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Properties

Label 200.2.q.a
Level $200$
Weight $2$
Character orbit 200.q
Analytic conductor $1.597$
Analytic rank $0$
Dimension $32$
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] = [200,2,Mod(9,200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(200, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("200.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 200.q (of order \(10\), degree \(4\), 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: \(1.59700804043\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{5} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{5} + 10 q^{9} + 6 q^{11} + 12 q^{15} - 6 q^{19} - 4 q^{21} - 30 q^{23} + 6 q^{25} - 2 q^{29} + 6 q^{31} + 8 q^{35} - 40 q^{37} - 12 q^{39} - 12 q^{45} - 20 q^{47} - 60 q^{49} - 60 q^{51} - 30 q^{53} - 28 q^{55} - 30 q^{59} + 14 q^{61} - 20 q^{63} - 26 q^{65} - 4 q^{69} + 12 q^{71} + 40 q^{73} + 16 q^{75} + 16 q^{79} - 52 q^{81} + 30 q^{83} + 60 q^{85} + 110 q^{87} + 24 q^{89} - 4 q^{91} + 68 q^{95} + 30 q^{97} + 92 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 0 −2.92097 + 0.949081i 0 −0.820266 + 2.08018i 0 − 4.49756i 0 5.20427 − 3.78112i 0
9.2 0 −2.57764 + 0.837527i 0 1.74676 − 1.39601i 0 − 0.760910i 0 3.51574 − 2.55433i 0
9.3 0 −0.811472 + 0.263663i 0 −0.390840 + 2.20165i 0 1.47738i 0 −1.83808 + 1.33545i 0
9.4 0 −0.685617 + 0.222770i 0 −2.21883 − 0.277142i 0 4.42421i 0 −2.00661 + 1.45789i 0
9.5 0 0.726954 − 0.236202i 0 −0.0130818 − 2.23603i 0 − 4.32704i 0 −1.95438 + 1.41994i 0
9.6 0 1.54478 − 0.501931i 0 2.03630 − 0.923846i 0 2.51754i 0 −0.292627 + 0.212606i 0
9.7 0 1.92858 − 0.626633i 0 1.20432 + 1.88404i 0 0.498275i 0 0.899692 − 0.653665i 0
9.8 0 2.79539 − 0.908276i 0 −2.16239 + 0.569263i 0 − 3.13612i 0 4.56217 − 3.31461i 0
89.1 0 −2.92097 − 0.949081i 0 −0.820266 − 2.08018i 0 4.49756i 0 5.20427 + 3.78112i 0
89.2 0 −2.57764 − 0.837527i 0 1.74676 + 1.39601i 0 0.760910i 0 3.51574 + 2.55433i 0
89.3 0 −0.811472 − 0.263663i 0 −0.390840 − 2.20165i 0 − 1.47738i 0 −1.83808 − 1.33545i 0
89.4 0 −0.685617 − 0.222770i 0 −2.21883 + 0.277142i 0 − 4.42421i 0 −2.00661 − 1.45789i 0
89.5 0 0.726954 + 0.236202i 0 −0.0130818 + 2.23603i 0 4.32704i 0 −1.95438 − 1.41994i 0
89.6 0 1.54478 + 0.501931i 0 2.03630 + 0.923846i 0 − 2.51754i 0 −0.292627 − 0.212606i 0
89.7 0 1.92858 + 0.626633i 0 1.20432 − 1.88404i 0 − 0.498275i 0 0.899692 + 0.653665i 0
89.8 0 2.79539 + 0.908276i 0 −2.16239 − 0.569263i 0 3.13612i 0 4.56217 + 3.31461i 0
129.1 0 −1.98040 + 2.72579i 0 −2.13417 − 0.667306i 0 0.794375i 0 −2.58089 − 7.94317i 0
129.2 0 −1.31712 + 1.81286i 0 2.19384 − 0.432536i 0 1.85909i 0 −0.624612 − 1.92236i 0
129.3 0 −0.403532 + 0.555415i 0 1.58398 − 1.57829i 0 − 3.54998i 0 0.781404 + 2.40491i 0
129.4 0 −0.371479 + 0.511297i 0 −2.16854 + 0.545356i 0 2.09441i 0 0.803623 + 2.47330i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 200.2.q.a ✓ 32
4.b odd 2 1 400.2.y.d 32
5.b even 2 1 1000.2.q.c 32
5.c odd 4 1 1000.2.m.d 32
5.c odd 4 1 1000.2.m.e 32
25.d even 5 1 1000.2.q.c 32
25.e even 10 1 inner 200.2.q.a ✓ 32
25.f odd 20 1 1000.2.m.d 32
25.f odd 20 1 1000.2.m.e 32
25.f odd 20 1 5000.2.a.q 16
25.f odd 20 1 5000.2.a.r 16
100.h odd 10 1 400.2.y.d 32
100.l even 20 1 10000.2.a.bq 16
100.l even 20 1 10000.2.a.br 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.q.a ✓ 32 1.a even 1 1 trivial
200.2.q.a ✓ 32 25.e even 10 1 inner
400.2.y.d 32 4.b odd 2 1
400.2.y.d 32 100.h odd 10 1
1000.2.m.d 32 5.c odd 4 1
1000.2.m.d 32 25.f odd 20 1
1000.2.m.e 32 5.c odd 4 1
1000.2.m.e 32 25.f odd 20 1
1000.2.q.c 32 5.b even 2 1
1000.2.q.c 32 25.d even 5 1
5000.2.a.q 16 25.f odd 20 1
5000.2.a.r 16 25.f odd 20 1
10000.2.a.bq 16 100.l even 20 1
10000.2.a.br 16 100.l even 20 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(200, [\chi])\).