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Properties

Label 8048.2.a.y
Level $8048$
Weight $2$
Character orbit 8048.a
Self dual yes
Analytic conductor $64.264$
Analytic rank $0$
Dimension $33$
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: \(33\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 33 q + 2 q^{3} + 12 q^{5} - 4 q^{7} + 43 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 33 q + 2 q^{3} + 12 q^{5} - 4 q^{7} + 43 q^{9} - 22 q^{11} + 25 q^{13} + 4 q^{15} + 17 q^{17} - 6 q^{19} + 18 q^{21} - 16 q^{23} + 47 q^{25} + 20 q^{27} + 47 q^{29} + 7 q^{31} - 6 q^{33} - 19 q^{35} + 75 q^{37} - 21 q^{39} + 22 q^{41} + 5 q^{43} + 33 q^{45} - 10 q^{47} + 31 q^{49} - 9 q^{51} + 64 q^{53} + 3 q^{55} + 5 q^{57} - 28 q^{59} + 49 q^{61} + 10 q^{63} + 46 q^{65} + 14 q^{67} + 30 q^{69} - 35 q^{71} + 19 q^{73} + 33 q^{75} + 32 q^{77} + 12 q^{79} + 57 q^{81} + 82 q^{85} + 5 q^{87} + 42 q^{89} + 15 q^{91} + 55 q^{93} - 33 q^{95} + 4 q^{97} - 22 q^{99}+O(q^{100}) \) 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 −3.29586 0 −1.31643 0 −2.99871 0 7.86271 0
1.2 0 −2.95302 0 0.296089 0 −1.50033 0 5.72032 0
1.3 0 −2.93503 0 3.00897 0 2.93938 0 5.61438 0
1.4 0 −2.66114 0 3.83009 0 −0.592406 0 4.08164 0
1.5 0 −2.60209 0 1.12945 0 1.15113 0 3.77086 0
1.6 0 −2.45195 0 0.279836 0 −3.64586 0 3.01204 0
1.7 0 −2.44174 0 −3.94731 0 −0.854513 0 2.96210 0
1.8 0 −1.79707 0 2.90936 0 −4.51295 0 0.229458 0
1.9 0 −1.66808 0 −3.27041 0 4.00229 0 −0.217521 0
1.10 0 −1.61089 0 −0.419657 0 5.20635 0 −0.405023 0
1.11 0 −1.23409 0 −1.43413 0 1.24533 0 −1.47702 0
1.12 0 −1.07783 0 −1.09164 0 −2.08785 0 −1.83829 0
1.13 0 −1.01188 0 3.66411 0 −1.84945 0 −1.97610 0
1.14 0 −0.368053 0 2.18272 0 −4.72026 0 −2.86454 0
1.15 0 −0.220191 0 1.20642 0 3.77303 0 −2.95152 0
1.16 0 −0.176667 0 2.93614 0 0.0214699 0 −2.96879 0
1.17 0 −0.0147975 0 −2.88853 0 1.61248 0 −2.99978 0
1.18 0 0.224625 0 −2.01216 0 −1.98253 0 −2.94954 0
1.19 0 0.254882 0 3.76559 0 2.29648 0 −2.93504 0
1.20 0 0.502871 0 −1.85170 0 −1.22131 0 −2.74712 0
See all 33 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.33
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.y 33
4.b odd 2 1 4024.2.a.f ✓ 33
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4024.2.a.f ✓ 33 4.b odd 2 1
8048.2.a.y 33 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8048))\):

\( T_{3}^{33} - 2 T_{3}^{32} - 69 T_{3}^{31} + 130 T_{3}^{30} + 2141 T_{3}^{29} - 3761 T_{3}^{28} + \cdots + 704 \) Copy content Toggle raw display
\( T_{5}^{33} - 12 T_{5}^{32} - 34 T_{5}^{31} + 915 T_{5}^{30} - 878 T_{5}^{29} - 30526 T_{5}^{28} + \cdots - 11206656 \) Copy content Toggle raw display
\( T_{7}^{33} + 4 T_{7}^{32} - 123 T_{7}^{31} - 513 T_{7}^{30} + 6578 T_{7}^{29} + 28866 T_{7}^{28} + \cdots - 56757896 \) Copy content Toggle raw display
\( T_{13}^{33} - 25 T_{13}^{32} + 54 T_{13}^{31} + 3522 T_{13}^{30} - 26723 T_{13}^{29} + \cdots + 186534937952 \) Copy content Toggle raw display