Properties

Label 200.2.k.d
Level $200$
Weight $2$
Character orbit 200.k
Analytic conductor $1.597$
Analytic rank $0$
Dimension $2$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [200,2,Mod(43,200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(200, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("200.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 200.k (of order \(4\), degree \(2\), 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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

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

\(f(q)\) \(=\) \( q + (i + 1) q^{2} + ( - 2 i + 2) q^{3} + 2 i q^{4} + 4 q^{6} + (2 i - 2) q^{8} - 5 i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (i + 1) q^{2} + ( - 2 i + 2) q^{3} + 2 i q^{4} + 4 q^{6} + (2 i - 2) q^{8} - 5 i q^{9} - 6 q^{11} + (4 i + 4) q^{12} - 4 q^{16} + (4 i + 4) q^{17} + ( - 5 i + 5) q^{18} - 2 i q^{19} + ( - 6 i - 6) q^{22} + 8 i q^{24} + ( - 4 i - 4) q^{27} + ( - 4 i - 4) q^{32} + (12 i - 12) q^{33} + 8 i q^{34} + 10 q^{36} + ( - 2 i + 2) q^{38} - 6 q^{41} + ( - 6 i + 6) q^{43} - 12 i q^{44} + (8 i - 8) q^{48} + 7 i q^{49} + 16 q^{51} - 8 i q^{54} + ( - 4 i - 4) q^{57} - 6 i q^{59} - 8 i q^{64} - 24 q^{66} + ( - 6 i - 6) q^{67} + (8 i - 8) q^{68} + (10 i + 10) q^{72} + ( - 12 i + 12) q^{73} + 4 q^{76} - q^{81} + ( - 6 i - 6) q^{82} + ( - 2 i + 2) q^{83} + 12 q^{86} + ( - 12 i + 12) q^{88} + 18 i q^{89} - 16 q^{96} + (12 i + 12) q^{97} + (7 i - 7) q^{98} + 30 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 4 q^{3} + 8 q^{6} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 4 q^{3} + 8 q^{6} - 4 q^{8} - 12 q^{11} + 8 q^{12} - 8 q^{16} + 8 q^{17} + 10 q^{18} - 12 q^{22} - 8 q^{27} - 8 q^{32} - 24 q^{33} + 20 q^{36} + 4 q^{38} - 12 q^{41} + 12 q^{43} - 16 q^{48} + 32 q^{51} - 8 q^{57} - 48 q^{66} - 12 q^{67} - 16 q^{68} + 20 q^{72} + 24 q^{73} + 8 q^{76} - 2 q^{81} - 12 q^{82} + 4 q^{83} + 24 q^{86} + 24 q^{88} - 32 q^{96} + 24 q^{97} - 14 q^{98}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
1.00000i
1.00000i
1.00000 1.00000i 2.00000 + 2.00000i 2.00000i 0 4.00000 0 −2.00000 2.00000i 5.00000i 0
107.1 1.00000 + 1.00000i 2.00000 2.00000i 2.00000i 0 4.00000 0 −2.00000 + 2.00000i 5.00000i 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
5.c odd 4 1 inner
40.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 200.2.k.d yes 2
4.b odd 2 1 800.2.o.a 2
5.b even 2 1 200.2.k.a 2
5.c odd 4 1 200.2.k.a 2
5.c odd 4 1 inner 200.2.k.d yes 2
8.b even 2 1 800.2.o.a 2
8.d odd 2 1 CM 200.2.k.d yes 2
20.d odd 2 1 800.2.o.d 2
20.e even 4 1 800.2.o.a 2
20.e even 4 1 800.2.o.d 2
40.e odd 2 1 200.2.k.a 2
40.f even 2 1 800.2.o.d 2
40.i odd 4 1 800.2.o.a 2
40.i odd 4 1 800.2.o.d 2
40.k even 4 1 200.2.k.a 2
40.k even 4 1 inner 200.2.k.d yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.k.a 2 5.b even 2 1
200.2.k.a 2 5.c odd 4 1
200.2.k.a 2 40.e odd 2 1
200.2.k.a 2 40.k even 4 1
200.2.k.d yes 2 1.a even 1 1 trivial
200.2.k.d yes 2 5.c odd 4 1 inner
200.2.k.d yes 2 8.d odd 2 1 CM
200.2.k.d yes 2 40.k even 4 1 inner
800.2.o.a 2 4.b odd 2 1
800.2.o.a 2 8.b even 2 1
800.2.o.a 2 20.e even 4 1
800.2.o.a 2 40.i odd 4 1
800.2.o.d 2 20.d odd 2 1
800.2.o.d 2 20.e even 4 1
800.2.o.d 2 40.f even 2 1
800.2.o.d 2 40.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(200, [\chi])\):

\( T_{3}^{2} - 4T_{3} + 8 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$3$ \( T^{2} - 4T + 8 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 8T + 32 \) Copy content Toggle raw display
$19$ \( T^{2} + 4 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 12T + 72 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 36 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 12T + 72 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 24T + 288 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 4T + 8 \) Copy content Toggle raw display
$89$ \( T^{2} + 324 \) Copy content Toggle raw display
$97$ \( T^{2} - 24T + 288 \) Copy content Toggle raw display
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