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Properties

Label 800.2.u.g
Level $800$
Weight $2$
Character orbit 800.u
Analytic conductor $6.388$
Analytic rank $0$
Dimension $24$
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] = [800,2,Mod(161,800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(800, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.161");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.u (of order \(5\), 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: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 5 q^{3} + 6 q^{5} - 8 q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 5 q^{3} + 6 q^{5} - 8 q^{7} - 9 q^{9} - 7 q^{11} + 17 q^{13} - 2 q^{15} - 9 q^{17} + 24 q^{19} - 5 q^{21} + 3 q^{23} + 4 q^{25} + 17 q^{27} + 3 q^{29} - 13 q^{31} - 11 q^{33} - 7 q^{35} - 14 q^{37} - 14 q^{39} + 5 q^{41} - 12 q^{43} + q^{45} - 9 q^{47} + 8 q^{49} + 18 q^{51} - 19 q^{53} - 29 q^{55} - 88 q^{57} + 10 q^{59} + 6 q^{61} + 48 q^{63} - 16 q^{65} + 25 q^{67} - 23 q^{69} + 69 q^{71} + 20 q^{73} - 78 q^{75} - q^{77} - 34 q^{79} - 27 q^{81} - 51 q^{83} - 3 q^{85} - 14 q^{87} + 43 q^{89} + 31 q^{91} - 54 q^{93} + 58 q^{95} + 10 q^{97} - 2 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1 0 −0.982553 + 3.02399i 0 −1.64106 − 1.51885i 0 −2.53328 0 −5.75204 − 4.17910i 0
161.2 0 −0.605029 + 1.86209i 0 2.17273 − 0.528439i 0 −0.0565786 0 −0.674264 − 0.489881i 0
161.3 0 −0.385847 + 1.18751i 0 −1.98681 + 1.02596i 0 2.49468 0 1.16574 + 0.846959i 0
161.4 0 0.180759 − 0.556318i 0 −0.804890 − 2.08618i 0 2.91336 0 2.15023 + 1.56224i 0
161.5 0 0.662517 − 2.03902i 0 1.34136 − 1.78906i 0 −3.82348 0 −1.29161 − 0.938412i 0
161.6 0 0.703102 − 2.16393i 0 1.30063 + 1.81889i 0 1.24136 0 −1.76118 − 1.27957i 0
321.1 0 −2.15974 − 1.56914i 0 2.09881 − 0.771367i 0 −0.173295 0 1.27521 + 3.92470i 0
321.2 0 −0.189423 − 0.137624i 0 −1.23608 + 1.86336i 0 −0.734169 0 −0.910110 − 2.80103i 0
321.3 0 0.0318873 + 0.0231675i 0 2.06729 + 0.852239i 0 −4.39229 0 −0.926571 − 2.85169i 0
321.4 0 0.780422 + 0.567010i 0 0.639317 − 2.14273i 0 3.42688 0 −0.639493 − 1.96816i 0
321.5 0 2.10936 + 1.53254i 0 1.01301 + 1.99344i 0 1.67419 0 1.17366 + 3.61216i 0
321.6 0 2.35455 + 1.71068i 0 −1.96431 − 1.06840i 0 −4.03739 0 1.69042 + 5.20257i 0
481.1 0 −2.15974 + 1.56914i 0 2.09881 + 0.771367i 0 −0.173295 0 1.27521 − 3.92470i 0
481.2 0 −0.189423 + 0.137624i 0 −1.23608 − 1.86336i 0 −0.734169 0 −0.910110 + 2.80103i 0
481.3 0 0.0318873 − 0.0231675i 0 2.06729 − 0.852239i 0 −4.39229 0 −0.926571 + 2.85169i 0
481.4 0 0.780422 − 0.567010i 0 0.639317 + 2.14273i 0 3.42688 0 −0.639493 + 1.96816i 0
481.5 0 2.10936 − 1.53254i 0 1.01301 − 1.99344i 0 1.67419 0 1.17366 − 3.61216i 0
481.6 0 2.35455 − 1.71068i 0 −1.96431 + 1.06840i 0 −4.03739 0 1.69042 − 5.20257i 0
641.1 0 −0.982553 − 3.02399i 0 −1.64106 + 1.51885i 0 −2.53328 0 −5.75204 + 4.17910i 0
641.2 0 −0.605029 − 1.86209i 0 2.17273 + 0.528439i 0 −0.0565786 0 −0.674264 + 0.489881i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.2.u.g yes 24
4.b odd 2 1 800.2.u.f ✓ 24
25.d even 5 1 inner 800.2.u.g yes 24
100.j odd 10 1 800.2.u.f ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.2.u.f ✓ 24 4.b odd 2 1
800.2.u.f ✓ 24 100.j odd 10 1
800.2.u.g yes 24 1.a even 1 1 trivial
800.2.u.g yes 24 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 5 T_{3}^{23} + 26 T_{3}^{22} - 94 T_{3}^{21} + 357 T_{3}^{20} - 946 T_{3}^{19} + 2870 T_{3}^{18} + \cdots + 16 \) acting on \(S_{2}^{\mathrm{new}}(800, [\chi])\). Copy content Toggle raw display