↑

Properties

Label 2008.4.a.d
Level $2008$
Weight $4$
Character orbit 2008.a
Self dual yes
Analytic conductor $118.476$
Analytic rank $0$
Dimension $54$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2008,4,Mod(1,2008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2008, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2008.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2008 = 2^{3} \cdot 251 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2008.a (trivial)

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: yes
Analytic conductor: \(118.475835292\)
Analytic rank: \(0\)
Dimension: \(54\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 54 q + 5 q^{3} + 33 q^{5} + 4 q^{7} + 641 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 54 q + 5 q^{3} + 33 q^{5} + 4 q^{7} + 641 q^{9} + 55 q^{11} + 71 q^{13} + 101 q^{15} + 301 q^{17} - 122 q^{19} + 195 q^{21} + 317 q^{23} + 2111 q^{25} + 245 q^{27} + 253 q^{29} - 31 q^{31} + 994 q^{33} + 602 q^{35} + 482 q^{37} - 157 q^{39} + 1049 q^{41} - 218 q^{43} + 942 q^{45} + 1349 q^{47} + 5000 q^{49} - 689 q^{51} + 1701 q^{53} - 162 q^{55} + 2429 q^{57} + 710 q^{59} + 539 q^{61} + 798 q^{63} + 2357 q^{65} + 130 q^{67} + 546 q^{69} + 1347 q^{71} + 3185 q^{73} - 316 q^{75} + 2862 q^{77} - 1470 q^{79} + 10610 q^{81} + 2366 q^{83} + 1376 q^{85} - 32 q^{87} + 3566 q^{89} - 1986 q^{91} + 2984 q^{93} + 1655 q^{95} + 4344 q^{97} + 308 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 0 −10.0770 0 −17.9311 0 −29.2373 0 74.5465 0
1.2 0 −9.76150 0 14.6842 0 −9.20861 0 68.2869 0
1.3 0 −9.26461 0 −15.5969 0 15.6220 0 58.8330 0
1.4 0 −8.85033 0 10.3613 0 −18.1180 0 51.3283 0
1.5 0 −8.84537 0 13.5736 0 32.7953 0 51.2406 0
1.6 0 −8.80286 0 13.0381 0 −10.8782 0 50.4903 0
1.7 0 −8.28381 0 −11.2785 0 12.3758 0 41.6215 0
1.8 0 −7.55960 0 8.77209 0 −29.9147 0 30.1476 0
1.9 0 −7.33297 0 21.7147 0 11.7137 0 26.7724 0
1.10 0 −7.31720 0 −12.9976 0 20.8518 0 26.5414 0
1.11 0 −6.83333 0 −7.99011 0 −14.8983 0 19.6944 0
1.12 0 −6.44865 0 −5.28955 0 −21.9141 0 14.5850 0
1.13 0 −6.33097 0 −6.05418 0 −10.0969 0 13.0811 0
1.14 0 −5.86273 0 −21.2394 0 15.5711 0 7.37159 0
1.15 0 −5.53287 0 9.84899 0 33.6362 0 3.61263 0
1.16 0 −3.98373 0 6.01518 0 16.2448 0 −11.1299 0
1.17 0 −3.75725 0 −14.5006 0 2.80900 0 −12.8831 0
1.18 0 −3.62657 0 −8.39243 0 5.94293 0 −13.8480 0
1.19 0 −3.55317 0 1.36564 0 9.75678 0 −14.3750 0
1.20 0 −3.25359 0 13.4101 0 −20.0899 0 −16.4142 0
See all 54 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.54
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(251\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2008.4.a.d ✓ 54
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2008.4.a.d ✓ 54 1.a even 1 1 trivial