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Properties

Label 8048.2.a.u
Level $8048$
Weight $2$
Character orbit 8048.a
Self dual yes
Analytic conductor $64.264$
Analytic rank $0$
Dimension $26$
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: \(26\)
Twist minimal: no (minimal twist has level 503)
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.

\(\operatorname{Tr}(f)(q) = \) \( 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9} + 17 q^{11} + 14 q^{13} - 18 q^{15} + 17 q^{17} + 22 q^{19} - 16 q^{21} - 27 q^{23} + 93 q^{25} - 31 q^{27} + 13 q^{29} - 26 q^{31} + 6 q^{33} + 22 q^{35} + 55 q^{37} + 15 q^{39} + 24 q^{41} - 20 q^{43} - 8 q^{45} + 25 q^{47} + 65 q^{49} - 7 q^{51} + 30 q^{53} - 25 q^{55} + 9 q^{57} + 26 q^{59} + 15 q^{61} + 19 q^{63} + 20 q^{65} + 20 q^{67} - 27 q^{69} + 35 q^{71} + 38 q^{73} - 2 q^{75} - 6 q^{77} - 21 q^{79} + 70 q^{81} + 48 q^{83} + 6 q^{85} + 9 q^{87} - 5 q^{89} + 24 q^{91} - 8 q^{93} - 43 q^{95} + 142 q^{97} + 50 q^{99}+O(q^{100}) \) 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 −3.34519 0 2.36922 0 0.269938 0 8.19027 0
1.2 0 −3.26409 0 1.15565 0 1.19216 0 7.65430 0
1.3 0 −3.12024 0 −4.14994 0 3.36863 0 6.73592 0
1.4 0 −2.94588 0 −4.09318 0 −3.85397 0 5.67824 0
1.5 0 −2.76613 0 3.34245 0 5.14301 0 4.65146 0
1.6 0 −2.32813 0 1.62138 0 −3.43484 0 2.42020 0
1.7 0 −2.27911 0 3.79116 0 −4.03115 0 2.19436 0
1.8 0 −1.74581 0 1.36534 0 0.0430977 0 0.0478420 0
1.9 0 −1.69727 0 0.763844 0 0.178001 0 −0.119277 0
1.10 0 −1.34769 0 4.23270 0 3.43664 0 −1.18373 0
1.11 0 −1.22555 0 −2.04978 0 −4.12271 0 −1.49802 0
1.12 0 −0.577924 0 2.10638 0 −1.73516 0 −2.66600 0
1.13 0 −0.249551 0 −3.41933 0 3.14640 0 −2.93772 0
1.14 0 0.205279 0 2.58453 0 2.90918 0 −2.95786 0
1.15 0 0.705033 0 −0.869095 0 −5.04945 0 −2.50293 0
1.16 0 1.04994 0 −1.98731 0 −2.61319 0 −1.89762 0
1.17 0 1.08298 0 3.81229 0 −4.19464 0 −1.82716 0
1.18 0 1.08803 0 3.67682 0 −1.62501 0 −1.81619 0
1.19 0 1.16576 0 −3.04539 0 −0.946556 0 −1.64102 0
1.20 0 1.65492 0 1.16845 0 5.22170 0 −0.261242 0
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.26
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.u 26
4.b odd 2 1 503.2.a.f ✓ 26
12.b even 2 1 4527.2.a.o 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
503.2.a.f ✓ 26 4.b odd 2 1
4527.2.a.o 26 12.b even 2 1
8048.2.a.u 26 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8048))\):

\( T_{3}^{26} + 4 T_{3}^{25} - 52 T_{3}^{24} - 211 T_{3}^{23} + 1175 T_{3}^{22} + 4814 T_{3}^{21} + \cdots - 113513 \) Copy content Toggle raw display
\( T_{5}^{26} - 9 T_{5}^{25} - 71 T_{5}^{24} + 873 T_{5}^{23} + 1333 T_{5}^{22} - 35769 T_{5}^{21} + \cdots + 7351042048 \) Copy content Toggle raw display
\( T_{7}^{26} + 11 T_{7}^{25} - 63 T_{7}^{24} - 1107 T_{7}^{23} + 43 T_{7}^{22} + 44446 T_{7}^{21} + \cdots + 5803441 \) Copy content Toggle raw display
\( T_{13}^{26} - 14 T_{13}^{25} - 120 T_{13}^{24} + 2386 T_{13}^{23} + 3454 T_{13}^{22} - 168492 T_{13}^{21} + \cdots - 4565810269 \) Copy content Toggle raw display