f 2
5.b even 2 1 200.3.l.c yes 2
5.c odd 4 1 inner 200.3.l.a ✓ 2
5.c odd 4 1 200.3.l.c yes 2
15.d odd 2 1 1800.3.v.g 2
15.e even 4 1 1800.3.v.a 2
15.e even 4 1 1800.3.v.g 2
20.d odd 2 1 400.3.p.c 2
20.e even 4 1 400.3.p.c 2
20.e even 4 1 400.3.p.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.3.l.a ✓ 2 1.a even 1 1 trivial
200.3.l.a ✓ 2 5.c odd 4 1 inner
200.3.l.c yes 2 5.b even 2 1
200.3.l.c yes 2 5.c odd 4 1
400.3.p.c 2 20.d odd 2 1
400.3.p.c 2 20.e even 4 1
400.3.p.f 2 4.b odd 2 1
400.3.p.f 2 20.e even 4 1
1800.3.v.a 2 3.b odd 2 1
1800.3.v.a 2 15.e even 4 1
1800.3.v.g 2 15.d odd 2 1
1800.3.v.g 2 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(200, [\chi])\):

\( T_{3}^{2} + 2T_{3} + 2 \) Copy content Toggle raw display
\( T_{7}^{2} + 14T_{7} + 98 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 14T + 98 \) Copy content Toggle raw display
$11$ \( (T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 24T + 288 \) Copy content Toggle raw display
$17$ \( T^{2} + 40T + 800 \) Copy content Toggle raw display
$19$ \( T^{2} + 784 \) Copy content Toggle raw display
$23$ \( T^{2} + 18T + 162 \) Copy content Toggle raw display
$29$ \( T^{2} + 1156 \) Copy content Toggle raw display
$31$ \( (T + 40)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 32T + 512 \) Copy content Toggle raw display
$41$ \( (T - 32)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 14T + 98 \) Copy content Toggle raw display
$47$ \( T^{2} - 62T + 1922 \) Copy content Toggle raw display
$53$ \( T^{2} - 104T + 5408 \) Copy content Toggle raw display
$59$ \( T^{2} + 1936 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 162T + 13122 \) Copy content Toggle raw display
$71$ \( (T + 112)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 88T + 3872 \) Copy content Toggle raw display
$79$ \( T^{2} + 5184 \) Copy content Toggle raw display
$83$ \( T^{2} - 98T + 4802 \) Copy content Toggle raw display
$89$ \( T^{2} + 9604 \) Copy content Toggle raw display
$97$ \( T^{2} - 88T + 3872 \) Copy content Toggle raw display
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