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Properties

Label 200.12.c.c
Level $200$
Weight $12$
Character orbit 200.c
Analytic conductor $153.669$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [200,12,Mod(49,200)]
 
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([0, 0, 1]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("200.49");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 200.c (of order \(2\), degree \(1\), not 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: \(153.668636112\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{109})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 55x^{2} + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{15} \)
Twist minimal: no (minimal twist has level 8)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - 14 \beta_1) q^{3} + ( - 42 \beta_{2} + 22764 \beta_1) q^{7} + (56 \beta_{3} - 270101) q^{9} + (693 \beta_{3} + 79540) q^{11} + (948 \beta_{2} - 262619 \beta_1) q^{13} + (10824 \beta_{2} + 357721 \beta_1) q^{17}+ \cdots + ( - 182725753 \beta_{3} - 4157458628) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 1080404 q^{9} + 318160 q^{11} + 43733200 q^{19} + 80105088 q^{21} + 457655400 q^{29} + 129444224 q^{31} - 1751818144 q^{39} + 2402428392 q^{41} - 3532138148 q^{49} - 19249975840 q^{51} + 12025853168 q^{59}+ \cdots - 16629834512 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 55x^{2} + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{3} + 56\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 64\nu^{3} + 5248\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 128\nu^{2} + 3520 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} - 32\beta_1 ) / 128 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 3520 ) / 128 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{2} + 656\beta_1 ) / 32 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
− 5.72015i
− 4.72015i
4.72015i
5.72015i
0 − 696.180i 0 0 0 73591.5i 0 −307519. 0
49.2 0 − 640.180i 0 0 0 − 17464.5i 0 −232683. 0
49.3 0 640.180i 0 0 0 17464.5i 0 −232683. 0
49.4 0 696.180i 0 0 0 − 73591.5i 0 −307519. 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 200.12.c.c 4
5.b even 2 1 inner 200.12.c.c 4
5.c odd 4 1 8.12.a.b ✓ 2
5.c odd 4 1 200.12.a.d 2
15.e even 4 1 72.12.a.e 2
20.e even 4 1 16.12.a.d 2
40.i odd 4 1 64.12.a.h 2
40.k even 4 1 64.12.a.k 2
60.l odd 4 1 144.12.a.p 2
80.i odd 4 1 256.12.b.h 4
80.j even 4 1 256.12.b.k 4
80.s even 4 1 256.12.b.k 4
80.t odd 4 1 256.12.b.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.12.a.b ✓ 2 5.c odd 4 1
16.12.a.d 2 20.e even 4 1
64.12.a.h 2 40.i odd 4 1
64.12.a.k 2 40.k even 4 1
72.12.a.e 2 15.e even 4 1
144.12.a.p 2 60.l odd 4 1
200.12.a.d 2 5.c odd 4 1
200.12.c.c 4 1.a even 1 1 trivial
200.12.c.c 4 5.b even 2 1 inner
256.12.b.h 4 80.i odd 4 1
256.12.b.h 4 80.t odd 4 1
256.12.b.k 4 80.j even 4 1
256.12.b.k 4 80.s even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 894496T_{3}^{2} + 198630662400 \) acting on \(S_{12}^{\mathrm{new}}(200, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + \cdots + 198630662400 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 16\!\cdots\!44 \) Copy content Toggle raw display
$11$ \( (T^{2} - 159080 T - 208087277936)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{2} + \cdots + 110143192291984)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 22\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( (T^{2} + \cdots + 11\!\cdots\!96)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + \cdots + 681230587110400)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{2} + \cdots + 17\!\cdots\!88)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 21\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 64\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 35\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( (T^{2} + \cdots + 75\!\cdots\!48)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots + 14\!\cdots\!20)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 22\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( (T^{2} + \cdots - 17\!\cdots\!60)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 66\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots - 89\!\cdots\!80)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 81\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{2} + \cdots - 24\!\cdots\!96)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 33\!\cdots\!36 \) Copy content Toggle raw display
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