Properties

Label 200.4.c.f
Level $200$
Weight $4$
Character orbit 200.c
Analytic conductor $11.800$
Analytic rank $0$
Dimension $2$
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] = [200,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("200.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(11.8003820011\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta q^{3} - 8 \beta q^{7} + 11 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta q^{3} - 8 \beta q^{7} + 11 q^{9} + 36 q^{11} - 21 \beta q^{13} + 55 \beta q^{17} + 116 q^{19} + 64 q^{21} + 8 \beta q^{23} + 76 \beta q^{27} - 198 q^{29} + 240 q^{31} + 72 \beta q^{33} + 129 \beta q^{37} + 168 q^{39} + 442 q^{41} - 146 \beta q^{43} - 196 \beta q^{47} + 87 q^{49} - 440 q^{51} + 71 \beta q^{53} + 232 \beta q^{57} + 348 q^{59} - 570 q^{61} - 88 \beta q^{63} - 346 \beta q^{67} - 64 q^{69} + 168 q^{71} - 67 \beta q^{73} - 288 \beta q^{77} - 784 q^{79} - 311 q^{81} + 282 \beta q^{83} - 396 \beta q^{87} - 1034 q^{89} - 672 q^{91} + 480 \beta q^{93} + 191 \beta q^{97} + 396 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 22 q^{9} + 72 q^{11} + 232 q^{19} + 128 q^{21} - 396 q^{29} + 480 q^{31} + 336 q^{39} + 884 q^{41} + 174 q^{49} - 880 q^{51} + 696 q^{59} - 1140 q^{61} - 128 q^{69} + 336 q^{71} - 1568 q^{79} - 622 q^{81} - 2068 q^{89} - 1344 q^{91} + 792 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.00000i
1.00000i
0 4.00000i 0 0 0 16.0000i 0 11.0000 0
49.2 0 4.00000i 0 0 0 16.0000i 0 11.0000 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 200.4.c.f 2
3.b odd 2 1 1800.4.f.d 2
4.b odd 2 1 400.4.c.h 2
5.b even 2 1 inner 200.4.c.f 2
5.c odd 4 1 40.4.a.b 1
5.c odd 4 1 200.4.a.d 1
15.d odd 2 1 1800.4.f.d 2
15.e even 4 1 360.4.a.f 1
15.e even 4 1 1800.4.a.h 1
20.d odd 2 1 400.4.c.h 2
20.e even 4 1 80.4.a.b 1
20.e even 4 1 400.4.a.p 1
35.f even 4 1 1960.4.a.e 1
40.i odd 4 1 320.4.a.e 1
40.i odd 4 1 1600.4.a.bk 1
40.k even 4 1 320.4.a.j 1
40.k even 4 1 1600.4.a.q 1
60.l odd 4 1 720.4.a.d 1
80.i odd 4 1 1280.4.d.d 2
80.j even 4 1 1280.4.d.m 2
80.s even 4 1 1280.4.d.m 2
80.t odd 4 1 1280.4.d.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.4.a.b 1 5.c odd 4 1
80.4.a.b 1 20.e even 4 1
200.4.a.d 1 5.c odd 4 1
200.4.c.f 2 1.a even 1 1 trivial
200.4.c.f 2 5.b even 2 1 inner
320.4.a.e 1 40.i odd 4 1
320.4.a.j 1 40.k even 4 1
360.4.a.f 1 15.e even 4 1
400.4.a.p 1 20.e even 4 1
400.4.c.h 2 4.b odd 2 1
400.4.c.h 2 20.d odd 2 1
720.4.a.d 1 60.l odd 4 1
1280.4.d.d 2 80.i odd 4 1
1280.4.d.d 2 80.t odd 4 1
1280.4.d.m 2 80.j even 4 1
1280.4.d.m 2 80.s even 4 1
1600.4.a.q 1 40.k even 4 1
1600.4.a.bk 1 40.i odd 4 1
1800.4.a.h 1 15.e even 4 1
1800.4.f.d 2 3.b odd 2 1
1800.4.f.d 2 15.d odd 2 1
1960.4.a.e 1 35.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(200, [\chi])\):

\( T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{2} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 256 \) Copy content Toggle raw display
$11$ \( (T - 36)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1764 \) Copy content Toggle raw display
$17$ \( T^{2} + 12100 \) Copy content Toggle raw display
$19$ \( (T - 116)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 256 \) Copy content Toggle raw display
$29$ \( (T + 198)^{2} \) Copy content Toggle raw display
$31$ \( (T - 240)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 66564 \) Copy content Toggle raw display
$41$ \( (T - 442)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 85264 \) Copy content Toggle raw display
$47$ \( T^{2} + 153664 \) Copy content Toggle raw display
$53$ \( T^{2} + 20164 \) Copy content Toggle raw display
$59$ \( (T - 348)^{2} \) Copy content Toggle raw display
$61$ \( (T + 570)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 478864 \) Copy content Toggle raw display
$71$ \( (T - 168)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 17956 \) Copy content Toggle raw display
$79$ \( (T + 784)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 318096 \) Copy content Toggle raw display
$89$ \( (T + 1034)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 145924 \) Copy content Toggle raw display
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