Properties

Label 200.2.m.a
Level $200$
Weight $2$
Character orbit 200.m
Analytic conductor $1.597$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [200,2,Mod(41,200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(200, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("200.41");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 200.m (of order \(5\), degree \(4\), minimal)

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: no
Analytic conductor: \(1.59700804043\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{3} + ( - 2 \zeta_{10}^{2} + \zeta_{10} - 2) q^{5} + (\zeta_{10}^{3} - \zeta_{10}^{2} + 1) q^{7} - 2 \zeta_{10} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{3} + ( - 2 \zeta_{10}^{2} + \zeta_{10} - 2) q^{5} + (\zeta_{10}^{3} - \zeta_{10}^{2} + 1) q^{7} - 2 \zeta_{10} q^{9} + ( - 2 \zeta_{10}^{3} + 2) q^{11} + ( - \zeta_{10}^{2} - 4 \zeta_{10} - 1) q^{13} + (5 \zeta_{10}^{3} - 5 \zeta_{10}^{2} + \cdots - 5) q^{15} + \cdots + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} - 4) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 5 q^{3} - 5 q^{5} + 6 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 5 q^{3} - 5 q^{5} + 6 q^{7} - 2 q^{9} + 6 q^{11} - 7 q^{13} - 5 q^{15} - 8 q^{17} - 9 q^{19} + 10 q^{21} + 9 q^{23} - 5 q^{25} + 5 q^{27} - 3 q^{29} + 13 q^{31} - 5 q^{35} + q^{37} - 25 q^{39} + 16 q^{41} + 22 q^{43} + 10 q^{45} + 4 q^{47} - 14 q^{49} - 20 q^{51} - 19 q^{53} - 20 q^{55} - 10 q^{57} - 6 q^{59} - 23 q^{61} + 2 q^{63} + 20 q^{65} - 22 q^{67} + 15 q^{69} + 10 q^{71} - 11 q^{73} + 4 q^{77} + 15 q^{79} + 11 q^{81} - q^{83} + 20 q^{85} - 30 q^{87} - 4 q^{89} + 2 q^{91} + 60 q^{93} + 35 q^{95} + 36 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(151\) \(177\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{10}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1
0.809017 0.587785i
−0.309017 + 0.951057i
−0.309017 0.951057i
0.809017 + 0.587785i
0 0.690983 + 2.12663i 0 −1.80902 + 1.31433i 0 0.381966 0 −1.61803 + 1.17557i 0
81.1 0 1.80902 1.31433i 0 −0.690983 + 2.12663i 0 2.61803 0 0.618034 1.90211i 0
121.1 0 1.80902 + 1.31433i 0 −0.690983 2.12663i 0 2.61803 0 0.618034 + 1.90211i 0
161.1 0 0.690983 2.12663i 0 −1.80902 1.31433i 0 0.381966 0 −1.61803 1.17557i 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
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 200.2.m.a 4
4.b odd 2 1 400.2.u.a 4
5.b even 2 1 1000.2.m.a 4
5.c odd 4 2 1000.2.q.a 8
25.d even 5 1 inner 200.2.m.a 4
25.d even 5 1 5000.2.a.c 2
25.e even 10 1 1000.2.m.a 4
25.e even 10 1 5000.2.a.a 2
25.f odd 20 2 1000.2.q.a 8
100.h odd 10 1 10000.2.a.i 2
100.j odd 10 1 400.2.u.a 4
100.j odd 10 1 10000.2.a.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.m.a 4 1.a even 1 1 trivial
200.2.m.a 4 25.d even 5 1 inner
400.2.u.a 4 4.b odd 2 1
400.2.u.a 4 100.j odd 10 1
1000.2.m.a 4 5.b even 2 1
1000.2.m.a 4 25.e even 10 1
1000.2.q.a 8 5.c odd 4 2
1000.2.q.a 8 25.f odd 20 2
5000.2.a.a 2 25.e even 10 1
5000.2.a.c 2 25.d even 5 1
10000.2.a.g 2 100.j odd 10 1
10000.2.a.i 2 100.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 5T_{3}^{3} + 15T_{3}^{2} - 25T_{3} + 25 \) acting on \(S_{2}^{\mathrm{new}}(200, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 5 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$5$ \( T^{4} + 5 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} - 3 T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 6 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( T^{4} + 7 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$17$ \( T^{4} + 8 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( T^{4} + 9 T^{3} + \cdots + 1681 \) Copy content Toggle raw display
$23$ \( T^{4} - 9 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$29$ \( T^{4} + 3 T^{3} + \cdots + 841 \) Copy content Toggle raw display
$31$ \( T^{4} - 13 T^{3} + \cdots + 1681 \) Copy content Toggle raw display
$37$ \( T^{4} - T^{3} + \cdots + 961 \) Copy content Toggle raw display
$41$ \( T^{4} - 16 T^{3} + \cdots + 1936 \) Copy content Toggle raw display
$43$ \( (T^{2} - 11 T + 29)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 4 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$53$ \( T^{4} + 19 T^{3} + \cdots + 6241 \) Copy content Toggle raw display
$59$ \( T^{4} + 6 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$61$ \( T^{4} + 23 T^{3} + \cdots + 14641 \) Copy content Toggle raw display
$67$ \( T^{4} + 22 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$71$ \( T^{4} - 10 T^{3} + \cdots + 3025 \) Copy content Toggle raw display
$73$ \( T^{4} + 11 T^{3} + \cdots + 14641 \) Copy content Toggle raw display
$79$ \( T^{4} - 15 T^{3} + \cdots + 3025 \) Copy content Toggle raw display
$83$ \( T^{4} + T^{3} + \cdots + 3721 \) Copy content Toggle raw display
$89$ \( T^{4} + 4 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$97$ \( T^{4} - 36 T^{3} + \cdots + 77841 \) Copy content Toggle raw display
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