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Properties

Label 8048.2.a.x
Level $8048$
Weight $2$
Character orbit 8048.a
Self dual yes
Analytic conductor $64.264$
Analytic rank $1$
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: \(1\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 33 q - 10 q^{3} - 12 q^{7} + 47 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 33 q - 10 q^{3} - 12 q^{7} + 47 q^{9} - 22 q^{11} - 17 q^{13} - 22 q^{15} + 9 q^{17} - 16 q^{19} + 6 q^{21} - 36 q^{23} + 47 q^{25} - 34 q^{27} + 13 q^{29} - 21 q^{31} + 14 q^{33} - 33 q^{35} - 55 q^{37} - 37 q^{39} + 42 q^{41} - 23 q^{43} + 5 q^{45} - 20 q^{47} + 55 q^{49} - 53 q^{51} - 32 q^{53} - 35 q^{55} + 21 q^{57} - 20 q^{59} - 15 q^{61} - 48 q^{63} + 34 q^{65} - 66 q^{67} - 4 q^{69} - 61 q^{71} + 19 q^{73} - 59 q^{75} + 2 q^{77} - 62 q^{79} + 77 q^{81} - 36 q^{83} - 14 q^{85} - 43 q^{87} + 34 q^{89} - 41 q^{91} - 11 q^{93} - 61 q^{95} - 8 q^{97} - 98 q^{99}+O(q^{100}) \) 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 −3.42195 0 −3.41581 0 −3.07051 0 8.70973 0
1.2 0 −3.41276 0 2.47839 0 2.17339 0 8.64692 0
1.3 0 −3.14063 0 2.58012 0 −1.35399 0 6.86353 0
1.4 0 −2.91492 0 1.91993 0 −3.81516 0 5.49674 0
1.5 0 −2.90120 0 −2.20357 0 1.74877 0 5.41693 0
1.6 0 −2.74246 0 2.09147 0 −3.15984 0 4.52108 0
1.7 0 −2.60958 0 −1.79981 0 2.96082 0 3.80991 0
1.8 0 −2.40618 0 1.57414 0 −2.69643 0 2.78968 0
1.9 0 −2.03324 0 4.46117 0 −0.828843 0 1.13405 0
1.10 0 −1.94894 0 2.86334 0 4.75146 0 0.798369 0
1.11 0 −1.78801 0 −0.988126 0 −3.73717 0 0.196987 0
1.12 0 −1.65470 0 −3.99833 0 4.68473 0 −0.261967 0
1.13 0 −1.05277 0 −1.97996 0 −4.35506 0 −1.89168 0
1.14 0 −0.940347 0 2.53897 0 −0.182218 0 −2.11575 0
1.15 0 −0.885832 0 −0.282055 0 2.95093 0 −2.21530 0
1.16 0 −0.649284 0 −2.92377 0 1.90965 0 −2.57843 0
1.17 0 −0.410265 0 −4.21207 0 −3.36092 0 −2.83168 0
1.18 0 −0.281703 0 −0.580455 0 −2.82149 0 −2.92064 0
1.19 0 −0.221515 0 4.40155 0 −5.16895 0 −2.95093 0
1.20 0 0.105343 0 −1.39521 0 1.93153 0 −2.98890 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.x 33
4.b odd 2 1 4024.2.a.g ✓ 33
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4024.2.a.g ✓ 33 4.b odd 2 1
8048.2.a.x 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8048))\):

\( T_{3}^{33} + 10 T_{3}^{32} - 23 T_{3}^{31} - 532 T_{3}^{30} - 529 T_{3}^{29} + 12061 T_{3}^{28} + \cdots + 116224 \) Copy content Toggle raw display
\( T_{5}^{33} - 106 T_{5}^{31} - 5 T_{5}^{30} + 5010 T_{5}^{29} + 446 T_{5}^{28} - 139884 T_{5}^{27} + \cdots - 9535488 \) Copy content Toggle raw display
\( T_{7}^{33} + 12 T_{7}^{32} - 71 T_{7}^{31} - 1361 T_{7}^{30} + 486 T_{7}^{29} + 66228 T_{7}^{28} + \cdots + 1527027904 \) Copy content Toggle raw display
\( T_{13}^{33} + 17 T_{13}^{32} - 96 T_{13}^{31} - 3138 T_{13}^{30} - 3085 T_{13}^{29} + \cdots - 116262986803488 \) Copy content Toggle raw display