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Properties

Label 800.2.y.f
Level $800$
Weight $2$
Character orbit 800.y
Analytic conductor $6.388$
Analytic rank $0$
Dimension $88$
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] = [800,2,Mod(101,800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(800, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.y (of order \(8\), 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: \(6.38803216170\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(22\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 88 q + 8 q^{4} - 8 q^{6} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 88 q + 8 q^{4} - 8 q^{6} + 8 q^{9} - 8 q^{11} + 24 q^{14} - 8 q^{16} + 8 q^{19} - 8 q^{21} + 16 q^{24} + 32 q^{26} + 8 q^{29} - 64 q^{31} + 24 q^{34} + 72 q^{36} + 8 q^{39} - 8 q^{41} + 8 q^{44} - 8 q^{46} - 48 q^{51} - 24 q^{54} - 56 q^{56} - 24 q^{59} + 24 q^{61} - 64 q^{64} - 8 q^{66} + 40 q^{69} - 40 q^{71} - 128 q^{74} - 8 q^{76} - 200 q^{84} + 24 q^{86} + 8 q^{89} - 8 q^{91} - 120 q^{94} - 56 q^{96} + 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1 −1.40863 − 0.125510i −0.772505 − 1.86499i 1.96849 + 0.353596i 0 0.854100 + 2.72405i 0.798508 + 0.798508i −2.72851 − 0.745154i −0.760109 + 0.760109i 0
101.2 −1.38439 − 0.288907i −0.0567199 − 0.136934i 1.83307 + 0.799919i 0 0.0389612 + 0.205956i −2.19453 − 2.19453i −2.30657 − 1.63698i 2.10579 − 2.10579i 0
101.3 −1.25386 + 0.654098i 0.596710 + 1.44059i 1.14431 − 1.64029i 0 −1.69047 − 1.41598i −1.37373 − 1.37373i −0.361890 + 2.80518i 0.402097 − 0.402097i 0
101.4 −1.24705 − 0.666991i 0.946090 + 2.28406i 1.11025 + 1.66354i 0 0.343631 − 3.47936i −0.233992 − 0.233992i −0.274967 − 2.81503i −2.20053 + 2.20053i 0
101.5 −1.14993 + 0.823203i −0.594494 − 1.43524i 0.644675 − 1.89325i 0 1.86512 + 1.16103i 3.08764 + 3.08764i 0.817197 + 2.70780i 0.414843 − 0.414843i 0
101.6 −0.914328 − 1.07889i 0.392381 + 0.947291i −0.328009 + 1.97292i 0 0.663258 − 1.28947i 3.09428 + 3.09428i 2.42847 − 1.45001i 1.37792 − 1.37792i 0
101.7 −0.876939 + 1.10949i −0.870016 − 2.10041i −0.461955 − 1.94592i 0 3.09334 + 0.876650i −3.08109 − 3.08109i 2.56409 + 1.19392i −1.53345 + 1.53345i 0
101.8 −0.803453 − 1.16381i −1.19554 − 2.88628i −0.708927 + 1.87014i 0 −2.39854 + 3.71037i −1.95573 − 1.95573i 2.74608 − 0.677509i −4.77998 + 4.77998i 0
101.9 −0.633607 + 1.26433i 1.12946 + 2.72675i −1.19708 − 1.60218i 0 −4.16316 − 0.299677i −0.911488 − 0.911488i 2.78418 − 0.498360i −4.03817 + 4.03817i 0
101.10 −0.377917 − 1.36278i 0.214640 + 0.518186i −1.71436 + 1.03004i 0 0.625059 − 0.488339i −0.589624 − 0.589624i 2.05161 + 1.94703i 1.89887 − 1.89887i 0
101.11 −0.0348706 − 1.41378i −0.277774 − 0.670605i −1.99757 + 0.0985990i 0 −0.938404 + 0.416096i 0.00242493 + 0.00242493i 0.209054 + 2.82069i 1.74877 − 1.74877i 0
101.12 0.0348706 + 1.41378i 0.277774 + 0.670605i −1.99757 + 0.0985990i 0 −0.938404 + 0.416096i −0.00242493 − 0.00242493i −0.209054 − 2.82069i 1.74877 − 1.74877i 0
101.13 0.377917 + 1.36278i −0.214640 − 0.518186i −1.71436 + 1.03004i 0 0.625059 − 0.488339i 0.589624 + 0.589624i −2.05161 − 1.94703i 1.89887 − 1.89887i 0
101.14 0.633607 − 1.26433i −1.12946 − 2.72675i −1.19708 − 1.60218i 0 −4.16316 − 0.299677i 0.911488 + 0.911488i −2.78418 + 0.498360i −4.03817 + 4.03817i 0
101.15 0.803453 + 1.16381i 1.19554 + 2.88628i −0.708927 + 1.87014i 0 −2.39854 + 3.71037i 1.95573 + 1.95573i −2.74608 + 0.677509i −4.77998 + 4.77998i 0
101.16 0.876939 − 1.10949i 0.870016 + 2.10041i −0.461955 − 1.94592i 0 3.09334 + 0.876650i 3.08109 + 3.08109i −2.56409 − 1.19392i −1.53345 + 1.53345i 0
101.17 0.914328 + 1.07889i −0.392381 − 0.947291i −0.328009 + 1.97292i 0 0.663258 − 1.28947i −3.09428 − 3.09428i −2.42847 + 1.45001i 1.37792 − 1.37792i 0
101.18 1.14993 − 0.823203i 0.594494 + 1.43524i 0.644675 − 1.89325i 0 1.86512 + 1.16103i −3.08764 − 3.08764i −0.817197 − 2.70780i 0.414843 − 0.414843i 0
101.19 1.24705 + 0.666991i −0.946090 − 2.28406i 1.11025 + 1.66354i 0 0.343631 − 3.47936i 0.233992 + 0.233992i 0.274967 + 2.81503i −2.20053 + 2.20053i 0
101.20 1.25386 − 0.654098i −0.596710 − 1.44059i 1.14431 − 1.64029i 0 −1.69047 − 1.41598i 1.37373 + 1.37373i 0.361890 − 2.80518i 0.402097 − 0.402097i 0
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
32.g even 8 1 inner
160.z even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.2.y.f 88
5.b even 2 1 inner 800.2.y.f 88
5.c odd 4 2 160.2.z.a ✓ 88
20.e even 4 2 640.2.z.a 88
32.g even 8 1 inner 800.2.y.f 88
160.u even 8 1 640.2.z.a 88
160.v odd 8 1 160.2.z.a ✓ 88
160.z even 8 1 inner 800.2.y.f 88
160.ba even 8 1 640.2.z.a 88
160.bb odd 8 1 160.2.z.a ✓ 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.z.a ✓ 88 5.c odd 4 2
160.2.z.a ✓ 88 160.v odd 8 1
160.2.z.a ✓ 88 160.bb odd 8 1
640.2.z.a 88 20.e even 4 2
640.2.z.a 88 160.u even 8 1
640.2.z.a 88 160.ba even 8 1
800.2.y.f 88 1.a even 1 1 trivial
800.2.y.f 88 5.b even 2 1 inner
800.2.y.f 88 32.g even 8 1 inner
800.2.y.f 88 160.z even 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{88} - 4 T_{3}^{86} + 8 T_{3}^{84} + 168 T_{3}^{82} + 18732 T_{3}^{80} - 62560 T_{3}^{78} + \cdots + 107495424 \) acting on \(S_{2}^{\mathrm{new}}(800, [\chi])\). Copy content Toggle raw display