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Properties

Label 8048.2.a.w
Level $8048$
Weight $2$
Character orbit 8048.a
Self dual yes
Analytic conductor $64.264$
Analytic rank $0$
Dimension $29$
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] = [8048,2,Mod(1,8048)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8048, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8048.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8048 = 2^{4} \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8048.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: \(64.2636035467\)
Analytic rank: \(0\)
Dimension: \(29\)
Twist minimal: no (minimal twist has level 4024)
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.

\(\operatorname{Tr}(f)(q) = \) \( 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9} + 27 q^{11} + 16 q^{13} + 14 q^{15} - 15 q^{17} + 14 q^{19} + q^{21} + 25 q^{23} + 21 q^{25} + 25 q^{27} - 13 q^{29} + 27 q^{31} - 9 q^{33} + 29 q^{35} + 35 q^{37} + 38 q^{39} - 30 q^{41} + 38 q^{43} + q^{45} + 35 q^{47} + 14 q^{49} + 21 q^{51} + 2 q^{53} + 25 q^{55} - 25 q^{57} + 40 q^{59} + 10 q^{61} + 56 q^{63} - 50 q^{65} + 31 q^{67} + 11 q^{69} + 65 q^{71} - 23 q^{73} + 32 q^{75} + 13 q^{77} + 44 q^{79} - 7 q^{81} + 41 q^{83} + 26 q^{85} + 25 q^{87} - 48 q^{89} + 44 q^{91} + 25 q^{93} + 75 q^{95} - 18 q^{97} + 80 q^{99}+O(q^{100}) \) 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 −3.11496 0 0.544032 0 4.06617 0 6.70298 0
1.2 0 −2.62135 0 −1.76626 0 −2.03701 0 3.87147 0
1.3 0 −2.38925 0 −0.218073 0 2.23687 0 2.70851 0
1.4 0 −2.22567 0 −3.22615 0 −0.00440441 0 1.95359 0
1.5 0 −2.04161 0 −4.02753 0 −1.43352 0 1.16816 0
1.6 0 −1.71451 0 −1.13020 0 1.00303 0 −0.0604433 0
1.7 0 −1.62590 0 3.78842 0 2.93814 0 −0.356460 0
1.8 0 −1.30647 0 0.939477 0 −2.17911 0 −1.29314 0
1.9 0 −1.29900 0 2.09702 0 3.13973 0 −1.31260 0
1.10 0 −1.28849 0 −1.09309 0 −0.799248 0 −1.33979 0
1.11 0 −0.936010 0 1.25557 0 −0.290471 0 −2.12388 0
1.12 0 −0.454325 0 0.622756 0 4.14589 0 −2.79359 0
1.13 0 0.137144 0 1.57958 0 −4.44075 0 −2.98119 0
1.14 0 0.278789 0 −0.533193 0 −1.93823 0 −2.92228 0
1.15 0 0.502218 0 −3.38571 0 −3.39706 0 −2.74778 0
1.16 0 0.582422 0 −3.18976 0 3.00819 0 −2.66078 0
1.17 0 0.782310 0 2.74195 0 3.13445 0 −2.38799 0
1.18 0 0.795174 0 3.83818 0 −0.297725 0 −2.36770 0
1.19 0 1.10456 0 −2.70098 0 −3.17910 0 −1.77994 0
1.20 0 1.15550 0 −3.26045 0 1.92982 0 −1.66482 0
See all 29 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.29
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(503\) \(-1\)

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 8048.2.a.w 29
4.b odd 2 1 4024.2.a.e ✓ 29
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4024.2.a.e ✓ 29 4.b odd 2 1
8048.2.a.w 29 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8048))\):

\( T_{3}^{29} - 7 T_{3}^{28} - 29 T_{3}^{27} + 295 T_{3}^{26} + 203 T_{3}^{25} - 5370 T_{3}^{24} + \cdots - 6975 \) Copy content Toggle raw display
\( T_{5}^{29} + 4 T_{5}^{28} - 75 T_{5}^{27} - 297 T_{5}^{26} + 2432 T_{5}^{25} + 9505 T_{5}^{24} + \cdots + 16384 \) Copy content Toggle raw display
\( T_{7}^{29} - 13 T_{7}^{28} - 24 T_{7}^{27} + 969 T_{7}^{26} - 1832 T_{7}^{25} - 29380 T_{7}^{24} + \cdots + 64747 \) Copy content Toggle raw display
\( T_{13}^{29} - 16 T_{13}^{28} - 47 T_{13}^{27} + 2000 T_{13}^{26} - 4659 T_{13}^{25} - 94601 T_{13}^{24} + \cdots - 3458323023 \) Copy content Toggle raw display