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Properties

Label 4017.2.a.l
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 5 q^{2} + 32 q^{3} + 45 q^{4} + q^{5} + 5 q^{6} + 11 q^{7} + 12 q^{8} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 5 q^{2} + 32 q^{3} + 45 q^{4} + q^{5} + 5 q^{6} + 11 q^{7} + 12 q^{8} + 32 q^{9} + 16 q^{10} + 3 q^{11} + 45 q^{12} - 32 q^{13} + 12 q^{14} + q^{15} + 75 q^{16} + 10 q^{17} + 5 q^{18} + 4 q^{20} + 11 q^{21} + 27 q^{22} + 53 q^{23} + 12 q^{24} + 67 q^{25} - 5 q^{26} + 32 q^{27} + 32 q^{28} + 6 q^{29} + 16 q^{30} + 12 q^{31} + 19 q^{32} + 3 q^{33} + 25 q^{34} + 16 q^{35} + 45 q^{36} + 36 q^{37} + 8 q^{38} - 32 q^{39} + 36 q^{40} - 19 q^{41} + 12 q^{42} + 43 q^{43} - 23 q^{44} + q^{45} + 11 q^{46} + 30 q^{47} + 75 q^{48} + 75 q^{49} + 28 q^{50} + 10 q^{51} - 45 q^{52} + 22 q^{53} + 5 q^{54} + 58 q^{55} + 60 q^{56} + 33 q^{58} - 24 q^{59} + 4 q^{60} + 47 q^{61} - 25 q^{62} + 11 q^{63} + 146 q^{64} - q^{65} + 27 q^{66} + 34 q^{67} + 58 q^{68} + 53 q^{69} - 35 q^{70} + 18 q^{71} + 12 q^{72} - 2 q^{73} - 20 q^{74} + 67 q^{75} + 24 q^{76} + 39 q^{77} - 5 q^{78} + 39 q^{79} + 2 q^{80} + 32 q^{81} + 64 q^{82} + 17 q^{83} + 32 q^{84} + 35 q^{85} - 13 q^{86} + 6 q^{87} + 55 q^{88} - 48 q^{89} + 16 q^{90} - 11 q^{91} + 39 q^{92} + 12 q^{93} + 58 q^{94} + 59 q^{95} + 19 q^{96} + 42 q^{97} + 16 q^{98} + 3 q^{99}+O(q^{100}) \) 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 −2.75387 1.00000 5.58381 3.90153 −2.75387 0.781129 −9.86935 1.00000 −10.7443
1.2 −2.67949 1.00000 5.17965 −3.25805 −2.67949 0.886771 −8.51984 1.00000 8.72990
1.3 −2.65538 1.00000 5.05105 −1.68203 −2.65538 −3.96430 −8.10171 1.00000 4.46642
1.4 −2.55932 1.00000 4.55014 −2.63190 −2.55932 2.75923 −6.52662 1.00000 6.73589
1.5 −1.93942 1.00000 1.76133 0.142452 −1.93942 3.99599 0.462872 1.00000 −0.276273
1.6 −1.85552 1.00000 1.44297 3.38665 −1.85552 −2.21342 1.03358 1.00000 −6.28400
1.7 −1.79771 1.00000 1.23177 3.18677 −1.79771 1.45324 1.38105 1.00000 −5.72890
1.8 −1.78664 1.00000 1.19208 −1.81642 −1.78664 −4.42230 1.44346 1.00000 3.24528
1.9 −1.48812 1.00000 0.214502 −3.58678 −1.48812 0.0236485 2.65704 1.00000 5.33757
1.10 −1.42411 1.00000 0.0280857 −0.882122 −1.42411 4.13066 2.80822 1.00000 1.25624
1.11 −1.08109 1.00000 −0.831238 −0.942662 −1.08109 −2.40511 3.06083 1.00000 1.01911
1.12 −0.772884 1.00000 −1.40265 −0.606956 −0.772884 1.27278 2.62985 1.00000 0.469107
1.13 −0.365601 1.00000 −1.86634 0.232769 −0.365601 −3.73203 1.41354 1.00000 −0.0851006
1.14 −0.238219 1.00000 −1.94325 2.19571 −0.238219 5.28038 0.939357 1.00000 −0.523060
1.15 −0.163737 1.00000 −1.97319 3.16137 −0.163737 4.62100 0.650558 1.00000 −0.517632
1.16 −0.0270033 1.00000 −1.99927 −4.41470 −0.0270033 1.75798 0.107994 1.00000 0.119212
1.17 0.310907 1.00000 −1.90334 2.19968 0.310907 −3.79510 −1.21358 1.00000 0.683896
1.18 0.609784 1.00000 −1.62816 2.30670 0.609784 −1.80421 −2.21240 1.00000 1.40659
1.19 0.934190 1.00000 −1.12729 −1.38248 0.934190 −2.39473 −2.92148 1.00000 −1.29149
1.20 1.11274 1.00000 −0.761813 −3.97825 1.11274 −2.83032 −3.07318 1.00000 −4.42675
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.l ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4017.2.a.l ✓ 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4017))\):

\( T_{2}^{32} - 5 T_{2}^{31} - 42 T_{2}^{30} + 241 T_{2}^{29} + 733 T_{2}^{28} - 5160 T_{2}^{27} + \cdots + 448 \) Copy content Toggle raw display
\( T_{23}^{32} - 53 T_{23}^{31} + 944 T_{23}^{30} - 1055 T_{23}^{29} - 190237 T_{23}^{28} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display