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Properties

Label 200.3.g.h
Level $200$
Weight $3$
Character orbit 200.g
Analytic conductor $5.450$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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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: \(8\)
Coefficient field: 8.0.53824000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 6x^{6} + 36x^{4} - 96x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 40)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + \beta_{6} q^{3} + (\beta_{5} + 2) q^{4} + ( - \beta_{7} + \beta_{5} - \beta_{2} + 2) q^{6} + (\beta_{4} + \beta_{3} - 2 \beta_1) q^{7} + ( - \beta_{6} - 3 \beta_{4} + \cdots + \beta_1) q^{8}+ \cdots + ( - 2 \beta_{7} + 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + \beta_{6} q^{3} + (\beta_{5} + 2) q^{4} + ( - \beta_{7} + \beta_{5} - \beta_{2} + 2) q^{6} + (\beta_{4} + \beta_{3} - 2 \beta_1) q^{7} + ( - \beta_{6} - 3 \beta_{4} + \cdots + \beta_1) q^{8}+ \cdots + ( - 3 \beta_{7} - 20) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{4} + 12 q^{6} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{4} + 12 q^{6} + 40 q^{9} + 32 q^{11} + 20 q^{14} - 72 q^{16} - 32 q^{19} - 112 q^{24} - 80 q^{26} - 152 q^{34} - 20 q^{36} + 192 q^{41} + 88 q^{44} - 60 q^{46} - 8 q^{49} + 128 q^{51} + 352 q^{54} + 224 q^{59} - 288 q^{64} - 152 q^{66} - 360 q^{74} + 152 q^{76} - 168 q^{81} + 760 q^{84} + 316 q^{86} - 112 q^{89} - 320 q^{91} + 300 q^{94} - 368 q^{96} - 160 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 6x^{6} + 36x^{4} - 96x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 2\nu^{3} - 4\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} + 2\nu^{4} - 12\nu^{2} ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 2\nu^{3} + 20\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 6\nu^{5} - 36\nu^{3} + 96\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + 6\nu^{4} - 36\nu^{2} + 64 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{7} + 10\nu^{5} - 28\nu^{3} ) / 64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{6} + 6\nu^{4} - 20\nu^{2} + 48 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - 2\beta_{5} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - 3\beta_{4} + 2\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{7} - 2\beta_{5} - 2\beta_{2} - 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{6} - 6\beta_{4} + 2\beta_{3} - 8\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{5} - 12\beta_{2} - 32 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -24\beta_{6} + 8\beta_{4} - 12\beta_{3} - 36\beta_1 \) 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
1.81907 − 0.831254i
1.81907 + 0.831254i
1.48020 − 1.34500i
1.48020 + 1.34500i
−1.48020 − 1.34500i
−1.48020 + 1.34500i
−1.81907 − 0.831254i
−1.81907 + 0.831254i
−1.81907 − 0.831254i 2.24849 2.61803 + 3.02422i 0 −4.09017 − 1.86907i − 6.40747i −2.24849 − 7.67752i −3.94427 0
51.2 −1.81907 + 0.831254i 2.24849 2.61803 − 3.02422i 0 −4.09017 + 1.86907i 6.40747i −2.24849 + 7.67752i −3.94427 0
51.3 −1.48020 − 1.34500i −4.79002 0.381966 + 3.98172i 0 7.09017 + 6.44256i 7.67752i 4.79002 − 6.40747i 13.9443 0
51.4 −1.48020 + 1.34500i −4.79002 0.381966 − 3.98172i 0 7.09017 − 6.44256i − 7.67752i 4.79002 + 6.40747i 13.9443 0
51.5 1.48020 − 1.34500i 4.79002 0.381966 − 3.98172i 0 7.09017 − 6.44256i 7.67752i −4.79002 − 6.40747i 13.9443 0
51.6 1.48020 + 1.34500i 4.79002 0.381966 + 3.98172i 0 7.09017 + 6.44256i − 7.67752i −4.79002 + 6.40747i 13.9443 0
51.7 1.81907 − 0.831254i −2.24849 2.61803 − 3.02422i 0 −4.09017 + 1.86907i − 6.40747i 2.24849 − 7.67752i −3.94427 0
51.8 1.81907 + 0.831254i −2.24849 2.61803 + 3.02422i 0 −4.09017 − 1.86907i 6.40747i 2.24849 + 7.67752i −3.94427 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.d odd 2 1 inner
40.e 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.h 8
4.b odd 2 1 800.3.g.h 8
5.b even 2 1 inner 200.3.g.h 8
5.c odd 4 2 40.3.e.c ✓ 8
8.b even 2 1 800.3.g.h 8
8.d odd 2 1 inner 200.3.g.h 8
15.e even 4 2 360.3.p.g 8
20.d odd 2 1 800.3.g.h 8
20.e even 4 2 160.3.e.c 8
40.e odd 2 1 inner 200.3.g.h 8
40.f even 2 1 800.3.g.h 8
40.i odd 4 2 160.3.e.c 8
40.k even 4 2 40.3.e.c ✓ 8
60.l odd 4 2 1440.3.p.g 8
80.i odd 4 2 1280.3.h.m 16
80.j even 4 2 1280.3.h.m 16
80.s even 4 2 1280.3.h.m 16
80.t odd 4 2 1280.3.h.m 16
120.q odd 4 2 360.3.p.g 8
120.w even 4 2 1440.3.p.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.3.e.c ✓ 8 5.c odd 4 2
40.3.e.c ✓ 8 40.k even 4 2
160.3.e.c 8 20.e even 4 2
160.3.e.c 8 40.i odd 4 2
200.3.g.h 8 1.a even 1 1 trivial
200.3.g.h 8 5.b even 2 1 inner
200.3.g.h 8 8.d odd 2 1 inner
200.3.g.h 8 40.e odd 2 1 inner
360.3.p.g 8 15.e even 4 2
360.3.p.g 8 120.q odd 4 2
800.3.g.h 8 4.b odd 2 1
800.3.g.h 8 8.b even 2 1
800.3.g.h 8 20.d odd 2 1
800.3.g.h 8 40.f even 2 1
1280.3.h.m 16 80.i odd 4 2
1280.3.h.m 16 80.j even 4 2
1280.3.h.m 16 80.s even 4 2
1280.3.h.m 16 80.t odd 4 2
1440.3.p.g 8 60.l odd 4 2
1440.3.p.g 8 120.w even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 28T_{3}^{2} + 116 \) acting on \(S_{3}^{\mathrm{new}}(200, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 6 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( (T^{4} - 28 T^{2} + 116)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 100 T^{2} + 2420)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 8 T - 4)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 320 T^{2} + 5120)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 368 T^{2} + 1856)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 8 T - 484)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 100 T^{2} + 500)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 1760 T^{2} + 37120)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2400 T^{2} + 928000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 3280 T^{2} + 1613120)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 48 T + 496)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 1372 T^{2} + 111476)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 5860 T^{2} + 444020)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 5760 T^{2} + 6635520)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 56 T - 1636)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 1080 T^{2} + 187920)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 12988 T^{2} + 41620916)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 12000 T^{2} + 35672320)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 7408 T^{2} + 3119936)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 22880 T^{2} + 129214720)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 1628 T^{2} + 116)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 28 T - 4924)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 44912 T^{2} + 344772416)^{2} \) Copy content Toggle raw display
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