Properties

Label 200.3.g.b
Level $200$
Weight $3$
Character orbit 200.g
Self dual yes
Analytic conductor $5.450$
Analytic rank $0$
Dimension $2$
CM discriminant -8
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [200,3,Mod(51,200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("200.51");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 200.g (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(5.44960528721\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = 2\sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + (\beta + 1) q^{3} + 4 q^{4} + ( - 2 \beta - 2) q^{6} - 8 q^{8} + (2 \beta + 16) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + (\beta + 1) q^{3} + 4 q^{4} + ( - 2 \beta - 2) q^{6} - 8 q^{8} + (2 \beta + 16) q^{9} + (3 \beta - 7) q^{11} + (4 \beta + 4) q^{12} + 16 q^{16} + ( - 6 \beta - 1) q^{17} + ( - 4 \beta - 32) q^{18} + ( - 3 \beta + 17) q^{19} + ( - 6 \beta + 14) q^{22} + ( - 8 \beta - 8) q^{24} + (9 \beta + 55) q^{27} - 32 q^{32} + ( - 4 \beta + 65) q^{33} + (12 \beta + 2) q^{34} + (8 \beta + 64) q^{36} + (6 \beta - 34) q^{38} + ( - 12 \beta + 23) q^{41} + 14 q^{43} + (12 \beta - 28) q^{44} + (16 \beta + 16) q^{48} + 49 q^{49} + ( - 7 \beta - 145) q^{51} + ( - 18 \beta - 110) q^{54} + (14 \beta - 55) q^{57} - 82 q^{59} + 64 q^{64} + (8 \beta - 130) q^{66} + ( - 21 \beta - 31) q^{67} + ( - 24 \beta - 4) q^{68} + ( - 16 \beta - 128) q^{72} + (6 \beta + 71) q^{73} + ( - 12 \beta + 68) q^{76} + (46 \beta + 127) q^{81} + (24 \beta - 46) q^{82} + ( - 9 \beta - 79) q^{83} - 28 q^{86} + ( - 24 \beta + 56) q^{88} + ( - 18 \beta - 73) q^{89} + ( - 32 \beta - 32) q^{96} - 94 q^{97} - 98 q^{98} + (34 \beta + 32) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 2 q^{3} + 8 q^{4} - 4 q^{6} - 16 q^{8} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} + 2 q^{3} + 8 q^{4} - 4 q^{6} - 16 q^{8} + 32 q^{9} - 14 q^{11} + 8 q^{12} + 32 q^{16} - 2 q^{17} - 64 q^{18} + 34 q^{19} + 28 q^{22} - 16 q^{24} + 110 q^{27} - 64 q^{32} + 130 q^{33} + 4 q^{34} + 128 q^{36} - 68 q^{38} + 46 q^{41} + 28 q^{43} - 56 q^{44} + 32 q^{48} + 98 q^{49} - 290 q^{51} - 220 q^{54} - 110 q^{57} - 164 q^{59} + 128 q^{64} - 260 q^{66} - 62 q^{67} - 8 q^{68} - 256 q^{72} + 142 q^{73} + 136 q^{76} + 254 q^{81} - 92 q^{82} - 158 q^{83} - 56 q^{86} + 112 q^{88} - 146 q^{89} - 64 q^{96} - 188 q^{97} - 196 q^{98} + 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
−2.44949
2.44949
−2.00000 −3.89898 4.00000 0 7.79796 0 −8.00000 6.20204 0
51.2 −2.00000 5.89898 4.00000 0 −11.7980 0 −8.00000 25.7980 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 \(\Q(\sqrt{-2}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 200.3.g.b 2
4.b odd 2 1 800.3.g.b 2
5.b even 2 1 200.3.g.d yes 2
5.c odd 4 2 200.3.e.b 4
8.b even 2 1 800.3.g.b 2
8.d odd 2 1 CM 200.3.g.b 2
20.d odd 2 1 800.3.g.d 2
20.e even 4 2 800.3.e.b 4
40.e odd 2 1 200.3.g.d yes 2
40.f even 2 1 800.3.g.d 2
40.i odd 4 2 800.3.e.b 4
40.k even 4 2 200.3.e.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.3.e.b 4 5.c odd 4 2
200.3.e.b 4 40.k even 4 2
200.3.g.b 2 1.a even 1 1 trivial
200.3.g.b 2 8.d odd 2 1 CM
200.3.g.d yes 2 5.b even 2 1
200.3.g.d yes 2 40.e odd 2 1
800.3.e.b 4 20.e even 4 2
800.3.e.b 4 40.i odd 4 2
800.3.g.b 2 4.b odd 2 1
800.3.g.b 2 8.b even 2 1
800.3.g.d 2 20.d odd 2 1
800.3.g.d 2 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2T_{3} - 23 \) acting on \(S_{3}^{\mathrm{new}}(200, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T - 23 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 14T - 167 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2T - 863 \) Copy content Toggle raw display
$19$ \( T^{2} - 34T + 73 \) 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
$41$ \( T^{2} - 46T - 2927 \) Copy content Toggle raw display
$43$ \( (T - 14)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( (T + 82)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 62T - 9623 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 142T + 4177 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 158T + 4297 \) Copy content Toggle raw display
$89$ \( T^{2} + 146T - 2447 \) Copy content Toggle raw display
$97$ \( (T + 94)^{2} \) Copy content Toggle raw display
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