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Properties

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

\(\operatorname{Tr}(f)(q) = \) \( 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9} + 9 q^{11} - 16 q^{13} + 22 q^{15} + q^{17} + 12 q^{19} - 2 q^{21} + 22 q^{23} + 18 q^{25} + 43 q^{27} - 13 q^{29} + 28 q^{31} + 3 q^{33} + 12 q^{35} - 39 q^{37} + 11 q^{39} + 4 q^{41} + 50 q^{43} - 6 q^{45} + 27 q^{47} + 16 q^{49} + 37 q^{51} - 24 q^{53} + 49 q^{55} - q^{57} + 22 q^{59} - 22 q^{61} + 49 q^{63} - 14 q^{65} + 62 q^{67} - 17 q^{69} + 21 q^{71} - 6 q^{73} + 52 q^{75} - 24 q^{77} + 65 q^{79} + 29 q^{81} + 18 q^{83} - 54 q^{85} + 31 q^{87} + q^{89} + 45 q^{91} - 26 q^{93} + 53 q^{95} - 2 q^{97} + 58 q^{99}+O(q^{100}) \) 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 −2.92891 0 −2.56957 0 1.38532 0 5.57849 0
1.2 0 −2.20385 0 −1.17070 0 −0.239315 0 1.85694 0
1.3 0 −2.09799 0 −0.681202 0 4.42487 0 1.40156 0
1.4 0 −1.85682 0 −3.94450 0 1.81629 0 0.447775 0
1.5 0 −1.37429 0 2.26464 0 3.08988 0 −1.11131 0
1.6 0 −1.22261 0 0.941648 0 1.12930 0 −1.50523 0
1.7 0 −0.954017 0 −2.55262 0 −2.39192 0 −2.08985 0
1.8 0 −0.301406 0 3.72608 0 3.06802 0 −2.90915 0
1.9 0 −0.0551142 0 0.393007 0 −3.87485 0 −2.99696 0
1.10 0 −0.0212378 0 −0.474891 0 0.949153 0 −2.99955 0
1.11 0 0.100719 0 −0.707548 0 −0.593317 0 −2.98986 0
1.12 0 0.972125 0 0.716085 0 −2.42323 0 −2.05497 0
1.13 0 1.41685 0 2.80516 0 3.87406 0 −0.992528 0
1.14 0 1.70783 0 −2.09685 0 −2.97923 0 −0.0833023 0
1.15 0 2.02067 0 −4.25326 0 0.423427 0 1.08311 0
1.16 0 2.08528 0 2.45828 0 0.831485 0 1.34837 0
1.17 0 2.48667 0 −2.90154 0 4.76998 0 3.18353 0
1.18 0 2.49728 0 1.49435 0 1.21857 0 3.23640 0
1.19 0 3.12082 0 −2.38568 0 −0.375935 0 6.73950 0
1.20 0 3.19390 0 3.60846 0 −4.05198 0 7.20102 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.t 21
4.b odd 2 1 2012.2.a.a ✓ 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2012.2.a.a ✓ 21 4.b odd 2 1
8048.2.a.t 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8048))\):

\( T_{3}^{21} - 10 T_{3}^{20} + 8 T_{3}^{19} + 219 T_{3}^{18} - 607 T_{3}^{17} - 1646 T_{3}^{16} + 7568 T_{3}^{15} + \cdots - 3 \) Copy content Toggle raw display
\( T_{5}^{21} + 3 T_{5}^{20} - 57 T_{5}^{19} - 161 T_{5}^{18} + 1349 T_{5}^{17} + 3571 T_{5}^{16} + \cdots + 82944 \) Copy content Toggle raw display
\( T_{7}^{21} - 15 T_{7}^{20} + 31 T_{7}^{19} + 571 T_{7}^{18} - 3045 T_{7}^{17} - 4680 T_{7}^{16} + \cdots - 64245 \) Copy content Toggle raw display
\( T_{13}^{21} + 16 T_{13}^{20} - 36 T_{13}^{19} - 1788 T_{13}^{18} - 4684 T_{13}^{17} + 73110 T_{13}^{16} + \cdots - 620548203 \) Copy content Toggle raw display