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Properties

Label 200.4.k.j
Level $200$
Weight $4$
Character orbit 200.k
Analytic conductor $11.800$
Analytic rank $0$
Dimension $32$
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] = [200,4,Mod(43,200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(200, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("200.43");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 200.k (of order \(4\), degree \(2\), 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: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{2} + 4 q^{3} - 16 q^{6} + 44 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{2} + 4 q^{3} - 16 q^{6} + 44 q^{8} - 8 q^{11} - 28 q^{12} + 72 q^{16} - 48 q^{17} + 278 q^{18} - 68 q^{22} - 92 q^{26} - 104 q^{27} - 620 q^{28} - 288 q^{32} + 112 q^{33} + 476 q^{36} - 636 q^{38} - 8 q^{41} - 1020 q^{42} + 868 q^{43} + 1328 q^{46} + 784 q^{48} + 1480 q^{51} + 1900 q^{52} - 2392 q^{56} - 104 q^{57} + 700 q^{58} + 2880 q^{62} - 4360 q^{66} + 1852 q^{67} - 1196 q^{68} - 5596 q^{72} + 744 q^{73} + 4312 q^{76} - 2240 q^{78} - 1240 q^{81} - 5828 q^{82} - 2676 q^{83} + 6976 q^{86} + 2864 q^{88} - 1704 q^{91} + 7500 q^{92} - 10656 q^{96} + 584 q^{97} + 3814 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −2.76244 − 0.607409i 2.02737 + 2.02737i 7.26211 + 3.35586i 0 −4.36904 − 6.83193i 1.63237 + 1.63237i −18.0227 − 13.6814i − 18.7795i 0
43.2 −2.59830 + 1.11752i −4.00710 − 4.00710i 5.50228 − 5.80731i 0 14.8897 + 5.93361i 10.5258 + 10.5258i −7.80676 + 21.2380i 5.11376i 0
43.3 −2.58698 + 1.14348i 4.39586 + 4.39586i 5.38489 − 5.91633i 0 −16.3986 − 6.34539i −18.5943 − 18.5943i −7.16538 + 21.4629i 11.6471i 0
43.4 −2.24907 − 1.71514i −3.49003 − 3.49003i 2.11662 + 7.71491i 0 1.86344 + 13.8352i 4.97302 + 4.97302i 8.47169 − 20.9817i − 2.63942i 0
43.5 −1.14348 + 2.58698i 4.39586 + 4.39586i −5.38489 − 5.91633i 0 −16.3986 + 6.34539i 18.5943 + 18.5943i 21.4629 − 7.16538i 11.6471i 0
43.6 −1.11752 + 2.59830i −4.00710 − 4.00710i −5.50228 − 5.80731i 0 14.8897 − 5.93361i −10.5258 − 10.5258i 21.2380 − 7.80676i 5.11376i 0
43.7 −0.995877 − 2.64731i 0.102537 + 0.102537i −6.01646 + 5.27278i 0 0.169333 − 0.373562i −15.5472 − 15.5472i 19.9503 + 10.6764i − 26.9790i 0
43.8 −0.589963 − 2.76621i 6.56128 + 6.56128i −7.30389 + 3.26393i 0 14.2790 − 22.0208i 8.83176 + 8.83176i 13.3377 + 18.2785i 59.1008i 0
43.9 0.583098 − 2.76767i −6.15076 − 6.15076i −7.31999 − 3.22764i 0 −20.6098 + 13.4368i 16.5614 + 16.5614i −13.2013 + 18.3773i 48.6638i 0
43.10 0.607409 + 2.76244i 2.02737 + 2.02737i −7.26211 + 3.35586i 0 −4.36904 + 6.83193i −1.63237 − 1.63237i −13.6814 − 18.0227i − 18.7795i 0
43.11 1.68522 − 2.27157i 1.56085 + 1.56085i −2.32005 − 7.65620i 0 6.17594 − 0.915194i 18.5221 + 18.5221i −21.3014 − 7.63226i − 22.1275i 0
43.12 1.71514 + 2.24907i −3.49003 − 3.49003i −2.11662 + 7.71491i 0 1.86344 − 13.8352i −4.97302 − 4.97302i −20.9817 + 8.47169i − 2.63942i 0
43.13 2.27157 − 1.68522i 1.56085 + 1.56085i 2.32005 − 7.65620i 0 6.17594 + 0.915194i −18.5221 − 18.5221i −7.63226 − 21.3014i − 22.1275i 0
43.14 2.64731 + 0.995877i 0.102537 + 0.102537i 6.01646 + 5.27278i 0 0.169333 + 0.373562i 15.5472 + 15.5472i 10.6764 + 19.9503i − 26.9790i 0
43.15 2.76621 + 0.589963i 6.56128 + 6.56128i 7.30389 + 3.26393i 0 14.2790 + 22.0208i −8.83176 − 8.83176i 18.2785 + 13.3377i 59.1008i 0
43.16 2.76767 − 0.583098i −6.15076 − 6.15076i 7.31999 − 3.22764i 0 −20.6098 − 13.4368i −16.5614 − 16.5614i 18.3773 − 13.2013i 48.6638i 0
107.1 −2.76244 + 0.607409i 2.02737 − 2.02737i 7.26211 − 3.35586i 0 −4.36904 + 6.83193i 1.63237 − 1.63237i −18.0227 + 13.6814i 18.7795i 0
107.2 −2.59830 − 1.11752i −4.00710 + 4.00710i 5.50228 + 5.80731i 0 14.8897 − 5.93361i 10.5258 − 10.5258i −7.80676 − 21.2380i − 5.11376i 0
107.3 −2.58698 − 1.14348i 4.39586 − 4.39586i 5.38489 + 5.91633i 0 −16.3986 + 6.34539i −18.5943 + 18.5943i −7.16538 − 21.4629i − 11.6471i 0
107.4 −2.24907 + 1.71514i −3.49003 + 3.49003i 2.11662 − 7.71491i 0 1.86344 − 13.8352i 4.97302 − 4.97302i 8.47169 + 20.9817i 2.63942i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
8.d odd 2 1 inner
40.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 200.4.k.j 32
5.b even 2 1 40.4.k.a ✓ 32
5.c odd 4 1 40.4.k.a ✓ 32
5.c odd 4 1 inner 200.4.k.j 32
8.d odd 2 1 inner 200.4.k.j 32
20.d odd 2 1 160.4.o.a 32
20.e even 4 1 160.4.o.a 32
40.e odd 2 1 40.4.k.a ✓ 32
40.f even 2 1 160.4.o.a 32
40.i odd 4 1 160.4.o.a 32
40.k even 4 1 40.4.k.a ✓ 32
40.k even 4 1 inner 200.4.k.j 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.4.k.a ✓ 32 5.b even 2 1
40.4.k.a ✓ 32 5.c odd 4 1
40.4.k.a ✓ 32 40.e odd 2 1
40.4.k.a ✓ 32 40.k even 4 1
160.4.o.a 32 20.d odd 2 1
160.4.o.a 32 20.e even 4 1
160.4.o.a 32 40.f even 2 1
160.4.o.a 32 40.i odd 4 1
200.4.k.j 32 1.a even 1 1 trivial
200.4.k.j 32 5.c odd 4 1 inner
200.4.k.j 32 8.d odd 2 1 inner
200.4.k.j 32 40.k even 4 1 inner

Hecke kernels

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

\( T_{3}^{16} - 2 T_{3}^{15} + 2 T_{3}^{14} + 52 T_{3}^{13} + 8068 T_{3}^{12} - 10656 T_{3}^{11} + \cdots + 165894400 \) Copy content Toggle raw display
\( T_{7}^{32} + 1559484 T_{7}^{28} + 916766007440 T_{7}^{24} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display