Properties

Label 8048.2.a.s
Level $8048$
Weight $2$
Character orbit 8048.a
Self dual yes
Analytic conductor $64.264$
Analytic rank $1$
Dimension $21$
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: \(21\)
Twist minimal: no (minimal twist has level 2012)
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.

\(\operatorname{Tr}(f)(q) = \) \( 21 q - 10 q^{3} + 3 q^{5} - 13 q^{7} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 21 q - 10 q^{3} + 3 q^{5} - 13 q^{7} + 21 q^{9} - 7 q^{11} + 12 q^{13} - 14 q^{15} + q^{17} - 14 q^{19} + 14 q^{21} - 26 q^{23} + 18 q^{25} - 37 q^{27} + 9 q^{29} - 28 q^{31} + 3 q^{33} - 20 q^{35} + 31 q^{37} - 29 q^{39} + 4 q^{41} - 38 q^{43} + 24 q^{45} - 9 q^{47} + 16 q^{49} - 15 q^{51} + 22 q^{53} - 35 q^{55} - q^{57} - 10 q^{59} + 22 q^{61} - 35 q^{63} - 14 q^{65} - 58 q^{67} + 15 q^{69} - 27 q^{71} - 6 q^{73} - 48 q^{75} + 16 q^{77} - 47 q^{79} + 29 q^{81} - 22 q^{83} + 14 q^{85} - 29 q^{87} + q^{89} - 51 q^{91} + 34 q^{93} - 23 q^{95} - 2 q^{97} - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −3.31593 0 −2.21364 0 1.66896 0 7.99538 0
1.2 0 −3.16551 0 −0.445451 0 −3.10818 0 7.02047 0
1.3 0 −3.07762 0 3.88947 0 0.0828849 0 6.47174 0
1.4 0 −2.69810 0 1.28348 0 −4.83591 0 4.27974 0
1.5 0 −2.44429 0 3.11768 0 1.05875 0 2.97454 0
1.6 0 −2.30121 0 1.38461 0 2.51996 0 2.29558 0
1.7 0 −1.93158 0 −3.25684 0 −2.49607 0 0.730999 0
1.8 0 −1.38206 0 −2.65136 0 −3.15617 0 −1.08990 0
1.9 0 −1.32334 0 2.33991 0 −2.12211 0 −1.24878 0
1.10 0 −1.10653 0 2.64866 0 −3.45308 0 −1.77559 0
1.11 0 −0.207528 0 −1.91018 0 4.59256 0 −2.95693 0
1.12 0 0.112957 0 0.539075 0 1.01688 0 −2.98724 0
1.13 0 0.175511 0 3.29961 0 −3.77232 0 −2.96920 0
1.14 0 0.273229 0 −1.56489 0 3.17063 0 −2.92535 0
1.15 0 0.388717 0 −4.11083 0 1.27214 0 −2.84890 0
1.16 0 1.27408 0 3.64848 0 0.903884 0 −1.37671 0
1.17 0 1.30141 0 −2.99143 0 −4.27894 0 −1.30633 0
1.18 0 1.91007 0 1.16052 0 −1.88100 0 0.648366 0
1.19 0 2.22301 0 −1.09600 0 1.95566 0 1.94176 0
1.20 0 2.41348 0 −0.469286 0 1.19654 0 2.82487 0
See all 21 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.21
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.s 21
4.b odd 2 1 2012.2.a.b 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2012.2.a.b 21 4.b odd 2 1
8048.2.a.s 21 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8048))\):

\( T_{3}^{21} + 10 T_{3}^{20} + 8 T_{3}^{19} - 221 T_{3}^{18} - 627 T_{3}^{17} + 1614 T_{3}^{16} + 7876 T_{3}^{15} - 2568 T_{3}^{14} - 44493 T_{3}^{13} - 22587 T_{3}^{12} + 129082 T_{3}^{11} + 122117 T_{3}^{10} - 191892 T_{3}^{9} + \cdots + 41 \) Copy content Toggle raw display
\( T_{5}^{21} - 3 T_{5}^{20} - 57 T_{5}^{19} + 167 T_{5}^{18} + 1337 T_{5}^{17} - 3785 T_{5}^{16} - 16871 T_{5}^{15} + 45604 T_{5}^{14} + 125170 T_{5}^{13} - 319041 T_{5}^{12} - 558421 T_{5}^{11} + 1327922 T_{5}^{10} + \cdots - 64512 \) Copy content Toggle raw display
\( T_{7}^{21} + 13 T_{7}^{20} + 3 T_{7}^{19} - 609 T_{7}^{18} - 1837 T_{7}^{17} + 10228 T_{7}^{16} + 47618 T_{7}^{15} - 80812 T_{7}^{14} - 561858 T_{7}^{13} + 334765 T_{7}^{12} + 3813428 T_{7}^{11} - 993830 T_{7}^{10} + \cdots - 1292009 \) Copy content Toggle raw display
\( T_{13}^{21} - 12 T_{13}^{20} - 64 T_{13}^{19} + 1204 T_{13}^{18} + 184 T_{13}^{17} - 46562 T_{13}^{16} + 71067 T_{13}^{15} + 894414 T_{13}^{14} - 2107392 T_{13}^{13} - 9123021 T_{13}^{12} + 26181456 T_{13}^{11} + \cdots + 2465001 \) Copy content Toggle raw display