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Properties

Label 8013.2.a.c
Level $8013$
Weight $2$
Character orbit 8013.a
Self dual yes
Analytic conductor $63.984$
Analytic rank $1$
Dimension $116$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8013,2,Mod(1,8013)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8013.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8013, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8013 = 3 \cdot 2671 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8013.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [116] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(63.9841271397\)
Analytic rank: \(1\)
Dimension: \(116\)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 116 q - 16 q^{2} - 116 q^{3} + 116 q^{4} - 20 q^{5} + 16 q^{6} - 33 q^{7} - 45 q^{8} + 116 q^{9} + 3 q^{10} - 57 q^{11} - 116 q^{12} + 6 q^{13} - 9 q^{14} + 20 q^{15} + 112 q^{16} - 30 q^{17} - 16 q^{18}+ \cdots - 57 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.78241 −1.00000 5.74180 −1.84035 2.78241 −4.26418 −10.4112 1.00000 5.12061
1.2 −2.77241 −1.00000 5.68625 1.80856 2.77241 −1.74273 −10.2198 1.00000 −5.01408
1.3 −2.74646 −1.00000 5.54305 −2.85945 2.74646 1.69114 −9.73084 1.00000 7.85338
1.4 −2.74034 −1.00000 5.50946 3.71802 2.74034 −2.61622 −9.61710 1.00000 −10.1886
1.5 −2.72179 −1.00000 5.40814 2.52515 2.72179 2.50598 −9.27623 1.00000 −6.87293
1.6 −2.71385 −1.00000 5.36498 −0.752198 2.71385 4.51557 −9.13205 1.00000 2.04135
1.7 −2.66795 −1.00000 5.11795 −3.13798 2.66795 0.0813556 −8.31854 1.00000 8.37196
1.8 −2.59210 −1.00000 4.71896 −1.33515 2.59210 −1.92470 −7.04781 1.00000 3.46084
1.9 −2.58282 −1.00000 4.67097 −1.48132 2.58282 0.817005 −6.89864 1.00000 3.82598
1.10 −2.57433 −1.00000 4.62715 −4.27877 2.57433 −4.90523 −6.76315 1.00000 11.0149
1.11 −2.49870 −1.00000 4.24352 −4.37212 2.49870 1.33064 −5.60588 1.00000 10.9246
1.12 −2.45463 −1.00000 4.02523 1.82739 2.45463 −2.98046 −4.97119 1.00000 −4.48558
1.13 −2.42944 −1.00000 3.90217 −3.31824 2.42944 4.69634 −4.62122 1.00000 8.06146
1.14 −2.39685 −1.00000 3.74487 3.43258 2.39685 −5.25948 −4.18219 1.00000 −8.22737
1.15 −2.30332 −1.00000 3.30530 0.401271 2.30332 −3.38368 −3.00653 1.00000 −0.924257
1.16 −2.24523 −1.00000 3.04104 −1.84042 2.24523 1.63491 −2.33738 1.00000 4.13216
1.17 −2.23895 −1.00000 3.01291 1.55531 2.23895 1.34297 −2.26786 1.00000 −3.48228
1.18 −2.20341 −1.00000 2.85502 −0.504029 2.20341 −3.35275 −1.88396 1.00000 1.11058
1.19 −2.18030 −1.00000 2.75372 1.60452 2.18030 4.21617 −1.64333 1.00000 −3.49835
1.20 −2.15908 −1.00000 2.66162 −0.295164 2.15908 3.97405 −1.42849 1.00000 0.637282
See next 80 embeddings (of 116 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.116
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(2671\) \( +1 \)

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 8013.2.a.c ✓ 116
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8013.2.a.c ✓ 116 1.a even 1 1 trivial