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Properties

Label 2008.4.a.a
Level $2008$
Weight $4$
Character orbit 2008.a
Self dual yes
Analytic conductor $118.476$
Analytic rank $1$
Dimension $40$
CM no
Inner twists $1$

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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: \(1\)
Dimension: \(40\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{3} - 32 q^{5} - 10 q^{7} + 200 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{3} - 32 q^{5} - 10 q^{7} + 200 q^{9} - 66 q^{11} - 72 q^{13} - 19 q^{15} - 209 q^{17} + 125 q^{19} - 57 q^{21} - 373 q^{23} + 186 q^{25} - 79 q^{27} - 298 q^{29} + 341 q^{31} - 590 q^{33} - 98 q^{35} - 295 q^{37} + 155 q^{39} - 1329 q^{41} + 169 q^{43} - 813 q^{45} - 907 q^{47} + 198 q^{49} + 25 q^{51} - 1426 q^{53} + 718 q^{55} - 1675 q^{57} - 529 q^{59} - 132 q^{61} - 714 q^{63} - 2193 q^{65} - 741 q^{67} - 972 q^{69} - 925 q^{71} - 2071 q^{73} - 991 q^{75} - 1912 q^{77} + 1690 q^{79} - 1864 q^{81} - 1867 q^{83} - 1174 q^{85} - 1424 q^{87} - 2842 q^{89} - 166 q^{91} - 3340 q^{93} - 1005 q^{95} - 3222 q^{97} - 1573 q^{99}+O(q^{100}) \) 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 −9.75914 0 2.49080 0 −5.59596 0 68.2408 0
1.2 0 −9.53286 0 −0.408602 0 27.1688 0 63.8754 0
1.3 0 −8.16921 0 −9.56525 0 −29.2699 0 39.7360 0
1.4 0 −7.90028 0 0.788612 0 10.9006 0 35.4144 0
1.5 0 −7.67803 0 −17.3282 0 0.712374 0 31.9522 0
1.6 0 −6.74127 0 −3.82368 0 10.2384 0 18.4447 0
1.7 0 −6.38113 0 9.95924 0 17.8666 0 13.7188 0
1.8 0 −6.33265 0 20.4845 0 −2.18125 0 13.1024 0
1.9 0 −6.23855 0 −2.40171 0 −10.1365 0 11.9195 0
1.10 0 −4.91735 0 11.6913 0 10.8338 0 −2.81964 0
1.11 0 −4.91169 0 10.3172 0 −32.0561 0 −2.87525 0
1.12 0 −4.35570 0 −8.91285 0 −31.3542 0 −8.02786 0
1.13 0 −3.89516 0 −21.6630 0 −15.1601 0 −11.8277 0
1.14 0 −3.11635 0 −13.9491 0 33.8550 0 −17.2884 0
1.15 0 −2.69611 0 12.6760 0 −8.37003 0 −19.7310 0
1.16 0 −2.17893 0 11.6879 0 10.1945 0 −22.2523 0
1.17 0 −2.13154 0 −4.81619 0 −16.9267 0 −22.4565 0
1.18 0 −1.87284 0 −14.6678 0 30.0049 0 −23.4925 0
1.19 0 −1.32857 0 −15.8143 0 −25.0725 0 −25.2349 0
1.20 0 −0.916396 0 4.67816 0 −10.2578 0 −26.1602 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.40
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.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2008.4.a.a ✓ 40 1.a even 1 1 trivial