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Properties

Label 4017.2.a.k
Level $4017$
Weight $2$
Character orbit 4017.a
Self dual yes
Analytic conductor $32.076$
Analytic rank $0$
Dimension $32$
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] = [4017,2,Mod(1,4017)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4017, 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("4017.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4017 = 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4017.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: \(32.0759064919\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 5 q^{2} + 32 q^{3} + 41 q^{4} + 7 q^{5} + 5 q^{6} + 25 q^{7} + 12 q^{8} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 5 q^{2} + 32 q^{3} + 41 q^{4} + 7 q^{5} + 5 q^{6} + 25 q^{7} + 12 q^{8} + 32 q^{9} + 2 q^{10} + 17 q^{11} + 41 q^{12} + 32 q^{13} - 10 q^{14} + 7 q^{15} + 51 q^{16} + 2 q^{17} + 5 q^{18} + 36 q^{19} - 2 q^{20} + 25 q^{21} - 3 q^{22} + 37 q^{23} + 12 q^{24} + 43 q^{25} + 5 q^{26} + 32 q^{27} + 54 q^{28} + 2 q^{29} + 2 q^{30} + 44 q^{31} + 19 q^{32} + 17 q^{33} + 27 q^{34} - 10 q^{35} + 41 q^{36} + 46 q^{37} - 6 q^{38} + 32 q^{39} - 6 q^{40} + 5 q^{41} - 10 q^{42} + 19 q^{43} + 37 q^{44} + 7 q^{45} + 23 q^{46} + 50 q^{47} + 51 q^{48} + 67 q^{49} - 4 q^{50} + 2 q^{51} + 41 q^{52} + 5 q^{54} + 18 q^{55} - 54 q^{56} + 36 q^{57} + 27 q^{58} + 26 q^{59} - 2 q^{60} + 23 q^{61} + 27 q^{62} + 25 q^{63} + 70 q^{64} + 7 q^{65} - 3 q^{66} + 30 q^{67} - 22 q^{68} + 37 q^{69} + 59 q^{70} + 34 q^{71} + 12 q^{72} + 54 q^{73} + 18 q^{74} + 43 q^{75} + 40 q^{76} - 5 q^{77} + 5 q^{78} + 35 q^{79} - 46 q^{80} + 32 q^{81} + 23 q^{83} + 54 q^{84} + 59 q^{85} - 5 q^{86} + 2 q^{87} - 13 q^{88} + 16 q^{89} + 2 q^{90} + 25 q^{91} + 101 q^{92} + 44 q^{93} - 16 q^{94} - q^{95} + 19 q^{96} + 44 q^{97} - 44 q^{98} + 17 q^{99}+O(q^{100}) \) 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 −2.81093 1.00000 5.90132 −1.40450 −2.81093 4.16883 −10.9663 1.00000 3.94794
1.2 −2.59984 1.00000 4.75917 1.47347 −2.59984 1.66117 −7.17341 1.00000 −3.83079
1.3 −2.39948 1.00000 3.75753 −1.39162 −2.39948 −0.807842 −4.21716 1.00000 3.33917
1.4 −2.23055 1.00000 2.97534 2.00994 −2.23055 −4.12307 −2.17553 1.00000 −4.48326
1.5 −2.21491 1.00000 2.90581 3.81514 −2.21491 4.67656 −2.00629 1.00000 −8.45019
1.6 −2.13224 1.00000 2.54643 −4.09989 −2.13224 4.42610 −1.16513 1.00000 8.74193
1.7 −1.74675 1.00000 1.05112 2.78436 −1.74675 −1.73589 1.65745 1.00000 −4.86356
1.8 −1.60531 1.00000 0.577033 −3.03050 −1.60531 3.88777 2.28431 1.00000 4.86490
1.9 −1.46454 1.00000 0.144881 1.31201 −1.46454 2.70261 2.71690 1.00000 −1.92149
1.10 −1.16220 1.00000 −0.649300 0.840727 −1.16220 −4.11692 3.07901 1.00000 −0.977089
1.11 −1.05981 1.00000 −0.876804 −3.52707 −1.05981 −1.13469 3.04886 1.00000 3.73802
1.12 −0.869920 1.00000 −1.24324 3.55798 −0.869920 3.07477 2.82136 1.00000 −3.09516
1.13 −0.628086 1.00000 −1.60551 −0.0316739 −0.628086 −0.526505 2.26457 1.00000 0.0198939
1.14 −0.166477 1.00000 −1.97229 3.74884 −0.166477 −0.361248 0.661293 1.00000 −0.624094
1.15 −0.0300119 1.00000 −1.99910 −0.348908 −0.0300119 −0.368688 0.120021 1.00000 0.0104714
1.16 0.135928 1.00000 −1.98152 −2.49648 0.135928 4.30504 −0.541201 1.00000 −0.339342
1.17 0.262075 1.00000 −1.93132 0.855445 0.262075 3.34709 −1.03030 1.00000 0.224191
1.18 0.397144 1.00000 −1.84228 −2.35712 0.397144 −3.48052 −1.52594 1.00000 −0.936118
1.19 0.921096 1.00000 −1.15158 −1.44579 0.921096 −0.134596 −2.90291 1.00000 −1.33171
1.20 1.15421 1.00000 −0.667790 1.41297 1.15421 4.47597 −3.07920 1.00000 1.63087
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.32
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(13\) \(-1\)
\(103\) \(-1\)

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 4017.2.a.k ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4017.2.a.k ✓ 32 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4017))\):

\( T_{2}^{32} - 5 T_{2}^{31} - 40 T_{2}^{30} + 231 T_{2}^{29} + 663 T_{2}^{28} - 4748 T_{2}^{27} + \cdots - 192 \) Copy content Toggle raw display
\( T_{23}^{32} - 37 T_{23}^{31} + 308 T_{23}^{30} + 4413 T_{23}^{29} - 78221 T_{23}^{28} + \cdots + 10\!\cdots\!52 \) Copy content Toggle raw display