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Properties

Label 200.3.g.f
Level $200$
Weight $3$
Character orbit 200.g
Analytic conductor $5.450$
Analytic rank $0$
Dimension $6$
CM no
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: no
Analytic conductor: \(5.44960528721\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.189974000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 8x^{4} - 8x^{3} + 23x^{2} + 3x + 40 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} - \beta_{2} q^{3} + (\beta_{5} - \beta_{2} - \beta_1) q^{4} + ( - \beta_{4} - \beta_{3} - \beta_{2} + \cdots + 1) q^{6}+ \cdots + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} - \beta_{2} q^{3} + (\beta_{5} - \beta_{2} - \beta_1) q^{4} + ( - \beta_{4} - \beta_{3} - \beta_{2} + \cdots + 1) q^{6}+ \cdots + ( - \beta_{3} + 20 \beta_{2} + \cdots + 45) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} - 2 q^{3} - 7 q^{4} + q^{6} - 3 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{2} - 2 q^{3} - 7 q^{4} + q^{6} - 3 q^{8} - 8 q^{9} + 30 q^{11} + 27 q^{12} + 4 q^{14} - 39 q^{16} + 2 q^{17} + 20 q^{18} - 2 q^{19} + 37 q^{22} + 59 q^{24} - 36 q^{26} - 62 q^{27} + 40 q^{28} - 47 q^{32} - 138 q^{33} + 89 q^{34} - 8 q^{36} + 113 q^{38} - 70 q^{41} - 220 q^{42} - 76 q^{43} - 77 q^{44} + 184 q^{46} - 57 q^{48} - 138 q^{49} + 114 q^{51} + 200 q^{52} - 69 q^{54} - 176 q^{56} + 78 q^{57} - 20 q^{58} - 44 q^{59} + 320 q^{62} + 113 q^{64} - 101 q^{66} - 18 q^{67} + 143 q^{68} - 240 q^{72} + 18 q^{73} + 204 q^{74} + 155 q^{76} + 180 q^{78} - 142 q^{81} - 323 q^{82} + 398 q^{83} - 400 q^{84} + 218 q^{86} - 317 q^{88} + 98 q^{89} + 288 q^{91} + 600 q^{92} - 316 q^{94} - 409 q^{96} + 76 q^{97} - 213 q^{98} + 304 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 8x^{4} - 8x^{3} + 23x^{2} + 3x + 40 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} - 8\nu^{3} + 4\nu^{2} - 7\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + 2\nu^{4} - 6\nu^{3} + 2\nu^{2} - 13\nu - 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{4} + 5\nu^{3} - 9\nu^{2} + 9\nu - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} - 8\nu^{3} + 12\nu^{2} - 15\nu + 8 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + \beta_{4} - 5\beta_{3} + 4\beta_{2} - 5\beta _1 - 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{5} - 4\beta_{4} - 12\beta_{3} + 12\beta_{2} - 15\beta _1 - 24 \) Copy content Toggle raw display

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

Character values

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

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

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
1.94755 + 1.76129i
1.94755 − 1.76129i
0.198648 + 1.83244i
0.198648 − 1.83244i
−0.646201 + 1.13579i
−0.646201 − 1.13579i
−0.947553 − 1.76129i 0.486535 −2.20429 + 3.33783i 0 −0.461018 − 0.856930i − 5.23644i 7.96757 + 0.719613i −8.76328 0
51.2 −0.947553 + 1.76129i 0.486535 −2.20429 − 3.33783i 0 −0.461018 + 0.856930i 5.23644i 7.96757 − 0.719613i −8.76328 0
51.3 0.801352 − 1.83244i −4.03404 −2.71567 − 2.93686i 0 −3.23269 + 7.39214i 11.1194i −7.55783 + 2.62284i 7.27349 0
51.4 0.801352 + 1.83244i −4.03404 −2.71567 + 2.93686i 0 −3.23269 − 7.39214i − 11.1194i −7.55783 − 2.62284i 7.27349 0
51.5 1.64620 − 1.13579i 2.54751 1.41995 − 3.73948i 0 4.19371 − 2.89344i − 8.05846i −1.90974 − 7.76871i −2.51021 0
51.6 1.64620 + 1.13579i 2.54751 1.41995 + 3.73948i 0 4.19371 + 2.89344i 8.05846i −1.90974 + 7.76871i −2.51021 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 200.3.g.f yes 6
4.b odd 2 1 800.3.g.f 6
5.b even 2 1 200.3.g.e ✓ 6
5.c odd 4 2 200.3.e.c 12
8.b even 2 1 800.3.g.f 6
8.d odd 2 1 inner 200.3.g.f yes 6
20.d odd 2 1 800.3.g.e 6
20.e even 4 2 800.3.e.c 12
40.e odd 2 1 200.3.g.e ✓ 6
40.f even 2 1 800.3.g.e 6
40.i odd 4 2 800.3.e.c 12
40.k even 4 2 200.3.e.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.3.e.c 12 5.c odd 4 2
200.3.e.c 12 40.k even 4 2
200.3.g.e ✓ 6 5.b even 2 1
200.3.g.e ✓ 6 40.e odd 2 1
200.3.g.f yes 6 1.a even 1 1 trivial
200.3.g.f yes 6 8.d odd 2 1 inner
800.3.e.c 12 20.e even 4 2
800.3.e.c 12 40.i odd 4 2
800.3.g.e 6 20.d odd 2 1
800.3.g.e 6 40.f even 2 1
800.3.g.f 6 4.b odd 2 1
800.3.g.f 6 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} + T_{3}^{2} - 11T_{3} + 5 \) acting on \(S_{3}^{\mathrm{new}}(200, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 3 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( (T^{3} + T^{2} - 11 T + 5)^{2} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 216 T^{4} + \cdots + 220160 \) Copy content Toggle raw display
$11$ \( (T^{3} - 15 T^{2} + \cdots + 53)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 616 T^{4} + \cdots + 3522560 \) Copy content Toggle raw display
$17$ \( (T^{3} - T^{2} - 321 T - 1055)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + T^{2} - 475 T + 373)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 3136 T^{4} + \cdots + 352256000 \) Copy content Toggle raw display
$29$ \( T^{6} + 2920 T^{4} + \cdots + 352256000 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 2201600000 \) Copy content Toggle raw display
$37$ \( T^{6} + 3336 T^{4} + \cdots + 220160 \) Copy content Toggle raw display
$41$ \( (T^{3} + 35 T^{2} + \cdots - 26447)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 38 T^{2} + \cdots - 20000)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 5176 T^{4} + \cdots + 465858560 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 4072079360 \) Copy content Toggle raw display
$59$ \( (T^{3} + 22 T^{2} + \cdots + 6752)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 176353664000 \) Copy content Toggle raw display
$67$ \( (T^{3} + 9 T^{2} + \cdots - 18155)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 110982656000 \) Copy content Toggle raw display
$73$ \( (T^{3} - 9 T^{2} + \cdots - 156935)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 10176896000 \) Copy content Toggle raw display
$83$ \( (T^{3} - 199 T^{2} + \cdots - 131155)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} - 49 T^{2} + \cdots + 97393)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 38 T^{2} + \cdots + 118520)^{2} \) Copy content Toggle raw display
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