Properties

Label 4029.2.a.k
Level $4029$
Weight $2$
Character orbit 4029.a
Self dual yes
Analytic conductor $32.172$
Analytic rank $0$
Dimension $31$
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] = [4029,2,Mod(1,4029)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4029, 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("4029.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4029 = 3 \cdot 17 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4029.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: \(32.1717269744\)
Analytic rank: \(0\)
Dimension: \(31\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 31 q + 4 q^{2} + 31 q^{3} + 34 q^{4} + 11 q^{5} + 4 q^{6} + 4 q^{7} + 12 q^{8} + 31 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 31 q + 4 q^{2} + 31 q^{3} + 34 q^{4} + 11 q^{5} + 4 q^{6} + 4 q^{7} + 12 q^{8} + 31 q^{9} + 5 q^{10} + 26 q^{11} + 34 q^{12} + 7 q^{13} + 19 q^{14} + 11 q^{15} + 40 q^{16} + 31 q^{17} + 4 q^{18} + 32 q^{19} + 23 q^{20} + 4 q^{21} + 2 q^{22} + 29 q^{23} + 12 q^{24} + 32 q^{25} + 13 q^{26} + 31 q^{27} - 13 q^{28} + 25 q^{29} + 5 q^{30} + 22 q^{31} + 28 q^{32} + 26 q^{33} + 4 q^{34} + 20 q^{35} + 34 q^{36} - 4 q^{37} + 19 q^{38} + 7 q^{39} - 3 q^{40} + 33 q^{41} + 19 q^{42} + 6 q^{43} + 30 q^{44} + 11 q^{45} - 11 q^{46} + 23 q^{47} + 40 q^{48} + 31 q^{49} + 6 q^{50} + 31 q^{51} - 7 q^{52} + 12 q^{53} + 4 q^{54} + 40 q^{56} + 32 q^{57} + 9 q^{58} + 27 q^{59} + 23 q^{60} - 4 q^{61} + 25 q^{62} + 4 q^{63} + 10 q^{64} + 54 q^{65} + 2 q^{66} + 34 q^{68} + 29 q^{69} - 59 q^{70} + 35 q^{71} + 12 q^{72} + 5 q^{73} + 48 q^{74} + 32 q^{75} + 32 q^{76} + 42 q^{77} + 13 q^{78} + 31 q^{79} + 24 q^{80} + 31 q^{81} + 5 q^{82} + 67 q^{83} - 13 q^{84} + 11 q^{85} - 20 q^{86} + 25 q^{87} - 7 q^{88} + 22 q^{89} + 5 q^{90} + 16 q^{91} + 57 q^{92} + 22 q^{93} + 45 q^{94} + 73 q^{95} + 28 q^{96} - 13 q^{97} - 19 q^{98} + 26 q^{99}+O(q^{100}) \) 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 −2.61491 1.00000 4.83775 −2.62328 −2.61491 −4.80522 −7.42046 1.00000 6.85964
1.2 −2.56879 1.00000 4.59868 3.14567 −2.56879 −0.137684 −6.67546 1.00000 −8.08058
1.3 −2.41826 1.00000 3.84800 1.43707 −2.41826 3.74160 −4.46895 1.00000 −3.47522
1.4 −2.28064 1.00000 3.20133 2.67486 −2.28064 −3.60745 −2.73981 1.00000 −6.10039
1.5 −2.22001 1.00000 2.92842 −1.63428 −2.22001 −1.90265 −2.06110 1.00000 3.62811
1.6 −1.88694 1.00000 1.56054 −0.846288 −1.88694 3.01609 0.829236 1.00000 1.59689
1.7 −1.73268 1.00000 1.00217 2.89501 −1.73268 3.30216 1.72891 1.00000 −5.01613
1.8 −1.49453 1.00000 0.233609 −1.69557 −1.49453 0.268778 2.63992 1.00000 2.53407
1.9 −1.27443 1.00000 −0.375837 −3.50336 −1.27443 −0.192219 3.02783 1.00000 4.46478
1.10 −1.26690 1.00000 −0.394972 3.60603 −1.26690 −2.46109 3.03418 1.00000 −4.56847
1.11 −0.719826 1.00000 −1.48185 1.82945 −0.719826 0.414755 2.50633 1.00000 −1.31688
1.12 −0.639237 1.00000 −1.59138 −1.53158 −0.639237 −2.74819 2.29574 1.00000 0.979040
1.13 −0.626356 1.00000 −1.60768 −1.73582 −0.626356 −3.52892 2.25969 1.00000 1.08724
1.14 −0.103363 1.00000 −1.98932 −2.25663 −0.103363 4.11147 0.412347 1.00000 0.233252
1.15 0.0488355 1.00000 −1.99762 −0.188126 0.0488355 3.46836 −0.195225 1.00000 −0.00918720
1.16 0.0509119 1.00000 −1.99741 3.95784 0.0509119 1.89285 −0.203515 1.00000 0.201501
1.17 0.308270 1.00000 −1.90497 2.48291 0.308270 3.17570 −1.20379 1.00000 0.765408
1.18 0.617959 1.00000 −1.61813 3.16624 0.617959 −0.196167 −2.23585 1.00000 1.95660
1.19 0.680303 1.00000 −1.53719 −1.16933 0.680303 −3.22223 −2.40636 1.00000 −0.795496
1.20 0.959467 1.00000 −1.07942 −3.93035 0.959467 −0.0159719 −2.95460 1.00000 −3.77104
See all 31 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.31
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(17\) \(-1\)
\(79\) \(-1\)

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 4029.2.a.k 31
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4029.2.a.k 31 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4029))\):

\( T_{2}^{31} - 4 T_{2}^{30} - 40 T_{2}^{29} + 172 T_{2}^{28} + 693 T_{2}^{27} - 3280 T_{2}^{26} - 6767 T_{2}^{25} + 36580 T_{2}^{24} + 40284 T_{2}^{23} - 264955 T_{2}^{22} - 144079 T_{2}^{21} + 1309226 T_{2}^{20} + 250363 T_{2}^{19} + \cdots - 16 \) Copy content Toggle raw display
\( T_{5}^{31} - 11 T_{5}^{30} - 33 T_{5}^{29} + 767 T_{5}^{28} - 751 T_{5}^{27} - 21949 T_{5}^{26} + 57042 T_{5}^{25} + 331924 T_{5}^{24} - 1299128 T_{5}^{23} - 2793266 T_{5}^{22} + 16257159 T_{5}^{21} + 11412047 T_{5}^{20} + \cdots - 245126 \) Copy content Toggle raw display