Properties

Label 4016.2.a.m
Level $4016$
Weight $2$
Character orbit 4016.a
Self dual yes
Analytic conductor $32.068$
Analytic rank $0$
Dimension $23$
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] = [4016,2,Mod(1,4016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4016, 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("4016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4016 = 2^{4} \cdot 251 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4016.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: \(32.0679214517\)
Analytic rank: \(0\)
Dimension: \(23\)
Twist minimal: no (minimal twist has level 2008)
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.

\(\operatorname{Tr}(f)(q) = \) \( 23 q - 2 q^{3} + 8 q^{5} - 2 q^{7} + 45 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 23 q - 2 q^{3} + 8 q^{5} - 2 q^{7} + 45 q^{9} - 8 q^{11} + 8 q^{13} - 7 q^{15} + 19 q^{17} + 9 q^{19} + 9 q^{21} - 21 q^{23} + 65 q^{25} - 5 q^{27} + 10 q^{29} + 9 q^{31} + 34 q^{33} - 12 q^{35} + 11 q^{37} + 9 q^{39} + 35 q^{41} + 9 q^{43} + 29 q^{45} - 37 q^{47} + 77 q^{49} + 17 q^{51} + 38 q^{53} + 20 q^{55} + 51 q^{57} - 17 q^{59} - 22 q^{63} + 41 q^{65} - 9 q^{67} + 8 q^{69} - 13 q^{71} + 41 q^{73} - 25 q^{75} + 36 q^{77} + 36 q^{79} + 127 q^{81} - 29 q^{83} + 34 q^{85} - 10 q^{87} + 36 q^{89} + 6 q^{91} + 36 q^{93} - 25 q^{95} + 40 q^{97} - 19 q^{99}+O(q^{100}) \) 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 −3.32587 0 −0.316897 0 −0.268063 0 8.06143 0
1.2 0 −3.26165 0 −4.10224 0 1.61221 0 7.63839 0
1.3 0 −3.15136 0 4.04418 0 −3.52087 0 6.93109 0
1.4 0 −2.67736 0 4.29688 0 4.01755 0 4.16827 0
1.5 0 −2.53334 0 1.50187 0 −0.478852 0 3.41783 0
1.6 0 −2.51727 0 −0.472191 0 −4.16870 0 3.33662 0
1.7 0 −1.96174 0 0.807414 0 −3.34299 0 0.848432 0
1.8 0 −1.63778 0 2.87336 0 2.53418 0 −0.317671 0
1.9 0 −1.39383 0 −3.54571 0 3.34862 0 −1.05724 0
1.10 0 −0.452441 0 1.91039 0 −1.96124 0 −2.79530 0
1.11 0 −0.259421 0 −1.96371 0 4.76268 0 −2.93270 0
1.12 0 0.0812979 0 4.08000 0 −4.55548 0 −2.99339 0
1.13 0 0.201914 0 −1.79819 0 −3.34181 0 −2.95923 0
1.14 0 0.347951 0 −1.66668 0 −3.92094 0 −2.87893 0
1.15 0 1.15579 0 −4.09592 0 0.621313 0 −1.66414 0
1.16 0 1.19399 0 −1.76720 0 4.20476 0 −1.57440 0
1.17 0 1.28390 0 3.72143 0 0.978625 0 −1.35160 0
1.18 0 2.13165 0 0.257423 0 −0.578263 0 1.54395 0
1.19 0 2.21536 0 3.09704 0 4.67984 0 1.90784 0
1.20 0 2.74485 0 2.14962 0 3.54966 0 4.53421 0
See all 23 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.23
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(251\) \(-1\)

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 4016.2.a.m 23
4.b odd 2 1 2008.2.a.d 23
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2008.2.a.d 23 4.b odd 2 1
4016.2.a.m 23 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{23} + 2 T_{3}^{22} - 55 T_{3}^{21} - 107 T_{3}^{20} + 1286 T_{3}^{19} + 2405 T_{3}^{18} - 16716 T_{3}^{17} - 29527 T_{3}^{16} + 132795 T_{3}^{15} + 215665 T_{3}^{14} - 668901 T_{3}^{13} - 957238 T_{3}^{12} + \cdots + 1408 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4016))\). Copy content Toggle raw display