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Properties

Label 200.2.k.f
Level $200$
Weight $2$
Character orbit 200.k
Analytic conductor $1.597$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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 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.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{2} + ( - \beta_{3} + \beta_{2} - 1) q^{3} + 2 \beta_{2} q^{4} + ( - \beta_{3} + \beta_1 - 2) q^{6} + (2 \beta_{2} - 2) q^{8} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{2} + ( - \beta_{3} + \beta_{2} - 1) q^{3} + 2 \beta_{2} q^{4} + ( - \beta_{3} + \beta_1 - 2) q^{6} + (2 \beta_{2} - 2) q^{8} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{9} + ( - \beta_{3} + \beta_1 + 3) q^{11} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{12} - 4 q^{16} + ( - 2 \beta_{2} - 3 \beta_1 - 2) q^{17} + (4 \beta_{3} - 2 \beta_{2} + 2) q^{18} + ( - 3 \beta_{3} + \beta_{2} - 3 \beta_1) q^{19} + (3 \beta_{2} + 2 \beta_1 + 3) q^{22} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1) q^{24} + (5 \beta_{2} - 3 \beta_1 + 5) q^{27} + ( - 4 \beta_{2} - 4) q^{32} - \beta_{3} q^{33} + ( - 3 \beta_{3} - 4 \beta_{2} - 3 \beta_1) q^{34} + (4 \beta_{3} - 4 \beta_1 + 4) q^{36} + ( - 6 \beta_{3} + \beta_{2} - 1) q^{38} + (4 \beta_{3} - 4 \beta_1 + 3) q^{41} + ( - 6 \beta_{2} + 6) q^{43} + (2 \beta_{3} + 6 \beta_{2} + 2 \beta_1) q^{44} + (4 \beta_{3} - 4 \beta_{2} + 4) q^{48} + 7 \beta_{2} q^{49} + ( - \beta_{3} + \beta_1 - 5) q^{51} + ( - 3 \beta_{3} + 10 \beta_{2} - 3 \beta_1) q^{54} + ( - 10 \beta_{2} + 7 \beta_1 - 10) q^{57} - 6 \beta_{2} q^{59} - 8 \beta_{2} q^{64} + ( - \beta_{3} + \beta_1) q^{66} + (3 \beta_{2} + 7 \beta_1 + 3) q^{67} + ( - 6 \beta_{3} - 4 \beta_{2} + 4) q^{68} + (4 \beta_{2} - 8 \beta_1 + 4) q^{72} + ( - \beta_{3} + 6 \beta_{2} - 6) q^{73} + ( - 6 \beta_{3} + 6 \beta_1 - 2) q^{76} + ( - 2 \beta_{3} + 2 \beta_1 - 13) q^{81} + (3 \beta_{2} - 8 \beta_1 + 3) q^{82} + (9 \beta_{3} + \beta_{2} - 1) q^{83} + 12 q^{86} + (4 \beta_{3} + 6 \beta_{2} - 6) q^{88} + (2 \beta_{3} - 9 \beta_{2} + 2 \beta_1) q^{89} + (4 \beta_{3} - 4 \beta_1 + 8) q^{96} + (12 \beta_{2} + 12) q^{97} + (7 \beta_{2} - 7) q^{98} + (4 \beta_{3} + 6 \beta_{2} + 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 4 q^{3} - 8 q^{6} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} - 4 q^{3} - 8 q^{6} - 8 q^{8} + 12 q^{11} - 8 q^{12} - 16 q^{16} - 8 q^{17} + 8 q^{18} + 12 q^{22} + 20 q^{27} - 16 q^{32} + 16 q^{36} - 4 q^{38} + 12 q^{41} + 24 q^{43} + 16 q^{48} - 20 q^{51} - 40 q^{57} + 12 q^{67} + 16 q^{68} + 16 q^{72} - 24 q^{73} - 8 q^{76} - 52 q^{81} + 12 q^{82} - 4 q^{83} + 48 q^{86} - 24 q^{88} + 32 q^{96} + 48 q^{97} - 28 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

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

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(151\) \(177\)
\(\chi(n)\) \(-1\) \(-1\) \(\beta_{2}\)

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
−1.22474 + 1.22474i
1.22474 − 1.22474i
−1.22474 − 1.22474i
1.22474 + 1.22474i
1.00000 − 1.00000i −2.22474 − 2.22474i − 2.00000i 0 −4.44949 0 −2.00000 − 2.00000i 6.89898i 0
43.2 1.00000 − 1.00000i 0.224745 + 0.224745i − 2.00000i 0 0.449490 0 −2.00000 − 2.00000i − 2.89898i 0
107.1 1.00000 + 1.00000i −2.22474 + 2.22474i 2.00000i 0 −4.44949 0 −2.00000 + 2.00000i − 6.89898i 0
107.2 1.00000 + 1.00000i 0.224745 − 0.224745i 2.00000i 0 0.449490 0 −2.00000 + 2.00000i 2.89898i 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.f yes 4
4.b odd 2 1 800.2.o.f 4
5.b even 2 1 200.2.k.e ✓ 4
5.c odd 4 1 200.2.k.e ✓ 4
5.c odd 4 1 inner 200.2.k.f yes 4
8.b even 2 1 800.2.o.f 4
8.d odd 2 1 CM 200.2.k.f yes 4
20.d odd 2 1 800.2.o.e 4
20.e even 4 1 800.2.o.e 4
20.e even 4 1 800.2.o.f 4
40.e odd 2 1 200.2.k.e ✓ 4
40.f even 2 1 800.2.o.e 4
40.i odd 4 1 800.2.o.e 4
40.i odd 4 1 800.2.o.f 4
40.k even 4 1 200.2.k.e ✓ 4
40.k even 4 1 inner 200.2.k.f yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.k.e ✓ 4 5.b even 2 1
200.2.k.e ✓ 4 5.c odd 4 1
200.2.k.e ✓ 4 40.e odd 2 1
200.2.k.e ✓ 4 40.k even 4 1
200.2.k.f yes 4 1.a even 1 1 trivial
200.2.k.f yes 4 5.c odd 4 1 inner
200.2.k.f yes 4 8.d odd 2 1 CM
200.2.k.f yes 4 40.k even 4 1 inner
800.2.o.e 4 20.d odd 2 1
800.2.o.e 4 20.e even 4 1
800.2.o.e 4 40.f even 2 1
800.2.o.e 4 40.i odd 4 1
800.2.o.f 4 4.b odd 2 1
800.2.o.f 4 8.b even 2 1
800.2.o.f 4 20.e even 4 1
800.2.o.f 4 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 \(S_{2}^{\mathrm{new}}(200, [\chi])\):

\( T_{3}^{4} + 4T_{3}^{3} + 8T_{3}^{2} - 4T_{3} + 1 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 4 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 6 T + 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 8 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$19$ \( T^{4} + 110T^{2} + 2809 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 6 T - 87)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 12 T + 72)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} - 12 T^{3} + \cdots + 16641 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 24 T^{3} + \cdots + 4761 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 4 T^{3} + \cdots + 58081 \) Copy content Toggle raw display
$89$ \( T^{4} + 210T^{2} + 3249 \) Copy content Toggle raw display
$97$ \( (T^{2} - 24 T + 288)^{2} \) Copy content Toggle raw display
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