Properties

Label 200.4.c.b
Level $200$
Weight $4$
Character orbit 200.c
Analytic conductor $11.800$
Analytic rank $1$
Dimension $2$
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: \(1\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 9 i q^{3} - 26 i q^{7} - 54 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 9 i q^{3} - 26 i q^{7} - 54 q^{9} - 59 q^{11} + 28 i q^{13} - 5 i q^{17} - 109 q^{19} + 234 q^{21} - 194 i q^{23} - 243 i q^{27} + 32 q^{29} + 10 q^{31} - 531 i q^{33} + 198 i q^{37} - 252 q^{39} + 117 q^{41} + 388 i q^{43} + 68 i q^{47} - 333 q^{49} + 45 q^{51} - 18 i q^{53} - 981 i q^{57} - 392 q^{59} - 710 q^{61} + 1404 i q^{63} + 253 i q^{67} + 1746 q^{69} - 612 q^{71} - 549 i q^{73} + 1534 i q^{77} - 414 q^{79} + 729 q^{81} - 121 i q^{83} + 288 i q^{87} + 81 q^{89} + 728 q^{91} + 90 i q^{93} + 1502 i q^{97} + 3186 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 108 q^{9} - 118 q^{11} - 218 q^{19} + 468 q^{21} + 64 q^{29} + 20 q^{31} - 504 q^{39} + 234 q^{41} - 666 q^{49} + 90 q^{51} - 784 q^{59} - 1420 q^{61} + 3492 q^{69} - 1224 q^{71} - 828 q^{79} + 1458 q^{81} + 162 q^{89} + 1456 q^{91} + 6372 q^{99}+O(q^{100}) \) 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 9.00000i 0 0 0 26.0000i 0 −54.0000 0
49.2 0 9.00000i 0 0 0 26.0000i 0 −54.0000 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.b 2
3.b odd 2 1 1800.4.f.w 2
4.b odd 2 1 400.4.c.b 2
5.b even 2 1 inner 200.4.c.b 2
5.c odd 4 1 200.4.a.b 1
5.c odd 4 1 200.4.a.j yes 1
15.d odd 2 1 1800.4.f.w 2
15.e even 4 1 1800.4.a.c 1
15.e even 4 1 1800.4.a.bh 1
20.d odd 2 1 400.4.c.b 2
20.e even 4 1 400.4.a.a 1
20.e even 4 1 400.4.a.t 1
40.i odd 4 1 1600.4.a.c 1
40.i odd 4 1 1600.4.a.bz 1
40.k even 4 1 1600.4.a.b 1
40.k even 4 1 1600.4.a.by 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.4.a.b 1 5.c odd 4 1
200.4.a.j yes 1 5.c odd 4 1
200.4.c.b 2 1.a even 1 1 trivial
200.4.c.b 2 5.b even 2 1 inner
400.4.a.a 1 20.e even 4 1
400.4.a.t 1 20.e even 4 1
400.4.c.b 2 4.b odd 2 1
400.4.c.b 2 20.d odd 2 1
1600.4.a.b 1 40.k even 4 1
1600.4.a.c 1 40.i odd 4 1
1600.4.a.by 1 40.k even 4 1
1600.4.a.bz 1 40.i odd 4 1
1800.4.a.c 1 15.e even 4 1
1800.4.a.bh 1 15.e even 4 1
1800.4.f.w 2 3.b odd 2 1
1800.4.f.w 2 15.d odd 2 1

Hecke kernels

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

\( T_{3}^{2} + 81 \) Copy content Toggle raw display
\( T_{7}^{2} + 676 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 81 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 676 \) Copy content Toggle raw display
$11$ \( (T + 59)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 784 \) Copy content Toggle raw display
$17$ \( T^{2} + 25 \) Copy content Toggle raw display
$19$ \( (T + 109)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 37636 \) Copy content Toggle raw display
$29$ \( (T - 32)^{2} \) Copy content Toggle raw display
$31$ \( (T - 10)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 39204 \) Copy content Toggle raw display
$41$ \( (T - 117)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 150544 \) Copy content Toggle raw display
$47$ \( T^{2} + 4624 \) Copy content Toggle raw display
$53$ \( T^{2} + 324 \) Copy content Toggle raw display
$59$ \( (T + 392)^{2} \) Copy content Toggle raw display
$61$ \( (T + 710)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 64009 \) Copy content Toggle raw display
$71$ \( (T + 612)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 301401 \) Copy content Toggle raw display
$79$ \( (T + 414)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 14641 \) Copy content Toggle raw display
$89$ \( (T - 81)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 2256004 \) Copy content Toggle raw display
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